A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium

Arbogast, Todd; Brunson, Dana

doi:10.1007/s10596-007-9043-0

Cited by 207 publications

(127 citation statements)

References 23 publications

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“…[2], [9], [10], [11], [13], [14], [18], [20], [21], [27], [28], [29], [33], [34], [35], [37], and the references therein). This fact has been motivated by the diverse applications of this coupled model (in petroleum engineering, hydrology, and environmental sciences, to name a few), and also by the increasing need of simpler, more accurate, and more efficient procedures to solve it.…”

Section: Introductionmentioning

confidence: 99%

Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Gatica¹,

Oyarzúa²,

Sayas

2011

Math. Comp.

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Abstract.In this paper we analyze fully-mixed finite element methods for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The fully-mixed concept employed here refers to the fact that we consider dual-mixed formulations in both media, which means that the main unknowns are given by the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium. In addition, the transmission conditions become essential, which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. We apply the Fredholm and Babuška-Brezzi theories to derive sufficient conditions for the unique solvability of the resulting continuous and discrete formulations. In particular, we show that the existence of uniformly bounded discrete liftings of the normal traces simplifies the derivation of the corresponding stability estimates. A feasible choice of subspaces is given by Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the Lagrange multipliers. Finally, several numerical results illustrating the good performance of the method with these discrete spaces, and confirming the theoretical rate of convergence, are provided.

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Section: Introductionmentioning

confidence: 99%

Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Gatica¹,

Oyarzúa²,

Sayas

2011

Math. Comp.

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“…Examples of direct numerical simulation of these equations in a non-vuggy context (i.e., in a medium with at most a few very large vugs) can be found in, e.g., [25,13,19]. In a vuggy context, see, e.g., [3].…”

Section: Introductionmentioning

confidence: 99%

Homogenization of a Darcy–Stokes system modeling vuggy porous media

2006

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Abstract. We derive a macroscopic model for single phase, incompressible, viscous fluid flow in a porous medium with small cavities called vugs. We model the vuggy medium on the microscopic scale using Stokes equations within the vugular inclusions, Darcy's law within the porous rock, and a Beavers-Joseph-Saffman boundary condition on the interface between the two regions. We assume periodicity of the medium, and obtain uniform energy estimates independent of the period. Through a two-scale homogenization limit as the period tends to zero, we obtain a macroscopic Darcy's law governing the medium on larger scales. We also develop some needed generalizations of the two-scale convergence theory needed for our bi-modal medium, including a two-scale convergence result on the Darcy-Stokes interface. The macroscopic Darcy permeability is computable from the solution of a cell problem. An analytic solution to this problem in a simple geometry suggests that: (1) flow along vug channels is primarily Poiseuille with a small perturbation related to the Beavers-Joseph slip, and (2) flow that alternates from vug to matrix behaves as if the vugs have infinite permeability.

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“…On the other hand, this FE space experiments a loss of accuracy in some meshes (see [2]). Finally, when dealing with a coupled StokesDarcy problem it is hard to find mixed FE methods that are stable for both the Stokes and the Darcy problems (see [1,23]). The FE spaces that satisfy these conditions are expensive and restricted to particular typologies of meshes that complicate their use in real applications.…”

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confidence: 99%

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

Badia

Codina

2010

Computer Methods in Applied Mechanics and Engineering

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Abstract. We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic the different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L 2 -norm.Key words. Darcy's problem, stabilized finite element methods, characteristic length scale, orthogonal subgrid scales AMS subject classifications. 65N30, 35Q301. Introduction. Darcy's problem governs the flow of an incompressible fluid through a porous medium. It is composed by the Darcy law that relates the fluid velocity (the flux) and the pressure gradient and the mass conservation equation. In flow in porous media, a proper functional setting for this problem is to consider the flux in H(div, Ω) and the pressure in L 2 (Ω). This yields a saddle-point problem that is well posed due to inf-sup conditions known to hold at the continuous level, and that allow one to obtain stability estimates for the pressure and the velocity divergence.The Galerkin approximation of this indefinite system is a difficult task, because the continuous inf-sup conditions are not naturally inherited by most finite element (FE) velocitypressure spaces. We can avoid these problems by invoking the Darcy law in the mass conservation equation, getting a pressure Poisson problem; this is an elliptic problem that can be easily approximated by the Galerkin technique and Lagrangian elements. The fluxes can be obtained as a postprocess by using a L 2 -projection. This approach is computationally appealing because pressure and velocity computations are decoupled and the implementation is easy. Unfortunately, this approach has two drawbacks: the loss of accuracy for the velocity and the very weak enforcement of the mass conservation equation. Improved post-processing techniques that reduce these problems can be found e.g. in [15,17]. This approach has been restricted to continuous (H 1 -conforming) pressure FE spaces. However, the continuous pressure admits discontinuities, e.g. in regions with jumps of the physical properties (conductivity), and this approach leads to poor accuracy in the vicinity of these regions.Th...

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A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium

Cited by 207 publications

References 23 publications

Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Homogenization of a Darcy–Stokes system modeling vuggy porous media

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

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