Modeling and Analysis of Flow Problems in Aquifer Systems Involving Nonlinear Source-terms: Application of Schauder’s Fixed Point Theorem

Abstract

This work deals with two mathematical aspects of subsurface flow problems within aquifer systems, namely Mathematical Modeling and Theoretical Analysis. Concerning the Mathematical Modeling, the classical challenges for this class of problems are a rigorous description of diverse interactions that may take place between different involved aquifers. Recall that the main challenge at this stage is a realistic description of the flows from one aquifer to another passing necessarily through an aquitard which is a porous layer with small permeability coefficient and small thickness (compared with the mean thickness of involved aquifers. In the same way as most of flow phenomena, the governing equations of subsurface flows are based upon conservation laws.) To address the mathematical modeling of water exchange between different aquifers separated by aquitards we expose a mathematical approach based upon the Taylor expansion. Introducing the concept of observers located inside the aquitard and the neighboring aquifers, the mass conservation law has been applied and has led to one mass balance equation for each aquifer. Thanks to this original approach we have recovered the well-known mass balance equations exposed in the literature for flow problems in aquifer systems. Due to the assumptions of small thickness and homogeneity of the absolute permeability of aquitards for our framework the water flow is supposed vertical in aquitards and so we deal with one-dimensional flows there. This is the reason why the Taylor expansion deployed there concerns only the vertical space variable. The flux continuity has been applied to get the coupling of flow equations in the two aquifers. Since the flows in aquifers are supposed horizontal it is clear that the interface aquitard/aquifer flux acts as an additional source-term for each aquifer (and not a boundary term). Concerning the Theoretical Analysis of the global system of elliptic equations (as the flow is supposed to be submitted to a steady state) the Schauder Fixed Point Theorem has been applied for facing the nonlinearity of the right-hand sides of the system. This is the way we have got the existence of a solution to the system, but not the uniqueness. Thanks to a monotonicy assumption on the right-hand side vector-function we get the uniqueness of the solution. Finally the stability of that solution has been established under appropriate conditions.