Constitutive relations for unsaturated flow in a fracture network

Modeling water flow in unsaturated fractured rocks is of interest to many areas of active research, including geological disposal of nuclear waste and subsurface contaminant transport. A commonly used approach for modeling water flow in unsaturated fractured rocks is the continuum approach, in which the constitutive relation models, originally developed for porous media, have often been borrowed to represent constitutive relations for the fracture continuum. While these models have been successfully used for soils and other porous media, their usefulness and limitations have not been investigated for the fracture continuum. The objectives of this study are: (1) to present an evaluation of the commonly used van Genuchten (VG) and Brooks–Corey models by mainly comparing these models with constitutive relations simulated from two-dimensional fracture networks, and (2) to develop improved constitutive relation models based on the evaluation results. We find that the VG model can reasonably well match the simulated water retention curves (except for a range of high saturations), but both the VG and Brooks–Corey models generally underestimate relative permeability values. Simulation results also indicate that a combination of a modified Brooks–Corey relative permeability–saturation relation (obtained by introducing a new expression for tortuosity) and the VG capillary pressure–saturation relation can generally represent the simulated curves, with no additional parameters for this combination needed other than those used in the VG model.