Fluid flow and conduction in two-dimensional fractures with rough, self-affine surfaces: A comparative study

We study fluid flow and conduction in a two-dimensional model of a fracture with rough, self-affine internal surfaces. The model consists of two parallel flat plates on which two rough, self-affine surfaces, characterized by a roughness exponent H, are superimposed. The methods that we use for computing the effective flow and transport properties of the fracture include the lattice Boltzmann method for computing the flow properties, a random walk method for determining the effective conductivity of the (fluid-saturated) fracture, and the Reynolds approximation. We also develop an asymptotic expression for the effective conductivity. The aperture of the fracture, as well as the roughness of its rough surface, are systematically varied in order to assess their effect on the effective permeability and conductivity of the fracture, and also test the accuracy and consistency of the methods. For large mean apertures, and all values of the roughness exponent H, all the methods yield essentially the same results. However, as the mean aperture decreases, the differences between the predictions of the methods increase significantly. We find that the Reynolds approximation provides relatively accurate estimates of the (hydraulic or electrical) apertures only if the fracture is at least moderately wide and that, similar to real three-dimensional fractures, the electrical aperture is always smaller than the hydraulic fracture.