A Non Iterative Technique for Solving Parabolic Partial Differential Equation Problems

Abstract

Abstract This paper presents a new noniterative difference technique which may be used to solve parabolic partial differential equation parabolic partial differential equation problems, including those arising in the prediction problems, including those arising in the prediction of fluid flow in a porous media. It is shown that this difference technique is unconditionally stable and is suitable to solve engineering problems. A comparison is made between the new problems. A comparison is made between the new method and two other stable noniterative difference techniques, the Saul'ev technique and the noniterative ADI technique. Of the two problems worked as examples, the new method is problems worked as examples, the new method is most efficient for the elongated grid problem, and noniterative ADI is most efficient for the square grid problem. Introduction Since the introduction of high speed computers, a reservoir engineering research goal has been to develop increasingly efficient methods of predicting reservoir fluid movement. Various methods have been developed or used by petroleum engineers to predict the performance petroleum engineers to predict the performance of an oil field. The major emphasis in the solution of partial differential equations simulating multiphase fluid flow has been the finite difference approach, which can be summarized by the following series of steps.The reservoir or section of a reservoir is characterized by a series of mesh points wit varying rock and fluid properties simulated at each point.The partial differential equations which describe the fluid flow are written.At each mesh point the partial differential equations are replaced by difference equations.The resulting equations are solved [sometimes iteratively to obtain an approximate solution to the original partial differential equation problem. This paper presents a new technique for replacing the differential equation at each mesh point by a difference equation. Difference equations are evaluated in terms of the stability, consistency and order of accuracy. Usually these items are discussed in regard to solution of the diffusivity equation, a linear partial differential equation for sore uniform rectangular mesh covering the region of interest. The same approach is used here. However, it should be noted that the conclusions reached considering a linear partial differential equation and a uniform rectangular mesh are not always applicable to petroleum problems where the differential equations are nonlinear and the mesh is often not uniform. DESCRIPTION OF THE DIFFERENCE TECHNIQUE In this paper a new two-step noniterative difference technique is presented.