Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media

Chainais-Hillairet, Claire; Krell, Stella; Mouton, Alexandre

doi:10.1002/num.21913

Cited by 23 publications

(24 citation statements)

References 35 publications

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“…In [7], the mixed finite volume (MFV) of [20] is adapted to the Peaceman model, its convergence analysed and numerical tests are provided; as shown in [21], this method can be embedded in a larger family, the hybrid mimetic method (HMM) family, that also contains the SUSHI scheme of [28] and the mixed-hybrid mimetic finite difference methods of [6]. Discrete duality finite volume (DDFV) methods are considered in [8,9]. HMM and DDFV are finite volume schemes with first-order approximation properties, and rely for the miscible displacement model on upwinding to stabilise the advective terms; this raises the concern of an over-diffusion of the transition layer between the invading solvent and the residing oil.…”

Section: Introductionmentioning

confidence: 99%

An Arbitrary-Order Scheme on Generic Meshes for Miscible Displacements in Porous Media

Anderson¹,

Droniou²

2018

SIAM J. Sci. Comput.

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We design, analyse and implement an arbitrary order scheme applicable to generic meshes for a coupled elliptic-parabolic PDE system describing miscible displacement in porous media. The discretisation is based on several adaptations of the Hybrid-High-Order (HHO) method due to Di Pietro et al. [Computational Methods in Applied Mathematics, 14(4), (2014)]. The equation governing the pressure is discretised using an adaptation of the HHO method for variable diffusion, while the discrete concentration equation is based on the HHO method for advection-diffusion-reaction problems combined with numerically stable flux reconstructions for the advective velocity that we have derived using the results of Cockburn et al. [ESAIM: Mathematical Modelling and Numerical Analysis, 50(3), (2016)]. We perform some rigorous analysis of the method to demonstrate its L 2 stability under the irregular data often presented by reservoir engineering problems and present several numerical tests to demonstrate the quality of the results that are produced by the proposed scheme.

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Section: Introductionmentioning

confidence: 99%

An Arbitrary-Order Scheme on Generic Meshes for Miscible Displacements in Porous Media

Anderson¹,

Droniou²

2018

SIAM J. Sci. Comput.

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“…The computations of the proof are similar to those present in [5] and [2]. In [5], the proof is given for finite volume methods; in [2], the proof is given for DDFV method but in the case of L 1 norm and with a different definition of u ∂ M∪∂ M * .…”

Section: Ddfv Schemes For the Stokes Equationmentioning

confidence: 87%

“…In [5], the proof is given for finite volume methods; in [2], the proof is given for DDFV method but in the case of L 1 norm and with a different definition of u ∂ M∪∂ M * . Moreover, our proof has been adapted to the vectorial case.…”

Section: Ddfv Schemes For the Stokes Equationmentioning

confidence: 99%

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Numerical Analysis of the DDFV Method for the Stokes Problem with Mixed Neumann/Dirichlet Boundary Conditions

Goudon¹,

Krell²,

Lissoni³

2017

Springer Proceedings in Mathematics &Amp; Statistics

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The aim of this work is to analyze "Discrete Duality Finite Volume" schemes (DDFV for short) on general meshes by adapting the theory known for the linear Stokes problem with Dirichlet boundary conditions to the case of Neumann boundary conditions on a fraction of the boundary. We prove well-posedness for stabilized schemes and we derive some error estimates. Finally, we illustrate some numerical results in which we compare stabilized and unstabilized schemes.

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“…Related to these are the so-called Eulerian-Lagrangian localised adjoint methods (ELLAMs) [9,61]. Finite volume (FV) and mixed finite volume (MFV) methods have been studied for the transport equation alone [2] (with a MFE method for the pressure equation) and the whole system [10], and also Discrete Duality Finite Volume (DDFV) methods [11,12]. Discontinuous Galerkin (dG) methods are also often employed in the numerical study of (1.1) [4,50,56,60,62].…”

Section: Introductionmentioning

confidence: 99%

Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

et al. 2018

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This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in L ∞ (0, T ; L 2 (Ω)) of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes are compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion.

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Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media

Cited by 23 publications

References 35 publications

An Arbitrary-Order Scheme on Generic Meshes for Miscible Displacements in Porous Media

An Arbitrary-Order Scheme on Generic Meshes for Miscible Displacements in Porous Media

Numerical Analysis of the DDFV Method for the Stokes Problem with Mixed Neumann/Dirichlet Boundary Conditions

Unified Convergence Analysis of Numerical Schemes for a Miscible Displacement Problem

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