Some Practical Considerations in the Consruction of a Semi-Implicit Simulator
Authors: Chappelear J.E. , Rogers W.L. 
Published: December 2007

Introduction
The semi-implicit reservoir simulator has become a very important part of the total simulation package necessary for the practicing reservoir engineer. Any multiphase simulation of a single well tom, problem (e.g., a study of water-oil coning) is very problem (e.g., a study of water-oil coning) is very expensive unless such a simulator is available. With a "standard" reservoir simulator, this difficulty arises because the relative permeabilities and capillary pressures, which depend upon saturations, lag one time step behind the pressure calculation. For this reason such a simulator is said to be implicit in pressure and explicit in saturation (abbreviated as IMPES). When "new" saturations are obtained, they are formed from "old" relative permeabilities and capillary pressures. The mathematical form of the equations is pressures. The mathematical form of the equations is such that an uncontrolled oscillation in the saturation values develops if the time step is too large. Only by taking smaller time steps can this oscillation be suppressed in an IMPES simulator, and very small time steps are then necessary for simulating coning behavior. The same problem can also appear in any production well model (in an IMPES simulator) that production well model (in an IMPES simulator) that distributes fluid production proportionally to phase mobilities. These saturation oscillations can be eliminated by making the well model "implicit in saturation." To overcome this instability, Blair and Weinaug developed a fully implicit simulator. All coefficients were updated iteratively until convergence occurred. It was necessary for them to stabilize their solution technique by the use of Newtonian iteration. Then it was found that a time-step limitation occurred because of nonlinearities, since the Newtonian iteration would not converge without a good initial estimate. Coats and MacDonald proposed an effective solution to this problem. They suggested estimating the relative permeabilities and capillary pressures by an extrapolation; e.g., pressures by an extrapolation; e.g.,(1)
A mathematical investigation showed that since Sn + 1, the saturation at be new time step, is found simultaneously with the pressures as part of the solution, the mathematical time-step limitation inherent in the IMPES technique as a result of using "old" relative permeabilities would not occur. They also suggested that the equations be linearized by dropping products of (Sn+1 - Sn) and (pn+1 - pn). The equations are then more nearly linear. Hence, the difficulties in convergence of the solution technique are greatly reduced (Newtonian iteration is not needed). Nolen and Berry showed that linearization of the accumulation terms was not necessarily the best strategy (in problems that have solution gas), because material-balance errors would result. They felt that linearization of the flux terms made little difference. Many questions still remain unanswered by these papers. Nonlinearities remain in the equations, papers. Nonlinearities remain in the equations, particularly when a phase is near its immobile particularly when a phase is near its immobile saturation. Because of the use of upstream weighting of the relative permeabilities, another type of nonlinearity (potential reversal) can occur. Furthermore, the question of a practical procedure for selecting the time step must be settled. Finally, there are nonlinearities in the well model, which can cause slow convergence, or failure to converge, especially when dealing with a well completed in several layers or with a well that changes constraints. The problems just mentioned are all more severe if large time steps are used. Reducing time-step size is expensive, and in many cases difficult to automate. In this paper we present our experience in treating or circumventing these problems. We have felt that an important principle to follow is to eliminate time-step limitations due to mathematical instabilities. Thus, one should be able to run steady-state problems with essentially unlimited time-step size. For transient problems, it is expected that time truncation errors would normally govern the time-step size. One final, practical goal was to avoid running problems that would be annoying and mysterious to the field reservoir engineer.
SPEJ
P. 216