Open access peer-reviewed chapter

Novel, Integrated and Revolutionary Well Test Interpretation and Analysis

By Freddy Humberto Escobar Macualo

Submitted: August 20th 2018Reviewed: August 22nd 2018Published: November 5th 2018

DOI: 10.5772/intechopen.81078

Downloaded: 1414

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.

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+cfthen,

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+jlnti0.2and 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.

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

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

Suffices

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

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.

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.

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.

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].

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]: