Upscaling two-phase flow in naturally fractured reservoirs

Simulation grid blocks of naturally fractured reservoirs contain thousands of fractures with variable flow properties, dimensions, and orientations. This complexity precludes direct incorporation into field-scale models. Macroscopic laws capturing their integral effects on multiphase flow are required. Numerical discrete fracture and matrix simulations show that ensemble relative permeability as a function of water saturation (R-ri[S-w]) water breakthrough, and cut depend on the fraction of the cross-sectional flux that occurs through the fractures. This fracture-matrix flux ratio (q(f)/q(m)) can be quantified by steady-state computation. Here we present a new semianalytical model that uses q(f)/q(m), and the fracture-related porosity (phi(f)) to predict k(ri)(S-w) capturing that, shortly after the first oil is recovered, the oil relative permeability (k(ro)) becomes less that that of water (k(rw)), and k(rw)/k(ro) approaches q(f)/q(m) as soon as the most conductive fractures become water saturated. To include a capillary-driven fracture-matrix transfer into our model, we introduce the nonconventional parameter A(f),(w)(S-w), the fraction of the fracture-matrix interface area in contact with the injected water for any grid-block average saturation. The A(f,w)(S-w) is used to scale the capillary transfer modeled with conventional transfer functions and expressed in terms of a rate- and capillary-pressure-dependent k(ro). All predicted parameters can be entered into conventional reservoir simulators. We explain how this is accomplished in both, single- and dual-continua formulations. The predicted grid-block-scale fractional flow (f(i)[S-w]) is convex with a near-infinite slope at the initial saturation. The upscaled flow equation therefore does not contain an S-w shock but a long leading edge, capturing the progressively widening saturation fronts observed in numerical experiments published previously.