Degenerate equations in a diffusion–precipitation model for clogging porous media

Abstract

In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

An adaptive multi-scale iterative scheme for a phase-field model for precipitation and dissolution in porous media

“…We assume that the mineral never dissolves entirely and that the void space is always connected; thus the porosity is never vanishing. We refer to [39,40] for the analysis of models, including vanishing porosity and to [41] for a comparison of different approaches used in the context near clogging.…”

Uniqueness of Degenerating Solutions to a Diffusion-Precipitation Model for Clogging Porous Media

“…The present study considers a model of [7,10] that describes the diffusive transport of a reactive substance in a saturated porous medium, including variable porosity described by a system of coupled (partial) differential equations. In [10], a two-scale asymptotic expansion in a level set framework was used to derive an effective, nonlinear diffusion equation coupled to an ordinary differential equation (ODE) for porosity change.…”

“…This modeled system of PDEs was also analyzed in [10], though clogging effects were excluded. Recently, the analysis of degenerating equations due to vanishing porosity was included in [7].…”

“…For readability, it is reasonable to assume that the diffusivity satisfies D(θ) = θ d for some d ≥ 1, i.e., α = 1, cf. [7].…”

“…In [7], the author has already analyzed the coupled degenerating PDE-ODE model (1.1). An adjusted Rothe method was used to verify the existence of weak solutions in weighted function spaces.…”