IMPES formulations of reservoir simulation equations solve for the pressure implicitly using time lagged mobility functions and subsequently update the concentrations explicitly. The Courant stability criterion for the explicit step is well known; however a new source of instability has been identified in the pressure equation when the mobility terms contain a pressure dependence which is time lagged. This can arise from shear dependent viscosities in polymer flooding or from the capillary number dependence of relative permeabilities in surfactant flooding. A stability criterion is derived for one dimensional problems and demonstration calculations for surfactant flooding are used to illustrate its application. If the criterion is violated stability cannot be restored by reducing the timestep.

Two new formulations of the pressure equation are derived to eliminate the instability and are implemented in one dimension using a modified Newton scheme implicit in the capillary number. One of these formulations uses an improved numerical representation of the capillary number and mobility terms, and is preferred because it reduces numerical dispersion and the solution time in the new pressure solver.

The new methods are generalised to two and three dimensions. This involves the introduction of more terms into the pressure matrix which is now non-symmetric. However to leading order the new representation of capillary number leaves the bandwidth the same as the conventional IMPES method, thus limiting the cost penalty of the improved scheme.

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