The AMR technique for simulating dissolution processes with a diffuse interface model

Abstract

Dissolution of solids or porous media, for example underground cavities, poses many practical engineering problems, which may be better understood through numerical simulations. There are mainly two ways for such simulations: first, a direct treatment of the moving interface, for instance using an ALE technique ; second, using a diffuse interface model (referred to as DIM in the paper sequel) to smooth the interface with continuous quantities . Recently, Luo et al. [3] presented a Darcy-scale local non-equilibrium diffuse interface model (DIM) obtained with the help of a volume averaging theory [1]. Numerical computations over various dissolution problems showed that it is more convenient and efficient to use a DIM model rather than an ALE technique, as the DIM model provides global control equations such that the ’artificial interfaces’ can move freely among different meshes without the need for re-meshing. Since there are sharp fronts in this problem, the further advantage of the DIM model is that it enables us to introduce adaptive mesh refinement (AMR) technique to this problem in a relative straightforward manner. The AMR technique [2] improves the computational efficiency by using an adaptive mesh system instead of a fixed fine grid. The grid system is automatically generated according to pre-defined refinement criteria, with fine grids near the fronts and coarse grids where the quantities vary slowly. In this paper, the AMR technique is applied to the simulation of solid/liquid dissolution problems with a DIM model. Besides algorithmic difficulties, the major problem is the design of accurate interpolation schemes between the different grids. For instance, the pressure interpolation at the interface must take into account gravity effects. Pressure values are therefore predicted by integration over the grid, based on the used momentum equation (Darcy’s law for a porous medium problem). In the following sections, we present the control equations of the DIM model, the AMR algorithm, and numerical examples showing its efficiency and accuracy.