Hydraulic conductivity of partially saturated fractured porous media: Flow in a cross-section

Standard models for hydraulic functions of partially saturated fractured porous media (FPM) often rely on macroscopic continuum representation and embrace constitutive relationships originally developed for homogeneous porous media to describe hydraulic behavior of dual (or multi) continua FPM. Such approaches lead to inconsistencies due to neglect of underlying physical processes governing liquid retention and flow in the vastly different pore spaces. We propose a framework that considers equilibrium liquid configurations in dual continuum pore space as the basis for calculation of liquid saturation and introduction of hydrodynamic considerations. FPM cross-sectional pore space is represented by a bimodal size distribution reflecting two disparate populations of matrix pores and fracture apertures (with rough-walled surfaces). Three laminar flow regimes are considered, flow in: (1) completely liquid filled pore spaces; (2) partially filled pores or grooves bounded by liquid-vapor interfaces; and (3) surface film flow. Assuming that equilibrium liquid-vapor interfaces remain stable under slow laminar flows, sample-scale unsaturated hydraulic conductivity is derived from average velocity expressions for each flow regime weighted by the appropriate liquid-occupied cross-sectional areas (neglecting 3-D network effects). A parameter estimation scheme was developed and evaluated using two data sets. The results point to the critical need for definitive data sets for improved understanding of flow in partially saturated FPM. Hydraulic conductivity functions for non-equilibrium conditions between matrix and fracture domains are discussed. Approximations for inclusion of network effects are proposed based on direct measurement of saturated hydraulic conductivity supplemented by theoretical considerations applying critical path analysis.