Darcy Problem Research Articles

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015

Weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non‐matching meshes

SummaryIn this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational domain does not match with the element boundaries. The problems considered are the Poisson problem, the Stokes problem, and the Darcy problem, the latter both in the primal and in the dual formulation. The formulation proposed is of variational type. The key idea is to start with the variational form that defines the problem and treat the boundary condition as a constraint. The particular feature is that the Lagrange multiplier is not defined on the boundary where the essential condition needs to be prescribed but is taken as a certain trace of a field defined in the computational domain, either in all of it or just in a region surrounding the boundary. When approximated numerically, this may allow one to condense the DOFs of the new field and end up with a problem posed only in terms of the original unknowns. The nature of the field used to weakly impose boundary conditions depends on the problem being treated. For the Poisson problem, it is a flux; for the Stokes problem, a stress; for the Darcy problem in primal form, a velocity field; and for the Darcy problem in dual form, it is a potential. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical analysis for the mixed Navier–Stokes and Darcy Problem with the Beavers–Joseph interface condition

In this article, we consider the coupled Navier–Stokes and Darcy problem with the Beavers–Joseph interface condition. With suitable restrictions of physical parameters α and ν, we prove the existence and local uniqueness of a weak solution. Then we propose a coupled finite element scheme and a decoupled and linearized scheme based on two‐grid finite element. Under suitable further restrictions, their optimal error estimates are obtained. Finally numerical experiments indicate the validity of the theoretical results as well as the efficiency and effectiveness of the decoupled and linearized two‐grid algorithm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1009–1030, 2015

3D robust iterative coupling of Stokes, Darcy and solid mechanics for low permeability media undergoing finite strains

Stokes, Darcy and solid mechanics coupling is a matter of interest in many domains of engineering such as soil mechanics, bio-mechanics, and composites. The aim of this paper is to present a robust iterative method to deal with this coupling for low permeability media within the framework of industrial simulation, and especially for composite manufacturing processes. Stokes and Darcy problems are solved using a mixed velocity–pressure finite element using a mini-element formulation, and coupled together by the so-called Beavers–Joseph–Saffman conditions through their interface. This fluid formulation is then coupled to a non-linear solid mechanics formulation in finite deformations using Terzaghi׳s law at the pore level, and an explicit dependence of permeability with respect to porosity that is exactly computed from the solid mechanics kinematics. Then, those formulations are validated with test-cases and by the Method of the Manufactured Exact Solution (MMES) (Knupp and Salari, 2003[1]). Finally, a 3D curved transient example of application is presented.

A two-grid decoupling method for the mixed Stokes–Darcy model

In this paper, we consider the mixed Stokes–Darcy problem which describes a fluid flow coupled with a porous media. We present a modified two-grid method for decoupling this mixed model. Stability is proved and optimal error estimates are derived. The numerical results show that the modified two-grid method is effective and has the same accuracy as the coupling scheme when we choose h=H2.

Algebraic multigrid techniques for discontinuous Galerkin methods with varying polynomial order

We present a parallel algebraic multigrid (AMG) algorithm for the implicit solution of the Darcy problem discretized by the discontinuous Galerkin (DG) method that scales optimally for regular and irregular meshes. The main idea centers on recasting the preconditioning problem so that existing AMG solvers for nodal lower order finite elements can be leveraged. This is accomplished by a transformation operator which maps the solution from a Lagrange basis representation to a Legendre basis representation. While this mapping function must be user supplied, we demonstrate how easily it can be constructed for somepopular finite element representations includingquadrilateral/hexahedral and triangular/tetrahedral DG formulations. Furthermore, we show that the mapping does not depend on the Jacobian transformation between reference and physical space and so it can be constructed with very limited mesh information. Parallel performance studies demonstrate the versatility of this approach.

Strong coupling of finite element methods for the Stokes-Darcy problem

The aim of this paper is to propose a systematic way to obtain convergent nite element schemes for the Darcy-Stokes ow problem by combining well-known mixed nite elements that are separately convergent for Darcy and Stokes problems. In the approach in which the Darcy problem is set in its natural H(div) formulation and the Stokes problem is expressed in velocity-pressure form, the transmission condition ensuring global mass conservation becomes essential. As opposed to the strategy that handles weakly this transmission condition through a Lagrange multiplier, we impose here this restriction exactly in the space of global velocity eld. Our analysis of the Galerkin discretization of the resulting problem reveals that, if the mixed nite element space used in the Darcy domain admits an H(div)-stable discrete lifting of the normal trace, then it can be combined with any stable Stokes mixed nite element of the same order to deliver a stable global method with quasi-optimal convergence rate. Finally, we present a series of numerical tests conrming our theoretical convergence estimates.

On classical iterative subdomain methods for the Stokes–Darcy problem

Within classical iterative subdomain methods, the problems in the subdomains are solved alternately by only using data on the interface provided from the other subdomains. Methods of this type for the Stokes–Darcy problem that use Robin boundary conditions on the interface are reviewed. Their common underlying structure and their main differences are identified. In particular, it is clarified that there are different updating strategies for the interface conditions. For small values of fluid viscosity and hydraulic permeability, which are relevant in applications from geosciences, it is shown in numerical studies that only one of these updating strategies leads to an efficient numerical method, if it is used with appropriate parameters in the Robin conditions.

Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems

Two parallel, non-iterative, multi-physics, domain decomposition methods are proposed to solve a coupled time-dependent Stokes-Darcy system with the Beavers-Joseph-Saffman-Jones interface condition. For both methods, spatial discretization is effected using finite element methods. The backward Euler method and a three-step backward differentiation method are used for the temporal discretization. Results obtained at previous time steps are used to approximate the coupling information on the interface between the Darcy and Stokes subdomains at the current time step. Hence, at each time step, only a single Stokes and a single Darcy problem need be solved; as these are uncoupled, they can be solved in parallel. The unconditional stability and convergence of the first method is proved and also illustrated through numerical experiments. The improved temporal convergence and unconditional stability of the second method is also illustrated through numerical experiments.

Equal order approximations enriched with bubbles for coupled Stokes–Darcy problem

As a remedy to the instability of the Galerkin finite element formulation, symmetric stabilization techniques such as the continuous interior penalty, the subgrid and local projection methods were proposed and analyzed by Burman and Hansbo (2006) [10], Badia and Codina (2009) [11], Becker and Braack (2001) [12], and Nafa and Wathen (2009) [13]. In this work we consider a coupled Stokes–Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous finite elements in the two parts, such as Taylor-Hood or MINI elements for the Stokes domain, and mixed elements of Raviart–Thomas elements type for the Darcy domain. Here, we discretize by standard equal-order finite elements enriched with bubbles functions and use local projection stabilization technique (LPS) to stabilize the method and control the fluctuation of the velocity divergence vector on the Darcy region. We also suggest a way to control the natural H(div) velocity.

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

We study the mixed formulation of the stochastic Hodge-Laplace problem dened on a $n$-dimensional domain $D (n ≥ 1)$, with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the $m-th$ moment $(m ≥ 1$) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.

Finite volume element approximation of the coupled continuum pipe‐flow/Darcy model for flows in karst aquifers

A finite volume element method is applied to approximate the continuum pipe‐flow/Darcy problem, which models the coupled conduit flow and porous media flow in Karst aquifers. A decoupled scheme is proposed for solving the coupled discretization problem. Optimal error estimates in L2 norm and H1 norm are given in this article. Some numerical examples are presented to verify the theoretical results and demonstrate the effectiveness of the decoupling approach. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 376–392, 2014

From Stokes to Darcy in infinite cylinders: do limits commute?

The Darcy flow problem in a porous medium in an infinite cylinder is looked at as a two-parameter limit problem, in terms of the characteristic pore size and the cylinder length. As the characteristic pore size tends to zero, the Stokes problem on the finite cylinder converges to a Darcy problem, and the Darcy problem in the infinite cylinder is obtained as its limit when the length of the cylinder goes to infinity. But one could do this in the opposite order: first consider the limit of the Stokes problem in an infinite cylinder and then consider the homogenized limit to obtain Darcy in an infinite cylinder. Would these two procedures yield the same result? In other words do the limits commute? The answer is shown to be affirmative.

Permeability coefficient in Darcy’s filtration law

Using the equation of living forces averaged over the volume of the filter and obtained from the Navier-Stokes equations of an incompressible fluid, an analytical formula for the permeability coefficient in the classical Darcy problem is derived to within the principal terms of the Poiseuille number. It is shown that the resulting permeability coefficient is inversely proportional to the square of the specific surface area and the dimensionless average dissipation rate of the kinetic fluid energy in the filter.

On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations

The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation of the problem involving as unknowns the main field in the interior of the element domains and its fluxes and traces on the element boundaries. Rather than working with this three-field formulation, fluxes are modeled using finite difference expressions and then the traces are determined by imposing continuity of fluxes, although other strategies could be devised. This procedure is applied to four elliptic problems, namely, the convection–diffusion equation (in the diffusion dominated regime), the Stokes problem, the Darcy problem and the Maxwell problem. We justify some well known dG methods with some modifications that in fact allow to improve the performance of the original methods, particularly when the physical properties are discontinuous.

A numerical method for two-phase flow in fractured porous media with non-matching grids

We propose a novel computational method for the efficient simulation of two-phase flow in fractured porous media. Instead of refining the grid to capture the flow along the faults or fractures, we represent the latter as immersed interfaces, using a reduced model for the flow and suitable coupling conditions. We allow for non matching grids between the porous matrix and the fractures to increase the flexibility of the method in realistic cases. We employ the extended finite element method for the Darcy problem and a finite volume method that is able to handle cut cells and matrix-fracture interactions for the saturation equation. Moreover, we address through numerical experiments the problem of the choice of a suitable numerical flux in the case of a discontinuous flux function at the interface between the fracture and the porous matrix. A wrong approximate solution of the Riemann problem can yield unphysical solutions even in simple cases.

Proper Generalized Decomposition method for incompressible Navier–Stokes equations with a spectral discretization

Proper Generalized Decomposition (PGD) is a method which consists in looking for the solution to a problem in a separate form. This approach has been increasingly used over the last few years to solve mathematical problems. The originality of this work consists in the association of PGD with a spectral collocation method to solve transfer equations as well as Navier–Stokes equations. In the first stage, the PGD method and its association with spectral discretization is detailed. This approach was tested for several problems: the Poisson equation, the Darcy problem, Navier–Stokes equations (the Taylor Green problem and the lid-driven cavity). In the Navier–Stokes problems, the coupling between velocity and pressure was performed using a fractional step scheme and a PN–PN-2 discretization. For all problems considered, the results from PGD simulations were compared with those obtained by a standard solver and/or with the results found in the literature. The simulations performed showed that PGD is as accurate as standard solvers. PGD preserves the spectral behavior of the errors in velocity and pressure when the time step or the space step decreases. Moreover, for a given number of discretization nodes, PGD is faster than the standard solvers.

Analysis of Long Time Stability and Errors of Two Partitioned Methods for Uncoupling Evolutionary Groundwater--Surface Water Flows

The most effective simulations of the multiphysics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes. Partitioned methods, which solve the coupled problem by successively solving the subphysics problems, have recently been studied for the Stokes--Darcy coupling with convergence established over bounded time intervals (with constants growing exponentially in $t$). This report analyzes and tests two such partitioned (noniterative, domain decomposition) methods for the fully evolutionary Stokes--Darcy problem. Under a modest time step restriction of the form $\Delta t\leq C$, where $C=C$ (physical parameters), we prove unconditional, long time (over $0\leq t<\infty $) stability of both partitioned methods. From this we derive an optimal error estimate that is uniform in time over $0\leq t<\infty$.

Partitioned Time Stepping Method for Fully Evolutionary Stokes--Darcy Flow with Beavers--Joseph Interface Conditions

In this report, a partitioned time stepping algorithm for transient flow in a matrix domain coupled to a free flow in embedded conduits is analyzed. The coupled flow is modeled by the fully evolutionary Stokes--Darcy problem. This method requires solving only one, uncoupled Stokes and Darcy subphysics and subdomain per time step. On the interface between the matrix and conduit, Beavers--Joseph interface conditions, instead of the simplified Beavers--Joseph--Saffman condition, are imposed. Under a modest time step restriction of the form $\triangle t\leq C$, where $C=C$ (physical parameters) we prove stability of the method. We also derive error estimates. Numerical tests illustrate the validity of the theoretical results.

Efficient and Long-Time Accurate Second-Order Methods for the Stokes--Darcy System

We propose and study two second-order in time implicit-explicit methods for the coupled Stokes--Darcy system that governs flows in karst aquifers and other subsurface flow systems. The first method is a combination of a second-order backward differentiation formula and the second-order Gear's extrapolation approach. The second is a combination of the second-order Adams--Moulton and second-order Adams--Bashforth methods. Both algorithms only require the solution of decoupled Stokes and Darcy problems at each time-step. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. The uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long time scales. Error estimates for fully discretized schemes using finite element spatial discretizations are derived. Nume...