Symmetric Positive Definite Flux-Continuous Full-Tensor Finite-Volume Schemes on Unstructured Cell-Centered Triangular Grids

Friis, Helmer André; Edwards, Michael G.; Mykkeltveit, Johannes

doi:10.1137/070692182

Cited by 63 publications

(99 citation statements)

References 38 publications

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“…Prior to the solution process, the auxiliary pressures are expressed algebraically in terms of the primal pressures via the local flux continuity conditions and lead to a locally conservative formulation with continuous fluxes only dependent on primal pressures. Both cell-centred and vertex-centred CVD-MPFA approximations are considered in this work [19,20,[22][23][24][25][26][27][28]. A comparison between cell-centred and vertex-centred CVD-MPFA schemes is presented here, in terms of computed flow fields resulting from the respective pressure equation approximations.…”

Section: Pressure Equationmentioning

confidence: 99%

“…For a given grid type, there are two basic types of CVD-MPFA formulation determined by the choice of basis functions: (a) triangle pressure support (TPS) with linear basis functions over subcell triangles leading to pointwise pressure continuity on control-volume sub-faces [19,20,23,24]; (b) full pressure support (FPS) with subcell bilinear basis functions, leading to full pressure continuity over control-volume sub-faces [22,25,28].…”

Section: Pressure Equationmentioning

confidence: 99%

“…The schemes employed here have been designed to overcome the TPFA limitations and involve families of control-volume distributed multi-point flux approximations (CVD-MPFA), with flow variables and rock properties being controlvolume distributed and assigned to share the same controlvolume locations. Both cell-vertex and cell-centred CVD-MPFA formulations [19][20][21][22][23][24][25][26][27][28] are compared in this work on unstructured grids. Related cell-centred MPFA methods are presented in [1-3, 44, 48, 80].…”

Section: Introductionmentioning

confidence: 99%

“…In the next section, description of the grid generation techniques together with generation of BAGs and WAGs is presented. This is followed by a brief description of the control-volume distributed multipoint flux approximation (CVD-MPFA) schemes, including the earlier piecewise-linear based formulations [19,20,23,24], and more robust piecewise-bilinear based formulations [22,25,27,28]. A comparison between the cell-centred and cell-vertex CVD-MPFA results and corresponding grids is presented, followed by conclusions.…”

Section: Introductionmentioning

confidence: 99%

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Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

et al. 2017

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Grid generation for reservoir simulation must honor classical key constraints and be boundary aligned such that control-volume boundaries are aligned with geological features such as layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. An unstructured grid generation procedure is proposed that automates control-volume and/or control point boundary alignment and yields a PEBI-mesh both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles, and to generate a dual-cell feature based grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom. The formulation of CVD-MPFA schemes in cellcentred and cell-vertex modes is analogous and requires switching control volume from primal to dual or vice versa together with appropriate data structures and boundaryThe original version of this article was revised: Due to an oversight, some author's corrections were not carried out during Performing proof corrections stage. The Publisher apologizes for these mistakes. conditions. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are made clear in terms of flow resolution and degrees of freedom required.

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Section: Pressure Equationmentioning

confidence: 99%

Section: Pressure Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

et al. 2017

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“…The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) [6] coupled with a higher resolution approximation for convective transport [1,5]. The symmetric CVD-MPFA method is used for Darcy-flux approximation including pressure, gravity, and capillary pressure flux components.…”

Section: Introductionmentioning

confidence: 99%

Higher resolution total velocity Vt and Va finite-volume formulations on cell-centred structured and unstructured grids

Xie

Edwards

2017

Comput Geosci

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Novel cell-centred finite-volume formulations are presented for incompressible and immiscible two-phase flow with both gravity and capillary pressure effects on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity, and capillary pressure flux operators. Two IMPES formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both Vt and Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However, when gravity is present, flow regimes are identified where the Va formulation becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first-order single-point upstream weighting.

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Positive‐definiteq‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

Edwards

Pal

2007

Numerical Methods in Fluids

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SUMMARYA new family of locally conservative cell-centred flux-continuous schemes is presented for solving the porous media general-tensor pressure equation. A general geometry-permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control-volume distributed subcell flux-continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994;Comput. Geosci. 1998; 2:259-290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive-definite (SPD) discretization matrix is obtained for the base member (q = 1) of the formulation. The physical-space schemes are shown to be non-symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non-symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical-space and subcell-space q-families of schemes. M-matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical-space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell-wise constant tensor schemes and that subcell tensor approximation using the control-volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested.

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Symmetric Positive Definite Flux-Continuous Full-Tensor Finite-Volume Schemes on Unstructured Cell-Centered Triangular Grids

Cited by 63 publications

References 38 publications

Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Higher resolution total velocity Vt and Va finite-volume formulations on cell-centred structured and unstructured grids

Positive‐definiteq‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

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