Coupling Fluid Flow with Porous Media Flow

Abstract

The transport of substances back and forth between surface water and groundwater is a very serious problem. We study herein the mathematical model of this setting consisting of the Stokes equations in the fluid region coupled with the Darcy equations in the porous medium, coupled across the interface by the Beavers-Joseph-Saffman conditions. We prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems. This is important because there are many "legacy" codes available which have been optimized for uncoupled porous media and fluid flow. Key words. coupled porous media and fluid flow, Stokes and Darcy equations, Beavers-JosephSaffman condition, weak solutions, finite element scheme, error estimates AMS subject classifications. 35Q35, 65N30, 65N15, 76D07, 76S05 PII. S00361429013927661. Introduction and the model. There are many serious problems currently facing the world in which the coupling between groundwater and surface water is important. These include questions such as predicting how pollution discharged into streams, lakes, and rivers makes its way into the water supply. This coupling is also important in technological applications involving filtration.The aim of our research is to begin the study of the following problem: an incompressible fluid in a region Ω 1 can flow both ways across an interface Γ I into a domain Ω 2 which is a porous medium saturated with the same fluid. The mathematical theory and numerical analysis of each subproblem is well developed, and reliable codes are available. Nevertheless, the mathematical theory of the coupled problem seems to be not completely understood. The model of this situation which is most accessible to large scale computations consists of the Navier-Stokes equations (or Stokes equations) in the fluid region coupled across an interface with the Darcy equations for the filtration velocity in the porous medium. This leads to mathematical difficulties arising from the coupled system of equations of different orders in different regions. See Jäger and Mikelić [16], Payne and Straughan [22] for the beginning of analytical studies of this problem. (For the Brinkman model of porous media flow this difficulty does not occur; see Jäger and Mikelić [17], Angot [1].) The second issue concerns the correct transmission conditions on the interface. The Beavers-Joseph-Saffman interface conditions [3, 25] are now well established. The third difficulty is technical: where the interface meets the other boundaries, there are incompatibilities between the imposed boundary conditions.

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

“…So far, several numerical methods have been developed to approximate the solution of the Stokes-Darcy coupled problem (see for instance [13,14,18,19,20,22,26,27,28,32,36,37,39,40,41,7]), most of them based on appropriate combinations of stable elements for both media. In this direction, the first theoretical results go back to [39] and [20].…”

“…So far, several numerical methods have been developed to approximate the solution of the Stokes-Darcy coupled problem (see for instance [13,14,18,19,20,22,26,27,28,32,36,37,39,40,41,7]), most of them based on appropriate combinations of stable elements for both media. In this direction, the first theoretical results go back to [39] and [20]. In [20] the authors introduce an iterative subdomain method employing the standard velocity-pressure formulation for the Stokes equation and the primal one in the Darcy domain, whereas in [39] the authors apply the velocitypressure formulation in the free fluid domain and the dual-mixed velocitypressure formulation in the porous medium, yielding the introduction of the trace of the porous medium pressure on the interface as an additional unknown.…”

“…In [20] the authors introduce an iterative subdomain method employing the standard velocity-pressure formulation for the Stokes equation and the primal one in the Darcy domain, whereas in [39] the authors apply the velocitypressure formulation in the free fluid domain and the dual-mixed velocitypressure formulation in the porous medium, yielding the introduction of the trace of the porous medium pressure on the interface as an additional unknown. Next, in [26] and [28] new mixed finite element discretizations of the variational formulation from [39] have been introduced and analyzed. The stability of a specific Galerkin method, employing the Bernardi-Raugel and Raviart-Thomas elements for the free fluid and the fluid in the porous medium, respectively, is the main result in [26].…”

“…They approximate the coupled nonlinear problem using conformal finite elements in both media and study the convergence properties of Newton-like iterative methods for solving this problem. We point out that, differently from [39] and [26], the approach adopted in [31] and [6] avoids the introduction of Lagrange multipliers to impose the continuity of the normal velocity of the fluid through the interface. Indeed, this condition is imposed weakly using the primal formulation of the Darcy problem for the sole pressure unknown.…”

Iterative Methods for Stokes/Darcy Coupling

“…, d − 1); α is a dimensionless coefficient depending essentially on ν and the hydraulic conductivity of the porous medium. For an extensive discussion about these coupling conditions we refer to Discacciati et al [2002], Jäger and Mikelić [1996], Layton et al [2003], Payne and Straughan [1998]. The mathematical analysis of the coupled problem has been addressed in previous works concerning both the continuous case and the finite element approximation (see Discacciati and Quarteroni [2003], Layton et al [2003]).…”