Coupling Stokes–Darcy Flow with Transport

Vassilev, Danail; Yotov, Ivan

doi:10.1137/080732146

Cited by 81 publications

(42 citation statements)

References 31 publications

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“…For the numerical experiments we take Ω f = (0, 1) × (0.5, 1), Ω p = (0, 1) × (0, 0.5), and I = {( x , 1/2): 0 < x < 1}. We use for the numerical experiments

c (x, y, t) = t (\cos (π x) + \cos (π y)) / π, and u (x, y, t) = [\begin{matrix} right \\ \sin (\frac{x}{G} + ω) e^{y / G} \\ - \cos (\frac{x}{G} + ω) e^{y / G} \end{matrix}],

where

G = 2 \sqrt{0.1}

, and ω = 1.05. Additionally, for the parameters in the modeling Equation , we use β = 1 and D = 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

Partitioned penalty methods for the transport equation in the evolutionary Stokes–Darcy‐transport problem

Ervin

Kubacki

Layton

et al. 2018

Numerical Methods Partial

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There has been a surge of work on models for coupling surface‐water with groundwater flows which is at its core the Stokes–Darcy problem, as well as methods for uncoupling the problem into subdomain, subphysics solves. The resulting (Stokes–Darcy) fluid velocity is important because the flow transports contaminants. The numerical analysis and algorithm development for the evolutionary transport problem has, however, focused on a quasi‐static Stokes–Darcy model and a single domain (fully coupled) formulation of the transport equation. This report presents a numerical analysis of a partitioned method for contaminant transport for the fully evolutionary system. The algorithm studied is unconditionally stable with one subdomain solve per step. Numerical experiments are given using the proposed algorithm that investigates the effects of the penalty parameters on the convergence of the approximations.

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“…For the numerical experiments we take Ω f = (0, 1) × (0.5, 1), Ω p = (0, 1) × (0, 0.5), and I = {( x , 1/2): 0 < x < 1}. We use for the numerical experiments

c (x, y, t) = t (\cos (π x) + \cos (π y)) / π, and u (x, y, t) = [\begin{matrix} right \\ \sin (\frac{x}{G} + ω) e^{y / G} \\ - \cos (\frac{x}{G} + ω) e^{y / G} \end{matrix}],

where

G = 2 \sqrt{0.1}

, and ω = 1.05. Additionally, for the parameters in the modeling Equation , we use β = 1 and D = 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

Partitioned penalty methods for the transport equation in the evolutionary Stokes–Darcy‐transport problem

Ervin

Kubacki

Layton

et al. 2018

Numerical Methods Partial

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“…The Darcy model occupies the region

{normalΩ}_{italicG} = (0, 1) \times (0, +1 \frac{1}{2})

, and the Stokes model occupies the region

{normalΩ}_{italicL} = (0, 1) \times (+1 \frac{1}{2}, 1)

. We devise an exact solution similar to those proposed in that satisfies the interfacial conditions between the two regions:

{italicv}_{italicGx} = sin (\frac{italicx}{italicG} + ω) {italice}^{y / G} + ω sin (italicωx)

{italicv}_{italicGy} = - cos (\frac{italicx}{italicG} + ω) {italice}^{y / G}

{italicp}_{italicG} = \frac{italicμG}{italicκ} cos (\frac{italicx}{italicG} + ω) {italice}^{y / G} + \frac{italicμ}{italicκ} cos (italicωx)

{italicv}_{italicLx} = sin (\frac{italicx}{italicG} + ω) {italice}^{y / G}

{italicv}_{italicLy} = - cos (\frac{italicx}{italicG} + ω) {italice}^{y / G}

…”

Section: Numerical Resultsmentioning

confidence: 99%

A heterogeneous modeling method for porous media flows

Hlepas

Truster

Masud

2014

Numerical Methods in Fluids

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SummaryThis paper presents a new heterogeneous multiscale modeling method for porous media flows. Physics at the global level is governed by one set of PDEs, while features in the solution that are beyond the resolution capacity of the global model are accounted for by the next refined set of governing equations. In this method, the global or coarse model is given by the Darcy equation, while the local or refined model is given by the Darcy–Stokes equation. Concurrent domain decomposition where global and local models are applied to adjacent subdomains, as well as overlapping domain decomposition where global and local models coexist on overlapping domains, is considered. An interface operator is developed for the case where global and local models commute along the common interface. For the overlapping decomposition, a residual‐based coupling technique is developed that consistently facilitates bottom‐up embedding of scale effects from the local Darcy–Stokes model into the global Darcy model. Numerical results are presented for nonoverlapping and overlapping domain decompositions for various benchmark problems. Computed results show that the hierarchically coupled models accurately account for the heterogeneity of the medium and efficiently incorporate local features into the global response. Copyright © 2014 John Wiley & Sons, Ltd.

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“…These include finite volume (FV), mixed finite element (MFE) or more advanced, locally conservative discontinuous Galerkin (DG) schemes, see [3][4][5][6][7][8][9][10] and references therein. These schemes require the knowledge of physical quantities like fluid viscosity, permeabilities and experimentally measured BJS interface parameters for approximating the Darcy-Stokes flow.…”

Section: Introductionmentioning

confidence: 99%

A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

Kumar

Luo

Gaspar

et al. 2018

Journal of Computational Physics

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a r t i c l e i n f o a b s t r a c t A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advectiondominated contaminant transport in a coupled Darcy-Stokes flow system is described.In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy-Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented.

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Coupling Stokes–Darcy Flow with Transport

Cited by 81 publications

References 31 publications

Partitioned penalty methods for the transport equation in the evolutionary Stokes–Darcy‐transport problem

Partitioned penalty methods for the transport equation in the evolutionary Stokes–Darcy‐transport problem

A heterogeneous modeling method for porous media flows

A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

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