A High-Stability Sequential Solution Approach to Reservoir Simulation
Authors: Spillette A.G. , Hillestad J.G. , Stone H.L. 
Published: December 2007

This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegas, Nev., Sept. 30-Oct. 3, 1973. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made.
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Abstract
Sequential solution of the reservoir flow equations has been made to yield the stability of calculations previously associated only with the simultaneous solution previously associated only with the simultaneous solution of the equations. Sequential solution is computationally more efficient than simultaneous solution, and it permits the use of simulation algorithms that are more dependable when applied to irregular-shaped, highly heterogeneous reservoirs. Such reservoirs are often troublesome to simulate. Coning, gas percolation, and other high-flow velocity problems have been made tractable through the use of semi-implicit saturation functions.

Introduction
Sequential solution of the flow equations in a reservoir simulator has been made to yield the calculational stability previously associated only with simultaneous solution of the equations. This new approach forms the basis of a general-purpose reservoir simulator, GPSIM.
Sequential solution is computationally more efficient than simultaneous solution, and it permits the use of algorithms that are more dependable when predicting the performance of irregular-shaped, highly heterogeneous performance of irregular-shaped, highly heterogeneous reservoirs. Such reservoirs, which are frequently encountered in practice, are often quite troublesome to simulate. Well coning problems and fieldwide, multiwell gas percolation problems have been made tractable through the optional use of semi-implicit saturation functions including capillary pressures and portions of the Buckley-Leverett fractional flow terms. The semi-implicit approach used in GPSIM is closely related to the semi-implicit relative permeability approach previously described in the literature.
Sequential solution conveniently separates the solution of the pressure equation, which is often predominantly elliptic in nature, from that of the saturation equations, which range between parabolic and hyperbolic in character. This permits approximation of the saturation equations by alternating-direction difference equations, which can be solved by direct elimination, thus avoiding an iterative solution. Optional use of the backwards-in-time difference formulation solved by strongly implicit iteration (SIP) or direct solutions methods is provided in GPSIM for use on especially difficult problems such as coning studies.