Bill Wooden Mehdi Azari Mohamed SolimanHalliburton Reservoir Services

Dallas

For modeling well test data more reliably, a F,new computer program easily and accurately inverts the Laplace transform.

Converting real time and space solution to Laplace space is often done in the petroleum industry and provides the vehicle to develop numerous new solutions. The Laplace space transform is frequently used in pressure transient analysis primarily because it can reduce or transform a highly difficult problem into a much simpler one.

Typically, a Laplace space equation can be manipulated by use of simple algebra to accomplish other desired ends, such as incorporating additional transformed equations to solve other aspects of the engineering problem.

Once the transformed equation is complete, it is then necessary to convert to real time and space.

This conversion is accomplished analytically by what is referred to as inverting the Laplace transform with sets of formulas and relationships between real and transformed space and time.

In many cases, this inversion is not easy or cannot be done by conventional analytic means. In those situations, an engineer requires a program that numerically inverts the transform.

The new program, the Azari-Wooden-Gaver, or AWG method, has this capability.

### BACKGROUND

For numerically inverting the Laplace transform of petroleum engineering problems probably Stehfest's 1 algorithm is the most common. Stehfest's method is based on the work by Gaver 2 that contains a rather complicated summation to reach the intended inversion.

In Gaver's method, the number of terms in the summation and the floating-point precision in the coefficients play an important role in the accuracy of the inversion.

The AWG algorithm overcomes the difficulties of Stehfest's and Gaver's methods by efficiently generating the precise coefficients and by allowing additional terms in the summation to be added.

Another difficulty with techniques such as Gaver's is instabilities in inversion of certain types of functions, in particular, transformed solutions that contain transcendentals and shifting functions. These problems were pointed out by Berger. 3

### AWG METHOD

This research for easily solving all classes of Laplace space transforms relevant to petroleum engineering is admittedly incomplete. What is presented here are informative guidelines for using the AWG program.

Current research indicates that the remaining problems of transcendental functions and shifting can be solved by employing rather complicated software if one is willing to state the problem in complex space. Software of this type is typically slow and ill suited to computer modeling.

Although these same difficulties exist in the AWG method, we are confident that further research into these areas will yield less cumbersome solutions.

### METHOD COMPARISON

Table 1 presents a numerical comparison of the AWG program's output to Stehfest's. This comparison clearly shows a marked improvement over Stehfest's coding of Gaver's method.

This enhancement is the result of improvements made in determining coefficients and by extending the summation past the limits of the Stehfest method.

Results of extensive testing against other published methods appear in Table 2.

Table 2 compares a range of functions in Laplace space judged against the exact analytic solution value. These comparisons include programs by Veillon, 4 Stehfest, 1 Dubner, 5 Talbot, 6 Piessens, 7 and Bellman. 8

The ranking of each program is based upon a comparison of the respective methods to the exact values for the functions. On average, the AWG program matched the exact solution to eight significant digits.

In cases where the new program failed, we discovered problems related to the Gaver solution method as opposed to either coefficient precision or summation error. In these situations, the methods of Talbot and Piessens invert the transform more reliably.

For a finite-conductivity, vertical fracture in an infinite reservoir, Fig. 1 plots dimensionless pressure and its derivative vs. dimensionless time of the Laplace space transform of Azari, et al. 9

This plot was generated by Stehfest's method with 10 summation terms. Note the oscillation in the dashed derivative curve near a dimensionless time of 10-5.

Because of precision limits of the coefficients, efforts to remove this oscillation by increasing the number of terms in the summation were unsuccessful.

Fig. 2 illustrates a similar plot as in Fig. 1 with the exception that Fig. 2 was generated with the AWG inversion method. The number of terms in the summation was 18. The result was a very smooth derivative curve throughout the dimensionless time. To date, we have successfully tested the method to 25 terms.

The accompanying box lists the computer program which numerically inverts a user-supplied Laplace transform at a specified time. The computational results are the inverted value, Pd, its derivative, Pdd, and the dimensionless flow rate, qd.

### METHOD LIMITATIONS

Calculation precision and efficiency have been optimized in the AWG method for a wide variety of functions.

The AWG method has been employed in numerous automated transient well-test analysis programs with great success. Yet, a need for an efficient and exact numerical inversion technique for several types of functions remains.

The program user for petroleum applications should note that an n value ranging from 8 to 10 is used for the radial cylinder source models wherein Bessel functions are included. We have found that higher n values destabilize the inversion as the oscillatory nature of the Bessel function becomes evident. Laplace space transforms not containing transcendentals, Bessel functions or shifting functions will benefit from increasing n values. These higher n values are a distinct advantage of the AWG program.

### REFERENCES

- Stehfest, H., "Algorithm 368: Numerical Inversion of Laplace Transform," Comm. ACM, Vol. 13, No. 1, January 1970, pp. 47-49.
- Gaver, D.P., "Observing Stochastic Processes, and Approximate Transform Inversion," Oper. Res., Vol. 14, No. 3, 1966, pp. 444-59.
- Berger, B.S., and Duangudom, S., "A Technique for Increasing the Accuracy of the Numerical Inversion of the Laplace Transform with Applications," ASME Journal of Applied Mechanics, Paper No. V3-WA/APM-1, December 1973.
- Veillon, F., "Algorithm 486: Numerical Inversion of Laplace Transform," Comm. ACM, Vol. 17, No. 10, October 1974, pp. 587-89.
- Dubner, H., and Abate, J., "Numerical Inversion of Laplace Transforms and the Finite Fourier Transforms," J. ACM, Vol. 15, No. 1, January 1968, pp. 115-23.
- Talbot, A., "The Accurate Numerical Inversion of Laplace Transforms," J. Inst. Math. Appl., Vol. 23, 1979.
- Piessens, R., and Huysmans, R., "Algorithm 619: Automatic Numerical Inversion of the Laplace Transform," ACM Trans. Math. Softw., Vol. 10, No. 3, 1984.
- Bellman, R.E., Kalaba, R.E., and Lockett, J., Numerical Inversion of the Laplace Transform, American Elsevier, New York, 1966.
- Azari, M., Wooden, W.O., and Coble, L.E., "A Complete Set of Laplace Transforms for Finite-Conductivity Vertical Fractures Under Bilinear and Trilinear Flows," Paper No. SPE 20556, 65th SPE Annual Technical Conference and Exhibition, New Orleans, Sept. 23-26, 1990.

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