Unfitted FEM for Modelling the Interaction of Multiple Fractures in a Poroelastic Medium

Giovanardi, Bianca; Formaggia, Luca; Scotti, Anna; Zunino, Paolo

doi:10.1007/978-3-319-71431-8_11

Cited by 21 publications

(22 citation statements)

References 22 publications

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“…For a poroelastic domain including fractures, different models for the contact problem are developed [35,34,20,17]. Most of these models, however, do not take into account the contact problem either by assuming the fractures stick together [35] or that the fluid pressure inside the fractures is so large that the fracture surfaces are never in contact [34,20]. The full contact problem for a fractured poroelastic domain is considered by Garipov et al [17], where they applied the penalty method to solve the nonlinear variational inequalities resulting from the contact problem.…”

Section: Introductionmentioning

confidence: 99%

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

Berge

Berre

Keilegavlen

et al. 2019

Numerical Meth Engineering

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A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law while slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2d and 3d, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.

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Section: Introductionmentioning

confidence: 99%

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

Berge

Berre

Keilegavlen

et al. 2019

Numerical Meth Engineering

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“…We mention that the fixed‐stress splitting scheme also can be applied to more involved extensions of Biot's equations, for example, including nonlinear water compressibility, unsaturated poroelasticity, the multiple‐network poroelasticity theory, finite‐strain poroplasticity, fractured porous media, and fracture propagation . For nonlinear problems, one combines a linearization technique, eg, the L ‐scheme, with the splitting algorithm; the convergence of the resulting scheme can be proved rigorously .…”

Section: Introductionmentioning

confidence: 99%

“…We mention that the fixed-stress splitting scheme also can be applied to more involved extensions of Biot's equations, for example, including nonlinear water compressibility, 32 unsaturated poroelasticity, 33,34 the multiple-network poroelasticity theory, 35,36 finite-strain poroplasticity, 37 fractured porous media, 38 and fracture propagation. 39,40 For nonlinear problems, one combines a linearization technique, eg, the L-scheme, 41,42 with the splitting algorithm; the convergence of the resulting scheme can be proved rigorously. 32,33 Finally, we would like to mention some valuable variants of the fixed-stress splitting scheme: the multirate fixed-stress method, 43 the multiscale fixed-stress method, 29 and the parallel-in-time fixed-stress method.…”

Section: Introductionmentioning

confidence: 99%

On the optimization of the fixed‐stress splitting for Biot's equations

Storvik

Both

Kumar

et al. 2019

Numerical Meth Engineering

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Summary In this work, we are interested in efficiently solving the quasi‐static, linear Biot model for poroelasticity. We consider the fixed‐stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well known that the convergence properties of the method strongly depend on the applied stabilization/tuning parameter. We show theoretically that, in addition to depending on the mechanical properties of the porous medium and the coupling coefficient, they also depend on the fluid flow and spatial discretization properties. The type of analysis presented in this paper is not restricted to a particular spatial discretization, although it is required to be inf‐sup stable with respect to the displacement‐pressure formulation. Furthermore, we propose a way to optimize this parameter that relies on the mesh independence of the scheme's optimal stabilization parameter. Illustrative numerical examples show that using the optimized stabilization parameter can significantly reduce the number of iterations.

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“…Such approaches have been proposed for both FV discretization of flow [7][8][9] and poromechanics [10][11][12] and as extended finite-element methods (XFEM) for flow, 13,14 mechanics, and poromechanics. [14][15][16][17] Recently, these methods have also been combined for simulation of coupled multiphase flow and mechanics in fractured porous media using a mixed FE/FV discretization. 18 In this work, we propose an FV discretization of the multiphase mass balance equations along with an FE scheme for the momentum balance equation.…”

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confidence: 99%

Simulation of coupled multiphase flow and geomechanics in porous media with embedded discrete fractures

Cusini

White

Castelletto

et al. 2020

Num Anal Meth Geomechanics

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In fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using uncoupled methods. In recent years, significant research has focused on discretization strategies for these coupled systems, particularly in the presence of complicated fracture network geometries. In this work, we explore a finitevolume discretization for the multiphase flow equations coupled with a finiteelement scheme for the mechanical equations. Fractures are treated as lower dimensional surfaces embedded in a background grid. Interactions are captured using the embedded discrete fracture model (EDFM) and the embedded finite element method (EFEM) for the flow and the mechanics, respectively. This nonconforming approach significantly alleviates meshing challenges. EDFM considers fractures as lower dimension finite volumes that exchange fluxes with the rock matrix cells. The EFEM method provides, instead, a local enrichment of the finite-element space inside each matrix cell cut by a fracture element. Both the use of piecewise constant and piecewise linear enrichments are investigated. They are also compared to an extended finite element approach. One key advantage of EFEM is the element-based nature of the enrichment, which reduces the geometric complexity of the implementation and leads to linear systems with advantageous properties. Synthetic numerical tests are presented to study the convergence and accuracy of the proposed method. It is also applied to a realistic scenario, involving a heterogeneous reservoir with a complex fracture distribution, to demonstrate its relevance for field applications.

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Unfitted FEM for Modelling the Interaction of Multiple Fractures in a Poroelastic Medium

Cited by 21 publications

References 22 publications

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

On the optimization of the fixed‐stress splitting for Biot's equations

Simulation of coupled multiphase flow and geomechanics in porous media with embedded discrete fractures

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