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 (kri[Sw]), water breakthrough, and cut depend on the fraction of the cross-sectional flux that occurs through the fractures. This fracture-matrix flux ratio (qf/qm) can be quantified by steady-state computation.
Here we present a new semianalytical model that uses qf/qm and the fracture-related porosity (ϕf) to predict kri(Sw) capturing that, shortly after the first oil is recovered, the oil relative permeability (kro) becomes less that that of water (krw), and krw/kro approaches qf/qm 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 Af,w(Sw), the fraction of the fracture-matrix interface area in contact with the injected water for any grid-block average saturation. The Af,w(Sw) is used to scale the capillary transfer modeled with conventional transfer functions and expressed in terms of a rate- and capillary-pressure-dependent kro. 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 (fi[Sw]) is convex with a near-infinite slope at the initial saturation. The upscaled flow equation therefore does not contain an Sw shock but a long leading edge, capturing the progressively widening saturation fronts observed in numerical experiments published previously.