Stable Cell-Centered Finite Volume Discretization for Biot Equations

Nordbotten, Jan Martin

doi:10.1137/15m1014280

Cited by 93 publications

(81 citation statements)

References 33 publications

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“…We note that the discretization of the solid pressure is identical to the MPSA discretization for Biot's equations in the limit of zero fluid compressibility and rock permeability. Indeed, although poro-elastic systems are outside the scope of this paper, we have verified that similar to MPSA-O, the weakly symmetric method can be extended to poro-elasticity by applying the same steps as introduced in [21].…”

Section: Coupling With Solid Pressurementioning

confidence: 64%

Finite volume methods for elasticity with weak symmetry

Keilegavlen

Nordbotten

2017

Numerical Meth Engineering

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We introduce a new cell-centered finite volume discretization for elasticity with weakly enforced symmetry of the stress tensor. The method is motivated by the need for robust discretization methods for deformation and flow in porous media and falls in the category of multi-point stress approximations (MPSAs). By enforcing symmetry weakly, the resulting method has flexibility beyond previous MPSA methods. This allows for a construction of a method that is applicable to simplexes, quadrilaterals, and most planar-faced polyhedral grids in both 2D and 3D, and in particular, the method amends a convergence failure in previous MPSA methods for certain simplex grids. We prove convergence of the new method for a wide range of problems, with conditions that can be verified at the time of discretization. We present the first set of comprehensive numerical tests for the MPSA methods in three dimensions, covering Cartesian and simplex grids, with both heterogeneous and nearly incompressible media. The tests show that the new method consistently is second order convergent in displacement, despite being the lowest order, with a rate that mostly is between 1 and 2 for stresses. The results further show that the new method is more robust and computationally cheaper than previous MPSA methods.

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Section: Coupling With Solid Pressurementioning

confidence: 64%

Finite volume methods for elasticity with weak symmetry

Keilegavlen

Nordbotten

2017

Numerical Meth Engineering

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“…More generally, we can conclude that the grids where the MPSA method and the VE method lock are dual grids of each other (not true for quadrilateral). As proven in [14], a practical advantage of the MPSA method is that when it is coupled with a finite volume discretization, which is the most common choice of discretization for the flow equation, the method will be stable independently of the time-step size, even in the limit of incompressible fluid. In the case of geological applications, the compressibility of water is about the same as of the rock, which means that locking is not happening.…”

Section: 32mentioning

confidence: 99%

“…The MPSA method is attractive from the physical point of view, due to the explicit treatment of the force continuity at the cell interfaces. The MPSA offers a natural stable coupling with poro-elasticity, see [14]. From the implementation point of view, the MPSA method is cell-centered and therefore shares the same grid structure as the MPFA method, which is also often the preferred convergent method for multi-phase flow.…”

Section: Casementioning

confidence: 99%

“…As in the MFD method, a complete finite element basis for a polyhedral cell is not computed, some of the basis elements become virtual. Both the MPSA and the VE methods for mechanics naturally define the divergence of the displacement on cells (see [14] for MPSA), which is also the natural coupling term between flow and mechanics, when flow is discretized with finite volume methods. As pointed out in [15], any attempt to extend the TPFA method to mechanics is bound to fail as the method already fails the local patch test.…”

Section: Introductionmentioning

confidence: 99%

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Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

2017

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Abstract. In this paper we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regards to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular we look how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics.

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“…The general discretization methodology chosen in this paper has recently been extended to mechanics and coupled mechanics and flow [21,26,27,37]. We expect that the preconditioning strategies developed herein can in principle be applied to the resulting discrete linear systems, however due to the presence of rigid body motions in mechanics, some additional developments will likely be needed (see, e.g., [9] for analogous extensions).…”

Section: Introductionmentioning

confidence: 99%

Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs

et al. 2017

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In simulation of fluid injection in fractured geothermal reservoirs, the characteristics of the physical processes are severely affected by the local occurence of connected fractures. To resolve these structurally dominated processes, there is a need to develop discretization strategies that also limit computational effort. In this paper, we present an upscaling methodology for geothermal heat transport with fractures represented explicitly in the computational grid. The heat transport is modeled by an advectionconduction equation for the temperature, and solved on a highly irregular coarse grid that preserves the fracture heterogeneity. The upscaling is based on different strategies for the advective term and the conductive term. The coarse scale advective term is constructed from sums of fine scale fluxes, whereas the coarse scale conductive term is constructed based on numerically computed basis functions. The method naturally incorporates the coupling between solution variables in the matrix and in the fractures, respectively, via the discretization. In this way, explicit transfer terms that couple fracture and matrix solution variables are avoided. Numerical results show that the upscaling methodology performs well, in particular for large upscaling ratios, and that it is applicable also to highly complex fracture networks.

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Stable Cell-Centered Finite Volume Discretization for Biot Equations

Cited by 93 publications

References 33 publications

Finite volume methods for elasticity with weak symmetry

Finite volume methods for elasticity with weak symmetry

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs

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