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

## 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 TDStechnique 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 TDStechnique 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 TDStechnique 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 TDStechnique, 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. TDStechnique 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 TDStechnique is the panacea for well test interpretation. TDStechnique is such an easy and practical methodology that his creator, Dr. Djebbar Tiab, when day said to me “I still don’t believe TDSworks!” 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.

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.

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.

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).

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

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, ξis 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 Tare 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/μzis constant; (b) μctis 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/μzremains 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 μzremains 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 μctremain 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 μgctin 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 fis 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:

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 rbfrom 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 sfunction 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/∂sgives:

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 dfand 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 fvariable 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.

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.

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 Eivalues 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 Eifunction 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 Eifunction 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 Eifunction code given in Figure 1.8.

2PrD2+1rDPrD=ϕμctrw2ktoPtDE1.92

Definition of torequires 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:

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 Eifunction. 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 xvalue 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 xvalue from Table 1.2. However, a trial‐and‐error procedure with the function given in Figure 1.8 was performed to find an xvalue 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 tDAvalue 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 Eiversus xplot 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

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 xwith 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.

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 xwith 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 PDvalue 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.

### 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= rwand 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].

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:

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

rsand ksare 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

### 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,” TDStechnique, [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, TDSis 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 Eifunction.

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 Eifunction 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 ΔutDand replacing uby 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:

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.

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]:

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

Let Xis 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 ψis 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]:

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 nnumber of wells:

ΔPN=i=1n141.2qμBkh[PD(rDi,tD)]E1.159

If point Nis 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 tand 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,

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:

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)= ΔPcaused by production from well 1 to well 1 + ΔPcaused 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 ΔPin 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 xare: x=0.02268, 0.2494, 2.268 × 10−6, and 8.316 × 10−6. Estimation of Eirequires 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 PDare 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 2dfrom 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

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

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.

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 Eiand 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

 A area, ft2 or Ac Bg gas volume factor, ft3/STB Bo oil volume factor, bbl/STB Bw oil volume factor, bbl/STB bx distance from closer lateral boundary to well along the x‐direction, ft by distance from closer lateral boundary to well along the y‐direction, ft c compressibility, 1/psia cf pore volume compressibility, 1/psia ct total or system compressibility, 1/psia d distance from a well to a fault, ft f a given function h formation thickness, ft k permeability, md ks permeability in the damage zone, md krf phase relative permeability, f = oil, water or gas L reservoir length, ft m slope m(P) pseudopressure function, psia2/cp M gas molecular weight, lb/lbmol P pressure dP/dr pressure gradient, psia/ft PD′ dimensionless pressure derivative PD″ dimensionless second pressure derivative PD dimensionless pressure Pi initial reservoir pressure, psia Pwf well flowing pressure, psia q flow rate, bbl/D. For gas reservoirs the units are Mscf/D Rs gas dissolved in crude oil, SCF/STB Rsw gas dissolved in crude water, SCF/STB rD dimensionless radius rDe dimensionless drainage radius = re/rw r radial distance, radius, ft re drainage radius, ft rs radius of the damage zone, ft rw well radius, ft Sf fluid saturation, f = oil, gas or water s skin factor T reservoir temperature, ºR t time, h ta pseudotime, psia h/cp to dummy time variable ur radial flow velocity, ft/h tD dimensionless time based on well radius tDA dimensionless time based on reservoir area tD*PD′ logarithmic pressure derivative V volume, ft3 z vertical direction of the cylindrical coordinate, real gas constant

## Greek

 Δ change, drop Δt shut‐in time, h ϕ porosity, fraction. Spherical coordinate λ phase mobility, md/cp η hydraulic diffusivity constant, md‐cp/psia ρ density, lbm/ft3 θ cylindrical coordinate μ viscosity, cp ζ time function

## Suffices

 1 hr reading at time of 1 h D dimensionless DA dimensionless with respect to area f formation g gas i initial conditions o oil, based condition w well, water p pore

## 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/qwhen the surface rate is changed from 0 to q, when C= 0, qsf/q= 1, while for C> 0, the relation qsf/qgradually 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. TDStechnique, 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 ΔVand ΔPdoes 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 Vuis the wellbore volume/unit length, bbl/ft, ris the density of the fluid in the wellbore, lbm/ft3, and Cis the wellbore storage coefficient, bbl/psia.

For injector wells or wells completely filled with fluids:

C=cwbVwbE2.3

where Cwbis the wellbore fluid compressibility = 1/Pwb, Vwbis the total wellbore volume, and Vucan 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, qsftends to qand 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

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: TDStechnique. 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 TDStechnique 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 PwDand from dimensionless time zero to tD, and taking logarithm to both terms, it yields:

logPD=logtDlogCDE2.20

Suffix wis 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 PDvs. tDand 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 ΔPversus time. Any point Nis taken from the unit‐slope straight line portion. The value of Cobtained 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 TDStechnique [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]:

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.

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

### 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.

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

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− Piterm. 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, rsand ksare 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 tSSLvalue 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$L=365q(OP)DFE2.56 where OPis oil price. Example 2.2 What will be the annual loss of a well that produces 500 BFD, which has a damage factor of 8, drains an area of 120 acres and has a radius of 6 inches? Assume circular reservoir area and a price of oil crude of USD$ 55/barrel.

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 (Cand 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 CAis 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)psswhich is found from:

(tDA)pss=0.1833m*mtpssE2.61

tpsscan 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

TDStechnique 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 TDSin 1995, several scenarios, reservoir geometries, fluid types, well configurations, and operation conditions. For instance, extension of TDStechnique 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 TDStechnique 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. TDStechnique 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]. TDStechnique 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 TDStechnique, and a more comprehensive state‐of‐the‐art on TDStechnique is given by [58]. This book revolves around this methodology; therefore, practically, the whole content of [71]—pioneer paper of TDStechnique—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 Cwill 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 rstands 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, pssstands for pseudosteady state, istands 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')rfalling 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

tDsris 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 istands 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, txand (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′)rfrom 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.

TDStechnique 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 sfrom the final replacement leads to:

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

being trany convenient time during the infinite‐acting radial flow regime throughout which a horizontal straight line should have been drawn. Read the ΔPrcorresponding 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, tpssand (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 TDStechnique (summary given in Table 2.2) and for conventional analysis are used for well‐drainage area determination in circular, square, and elongated systems.

TDStechnique has certain step‐by‐step procedures which not necessarily are to be followed since the interpreter is welcome to explore and use TDSas 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 ft q= 250 BPD μ= 1.2 cp ct= 26.4 × 10−5psi−1 h= 16 pies ϕ= 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 TDStechnique:

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