Coupling Biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction

Badia, Santiago; Quaini, Annalisa; Quarteroni, Alfio

doi:10.1016/j.jcp.2009.07.019

Cited by 129 publications

(135 citation statements)

References 63 publications

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“…Fixed-point of Newton methods can be used for the linearization of the Navier-Stokes/Biot coupled system. To solve the linear fluid-poroelastic-structure interaction problem, in [107,18] several methods are proposed based both on the so-called monolytic approach and on domain decomposition methods. In the first case the linear system obtained after time and space discretizations is solved using direct or iterative methods with ad-hoc preconditioners (see [108,17,16]).…”

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confidence: 99%

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Navier-Stokes/darcy coupling: modeling, analysis, and numerical approximation

2009

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confidence: 99%

“…8.4 and 10, respectively, for fluid-groundwater coupling. We refer to [107,18] for an extensive analysis of these methods and for several numerical results.…”

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confidence: 99%

Navier-Stokes/darcy coupling: modeling, analysis, and numerical approximation

2009

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“…In particular, the emergence of strongly engineered geological systems such as CO 2 storage [31, 33], geothermal energy [35], and shale-gas extraction all require analysis of the coupling of fluid flow and deformation. Beyond the subsurface, physiological processes are increasingly simulated, exploiting the Biot models [8].With this motivation, we consider the following model problem poroelastic media [9]: …”

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confidence: 99%

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Nordbotten¹

2016

SIAM J. Numer. Anal.

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Abstract. In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate subproblems. The coupled discretization has the following key properties, the combination of which is novel: (1) The variables for the pressure and displacement are co-located and are as sparse as possible (e.g., one displacement vector and one scalar pressure per cell center). (2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. (3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces and mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves convergence of the method. Finally, we give numerical examples verifying both the analysis and the convergence of the method. 1. Introduction. Deformable porous media are becoming increasingly important in applications. In particular, the emergence of strongly engineered geological systems such as CO 2 storage [31, 33], geothermal energy [35], and shale-gas extraction all require analysis of the coupling of fluid flow and deformation. Beyond the subsurface, physiological processes are increasingly simulated, exploiting the Biot models [8].With this motivation, we consider the following model problem poroelastic media [9]:

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“…In Fig. 6, for different 6,7,8), the robustness of the proposed multigrid method with the Vanka smoother is displayed for different values of K and ν and different grid sizes. The maximum norm of the residual divided by the maximum norm of the right-hand side is plotted in logarithmic scale against the number of multigrid cycles necessary to fulfill the stopping criterion.…”

Section: Multigrid Convergence With Different Parametersmentioning

confidence: 99%

“…We take the data from the experiment in [7]. The final time is T = 0.01 and the time step is τ = 1.25 × 10 −3 .…”

Section: Two-block Realistic Testmentioning

confidence: 99%

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

Luo

Rodrigo

Gaspar

et al. 2018

Journal of Computational Physics

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The interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem, which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with either a coupled Vanka smoother or a decoupled Uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids. A specialty in our modeling approach is that at the interface of the fluid and poroelastic medium, two unknowns from the different subsystems are defined at the same grid point. We propose a special discretization at and near the points on the interface, which combines the approximation of the governing equations and the considered interface conditions. In the decoupled Uzawa smoother, Local Fourier Analysis (LFA) helps us to select optimal values of the relaxation parameter appearing. To implement the monolithic multigrid method, grid partitioning is used to deal with the interface updates when communication is required between two subdomains. Numerical experiments show that the proposed numerical method has an excellent convergence rate. The efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients.

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Coupling Biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction

Cited by 129 publications

References 63 publications

Navier-Stokes/darcy coupling: modeling, analysis, and numerical approximation

Navier-Stokes/darcy coupling: modeling, analysis, and numerical approximation

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

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