Effective Relative Permeabilities for Viscously-Dominated Flow Through Cross-Bedded Units

Abstract

ABSTRACT The size of the local heterogeneity of most naturally-occurring permeable media is too small to be included in a numerical simulation. However, we can characterize the medium with prototype regularities in the heterogeneity and express the effects of these prototypes through effective relative permeability functions. Cross-bedded flow units, found mainly in sandstone reservoirs, are good candidates for this procedure. This study describes an analytic method to generate effective relative permeabilities that account for the effects of heterogeneity on a miscible displacement in viscously dominated (gravity-free) flow. The method consists of discretizing a flow unit into subunits to which is assigned an effective permeability tensor that resolves cross bedding and cross-bed orientation. The effective permeability is, in general, a tensor that has nonzero off-diagonal elements. We then analytically generate effective (pseudo) relative permeabilities to account for vertical efficiency sweep caused by differences between the subunits. We validate the proposed procedure with results that model permeability laminations explicitly on a very fine scale with a finite element simulator. The comparison indicates that analytically generated effective relative permeabilities can accurately account for small heterogeneities in numerical simulations. Thus, effective relative permeabilities, an immiscible flow concept, are used to represent heterogeneities in a miscible displacement. The procedure is limited, however, to viscously-dominated flow.