Numerical Solution of Slightly Compressible Single Phase Transient Flow Problems by Spline Functions

Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract Two dimensional spline functions were utilized in the solution of the linear diffusion equation both with and without a sink within the domain. A collocation method of solution was used and several types of spline representations were investigated. Results were compared with finite difference solutions and with the analytical solutions. B-splines with an extended region of definition were concluded to be superior. For problems with a sink the B-splines alone were not accurate, however, supplementing the splines with a log term produced very good results. Introduction In engineering design, ship builders and others often use a spline, an elastic ruler which can be bent so that it passes through a given set of points. Under certain assumptions the curve can be approximately described as being built up of different third degree polynomials (cubic polynomials) in such a way polynomials (cubic polynomials) in such a way that the function and its first two derivatives are everywhere continuous. The third derivative however, can have discontinuities at the given points. Such a function is called cubic spline. In recent years, spline functions have found wide applications in approximation theory, data fitting, interpolation and so on 4, 9, 15, 16, 17. A smaller amount of work has been devoted to the solution of ordinary and partial differential equations. partial differential equations. Application of spline functions in reservoir problems was initiated by Culham and Varga. problems was initiated by Culham and Varga. These authors used spline functions to approximate the space derivative in a one dimensional nonlinear parabolic partial differential equation which describes the real gas flow in a porous media. The time derivative, however, was approximated by means of various finite difference schemes. They employed Galarkin's and some non-Galarkin type methods to obtain the solutions. Cubic spline basis functions were recommended as the optimum choice for a single-phase flow. When a spline function is used to approximate the solution of a differential equation, two different methods may be used to eliminate the space dependency. These are namely, Galarkin's method (or Integral Methods) and collocation method. Comparisons of these two different solution methods were made by Panton and Salle on a one dimensional heat Panton and Salle on a one dimensional heat conduction equation. They reported that integral methods give higher accuracy in the solution but programming is quite laborous.