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.
Highlights
In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity
We consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity described by a system of coupled partial differential equations (PDEs)
Modelling substrate transporting the effective diffusivity D is the essential input since it contains all the information that is specific for the considered porous medium
Summary
A porous medium is characterised by a rigid porous matrix. Recently, the integration of an evolving porous matrix caused by diverse heterogeneous reactions, e.g., crystal precipitation or biofilm growth, has attracted increased interest. In [12], an effective, non-linear diffusion equation with degenerating coefficients coupled to an ODE for the porosity was derived by two-scale asymptotic expansion in a level set framework. This modelled system of PDEs is analysed in the following. Modelling substrate transporting the effective diffusivity D is the essential input since it contains all the information that is specific for the considered porous medium This effective tensor is very difficult to characterise in (natural) porous media – even if they are assumed to be represented by scalars. In contrast to [12], the model (1.1) is solved even for substantially degenerating hydrodynamic parameters
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