A multiscale quasilinear system for colloids deposition in porous media: Weak solvability and numerical simulation of a near-clogging scenario

Abstract

We study the weak solvability of a quasilinear reaction-diffusion system nonlinearly coupled with an linear elliptic system posed in a domain with distributed microscopic balls in 2D. The size of these balls are governed by an ODE with direct feedback on the overall problem. The system describes the diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles of various sizes inside a porous media made of prescribed arrangement of balls. The mathematical analysis of the problem relies on a suitable application of Schauder's fixed point theorem which also provides a convergent algorithm for an iteration method to compute finite difference approximations of smooth solutions to our multiscale model. Numerical simulations illustrate the behavior of the local concentration of the colloidal populations close to clogging situations.

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

“…As the mapping is linear with respect to the first variable, we again obtain C m -regularity for F . More precisely, the following theorem holds extending the smooth dependence results for simple parametric families of shapes as derived in [5,30]. F (u, 0) = (0, 0), i.e.…”

“…In this case, a smooth diffeomorphism on Y can be easily constructed by radially compressing/expanding annuli within a compact subset of Y . Given two radii 0 < r 1 ≤ r 2 < 1 2 , a deformation mapping a circle of radius r 2 to a circle of radius r 1 is defined by [5]. As such, the domain remains unchanged for |y| > 1 2 and is uniformly contracted for |y| ≤ r 2 with a smooth convex-combination layer in between.…”

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

“…Yet, in the case of pure Dirichlet boundaries, it is straightforward to show that the above estimate indeed can be refined to hold in L ∞ ( ) by standard elliptic theory [10]. We capture this observation in the following corollary.…”

“…the characteristics admit a strictly monotone distance to the set of prescribed initial data Ȳ × {0}. As such, equation (2.7) admits a unique local-in-time C 2 -solution by standard theory [10]. Furthermore, the parameterised trajectories associated with x induce a smooth path of diffeomorphisms and the artificial parameter s coincides with the actual physical time t. More precisely, we have the following statement: Proof.…”

“…Due to the finite propagation speed of information in (2.7), cf. [10], we refrain from posing explicit boundary conditions on (0, T) × × ∂Y here, see also Section 3.…”