American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.
THIS PAPER IS SUBJECT TO CORRECTION
This study presents a development of an Alternating Direction Galerkin (ADG) method following some theoretical work of Douglas and Dupont. The technique is applied to the construction of a two-phase coning simulator in cylindrical coordinates. Solutions are approximated in smooth bicubic Hermite subspaces. The non-linear coefficients in the differential equations are interpolated into a Hermite subspace with a mesh three times finer than the solution mesh. This interpolation scheme provides an order of accuracy consistent with and as high as that of the method of solution. Also it makes a substantial reduction in computer time. A novel method has been devised to simplify the notation and programming effort compared to work previous done in this area. In addition, a permutation matrix is employed to bring the matrix problem for each sweep into a simpler consistent form such that a block solution technique can be readily applied.
Finally, some computational results for hypothetical coning problems are presented and discussed. These are compared to those problems are presented and discussed. These are compared to those obtained from a non-alternating Galerkin formulation and comparable finite difference model.
For the past two decades reservoir engineers have been confronted with the problem of describing and predicting reservoir behavior. The lack of analytical solutions to the partial differential equations generated by such problems, along with recent advances in high speed electronic computers, has led to numerical methods using finite difference techniques. Despite the relative success of these methods, it was soon discovered that not all problems are adequately amenable to such treatment. Certain reservoir phenomena involving sudden changes in the dependent variables are poorly simulated by finite difference techniques. For example, to describe the zone of rapid change occurring in Buckley-Leverett type displacement, one must resort to excessive grid refinement. Computation of pressure and saturation distributions around wellbores must be approached in the same manner. The main drawback is that only discrete solutions in time and space are obtained by finite difference methods. Furthermore, these techniques are frequently limited by stability and convergence considerations.
Because of these problems attention has shifted to numerical methods which do not require spatial discretization of derivatives. Galerkin's method is one of these. Combined with piecewise polynomial approximations, Galerkin's process yields results polynomial approximations, Galerkin's process yields results superior, to those obtained by the finite difference technique.
