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 (k_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 (_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.