Flow in unsaturated fractured porous media: Hydraulic conductivity of rough surfaces

The general trend in models for flow in unsaturated fractured porous media is to regard desaturated fractures as nonparticipating elements that impede flow. Mounting experimental and theoretical evidence shows that fractures retain and conduct liquid in the form of film and partially filled corner flow to a relatively low degree of saturation. A simple geometrical model for rough fracture surfaces is developed offering a tractable geometry for calculations of surface liquid storage due to adsorbed films and capillary menisci. Assuming that under slow laminar flow the equilibrium liquid configurations on the fracture surface are not modified significantly, the average hydraulic conductivities for film and corner flows were derived and used as building blocks for a representative fracture roughness element and an assemblage of statistically distributed surface roughness elements. Calculations for a single representative element yielded excellent agreement with surface storage and unsaturated hydraulic conductivity measurements of Tokunaga and Wan [1997]. A statistical representation of surface roughness using a gamma distribution of pit depths resulted in closed-form expressions for unsaturated hydraulic conductivity averaged across the fracture length (transverse to flow) or weighted by the liquid cross section occupying the fracture surface. An important attribute of the surface roughness model is the direct link between fracture surface and matrix processes unified by the matric potential. The proposed model represents a first step toward development of a comprehensive approach for liquid retention and hydraulic conductivity of unsaturated fractured porous media based on details of liquid configuration for different matric potentials.