An Application of the Product Solution Principle for Instantaneous Source and Green's Functions

Abstract

Summary. Basic building blocks to apply the product-solution method in the Laplace domain are derived. Example applications illustrate the utility of these building blocks. These solutions should be useful in applying computer algebra to develop pressure distribution in reservoirs. Introduction Newman's product method for certain initial and boundary conditions allows us to construct solutions of multidimensional problems of the diffusivity equation directly from appropriate 1D problems of the diffusivity equation directly from appropriate 1D solutions. For example, the solution to a 2D problem, f(x, y, t), may be of the form of (1) Gringarten and Ramey exploited this technique by noting that source and Green's functions for multidimensional flows through porous media can be constructed as a product of the respective ID porous media can be constructed as a product of the respective ID functions. They showed that solutions to a wide range of problems of interest to petroleum engineers can be obtained systematically. Unfortunately, the product-solution technique is applicable only in the time domain. As Ozkan and Raghavan noted, the Utility of the solution procedure that Gringarten and Ramey presented becomes limited if one desires to transform their solutions to the Laplace domain. Thus, it is not possible to extend the solutions of Gringarten and Ramey readily With the Stehfest or other algorithms to incorporate the influences of variable sandface-rate production, wellbore-storage effects, and dual porosity or to use their solutions in history-matching algorithms. The problem that arises in applying the product-solution principle in Laplace domain to the 2D systems considered in Eq. 1 can principle in Laplace domain to the 2D systems considered in Eq. 1 can be stated as (2) The limitation noted in Eq. 2 has essentially stymied progress in the use of the product-solution method, although it may be possible to obtain the expression for the function f(x, y, t) and then obtain possible to obtain the expression for the function f(x, y, t) and then obtain [f(x, y, t)] (e.g., Refs. 5 and 6). This approach has three principal disadvantages:such a procedure is not efficient because additional steps are required to obtain [f(x, y, t)];in some cases, obtaining [f(x, y, t)] for some of the problems of interest is not easy 3;because we have to obtain f(x, y, t) and [f(x, y, t)] for each problem on a case-by- case basis, implementation of the product-solution method becomes unsuitable for developing algorithms to calculate pressure distributions in the reservoir. The major objective of this study is to explore the possibility of applying the product method (or a similar principle) to the Laplace transform of instantaneous source and Green's functions. Symbolically, we solve the following problem (for a general presentation, the space variables are dropped): (3) where U{...} is an operator that preserves the equality in Eq. 3. If this can be done, then a Laplace-transform solution for a complex system can be directly constructed from the Laplace-transform building-block functions. In this way, all the advantages of using the product-solution principle and instantaneous source and Green's functions product-solution principle and instantaneous source and Green's functions in the time domain are preserved in the Laplace domain. On analyzing the structure of the product-solution procedure and the characteristics of some of the basic source and Green's functions, we have developed, for certain conditions, an analog product-solution procedure in the Laplace domain. The procedure presented product-solution procedure in the Laplace domain. The procedure presented here is applicable to reservoirs bounded by linear boundaries in at least one direction. The end product of our work is presented in tabular form for convenience. Theoretical Considerations We briefly discuss the product-solution method in time and Laplace domains and document a summary of our procedure. Example applications of our method are discussed in the next section. The product method may be illustrated by the following example. If the pressure distribution in a reservoir is governed by the following 3D equation, (4) where a, less than × less than b (r = 1,2,3), then the solution to Eq. 4 can be expressed as a product of the solutions of three single-space variable problems, provided product of the solutions of three single-space variable problems, provided that the initial and boundary conditions satisfy certain conditions. Thus, the solution can be written directly if 1D solutions are available. The consists imposed on this procedure are that (1) the initial pressure distribution is expressed as a product of the single-space variable functions (including the case of uniform initial pressure condition) and (2) the boundary conditions are homogeneous. For our particular case, if it is assumed that the pressure drop P()=[ )]=0 initially, then it can be shown that (5) where (x, t), r= 1,2,3, satisfies (6) and the corresponding homogeneous boundary conditions (sealed, constant pressure, or mixed), and ()=0 at t=O. Although pressure, or mixed), and ()=0 at t=O. Although a finite system is considered in the above example, the product method is also applicable to infinite and semi-infinite systems. The same principle is applicable to cylindrical reservoirs with mixed principle is applicable to cylindrical reservoirs with mixed (radial/linear) boundaries. Multidimensional instantaneous Green's and source functions can be constructed as a product of 1D instantaneous Green's and/or source functions, as illustrated in Refs. 2 and 7. Symbolically, following the ideas in Ref. 2 for 2D flow in Cartesian coordinates, the pressure drop, (x, y, t), for production from a continuous source is given by (7) In Eq. 7, X(x, x, t) and Y(y, y, t) represent the appropriate instantaneous source functions for the appropriate boundary conditions, and q(t) is withdrawal rate per unit length or area of the source. *Now at Phillips Petroleum Co. Copyright 1991 Society of Petroleum Engineers P. 161