The total velocity sequential semi-implicit method has received significant use since it was originally developed in about 1970 by Spillette, Hillestad, and Stone1. Its advantage is a reduction in the per-timestep computational requirement. It comprises two major steps: a solution of the IMPES pressure equation, then a solution of coupled saturation equations. Since the pressure solution is for a single unknown at each gridblock, it requires less work than solving fully implicit equations. Since the set of saturation equations does not have the elliptic nature of the pressure equation or of the fully implicit coupled set of equations, the saturation solution converges rapidly. Overall, the per-timestep computational work required is typically a half to a fifth that required for fully implicit computations.

The sequential semi-implicit method is far more stable than IMPES, and it can be effective in a wide variety of reservoir problems. However, there are problems in which it does not work as well as the fully implicit method, its per-timestep economy being more than offset by a requirement to perform more timesteps.

This paper describes an attempt to get the best of both worlds by using the total velocity equations in a preconditioner for the implicit matrix equation. The resulting method, which is similar to CPR2 , has performed effectively in a variety of problems. However, its performance has not yet been compared to that of competing methods.

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