Flow Simulation in Heterogeneous Porous Media With the Moving Least-Squares Method

White, Laurent; Panchadhara, Rohan; Trenev, Dimitar

doi:10.1137/16m1070840

Cited by 5 publications

(4 citation statements)

References 38 publications

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“…This finite volume scheme uses a moving least-squares reconstruction that makes it suitable for arbitrary meshes. This least-squares approach has been applied previously to shallow water flows [40], aeronautic [41] and porous media [42] simulations.…”

Section: Introductionmentioning

confidence: 99%

Multidimensional method-of-lines transport for atmospheric flows over steep terrain using arbitrary meshes

Shaw

Weller

Methven

et al. 2017

Journal of Computational Physics

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Including terrain in atmospheric models gives rise to mesh distortions near the lower boundary that can degrade accuracy and challenge the stability of transport schemes. Multidimensional transport schemes avoid splitting errors on distorted, arbitrary meshes, and method-of-lines schemes have low computational cost because they perform reconstructions at fixed points. This paper presents a multidimensional method-of-lines finite volume transport scheme, "cubicFit", which is designed to be numerically stable on arbitrary meshes. Constraints derived from a von Neumann stability analysis are imposed during model initialisation to obtain stability and improve accuracy in distorted regions of the mesh, and near steeply-sloping lower boundaries. Reconstruction calculations depend upon the mesh only, needing just one vector multiply per face per time-stage irrespective of the velocity field.The cubicFit scheme is evaluated using three, idealised numerical tests. The first is a variant of a standard horizontal transport test on severely distorted terrain-following meshes. The second is a new test case that assesses accuracy near the ground by transporting a tracer at the lower boundary over steep terrain on terrain-following meshes, cut-cell meshes, and new, slanted-cell meshes that do not suffer from severe time-step constraints associated with cut cells. The third, standard test deforms a tracer in a vortical flow on hexagonal-icosahedral meshes and cubed-sphere meshes. In all tests, cubicFit is stable and largely insensitive to mesh distortions, and cubicFit results are more accurate than those obtained using a multidimensional linear upwind transport scheme. The cubicFit scheme is second-order convergent regardless of mesh distortions.

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Section: Introductionmentioning

confidence: 99%

Multidimensional method-of-lines transport for atmospheric flows over steep terrain using arbitrary meshes

Shaw

Weller

Methven

et al. 2017

Journal of Computational Physics

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“…Nevertheless, it should be noted that the pore pressure solutions may exhibit discontinuities if the permeability variations in the porous medium are too drastic. [22][23][24]…”

Section: Mathematical Derivationmentioning

confidence: 99%

High‐order ADE scheme for solving the fluid diffusion equation in non‐uniform grids and its application in coupled hydro‐mechanical simulation

Prassetyo

Gutierrez

2018

Num Anal Meth Geomechanics

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Summary To improve the stability and efficiency of explicit technique, one proposed method is to use an unconditionally stable alternating direction explicit (ADE) scheme. However, the standard ADE scheme is only moderately accurate and restricted to uniform grids. This paper derives a novel high‐order ADE scheme capable of solving the fluid diffusion equation in non‐uniform grids. The new scheme is derived by performing a fourth‐order finite difference approximation to the spatial derivatives of the diffusion equation in non‐uniform grid. The implicit Crank‐Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps, giving rise to a new high‐order ADE scheme. Because the new scheme can be potentially applied in coupled hydro‐mechanical (H‐M) simulation, the pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer program Fast Lagrangian Analysis of Continua. This coupling procedure is called the sequentially explicit coupling technique based on the fourth‐order ADE scheme (SEA‐4). Verifications of well‐known consolidation problems showed that the new ADE scheme and SEA‐4 can reduce computer runtime by 46% to 75% to that of Fast Lagrangian Analysis of Continua's basic scheme. At the same time, the techniques still maintained average percentage error of 1.6% to 3.5% for pore pressure and 0.2% to 1.5% for displacement solutions and were still accurate under typical grid non‐uniformities. This result suggests that the new high‐order ADE scheme can provide an efficient explicit technique for solving the flow equation of a coupled H‐M problem, which will be beneficial for large‐scale and long‐term H‐M problems in geoengineering.

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“…The proposed approximation is based on standard assumptions on the continuity of the solution and the diffusive flux at the faces of computational cells, even if the diffusion coefficient is discontinuous across the face. Furthermore, inspired by [30], we apply a restricted least-squares approximation of gradients that respects the discontinuity of the diffusion coefficient and leads to a distance weighted harmonic mean of diffusion coefficient. Finally, we use an iterative method in a deferred correction manner [6,11,3] to use a 1-ring neighborhood assumption when a local stiffness matrix is assembled.…”

mentioning

confidence: 99%

“…This is not the case in general if the diffusion coefficient has a discontinuity in the same neighborhood. Inspired by [30], we use in this case a restricted least-squares gradient ∇φ p by restricting the set N p in (8) only to the neighborhood where the diffusion coefficient is continuous.…”

mentioning

confidence: 99%

Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh

Frolkovič¹,

Mikula²,

Hahn³

et al. 2021

Discrete &Amp; Continuous Dynamical Systems - S

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A numerical method for solving diffusion problems on polyhedral meshes is presented. It is based on a finite volume approximation with the degrees of freedom located in the centers of computational cells. A numerical gradient is defined by a least-squares minimization for each cell, where we suggest a restricted form in the case of discontinuous diffusion coefficient. The flux balanced approximation is proposed without numerically computing the gradient itself at the faces of computational cells in order to find a normal diffusive flux. To apply the method for parallel computations with a 1-ring neighborhood, we use an iterative method to solve the obtained system of algebraic equations. Several numerical examples illustrate some advantages of the proposed method.

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Flow Simulation in Heterogeneous Porous Media With the Moving Least-Squares Method

Cited by 5 publications

References 38 publications

Multidimensional method-of-lines transport for atmospheric flows over steep terrain using arbitrary meshes

Multidimensional method-of-lines transport for atmospheric flows over steep terrain using arbitrary meshes

High‐order ADE scheme for solving the fluid diffusion equation in non‐uniform grids and its application in coupled hydro‐mechanical simulation

Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh

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