Existence and uniqueness of solutions to a flow and transport problem with degenerating coefficients

Abstract

Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, related analytical investigations remain challenging due to potentially vanishing coefficients in the respective systems of partial differential equations. In this research, we apply an appropriate scaling of the unknowns and work with porosity-weighted function spaces. This enables us to prove existence, uniqueness and non-negativity of weak solutions to a combined flow and transport problem with vanishing, but prescribed porosity field, permeability and diffusion.

Local existence of strong solutions to micro–macro models for reactive transport in evolving porous media

“…This research is based on a micro-macro model for reactive flow and transport in evolving porous media introduced by [42] where its derivation from a detailed pore-scale model by formal homogenisation arguments was performed in two spatial dimensions. A generalisation to three dimensions was derived in [34] by modification of the original deduction. More precisely, the model under consideration consists of a set of coupled PDEs describing solute transport, fluid flow and geometry evolution.…”

“…As the term ∂ t (φc) in (2.1) is difficult to handle analytically, it is shifted to the right-hand side, cf. [34]. Therefore, we write is maintained by an appropriate choice of initial conditions.…”

“…Thereby, we extend similar results known in the literature for simple and restrictive geometries like cubes or spheres, cf. [34]. Moreover, we discuss sufficient conditions for the long-term existence of solutions in the presented framework.…”

“…However, the analysis performed requires the smoothness of the effective diffusion tensor as an additional assumption (partial coupling) or neglects its evolution with time completely (full coupling). Likewise, in [34], fully coupled systems with diffusion-driven transport are investigated. Furthermore, [33] considered partially coupled systems with advection including degenerating hydrodynamic parameters.…”

Local existence of strong solutions to micro-macro models for reactive transport in evolving porous media

“…Likewise, in [30], fully coupled systems with diffusion-driven transport are investigated. Furthermore, [29] considered partially coupled systems with advection including degenerating hydrodynamic parameters. Yet, a key assumption is the a-priori knowledge of the relation between effective parameters and the porosity which plays the role of an order parameter.…”