Sand erosion as an internal boundary value problem

Abstract

The hydraulic erosion of a fine particle fraction (-fines-) from a granular layer is investigated analytically and computationally. The erosion is assumed to occur instantaneously leading to an internal boundary value problem. A low-porosity domain is separated from a high-porosity domain by a propagating, spatial discontinuity surface. The level set method is applied to computationally simulate the propagation of the discontinuity surface or interface over time. Examples for such interfaces are ubiquitous in geomechanics and include reaction fronts, free surfaces in Eulerian formulations, and infiltration and injection fronts. First, a linear instability analysis is conducted and the resulting dispersion relationship is discussed. The analysis indicates that the interface problem is ill-posed. A stabilization term proportional to the curvature of the interface is introduced into the Stefan condition restoring well posedness. Subsequently, a finite element-based level set technique is applied to computationally investigate the propagation and stability of the discontinuity surface. The mesh independence of the solution upon introduction of the regularizing curvature term is also demonstrated. Techniques are presented for the treatment of the hyperbolic differential system at the core of the interface dynamics. These techniques are based upon an algorithm developed in the finite difference context, but are modified to take advantage of the robustness and flexibility of the finite element method.