| 1. |
- Almani, T., et al.
(författare)
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Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics
- 2016
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 311, s. 180-207
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Tidskriftsartikel (refereegranskat)abstract
- We consider multirate iterative schemes for the Biot system modeling coupled flow and geomechanics in a poro-elastic medium. The multirate iterative coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step. We adapt the fixed stress split algorithm that decouples the flow and mechanics equations for the multirate case and perform an iteration between the two problems until convergence. We provide a fully discrete scheme that uses Backward Euler time discretization and mixed spaces for flow and conformal Galerkin for mechanics. Our analysis is based on studying the equations satisfied by the difference of iterates and using Banach contraction argument to prove that the corresponding scheme is a fixed point contraction. The analysis provides the value of an adjustable coefficient used in the proposed iterative coupling algorithms. Furthermore, we show that the converged quantities satisfy the variational weak form for the coupled discrete system. (C) 2016 Elsevier B.V. All rights reserved.
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| 2. |
- Almani, T., et al.
(författare)
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Convergence and error analysis of fully discrete iterative coupling schemes for coupling flow with geomechanics
- 2017
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Ingår i: Computational Geosciences. - : Springer. - 1420-0597 .- 1573-1499. ; 21:5-6, s. 1157-1172
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider an iterative coupling scheme for solving a fully discretized Biot system based on the fixed-stress split coupling algorithm. Specifically, we derive a priori error estimates for quantifying the error between the solution obtained at any iterate and the true solution. Our approach is based on studying the equations satisfied by the difference of iterates and utilizing a Banach contraction argument to show that the corresponding scheme is a fixed point iteration. Obtained contraction results are then used to derive theoretical convergence error estimates for the single rate iterative coupling scheme. We compare our numerical computations against the theoretically derived contraction estimates and show a good agreement with theory.
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| 3. |
- Almani, T., et al.
(författare)
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Convergence of the undrained split iterative scheme for coupling flow with geomechanics in heterogeneous poroelastic media
- 2019
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Ingår i: Computational Geosciences. - : Springer. - 1420-0597 .- 1573-1499. ; 24:2, s. 551-569
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Tidskriftsartikel (refereegranskat)abstract
- Recently, an accurate coupling between subsurface flow and reservoir geomechanics has received more attention in both academia and industry. This stems from the fact that incorporating a geomechanics model into upstream flow simulation is critical for accurately predicting wellbore instabilities and hydraulic fracturing processes. One of the recently introduced iterative coupling algorithms to couple flow with geomechanics is the undrained split iterative coupling algorithm as reported by Kumar et al. (2016) and Mikelic and Wheeler (Comput. Geosci. 17: 455–461 2013). The convergence of this scheme is established in Mikelic and Wheeler (Comput. Geosci. 17:455–461 2013) for the single rate iterative coupling algorithm and in Kumar et al. (2016) for the multirate iterative coupling algorithm, in which the flow takes multiple finer time steps within one coarse mechanics time step. All previously established results study the convergence of the scheme in homogeneous poroelastic media. In this work, following the approach in Almani et al. (2017), we extend these results to the case of heterogeneous poroelastic media, in which each grid cell is associated with its own set of flow and mechanics parameters for both the single rate and multirate schemes. Second, following the approach in Almani et al. (Comput. Geosci. 21:1157–1172 2017), we establish a priori error estimates for the single rate case of the scheme in homogeneous poroelastic media. To the best of our knowledge, this is the first rigorous and complete mathematical analysis of the undrained split iterative coupling scheme in heterogeneous poroelastic media.
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| 4. |
- Almani, T., et al.
(författare)
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Stability of multirate explicit coupling of geomechanics with flow in a poroelastic medium
- 2019
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Ingår i: Computers and Mathematics with Applications. - : Elsevier. - 0898-1221 .- 1873-7668. ; 78:8, s. 2682-2699
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Tidskriftsartikel (refereegranskat)abstract
- We consider single rate and multirate explicit schemes for the Biot system modeling coupled flow and geomechanics in a poro-elastic medium. These schemes are widely used in practice that follows a sequential procedure in which the flow and mechanics problems are fully decoupled. In such a scheme, the flow problem is solved first with time-lagging the displacement term followed by the mechanics solve. The multirate explicit coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step. We provide fully discrete schemes for both the single and multirate approaches that use Backward Euler time discretization and mixed spaces for flow and conformal Galerkin for mechanics. We perform a rigorous stability analysis and derive the conditions on reservoir parameters and the number of finer flow solves to ensure stability for both schemes. Furthermore, we investigate the computational time savings for explicit coupling schemes against iterative coupling schemes.
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| 5. |
- Bogers, J., et al.
(författare)
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A multiscale domain decomposition approach for chemical vapor deposition
- 2013
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Ingår i: Journal of Computational and Applied Mathematics. - Amsterdam, Netherlands : Elsevier. - 0377-0427 .- 1879-1778. ; 246, s. 65-73
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Tidskriftsartikel (refereegranskat)abstract
- We consider the process of chemical vapor deposition on a trenched Si substrate. To understand the process (including e.g. the layer conformality) at the trench scale (microscale), we need solutions at both the trench and reactor scales (macroscale). Due to the huge difference in size of these scales, straightforward numerical computations are very challenging. To overcome this difficulty, we consider a multiscale approach by introducing an intermediate scale (the mesoscale). We start with a time-continuous model describing the transport processes and then perform time discretization. At each time step, using the ideas of domain decomposition inspired from Lions (1988) [4], we provide iterative coupling conditions for these three different scales. Using a weak formulation for the time-discrete equations, we prove the convergence of this iterative scheme at each time step. The approach also provides an alternative proof for the existence of the solutions for the time-discrete formulation. (C) 2012 Elsevier B.V. All rights reserved.
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| 6. |
- Borregales, Manuel, et al.
(författare)
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Robust iterative schemes for non-linear poromechanics
- 2018
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Ingår i: Computational Geosciences. - Dordrecht : Springer. - 1420-0597 .- 1573-1499. ; 22:4, s. 1021-1038
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Tidskriftsartikel (refereegranskat)abstract
- We consider a non-linear extension of Biot's model for poromechanics, wherein both the fluid flow and mechanical deformation are allowed to be non-linear. Specifically, we study the case when the volumetric stress and the fluid density are non-linear functions satisfying certain assumptions. We perform an implicit discretization in time (backward Euler) and propose two iterative schemes for solving the non-linear problems appearing within each time step: a splitting algorithm extending the undrained split and fixed stress methods to non-linear problems, and a monolithic L-scheme. The convergence of both schemes are shown rigorously. Illustrative numerical examples are presented to confirm the applicability of the schemes and validate the theoretical results.
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| 7. |
- Both, Jakub W., et al.
(författare)
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Iterative Methods for Coupled Flow and Geomechanics in Unsaturated Porous Media
- 2017
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Ingår i: Poromechanics VI. - : American Society of Civil Engineers (ASCE). - 9780784480779 ; , s. 411-418
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Konferensbidrag (refereegranskat)abstract
- This work concerns the linearization of a three-field discretization of generalized Biot's equations describing coupled fluid flow and mechanical deformation in unsaturated porous media. The model of interest employs the effective stress based on the so-called equivalent pore pressure and can be interpreted as linear mechanics nonlinearly coupled with Richards' equation. As linearization, we apply simultaneously the L-scheme and the Fixed Stress Splitting scheme, which have been established and analyzed for Richards' equation and the linear Biot's equations, respectively. Numerical results demonstrate robustness and mesh independent convergence rates, whereas the popular, locally convergent Newton's method does not display robust convergence for the numerical examples we present.
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| 8. |
- 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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| 9. |
- Bringedal, Carina, et al.
(författare)
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Effective Behavior Near Clogging in Upscaled Equations for Non-isothermal Reactive Porous Media Flow
- 2017
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Ingår i: Transport in Porous Media. - : Springer Science and Business Media LLC. - 0169-3913 .- 1573-1634. ; 120:3, s. 553-577
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Tidskriftsartikel (refereegranskat)abstract
- For a non-isothermal reactive flow process, effective properties such as permeability and heat conductivity change as the underlying pore structure evolves. We investigate changes of the effective properties for a two-dimensional periodic porous medium as the grain geometry changes. We consider specific grain shapes and study the evolution by solving the cell problems numerically for an upscaled model derived in Bringedal et al. (Transp Porous Media 114(2):371-393, 2016. doi 10.1007/s11242-015-0530-9). In particular, we focus on the limit behavior near clogging. The effective heat conductivities are compared to common porosity-weighted volume averaging approximations, and we find that geometric averages perform better than arithmetic and harmonic for isotropic media, while the optimal choice for anisotropic media depends on the degree and direction of the anisotropy. An approximate analytical expression is found to perform well for the isotropic effective heat conductivity. The permeability is compared to some commonly used approaches focusing on the limiting behavior near clogging, where a fitted power law is found to behave reasonably well. The resulting macroscale equations are tested on a case where the geochemical reactions cause pore clogging and a corresponding change in the flow and transport behavior at Darcy scale. As pores clog the flow paths shift away, while heat conduction increases in regions with lower porosity.
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| 10. |
- Endo Kokubun, M. A., et al.
(författare)
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A pore-scale study of transport of inertial particles by water in porous media
- 2019
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Ingår i: Chemical Engineering Science. - : Elsevier. - 0009-2509 .- 1873-4405. ; 207, s. 397-409
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Tidskriftsartikel (refereegranskat)abstract
- We study the transport of inertial particles in water flow in porous media. Our interest lies in understanding the accumulation of particles including the possibility of clogging. We propose that accumulation can be a result of hydrodynamic effects: the tortuous paths of the porous medium generate regions of dominating strain, which favour the accumulation of particles. Numerical simulations show that essentially two accumulation regimes are identified: for low and for high flow velocities. When particles accumulate at the entrance of a pore throat (high-velocity region), a clog is formed. This significantly modifies the flow, as the partial blockage of the pore causes a local redistribution of pressure, which diverts the upstream water flow into neighbouring pores. Moreover, we show that accumulation in high velocity regions occurs in heterogeneous media, but not in homogeneous media, where we refer to homogeneity with respect to the distribution of the pore throat diameters.
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