On the causes of pressure oscillations in low‐permeable and low‐compressible porous media

Haga, Joachim Berdal; Osnes, Harald; Langtangen, Hans Petter

doi:10.1002/nag.1062

Cited by 110 publications

(104 citation statements)

References 28 publications

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“…Here the discrete flux and traction are defined in (17) and (20), utilizing that due to (18) and (21) we can (up to the sign) evaluate the expression for either K ∈ T σ . In order to show stability according to Definition 1 and verify Lemma 23, we combine the above errors to the stable error…”

Section: Convergence Resultsmentioning

confidence: 99%

“…In contrast, the solid subproblem does not lend itself trivially to mixed finite elements, and although much recent literature relates to this problem, simple low-order element combinations appear to be impossible to devise [6]. A thorough numerical investigation of various combinations of mixed methods for the full Biot equations has been reported [18]. An alternative to mixed finite elements (but similar in spirit) is to use staggered grids for the mechanics and flow equations [37,17].…”

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

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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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Section: Convergence Resultsmentioning

confidence: 99%

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Nordbotten¹

2016

SIAM J. Numer. Anal.

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“…It is well-known that the standard finite element solution of the poroelasticity equations can present strong nonphysical oscillations in the fluid pressure, see for instance [1,7,8,15,29]. For example, this is the case when linear finite elements are used to approximate both displacement and pressure unknowns.…”

Section: Stable Discretization Of the Modelmentioning

confidence: 99%

“…We consider a porous material on which a low-permeable layer (K = 10 −8 ) is placed between two layers with unit permeability (K = 1), as shown in Fig. 4, see [15]. The boundary of the squared domain is split in two disjoint subsets 1 and 2 on which we assume the following boundary conditions: on the top, which is free to drain, a uniform load is applied, that is, whereas at the sides and bottom that are rigid the boundary is considered to be impermeable, that is,…”

Section: One-dimensional Problemmentioning

confidence: 99%

Poroelasticity problem: numerical difficulties and efficient multigrid solution

Rodrigo

2015

SeMA

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This work contains some of the more relevant results obtained by the author regarding the numerical solution of the Biot's consolidation problem. The emphasis here is on the stable discretization and the highly efficient solution of the resulting algebraic system of equations, which is of saddle point type. On the one hand, a stabilized linear finite element scheme providing oscillation-free solutions for this model is proposed and theoretically analyzed. On the other hand, a monolithic multigrid method is considered for the solution of the resulting system of equations after discretization by using the stabilized scheme. Since this system is of saddle point type, special smoothers of "Vanka"-type have to be considered. This multigrid method is designed with the help of an special local Fourier analysis that takes into account the specific characteristics of the considered block-relaxations. Results from this analysis are presented and compared with those experimentally computed.

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“…It has been suggested that this problem is caused by the saddle point structure in the coupled equations resulting in a violation of the famous LadyzhenskayaBabuska-Brezzi (LBB) condition, thus highlighting the need for a stable combination of mixed finite elements [23].…”

Section: Introductionmentioning

confidence: 99%

A stabilized finite element method for finite-strain three-field poroelasticity

Berger¹,

Bordas

Kay

et al. 2017

Comput Mech

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We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.

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On the causes of pressure oscillations in low‐permeable and low‐compressible porous media

Cited by 110 publications

References 28 publications

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Poroelasticity problem: numerical difficulties and efficient multigrid solution

A stabilized finite element method for finite-strain three-field poroelasticity

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