Analytic methods to calculate an effective permeability tensor and effective relative permeabilities for cross-bedded flow units

Most naturally-occurring permeable media are heterogeneous on too small of a scale to include all the detailed heterogeneity into a numerical simulation. Instead, lumping the effects of those heterogeneities in a form that can be easily inserted into simulators is an alternative. Many of the effects of those heterogeneities can be quantified analytically by calculating an effective permeability tensor, with non-zero off-diagonal terms, when the heterogeneity is non-uniform. If there exist some prototype regularities, in addition to the effective permeability tensor, effective relative permeabilities can be generated to account for an uneven displacement front in the direction normal to the main flow in viscously dominated flows.; For non-uniform heterogeneities, an analytic method to calculate effective cell permeabilities as a tensor based on geometry, size of the numerical cell, tensorial local permeabilities and geology within the cell is proposed. The method is based on flow through parallel and serial cross-beds which is subsequently rotated to arrive at tensorial permeabilities having non-zero off-diagonal terms.; The procedure is applied to a simulation of flow through an outcrop of the eolian Page Sandstone. The results of the fluid flow simulations show that the relative positions of the main geologic features and the ratio between the grainflow and windripple permeabilities are more important than bounding surfaces, cross-bedding and dispersion in determining flow behavior.; For uniform heterogeneities, an analytical method to generate effective relative permeabilities which account for an uneven displacement front is proposed. The procedure considers only viscously-dominated flows and consists of discretizing the flow unit into subunits and homogenizing each subunit by calculating an effective permeability tensor which resolves cross-bedding and cross-bed orientation. Effective relative permeabilities are then generated analytically to account for differences in sweep between the subunits.; The method is applied to one-dimensional simulations of fluid flow in the C2 and B units of the Page Sandstone with less detail (36 elements, instead of 11520 elements of the detailed simulations). The resulting recovery predictions for different mobility ratios are compared with the ones from the detailed simulations. The comparisons of the recovery predictions indicate that the calculated effective relative permeabilities can capture the effect of heterogeneity on the sweep efficiency.; Both methods have been validated using a finite element numerical simulator which models the permeability discontinuities explicitly. Comparison of analytical and numerical effective permeability and effective relative permeabilities indicate that the analytically calculated effective permeabilities and generated effective relative permeabilities are valid, easy to implement, and are practical alternatives to account for detailed heterogeneities in numerical simulations.