A Further Stability Analysis for the ADI Procedure as Applied to Singular Matrices

Abstract

American Institute of Mining, Metallurgical and Petroleum Engineers Inc. Abstract Solving =, by numerical techniques requires application of ADI procedures to singular matrices which leads to numerical instability for small values of the ADI parameter w. A study of this problem demonstrates that restriction of IRml/w (where Rm is the largest residual) will control these instabilities. Applications to compressible and incompressible Darcy flow are discussed. Introduction The equations of reservoir flow commonly have the following form V (KVP) Q where in two dimensions, ..........................................(1) The ADI (or Peaceman-Rachford) procedure is one of the methods frequently used to solve the associated difference equations. In our mathematical discussion we will write these difference equations approximating the above differential equation In matrix notation as Au = q where u is a vector representing pressure or potential at node points, q is a vector of production rates and points, q is a vector of production rates and A is the matrix of coefficients which depend on the permeabilities or mobilities. In using this procedure numerical instabilities often occur for small values of the parameter w,, particularly for incompressible parameter w,, particularly for incompressible flow with no-flow boundaries. These instabilities have been observed often and techniques to avoid them have been suggested. In this paper we will consider the mathematical and computational origin of these instabilities. This study will allow us to suggest improved methods of handling such problems and will give us a better understanding of the roll of the parameter w in the iteration process. process. MATRIX DESCRIPTION OF THE ADI PROCEDURE Consider three symmetric semi-definite n × n matrices A, H, and V such that ..........................................(2) Let D be a symmetric, positive definite normalizing matrix, q a given vector, Uo an arbitrary initial vector, and w-a number for any positive integer k. Then the ADI procedure can be defined by procedure can be defined by ..........................................(3) and ..........................................(4) Here the vectors will approximate a solution to the equation ..........................................(5)