A continuum mathematical model for erosion and deposition in a porous medium*

In this work, we investigate the dynamic processes of erosion and deposition in a porous medium that occur when the solid internal morphology of the porous medium interacts with fluids at its contact interface. These phenomena are encountered both in natural settings, such as soil erosion, and in various industrial applications, like water-filtration devices. The focus of our research is to develop a comprehensive two-dimensional continuum model that accurately describes how erosion and deposition influence the internal morphology of the porous medium under a fluid flow. To achieve this goal, we utilize first-principle equations, including the Darcy and continuity equations, to model the fluid flow. The Navier-Cauchy equations are adapted to describe the deformation of the elastic porous medium due to the flow shear stress. Further, we incorporate the advection-diffusion-reaction equation to study the mass transport of particles within the porous medium. By integrating an erosion and deposition evolution model, we effectively monitor how particle concentration of the fluid and porosity of the porous medium evolve together. To simplify our model, we employ asymptotic analysis, based on the porous medium small aspect ratio, to derive a reduced model. As a result of the erosion and deposition model, the porous medium expands and shrinks due to erosion and deposition, respectively.