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.
- TDS technique
- well‐drainage area
- flow regimes
- intersection points
- transient pressure analysis
- conventional analysis
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 (
The book contains the application and detailed examples of the
My book entitled “
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
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.
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
1.2.2 Pressure tests run in injector wells
1.2.3 Other 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 , 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 . 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 :
According to the volume element given in Figure 1.5,
The right‐hand side part of Eq. (1.1) corresponds to the mass accumulated in the volume element. Darcy’s law for radial flow:
The cross‐sectional area available for flow is provided by cylindrical geometry, 2π
Replacing Eq. (1.3) into (1.1) yields:
If the control volume remains constant with time, then, Eq. (1.4) can be rearranged as:
Rearranging further the above expression:
The left‐hand side of Eq (1.6) corresponds to the definition of the derivative; then, it can be rewritten as:
The definition of compressibility has been widely used;
By the same token, the pore volume compressibility is given by:
The integration of Eq. (1.8) will lead to obtain:
The right‐hand side part of Eq. (1.7) can be expanded as:
Considering that the total compressibility,
The gradient term can be expanded as:
After simplification and considering permeability and viscosity to be constant, we obtain:
The hydraulic diffusivity constant is well known as
In expanded form:
The final form of the diffusivity equation strongly depends upon the flow geometry. For cylindrical, [11, 14], spherical , and elliptical coordinates , the diffusivity equation is given, respectively,
1.4. Limitations of the diffusivity equation
Isotropic, horizontal, homogeneous porous medium, permeability, and constant porosity
A single fluid saturates the porous medium
Constant viscosity, incompressible, or slightly compressible fluid
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:
For practical purposes, Eq. (1.25) can be expressed as:
Perrine method assumes negligible pressure and saturation gradients. Martin  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 , (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 .
Starting from the equation of continuity and the equation of Darcy:
Combining the above three equations:
Applying the differentiation chain rule to the right‐hand side part of Eq. (1.32) leads to:
Expanding and rearranging,
Using the definition of compressibility for gas flow:
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)
1.6.1 The equation of diffusivity in terms of pressure
Assuming the term
1.6.2 The equation of diffusivity in terms of pressure squared
Assuming the term μ
This expression is similar to Eq. (1.37), but the dependent variable is
1.6.3 Gas diffusivity equation in terms of pseudopressure,
The diffusivity equation in terms of
Taking the derivative with respect to both time and radius and replacing the respective results in Eq. (1.37), we obtain:
Expanding the above equation and expressing it in oilfield units:
The solution to the above expression is similar to the solution of Eq. (1.17), except that it is now given in terms of
With this criterion, the diffusivity equation for gases is:
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, . The normalized pseudovariables are:
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.  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 . This takes a function
The diffusivity equation is:
Eq. (1.51) is subjected to the following initial and boundary conditions:
Multiplying the Eq. (1.51) by
The new derivatives are obtained from Eq. (1.50):
Replacing the above derivatives into Eq. (1.56) and rearranging:
Comparing the term enclosed in square brackets with Eq. (1.50) shows that
From Eq. (1.61) follows
The term enclosed in square brackets is a constant that is assumed equal to 1 for convenience. Since
Writing as an ordinary differential equation:
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
Darcy’s law is used to convert the internal boundary condition. Eq. (1.54) multiplied by
Applying this definition into the ordinary differential expression given by Eq. (1.65), it results:
Integration of the above expression leads to:
Eq. (1.75) cannot be analytically integrated (solved by power series). Simplifying the solution:
This can be further simplified to:
The integral given in Eq. (1.80) is well known as the exponential integral,
In dimensionless form,
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
The exponential function can be evaluated by the following formula, , for
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.
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
Define dimensionless radius, dimensionless time, and dimensionless pressure as:
For pressure drawdown tests, Δ
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.
Replacing the above derivatives into Eq. (1.20),
Replacing this definition into Eq. (1.88) and solving for the dimensionless time (oilfield units),
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,
Taking twice derivative to Eq. (1.87), excluding the conversion factor, will provide:
If the characteristic length is the area, instead of wellbore radius, Eq. (1.92) can be expressed as:
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:
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
Using Eq. (1.101) with the above given reservoir and well data:
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:
Eq. (1.82) allows finding
The radial distance from the well to the nearest boundary corresponds to one half of the square side, the
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 1.11–1.14). To do so, Eq. (1.98) is employed for the whole reservoir area:
Using Eq. (1.102) will result:
The well‐flowing pressure is estimated with Eq. (1.87); thus,
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
this leads to,
Replacing this new definition into Eq. (1.82) will result in:
At the well
The above indicates that the well pressure behavior obeys a semi‐logarithmic behavior of pressure versus time.
A well and infinite reservoir has the following characteristics:
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.
For the wellbore radius, find
Using the function given in Figure 1.9 or Eq. (1.103), a value of
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.
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?
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.
1.10. Pressure distribution and skin factor
Once the dimensionless parameters are plugged in Eq. (1.82), this yields:
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:
Invasion of drilling fluids
Partial well penetration
Blocking of perforations
Inadequate drilling density or limited drilling
Dispersion of clays
Presence of cake and cement
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:
From the above expression can be easily obtained:
Therefore, the skin factor pressure drop is given by:
Assuming steady state near the wellbore and the damage area has a finite radius,
Taking natural logarithm to 0.0002637 and adding its result to 0.80908 results in:
Multiplying and dividing by the natural logarithm of 10 and solving for the well‐flowing pressure:
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,
Eq. (1.99) is now subjected to the following initial and boundary conditions:
The pseudosteady‐state period takes place at late times (
At the well,
This can be approximated to:
Invoking Eq. (1.98) for a circular reservoir area,
It follows that;
The derivative with respect to time of the above equation in dimensional form allows obtaining the pore volume:
An important feature of this period is that the rate of change of pressure with respect to time is a constant, that is,
Similar to the pseudosteady‐state case, steady state takes place at late times. Now, its initial, external, and internal boundary conditions are given by:
The solution to the steady‐state diffusivity equation is :
As time tends to infinity, the summation tends to infinity, then:
In dimensional terms, the above expression is reduced to Darcy’s equation. The dimensionless pressure function for linear flow is given by:
Steady state can occur in reservoirs only when the reservoir is fully recharged by an aquifer or when injection and production are balanced. However, a reservoir with a very active aquifer will not always act under steady‐state conditions. First, there has to be a period of unsteady state, which will be followed by the steady state once the pressure drop has reached the reservoir boundaries. Extraction of fluids from a pressurized reservoir with compressible fluids causes a pressure disturbance which travels throughout the reservoir. Although such disturbance is expected to travel at the speed of sound, it is rapidly attenuated so that for a given duration of production time, there is a distance, the drainage radius, beyond which no substantial changes in pressure will be observed. As more fluid is withdrawn (or injected), the disturbance moves further into the reservoir with continuous pressure decline at all points that have experienced pressure decline. Once a reservoir boundary is found, the pressure on the boundary continues to decline but at a faster rate than when the boundary was not detected. On the other hand, if the pressure transient reaches an open boundary (water influx), the pressure remains constant at some point; the pressure closest to the well will decline more slowly than if a closed boundary were found. Flow changes or the addition of new wells cause additional pressure drops that affect both the pressure decline and the pressure distribution. Each well will establish its own drainage area that supplies fluid. When a flow boundary is found, the pressure gradient—not the pressure level—tends to stabilize after sufficiently long production time. For the closed boundary case, the pressure reaches the pseudosteady state with a constant pressure gradient and general pressure drop everywhere, which is linear over time. For constant‐pressure boundaries, steady state is obtained; both the pressure and its gradient remain constant over time.
1.12. The pressure derivative function
Pressure derivative has been one of the most valuable tools ever introduced to the pressure transient analysis field. In fact,  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,”
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,  in 1961, tried to approach the rate of pressure change with time for detection of reservoir boundaries. Later, in 1965,  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.  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
Taking the derivative Δ
From inspection of Eq. (1.81) results:
In oilfield units,
At the well,
The derivative of equation (1.145) is better known as the Cartesian derivative. The natural logarithmic derivative is obtained from:
Later on,  use the natural logarithmic derivative to develop a type‐curve matching technique.
Appendix C in  also provides the derivation of the second pressure derivative:
Conversion of Eq. (1.145) to natural logarithmic derivative requires multiplying both sides of it by
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:
The above expression corresponds of a straight line with negative unit slope. In dimensional form:
Taking logarithm to both sides of the above expression:
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.  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.  also found the best procedure for data analysis of pressure against time is to differentiate and then smooth the data.
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.
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,
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, . 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 , the superposition principle is defined by:
1.13.1 Space superposition
If reservoir and fluid properties are considered constant, then, Eq. (1.87) can be applied to the above expression, so that:
The dimensionless radii are defined by:
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, , a single well is visualized as if there were two wells at the same point, one with a production rate of
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.
Using Eq. (1.101) for the well,
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:
Using Eq. (1.101), the four respective values of
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,
For the constant‐pressure boundary, lower part in Figure 1.21, the dimensionless pressure can be expressed as:
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:
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),
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.
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:
(b) What would be the well‐flowing pressure after a week of production if the well were in an infinite reservoir?
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:
By symmetry, the above expression becomes:
Using Eq. (1.101) for the well:
Estimation for the image wells are given below. In all cases, x > 0.0025, then, Table 1.2 is used to find
Then, the pressure drop in A will be:
If the well were located inside an infinite reservoir, the pressure drop would not include imaginary wells, then:
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.
|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|
|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|
|ks||permeability in the damage zone, md|
|krf||phase relative permeability, f = oil, water or gas|
|L||reservoir length, ft|
|m(P)||pseudopressure function, psia2/cp|
|M||gas molecular weight, lb/lbmol|
|dP/dr||pressure gradient, psia/ft|
|PD′||dimensionless pressure derivative|
|PD″||dimensionless second pressure derivative|
|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|
|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|
|T||reservoir temperature, ºR|
|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|
|z||vertical direction of the cylindrical coordinate, real gas constant|
|Δt||shut‐in time, h|
|ϕ||porosity, fraction. Spherical coordinate|
|λ||phase mobility, md/cp|
|η||hydraulic diffusivity constant, md‐cp/psia|
|1 hr||reading at time of 1 h|
|DA||dimensionless with respect to area|
|o||oil, based condition|
Pressure Drawdown Testing
As can be seen in
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 (
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 .
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:
As commented by Earlougher , 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
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 . 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.
For injector wells or wells completely filled with fluids:
When opening a well, see Figure 2.3, the oil production will be given by the fluid that is stored in the well,
The flow rate is given by:
The rate of volume change depends upon the difference between the subsurface and surface rates:
Taking the derivative to Eq. (2.8),
Defining the dimensionless wellbore storage coefficient;
Rewriting Eq. (2.14);
The main advantage of using downhole shut‐in devices is the minimization of wellbore storage effects and after‐flow duration.
Rhagavan  presents the solution for the radial flow diffusivity equation considering wellbore storage and skin effects in both Laplace and real domains, respectively:
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:
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
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.
By integration between 0 and a given
Eq. (2.21) serves to determine the storage coefficient from data from a pressure decline test using a log‐log plot of Δ
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 . 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,
After which time, the test is free of wellbore storage effects [2, 5, 6]. Along with
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.
Place the fdp on the master curve so that the axes are parallel.
Find the best match with one of the curves in Figure 2.4.
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).
Estimate permeability, porosity, and wellbore storage coefficient, respectively:E2.23E2.24E2.25
The results from the Ramey’s type curve must be verified with some other type curve. For instance, Earlougher and Kersch , formulated another type curve, Figure 2.5, which result should agree with those using Ramey method. The procedure for this method  is outlined as follows:
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:
Another important type curve that is supposed to provide a better match was presented by Bourdet et al. , Figure 2.6. This includes both pressure and pressure derivative curves. The variables to be matched are Δ
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.
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
18.104.22.168 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);
Solving for the well‐flowing pressure;
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;
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,
Since the slope possesses a negative signed, so does the
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:
Eq. (1.110) is useful to find either skin factor,
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  defined the apparent or effective wellbore radius,
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?
Application of Eq. (2.42) for the damaged‐well case gives:
Application of Eq. (2.42) for the damaged‐well case gives:
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 (
The starting time of the semilog straight line defined by Ramey  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 :
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
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:
For any producing time,
The point reached by the disturbance does not imply fluid movement occurs there. The drainage radius is about 90% that value, then
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,
Increasing the permeability in the zone near the well—hydraulic fracturing;
Reduce viscosity—steam injection, dissolvent, or in situ combustion;
Increase well penetration;
Reduce volumetric factor—choosing correct surface separators.
Other parameters to quantify well damage are :
Damage ratios less than the unity indicate stimulation.
Negative values of damage factors indicate stimulation. The damage factor can also be estimated from :
Eq. (2.54) applies to circular‐shaped reservoir.
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.
120 acres = 5,227,200 ft2. If the area is circular, then:
Find the yearly loss income using Eq. (2.56)
This indicates that the well requires immediate stimulation.
22.214.171.124 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 (
Once the value of
2.2.4 Tiab’s direct synthesis (
The starting point is the definition of the dimensionless pressure derivative from Eq. (1.89);
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  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:
Combination of Eqs. (1.94) and (2.15) results in:
Replacing Eq. (1.89) in the above expression yields;
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
Where the derivative of the dimensionless pressure is:
Multiplying and dividing by 0.8935;
Recalling Eq. (2.15), the above becomes:
Since the unit slope is one, then
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:
Combining the above equation with Eq. (2.73) results the best expression to estimate reservoir permeability:
It is recommended to draw a horizontal line throughout the radial flow regime and choose one convenient value of (
Tiab  also obtained the start time of the infinite line of action of the pressure derivative is:
The point of intersection,
For the unit‐slope line, the pressure curve is the same as for the pressure derivative curve. Then, at the intersection point:
Tiab  correlated for
Either formation permeability or wellbore storage coefficient can be determined using the coordinates of the peak,
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 (
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.
This last expression is useful to find wellbore storage coefficient when the early unitary slope line is absent.
Then, Tiab  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:
The above expression uses any convenient point,
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
Find permeability, skin factor, drainage area, and flow efficiency by conventional analysis. Find permeability, skin factor, and three values of drainage area using
Find the pressure loss due to skin factor with Eq. (2.40);
Since the average reservoir pressure is not reported, then, the initial pressure value is taken instead. Eq. (2.51) allows estimating the flow efficiency.
From the Cartesian plot, Figure 2.10, is read the following data:
Use Eq. (2.59) to find well drainage area:
Find the Dietz shape factor with Eq. (2.60);
As observed in Table 2.1, there exist three possible well drainage area geometry values (hexagon, circle, and square) close to the above value. To discriminate which one should be the appropriate system geometry find the dimensionless time in which pseudosteady‐state period starts by using Eq. (2.61):
Even, more parameters can be reestimated with
Taken from  with the data from the previous example, Example 2.3, determine
Solving for annulus capacity from Eq. (2.5);
The theoretical capacity is found with Eq. (2.45), so:
This leads to the conclusion that the annular liquid is falling.
2.3. Multiphase flow
According to Perrine , the single fluid flow may be applied to the multiple fluid flow systems when the gas does not dominate the pressure tests, it means liquid production is much more relevant than gas flow. Under this condition, the diffusivity equation, Eq. (1.27), will result and the total fluid mobility is determined by Eq. (1.24). We also mentioned in Chapter 1 that Martin  provided some tips for a better use of Perrine method. Actually, Perrine method works very well in liquid systems.
The semilog equations for drawdown and build tests are, respectively, given below:
The flow rate is estimated by:
Eq. (2.107) is recommended when oil flow dominates the test. It is removed from the denominator, otherwise. It advised to use consistent units in Eq. (2.107) meaning that the gas flow rate must be in Mscf/D and the gas volume factor bbl/SCF.
Once the semilog slope has been estimated, the total mobility, the phase effective permeabilities, and the mechanical skin factor are found from:
The best way of interpreting multiphasic flow tests in by using biphasic and/or triphasic pseudofunctions. Normally, well test software uses empirical relationships to estimate relative permeability data. The accuracy of the following expression is sensitive to the relative permeability data:
The expressions used along this textbook for reservoir characterization may apply for both single fluid and multiple fluid production tests. Single mobility has to be changed by total fluid mobility and individual flow rate ought to be replaced by the total fluid rate. Just to cite a few of them, Eqs. (2.66), (2.76), (2.85), (2.91), (2.92), and (2.97) become:
Also, the effective liquid permeabilities are found using the individual viscosity, rate, and volume factor. Then, Eq. (2.76) applied to oil and water will yield:
However, from a multiple fluid test, it is a challenge to find the reservoir absolute permeability. Several methods have been presented. For instance, Al‐Khalifah et al.  presented a sophisticated method applied to either drawdown or multiple rate tests. Their method even includes the estimating of the saturation change respect to pressure. However, we presented the method by Kamal and Pan  which applies well for liquid fluid. Relative permeabilities must be known for its application. Once effective permeabilities are found, let us say from Eqs. (2.119) and (2.120), estimate the permeability ratio
2.4. Partial penetration and partial completion
When a well penetrates a small part of the formation thickness, hemispherical flow takes place. See Figure 2.13 top. When the well is cased above the producer range and only a small part of the casing is perforated, spherical flow occurs in the region near the face of the well. See Figure 2.13 bottom. As the transient moves further into depth of the formation, the flow becomes radial, but if the test is short, the flow will be spherical. Both types of flow are characterized by a slope of −1/2 in the log‐log plot of pressure derivative versus time [90, 91]. Theoretically, before either hemispherical or spherical flow takes place, there exists a radial flow regime occurring by fluids withdrawn from the formation thickness that is close in height to the completion interval. This represents the transmissibility of the perforated interval. Actually, this flow regime is unpractical to be seen mainly because of wellbore storage effects. We will see further in this chapter that there are especial conditions for hemispherical/spherical flow to be observed which occur later that the completion‐interval‐limited radial flow regime. Both hemispherical and spherical flow vanished when the top and bottom boundaries have been fully reached by the transient wave; the true radial flow is developed throughout the full reservoir thickness.
The apparent skin factor,
According to Cinco et al. , the above equation is valid for 0° ≤
Taken from . A directional well which has an angle to the vertical of 24.1° has a skin factor
The deviation angle affected by the anisotropy is estimated with Eq. (2.126);
The pseudoskin factor caused by well deviation is found from Eq. (2.127):
From Eq. (2.122);
Therefore, 66.1 % of the skin factor is due to the well deviation.
2.4.1 Conventional analysis for spherical flow
The diffusivity equation for spherical flow assuming constant porosity, compressibility, and mobility is given by Abbott et al. :
The physical system is illustrated in Figure 2.14, right. This region is called a “spherical sink.”
The spherical flow equations for pressure drawdown and pressure buildup when the flow time is much longer than the shut‐in time were presented by :
The spherical pressure buildup equation when the flow time is shorter than the shut‐in time:
Then, from a Cartesian plot of
2.4.2 Conventional analysis for hemispherical flow
The model for hemispheric flow is very similar to that of spherical flow . The difference is that a boundary condition considers half sphere. Figure 2.14 (left) outlines the geometry of such system. The drawdown and pressure equations are given below :
As for the spherical case, from a Cartesian plot of
TDS for spherical flow
Moncada et al.  presented the expressions for interpreting both pressure drawdown or buildup tests in either gas or oil reservoirs using the
The total skin,
If the radial flow were seen, the horizontal permeability can be estimated from:
Moncada et al.  observed that the value of the derivative for the late radial flow in spherical geometry is equivalent to 0.0066 instead of 0.5 as of the radial system. In addition, the slope line −½ corresponding to the spherical flow and the late radial flow line of the curve of the dimensionless pressure derivative in spherical symmetry intersect in:
Replacing the dimensionless time results:
In the above equation, suffix
Replacing the dimensionless time will give:
TDS for hemispherical flow
Here the same considerations are presented in Section 2.4.3. Using a pressure and a pressure derivative value reading at any time during hemispherical flow allows finding hemispherical permeability and partial penetration skin ,
Moncada et al.  also found that the derivative in spherical geometry of the late radial flow corresponds to 0.0033 instead of 0.5 as of the radial system. This time the line of radial flow and hemispheric flow, in hemispherical symmetry, intersect in:
As for the spherical case, there exists an expression to define the intersection time of the −½ slope line of the hemispherical flow regime pressure derivative and the late radial flow line pressure derivative but, now, in radial symmetry:
This point of intersection in radial symmetry gives the following equation:
Skin factors are estimated in a manner similar to Section 2.4.3.
2.4.5 Wellbore storage and perforation length effects on hemispherical/spherical flow
It is important to identify the range of WBS values, which can influence the interpretation of the spherical and hemispheric flow regime. Figure 2.15 is a plot of
The length of the completed interval or the length of the partial penetration,
Abbott et al.  presented pressure‐time data for a pressure drawdown test. Well no. 20 is partially completed in a massive carbonate reservoir. The well was shut‐in for stabilization and then flowed to 5200 BOPD for 8.5 h. The pressure data are given in Table 2.4 and reservoir and fluid properties are given below:
Figure 2.18 contains a Cartesian graph of
The following data points were read from Figure 2.19.
Wellbore storage coefficient is found from Eq. (2.66)
Vertical permeability is determined from Eq. (2.130);
Table 2.5 presents the comparison of the results obtained by the conventional method and
2.5. Multirate testing
So far, the considerations revolve around a single flow test, meaning the production rate is kept constant for the application of the solution of the diffusivity equation. However, there are cases in which the flow rate changes; in such cases, the use of the solution to the diffusivity equation requires the application of the time superposition principle already studied in Section 1.14.2. Some reasons for the use of multirate testing are outlined as follows:
It is often impractical to keep a constant rate for a long time to perform a complete pressure drawdown test.
When the well was not shut‐in long enough to reach the static pressure before the pressure drawdown test started. It implies superposition effects.
When, it is not economically feasible shutting‐in a well to run a pressure buildup test.
Whether the production rates are constant or not during the flow periods, there are mainly four types of multirate tests:
Uncontrolled variable flow rate;
Series of constant flow rates;
Variable flow rate while keeping constant bottom‐hole pressure, Pwf. This test is common in gas wells producing very tight formations and more recently applied on testing of shale formations;
Pressure buildup (fall‐off) tests.
Actually, a holistic classification of transient well testing is given in
2.5.1 Conventional analysis
Next step is to replace
Coming back to Eq. (2.161), the assumed superposition function to be used is the radial one; then, this equation becomes:
Since it is uneasy to find natural log paper in the stationary shops, then, dividing for the natural log of 10 is recommended to express Eq. (2.162) in decadic log; then,
Solving for skin factor from Eq. (2.165);
which indicates that a Cartesian plot of Δ
And the equivalent time is then defined by,
For a two‐rate case, Russell  developed the governing well‐flowing pressure equation, as follows:
Therefore, the slope,