Simultaneous Solution of Multiphase Reservoir Flow Equations
Authors: Weinstein H.G. , Stone H.L. , Kwan T.V. 
Published: December 2007

Abstract
A strongly implicit iterative procedure has been developed to solve systems of equations arising in multiphase, two-dimensional reservoir flow problems. The two-dimensional, two-phase and problems. The two-dimensional, two-phase and two-dimensional, three-phase algorithms have been evaluated by several test problems and compared with the corresponding alternating direction iterative routines. The strongly implicit procedure (SIP) has been found to have several advantages in the solution of reservoir problems. It is fast, and in problems with extreme anisotropy in the transmissibilities and/or highly irregular geometries it can obtain a solution where the alternating direction procedure many times cannot. For the problems tested, it bas been found that a reliable set of iteration parameters is easily calculated from the coefficient matrix. Finally, SIP appears to be relatively insensitive to the rounding errors inherent in machine computations.

Introduction
The efficient solution of multidimensional reservoir problems involving the flow of two- or three-fluid phases is essential in petroleum reservoir simulation. Because of nonlinearities and generally complex geometries and boundary conditions, analytic solutions of the differential equations are at present impossible. One must, instead, seek solutions of the finite difference approximations of the equations through iterative techniques. Many iterative methods have been developed. Most of these, including relaxation and successive overrelaxation techniques, require excessive computer effort because they converge rather slowly or fail to converge. The more implicit alternating direction iteration procedure (ADIP) converges faster than the relaxation and overrelaxation schemes and, in general, requires less computational work. More recently, a new iterative technique has been developed. This technique is called the strongly implicit procedure, or simply SIP. It was demonstrated by Scone that SIP achieved greater rates of convergence than ADIP on all problems tested except the simple model problem in which the coefficients in the difference equation were constant and isotropic. Furthermore, the advantage of SIP over ADIP appears to increase as the complexity of the problem increases. SIP was originally developed and tested for the solution of a single equation in two-space dimensions. Its improved convergence over ADIP in this case led naturally to the development of SIP for the simultaneous solution of two or three coupled equations in two dimensions, such as arise in the simultaneous-solution approach to multiphase two-dimensional flow problems. SIP has also been extended to the solution of multiphase reservoir flow problems in three-space dimensions. The development and testing of the latter procedure is discussed elsewhere. In this paper, the SIP algorithms for two-dimensional problems are presented. The algorithms have been evaluated by presented. The algorithms have been evaluated by several test problems and compared with the corresponding ADIP routines.

TWO DIMENSIONS: THE MULTIPHASE PROBLEM
For purposes of generality, the system to be studied is comprised of coupled, two-dimensional parabolic equations. Employing this system will parabolic equations. Employing this system will facilitate investigation of any problem related to the two-dimensional flow of several fluids in a porous medium. porous medium. SPEJ
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