Comparative Evaluation of the Strongly Implicit Procedure (SIP) For Gas Reservoir Simulation
Authors: Steen H. Dale , Ali S.M. Farouq 
Published: March 2010

Abstract
The present work was principally concerned with an evaluation of the new SIP algorithm for solving the matrix problem arising from the application of the various implicit difference techniques to the nonlinear partial differential equation for the flow of a real gas in two dimensions, expressed ill two different forms, utilizing the original and the transformed variables, respective. A number of con,- emotional difference methods were also tested for comparison purposes.
The equations resulting from the application of the completely implicit scheme were solved by both SIP and SOR (successive over-relaxation), in order to evaluate the speed of convergence using the two methods. In addition, a series of numerical experiments was carried out using the SIP algorithm in order to study the effect of the number of iteration parameters on the number of iterations required for convergence, the rate of convergence and the material balance error induced.
It was found that the explicit schemes tested suffered from time step size limitations and increasing material balance errors with time. The implicit schemes, in particular the modified implicit scheme, is therefore recommended. Computational results show that the SIP algorithm is three times as fast as SOR. The optimized SIP algorithm, although involving more computational work per iteration requires fewer iterations to converge than SOR, for a given accuracy. The number of SIP iteration parameters should be optimized where feasible.

INTRODUCTION
NUMEROUS AUTHORS have discussed gas reservoir simulation schemes, involving both explicit and implicit differencing methods. The present study gives a comparison of some of the schemes, and was principally concerned with an evaluation of the Strongly Implicit Procedure (SIP) recently proposed by Weinstein, Stone and Kwan(1) for solving the equations resulting from the conventional as well as from the completely implicit scheme due to Blair and Weinaug(2).

Mathematical Model
The equation describing the flow of a gas in a two dimensional porous medium can be derived by a consideration of the continuity and Darcy's equations, and is as follows:
(Equation Available In Full Paper)
where -q represents the production term; the remaining symbols are explained in the Nomenclature.
Equation (1) can be rendered quasilinear by use of the gas potential, P*, proposed by AI-Hussainy, Ramey and Crawford(3) and by Russell, Goodrich, Perry and Bruskotter(4):
(Equation Available In Full Paper)
By use of Eq. (2), Eq. (1) can be transformed to the following form (later referred to as Model 1) :
(Equation Available In Full Paper)
and may be expressed as a function of P*. In the same manner, Eq. (4) can be used to transform Eq. (1) as follows (later referred to as Model 2):
(Equation Available In Full Paper)
Eqs. (3) and (5) were solved in the present study by use of a number of explicit and implicit differencing schemes. The hypothetical reservoir employed for numerical computations was a 320-acre square (side = 3,734 ft) with a formation thickness of 20 ft, porosity of 25 per cent and permeability of 100 md (in the homogeneous case;