Analysis of Model Equations for Stress-Enhanced Diffusion in Coal Layers. Part I: Existence of a Weak Solution

This paper is motivated by the study of the sorption processes in the coal. They are modelled by a nonlinear degenerate pseudo-parabolic equation for stress enhanced diffusion of carbon dioxide in coal ∂tφ = ∂x { D (φ) ∂xφ + D (φ) φ B ∂x ( e∂tφ )} , where B, m are positive constants and the diffusion coefficient D (φ) has a small value when the CO2 volume fraction φ is 0 ≤ φ < φc, representative of coal in the glass state and orders of magnitude higher value for φ > φc, when coal is in the rubber like state. These type of equations arise in a number of cases when non-equilibrium thermodynamics or extended non-equilibrium thermodynamics is used to compute the flux. For this equation existence of the travelling wave type solutions was extensively studied. Nevertheless, the existence seems to be known only for sufficiently short time. We use the corresponding entropy functional in order to get existence, for any time interval, of an appropriate ∗The research of A.M. was partially supported by the GDR MOMAS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN) as a part of the project ”Changements d’échelle dans la modélisation du transport multiphasique et réactif en milieux poreux : application aux milieux fracturés et aux argiles. ” . It was initiated during the sabbatical visit of A.M. to the TU Eindhoven in Spring 2006, supported by the Visitors Grant B-61-602 of the Netherlands Organisation for Scientific Research (NWO). †The research of H.B. was supported by the Shell International Exploration and Production B.V., NWO-Novem and the Greenhouse-gas Removal Apprenticeship and Student Program (GRASP) funded by the EC, The idea of stress enhanced diffusion [30] [31] for coal, worked out in the doctoral thesis of Saikat Mazumder [24], was the inspiration to this work. 1 weak solution with square integrable first derivatives and satisfying uniform L∞-bounds. Due to the degeneracy, we obtain square integrability of the mixed second order derivative only in the region where the concentration φ is strictly positive. In obtaining the existence result it was crucial to have the regularized entropy as unknown for the approximate problem and not the original unknown (the concentration).