Efficient 2-phase upscaling in 2D

摘要

In heterogeneous reservoir containing 2 liquid phases, e.g., oilrnand water, the flow rate in combination with small to mediumrnscale heterogeneities in both absolute permeability andrncapillarity defines dynamic capillary entrapment of oil andrnwater. Since those effects involving both viscous and capillaryrnforces happen on the lengths of cm to meters scale a simulatorrncan only capture them if the grid is sufficiently refined. Withrntypical size of the reservoir model grid blocks of the order ofrn100m in horizontal directions, and of the order of meters in thernvertical direction, the small-scale heterogeneities have to bernaccounted for through the so-called effective absolute andrnrelative permeabilities.rnGenerally, the effective relative permeabilities are ratedependent[rn17],[13],[8],[3]. We have developed a software packagernwhich computes the effective absolute and also relativernpermeabilities in 2D in 2 "easy" cases, at capillary limitrn(which is also called capillary-gravity equilibrium), I.e., at lowrnrate, and at viscous limit, I.e. high rate limit. Both ensuingrneffective properties, the absolute and the relativernpermeabilities, are generally full tensors. The computationalrnprocedure includes numerical finite difference solution of thernelliptic equation with periodic or classic boundary conditions.rnWe present solutions of the problem for a number ofrnsimple heterogeneities and compare it with solutions obtainedrnby other numerical and analytical procedures including the socalledrnself-consistent approach. The comparison revealsrnquantitative estimates of the accuracy and the computationalrnefficiency of the developed software. In addition, a number ofrnrealistic heterogeneities are considered when only numericalrnupscaling is feasible.