Simultaneous Direct And Semi-direct Solutions In Reservoir Simulation
Authors: Burgess K.A. , Ali S.M. Farouq 
Published: March 2010

Introduction
An efficient simultaneous semi-direct iterative method, which improves upon the capability of iterative methods, is described. The semi-direct method presents the advantages of including additional diagonals in the "triangular decomposition process, with a different internal approximation to Lelkeman"s approach. This investigation includes a study of the computational storage and work estimates, and the computer time per step for the improved semi-direct method, vis-a-vis three other schemes, all for simultaneous solution formulation. The paper gives the solution algorithm for the semi-direct method developed, and discusses the application of the method to the cyclic steam stimulation process. A comparison shows that the semi-direct method requires the least computer time while converging faster than the strongly implicit procedure (SIP). However it requires about a third more storage than SIP.

INTRODUCTION
In many oil reservoir simulation problems, finite-difference approximations to the system of partial differential equations generate a linear system of simultaneous equations that is generally solved by iterative(1–5) or direct(6–8) solution techniques. This paper presents an algorithm that is competitive with the strongly implicit, procedure (SIP) of Stone(4), and the alternate diagonal (D-4) method of Price and Coats(6), by augmenting the number of diagonals in the L/U decomposition process.
The choice of an iterative technique over a direct solution scheme encompasses questions on the number of iterations required for convergence, the computer time required per iteration and the selection of "best" acceleration parameters. Although iterative solution schemes have been known to fail to converge in the more complex compositional simulations, iterative and IMPES techniques are more generally used in conventional reservoir models. This is due to the ease of formulation and implementation, and the smaller requirements of storage locations in two and three dimensions.
On the other hand, in order to make the best use of direct solution schemes, reordering of the grid elements as preseuted in Ogbuobire et al(9)and Price and Coats(6) improve on computer storage and computer time requirements by capitalizing on matrix sparsity(7,8) and most recently by utilizing vector operations(10,11). For an asymmetrio coefficient matrix, which generally occurs in reservoir simulation problems; direct methods are efficient for small grid systems, but: computer time requirements become excessive if sparsity is not exploited.
In this paper, a modified semi-direct iterative algorithm is presented. This algorithm employs a methodology similar to that used by Letkeman(6), but with a different internal approximation of the diagonals. Comparison of the work estimates and the computer storage requirements are discussed for the improved algorithm, SIP, and the alternate diagonal, D-4, scheme. The application of these solution methods to the cyclic steam stimulation process is also discussed in a simultaneous form.

The General Reservoir Simulation Problem
Common finite-difference approximation techniques used to solve the cyclic steam stimulation process, given below, lead to a system of linear equations:

(Equation in Full Paper)
In the case of the cyclic steam stimulation process, the complete implicit procedure gives X, which consists of unknown vapour saturation, water saturation, water phase pressure and temperature.