Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media

Dana, Saumik; Wheeler, Mary F.

doi:10.1016/j.cma.2018.07.018

Cited by 43 publications

(32 citation statements)

References 16 publications

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“…Borregales [8] extended the fixed-stress split to a nonlinear case. Dana et al [14,15] studied a multiscale extension of the fixed-stress split to a poroelastic-elastic system where the poromechanics equation is solved on a larger domain with a coarse grid and the flow equation is solved on a small domain with finer grid. Moreover, Bause et al [6] and Borregales et al [7] explored space-time methods of the fixed-stress split, and the work of Rodrigo et al [41] considered the stability analysis of the discretization schemes.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for Biot system using Enriched Galerkin for flow

Girault

Lü

Wheeler

2020

Computer Methods in Applied Mechanics and Engineering

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We analyze the Biot system solved with a fixed-stress split, Enriched Galerkin (EG) discretization for the flow equation, and Galerkin for the mechanics equation. Residual-based a posteriori error estimates are established with both lower and upper bounds. These theoretical results are confirmed by numerical experiments performed with Mandel's problem. The efficiency of these a posteriori error estimators to guide dynamic mesh refinement is demonstrated with a prototype unconventional reservoir model containing a fracture network. We further propose a novel stopping criterion for the fixed-stress iterations using the error indicators to balance the fixed-stress split error with the discretization errors. The new stopping criterion does not require hyperparameter tuning and demonstrates efficiency and accuracy in numerical experiments.

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Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for Biot system using Enriched Galerkin for flow

Girault

Lü

Wheeler

2020

Computer Methods in Applied Mechanics and Engineering

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“…With that in mind, we define a measure of mean stress which equates to the actual mean stress only as a special case. As a result, the staggering in this work is a generalization of the fixed stress split staggering that was studied in Mikelić and Wheeler [18], Almani et al [2] and Dana and Wheeler [7]. This paper is structured as follows: Section 2 presents the model equations for flow and poromechanics, Section 3 presents the statement of contraction of the two-grid fixed stress split iterative scheme, Section 4 presents the details of how the statement of contraction is used to arrive at restriction and prolongation operators as well as the effective coarse scale moduli, Section 5 presents the two-grid fixed stress split algorithm and Section 6 discusses the link between the decoupling constraint and the Voigt and Reuss bounds.…”

Section: Introductionmentioning

confidence: 92%

“…The measure of mean stress that remains fixed during the flow solve is hydrostatic part of the total stress, also refered to as the mean stress. The interesting result of the work of Dana and Wheeler [7] is that the convergence analysis lends itself to an expression for coarse scale bulk moduli in terms of fine scale bulk moduli, and further the coarse scale moduli are a harmonic mean of the fine scale moduli. The harmonic mean is exactly the Reuss bound (see Saeb et al [22]).…”

Section: Introductionmentioning

confidence: 99%

The Correspondence between Voigt and Reuss Bounds and the Decoupling Constraint in a Two-Grid Staggered Algorithm for Consolidation in Heterogeneous Porous Media

Dana¹,

Ita²,

Wheeler³

2020

Multiscale Model. Simul.

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We perform a convergence analysis of a two-grid staggered solution algorithm for the Biot system modeling coupled flow and deformation in heterogeneous poroelastic media. The algorithm first solves the flow subproblem on a fine grid using a mixed finite element method (by freezing a certain measure of the mean stress) followed by the poromechanics subproblem on a coarse grid using a conforming Galerkin method. Restriction operators map the fine scale flow solution to the coarse scale poromechanical grid and prolongation operators map the coarse scale poromechanical solution to the fine scale flow grid. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The analysis is based on studying the equations satisfied by the difference of iterates to show that the two-grid scheme is a contraction map under certain conditions. Those conditions are used to construct the restriction and prolongation operators as well as arrive at coarse scale elastic properties in terms of the fine scale data. We show that the adjustable parameter in the measure of the mean stress is linked to the Voigt and Reuss bounds frequently encountered in computational homogenization of multiphase composites.

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“…As shown in Fig. 2b, the two-grid approach is built on top of a staggered solution algorithm in which the flow and geomechanics subproblems are solved sequentially and iteratively using a fixed stress split iterative scheme [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,21,46,47,48,49,50,51,52]. The accuracy of the two-grid method has been demonstrated for the classical Mandel's problem [53,54] in our earlier work [21].…”

Section: Motivationmentioning

confidence: 99%