Mathematical modeling of erosion and deposition in porous media

Erosion and deposition are represented as the evolution of solid bodies due to the forces exerted by the fluid or air on the contact surfaces, which both often lead to reconfiguration and change of the topology and structure of the porous media. These processes are notably very complicated and challenging to study. In this work, we formulate simplified and idealized mathematical models to examine the internal evolution of flow networks in the setting of cylindrical channels, undergoing a unidirectional flow, by using asymptotic and numerical techniques. Starting from the Stokes equations combined with the advectiondiffusion equation for solid transport, we propose a model to construct a complete analysis of both the erosion and deposition. The considered approach is of the form of threshold laws: the fluid-solid interface erosion and deposition occur when the total shear stress is, respectively, greater or lower than some specified critical values, depending on the solid material. As a consequence of the erosion and deposition, the radii of the channels in the structure expand and shrink, respectively, due to several key parameters, which we find and investigate in this paper. We also perform a parametric study to quantify the correlation between these threshold values and the particle concentration in the flow. A comprehensive parametric study of the constructed model reveals that the final configuration of the structure can be predicted from the system parameters.