| 1. |
- Both, Jakub Wiktor, et al.
(författare)
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Robust fixed stress splitting for Biot’s equations in heterogeneous media
- 2017
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Ingår i: Applied Mathematics Letters. - Amsterdam, Netherlands : Elsevier. - 0893-9659 .- 1873-5452. ; 68, s. 101-108
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Tidskriftsartikel (refereegranskat)abstract
- We study the iterative solution of coupled flow and geomechanics in heterogeneous porous media, modeled by a three-field formulation of the linearized Biot's equations. We propose and analyze a variant of the widely used Fixed Stress Splitting method applied to heterogeneous media. As spatial discretization, we employ linear Galerkin finite elements for mechanics and mixed finite elements (lowest order Raviart Thomas elements) for flow. Additionally, we use implicit Euler time discretization. The proposed scheme is shown to be globally convergent with optimal theoretical convergence rates. The convergence is rigorously shown in energy norms employing a new technique. Furthermore, numerical results demonstrate robust iteration counts with respect to the full range of Lame parameters for homogeneous and heterogeneous media. Being in accordance with the theoretical results, the iteration count is hardly influenced by the degree of heterogeneities.
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| 2. |
- Kumar, Kundan, et al.
(författare)
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Formal upscaling and numerical validation of unsaturated flow models in fractured porous media
- 2020
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 407, s. 1-21
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we consider a mathematical model for describing flow in an unsaturated porous medium containing a fracture. Both the flow in the fracture as well as in the matrix blocks are governed by Richards' equation coupled by natural transmission conditions. Using formal asymptotics, we derive upscaled models as the limit of vanishing epsilon, the ratio of the width and length of the fracture. Our results show that the ratio of porosities and permeabilities in the fracture to matrix determine, to the leading order of approximation, the appropriate effective model. In these models the fracture is a lower dimensional object for which different transversally averaged models are derived depending on the ratio of the porosities and permeabilities of the fracture and respective matrix blocks. We obtain a catalogue of effective models which are validated by numerical computations. (C) 2019 Published by Elsevier Inc.
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| 3. |
- List, Florian, et al.
(författare)
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Rigorous upscaling of unsaturated flow in fractured porous media
- 2020
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Ingår i: SIAM Journal on Mathematical Analysis. - : SIAM PUBLICATIONS. - 0036-1410 .- 1095-7154. ; 52:1, s. 239-276
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we consider a mathematical model for flow in an unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called epsilon > 0. Then we take the limit epsilon -> 0 and give a rigorous proof for the convergence toward effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively, in the matrix, scale in terms of epsilon, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.
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| 4. |
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| 5. |
- Radu, Florin Adrian, et al.
(författare)
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A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media
- 2015
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 289, s. 134-141
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Tidskriftsartikel (refereegranskat)abstract
- In this work we consider a mathematical model for two-phase flow in porous media. The fluids are assumed immiscible and incompressible and the solid matrix non-deformable. The mathematical model for the two-phase flow is written in terms of the global pressure and a complementary pressure (obtained by using the Kirchhoff transformation) as primary unknowns. For the spatial discretization, finite volumes have been used (more precisely the multi-point flux approximation method) and in time the backward Euler method has been employed. We present here a new linearization scheme for the nonlinear system arising after the temporal and spatial discretization. We show that the scheme is linearly convergent. Numerical experiments are presented that sustain the theoretical results.
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| 6. |
- Reveron, Manuel Antonio Borregales, et al.
(författare)
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Iterative solvers for Biot model under small and large deformations
- 2021
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Ingår i: Computational Geosciences. - : Springer. - 1420-0597 .- 1573-1499. ; 25, s. 687-699
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Tidskriftsartikel (refereegranskat)abstract
- We considerL-scheme and Newton-based solvers for Biot model under large deformation. The mechanical deformation follows the Saint Venant-Kirchoff constitutive law. Furthermore, the fluid compressibility is assumed to be non-linear. A Lagrangian frame of reference is used to keep track of the deformation. We perform an implicit discretization in time (backward Euler) and propose two linearization schemes for solving the non-linear problems appearing within each time step: Newton's method andL-scheme. Each linearization scheme is also presented in a monolithic and a splitting version, extending the undrained split methods to non-linear problems. The convergence of the solvers, here presented, is shown analytically for cases under small deformation and numerically for examples under large deformation. Illustrative numerical examples are presented to confirm the applicability of the schemes, in particular, for large deformation.
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| 7. |
- Vasilyev, Leonid, et al.
(författare)
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On the Properties of the Parameter Space of the Generalized Continuum Transport Model for Description of Fluid Flow in Porous Networks
- 2017
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Ingår i: Transport in Porous Media. - : Springer Science and Business Media LLC. - 0169-3913 .- 1573-1634. ; 119:3, s. 673-688
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Tidskriftsartikel (refereegranskat)abstract
- Generalized transport models, such as Dual and Multiple Continua Models, Global Random Walk, Multirate Mass Transfer and Continuous Time Random Walk are widely used for description of anomalous transport in fractured and porous media. For these models the form of the parameter space is crucial for the most accurate description of anomalous effects as well as the mean transport phenomenon. Constraining of the parameter space is required for the proper interpretation of the physical properties taking place. In this study the Generalized Continuum Transport model is considered as a versatile tool for the parameter space selection as well as better quantification of anomalous (non-Fickian) transport. Different variants of the parameter space are applied to the GCT model and the breakthrough curves obtained from the pore-network models with strong anomalities are fitted. Flexibility of the model is demonstrated through its static and dynamic adaptivity to network structure and transport complexity. The beneficial results of the curve fitting are also compared with the classical models. It is thus demonstrated that the complexity of the model as well as the model parameters can be directly determined based on fine-scale simulations.
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