Direct Methods in Reservoir Simulation
Authors: Price H.S. , Coats K.H. 
Published: December 2007

Abstract
During the past decade, efforts in reservoir modeling have focused on the three areas of capability efficiency, and reliability. Capability means the ability to handle larger and more complex problems where complexity includes physical problems where complexity includes physical phenomena, such as gas percolation and variable phenomena, such as gas percolation and variable PVT properties, and severe heterogeneity due to PVT properties, and severe heterogeneity due to property variation or geometry, or both. Efficiency property variation or geometry, or both. Efficiency is increased by improving model formulations anti solution techniques to increase tolerable time-step size and reduce computer time per time step. Reliability refers to ease of use and minimum burden in selecting or experimenting with time-step size, solution technique options, iteration parameters, and closure tolerances. parameters, and closure tolerances. The single facet of a reservoir simulator that has the greatest combined influence in all three categories is the technique used to solve the large systems of equations arising from the numerical approximation of the nonlinear fluid flow equations. Available techniques include both direct solution and iterative methods such as ADIP, SOR, and SIP. Iterative methods are currently used almost to the exclusion of direct solution because of the significantly higher computer storage and time requirements of the latter. This paper describes some new ordering schemes for Gaussian elimination that reduce computing time and storage requirements by factors as large as 6 and 3, respectively, relative to more standard orderings. Computational work estimates are given for these methods, for the standard Gaussian ordering, and for several iterative methods. These work estimates are checked by comparisons of actual run times using different solution techniques. Numerical examples are given to illustrate the increased efficiency and reliability that can be achieved in many cases through use of the new direct solution methods.

Introduction
It is well known that the way we number or order the unknowns of a sparse system of linear algebraic equations can drastically affect the amount of computation and storage for a direct solution. However, until recently the best ordering scheme that appeared in the literature numbered the points of a three-dimensional grid first along the shortest direction - i.e., the dimension with the fewest number of grid points - then in the next shortest direction, and finally in the longest direction. This ordering, which we shall call the standard ordering for Gaussian elimination is still widely used even though it is substantially slower than many other orderings. Ogbuobiri et al. present a survey of the literature related to ordering schemes that exploit matrix sparsity. These schemes are grouped into the two classes of matrix-banding schemes and optimal or pseudo-optimal schemes. The latter schemes pseudo-optimal schemes. The latter schemes purport to yield generally greater efficiency. purport to yield generally greater efficiency. In a recent paper, Georges has shown that for five-point difference approximations on square n x n two-dimensional grids, the total work for certain orderings of the grid points is less than C1n3 and the storage is less than C2n2 log n, compared with n4 and n3, respectively, for the standard ordering. Moreover, George has shown that no ordering scheme can require less work than the order of n3. For the special case of n - 21 he shows that work W is less than 10n3 and the storage S is less than 8ln2 for symmetric matrices. For nonsymmetric matrices these results become W less than 20n3 and S less than 16ln2, respectively. In this paper we describe some specific orderings in the matrix-banding class. Analyses of work and storage requirements are given for these orderings as applied to the diffusivity-type pressure equation that arises in reservoir simulation problems. These work and storage requirements are compared with those of the standard Gaussian ordering and of some iterative methods. These comparisons are performed for problems ranging from simple performed for problems ranging from simple homogeneous squares to practical reservoir problems of typical heterogeneity and irregular problems of typical heterogeneity and irregular geometry. The work requirements of the orderings presented here are also compared experimentally with those of one of the leading pseudo-optimal schemes.
SPEJ
P. 295