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Novel, Integrated and Revolutionary Well Test Interpretation and Analysis

Written By

Freddy Humberto Escobar Macualo

Submitted: 20 August 2018 Reviewed: 22 August 2018 Published: 05 November 2018

DOI: 10.5772/intechopen.81078

From the Monograph

Novel, Integrated and Revolutionary Well Test Interpretation and Analysis

Authored by Freddy Humberto Escobar Macualo

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Abstract

Well test interpretation is an important tool for reservoir characterization. There exist four methods to achieve this goal, which are as follows: type‐curve matching, conventional straight‐line method, non‐linear regression analysis, and TDS technique. The first method is basically a trial‐and‐error procedure; a deviation of a millimeter involves differences up to 200 psi and the difficulty of having so many matching charts. The second one, although very important, requires a plot for every flow regime, and there is no way for verification of the calculated parameters, and the third one has a problem of diversity of solutions but is the most used by engineers since it is automatically made by a computer program. This book focuses on the fourth method that uses a single plot of the pressure and pressure derivative plot for identifying different lines and feature for parameter estimation. It can be used alone and is applied practically to all the existing flow regime cases. In several cases, the same parameter can be estimated from different sources making a good way for verification. Combination of this method along with the second and third is recommended and widely used by the author.

Keywords

  • TDS technique
  • permeability
  • well‐drainage area
  • flow regimes
  • intersection points
  • transient pressure analysis
  • conventional analysis

Introduction

Well testing is a valuable and economical formation evaluation tool used in the hydrocarbon industry. It has been supported by mathematical modeling, computing, and the precision of measurement devices. The data acquired during a well test are used for reservoir characterization and description. However, the biggest drawback is that the system dealt with is neither designed nor seen by well test interpreters, and the only way to make contact with the reservoir is through the well by making indirect measurements.

Four methods are used for well test interpretation: (1) The oldest one is the conventional straight‐line method which consists of plotting pressure or the reciprocal rate—if dealing with transient rate analysis—in the y‐axis against a function of time in the x‐axis. This time function depends upon the governing equation for a given flow. For instance, radial flow uses the logarithm of time and linear flow uses the square root of time. The slope and intercept of such plot are used to find reservoir parameters. The main disadvantage of this method is the lack of confirmation and the difficulty to define a given flow regime. The method is widely used nowadays. (2) Type‐curve matching uses predefined dimensionless pressure and dimensionless time curves (some also use dimensionless pressure derivative), which are used as master guides to be matched with well pressure data to obtain a reference point for reservoir parameter determination. This method is basically a trial‐and‐error procedure which becomes into its biggest disadvantage. The method is practically unused. (3) Simulation of reservoir conditions and automatic adjustment to well test data by non‐linear regression analysis is the method widely used by petroleum engineers. This method is also being widely disused since engineers trust the whole task to the computer. They even perform inverse modeling trying to fit the data to any reservoir model without taking care of the actual conditions. However, the biggest weakness of this method lies on the none uniqueness of the solution. Depending on the input starting values, the results may be different. (4) The newest method known as Tiab’s direct synthesis (TDS) [1, 2] is the most powerful and practical one as will be demonstrated throughout the book. It employs characteristic points and features found on the pressure and pressure derivative versus time log‐log plot to be used into direct analytic equations for reservoir parameters’ calculation. It is even used, without using the original name, by all the commercial software. One of them calls it “Specialized lines.” Because of its practicality, accuracy and application is the main object of this book. Conventional analysis method will be also included for comparison purposes.

The TDS technique can be easily implemented for all kinds of conventional or unconventional systems. It can be easily applied on cases for which the other methods fail or are difficult to be applied. It is strongly based on the pressure derivative curve. The method works by sector or regions found on the test. This means once a given flow regime is identified, a straight line is drawn throughout it, and then, any arbitrary point on this line and the intersection with other lines as well are used into the appropriate equations for the calculation of reservoir parameters.

The book contains the application and detailed examples of the TDS technique to the most common or fundamental reservoir/fluid scenarios. It is divided into seven chapters that are recommended to be read in the other they appear, especially for academic purposes in senior undergraduate level or master degree level. Chapter 1 contains the governing equation and the superposition principle. Chapter 2 is the longest one since it includes drawdown for infinite and finite cases, elongated system, multi‐rate testing, and spherical/hemispherical flow. All the interpretation methods are studied in this chapter which covers about 45% of the book. Chapter 3 deals with pressure buildup testing and average reservoir pressure determination. Distance to barriers and interference testing are, respectively, treated in Chapters 4 and 5. Since the author is convinced that all reservoirs are naturally fractured, Chapter 6 covers this part which is also extended in hydraulically fractured wells in Chapter 7. In this last chapter, the most common flow regime shown in fractured wells: bilinear, linear, and elliptical are discussed with detailed for parameter characterization. The idea is to present a book on TDS technique as practical and short as possible; then, horizontal well testing is excluded here because of its complexity and extension, but the most outstanding and practical publications are named here.

My book entitled “Recent Advances in Practical Applied Well Test Analysis,” published in 2015, was written for people having some familiarity with the TDS technique, so that, it can be read in any order. This is not the case of the present textbook. It is recommended to be read in order from Chapter 1 and take especial care in Chapter 2 since many equations and concepts will be applied in the remaining chapters. TDS technique applies indifferently to both pressure drawdown and pressure buildup tests.

Finally, this book is an upgraded and updated version of a former one published in Spanish. Most of the type curves have been removed since they have never been used by the author on actual well test interpretations. However, the first motivation to publish this book is the author’s belief that TDS technique is the panacea for well test interpretation. TDS technique is such an easy and practical methodology that his creator, Dr. Djebbar Tiab, when day said to me “I still don’t believe TDS works!” But, it really does. Well, once things have been created, they look easy.

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Fundamentals

1.1. Basic concepts

Pressure test fundamentals come from the application of Newton’s law, especially the third one: Principle of action‐reaction, since it comes from a perturbation on a well, as illustrated in Figure 1.1.

Figure 1.1.

Diagram of the mathematical representation of a pressure test.

A well can be produced under any of two given scenarios: (a) by keeping a constant flow rate and recording the well‐flowing pressure or (b) by keeping a constant well‐flowing pressure and measuring the flow rate. The first case is known as pressure transient analysis, PTA, and the second one is better known as rate transient analysis, RTA, which both are commonly run in very low permeable formations such as shales.

Basically, the objectives of the analysis of the pressure tests are:

  • Reservoir evaluation and description: well delivery, properties, reservoir size, permeability by thickness (useful for spacing and stimulation), initial pressure (energy and forecast), and determination of aquifer existence.

  • Reservoir management.

There are several types of tests with their particular applications. DST and pressure buildup tests are mainly used in primary production and exploration. Multiple tests are most often used during secondary recovery projects, and multilayer and vertical permeability tests are used in producing/injectors wells. Drawdown, interference, and pulse tests are used at all stages of production. Multi‐rate, injection, interference, and pulse tests are used in primary and secondary stages [3, 4, 5, 6, 7].

Pressure test analysis has a variety of applications over the life of a reservoir. DST and pressure buildup tests run in single wells are mainly used during primary production and exploration, while multiple tests are used more often during secondary recovery projects. Multilayer and vertical permeability tests are also run in producing/injectors wells. Drawdown, buildup, interference, and pulse tests are used at all stages of production. Multi‐rate, injection, interference, and pulse testing are used in the primary and secondary stages. Petroleum engineers should take into account the state of the art of interpreting pressure tests, data acquisition tools, interpretation methods, and other factors that affect the quality of the results obtained from pressure test analysis.

Once the data have been obtained from the well and reviewed, the pressure test analysis comprises two steps: (1) To establish the reservoir model and the identification of the different flow regimes encountered during the test and (2) the parameter estimation. To achieve this goal, several plots are employed; among them, we have log‐log plot of pressure and pressure derivative versus testing time (diagnostic tool), semilog graph of pressure versus time, Cartesian graph of the same parameters, etc. Pressure derivative will be dealt later in this chapter.

The interpretation of pressure tests is the primary method for determining average permeability, skin factor, average reservoir pressure, fracture length and fracture conductivity, and reservoir heterogeneity. In addition, it is the only fastest and cheapest method to estimate time‐dependent variables such as skin factor and permeability in stress‐sensitive reservoirs.

In general, pressure test analysis is an excellent tool to describe and define the model of a reservoir. Flow regimes are a direct function of the characteristics of the well/reservoir system, that is, a simple fracture that intercepts the well can be identified by detection of a linear flow. However, whenever there is linear flow, it does not necessarily imply the presence of a fracture. The infinite‐acing behavior occurs after the end of wellbore storage and before the influence of the limits of the deposit. Since the boundaries do not affect the data during this period, the pressure behavior is identical to the behavior of an infinite reservoir. The radial flow can be recognized by an apparent stabilization of the value of the derivative.

1.2. Type of well tests

Well tests can be classified in several ways depending upon the view point. Some classifications consider whether or not the well produces or is shut‐in. Other engineers focus on the number of flow rates. The two main pressure tests are (a) pressure drawdown and (b) buildup. While the first one involves only one flow rate, the second one involves two flow rates, one of which is zero. Then, a pressure buildup test can be considered as a multi‐rate test.

1.2.1 Pressure tests run in producer wells

Drawdown pressure test (see Figure 1.2): It is also referred as a flow test. After the well has been shut‐in for a long enough time to achieve stabilization, the well is placed in production, at a constant rate, while recording the bottom pressure against time. Its main disadvantage is that it is difficult to maintain the constant flow rate.

Figure 1.2.

Schematic representation of pressure drawdown and pressure buildup tests.

Pressure buildup test (see Figure 1.2): In this test, the well is shut‐in while recording the static bottom‐hole pressure as a function of time. This test allows obtaining the average pressure of the reservoir. Although since 2010, average reservoir pressures can be determined from drawdown tests. Its main disadvantage is economic since the shut‐in entails the loss of production.

1.2.2 Pressure tests run in injector wells

Injection test (see Figure 1.3): Since it considers fluid flow, it is a test similar to the pressure drawdown test, but instead of producing fluids, fluids, usually water, are injected.

Figure 1.3.

Injection pressure test (left) and falloff test (right).

Falloff test (see Figure 1.3): This test considers a pressure drawdown immediately after the injection period finishes. Since the well is shut‐in, falloff tests are identical to pressure buildup tests.

1.2.3 Other tests

Interference and/or multiple tests: They involve more than one well and its purpose is to define connectivity and find directional permeabilities. A well perturbation is observed in another well.

Drill stem test (DST): This test is used during or immediately after well drilling and consists of short and continuous shut‐off or flow tests. Its purpose is to establish the potential of the well, although the estimated skin factor is not very representative because well cleaning can occur during the first productive stage of the well (Figure 1.4).

Figure 1.4.

Well test classification based on the number of flow rates.

Short tests: There are some very short tests mainly run in offshore wells. They are not treated in this book. Some of them are slug tests, general close chamber tests (CCTs), surge tests, shoot and pool tests, FasTest, and impulse tests.

As stated before, in a pressure drawdown test, the well is set to a constant flow rate. This condition is, sometimes, difficult to be fulfilled; then, multi‐rate tests have to be employed. According to [8], multi‐rate tests fit into four categories: (a) uncontrolled variable rate [9, 10], series of constant rates [11, 12], pressure buildup testing, and constant bottom‐hole pressure with a continuous changing flow rate [13]. This last technique has been recently named as rate transient analysis (RTA) which is included in PTA, but its study is not treated in this book.

1.3 Diffusivity equation

At the beginning of production, the pressure in the vicinity of the well falls abruptly and the fluids near the well expand and move toward the area of lower pressure. Such movement is retarded by friction against the walls of the well and the inertia and viscosity of the fluid itself. As the fluid moves, an imbalance of pressure is created, which induces the surrounding fluids to move toward the well. The process continues until the pressure drop created by the production dissipates throughout the reservoir. The physical process that takes place in the reservoir can be described by the diffusivity equation whose deduction is shown below [5]:

According to the volume element given in Figure 1.5,

[Mass entering the element][Mass coming out from the element]=[System accumulation rate]E1.1

Figure 1.5.

Radial volume element.

The right‐hand side part of Eq. (1.1) corresponds to the mass accumulated in the volume element. Darcy’s law for radial flow:

q=kAμdPdrE1.2

The cross‐sectional area available for flow is provided by cylindrical geometry, 2πrh. Additionally, flow rate must be multiplied by density, ρ, to obtain mass flow. With these premises, Eq. (1.2) becomes:

q=kμ2πrhPrE1.3

Replacing Eq. (1.3) into (1.1) yields:

kρμ(2πrh)Pr|r+kρμ(2πrh)Pr|r+dr=t([2πrhdrϕ]ρ)E1.4

If the control volume remains constant with time, then, Eq. (1.4) can be rearranged as:

2πhkρμrPr|r+2πhkρμrPr|r+dr=2πrhdrt(ϕρ)E1.5

Rearranging further the above expression:

1r[kρμrPr|r+drkρμrPr|r]dr=t(ϕρ)E1.6

The left‐hand side of Eq (1.6) corresponds to the definition of the derivative; then, it can be rewritten as:

1rr(kρμrPr)=t(ϕρ)E1.7

The definition of compressibility has been widely used;

c=1VVP=1ρρPE1.8

By the same token, the pore volume compressibility is given by:

cf=1ϕϕPE1.9

The integration of Eq. (1.8) will lead to obtain:

ρ=ρoec(PPo)E1.10

The right‐hand side part of Eq. (1.7) can be expanded as:

t(ϕρ)=ϕtρ+ρtϕ=ϕρt+ρϕPPρρtE1.11

Using the definitions given by Eqs. (1.9) and (1.10) into Eq. (1.11) leads to:

t(ϕρ)=ϕρt+ρϕcfcρρt=ϕρt[1+cfc]=ϕc[cf+c]ρtE1.12

Considering that the total compressibility, ct, is the result of the fluid compressibility, c, plus the pore volume compressibility, cf, it yields:

1rr(kρμrPr)=ϕctcρtE1.13

The gradient term can be expanded as:

Pr=Pρρr=1cρρrE1.14

Combination of Eqs. (1.14) and (1.13) results in:

1rr(krμcρr)=ϕcctρtE1.15

Taking derivative to Eq. (1.10) with respect to both time and radial distance and replacing these results into Eq. (1.15) yield:

1rr(krμcρoec(PPo)cPr)=ϕcctρoec(PPo)cPtE1.16

After simplification and considering permeability and viscosity to be constant, we obtain:

1rkμr(rPr)=ϕctPtE1.17

The hydraulic diffusivity constant is well known as

1η=ϕμctkE1.18

Then, the final form of the diffusivity equation in oilfield units is obtained by combination of Eqs. (1.17) and (1.18):

1rr(rPr)=ϕμct0.0002637kPt=1ηPtE1.19

In expanded form:

2Pr2+1rPr=10.0002637ηPtE1.20

The final form of the diffusivity equation strongly depends upon the flow geometry. For cylindrical, [11, 14], spherical [14], and elliptical coordinates [15], the diffusivity equation is given, respectively,

2Pr2+1rPr+kθkr1r22Pθ2+kzkr2Pz2=ϕμctkrPtE1.21
1r[r(r2Pr)+1sinθθ(sinθPθ)+1sin2θ2Pϕ2]=ϕcμkPtE1.22
2Pξ2+2Pη2=12a2(cosh2ξcos2η)ϕcμkPtE1.23

Here, ξ is a space coordinate and represents a family of confocal ellipses. The focal length of these ellipses is 2a. The space coordinate, η, represents a family of confocal hyperbolas that represent the streamlines for elliptical flow. These two coordinates are normal to each other.

1.4. Limitations of the diffusivity equation

  1. Isotropic, horizontal, homogeneous porous medium, permeability, and constant porosity

  2. A single fluid saturates the porous medium

  3. Constant viscosity, incompressible, or slightly compressible fluid

  4. The well completely penetrates the formation. Negligible gravitational forces

The density of the fluid is governed by an equation of state (EOS). For the case of slightly compressible fluid, Eq. (1.8) is used as the EOS.

1.5. Multiphase flow

Similar to the analysis of gas well tests as will be seen later, multiphase tests can be interpreted using the method of pressure approximation (Perrine method), [6, 7, 16], which is based on phase mobility:

λt=koμo+kgμg+kwμw=kroμo+krgμg+krwμwE1.24

The total compressibility is defined by [17, 18]:

ct=coSo+cgSg+cwSw+cf+SoBg5.615BoRsP+SwBg5.615BwRswPE1.25

For practical purposes, Eq. (1.25) can be expressed as:

ctcoSo+cgSg+cwSw+cfE1.26

As commented before Eq. (1.19) is limited to a single fluid. However, it can be extended to multiphase flow using the concept expressed by Eq. (1.24):

1rr(rPr)=ϕct0.00026371λtPtE1.27

Perrine method assumes negligible pressure and saturation gradients. Martin [19] showed that (a) the method loses accuracy as the gas saturation increases, (b) the estimation of the mobility is good, and (c) the mobility calculations are sensitive to the saturation gradients. Better estimates are obtained when the saturation distribution is uniform and (d) underestimates the effective permeability of the phase and overestimates the damage factor.

1.6. Gas flow

It is well known that gas compressibility, gas viscosity, and gas density are highly dependent pressure parameters; then, the liquid diffusivity equation may fail to observe pressure gas behavior. Therefore, there exist three forms for a better linearization of the diffusivity equation to better represent gas flow: (a) the pseudopressure approximation [20], (b) the P2 approximation, and (c) linear approximation. The first one is valid for any pressure range; the second one is valid for reservoir pressures between 2000 and 4000 psia, and the third one is for pressures above 4000 psia [20].

Starting from the equation of continuity and the equation of Darcy:

1rr(rρur)=t(ϕρ)E1.28
ur=kμPrE1.29

The state equation for slightly compressible liquids does not model gas flow; therefore, the law of real gases is used [21, 22]:

ρ=PMzRTE1.30

Combining the above three equations:

1rr(rkPMμzRTpt)=t(ϕPMzRT)E1.31

Since M, R, and T are constants and assuming that the permeability is constant, the above equation reduces to:

1rr(rPμzPr)=1kt(ϕPz)E1.32

Applying the differentiation chain rule to the right‐hand side part of Eq. (1.32) leads to:

1rr(rPμzPr)=1k[Pzϕt+ϕt(Pz)]E1.33

Expanding and rearranging,

1rr(rPμzPr)=PϕzkPt[1ϕϕP+zPP(Pz)]E1.34

Using the definition of compressibility for gas flow:

cg=1ρρP=zRTPMP(PMzRT)=zPP(Pz)E1.35

Using Eqs. (1.9) and (1.35) into Eq. (1.34),

1rr(rPμzPr)=PϕzkPt(cf+cg)E1.36

If ct=cg+cf then,

1rr(rPμzPr)=PϕctzkPtE1.37

The above is a nonlinear partial differential equation and cannot be solved directly. In general, three limiting assumptions are considered for its solution, namely: (a) P/μz is constant; (b) μct is constant; and (c) the pseudopressure transformation, [20], for an actual gas.

1.6.1 The equation of diffusivity in terms of pressure

Assuming the term P/μz remains constant with respect to the pressure, Eq. (1.17) is obtained.

1.6.2 The equation of diffusivity in terms of pressure squared

Eq. (1.37) can be written in terms of squared pressure, P2, starting from the fact that, [3, 4, 5, 6, 7, 9, 17, 21, 22]:

PPr=12P2rE1.38
PPt=12P2tE1.39
1rr(rμzP2r)=ϕctkzP2tE1.40

Assuming the term μz remains constant with respect to the pressure, and of course, the radius, then the above equation can be written as:

1rr(rP2r)=ϕμctkP2tE1.41

This expression is similar to Eq. (1.37), but the dependent variable is P2. Therefore, its solution is similar to Eq. (1.17), except that it is given in terms of P2. This equation also requires that μct remain constant.

1.6.3 Gas diffusivity equation in terms of pseudopressure, m(P)

The diffusivity equation in terms of P2 can be applied at low pressures, and Eq. (1.17) can be applied at high pressures without incurring errors. Therefore, a solution is required that applies to all ranges. Ref. [20] introduced a more rigorous linearization method called pseudopressure that allows the general diffusivity equation to be solved without limiting assumptions that restrict certain properties of gases to remain constant with pressure [3, 4, 5, 6, 7, 9, 17, 20, 21, 22]:

m(P)=2P0PPμzdPE1.42

Taking the derivative with respect to both time and radius and replacing the respective results in Eq. (1.37), we obtain:

1rr[rPμz(μz2Pm(P)r)]=Pϕctzk(μz2Pm(P)t)E1.43

After simplification,

1rr(rm(P)r)=ϕμctkm(P)tE1.44

Expanding the above equation and expressing it in oilfield units:

2m(P)r2+1rm(P)r=ϕμgict0.0002637kgim(P)tE1.45

The solution to the above expression is similar to the solution of Eq. (1.17), except that it is now given in terms of m(P) which can be determined by numerical integration if the PVT properties are known at each pressure level.

For a more effective linearization of Eq. (1.45), [23] introduced pseudotime, ta, since the product μgct in Eq. (1.45) is not constant:

ta=20tdςμctE1.46

With this criterion, the diffusivity equation for gases is:

1rr(rm(P)r)=2ϕ(cf+cg)kcgm(P)taE1.47

The incomplete linearization of the above expression leads to somewhat longer semilog slopes compared to those obtained for liquids. Sometimes it is recommended to use normalized variables in order to retain the units of time and pressure, [6]. The normalized pseudovariables are:

m(P)n=Pi+μiρiP0Pρ(ς)μ(ς)dςE1.48
tan=μicti+0tdςμ(ς)Z(ς)E1.49

1.7. Solution to the diffusivity equation

The line‐source solution: The line‐source solution assumes that the wellbore radius approaches zero. Furthermore, the solution considers a reservoir of infinite extent and the well produces as a constant flow rate. Ref. [4] presents the solution of the source line using the Boltzmann transform, the Laplace transform, and Bessel functions. The following is the combinations of independent variables method, which is based on the dimensional analysis of Buckingham’s theorem [24]. This takes a function f = f(x, y, z, t), it must be transformed into a group or function containing fewer variables, f = f(s1,s2…). A group of variables whose general form is proposed as [24]:

s=arbtcE1.50

The diffusivity equation is:

1rr(rfr)=ftE1.51

where f is a dimensionless term given by:

f=PPwfPiPwfE1.52

Eq. (1.51) is subjected to the following initial and boundary conditions:

f=0,0r,t=0E1.53
rfr=1,r=0,t>0E1.54
f=0,r,t>0E1.55

Multiplying the Eq. (1.51) by ∂s/∂s:

1rssr(rssfr)=ssftE1.56

Exchanging terms:

1rsrs(rsrfs)=stfsE1.57

The new derivatives are obtained from Eq. (1.50):

sr=abrb1tcE1.58
st=acrbtc1E1.59

Replacing the above derivatives into Eq. (1.56) and rearranging:

1ra2b2rbrt2cs(rrbrfs)=acrbtc1fsE1.60

Solving from rb from Eq. (1.50) and replacing this result into Eq. (1.6). After rearranging, it yields:

s(sfs)=cb2[r2t1]sfsE1.61

Comparing the term enclosed in square brackets with Eq. (1.50) shows that b = 2, c = −1, then

s=ar2tE1.62

From Eq. (1.61) follows r2t‒1 = s/a, then

s(sfs)=[cb2a]sfsE1.63

The term enclosed in square brackets is a constant that is assumed equal to 1 for convenience. Since c/(b2a) = 1, then a = −1/4. Therefore, the above expression leads to:

s(sfs)=sfsE1.64

Writing as an ordinary differential equation:

dds(sdfds)=sdfdsE1.65

The differential equation is now ordinary, and only two conditions are required to solve it. Applying a similar mathematical treatment to both the initial and boundary conditions to convert them into function of s. Regarding Eq. (1.62) and referring to the initial condition, Eq. (1.53), when the time is set to zero; then, then s function tends to infinite:

att=0,f=0whensE1.66

Darcy’s law is used to convert the internal boundary condition. Eq. (1.54) multiplied by ∂s/∂s gives:

rfssr=1E1.67

Replacing Eqs. (1.57) in the above equation; then, replacing Eq. (1.62) into the result, and after simplification, we obtain

fsabsatctc=1E1.68

Since b = 2, then,

sfs=12E1.69

For the external boundary condition, Eq. (1.55), consider the case of Eq. (1.62) when r → ∞ then:

s=ar2t;f=0,sE1.70

Then, the new differential equation, Eq. (1.65) is subject to new conditions given by Eqs. (1.66), (1.69), and (1.70). Define now,

g=sdfdsE1.71

Applying this definition into the ordinary differential expression given by Eq. (1.65), it results:

ddsg=gE1.72

Integration of the above expression leads to:

lng=s+c1E1.73

Rearranging the result and comparing to Eq. (1.71) and applying the boundary condition given by Eq. (1.69):

g=c1es=sdfds=12E1.74

Solving for df and integrating,

df=c1essdsE1.75

Eq. (1.75) cannot be analytically integrated (solved by power series). Simplifying the solution:

f=c1essds+c2E1.76

When s = 0, es = 0, then c1 = ½ and Eq. (1.76) becomes:

f=120sessds+c2E1.77

Applying the external boundary condition, Eq. (1.69), when s → ∞, f = 0, therefore, Eq. (1.77) leads,

c2=120essdsE1.78

Replacing c1 and c2 into Eq. (1.76) yields:

f=120sessds120essdsE1.79

This can be further simplified to:

f=12sessdsE1.80

The integral given in Eq. (1.80) is well known as the exponential integral, Ei(−s). If the f variable is changed by pressure terms:

P(r,t)=12Ei(r24t)E1.81

In dimensionless form,

PD(rD,tD)=12Ei(rD24tD)=12Ei(x)E1.82

The above equation is a very good approximation of the analytical solution when it is satisfied (Mueller and Witherspoon [2, 9, 18, 19, 25, 26]) that rD ≥ 20 or tD/rD2 ≥ 0.5, see Figure 1.6. If tD/rD2≥ 5, an error is less than 2%, and if tD/rD2 ≥ 25, the error is less than 5%. Figure 1.7 is represented by the following adjustment which has a correlation coefficient, R2 of 0.999998. This plot can be easily rebuilt using the algorithm provided in Figure 1.8. The fitted equation was achieved with the data generated from simulation.

Figure 1.6.

Dimensionless pressure for different values of the dimensionless radius, taken from [9, 25].

Figure 1.7.

Dimensionless well pressure behavior for a well without skin and storage effects in an infinite reservoir, taken from [9, 25].

Figure 1.8.

BASIC code function to calculate Ei function, taken from [29].

PD=100.2820668952451542+0.4472760048082251x+0.2581584173632316x2+0.04998332927590892x31+1.047015081287319x+0.3493329681392351x2+0.02955955788180784x30.000163604729430738x4E1.83

being x = log(tD/rD2) > −1.13.

The exponential function can be evaluated by the following formula, [27], for x ≤ 25:

Ei(x)=0.57721557+lnxx+x222!x333!+x444!….E1.84

Figure 1.8 shows a listing of a program code in Basic, which can be easily added as a function in Microsoft Excel to calculate the exponential function. Figure 1.9 and Table 1.1, 1.2, 1.3, and 1.4 present solutions of the exponential function.

Figure 1.9.

Values of the exponential integral for 1 ≤ x ≤ 10 (left) and 0.0001 ≤ x ≤ 1 (right).

abcdef
—0.09067656735636530.5133959845491270—0.0243644307428167—0.0000014346860800—0.4865489789766050
0.74802029191995701.3629598993866700—0.59600919611684000.0275653486990893—0.7768782064908800−0.0010740336145794

Table 1.1.

Constants for Eqs. (1.85) and (1.86).

x0123456789
0.0008.633227.940187.534817.247237.024196.841976.687916.554486.43680
0.0016.331546.236336.149426.069485.995475.926575.862145.801615.744555.69058
0.0025.639395.590705.544285.499935.457475.416755.377635.339995.303725.26873
0.0035.234935.202245.170595.139915.110165.081275.053205.025904.999344.97346
0.0044.948244.923654.899654.876224.853334.830964.809084.787674.766724.74620
0.0054.726104.706394.687074.668134.649534.631284.613374.595774.578474.56148
0.0064.544774.528344.512184.496284.480634.465234.450064.435124.420414.40591
0.0074.391624.377534.363654.349954.336454.323124.309984.297004.284204.27156
0.0084.259084.246764.234594.222574.210694.198964.187364.175904.164574.15337
0.0094.142294.131344.120524.109804.099214.088734.078354.068094.057934.04788
0.014.037933.943613.857603.778553.705433.637433.573893.514253.458093.40501
0.023.354713.306913.261383.217913.176343.136513.098283.061523.026142.99203
0.032.959122.927312.896552.866762.837892.809892.782702.756282.730602.70560
0.042.681262.657552.634432.611882.589872.568382.547372.526852.506772.48713
0.052.467902.449072.430632.412552.394842.377462.360412.343692.327272.31114
0.062.295312.279752.264462.249432.234652.220112.205812.191742.177892.16426
0.072.150842.137622.124602.111772.099132.086672.074392.062282.050342.03856
0.082.026942.015482.004171.993011.981991.971121.960381.949781.939301.92896
0.091.918741.908651.898681.888821.879081.869451.859941.850531.841221.83202
0.101.822921.813931.805021.796221.787511.778891.770361.761921.753561.74529
0.111.737111.729001.720981.713041.705171.697381.689671.682031.674461.66697
0.121.659541.652191.644901.637671.630521.623431.616401.609431.602531.59568
0.131.588901.582171.575511.568901.562341.555841.549401.543011.536671.53038
0.141.524151.517961.511831.505741.499701.493711.487771.481881.476031.47022
0.151.464461.458751.453071.447441.441861.436311.430801.425341.419921.41453
0.161.409191.403881.398611.393381.388191.383031.377911.372821.367781.36276
0.171.357781.352841.347921.343041.338201.333391.328601.323861.319141.31445
0.181.309801.305171.300581.296011.291471.286971.282491.278041.273621.26922
0.191.264861.260521.256211.251921.247661.243431.239221.235041.230891.22676
0.21.222651.218571.214511.210481.206471.202481.198521.194581.190671.18677

Table 1.2.

Values of the exponential integral for 0.0001 ≤ x ≤ 0.209.

x0123456789
437.794000037.792753033.488805229.687620926.329119223.360100520.734007818.410058416.352495014.5299393
511.483904911.482955710.21300089.08621588.08608307.19804426.40926035.70840155.08546474.5316127
63.60177353.60082453.21087032.86376342.55471432.27947962.03429871.81583741.62113851.4475779
71.15576631.15481731.03171270.92188120.82387250.73639720.65830890.58858770.52632610.4707165
80.37760520.37665620.33699510.30154860.26986410.24153820.21621120.19356250.17330600.1551866
90.12542260.12447350.11149540.09988070.08948490.08017900.07184770.06438830.05770860.0517267
100.04251870.04156970.03727040.03341860.02996730.02687470.02410310.02161910.01939250.0173966
110.01495200.01400300.01256450.01127460.01011780.00908040.00814980.00731510.00656630.0058946
120.00570010.00475110.00426580.00383030.00343950.00308880.00277390.00249130.00223770.0020099
130.00257090.00162190.00145700.00130900.00117610.00105670.00094950.00085320.00076670.0006890
140.00150560.00055660.00050020.00044960.00040420.00036330.00032660.00029360.00026400.0002373
150.00114090.000191860.000172510.000155130.000139500.000125450.000112820.000101469.1257E−058.2079E−05
160.00101556.6405E−095.9732E−095.3732E−094.8336E−094.3483E−093.9119E−093.5194E−093.1664E−092.8489E−09
170.00097252.3064E−092.0754E−091.8675E−091.6805E−091.5123E−091.3609E−091.2248E−091.1022E−099.9202E−10
180.00095638.0361E−107.2331E−106.5105E−105.8603E−105.2752E−104.7486E−104.2747E−103.8482E−103.4643E−10
190.00095112.8078E−102.5279E−102.2760E−102.0492E−101.8451E−101.6613E−101.4959E−101.3470E−101.2129E−10
200.00095269.8355E−118.8572E−117.9764E−117.1833E−116.4692E−115.8263E−115.2473E−114.7260E−114.2566E−11
210.00092483.4532E−113.1104E−112.8017E−112.5237E−112.2733E−112.0478E−111.8447E−111.6617E−111.4970E−11
220.00091831.2149E−111.0945E−119.8610E−128.8842E−128.0043E−127.2117E−126.4976E−125.8544E−125.2750E−12
230.00094644.2827E−123.8590E−123.4773E−123.1334E−122.8236E−122.5444E−122.2929E−122.0663E−21.8621E−12
240.00093161.5123E−121.3629E−121.2283E−121.1070E−129.9772E−138.9922E−138.1046E−137.3048E−136.5839E−13
250.00007795.3489E−134.8213E−134.3458E−133.9172E−133.5310E−133.1829E−132.8692E−132.5864E−132.3315E−13

Table 1.3.

Values of the exponential integral, Ei(−x) × 10−4, for 4 ≤ x ≤ 25.9.

x0123456789
0.201.2226511.1829021.1453801.1098831.0762361.0442831.0138890.9849330.9573080.930918
0.300.9056770.8815060.8583350.8361010.8147460.7942160.7744620.7554420.7371120.719437
0.400.7023800.6859100.6699970.6546140.6397330.6253310.6113870.5978780.5847840.572089
0.500.5597740.5478220.5362200.5249520.5140040.5033640.4930200.4829600.4731740.463650
0.600.4543800.4453530.4365620.4279970.4196520.4115170.4035860.3958530.3883090.380950
0.700.3737690.3667600.3599180.3532370.3467130.3403410.3341150.3280320.3220880.316277
0.800.3105970.3050430.2996110.2942990.2891030.2840190.2790450.2741770.2694130.264750
0.900.2601840.2557140.2513370.2470500.2428510.2387380.2347080.2307600.2268910.223100
1.000.21938400.21574170.21217120.20867070.20523840.20187290.19857240.19533550.19216060.1890462
1.100.18599100.18299360.18005260.17716670.17433470.17155540.16882760.16615010.16352180.1609417
1.200.15840850.15592140.15347930.15108130.14872630.14641350.14414190.14191070.13971910.1375661
1.300.13545110.13337310.13133140.12932530.12735410.12541690.12351320.12164230.11980340.1179960
1.400.11621940.11447300.11275620.11106840.10940900.10777750.10617340.10459600.10304500.1015197
1.500.10001970.09854450.09709360.09566650.09426290.09288220.09152410.09018800.08887370.0875806
1.600.08630840.08505680.08382520.08261340.08142110.08024770.07909310.07795680.07683850.0757379
1.700.07465470.07358860.07253920.07150630.07048960.06948880.06850350.06753360.06657880.0656387
1.800.06471320.06380200.06290480.06202140.06115160.06029510.05945160.05862110.05780320.0569977
1.900.05620450.05542320.05465380.05389600.05314960.05241450.05169040.05097710.05027450.0495824
2.000.04890060.04822900.04756730.04691550.04627330.04564070.04501730.04440320.04379810.0432019
2.100.04261440.04203560.04146520.04090320.04034930.03980360.03926570.03873570.03821330.0376986
2.200.03719120.03669120.03619840.03571270.03523400.03476220.03429710.03383870.03338680.0329414
2.300.03250240.03206960.03164290.03122230.03080770.03039900.02999610.02959880.02920720.0288210
2.400.02844040.02806500.02769500.02733010.02697040.02661570.02626590.02592100.02558100.0252457
2.500.02491500.02458900.02426740.02395040.02363770.02332940.02302530.02272540.02242960.0221380
2.600.02185030.02156660.02128680.02101090.02073870.02047020.02020540.01994430.01968670.0194326
2.700.01918200.01893480.01869090.01845040.01821310.01797900.01774810.01752040.01729570.0170740
2.800.01685540.01663970.01642690.01621690.01600980.01580550.01560390.01540500.01520870.0150151
2.900.01482410.01463560.01444970.01426620.01408520.01390660.01373030.01355640.01338490.0132155
3.000.01304850.01288360.01272090.01256040.01240200.01224570.01209150.01193920.01178900.0116408
3.100.01149450.01135020.01120770.01106710.01092830.01079140.01065620.01052290.01039120.0102613
3.200.01013310.01000650.00988160.00975840.00963670.00951660.00939810.00928110.00916560.0090516
3.300.00893910.00882810.00871850.00861030.00850350.00839810.00829400.00819130.00808990.0079899
3.400.00789110.00779350.00769730.00760220.00750840.00741580.00732440.00723410.00714500.0070571
3.500.00697020.00688450.00679990.00671630.00663380.00655240.00647200.00639260.00631430.0062369
3.600.00616050.00608510.00601060.00593710.00586450.00579290.00572210.00565230.00558330.0055152
3.700.00544790.00538150.00531600.00525120.00518730.00512420.00506190.00500030.00493960.0048796
3.800.00482030.00476180.00470410.00464700.00459070.00453510.00448020.00442590.00437240.0043195
3.900.00426720.00421570.00416470.00411440.00406480.00401570.00396730.00391940.00387220.0038255
4.000.00377940.00373390.00368900.00364460.00360080.00355750.00351480.00347250.00343080.0033896

Table 1.4.

Values of the exponential integral for 0.1 ≤ x ≤ 4.09.

1.8. Dimensionless quantities

Dimensional parameters do not provide a physical view of the parameter being measured but rather a general or universal description of these parameters. For example, a real time of 24 hours corresponds to a dimensionless time of approximately 300 hours in very low permeability formations or more than 107 in very permeable formations [3, 9, 21, 25, 28].

A set number of Ei values for 0.0001 ≤ x ≤ 25 with the aid of the algorithm given in Figure 1.8. Then, a fitting of these data was performed to obtain the polynomials given by Eqs. (1.85) and (1.90). The first one has a R2 of 1, and the second one has a R2 of 0.999999999 which implies accuracy up to the fifth digit can be obtained.

Ei(x)=a+bx+cx2.5+dlnx+eexp(x);x1E1.85
lnEi(x)=a+cx+ex21+bx+dx2+fx3;x>1E1.86

Adapted from [29] and generated with the Ei function code given in Figure 1.8.

Define dimensionless radius, dimensionless time, and dimensionless pressure as:

rD=r/rwE1.87
tD=ttoE1.88
PD=kh(PiP)141.2qμBE1.89

Adapted from [29] and generated with the Ei function code given in Figure 1.8.

For pressure drawdown tests, ΔP = PiPwf. For pressure buildup tests, ΔP = PwsPwft = 0).

This means that the steady‐state physical pressure drop for radial flow is equal to the dimensionless pressure multiplied by a scalable factor, which in this case depends on the flow and the properties of the reservoir, [3, 4, 5, 6, 7, 9, 21, 26, 30]. The same concept applies to transient flow and to more complex situations, but in this case, the dimensionless pressure is different. For example, for transient flow, the dimensionless pressure is always a function of dimensionless time.

Taking derivative to Eqs. (1.87) and (1.88),

r=rwrDE1.90
t=totDE1.91

Replacing the above derivatives into Eq. (1.20),

Adapted from [5] and generated with the Ei function code given in Figure 1.8.

2PrD2+1rDPrD=ϕμctrw2ktoPtDE1.92

Definition of to requires assuming ϕμctrw2kto = 1, [24], then;

to=ϕμctrw2kE1.93

Replacing this definition into Eq. (1.88) and solving for the dimensionless time (oilfield units),

tD=0.0002637ktϕμctrw2E1.94

Replacing Eq. (1.93) in Eq. (1.92) leads, after simplification, to:

2PrD2+1rDPrD=PtDE1.95

The dimensionless pressure is also affected by the system geometry, other well systems, storage coefficient, anisotropic characteristics of the reservoir, fractures, radial discontinuities, double porosity, among others. In general, the pressure at any point in a single well system that produces the constant rate, q, is given by [25]:

[PiP(r,t)]=qBμkhPD(tD,rD,CD,geometry,….)E1.96

Taking twice derivative to Eq. (1.87), excluding the conversion factor, will provide:

PD=khqBμPE1.97
2PD=khqBμ2PE1.98

Replacing Eqs. (1.97) and (1.98) in Eq. (1.95) and simplifying leads to:

2PDrD2+1rDPDrD=1rDrD(rDPDrD)=PDtDE1.99

If the characteristic length is the area, instead of wellbore radius, Eq. (1.92) can be expressed as:

tDA=0.0002637ktϕμctA=tD(rw2A)E1.100

Example 1.1

A square shaped reservoir produces 300 BPD through a well located in the center of one of its quadrants. See Figure 1.10. Estimate the pressure in the well after 1 month of production. Other relevant data:

Figure 1.10.

Geometry of the reservoir for example 1.1.

Pi = 3225 psia,            h = 42 ft

ko = 1 darcy,         ϕ = 25%

μo = 25 cp,           ct = 6.1 × 10−6/psia

Bo = 1.32 bbl/BF, rw = 6 in

A = 150 Acres,  q = 300 BPD

Solution

Assuming the system behaves infinitely, it means, during 1 month of production the transient wave has not yet reached the reservoir boundaries, the problem can be solved by estimating the Ei function. Replacing Eqs. (1.82) and (1.92) into the argument of Eq. (1.82), it results:

x=rD24tD=948ϕμctr2ktE1.101

Using Eq. (1.101) with the above given reservoir and well data:

x=948(0.25)(25)(6.1×106)(0.52)(1000)(720)=1.25×108

This x value allows finding Ei(−x) = 17.6163 using the function provided in Figure 1.8. From the application of Eq. (82), PD = 8.808. This dimensionless pressure is meaningless for practical purposes. Converting to oilfield units by means of Eq. (1.87), the well‐flowing pressure value after 1 month of production is given as:

8.808=(1000)(42)(141.2)(300)(1.32)(25)(3225Pwf)

Pwf = 2931.84 psia.

How it can be now if the example was correctly done? A good approximation consists of considering a small pressure drop; let us say ± 0.002 psia (smallest value that can be read from current pressure recorders) at the closest reservoir boundary. Use Eq. (1.87) to convert from psia to dimensionless pressure:

PD=(1000)(42)(141.2)(300)(1.32)(25)(0.002)=6.0091×105

Eq. (1.82) allows finding Ei(−x) = 0.00012. This value can be used to determine an x value from Table 1.2. However, a trial‐and‐error procedure with the function given in Figure 1.8 was performed to find an x value of 6.97. Then, the time at which this value takes place at the nearest reservoir boundary is found from Eq. (1.101). The nearest boundary is obtained from one‐fourth of the reservoir size area (3.7 Ac or 1663500 ft2). Then, for a square geometry system (the system may also be approached to a circle):

L=1663500=1278.09ft

The radial distance from the well to the nearest boundary corresponds to one half of the square side, the r = 639.04 ft. Solving for time from Eq. (1.101);

t=948ϕμctr2kx=948(0.25)(25)(6.1×106)(639.042)(1000)(6.97)=2.118h

This means that after 2 h and 7 min of flow, the wave has reached the nearest reservoir boundary; therefore, the infinite‐acting period no longer exists for this reservoir, then, a pseudosteady‐state solution ought to be applied (Figures 1.111.14). To do so, Eq. (1.98) is employed for the whole reservoir area:

tDA=(0.0002637)(1000)(720)(0.25)(25)(6.1×106)(6534000)=0.76

With this tDA value of 0.76, the normal procedure is to estimate the dimensionless pressure for a given reservoir‐well position configuration, which can be found in Figures C.13 through C.16 in [25] for which data were originally presented in [31]. These plots provide the pressure behavior for a well inside a rectangular/square no-flow system, without storage wellbore and skin factor; A0.5/rw = 2000 can also be found in [3, 9, 26]. This procedure is avoided in this textbook. Instead new set of data was generated and adjusted to the following polynomial fitting in which constants are reported in Table 1.5:

PD=a+b*tDA+c*tDA2+d*tDA0.5lntDA+etDA0.5E1.102

Using Eq. (1.102) will result:

PD=4.4765+9.3437(12)0.2798(122)2.751612ln(12)0.01609812

Table 1.5.

Constants for Eq. (1.102).

PD = 12.05597.

The well‐flowing pressure is estimated with Eq. (1.87); thus,

12.056=(1000)(42)(141.2)(300)(1.32)(25)(PiPwf)

Pwf = 2823.75 psia.

1.9. Application of the diffusivity equation solution

A straight‐line behavior can be observed in mostly the whole range on the right‐hand plot of Ei versus x plot given in Figure 1.9. Then, it was concluded, [3, 4, 5, 6, 7, 9, 11, 19, 21, 26, 30], when x < 0.0025, the more complex mathematical representation of Eq. (1.82) can be replaced by a straight line function, given by:

Ei(x)=ln(1.781x)E1.103

this leads to,

Ei(x)=lnx+0.5772E1.104

Replacing this new definition into Eq. (1.82) will result in:

PD=12[ln(rD24tD)+0.5772]E1.105

At the well rD = 1, after rearranging,

PD=12[lntD+0.80907]E1.106

The above indicates that the well pressure behavior obeys a semi‐logarithmic behavior of pressure versus time.

Example 1.2

A well and infinite reservoir has the following characteristics:

q = 2000 STB/D,   μ = 0.72 cp,   ct = 1.5 × 10−5 psia−1

ϕ = 23%,          Pi = 3000 psia,         h = 150 ft

B = 1.475 bbl/STB,     k = 10 md,       rw = 0.5 ft

Estimate the well‐flowing pressure at radii of 0.5, 1, 5, 10, 20, 50, 70, 100, 200, 500, 1000, 2000, 2500, 3000, and 4000 feet after 1 month of production. Plot the results.

Solution

For the wellbore radius, find x with Eq. (1.101);

x=948(0.23)(0.72)(1.5×105)(0.52)(10)(720)=8.177×108

Using the function given in Figure 1.9 or Eq. (1.103), a value of Ei(−x) of 15.7421 is found. Then, Eq. (1.82) indicates that PD = 7.871. Use of Eq. (1.87) allows estimating both pressure drop and well‐flowing pressure:

ΔP=PiPwf=141.2qμBkhPD=141.2(2000)(0.72)(1.475)(10)(150)7.871=1573.74 psia

The remaining results are summarized in Table 1.6 and plotted in Figure 1.11. From this, it can be inferred that the highest pressure drop takes place in the near‐wellbore region which mathematically agrees with the continuity equation stating that when the area is reduced, the velocity has to be increased so the flow rate can be constant. The higher the fluid velocity, the higher the pressure drops.

r, ftxEi(−x)P, psiaPwf, psia
0.58.18E−0815.74211537.741462.26
13.27E−0714.35581435.151564.85
58.18E−0611.1371113.361886.64
103.27E−049.75974.782025.22
201.31E−048.365836.22163.8
508.18E−046.533653.072346.93
701.60E−035.86585.872414.13
1003.27E−035.149514.722485.28
2001.31E−023.772377.112622.89
5008.17E−022.007200.6162799.384
10003.27E−010.842584.2252915.775
20001.31E+000.133713.3682986.632
25002.04E+000.0464.62995.4
30002.94E+000.0141.4012998.599
40005.23E+000.00090.0872999.913

Table 1.6.

Summarized results for example 1.2.

Figure 1.11.

Pressure versus distance plot for example 1.2.

Example 1.3

Re‐work example 1.2 to estimate the sand‐face pressure at time values starting from 0.01 to 1000 h. Show the results in both Cartesian and semilog plots. What does this suggest?

Solution

Find x with Eq. (1.101);

x=948(0.23)(0.72)(1.5×105)(0.52)(10)(0.01)=0.000948

A value of Ei(−x) of 6.385 is found with Eq. (1.103). Then, Eq. (1.82) gives a PD value of 3.192 and Eq. (1.87) leads to calculate a well‐flowing pressure of;

Pwf=Pi141.2qμBkhPD=3000141.2(2000)(0.72)(1.475)(10)(150)3.192=2361.71psia

The remaining well‐flowing pressure values against time are given in Table 1.7 and plotted in Figure 1.12. The semilog behavior goes in the upper part of the plot (solid line), and the Cartesian plot corresponds to the lower dashed line. The semilog line behaves linearly while the Cartesian curve does not. This situation perfectly agrees with Eq. (1.106), which ensures that the behavior of pressure drop versus time obeys a semilog trend. In other word, in a transient radial system, pressure drops is a linear function of the logarithm of time.

t, hxEi(−x)PDPwf, Psiat, hxEi(−x)PDPwf, psia
0.019.480E−046.3853.1922361.7161.580E−0612.7816.3901722.30
0.024.740E−047.0783.5392292.4671.354E−0612.9356.4681706.89
0.033.160E−047.4833.7412251.9481.185E−0613.0696.5341693.54
0.042.370E−047.7703.8852223.1991.053E−0613.1866.5931681.77
0.051.896E−047.9943.9972200.89109.480E−0713.2926.6461671.23
0.061.580E−048.1764.0882182.66204.740E−0713.9856.9921601.94
0.071.354E−048.3304.1652167.25303.160E−0714.3907.1951561.41
0.081.185E−048.4644.2322153.91402.370E−0714.6787.3391532.65
0.091.053E−048.5814.2912142.13501.896E−0714.9017.4511510.34
0.19.480E−058.6874.3432131.60601.580E−0715.0837.5421492.11
0.24.740E−059.3804.6902062.31701.354E−0715.2387.6191476.70
0.33.160E−059.7854.8932021.78801.185E−0715.3717.6861463.35
0.42.370E−0510.0735.0361993.02901.053E−0715.4897.7441451.58
0.51.896E−0510.2965.1481970.711009.480E−0815.5947.7971441.05
0.61.580E−0510.4785.2391952.492004.740E−0816.2878.1441371.75
0.71.354E−0510.6325.3161937.083003.160E−0816.6938.3461331.22
0.81.185E−0510.7665.3831923.734002.370E−0816.9818.4901302.46
0.91.053E−0510.8845.4421911.955001.896E−0817.2048.6021280.15
19.480E−0610.9895.4951901.426001.580E−0817.3868.6931261.92
24.740E−0611.6825.8411832.137001.354E−0817.5408.7701246.51
33.160E−0612.0886.0441791.598001.185E−0817.6748.8371233.17
42.370E−0612.3756.1881762.849001.053E−0817.7928.8961221.39
51.896E−0612.5996.2991740.5310009.480E−0917.8978.9481210.86

Table 1.7.

Summarized results for example 1.3.

Figure 1.12.

Pressure versus time plot for example 1.3.

1.10. Pressure distribution and skin factor

Once the dimensionless parameters are plugged in Eq. (1.82), this yields:

P(r,t)=Pi70.6qBμkhEi{948ϕμctr2kt}E1.107

At point N, Figure 1.13, the pressure can be calculated by Eq. (1.107). At the wellbore rD = r/rw = 1, then, r = rw and P(r,t) = Pwf. Note that application of the line‐source solution requires the reservoir to possess an infinite extent, [3, 9, 18, 21, 25, 26].

Figure 1.13.

Pressure distribution in the reservoir.

There are several ways to quantify damage or stimulation in an operating well (producer or injector). These conditions are schematically represented in Figure  1.14. The most popular method is to represent a well condition by a steady‐state pressure drop occurring at the wellbore, in addition to the transient pressure drop normally occurring in the reservoir. This additional pressure drop is called “skin pressure drop” and takes place in an infinitesimally thin zone: “damage zone,” [4, 5, 9, 11, 19, 30]. It can be caused by several factors:

Figure 1.14.

Skin factor influence.

  1. Invasion of drilling fluids

  2. Partial well penetration

  3. Partial completion

  4. Blocking of perforations

  5. Organic/inorganic precipitation

  6. Inadequate drilling density or limited drilling

  7. Bacterial growth

  8. Dispersion of clays

  9. Presence of cake and cement

  10. Presence of high gas saturation around the well

Skin factor is a dimensionless parameter; then, it has to be added to the dimensionless pressure in Eq. (1.87), so that:

PiPwf=141.2qμBkh(PD+s)E1.108

From the above expression can be easily obtained:

PiPwf=141.2qμBkhPD+141.2qμBkhsE1.109

Therefore, the skin factor pressure drop is given by:

ΔPs=141.2qμBkhsE1.110

Assuming steady state near the wellbore and the damage area has a finite radius, rs, with an altered permeability, ks, the pressure drop due to the damage is expressed as the pressure difference between the virgin zone and the altered zone, that is to say:

ΔPs=141.2qμBkshlnrsrw141.2qμBkhlnrsrwE1.111

Rearranging;

ΔPs=141.2qμBkh(kks1)lnrsrwE1.112

Comparing Eqs (1.112) and (1.107), the following can be concluded:

s=(kks1)lnrsrwE1.113

rs and ks are not easy to be obtained.

Equation (1.82) and (1.106) can be respectively written as:

PD+s=12Ei(x)E1.114
PD+s=12[lntD+0.80907]E1.115

Replacing the dimensionless quantities given by Eqs. (1.87) and (1.95) in Eq. (1.115) will result:

Pi=Pwf+70.6qμBkh[ln(0.0002637ktϕμctrw2)+0.80908+2s]E1.116

Taking natural logarithm to 0.0002637 and adding its result to 0.80908 results in:

Pi=Pwf+70.6qμBkh[7.4316+ln(ktϕμctrw2)+2s]E1.117

Multiplying and dividing by the natural logarithm of 10 and solving for the well‐flowing pressure:

Pwf=Pi162.6qμBkh[log(ktϕμctrw2)3.2275+0.8686s]E1.118

Thus, a straight line is expected to develop from a semilog plot of pressure against the time, as seen on the upper curve of Figure 1.12.

1.11. Finite reservoirs

In closed systems, the radial flow is followed by a transition period. This in turn is followed by the pseudosteady, semi‐stable, or quasi‐stable state, which is a transient flow regime where the pressures change over time, dP/dt, is constant at all points of the reservoir:

dPdt=qcVpE1.119

Eq. (1.99) is now subjected to the following initial and boundary conditions:

PD(rD,tD=0)=0E1.120
(PDrD)reD=0E1.121
(PDrD)rD=1=1E1.122

Which solution is [9, 30]:

PD(rD,tD)=2(reD21)(rD24+tD)reD2lnrD(r2eD1)(3reD44reD4lnreD2reD21)4(reD21)2+πn=1{ean2tDJ12(anreD)[J1(an)Y0(anrD)Y1(an)(J0)(anrD)]an[J12(anreD)J12(an)]}E1.123

The pseudosteady‐state period takes place at late times (t > 948ϕμctre2/k), so that as time tends to infinity, summation tends to zero, then:

PD(rD,tD)=2(reD21)(rD24+tD)reD2lnrD(reD21)(3reD44reD4lnreD2reD21)4(reD21)2E1.124

At the well, rD = 1 and as reD >>>> 1, the above expression is reduced to:

PD(tD)=2reD2+2tDreD234+lnreD+12reD214reD4E1.125

This can be approximated to:

PD(tD)2tDreD2+lnreD34E1.126

Invoking Eq. (1.98) for a circular reservoir area,

tDA=tDrw2πre2=tDπreD2E1.127

It follows that;

πtDA=tDreD2E1.128

The final solution to the pseudosteady‐state diffusivity equation is obtained from using the definition given by Eq. (1.128) in Eq. (1.129):

PD(tD)=2πtDA+lnreD34E1.129

The derivative with respect to time of the above equation in dimensional form allows obtaining the pore volume:

dP(r,t)dt=1.79qBhϕctre2E1.130

An important feature of this period is that the rate of change of pressure with respect to time is a constant, that is, dPD/dtDA = 2π.

When the reservoir pressure does not change over time at any point, the flow is said to be stable. In other words, the right side of Eq. (1.99) is zero, [3]:

1rDrD(rDPDrD)=0E1.131

Similar to the pseudosteady‐state case, steady state takes place at late times. Now, its initial, external, and internal boundary conditions are given by:

PD(rD,tD=0)=0E1.132
PD(rDe,0)=0E1.133
(PDrD)rD=1=1E1.134

The solution to the steady‐state diffusivity equation is [3]:

PD(rD,tD)=lnreD2n=1{eβn2tDnJ02(βnreD)βn2[J12(βn)J02(βnreD)]}E1.135

As time tends to infinity, the summation tends to infinity, then:

(PD)ssr=lnreD=lnrerwE1.136

In dimensional terms, the above expression is reduced to Darcy’s equation. The dimensionless pressure function for linear flow is given by:

(PD)ssL=2πLhAE1.137

Steady state can occur in reservoirs only when the reservoir is fully recharged by an aquifer or when injection and production are balanced. However, a reservoir with a very active aquifer will not always act under steady‐state conditions. First, there has to be a period of unsteady state, which will be followed by the steady state once the pressure drop has reached the reservoir boundaries. Extraction of fluids from a pressurized reservoir with compressible fluids causes a pressure disturbance which travels throughout the reservoir. Although such disturbance is expected to travel at the speed of sound, it is rapidly attenuated so that for a given duration of production time, there is a distance, the drainage radius, beyond which no substantial changes in pressure will be observed. As more fluid is withdrawn (or injected), the disturbance moves further into the reservoir with continuous pressure decline at all points that have experienced pressure decline. Once a reservoir boundary is found, the pressure on the boundary continues to decline but at a faster rate than when the boundary was not detected. On the other hand, if the pressure transient reaches an open boundary (water influx), the pressure remains constant at some point; the pressure closest to the well will decline more slowly than if a closed boundary were found. Flow changes or the addition of new wells cause additional pressure drops that affect both the pressure decline and the pressure distribution. Each well will establish its own drainage area that supplies fluid. When a flow boundary is found, the pressure gradient—not the pressure level—tends to stabilize after sufficiently long production time. For the closed boundary case, the pressure reaches the pseudosteady state with a constant pressure gradient and general pressure drop everywhere, which is linear over time. For constant‐pressure boundaries, steady state is obtained; both the pressure and its gradient remain constant over time.

1.12. The pressure derivative function

Pressure derivative has been one of the most valuable tools ever introduced to the pressure transient analysis field. In fact, [32] affirms that pressure derivative and deconvolution have been the best elements added for well test interpretation. However, here it is affirmed that besides these two “blessings,” TDS technique, [1, 2], is the best and practical well test interpretation method in which application will be very devoted along this textbook. Actually, in the following chapters, TDS is extended for long, homogeneous reservoirs, [33], interference testing [34], drainage area determination in constant‐pressure reservoirs, [35], and recent applications on fractured vertical wells, [36], among others. More complex scenarios, for instance finite‐conductivity faults, [37], are treated extensively in [38].

Attempts to introduce the pressure derivative are not really new. Some of them try to even apply the derivative concept to material balance. Just to name a few of them, [39] in 1961, tried to approach the rate of pressure change with time for detection of reservoir boundaries. Later, in 1965, [40] presented drawdown curves of well pressure change with time for wells near intersecting faults (36 and 90°). These applications, however, use numerical estimations of the pressure rate change on the field data regardless of two aspects: (1) an understanding of the theoretical situation behind a given system and (2) noise in the pressure data.

Between 1975 and 1976, Tiab’s contributions on the pressure derivative were remarkable. Actually, he is the father of the pressure derivative concept as used nowadays. Refs. [41, 42] include detailed derivation and application of the pressure derivative function. These results are further summarized on [41, 42, 43, 44, 45]. Ref. [46] applied Tiab’s finding to provide a type‐curve matching technique using the natural logarithm pressure derivative.

It was required to obtain the pressure derivative from a continuous function, instead of attempting to work on discrete data in order to understand the pressure derivative behavior in an infinite system. Then, Tiab decided to apply the Leibnitz’s rule of derivation of an integral to the Ei function.

xf(x)h(x)g(u)du{g[h(x)][h(x)]xg[f(x)][f(x)]x}E1.138

Applying Leibnitz’s rule to the Ei function in Eq. (1.81) to differentiate with respect to tD (see Appendix B in [42]),

ΔΔtD[Ei(rD24tD)]=rD24tDeuuΔu=euuΔuΔtD|rD24tDE1.139

Taking the derivative ΔutD and replacing u by rD2/4tD,

ΔΔtD[Ei(rD24tD)]=e(rD2/4tD)rD2/4tD(rD24tD2)E1.140

After simplification,

ΔΔtD[Ei(rD24tD)]=1tDe(rD2/4tD)E1.141

From inspection of Eq. (1.81) results:

PDtD=121tDerD24tDE1.142

In oilfield units,

ΔPwfΔt=70.6qμBkhte(948ϕμctkt)E1.143

At the well, rD = 1, then, Eq. (1.142) becomes:

PD=12tDe14tDE1.144

For tD > 250, e−1/4tD = 1; then, Eq. (1.144) reduces to

PD=12tDE1.145

The derivative of equation (1.145) is better known as the Cartesian derivative. The natural logarithmic derivative is obtained from:

tD*PD=tDPDtD=tDPD(lntD)/tD=PDlntDE1.146

Later on, [46] use the natural logarithmic derivative to develop a type‐curve matching technique.

Appendix C in [42] also provides the derivation of the second pressure derivative:

PD=PD1tD(rD24tD1)E1.147

Conversion of Eq. (1.145) to natural logarithmic derivative requires multiplying both sides of it by tD; then, it results:

tD*PD=12E1.148

Eq. (1.148) suggests that a log‐log plot of dimensionless pressure derivative against dimensionless time provides a straight line with zero slope and intercept of ½. Taking logarithm to both sides of Eq. (1.145) leads to:

logPD=logtD0.301E1.149
Pwf=Pwft=1t(70.6qμBkh)E1.150

The above expression corresponds of a straight line with negative unit slope. In dimensional form:

Taking logarithm to both sides of the above expression:

logPwf=logt+log(70.6qμBkh)E1.151

As shown in Figure 1.15, Eq. (1.151) corresponds to a straight line with negative unit slope and intercept of:

Figure 1.15.

Log‐log plot of Pwf′ against t.

P1hr=70.6qμBkhE1.152

Eq. (1.152) is applied to find permeability from the intersect plot of the Cartesian pressure derivative versus time plot. This type of plot is also useful to detect the presence of a linear boundary (fault) since the negative unit slope line displaces when the fault is felt as depicted in Figure 1.16.

Figure 1.16.

Fault identification by means of a log‐log plot of PD′ vs. tD.

The noise that occurs in a pressure test is due to such factors as (1) turbulence, (2) tool movements, (3) temperature variations, (4) opening and closing wells in the field, and (5) gravitational effects of the sun and moon on the tides (near the great lakes the noise is about 0.15 psia and offshore up to 1 psia).

The estimation of the pressure derivative with respect to time to actual data, of course, must be performed numerically since data recorded from wells are always discrete. During the derivative calculation, the noise is increased by the rate of change that the derivative imposes, so it is necessary to soften the derivative or to use smoothing techniques. The low resolution of the tool and the log‐log paper also increase or exaggerate the noise. Therefore, calculating the derivative of pressure requires some care because the process of data differentiation can amplify any noise that may be present. Numerical differentiation using adjacent points will produce a very noisy derivative, [8, 47, 48].

Ref. [8] conducted a comparative study of several algorithms for estimation of the pressure derivative. They obtained synthetic pressure derivatives for seven different reservoir and well configuration scenarios and, then, estimated the pressure derivative using several comparative methods. They found that the Spline algorithm (not presented here) is the best procedure to derive pressure versus time data since it produces minimal average errors. It is the only algorithm of polynomial character that to be continuous can be smoothed during any derivation process and the form of the curve obtained is in agreement with the worked model. The Horne and Bourdet algorithms when the smoothing window is of either 0.2 or 0.4 are good options for derivation processes. Ref. [8] also found the best procedure for data analysis of pressure against time is to differentiate and then smooth the data.

By itself, the central finite difference formula fails to provide good derivative computation. Instead, some modifications are introduced by [18, 20, 46], respectively:

Horne equation [32]:

t(Pt)i=t(Plnt)i={ln(ti/tik)ΔPi+jln(ti+j/ti)ln(ti+j/tik)+ln(ti+jtik/ti2)ΔPiln(ti+1/ti)ln(ti/ti1)ln(ti+j/ti)ΔPi1ln(ti/tik)ln(ti+j/tik)}E1.153

lnti+jlnti0.2 and lntilntik0.2

When the data are distributed in a geometrical progression (with the time difference from one point to the next much larger as the test passes), then the noise in the derivative can be reduced using a numerical differentiation with respect to the logarithm of time. The best method to reduce noise is to use data that is separated by at least 0.2 logarithmic cycles, rather than points that are immediately adjacent. This procedure is recognized as smoothing and is best explained in Figure 1.17.

Equation of Bourdet et al. [46]:

Figure 1.17.

Smoothing diagram.

(dPdx)i=PiPi1XiXi1(Xi+1Xi)+Pi+1PiXi+1Xi(XiX1i)Xi+1Xi1E1.154

Let X is the natural logarithm of the time function.

This differentiation algorithm reproduces the test type curve over the entire time interval. It uses a point before and a point after the point of interest, i, to calculate the corresponding derivative and places its weighted mean for the objective point. Smoothing can also be applied.

1.13. The principle of superposition

This principle is not new. It was first introduced to the petroleum literature by van Everdingen and Hurst in 1949, [49]. However, its application is too important and many field engineers fail or neglect to use it. Superposition is too useful for systems having one well producing at variable rate or the case when more than one well produces at different flow rates.

As quoted from [25], the superposition principle is defined by:

Adding solutions to the linear differential equation will result in a new solution of that differential equation but for different boundary conditions,” which mathematically translates to:

ψ=ψ1f1+ψ2f2+ψ3f3E1.155

where ψ is the general solution and ψ1 f1, ψ2 f2 and ψ3 f3… are the particular solutions.

1.13.1 Space superposition

If the wells produce at a constant flow rate, the pressure drop at point N, Figure 1.18, will be [3, 9, 19, 21, 25]:

Figure 1.18.

Pressure at the point N.

ΔPN=ΔPN,1+ΔPN,2+ΔPN,3E1.156

If reservoir and fluid properties are considered constant, then, Eq. (1.87) can be applied to the above expression, so that:

ΔPN=141.2μkh[(qBo)1PD(rD1,tD)+(qBo)2PD(rD2,tD)+(qBo)3PD(rD3,tD)]E1.157

The dimensionless radii are defined by:

rDn=rnrw;n=1,2,3E1.158

Extended to n number of wells:

ΔPN=i=1n141.2qμBkh[PD(rDi,tD)]E1.159

If point N is an active well, its contribution to the total pressure drop plus the skin factor pressure drop, Eq. (1.108), must be included in Eq. (1.159), then,

ΔPN=i=1n141.2qμBkh[PD(rDNi,tD)]+141.2qμBkhs|NE1.160

Notice that in Eqs. (1.159) and (1.160), changes of pressures or dimensionless pressures are added. If the point of interest is a well in operation, the damage factor should be added to the dimensionless pressure of that well only.

1.13.2 Time superposition

Sometimes there are changes in flow rate when a well produces as referred in Figures 1.19 and 1.22. Then, the superposition concept must be applied. To do this, [25], a single well is visualized as if there were two wells at the same point, one with a production rate of q1 during a time period from t = 0 to t and another imaginary well with a production rate of q2q1 for a time frame between t1 and tt1. The total rate after time t1 is q1+ (q2q1) = q2. The change in well pressure due to the rate change [19, 25] is,

Figure 1.19.

Time superposition.

ΔP=141.2μBkh[q1PD(rD,tD1)+(q2q1)PD(rD,tD2+s)]E1.161

where tD2 = (tt1)D. If there are more variations in flow rate,

ΔP=141.2μkhi=1n[(qB)i(qB)i1]{PD(rD,(tti)D+s}E1.162

Example 1.4

This example is taken [25]. The below data and the schematic given in Figure 1.20 correspond to two wells in production:

Figure 1.20.

Flow rate changes for example 1.4.

k = 76 md,    ϕ = 20 %,  B = 1.08 bbl/STB

Pi = 2200 psia, μ = 1 cp,  ct = 10 × 10−6/psia

h = 20 ft

Calculate the pressure in (a) well 1 after 7 h of production and (b) in well 2 after 11 h of production. Assume infinite behavior.

Solution

Part (a):

ΔP(7 hr)= ΔP caused by production from well 1 to well 1 + ΔP caused by production from well 2 to well 1. Mathematically,

ΔP7hr@well1=141.2μq1Bkh[PD(rD1,tD)+s]+141.2μq2Bkh(PD(rD2,tD))

Using Eq. (1.101) for the well,

x=948ϕμctr2kt=948(0.2)(1)(1×105)12(76)(7)=3.56×106

Since x <<<< 0.0025, it implies the use of Eq. (1.82) with Eq. (1.103); then,

PD(rD,tD)=12ln(1.781x)E1.163
PD=12|[ln(1.781*3.56×106)]|=5.98

In well 2, x = 0.03564 from Eq. (1.101). Interpolating this value in Table 1.2, Ei(−x) = 2.7924; then, PD ≅ 1.4. Estimating ΔP in well 1 will result:

ΔP7hr,rD=1=141.2(100)(1.08)(1)(76)(20)(5.98+5)+141.2(100)(1.08)(1)(76)(20)(1.4)=113.7

Pwf @ well1 = 2200−113.7 = 2086.4 psia (notice that skin factor was only applied to well 1)

Part (b);

At 11 h, it is desired to estimate the pressure in well 2. Two flow rates should be considered for in each well. Then, the use of Eq. (1.162) will provide:

ΔP(11hr,well2)=ΔPwell1Well2,t=11hr,q=100BPD,rD=100+ΔPwell1Well2,t=(1110)hr,q=(50‐100)BPD,rD=100ΔPwell1Well1,t=11hr,q=25BPD,rD=1,s2+ΔPwell1Well1,t=(118)hr,q=(100‐25)BPD,rD=1,s2

Using Eq. (1.101), the four respective values of x are: x =0.02268, 0.2494, 2.268 × 10−6, and 8.316 × 10−6. Estimation of Ei requires the use of Table 1.2 for the first two values and use of Eq. (1.103) for the last two values. The four values of Ei(−x) are: 0.0227, 0.811, 12.42, and 11.12. Therefore, the respective values of PD are 1.605, 0.405, 6.209, and 5.56. The total pressure drop is found with Eq. (1.161) as follows:

ΔPwell2,11hr)=141.2(1)(1.08)(76)(20){(100)(1.605)+(50100)(0.405)+(25)(6.209+1.7)+(10025)(5.56+1.7)}=87.75psia

Pwf @ well2 = 2200 − 87.75 = 2112.25 psia

1.13.3 Space superposition—method of images

The method of images applies to deal with either no‐flow or constant‐pressure boundaries. If a well operates at a constant flow rate at a distance, d, from an impermeable barrier (fault), the systems acts as if there were two wells separated 2d from each other [3, 25]. For no‐flow boundaries, the image well corresponds to the same operating well. For constant‐pressure boundary, the resulting image corresponds to an opposite operating well. In other words, if the well is a producer near a fault, the image well corresponds to an injector well. These two situations are sketched in Figure 1.21. For the no‐flow boundary, upper system in Figure 1.21, the dimensionless pressure can be expressed as:

PDatrealwell=PDatrealwell,rD=1,s+PDatimagewellrealwell,rD= 2d/rwE1.164

Figure 1.21.

Well near a linear barrier.

For the constant‐pressure boundary, lower part in Figure 1.21, the dimensionless pressure can be expressed as:

PDatrealwell=PDatrealwell,rD=1,sPDatimagewellrealwell,rD= 2d/rwE1.165

The negative sign in Eq. (1.165) is because of dealing with an imaginary injector well.

For the case of two intersecting faults, the total number of wells depends on the value of the angle formed by the two faults, thus:

nwells=360θE1.166

The image method is limited to one well per quadrant. If this situation fails to be fulfilled, then, the method cannot be applied. In the system of Figure 1.22, an angle of 90° is formed from the intersecting faults. According to Eq. (1.166), nwells = 360/90 = 4 wells, as shown there. The ratio of the distances from the well to each fault is given by:

yD=by/bxE1.167

Figure 1.22.

Well between two intersecting faults.

The practical way to apply space superposition for generating the well system resulting from two intersecting faults consist of extending the length of the faults and setting as many divisions as suggested by Equation (1.166); that is, for example, 1.5, Figure 1.23 left, six well spaces are obtained. Then, draw a circle with center at the fault intersection and radius at well position. This guarantees that the total length corresponds to the double length value from the well to the fault. Draw from the well a line to be perpendicular to the nearest fault and keep drawing the line until the circle line has been reached. See Figure 1.24 left. Set the well. A sealing fault provides the same type of well as the source well, that is, a producing well generates another producing well to the other side of the fault. A constant‐pressure boundary provides the opposite well type of the source well, that is, a producing well generates an injector well on the other side of the line. Draw a new line from the just drawn imaginary well normal to the fault and keep drawing the line until the line circle is reached. See Figure 1.24 right. Repeat the procedure until the complete well set system has been drawn.

Figure 1.23.

Location of well A and resulting well number system for example 1.5.

Figure 1.24.

Generating the well system for two intersecting faults.

For more than six well spaces generated, that is angles greater than 60°, as the case of example 1.5, when a fault intersects a constant‐pressure boundary injector and producer imaginary wells ought to be generated. What type of line should be drawn? A solid line representing a sealing fault, or a dash line, representing a constant‐pressure boundary? The answer is any of both. The lines should be drawn alternatively and as long as the system closes correctly, superposition works well.

Example 1.5

Well A in Figure 1.23 has produced a constant rate of 380 BPD. It is desired (a) to estimate the well‐flowing pressure after one week of production. The properties of the reservoir, well and fluid are given as follows:

Pi = 2500 psia,  B = 1.3 bbl/STB,   μ = 0.87 cp

h = 40 ft,      ct = 15×10−6/psia,    ϕ = 18 %

rw = 6 in,       k = 220 md, s = −5

(b) What would be the well‐flowing pressure after a week of production if the well were in an infinite reservoir?

Solution

Part (a)

The pressure drop in well A is affected by its own pressure drop and pressure drop caused by its well images. The distance from well A to its imaginary wells is shown in Figure 1.23 (right‐hand side). The total pressure drop for well A is:

ΔPA=ΔPA,r=rw+ΔPimage1wellA,r=500ft+ΔPimage2wellA,r=866ft+ΔPimage3wellA,r=1000ft+ΔPimage4wellA,r=866ft+ΔPimage5wellA,r=500ft

By symmetry, the above expression becomes:

ΔPA=ΔPA,r=rw+2ΔPimage1wellA,r=500ft+2ΔPimage2wellA,r=866ft+ΔPimage3wellA,r=1000ft

Using Eq. (1.101) for the well:

948(0.18)(0.87)(1.5×105)(0.5)2(220)(168)=1.5×108

Since x <<<< 0.0025, Eq. (1.163) applies:

PD(rD,tD)=12|ln[1.781(1.5×108)]|=8.72

Estimation for the image wells are given below. In all cases, x > 0.0025, then, Table 1.2 is used to find Ei and the resulting below divided by 2 for the estimation of PD,

ximagewell1or5=948(0.18)(0.87)(1.5×105)(500)2(220)(168)=0.015,PD=1.816
ximagewell2or4=948(0.18)(0.87)(1.5×105)(8662)(220)(168)=0.0452,PD=1.282
ximagewell3=948(0.18)(0.87)(1.5×105)(10002)(220)(168)=0.06,PD=1.145

Then, the pressure drop in A will be:

ΔPA=141.2qμBkh[(PDA,r=rw+s)+2PDimage1,r=500ft+2PDimage2,r=866ft+PDimage3,r=1000ft]
ΔPA=141.2(380)(0.87)(1.3)(220)(40)[(8.725)+2(1.816)+2(1.282)+1.145]=76.3psia

Pwf @ well A = 2500 − 76.3 = 2423.7 psia

Part (b)

If the well were located inside an infinite reservoir, the pressure drop would not include imaginary wells, then:

ΔPA=141.2qμBkh[PD,r=rw+s]
ΔPA=141.2(380)(0.87)(1.3)(220)(40)[8.725]=25.63psia

The well‐flowing pressure would be (2500 − 25.3) = 2474.4 psia. It was observed that the no‐flow boundaries contribute with 66.4% of total pressure drop in well A.

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Nomenclature

Aarea, ft2 or Ac
Bggas volume factor, ft3/STB
Booil volume factor, bbl/STB
Bwoil volume factor, bbl/STB
bxdistance from closer lateral boundary to well along the x‐direction, ft
bydistance from closer lateral boundary to well along the y‐direction, ft
ccompressibility, 1/psia
cfpore volume compressibility, 1/psia
cttotal or system compressibility, 1/psia
ddistance from a well to a fault, ft
fa given function
hformation thickness, ft
kpermeability, md
kspermeability in the damage zone, md
krfphase relative permeability, f = oil, water or gas
Lreservoir length, ft
mslope
m(P)pseudopressure function, psia2/cp
Mgas molecular weight, lb/lbmol
Ppressure
dP/drpressure gradient, psia/ft
PD′dimensionless pressure derivative
PD″dimensionless second pressure derivative
PDdimensionless pressure
Piinitial reservoir pressure, psia
Pwfwell flowing pressure, psia
qflow rate, bbl/D. For gas reservoirs the units are Mscf/D
Rsgas dissolved in crude oil, SCF/STB
Rswgas dissolved in crude water, SCF/STB
rDdimensionless radius
rDedimensionless drainage radius = re/rw
rradial distance, radius, ft
redrainage radius, ft
rsradius of the damage zone, ft
rwwell radius, ft
Sffluid saturation, f = oil, gas or water
sskin factor
Treservoir temperature, ºR
ttime, h
tapseudotime, psia h/cp
todummy time variable
urradial flow velocity, ft/h
tDdimensionless time based on well radius
tDAdimensionless time based on reservoir area
tD*PD′logarithmic pressure derivative
Vvolume, ft3
zvertical direction of the cylindrical coordinate, real gas constant
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Greek

Δchange, drop
Δtshut‐in time, h
ϕporosity, fraction. Spherical coordinate
λphase mobility, md/cp
ηhydraulic diffusivity constant, md‐cp/psia
ρdensity, lbm/ft3
θcylindrical coordinate
μviscosity, cp
ζtime function
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Suffices

1 hrreading at time of 1 h
Ddimensionless
DAdimensionless with respect to area
fformation
ggas
iinitial conditions
ooil, based condition
wwell, water
ppore
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Pressure Drawdown Testing

As can be seen in Figure 1.4, well pressure test analysis (PTA) considers this as the most basic and simple test, which does not mean that it is not important. In these tests, bottom‐hole well‐flowing pressure, Pwf, is continuously recorded keeping the flow constant. These tests are also referred as flow tests. Similar to an injection test, these tests require either production/injection from/into the well.

These tests are performed with the objective of (a) obtaining pore volume of the reservoir and (b) determining heterogeneities (in the drainage area). In fact, what is obtained is (a) transmissibility and (b) porous volume by total compressibility. In fact, a recent study by Agarwal [1] allows using drawdown tests to estimate the average permeability in the well drainage area. To run a pressure decline test, the following steps are generally followed:

  • The well is shut‐in for a long enough time to achieve stabilization throughout the reservoir, if this is not achieved, multirate testing is probably required;

  • The recording pressure tool is lowered to a level immediately above the perforations. This is to reduce Joule‐Thompson effects. It is important to have at least two pressure sensors for data quality control purposes;

  • The well opens in production at constant flow and in the meantime the well‐flowing pressure is continuously recorded.

Ideally, the well is closed until the static reservoir pressure. The duration of a drawdown test may last for a few hours or several days, depending upon the test objectives and reservoir characteristics. There are extensive pressure drawdown tests or reservoir limit tests (RLT) that run to delimit the reservoir or estimate the well drainage volume. Other objectives are the determination of: well‐drainage area permeability, skin factor, wellbore storage coefficient (WBS), porosity, reservoir geometry, and size of an adjacent aquifer.

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2.1. Wellbore storage coefficient

It is the continuous flow of the formation to the well after the well has been shut‐in for stabilization. It is also called after‐flow, postproduction, postinjection, loading, or unloading (for flow tests). The flow occurs by the expansion of fluids in the wellbore. In pressure buildup tests, after‐flow occurs. Figure 2.1 illustrates the above [2].

Traditional pressure tests had to be long enough to cope with both wellbore storage and skin effects so that a straight line could be obtained indicating the radial flow behavior. Even this approach has disadvantages since more than one apparent line can appear and analysts have problems deciding which line to use. In addition, the scale of the graph may show certain pressure responses as straight lines when in fact they are curves. To overcome these issues, analysts developed the method the type‐curve matching method.

There is flow in the wellbore face after shutting‐in the well in surface. Wellbore storage affects the behavior of the pressure transient at early times. Mathematically, the storage coefficient is defined as the total volume of well fluids per unit change in bottom‐hole pressure, or as the capacity of the well to discharge or load fluids per unit change in background pressure:

C=ΔVΔPE2.1

As commented by Earlougher [2], wellbore storage causes the flow rate at the face of the well to change more slowly than the surface flow rate. Figure 2.2 schematizes the relation qsf/q when the surface rate is changed from 0 to q, when C = 0, qsf/q = 1, while for C > 0, the relation qsf/q gradually changes from 0 to 1. The greater the value of C, the greater the transition is. As the storage effects become less severe, the formation begins to influence more and more the bottom‐hole pressure until the infinite behavior is fully developed. Pressure data that are influenced by wellbore storage can be used for interpretation purposes since fluids unload or load has certain dependence on reservoir transmissibility; however, this analysis is risky and tedious. TDS technique, presented later in this chapter, can provide a better solution to this problem.

Typically, the flow rate is surface‐controlled (unless there is a bottom shut‐in tool), the fluids in the well do not allow an immediate transmission of the disturbance from the subsurface to the surface, resulting in uneven surface and wellbore face flow [2, 3, 4, 5, 6, 7]. Wellbore storage can change during a pressure test in both injector and producer wells. Various circumstances cause changes in storage, such as phase redistribution and increase or decrease in storage associated with pressure tests in injector wells. In injector wells, once the well is closed, the surface pressure is high but could decrease to atmospheric pressure and go to vacuum if the static pressure is lower than the hydrostatic pressure. This causes an increase in storage (up to 100 times) of an incompressible system to one in a system where the liquid level drops [2]. The inverse situation occurs in injector wells with a high level of increase of liquid storage level and in producing wells with a high gas‐oil ratio or by redissolution of the free gas. Both for increase or decrease of storage, the second storage coefficient determines the beginning of the semilogarithmic straight line.

When the relationship between ΔV and ΔP does not change during the test, the wellbore storage coefficient is constant and can be estimated from completion data [2, 3, 4].

C=(144ρ)VuE2.2

where Vu is the wellbore volume/unit length, bbl/ft, r is the density of the fluid in the wellbore, lbm/ft3, and C is the wellbore storage coefficient, bbl/psia.

Figure 2.1.

Effects of wellbore storage on buildup and drawdown tests, taken from [2].

Figure 2.2.

Effect of storage on the flow rate at the face of the well, C3>C2>C1, taken from [2].

For injector wells or wells completely filled with fluids:

C=cwbVwbE2.3

where Cwb is the wellbore fluid compressibility = 1/Pwb, Vwb is the total wellbore volume, and Vu can be estimated with internal casing, IDcsg, and external tubing, ODtbg, diameters.

Vu=0.0009714(IDcsg2ODtbg2)E2.4

When opening a well, see Figure 2.3, the oil production will be given by the fluid that is stored in the well, qsf = 0. As time goes by, qsf tends to q and storage is neglected and the amount of liquid in the wellbore will be constant. The net accumulation volume will be (assuming constant B) [3, 5]:

Vwb=Awb(Z)E2.5

Figure 2.3.

Schematic representation of wellbore storage, taken from [3].

The flow rate is given by:

dVwbdt=AwbdZdtE2.6

The rate of volume change depends upon the difference between the subsurface and surface rates:

dVwbdt=(qsfq)B=245.615AwbdZdtE2.7

Since (assuming g/gc = 1):

PwPt=ρZ144E2.8

Taking the derivative to Eq. (2.8),

ddt(PwPt)=ρ144dZdtE2.9

Combining Eqs. (2.7) and (2.9) will result:

(qsfq)B=24(144)5.615Awbρd(PwPt)dtE2.10

Define

C=1445.615AwbρE2.11

Assuming constant, Pt, replacing the definition given by Eq. (2.11) and solving for the wellbore face flow rate, qsf, leads to:

qsf=q+24CBdPwdtE2.12

Taking derivative to Eqs. (1.89) and (1.94) with respect to time and taking the ratio of these will yield:

dPwdt=(0.0373qBϕhctrw2)dPDdtDE2.13

Combining Eqs. (2.12) and (2.13);

qsf=q0.894qCϕcthrw2dPwDdtDE2.14

Defining the dimensionless wellbore storage coefficient;

CD=0.894Cϕcthrw2E2.15

Rewriting Eq. (2.14);

qsfq=1CDdPwDdtDE2.16

The main advantage of using downhole shut‐in devices is the minimization of wellbore storage effects and after‐flow duration.

Rhagavan [5] presents the solution for the radial flow diffusivity equation considering wellbore storage and skin effects in both Laplace and real domains, respectively:

P¯D=K0(u)+suK1(u)u{uK1(u)+CDu[K0(u)+suK1(u)]}E2.17
PD=4π201ex2tDx3{[vCDJ0(x)f(x)J1(x)]2+[xCDY0(x)f(x)Y1(x)]2}dvE2.18

where f(x) = 1−CD(s) x2, and K0, K1, J0, J1,Y0, and Y1 are Bessel functions.

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2.2. Well test interpretation methods

There exist four methods for well test interpretation as follows: (a) conventional straight‐line, (b) type‐curve matching, (c) regression analysis, and (d) modern method: TDS technique. Although they were named chronologically, from oldest to most recent, they will be presented in another way:

2.2.1 Regression analysis

This is the most widely used method. It consists of automatically matching the pressure versus time data to a given analytical solution (normally) of a specific reservoir model. The automatic procedure uses nonlinear regression analysis by taking the difference between a given matching point and the objective point from the analytical solution.

This method has been also widely misused. Engineers try to match the data with any reservoir model without considering the reservoir physics. The natural problem arid=sing with this method is the none‐uniqueness of the solution. This means that for a given problem, the results are different if the starting simulation values change. This can be avoided if the starting values for the simulation values are obtained from other techniques, such as TDS technique or conventional analysis, and then, the range of variation for a given variable is reduced. This technique will not be longer discussed here since this book focused on analytical and handy interpretation techniques.

2.2.2 Type‐curve matching

As seen before, this technique was the second one to appear. Actually, it came as a solution to the difficulty of identity flow regimes in conventional straight‐line plots. However, as observed later, the technique is basically a trial‐and‐error procedure. This makes the technique tedious and risky to properly obtain reservoir parameters.

The oldest type‐curve method was introduced by Ramey [2, 8, 9]. If CD = 0 in Eq. (2.16), then, qsf = q. Therefore;

1CDdPDdtD=0E2.19

By integration between 0 and a given PwD and from dimensionless time zero to tD, and taking logarithm to both terms, it yields:

logPD=logtDlogCDE2.20

Suffix w is used to emphasize that the pressure drop takes place at the wellbore bottom‐hole. This will be dropped for practical purposes. It is clearly observed in Eq. (2.19) that the slope is one. Then in any opportunity that is plotted PD vs. tD and a straight line with a unitary slope is observed at early times, is a good indication that storage exists. Substituting the dimensionless quantities given by Eqs. (1.89), (1.94) and (2.15) in Eq. (2.20), we have:

C=qB24tΔP=qB24tN(PiPwf)NE2.21

Eq. (2.21) serves to determine the storage coefficient from data from a pressure decline test using a log‐log plot of ΔP versus time. Any point N is taken from the unit‐slope straight line portion. The value of C obtained using Eq. (2.21) must match the value obtained from Eq. (2.5). Otherwise, there may be an indication that the liquid level is going down or rising inside the well. The reasons most commonly attributed to this phenomenon are high gas‐oil ratios, highly stimulated wells, exhaust gaskets or spaces in the well connections caused by formation collapse or poor cementation and wells used for viscous fluid injection. In conclusion, the properties of Ramey's type curves allow (a) a unitary slope to be identified which indicates wellbore storage and (b) the fading of wellbore storage effects.

It can also be seen that each curve deviates from the unitary slope and forms a transition period lasting approximately 1.5 logarithmic cycles. This applies only to constant wellbore storage, otherwise, refer to [10]. If every ½ cycle is equal to (100.5 = 3.1622), it means that three half cycles (3.16223 = 31.62) represent approximately a value of 30. That is to say that a line that deviates at 2 min requires 1 h forming the transient state or radial flow regime. In other words, the test is masked for 1 h by wellbore storage effects [2, 5, 11]. It is also observed that a group of curves that present damage are mixed at approximately a dimensionless time,

tD(60+3.5s)CDE2.22

After which time, the test is free of wellbore storage effects [2, 5, 6]. Along with TDS technique [10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73] which will be discussed later in this chapter, type‐curve matching is the only manual procedure that can be applied in short tests where radial flow has not been developed (semilog line). However, type‐curve matching is risky because it is a trial‐and‐error technique, but can provide approximate results even when conventional methods fail. One millimeter shifting can cause pressure differences of up to 200 psia. The procedure is as follows [2, 9]:

Figure 2.4.

Type curve of dimensionless pressure against dimensionless time for a well in an infinite reservoir (wellbore storage and skin), taken from [2, 9].

  1. Prepare a plot of DP vs. t on logarithmic paper using the same scale as the master curve given in Figure 2.4. This is recognized as the field data plot, fdp.

  2. Place the fdp on the master curve so that the axes are parallel.

  3. Find the best match with one of the curves in Figure 2.4.

  4. Choose a suitable match point and read the corresponding coordinates DPM, tM, PDM, tDM, and CDM. The two first parameters are read from the fdp. The remaining from the type‐curve (Figure 2.4).

  5. Estimate permeability, porosity, and wellbore storage coefficient, respectively:

    k=141.2qμBh(PDMΔPM)E2.23
    ϕ=0.0002637kμctrw2(tMtDM)E2.24
    C=ϕcthrw20.8936CDME2.25

The results from the Ramey’s type curve must be verified with some other type curve. For instance, Earlougher and Kersch [8], formulated another type curve, Figure 2.5, which result should agree with those using Ramey method. The procedure for this method [8] is outlined as follows:

  1. Plot ΔP/t vs. t (fdp) on logarithmic paper using the same scale as the master curve given in Figure 2.5. Match the plotted curve, fdp, with the appropriate curve of Figure 2.5. Choose any convenient point and read from the master graph (CDe2s)M, (ΔP/t 24C/qB)M and (kh/µ t/C)M. Read from the fdp: (ΔP/t)M and tM.

Figure 2.5.

Earlougher and Kersch type‐curve for a well in infinite reservoir with wellbore storage and skin, taken from [2, 8].

Find wellbore storage coefficient, formation permeability, and skin factor using, respectively, the below expressions:

C=qB24(ΔPt24CqB)M/(ΔPt)ME2.26
k=μCh(khμtc)M/tME2.27
s=12ln[ϕμcthrw20.89359C(CDe2s)M]E2.28

Another important type curve that is supposed to provide a better match was presented by Bourdet et al. [73], Figure 2.6. This includes both pressure and pressure derivative curves. The variables to be matched are ΔPM, (tP′)M, (PD)M, [(tD/CD)PD′]M, tM, (tD/CD)M, and (CDe2s)M. The equations use after the matching are [73]:

k=141.2qμBhPDMΔPME2.29
C=(0.000295khμ)tM(tD/CD)ME2.30
k=1412qμBh[(tD/CD)PD]M(t*ΔP)ME2.31
s=12ln(CDe2s)MCE2.32

Figure 2.6.

Bourdet et al. [73] pressure and pressure‐derivative versus time‐type curve.

2.2.3 Straight‐line conventional analysis

The conventional method implies plotting either pressure or pressure drop against a given time function. The intercept and slope of such plot is used for reservoir and well parameters estimation. When the fluid initiates its path from the farthest reservoir point until the well head, several states and flow regimes are observed depending on the system geometry. For instance, if the reservoir has an elongated shape, probably linear flow will be observed. Linear flow obeys a pressure dependency on the square‐root of time, or, if the fluid experiences radial flow regime, the relation between pressure and time observes a semilog behavior, or, either inside the well or the limitation of the reservoir boundaries imply a pseudosteady‐state condition, then, pressure is a linear function of time.

Pαf(t)E2.33

The time function depends on the system geometry and could be any of the kinds described by Eq. (2.34).

Normally, the pressure or pressure drop are plotted in Cartesian coordinates, except certain few cases as for the Muskat method, see Chapter 3, which requires a potential plot, meaning, logarithm scale of pressure drop in the y‐axis and Cartesian scale for time in the x‐direction.

2.2.3.1 Semilog analysis

It is commonly referred as the “semilog method” since the radial flow is the most important regime found on a pressure test. Then, a semilogarithm plot is customary used in well test analysis.

f(t)={log tRadialflowlogtp+ΔtΔtRadialflow(Hornerplot)tPseudosteadystatet0.135Ellipsoidalflowt0.25Bilinearflowt0.36Birradialflowt0.5Linearflow1/t0.5Spherical/Hemisphericalflow(tp+Δt)ξΔtξ;ξ=0.135Ellipsoidal,0.25,0.36,0.51/tp+1/Δt1/tp+ΔtSpherical/Hemisphericalflow1/Δt1/tp+ΔtSpherical/HemisphericalflowE2.34

Starting by including the skin factor in Eq. (1.106);

PD=12[lntD+0.80907]+2sE2.35

Replacing the dimensionless terms given by Eqs. (1.89) and (1.94) into Eq. (2.35) and dividing both terms by ln 10 will lead to:

kh(PiPwf)162.6qμB=[log(0.0002637ktϕμctrw2)+0.3514+0.8686s]E2.36

Solving for the well‐flowing pressure;

Pwf=Pi162.6qμBkh[log(ktϕμctrw2)3.2275+0.8686s]E2.37

Figure 2.7.

Behavior of the well‐flowing pressure observed in a semilog graph, taken from [68].

Eq. (2.37) suggests a straight‐line behavior which is represented in the central region of Figure 2.7. The other two regions are affected by wellbore storage and skin effects, at early times and boundary effects at late times. Reservoir transmissivity, mobility, or permeability can be determined from the slope;

m=T=khμ=|162.6qBm|E2.38

The intercept of Eq. (2.34) is used for the determination of the mechanical skin factor. For practical purposes, the well‐flowing pressure at time of 1 h, P1hr, is read from the straight‐line portion of the semilog behavior, normally extrapolated as sketched in Figure 2.7, so solving for skin factor, s, from Eq. (2.34) results:

s=1.1513[P1hrPimlog(kϕμctrw2)+3.23]E2.39

Since the slope possesses a negative signed, so does the P1hr− Pi term. Therefore, the first fractional in the above equation is always positive unless the well is highly stimulated.

According to Eq. (2.39), the contribution to the pressure drop caused by the mechanical skin factor is included to the last term: 0.8686s multiplied by the slope. Then:

ΔPs=|0.87(m)|s,{ifs>0ΔPs>0ifs<0ΔPs<0E2.40

Eq. (2.40) is similar to Eq. (1.110) and works for either pressure drawdown or pressure buildup tests.

ΔPs=0.87(m){kks1}lnrsrwE2.41

Eq. (1.110) is useful to find either skin factor, s, formation damaged permeability, ks, or the damaged or affected skin zone radius, rs. However, since the skin zone covers an infinitely thin area and the pressure wave travels at high speed, it is difficult to detect transmissivity changes, then, rs and ks are difficult to be measured.

Eqs. (1.110) and (2.37) imply the skin factor along flow rate just increases or decreases the well pressure drop. However, this occurs because the well radius behaves as if its radius was modified by the value of the skin factor. Brons and Miller [74] defined the apparent or effective wellbore radius, rwa, to be used in Eqs. (1.89), (1.94), and (1.100)

rwa=rwesE2.42

Example 2.1

A well with a radius of 0.25 ft was detected to have a skin factor of 2. A skin factor of −2 was obtained after a stimulation procedure. Find the apparent radii and the percentage of change in the radius due to the stimulation. What conclusion can be drawn?

Solution

Application of Eq. (2.42) for the damaged‐well case gives:

rwa=rwes=0.25e2=0.034 ft=0.406 in

Application of Eq. (2.42) for the damaged‐well case gives:

rwa=rwes=0.25e(2)=1.848ft=22.17in

It can be observed that 1.847 × 100/0.034 ≅ 5460%, meaning that the stimulation helps the well to increase its radius 55 times. It can be concluded from the example that for positive skin factor values, the effective wellbore radius decreases (rwa<rw) and for negative skin factor values, the effective wellbore radius increases (rwa>rw).

The starting time of the semilog straight line defined by Ramey [9] in Eq. (2.22) allows determining mathematically where the radial flow starts, i.e., the moment wellbore storage effects no longer affect the test. Replacing into Eq. (2.22) the dimensionless parameters given by Eqs. (1.94) and (2.15) results [2]:

tSSL=(200000+12000s)μCkhE2.43

The application of Eq. (2.40) is twofolded. (1) It can be used for test design purposes. The duration of a pressure drawdown test should be last 10 times the value of tSSL, so a significant portion of the radial flow regime can be observed and analyzed and (2) finding the semilog slope can be somehow confusing. Once the semilog line is drawn and permeability, skin factor, and wellbore storage are calculated, then, Eq. (2.40) can be used to find the starting point of the radial flow regime. Radial flow is correctly found if the tSSL value agrees with the one chosen in the plot. This last situation is avoid if the pressure and pressure derivative plot is available since radial flow is observed once the pressure derivative curve gets flat as seen in Figure 2.6.

The declination stabilization time (time required to reach the boundaries and develop the pseudosteady‐state period) during the test can be from the maximum time at which the maximum pressure drops (not shown here) take place. This is:

tmax=948ϕμctr2kE2.44

From which;

tpss=948ϕμctre2kE2.45

For square or circular geometries, tDA = 0.1 from Table 2.1. Replacing this value in Eq. (1.100) and solving for time leads to:

tpss=1190ϕμctre2kE2.46

from

rinv=0.0325ktpμϕctE2.47

For any producing time, tp, the radius of investigation—not bigger than re—can be found.

The point reached by the disturbance does not imply fluid movement occurs there. The drainage radius is about 90% that value, then

rd=0.029ktpμϕctE2.48

Skin factor is a dimensionless quantity. This does not necessarily reflect the degree of either damage or stimulation of a well. Then, more practical measurement parameters ought to be used. One of this is the flow efficiency, FE, which implies what percentage of the total pressure drawdown is due to skin factor. The flow efficiency is defined as the ratio between the actual productivity index, J, and the ideal productivity index. The productivity index involves money since it is defined as the amount of pressure drop needed to produce a barrel of fluid per day. In other words, it is the energy required to produce one BPD. Mathematically;

J=qP¯PwfE2.49
Jideal=qP¯PwfΔPsE2.50
FE=JJideal=1ΔPsP¯PwfE2.51

Table 2.1.

Shape factors for different drainage areas, taken from [8, 75].

FE < 1 is an indication that well damage exists, otherwise there is stimulation. The productivity index can be increased by:

  • Increasing the permeability in the zone near the well—hydraulic fracturing;

  • Reduce viscosity—steam injection, dissolvent, or in situ combustion;

  • Damage removing—acidification;

  • Increase well penetration;

  • Reduce volumetric factor—choosing correct surface separators.

Other parameters to quantify well damage are [68]:

Damage ratio, DR

DR=1/FEE2.52

Damage ratios less than the unity indicate stimulation.

Damage factor, DF

DR=1FEE2.53

Negative values of damage factors indicate stimulation. The damage factor can also be estimated from [68]:

DF=ss+ln(re/rw)=1q(actual)q(ideal)E2.54

Eq. (2.54) applies to circular‐shaped reservoir.

Productivity ratio, PR

PR=qqa=q(ideal)q(actual)=ln(re/rw)ln(re/rw)+sE2.55

Annual loss income, FD$L (USD$)

FD$L=365q(OP)DFE2.56

where OP is oil price.

Example 2.2

What will be the annual loss of a well that produces 500 BFD, which has a damage factor of 8, drains an area of 120 acres and has a radius of 6 inches? Assume circular reservoir area and a price of oil crude of USD $ 55/barrel.

Figure 2.8.

Characteristics found in the Cartesian graph, taken from [68].

Solution

120 acres = 5,227,200 ft2. If the area is circular, then: r = 1290 ft. Find the damage factor from Eq. (2.54);

DF=ss+ln(re/rw)=88+ln(1290/0.5)=0.5046

Find the yearly loss income using Eq. (2.56)

FD$L=365q(OP)DF=(365)(500)(55)(0.5046)=USD$ 5064922

This indicates that the well requires immediate stimulation.

2.2.3.2 Reservoir limit test, RLT

It is a drawdown test run long enough to reach the reservoir boundaries. Normal pressure drawdown tests, during either radial flow or transient period test, are used to estimate formation permeability and artificial well conditions (C and s), while an RLT test—introduced by [76]—deals with boundaries and is employed to determine well drainage area or well drainage pore volume. In a Cartesian graph for a closed boundary system, Figure 2.8, three zones are distinguished [8, 68]: (i) skin and wellbore storage dominated zone, (ii) transient zone (radial flow), and (iii) pseudosteady‐state zone. As indicated by Eq. (1.129), the pressure drop is a linear function of time. Eq. (1.129) is given for circular reservoir geometry. For any geometry, the late time pseudosteady‐state solution involves the Dietz shape factor, [75], to extent the use of Eq. (1.129) for other reservoir geometries, as described in Table 2.1. Under this condition, Eq. (1.129) becomes [77]:

PD=2πtDA+12ln(Arw2)+ln(2.2458CA)E2.57

Replacing in the above expression the dimensionless quantities given by Eqs. (1.89) and (1.94), it results:

Pwf=[0.23395qBϕctAh]t+Pi70.6qμBkh[lnArw2+ln(2.2458CA)+2s]E2.58

From the slope, m*, and intercept, PINT, of Eq. (2.58), the reservoir pore volume and Dietz shape factor [74] can be obtained from either:

Vp=0.23395qBctm*E2.59
CA=5.456mm*e2.303P1hrPINTmE2.60

Once the value of CA is obtained from Eq. (2.60), the reservoir geometry can be obtained from Table 2.1 by using the closest tabulated value (“exact for tDA”) and confront with the time to develop pseudosteady‐state regime, (tDA)pss which is found from:

(tDA)pss=0.1833m*mtpssE2.61

tpss can be read from the Cartesian plot. However, this reading is inexact; therefore, it is recommended to plot the Cartesian pressure derivative and to find the exact point at which this becomes flat.

2.2.4 Tiab’s direct synthesis (TDS) technique

TDS technique is the latest methodology for well test interpretation. Its basis started in 1989 [70]. TDS’ creator was Tiab [71], who provided analytical and practical solutions for reservoir characterization using characteristic points or features—called by him “fingerprints”—read from a log‐log plot of pressure and pressure derivative [15], versus time. Since the introduction of TDS in 1995, several scenarios, reservoir geometries, fluid types, well configurations, and operation conditions. For instance, extension of TDS technique to elongated systems can be found in [13, 14, 16–19, 23, 24, 28, 30, 31]. Some applications of conventional analysis in long reservoirs are given in [20, 29, 38, 54]. For vertical and horizontal gas wells with and without use of pseudotime, refer to [22, 36, 39]. Special cases of horizontal wells are found in [12, 47]. For transient rate analysis, refer to [27, 35, 49]. Applications on heavy oil (non‐Newtonian fluids) can be found in [32, 34, 41, 42, 45, 52, 56, 62, 64]. For cases on shales reservoirs, refer to [49, 51, 56, 78]. Well test analysis by the TDS technique on secondary and tertiary oil recovery is presented by [25, 33, 60, 79]. For multirate testing in horizontal and vertical wells, refer, respectively, to [65, 67]. References [43, 46] are given for conductive faults. For deviated and partially penetrated wells, refer to [37, 64], respectively. TDS technique extended to multiphase flow was presented by [26]. Wedged and T‐shaped reservoirs can be found in [48] and coalbed‐methane reservoirs with bottom water drive are given in [53]. TDS technique is excellent for interpreting pressure test in hydraulically fractured vertical wells since unseen flow regimes can be generated [50, 69, 80]. The first publications on horizontal wells in naturally fractured and anisotropic media are given in [81, 82]. The threshold pressure gradient is dealt by [57, 72]. For vertical wells in double porosity and double permeability formations, refer, respectively, to [41, 83]. A book published by Escobar [56] presents the most recent topics covered by the TDS technique, and a more comprehensive state‐of‐the‐art on TDS technique is given by [58]. This book revolves around this methodology; therefore, practically, the whole content of [71]—pioneer paper of TDS technique—will be brought here:

The starting point is the definition of the dimensionless pressure derivative from Eq. (1.89);

tD*PD=kh(t*ΔP)141.2qμBE2.62

By looking at Eqs. (2.17) and (2.18), we can conclude the difficulty of using hand mathematical operations with them. Instead of using these general solutions, Tiab [71] obtained partial solutions to the differential equation for each flow regime or time period. For instance, during early pseudosteady‐state, the governing equation reduces to:

PD=tDCDE2.63

Combination of Eqs. (1.94) and (2.15) results in:

tDCD=(2.95×104hμ)tCE2.64

Replacing Eq. (1.89) in the above expression yields;

(kh141.2qμB)ΔP=(2.95×104khμ)tCE2.65

Solving for C;

C=(qB24)tΔPE2.66

The pressure derivative curve also has a straight line of unitary slope at early times. The equation of this line is obtained by taking the derivative of Eq. (2.63) with respect to the natural logarithm of tD/CD. So:

(tDCD)PD=tDCDE2.67

Where the derivative of the dimensionless pressure is:

PD=dPD/dtD=(kh141.2qμB)dP/(0.0002637kϕμctrw2)dtE2.68

Rearranging;

PD=(26.856rw2ϕcthqB)ΔPE2.69

Converting to dimensional form, the left‐hand side of Eq. (2.67) by using the definitions given by Eqs. (2.64) and (2.68):

(tDCD)PD=0.00792252(khqμB)(ϕcthrw2C)(t*ΔP)E2.70

Multiplying and dividing by 0.8935;

(tDCD)PD=0.007087(khqμB)(ϕcthrw20.8935C)(t*ΔP)E2.71

Recalling Eq. (2.15), the above becomes:

(tDCD)PD=0.007087(khqμB)(1CD)(t*ΔP)E2.72

Since the unit slope is one, then CD = 1, thus;

(tDCD)PD=kh(t*ΔP')141.2qμBE2.73

From looking at Figure 2.6, both pressure and pressure derivative curves display a unitary slope at early times. Replacing Eqs. (2.64), (2.73) in (2.67) and solving for C will result:

C=(qB24)tt*ΔPE2.74

As seen in Figure 2.6, the infinitely acting radial flow portion of the pressure derivative is a horizontal straight line with intercept of 1/2. The governing equation is:

[(tDCD)PD]r=12E2.75

Combining the above equation with Eq. (2.73) results the best expression to estimate reservoir permeability:

k=70.6qμBh(t*ΔP)rE2.76

Subscript r stands for radial flow line. A customary use of TDS, as established by Tiab [71], is to provide suffices to identify the different flow regimes. For instance, pss stands for pseudosteady state, i stands for either initial or intercept, etc. In terms of pressure, the equation of this line is:

PDr=0.5{ln(tDCD)r+0.80907+ln[CDe2s]}E2.77

It is recommended to draw a horizontal line throughout the radial flow regime and choose one convenient value of (tP')r falling on such line.

Tiab [71] also obtained the start time of the infinite line of action of the pressure derivative is:

(tDCD)sr=10log(CDe2s)10E2.78

Replacing Eqs. (1.92) and (2.15) in the above equation will yield:

tsr=μC6.9×105kh[ln(0.8935Cϕcthrw2)+2s]E2.79

A better form of Eq. (2.78) was given by [84];

(tDCD)sr=1α[ln(CDe2s)+ln(tDCD)SR]E2.80

Setting a = 0.05 in the above equation and solving for C:

C=0.056ϕcthrw2(tDsr2s+lntDsr)E2.81

tDsr is calculated with Eq. (1.94) letting t = tsr.

The point of intersection, i, between the early time unit‐slope line defined by Eqs. (2.63) and (2.67) and the late‐time infinite‐acting line of the pressure derivative, defined by Eq. (2.75), is given by:

(tDCDPD)i=0.5E2.82
(tDCD)i=0.5E2.83

where i stands for intersection. After replacing the definitions given by Eqs. (1.94), (2.15), and (2.72) will, respectively, provide:

(t*ΔP)i=70.6qμBkhE2.84
k=1695μChtiE2.85

For the unit‐slope line, the pressure curve is the same as for the pressure derivative curve. Then, at the intersection point:

(ΔP)i=(t*ΔP)i=(t*ΔP)rE2.86

Tiab [71] correlated for CDe2s > 100 permeability, wellbore storage coefficient, and skin factor with the coordinates of the maximum point—suffix x—displayed once the “hump” observed once wellbore storage effects start diminishing. These correlations are given as follows:

(tDCDPD)x=0.35717(tDCD)x0.50E2.87
log(CDe2s)=0.35(tDcD)1.24E2.88
log(CDe2s)=1.71(tDCDPD)1.24E2.89

Replacing Eqs. (2.64) and (2.73) into Eq. (2.87) leads to:

(t*ΔP)x=(0.015qBC)tx0.42(141.2qμBkh)E2.90

Either formation permeability or wellbore storage coefficient can be determined using the coordinates of the peak, tx and (tP′)x. Solving for both of these parameters from Eq. (2.90) results:

k=(70.6qμBh)1(0.014879qB/C)tx(t*ΔP)xE2.91
C=0.014879qBtx(t*ΔP)x+(t*ΔP)rE2.92

The constants in Eqs. (2.91) and (2.92) are slightly different as those in [58]. These new unpublished versions were performed by TDS’ creator.

Eq. (2.91) is so helpful to find reservoir permeability in short test when radial flow is absent which is very common in fall‐off tests. Once permeability is found from Eq. (2.91), solved for (tP′)r from Eq. (2.76) and plot on a horizontal line throughout this value. Then, compare with the actual derivative plot and use engineering criterion to determine if the permeability value is acceptable. This means, if the straight line is either lower or higher than expected. Otherwise, new coordinates of the peak ought to be read for repeating the calculations since the hump should look some flat.

Substitution of Eqs. (2.64) and (2.73) in Eqs. (2.88) and (2.89) allows obtaining two new respective correlations for the determination of the mechanical skin factor:

s=0.171(txti)1.240.5ln(0.8935Cϕhctrw2)E2.93
s=0.921((t*ΔP)x(t*ΔP)i)1.10.5ln(0.8935Cϕhctrw2)E2.94

Sometimes, the reading of the peak coordinates may be wrong due to the flat appearance of it. Then, it should be a good practice to estimate the skin factor using both Eqs. (2.93) and (2.94). These values should match each other.

Divide Eq. (2.87) by Eq. (2.75); then, in the result replace Eqs. (2.64) and (2.73) and solve for both permeability and wellbore storage:

k=4745.36μChtx{(t*ΔP)x(t*ΔP)r+1}E2.95
C=0.014879qBtx(t*ΔP')x+(t*ΔP)rE2.96

This last expression is useful to find wellbore storage coefficient when the early unitary slope line is absent.

TDS technique has a great particularity: for a given flow regime, the skin factor equation can be easily derived from dividing the dimensionless pressure equation by the dimensionless derivative equation of such flow regime. Then, the division of Eq. (2.77) by Eq. (2.75) leads to the below expression once the dimensionless parameters given by Eqs. (1.89), (1.94), and (2.73) are replaced in the resulting quotient. Solving for s from the final replacement leads to:

s=0.5(ΔPr(t*ΔP)rln[ktrϕμctrw2]+7.43)E2.97

being tr any convenient time during the infinite‐acting radial flow regime throughout which a horizontal straight line should have been drawn. Read the ΔPr corresponding to tr. Comparison between Eqs. (2.38) and (2.76) allows concluding:

m=2.303(t*ΔP)r=ln(10)(t*ΔP)rE2.98

which avoids the need of using the semilog plot if the skin pressure drop is needed to be estimated by Eq. (2.40), otherwise, Eq. (2.40) becomes:

ΔPs=|2(t*ΔP)r|s,{ifs>0ΔPs>0ifs<0ΔPs<0E2.99

For the determination of well‐drainage area, Tiab [69] expressed Eq. (2.75) as:

(tD*PD)r=12E2.100

Also, Tiab [69] differentiated the dimensionless pressure with respect to dimensionless time in Eq. (2.57), so:

tD*PD=2πtDAE2.101

Then, Tiab [69] based on the fact that two given flow regime governing equations can be intersected each other, regardless the physical meaning of such intersection, and solving for any given parameter, intercepted Eqs. (2.100) with (2.101), then, replaced in the resulting expression the dimensionless quantities given by Eqs. (2.92), (2.97), and (2.62) and solved for the area given in ft2:

A=ktrpssi301.77ϕμctE2.102

Furthermore, Chacon et al. [85] replaced the dimensionless time given by Eq. (1.100) and the dimensionless pressure derivative of Eq. (2.62) into Eq. (2.102) and also solved for the area in ft2:

A=0.234qBtpssϕcth (t*ΔP)pssE2.103

The above expression uses any convenient point, tpss and (tP’)pss, during the late time pseudosteady‐state period. Because of noisy pressure derivative data, the readings of several arbitrary points may provide, even close, different area values. Therefore, it is convenient to use an average value. To do so, it is recommended to draw the best late‐time unit‐slope line passing through the higher number of pressure derivative points and extrapolate the line at the time of 1 h and read the pressure derivative value, (tP')pss1. Under these circumstances, Eq. (2.103) becomes:

A=0.234qBϕcth (t*ΔP)pss1E2.104

Eqs. (2.102) through (2.104) apply only to closed‐boundary reservoirs of any geometrical shape. For constant‐pressure reservoirs, the works by Escobar et al. [28, 54] for TDS technique (summary given in Table 2.2) and for conventional analysis are used for well‐drainage area determination in circular, square, and elongated systems.

TDS technique has certain step‐by‐step procedures which not necessarily are to be followed since the interpreter is welcome to explore and use TDS as desired. Then, they are not provided here but can be checked in [69, 71].

Example 2.3

Taken from [68]. The pressure and pressure derivative data given in Table 2.3 corresponds to a drawdown test of a well. Well, fluid, and reservoir data are given below:

rw = 0.267 ftq = 250 BPDμ = 1.2 cp
ct = 26.4 × 10−5psi−1h = 16 piesϕ = 18%B = 1.229 bbl/BF

Find permeability, skin factor, drainage area, and flow efficiency by conventional analysis. Find permeability, skin factor, and three values of drainage area using TDS technique:

Solution

Conventional analysis. Figure 2.9 and 2.10 present the semilog and Cartesian plots, respectively, to be used in conventional analysis. From Figure 2.9, the semilog slope, m, is of 18 psia/cycle and P1hr = 2308 psia. Permeability and skin factor are calculated using Eqs. 2.38 and 2.39, respectively, thus:

k=|162.6(250)(1.2)(1.229)(18)(16)|=208md
s=1.1513[2308273318log(2080.18(1.2)(26.4×105)(0.267)2)+3.23]=22.15

Find the pressure loss due to skin factor with Eq. (2.40);

ΔPs=|0.87(18)|22.15=346.7psia

Since the average reservoir pressure is not reported, then, the initial pressure value is taken instead. Eq. (2.51) allows estimating the flow efficiency.

Table 2.2.

Summary of equations, taken from [28].

t, hPwf, psiaDP, psiat*DP′, psia/ht, hPwf, psiaDP, psiat*DP′, psia/h
0.00273305231242165.42
0.1027033031.057229344035.32
0.2026726158.959.622914425.86
0.3026448984.141222904435.85
0.402616117106.3016.822874467.63
0.652553180129.7033.622824517.99
1.002500233135.155022794547.94
1.502440293151.9072227645710.50
2.002398335127.2685227445912.18
3.002353380102.10100227246113.36
4.00232940481.44

Table 2.3.

Pressure and pressure derivative versus time data for example 2.3.

Figure 2.9.

Semilog plot for example 2.3.

Figure 2.10.

Cartesian plot for example 2.3.

Figure 2.11.

Pressure and pressure derivative plot for example 2.3.

FE=1346.927332272=24.75%

From the Cartesian plot, Figure 2.10, is read the following data:

m* = −0.13 psia/hPINT = 2285 psiatpss ≈ 50 h

Use Eq. (2.59) to find well drainage area:

A=0.234qBϕhctm*=0.234(250)(1.229)(0.18)(16)(26.4×105)(0.13)=727391.1ft2=16.7Ac

Find the Dietz shape factor with Eq. (2.60);

CA=5.456180.13e[2.303(23082285)18]=39.82

As observed in Table 2.1, there exist three possible well drainage area geometry values (hexagon, circle, and square) close to the above value. To discriminate which one should be the appropriate system geometry find the dimensionless time in which pseudosteady‐state period starts by using Eq. (2.61):

(tDA)pss=0.18330.131850=0.0660.1

TDS technique. The following are the characteristic points read from Figure 2.11:

(tP')r = 7.7 psiatr = 33.6 hΔPr = 451 psia
tpss = 85 h(tP')pss = 12.18 psiatrpi = 58 h
(tP')pss1 = 0.14 psia

Find permeability and skin factor with Eqs. (2.76) and (2.97), respectively:

k=70.6qμBh(t*ΔP')r=70.6(250)(1.2)(1.229)(16)(7.7)=211.3md
s=0.5[4517.7ln(211.3(33.6)0.18(1.2)(26.4×105)(0.2672))+7.43]=22.4

Determine the well drainage area with Eqs. (2.102) and (2.103), thus;

A=211.3(58)301.77(0.18)(1.2)(26.4×105)(43560)=16.35Ac
A=0.234(250)(1.229)(85)(0.18)(26.4×105)(16)(12.18)(43560)=15.15Ac
A=0.234(250)(1.229)(0.18)(26.4×105)(16)(0.14)(43560)=15.5Ac

Even, more parameters can be reestimated with TDS technique for verification purposes but it will not be performed for saving‐space reasons. However, the reader is invited to read the coordinates of the peak and the intersection point of the wellbore storage and radial flow lines. Then, estimate formation permeability with Eqs. (2.84), (2.85), and (2.91). Also, find the wellbore storage coefficient using Eqs. (2.74), (2.81), (2.92), and (2.96) and skin factor with Eqs. (2.93) and (2.94).

Example 2.4

Taken from [68] with the data from the previous example, Example 2.3, determine tSSL and find if the well fluid level is increasing or decreasing in the annulus if the well has a drill pipe with 2 in external diameter inside a liner with 5 in of inner diameter including joint gaskets. The density of the wellbore fluid is 42.5 lbm/ft3.

Solution

From Figure 2.11, a point is chosen on the early unit‐slope line. This point has coordinates: DP = 59 psia and t = 0.2 h. Wellbore storage coefficient is found with Eq. (2.21):

C=(250)(1.229)240.259=0.0434bbl/psia

Solving for annulus capacity from Eq. (2.5);

Vu=(ρ144)C=(42.5144)0.0434=0.0128bbl/ft

The theoretical capacity is found with Eq. (2.45), so:

Vu=0.0009714(5222)=0.0204bbl/ft

This leads to the conclusion that the annular liquid is falling.

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2.3. Multiphase flow

According to Perrine [86], the single fluid flow may be applied to the multiple fluid flow systems when the gas does not dominate the pressure tests, it means liquid production is much more relevant than gas flow. Under this condition, the diffusivity equation, Eq. (1.27), will result and the total fluid mobility is determined by Eq. (1.24). We also mentioned in Chapter 1 that Martin [63] provided some tips for a better use of Perrine method. Actually, Perrine method works very well in liquid systems.

The semilog equations for drawdown and build tests are, respectively, given below:

Pwf=Pi162.6qtλth(logλtt1688ϕctrw2+0.869s)E2.105
Pws=Pi162.6qtλthlog(tp+ΔtΔt)E2.106

The flow rate is estimated by:

qt=(qoBo+(qgqoRs/1000)Bg+qwBw)/BoE2.107

Eq. (2.107) is recommended when oil flow dominates the test. It is removed from the denominator, otherwise. It advised to use consistent units in Eq. (2.107) meaning that the gas flow rate must be in Mscf/D and the gas volume factor bbl/SCF.

Once the semilog slope has been estimated, the total mobility, the phase effective permeabilities, and the mechanical skin factor are found from:

λt=162.6qtmhE2.108
kL=162.6qLBLμLmh;L=wateroroilE2.109
kg=162.6(qgqoRs/1000)BgμgmhE2.110
s=1.1513(PwfP1hrmlog(λtϕctrw2)+3.23)E2.111

The best way of interpreting multiphasic flow tests in by using biphasic and/or triphasic pseudofunctions. Normally, well test software uses empirical relationships to estimate relative permeability data. The accuracy of the following expression is sensitive to the relative permeability data:

m(P)=P0PkroμoBodPE2.112

The expressions used along this textbook for reservoir characterization may apply for both single fluid and multiple fluid production tests. Single mobility has to be changed by total fluid mobility and individual flow rate ought to be replaced by the total fluid rate. Just to cite a few of them, Eqs. (2.66), (2.76), (2.85), (2.91), (2.92), and (2.97) become:

C=(qt24)tΔPE2.113
λt=kμ|t=70.6qth(t*ΔP')rE2.114
(kμ)t=1695ChtiE2.115
λt=(70.6qth)1(0.014879qt/C)tx(t*ΔP)xE2.116
C=0.014879qttx(t*ΔP')x+(t*ΔP)rE2.117
s=0.5[ΔPr(t*ΔP)rln(λttrϕctrw2)+7.43]E2.118

Also, the effective liquid permeabilities are found using the individual viscosity, rate, and volume factor. Then, Eq. (2.76) applied to oil and water will yield:

ko=70.6qoμoBoh(t*ΔP)rE2.119
kw=70.6qwμwBwh(t*ΔP')rE2.120

However, from a multiple fluid test, it is a challenge to find the reservoir absolute permeability. Several methods have been presented. For instance, Al‐Khalifah et al. [87] presented a sophisticated method applied to either drawdown or multiple rate tests. Their method even includes the estimating of the saturation change respect to pressure. However, we presented the method by Kamal and Pan [88] which applies well for liquid fluid. Relative permeabilities must be known for its application. Once effective permeabilities are found, let us say from Eqs. (2.119) and (2.120), estimate the permeability ratio ko/kw and find the water saturation from the relative permeability curves as schematically depicted in Figure 2.12 (left). Then, using the estimated water saturation value, enter Figure 2.12 (right) and read a value from a relative permeability curve. Use the most dominant flow curve. The dominant phase is assumed to be oil for the example in Figure 2.12. Since both phase effective permeability and phase relative permeability are known, the absolute permeability is found from the definition of relative permeability:

Figure 2.12.

Determination of absolute permeability as outlined by Kamal and Pan [88].

k=kokroE2.121

Further recommendations for handling multiphase flow tests are presented by Al‐Khalifah et al. [89] and are also reported by Stanislav and Kabir [7].

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2.4. Partial penetration and partial completion

When a well penetrates a small part of the formation thickness, hemispherical flow takes place. See Figure 2.13 top. When the well is cased above the producer range and only a small part of the casing is perforated, spherical flow occurs in the region near the face of the well. See Figure 2.13 bottom. As the transient moves further into depth of the formation, the flow becomes radial, but if the test is short, the flow will be spherical. Both types of flow are characterized by a slope of −1/2 in the log‐log plot of pressure derivative versus time [90, 91]. Theoretically, before either hemispherical or spherical flow takes place, there exists a radial flow regime occurring by fluids withdrawn from the formation thickness that is close in height to the completion interval. This represents the transmissibility of the perforated interval. Actually, this flow regime is unpractical to be seen mainly because of wellbore storage effects. We will see further in this chapter that there are especial conditions for hemispherical/spherical flow to be observed which occur later that the completion‐interval‐limited radial flow regime. Both hemispherical and spherical flow vanished when the top and bottom boundaries have been fully reached by the transient wave; the true radial flow is developed throughout the full reservoir thickness.

Figure 2.13.

Ideal flow regimes in partial penetration (top) and partial completion (bottom) systems, after [66].

The apparent skin factor, sa, obtained from pressure transient analysis is a combination of several “pseudoskin” factors such as [91]:

sa=s+sp+sθ+scp+……E2.122

where s is the true damage factor caused by damage to the well portion, sp is the pseudoskin factor due to the restricted flow entry, sq is the pseudoskin factor resulting from a well deviation angle, and scp is the pseudoskin due to a change in permeability near the face of the well. sp can be estimated from [92]:

sp=(hhp1)lnhDE2.123

hp = length of perforated or open interval. The equations of dimensionless thickness, hD, for hemispherical and spherical flow, respectively:

hD=hrwkhkzE2.124
hD=h2rwkhkzE2.125

where kh is the horizontal permeability, kz = kv is the vertical permeability. The contribution of the pseudoskin of an inclined well is given by Cinco et al. [92]:

ψ=tan1(kzkhtanψ)E2.126
sθ=(ψ41)2.06(ψ56)1.865×log(h100rw)E2.127

According to Cinco et al. [92], the above equation is valid for 0° ≤ q ≤ 75°, h/rw > 40, and tD > 100. Note that Eq. (2.127) could provide a negative value. This is because the deviation at the face of the well increases the flow area or presents reservoir pseudothickness. The pseudoskin responding for permeability changes near wellbore is given by [93]:

scp=hhp[10.2(rsrwhp)](kksks)lnrsrwE2.128

Example 2.4

Taken from [91]. A directional well which has an angle to the vertical of 24.1° has a skin factor s = −0.8. The thickness of the formation is 100 ft, the radius of the wellbore is 0.3 ft, and the horizontal to vertical permeability ratio is 5. Which portion of the damaged corresponds to the deviation of the well?

Solution

The deviation angle affected by the anisotropy is estimated with Eq. (2.126);

ψ=tan1(5tan(24.1))=45°

The pseudoskin factor caused by well deviation is found from Eq. (2.127):

sθ=(4541)2.06(4556)1.865log(100100(0.3))=1.56

From Eq. (2.122);

sa=s+sθ=0.81.56=2.36

Therefore, 66.1 % of the skin factor is due to the well deviation.

2.4.1 Conventional analysis for spherical flow

The diffusivity equation for spherical flow assuming constant porosity, compressibility, and mobility is given by Abbott et al. [90]:

1r2r(r2Pr)=ϕμctkspPtE2.129

where ksp is the spherical permeability which is defined as the geometrical mean of the vertical and horizontal permeabilities:

ksp=kvkh23=khsE2.130

The physical system is illustrated in Figure 2.14, right. This region is called a “spherical sink.” rsw is given by:

rsw=hp2ln(hprw)E2.131

The spherical flow equations for pressure drawdown and pressure buildup when the flow time is much longer than the shut‐in time were presented by [94]:

Pwf=Pi70.6qμBksprsw(1+ssp)+2453qμBksp3/2ϕμct1tE2.132
Pws=Pwf+70.6qμBksprsw(1+ssp)2453qμBksp3/2ϕμct[1tp+1Δt1tp+Δt]E2.133

Figure 2.14.

Cylindrical, hemispherical, and spherical sinks, after [66].

The spherical pressure buildup equation when the flow time is shorter than the shut‐in time:

Pws=Pi2453qμBksp3/2ϕμct[1Δt1tp+Δt]+70.6qμBksprswsspE2.134

Then, from a Cartesian plot of Pwf as a function of t −1/2, for drawdown, or Pws as a function of either [tp−1/2 + Δt−1/2−(tp + Δt)−1/2] or [Δt−1/2−(tp + Δt)−1/2] for buildup, we obtain a line which slope, m, and intercept, I, can be used to estimate tridimensional permeability and geometrical (spherical) skin factor.

ksp=(2453qμBmϕμct)2/3E2.135
ssp=(IPwf)ksprsw70.6qμB1E2.136

Once the spherical permeability is known, we solve for the vertical permeability from Eq. (2.130), and then, estimate the value of skin effects due to partial penetration [94]:

sc=(1b1)[lnhDG]E2.137

where b = hp/h. hD can be estimated from Eq. (2.125), and G is found from [94]:

G=2.9487.363b+11.45b24.576b3E2.138

2.4.2 Conventional analysis for hemispherical flow

The model for hemispheric flow is very similar to that of spherical flow [94]. The difference is that a boundary condition considers half sphere. Figure 2.14 (left) outlines the geometry of such system. The drawdown and pressure equations are given below [94]:

Pwf=Pi141.2qμBksprsw(1+ssp)+4906qμBksp3/2ϕμct1tE2.139
Pws=Pwf+141.2qμBksprsw(1+ssp)4906qμBksp3/2ϕμct[1tp+1Δt1tp+Δt]E2.140
Pws=Pi4906qμBksp3/2ϕμct[1Δt1tp+Δt]+141.2qμBksprswsspE2.141

As for the spherical case, from a Cartesian plot of Pwf as a function of t−1/2, for drawdown, or Pws as a function of either [tp−1/2 + Δt−1/2−(tp + Δt)−1/2] or [Δt−1/2− (tp + Δt)−1/2] for buildup, we obtain a line which slope, m, and intercept, I, can be used to estimate spherical permeability and geometrical (spherical) skin factor.

ksp=(4906qμBmϕμct)2/3E2.142
ssp=(IPwf)ksprsw141.2qμB1E2.143

2.4.3 TDS for spherical flow

Moncada et al. [66] presented the expressions for interpreting both pressure drawdown or buildup tests in either gas or oil reservoirs using the TDS methodology. Spherical permeability is estimated by reading the pressure derivative at any arbitrary time during which spherical flow can calculate spherical permeability and the spherical skin factor also uses the pressure reading at the same chosen time:

ksp=(1227qBμ(t*ΔP')spϕμcttsp)2/3E2.144
ssp=34.74ϕμctrsw2ksptsp[(ΔP)sp2(t*ΔP)sp+1]1E2.145

The total skin, st, is defined as the sum of all skin effects at the well surroundings:

st=sb+sc+sspE2.146

If the radial flow were seen, the horizontal permeability can be estimated from:

kH=k=70.6qBμhp(t*ΔP)r1E2.147

The suffix r1 implies the first radial flow.

Moncada et al. [66] observed that the value of the derivative for the late radial flow in spherical geometry is equivalent to 0.0066 instead of 0.5 as of the radial system. In addition, the slope line −½ corresponding to the spherical flow and the late radial flow line of the curve of the dimensionless pressure derivative in spherical symmetry intersect in:

(tD*PD)i=12πtDsp1/2=0.0066E2.148

Replacing the dimensionless time results:

ti=6927748.85φμctrsw2kspE2.149

In the above equation, suffix i denotes the intersection between the spherical flow and the late radial flow. If the radial flow is not observed, this time can give an initial point to draw the horizontal line corresponding to the radial flow regime, from which horizontal permeability is determined. This point can also be used to verify spherical permeability, ksp. Another equation defining the mentioned dimensionless time can be found from the intersection of the slope line −½ (spherical flow) with the radial line of late radial flow but in radial symmetry, knowing that:

(tD*PD')i=k3/2h4ksp3/2πrsw21tD=0.5E2.150

Replacing the dimensionless time will give:

ti=301.77k2h2ϕμctksp3E2.151

Combining Eqs. (2.149) and (2.151), an expression to find the spherical wellbore radius, rsw:

rsw=0.0066khkspE2.152

2.4.4 TDS for hemispherical flow

Here the same considerations are presented in Section 2.4.3. Using a pressure and a pressure derivative value reading at any time during hemispherical flow allows finding hemispherical permeability and partial penetration skin [66],

khs=(2453qBμ(t*ΔP')hsϕμctths)2/3E2.153
shs=34.74ϕμctrsw2khsths[(ΔP)hs2(t*ΔP')hs+1]1E2.154

Moncada et al. [66] also found that the derivative in spherical geometry of the late radial flow corresponds to 0.0033 instead of 0.5 as of the radial system. This time the line of radial flow and hemispheric flow, in hemispherical symmetry, intersect in:

(tD*PD')i=12πtDsh=0.0033E2.155

From where,

ti=27710995.41ϕμctrsw2khsE2.156

As for the spherical case, there exists an expression to define the intersection time of the −½ slope line of the hemispherical flow regime pressure derivative and the late radial flow line pressure derivative but, now, in radial symmetry:

(tD*PD')i=k3/2h2khs3/2πrsw21tD=0.5E2.157
ti=1207.09k2h2ϕμctkhs3E2.158

This point of intersection in radial symmetry gives the following equation:

rsw=0.0033khkhsE2.159

Skin factors are estimated in a manner similar to Section 2.4.3.

2.4.5 Wellbore storage and perforation length effects on hemispherical/spherical flow

It is important to identify the range of WBS values, which can influence the interpretation of the spherical and hemispheric flow regime. Figure 2.15 is a plot of PD vs. tD providing an idea of the storage effect. As can be seen, the pressure response for several CD values can be distinguished when storage is low (<10), whereas for larger CD values, the response is almost identical. For CD < 10, the slope of −½ that characterizes both spherical and hemispherical flow is well distinguished. For values of 10 < CD < 100, the slope of −½ is more difficult to identify. For values of CD > 100, the spherical flow regime has been practically masked by storage, which makes it impossible to apply the technique presented above to estimate the vertical permeability. Then, to ensure there is no CD masking, it should be less than 10 [66].

The length of the completed interval or the length of the partial penetration, hp, plays an important role in defining the spherical/hemispherical flow. The presence of spherical or hemispheric flow is characterized by a slope of −½. This characteristic slope of −½ is absent when the penetration ratio, b = hp/h, is greater than 20% [66], as shown in Figure 2.16.

Example 2.5

Abbott et al. [90] presented pressure‐time data for a pressure drawdown test. Well no. 20 is partially completed in a massive carbonate reservoir. The well was shut‐in for stabilization and then flowed to 5200 BOPD for 8.5 h. The pressure data are given in Table 2.4 and reservoir and fluid properties are given below:

h = 302 ftrw = 0.246 fPi = 2298 psia
hp = 20 ftq = 5200 BPDB = 1.7 bbl/STB
φ = 0.2μ = 0.21 cpct = 34.2 × 10−6 psia−1

Figure 2.15.

Pressure derivative spherical source solution for a single well in an infinite system including WBS and no skin, after [66].

Figure 2.16.

Pressure derivative behavior for a single well in an infinite reservoir with different partial penetration lengths (CD = 0, s = 0), after [66].

Solution by conventional analysis

Using the slope value of −122 psia/cycle read from the semilog plot of Figure 2.17, the reservoir permeability is calculated with Eq. (2.38);

k=162.6qBμmh=|162.6(5200)(1.7)(0.21)(122)(302)|=8.19md

The mechanical skin factor is determined with Eq. (2.39) once the intercept of 2252 psia is read from Figure 2.17.

s=1.1513[22522298122log(8.19(0.2)(0.21)(34.2×106)(0.246)2)+3.2275]=5.03
t, ht−0.5, h−0.5Pwf, psiaΔP, psiatP′, psia
0.022660
0.51.41422551111.5
1.01.00022432324.5
1.60.79122283840.0
2.00.70722184845.0
2.50.63222085852.5
3.00.57721976969.0
3.50.53521858166.5
4.00.50021788860.0
4.50.47121709656.3
5.50.426216110546.8
6.00.408215710948.0
6.50.392215311352.0
7.00.378214911749.0
7.50.365214612052.5
8.00.354214212448.0
8.50.3432140126

Table 2.4.

Pressure and pressure derivative versus time data for example 2.5.

Figure 2.18 contains a Cartesian graph of Pwf as a function of t−1/2. From there, the observed slope is m = 250 psia (h−1/2) and intercept, I = 2060 psia, spherical permeability, and spherical skin factors are calculated using Eqs. (2.239) and (2.240), respectively:

ksp=(2453qμBmϕμct)2/3=(2453(5200)(0.21)(1.7)250(0.2)(0.21)(34.2×10‐6)2/3=7.81md
ssp=(PiI)ksprsw70.6qμB1=(22982060)(7.81)(9.69)70.6(5200)(0.21)(1.7)1=0.86

Vertical permeability and spherical wellbore radius are found with Eq. (2.130) and (2.131), respectively,

kv=ksp3kh2=7.8138.192=7.1md
rsw=b2ln(brw)=1202ln(1200.246)=9.69ft

With the value of the vertical permeability, it is possible to estimate the skin factor caused by partial penetration with Eqs. (2.125), (2.138), and (2.137), thus:

Figure 2.17.

Semilog plot for well no. 20.

Figure 2.18.

Cartesian spherical plot for well no. 20.

Figure 2.19.

Pressure and pressure derivative versus time log‐log plot for well no. 20.

hD=khkv(hrw)=8.267.1(3020.246)=1324.1
G=2.9487.363b+11.45b24.675b3=2.9487.363(120302)+11.45(120302)24.675(120302)3=1.57
sc=(1hp/h1)[lnhDG]=(1120/3021)[ln1318.51.57]=8.51

Solution by TDS technique

The following data points were read from Figure 2.19.

tN = 1 hΔP = 23 psia
(tP′)sp = 56.25 psiaΔPs = 96 psiatsp = 4.5 h
(tP′)r2 = 52.5 psiaΔPr2 = 96 psiatr2 = 7.5 h

Wellbore storage coefficient is found from Eq. (2.66)

C=(qB24)tN(ΔP)N=((5200)(1.7)24)123=16.01 bbl/psi

From the spherical flow pressure derivative line, m = −1/2, the spherical permeability and mechanical spherical skin factor are, respectively, estimated by Eqs. (2.144) and (2.145);

ksp=(1227qBμ(t*ΔP')spϕμcttsp)2/3=(1227(5200)(1.7)(0.21)56.25(0.2)(0.21)(34.2×10‐6)4.5)2/3=8.05md
ssp=34.74ϕμctrsw2ksptsp[(ΔPw)sp2(t*ΔP')sp]1ssp=34.74(0.2)(0.21)(34.2 x 10‐6)(9.692)(8.05)(4.5)[(96)2(56.25)]1=0.93

The horizontal permeability and mechanical skin are found during the late radial flow using Eqs. (2.76) and (2.97), respectively;

kr=70.6qBμh(t*ΔP')r=70.6(5200)(1.7)(0.21)(302)(52.5)=8.26md
s=0.5[ΔPr(t*ΔP')rln(krtrϕμctrw2)+7.43]=0.5[12052.5ln((8.26)(7.5)(0.2)(0.21)(34.2×10‐6)(0.2462))+7.43]=5.33

Vertical permeability is determined from Eq. (2.130);

kv=ksp3kh2=8.0538.262=7.65md

Table 2.5 presents the comparison of the results obtained by the conventional method and TDS technique.

ParameterConventionalTDS
ksp, md7.018.05
ssp−0.86−0.93
k¯, md8.198.26
sr−5.03−5.53
kv, md7.107.65

Table 2.5.

Comparison of results.

2.5. Multirate testing

So far, the considerations revolve around a single flow test, meaning the production rate is kept constant for the application of the solution of the diffusivity equation. However, there are cases in which the flow rate changes; in such cases, the use of the solution to the diffusivity equation requires the application of the time superposition principle already studied in Section 1.14.2. Some reasons for the use of multirate testing are outlined as follows:

  • It is often impractical to keep a constant rate for a long time to perform a complete pressure drawdown test.

  • When the well was not shut‐in long enough to reach the static pressure before the pressure drawdown test started. It implies superposition effects.

  • When, it is not economically feasible shutting‐in a well to run a pressure buildup test.

Whether the production rates are constant or not during the flow periods, there are mainly four types of multirate tests:

  1. Uncontrolled variable flow rate;

  2. Series of constant flow rates;

  3. Variable flow rate while keeping constant bottom‐hole pressure, Pwf. This test is common in gas wells producing very tight formations and more recently applied on testing of shale formations;

  4. Pressure buildup (fall‐off) tests.

Actually, a holistic classification of transient well testing is given in Figure 1.4. It starts with PTA which is known in the oil argot as pressure transient analysis. As seen in the figure, it is divided in single well tests, normally known as drawdown (flow) tests for production cases or injection tests for injection fluid projects. Our field of interest focuses on more than one rate operation (multirate testing) which includes all the four types just above described. It is worth to mention types 3 and 4. Type three is also known as rate transient analysis (RTA) which has been dealt with in a full chapter by this book's author in reference [56]. As far as case 4 is concerned, pressure buildup testing is the most basic multirate test ever existed since it comprises two flow rates: (1) one time period at a given q value different than zero and (2) another time period with a zero flow rate. This is because when a well is shut‐in, the flow stops at surface by the formation keeps still providing fluid to the well due to inertia.

2.5.1 Conventional analysis

Considering the sketch of Figure 2.20, application of the superposition principle [2, 3, 4, 6, 7, 11, 27, 44, 56, 60, 65, 67, 95, 96, 97] leads to:

Pwf(t)=Pi141.2μBkh{q1[PD(tD)+s]+(q2q1)[PD([tt1]D)+s]+(q3q2)[PD([tt2]D)+s]+(q4q3)[PD([tt3]D)+s]+….+(qNqN1)[PD([ttN]D)+s]}E2.160

Rearranging;

Pwf(t)=Pi141.2μBkh{q1{PD(tD)PD([tt1]D}+q2{PD([tt1])DPD([tt2])D}++qN1{PD([ttN2])DPD([ttN1])D}++qN{PD([ttN1])D}+s}E2.161

Next step is to replace PD by an appropriate diffusivity equation solution which depends upon the flow regime dealt with. Figure 2.21 presents the most typical superposition functions applied to individual flow regimes. The normal case is to use radial flow, top function in Figure 2.21. However, Escobar et al. [44] presented the inconvenience of not applying the appropriate superposition function for a given flow regime. They found, for instance, that if the radial superposition is used, instead of the linear, for characterization of an infinite‐conductivity hydraulic fracture, the estimated half‐fracture length would be almost three times longer than the actual one.

Coming back to Eq. (2.161), the assumed superposition function to be used is the radial one; then, this equation becomes:

Pwf(t)=Pi70.6μBkh{q1ln(ttt1)+q2ln(tt1tt2)+q3ln(tt2tt3)+qN1ln(ttN2ttN1)+qN{ln(ttN1)}+lnkϕμctrw27.4316+2s}E2.162

Figure 2.20.

Schematic representation of a multirate test (typ. 1).

Figure 2.21.

Flow regime superposition functions.

Since it is uneasy to find natural log paper in the stationary shops, then, dividing for the natural log of 10 is recommended to express Eq. (2.162) in decadic log; then,

Pwf(t)=Pi162.6μBkh{j=1N1qjlog(ttj1ttj)+qN{log(ttN1)}+logkϕμctrw23.2275+0.8686s}E2.163

Simplifying;

PiPwf(t)qN=162.6μBkh{j=1N(qjqj1qN)log(ttj1)+logkϕμctrw23.2275+0.8686s}E2.164

Let;

s'=logkϕμctrw23.23+0.87sE2.165
m'=162.6μBkhE2.166

Solving for skin factor from Eq. (2.165);

s=1.1513[b'm'logkϕμctrw2+3.23]E2.167
Xn=i=1n(qiqi1qn)log(tti1)E2.168

Plugging Eqs. (2.165), (2.166), and (2.168) into Eq. (2.164) will lead to:

PiPwf(t)qn=m'Xn+m's'E2.169

which indicates that a Cartesian plot of ΔP/qn against the superposition time, Xn, provides a straight line which slope, m', and intercept, m'b’ allows finding reservoir permeability and skin factor using Eqs. (2.166) and (2.167), respectively. However, it is customary for radial flow well interpretation to employ a semilog plot instead of a Cartesian plot. This issue is easily solved by taking the antilogarithm to the superposition function resulting into the equivalent time, teq. Under this situation, Eq. (2.169) becomes:

PiPwf(tn)qn=mnlogteq+bnE2.170

And the equivalent time is then defined by,

PiPwf(tn)qn=mnlogteq+bnE2.171
teq=i=1n(tnti1)(qiqi1)/qn=10XnE2.172

For a two‐rate case, Russell [96] developed the governing well‐flowing pressure equation, as follows:

Pwf=m'1[log(t1+ΔtΔt)+q2q1log(Δt)]+PINTE2.173

Therefore, the slope, m'1, and intercept, PINT, of a Cartesian plot of Pwf versus log[(t1t)/Δt] + (q2/q1)log(Δt) allows finding permeability and skin factor from the following relationships:

k=162.6q1μBm1'hE2.174
s=1.1513[q1q1q2(Pwf(Δt=0)P1hrm'1)logkϕμctrw2+3.23]E2.175

In general, the lag time, tlag, transition occurred during the rate change, is shorter when there is a rate reduction than a rate increment, i.e., if q2 < q1, then the tlag will be short and if q2 > q1, then the tlag will be longer due to wellbore storage effects.

The pressure drop across the damage zone is:

ΔPs(q1)=0.87(m'1)sE2.176
ΔPs(q2)=0.87q2q1(m'1)sE2.177

And;

P=Pintq1(q2q1)[Pwf(Δt=0)P1hr]E2.178

P* is known as “false pressure” and is often used to estimate the average reservoir pressure which is treated in Chapter 3.

2.5.2 TDS technique

The mathematical details of the derivation of the equations are presented in detail by Perrine [86]. Application of TDS technique requires estimating the following parameters:

ΔPq=PiP(tn)qnE2.179
tn=tn1+ΔtE2.180

And equivalent time, teq, estimation is achieved using Eq. (2.172). Mongi and Tiab [67] suggested for moderate flow rate variation, to use real time rather than equivalent time with excellent results. In contrast, sudden changes in the flow rate provide unacceptable results. However, it is recommended here to always use equivalent time as will be demonstrated in the following exercise where using equivalent time the pressure derivative provides a better description. Mongi and Tiab [67] also recommended that test data be recorded at equal intervals of time to obtain smoother derivatives. However, it is not a practical suggestion since derivative plot is given in log coordinates. TDS is also applicable to two‐rate tests and there is also a TDS technique where there is a constant flow rate proceeded by a variable flow rate. For variable injection tests, refer to [60].

With the equivalent time, Eq. (2.172) determines the pressure derivative, teq*(DP/q)', and plot the derivative in a similar fashion as in Section 2.2.4; wellbore storage coefficient can be obtained by taking any point on the early‐time unit‐slope line by:

C=(B24)(tΔPq)E2.181

Permeability and mechanical skin factor are estimated from:

k=70.6μBh(teq*ΔPq)rE2.182
s=0.5[(ΔPq)r(teq*ΔPq')rln(k(teq)rϕμctrw2)+7.43]E2.183

Once again, rigorous time instead of equivalent time can be used in Eqs. (2.182) and (2.183); however, a glance to Figure 2.23 and 2.24 tells us not to do so.

Example 2.6

Earlougher and Kersch [8] presented an example to estimate permeability using a Cartesian plot of flowing pressure, Pwf, versus superposition time, Xn, and demonstrated the tedious application of Eq. (2.168). A slope of 0.227 psia/(BPD/cycle) was estimated which was used in Eq. (2.166) to allow finding a permeability value of 13.6. We determined an intercept of 0.5532 psia/(BPD/cycle) which led us to find a skin factor of −3.87 with Eq. (2.167).

Use semilog conventional analysis and TDS technique to find reservoir permeability and skin factor, as well. Pressure and rate data are given in Table 2.6 along another parameters estimated here. Reservoir, fluid, and well parameters are given below:

Pi = 2906 psiaB = 1.27 bbl/STBµ = 0.6 cp
h = 40 ftrw = 0.29 ftφ = 11.2%ct = 2.4 × 10−61/psia

Solution by semilog conventional analysis

Figure 2.22 is a semilog graph of [PiPwf(t)]/qn versus t and teq. The purpose of this graph is to compare between the rigorous analysis using equivalent time, teq, and analysis using the real time of flow, t. Note that during the first cycle, the graphs of t and teq are practically the same. By regression for the real‐time case gave a slope m' = 0.2411 psia/BPD/cycle and intercept ΔP/q(1hr) = 0.553 psia/BPD/cycle. Permeability and skin factors are calculated with the Eqs. (2.166) and (2.167), respectively:

k=162.6μBm'h=162.6(1.27)(0.6)0.2411(40)=12.84md
s=1.1513[0.5530.2411log(12.84(0.112)(0.6)(2.4×106)(0.292))+3.23]=3.98

The straight line with teq has a slope m’ = 0.2296 psia/BPD/cycle, and intercept ΔP/q(1hr) = 0.5532 psia/BPD/cycle. Then, permeability and skin factor estimated by Eqs. (2.166) and (2.167) are 13.49 md and −3.87, respectively.

Solution by TDS technique

The derivative of normalized pressure is also reported in Table 2.6. Figure 2.23 illustrates a log‐log plot of ΔPq versus teq and (t*ΔP'q) and (teq*ΔP'q) versus t and teq. Both derivatives were estimated with a smooth value of 0.5. During the first cycle, the two sets of data have roughly the same trend; also the flow regimes are quite different. Also, the equivalent normalized pressure derivative suggests a faulted system and possibly the pseudosteady‐state period has been reached. This last situation is unseen in the normalized pressure derivative. From this graph, the following values are read:

(t*ΔP'q)r = 0.097 psia/BPD/cycle(ΔPq)r = 0.693 psia/BPD/cycle(teq)r = 4.208 h

Permeability and skin factor are estimated, respectively, using Eqs. (2.182) and (2.183):

k=70.6μBh(t*ΔPq)r=(70.6)(1.27)(0.6)0.097(40)=13.86md
s=0.5[0.6930.097ln((13.86)(4.208)(0.112)(0.6)(2.4×106)(0.292))+7.43]=3.804

The comparison of the results obtained by the different methods is summarized in Table 2.7. The permeability absolute deviation with respect to arithmetic mean is less than 5% using actual time. Note that all results agree well. Even though, when Earlougher and Kersch [8] written, pressure derivative function was still in diapers; then, it was not possible to differentiate the second straight‐line which for Earlougher and Kersch [8] corresponded to pseudosteady‐state period instead of a fault as clearly seen in Figure 2.23. Also, the absolute deviation of the flow rate (referred to the first value) is less than 10% during radial flow regime. However, when using real time, the radial flow regime is different; then, the recommendation is to always use equivalent time.

nt, hq, BPDPwf, psiaΔP, psiaΔP /q, psia/BPDXnteq, ht*(ΔP/q)', psia/BPDteq*(ΔP/q)', psia/BPD
02906
11158020238830.5590.0001.0000.5590.261
11.5158019689380.5940.1761.5000.5940.131
11.89158019419650.6110.2761.8900.6110.102
12.41580
231490189210140.6810.5193.3060.6810.099
23.451490188210240.6870.5693.7070.6870.103
23.981490187310330.6930.6244.2080.6930.099
24.51490186710390.6970.6734.7120.6970.095
24.81490
35.51440185310530.7310.7876.1240.7310.104
36.051440184310630.7380.8196.5960.7380.111
36.551440183410720.7440.8497.0560.7440.120
371440183010760.7470.8747.4810.7470.128
37.21440
47.51370182710790.7880.9749.4120.7880.148
48.951370182110850.7921.00910.2120.7920.154
49.61370
5101300181510910.8391.12413.3110.8390.192
5121300179711090.8531.15314.2390.8530.188
614.41260
7151190177511310.9501.33721.7460.9500.205
7181190177111350.9541.35522.6620.9540.206
719.21190
8201160177211340.9781.42326.4570.9780.208
821.61160
9241137175611501.0111.48530.5531.0110.208
1028.81106
11301080175111551.0691.60740.4261.0690.248
1133.61080
12361000
1336.2983175611501.1701.78861.4141.1700.447
1348983174311631.1831.79963.0201.1830.463

Table 2.6.

Pressure and rate data for example 2.6, after [8].*

The three last columns are not given in [8].


2.6. Pressure drawdown tests in developed reservoirs

Slider [11, 98, 99] suggested a methodology to analyze pressure tests when there are no constant conditions prior to the test. Figure 2.24 schematizes a well with the shutting‐in pressure declining (solid line) before the flow test started at a time t1. The dotted line represents future extrapolation without the effect of other wells in the reservoir. The production starts at t1 and the pressure behaves as shown by the solid line [11].

Figure 2.22.

Semilog of normalized pressure versus actual and equivalent time for example 2.6.

Figure 2.23.

Normalized pressure and pressure derivative versus time and equivalent time log‐log plot for example 2.6.

2.6.1 Conventional analysis

The procedure suggested by Slider [11, 99] to correctly analyze such tests is presented below:

  1. Extrapolate the shutting‐in pressure correctly (dotted line in Figure 2.24).

  2. Estimate the difference between the observed well‐flowing pressure and the extrapolated pressure, ΔPΔt.

  3. Graph ΔPΔt vs. Log Δt. This should give a straight line which slope and intercept can be used for estimation of permeability and skin factor using Eqs. (2.38) and (2.39), respectively. For this particular case, Eq. (2.39) is rewritten as:

    s=1.1513[(ΔPΔt)1hrmlog(kϕμctrw2)+3.23]E2.184

However, this analysis could be modified as follows [8, 11, 21, 98, 99]. Consider a shut‐in developed with other wells in operation. There is a pressure decline in the shut‐in well resulting from the production of the other wells (superposition). After the test, well has been put into production at time t1, its pressure will be:

Pwf=Pi141.2qμBkh[PD(ΔtD,rD=1,)+s]ΔPow(t)E2.185

According to Figure 2.24, ΔPwo(t) is the pressure drop referred to Pi caused by other wells in the reservoir and measured at a time t = t1 + Δt. ΔPwo(t) can be estimated by superposing by:

ΔPow(t)=PiPw(t)=141.2μkhj=2nqjBjPD(tD,rDj)E2.186

Eq. (2.186) assumes that all wells start to produce at t = 0. This is not always true. Including wells that start at different times require a more complex superposition. If the other wells in the reservoir operate under pseudosteady‐state conditions, as is usually the case, Eq. (2.152) becomes:

ΔPow(t)=bm*tE2.187

The slope, m*, is negative when ΔPwo(t) vs. t is plotted. Instead, it is positive, if Pw vs. t is plotted. m* is estimated before the test well is opened in production at the pressure decline rate:

m*=dPwsdt=(Pws)2(Pws)1t2t1E2.188

Figure 2.24.

Behavior of a declination test in a depleted well, after [11, 21].

If pressure data is available before the test, m* can be easily estimated. Also, it can be estimated by an equation resulting from replacing Eq. (2.57) in (2.186):

m*=0.23395ϕcthAj=2nqjBjE2.189

The reservoir volume is given in ft3. Combining Eq. (1.106) with rD = 1, (1.94), (2.185), and (2.187), results:

Pwfm*Δt=mlogΔt+ΔP1hrE2.190

Eq. (2.190) indicates that a graph of Pwfmt vs. log Δt gives a straight line of slope m and intercept ΔP1hr at Δt = 1 h. The permeability can be found from Eq. (2.38). The skin is estimated from an arrangement of Eq. (2.39):

s=1.1513[ΔP1hrPws(Δt=0)mlog(kϕμctrw2)+3.23]E2.191

2.6.2 TDS technique

TDS technique for developed reservoirs was extended by Escobar and Montealegre [21]. Escobar and Montealegre [21] showed that the technique could be applied taking the derivative to the pressure in a rigorous way, that is to say, without considering the effect of the production of other wells. As it will be seen in the example 2.7, this is not recommended since the derivative is not correctly defined and, therefore, the results could include deviations above 10%. In this case, it is advisable to correct or extrapolate the pressure by means of Eq. (2.192) and, then, take the extrapolated pressure derivative and apply the normal equations of the TDS technique given in Section 2.2.4. Needless to say that any TDS technique equation can also be used once the pressure derivative has been properly estimated with the extrapolated pressure:

Pext=Pwfm*ΔtE2.192

Example 2.7

Escobar and Montealegre [21] presented a simulated pressure test of a square‐shaped reservoir with an area of 2295.7 acres having a testing well 1 in the center and another well 2 at 1956 ft north of well 1. Well 2 produced at a rate of 500 BPD during 14000 h. After 4000 h of flow, well 1 was opened at a flow rate of 320 BPD to run a pressure drawdown test which data are reported in Table 2.8 and Figure 2.26. The data used for the simulation were:

Methodologyk, mds
Superposition time, Cartesian plot13.6−3.87
Equivalent time, semilog plot13.49−3.87
Actual time, semilog plot12.84−3.98
TDS13.86−3.794
Average13.45−3.88

Table 2.7.

Comparison of estimated results of example 2.6.

t, hPwf, psiat, hPwf, psiat, hPwf, psia
050004000.004278.937091.282007.41
4.515000.00014000.104134.447511.281899.99
10.104999.984000.204015.567931.281792.61
56.794991.084000.403830.828351.281685.19
100.984970.974000.643676.328771.281577.72
201.484926.984001.133478.409191.281470.25
319.334887.164001.803345.409611.281362.85
402.024864.594005.063166.1110031.281255.45
506.114840.134017.963039.9010451.281148.04
637.154813.274090.002891.6510871.281040.57
802.134782.994201.482807.8511291.28933.06
1009.824747.744402.022720.0011711.28825.63
1271.284705.414637.152644.7012131.28718.26
1551.284661.105009.822542.1612551.28610.85
2111.284573.465411.282437.6112971.28503.40
2671.284486.125831.282329.7013391.28395.95
3091.284420.636251.282222.2613811.28288.51
3511.284355.136671.282114.8514000.00240.23

Table 2.8.

Pressure data of a developed reservoir in example 2.7, after [21].

rw = 0.3 pieμ = 3 cpct = 3 × 10−6 psia−1h = 30 pies
ϕ = 10%B = 1.2 bbl/BFk = 33.33 mds = 0

Interpret this test using conventional and TDS techniques considering and without considering the presence of well 2.

Solution by conventional analysis

A pressure change is observed in well 1 up to a time of 4000 h, after which it is put into production for the declination test, as shown in Figure 2.25. Figure 2.26 presents a plot of Pwf vs. log Δt obtained with the information in Table 2.9. Hence, the slope and intercept are, respectively, −230 psia/cycle and 3330.9 psia. Permeability and skin factor are, respectively, estimated from Eqs. (2.38) and (2.191):

Δt, hPwf, psiaΔPwf, psiatPwf′, psiaPext, psiaΔPext, psiatPext′, psia
0.004278.930.000.004278.930.000.00
0.014263.1715.7616.174263.1715.7616.17
0.024247.8031.1331.504247.8031.1331.50
0.034232.7746.1646.154232.7746.1646.14
0.054203.6575.2873.434203.6675.2773.42
0.064189.5389.4086.244189.5489.3986.23
0.084162.11116.83110.144162.12116.81110.13
0.1134118.69160.24145.704118.71160.22145.69
0.1604062.04216.89187.894062.06216.87187.86
0.2263989.51289.42234.793989.55289.38234.75
0.3193899.81379.12281.463899.86379.07281.41
0.4513793.92485.01319.723793.99484.94319.65
0.6373676.32602.61339.603676.42602.51339.50
0.9003555.47723.47333.333555.61723.33333.19
1.2713442.11836.82300.483442.31836.63300.27
1.7963345.40933.53250.083345.68933.25249.79
2.5373269.251009.68196.693269.651009.28196.28
3.5833211.351067.58152.723211.911067.02152.14
5.0613166.111112.82123.223166.901112.03122.39
7.1493128.101150.84106.393129.211149.72105.23
10.0983093.701185.2397.753095.281183.6596.11
16.0053050.461228.4792.623052.961225.9790.01
22.613018.941259.9991.113022.471256.4687.43
31.932987.691291.2590.862992.671286.2685.67
45.112956.321322.6191.762963.371315.5784.42
63.722924.461354.4793.892934.411344.5283.53
90.002891.651387.2897.542905.701373.2382.89
127.132857.331421.60103.152877.171401.7682.47
179.572820.741458.19111.392848.771430.1682.17
253.652780.811498.12123.242820.411458.5281.97
358.302736.241542.70140.712792.171486.7682.41
506.112684.751594.18167.762763.761515.1785.41
714.902622.341656.59211.072733.941544.9994.76
1009.822542.161736.77279.692699.801579.13115.40
1411.282437.611841.32380.452657.931621.01150.84
1831.282329.701949.23489.812615.591663.35191.86
2251.282222.262056.67600.722573.711705.22234.44
2811.282079.042199.90749.482517.911761.03292.10
3371.281935.782343.15898.682462.071816.86350.18
4071.281756.822522.111085.262392.381886.55422.88
4771.281577.722701.211271.942322.571956.36495.67
5611.281362.852916.081495.852238.832040.10582.91
6451.281148.043130.891719.922155.152123.78670.32
6591.281112.233166.711757.272141.192137.74684.89
7711.28825.633453.312055.972029.432249.50801.38
8831.28539.223739.712354.681917.882361.06917.87
9951.28252.694026.242653.471806.192472.741034.44
10000.00240.234038.702666.471801.332477.601039.51

Table 2.9.

Data of Pwf, Pext = Pwf−m*Δt, t*ΔPwf′, t*ΔPext′ for example 2.7, after [21].

Figure 2.25.

Cartesian plot of pressure versus time data simulated for well 1, after [21].

k=162.2qμBhm=162.6(320)(3)(1.2)30(230)=27.15md
s=1.1513[3330.94278.93230log(27.15(0.1)(3)(3×106)(0.32))+3.23]=1.35

Table 2.9 also reports the data of Pwfmt. Figure 2.26 presents, in addition, the plot of Pwfmt vs. log Δt. Now, the slope and intercept are, respectively, 193.9 psia/cycle and 3285.9 psia. A permeability of 32.2 md is found from Eq. (2.38) and a skin factor of −0.28 is estimated from Eq. (2.191).

Figure 2.26.

Semilog plot for example 2.7, after [21].

Figure 2.27.

Log‐log plot of pressures and pressure derivatives versus time for example 2.7, after [21].

From the derivative plot, Figure 2.27, we can observe that the pseudosteady‐state period has been perfectly developed; as a consequence, we can obtain the Cartesian slopes performing a linear regression with the last 10 pressure points, namely: m* (Pwf vs. Δt) = −0.256 psia/h and m* (Pext vs. Δt) = −0.0992 psia/h. Eq. (2.59) allows obtaining the well drainage area of well 2:

A(Pwf)=0.23395qBϕcthm*=0.23395(320)(1.2)(0.1)(3×106)(30)(0.256)(43560)=895.2Ac
A(Pext)=0.23395qBϕcthm*=0.23395(320)(1.2)(0.1)(3×106)(30)(0.0992)(43560)=2310Ac

Solution by TDS technique

Application of TDS, the pressure derivative is initially taken to the well‐flowing pressure data, see Table 2.9. Then, the derivative is taken to the corrected pressure, Pwfmt. Both pressure derivatives are reported in Figure 2.27. For the uncorrected pressure, the following information was read from Figure 2.27:

tr = 35.826 h(tP′)r = 90.4 psiaΔPr = 1301.7 psia

Permeability and skin factor are calculated with Eqs. (2.76) and (2.97);

k=70.6qμBh(t*ΔP')r=70.6(320)(3)(1.2)30(90.4)30md
s=0.5[1301.790.4ln((30)(35.826)(0.1)(3)(3×106)(0.32))+7.43]=0.74

Then, for the corrected pressure case, the following data were read from Figure 2.27;

tr = 319.3321 h(tP′)r = 82.1177 psiaΔPr = 1477.3508 psia

With these data, Eq. (2.76) provided a permeability value of 33.07 md and Eq. (2.97) allows estimating a skin factor of −0.087. Eq. (2.102) is used to find the well drainage area using trpi = 376.6049 h (uncorrected pressure) and trpi = 800.5503 h (corrected pressure) read from Figure 2.28, then,

APwf=ktrpssi301.77ϕμct=(30)(376.604)301.77(0.1)(3)(3×106)143560=955 Ac
APext=ktrpssi301.77ϕμct=(33.07)(800.5503)301.77(0.1)(3)(3×106)143560=2237.8 Ac
Methodk, mdAbs. error, %sAbs. error, %
Simulation33.330
Semilog with Pwf27.1518.54−1.35135
Semilog with Pext32.23.39−0.2929
TDS with Pwf309.99−0.7474
TDS with Pext33.070.78−0.0878.7

Table 2.10.

Permeability and skin factor results for example 2.7, after [21].

Figure 2.28 provides a comparison of the derivative of the flowing bottom pressure ignoring the effect of well 2 and the pressure derivative including the effect of well 2. It is noted there that the radial flow zone is shorter and less defined. On the other hand, the pseudosteady‐state zone appears first when the effect of the adjacent well is not included, indicating that the well drainage area, and therefore, the reserves present therein will be substantially underestimated. Table 2.10 shows all the permeability and skin factor values obtained for this example with their respective absolute errors with reference to the input simulation value. TDS when corrected pressure is taken gives the best results.

2.7. Elongated systems

These deposits can be approximated to the geometry described by Figure 2.28. They mainly result from fluvial depositions (deltaic), commonly called channels, terrace faulting, and carbonate reefs. The possible flow regimes when the well is completely off‐center are presented in Figure 2.28b when the parallel reservoir boundaries are no‐flow type (closed). Once radial flow vanishes, two linear flows take place at both sides of the reservoir. This flow regime is normally known as linear flow regime, see Figure 2.27b; actually, it consists of two linear flow regimes forming a 180° angle between each other. Therefore, Escobar et al. [19] named it dual‐linear flow. Once the shorter reservoir boundary has been reached by the transient wave, only a unique linear flow is kept and lasts until the other boundary is reached. This unique flow is referred as single‐linear flow by Escobar et al. [19]. However, since linear flow is taken on one side of the reservoir, it is also known as hemilinear flow regime. This is the only linear flow taken place in the system depicted in Figure 2.28c.

Both linear flows are characterized by a slope of 0.5 in the pressure derivative curve. Figure 2.29 sketches the pressure derivative behavior of the mentioned systems.

2.7.1 TDS technique

The governing pressure and pressure derivative equations for the single‐linear and dual‐linear flow regimes are, respectively, given below [13, 16, 17, 18, 19, 20, 23, 24, 28, 29, 31, 35, 38, 55, 56]:

(PD)L=2πtDL+sL=2πtDWD+sLE2.193
(tD*PD')L=πtDWDE2.194
(PD)DL=2πtDWD+sDLE2.195
(tD*PD')DL=πtDWDE2.196

Being sL is the geometrical skin factor caused by converging from either radial to linear flow regime (well located at one end of reservoir sides, Figure 2.28c or from dual‐linear to linear flow (well off‐center well). sDL is the geometrical skin factor caused by converging from either radial to linear flow regime. The dimensionless parameters are defined by Escobar et al. [19] as:

Figure 2.28.

Reservoir geometry and description of flow regimes. (a) Reservoir approximated geometry, (b) Dual linear flow, (c) Single linear or hemilinear flow.

WD=YErwE2.197
tDL=tDWD2E2.198

The dimensionless distances are given by:

XD=2bxXEE2.199
YD=2byYEE2.200

Variables bx and by correspond to the nearest distances from the well to the reservoir boundaries in the directions x and y, respectively. See Figure 2.28a. Replacing Eqs. (1.94), (2.62) and (2.197) in Eq. (2.194) and solving for the root product of permeability by the reservoir width, YE, will yield:

kYE=7.2034qBh(t*ΔP')LtLμϕctE2.201

Since, TDS equations apply to either drawdown or buildup tests; then, when either t or Δt = 1 h, Eq. (2.200) becomes:

Figure 2.29.

Dimensionless well pressure derivative versus time behavior for a rectangular reservoir with the well located off‐center, after [19].

kYE=7.2034qBh(t*ΔP')L1μϕctE2.202

The root product of permeability by the reservoir width can be also calculated from the dual‐linear flow, DL. This can be performed by replacing also Eqs. (1.94), (2.62), and (2.197) into the dimensionless pressure derivative equation into Eq. (2.196) leading to:

kYE=4.064qBh(t*ΔP')DLtDLμϕctE2.203

Again at either t or Δt = 1 h, the above equation becomes:

kYE=4.064qBh(t*ΔP')DL1μϕctE2.204

2.7.1.1 Intersection points

For long production times, the pseudosteady‐state period is reached. Both pressure and pressure derivative are joined into a unit‐slope line, we obtain a straight line. The governing pressure derivative equation at this time is given by Eq. (2.101). For the systems dealt with in this section, Eq. (2.102) which uses the point of intersection radial‐pseudosteady state, Eq. (2.103) and (2.104) also apply. The straight line given by Eq. (2.101) also intersects the lines given by Eqs. (2.96) and (2.98); then, reservoir area can be found from such intersection times, thus, [13, 19]:

A=ktDLpssiYE2301.77ϕμctE2.205
A=ktLpssiYE2948.047ϕμctE2.206

Likewise, the intersection times of the line of infinite radial behavior of the pressure derivative (horizontal straight line) with the hemilinear and dual‐linear flow regimes lead to obtain reservoir width from:

YE=0.05756ktrDLiϕμctE2.207
YE=0.102ktrLiϕμctE2.208

As indicated by Tiab [71], the geometrical skin factors, or any skin factor, can be obtained by dividing the pressure equation by its derivative equation and solving for the skin factor. Following this, Escobar et al. [19] divided Eqs. (2.195) and (2.193) by Eqs. (2.196) and (2.194), respectively, after replacing the dimensionless quantities given by Eqs. (1.94), (1.89), (2.62), and (2.197) and solving for the geometrical skin factor will provide:

sDL=(ΔPDL(t*ΔP')DL2)119.601YEktDLϕμctsE2.209
sL=(ΔPL(t*ΔP')L2)134.743YEktLϕμctsDLE2.210

where both tDL and tL are read at any convenient point during each respective linear flow regime. The pressure and pressure derivative values, ΔPDL, tPDL′, ΔPL, and tPL′, used in either Eqs. Eqs. (2.209) or (2.210) are read at these arbitrary times. The characteristic points used so far in this section are better explained in Figure 2.29.

In linear deposits, when the well is off‐centered and there is a simultaneous action of the linear flow on one reservoir side and the steady state on the other side, a slope flow of −1/2 develops, which does not correspond to either spherical or hemispherical flows, see Figure 2.30. Given the isobaric geometry, this flow regime is called parabolic flow [19]. Although Sui et al. [100] called it dipolar flow, Escobar et al. [16, 17] performed numerical simulation and plotted the isobaric lines and found that the closest geometry shape corresponds to a parabola. The governing equations of this flow regime are:

PD=(WD)(XD)2(XEYE)2tD0.5+sPBE2.211
tD*PD'=WD2(XD)2(XEYE)2tD0.5E2.212

Once the division of the pressure equation by the pressure derivative equation is attained and the appropriate dimensionless expressions are replaced in the resulting division, the parabolic skin factor equation is obtained:

sPB=(ΔPPB(t*ΔP')PB+2)123.16bxYEϕμctktPBsDLE2.213

Also, by substituting the dimensionless quantities into Eq. (2.212), the following equation is derived:

k1.5YEbx2=17390[qμBh(t*ΔP')PB][ϕμcttPB]0.5E2.214

In the above two equations, the pressure and pressure derivative values are read to a convenient or arbitrary point, tPB.

The total skin factor for this type of reservoir is evaluated according to the flow regimes that are presented:

  • Well near a closed boundary. In this case, radial, dual‐linear, and hemilinear flows are presented.

    s=sr+sDL+sLE2.215

  • Well near an open boundary. In this case, radial, dual‐linear, and parabolic flows are presented.

    st=s+sDL+sPBE2.216

If dual‐linear is unseen, as presented in Figure 2.28c, Eq. (2.215) reduces to;

st=s+sLE2.217

Escobar and Montealegre [18] performed a detailed analysis of the geometrical skin factor causes.

Figure 2.30.

Dimensionless pressure derivative versus time behavior for a well displaying parabolic flow regime, after [19].

The points of intersection, see Figures 2.30 and 2.31, found between the different lines of the pressure derivative curve allows developing the following equations:

bx=165.41ktDLPBiϕμctE2.218
bx=(YE246.32)*(ktrPBiϕμct)0.5E2.219

For steady‐state cases, a negative unit‐slope line, SS1, tangent to the pressure derivative curve during late time is drawn. This occurs when the far boundary is at constant pressure. Its intercept with the parabolic flow straight line makes it possible to estimate the length of the reservoir, see Figure 2.30.

XE3=177.9(ktPBSS1iϕμct)bxE2.220

Several scenarios arise for cases of lateral constant‐pressure boundaries:

  • Intersection of the −1‐slope line with the dual‐linear flow regime line:

    XE3=(11.426×109)(ktDLSS1iϕμct)3(1bx3)E2.221

  • Intersection of the −1‐slope line with the radial flow regime line:

    XE3=(14.72×106)(ktrSS1iϕμct)2(YE2bx3)E2.222

  • Intersection of the −1‐slope line with the parabolic flow regime line:

    XE3=177.9(ktPBSS1iϕμct)bxE2.223

Again, a negative unit‐slope line, SS2, tangent to the pressure derivative curve during late time is drawn. This takes place when a no‐flow far boundary exists. Its intercept with the dual‐linear, radial, and parabolic flow straight lines can provide equations to estimate the length of the reservoir, see Figure 2.30.

  • Intersection of the −1‐slope line with the dual‐linear flow regime line:

    XE3=(11.42×1010)(ktDLSS2iϕμct)3(1bx3)E2.224

  • Intersection of the −1‐slope line with the radial flow regime line:

    XE3=(14.66×107)(ktrSS2iϕμct)2(YE2bx3)E2.225

  • Intersection of the −1‐slope line with the parabolic flow regime line:

    XE3=1768.4(ktPBSS2iϕμct)bxE2.226

From the inflection point between linear and dual‐linear flow, the position of the well can be obtained by any of the following relationships:

bx=ktF5448.2ϕμctE2.227
bx=khYE(t*ΔP')F415.84qμBE2.228

2.7.1.2 Maximum points

As seen in Figure 2.30, when the well is located near a constant‐boundary pressure but the far boundary has no‐flow, both parabolic flow regime and a maximum point, X1 (between dual linear and parabolic) are observed. If the far boundary is at constant pressure, another maximum, X2, can be developed. The first maximum is governed by:

(tD*PD')X1=23πWDtDX10.5E2.229
XEYE=23(πWDXD)tDX10.5E2.230
XEYE=23(πWDXD)tDX10.5E2.231

From which it is obtained:

bx=(158.8)(ktX1ϕμct)E2.232
bx=khYE(t*ΔP')X1159.327qμBE2.233

The second maximum has a governing equation given by:

(tD*PD')X2=πWD(XD2)tDX20.5E2.234
XEYE=(π2WD)tDX20.5E2.235
XEYE=(π2XD2)(tD*PD')X2E2.236

From which is obtained:

XE=637.3(bx2YE)(qμBkh)(1(t*ΔP')X2)E2.237
XE=139.2(ktX2ϕμct)0.5E2.238

When a rectangular reservoir has mixed boundaries and the well is near the no‐flow boundary, see Figure 2.31, another maximum point, X3, can be displayed on the pressure derivative once the constant‐pressure boundary is felt. The governing equation for this maximum point is:

XEYE=π1.54(1WD)tDX30.5E2.239

After replacing the dimensionless parameters and solving for the reservoir length, it will result:

XE=144.24ktX3ϕμctE2.240

Another steady‐state period is depicted in Figure 2.31 when the well is near a no‐flow boundary and the farther one is at constant pressure. Again, one negative‐unit‐slope line is drawn tangent to the pressure derivative curve. In this case, both dual‐linear flow and single linear flow regimes are developed. This is followed by a maximum. The governing equation of the mentioned negative slope line is:

tD*PD'=(XEYE)3WD2tDE2.241

Equating Eq. (2.75) with Eq. (2.241), an equation will be obtained that uses the radial and steady‐state intercept to find reservoir length:

XE=ktrSSiYE7584.2ϕμct3E2.242

If it is assumed that the area is obtained from the product of the width by the length of the reservoir, A = XEYE, then,

A=ktrSSiYE47584.2ϕμct3E2.243

When the well is centered along the rectangular reservoir, different behavior occurs if one or both boundaries are at constant pressure as seen in Figure 2.32. The equations of the straight line with unit slope passing tangentially to the pressure derivative curve are, respectively, given by the following expressions:

Figure 2.31.

Dimensionless pressure derivative versus time behavior for an off‐centered well near a no‐flow boundary and the far boundary is either at constant pressure or no flow, after [19].

tD*PD'=32WD219π(XEYE)3tD1E2.244
tD*PD'=WD25π(XEYE)3tD1E2.245

The equations to estimate the drainage area is obtained from the intercept of Eqs. (2.140) and (2.141) with Eq. (2.75). After replacing the dimensionless parameters and assuming perfect rectangular geometry, we, respectively, have:

A=ktrSSiYE44066ϕμct3E2.246
A=ktrSSiYE4482.84ϕμct3E2.247

The maximum point when one of the two boundaries is at constant pressure is given by:

XEYE=1532π(1WD)tDxc0.5E2.248

And, for the other case, when both extreme boundaries are subjected to a constant pressure:

XEYE=1516π(1WD)tDxc0.5E2.249

After replacing the dimensionless quantities in Eqs. (2.248) and (2.249), it is possible to find expressions to determine reservoir length and area, respectively:

Figure 2.32.

Dimensionless pressure derivative versus time behavior for a centered well when one or both boundaries are at constant pressure, after [28].

XE=141.82(ktxcϕμct)0.5E2.250
A=YE41.82(ktxcϕμct)0.5E2.251
A=YE41.82(ktxcϕμct)0.5E2.252
XE=120.91(ktxcϕμct)0.5E2.253
A=YE20.91(ktxcϕμct)0.5E2.254

Table 2.11.

Equations for area determination in constant‐pressure systems, after [28].

Escobar et al. [28] determined the pressure derivative governing equation for constant pressure both circular or square systems;

tD*PD'π84tDAE2.255

which intercept with the radial flow pressure derivative equation, Eq. (2.75), allows finding the well‐drainage area:

A=ktrSSi283.66ϕμctE2.256

Table 2.11 presents a summary of the different equations to determine the drainage area in constant‐pressure systems, since Escobar et al. [28] showed that Eq. (2.102) hugely fails in constant‐pressure systems.

Escobar et al. [23] presented TDS technique for long reservoirs when the width is known from another source, like seismic. Under this condition, the reservoir areal anisotropy and even the anisotropy angle can be determined. Later, Escobar et al. [29, 30] presented conventional analysis and TDS technique, respectively, when changes in either reservoir width or facies are seen in elongated systems.

2.7.2 Conventional method

The dimensional pressure governing equation for dual‐linear flow regime is [20, 24]:

ΔP=8.1282YEqBkh(μϕctk)0.5t+141.2qμBkh(sDLs)E2.257

For pressure buildup tests, the superposition principle leads to find:

ΔP=8.1282YEqBkh(μϕctk)0.5(tp+ΔtΔt)E2.258

Eqs. (2.257) and (2.258) indicate that a linear plot of pressure drop or pressure versus either t0.5 (for drawdown tests) or (tp+ Δt)0.5 − Δt0.5 (for buildup tests), tandem square root, will yield a straight line which slope, mDLF, and intercept, bDLF, are used to, respectively, find reservoir width, YE, and dual‐linear (geometrical) skin factor, sDL;

YE=8.1282qBmDLFh[μkϕct]0.5E2.259
sDL=khbDLF141.2qμBsE2.260

Wong et al. [101] presented another version of the skin equation:

sDL=12[khbDLF141.2qμB+ln(rwYE)]E2.261

Escobar and Montealegre [20] found that Eq. (2.260) compared quite well with the results of [59, 102]. The governing equations for drawdown and buildup, respectively, for hemilinear flow regime are [20, 24, 28]:

ΔPwf=14.407YEqμBkhktϕμct+141.2qμBkhsLE2.262
ΔPws=14.407YEqμBkhkϕμct(tp+ΔtΔt)E2.263

Similar to the dual‐linear case, when plotting in Cartesian coordinates either P or ΔP versus either t0.5 (for drawdown tests) or (tp+ Δt)0.5−Δt0.5 (for buildup tests), a straight line influenced by the linear flow will be obtained. Its slope, mLF, and intercept, bLF, are used, respectively, to estimate reservoir width, YE, and skin factor, sL.

YE=14.407mLFqBh(μϕctk)0.5E2.264
sL=khbLF141.2qμBsDLE2.265

The governing equations for parabolic flow regime under drawdown and buildup conditions are given by [13, 19, 20, 24]:

ΔPwf=34780.8qBbx2ϕcthYE(μk)1.51t+141.2qμBkhsPBE2.266
ΔPws=34780.8qBbx2ϕcthYE(μk)1.5(1tp+Δt1Δt)E2.267

A straight line will be observed on a Cartesian plot of either P or ΔP versus either 1/t0.5 (for drawdown tests) or 1/(tpt)0.5−1/Δt0.5 (for buildup tests). Its mPB and intercept, bPB, lead to obtain well position along the x‐direction, bx, and parabolic skin factor, sPB, respectively, from:

bx=mPBhYE34780.8qBϕct(kμ)1.5E2.268
sPB=khbPB141.2qμBsDLE2.269

Area, A = XEYE, can be found from the Cartesian plot of pressure versus time using Eq. (2.59).

Example 2.8

Escobar et al. [19] presented a pressure test run in a reservoir in South America. Test data are given in Table 2.12 and other relevant information is given below:

q = 1400 BPDh = 14 ftct = 9 × 10−6 psia−1Pi = 1326.28 psia
rw = 0.51 piesϕ = 24%B = 1.07 bbl/STBμ = 3.5 cp

It is required to conduct the interpretation of the test by TDS and conventional analysis.

Solution by TDS technique

The following information was read from Figure 2.33;

(tP′)r = 60 psiaΔPr = 122.424 psiatDL = 2 h
(t*ΔP′)DL = 105.81 psiaΔPDL = 265.942 psiatRDLi = 0.7 h
tPB = 10.157 h(t*ΔP′)PB = 132.873 psiaΔPPB = 458.466 psia
tPBDLi = 6 htPBRi = 50 htDLSS1i = 7.5 h
tRSS1i = 24 htPBSS1i = 12 h

Permeability is obtained from Eq. (2.76) and reservoir width with Eq. (2.203), respectively:

k=70.6qμBh(t*ΔP')r=70.6(1400)(3.5)(1.07)(14)(60)=440.7md
YE=4.064qBkh(t*ΔP')DLΔtDLμϕct=4.064(1400)(1.07)440.7(14)(105.81)(2)(3.5)(0.24)(9×106)=352.4ft

Verify YE with Eq. (2.207):

YE=0.05756ktRDLiϕμct=0.05756(440.7)(0.7)(0.24)(3.5)(9×106)=367.7ft

The well position along the reservoir is found with Eq. (2.214):

bx=440.71.5(367.8)17390[1400(3.5)(1.07)14(132.873)][0.24(3.5)(9×106)10.157]0.5=283.7ft

Verify bx of Eqs. (2.218) and (2.219):

bx=165.41ktDLPBiϕμct=165.41440.7*60.24(3.5)(9×106)=285.9ft
bx=(YE246.32)*(ktrPBiϕμct)0.5=(367.7246.32)(440.7(50)0.24(3.5)(9×106))0.5=283.9ft
t, hΔP, psiatP′, psiat, hΔP, psiatP′, psiat, hΔP, psiatP′, psia
0.16549.0051.004.331331.24147.659.824458.47136.70
0.33299.0067.004.498336.89149.1710.157463.23132.87
0.498122.4260.384.665342.30151.3910.490467.44134.03
0.665140.4966.574.831347.70152.7910.824471.48132.58
0.831156.0773.694.998352.84154.0911.157475.61130.61
0.998170.1480.125.165358.01155.8311.490479.47127.78
1.165182.9284.745.331362.96156.8711.824483.19126.71
1.331194.4888.675.498367.77157.8512.157486.70125.21
1.498205.1792.445.665372.54159.5112.490489.94122.97
1.665215.0996.635.831377.15159.5512.824493.12119.84
1.831224.53101.165.998381.67159.9413.157496.26117.32
1.998233.54105.286.165386.10161.2513.490499.19115.01
2.165242.11109.196.331390.50161.3113.824502.05113.78
2.331250.33113.126.498394.60161.7414.157504.71111.16
2.498258.24116.376.665398.63161.5814.490507.15109.55
2.665265.94120.686.831402.76161.8814.990510.78106.05
2.831273.23124.156.998406.64161.6615.490514.29102.52
2.998280.57126.977.165410.42161.9015.990517.4599.40
3.165287.49130.917.331414.19161.6616.490520.5997.21
3.331294.22133.247.657421.18161.2116.990523.4893.62
3.498300.85136.377.990428.16160.7317.490526.1090.44
3.665307.28138.938.324434.62153.7717.990528.5786.87
3.831313.54141.428.657440.94149.9920.474470.7950.29
3.998319.60143.738.990446.87146.5922.640538.4210.60
4.165325.50145.489.490453.81140.47

Table 2.12.

Pressure and pressure derivative data versus time for example 2.8.

Observe on the pressure derivative curve that once the parabolic flow is finished, before falling, it rises a little, from which it is inferred that the far boundary is of no flow, this maximum point is not observed with much clarity; then, Eqs. (2.224), (2.226), and (2.225), using the intersection of the −1‐slope line with the dual linear, parabolic, and radial flow lines are used:

XE3=(11.41×1010)(440.7×7.50.24×3.5×(9×106))3(12843)=637.2ft
XE3=(14.66×107)(440.7×240.24×3.5×(9×106))2(367.722843)=628.2ft
XE=(1768.4(440.7×120.24×3.5×(9×106))×284)1/3=637.1ft

Skin factor are found with Eqs. (2.97), (2.209), (2.210), and (2.213):

Figure 2.33.

Log‐log plot of pressure and pressure derivative versus time for example 2.8, after [19].

s=0.5[122.42460ln(440.7×0.50.24×3.5×9×106×0.332)+7.43]=4.9
sDL=(265.942105.812)134.743×367.7440.7×20.24×3.5×9×106=0.4+4.9=5.3
sPB=(458.466132.873+2)123.16(283.72)352.4(0.24)(3.5)(9×106)(440.7)(10.157)=6.35.3=1

The total skin factor is calculated from the sum of the partial skin factors, Eq. (2.216):

s = sr + sDL + sPB = −4.9 + 5.3 + 1 = 1.4

Solution by conventional analysis

The following information was read from Figure 2.34, 2.35, and 2.36;

m = 140 psia/cyclemDLF = −150.8bDLF = 19.4
mPB= −851.6bPB = 730.64P1hr = 1158 psia

Permeability is determined using the slope of the semilog plot, m, by means of Eq. (2.38) and mechanical skin factor with Eq. (2.39);

Figure 2.34.

Semilog plot for example 2.8, after [20].

Figure 2.35.

Cartesian plot of ΔP vs. t0.5 for example 2.8, after [20].

k=|162.6qμBhm|=|162.6(1400)(3.5)(1.07)(14)(140)|=434.96md
s=1.1513[1158140log(434.96(0.24)(3.5)(9×106)(0.51)2)+3.2275]=4.6

Using mDLF the reservoir width value is calculated Eq. (2.259);

Figure 2.36.

Cartesian plot of ΔP vs. 1/t−0.5 for example 2.8, after [20].

YE=8.1282(1400)(1.07)(150.8)(14)[(3.5)(434.96)(0.24)(9×106)]0.5=350ft

bx is found from Eq. (2.268):

bx=851.6(14)(350)34780.8(1400)(1.07)(0.24)(9×106)(434.963.5)1.5=277.4ft

Geometrical skin factor is found from Eqs. (2.260) and (2.269), thus:

sDL=khbDLF141.2qμB+4.6=(434.96)(14)(19.4)141.2(1400)(3.5)(1.07)=4.8
sPB=khbPB141.2qμBsDL=(434.96)(14)(730.64)141.2(1400)(3.5)(1.07)4.8=1.3

When comparing with the results of the simulation with those obtained by the TDS technique and those of the conventional method no greater difference is found.

2.8. Determination of average reservoir pressure from flow tests

Until 2010, pressure buildup tests, chapter 3, were the only means to determine the average pressure of a reservoir. However, in 2010, Agarwal [1] presented a methodology to obtain the average pressure from drawdown tests, using the following expression:

P¯=Pwf+887.18q B μkhE2.270

This does not consider the Dietz shape factor, but conditions that the well‐flowing pressure is determined at the point where the late pseudosteady‐state period develops. According to Agarwal [1], this point is determined using the arithmetic derivative (not multiplied by time). In the arithmetic derivative, the radial flow is presented with a slope of −1. The pseudosteady‐state period (postradial) takes place when the slope of the arithmetic derivative becomes zero (flat). That is the right moment where the well‐flowing pressure, Pwf, is read.

Nomenclature

Aarea, ft2
Bggas volume factor, ft3/STB
Booil volume factor, bbl/STB
Bwoil volume factor, bbl/STB
bfraction of penetration/completion
bDLFintercept of P vs t0.25 plot during dual‐linear flow, psia0.5
bLFintercept of P vs t0.5 plot during hemilinear flow, psia0.5
bPBintercept of P vs 1/t0.5 plot during hemilinear flow, h−1
bxdistance from closer lateral boundary to well along the x‐direction, ft
bydistance from closer lateral boundary to well along the y‐direction, ft
ccompressibility, 1/psia
Cwellbore storage coefficient, bbl/psia
CAreservoir shape factor
cttotal or system compressibility, 1/psia
DFdamage factor
DRdamage ratio
FEflow index
f(t)time function
hformation thickness, ft
hplength of perforations, ft
Iintercept
Jproductivity index, bbl/psia
kpermeability, md
kggas effective permeability, md
kooil effective permeability, md
kwwater effective permeability, md
IDcsginternal casing diameter, in
mslope of P vs log t plot, psia/h/cycle
m*slope of P vs t plot, psia/h
mDLFslope of P vs t0.25 plot during dual‐linear flow, psia0.5/h
mLFslope of P vs t0.5 plot during hemilinear flow, psia0.5/h
mPBslope of P vs 1/t0.5 plot during hemilinear flow, (psia0.5/h)−1
m′slope of superposition or equivalent time plot, psia/BPD/cycle
m′b′intercept of superposition or equivalent time plot, psia/BPD/cycle
m(P)pseudopressure, psia/cp
ODcsgexternal casing diameter, in
Ppressure, psia
average reservoir pressure, psia
PD′dimensionless pressure derivative
PDdimensionless pressure
Piinitial reservoir pressure, psia
PRproductivity ratio
Ptshut‐in casing pressure, psia
Pwfwell flowing pressure, psia
Pwswell shut‐in or static pressure, psia
P1hrintercept of the semilog plot, psia
P*false pressure, psia
ΔPspressure drop due to skin conditions, psia
qliquid flow rate, bbl/D
qscgas flow rate, Mscf/D
rDdimensionless radius
rradius, ft
redrainage radius, ft
Rsgas dissolved in oil, scf/STB
rwwell radius, ft
sskin factor
scskin due to partial penetration
scpskin due to a change in permeability
sDLgeometrical skin factor converging from radial to dual‐linear flow
sLgeometrical skin factor converging from dual‐linear to linear flow
spskin factor due to the restricted flow entry
sPBgeometrical skin factor converging from dual‐linear to parabolic flow
sttotal skin factor
skin factor resulting from a well deviation angle
Treservoir temperature, ºR, Transmissivity, md‐ft/cp
ttime, h
tpproduction (horner) time before shutting‐in a well, h
tDdimensionless time based on well radius
tDAdimensionless time based on reservoir area
t*DP′pressure derivative, psia
Vvolume, ft3
Vuwellbore volume/unit length, bbl/ft
Xdistance in the x‐direction
XEreservoir length, ft
XNsuperposition time, h
Ydistance in the y‐direction
YEreservoir width, ft
WDdimensionless reservoir width
Zheight, ft

Greek

Δchange, drop
Δtshut‐in time, h
φporosity, fraction
λmobility, md/cp
ρfluid density, lbm/ft3
θdeviation angle, °
ψmeasured deviation angle, °
ψ'corrected deviation angle, °
μviscosity, cp

Suffices

0base conditions
1hrtime of 1 h
aactual
ddrainage
Ddimensionless
DAdimensionless with respect to area
DLdual linear flow
DL1dual linear flow at 1 h
DLpssiintersection of pseudosteady‐state line with dual‐linear line
DLSS1intercept between dual‐linear line and the −1‐slope line (SS1)
DLSS2intercept between dual‐linear line and the −1‐slope line (SS2)
eqequivalent
Finflection
ggas
hhorizontal
hshemispherical
iintersection or initial conditions
idealideal
INTintercept
invinvestigation
Llinear or hemilinear flow
L1linear flow at 1 h
laglag
Lpssiintercept of linear and pseudosteady state lines
Nan arbitrary point during early pseudosteady‐state period
maxmaximum
ooil
OPoil price, US$/STB
pproduction, porous
PBparabolic flow
PBSS1intercept between parabolic line and the −1‐slope line (SS1)
PBSS2intercept between parabolic line and the −1‐slope line (SS2)
psspseudosteady state
pss1pseudosteady state at 1 h
rradial flow
rDLiintercept of radial and dual linear lines
r1radial flow before spherical/hemispherical flow
rgrelative to gas
rLiintercept of radial and linear lines
rorelative to oil
rpssiintersection of pseudosteady‐state line with radial line
rSSiintersection between the radial line and the −1‐slope line
rSS1iintersection between the radial line and the −1‐slope line (SS1)
rSS2iintersection between the radial line and the −1‐slope line (SS2)
rwrelative to water
sskin
sfsandface
spspherical
SSsteady
SSLstart of semilog line (radial flow)
SS1−1‐slope line formed when the parabolic flow ends and steady‐state flow regime starts. Well is near the open boundary and the far boundary is opened
SS2−1‐slope line formed when the parabolic flow ends and steady‐state flow regime starts. Well is near the open boundary and the far boundary is closed
swspherical/hemispherical wellbore
wwell, water
waapparent wellbore
wbwellbore
wDdimensionless emphasizing at wellbore
wfwell flowing
wswell shut‐in
xmaximum point (peak) during wellbore storage
xcmaximum point for centered wells
X1maximum point between dual linear and parabolic lines
X2maximum point between parabolic and negative unit slope lines
X3maximum point between hemilinear and negative unit slope lines
zvertical direction

Pressure Buildup Testing

A pressure buildup test has been a very popular technique used in the hydrocarbon industry. Several reasons have become a very popular test, some of these are: (a) it does not require very detailed supervision and (b) permeability and skin factor can be determined from both pressure buildup and drawdown tests. However, as studied in Section 2.8, until 2010, a flow test did not allow estimating the average reservoir pressure, while a pressure test does [7, 8, 9, 10, 25, 26, 28, 30].

Figure 3.1.

Schematic representation of pressure restoration.

Figure 3.1 shows a plot of an ideal pressure buildup test. In general terms, it requires shutting‐in a producer well after it has produced for some time, tp, with a stable flow rate. A pressure buildup test is run as follows:

  1. Place the pressure sensors in the selected site. It is recommended as close as possible to the perforations.

  2. Stabilize the well to a constant production rate, q.

  3. Close the well and record the Pwf value (just before closing).

  4. Read the well bottom-hole pressure, Pws, at short time intervals of 15 s for the first few minutes (10–15 min), then I could be every 10 min for the first hour. During the next 10 h, hourly pressure readings should be taken. When the test progresses, the time intervals can be expanded to 5 h. With recently introduced pressure recorders, the readings can be taken at shorter intervals. It can start reading every second or less.

To run a pressure buildup test, the well produces a constant rate for a period of time, tp. A pressure recorder is lowered to the well immediately before closing. tp should not be too small to avoid problems associated with superposition and investigation radius [12].

3.1. Superposition principle

Suppose that after the well has produced a constant rate for a time period, tp, it is decided to shut‐in the well to obtain a pressure buildup test. Intuitively, fluid movement is expected at the reservoir after the well is shut‐in, but at surface q = 0. A similar situation arises in fall‐off testing, but injections takes place instead of production. An analogy is made to the fluid movement at the reservoir [10, 11, 12, 28, 37, 40] as follows: the well is allowed to produce indefinitely at a flow rate, q, and at the instant of shutting‐in the well, the same flow rate, q, is injected into the same well, and then the pressure drop is added due to the production of q and same pressure data multiplied by −1 and displaced at the time the well is shut‐in. This, however, is not easy to understand. The better way is to understand, refer to Figure 3.1, is to estimate the well pressure drop at a time, tp + Δt, and then subtract the pressure drop during a time, Δt. Mathematically;

PDws=PD(tp+Δt)DPD(Δt)DE3.1

If tp is not given, it can be estimated if cumulative production, Np, is known,

tp=24NpqE3.2

Assuming wellbore storage is neglected and the reservoir is of infinite size; then, Eq. (1.115) applies:

PD(tp+Δt)D=12[ln(tp+Δt)D+0.80907]sE3.3
PD(Δt)D=12[ln(Δt)D+0.80907]sE3.4

Combining Eqs. (3.3) and (3.4) in Eq. (3.1), then replacing in the resulting combination the dimensionless parameter given by Eqs. (1.89) and (1.94) yields:

Pws=Pi162.6qμBkhlog(tp+ΔtΔt)E3.5

This is known as Horner equation. As a result of the application of the superposition principle is that the skin factor, s, disappears in the Horner’s simplified equation. That means the slope of the Horner plot is not affected by the skin factor. However, the skin factor alters, even greater than in flow tests, the shape of the pressure buildup curve. The skin factor affects the buildup test more than the drawdown test because wellbore storage persists.

3.2. Buildup test methods

3.2.1 Horner method

Eq. (3.5) suggests that a semi‐log plot of well‐shut‐in pressure versus (tp + Δt)/Δt will yield a straight line which slope allows finding the permeability from Eq. (2.34). Estimating the Horner time, (tpt)/Δt was tedious before 1970 when computer power was limited which is not today’s case. When superposition is overcome, the semi‐log plot of Pws versus Δt can be applied. In pressure buildup testing, this semi‐log plot is rather known as MDH plot, as shown later.

Horner plot is generally not preferred, since it requires more work than MDH. It is strongly recommended to be used when tp < tpss [12, 26]. This is because superposition effects make the semi‐log straight line difficult to identify. Actually, Horner plot virtually increases about four times the length of the semi‐log slope. If tp is at least twice the size of tpss, it is then justified to plot using tpss instead of tpss in finite systems [12, 26], since the Horner plot tends to prolong the semi‐log line. Preparing a Horner plot with tpss instead of tp has meaning to minimize errors in the estimation of the average pressure. However, with the advent of the pressure derivative function, the identification of radial flow became easier [6, 13, 25].

Just to look alike a MDH plot, Horner plot uses inverted abscissa scale as shown in Figure 3.2. For long producing times, a slight variation of Eq. (2.34) is used to find skin factor when tp> 1;

s=1.1513[P1hrPwfmlog(kϕμctrw2)+3.23]E3.6

Figure 3.2.

Typical Horner plot, tp = 83 hr.

Here Pwf is used instead of Pi. Pwf is the registered pressure just before shutting‐in the well. Finding P1hr requires using tp and adding one to that value. Use that estimated (tp + Δt)/Δt value and enter the Horner plot and read on the semi‐log line the value of P1hr. It is meaningless to estimate (tp + Δt)/Δt = 1. However, such reading will be used later to estimate the average reservoir pressure. When tp < 1, the following expression ought to be used to find skin factor:

s=1.1513[P1hrPwfm+log(1+1tp)log(kϕμctrw2)+3.2275]E3.7

Once the skin factor is estimated, the skin pressure drop and flow efficient can be found using Eqs. (2.35) and (2.46).

When the well is shut‐in for a buildup test, the formation fluid keeps flowing, even though using downhole devise shutting. Again, this after‐flow duration can be determined easily from the pressure derivative plot once radial flow starts. This was not the case before 1980s. This after‐flow rate, qaf, due to the wellbore storage, has a significant influence on the pressure data. This occurs because the head pressure is not equal to the bottom shut‐in pressure, and therefore the fluid continues to flow from the formation to the well. Then the pressure does not recover as fast as expected. As the flow rate tends to zero, the pressure increases rapidly. The semi‐log graph is pronounced and linear in this period and can be confused with the semi‐log slope [10, 12, 26, 28, 37, 40].

qaf=24CVwBdPwsd(Δt)E3.8

C is found from transient pressure analysis using Eq. (2.18). For producer and injector wells, respectively, the after‐flow duration can be estimated from [40];

Δtaf=204(CBJ)E3.9
Δtaf204(CVuBJ)E3.10

where J is the productivity index, Eq. (2.44), B is the volume factor and, Vu, the wellbore capacity, Eq. (2.4), and C is the wellbore storage coefficient found from Eq. (2.18). When qaf/q < 0.01, it is concluded that wellbore storage does not affect the semi‐log slope. In other words, after this time, WBS effects are negligible.

Because of superposition, skin and wellbore storage effects, the start time of the semi‐log slope, ΔtSSL, is given by [12],

(ΔtD)SSL=50CDe0.14sE3.11

By taking a glance to Eq. (2.19), it is appreciated a higher effect of skin factor and wellbore storage in the above expression. After replacing the dimensionless parameters, Eqs. (2.14) and (1.94), in Eq. (3.11), it results:

ΔtSSL=170000μCe0.14skhE3.12

Eq. (3.5) applies to infinite‐size reservoir. For finite reservoirs, Eq. (3.59), becomes [12, 40],

Pws=P*162.6qμBkhlog(tp+ΔtΔt)=P*mlog(tp+ΔtΔt)E3.13

However, this equation applies similar to Eq. (2.5). The false pressure, P*, is read at a Horner time, (tp + Δt)/Δt = 1, and does not have physical meaning, but is useful to determine the average reservoir pressure [28].

3.2.2 Miller‐Dyes‐Hutchinson (MDH) method

This is based on the assumption that the production time, tp, is long enough to reach the pseudo‐steady‐state period; then, it is more representative to use average pressure than initial pressure. The MDH method is preferred in old wells or depleted formations, which would make it difficult to obtain stabilization before shutting‐in [40]. The Horner plot can be simplified [12, 28, 40], if Δttp, then:

tp+ΔttpE3.14

Then,

log(tp+ΔtΔt)logtplogΔtE3.15

Combining Horner equation, Eq. (3.13) with Eq. (3.15), it yields [12, 41]:

Pws=P*mlogtp+mlogΔtE3.16

If P* − m log tp = constant = intercept; then,

Pws=P1hr+162.6qμBkhlogΔtE3.17

This suggests that a semi‐log plot of Pws versus Δt will yield a straight line which slope, m, and intercept, P1hr, are used to find reservoir permeability with Eq. (2.33) and skin factor with Eq. (3.6).

Some expressions and plots [12] can be used to estimate the end of the semi‐log straight line. However, the use of the pressure derivative [6, 13, 25] avoids using them. Therefore, they are omitted in this chapter.

3.2.3 Extended Muskat method

It is a trial‐and‐error method that is more attractive in cases of constant pressure or water injection systems (filling) because in these cases, the straight line would be longer and, therefore, easier to identify [12, 32]. Muskat [32] proposed to build a potential plot (log(ΔP) versus Δt), see Figure 3.3. Cobb and Smith [8] and Ramey and Cobb [39] recommended using it only as a method of late‐time analysis. For the application of the method, the average reservoir pressure is assumed as many times as a straight line results in the plot. This author found that changing the average reservoir pressure in a range of 30 psia above or below the target average pressure always provides a straight line. Permeability and skin factor are found from the intercept of such plot, ΔPMint, read at Δt = 0:

k=141.2qμBhPD(tDA)intΔPMintE3.17.1
s=(PD(tDA)intΔPint)[P¯Pwf(Δt=0)]lnrerw+0.75E3.18

Figure 3.3.

Schematic representation of the Muskat plot for the analysis of pressure buildup tests, after [12].

PD(tpDA) is normally found from plots [12]. However, this author fitted the curves of such plots to polynomials. For a well within a square shaped reservoir—constant pressure case:

PDMint=0.0118157+1.3509395(1exp(21.692995tpDA)E3.18.1

For a well within a square geometry reservoir—no‐flow boundary case.

PDMint=0.02056+0.682297(1exp(50.7038508tpDA)E3.19

where

tpDA=(0.0002637ktpϕμctA)E3.20

The slope of the Muskat, mM, plot can be used to find the drain area:

A=(kϕμctmM)MSFE3.21

For the values of tpDA > 1, the Musk shape factor, MSF, is 0.67, 1.34, and 0.84 for no‐flow boundary square reservoirs, constant‐pressure boundary square reservoirs and no‐flow boundary circular reservoirs, respectively, with a unique well in the center of such systems [12].

If A is known, then,

ϕcth=St=kh43560μMSFmMA=TMSF43560mMAE3.22

It can be concluded that MDH is generally preferred because it is easy to use. For short production times, it is recommended to use the Horner method since the semi‐log line is longer than that provided by MDH. Earlougher [12] and Tiab [40] recommend the following aspects:

  1. The Horner method could be used to analyze pressure buildup data, assuming tp is known. However, MDH, and then Horner, are usually used as the first choice. If tp is unknown, then use MDH.

  2. Use MDH as the first test unless tp < tpss (reservoir acting as infinity, then Horner is applied) or unless the well is in the center of a square shaped reservoir with open boundaries, such as an injection pattern of five points.

  3. The Muskat method is used as a last option. It also provides the determination of the drainage area.

As for the MDH case, the starting and the end of the Muskat straight line can be estimated only for square shape reservoirs with the well at the center within it. For this reason and the help of the pressure derivative, it is not presented here. However, these procedures can be found in [12, 39].

3.2.4 Type‐curve matching

Such type curves as the given in Figure 2.4 and 2.6 and their accompanying equations also apply for buildup tests. However, superposition may cause trouble as can be seen in Figure 3.4. This was reported by Gringarten [23] when demonstrating the importance of deconvolution. Then, both pressure and pressure derivative must be corrected [3], before applying type‐curve matching. To overcome this issue, Agarwal [1] introduced the equivalent time, given by:

Δte=tpΔtΔt+tpE3.23

Figure 3.4.

Drawdown versus buildup log‐log derivative shapes, after [23].

Eq. (3.23) is the most common equivalent‐time equation. However, it was developed only for radial flow regime; therefore, it may fail providing good results if applied to other flow regimes. Then, the equivalent‐time equations for bilinear [38], linear, [38], birradial (elliptical), and spherical/hemispherical/parabolic flow regimes are, respectively, given as follows:

ΔteBL=[tp4+Δt4tp+Δt4]4E3.24
ΔteL=[tp+Δttp+Δt]2E3.25
ΔteBR=[tp925+Δt925(tp+Δt)925]25/9E3.26
ΔteSP=1[1tp+1Δt1tp+Δt]2E3.27

Once, the equivalent time is determined, then, the equivalent pressure derivative is estimated by:

Δte*ΔP=Δte[d(ΔP)d(Δte)]E3.28

Since Eq. (3.23) is normally used for time corrections, possibly, some recommendations given by [38] regarding the use of equivalent time were provided:

  • teq is primarily useful for homogeneous infinite‐acting radial flow systems when tp >> Δt.

  • teq is not recommended for fractured wells where linear flow dominates early time.

  • teq should not be used if multiphase flow is dominant.

  • Pressure data affected by boundaries are usually better plotted with Δt.

3.2.5 TDS technique

Good news! TDS technique applies to drawdown, buildup and, of course, drill stem tests. The equations already seen in Chapter 2 also apply here. Care must be taken while taking the pressure derivative. If superposition effects are observed; then, it is recommended to use equivalent time, Section 3.2.4, for the pressure derivative estimation. Drawdown pressure derivative may be taken, otherwise.

Once the derivative is estimated and the log‐log of pressure and pressure derivative versus time is built, Equations provided in Chapter 2 also apply for pressure buildup test analysis. Just to name a few references [15, 16, 17, 18, 19, 20, 21, 22, 31, 41, 42] also apply here.

3.3. Pressure buildup tests in developed reservoirs

The methods presented above may yield erroneous results when the test well produces under pseudo‐steady‐state conditions before shutting‐in for a pressure buildup test or undergoes a pressure drawdown due to the production of adjacent wells in the reservoir. In such cases, it is better to use Eq. (3.1) in a more general way. Slider [34, 35, 36, 37] has suggested a technique to treat the case of pressure tests in wells where the pressure drop contains the contribution of nearby wells. A procedure similar to that presented for the case of pressure drawdown, Section 2.6, is presented.

3.3.3 Conventional buildup analysis for developed reservoirs

It is required to extrapolate the well‐flowing pressure over the pressure buildup period to estimate, Pw ext, see Figure 3.5. Then, find the difference between the observed shut‐in pressure and the extrapolated well‐flowing pressure, ΔPΔt, and plot this as a function of Δt. The data should be adjusted to the following equation [34, 35, 36, 37]:

ΔPΔt=PwsPwext=ΔP1hr+mlogΔtE3.29

Figure 3.5.

Schematization of pressure buildup in a developed reservoir, after [34, 35, 36, 37].

A straight line on this plot gives a slope m given by Eq. (2.33) and intercept:

ΔP1hr=162.6qμBkh[log(kϕμctrw2)3.2275+0.86859s]E3.30

The permeability is found with Eq. (2.33) and skin factor with a modified version of Eq. (3.6) resulting from changing P1hr by ΔP1hr*.

s=1.1513[ΔP1hr*mlog(kϕμctrw2)+3.2275]E3.31

If the pressure drop is linear before shutting‐in the well, which normally occurs because of the existence of the pseudo‐steady state, Eq. (3.29) becomes:

Pwsm*Δt=ΔP1hr*+mlogΔtE3.32

where m*, usually has a negative value, is the linear change of pressure drop before shutting‐in the well:

m*=dPwfdtwhent<tpE3.33

Normally, m* is negative. The value of ΔP*1hr in Eq. (3.32) is derived from Eq. (3.30) for the extrapolated linear behavior [37], which is:

ΔP1hr*=Pwf(Δt=0)+m[log(kϕμctrw2)+3.22750.86859s]E3.34

So, when the pressure declines linearly before the test, a plot of (Pwsmt) vs. log Δt should give a straight line. The permeability is calculated with Eq. (2.33) and the skin factor with Eq. (3.31) by changing P1hr instead of ΔP1hr*. Usually, production occurs under pseudo‐steady‐state conditions; therefore, the pressure that the well would have if production were to continue would be given by:

Pext=Pwf(Δt=0)m*ΔtE3.35

And, ΔP* is calculated as the difference between the observed pressure and the extrapolated pressure:

ΔP*=PwsPextE3.36

3.3.4. TDS buildup analysis for developed reservoirs

As demonstrated by Escobar and Montealegre [14], TDS technique is also applicable to developed reservoirs deriving DP* and using the traditional equations of the technique.

Example 3.1

Slider [36, 37] first presented this example and then Escobar and Montealegre [14] reworked by TDS technique. A well drilled in a field with a uniform spacing of 40 acres has produced an average flow rate of 280 STB/D for 10 days. The well is shut‐in for a pressure buildup study. In the five days prior to shutting‐in, the flow pressure at the wellhead drops to around 24 psia/day (1 psia/hr). The oil‐gas ratio remained constant during production. The test data are reported in Table 3.1. The following information is also available:

t, hrPws, psiaPext, psiaΔP*, psia[t*(ΔP*)’], psia
0112311230
2229011211169606.577
4251411191395225.854
825841115146997.372
1226121111150183.543
1626321107152573.916
2026431103154070.157
2426501099155170.988
3026581093156577.176

Table 3.1.

Pressure data for Example 3.1 of developed reservoir, after [14] and [35].

B= 1.31 rb/STB,               μ = 2 cp,  h = 40 ft,  rw = 0.33 ft

Solution by conventional analysis

Estimate Pext by means of Eq. (3.35),

Pext=Pwf(Δt=0)m*Δt=1123(1)(0)=1123psia

Estimate ΔP* using the observed pressure minus the extrapolated pressure, Eq. (3.36);

ΔP*=PwsPext=11231123=0psia

The remaining estimated values are given in Table 3.1.

Figure 3.6 shows a graph of ΔP* against the log Δt, from which a slope of 192.92 psia/cycle is obtained, which allows estimating the permeability value with Eq. (2.33):

k=162.2qμBhm=162.6(280)(2)(1.31)40(192.92)=15.42md

Figure 3.6.

Semi‐log plot of DP* against Dt.

As dP/dt is known, assuming that the drainage area approaches a circle, (re = 745 ft) product ϕct is solved from Eq. (1.130):

ϕct=1.79qBhre2(dP/dt)=1.79(280)(1.31)(40)(7452)(24)=1.24×106/psia

It is seen from Figure 3.6 that the intercept, ΔP*1hr = 1287.6 psia. The skin factor is calculated from Eq. (3.31):

s=1.1513[1287.6192.92log(15.42(2)(1.24×106)(0.332))+3.2275]=2.47

Solution by TDS technique

To apply TDS technique to this example, derivative of ΔP*, see the last column in Table 3.1. The pressure and pressure derivative plot is built and given in Figure 3.8. Read from this plot the characteristic points, namely, tr = 24 hr, [t*(ΔP*)′]r = 70.998 psia, and (ΔP*)r = 1551 psia. The permeability and the skin factor are found with Eqs. (2.71) and (2.92), respectively:

k=70.6qμBh(t*[ΔP*])r=70.6(280)(2)(1.31)40(84)=15.4md
s=0.5[155170.998 ln((24)(15.4)(2)(1.24×108)(0.332))+7.43]=1.81

A good approximation to the data estimated by the two methods is observed in Example 3.1.

3.4. Average reservoir pressure

The average pressure for a reservoir without water intrusion is the pressure that the reservoir would reach if all the wells shut‐in for infinite time. The average pressure is useful for [10, 12, 22, 28, 37, 40] (Figure 3.7):

  1. For reservoir characterization:

    1. ΔP = P¯Pwf is small per unit of production, what is known as productivity index, J, indicates that there is an active water influx or a very large reservoir.

    2. If P is large per unit of production, it involves drainage from a small reservoir, sand lens, or faulted reservoir.

  2. To calculate in‐site oil.

  3. For ultimate reservoir recovery.

  4. The average pressure is a fundamental parameter that must be understood in processes of primary, secondary, and pressure maintenance projects.

Figure 3.7.

Pressure and pressure derivative of ΔP* versus Δt.

The average reservoir pressure in the drainage region can be obtained by using well pressure test analysis. Most of the methods to estimate this parameter will be presented now.

3.4.3 Matthew‐Bronz & Hazebrock (MBH) method

This method is considered the most accurate [12, 28] and was corrected by Odeh [37]. Use a Horner plot. It is applied in most situations where it is desired to find the average pressure in a closed reservoir for any well location within a variety of drain forms. The method assumes that there are no variations in fluid mobilities or fluid compressibilities within the drain region. This limitation can be overcome by using a production time tp equal tpss. The procedure is outline below:

  1. If not given, calculate the Horner time, tp, with Eq. (3.2).

  2. The tp value must be compared with the time required to reach the pseudo‐steady‐state conditions. Therefore obtain (tDA)pss from Table 2.1, from the column “Exact for tDA >”. For this, the reservoir geometrical shape must be previously known.

  3. Calculate the time to reach the pseudo‐steady state, tpss:

    tpss=ϕμctA(tDA)pss0.0002637kE3.37

  4. Estimate the ratio α, α = tp /tpss. If α > 2.5, then, set t = tpss. If α < 2.5 (for very high flow rates, the improvement in the average pressure calculation is significant when α lies between 2.5 and 5), then, set t = tp. Build a plot of Pws versus (t + Δt)/Δt. As seen earlier, the use of tpss in the Horner method can increase the length of the semi‐log line, contrary to the MDH plot.

  5. With time, t, defined in the previous step, determine tpDA.

    tpDA=0.0002637kϕμctAtE3.38

  6. Extrapolate the semi‐log line of the Horner graph and find P*. See Figure 3.2.

  7. Using the tpDA calculated in step 5, determine PDMBH from the following equations and tables. Notice that normally, PDMBH is found from charts [12, 28]. However, the appropriate charts provided by [10, 28] were adjusted to polynomials with correlation coefficients greater than 0.999 (Table 3.2).

    PDMBH2=a+blog(tpDA)+c[log(tpDA)]2+d[log(tpDA)]3+e[log(tpDA)]4E3.39
    PDMBH2=a+blog(tpDA)+c[log(tpDA)]2+d[log(tpDA)]3+e[log(tpDA)]4+f[log(tpDA)]5E3.40
    PDMBH2=a+blog(tpDA)+c[log(tpDA)]3+delog(tpDA)+eelog(tpDA)E3.41
    PDMBH=a+clog(tpDA)+e[log(tpDA)]2+g[log(tpDA)]31+blog(tpDA)+d[log(tpDA)]2+f[log(tpDA)]3E3.42
    PDMBH=a+clog(tpDA)+e[log(tpDA)]21+blog(tpDA)+d[log(tpDA)]2+f[log(tpDA)]3E3.43
    PDMBH=a+blog(tpDA)+c[log(tpDA)]2+d[log(tpDA)]3+e[log(tpDA)]4+f[log(tpDA)]5+g[log(tpDA)]6+h[log(tpDA)]7+i[log(tpDA)]80.082557382[log(tpDA)]90.012745849[log(tpDA)]10E3.44

  8. Calculate the average reservoir pressure from:

P¯=P*(m2.3025)PDMDHE3.45
Reservoir geometryabcde
Hexagon and circle12.071926216.99987096.07856232−0.7618991−0.5297593
Square2.066524216.991635680.082030880.754017370.52737147
Equilateral triangle10.162067814.95528626.630900110.42119362−0.2122641
Rhombus9.8952639114.57565396.492690930.434827−0.2016777
Right triangle8.6812135213.25261166.601472380.93940158−0.0660697

Table 3.2.

Parameters for Eqs. (3.39).

Due to the compensation factors (low values of P* with corresponding small corrections), any value of tp used with the MBH method will theoretically give identical results for average reservoir pressure. Practically, a relatively short tp can eliminate serious numerical problems in the calculation of average pressure. This includes errors caused by long extrapolations and deviations from theoretical assumptions: (1) lack of stabilization of the flow rate prior to closure, (2) migration and change of drainage areas in reservoirs with multiple wells and (3) variations in the compressibility of the system and mobility [12, 28].

3.4.4 Dietz method

This method [12, 28]‐assumes that the well flowed sufficiently until it reached the pseudo‐steady‐state period before shutting‐in and that the semi‐log straight developed properly. This method is simple and is usually preferred in wells without significant skin factor, s > −3 or rw′ = 0.05 re. The procedure for this method is:

  1. Knowing the reservoir shape and the well location, read CA from Table 2.1.

  2. Calculate the Dietz shutting‐in time, (Δt)P¯.

    ΔtP¯=ϕμctA0.0002637CAkE3.46

  3. Prepare a MDH plot (optionally find k and s).

  4. Enter the MDH plot with the Dietz shutting‐in time calculated in step 2 and read the corresponding average reservoir pressure value on the semi‐log straight line.

3.4.5 Miller‐Dietz‐Hutchinson (MDH) method

This was elaborated to estimate the average pressure in circular or square reservoirs. It is applied only in wells that operate under pseudo‐steady‐state conditions [8, 28]. The procedure is presented as follows:

  1. On an MDH graph, choose any point on the semi‐log trend and read its coordinates, (Pws)N and ΔtN.

    ΔtDA|N=0.0002637kϕμctAΔtNE3.47

  2. Calculate ΔtDA.

  3. Determine PDMDH corresponding to (ΔtDA)N. This was traditionally done on charts. Again, fitted polynomials are presented here.

    PDMDH=a+bΔtDA+cΔtDA2ln(ΔtDA)+d[ln(ΔtDA)]2+eln(ΔtDA)E3.48

  4. Calculate the average reservoir pressure from Table 3.3:

P¯=Pws|N+(m1.1513)PDMDHE3.49

Table 3.3.

Parameters for Eqs. (3.39)(3.41).

3.4.6 Ramey‐Cobb method

They presented a method to extrapolate the average pressure of a Horner plot when ttpss|. This method [12, 28, 40] requires information on the shape of the drainage area, the location of the well, and the confirmation that the boundaries are closed. The Ramey‐Cobb procedure is (Table 3.4):

  1. Knowing the reservoir shape and the well location, obtain (tDA)pss, and calculate tp and tpss.

    tpss=ϕμctA(tDA)pss0.0002637kE3.50

  2. If tp < tpss, then, the method is not reliable. Calculate the Horner time corresponding to the average reservoir pressure.

    (tp+ΔtΔt)P¯=0.0002637kCAϕμctAtP=CAtpDAE3.51

    When (tp + Δt) = tp, Eq. (3.51) reduces to Eq. (3.46).

  3. Prepare a Horner plot (optionally find k and s) (Tables 3.5 and 3.6).

  4. Enter the Horner plot with the result from Eq. (3.51) and read the average reservoir pressure on the straight line trend.

Table 3.4.

Parameters for Eqs. (3.39)(3.42).

Table 3.5.

Parameters for Eq. (3.42)(3.44).

Table 3.6.

Parameters for Eq. (3.48).

3.4.7 Arari or direct method

Arari [4] presented in 1987 a simple method to calculate the average reservoir pressure during production or buildup without the help of any graph. This method requires knowing the distance from the well to which the reservoir pressure is the same average pressure. For no‐flow boundary reservoirs:

P¯ Pwf+162.6qμBkh(2logrerw0.5203+0.87s)E3.52
P¯=Pwf+162.6qμBkh(logArw21.1224+0.87s)E3.53

For constant‐pressure boundary reservoirs:

P¯=Pwf+162.6qμBkh(2logrerw0.4342+0.87s)E3.54
P¯=Pwf+162.6qμBkh(logArw21.036+0.87s)E3.55

In order to consider different well positions and different reservoir geometries, the flow equations were developed by introducing the Dietz shape geometrical factors in Eqs. (3.539) and (3.55) which are, respectively, transformed into:

P¯=Pwf+162.6qμBkh(logACArw2+0.368+0.87s)E3.56
P¯=Pwf+162.6qμBkh(logACArw2+0.454+0.87s)E3.57

3.4.8 TDS technique

3.4.8.1 Circular reservoirs

For a well in the center of a circular reservoir, the average reservoir pressure is obtained from a log‐log plot of pressure and pressure derivative versus time according to the following expression [7, 29]:

P¯ = Pi141.2 qμBkh[((t*ΔP)pss(ΔP)pss(t*ΔP)pss)ln(rerw34)]E3.58

where Piis the initial pressure (in some cases, it can approximate P*), (ΔP)pss and (t*ΔP')pss are the values of (ΔP) and (t*ΔP') in the late straight line of pseudo‐steady‐state period.

3.4.8.2 Naturally fractured reservoirs

For naturally fractured reservoirs with the dimensionless average pressure and the average reservoir pressure are defined as [29]:

P¯D=kh141.2qμB(P¯Pws)E3.59
P¯=Pwf+ΔPpss+(t*ΔP)pss(1+3792.2ϕμctrw2(1ω)2λktpss)E3.60

Being ω and λ the naturally fractured reservoir parameters which will be discussed about in Chapter 6.

P¯=Pwf+ΔPpss+141.2qμBkh[((t*ΔP)pssΔPpss(t*ΔP)pss)(lnrerw34+2πrw2(1ω)2λA)]E3.61
P¯=Pwf+ΔPpss+141.2qμBkh[((t*ΔP)pssΔPpss(t*ΔP)pss)(lnrerw34+0.1987CArw2(1ω)2λA)]E3.62

If the dimensionless average pressure is defined as [29]:

P¯D=kh(PiP¯)141.2qμBE3.63

The average reservoir pressure is given by [29]:

P¯=Pi(t*ΔP)pss(1+3792.2ϕμctrw2(1ω)2λktpss)E3.64
P¯=Pi141.2qμBkh[((t*ΔP)pssΔPpss(t*ΔP)pss)(lnrerw34+2πrw2(1ω)2λA)]E3.65
P¯=Pi141.2qμBkh[((t*ΔP)pssΔPpss(t*ΔP)pss)(lnrerw34+0.1987CArw2(1ω)2λA)]E3.66

The Dietz shape factor can be estimated by [7]:

CA2.2458Arw2 {ektpss301.77ϕμctA ((ΔP)pss(t*ΔP)pss  1)}1E3.67

For a circular geometry with the well at the center, [26×] arrived to the following expression:

P¯=Pi1.26(t*ΔP)r2(t*ΔP)r[(2πrw2(1ω)2λA)]E3.68

They also assumed that:

P¯D=2πtDAE3.69

and,

2πtDA>>2πrw2(1ω)2λAE3.70

After some manipulations, Igbokoyi and Tiab [27] also obtained an expression free of the naturally fractured reservoir parameters:

P¯ =Pi(t*ΔP)r[((t*ΔP)pss(ΔPw)pss(t*ΔP)pss)ln(2.2458 ACArw2)+2s]E3.71

3.4.8.3 Bounded elongated systems

The shape factor is given by Eq. (3.67), and the average reservoir pressure equation for these systems is given below [7]:

P¯ =Pi70.6qμBkh[((t*ΔP)pss(ΔPw)pss(t*ΔP)pss)ln(2.2458 ACArw2)]E3.72

3.4.8.4 Hydraulically fractured vertical well in no‐flow boundary reservoirs

The shape factor is estimated with Eq. (3.67), and the average reservoir pressure with Eq. (3.69) [7]:

P¯=PiqμBkh{0.23373ktpssϕμctA((ΔP)pss(t*ΔP)pss)70.6ln[(xexf)2(2.2458CA)]}E3.73

When birradial flow occurs, the area and the average reservoir pressure can be determined from the following equations [7]:

A=kBRpssi142.43ϕμct(xe/xf)1.123E3.74
P¯=Pi5.64[qμBkh(xexf)0.72(kϕμctA)0.36] tBRpssi0.36E3.75

For uniform flow fractures and when xe/xf < 8, birradial flow is difficult to be observed, then the intersection between the linear flow and the pseudo‐steady state line, tLpssi, is used. Then, the area and the average pressure are obtained from:

A=0.0033144[(kϕμct)(xfxe)2] tLpssiE3.76
P¯=Pi4.06[qBμh ϕctk(xexf)]tLpssiE3.77

3.4.8.5 Hydraulically fractured vertical wells in elongated systems

For these systems, the transition between the line of infinite behavior and that of the pseudo‐steady‐state period is longer compared to the case of square systems in both cases of fracture: infinite conductivity and uniform flux. When the birradial flow line is difficult to be observed, such is the case of xe/xf < 8, the following equation is used to determine permeability [7]:

k=(μϕ ctA)[8.128 qBh (t*ΔP)DL1]2E3.78

Since there may be two linear flow regimes, once before radial flow corresponding to flow from the formation to the fracture and the other once radial vanishes, then, (tP')DL1 is the value of (tP') at t = 1 hr on the dual‐linear flow regime—solving for area it yields [7]:

A=(μϕctk)[8.128 qBh (t*ΔP)DL1]2E3.79

The point of intersection between the closest parallel‐linear line flow: the second linear flow regime, for example, dual‐linear and the pseudo‐steady‐state line, tDLpssi, is unique. With this point, determine the area of the following equation [7]:

A = ktDLpssi1207.09ϕμct E3.80

This equation should be used for verification purposes of the permeability and area values obtained by Eqs. (3.78) and (3.79). The average reservoir pressure is obtained from [7]:

P¯PiqBh (μ11913.6ϕctkA)tDLpssiE3.81

This equation should be used if k and A can be determined from the nearest parallel boundary; in other words, from the dual‐linear flow regime.

3.4.9 Total average reservoir pressure

Golan and Whitson [24] presented a method to estimate the drainage area of wells that produce from a common reservoir. They assumed that the volume drained by a well is proportional to its flow rate. If the properties of the reservoirs are constant and uniform:

Aw=AT(qwqT)E3.82

All of the above studied methodologies give the value of the average reservoir pressure in the well drainage area. If a number of wells produce from the same reservoir, each well is analyzed separately to give the average reservoir pressure for its own drainage area. The average reservoir pressure can be estimated from the individual average pressures by (possibly from [38] less probably from [9, 24], the author does not remember the exact reference):

P=i[P¯iΔ(F)/ΔP¯]ii[Δ(F)/ΔP¯]iE3.83
Δ(F)=Ft+Δt+FtE3.84
Ft+Δt=0t+Δt[qoBo+qwBw+(qgqoRsqwRsw)Bg]dtE3.85
Ft=0t[qoBo+qwBw+(qgqoRsqwRsw)Bg]dtE3.86

Bossie‐Codreanu [5] suggest that the drainage area can be determined from a Horner or MDH plot by selecting the 3‐point coordinates in the straight section of the semi‐log graph to determine the slope of the pseudo‐steady‐state period line, m*:

  • Shutting‐in time Δt1 with corresponding shutting‐in pressure Pws1

  • Shutting‐in time Δt2 with corresponding shutting‐in pressure Pws2

  • Shutting‐in time Δt3 with corresponding shutting‐in pressure Pws3

The selected shutting‐in times satisfy t1 <t2 < t3. Then, m* is approximated by:

m*=(Pws2Pws1)log(Δt3/Δt1)(Pws3Pws1)log(Δt2/Δt1)(Δt3Δt1)log(Δt2Δt1)(Δt2Δt1)log(Δt3Δt1)E3.87

Example 3.2

The data of a pressure buildup test, taken from [40], are reported in Table 3.7, along the Horner time and the pressure derivative estimated (using equivalent time, Eq. (3.23)) with a smooth value of 0.1 cycles. The reservoir properties were obtained from a well located in the center of a square shaped reservoir. Given the following data:

Δt, hrPws, psia(tp + Δt)/Δt(tpss + Δt)/ΔtΔP, psiatP', psia
0298000
0.131003213.00807.45012083.41
0.231501607.00404.225170100.23
0.332001071.67269.817220110.92
0.53250643.40162.29027085.89
0.753275429.27108.52729559.48
13290322.2081.64531041.84
23315161.6041.32333534.48
33325108.0727.88234522.35
4333081.3021.16135020.29
5333565.2417.12935521.45
7334246.8912.52136221.96
10335033.129.06537023.97
15336022.416.37638021.42
20336417.065.03238414.87
30337011.713.68839012.65
4033729.033.0163929.02
5033747.422.6133948.47
6033756.352.3443957.14
7033765.592.1523968.55
8033775.022.0083979.76

Table 3.7.

Pressure buildup test data.

rw = 4 in,     h = 44 ft,     ϕ = 12%

μ = 0.76 cp,        B = 1.24 rb/STB,  Np = 4550 STB

A = 40 acres,       q = 340 BPD,        ct= 36 × 10−6 psia−1

Pwf = 2980 psia

It is required to estimate reservoir permeability and skin factor. Then, find the average reservoir pressure using all the studied methods.

Solution

Find tp with Eq. (3.2);

tp=24Npq=(24)(4550)340=321.176hr

Estimate the Horner time, (tpt)/Δt, for each pressure value. This is reported in the third column of Table 3.7 and builds the Horner plot given in Figure 3.8. From the Horner plot given in Figure 3.8, the slope and intercept are read to be 44 psia/cycle and 3306 psia. They are, respectively, used to find permeability, Eq. (2.33), and skin factor, Eq. (3.6), thus:

k=162.6qμBmh=(162.6)(340)(0.76)(1.24)(44)(44)=26.91md
s=1.1513[33062980(44)log(26.91(0.12)(0.76)(36×106)(0.333)2)+3.2275]=3.18

Figure 3.8.

Horner plot for Example 3.2.

Average reservoir pressure by MBH method

Determine (tDA)pss from Table 2.1 for square shaped reservoirs. It is read (tDA)pss = 0.1. Calculate tpss with Eq. (3.37):

tpss=ϕμctA(tDA)pss0.0002637k=(0.12)(0.76)(36×106)(40)(43560)(0.0002637)(26.91)(0.1)=80.645hr

Estimate the ratio a = tp /tpss = 312.176/80.645 = 3.982. Since α > 2, then t = tpss. Rebuild the Horner plot as Pws vs. log(tpss + Δt)/Δt, (see Table 3.7). In Figure 3.9, draw a straight line along the infinite‐acting period (radial flow) and extrapolate to a Horner time of 1. Read the false pressure value, P* = 3398 psia.

Find the dimensionless production time using Eq. (3.38):

tpDA=0.0002637ktϕμctA=(0.0002637)(26.91)(80.645)(0.12)(0.76)(36×106)(4043560)=0.09990.1

Figure 3.9.

Horner plot with tpss for Example 3.2.

Using Eq. (3.39) with data from Table 3.3 (first row), the MBH dimensionless pressure is PDMBH=1.152. Then, estimate the average reservoir pressure with Eq. (3.45).

P¯=P*mPDMBH2.303=3398(44)(1.152)2.303=3376psia

Average reservoir pressure by Dietz method

Prepare a MDH, Pws vs. log(Δt). See Figure 3.10. Determine the shape factor CA from Table 2.1 for a well at the center of a square reservoirs. CA = 30.8828. Find Dietz shutting‐in time with Eq. (3.46),

ΔtP=ϕμctA0.0002637kCA=(0.12)(0.76)(36×106)(40)(43560)(0.0002637)(26.91)(30.8828)=26.1136hr

Figure 3.10.

MDH plot for Example 3.2.

Enter with this value in Figure 3.10 and read an average reservoir pressure of 3368 psia.

Average reservoir pressure by Ramey‐Cobb method

Having tp, tpss, and CA from previous methods and since tp >> tpss, then estimate Ramey‐Cobb shutting‐in time from Eq. (3.51);

(t+ΔtΔt)P¯=(0.0002637)(26.91)(30.8828)(312.176)(0.12)(0.76)(36×106)(40)(43560)=12.299

Enter with this value in the Horner plot, Figure 3.8, and read an average pressure value of 3368 psia.

Average reservoir pressure by MDH method

Prepare a MDH plot, Figure 3.10 and choose any convenient point on the semi‐log straight line. For this case, ΔtN = 10 hr and (Pws)N = 3350 psia were chosen. Calculate the dimensionless shutting‐in time using the chosen time in Eq. (3.47):

ΔtDA=(0.0002637kφμctA)ΔtN=(0.0002637)(26.91)(0.12)(0.76)(36×106)(40)(43560)(10)=0.0124

Use this value of ΔtDA and estimate PDMDH from Eq. (3.48) and Table 3.6 for a no‐flow boundary square reservoir (first row), this gives PDMDH = 0.6. Estimate the average reservoir pressure with Eq. (3.49);

P¯=PwsN+m(PDMDH)1.1513=3350+44(0.6)1.1513=3372.9psia

Average reservoir pressure by direct (Arari) method

Using Eq. (3.53), for no‐flow boundaries;

P¯=2980+[162.6(340)(0.76)(1.24)(23.52)(44)(log40(43560)0.33321.1224+0.87(1.93))]=3365.2psia

Average reservoir pressure by TDS technique

The pressure and pressure derivative versus time log‐log plot for Example 3.2 is given in Figure 3.11. Notice that after radial flow, the pressure derivative takes a slope of negative one. This may be due to the changes in transmissibility. Anyhow, the pseudo‐steady‐state period starts at 60 hr. On that line, a point will be chosen for the estimation of the average reservoir pressure. From this plot, the following data are read:

tr = 7 hr,     ΔPr = 396 psia,    (tP′)pss = 8.55 psia

Figure 3.11.

Log‐log plot of pressure and pressure derivative against time for Example 3.2.

ΔPpss = 384 psia,  tpss = 70 h,      (tP′)r = 21.86 psia

Estimate permeability and skin factor with Eqs. (2.71) and (2.92),

k=70.6qμBh(t*ΔP)r=(70.6)(340)(0.76)(1.24)44(21.86)=23.52md
s=0.5[36221.86ln(23.52(7)(0.12)(0.76)(36×106)(0.32))+7.43]=1.93

Taking the case of no‐flow boundary rectangular system, Eqs. (3.67) and (3.72), find the shape factor and the average reservoir pressure. Here, the last pressure value, that is, 3377 psia, is taken as Pi.

CA=2.2458 (40)(43560)0.3332{exp[0.003314(23.52)(20)0.12(0.76)(30×106)(40)(43560)(3968.551)]}1=12.943
P¯ =337770.6(340)(0.76)(1.24)(23.52)(44)[(8.553968.55)ln(2.2458(40)(43560)12.943(0.3332))]=3369.9psia

The results of the estimation of the average reservoir pressure are reported in Table 3.8.

MethodAverage reservoir pressure, psia
Ramey & Cobb3368
MBH3376
MDH3372.9
Dietz3368
TDS3371.4
Arari3365.2
Average3370.25

Table 3.8.

Summary of average reservoir pressure results.

3.4.10 Average reservoir pressure in naturally fractured reservoirs from transient‐rate analysis

Amin et al. [2] follow the philosophy of the TDS technique to determine the average reservoir pressure from TRA (even though, this book does not include such analysis) by means of the following expression (slight simplification is shown here):

P¯=Pi887.186qμBkh[(kftpss3792.19(ϕct)m+fμ)(+rw2(1ω)2λA)]E3.88

Amin et al. [2] pointed out that from a curve of production rate versus time, a point qpss and tpss that satisfy tD > ω(1−ω)/λ (pseudo‐steady‐state period). In addition, qpss and tpss selected should be those when the flow rate becomes almost constant.

3.4.11 Average reservoir pressure from two‐rate tests

Sabet [33] and Dake [10] presented the mathematical development to find the average reservoir pressure from two‐rate tests. The final equation uses the value of the first semi‐log straight line and the well‐flowing pressure after the flow rate has been changed. It is given by:

P¯=2m1{log(2.241ACArw2)2+0.435s}+Pwf@Δt'=0E3.89

Example 3.3

Sabet [34] presented a two‐rate test which pressure versus time values are shown in Table 3.9. To interpret the test, the following reservoir information, PVT, and flow parameters are given:

Before flow rate changeAfter flow rate change
t, hrPwf, psiat, hrPwf, psiaDt
2881607.50.3324752.72
2891607.20.4224822.67
2901606.80.524872.63
2911606.40.5824922.6
2921606.10.6724972.56
2931605.70.7525002.54
2941605.40.8325022.52
29516050.9225052.5
2961604.6125082.48
2971604.31.2525142.43
2981603.91.525202.39
2991603.61.7525252.36
3001603.2225312.33
2.525422.28
325522.24
425682.18
525822.13
625902.1
726002.06
7.526042.05

Table 3.9.

Two‐rate test data for Example 3.3, after [34].

re = 745 ftrw = 0.25 ftf = 15%h = 20 ft
m = 1.2 cpct = 20 × 10−6 psia−1B = 1.25 bbl/STBtp = 300 hr
q1 = 100 STB/Dq2 = 50 STB/DPwf@Dt'=0 = 1603.2 psia

Determine average reservoir pressure, skin factor, and demonstrate that the test has not reached the transient period when the flow rate was changed.

Solution

A value of m1 = 274 psia/cycle and P1hr = 2485 psia were read from the Cartesian graph presented in Figure 3.12. Permeability and skin factors are calculated with Eqs. (2.271) and (2.272), respectively, and were found to be 4.5 md and approximately 3.0. The time to reach the pseudo‐steady‐state period in the test is estimated with Eq. (2.40) with r = re, as follows:

tpss=948ϕμctr2k=948(0.15)(1.2)(20×106)(7452)4.5=420hr

Figure 3.12.

Cartesian plot for two‐rate test of Example 3.3.

Assuming the well is in the center of a rectangular or circular reservoir, it is possible to appreciate the change (t = 300 hr) occurred before reaching the pseudo‐steady‐state conditions. Eq. (3.89) is used to calculate the average reservoir pressure, thus:

P¯=325.2(100)(1.2)(1.25)(84.5)(20)log0.472(745)0.33+0.87(274)(3)+1603.2=2405.8psi

Note that the average pressure value is not correct (it was obtained for explanatory effects) because the well was not producing under pseudo‐steady‐state period before changing to the second rate.

3.4.12 Average reservoir pressure from multi‐rate tests

For this type of tests, it is necessary to construct a Cartesian graph of pressure against the superposition time, Xn, Eq. (2.267), to obtain permeability and damage. Then, an MDH graph is made, and the Dietz method is applied. For this, it is necessary to determine the Dietz shutting‐in time by Eq. (3.46). With this value, the average reservoir pressure is read from the MDH plot. Actually, it is possible, then, to apply any of the average pressure methods seen in this chapter. However, for some of the methods, that is, MBH and Ramey‐Cobb, the production time, tp, is required. By definition, it refers to a constant flow rate before shutting‐in the well. It does not exist in this case. So, as a recommendation, the equivalent time should be estimated with Eq. (2.269) and this can be used as tp, and the flow rate is weighted with each period of duration.

However, TDS technique plays an important role. Escobar [15] obtained the average pressure equation for circular systems as given below.

P¯= Pi141.2qnμBkh[((t*ΔPq)pss(ΔPq)pss(t*ΔPq)pss)(lnrerw0.75)]E3.90

where Piis the initial pressure. (ΔPq)pss and (t*ΔPq')pss are the corresponding normalized pressure values and their derivative given at an arbitrary time, tpss. For any geometry, the resulting equation is [15]:

P¯=Pi70.6qnμBkh[((t*ΔPq)pss(ΔPq)pss(t*ΔPq)pss)ln(2.2458ACArw2)]E3.91

where CA is found from a slight modification of Eq. (3.67):

CA2.2458Arw2 {ektpss301.77ϕμctA ((ΔPq)pss(t*ΔPq)pss  1)}1E3.92

For multi‐rate tests in naturally fractured reservoirs, the respective equations [15] are:

P¯ = Pi141.2qnμBkh[(t*ΔPq)pss(ΔPq)pss(t*ΔPq)pss(lnrerw0.75+2rw2(1ω)2λre2)]E3.93
P¯ = Pi141.2qnμBkh[(t*ΔPq)pss(ΔPq)pss(t*ΔPq)pss(lnrerw0.75+0.198CArw2(1ω)2λA)]E3.94

Example 3.4

This actual field example presented by Escobar [15] comprises two pressure tests performed on an exploratory well in a reservoir which is believed to possess an approximated circular shape. A pressure buildup test, Table 3.10, was run so that the average reservoir pressure can be determined by conventional methods. Production was inactive during the following eight months; then, a multi‐rate test was performed. Its data are given in Table 3.11. The well had produced 190000 STB at a flow rate of 305 BPD before shut‐in it for the pressure buildup test. The well data and properties of rock and fluid are as follows:

Δt, hrPws, psiaΔPws, psiat + tp)/Δt(tpss + Δt)/Δt
02143.40
0.0132167.824.41150062.5413793.31
0.0192195.552.1786885.219437.84
0.0282229.886.4533958.146404.57
0.0372265.6122.2404076.684846.95
0.0562302.3158.9266979.573202.79
0.0672323.5180.1223147.272677.12
0.0792341.4198.0189251.632270.62
0.1122361.0217.6133490.291601.89
0.1532375.6232.397718.651172.90
0.2142389.5246.169864.55838.85
0.3292401.7258.345444.16545.98
0.4792411.5268.131213.53375.32
0.6082416.8273.424591.13295.90
0.8402425.4282.017799.57214.45
1.0992432.7289.313605.00164.15
1.4862437.6294.210062.10121.66
1.9882444.9301.57521.5291.19
2.6602448.2304.85621.6068.41
3.4822455.0311.64294.7452.49
4.5102465.3321.93316.0340.76
5.5352472.6329.22702.1433.39
6.5062479.8336.42299.0028.56
7.6482487.3343.91955.8624.44

Table 3.10.

Pressure buildup test data for Example 3.4.

t, hrteq, hrΔPq, psia/(STB/D)t*ΔP’q, psia/(STB/D)teq*ΔP’q, psia/(STB/D)qn,STB/D
0.2030.2141.79071.24851.2969296
0.2530.2622.29851.60261.6819295
0.3280.3142.95042.05712.0251293
0.4360.4564.35072.23562.1813291
0.6020.6215.74162.23562.0251292
0.8750.9027.16841.55871.5046290
1.2891.3278.23501.11741.0771286
1.9241.9039.20140.80100.7159285
2.6222.69310.00000.57420.4938281
3.6213.76210.28130.44730.3808279
5.3355.53410.28130.31190.2936278
8.0648.24510.28130.20010.1951274
13.69114.33610.57050.11490.1000270
19.91019.77410.57050.07790.0716266
29.71030.23110.57050.09200.0895265
44.90944.46810.57050.07580.0743263
61.22362.12910.57050.08700.0743258
89.02989.06810.86790.08230.0831256
124.545123.50010.57050.11170.1136255
161.241161.23010.86790.13570.1296253
211.463211.96310.86790.16480.1548251

Table 3.11.

Multi‐rate test data for Example 3.4.

rw = 0.33 ft,   ϕ = 13%,        h = 80 ft

μ = 0.9 cp,   ct = 1.9 × 10−5 psia−1   B = 1.3 bbl/STB

A = 62 Ac,   Pwf (t = 0) = 2143.4 psia, for the pressure buildup test

Pi = 2554 psia, for the multi‐rate test.

It is required to estimate the average pressure test from the buildup test using the MBH method and from the multi‐rate test using the TDS technique.

Solution by MBH method

The production time, tp, resulted to be 14950.8 hr with Eq. (3.2). From the Horner plot given in Figure 3.13, the slope and intercept are read to be 57 psia/cycle and 2427 psia. Permeability is found with Eq. (2.33) and skin factor with Eq. (3.6),

k=162.6qμBmh=(162.6)(305)(0.9)(1.3)(57)(80)=26.72md
s=1.1513[24272143.457log(12.7(0.13)(0.9)(1.9×105)(0.33)2)+3.2275] =0.56

Figure 3.13.

Horner plot for Example 3.4.

For the given reservoir, (tDA)pss from Table 2.1 is 0.1. Next, tpss = 179.3 hr from Eq. (3.37) and α ratio is, α = tp/tpss = 14950.8/179.3= 80.3; then, set t = tpss = 179.3. With this value, a new Horner plot is built and provided in Figure 3.14, from which P* = 3580 psia. Now, determine the dimensionless production time using Eq. (3.38):

tpDA=0.0002637ktϕμctA=(0.0002637)(12.7)(179.3)(0.13)(0.9)(1.9×105)(62)(43560)=0.1

Figure 3.14.

Horner plot with tpss for Example 3.4.

Using Eq. (3.39) with data from Table 3.3 (first row), the MBH dimensionless pressure is PDMBH =1.175. The average reservoir pressure is found with Eq. (3.45).

P¯=P*mPDMBH2.303= 2580 ‐ (57)(1.175)2.303  =2550.9psia

Solution by the TDS technique

The following information is read from the pressure derivative plot, Figure 3.15,

Figure 3.15.

Normalized pressure and pressure derivative versus equivalent time log‐log plot for Example 3.4.

(t*ΔP'q)r = 0.087 psia/BPD,   (ΔPq)r = 10.86 psia/BPD,  (teq)pss = 211.963 hr

(t*ΔP'q)pss = 0.1548 psia/BPD,  (ΔPq)pss = 10.86 psia/BPD

Permeability is found from Eq. (2.279), and the average reservoir pressure with Eq. (3.90);

k=70.6μBh(t*ΔPq)r=70.6(0.9)(1.3)(80)(0.087)=11.8md
P¯=2554141.2(251)(0.9)(1.3)(11.8)(80)[(0.164810.860.1648)(ln(927.180.33)34)]=2549.1psia

Escobar [15] reports the results of this example, Table 3.12, along with those from other methods already seen in this chapter. It is observed a very close agreement among the results.

Type of testMethodAverage reservoir pressure, psia
BuildupMBH
(Matthews‐Brons‐Hazebroek)
2550.9
BuildupDietz2542.6*
BuildupRamey‐Cobb2538.2*
BuildupMDH
(Miller‐Dyes‐Hutchinson)
2537.3*
BuildupAzari2535.4*
Multi‐rateTDS2549.1

Table 3.12.

Comparison of results for Example 3.4, after [15].

Reported in [15].


3.4.13. Other methods for estimating the average reservoir pressure

The average pressure can also be estimated using material balance [37],

P¯=Pi5.615qtctVp;Vpinft3E3.95

Another formulation to calculate the average reservoir pressure [37] is based upon integrating the reservoir pressure and the volume, thus:

P¯=PrΔVVE3.96

If ΔV is replaced as 2πrΔrhϕ and the pseudo‐steady‐state solution of the radial flow equation and after several manipulations will give:

P¯=Pw+0.8687m(lnrerw0.75)+ΔPsE3.97

For wells that are in steady state at the time of shutting‐in, constant 0.75 is changed by 0.5.

Slider [37] proposed an equation for the case where there is interference with other wells which includes obtaining the static pressure by extrapolating or correcting the pressure at a time equal to the stabilization time or time to reach the pseudo‐steady state, so that:

ΔPq=pwPwf+m*ΔtE3.98

According to Eqs. (1.130) and (2.40) (with the permeability in Darcies), the above equation becomes:

P¯=Pwf0.04ϕμctre2k1.79qϕhctre2+(ΔPq)tpssE3.99

Rearranging:

P¯=Pwf0.439m+(ΔPq)tpssE3.100

In summary, Slider [37] recommends the following methods for the determination of the average reservoir pressure at the shut‐in time:

  1. If the well is not acting under either steady or pseudo‐steady state, use material balance, Eq. (3.95). This includes wells under infinite behavior or in transition between infinite behavior and steady or pseudo‐steady state.

  2. If the well is in the center of its drainage area and is in either a steady or pseudo‐steady state, use Eq. (3.97), which does not require knowing either the porosity or the compressibility.

  3. If the well is located near the center of its drain area and is operating under pseudo‐steady state and the change in pressure with respect to time, m*, is known, Eq. (3.100) can be used.

  4. If the well is off‐center within the drain area and operates under pseudo‐stable state but m* is unknown, the MBH method is recommended by Slider [37].

  5. None of the methods are recommended for a well that is off‐center and operates in steady‐state conditions at shut‐in time.

  6. The above recommendations were produced when the pressure derivative did not exist. As could be seen, TDS technique is much more practical, and it is not limited to a few shape factors since this parameter is easily found with the technique. Also, TDS applies involve equations for fractured wells, horizontal wells (although not given here), and naturally fractured reservoirs. Since drill steam testing, DST consists of some small periods of buildup and drawdown, Chapters 2 and 3 apply to DST.

Nomenclature

Aarea, ft2
ATtotal field drainage area, Acres
Awwell drainage area, Acres
Bggas volume factor, ft3/STB
Booil volume factor, bbl/STB
Bwoil volume factor, bbl/STB
bfraction of penetration/completion
bDLFintercept of P‐vs.‐t0.25 plot during dual‐linear flow, psia0.5
bLFintercept of P‐vs.‐t0.5 plot during hemilinear flow, psia0.5
bPBintercept of P‐vs‐1/t0.5 plot during hemilinear flow, hr−1
bxdistance from closer lateral boundary to well along the x‐direction, ft
bydistance from closer lateral boundary to well along the y‐direction, ft
ccompressibility, 1/psia
Cwellbore storage coefficient, bbl/psia
CAreservoir shape factor
cttotal or system compressibility, 1/psia
DFdamage factor
DRdamage ratio
FEflow index
f(t)time function
hformation thickness, ft
hplength of perforations, ft
Iintercept
Jproductivity index, bbl/psia
kpermeability, md
kggas effective permeability, md
kooil effective permeability, md
kwwater effective permeability, md
IDcsginternal casing diameter, in
Npoil produced since last stabilization, bbl
mslope of P‐vs.‐log t plot, psia/hr/cycle
m*slope of P‐vs.‐t plot, psia/hr
mDLFslope of P‐vs.‐t0.25 plot during dual‐linear flow, psia0.5/hr
mLFslope of P‐vs.‐t0.5 plot during hemilinear flow, psia0.5/hr
mPBslope of P‐vs.‐1/t0.5 plot during hemilinear flow, (psia0.5/hr)−1
m'slope of superposition or equivalent time plot, psia/BPD/cycle
m‘b’intercept of superposition or equivalent time plot, psia/BPD/cycle
m(P)pseudopressure, psia/cp
ODcsgexternal casing diameter, in
Ppressure, psia
Paverage reservoir pressure, psia
PD′dimensionless pressure derivative
PDdimensionless pressure
Piinitial reservoir pressure, psia
PRproductivity ratio
Prreservoir pressure, psia
Ptshut‐in casing pressure, psia
Pwfwell flowing pressure, psia
Pwswell shut‐in or static pressure, psia
P1hrintercept of the semi‐log plot, psia
P*false pressure, psia
ΔPspressure drop due to skin conditions, psia
qliquid flow rate, BPD
qTtotal field flow rate, BPD
qwwell flow rate, BPD
qscgas flow rate, Mscf/D
rDdimensionless radius
rradius, ft
redrainage or external radius, ft
Rsgas dissolved in oil, scf/STB
rwwell radius, ft
sskin factor
scskin due to partial penetration
scpskin due to a change in permeability
sDLgeometrical skin factor converging from radial to dual‐linear flow
sLgeometrical skin factor converging from dual‐linear to linear flow
spskin factor due to the restricted flow entry
sPBgeometrical skin factor converging from dual‐linear to parabolic flow
sttotal skin factor
skin factor resulting from a well deviation angle
Treservoir temperature, ºR, transmissivity, md‐ft/cp
ttime, hr
tpproduction (Horner) time before shutting‐in a well, hr
tDdimensionless time based on well radius
tDAdimensionless time based on reservoir area
t*ΔP′pressure derivative, psia
Vvolume, ft3
Vuwellbore volume/unit length, bbl/ft
Xdistance in the x‐direction
XEreservoir length, ft
XNsuperposition time, hr
Ydistance in the y‐direction
YEreservoir width, ft
WDdimensionless reservoir width
Zheight, ft

Greek

Δtchange, drop, hut‐in time, hr
Δt′flow time after rate change in two‐rate tests
ϕporosity, fraction
λmobility, md/cp. Also, interporosity flow coefficient
ρfluid density, lbm/ft3
θdeviation angle, o
ψmeasured deviation angle, o
ψ′corrected deviation angle, o
μviscosity, cp
ωdimensionless storativity ratio

Suffices

0base conditions
1hrtime of 1 h
aactual
afafter flow
BRpssibirradial and pseudo‐steady‐state lines intersection
ddrainage
Ddimensionless
DAdimensionless with respect to area
DLdual linear flow
DL1dual linear flow at 1 hr
DLpssiintersection of pseudo‐steady‐state line with dual‐linear line
DLSS1intercept between dual‐linear line and the −1‐slope line (SS1)
DLSS2intercept between dual‐linear line and the −1‐slope line (SS2)
eqequivalent
eBLequivalent bilinear
eBRequivalent birradial
eLequivalent linear
eSPequivalent spherical
Finflection
ggas
hhorizontal
hshemispherical
iintersection or initial conditions
idealideal
INTintercept
invinvestigation
Llinear or hemilinear flow
L1linear flow at 1 hr
laglag
Lpssiintercept of linear and pseudo‐steady state lines
Nan arbitrary point during early pseudo‐steady‐state period
Mmuskat
Mintintercept of Muskat
maxmaximum
ooil
pproduction, porous
pDAdimensionless based on area and production time
PBparabolic flow
PBSS1intercept between parabolic line and the −1‐slope line (SS1)
PBSS2intercept between parabolic line and the −1‐slope line (SS2)
psspseudo‐steady state
pss1pseudo‐steady state at 1 hr
rradial flow
rDLiintercept of radial and dual linear lines
r1radial flow before spherical/hemispherical flow
rgrelative to gas
rLiintercept of radial and linear lines
rorelative to oil
rpssiintersection of pseudo‐steady‐state line with radial line
rSSiintersection between the radial line and the −1−slope line
rSS1iintersection between the radial line and the −1‐slope line (SS1)
rSS2iintersection between the radial line and the −1‐slope line (SS2)
rwrelative to water
sskin
sfsandface
spspherical
SSsteady
SSLstart of semi‐log line (radial flow)
SS1−1‐slope line formed when the parabolic flow ends and steady‐state flow regime starts. Well is near the open boundary, and the far boundary is opened
SS2−1‐slope line formed when the parabolic flow ends and steady‐state flow regime starts. Well is near the open boundary, and the far boundary is closed
swspherical/hemispherical wellbore
wwell, water
waapparent wellbore
wbwellbore
wDdimensionless emphasizing at wellbore
wfwell flowing
wf@Δt′=0well flowing at flow rate change
wswell shut‐in
xmaximum point (peak) during wellbore storage
xcmaximum point for centered wells
X1maximum point between Dual linear and parabolic lines
X2maximum point between parabolic and negative unit slope lines
X3maximum point between hemilinear and negative unit slope lines
zvertical direction

Distance to Linear Discontinuities

The available pressure analysis methods are based on assumptions of Darcy’s law, for example, a homogeneous and horizontal formation of uniform thickness, with isotropic and constant porosity and permeability distributions. The issue of pressure behavior in heterogeneous reservoirs has received considerable attention in recent years. The main reason for this is the need for greater accuracy in reservoir description, which has a significant effect on the design, operation, and therefore, the economic success of the projects involved. Since these methods can be applied only once to the reservoir, the need for a reliable description of the reservoir is obvious [5, 6, 13, 22].

Two techniques can be used in the fields to describe reservoirs: radioactive tracers and pressure transient tests. Pressure transient tests have been used more (and with better results) than tracers. The determination of the volumetric swept efficiency is a problem that has a better potential to be solved by the tracers. Currently, the description of the heterogeneity of the reservoir by adjusting the tracer behavior is affected by the lack of adequate numerical models, the long time spent to obtain the results, and the dependence of the adjustment to the additional parameters that are introduced by tracers themselves (e.g., dispersion coefficients, tracer retention, etc.). It is quite possible that tracers and future pressure transient tests will be used at the same time for the description of the reservoir [6].

Normally, any type of flow barrier cannot be seen in a DST since the time is too short to affect deep the reservoir. However, in cases where flow periods are so long to observe deviations from the semilog slope or deviation from the flat trend of the pressure derivative, which reflects changes in reservoir transmissibility, faults, discontinuities, boundary conditions, or reservoir geometry as illustrated in Figure 4.1. Some of the methods for estimating distance to linear boundaries will be shown later [4, 6, 14, 15].

4.1. Types of reservoir discontinuities

The heterogeneities of the reservoir (see Figure 4.1) are variations in rock and fluid properties resulting from deposition, folding, faulting, postdepositional changes in reservoir lithology, and changes in properties or types of fluids. The reservoir heterogeneities of the deposit may be small scale, as in carbonate reservoirs where the rock has two constituents, matrix and fractures, and cavities and caves. These can also be larger scale, such as physical barriers, faults, fluid‐fluid contacts, thickness changes, lithology changes, several layers with different properties in each layer, etc. In addition to these natural heterogeneities, man can induce artificial heterogeneities around wellbore during drilling (mud invasion), hydraulic fracturing, or fluid injection [6].

Figure 4.1

Types of discontinuities: (a) no‐flow boundary (fault), (b) change of fluid type, (c) change of formation thickness, and (d) permeability change (facies), after [6].

Another related feature is the anisotropy in the permeability, when this property varies with flow direction. Anisotropy can also be caused by sedimentary processes (filled cannel deposits) or by tectonism (fractures orientated parallel). Anisotropy takes place in both homogeneous and heterogeneous reservoirs. Therefore, anisotropy does not imply heterogeneity. Most reservoirs have vertical permeability less than horizontal, so there is anisotropy in that sense [5, 6].

4.2. Single‐boundary systems

Large‐scale heterogeneities can be detected by seismic. However, this technique can be up to one mile or more in error when estimating the well‐fault distance. Transient pressure analysis is the cheapest and most accurate form to obtain the distance from a well to a given barrier. In general, to locate faults [23, 24], a test long enough to explore the reservoir in depth is required, at least four times the distance to the fault.

4.2.1 Well‐fault distance from pressure buildup tests

Applying the superposition principle, the dimensionless shut‐in pressure for a well near a p boundary is given, respectively, by [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]:

PDS=[PD(1,(tp+Δt)D)+s][PD(1,ΔtD)+s]+PD(2drw,(tp+Δt)D)PD(2drw,ΔtD)E4.1

The Ei(−s) easily applies to find wellbore pressure, practically, at any time. Then, it can be applied to the two first terms of Eq. (4.1). For a production time, tp, long enough and for testing times very close to the shut‐in time, the Ei(−) solution can be fully applied to find the dimensionless pressure drops of Eq. (4.1), so that:

PiPws=70.6qμBkh[Ei(3792ϕμctd2kΔt)+Ei(3792ϕμctd2k(tp+Δt))Ei(3792ϕμctd2ktp)]E4.2

d can be calculated by a trial‐and‐error procedure using the above equation. This is exact but tedious. However, whenever tp >> Δt, the logarithmic approach for Ei(−x) function applies; therefore, Eq. (4.2) becomes:

Pws=Pi325.2qμBkhlog(tp+ΔtΔt)E4.3

Comparison between Eqs. (3.6) and (5.3) shows a doubled slope in the last one. It means that the detection of a fault leads to observe a doubled slope in the Horner plot since Eq. (4.3) can be expressed as:

Pws=Pi2mlog(tp+ΔtΔt)E4.4

Once the slope is folded, d can be easily calculated by reading the intercept time of the straight line of slope m with the slope straight line 2m, as illustrated in Figure 4.2(b) and (c). This behavior is also presented in a graph of drawdown pressure test indicated in Figure 4.2(a). However, the slope of a normal pressure buildup plot will not change at early times. Thus, this early straight line portion with slope m can be used to calculate k, s, and C as discussed in Chapter 3 section 3.2. Extrapolation of the double slope straight line is used for the estimation of the average reservoir pressure as studied in Chapter 3. Care must be taken since a similar behavior, double slope, is presented in multirate tests, injection tests, drawdown tests, etc. Different characteristics of the pressure transient may occur when a well is near multiple barriers. For example, when there are two faults intersecting at right angles to one well (one closer to the other), the slope will double and then doubled again. In general terms, the slopes that are obtained are function of the angles of intersection given by the following equation [20]:

Figure 4.2

Identification of linear boundaries from conventional (semilog) plots: (a) drawdown test—semilog plot, (b) buildup test—Horner plot, (c) buildup test—MDH plot, after [13].

Newslope=360θmE4.5

Similarly, the flat lines of radial flow regime will provide another flat pressure derivative line given by:

(t*ΔP)New=3602θ(t*ΔP)rE4.6

4.2.2 Methods for estimating the distance from a well to a discontinuity

Table 4.1 summarizes the available methods and provides some comments. Notice that developing the double slope takes a long time, actually, more than two log cycles.

MethodEquationEquation number and comment
Horner [5, 6]
d=ktpϕμct1.48109×104[(tp+Δt]/Δt)]xE4.7
It applies for ΔtD > 25. Less accurate for small tp values.
ΔtD=0.0002637kΔtϕμctd2E4.8
Earlougher and Kazemi, [7, 20, 25]
d=0.01217kΔtxϕμctE4.9
It uses either MDH or Horner plot. Strictly valid for [(tpt)/Δt]x ≥ 30. It requires long times for the slope to be double. Use Eq. (4.10) to find such time.
Δt=380000ϕμctd2kE4.10
Kucuk and Kabir [16]
d=0.00431kts2rϕμctE4.11
They modified Equation (4.9). ts2r is the beginning of the second semilog line. It is better found on the graph of the derivative.
Earlougher [5, 6, 18]
d=0.008119ktpϕμct(tD/rD2)E4.12
It is accurate for any time. PD is found with Eq. (4.13). It fails for tp >> Dt.
PD=12ln(tp+ΔtΔt)xE4.13
Find (tD/rD2) from Figure 1.7 or from Eq. (4.14) with x=log(PD).
tDrD2=100.53666069+1.843195406x10.8502854913x+0.1199676223x2E4.14
MDH [20, 21]
d=0.01217ktpϕμctΔtsE4.15
Dts is the time found by extrapolating the first slope to the value of Pi.
Sabet [20]
d=0.5×10{P2mPwf|2m|logt2mlogkϕμctrw2+3.230.435s}E4.16
Gray [12], Martinez and Cinco‐Ley [17]
d=tdvln(0.86859m1)3792.19ϕμctE4.17
TDS [14], and [15]
d=0.000422ktreϕμctE4.18
Corrected in this book
d=0.01217ktFϕμctE4.19

Table 4.1.

Methods to determine distance well‐discontinuity.

Example 4.1

Taken from [25]. The following pressure data were obtained from the Bravo‐1 well in West‐Texas. This is a limestone reservoir with water influence only in the southern portion. Geological data indicate the presence of a fault (Raven) to the east of the well. See the pressure buildup along with pressure derivative and Horner time data in Table 4.2. The properties of rock and fluid are as follows:

Δt, hrPws, psia(tpt)/ΔtΔP, psiatP′, psia
029000
0.530902401190119.12
0.731181715.29218111.80
1.131701091.9127095.56
1.6319975129991.80
2.5324048134089.52
3.53278343.8637872.10
5329024139046.45
73302172.4340232.60
93310134.3341030.91
13332093.3142029.23
2033336143327.72
3033434144333.68
4033503145042.54
5033632546348.26
70338218.1448257.39
10034001350058.58
1503423955062.50
25034505.855080.31

Table 4.2.

Pressure buildup test data.

rw = 5 in, h = 18 ft, ϕ = 14%, ct = 22 × 10−5/psia, μ = 1.8 cp, B = 1.31 bbl/STB

Pi = 3750 psia, r = 56.8 lbm/ft3, q = 180 BPD, Np = 9000 STB

Find permeability of the reservoir, flow efficiency, and distance to the Raven fault, using the methods of Horner, Earlougher and Kazemi, Kucuk and Kabir, Earlougher, Gray‐Martinez and Cinco‐Ley, and TDS technique [25].

Solution

Reservoir permeability. A Horner plot is given in Figure 4.2(a) Horner graph is given (semilog of Pws vs. (tpt)/Δt), where tp = 24Np/q = (24)(9000)/80 = 1200 hr, Eq. (3.2). Take the straight‐line portion with slope m = 66 psia/cycle (infinite behavior line). The permeability of the straight line is estimated with Eq. (2.33):

k=162.6qμBmh=162.6(180)(1.8)(1.31)66(18)=58.1mdE4.50

From Figure 4.2, P1hr = 3245 psia. Therefore, the mechanical skin factor is obtained from Eq. (3.6), thus:

s=1.1513[3245290066log(58.10.14(1.8)(22×105)(0.4172))+3.23]=1.7E4.51

The skin pressure drop is found with Eq. (2.35):

ΔPs=0.87(66)(1.7)psiaE4.52

P* = 3435 psia from the Horner plot, then flow efficiency is estimated with Eq. (2.46):

FE=197.634352900=81.8%E4.53

This means that stimulation is necessary. The distance to the linear boundary is found from the following methods:

Horner method

From the Horner plot, a value of (tpt)/Δt)x was found to be 27. Then, using it into Eq. (4.6) will provide:

d=0.01217kΔtpϕμct1[tp+Δt/Δt]x=0.0121758.1(1200)(0.14)(1.8)(22×105)(27)=83 ftE4.54

Earlougher and Kazemi method

From the MDH plot, a value of Δtx of 45 hr is read (Figure 4.3). Using it in Eq. (4.9),

Figure 4.2

Horner plot for Example 4.1.

Figure 4.3

MDH plot for Example 4.1.

d=0.01217kΔtxϕμct=0.0001481(58.1)(45)(0.14)(1.8)(22×105)=83.6ftE4.55

This method is supposed to work for [(tpt)/Δt]x ≥ 30 and [(tpt)/Δt]x = 27. Use Eq. (4.10) to find the time for the slope to be double:

Δt=380000ϕμctd2k=380000(0.14)(1.8)(22×105)(83.6)258.1=253.4hrE4.56

From the pressure derivative plot (Figure 4.4), it is possible to see the second plateau at about 70 hr.

Figure 4.4

Pressure and pressure derivative versus time log‐log plot for Example 4.1.

Kucuk and Kabir method

The pressure derivatives indicate that ts2r is about 70 hr. Using Eq. (4.11) gives:

d=0.00431kts2rϕμct=(58.1)(70)(0.14)(1.8)(22×105)=116.73ftE4.57

Earlougher method

It is obtained from Eq. (4.13):

PD=12ln(tp+ΔtΔt)x=12ln(27)=1.648E4.58

Then, x=log(PD)=0.217. Use this value in Eq. (4.14),

tDrD2=100.53666069+1.843195406(0.217)10.8502854913(0.217)+0.1199676223(0.217)2=13.82E4.59

This can be verified in Figure 1.7. Use the tD/rD2 value in Eq. (4.12):

d=0.008119ktpϕμct(tD/rD2)=0.008119(58.1)(1200)(0.14)(1.8)(22×105)(13.82)=77.5ftE4.60

Gray, Martinez, and Cinco‐Ley method

tdv = tre = 20 hr from the derivative plot (Figure 4.4). Use of Eq. (4.17) leads to:

d=tdvln(0.86859m1)3792.19ϕμct=20ln(0.86859[66])3792.19(0.14)(1.8)(22×105)=19.62E4.61

TDS technique

The buildup pressure derivative with a production time, tp, of 1200 hr was estimated and reported in Table 4.2. The second pressure derivative (not shown in here) was used to better define the inflection point. The pressure derivative was obtained and given in Figure 4.4 from which the following information was read:

tr = 9 hr, (tP′)r = 29.3 psia, ΔPr = 420 psia, tre = 20 hr, tre = 45.39 hr

Find permeability and skin factor with Eqs. (2.71) and (2.92), respectively:

k=70.6qμBh(t*ΔP)r=70.6(180)(1.8)(1.31)18(29.3)=56.82mdE4.62
s=0.5(42029.3ln[(56.82)(9)(0.14)(1.8)(22×105)(0.4172)]+7.43)=1.98E4.63

Use Eqs. (4.18) and (4.19) to find the distance from Bravo‐1 well to the Raven fault:

d=0.000422ktreϕμct=0.000422(56.82)(20)(0.14)(1.8)(22×105)=92.95ftE4.64
d=0.01217ktFϕμct=0.01217(56.82)(45.39)(0.14)(1.8)(22×105)=83ftE4.65

4.2.3 Methods for estimating the distance well‐discontinuity from DST

Table 4.3 presents some of the available methods to find linear discontinuity‐to‐well distance from drill stem tests.

MethodEquationEquation number and comment
Horner [4]Ei(948ϕμctd2ktp)=ln(tp+ΔtΔt)x(4.20)
For buildup
Dolan, Einarsen, and Hill [13]d=0.024337ktpϕμct[(tp+Δt]/Δt)]x(4.21)
For buildup
Ishteiwy and Van Poollen [13]d=0.015276ktpϕμct[(tp+Δt]/Δt)]x(4.22)
For buildup
(tp+ΔtΔt)x=tpD1.13(4.23)
tpD=0.0002637ϕμctd2(4.24)
Bixel, Larkin, and Van Poolen [2] and Bixel and Van Poolen [3]d=0.0307ktxϕμct(4.25)
For drawdown

Table 4.3.

Methods to determine distance well‐discontinuity from DST.

Example 4.2

Earlougher [6] presented DST data from the Red Formation of Major County, Oklahoma. The well was treated for completion work with approximately 480 bbl. The Horner chart is provided below.

q = 118 BPD,  ct = 8.2 × 10−6 1/psia,  μ = 1.3 cp, ϕ = 12%,           B = 1.1 bbl/STB,          m = 1321 psia/cycle, tp = 4 hr

  • Determine the permeability.

  • Determine the distance to the discontinuity using the methods mentioned.

  • Since there is no fault near the well, which suggests that it may be discontinuity?

Solution

(a) Permeability. Since the slope is given, permeability is found from Eq. (2.33):

k=162.6qμBmh=162.6(118)(1.1)(1.3)(15)(1321)=1.38mdE4.66

(b) Distance to the discontinuity

Horner method

From Figure 4.5, [(tpt)/Δt]x = 1.55. Using Eq. (4.20),

Figure 4.5

Horner plot for Example 4.2, after [13].

Ei(948(0.12)(1.3)(8.2×106)d21.38(4))=ln1.55E4.67
Ei[(2.1969×104)d2]=0.4383E4.68

Interpolating from Table 1.4,

2.1969×104d2=0.618E4.69

Then, d = 53 ft

Dolan, Einarsen, and Hill method

Using Eq. (4.21),

d=0.00059229ktpϕμct[(tp+Δt]/Δt)]x=0.00059229(1.38)(4)(0.12)(1.3)(8.2×106)(1.55)=40.7ftE4.70

Ishteiwy and van Poollen method

Using Eq. (4.22),

d=0.0002484ktpϕμct[(tp+Δt]/Δt)]x=0.0002484(1.38)(4)(0.12)(1.3)(8.2×106)(1.55)=25.5ftE4.71

(c) Since there is no fault near the well, which suggests that it may be discontinuity?

The slope decreases; however, there is evidence of a “constant pressure boundary.” The improvement in the transmissibility could be due to the treatment that well received before the test.

4.3. Leaky faults

There are cases in which the fault does not fully seal. A nonsealing fault allows the transient wave to cross over the fault and keep traveling. There probably exists a contrast in mobility. Meaning the formation on the other side of the fault may have or not the same properties. There are some other cases where the reservoir has a linear constant‐pressure boundary. The aquifer may act either fully or partially.

4.3.1 Nonsealing fault

Escobar et al. [8] used the dimensionless conductivity of the fault/boundary as:

FCD=kfkwfLE4.25

The FCD value typically ranges from 0 to 1.0 or more. A value of zero indicates a sealed boundary or absence of the boundary, and an infinite value indicates a constant pressure or a completely sealed fault.

4.3.1.1 Partially active aquifer

The expected behavior is given in Figure 4.6. At late time, the pressure derivative will display a negative slope of 1 (radial stabilization), which reduces its value as τ decreases. A second derivative in this zone will provide a maximum value.

A scalable dimensionless conductivity of the boundary, τ, is defined as [9]:

τ=eFCD1;1<τ<0E4.26

Negative values of τ indicate the presence of an act aquifer. Note that when τ = 0, FCD = 0 indicating that L = 0, and when t = −1, FCD= ∞, indicating that the boundary conductivity is infinite. Escobar et al. [8] presented the following correlation to find the acting degree of the aquifer.

τ=0.003759023.1638126(t*ΔP)2x(t*ΔP)rE4.27

If exists a second‐flat line below the radial flow pressure derivative line, the following expression applies:

τ=0.9833960.98603107(t*ΔP)2x(t*ΔP)rE4.28

Figure 4.6

Pressure and pressure derivative behavior for a partially active aquifer, after [9].

Figure 4.7

Pressure and pressure derivative behavior for a leaky fault, after [9].

4.3.1.2 Partially sealing (leaky) fault

This case is given in Figure 4.7 assuming there is no permeability contrast. This may be an explanation of why sometimes the second slope on the semilog plot is not doubled. A scalable dimensionless conductivity of the boundary (fault transparency), τ, is defined as [9]:

τ=1FCD;0<τ<1E4.29

Positive values of τ indicate the presence of no‐flow boundaries. A value of zero indicates that there is no fault/boundary, so the permeability on both sides of the border is the same. Note that when τ = 0, FCD= 1, and when τ = 1, FCD = 0, which indicate that the barrier has a permeability of zero. Escobar et al. [9] presented the following expressions to determine the fault transparency:

Figure 4.8

Pressure, pressure derivative, and second pressure derivative against time log‐log plot for Example 4.3, after [9].

τ=3.173803(t*ΔP)2x(t*ΔP)r0.0015121E4.30
τ=1.01338389(t*ΔP)r2(t*ΔP)r1.0146535E4.31

Example 4.3.

Escobar et al. [9] presented an example of a well near a leaky fault and it is required to confirm the transparency fault value. Pressure, pressure derivative, and second‐pressure derivative versus time data are given in Figure 4.8. Other relevant information is found below:

k = 10 md, h = 50 ft, d = 730 ft, τ = 0.22, ϕ = 23%, ct = 1.38 × 10−5 1/psia, μ = 0.3 cp, q = 200 BPD, B = 1.48 rb/STB, rw = 0.4 ft, CD = 70, s = 2, Pi = 5200 psia

Solution

The following information was read from Figure 4.8.

(tP′)r = 12.6 psia, (tP′)2r = 15 psia, (tP′)′2x = 0.865 psia

Use of Eqs. (4.30) and (4.31) allows finding the fault transparency:

τ=3.1738030.86512.60.0015121=0.217E4.72
τ=1.013383891512.61.0146535=0.192E4.73

Notice that both values closely match the original transparency fault value of 0.22.

4.3.2 Finite‐conductivity faults

Escobar et al. [10, 11] presented TDS technique extension for finite‐conductivity faults with mobility contrast and without it, respectively. The nonmobility contrast will be shortly discussed here. In these systems in which model was presented by Rahman et al. [19], the reservoir permeability has a lower permeability than that of the fault. Fluid flow takes place both across and along the fault plane (see Figure 4.9). The fault enhances the drainage reservoir capacity. It is observed in Figure 4.9, a normal flow radial regime is developed at early time around the well. When the transient reaches the fault (the fault may have some damage, sf), the pressure derivative declines along a negative‐unit slope. At this moment, the fault acts as constant pressure linear boundary. Then, as the pressure drops in the fault, a bilinear flow regime results when the flow is established in the fault plane thickness as shown in Figure 4.10. Finally, the pressure derivative response comes back to a plateau when radial is restored. This part is not shown in Figure 4.10.

New dimensionless quantities are presented here [10]:

tDf=0.0002637ktϕμctd2E4.32
hD=hdE4.33
FCD=kfwfkdE4.34

4.3.2.1 TDS technique

For the bilinear flow case, Escobar et al. [10] presented the following expressions for finding the fault conductivity, fault skin factor, and distance from well to fault. Escobar et al. [10] also presented equations for gas flow. Not all the expressions developed by Escobar et al. [10] are reported here.

d=0.0325ktreϕμctE4.35

Figure 4.9

Schematic of a typical fault system and flow lines, after [1].

Figure 4.10

Dimensionless pressure derivative for a well near finite‐conductivity fault. sf = 0 and 20, after [10].

d=0.0002637ktrssiϕμctsfhE4.36
sf=dh[(3.7351×106k2htss(t*ΔP)ssqμ2Bϕctd2)1]E4.37
kfwf=121.461(qμBh(t*ΔP)BL)2(tBLkϕμct)0.5E4.38
kfwf=1.694×109kd(ktBLssiϕμctLF2)2.51/(1+sfhLF)4E4.39

If the dimensionless fault conductivity is larger than 2.5 × 108, linear flow will be developed instead of bilinear and the fault has infinite conductivity and the distance from the well to it is found from:

d=6.42×106qBh(t*ΔP)LμtLkϕctE4.40

4.3.2.2 Conventional analysis

The bilinear flow model is given by:

ΔP=44.1qμBt1/4hkfwfϕμctk4+141.2qμBkhsBLE4.41

The above equation suggests that a Cartesian plot of ΔPwf versus t0.25 gives a linear trend in which slope allows for the estimation of the fault conductivity:

kfwf=[44.1qμBmBLhϕμctk4]2E4.42

The above equation was also found by Trocchio [28].

The pressure equation for the linear flow is:

ΔP=1.33×105qμBt1/2hd(ϕμctk)1/2+141.2qμBkhsLE4.43

As indicated before, Eq. (4.32) suggests that a Cartesian plot of ΔPwf versus t0.5 gives a linear trend in which slope allows the estimation for the distance from the well to the fault:

d=1.33×105qμBmLhϕμctkE4.44

The governing pressure equation for the steady state caused by the fault is:

ΔP=qμ2Bϕctd23.7351×106k2h(1+sfhd)21t+70.6qμBkhln(4d2rw2+8×105(sshd)2)E4.45

Also a Cartesian plot of ΔPwf versus 1/t gives a linear trend in which slope allows the estimation of fault skin factor:

sf=dh[(3.7351×106k2hmssqμ2Bϕctd2)1]E4.46

Trocchio [28] presented minimum fault length, xfmin, the following expression to find the dimensionless end time of the bilinear flow regime and the minimum fracture length:

xfmin=(2.54.55kkfwf±ϕμct0.0002637ktebf4)2E4.47

Example 4.4

Escobar et al. [10] presented a pressure test of a well inside finite‐conductivity faulted reservoir. Pressure and pressure derivative data are reported in Figure 4.11. It is required to estimate distance to fault and fault conductivity using both TDS and conventional methodologies.

k = 100 md, h = 100 ft, d = 730 ft, ϕ = 25%, ct = 1.3792 × 10−5 1/psia, μ = 0.7747 cp, q = 100 BPD, B = 1.553 rb/STB, rw = 0.3 ft, d = 250 ft

Solution

The log‐log plot of pressure and pressure derivative against production time is given in Figure 4.11 from which the following information was read:

(tP′)r = 8.474 psia, ter = 15.5 hr, tss = 551.93 hr, (tP′)ss = 0.955 psia, tBL = 983010 hr, (tP′)BL = 0.172 psia, trssi = 60 hr, tssBLi= 10000 hr, (t*ΔP′)min = 0.0912 psia

With the read parameters from Figure 4.11, the parameters were estimated and reported in Table 4.4. For straight‐line conventional analysis, only the bilinear flow regime part is plotted in Figure 4.12 from which a slope value of 0.0719 psia/hr0.25 was estimated. Then, Eq. (4.42) to find a finite conductivity of 1.128 × 109 md‐ft, which is also reported in Table 4.4.

Figure 4.11

Pressure and pressure derivative plot for Example 4.4, after [10].

Figure 4.12

Cartesian plot of pressure drop versus the fourth root of time for Example 4.4, after [10].

ParameterEquation usedResult
d, ft(4.35)247.85
d, ft(4.36)243.43
sf(4.37)0.00225
kfwf, md‐ft(4.38)1.14 × 109
kfwf, md‐ft(4.3)1.227 × 109
kfwf, md‐ft(4.42)*1.128 × 109
FCD(4.34)458900.15

Table 4.4.

Summary of results for Example 4.4, after [10].

Conventional analysis.


Nomenclature

Boil volume factor, bbl/STB
ccompressibility, 1/psia
ddistance from well to linear boundary, ft
cttotal or system compressibility, 1/psia
FCDfault dimensionless conductivity
FEflow index
hformation thickness, ft
kpermeability, md
kf wffault conductivity, md‐ft
Npoil produced since last stabilization, bbl
mslope of P‐vs‐log t plot, psia/hr/cycle
m1slope of first semilog straight line, psia/hr/cycle
Ppressure, psia
PD′dimensionless pressure derivative
PDdimensionless pressure
Piinitial reservoir pressure, psia
Pwfwell flowing pressure, psia
Pwswell shut‐in or static pressure, psia
P1hrintercept of the semilog plot, psia
P*false pressure, psia
ΔPspressure drop due to skin conditions, psia
qliquid flow rate, BPD
rradius, ft
rwwell radius, ft
sskin factor
ttime, hr
t*ΔP′pressure derivative, psia
t2*ΔP″second pressure derivative, psia
tpproduction (Horner) time before shutting‐in a well, hr
tDdimensionless time based on well radius
tdvtime at which either pressure or derivative deviate from first radial line, hr

Greek

Δchange, drop
Δtshut‐in time, hr
ϕporosity, fraction
ρfluid density, lbm/ft3
τscalable dimensionless conductivity
μviscosity, cp

Suffices

1hrtime of 1 hr
2msecond semilog straight line
2xmaximum in the second pressure derivative
BLbilinear flow regime
BLssiintercept of bilinear and steady‐state lines
ddistance
Ddimensionless
DAdimensionless with respect to area
Dfdimensionless with respect to fault length
ffault
ebfend of bilinear flow
Finflection, better found from second derivative
iintersection or initial conditions
Llinear flow regime
pproduction, porous
rradial flow
reend of radial flow regime
r2second plateau, hemiradial flow
rssiintercept of radial and steady‐state lines
sskin
ssa point on the steady‐state period
s2rstart of second semilog straight line
wwell
wfwell flowing
wswell shut‐in

Multiple Well Testing

The simplest form of interference testing involves two wells: a producer (or injector) and an observation well. The general idea is to produce in one well and observe the pressure drop in another. Multi‐interference testing usually involves a producer (or injector) and several observation wells. This is helpful to find horizontal anisotropy as explained by Earlougher and Kazemi [1] by type‐curve matching and [2] using TDS technique. To perform an interference test, all wells involved shut‐in to stabilize their bottom pressures. Then, the pressure recording tools are lowered into the observation well, and the producer (or injector) is opened to production (injection). If there is interference, a pressure drop is recorded in the observation well(s) within a reasonable length of time. Most of the multiple tests are performed in closed reservoirs [1, 3]. Multiple tests are performed for a number of reasons:

  • Search for reservoir connectivity and/or continuity of the reservoir [1, 4].

  • Detecting directional permeability and other heterogeneities [1].

  • Estimate reservoir volume [1].

  • Orientation (azimuth) of hydraulic fractures [5, 6].

For a two‐well system, the radius of investigation is given by [1]:

rinv=0.029ktϕμctE5.1

The skin in the active well does not affect the pressure in the observation well. There are two types of tests: interference and pulse.

5.1. Interference testing

These are used to determine [1, 7]:

  • Connectivity of the reservoir and transmissibility.

  • Direction of flow patterns. This is done by selective opening of wells around the observation well.

  • Storage capacity (storage factor) = St = ϕ ct h.

  • Determination of the nature and magnitude of the anisotropy. The permeability of the reservoir is found in all directions and the direction, θ, of the anisotropy angle [1, 2].

5.1.1 Conventional analysis

5.1.1.1 Earlougher method

Two wells: One active (injector or producer) and the other one of observation preferably shut‐in. The pressure in the observation well is [1]:

Pws=P1hr+mlogtE5.2

When t = 1 h, Pws ≈ P1hr ≈ Pi for new reservoirs. Eq. (5.2) is valid if tD/rD2 > 100 (x < 0.0025), being r the distance between wells. The restriction of tD/rD2 > 100 is applied with a 1% error [1].

When a plot of Pws versus log t is built, one should obtain a straight line which slope and intercept gives the transmissibility, Eq. (2.33) and porosity, respectively,

St=ϕctμ=Tr2e[2.302PiP1hrm7.41316]E5.3

Note that the skin factor does not appear in this equation since there is only fluid flow in the active well and not in the observation well. However, there are exceptions when the well is highly stimulated. Wellbore storage is also minimized in multiple tests but not entirely [1].

Two shut‐in wells

The buildup equation is given by [1]:

Pws=Pi+mlogt+ΔtΔtE5.4

t is the total production time (same as tp in normal buildup) in the active well. Prepare a Horner graph and using the slope value find the transmissibility with Eq. (5.3). Find porosity from:

St=Tr2e[2.302PiPwf(Δt=0)mln(1+1t)7.41316]E5.5

5.1.1.2 Ramey method

It includes one active well (producer or injector) and the one of observation preferably shut‐in. This method requires type‐curve matching with Figure 1.7. Once ΔPws = Pi − Pws versus at the observation well has been plotted and the best match is obtained:

(PD)M, (tD/rD2)M, ΔPM, tM

Use the following equations:

T=162.6qBPDMΔPME5.6
St=0.0002637Tr2tM(tD/rD2)ME5.7

Limitations:

rD > 20 (see Figure 1.6)

tD/rD2 >50 or 100

5.1.1.3 Tiab and Kumar method

As sketched in Figure 5.1, it uses some specific points:

P′m = the maximum value of the pressure derivative in the observation well which is placed at a distance r from the active well [8]. The units are psia/h.

tm = The time at which P′m occurs, h

Procedure:

  1. Obtain ΔP versus time in the observation well which is preferably shut‐in.

  2. Calculate P′ = Δ(ΔP)/Δt = change of ΔP/change in test time (later, it was known as the arithmetic derivative).

  3. Graph P′ versus t in log‐log paper, see Figure 5.2.

  4. Calculate St and transmissibility:

    St=0.0274qBr2(1Pm)E5.8
    T=948Str21tmE5.9

From the Cartesian plot (verification purposes):

T=382.2Str21toE5.10

It is very difficult to obtain the P’ due to noisy pressure values. Then, it is recommended to use the Cartesian plot. Select the inflection point there. Extrapolate the line and read the value of t0 as sketched in Figure 5.2.

Figure 5.1.

Log‐log plot of the arithmetic derivative, after [7, 8].

Figure 5.2.

Cartesian plot to find the inflection point, after [7, 8].

5.1.2 TDS technique

When plotting dimensionless pressure and pressure derivative versus dimensionless time divided by the dimensionless radius squared, a single profile will always be obtained as shown in Figure 5.3. This gives two characteristic features: (1) the radial flow regime is similar to that of a wellbore test with a flat derivative with a value of 0.5, Eq. (2.70), which allows finding the permeability from Eq. (2.71) and (2) a unique intersection point between the pressure and the pressure derivative that takes place before the actual flow regime is seen. These features allow obtaining the following observations [9]:

(tD/rD2)int=0.574952929E5.11

Figure 5.3.

Log‐log plot of PD and tD*PD′ versus tD/rD2 for an infinitive reservoir (line source) [2, 9].

where suffix int denotes intersection. The corresponding values of dimensionless pressure and the dimensionless derivative at this point of intersection are:

(PD)tD/rD2=0.57495=0.32369E5.12
(tD*PD)tD/rD2=0.57495=0.32369E5.13

Replacing Eqs. (1.89), (1.94), and (2.57) into the above expressions leads to obtain the following expressions:

T=khμ=45.705qB(ΔP)intE5.14
T=khμ=45.705qB(t*ΔP)intE5.15
St=ϕcth=0.000458646Tr2tintE5.16

And,

T=khμ=70.6qB(t*ΔP)r

Ref. [2] extended the application of the TDS technique in interference testing to determine areal anisotropy.

Example 5.1

Taken from [7], during an interference test, 3125 STB (stock-tank barrel) of oil was produced by well A. The pressure response was observed at well B, 138 ft away from well A for 300 hr. Test data are reported in Tables 5.1 and 5.2. Then, well A was shut‐in too, and the pressure response was observed at well B for 100 hr. Additionally, the following data are given:

μ = 1.3  cp,                   B = 1.14  bbl/STB,                   h = 31  ft

Pi = 2600 psia,           ρ = 55.4  lbm/ft3,                     s = −2.2 (well A)

ct = 16 × 10−6/psia,     Vu = 0.00697  bbl/ft

  1. Calculate permeability and porosity using: (A) Earlougher's method when well A is active and shut‐in, (B) the method of Tiab and Kumar, and (C) TDS technique

  2. Show that the wellbore storage effects are not important at well A.

t, hP, psiaΔP, psiat*ΔP, psiat, hP, psiaΔP, psiat*ΔP, psia
1.12595.64.45.15102575.524.511.19
1.52593.56.56.291525712911.60
22591.48.68.252525653511.39
2.52590108.683525613911.71
32587.512.59.056025554511.50
42585159.7610025495112.74
52583179.461502543.556.515.61
7.525792110.5330025307028.14

Table 5.1.

Pressure and pressure derivative data for well B (active).

t, hPws, psia(t1 + Δt)/Δtt, hPws, psia(t1 + Δt)/Δt
1.02541.0301.0010.02559.031.00
2.02544.0151.0015.02563.521.00
3.52547.085.7125.02569.013.00
5.02551.061.0040.02574.07.50
7.02555.043.8660.02577.05.00
100.02580.04.00

Table 5.2.

Pressure response at well B (shut‐in).

Solution

1. Calculate permeability and porosity using A) The Earlougher Method: Well A is active.

It is necessary to construct a graph in semilog of shut‐in pressure against time (see Figure 5.8). In this graph, a straight line is drawn whose slope, m = −25.517 psia/cycle. Since 3125 STB of oil were recovered during 300 hours of production, then flow rate, q, is 250 BPD. The permeability was then calculated using Eq. (2.33):

k=162.6qμBmh=162.6(250)(1.3)(1.14)(25.518)(31)=76.15 md

By linear regression analysis, we find that P1hr = 2600.53 psia. Use Eq (5.2) to find porosity (Figure 5.4):

ϕ=76.15(1.3)(1382)(16×106)e(2.302(26002600.53)25.5187.4316)=11.94%

Figure 5.4.

Semilog plot of Pwf versus Δt, after [7].

Earlougher method: well A is shut‐in

Figure 5.5 presents a semilog graph of Pws versus (t1t)/Δt. From the straight line, we have: m = −25.749 psia/cycle and P1hr = 2532.55 psia. Again, permeability is estimated with Eq. (2.33):

k=162.6(250)(1.3)(1.14)(26.749)(31)=72.65 md

Figure 5.5.

Horner plot of Pws versus (t1t)/Δt, after [7].

Find porosity with Eq. (5.5), thus:

ϕ=72.65(1.3)(1382)(16×106)e(2.303(2532.552530)26.749ln(1+1300)7.431)=8.7%

Tiab‐Kumar method

The derivative has a smooth of 0.5 cycles; then, it is smoothed possible derivative value. Figure 5.6 shows P'm = 4.19 psia and tm = 1.5 h. Eqs. (5.8) and Eqs. (5.9) are used to find porosity and permeability, respectively:

ϕ=0.0274qBhr2ct(1Pm)=0.0274(250)(1.14)(31)(1382)(16×106)(14.19)=19.7%
k=948ϕctμr2(1tm)=948(0.1574)(16×106)(1.3)(1382)(11.5)=39.4 md

Figure 5.6.

Arithmetic pressure derivative versus time log‐log plot for Example 5.1, after [7].

TDS technique

The pressure derivative plot gives a better understanding of the reservoir model. A very clear radial flow regime is seen, and actually, it is possible to observe late pseudosteady‐state period meaning that the reservoir boundaries have been felt. Since actual radial flow regime is observed, the permeability value found from there should be the most accurate one. The following information was read from Figure 5.7:

tint = 1.7 h, (tP')r = 11.5 psia, ΔPint = (tP')int = 7.5 psia

Figure 5.7.

Pressure and pressure versus time log‐log plot for Example 5.1, after [7].

Find permeability from Eqs. (2.76) and (5.14/5.15) and porosity from Eq. (5.16):

k=70.6qμBh(t*ΔP)r=70.6(250)(1.2)(1.14)31(11.6)=73.37 md
k=45.705qμBh(t*ΔP)int=45.705(250)(1.2)(1.14)(31)(7.5)=72.83 md
ϕ=0.000458646ktintμctr2=0.000458646(72.83)(1.7)(1.3)(12×106)(1382)=19.1%

2. Show that the wellbore storage effects are not important at well A

As seen in Section 3.2.1, if qaf/q < 0.01, it can be concluded that the afterflow or wellbore storage is not affecting the pressure data. To calculate qaf, find wellbore storage coefficient with Eq. (2.5) and then qaf with Eq. (3.8):

C=144Vuρ=1440.0069756.4=0.0178 bbl/psi
qaf=24CBdPwsdΔt=24(0.0178)1.144.682=1.755 BPD

The remaining calculations are shown in Table 5.3. In this table, it can be seen that the condition qaf/q < 0.01 is always fulfilled, so the effects of wellbore storage are not important.

t, hP′, psia/hqaf, BPDqaf/qt, hP′, psia/hqaf, BPDqaf/q
1.14.6801.7540.00702101.1190.4190.00168
1.54.1901.5700.00628150.7730.2900.00116
24.1251.5460.00618250.4560.1710.00068
2.53.4701.3000.00520350.3350.1250.00050
33.0171.1300.00452600.1920.0720.00029
42.4400.9140.003661000.1270.0480.00019
51.8920.7090.002841500.1040.0390.00016
7.51.4040.5260.002103000.0940.0350.00014

Table 5.3.

Arithmetic pressure derivative and afterflow data.

A summary of the results of porosity and permeability is given in Table 5.4 for comparison purposes. Definitely the radial flow is observed and provided a permeability of 73.37 md from Eq. (2.71). This value closely matches with those from conventional analysis, Eq. (2.33). The intersection point, Eq. (5.14), provided an excellent permeability value which means that that point was properly selected; therefore, the porosity should be about 19% which is well‐reported by TDS technique and Tiab‐Kumar method, but far from conventional analysis. Actually, from the derivative plot, Figure 5.7, the intersection point does not coincide with any datum in the test, but it was easily eyed interpolated which is not the case for the Tiab‐Kumar method which cannot be either interpolated or extrapolated. This old test does not have enough points but recently pressure well tests data have thousands of data points which enabled the use of Tiab‐Kumar method.

MethodPermeability, mdPorosity, %Equation number
Earlougher—Active76.1511.94(2.33) and (5.2)
Earlougher—shut‐in72.658.7(2.33) and (5.5)
Tiab‐Kumar39.419.7(5.9) and (5.8)
TDS73.37(2.71)
TDS72.8319.1(5.14/5.15) and (5.16)

Table 5.4.

Comparison of results of Example 5.1.

5.2. Pulse testing

This technique uses a series of short pulses of the flow rate. The pulses are alternating periods of production (or injection) and shut‐in with the same flow rate in each production. The pressure response to the pulses is measured in the observation well. The main advantage of pulse testing is the short duration of the pulse. A pulse can last for a few hours or a few days, which disrupts normal operation slightly compared to interference tests [1]. Besides determining conductivity (then, transmissibility and porosity), pulse testing has several applications, that is [6] use them to find the azimuth of a hydraulic fracture ([5] does the same with interference testing) and [10] for estimating permeability distributions.

The nomenclature of a pulse test is given in Figure 5.8. The following variables are defined as [1]:

  1. tL (time lag), is the time between the end of the pulse and the pressure peak caused by the pulse.

  2. ΔP/q (amplitude). The vertical distance between the tangent to two consecutive peaks and the line parallel to that tangent at the peak of the pulse to be measured, psia

  3. Δtc, pulse cycle. Time from start to end of a flow period, h.

  4. Δtp, pulse shut‐in period, h.

Figure 5.8.

Sketch of a pulse test, after [1].

The sign convention for ΔP is [1, 7]:

  1. ΔP > 0 if q > 0 (active producer well), ΔP/q > 0

  2. ΔP < 0 if q < 0 (active injector well), ΔP/q > 0

  3. ΔP < 0 for odd peaks

  4. ΔP > 0 for even peaks

5.2.1 Interpretation methods

5.2.1.1 Kamal‐Birgham method

Although the methodology was presented by [11], the charts and some equations were corrected later by [12]. The procedure is outline below:

  1. Plot ΔP/q versus t on Cartesian paper

  2. From this plot obtain the values of tL, Δtc and Δtp.

  3. Calculate the relation tLtc and F’ = Δtptc.

  4. Find [ΔPD(tLtc)2] from Figures 5.95.12, depending on the pulse, corresponding to F’ and tLtc from step 3 and calculate transmissibility, T, from:

T=141.2B(ΔP/q)(tL/Δtc)2[ΔPD(tL/Δtc)2]E5.17

Figure 5.9.

Relationship between transition time and amplitude response for the first odd pulse, after [12].

Figure 5.10.

Relationship between transition time and amplitude response for the first even pulse, after [12].

Figure 5.11.

Relationship between transition time and amplitude response for the all even pulses but first, after [12].

Figure 5.12.

Relationship between transition time and amplitude response for the all odd pulses but first, after [12].

Analyze all pulses since the first one may be affected by wellbore storage.

  1. Determine tLD/rD2, dimensionless time lag, from Figures 5.135.16 corresponding to F’ y tLtc obtained in step 3.

  2. Calculate St:

    St=0.000263(Tr2)tL(tLD/rD2)E5.18

Figure 5.13.

Relationship between transition time and cycle length for the first even pulse, after [12].

Figure 5.14.

Relationship between transition time and cycle length for the first odd pulse, after [12].

Figure 5.15.

Relationship between transition time and cycle length for all even pulses but first, after [12].

Figure 5.16.

Relationship between transition time and cycle length for all odd pulses but first, after [12].

Wellbore storage effects in the observation well increase with lag time and tend to reduce the amplitude of the first pulses. However, if r > 32(C/St)0.54 in the response well, storage effects are less than 5% of increase in the transition time and will not affect the amplitude. This is valid if [13]:

tDrD2>(230+15s)(CDrD2)0.86E5.19

Example 5.2

Taken from [7]. The pressure response data given in Table 5.4 were obtained from a producer well during a multiple test. Additional data concerning this test are shown below (Table 5.5):

t, hΔP, psiaΔP/q, psia/BPDt, hΔP, psiaΔP/q, psia/BPD
0.250.1750.00052.751.9250.0055
0.500.5600.00163.002.9750.0085
0.751.4000.00403.253.8500.0110
1.002.6250.00753.504.2700.0122
1.253.1500.00903.754.0600.0116
1.502.9400.00844.003.3600.0096
1.751.8900.00544.252.5900.0074
2.001.4000.00404.502.1000.0060
2.251.2600.00364.752.1000.0060
2.501.5050.00435.002.5550.0073

Table 5.5.

Pulse test data.

μ = 2.8 cp, B = 1.20 bbl/STB, h = 30 ft

ct = 12 × 10−6/psia, C = 0.002 bbl/psia at observation well

Shut‐in period = 0.7 h, flow period = 1.63 h

Well distance = 140 ft, q = 350 STB/D

  1. Calculate the formation permeability and porosity from the third pulse

  2. Recalculate k and φ with the other two pulses and compare. Explain the difference.

Figure 5.17.

Cartesian plot for Example 5.2.

Solution

The following information was read from Figure 5.17:

  1. ΔP1/q = 0.007041 psia/BPD, tL1 = 0.55 h

  2. ΔtC1 = ΔtC2 = ΔtC3 = 2.33 h, Δtp1 = Δtp2 = Δtp3 = 0.7 h

  3. ΔP2/q = 0.0066992 psia/BPD, tL2 = −0.0799 h

  4. ΔP3/q = 0.007455 psia/BPD, tL3 = 0.47 h

  1. Calculate the formation permeability and porosity from the third pulse

    First the ratios tLtC and F’ = ΔtptC are estimated to be tL3tC = 0.47/2.33 = 0.2017 and F’ = 0.7/2.33 = 0.3004. With these values, enter Figure 5.11 and read [ΔPD (tLtC)2] = 0.0033. Calculate permeability with Eq. (5.17),

    k=141.2μB[ΔPD(tL/Δtc)2](ΔP/q)(tL/Δtc)2h=141.2(2.8)(1.2)(0.0033)(0.007455)(0.20172)(30)=172.07 md

    The dimensionless time lag divided by the dimensionless squared radius is found from Figure 5.15 to be tLD/rD2 = 0.52. Estimate porosity with Eq. (5.18):

    ϕ=0.0002637kμr2cttL(tLD/rD2)=0.0002637(172.07)(2.8)(1402)(12×106)(0.47)(0.52)=6.22%

  2. Recalculate k and φ with the other two pulses and compare. Explain the difference.

    For pulse 1, tLtC = 0.55/2.33 = 0.236 and F’ = 0.7/2.33 = 0.3004. From Figure 5.10, [ΔPD (tLtC)2] = 0.0037. Calculate permeability with Eq. (5.17),

    k=141.2μB[ΔPD(tL/Δtc)2](ΔP/q)(tL/Δtc)2h=141.2(2.8)(1.2)(0.0037)(0.007041)(0.2362)(30)=149.21 md

    From Figure 5.14, tLD/rD2 = 0.25. Porosity is then estimated with Eq. (5.18),

    ϕ=0.0002637(55.98)(2.8)(1402)(12×106)(0.55)(0.26)=12.64%

For Pulse 2: Since the transition time, tlag, is negative (see Figure 5.17), which implies that the pressure is beginning to increase after the well is shut‐in, as shown in the graph. This behavior is not physically logical and may be caused by some error that occurred during the test.

At the first pulse, the permeability was reduced by 87%, and the porosity was increased by 203%. This was due to an increase in tL and reduction in the value of pulse amplitude. This can be caused by wellbore storage.

5.2.1.2 TDS technique

Figure 5.18 was built for different distance between wells and different ratios of production‐shut‐in periods. Comparing to Figure 5.3, the same intersection point is given. Also, the radial flow displays the same behavior plus some times this behavior is repeated among the pulses. Based on the above, it is concluded that Eqs. (5.14)(5.16) and (2.71) also work for pulse testing.

Figure 5.18.

Dimensionless pressure and pressure derivative against dimensionless both time and radius squared, finite‐source solution.

Example 5.3

A synthetic pulse test is presented in Table 5.6 and Figure 5.19. The below data were used to generate the test.

t, hΔP, psiat*ΔP, psiat, hΔP, psiat*ΔP, psiat, hΔP, psiat*ΔP, psia
0.0010.0116.0006.0651.46810.6851.3151.095
0.0140.0040.0246.0156.0620.03010.7201.5091.149
0.0160.0090.0466.0176.0560.05510.7621.7131.197
0.0190.0200.0796.0216.0440.09110.8121.9241.238
0.0230.0380.1256.0246.0240.13910.8712.1421.274
0.0270.0650.1836.0295.9950.19910.9412.3661.304
0.0320.1030.2546.0345.9550.27011.0252.5961.330
0.0390.1560.3366.0405.9020.35011.1252.8291.353
0.0460.2230.4266.0485.8350.43611.2443.0661.372
0.0550.3050.5206.0575.7540.52611.3853.3071.388
0.0650.4050.6166.0675.6570.61611.5533.5501.402
0.0770.5200.7106.0795.5460.70411.7533.7961.414
0.0920.6510.8016.0945.4210.78811.9924.0431.425
0.1100.7980.8856.1115.2830.86512.2754.2921.434
0.1310.9580.9636.1315.1320.93412.6124.5431.441
0.1551.1311.0336.1564.9700.99613.0144.7941.448
0.1851.3151.0956.1844.7991.05013.4925.0471.453
0.2201.5091.1496.2184.6191.09514.0605.3001.458
0.2621.7131.1976.2584.4331.13314.7375.5551.462
0.3121.9241.2386.3054.2401.16215.5425.8091.465
0.3712.1421.2746.3614.0431.18516.5006.0651.468
0.4412.3661.3046.4273.8421.20116.7585.8851.170
0.5252.5961.3306.5053.6401.21116.8055.6851.207
0.6252.8291.3536.5983.4351.21416.8615.4801.237
0.7443.0661.3726.7073.2311.21216.9275.2701.261
0.8853.3071.3886.8373.0281.20317.0055.0561.281
1.0533.5501.4026.9902.8261.18917.0984.8391.296
1.2533.7961.4147.1712.6271.17017.2074.6201.306
1.4924.0431.4257.3862.4321.14517.3374.4001.312
1.7754.2921.4347.6402.2411.11417.4904.1781.314
2.1124.5431.4417.9402.0571.07817.6713.9571.312
2.5144.7941.4488.2961.8781.03717.8863.7361.306
2.9925.0471.4538.7171.7070.99118.1403.5171.296
3.5605.3001.4589.2141.5440.94118.7963.0851.264
4.2375.5551.4629.8031.3900.88819.7142.6671.214
5.0425.8091.46510.5001.2460.83121.0002.2691.146

Table 5.6.

Pressure and pressure derivative versus time data for Example 5.3.

Figure 5.19.

Pressure and pressure derivative versus time for simulated pulse test of Example 5.3.

μ = 2 cp, B = 1.2 bbl/STB, h = 100 ft

Pi = 3200 psia, r = 100 lbm/ft3, q = 350 BPD

ct= 1.2 × 10−5/psia, rw= 0.3 ft, φ = 10%

k = 400 md

Find permeability and porosity for this example using TDS technique.

Solution

The following characteristic features were read from Figure 5.19:

tint = 0.131 h, (tP′)r = 1.5 psia, ΔPint = (tP′)int = 0.97 psia

Find permeability from Eq. (2.71) and (5.14/5.15) and porosity from Eq. (5.16):

k=70.6qμBh(t*ΔP)r=70.6(350)(2)(1.2)100(1.5)=395.4 md
k=45.705qμBh(t*ΔP)int=45.705(350)(2)(1.2)100(0.97)=395.8 md
ϕ=0.000458646ktintμctr2=0.000458646(395.4)(0.131)(2)(1.2×105)(1002)=9.9%

The results match quite well with the given porosity and permeability values.

Nomenclature

Boil volume factor, bbl/STB
ccompressibility, 1/psia
Cwellbore storage coefficient, bbl/psia
ddistance from well to linear boundary, ft
cttotal or system compressibility, 1/psia
FCDfault dimensionless conductivity
FEflow index
hformation thickness, ft
kpermeability, md
kfwffault conductivity, md‐ft
Npoil produced since last stabilization, bbl
mslope of P‐vs‐log t plot, psia/h/cycle
m1slope of first semilog straight line, psia/h/cycle
Ppressure, psia
PD′dimensionless pressure derivative
PDdimensionless pressure
Piinitial reservoir pressure, psia
P'mmaximum arithmetic pressure derivative, psia/h
P'arithmetic pressure derivative, psia/h
Pwfwell flowing pressure, psia
Pwswell shut‐in or static pressure, psia
P1hrintercept of the semilog plot, psia
P*false pressure, psia
ΔPspressure drop due to skin conditions, psia
qliquid flow rate, BPD
rradius, ft
rwwell radius, ft
sskin factor
Streservoir storativity, ft/psia
ttime, h
tLlag time, h
tmtime at which P'm occurs
t0extrapolated time for the inflection point, h
t*ΔP′pressure derivative, psia
t2*ΔP″second pressure derivative, psia
tpproduction (Horner) time before shutting‐in a well, h
tDdimensionless time based on well radius
tdvtime at which either pressure or derivative deviate from first radial line, hr
Treservoir transmissibility, md‐ft/cp

Greek

ΔP/qpulse amplitude
Δtshut‐in time, h
Δtcpulse cycle, h
Δtppulse shut‐in period
Δdrop, change
φporosity, fraction
ρfluid density, lbm/ft3
τscalable dimensionless conductivity
μviscosity, cp

Suffices

1hrtime of 1 h
2msecond semilog straight line
2xmaximum in the second pressure derivative
BLbilinear flow regime
BLssiintercept of bilinear and steady‐state lines
ddistance
Ddimensionless
DAdimensionless with respect to area
Dfdimensionless with respect to fault length
ffault
ebfend of bilinear flow
FInflection. Better found from second derivative
iintersection or initial conditions
intintersection between pressure and pressure derivative before radial flow
Llinear flow regime
Mmatching point
pproduction, porous
rradial flow
reend of radial flow regime
r2second plateau, hemi radial flow
rssiintercept of radial and steady‐state lines
sskin
ssa point on the steady‐state period
s2rstart of second semilog straight line
wwell
wfwell flowing
wswell shut‐in

Naturally Fractured Reservoirs

In these reservoirs, two different types of porosity are observed. The matrix has less permeability, and its porosity is small compared to that of the fractures, which also has high permeability. However, there are cases where the matrix has zero porosity and permeability, so the flow only occurs from the fractures. This type of behavior occurs in reservoirs with igneous or metamorphic rocks [4].

Naturally, fractured deposits have fractures with permeability, kf, and porosity, ϕf, and a matrix with permeability, km, and porosity, ϕm. Some reservoirs act as if they were naturally fractured, but they are not. This is the case of dissolved channels, interlayered systems with different permeability (interlayer dolomitic with limestones which have less density or interstratified sandstones with other limolites and fine‐grained sandstones). However, naturally fractured models can be applied to these types of reservoirs [29].

In this class of naturally fractured deposits, the two different types of porosity are found as shown on the left side of Figure 6.1. A very low porosity is presented in the fine pores, and another high porosity is represented by fissures, cavities, and fractures [46].

Figure 6.1.

Illustration of a naturally fractured deposit and its ideal representation [46].

When the classification of the naturally fractured reservoirs from the point of view of the flow (engineering) is carried out, the permeability and the porosity of the fracture must be taken into account and a comparison with the permeability and porosity of the matrix must be made. According to the above, Nelson [33] talks about the four types of fracture systems. Type I consists of those fractures that provide the storage capacity and permeability of the reservoir. Type II is that group of fractures that has a better permeability than that of the matrix. Type III is composed of those fractures in which the permeability is negligible, but the storage capacity of hydrocarbons is high. Finally, Type IV corresponds to those in which the fractures are filled with minerals, and it is generally not very feasible for the flow to develop [33].

Because of the above, this type of reservoirs is normally known as double porosity reservoirs. Their matrix permeability is negligible compared to that of the bulk fractured systems. Then, it is expected that the well to be fed only by the fractures as sketched in Figure 6.2.a. This chapter will devote on this type of systems. There is another kind in which the matrix porosity is not negligible and once depletion caused by fluid withdrawal takes place inside the fracture system, some fluid to the well once comes from the matrix as schematically shown in Figure 6.2.b. They are called double‐porosity double‐permeability reservoirs and TDS technique for this type of deposits is provided in [19].

As this point, the reader ought to be aware of one important issue. Most reservoirs, not all of them, are heterogeneous since porous media have chaotic and random distribution. However, as seen in Figure 6.2, the fluid comes from one or two media: either matrix or fractures. From the well testing point of view, when a unique system acts, then, the reservoir is recognized as a homogeneous, even though it is really heterogeneous. When two, as depicted in Figure 6.2, or more systems act, then the reservoir is meant to be heterogeneous.

Figure 6.2.

Schematic representation of (a) double‐porosity and (b) double‐permeability systems, after [27].

Based upon the above, naturally fractured reservoirs are heterogeneous. The idea of a homogeneous channel occurs outside of reality. However, the rock is fractured homogeneously, the percolation of the water causes mineral deposition, which reduces the permeability or completely blocks the channels of the fluid. Therefore, the fractures of homogeneous character change over time, and a heterogeneous rock is obtained. The porosity of the fracture is rarely greater than 1.5 or 2%. Usually, this is less than 1%. The storage capacity of the fracture, Sf = ϕfcfhf, is very small, because ϕf is small and hf is extremely low. In contrast, kf is very high. The storage capacity of the matrix, Sm = ϕmcmhm, is greater than the storage capacity of the fractures. Normally, the permeability of the matrix is less than the permeability of the fractures. If these have the same value, the system behaves as homogeneous and without fractures. If the permeability of the matrix is zero and the fractures are randomly distributed, the system has a homogeneous behavior. However, if the permeability of the matrix is zero, but the fractures have a preferential direction, then there is a linear flow. In addition, if the permeability of the matrix is small (usually less than 0.01 md) and the reservoir is widely fractured, the system behaves as homogeneous and without fractures. From the well testing point of view, three conditions must be met to determine if it is actually a naturally fractured deposit [34]:

  1. the porosity of the matrix is greater than the porosity of the fractures;

  2. the permeability of the matrix is not zero, but its permeability is much smaller than the permeability of the fractures; and

  3. the well intercepts the fractures.

6.1. Conventional analysis for characterization of naturally fractured reservoirs

Mavor and Cinco‐Ley [31] defined two parameters to characterize naturally fractured formations: the dimensionless storativity ratio, ω, and the interporosity flow parameter or flow capacity ratio, λ. As defined by Eq. (2.1), ω gives what fraction of the total porosity is provided by the fractures, and λ, Eq. (2.2), describes the matrix flow capacity available to the fractures.

ω=(ϕct)f(ϕct)f+(ϕct)mE6.1
λ=4n(n+2)kmrw2kfhm2E6.2

n in Eq. (6.2) depends on the model, n = 1 for strata mode, n = 2 for matchsticks model, and n= 3 for cubic model.

Odeh [34] examined several theoretical models and concluded that fractured deposits (especially with secondary porosity) generally behave as homogenous reservoirs. According to Warren and Root [46], a closure versus log pressure graph (tpt)/Δt will yield two portions of parallel straight lines as shown in Figure 6.3. The first straight line portion, if seen, can be used to calculate the total product kh by the conventional Horner method. Note that P1hr is taken from the second straight line. The average reservoir pressure is estimated by extrapolating, also, the second line to (tpt)/Δt = 1 to obtain P* and, then, using conventional techniques. Also, TDS technique is recommended [28, 32], as studied in Chapter 3. The vertical distance between the two semilog straight lines, see Figure 6.3, identified as ∂P can be used to estimate dimensionless storativity ratio [46]:

ω=e2.303Pm=10PmE6.3

Figure 6.3.

Horner plot for a naturally fractured reservoir [31].

From the above equation, if ∂P < 100, the storage capacity parameter, ω, may be in error. The Warren‐and‐Root parameter can also be estimated by reading the intersection points among the lines:

ω=t1/t2E6.4
λ=(ϕct)fμrw21.781kt1=(ϕct)f+mμrw21.781kt2E6.5

when ω approaches zero and λ ≤ 1×10−9, all permeability comes from the fractures. Figure 6.3 can be used to further understand the flow mechanics in naturally fractures formations. At the beginning of the flow, neglecting wellbore storage effects, all the flow comes from the fracture to the well in a radial flow manner; therefore, a flat line will be displayed in the pressure derivative curve. This is labeled as number 1 in Figure 6.3. Since the fracture porosity and height are small, it is expected that the fluid depletes inside the fracture system and pressure decline inside the fracture forces the fluid to come from the matrix to the fracture. That transition period is labeled as number 2 and is reflected as a “v” shape in the pressure derivative curve. At this point, it is good to know that flow from matrix to fractures can flow under pseudosteady‐state (mentioned “v”) of transient conditions. Once the fractures are filled with fluid, the radial flow regime (horizontal line on the pressure derivative curve) develops and the system behaves as homogeneous.

It is customary to assume that (ct)m = (ct)f and that is what is going to be treated in this book, just for academic purposes. However, Tiab et al. [43] demonstrated that the fracture compressibility is at least one order of magnitude higher than the matrix compressibility. Neglecting this reality can lead to a huge overestimation of fracture porosity. Actually, the determination of fracture compressibility is a laboratory challenge. The recommended way is using transient pressure analysis as demonstrated by Tiab et al. [43].

The interporosity flow parameter, λ, is a function of the ratio between the matrix permeability and the permeability of the fractures, the shape factor and wellbore radius [38, 46].

λ=α(kmkf)rw2E6.6

The α factor is the block shape parameter that depends on the geometry and the shape characteristics of the matrix‐fissures system, and is defined by:

α=AVxE6.7

where A = surface area of matrix block, ft2; V = matrix block volume, ft3; and x = characteristic length of the matrix block.

For the case of cubic block matrix separated by fractures, λ is given by:

λ=60lm2(kmkf)rw2E6.8

Being lm the length of the side of the block. For the case of spherical block matrix separated by fractures, λ is given by:

λ=15rm2(kmkf)rw2E6.9

where rm = radius of the sphere and finally, when the matrix is of blocks of horizontal strata (rectangular slab) separated by fractures, λ is given by (Figure 6.4):

λ=12hf2(kmkf)rw2E6.10

Figure 6.4.

Definition of the intersection point.

hf = thickness of a particular fracture or a high permeability layer.

Another method for estimating the interporosity parameter, λ, was proposed by Uldrich and Ershaghi [45]. This method used the inflection point time described in Figure 6.3. However, it is considered here not of practical use since it requires estimation of the Ei(‐x) function and chart‐information reading.

Eqs. (6.3)(6.5) has a strong drawback. Since the fractures promote increasing wellbore storage than the expected for a homogeneous system; then, the first or early semilog line is usually masked by wellbore storage. Therefore, conventional analysis cannot be used. To overcome this issue, Tiab and Bettam [42] provided an equation, Eq. (6.11), to find the interporosity flow parameter form the inflection point as Uldrich and Ershaghi [45] did. This equation is applicable to both drawdown and buildup tests. Once λ is known, t1 of Eqs. (6.4) and (6.5) can be known. Actually, Tiab et al. [43] demonstrated that wellbore storage affects more the estimation of ω than λ. In other words, the though occurred during the transition is more affected in the pressure scale than in the time scale; therefore, the estimation of the interporosity flow parameter may be acceptable.

λ=3792(ϕct)f+mμrw2kfbΔtF[ωln(1ω)]E6.11

Another good approximation for finding the interporosity flow parameter is presented by Stewart [38] with the aid of a MDH plot (although a Horner plot can also be taken) as the one given in Figure 6.5. A horizontal line passing throughout the transition period is drawn. The intersection of this line with the second semilog line provides tI which is used in the following expression:

λ=1.732(ϕct)m+fμrw2ktIE6.12

Figure 6.5.

MDH plot well R‐6, Example 6.1, after [36].

The beginning of the second semilog straight line, tb2, actual total system behavior response, can also be used. Under this condition, Eq. (6.11) becomes:

λ=4(ϕct)m+fμrw2ktb2E6.13

Bulk‐fracture permeability is found from an expression similar to Eq. (2.33);

kfb=162.6qμBmhE6.14

The skin factor can be determined from the first and second semilog line (recommended), respectively:

s=1.1513[P1hrPimlogt(kϕfcfμrw21ω)3.23]E6.15
s=1.1513[P1hrPimlogt(k(ϕct)f+mμrw2)3.23]E6.16

For pressure buildup analysis, change Pi by Pwf in Eqs. (6.15) and (6.16).

Example 6.1

Determine the bulk fracture permeability and ω and λ from a pressure test run in well R‐6 [36], according to the information given below and Table 6.1.

Δt, hrPws, psiatP′, psiaΔt, hrPws, psiatP′, psia
0.00052231.452699.87
0.01052325.612.052729.40
0.02352399.562.4527411.72
0.05852508.482.7527510.81
0.23052565.063.4552778.54
0.78052638.803.752817.78
1.40052699.874.052816.91

Table 6.1.

Pressure and pressure derivative data for well R‐6, after [36].

h = 1150 ft, rw= 0.292 ft, q = 17000 STB/D

Pwf = 5223 psia, tp = 408000 h, B = 1.74 rb/STB

μ = 0.47 cp, (ϕ ct)m+f = 1.4 × 10−06 psia−1, km = 0.148 md

Solution

The MDH graph given in Figure 6.6 confirms the existence of a system with double porosity (also, it can be verified in Figure 6.7). The following is read from there:

Figure 6.6.

Pressure and pressure derivative plot for well R‐6, Example 6.1, after [36].

Figure 6.7.

Points and characteristic lines of a naturally fractured reservoir with pseudosteady‐state interporosity flow, ω = 0.01, λ = 1 × 10−6, after [6].

m = 25.35 psia/cycle, ΔtF = 0.23 h, ∂P = 17.55 psia

The storativity capacity, ω, is estimated from the separation of the parallel lines using Eq. (6.3):

ω=exp(2.303Pm)=exp(2.30317.5525.35)=0.2031

Use Eq. (6.14) to find permeability,

kfb=141.2qμBmh=162.6(17000)(0.47)(1.74)(22)(1150)=89.4md

Find the interporosity flow parameter from Eq. (6.11):

λ=3792(ϕc)f+mμrw2kfbΔtF[ωln(1ω)]
λ=3792(1.4×106)(0.47)(0.2922)(77.54)(0.23)[0.2031ln(10.2031)]=3.86×106

Stewart and Asharsobbi [36] found a value of λ = 3.1×10−6.

6.2. Type‐curve matching for heterogeneous formations

There are several kinds of type curves available for naturally fractured reservoirs.

Some of the pressure type curves are free of wellbore storage effects along with their equation can be found in [29, 34, 39]. Some of them including the pressure derivative function can be found in [31, 35]. However, neither the equations nor the type curves are presented here since TDS technique precisely avoids them. Actually, the purpose of this book is to compile TDS technique application to several scenarios.

6.3. TDS technique for naturally fractured formations

Warren and Root [46] used this approach to develop an integrated and applicable solution for drawdown and buildup pressure tests in naturally fractured reservoirs with double porosity. From his work can be identified several flow regimes of the semilog analysis. In chronological order, there is a straight line in near time representing only fracture depletion and a straight line in final times, which corresponds to the time when the whole deposit produces as an equivalent homogeneous deposit. At these final times, the semilog straight line is parallel to the first straight line.

New developments by Mavor and Cinco‐Ley [31] included wellbore storage and skin effects for the interporosity flow parameter under pseudosteady‐state conditions in a naturally fractured reservoir. The solution was given in Laplace space and then was carried out in Laplacian space and inverted numerically using the Stehfest algorithm [37]. As a direct consequence, type curves were developed by Bourdet et al. [3], which included wellbore storage and skin in naturally fractured deposits. Subsequently, reservoir parameters could be estimated when storage would dominate pressure data at early times. An advance in the curves type of naturally fractured deposits occurred with the addition of the derivative curve [3]. Increasing the sensitivity of the derivative curve [9] for naturally fractured deposits results in a better accuracy when applying type‐curve matching.

Unfortunately, type‐curve matching is a trial‐and‐error method, which often provides nonunique responses. Besides, it could be really difficult to have all the type curves for all the emerged cases. Therefore, the Tiab’s direct synthesis technique [40, 41] extended for naturally fractured formations by Engler and Tiab [6] is presented in this chapter. Actually, a more extensive great work on the subject was performed by Engler [5] who also developed TDS technique for horizontal wells in anisotropic formations [7] and naturally fractured deposits [8]. As originally exposed by Tiab [40], this method combines the characteristic points and slopes of a log‐log plot of pressure and pressure‐derived versus time data with the exact analytical solutions to obtain expressions for reservoir characterization considering that flow from matrix to fractures take place under pseudosteady‐state situation.

6.3.1 Mathematical model

An actual naturally fractured formation consists of a heterogeneous system of vugs, fractures, and matrix, which are random in nature. To model this system, it is assumed that the reservoir consists of discrete matrix block elements separated by an orthogonal system of uniform and continuous fractures [46]. These fractures are oriented parallel to the main axes of permeability. Two geometries are commonly assumed, for example, layers and cubes of sugar. The flow between the matrix and the fractures is governed by a pseudosteady‐state condition, but only the fluid entering the well comes from the fracture network reach at a constant rate. It is assumed that the fluid is a single phase and slightly compressible. The dimensionless well pressure solution in a reservoir of infinite action along with its dimensionless pressure derivative, with the previous assumptions is given by [46], and also presented by Engler [5] and Engler and Tiab [6]:

PD=12[lntD+0.80908+Ei(λtDω(1ω))Ei(λtD1ω)]+sE6.17
tD*PD=12[1exp(λtD1ω)+exp(λtDω(1ω))]E6.18

The dimensionless quantities given by Eqs. (1.94), (1.89), and (2.57) are now rewritten as:

tD=0.0002637kfbtμ(ϕct)m+frw2E6.19
PD=kfbh(PiP)141.2qμBE6.20
tD*PD=kfbh(t*ΔP)141.2qμBE6.21

6.3.2 Characteristic points and lines

Refer to Figure 6.7 and notice that the radial flow has been interrupted by the transition period during which fractured are fed from the matrix. Each radial flow is labeled r1 and r2 for distinguishing purposes. As for the case of a homogeneous system, Eq. (2.70) applies (do not consider the dimensionless wellbore storage coefficient), so that Eq. (2.71) applies, now rewritten as:

kfb=70.6qμBh(t*ΔP)r1=70.6qμBh(t*ΔP)r2E6.22

Needless to say that in case that both radial flow regimes are seen, bulk fracture permeability can be obtained from any of them using the above expression.

The transition period is affected by the dimensionless storativity coefficient, but independent of the interporosity flow parameter. Engler [5] and Engler and Tiab [6] found an analytical expression for the minimum coordinates by taking the derivative of Eq. (6.18) and equating the result to zero. Subsequently, the dimensionless minimum coordinates are given by:

(tD)min=ωλln1ωE6.23

and,

(tD*PD)min=0.5(1+ω1/(1ω)ωω/(1ω))E6.24

Eq. (6.23) was based for [42] to derive Eq. (6.11) which is now rewritten as:

λ=3792(ϕct)f+mμrw2kfbΔtmin[ωln(1ω)]E6.25

To set Eq. (6.24) in oil‐field unit, Engler [5] and Engler and Tiab [6] developed a form to normalize it by division with the radial flow derivative, Eq. (2.70), to yield:

(t*ΔP)min/(t*ΔP)min=(1+ω1/(1ω)ωω/(1ω))E6.26

Engler [5] and Engler and Tiab [6] developed the following empirical correlation:

ω=0.15866{(t*ΔP)min(t*ΔP)r}+ 0.54653{(t*ΔP)min(t*ΔP)r}2E6.27

which is valid from 0 ≤ ω ≤ 0.10 with an error less than 1.5%. An alternative method for determining ω arises from the defined characteristic times of pressure derivative curve shown in Figure 6.7. These include the end of the first horizontal straight line, tDe1, the start of the second horizontal straight line, tDb2, and the time corresponding to the minimum derivative, tDmin. Engler and Tiab [6] developed several empirical correlations by relating such times:

λ=ω(1ω)50βte1=ωln(1/ω)βtmin=1βtusi=5(1ω)βtb2E6.28

where

β=0.0002637kfb(ϕct)f+mμrw2E6.29
ω=exp[10.9232(tmin50te10.4386)]E6.30

The correlation for the ratio of the minimum time to the time for the end of the first straight line has an error less than 5% [6].

ω=0.19211{5tmintb2}+0.80678{5tmintb2}2E6.31

The correlation using the ratio of the minimum time to the start time of the second straight line, valid for ω ≤ 0.1, has with an error less than 2%.

For a given dimensionless storativity coefficient, the minimum dimensionless pressure coordinate is independent of the interporosity flow parameter, while the minimum dimensionless time coordinate is a function of λ. Subsequently, Engler and Tiab [6] found that a log plot (tD*P'D)min versus log (λtD)min results in a straight line with unit slope. The corresponding empirical equation is:

ln(tD*PD)min=ln(λtDmin)+ln(0.63)E6.32

From which was obtained:

λ=[42.5h(ϕct)f+mrw2qB](t*ΔP)mintminE6.33

An alternative method for determining λ can be carried out by observing a straight line with unit slope characteristic during the last transition period. The smaller dimensionless storativity coefficient (lowest point of the curve) adjusts the data more exactly to the unit slope line. A ω < 0.05 results in a more accurate estimate of λ. For ω > 0.05, λ will be overestimated. The analytical equation for this behavior of the last transition time is [6]:

ln(tD*PD)us=ln(λtDus/2)E6.34

The intersection of the unit slope line of the transition period with the line of the radial flow regime pressure derivative, Eq. (2.70), shown in Figure 6.7, allowed finding a simple expression to determine λ [6]:

λ=1/tDusiE6.35

Replacing Eq. (1.94) in the above expression leads to:

λ=((ϕct)f+mμrw20.0002637kfb)1tusiE6.36

As for the case of Eq. (2.92), the mechanical skin factor, expected always to be negative for naturally fractured formations, is, found for each one of the radial flow regimes, r1 and r2, thus, [6]:

s=12[(ΔPt*ΔP)r1ln(kfbtr1(ϕct)fμrw2ω)+7.43]E6.37
s=12[(ΔPt*ΔP)r2ln(kfbtr2(ϕct)f+mμrw2)+7.43]E6.38

Eq. (6.37) may be of not practical use since as commented before on the difficulty to obtain a representative value of the fracture compressibility.

6.3.3 Wellbore storage effects

As discussed earlier in this chapter, a direct consequence of wellbore storage is the tendency to mask the early time radial flow period. Therefore, the late or second radial flow line of infinite action is essential for estimating the skin factor and the permeability of the net of fractures. If wellbore storage is presented, it can be obtained, from the early unit slope, by using Eqs. (2.61), (2.69), and (2.80). We must be aware that Eqs. (2.81) and (2.87) were developed for higher skin factors, then, they are not recommended to apply in naturally fractured reservoirs.

The influence of wellbore storage on minimum coordinates is of great importance in the analysis. As Figure 6.8 shows, the dilemma is whether the minimum observed point is the actual minimum or a “pseudo‐minimum” as a direct result of wellbore storage. Engler [5] and Engler and Tiab [6] have shown that the minimum point is not affected by storage for all ω and λ, provided that,

(tD)min,o(tD)x10E6.39

Figure 6.8.

Wellbore storage effect on the minimum value of the pressure derivative, ω = 0.05 and s = 0, after [6].

Accordingly, the procedures described above are valid. When the ratio of the minimum time to the time at the peak is less than the limit defined by Eq. (6.39), a “pseudo‐minimum” occurs in the curve of the pressure derivative. An empirical correlation generated during this region provides a method to calculate the interporosity flow parameter [6],

[λlog(1/λ)]min=1CD[5.565txtmin,o]10E6.40

where,

λ=([λlog(1/λ)]min1.924)1.0845E6.41

The corrected ω is found from Figure 6.9.

Figure 6.9.

Determination of the dimensionless storativity coefficient using the ratio of the radial with the minimum pressure derivatives, after [6].

An alternative method for determining λ is based on the ratio of the coordinate of the minimum pressure derivative to the coordinate of the pressure derivative at the peak. This correlation is valid only for CDλ > 0.001 [6], and CD is found from Eq. (2.14);

λ=110CD(t*ΔP)min(t*ΔP)xE6.42

Tiab et al. [43] determined that the minimum is not affected by wellbore storage for any value w, provided the conditions given in Table 6.2 are fulfilled. Then, they proposed a better expression for correcting the minimum:

(t*ΔP)min=(t*ΔP)r+(t*ΔP)minO(t*ΔP)r[1+2D1D2]1+D2[ln(C(ϕct)f+mhrw2)+2s0.8801]E6.43
λCD
10−4CD > 10
10−5CD > 100
10−6CD > 103
10−7CD > 104
10−8CD > 105

Table 6.2.

Conditions for the minimum pressure derivative being affected by wellbore storage, after [43].

where;

D1=[ln(qBtminO(t*ΔP)r(ϕct)f+mhrw2)+2s4.17]E6.44

and;

D2=48.02CqB((t*ΔP)rtminO)E6.45

Being tminO and (tP')minO, the value of the coordinates of the minimum point in the derivative without making any correction (observed) when there is wellbore storage effect. Once corrected, the following expression can be applied:

ω=(2.9114+4.5104(t*ΔP)r(t*ΔP)min6.5452e0.7912(t*ΔP)r(t*ΔP)min)1E6.46

Tiab et al. [43] also provided another expression for ω;

ωω=eλtDminE6.47

where tDmin is found using Eq. (1.94) rewritten as;

tDmin=0.0002637ktmin(ϕct)f+mμrw2E6.48

For values of ω less than 0.5 the solution of Eq. (4.47) is:

ω=(2.9114+3.5688λtDmin+6.5452λtDmin)1E6.49

Although Engler [5] and Engler and Tiab [6] provided step‐by‐step procedures for the application of TDS technique, they are omitted here since it is not mandatory to follow such procedures.

Example 6.2

Tiab e al. [43] presented the derivative plot, Figure 6.10, for a pressure test run in a heterogeneous formation. Other relevant information is given below:

Figure 6.10.

Pressure and pressure derivative against time log‐log plot for Example 6.2, after [43].

q = 960 BPD,  B = 1.28 rb/stb,   μ =1.01 cp

h = 36 ft,     rw = 0.29 ft,     (ϕct)m+f = 0.7 × 10−61/psia

It is required the interpretation of this test to provide permeability, skin factor, and the Warren‐and‐Root parameters.

Solution

As observed in the pressure derivative plot, Figure 6.10, this pressure test may not be interpretable using conventional analysis since the first radial flow regime is absolutely masked by wellbore storage. The following information is read from such plot:

tN = 0.00348 h, ΔPN = 11.095 psia, tr2 = 1.8335 h

ΔPr2 = 61.5 psia, (tP′)r2 = 10.13 psia, tmin,o = 0.07 h, (tP')minO = 5.32 psia

Permeability is found from Eq. (6.22);

kfb=70.6qμBh(t*ΔP)r2=70.6×960×1.01×1.2810.13×36=238md

The wellbore storage coefficient and the dimensionless wellbore storage coefficient are determined with Eqs. (2.61) and (2.14) to be:

C=qB24(tΔP)N=(960(1.28)24)0.0034811.095=0.0161bbl/psia
CD=0.894C(ϕct)f+mhrw2=0.894(0.0161)0.7×106(36)(0.292)=6792

Estimate skin factor from Eq. (6.38):

s=12[61.510.13ln((238)(1.8335)0.7×106(1.01)(0.292))+7.43]=4.6

The interporosity flow parameter is found with Eq. (6.33)

λ=[42.5h(ϕct)f+mrw2qB](t*ΔP)mintmin=[42.5(37)0.7×106(0.292)960(1.28)]5.320.07=5.57×106

Since λ = 5.57 × 10−6 and CD = 6792, by looking at the third row in Table 6.2, the coordinates of the minimum point must be corrected before finding ω. Therefore, use Eqs. (6.44), (6.45), and (6.43), so that:

D1=[ln(960(1.28)(0.07)(10.13))+2(4.6)4.17]=1.8334
D2=48.02(0.0161)960(1.28)(10.130.07)=0.09105
(t*ΔP)min=10.13+5.32(10.13)[1+2(1.8334)(0.09105)]1+0.09105[ln(0.01610.7×106(36)(0.292))+2(4.6)0.8801]=0.9849

Then, ω is found from Eq. (6.46), and the correlation is given by Eq. (6.27):

ω=(2.9114+4.510410.130.98496.5452e0.791210.130.9849)1=0.024
ω=0.15866{0.984910.13}+ 0.54653{0.984910.13}2=0.0206

Without correction, Eq. (6.27) would have given a value of ω = 0.234.

Example 6.3

This was also worked by Tiab et al. [43]. Pressure and pressure derivative against time data are given in Table 6.3 and plotted in Figure 6.11 for its interpretation. Other important data are given below:

Figure 6.11.

Pressure and pressure derivative against time log‐log plot for Example 6.3, after [43].

t, hPwf, psiatP', psiat, hPwf, psiatP', psiat, hPwf, psiatP', psia
0.09334373.484.4731.09304060.387.23412.433824.2137.651
0.17664299.1133.4831.264043.184.38414.433804.1136.857
0.26004246.1146.7761.4274032.276.71920.433758.7138.810
0.34334203.6151.5952.4274002.875.40126.433720.3135.210
0.42664173.8157.6183.4273971.390.50232.433695.1134.790
0.51004139.7150.2954.4273948.387.16838.433674.6134.116
0.59334118.5141.3555.4273931.695.59544.433652.4156.278
0.67664103.5111.6766.4273917.1108.30350.433636.9183.611
0.76004086.499.6947.4273898.4122.33653.433625.2196.734
0.92664075.495.7209.4273865.3142.426

Table 6.3.

Pressure and pressure derivative data against time for Example 6.3, after [43].

q = 3000 BPD,     ϕ = 0.10,     μ = 1.0 cp

ct = 3.0 × 10−5 psia−1, B = 1.25 bbl/stb,  h = 100 ft

rw = 0.40 ft,     Pi(t=0) = 4473 psia

Calculate the permeability of the fractured system, skin factor, wellbore storage coefficient, interporosity flow parameter, and dimensionless storativity coefficient.

Solution

The following information was read from Figure 6.11:

ΔPN = 99.6 psia,    (tP′)N = 116.4 psia,   tN = 0.093 h

tb2 = 14.4 h,      tr2 = 20.43 h,           ΔPr2 = 714.3 psia

(t*ΔP′)r2= 138.5 psia,  tx = 0.43,        tmin,o = 2.427 h

tmin,o/tx = 5 < 10,         ti = 0.14 h,           (tP′)min = 72.087 psia

Solution

Use Eq. (6.22) to estimate permeability

kfb=70.6qμBh(t*ΔP)r2=70.6(3000)(1.0)(1.25)100(138.5)=19.1md

Find the wellbore storage coefficient using Eqs. (2.61), (2.69), and (2.80).

C=(qB24)tNΔPN=(3000(1.25)24)0.09399.6=0.146bbl/psia
C=(qB24)tN(t*ΔP)N=(3000(1.25)24)0.093116.4=0.129bbl/psia
C=kfbhti1695μ=19.1(100)(0.14)1695μ=0.158bbl/psia

Estimate skin factor from Eq. (6.38):

s=12[714.3138.5ln19.1(20.43)3×106(0.1)(1)(0.42)+7.43]=5.13

The ratio between the minimum time and the time in the peak suggests that the minimum coordinates are influenced by wellbore storage. Then, find the dimensionless wellbore storage using Eq. (2.14) with an average C of 0.143 bbl/psia:

CD=0.8935Cϕhctrw2=0.8935(0.143)0.1(100)(3×106)(0.42)=26618

Eq. (6.40) leads to find:

[λlog(1/λ)]min=1CD[5.565txtmin,o]10=126618[5.5650.432.427]10=0.000032

Then, λ = 5.2 × 10−7. Find again λ with Eq. (6.42):

λ=110CD(t*ΔP)min(t*ΔP)x=110(26618)72.087151.94=1.78×106

For the determination of ω, estimate the ratio between the pressure derivatives of minimum point and radial flow regime, (tP′)min/(tP′)r = 72.078/138.5 = 0.52. Then, find the parameter needed to enter in Figure 6.9, thus:

CDλ[log(1λ)+0.8686s]=26618(1.78×106)[log(11.78×106)+0.8686(5.13)]=0.061

From Figure 6.9, it is found that ω = 0.07.

The reader may think that the subject covered by the TDS technique in this chapter is the only one as far as naturally fractured reservoirs are concerned. The material exposed in this chapter was the first one ever introduced. TDS technique is certainly rich in applications. Regarding naturally fractured systems can be named: double porosity and double permeability for vertical wells [19], already mentioned at the beginning of this chapter, and for horizontal wells [30]. For triple porosity, reservoirs referred to [10]. For horizontal wells in both homogeneous and heterogeneous deposits including the effect of the threshold pressure gradient, the reader is invited to read [21]. The work originally presented for long homogeneous reservoirs [11] was extended to naturally fractured deposits by Escobar et al. [13, 14]. For hydraulically fractured wells, draining heterogeneous formations refer to [13, 20, 26, 44]. Escobar et al. [15] presented TDS technique for gas wells in naturally fractured systems. The effect of pseudotime on the Warren‐and‐Root parameters was observed by [18] for vertical wells and [17] for horizontal wells. TS technique for rate transient analysis of homogeneous and heterogeneous formations was presented by [1] and extended to long reservoirs by [16] and gas bearing long fractured formations by [12]. However, there are publications written by some other researchers and are not reported here.

Lately, shale reservoirs have been the target of many oil and gas companies. Gas reservoirs must be hydraulically fractured for production to occur. Their behavior normally follows those of naturally fractured formations. Some publications regarding shales tested either at constant bottom‐hole pressure or constant rate using TDS technique are [2, 22, 23, 24, 25].

Nomenclature

Boil volume factor, bbl/STB
bfraction of penetration/completion
ccompressibility, 1/psia
Cwellbore storage coefficient, bbl/psia
D1minimum point correction parameter
D2minimum point correction parameter
cttotal or system compressibility, 1/psia
hformation thickness, ft
kefbulk fractured network permeability, md
mslope of P‐vs‐log t plot, psia/h/cycle
Ppressure, psia
PD′dimensionless pressure derivative
PDdimensionless pressure
Piinitial reservoir pressure, psia
Pwfwell flowing pressure, psia
Pwswell shut‐in or static pressure, psia
P1hrintercept of the semilog plot, psia
P*false pressure, psia
qliquid flow rate, BPD
rDdimensionless radius
rradius, ft
rwwell radius, ft
sskin factor
ttime, h
tpproduction (Horner) time before shutting‐in a well, h
tDdimensionless time based on well radius
t*DP′pressure derivative, psia
Vvolume, ft3

Greek

αshape factor
Δchange, drop
Δtshut‐in time, h
∂ Pparallel difference between the two radial flow regime slopes, psia
ϕporosity, fraction
λinterporosity flow coefficient
μviscosity, cp
ρfluid density, lbm/ft3
ωdimensionless storativity coefficient

Suffices

1intercept between first semilog radial flow and the transition line
1hrtime of 1 h
2, Iintercept between second semilog radial flow and the transition line
b2start of second radial flow regime
Ddimensionless
e1end of first radial flow regime
Finflection
ffracture network
iintersection or initial conditions
Nan arbitrary point during early pseudosteady‐state period
mmatrix
maxmaximum point
minminimum point
minOobserved minimum point
rradial flow
r1radial flow before transition period
r2radial flow after transition period
sskin
usiintersect of the pressure derivative lines of the unit‐slope line during the transition and second radial flow regime
wwell
wfwell flowing
wswell shut‐in

Hydraulically Fractured

Hydraulic fracturing is a technique consisting of cracking the rock with a liquid fluid, normally of non‐Newtonian nature, which carries some solid particles (sand or synthetic material) so that reservoir fluids can move easier toward the well. The fluid has three main uses: (1) as pressuring fracturing tool by overpassing the formation fracture gradient, (2) as carrying agent to transport the solid material, called proppant, to the fracture face so, when reducing the pressure, avoids the fracture to close completely and provides flow capability to the fracture, called conductivity, and (3) as lubrication agent. In unconventional fracturing, the fracturing fluid is water and about 4% of the injected mass corresponds to sand.

The orientation of the hydraulic fractures is a function of the distribution of stress in the formation [3, 14]. If the least stress in the formation is horizontal, then a vertical fracture will be obtained. On the other hand, if the least important stress is vertical, then a horizontal fracture will occur [4, 8, 30]. However, there is a general belief that vertical fractures are obtained at depths greater than 3000 ft.

Figure 7.1 is a plane of a bounded circular system, which is a well with a vertical fracture. The fracture length has been exaggerated for explanatory purposes. Generally, the fluid enters the fracture at a uniform flow rate per unit area of the face of the fracture so that there is a pressure drop in the fracture. In this case, the fracture refers to a “uniform flow fracture.” However, for some fractures that have infinite permeability (conductivity), the pressures are uniform throughout. Except for fractures with a high content of support material and conductive fractures, it is thought that the uniform flow fracture represents much better reality than the fracture of infinite conductivity [7, 12, 39].

7.1. Well drawdown pressure behavior

The dimensionless pressure in the well for the case of a uniform flow fracture is [4, 5, 7, 8]:

PD=πtDxferf(1/2tDxf)0.5Ei(1/4tDxf)E7.1

And for the case of an infinite conductivity, fracture is:

PD=0.5πtDxf[erf(0.134/tDxf+erf(0.866/tDxf)][0.067Ei(0.018/tDxf)0.433Ei(0.75/tDxf)]E7.2

Figure 7.1

Schematic representation of a vertical fracture, after [6].

where

tDxf=tD(rw/xf)2E7.3

If tDxf < 0.1 in Eq. (7.1) and tDxf < 0.1 in Eq. (7.2), these two equations become:

PD=πtDxfξE7.4

It is important to note that Eq. (7.4) is a new version proposed by Bettam et al. [2], which considers both homogeneous deposits, ξ = 1, and heterogeneous (naturally fractured double porosity) deposits, ξ = ω (dimensionless storage coefficient). Eq. (7.4) also indicates that at early times the flow within the fracture is linear. In real units, Eq. (7.4) can be written as [8]:

Pwf=PimlftE7.5

where

mlf=4.064qBhμkξϕctxf2E7.6

mvf is the slope of the Cartesian plot of Pwf against t1/2, which can be used to calculate:

kxf2=(4.064qBhmlf)2μξϕctE7.7

If certainty exists in the determination of the linear flow regime, let us say the pressure derivative is available and reservoir permeability is accurate (it does not need correction), then use Eq. (7.7) directly and find the half‐fracture length. The following procedure should be flowed, otherwise. In which the conventional semilog analysis applies to tDxf > 10. Eqs. (7.1) and (7.2) are, respectively, converted to [5, 6, 26, 33]:

PD=12[ln(tDxf)+2.80907]E7.8
PD=12[ln(tDxf)+2.2]E7.9

These two equations give the dimensionless pressure for pseudoradial flow, as long as the boundary effects are not found. In a bounded reservoir, the period of infinite action pseudoradial flow only develops completely if xe/xf > 5.

There is an approximate relationship between the pressure change at the end of the linear flow period, ΔPel, and at the beginning of the semilog straight line, Pbsl, [26, 33]:

ΔPsbl2ΔPelE7.10

If this relationship is not met, it is because the linear flow period or the radial flow period was incorrectly selected. A couple of pertinent observations:

  • A graph of ΔP versus time on a log‐log paper will produce a straight line of mean slope during the linear period.

  • The above analysis is valid for pressure decline and injection tests.

7.2. Conventional analysis

In vertically fractured wells, pressure buildup and falloff tests are similar to nonfractured wells. As shown in Figure 7.2, the semilog slope, m, obtained from traditional analysis of a fractured well is erroneously very small and the value of m decreases progressively as xf increases [32]. In other words, the fracture presence partially masks the radial flow regime. The pressure derivative certainly allows finding the true start of the semilog line. Because of that fracture effect, permeability, estimated from the Horner or MDH graph, should be corrected as follows:

k=kcFcorE7.11

where

Fcor=(kh)true/(kh)apparentE7.12

Figure 7.2

Effect of fracture length on the semilog slope [32].

And the uncorrected permeability is found from Eq. (2.33),

kc=162.6qμBmhE7.13

The correction factor, Fcor, is read from Figure 7.3. The Horner plot is strongly recommended for data analysis of vertically fractured wells. xe/xf must be known (normally assumed by a trial‐and‐error procedure) to use Figure 7.3. xf is simply the half‐fracture length, which can be estimated from the slope of the Cartesian plot of pressure versus square root of time, using Eq. (7.7).

Figure 7.3

Correction factor for kh estimated from pressure buildup tests in vertically fractured wells, after [31].

Fcor can be estimated iteratively as follows:

  1. Estimate kxf2 from Eq. (7.7).

  2. Estimate kc using Eq. (7.13).

  3. Calculate k from Eq. (7.11) using a reasonable assumed value of xe/xf (As a first try, assume xf = 0.5xe) in Figure 7.3.

  4. Use the value of k from step 3 to estimate xf with Eq. (7.7).

  5. Find xe with Eq. (7.14).

    xe=0.029ktpϕμctE7.14

  6. This new value of xf is used to compute a new value of xe/xf to be used in Figure 7.3. This would improve the estimation of k.

  7. This process continues until two successive xe/xf values are equal.

Example

The pressure buildup data obtained after a hydraulic fracturing treatment are shown in Table 7.1. The characteristics of the reservoir and well for this test are as follows:

Δt, hrPws, psiaΔt0.5, hr0.5ΔP, psia(tpt)/ΔttP’, psia
0.011700.000
0.513290.707159729.00101.02
1.013881.000218365.00134.47
1.514641.225294243.67157.31
215011.414331183.00170.63
315701.732400122.33204.95
416392.00046992.00243.36
617482.44957861.67285.81
1018993.16272937.40300.65
1820754.24390521.22333.95
2722095.196103914.48357.02
3623046.000113411.11362.66
4523756.70812059.09360.86
5424347.34812647.74375.13
6324817.93713116.78391.74
7125168.42613466.13415.29

Table 7.1.

Pressure, pressure drop, and pressure derivative versus time data for Example 7.1, after [34]*.

Pressure derivative was not given in [34].


q = 101 BPD, rw = 0.198 ft, h = 42 ft, φ = 8%, μ = 0.45 cp, B = 1.507 bbl/STB, ct = 17.7 × 10−6/psia, tp = 364 hr

Find permeability, skin factor, and half‐fracture length using conventional analysis.

Solution

Taken from [34]. In this example, ξ = 1 because it is a homogenous deposit. A Horner graph for the data given in Table 7.1 is presented in Figure 7.4. The pressure and pressure derivative plot was just built for verification purposes (Figure 7.5). In fact, an early slope of ½ is shown in such plot. An infinite‐conductivity or uniform‐flux fracture is dealt with since during that period the pressure doubles the pressure derivative. The slope is –510 psia/cycle. The permeability can be estimated from the slope of the semilog straight line, using Eq. (7.13):

kc=162.6qμBmh=(162.6)(101)(0.45)(1.507)(510)(42)=0.52mdE7.150

Figure 7.4

Horner plot for Example 7.1, after [34].

Figure 7.5

Log‐log plot of pressure and pressure derivative against time.

Figure 7.6

Cartesian plot of Pws against t0.5, after [34].

Figure 7.6 contains a Cartesian plot of pressure versus the square root of time (actually, a tandem time function is recommended to be used instead of a normal square root of time function). To estimate permeability, the trial‐and‐error process is evoked, so that:

  1. Assume xf/xe = 0.5.

  2. With the value of step 1, a correction factor, Fcor=0.46, was read from Figure 7.3 (Horner curve).

  3. From Figure 7.6, Cartesian plot of Pws versus t0.5, the slope mlf is 141.3 psia/h0.5.

  4. Estimate the kxf2 from Eq. (7.7):

    kxf2=((4.064)(101)(1.507)(42)(141.3))2(0.45)(0.08)(17.7×106)=3452.6mdft2E7.151

  5. Apply the correction factor on Eq. (7.11) to find:

    k=kcFcor=(0.52)(0.46)=0.2392mdE7.152

  6. Estimate the half‐fracture length:

    xf=kxf2k=3452.60.2392=120.14ftE7.153

  7. Find xe from Eq. (7.14):

    xe=0.029ktpϕμct=0.029(0.2392)(364)(0.08)(0.45)(17.7×106)=339ftE7.154

  8. Calculate the ratio xf/xe:

xfxe=120.14339=0.356E7.155

Repeat step. 2 through 7 for n iterations until xf(i) ≈ xf(i‐1). After 12 steps, the permeability is k = 0.475493 md. A computer program was written for this purpose. The remaining results are reported in Table 7.2. Once the final iteration is achieved, the half‐fracture length is found from:

xf=kxf2k=3452.60.4754=85.2ftE7.156

Assumed xf/xexe, ftFcork, mdxf, ftNew xf/xe
0.5337.98970.457280.237785120.49820.356514
0.356514385.99660.5964060.310131105.51170.273349
0.273349449.98460.8105330.42147790.507850.201135
0.201135471.30040.8891410.46235486.414390.183353
0.183353476.48180.9087990.47257585.474720.179387
0.179387477.63160.9131900.47485985.268940.178525
0.178525477.88140.9141460.47535685.224370.178338
0.178338477.93550.9143520.47546385.214730.178298
0.178298477.94710.9143970.47548785.212650.178289
0.178289477.94970.9144070.47549285.21220.178287
0.178287477.95020.9144090.47549385.21210.178287
0.178287477.95030.9144090.47549385.212080.178287

Table 7.2.

Summary of iterations for Example 7.1.

Since P1h = 1138 psia and using Pwf = 1170 psia known from Table 7.1, the skin factor is estimated with Eq. (3.6):

s=1.1513[11701138320log(0.671(0.08)(0.45)(17.7×106)(0.282))+3.23]=4.37E7.157

As expected, the well is stimulated.

7.3. Type‐curve matching

As mentioned before, type‐curve matching is attempted to be avoided with TDS technique. However, for the interest of the reader, several type‐curves have been presented. For this goal, the reader may refer to [5, 7, 29, 31, 32, 39]. The last one included pressure derivative and wellbore storage.

7.4. Fracture conductivity

The product of fracture permeability, kf, and fracture width, wf, is known as fracture conductivity, kfwf,. The conductivity of the dimensionless fracture is expressed mathematically as [6]:

CfD=kfwfkxfE7.15

The above expression can also be found multiplied by π. However, it is customary, in well test analysis, to be used as given in Eq. (7.15). The uniform flow fracture [25, 26, 35] is one of the concepts introduced in the literature for the interpretation of well test data in fractured wells. This type of conductivity assumes that the flow from the reservoir to the fracture is uniform and a small pressure drop occurs along the fracture [25]. This type of conductivity can be observed in fractures with high damage caused by a zone of low permeability around the fracture. An infinite‐conductivity fracture has a conductivity such that the pressure drop along the fracture is considered to be zero. In a log‐log plot, this type of fracture is identified by a half slope on the pressure and pressure‐derived early data. A fracture is considered to have infinite conductivity and the separation between these two curves should be two times. When its dimensionless fracture conductivity is greater than 300, the fracture has finite conductivity, otherwise [6], which is identified in a log‐log plot by a slope of ¼ of the early data on both pressure and pressure derivative. The separation between these two curves should be four times. If this number is higher than four, possible, pseudoskin due to high gas flow rate is presented. A slope ½ may or may not be displayed later. A finite‐conductivity fracture involves a pressure drop along the fracture. This pressure drop contributes to the formation of a simultaneous linear flow in the fracture and a linear flow from the formation to the fracture, called bilinear. The long duration of the bilinear flow is a consequence of low fracture conductivity. Figure 7.7 clearly explains how the flux from formation to the fracture is. There exists a linear flow inside the fracture for finite‐conductivity fracture cases.

In infinite‐conductivity fractures, Tiab [35] showed that the ratio of the length, xe, with the fracture length, xf, has some influence on the flow pattern (see Figure 7.10). Theoretically, if xe = xf, only a slope of ½ will be observed indicating the presence of pure linear flow in the formation. However, as the increase of xe/xf ≥ 16 in the straight line of unit slope is short, then only a slop. 0.36 is formed. This is due to the biradial flow, Tiab [35] calls it. Other authors have called it elliptical flow. When the relation xe/xf ≥ 16, only the slop. 0.36 is developed and observed.

7.5. Cartesian plot of pressure against one‐fourth root of time

The modified bilinear flow equation [2] (originally presented by Cinco et al. [6]), to respond for homogeneous and heterogeneous reservoirs, is presented below:

PD=2.45CfD1/2tDxfξ4E7.16

After replacing the dimensionless variables (Eqs. 1.89, 7.15, and 7.45), the following expression is obtained:

ΔP=44.1qμBhfkfwf(ξϕμctk)0.25t0.25E7.17

The fracture conductivity can be found from the slope, mbf, so that:

kfwf=44.1qμBmbf(ξϕμctk)0.25hft0.25E7.18

For this case, xf can also be expressed using the abovementioned test and error procedure. When the bilinear flow ends, the graph exhibits curvature either concave up or down depending on CfD. If CfD < 1.6, there will be concavity downward. If CfD > 1.6, the concavity up toward that indicates that the tip of the fracture begins to affect the pressure well behavior.

If the test is not run long enough to terminate the bilinear flow when CfD > 1.6, it is not possible to determine the half‐fracture length. When CfD < 1.6, the flow of fluid in the reservoir has changed from a predominant one‐dimensional linear flow and a two‐dimensional flow regime. In this case, it is not possible to properly determine xf even if the bilinear flow ends during the test. These rules can be avoided with TDS technique and the pressure derivative curve.

Cinco et al. [6] indicated that CfD can be estimated from a Cartesian graph of P versus t1/4 reading the value of ΔP when bilinear flow ends, ΔPebf by:

CfD194.9qμBkhΔPebfE7.19

Figure 7.7

Flow distribution along the fracture, after [6].

Cinco et al. [6] also showed that the end of the bilinear flow line “ebf” depends on CfD and can be estimated from:

tDebf0.1CfD2;CfD>3E7.20
tDebf0.0205[CfD1.5]1.53;1.6CfD3E7.21
tDebf[4.55CfD2.5]4;CfD1.6E7.22

Since CfD and kfwf are known, then xf can be estimated from the definition of CfD.

7.6. Cartesian plot of pressure against the square root of time

Cinco et al. [6] and Cinco‐Ley and Samaniego [7] presented the following expressions:

ΔP=4.064qBhfxfμtξϕctkE7.23

Eq. (7.23) has the modification given by Bettam et al. [2]. The slope mlf is obtained from the Cartesian plot and is useful to find the half‐fracture length:

xf=4.064qBmlfhfμξϕctkE7.24

The outer boundary can distort the semilog line if xf > xe/3. The pressure behavior during the infinite‐acting period is very dependent on xf. For relatively short fractures, the flow is radial but becomes linear as xf grows and reaches xe. The m (semilog) obtained from conventional analysis of a fractured well is erroneously very small and the value of m decreases progressively as xf increases [6, 7, 26, 33], and hence the calculation of the half‐fracture length requires trial and error.

The smaller the flow capacity, the longer the curved portion. The beginning of the linear flow in the formation “blf” depends on CfD and can be approximated by [6, 7, 26, 33]:

tDblf=100(CfD)2E7.25

And at the end of the linear flow period, “elf” occurs approximately at:

tDblf=0.016E7.26

telf and tblf are the times for the end and beginning of the linear flow regime and serve to determine the dimensionless conductivity of fracture.

CfD=0.0125telftblfE7.27

The linear flow in the fracture ends as a function of the value of the dimensionless hydraulic diffusivity of the fracture [7] ηfD:

tDxf=0.01(CfD)2(ηfD)2E7.28
ηfD=kfϕcftkfϕctE7.29

The pressure data during the transition period show a curved portion before the line representing the linear flow is obtained. The duration of the curved part represents the transition and depends on the flow capacity of the fracture. For CfD > 0.5, the start time of the linear flow regime is governed by:

tblf=227.8μctxf2k(CfD1.39)E7.30

The linear flow ends at a dimensionless time of approximately 0.016 and the pseudoradial flow starts at a tD of about 3 and continues until the boundaries have been felt. The pseudoradial flow does not appear if the distance to the border is 10 times smaller than xf. The equation that approximates this flow regime is [26, 33]:

PDf=0.5lntDlf+1.1E7.31

The uniform flux has less duration than the linear. In the linear, the pseudoradial period is achieved earlier at a tD ≈ 1. For uniform flow, the fracture length is:

xf=rwes+1E7.32
s=lnxfrw1E7.33

In fractured wells, the pseudoradial flow is governed by:

PDf=0.5lntDf+f(CfD)E7.34

The pseudoradial flow period is identical to the radial flow of an unfractured well but with a negative damage factor caused by the influence of the fracture. During this period, the behavior of the pressure is described by:

PDf=0.5ln(xf2tDxfrw2)+0.404+sE7.35

The start of the semilog line is given by [6]:

tDssl=5exp[0.5(CfD0.6)]E7.36

There is an approximate relationship [6, 26, 33] between ΔPelf and ΔPbrf:

ΔPbrf=2ΔPelfE7.37

Figure 7.8

Determination of n, after [6, 7, 26].

This rule is known as the “double P rule.” For fractured wells, twice the Pelf marks the beginning of the pseudoradial flow. Equivalently, a time rule referred to as “rule 10t” can be applied at the beginning of the pseudoradial flow, tbrf, by:

tbrf=10telfE7.38

Another approach that can be used to mark the beginning of the radial flow for finite‐conductivity fractures is:

tDbrf5exp(CfD0.6);CfD>0.1E7.39

The fracture length can be determined with the following expressions and the aid of Figure 7.8 for the determination of n [6, 7, 26].

xf=nrwesE7.40
s=lnxfnrwE7.41

Alternatively, instead of using Figure 7.8, the below polynomial fit can be used:

y=0.9581692210.414066786x+0.308171775x2+0.05438571x30.197400959exE7.42
x=log(CfD)E7.43
n=10yE7.44

Theoretically, bilinear flow regime takes place at a dimensionless time given by:

tDxf=0.1CfD2;CfD>16E7.45
tDxf=(4.55CfD1/22.5)4;CfD<16E7.46

On the other hand, the occurrence of linear flow formation is characterized by a slope of 1/2 in the graph log‐log of pressure and pressure derivative. This flow regime will normally be evident and analyzable for fractures with high conductivity (CfD > 100). The beginning of the linear flow regime occurs in:

tDfCfD2=100E7.47

To verify that the data used for the analysis actually represent linear flow, Eq. (7.7) was properly applied; the valid range of data occurs during:

100CfD2<tDxf<0.016E7.48

Based on the time at which the linear flow ends, tDxf = 0.016, it is possible to estimate the permeability of the formation. At the end of the linear flow, the data of P versus Dt1/2 deviate from the straight line. Using the time of this deviation with Eqs. (7.15) and (7.50) will yield:

k=101.1qμBhmvftelfE7.49

In Eq. (7.49), telf represents the end of the linear flow regime.

Further in this chapter, the biradial flow will be characterized (Figure 7.11). The pressure equation for such flow was presented by [16]:

PD=2.14(xexf)0.72(tDAξ)0.36E7.50

After replacing the dimensionless quantities in the above expression:

ΔP=mellt0.36E7.51

where

mell=15.53(qBμkh)(xexf)0.72(kξμϕctA)0.36E7.52

which indicates that a straight line will be obtained from a Cartesian plot of ΔP versus t0.36 (for drawdown) or ΔP versus [(tpt)0.36 − Δt0.36] (for buildup). The slope, mell, of such line will be useful to find the half‐fracture length:

xf=45.124xe(qBμkhmell)25/18kξμϕctAE7.53

7.7. TDS technique for hydraulically fractured vertical wells

This section deals with the analysis of test data from wells that have been fractured hydraulically. Initially, hydraulic fracturing became a good way to increase the productivity of completed wells in low permeability reservoirs. However, lately, it has become a common practice thanks to its impact to increase well productivity and remove damage. The purpose of fracture well tests is to determine fracture and reservoir properties to provide an effective assessment of fracture treatment and to predict long‐term productivity for the reservoir. The fracture does not alter the permeability of the reservoir but it alters the average permeability of the system. Basically, fracturing increases the effective radius of the face of the well:

rwa=xf/2=rwesE7.54

After a well has been fractured, a new group of flow regimes is formed. The main flow regimes are presented in Figure 7.9 and are as follows [6, 26, 33].

  • Linear flow in the fracture

  • Bilinear flow (fracture and formation)

  • Linear flow in the formation (or elliptical)

  • Pseudoradial flow

For infinite conductivity and uniform flow fracture systems, only the third and fourth flow regimes can be seen in the pressure data. Linear flow usually occurs at a very early time, since it is normally masked by wellbore storage effects. The onset of pseudoradial flow can occur at a time that is economically unachievable and therefore cannot occur at any time during a well test. To determine kh of the reservoir, it is necessary that the reservoir is in radial flow, unless the interpretation is conducted by TDS technique that, in most cases, may be successfully interpreted with having the radial flow regime. A typical case of this is when the fracture is of finite conductivity and the slopes of a half and a quarter are observed; it is possible to obtain the permeability of the point of intersection between these lines. So whenever an analysis of a fractured well test is required, it is important that a prefracture test is involved to determine the kh of the reservoir, if conventional methods or type curves are used. If this does not occur, a unique analysis of the data may not be possible, since there are two unknowns: reservoir permeability and fracture length [35, 36].

Figure 7.9

Flow regimes governing pressure behavior in a with a finite‐conductivity fracture [6, 36].

Wellbore storage may mask the first of the three flow regimes. If this occurs, analysis to determine fracture length is not possible. In this case, the success of the fracture treatment will have to be determined using the calculated skin factor. As a general rule, a fracture is successful if the skin factor is reduced to less than −3. If the effects of storage are short‐lived, then bilinear flow or linear flow can be analyzed to determine fracture length and conductivity. For analysis of fractured wells, a new set of dimensionless parameters is used. These are the dimensionless time for a fractured well, tDxf, (Eq. 7.45) and the dimensionless fracture conductivity, CfD (Eq. 7.15).

tDxf=0.0002637ktϕμctxf2E7.55

7.7.1 Hydraulic fractured wells in bounded systems

For the case of a uniform‐flux fracture, the pressure derivative plots for various xe/xf ratios reveal three dominant flow periods. During early times, the flow of fluids is linear and can be identified by a straight line of a slope of 0.5. The linear flow line is used to calculate the average half‐fracture length. The infinite‐action radial flow regime, which can be identified by a horizontal straight line, is dominated by xe/xf > 8. This flow regime is used to calculate permeability and skin factor. The third straight line, which corresponds to the pseudosteady‐state period, has a unit slope. This line is used to calculate the drainage area and the shape factor. For the case of infinite‐conductivity fracture, pressure derivative plots show a fourth dominant flow regime, referred to here as biradial flow. This flow regime, which can be identified by a straight line of slop. 0.36, can also be used to calculate the half‐fracture length and permeability [35].

7.7.2 Characteristics of uniform‐flux fracture

Figure 7.10 shows a log‐log plot of pressure and the pressure derivative group versus dimensionless time for three values of xe /xf. These curves have several unique characteristics, which can be used to interpret pressure transient tests in fractured wells without using type‐curve matching [35].

(1) For short production times, the flow in the fracture is linear. The duration of this flow regime is a function of the penetration ratio xe /xf. The equation corresponding to this straight line at early times is:

tDA*PD=1.772xexftDAξE7.56

Taking logarithm at both sides, it yields:

log(tDA*PD)=0.5log(tDAξ)+log(πxexf)E7.57

The slope of this straight line is 0.5, which in itself is a unique feature of the linear flow regime. Substituting the dimensionless quantities in Eq. (7.4) and solving for the well pressure derivative, the following is obtained:

t*ΔP=0.5mLtE7.58

where

mL=4.064qBhμξϕctkxf2E7.59

Taking logarithm at both sides of the above expression:

log(t*ΔP)=0.5log(t)+log(0.5mL)E7.60

This expression shows that a graph of tP′ versus time in a log‐log graph will produce a straight line of slop. 0.5 if the linear flow regime is dominant. Let (tP′)L1 be the value of (tP′)L1 at a time t = 1 hr in the straight line of the linear flow regime (extrapolated, if necessary). Then, combining Eqs. (7.59) and (7.60) and solving for the half‐fracture length, xf, gives [23] and [35]:

xf=2.032qBh(t*ΔP)L1μξϕctkE7.61

Figure 7.10

Pressure derivative behavior for a uniform‐flux fracture inside a square reservoir, after [35].

The equation of the linear flow line portion of the pressure curve is:

ΔP=mLtE7.62

Let (ΔP)L1 be the value of ΔP in the straight line (extrapolated if necessary) at time t = 1 hr. Thus, after substituting for mL of Eq. (7.62), it results:

xf=4.064qBh(ΔP)L1μξϕctkE7.63

(2) After the linear flow regime, radial flow is developed. It is used as seen in Chapter 2. Then, Eqs. (2.71) and (2.92) apply for the estimation of permeability and skin factor.

(3) For long production times, the pressure derivative function will produce a unit‐slope straight line. This line corresponds to the pseudosteady‐state period, starting at a tDA value of approximately 0.2. The equation of this straight line is given by Eq. (2.96) and it is useful to estimate the drainage area. If the dimensionless quantities are substituted in Eq. (2.96), and solving for ΔP will yield,

t*ΔP=(qB4.27ϕct)tE7.64

This expression leads to find Eqs. (2.98) and (2.99).

(4) The dimensionless pressure during pseudosteady‐state period is a linear function of the dimensionless time. The equation corresponding to this period is [35]:

PD=2πtDA+ln(xe/xf)ln2.2458CAE7.65

Dividing Eq. (7.65) by Eq. (7.56),

PDtDA*PD=1+12πtDln(xexf2.2458CA)E7.66

From which is obtained after replacing the dimensionless variables:

CA=2.2458(xexf)2exp[0.000527ktpsϕμctA(1(ΔP)ps(t*ΔP)ps)]E7.67

or

CA=2.2458(xe/xf)2E7.68

If (ΔP)ps = (tP′)pss

(5) The point of intersection of the linear flow line and the infinite‐action radial flow line is unique. The coordinates of this point can be obtained by setting Eq. (7.56) to 0.5 and solving for the dimensionless intersection time:

tDALri=14π(xfxe)2E7.69

Substituting Eq. (1.100), setting A=4xe2, and solving for xf2/k, it yields:

xf2k=tLri1207ξϕμctE7.70

(6) The linear flow line and the pseudosteady‐state line intercept at:

tDALpssi=14πxexfE7.71

Substituting Eq. (1.100),

kxf2=7544ξϕμctA2tLpssiE7.72

This equation can be used for verification purpose or to calculate k given that xf is known.

(7) Combining Eqs. (7.69), (7.71), and the time of intercept of the pseudosteady‐state with the radial lines provides:

tLritrppsi=trppsitLpssi=tLritLpssi=(xe/xf)2E7.73

This expression can be used for verification purposes. It is also used when designing a pressure test in a well intercepted by a vertical fracture.

7.7.3 Characteristics of infinite‐conductivity fractures

Figure 7.11 is a graph of pressure dimensionless and pressure derivative versus dimensionless time based on area for a vertical fracture of infinite conductivity within a square system. This figure shows the existence of four straight lines: (a) half‐slope linear flow line, (b) 0.36‐slope biradial flow line, (c) infinity‐acting radial flow line (horizontal line), and (d) unit‐slope pseudosteady‐state flow line. For xe/xf > 8, the linear flow regime is almost nonexistent, and the biradial flow line is observed first. For xe/xf < 8, the biradial flow line disappears [34]. Only the characteristics of the biradial flow regime will be discussed here. The characteristics and interpretation of the other three flow regimes (linear, radial, and pseudosteady state) are the same as discussed above for uniform‐flow fracture.

(1) The equation of the biradial flow regime line introduced by [35] and modified by Bettam et al. [2] is:

tDA*PD=0.769(xexf)0.72(tDAξ)0.36E7.74

Taking logarithm to both members of the above equation leads to:

log(tDA*PD)=0.36log(tDAξ)+log[0.769(xexf)0.72]E7.75

In dimensional form, Eq. (7.74) becomes:

t*ΔP=0.769CBR(xe/xf)2t0.36E7.76

Figure 7.11

Pressure derivative behavior for an infinite‐conductivity fracture inside a square reservoir, after [35].

where

CBR=7.268qμBkh(kξϕμctA)9/25E7.77

Taking logarithm to both sides of Eq. (7.76) yields:

log(t*ΔP)=0.36logt+log(0.7699CBR(xexf)0.72)E7.78

Thus, the biradial flow line can be identified by its slope of 0.36. Let (tP′)BR1 be the value of pressure derivative at a time t = 1 hr in the straight line (extrapolated if necessary). An expression to find the half‐fracture length is found from Eq. (7.76) when linear flow regime is absent:

xf=0.694xe[CBR(t*ΔP)BR1]1.388E7.79

CBR is found from Eq. (7.77).

(2) The time of intersection between the linear flow and biradial flow regimes is given by Eqs. (7.56) and (7.74):

tDALBRi=0.00257(xfxe)2E7.80

Substituting the dimensionless time and solving for xf2/k yields:

xf2k=tLBRi39ξϕμctE7.81

If the radial flow is too short, permeability can be found from an expression obtained by combining the above expression and Eq. (7.61):

k=(12.67qμBh(t*ΔP)L1)1tLBRiE7.82

(3) The time of intersection between the biradial and radial flow regime lines can be used to verify k and xf:

tDArBRi=0.3023(xfxe)2E7.83

Substituting the dimensionless time in the above expression and solve for xf2/k:

xf2k=tBRri4587ξϕμctE7.84

(4) The time of intersection between the biradial flow regime line and the pseudosteady‐state line (Eqs. 2.96 and 7.74) provides:

tDABRpssi=0.03755(xexf)1.125E7.85

After substituting the dimensionless time based on area, Eq. (1.100) in Eq. (7.85) leads to:

k=142.3ξϕμctAtBRpssi(xexf)1.125E7.86

(5) Combination of Eqs. (7.81) with (7.84) and (7.83) with (7.85) will, respectively, yield:

tBRri=117.6tLBRiE7.87
tBRri=8(xfxe)2.125tBRpssiE7.88

which can be used for either verification or test design purposes.

7.7.4 Rectangular systems

For both types of fractures in rectangular systems, the transition between the infinite‐action radial flow and the pseudosteady‐state period is much longer than for a square system since, in the first one, formation linear flow exists as described in Section 2.7. The equation of this straight line proposed by Tiab [35] and modified by Escobar et al. [16] to include naturally fractured formations is:

tDA*PD=3.545tDAξE7.89

Substituting the dimensionless terms:

t*ΔP=mCBtE7.90

where

mCB=8.128qBh(μξϕctA)0.5E7.91

where (tP′)CB1 is the value of (tP′) at a time t = 1 hr on the formation linear flow straight line (extrapolated if necessary). Permeability can be solved from the above equation, so that:

k=66.0712(qBh(t*ΔP)CB1)2μξϕctAE7.92

Tiab [35] presents step‐by‐step procedures for the interpretation of pressure tests in fractured wells. These procedures are not included here.

Example 7.2

Tiab [35] presented an example of a pressure test in a highly productive fractured well. Pressure and pressure derivative [13] data versus time are reported in Figure 7.12 and Table 7.2. Other relevant data are given below:

q = 2000 STB/D, ϕ = 0.24, m = 0.3 cp, ct = 14.8 × 10−6 psia−1, B = 1.5 bbl/STB, h = 50 ft, rw = 0.4 ft, Pi = 5200 psia

Find permeability, skin factor, and half‐fracture length. Verify the value of the half‐fracture length.

Solution

Since it is a homogenous reservoir, then ξ = 1. The following characteristic points were read from Figure 7.12 and Table 7.3:

t, hrP, psiat*DP’, psiat, hrP, psiat*DP’, psia
0.0173785174.32212.6051.445445019.6868.901
0.0251195169.1615.2872.0892964992.60676.039
0.0363085163.03318.0273.0199524962.87983.251
0.0524815155.83621.0394.3651584930.87488.885
0.0758585147.4824.5416.3095734897.02593.405
0.1096485137.86828.0449.1201094861.74998.014
0.1318265132.55629.70513.182574825.404100.147
0.1905465120.82833.79215.848934806.92299.820
0.2754235107.46738.78722.908684769.489100.962
0.3981075092.23244.29233.113114731.58103.280
0.575445074.81550.62747.863014693.335109.456
0.8317645054.87958.30669.18314654.853109.456
1.2022645032.14265.7461004616.205109.456

Table 7.3.

Pressure and pressure derivative against time data of Example 7.3.

tr = 48 hr, ΔPr = 507 psia, (tP′)r =105.5 psia, tLRi = 1.2 hr, tLBRi = 0.047 hr, trBRi = 4.5 hr

Estimate permeability using Eq. (2.71) and skin factor with Eq. (2.92):

k=70.6qμBh(t*ΔP)r=70.6(2000)(0.3)(1.5)50(105.5)=12mdE7.158
s=0.5[507105.5ln(1248(0.24)(0.3)(14.8×106)(0.4)2)+7.43]=4.85E7.175

Find half‐fracture length with Eq. (7.63):

xf=2.032qBh(t*ΔP)L1μϕctk=2.032(2000)(1.5)50(97)0.30.24(14.8×106)(12)=105.4ftE7.159

Recalculate the half‐fracture length with Eqs. (7.70), (7.81), and (7.84), respectively.

xf=tLrik1207ξϕμct=(1.2)(12)1207(0.24)(0.3)(14.8×106)=105.81ftE7.160
xf=tBRLik39ξϕμct=(0.047)(12)39(0.24)(0.3)(14.8×106)=116.5ftE7.176
xf=tBRrik4587ξϕμct=(5.5)(12)4587(0.24)(0.3)(14.8×106)=116.2ftE7.161

Figure 7.12

Pressure and pressure derivative against time log‐log plot of Example 7.3.

7.7.5 Finite‐conductivity fractured vertical wells

A log‐log plot of pressure and pressure derivative versus test time for a fractured well in a closed system may reveal the presence of several straight lines corresponding to different flow regimes, excluding wellbore storage, such as (a) bilinear flow characterized by a slope of ¼ in the pressure and pressure derivative curve, (b) linear flow, (c) infinite‐action radial flow, and (d) pseudosteady‐state period. The slopes and points of intersection of these straight lines are unique and were used by Tiab et al. [36, 38] to find expressions for well test interpretation.

The characteristics of bilinear flow were first discussed by Cinco et al. [6]. It is called bilinear flow because it is the result of two linear flow regimes. A flow regime is the incompressible linear flow of the fracture and the other flow regime is the compressible linear flow in the formation, as shown in Figure 7.9. They showed mathematically that bilinear flow exists whenever (a) most of the fluid entering the well face comes from the formation and (b) the effects of the fracture do not affect well behavior.

During the bilinear flow regime, the behavior of the dimensionless pressure of the well given by Tiab et al. [36] and modified by Bettam et al. [2] is:

PD=2.45CfD1/2tDxfξ4E7.192

Replacing the dimensionless parameters given by Eqs. (1.89), (7.3), and (7.15), Eq. (7.92) becomes:

ΔP=mBLt4E7.93
mBL=44.13(ξϕμctk)0.25qμBhfkfwfE7.94

The fracture conductivity is solved from Eq. (7.93):

kfwf=1947.461ξϕμctk(qμBh(ΔP)BL1)2E7.95

The derivative of Eq. (7.92) is:

tD*PD=0.6125CfD1/2tDxfξ4E7.96

Replacing the dimensionless quantities, given by Eqs. (2.57), (7.3), and (7.15), in Eq. (7.96) and solving for the fracture conductivity:

kfwf=121.74ξϕμctk(qμBh(t*ΔP)BL1)2E7.97

Since this is linear flow, Eqs. (7.61) and (7.63) also apply for finite‐conductivity fractures.

The intercept between linear flow and bilinear flow lines given by the governing pressure derivative solutions (Eqs. 7.4 and 7.92) leads to:

tBLLi=13910ξϕμct(xf2kkfwf)E7.98

Solving for k,

k=(kfwfxf2)2tBLLi13910ξϕμctE7.99

The pressure derivative of Eq. (7.4) is:

tD*PD=12πtDxfξE7.100

The intercept between linear flow and bilinear flow lines given by the governing pressure derivative solutions (Eqs. 7.100 and 7.96) leads to:

k=(kfwfxf2)216tBLLi13910ξϕμctE7.101

Eqs. (7.99) and (7.101) can be used for verification purposes, if all three flow regimes are observed. If the test is too short to observe the radial flow line, or a prefracture test is not possible as in the low permeability formation, then Eqs. (7.99) and (7.101) can be used to calculate the permeability of the formation. Also, needless to say that Eqs. (2.71) and (2.93), along with many other relationships in Chapters 2 and 3, will apply to fractured wells.

The intersection of Eq. (2.70), neglect CD, and Eq. (7.96) leads after rearranging:

0.25mBLtBLri0.25=70.6qμBkhE7.102

Solving for the intersection time,

tBLri=1677ξϕμctk3(kfwf)2E7.103

which Tiab et al. [36] recommend to be used for verification purpose.

The intersection of the biradial flow regime pressure derivative (Eq. 7.74) with the bilinear flow regime pressure derivative (Eq. 7.96), lines will result in:

tBLBRi0.11=0.25mBL0.7699CBR(xexf)0.72=1.197k0.39(ξϕμctk)0.11xf0.72kfwf(xexf)0.72E7.104

being mBL and CBR defined by Eqs. (7.94) and (7.77), respectively. Either half‐fracture length or conductivity can be solved from Eq. (7.104).

The intersect of the pressure derivative bilinear governing expression (Eq. 7.96) with the pressure derivative pseudosteady‐state period Eq. (2.96) will lead to:

kfwf=2220.603A2k(ξϕμctktBLpssi)3/2E7.105

7.7.6 Special Cases

As also mentioned by Tiab et al. [36, 38], the above assumption assumes that all three flow regimes (bilinear, linear formation, and radial) are observed during the pressure test and that these are well defined in the pressure derivative curve. In many instances, at least one of the flow regimes is not observed or defined. For example, when the fracture has low conductivity, let us say, CfD < 5, probably linear flow regime is not seen. In the contrary case, when CfD > 50, probably bilinear flow is absent or maybe masked by wellbore storage effects. In such cases, the below correlations [37], which are excellent, can be used to find one parameter (Eq. 7.107) as a function of the other one or vice versa (Eq. (7.108)).

xf=1.921731/rwa3.31739k/kfwfE7.106

where rwa is the effective wellbore radius given by Eq. (7.54), so Eq. (7.106) can be rewritten as:

xf=1.92173es/rw3.31739k/kfwfE7.107
kfwf=3.31739kes/rw1.92173/xfE7.108

When radial flow is absent, this can be artificially from Eq. (2.71) by solving for the pressure derivative during the radial flow regime. Once estimated, a horizontal line can be drawn through this value, which corresponds to the place where radial flow really exists. Intersection of this line with others can be used without any problem. However, to find skin factor, the below correlation developed by Economides et al. [9] is recommended to be used:

s=ln[rw(1.92173xf)3.31739kfwf]E7.109

Figure 7.13

Effect of skin factor on fracture conductivity, after [9].

The internal result between parentheses may be considered in absolute value. Skin factor can also be estimated by a graphical procedure formulated by Cinco‐Ley and Samaniego [7], type‐curve matching, or the following correlation [9]:

s=lnrwxf+1.650.32u+0.11u21+0.18u+0.064u2+0.005u3E7.110

where

u=ln(kfwfkxf)=lnCfDE7.111

Finally, Tiab et al. [36, 38] also provided more relationships, which are not reported because of their relevance. Neither the step‐by‐step procedures are reported.

Alternatively, Eqs. (7.40), (7.41), and Figure 7.8 can be used.

Fracture conductivity can be found by a graphical correlation (Figure 7.13), given by Economides et al. [9], which polynomial fitting is given here:

x=s+ln(xfrw);0.67x2.8E7.112
CfD=10(0.592228061.77955x+0.86571983x211.5944514x+0.010112x2)E7.113

Example 7.4

Tiab et al. [36, 38] presented pressure data for a buildup test run in a fractured well. Pressure and pressure derivative data are reported in Table 7.4 and Figure 7.14. Other important information concerning this test is given below. Find the fracture and reservoir parameters for this well.

t, hrDP, psiat*DP', psiat, hrDP, psiat*DP', psia
0.2310226.315390117
0.391153020423112
0.613035.825446120
114540.830471141
1.818357.235493136.5
2.41956740510132
3.826083.345526135
4.126569.250540150
4.9628096.955556137.5
6.2308102.360565144
8.5320103.365580121.1
1034514971583

Table 7.4.

Pressure data for Example 7.4. Derivative digitized from [38].

q = 101 STB/D, ϕ = 0.08, μ = 0.45 cp, ct = 17.7 × 10−6 psia−1, B = 1.507 bbl/STB, h = 42 ft, rw = 0.28 ft, tp = 2000 hr, Pi = 2200 psia, ξ = 1

Solution

The following characteristic features were read from Figure 7.14:

tr = 30 hr, ΔPr = 471 psia, (tP’)r = 150 psia, (tP’)BL1 = 160 psia, ΔPBL1 = 40 psia, ΔPL1 = 120 psia, tLri = 8.2 hr, tBLri = 195 hr

Estimate permeability and skin factor from Eqs. (2.71) and (2.92):

k=70.6qμBh(t*ΔP)r=(70.6)(101)(0.45)(1.507)(42)(150)=0.76mdE7.162
s=12[471150ln((0.76)(30)(0.08)(0.45)(17.7×106)(0.282))+7.43]=4.68E7.177

Estimate fracture conductivity using Eqs. (7.95) and (7.97):

kfwf=1947.46(0.08)(0.45)(17.7×106)(0.76)((101)(0.45)(1.507)(42)(160))2=290.7md‐ftE7.163
kfwf=121.74(0.08)(0.45)(17.7×106)(0.76)((101)(0.45)(1.507)(42)(40))2=290.77md‐ftE7.178

Figure 7.14

Pressure and pressure derivative against time log‐log plot for Example 7.4.

Find the intercept between bilinear and biradial flow regimes with Eq. (7.103):

tBLri=1677(0.08)(0.45)(17.7×106)(0.76)3(310.8)2=235hrE7.164

This is in the range of 195 hr read from Figure 7.13. Use Eqs. (7.63) and (7.70) to find half‐fracture length:

xf=4.064((101)(1.507)(42)(120))0.45(0.08)(17.7×106)(0.76)=79ftE7.165
xf=ktLri1207ξϕμct=(0.76)(10)1207(0.08)(0.76)(17.7×106)=76.5ftE7.179

From Eq. (7.98), the time intercept of bilinear and linear flow regimes is found to be:

tBLLi=13910(0.08)(0.45)(17.7×106)(7920.76290.7)2=1.48hrE7.166

This is very close to the value of 1.4 hr found from the derivative plot. Use Eq. (7.107) to estimate the half‐fracture length:

xf=1.92173esrw3.31739kwfkf=1.92173e4.68440.283.31739(0.76)290.7=79ftE7.167

And the dimensionless fracture conductivity is found from Eq. (7.15), so that:

CfD=wfkfxfk=290.779(0.76)=4.8E7.168

Example 7.5

Tiab et al. [36, 38] presented a short buildup test run in a fractured well. Radial flow was not developed, but the reservoir permeability was measured from another to be 12.4 md. The pressure and pressure derivative [13] data are reported in Table 7.5 and Figure 7.15. Additional data:

t, hrDP, psiat*DP′, psiat, hrDP, psiat*DP′, psia
0.01727.4526.623.7893.5924.85
0.01942.3924.984.7899.5626.61
0.08248.55.195.78104.2628.49
0.2856.1810.507.78113.3630.37
0.3361.879.679.78121.0433.32
0.7863.7210.9111.78126.8735.17
1.0872.1113.1813.78131.8535.67
1.7876.3815.3617.78142.6638.51
2.7886.3422.5819.78146.0740.70

Table 7.5.

Pressure data for Example 7.5. Derivative digitized from [38].

q = 411.98 STB/D, f = 0.2, m = 0.53 cp, ct = 101 × 10−6 psia−1, B = 1.258 bbl/STB, h = 21 ft, rw = 0.689 ft, tp = 3000 hr, Pi = 479.61 psia, ξ = 1

Solution

Since permeability is known, the pressure derivative during infinite‐acting radial flow is found from Eq. (2.71):

(t*ΔP)r=70.6qμBhk=(70.6)(411.98)(0.53)(1.258)(21)(12.4)=74.5psiaE7.169

A horizontal line is drawn throughout (t*ΔP')r of 74.5 psia. This corresponds to an arterially created radial flow regime line. The following data were then read from Figure 7.15:

Figure 7.15

Pressure and pressure derivative against time log‐log plot for Example 7.5.

tLri = 75 hr, tBLLi = 14 hr, (tP′)BL1 = 18 psia, ΔPBL1 = 72 psia, ΔPL1 = 10 psia,

Use Eqs. (7.95) and (7.97) to determine fracture conductivity:

kfwf=1947.46(0.2)(0.53)(101×106)(12.4)((411.98)(0.53)(1.258)(21)(72))2=5578.35md‐ftE7.170
kfwf=121.74(0.2)(0.53)(101×106)(12.4)((411.98)(0.53)(1.258)(21)(18))2=5579.44md‐ftE7.181

Find the half‐fracture length with Eqs. (7.61) and (7.71):

xf=4.064(411.98(1.258)(21)(10))0.53(0.2)(101×106)(12.4)=260.5ftE7.171
xf=ktLri1207ξϕμct=(12.4)(75)1207(0.2)(0.53)(101×106)=268.3ftE7.180

Use Eq. (7.109) to find skin factor:

s=ln[|0.689(1.921732643.31739(12.4)5578.9)|]=9.6E7.172

Estimate the dimensionless fracture conductivity by means of Eq. (7.15):

CfD=kfwfkxf=5578.912.4(264)=1.7E7.173

7.8. New elliptical or biradial flow model

It was not possible to use Eq. (7.79) to find half‐fracture length in Example 7.2. This is because Eq. (7.74) depends on area, which should not be the case since this causes the test to be very long and therefore costly, which is not well accepted by most operators who in many circumstances do not allow fractured wells to develop radial flow during a well test. This implies the impossibility of determining the mean fracture length by means of Eq. (7.79) when at early times only biradial or elliptical flow is observed. In cases where the radial flow is observed or the permeability is known, it is possible to determine the mean fracture length using Eq. (7.84).

To overcome the above issue, Escobar et al. [23] presented a new model (see Figure 7.17) for biradial/elliptical flow, which excludes the drainage area and is presented below for homogeneous reservoirs (ξ = 1) or heterogeneous reservoirs (ξ= ω):

PD=259(πtDxf26ξ)0.36E7.114

which pressure derivative is given by:

tD*PD=(πtDxf26ξ)0.36E7.115

7.8.1 TDS technique for the new biradial flow model

Once the dimensionless parameters given by Eqs. (1.89), (7.55) and (2.57) in the above expressions solve for the half‐fracture length, it yields:

xf=22.5632(qBh(ΔP)BR)1.3889tBRξϕct(μk)1.778E7.116
xf=5.4595(qBh(t*ΔP)BR)1.3889tBRξϕct(μk)1.778E7.117

Figure 7.16

Pressure and pressure derivative behavior against dimensionless time for a vertical well with infinite‐conductivity fracture in a heterogeneous reservoir with λ = 1 × 10−8 and ω = 0.1, after [23].

Normally, the well test data are affected by noise, so it is recommended to draw the best line on the dealt flow (in this case, biradial) and read the value on that straight line at a time t = 1 hr (extrapolated if required), which leads to Eqs. (7.116) and (7.117) being rewritten as:

xf=22.5632(qBh(ΔP)BR1)1.38891ξϕct(μk)1.778E7.118
xf=5.4595(qBh(t*ΔP)BR1)1.38891ξϕct(μk)1.778E7.119

The intercept between the straight lines of the derivatives of bilinear and biradial flows tBLBRi (Eqs. 7.96 and 7.115) allows obtaining an expression to determine the half‐fracture length, xf,

kfwf=10.5422(ξϕμctk3.5454xf6.5454tBLBRi)0.22E7.120

The intercept between the straight lines of the derivatives of linear and biradial flows tLBRi (Eqs. 7.89 and 7.115) also allows obtaining an expression to determine the half‐fracture length, xf,

xf=ktBRLi39.044ξϕμctE7.121

Another way to obtain the half‐fracture length is the intercept of the straight lines of the radial flow derivatives (Eq. 2.70) and biradial flow regime (Eq. 7.115) tBRri,

xf=14584.16ktBRriξϕμctE7.122

The intersection formed by the line of the derivative of the biradial flow with the line of the derivative of pseudosteady state (Eq. 2.96), called tBRpssi, leads to:

xf=41.0554A1.3889(ξϕμctktBRpssi)0.8889E7.123

For circular/square constant pressure systems which pressure derivative is governed by Eq. (2.349), when intercepts with Eq. (7.115), called tBRSSi, also leads to:

xf=14247.92A25/18ω(ktBRSSiϕμct)179E7.124

Other recent publications dealing with elliptical/biradial flow regime in horizontal and vertical wells can be found in Refs. [15, 16, 17, 18, 28].

7.8.2 Conventional analysis for the new biradial flow model

After replacing Eqs. (1.89) and (7.55), kin Eq. (7.114) leads to:

ΔP=9.4286qμBkh(kξϕμctxf2)0.36t0.36E7.125

or

ΔP=mbirt0.36E7.126

Eq. (7.126) implies that a Cartesian graph of ΔP versus t0.36 (for drawdown tests) or ΔP versus ΔP versus [(tpt)0.36 − Δt0.36] (for buildup tests) provides a line which slope, mell, allows obtaining the half‐fracture length,

xf=9.4286qμBkhmbir(kξϕμct)0.36E7.127

Example 7.6

Determine the half‐fracture length for a pressure test which data are reported in Table 7.6 and plotted in Figure 7.17 for a hydraulically fractured well in a heterogeneous reservoir. Other relevant information for this test is given below:

t, hrΔP, psiatP, psiat, hrΔP, psiatP, psiat, hrΔP, psiatP, psia
0.0011.2460.6080.1018.1592.6248.02119.5661.680
0.0021.7220.7970.1278.7752.69610.09819.9321.484
0.0032.0630.9200.1609.4052.75314.26420.3971.234
0.0042.3361.0150.20110.0482.79820.14820.7771.074
0.0052.5691.0880.25410.6972.83228.46021.1081.012
0.0062.7731.1660.31911.3542.85240.20221.4421.093
0.0072.9561.2240.40212.0152.86355.10721.7971.283
0.0083.1221.2860.50612.6772.86370.10722.1211.474
0.0093.2771.3410.63713.3362.84985.10722.4231.678
0.0103.4331.3980.80213.9922.825100.10722.7071.838
0.0133.7681.5211.01014.6412.788140.10723.3902.200
0.0164.1321.6501.27115.2802.737185.10724.0512.472
0.0204.5281.7901.60015.9062.671240.10724.7342.691
0.0254.9561.9282.01516.5142.590330.10725.6372.903
0.0325.4182.0662.53717.1022.488420.10726.3552.999
0.0405.9112.2003.19317.6672.367545.10727.1473.009
0.0516.4342.3244.02018.1982.226685.10727.8523.194
0.0646.9852.4375.06118.6952.063985.10728.9733.194
0.0807.5622.5386.37219.1531.879

Table 7.6.

Pressure and pressure derivative versus time data for Example 7.6, after [16].

B = 1.25 bbl/STB, q = 350 STB/D, h = 100 ft, μ = 3 cp, rw = 0.4 ft, ct = 1 × 10−5 psia−1, Pi = 5000 psia, φ = 20%, k = 300 md, ω = 0.1, λ = 1 × 10−7, xf = 100 ft

Figure 7.17

Pressure and pressure derivative against time log‐log plot for Example 7.6, after [18].

Solution

The pressure derivative value during biradial flow regime at a time of 1 hr, (tP′)BR1 = 7 psia. Use Eq. (7.119) to find the half‐fracture length:

xf=5.4595(350(1.25)100(7))1.38891(0.1)(0.2)(1×105)(3300)1.778=105.96ftE7.174

The estimation of the naturally fractured parameters can be found in [18].

7.9. Horizontal wells

This last topic was left last since horizontal wells and fractured wells behave similarly. Actually, Escobar et al. [19] presented an approach for estimating the average reservoir pressure for horizontal wells under multirate testing using the mathematical solution of a vertical fractured well. The use of the TDS technique for horizontal wells is so extensive and deserves more than a chapter, for that reason it is only mentioned here. The pioneer papers on TDS technique for horizontal wells where presented by Engler and Tiab for naturally fractured deposit [10] and for anisotropic homogeneous formations [11]. The reader may not understand later publications without going them first. TDS technique for horizontal wells is so rich. Just to name so few cases, let us refer to treatment of zonal isolations by Al Rbeawi and Tiab [1] that even has conventional analysis by Escobar et al. [21]. Lu et al. [27] dealt with double permeability systems, and Escobar et al. [22] presented TDS technique for heterogeneous and homogeneous formations when the threshold gradient plays an important role for the flow to start flowing. Some applications on shale formations are summarized by Escobar [24].

Nomenclature

Aarea, ft2
Boil volume factor, bbl/STB
bfraction of penetration/completion
ccompressibility, 1/psia
Cwellbore storage coefficient, bbl/psia
CAreservoir shape factor
CfDdimensionless fracture conductivity
cttotal or system compressibility, 1/psia
Fcorcorrection factor
hformation thickness, ft
kpermeability, md
khreservoir flow capacity, md‐ft
hffracture height, ft
kcuncorrected reservoir permeability, md
kffracture permeability, md
kfwffracture conductivity, md‐ft
mslope of P versus log t plot, psia/hr/cycle
mbirslope of P versus t0.36 plot during elliptical/biradial flow, psia0.36/hr
mlfslope of P versus t0.5 plot during linear flow, psia0.5/hr
mvfslope of P versus t0.5 plot during linear flow, psia0.5/hr
mbfslope of P versus t0.25 plot during bilinear flow, psia0.25/hr
Ppressure, psia
PD′dimensionless pressure derivative
PDdimensionless pressure
Piinitial reservoir pressure, psia
Pwfwell flowing pressure, psia
Pwswell shut‐in or static pressure, psia
P* false pressure, psia
ΔPspressure drop due to skin conditions, psia
qliquid flow rate, bbl/D
qfDdimensionless flow rate
rDdimensionless radius
rradius, ft
redrainage radius, ft
rwwell radius, ft
rwaapparent wellbore radius, ft
sskin factor
ttime, hr
tpproduction (Horner) time before shutting‐in a well, hr
tDdimensionless time based on well radius
tDAdimensionless time based on reservoir area
tDxfdimensionless time based on half‐fracture length
tpDAdimensionless Horner time based on area
Xdistance along the x direction
xehalf‐reservoir side, ft (square system)
xfhalf‐fracture length, ft
t*ΔP′pressure derivative, psia
wffracture width, ft

Greek

Δchange, drop
Δtshut‐in time, hr
ηdiffusivity constant, φµct/k
ξindicator of either heterogeneous, ξ = ω, or homogeneous, ξ = 1, reservoir
φporosity, fraction
ρfluid density, lbm/ft3
μviscosity, cp
ωdimensionless storativity coefficient

Suffices

1hread at a time of 1 hr
Ddimensionless
DAdimensionless with respect to area
Dxfdimensionless with respect to area
BLbilinear flow
BL1bilinear flow at 1 hr
BLLiintercept of bilinear and linear lines in pressure curve
BLLiintercept of bilinear and linear lines in pressure derivative curve
BLBRiintercept of bilinear and biradial lines in pressure curve
BLriintercept of bilinear and radial lines in pressure curve
blfbeginning of linear flow
brfbeginning of radial flow
bslbeginning of semilog line
BRLiintercept of biradial and linear lines
BRriintercept of biradial and radial lines
BRSSiintercept of biradial and steady‐state lines
BRpssiintercept of biradial and pseudosteady‐state lines
CBformation linear flow regime
ebfend of bilinear flow
elend of linear flow
ffracture
iintersection or initial conditions
Llinear flow
L1linear flow at 1 hr
Lpssiintercept of linear and pseudosteady state lines
Lriintercept of linear and radial lines
pproduction
psspseudosteady state
pss1pseudosteady state at 1 hr
rradial flow
Lriintercept of linear and radial lines
rSSiintersection between the radial line and the −1‐slope line
sskin
SSsteady
vfvertical fracture
wwell, water
waapparent wellbore
wfwell flowing
wswell shut‐in

References

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Written By

Freddy Humberto Escobar Macualo

Submitted: 20 August 2018 Reviewed: 22 August 2018 Published: 05 November 2018