A Semianalytical Method for the Mathematical Solution of a Moving Boundary Problem
Authors: Bloom Sanford G. 
Published: December 2007

This paper was prepared for the 42nd Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Houston, Tex., Oct 1–4, 1967. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made.
Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers Office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
A semianalytical method is presented for the solution of certain types of partial differential equations. In this method, the solution, or a significant portion of it, is expressed as a finite series of functions with unknown coefficients. The functions must be determined by a separate physical or mathematical analysis of the problem and the coefficients are determined by numerically fitting the series to specified boundary conditions. For some problems, the use of this series can result in a considerable saving in computer time in comparison to a finte-difference method.
The method is applied to the sharp interface model of fluid displacement in a reservoir. A computer program was written to solve the resulting equations and results are given for various values of inplace to injected fluid viscosity ratios, gravity to viscous force ratios, and vertical to horizontal permeability ratios.

Introduction
Depending on what assumptions are made, the mathematical formulation of a fluid displacement problem can lead to a system of partial differential equations. Various methods can be used for obtaining the solutions to these equations and these methods may be grouped into either analytical, finite-difference, or seimanalytical techniques. It is the purpose of this paper, to demonstrate the application of a semianalytical technique to a moving boundary problem which is of so me interest to the petroleum industry.
Analytical techniques are generally the most desirable methods because analytical solutions can usually provide the most insight into the behavior of the solutions. However, analytical solutions are known for only the simpler equations at present. In contrast to analytical techniques, finite-difference techniques can be applied to almost any system of partial differential equations. However, the use of these methods can be expensive in terms of computer storage and time. Also, with finite-difference techniques, the space or time interval is often limited by a particular part of the problem whose accuracy has little effect on the rest of the problem. An example of this difficulty arises in the moving boundary problem which will be treated in this paper. In order to accurately position the boundary, it is necessary to have the space intervals very small in the vicinity of the boundary. Since the boundary moves, these small intervals are normally used over the entire region of interest.