by

### Hans Muhlhaus
^{1}
, Lutz Gross
^{1}
and Alex Scheuermann
^{2}

^{1}The University of Queensland, School of Earth Sciences, ESSCC

^{2}The University of Queensland, School of Civil Engineering

# ABSTRACT:

Porous media are ubiquitous in Geology and geomechanics. On a plate thickness scale the mathematical theory of porous media flow is used to describe the localised intrusion of lava into the upper lithosphere e.g. around spreading centres and subduction zones. In geological and geotechnical engineering the interaction of a flowing fluid with granular matter leading to a suspension consisting of fluidized particles is of great importance in geology but also in engineering disciplines such as chemical engineering and minerals processing. On the one hand, the forces of liquid particles or liquid matrix interactions can be used very efficiently for example in industrial processes – in particular in minerals processing – involving size and weight segregation, mixing and transport. On the other hand, the interaction of soil and water can be the trigger of destruction, destabilizing the integrity of engineering structures leading to the collapse and failure of slopes, dams and foundations. Likewise the propagation and penetration of the localised lava channels to the surface is a necessary accompaniment of volcanic eruptions.

I will begin with an introduction into the theory of porous media flow (notations, balance equations Darcy’s law, propagation modelling: Stefan condition), examples.

In the main part of my talk I will concentrate on a representative example of porous media modelling, namely *sand erosion*. In general, sand erosion involves a combination of hydraulic and solid deformation –soil plasticity type-mechanisms. In our contribution we shall show, that under certain conditions, sand erosion can be formulated as an internal boundary value problem, a so called Stefan problem. We assume that the number of eroded particles-determining the local porosity change caused by erosion-is determined by the magnitude of the local pore fluid velocity. We focus on purely hydraulic erosion mechanisms, neglect solid deformations, and assume that the concentration c of the eroded particles per volume pore space, which means the particles in suspension is very low i.e. c<<1. As a consequence, for the situation of a fluidized bed, an initially homogeneous porosity distribution, upon change of pressure gradient, is modified from a constant value in front- to a different value behind a propagating discontinuity line (2D) or surface (3D).

In the following section we give an outline of the governing conservation laws, Darcy’s law and the erosion model (Scheuermann et al., 1999, Vardoulakis, 2004). Subsequently a simplified model for the erosion process leading to an internal boundary value problem with discontinuous porosity alteration (Muhlhaus and Gross, 2013) is presented. In section 3 we derive a closed form expression for the propagation speed of a 1D porosity discontinuity. We then linearise the governing equations and conduct a linear instability analysis. Analogies to the chemical infiltration model by Orteleva et al. (1986) are pointed out. In the remainder of the paper we employ a finite element model (Gross et al. 2006 and 2007) to investigate nonlinear regime of the governing equations. In our numerical model the discontinuity surface is modeled by means of the level set method (Tornberg and Engquist 2000).