On Existence, Uniqueness and Two-Scale Convergence of a Model for Coupled Flows in Heterogeneous Media
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On Existence, Uniqueness and Two-Scale Convergence of a Model for Coupled Flows in Heterogeneous Media Michal Beneš1
Received: 7 February 2020 / Accepted: 4 December 2020 © The Author(s), under exclusive licence to Springer Nature B.V. part of Springer Nature 2020
Abstract This paper is concerned with the global existence, uniqueness and homogenization of degenerate partial differential equations with integral conditions arising from coupled transport processes and chemical reactions in three-dimensional highly heterogeneous porous media. Existence of global weak solutions of the microscale problem is proved by means of semidiscretization in time deriving a priori estimates for discrete approximations needed for proofs of existence and convergence theorems. It is further shown that the solution of the microscale problem is two-scale convergent to that of the upscaled problem as the scale parameter goes to zero. In particular, we focus our efforts on the contribution of the so-called first order correctors in periodic homogenization. Finally, under additional assumptions, we consider the problem of the uniqueness of the solution to the homogenized problem. Keywords Nonlinear degenerate parabolic system · Global existence and uniqueness of weak solutions · Qualitative properties · Mixed boundary conditions · Two-scale convergence · First order correctors · Homogenization · Asymptotic analysis · Coupled heat and mass transport
1 Introduction Coupled heat, moisture and pollutant transport through porous materials occurs in various real-world phenomena. Well-studied problems cover the analysis of underground contaminant transport through unsaturated soils [17], self-heating and self-desiccation of concrete at the early age and beyond [9], spalling of concrete walls at high temperatures [7, 18], among many other interesting applications. The mass conservation equation for the water becomes ∂t [ΦSw w ] + ∇ · (ΦSw w vw ) = fw .
B M. Beneš
[email protected]
1
Faculty of Civil Engineering, Department of Mathematics, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic
(1.1)
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M. Beneš
Here, Φ is the porosity, Sw represents the water saturation, w stands for the water density and fw is a production term. Further, vw is the water velocity, which is characterized by Darcy’s law ΦSw vw = −
ks Krw (Sw ) (∇pw − w g), μw
Sw = Sw (pw ),
(1.2)
where ks is the saturated hydraulic conductivity, Krw is the relative hydraulic conductivity, μw stands for the dynamic viscosity of the pore water, pw is the unknown water pressure and g is the gravitational force. The mass balance equation for the contaminant in the water can be given in the form [14, 15] ∂t [ΦSw mc ] + ∇ · Jp + ∇ · (mc ΦSw vw ) = fp . Here, mc stands for the unknown concentration of contaminant and Jp is the diffusion flux. We consider that the right-hand side fp is a function of the concentration of contaminant mc dissolved in the water and mass of the adsorbed contaminant per unit mass of porous media whic
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