In heterogeneous reservoir containing 2 liquid phases, e.g., oil and water, the flow rate in combination with small to medium scale heterogeneities in both absolute permeability and capillarity defines dynamic capillary entrapment of oil and water. Since those effects involving both viscous and capillary forces happen on the lengths of cm to meters scale a simulator can only capture them if the grid is sufficiently refined. With typical size of the reservoir model grid blocks of the order of 100m in horizontal directions, and of the order of meters in the vertical direction, the small-scale heterogeneities have to be accounted for through the so-called effective absolute and relative permeabilities.

Generally, the effective relative permeabilities are rate-dependent[17],[13],[8],[3] . We have developed a software package which computes the effective absolute and also relative permeabilities in 2D in 2 "easy" cases, at capillary limit (which is also called capillary-gravity equilibrium), i.e., at low rate, and at viscous limit, i.e. high rate limit. Both ensuing effective properties, the absolute and the relative permeabilities, are generally full tensors. The computational procedure includes numerical finite difference solution of the elliptic equation with periodic or classic boundary conditions.

We present solutions of the problem for a number of simple heterogeneities and compare it with solutions obtained by other numerical and analytical procedures including the so-called self-consistent approach. The comparison reveals quantitative estimates of the accuracy and the computational efficiency of the developed software. In addition, a number of realistic heterogeneities are considered when only numerical upscaling is feasible.

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