USMS 024409 I n v e s t i g a t i o n o f the D i f f u s i v i t y Equat ion With General

Pressure-Dependent Rock and F l u i d P rope r t i es as Appl ied t o Wel l Tes t i ng P.S. F a i r , S h e l l Development Co.

Copyright /pfL Society of Petroleum Engineers This manuscript was provided to the Society of Petroleum Engineers for distribution and possible publication in an SPE journal. The material is subject to correction by the author(s). Permission to copy is restricted to an abstract of not more than 300 words. Write SPE Book Order Dept., Library Technician, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex 730989 SPEDAL.

Submitted to the SPE as an unsolicited paper, December 1991

Technical Publications INVESTIGATION OF THE DIFFUSIVITY EQUATION

WITH GENERAL PRESSURE-DEPENDENT ROCK AND FLUID PROPERTIES

AS APPLIED TO WELL TESTING

P. S. Fair

Shell Development Company

Houston, Texas

INTRODUCTION

The general treatment of well testing for reservoirs, where the rock and fluid properties may vary

with pressure, has focused on transforming the problem to an equivalent solution of the diffusivity equation

for reservoirs of small and constant compressibility. Although it is desirable to be able to use the classical

solution for liquid flow for all problems, this can only be accomplished if the diffusivity equation transforms

to a linear partial differential equation.

This paper starts with the fundamental equations governing fluid flow in porous media and presents a form of the diffusivity equation written in terms of the porosity-density product. This form of the diffusivity equation is analogous to the equation for mass diffusion with a concentration-dependent diffusion coefficient. An alternate approach of defining an integral transform as a general pseudopressure variable is also presented. The forms are equivalent, nonlinear partial differential equations. They are derived for general geometry, and rock and fluid properties may be functions of pressure. Thus, the equations are valid

for reservoirs containing oil, water, gas or any other fluid whose density and viscosity are uniquely known for every pressure.

For the case of radial flow to a constant rate line source well in an infinite reservoir, a semi-analytical solution can be obtained by applying the Boltzmam transformation. The numerical

evaluation of'the solution is iterative because of the nonlinearity of the equation. The solution can be extended to account for multiple flow rates. This extension is not equivalent to superposition unless the hydraulic diffusivity is a constant. For the cases where pseudopressure is required to account for the change

of rock and fluid properties with pressure, the hydraulic diffusivity is not, in general, a constant.

To extend the solution to the case of wellbore damage, a mass rate-based skin can be defined. The distinction that the skin be defined on a mass flux basis is important to assure that the skin is not a function of pressure as it would be in a volumetric-based system. The use of pressure formulated solutions,

such as the pressure-squared and pressure approaches for gas wells, may yield skin values which change as the density of the fluid changes across the skin. This could occur with either increased drawdown or

reservoir depletion.

Finally, the error introduced by neglecting the nonlinearity is quantified. The deviation from the constant hydraulic diffusivity case grows as the pressure change increases for any test. The deviation is

related to the amount that the hydraulic diffusivity has changed from its initial value and so the error is pressure-sensitive. Indirectly, the error could appear rate or time-sensitive, since these parameters tend to

influence the pressure. The extension of this approach to multirate testing shows that the value of normalized time and pseudotime is limited to buildup (or falloff) analyses, where the hydraulic diffusivity

converges to its original value. In general, rigorous solution of the nonlinear partial differential equation must be obtained if the effects of non-Darcy flow, multiple rates, and wellbore damage are to be determined for reservoirs with pressure-dependent rock and fluid properties.

DISCUSSION OF LITERATURE

The purpose of this paper is not to present the history of well testing; however, a short discussion of

the evolution of the theory of well testing is in order. For this paper, the view of well testing will not be

focused on pressure transient analysis techniques. From a broader viewpoint, the equations and techniques

developed for predicting reservoir and well performance are fundamentally the same as those for pressure

transient analysis. The simplifying assumptions and the variables in which the solutions are expressed distinguish the differences between the solutions of pressure transient equations and reservoirlwell performance equations.

The early mathematical treatments of fluid flow in porous medial-7 expressed the diffusivity equation, with both initial and boundary conditions, as a linear partial differential equation so that analytical solutions could be obtained. Since the primary concern was to obtain analytical solutions which could be evaluated without resorting to numerical techniques, problems were usually limited to steady state flow, pseudosteady state flow, incompressible flow, or flow in reservoirs of small and constant compressibility. Thus, it appears that the early works were limited not by a lack of understanding of the physics but by

inability to evaluate the more general nonlinear problems.

Other investigators have suggested and used integral transforms to handle the nonl inear i t ie~~-~~ that occur when studying problems involving flow of compressible fluids, multiphase flow, compaction, and

other problems where the fluid andlor the rock properties may be functions of pressure. The best-known of these studies12 introduces the real gas pseudopressure for analyzing pressure tests in natural gas reservoirs. Although the investigators point out that the use of the real gas pseudopressure does not fully linearize the diffusivity equation, they justify its use with comparisons to numerically simulated data. They also point out that some correction to the dimensionless time terms may be required for multirate solutions based on the superposition of small and constant compressibility solutions, although it was their opinion that such a correction was not necessary for engineering calculations. The introduction of p s e u d ~ t i m e l ~ * ~ ~ approximately a decade later was intended to correct the dimensionless time for buildup tests.

With the evolution of better computer technology, pressure transient analysis has developed from

pencil, paper, and straight-line analysis techniques to sophisticated analytic simulators with computer

graphics. The concepts of pseudopressure and pseudotime may be incorporated into such a simulator which utilizes a scheme similar to superposition. The definition of pseudotime is not the same as is commonly

found in literature. Within this analytic scheme, the definition of pseudotime depends on the rate history of the well. For a finite rate, the value of pseudotime is calculated with properties based on the average reservoir pressure. For shut-in wells, the properties at the wellbore sandface are used. To include the

effects of wellbore storage and skin, the procedure must be modified so that the properties at the well bore sandface without the effects of wellbore damage are used for shut-in wells (or zero-rate flow periods). Empirically, this approach has been found to work well for many cases of interest. Figure 1 is provided as an example of the merits of the pseudotime approach. The solid lines represent data for a well producing in

a low pressure gas reservoir generated with a numerical simulator70 and matched by the analytic simulator. The dashed lines represent data generated analytically without the use of pseudotime.

No further discussion of previous work is warranted for the purpose of this paper, except that the introduction of pseudotime as a variable did not linearize the diffusivity equation. The question remains: Why does the use of pseudotime and pseudopressure allow more accurate analyses of some pressure

transient tests? One of the purposes of this study is to answer this question. For this reason, the fundamental

equations for well testing are derived in their most general form.

GENERAL DEVELOPMENT OF THE DIFFUSIVITY EOUATION

The development of the fundamental equations for well testing starts with the continuity equation,

and Darcy's law is used to express the flux term. By writing the flux term as

a form of the diffusivity equation which is implicit in pressure can be obtained:

where 17 is given by:

All of the terms in 17 can be pressure-dependent, therefore Equation (3) represents a general form of the diffusivity equation with pressure-dependent reservoir and fluid properties. Equation (3) is analogous to the mass transport form of the diffusion equation:'l

ac V . (DVC) =- ,

at

where C is the concentration and D is the diffusion coefficient. Concentration, which represents the amount of one substance stored in another substance, is analogous to the porosity-density product in general, but it is not analogous to pressure. The porosity-density product is the storage capacity of the reservoir-fluid

system, and 7 is the general form of the hydraulic diffusivity coefficient.

Another popular form of the diffusivity equation involves the use of integral transforms to linearize

the diffusivity equation. By starting with the continuity Equation (1) and applying Darcy's law in its traditional form,

the diffusivity equation becomes

From Equation (7). the general definition of pseudopressure is apparent:

Substitution into Equation (7) followed by subsequent simplification yields

(Appendix A shows that other forms of the pseudopressure can be obtained from this general definition.)

No further integral transforms are obvious, although the equivalent of pseudotime would be

.t= qdt' . i ~ e e ~ 5 showed that pseudotime did not linearize the diffusivity equation. In a later section, the

role of pseudotime in pressure transient analysis will be examined.

If one compares the two forms of the diffusivity equation, one finds that both Equations (3) and (9) are nonlinear partial differential equations. Working with the porosity-density product as a function of pressure should be no better (or worse) than working with pseudopressure as a function of pressure. The

location of the pressure-dependent hydraulic diffusion coefficient in Equations (3) and (9) appears to be

the only difference in the two formulations. At steady-state conditions, the pseudopressure formulation is a linear equation, while the porosity-density product formulation is a nonlinear equation. For this reason, the

pseudopressure formulation will be used.

RADIAL 1-D FORM OF THE DIFFUSIVITY EOUATION

For well testing, a classical solution to the diffusivity equation is for single-phase, one-dimensional radial flow to a well. The one-dimensional radial form of Equation (9) is

The initial and boundary conditions for a single well in an infinite reservoir are

Initial Condition m(r, t = 0) = mo ,

Boundary Condition 1 m(r+ a p t ) =m, ,

Boundary Condition 2

The second boundary condition represents a cylindrical source well producing at a constant mass

rate.

For convenience, Equations (1 1) through (14) can be converted to dimensionless variables. The

definition of dimensionless radius remains

r rD=- .

rw (15)

The hydraulic diffusivity is made dimensionless by normalizing it to the value at the initial

condition:

The pseudopressure is made dimensionless by using wellbore properties and constant mass rate:

(The subscript s refers to standard or stock tank conditions.)

Finally, time can be made dimensionless by multiplying by the initial hydraulic diffusivity and

dividing by the square of the wellbore radius:

The radial form of the diffusivity equation and the associated boundary conditions become

Initial Condition

Boundary Condition 1

and

Boundary Condition 2

Equations (19) through (22) are the general equations to solve for the radial one-dimensional

flow of fluids at a constant mass rate.

A semi-analytical solution for the radial flow form of the diffusivity equation for pressure dependent rock and fluid properties may be obtained for the constant mass rate case by applying the Boltzmann transform to Equations (19) through (22). The Boltzmann transform for the radial flow problem is

In applying the Boltzmann transform, the requirement is that the boundary conditions and initial

conditions can be written in terms of x alone and not in terms of rD and tD separately. This requires that the

initial condition and the infinite reservoir boundary condition collapse to a single boundary condition for the transformed equations. Likewise, the line source approximation for the well is required so that the other boundary condition is not at a specific radius. The transformed equations become

and

Separating the variables in Equation (24) and integrating yields

where C is an integration constant. The integration constant is determined from Equation (26) to be -112. The equation can then be integrated again:

The limits of integration are provided by Equation (25). The general solution is

By making the assumption that qD = 1, the traditional solution is obtained:

Traditionally qD is assumed to be constant, so Equation (24) becomes a linear equation and the

solutions can be superposed. This assumption may not be valid for real gases. However, Equation (29) may

be solved numerically. Crank71 suggested integrating with any of the standard integration formulas. An

alternative solution leaves the equation in the form of Equation (27). The integral can be evaluated numerically and then a fourth-order Runge-Kutta technique may be used to solve the first-order

differential equation. Either method is iterative, since qD is a function of mD . The initial estimate for qD

comes from the exponential integral solution [Equation (30)l. The porosity-density product equivalent

equations and solution are presented in Appendix B.

The author has programmed both methods for an IBM PCAT. As one would expect, the fourth-order Runge-Kutta gave results with higher accuracy than the integral method. In addition, the equations solved in terms of pseudopressure converged approximately twice as fast as the equivalent

porosity-density product solution. In all cases, the run times were on the order of several minutes.

The line source solution for radial flow may be generalized for multirate flow. The derivation is provided in Appendix C. The results are presented here. If there are a series of constant rates given by

where S(t) is the unit step function, the solution is given by

where

and

(see Figure 2). When q D = 1, the form of Equations (32) through (34) degenerates to the addition of

exponential integral terms, which is commonly thought of as the principal of superposition applied to varying rates. However, in the general case, qD is a function of pseudopressure.

Fig. 2 - Flow sequence for a multirate test.

WELLBORE DAMAGE AS SKIN EFFECT

Since real reservoirs seldom perform according to the idealized theories, a skin can be added to

account for certain nonidealities. For convenience, a skin located on the wellbore sandface can be applied similar to the method of van ~ v e r d i n ~ e n ~ ~ and ~ u r s t . ~ ~ The skin represents an additional steady state

pseudopressure drop as given by

This definition of skin is based on a steady state mass flux. Appendix D presents the justification for a mass

flux based skin.

This definition of skin allows the general solution for the transient to be calculated by

Equation (29) for the constant rate case or by Equations (32-34) for the multirate case. The gradient term

in Equation (35) is the wellbore boundary condition. For the constant rate case, this is given by Equation (23) and can be expressed as

for the more general multirate case. In Equation (36), q(t) is defined by Equation (3 1). For the constant rate case, Equation (35) merely becomes

This simple definition of skin provides a convenient method of obtaining wellbore pressures from

generated reservoir data; however, the inverse problem of analyzing well data is not as trivial a problem for

the general case, where the nonlinear equation must be used to describe flow in the reservoir. The problem

arises when the data are collected in the presence of both wellbore storage and skin. The problem is that

pseudotime is defined in the reservoir and not in the wellbore. In the presence of both wellbore storage and

skin, the time transformation requires pressure-dependent properties of the reservoir and fluids at the

reservoir sandface (the other side of the skin). However, this pressure is not known. The pressure is

measured only in the wellbore. It appears that even if linearization of both the wellbore storage equation

and the reservoir flow equation were valid, the two equations may only be coupled directly for a simple

noniterative well test analysis if the skin is not present. History matching of mwD by tuning the permeability

and skin in a numerical model presents the most reliable method of analysis of pressure tests.

Going back to the relation defining skin for the pseudopressure form of the solution, the skin

effect is clearly defined for a steady state mass vis-a-vis volumetric flux across the skin. This distinction was

not discussed in van Everdingen's and Hurst's papers72p73 on wellbore damage. Clearly, for the case of a

small and constant compressibility fluid, the volumetric and mass flux are equivalent. However, the use of

pressure and pressure-squared solutions for gas wells points out an inconsistency in the definitions of skin.

Comparing the skin for the pressure formulation to that given by Equation (35) gives the relationship that

where Sp is the skin used in the pressure formulation. It is readily seen that Sp is a constant only if the

properties are not pressure dependent. Application of the pressure and pressure-squared techniques may

appear to have rate sensitive skins, since the pressure at the wellbore will vary with changing rates. This

occurs because mass is not conserved across the skin.

NONLINEAR EFFECTS ON WELL TEST ANALYSES

Several previous investigators have pointed out that the error introduced by solving the diffusion

equation for a drawdown test neglecting the nonlinearity of the hydraulic diffusivity term can be handled as

additional skin.41175 This conclusion does not appear to be rigorously true. If the error term (6) is defined

as by Kale and ~ a t t a r , ~ l the skin term would be this error and could be calculated from

By taking the derivative of 6 with respect to x, one should be able to determine if 6 will be a

constant in the limit of long time:

By assuming that x is a small value (which occurs at long time), we can approximate Equation (4 1)

as

Since Equation (42) does not equal zero unless q D = 1, then b cannot converge to a constant.

However, the conclusion that the error does appear constant for long times is correct. This can be

seen by converting Equation (42) from the Boltzmann variable. This yields

dd 12 --L[;- dt, - 2tD

- ;2]

As dimensionless time grows to large values, the value of the derivative of 6 with respect to

dimensionless time approaches zero. Although 6 may not actually be constant, the change with time may be so small as to appear invariant.

Looking at Equation (42), it is obvious that the deviation from the classical q D = 1 solution is

affected by pressure, since the hydraulic diffusivity is pressure dependent. Clearly, both rate and time

directly affect the magnitude of qD. A well produced at a given rate will show a deviation from the classical

solution, which is different from the same well produced at twice the rate for the same length of time. This deviation would be lumped into a rate dependent skin term and added to the non-Darcy flow coefficient under classical pressure transient analyses. In addition, the magnitude and sign of 6 will depend on whether

the well is an injector or producer. For a constant-rate producer, the pressure will only decrease with time, so qD can only vary between zero and one. However, qDcan vary between one and infinity for an injector.

The practical limit is detennined by either artificial lift limitations or fracture gradient. It is this difference in

the behavior of q ,,with pressure that makes simple superposition invalid and causes the need for some type

of time transformation (i.e., pseudotime) to be used in pressure buildup analyses.

The role of pseudotime and normalized time have been suggested for pressure buildup analysis.

~ a n o n s e n ~ 9 ~ showed that the linearization of the diffusivity equation is sufficiently valid for buildups in spite - '

of the fact that it is incorrect for single-rate cases, using a perturbation technique. Perturbation techniques

require that the higher-order terms in the series expansion are insignificant enough to be neglected. By starting with Equation (33), it can be shown without series approximations that buildup (and falloff) tests are the only place where such time transformations are valid.

The purpose of the time transformations is to allow us to approach the nonlinear problem using

classical, slightly compressible fluid solutions. If a pseudo-Boltzmann transform y could be defined, then

Equation (33) would become

which is clearly the exponential integral solution. If Equation (44) is compared to

(which comes after applying the long-time approximation for the inside integral), it would appear that

defining

and

would transform Equation (45) to Equation (44). Note that Equation (46) represents the inverse of

normalized time59.63 as currently used in pressure transient analysis. The inconsistency of this approach is

that Equation (47) is incorrect and should actually be

From this relation, Equation (47) can only be made valid when the last term in Equation (48) is

zero. The relation

holds when q D does not vary with time. This implies that the pressure is constant. As pointed out

previously, the value of q D may vary from zero to infinity, depending on whether the pressure is decreasing - .

or increasing as a result of production or injection. However, the value of q D will return to one after some

time if the production or injection ceases. It is for this reason that the pseudo-Boltzmann transform works for both buildup and falloff tests. As expected, normalized time would tend to correct the late time behavior

for these tests.

If a normalized time

works in the limit of long time for a buildup, then it can be shown that pseudotime defined as Equation (10)

will actually approach the normalized time for the same limit. It should be noted again that b.oth normalized

time and pseudotime are only valid in the reservoir and can be determined directly from measured data only

after wellbore storage and skin have died out.

~ e e ~ ~ has shown that if the definition of pseudotime is based on the pressure in the wellbore, the

material balance on the wellbore is exact for the calculation of wellbore storage. As mentioned previously,

this value of pseudotime is not valid for the reservoir equations in the presence of wellbore storage and skin.

However, the value of pseudotime based on the pressure in the wellbore does approach the desired value for

the reservoir equations at late time in both buildup and falloff tests when the effects of wellbore storage and

skin have diminished. Thus, the use of pseudotime for both falloff and buildup tests gives the appearance of

fully linearizing the pressure transient equations, if the comparison of field data or simulated data plotted

with pseudotime to Ramey typecurves for reservoirs with fluids of small and constant compressibility is the

test. Clearly, pseudotime allows interpolation between the two limiting solutions - wellbore storage and late

time radial flow. Its basis for existence is empirical and not analytical. Since normalized time does not

conserve mass in the wellbore, it should not be expected to match the wellbore storage-dominated data at

all. Thus, normalized time has not received the attention in literature that pseudotime has received.

CONCLUSIONS

The conclusions of this study are general for reservoirs regardless of fluid type (oil, gas, or water).

Only the severity of the effects depends on the actual fluid and rock properties. The conclusions are as

follows.

1. The natural variable for the general diffusivity equation for flow of fluid in porous media with

pressure-dependent rock and fluid properties is the porosity-density product instead of pressure.

Use of the natural variable yields a form of the diffusion equation which is analogous to that for

mass diffusion, with a concentration-dependent diffusion coefficient.

2. The introduction of integral transforms allows a general pseudopressure variable to be defined,

which is a function of only rock and fluid properties. Specific forms of the pseudopressure, such

as the real gas pseudopressure, can be derived from this general definition. Valid forms of

pseudopressure are porosity- and geometry-independent.

3. In general, the diffusivity equation formulated with either the porosity-density product or

pseudopressure as the independent variable remains a nonlinear partial differential equation. The need to use pseudopressure to linearize the equation requires that the diffusivity equation be

treated as a nonlinear partial differential equation, since the hydraulic diffusivity contains the same rock and fluid properties as used to define pseudopressure. Solution of either form of the

diffusivity equation can yield the other, since they are mathematically equivalent.

4. Using the Boltzmann transform, a semi-analytical solution for the problem of one-dimensional

radial flow to a constant rate line source well in an infinite reservoir is derived. Numerical

evaluation of the solution is iterative due to the nonlinear character of the problem.

5. The extension t :o multiple consta ~ n t flow rates is provided. This solution is not equivalent to

superposition unless the hydraulic diffusivity is constant.

6. Wellbore damage or improvement may be handled by defining a skin which adds an additional

pseudopressure drop proportional to the mass flow rate across an infinitesimally small distance.

Skins used in the pressure or pressure-squared formulations may appear to change with rate and

time as the density of fluid at the skin varies.

7. For a constant mass flow rate case, the error obtained by neglecting the nonlinearity and assuming

the hydraulic diffusivity to be constant grows as the pressure deviates from its initial value. In

general, this error will not be the same magnitude for injection tests and drawdown tests of the

same rate. For this reason, superposition may yield unpredictable results depending on how

pressure dependent the rock and fluid properties are.

8. The use of normalized time and pseudotime should be restricted to falloffs and buildups where

they are approximately correct at long times. In the presence of both wellbore storage and skin,

the actual calculation of either normalized time or pseudotime from well test data is complicated

by the fact that they are only defined in the reservoir on the opposite side of the skin from the

wellbore.

Finally, it should be noted that the use of corrections, such as the Matthews-Brons-Hazebroek

technique,77 to correct the extrapolated pressure obtained from Homer plots of gas well pressure buildup

tests to average reservoir pressure should be used with caution. This method is based on the method of

imaging. This method applies the principle of superposition in space to obtain the corrections to the

extrapolated pressure. Since the principle of superposition is not valid, one might expect that the correction

is not completely accurate when pseudopressure is used in the analyses. In spite of this drawback, the MBH method is still the best method to use in the absence of an accurate simulation model.

ACKNOWLEDGMENT

The author would like to thank Shell Development Co. for the release of technical information.

In addition, it would be appropriate to acknowledge W. B. Fair Jr. and D. H. Welty for their frequent

discussions which helped guide this work. Finally, the author wishes to thank C. D. White and

J. G . "Sprat" Crump I11 for their editorial comments and technical review.

NOMENCLATURE

concentration permeability reduction factor

porosity reduction factor rock compressibility

total compressibility

water compressibility

diffusion coefficient exponential exponential integral permeability molecular weight pseudopressure dimensionless pseudopressure

dimensionless wellbore pseudopressure

pressure flowing wellbore pressure

volumetric flow rate universal gas constant radius dimensionless radius

wellbore radius

mass based skin volumetric based skin

unit step function

water saturation

irreducible water saturation

temperature time dimensionless time

normalized time

flow given by Darcy flow equation Boltzman variable pseudo-Boltzman variable real gas compressibility factor

turbulence factor hydraulic diffusivity dimensionless hydraulic diffusivity

potential porosity density pseudotime viscosity

REFERENCES

1. Muskat, M. (1937), The Flow of Homogeneous Fluids through Porous Media, McGraw-Hill Book Company, New York.

2. - (1937), Use of Data on the Build-Up of Bottom-Hole Pressures, Trans. Am. Inst. Min. Metall. Eng., v. 123, 44-48.

3. - (1949), Physical Principles of Oil Production. McGraw-Hill Book Company. -New York.

4. Van Everdingen, A. F. and Hurst, W. (1949). The Application of the Laplace Transformation to Flow Problems in Reservoirs, Trans. Am. Inst. Min. Metall. Eng., v. 186, 305-324.

5. Miller, C. C., Dyes, A. B.. and Hutchinson, C. A., Jr. (1950), Estimation of Permeability and Reservoir Pressure from Bottom-Hole Pressure Build-Up Characteristics, Trans. Am. Inst. Min. Metall. Eng., v. 189, 91-104.

6. Homer, D. R. (1951). Pressure Build-Up in Wells, paper Section 11, Preprint 7, presented at the Third World Pet. Cong., The Hague, Netherlands.

7. Matthews, C. S. and Russell, D. G. (1967), Pressure Buildup and Flow Tests in Wells, Mon. Vol. 1. Soc. Pet. Eng. AIME.

8. Aanonsen, S. I. (l985), Nonlinear Effects During Transient Fluid Flow in Reservoirs as Encountered in Well Test Analysis, Report No. 79, Dept. Appl. Math., Univ. Bergen, Bergen, Norway, March.

9. - (1985). Application of Pseudotirne to Estimate Average Reservoir Pressure, SPE 14256, presented at the 60th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 22-25.

10. Agarwal, R. G. (1979) Real Gas Pseudo-Time - A New Function for Pressure Buildup Analysis of MHF Gas Wells, SPE 8279, presented at the 54th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 23-26.

11. Al-Hussainy, R. and Ramey, H. J., Jr. (1966). Application of Real Gas Flow Theory to Well Testing and Deliverability Forecasting, J. Pet. Technol., May, pp. 637-642.

12. , Ramey, H. J., Jr., and Crawford, P. B. (1966). The Flow of Real Gases Through Porous Media, J. Pet. Technol., May, pp. 624-636.

13. Al-Khalifah, A-J. A., Aziz, K., and Home, R. N. (1987), A New Approach to Multiphase Well Test Analysis, SPE 16743, presented at the 62nd Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Dallas. TX, September 27-30.

14. , Horne, R. N., and Aziz, K. (1987), In-Place Determination of Reservoir Relative Permeability Using Well Test Analysis, SPE 16774, presented at the 62nd Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Dallas, TX, September 27-30.

15. Ayan, C. and Lee, W. J. (1988), The Effects of Multiphase Flow on the Interpretation of Pressure-Buildup Tests, SPE Formation Evaluation, June, pp. 459-466.

16. and Lee, W. J. (1988), Multiphase Pressure Buildup Analysis: Field Examples, SPE 17412, presented at the California Reg. Meet., Soc. Pet. Eng. AIME, Long Beach. CA, March 23-25.

17. Aziz, K., Mattar, L., KO. S., and Brar, G. S. (1976), Use of Pressure, Pressure-Squared or Pseudo-Pressure in the Analysis of Transient Pressure Drawdown Data From Gas Wells, J. Can. Pet. Technol., April-June, pp. 5 8-65.

18. Bayles, G. A. and Reznik, A. A. (1986), Pressure Transient Analysis of Methane Production From Coal Beds: An Analytical Approach, SPE 15226, presented at the Unconventional Gas Technology Symposium, Soc. Pet. Eng. AIME, Louisville, KY, May 18-2 1.

Boe, A., Skjaeveland, S. M., and Whitson, C. H. (1981). Two-Phase Pressure Test Analysis, SPE 10224, presented at the 56th Ann. Fall Tech. Conf. Exhib.. Soc. Pet. Eng. AIME, San Antonio. TX, October 5-7.

Camacho-V.. R. G. and Raghavan, R. (1987). Some Theoretical Results Useful in Analyzing Well Performance Under Solution Gas Drive, Unsolicited SPE 16580, Soc. Pet. Eng. AIME. Richardson, TX, June.

and Raghavan, R. (1987), Performance of Wells in Solution Gas-Drive Reservoirs, SPE 16745, presented at the 62nd Ann. Fall Tech. Conf. Exhib.. Soc. Pet. Eng. AIME, Dallas, TX, September 27-30.

(1988), Constant Pressure Production in Solution Gas-Drive Reservoirs: Transient Flow, Unsolicited SPE 17118, Soc. Pet. Eng. AIME, Richardson, TX, March.

Carter, R. D. (1962), Solutions of Unsteady-State Radial Gas Flow, J. Pet. Technol., May, pp. 549-554.

Chen, H. Y. and Poston, S. W. (1987), Application of a Pseudo-Time Function to Permit Better Decline Curve Analysis, SPE 17051, presented at the Eastern Reg. Meet., Soc. Pet. Eng. AIME, Pittsburgh, PA, October 2 1-23.

Clegg, M.. W. (l968), The Flow of Real Gases in Porous Media, SPE 209 1, presented at the 43rd Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Houston, TX, September 29-October 2.

Corbett, T. G. and Wattenbarger, R. A. (1985). An Analysis of and Correction Method for Gas Deliverability Curves, SPE 14208, presented at the 60th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 22-25.

Earlougher, R. C., Jr. (1977), Advances in Well Test Analysis, Mon. Vol. 5, Soc. Pet. Eng. AIME.

Econornides, M. J., Dehghani, K., Ogbe, D. 0.. and Ostermann, R. D. (1987), Hysteresis Effects for Gas Condensate Wells Undergoing Buildup Tests Below the Dew Point Pressure, SPE 16748, presented at the 62nd Ann. Fall Tech. Conf. Exhib.. Soc. Pet. Eng. AIME, Dallas, TX, September 27-30.

Evers, J. F. and Soeiinah, E. (1977), Transient Tests and Long-Range Performance Predictions in Stress-Sensitive Gas Reservoirs, J. Pet. Technol., August, pp. 1025-1030.

Fay, C. H. and hats, M. (1951). The Application of Numerical Methods to Cycling and Flooding Problems, Proc. Third World Pet. Cong., Section 11, pp. 555-564.

Fetkovich, M. J. (1973), The Isochronal Testing of Oil Wells, SPE 4529, presented at the 48th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 30-October 3.

, Vienot, M. E., Johnson. R. D., and Bowman, B. A. (1985). Case Study of a Low-Permeability Volatile Oil Field Using Individual Well Advanced Decline Curve Analysis, SPE 14237, presented at the 60th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 22-25.

Finjord, J. (1983). A Study of Pseudotime, Unsolicited SPE 12577, Soc. Pet. Eng. AIME, Dallas, TX, August.

(1985). An Analytical Study of Pseudotime for Pressure Drawdown in a Gas Reservoir, Unsolicited SPE 15205, Soc. Pet. Eng. AIME, Richardson, TX, April.

and Yue-Iu, Y. (1985), A Numerical Study of the Pseudotime Transformation for Drawdown in the Infinite-Acting Period, Unsolicited SPE 15206, Soc. Pet. Eng. AIME. Richardson, TX, May.

36. (1985), Non-Perturbative Approximate Analytical Solution in Pseudotime for Pressure Drawdown in a Gas Reservoir, Unsolicited SPE 15207, Soc. Pet. Eng. AIME, Richardson, TX, October.

37. Fraim, M. L. and Lee, W. J. (1987). Determination of Formation Properties from Long-Term Gas Well Production Affected by Non-Darcy Flow. SPE 16934, presented at the 62nd Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Dallas, TX, September 27-30.

38. , Lee, W. J., and Gatens. J. M., I11 (1986). Advanced Decline Curve Analysis Using Normalized-Time and Type Curves for Vertically Fractured Wells, SPE 15524, presented at the 61st Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, New Orleans, LA, October 5-8.

39. and Wattenbarger, R. A. (1988). Decline Curve Analysis for Multiphase Flow, SPE 18274, presented at the 63rd Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Houston, TX, October 2-5.

40. and Wattenbarger, R. A. (1987). Gas Reservoir Decline-Curve Analysis Using Type Curves with Real Gas Pseudo-Pressure and Normalized Time, SPE Formation Evaluation, December, pp. 671-682.

41. Kale, D. and Mattar, L. (1980). Solution of a Non-Linear Gas Flow Equation by the Perturbation Method, J. Can. Pet. Technol., October-December, pp. 63-67.

42. Kelkar, B. G. and Cox, R. (1985), Unified Relationship to Predict Future IPR Curves for Solution Gas-Drive Reservoirs, SPE 14239, presented at the 60th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 22-25.

43. Lee, R. L., Logan, R. W., and Tek, M. R. (1987), Effect of Turbulence on Transient Flow of Real Gases through Porous Media, SPE Formation Evaluation, March, pp. 108-120.

44. Lee, W. J. (1982). Well Testing, SPE Textbook Series, v. 1, Soc. Pet. Eng. AIME, Dallas, TX.

45. and Holditch. S. A. (1982), Application of Pseudo-Time to Buildup Test Analysis of Low Permeability Gas Wells with Long-Duration Wellbore Storage Distortion, J. Pet. Technol., December. pp. 2877- 2887.

46. Levine, J .S. and Prats, M. (1961), The Calculated Performance of Solution-Gas-Drive Reservoirs, Soc. Pet. Eng. J., September, pp. 142-152.

47. Meehan, D. N. and Schell, E. J. (1983). An Analysis of Rate-Sensitive Skin in Gas Wells, SPE 12 176, presented at the 58th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, San Francisco, CA, October 5-8.

48. Meunier, D. F.. Kabir, C. S., and Witunann, M. J. (1987), Gas Well Test Analysis: Use of Normalized Pseudovariables, SPE Formation Evaluation, December, pp. 629-636.

49. Muskat, M. and Meres, M. W. (1936), The Flow of Heterogeneous Fluids Through Porous Media, Physics. v. 7, September, 346-363.

50. Noman, R., Shrimanker, N., and Archer, J. S. (1985), Estimation of the Coefficient of Inertial Resistance in High-Rate Gas Wells, SPE 14207, presented at the 60th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 22-25.

51. Ostensen, R. W. (1986), The Effect of Stress-Dependent Permeability on Gas Production and Well Testing, SPE Formation Evaluation, June, 227-235.

52. Pedrosa, 0 . A. (l986), Pressure Transient Response in Stress-Sensitive Formations, SPE 15 115, presented at the 56th California Reg. Meet., Soc. Pet. Eng. AIME, Oakland, CA, March 2-4.

53. Raghavan, R. (1976). Well Test Analysis: Wells Producing by Solution Gas Drive, Soc. Pet. Eng. J., August, pp. 196-208.

54. (1986), Well Test Analysis for Multiphase Flow, SPE 14098, presented at the 1986 Int. Meet. Pet. Eng., Soc. Pet. Eng. AIME, Beijing, China, March 17-20.

55. , Scorer, J. D. T., and Miller, F. G. (1972), An Investigation by Numerical Methods of the Effect of Pressure-Dependent Rock and Fluid Properties on Well Flow Tests, Soc. Pet. Eng. J., June, pp. 267-275.

56. Reynolds, A. C., Bratvold, R. B., and Ding, W. (1987), Semilog Analysis of Gas Well Drawdown and Buildup Data. SPE Formation Evaluation, December, pp. 657-670.

57. Russell, D. G.. Goodrich, J. H., Perry, G. E., and Bruskotter, J. F. (1966), Methods for Predicting Gas Well Performance, J. Pet. Technol., January, pp. 99-108.

58. ~amanie~o-v. . F., Brigham, W. E.. and Miller. F. G. (1977), An Investigation of Transient Flow of Reservoir Fluids Considering F'ressure-Dependent Rock and Fluid Properties, Soc. Pet. Eng. J., April, pp. 140-150.

59. Scott, J. 0. (1979), Application of a New Method for Determining Flow Characteristics of Fractured Gas Wells in Tight Sands, SPE 793 1, presented at the Symposium on Low-Permeability Gas Reservoirs, Soc. Pet. Eng. AIME, Denver, CO, May 20-22.

60. Serra, K., Peres, A., and Reynolds, A. C. (1987), Well Test Analysis for Solution-Gas-Drive Reservoirs - Part 1: Determination of Relative and Absolute Permeabilities, Unsolicited SPE 17020, Soc. Pet. Eng. AIME, Richardson, TX, June.

61. Smith, G. E. (1986). Fluid Flow and Sand Production in Heavy Oil Reservoirs Under Solution Gas Drive. SPE 15094, presented at the 56th California Reg. Meet., Soc. Pet. Eng. AIME, Oakland, CA. March 2-4.

62. Spivey, J. P. and Lee, W. J. (1986). The Use of Pseudotime: Wellbore Storage and the Middle Time Region, SPE 15229, presented at the Unconventional Gas Technology Symposium, Soc. Pet. Eng. AIME, Louisville, KY, May 18-21.

63. and Lee, W. J. (1986), A Comparison of the Use of Pseudotime and Normalized Time for Gas Well Buildup Analysis in Various Geometries, SPE 15580, presented at the 6 1st Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, New Orleans, LA, October 5-8.

64. Stright, D. H., Jr. (1979), DST Analysis with Pressure Dependent Rock and Fluid Properties, SPE 8352, presented at the 54th Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Las Vegas, NV, September 23- 26.

65. Vairogs, J. and Rhoades, V. W. (1973). Pressure Transient Tests in Formations Having Stress-Sensitive Permeability, J. Pet. Technol., August, pp. 965-970.

66. Vo, D. T., Jones, J. R., and Raghavan, R. (1987). Performance Predictions for Gas Condensate Reservoirs, SPE 16984, presented at the 62nd Ann. Fall Tech. Conf. Exhib., Soc. Pet. Eng. AIME, Dallas, TX, September 27-30.

67. Wasson, J. and Mattar, L. (1981). Problem Gas Well Build-Up Tests - A Field Case Illustration of Solution Through the Use of Combined Techniques. Paper No. 8 1-32-49. presented at the 32nd Ann. Tech. Meeting of the Pet. Soc. of CIM, Calgary. Canada, May 3-6.

68. Wattenbarger, R. A. and Ramey, H. J., Jr. (1968), Gas Well Testing with Turbulence, Damage and Wellbore Storage, J. Pet. Technol., August, pp. 877-887.

69. Weir, G. J. and Kissling, W. M. (1984). "Scaled-Time" - A New Pseudo-Time Function for Approximately Linearising Gas Reservoir Models, Unsolicited SPE 13418, Soc. Pet. Eng. AIME, Dallas. TX, August.

70. Welty, D. H. and Meijerink, J. A. (1981), An Improved Formulation and Solution Method for Single-Phase Flow Problems, Soc. Pet. Eng. J., June, pp. 289-295.

71. Crank, J. (1956), The Mathematics of Diffusion, Oxford University Press, New York.

72. Van Everdingen, A. F. (1953), The Skin Effect and Its Influence on the Productive Capacity of a Well, Trans. Am. Inst. Min. Metall. Eng.. v. 198, 171-176.

73. Hurst, W. (1953), Establishment of the Skin Effect and Its Impediment to Fluid Flow into a Well Bore, Pet. Eng., October, pp. B-6 to B-16.

74. Aadnoy. B. S. and Finjord, J. (198 6). Perturbative Analytical Solution of the Boltzmann Transient Line Sink for an Oil Reservoir with Pressure Dependent Fluid and Formation Properties, Unsolicited SPE 15970, Soc. Pet. Eng. AIME, Richardson, TX, June.

75. Finjord, J. and Aadnoy, B. S. (1986), Effects of Nonlinear Gradient Term in Exact Analytical Solutions of the Radial Flow Equation for Oil in a Reservoir. Unsolicited SPE 15969, Soc. Pet. Eng. AIME, Richardson, TX. February.

76. Odeh. A. S. and Babu D. K. (1987). Comparison of Solutions of the Nonlinear gnd Linearized Diffusion Equation, Unsolicited SPE 17270, Soc. Pet. Eng. AIME, Richardson, TX, September.

77. Matthews, C. S., Brons, F., and Hazebroek, P. (1954), A Method for the Determination of Average Pressure in a Bounded Reservoir, Trans. Am. Inst. Min. Metall. Eng., v. 20 1, 182-189.

DERIVATION OF ALTERNATIVE FORMS OF PSEUDOPRESSURE

The use of integral transforms in the solution of the diffusivity equation has given rise to various definitions of pseudopressure for different reservoir fluid and porous media systems. Real gases, oils with

pressure-dependent properties, compaction, and multiphase flow have been handled with pseudopressures

defined specifically for each system. In general, valid forms of the pseudopressure variable can be shown to be special cases of the pseudopressure defined as

with the diffusivity equation written as

Perhaps the most common form of pseudopressure is that proposed for real gases by A1 Hussainy,

Ramey et al.:12

PO

If the real gas equation of state is applied to Equation (A-1), pseudopressure becomes

k Mpf m - ( p ZRT dp' .

PO

Assuming that the flow of gas in the reservoir is isothermal and permeability is constant, Equation

(A-4) can be written as

where

Now consider a new variable m ' which can be any linear function of m:

then

and

am 1 am' - at a at '

Substitution back into Equation (A-2) gives

1 a2m' 1 1 am' 1 1 am' -- +--- = a & a r a r a q d t

(A- 10)

(A- 1 1)

Thus, the expression for m given by Equations (A-5) and (A-6) can be any linear function and

still apply. The constant c ' is chosen so that c ' x c = 2. The resulting expression for pseudopressure is

Equation (A-3).

Another common form is for stress-dependent oil reservoirs where compaction may be

significant. In this case, the oil density may be handled with the formation volume factor. The equation for

pseudopressure becomes

P k

m = @TO r ~ , dp'

and any linear function of Equation (A-12) is a valid definition.

(A- 12)

Forms of the pseudopressure containing porosity or zone thickness as a function of pressure are

not valid without a modification to the diffusivity equation. In fact, if one specifies that the flow is to be one-dimensional radial flow, then the reservoir thickness may not be pressure-dependent. This is true

because some flow would have to occur in the Z direction if there is a thickness gradient.

SOLUTION TO THE DIFFUSIVITY EOUATION IN TERMS OF THE POROSITY-DENSITY PRODUCT

Although the generally accepted variable for well testing when rock and fluid properties may vary

with pressure is the pseudopressure, the equations governing pressure transient analysis may be developed in

terms of the porosity-density product. For the case of one-dimensional radial flow to a constant rate well in

an infinite reservoir, the problem may be formulated as

Initial Condition:

Boundary Condition 1:

Boundary Condition 2:

The solution for a line source well becomes:

where the Boltzmann transform x is:

(B- 10)

Comparison of Equation (B-9) to Equation (30) in the text reveals the similarity of the solutions.

Because the hydraulic diffusivity appears twice in Equation (B-9). the numerical evaluation of the solution

in terms of the porosity-density product may require more iterations than the solution in terms of

pseudopressure. However, the two solutions are mathematically equivalent.

JvIULTIRATE SOLUTION TO THE DIFFUSIVITY EOUATION

The need to handle variable flow rates frequently arises in well testing. Two of the more common

pressure transient tests, falloff and buildup tests, appeal because of the ability to ensure that the second rate

is constant (and zero) at the surface. Under classical well test analyses, multiple flow rates are approximated as a series of constant rate flow periods. Since classical techniques assume that the partial differential

equations are linear, the principle of superposition may be used to handle the effect of multiple flow rates

during well tests. In general, the nonlinearity of the parrial differential equations prevents superposition

from being valid, so a rigorous solution must be derived to handle multiple flow rates.

The problem of variable flow rates can be handled as a series of constant rate flow periods

described by

where

and

Since the flow of fluids in the reservoir is still governed by the general diffusivity equation, the

problem becomes one of incorporating the relations (C- 1) through (C-3) into the boundary conditions and then being able to solve the more complicated nonlinear problem. To do this, the solution in terms of

dimensionless pseudopressure can be formulated:

where

and pi is the pressure at the end of the last flow rate. Substituting Equation (C-4) into the diffusivity

equation, one obtains

The well sourcelsink term becomes

If Ati is defined as

then the equations to solve become

and

(C-9)

(C- 10)

(C- 1 1)

(C- 12)

The apparent decoupling of the problem into components of the flow rate intervals is not as complete as one would hope. In Equation (C- 9), the hydraulic diffusivity is based on mD and not mDi for

the case where the hydraulic diffusivity is a function of pressure.

Now that the problem has been derived for the case of multiple constant flow rates, the line source

solution can be obtained similar to the constant rate case by defining a Boltzmann transform variable:

and

The transformed equations are

(C- 13)

(C- 14)

(C-15)

(C- 16)

The solution for m ~ i becomes

(C- 17)

In order for a consistent solution to be obtained, the calculation of mDi for each individual flow

period assumes that this flow period will continue to infinite time. If the hydraulic diffusivity is assumed to be

a constant, this solution degenerates to the classical superposition solution.

APPENDIX D

JUSTIFICATION FOR A MASS FLUX BASED SKIN

Since Darcy's Law states that the volumetric flux is proportional to the pressure differential at a

point, it appears inconsistent to base an infinitesimally small thickness skin on mass flux which yields a skin

equivalent to a constant pseudopressure drop for a given surface flow rate. However, the correct flux basis

for the definition of skin may be determined by considering the derivation of a finite thickness skin. The

model to consider is a cylinder of porous media around the well bore (r,) of thickness r,. This cylinder is

the skin zone. If the skin zone has the same permeability as the reservoir, the skin will be defined as zero.

Otherwise, the skin will have a finite value which depends on the permeability and radius of the skin zone.

To determine the pressure drop across the skin zone, Darcy's Law is applied:

(D- 1 a)

where

For steady state flow through the skin zone, mass is conserved. This leads to

where ( q ~ ) , is a constant flow rate at standard conditions. If pseudopressure is defined as Equation (8) in

the text.

then Equation (D-2) can be rewritten as

To obtain the pseudopressure drop across the skin zone, separation of variables and integration may be

applied to Equation (D-3) to yield

- 2nh r s [mrs - mw] = ln-

r w

As stated previously, a zero skin will be defined for the case of the skin zone having the same

permeability as the reservoir. Therefore, the value of skin can be obtained for other cases by determining the difference in pressure drop through the skin zone and through the same size skin zone with reservoir

permeability. This leads to

and

where the superscripts R and S refer to reservoir and skin, respectively. This distinction is necessary since

the pseudopressure contains permeability within the integral. Since the desired result is a difference

between two pseudopressure drops, the pseudopressures in Equation (D-5b) need to be converted into ones consistent with Equation (D-5a). In order for the result to end up as a simple relation, a restriction is added at this point. This restriction is that the ratio of k, to k, is a constant. Although both k, and k, may be functions of pressure.

Applying Equation (D-6) to Equation (D-5b) yields

where the pseudopressure superscript SR refers to the altered permeability skin pseudopressure converted to

the reservoir permeability basis. Manipulating Equations (D-5a) and (D-7) yields

Since the reservoir provides fluid to the outer boundary of the skin zone at the same pressure,

which simplifies Equation (D-8) to

The left-hand side of Equation (D-10) is a dimensionless pseudopressure drop which is the definition of skin:

The final result,

(D- 1 1)

is identical to the result. for a finite thickness skin derived for reservoirs of small and constant compres~ibility.~~ The extensions to the infinitesimally small thickness skin zone would suggest that the skin

be a dimensionless pseudopressure drop to ensure that mass is conserved across the skin.

Incorrect Response (WITHOUT Pseudotime) pressure

Correct Response (WITH Pseudotime)

pressure ------------- derivative

1

Delta Time (hr)

Fig. 1 - Pressure responses with and without the use of pseudotime.