A New Approach To The Application Of Direct Methods In Petroleum Reservoir Simulation
Authors: Al-Marhoun M.A. 
Published: December 2007

Abstract
A new approach to the application of direct method in petroleum reservoir simulation is introduced, This new approach is based on a modified form of the alternating diagonal ordering scheme that will rearrange each type of elements along different secondary diagonals in the coefficient matrix. In a typical 2-dimensional 3-phase petroleum reservoir simulation problem of 345 grid system, the new method is substantially faster than the band algorithm applied to the matrix resulting from the standard ordering.

Introduction
In petroleum reservoir simulation, a linear system of equations is obtained. The coefficient matrix of such a linear system takes different forms depending on the ordering of the grid system. The resulting system of equations can be solved either by direct methods or by iterative methods. Direct solutions are more reliable, but they normally require larger storage space and more computer time. Therefore, a development of a better direct method approach is needed.
The ordering technique can be improved by the use of processes which are geared specifically to the direct solution. The method is the generate-and-solve algorithm used by Woo, et al. The drawback of this technique is the fact it requires large computer storage and overhead work to store and generate the nonlooping code.
The yet known systematic optimal ordering scheme in petroleum reservoir simulation is the alternating diagonal scheme developed by Price and Coats. However, there is a major drawback to this ordering scheme. In the alternating diagonal scheme, the elements in the upper right and lower left corners of the coefficient matrix do not form perfect secondary diagonals. Thus, the solution of the resulting matrix requires band algorithm which treats the zeros within the band as nonzeros. The treatment of the zeros as nonzeros makes the band matrix technique require larger storage space and more computational labor.
In this paper a modification to the alternating diagonal ordering is proposed. This modification will rearrange each type of elements along different secondary diagonals in the coefficient matrix. By this rearrangement no new element will be introduced in the process of factorization except in the lower right process of factorization except in the lower right corner of the composite matrix. Also the location of the original and new elements are known beforehand. This enables us to develop a general procedure to solve the system of equations in reservoir simulation. This general procedure eliminates the overhead calculations and extra storage needed for the generate-and-solve algorithm. Also this general procedure would eliminate the extra calculations required procedure would eliminate the extra calculations required on the zero elements in the band matrix algorithm.

A NEW DIRECT SOLUTION APPROACH
The new direct solution approach will be outlined to solve the following system of equations
(1)
The new approach begins with the definition of a new ordering scheme. Then the LU factorization will be applied to this new ordering. Finally the implementation of the new ordering scheme in typical reservoir simulators will be outlined.

RESTRICTED ALTERNATING DIAGONAL ORDERING
Here a modification to the alternating diagonal ordering that produces perfect secondary diagonals in the original A matrix is proposed. This ordering will be called restricted alternating diagonal (RAD). Figure 1 depicts the RAD ordering. Figure 2 shows the resulting coefficient matrix corresponding to RAD ordering scheme.
The numbering of this scheme must go along the shortest diagonal to reduce the band width of the new nonzero elements that will be generated later in the process of factorization. process of factorization.