Anil K. AmbasthaUniversity of Alberta

Edmonton

Edmond GomesBangladesh University of Engineering & Technology

Dacca, Bangladesh

Equations For Calculating Pseudo Skin (35842 bytes)

A simple, easy-to-use Fortran program calculates pseudoskin for a vertical well partially penetrating a multilayer reservoir under pseudosteady-state interlayer crossflow conditions.

Expressions for pseudoskin are listed for conditions covering both closed top and bottom boundaries and the existence of either a gas cap or bottom water zone. The Fortran program is based on these simple expressions for pseudoskin.

The program allows for any arbitrary number of layers in the well.

### PSEUDOSKIN

Often, a well is completed in only a limited number of the layers in a reservoir. For these wells it is important to calculate the pseudoskin under partial penetration.

Many papers discuss semi-empirical, analytical, and graphical correlations to obtain the pseudoskin for specialized situations.1-11 Most of these studies are restricted to either a single-layer reservoir1-3 5 8 10 11 or a two-layer reservoir.4 6

Pseudoskin for partially penetrated, multilayer reservoirs has been previously addressed either by using a numerical simulator with graphical presentation of pseudoskin correlations, or by extending pseudoskin expressions for a single-layer case to multilayer situation by redefining some parameters.9

The importance of the technique discussed in this article stems from the fact that we derived simple expressions for pseudoskins covering a wide variety of single and multilayer reservoir situations on the basis of rigorous, theoretical solutions for layered reservoirs.

To the best of our knowledge, such widely encompassing, simple expressions for pseudoskins have not appeared before in the literature.

### CLOSED TOP AND BOTTOM

Equation 1 (see equation box), containing two dimensionless parameters (k and lambda A), represents a simple expression to compute pseudoskin for a well partially penetrating a multilayer reservoir with closed top and bottom boundaries. Any arbitrary number of layers may be penetrated by the well.

In Equation 1, K0 and K1 are the modified Bessel functions of second kind of order zero and one, respectively. The variables K and lambda A are the total mobility-thickness ratio of the open (perforated) interval and average crossflow parameter, respectively, given by Equations 2 through 5.

The variables, k, kv, h, m, j, N, rw, and XA represent horizontal permeability, vertical permeability, thickness, fluid viscosity, layer number, total number of layers, wellbore radius, and semipermeability between layers j and j+1, respectively. Pseudoskin, Sb, due to partial penetration is a dimensionless variable proportional to the real pressure drop, _p, caused by partial penetration, and is defined by Equation 6, written in consistent SI units.

In Equation 6, q and B refer to production rate and formation volume factor, respectively.

Equation 1 has been developed on the basis of a limiting solution for a two-layer reservoir with pseudosteady-state interlayer crossflow. Readers interested in the theoretical details regarding the development of Equations 1 and 7 are referred to Reference 12.

### GAS CAP OR BOTTOM WATER

Equation 7, containing three dimensionless parameters (K, lambda A, and lambda C), represents a simple expression to compute pseudoskin for a well partially penetrating a multilayer reservoir, when either the top (or the bottom) boundary is a constant-pressure boundary due to the existence of a gas cap (or bottom water) zone.

In Equation 7, variables, s1, s2, a1, a2, and b, are functions of K, lambda A, and lambda C, and are given by Equations 8 12.

For Equations 7-12, the definitions of K and lambda A are given by Equations 2 and 4, respectively. However, the crossflow parameter, lambda C, for the layer adjacent to the gas cap (or bottom water) zone is given by Equation 13.

In Equation 13 the subscript j denotes the layer number for the layer adjacent to the gas cap (or bottom water) zone.

### COMPUTER PROGRAM

A Fortran program has been written to compute pseudoskin based on Equations 1 and 7. The program allows input of either the dimensionless parameters themselves or the necessary dimensional variables (such as well bore radius, layer properties, fluid viscosity, etc.) for a given situation.

The program allows for any arbitrary set of layers to be penetrated by the well. The output from the program is the pseudoskin value for the situation under consideration. Bessel function routines needed for the development of the computer program were taken from Reference 13.

The results from this computer program have been successfully compared with the various papers dealing with the subject of pseudoskin computation.1-11 These comparisons are documented in Reference 12. Additional comparisons are presented in Reference 14.

### EXAMPLE APPLICATION

A five-layer situation has been considered with Layers 2 and 4 from the bottom being perforated. Layer properties and other input parameters appear in Table 1 (12912 bytes). Both top and bottom boundaries are considered to be closed boundaries. Pseudoskin for this five-layer case is computed to be 1.7.

If this five-layer case were simplified to an equivalent homogeneous, but anisotropic, reservoir case, then Equation 3 yields a reservoir permeability of 81.9 rod using a total formation thickness of 105 ft.

When all layers are assigned the permeability of 81.9 md, with all other Parameters being the same as before, the computer program yields a pseudoskin of 7. Thus, this example illustrates the importance of rigorously computing pseudoskin for the appropriate reservoir situation.

### ACKNOWLEDGMENTS

E. Gomes acknowledges the financial support of the University of Alberta-BUET (Bangladesh University of Engineering & Technology)-CIDA (Canadian International Development Agency) Linkage Project. The Department of Mining, Metallurgical, and Petroleum Engineering at the University of Alberta provided the computer facilities for this work.

Editor's note: To obtain a compiled version of the Fortran program, Pskin.exe, and a Fortran listing, Journal subscribers can send a blank 5 1/4, or 3 1/5 diskette formatted to MS DOS and a self-addressed, postage paid or stamped return diskette mailer to: Production Editor, Oil & Gas Journal, 3050 Post Oak Blvd., Suite 200, Houston, TX 77056.

Subscribers outside the U.S. should send the diskette and return mailer without return postage to the same address. This mail offer will expire Dec. 31, 1995.

### REFERENCES

- Brons, F., and Marting, V.E., "The Effect of Restricted Flow Entry on Well Productivity," JPT, February 1961, pp. 172-4.
- Bilhartz, H.L., and Ramey, H.J. Jr., "The Combined Effect of Storage, Skin, and Partial Penetration on Well Test Analysis," paper SPE 6753, Annual Technical Meeting of SPE of AIME, Denver, Oct. 9-12, 1977.
- Streltsova-Adams, T.D., "Pressure Drawdown in a Well with Limited Flow Entry," JPT, November 1978, pp. 1469-76.
- Reynolds, A.C. et al., "Pseudoskin Factor Caused by Partial Penetration," JPT, December 1984, pp. 2197-210
- Papatzacos, P, "Approximate Partial-Penetration Pseudoskin for Infinite Conductivity Wells," SPERE, May 1987 , pp. 227 34.
- Olarewaju, J.S. and Lee, W.J., "Pressure Buildup Behavior of Partially Completed Wells in Layered Reservoirs", paper SPE 18876, Production Operations Symposium, Oklahoma City, Mar. 13-14, 1989.
- Yeh, N.S., and Reynolds, A.C., "Computation of Pseudoskin Factor Caused by a Restricted-Entry Well Completed in a Multilayer Reservoir," SPEFE, June 1989, pp. 253-63.
- Vrbik, J., "A Simple Approximation to the Pseudoskin Factor Resulting from Restricted Entry," SPEFE, December 1991, pp 444-46.
- Ding, W., and Reynolds, A.C., "Computation of the Pseudoskin Factor for a Restricted-Entry Well," SPEFE, March 1994, pp. 9-14.
- Odeh, A.S., "Steady-State Flow Capacity of Wells with Limited Entry to Flow," SPEJ, March 1968, pp. 43-51.
- Vrbik, J., "Calculating the Pseudo-Skin Factor due to Partial Well Completion," JCPT, September-October 1986, pp, 57-61.
- Gomes, E., and Ambastha, A.K., "Analytical Expressions for Pseudoskin for Partially Penetrating Wells under Various Reservoir Conditions," paper SPE 26484, 68th Annual Technical Conference and Exhibition of SPE of AIME, Houston, Oct. 3-6, 1993.
- Press, W.H., et al., Numerical Recipes The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.
- Gomes, H., "Well Test Analysis for Multilayered Composite Systems," Ph.D. dissertation, University of Alberta, Edmonton, April 1944.

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