The Use of Integral Equations in the Numerical Solution of Complex Reservoir Engineering Problems
Authors: Dumore J.M. , Marsal D. 
Published: December 2007

Abstract
Quite a number of problems that cannot be bandied economically with the methods usually applied in reservoir engineering can be solved by the theory of integral equations. The main purpose of this paper is to draw attention once more to the usefulness of integral equations. Their benefit is shown by considering a typical example; namely, the behavior of a horizontal multilayered system of oil reservoirs bounded by aquifers.

Introduction
A number of reservoir engineering problems involving the flow of compressible fluids in porous media are so complex that they cannot be solved analytically, yet the usual numerical techniques require excessive computer time. An example of such a problem is simultaneous production from a horizontal multilayered system of production from a horizontal multilayered system of separate oil reservoirs, each reservoir being bounded by its own aquifer but sharing production wells with the other reservoirs. The difficulties in obtaining an analytical solution to this problem are insuperable; consequently, Tempelaar-Lietz and Lefkovits et al have restricted their studies to particular aspects under simplified conditions. The modern approach is to simulate the flow problem numerically. This implies that the differential equations that describe the problem are approximated by finite-difference equations, problem are approximated by finite-difference equations, which are subsequently solved by a computer. As both the aquifers and the reservoirs must be simulated, the numerical calculations require considerable computer time and storage capacity. In this paper it is shown that problems involving the influx of fluids (e.g., fluid influx from pay zones into wellbores, or water influx from aquifers into reservoirs) can often be described by a system of intergral equations that require much less computer time for solution. These integral equations can be derived only if the motion of the inflowing fluids can be described by linear and homogeneous differential equations, so that the superposition principle applies. The fluid influx can then be principle applies. The fluid influx can then be calculated from integral expressions that contain standard functions, This obviates the need for numerical simulation of those parts of the reservoir from which the influx originates. The standard functions are the cumulative influx as a function of time for a unit pressure drop at the influx boundary, and the pressure drop at the influx boundary as a function of time for a unit influx rate. These functions are interrelated so that if one is known the other can be calculated. For homogeneous and isotropic flow regions of linear or radial geometry, the standard functions have been evaluated in dimensionless form by van Everdingen and Hurst. For other flow regions, they can be evaluated with an auxiliary computer program. The integral equations presented here are by no means new. They are special cases of Duhamel's theorem (see also Carslaw and Jaeger, p. 30), and one of them is explicitly formulated by Samara in his analysis of pressure decline.