| 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)
-
Convergence and error analysis of fully discrete iterative coupling schemes for coupling flow with geomechanics
- 2017
-
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
-
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
-
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
-
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
-
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
-
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
-
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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| 11. |
- Ganis, Benjamin, et al.
(författare)
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A global Jacobian method for mortar discretizations of a fully implicit two-phase flow model
- 2014
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Ingår i: Multiscale Modeling & simulation. - : SIAM Publications. - 1540-3459 .- 1540-3467. ; 12:4, s. 1401-1423
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Tidskriftsartikel (refereegranskat)abstract
- We consider a fully implicit formulation for two-phase flow in a porous medium with capillarity, gravity, and compressibility in three dimensions. The method is implicit in time and uses the multiscale mortar mixed finite element method for a spatial discretization in a nonoverlapping domain decomposition context. The interface conditions between subdomains are enforced in terms of Lagrange multiplier variables defined on a mortar space. The novel approach in this work is to linearize the coupled system of subdomain and mortar variables simultaneously to form a global Jacobian. This algorithm is shown to be more efficient and robust compared to previous algorithms that relied on two separate nested linearizations of subdomain and interface variables. We also examine various upwinding methods for accurate integration of phase mobility terms near subdomain interfaces. Numerical tests illustrate the computational benefits of this scheme.
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| 12. |
- Girault, Vivette, et al.
(författare)
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Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium
- 2016
-
Ingår i: Computational Geosciences. - Dordrecht, Netherlands : Springer Science and Business Media LLC. - 1420-0597 .- 1573-1499. ; 20:5, s. 997-1011
-
Tidskriftsartikel (refereegranskat)abstract
- We consider an iterative scheme for solving a coupled geomechanics and flow problem in a fractured poroelastic medium. The fractures are treated as possibly non-planar interfaces. Our iterative scheme is an adaptation due to the presence of fractures of a classical fixed stress-splitting scheme. We prove that the iterative scheme is a contraction in an appropriate norm. Moreover, the solution converges to the unique weak solution of the coupled problem.
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| 13. |
- Gjerde, Ingeborg G., et al.
(författare)
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A singularity removal method for coupled 1D-3D flow models
- 2020
-
Ingår i: Computational Geosciences. - : Springer. - 1420-0597 .- 1573-1499. ; 24:2, s. 443-457
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Tidskriftsartikel (refereegranskat)abstract
- In reservoir simulations, the radius of a well is inevitably going to be small compared to the horizontal length scale of the reservoir. For this reason, wells are typically modelled as lower-dimensional sources. In this work, we consider a coupled 1D-3D flow model, in which the well is modelled as a line source in the reservoir domain and endowed with its own 1D flow equation. The flow between well and reservoir can then be modelled in a fully coupled manner by applying a linear filtration law. The line source induces a logarithmic-type singularity in the reservoir pressure that is difficult to resolve numerically. We present here a singularity removal method for the model equations, resulting in a reformulated coupled 1D-3D flow model in which all variables are smooth. The singularity removal is based on a solution splitting of the reservoir pressure, where it is decomposed into two terms: an explicitly given, lower-regularity term capturing the solution singularity and some smooth background pressure. The singularities can then be removed from the system by subtracting them from the governing equations. Finally, the coupled 1D-3D flow equations can be reformulated so they are given in terms of the well pressure and the background reservoir pressure. As these variables are both smooth (i.e. non-singular), the reformulated model has the advantage that it can be approximated using any standard numerical method. The reformulation itself resembles a Peaceman well correction performed at the continuous level.
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| 14. |
- Gjerde, Ingeborg G., et al.
(författare)
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Splitting method for elliptic equations with line sources
- 2019
-
Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 53:5, s. 1715-1739
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain omega when the right-hand side is a (1D) line source ?. The analysis and approximation of such problems is known to be non-standard as the line source causes the solution to be singular. Our main result is a splitting theorem for the solution; we show that the solution admits a split into an explicit, low regularity term capturing the singularity, and a high-regularity correction term w being the solution of a suitable elliptic equation. The splitting theorem states the mathematical structure of the solution; in particular, we find that the solution has anisotropic regularity. More precisely, the solution fails to belong to H-1 in the neighbourhood of ?, but exhibits piecewise H-2-regularity parallel to ?. The splitting theorem can further be used to formulate a numerical method in which the solution is approximated via its correction function w. This recasts the problem as a 3D elliptic problem with a 3D right-hand side belonging to L-2, a problem for which the discretizations and solvers are readily available. Moreover, as w enjoys higher regularity than the full solution, this improves the approximation properties of the numerical method. We consider here the Galerkin finite element method, and show that the singularity subtraction then recovers optimal convergence rates on uniform meshes, i.e., without needing to refine the mesh around each line segment. The numerical method presented in this paper is therefore well-suited for applications involving a large number of line segments. We illustrate this by treating a dataset (consisting of similar to 3000 line segments) describing the vascular system of the brain.
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| 15. |
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| 16. |
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| 17. |
- Kumar, Kundan, et al.
(författare)
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Convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media
- 2014
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Ingår i: Numerische Mathematik. - Heidelberg, Germany : Springer. - 0029-599X .- 0945-3245. ; 127:4, s. 715-749
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we discuss the numerical analysis of an upscaled (core scale) model describing the transport, precipitation and dissolution of solutes in a porous medium. The particularity lies in the modeling of the reaction term, especially the dissolution term, which has a multivalued character. We consider the weak formulation for the upscaled equation and provide rigorous stability and convergence results for both the semi-discrete (time discretization) and the fully discrete schemes. In doing so, compactness arguments are employed.
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| 18. |
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| 19. |
- Kumar, Kundan, et al.
(författare)
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Effective dispersion equations for reactive flows involving free boundaries at the microscale
- 2011
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Ingår i: Multiscale Modeling & simulation. - : Society for Industrial and Applied Mathematics. - 1540-3459 .- 1540-3467. ; 9:1, s. 29-58
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Tidskriftsartikel (refereegranskat)abstract
- We consider a pore-scale model for reactive flow in a thin two-dimensional strip, where the convective transport dominates the diffusion. Reactions take place at the lateral boundaries of the strip (the walls), where the reaction product can deposit in a layer with a nonnegligible thickness compared to the width of the strip. This leads to a free boundary problem, in which the moving interface between the fluid and the deposited (solid) layer is explicitly taken into account. Using asymptotic expansion methods, we derive an upscaled, one-dimensional model by averaging in the transversal direction. The result is consistent with (Taylor dispersion) models obtained previously for a constant geometry. Finally, numerical computations are presented to compare the outcome of the effective (upscaled) model with the transversally averaged, two-dimensional solution.
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| 20. |
- Kumar, Kundan, et al.
(författare)
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Formal upscaling and numerical validation of unsaturated flow models in fractured porous media
- 2020
-
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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| 21. |
- Kumar, Kundan, et al.
(författare)
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Homogenization of a pore scale model for precipitation and dissolution in porous media
- 2016
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Ingår i: IMA Journal of Applied Mathematics. - : Oxford University Press. - 0272-4960 .- 1464-3634. ; 81:5, s. 877-897
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we employ homogenization techniques to provide a rigorous derivation of the Darcy scale model for precipitation and dissolution in porous media. The starting point is the pore scale model in van Duijn & Pop (2004), which is a coupled system of evolution equations, involving a parabolic equation which models ion transport in the fluid phase of a periodic porous medium, coupled to an ordinary differential equations modelling dissolution and precipitation at the grains boundary. The main challenge is in dealing with the dissolution and precipitation rates, which involve a monotone but possibly discontinuous function. In order to pass to the limit in these rate functions at the boundary of the grains, we prove strong two-scale convergence for the concentrations at the microscopic boundary and use refined arguments in order to identify the form of the macroscopic dissolution rate, which is again a discontinuous function. The resulting upscaled model is consistent with the Darcy scale model proposed in Knabner et al. (1995).
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| 22. |
- Kumar, Kundan, et al.
(författare)
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Multirate Undrained Splitting for Coupled Flow and Geomechanics in Porous Media
- 2016
-
Ingår i: NUMERICAL MATHEMATICS AND ADVANCED APPLICATIONS (ENUMATH 2015). - Cham : Springer Publishing Company. - 9783319399294 - 9783319399270 ; , s. 431-440
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Konferensbidrag (refereegranskat)abstract
- We consider a multirate iterative scheme for the quasi-static Biot equations modelling the coupled flow and geomechanics in a porous medium. The iterative scheme is based on undrained splitting where the flow and mechanics equations are decoupled with the mechanics solve followed by the pressure solve. The multirate scheme proposed here uses different time steps for the two equations, that is, uses q flow steps for each coarse mechanics step and may be interpreted as using a regularization parameter for the mechanics equation. We prove the convergence of the scheme and the proof reveals the appropriate regularization parameter and also the effect of the number of flow steps within coarse mechanics step on the convergence rate.
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| 23. |
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| 24. |
- Kumar, Kundan, et al.
(författare)
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Reactive Flow And Reaction-Induced Boundary Movement In A Thin Channel
- 2013
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM PUBLICATIONS. - 1064-8275 .- 1095-7197. ; 35:6, s. B1235-B1266
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Tidskriftsartikel (refereegranskat)abstract
- We study the reactive flow in a thin strip where the geometry changes take place due to reactions. Specifically, we consider precipitation-dissolution processes taking place at the lateral boundaries of the strip. The geometry changes depend on the concentration of the solute in the bulk (trace of the concentration), which makes the problem a free-moving boundary problem. The numerical computations are challenging in view of the nonlinearities in the description of the reaction rates. In addition to this, the movement of the boundary depends on the unknown concentration (and hence part of the solution), and the computation of the coupled model remains a delicate issue. Our aim is to develop appropriate numerical techniques for the computation of the solutions of the coupled convection-diffusion problem and the equation describing the geometry changes. The performance is demonstrated with the help of several numerical tests.
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| 25. |
- Kumar, Kundan, et al.
(författare)
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Rigorous upscaling of rough boundaries for reactive flows
- 2014
-
Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : John Wiley & Sons. - 0044-2267 .- 1521-4001. ; 94:7-8, s. 623-644
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Tidskriftsartikel (refereegranskat)abstract
- We consider a mathematical model for reactive flow in a channel having a rough (periodically oscillating) boundary with both period and amplitude ε. The ions are being transported by the convection and diffusion processes. These ions can react at the rough boundaries and get attached to form the crystal (precipitation) and become immobile. The reverse process of dissolution is also possible. The model involves non‐linear and multi‐valued rates and is posed in a fixed geometry with rough boundaries. We provide a rigorous justification for the upscaling process in which we define an upscaled problem defined in a simpler domain with flat boundaries. To this aim, we use periodic unfolding techniques combined with translation estimates. Numerical experiments confirm the theoretical predictions and illustrate a practical application of this upscaling process.
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| 26. |
- Kumar, Kundan, et al.
(författare)
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Upscaling Of Reactive Flows In Domains With Moving Oscillating Boundaries
- 2014
-
Ingår i: Discrete and Continuous Dynamical Systems. Series S. - : American Institute of Mathematical Sciences. - 1937-1632 .- 1937-1179. ; 7:1, s. 95-111
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Tidskriftsartikel (refereegranskat)abstract
- We consider the flow and transport of chemically reactive substances (precursors) in a channel over substrates having complex geometry. In particular, these substrates are in the form of trenches forming oscillating boundaries. The precursors react at the boundaries and get deposited. The deposited layers lead to changes in the geometry and are explicitly taken into account. Consequently, the system forms a free boundary problem. Using formal asymptotic techniques, we obtain the upscaled equations for the system where these equations are defined on a domain with flat boundaries. This provides a huge gain in computational time. Numerical experiments show the effectiveness of the upscaling process.
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| 27. |
- Landa-Marban, David, et al.
(författare)
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An Upscaled Model for Permeable Biofilm in a Thin Channel and Tube
- 2020
-
Ingår i: Transport in Porous Media. - : Springer. - 0169-3913 .- 1573-1634. ; 132:1, s. 83-112
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we derive upscaled equations for modeling biofilm growth in porous media. The resulting macroscale mathematical models consider permeable multi-species biofilm including water flow, transport, detachment and reactions. The biofilm is composed of extracellular polymeric substances (EPS), water, active bacteria and dead bacteria. The free flow is described by the Stokes and continuity equations, and the water flux inside the biofilm by the Brinkman and continuity equations. The nutrients are transported in the water phase by convection and diffusion. This pore-scale model includes variations in the biofilm composition and size due to reproduction of bacteria, production of EPS, death of bacteria and shear forces. The model includes a water-biofilm interface between the free flow and the biofilm. Homogenization techniques are applied to obtain upscaled models in a thin channel and a tube, by investigating the limit as the ratio of the aperture to the length epsilon of both geometries approaches to zero. As epsilon gets smaller, we obtain that the percentage of biofilm coverage area over time predicted by the pore-scale model approaches the one obtained using the effective equations, which shows a correspondence between both models. The two derived porosity-permeability relations are compared to two empirical relations from the literature. The resulting numerical computations are presented to compare the outcome of the effective (upscaled) models for the two mentioned geometries.
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| 28. |
- 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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| 29. |
- Pop, Iuliu Sorin, et al.
(författare)
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Analysis and Upscaling of a Reactive Transport Model in Fractured Porous Media with Nonlinear Transmission Condition
- 2017
-
Ingår i: Vietnam Journal of Mathematics. - Singapore, Singapore : Springer. - 2305-221X .- 2305-2228. ; 45:1-2, s. 77-102
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Tidskriftsartikel (refereegranskat)abstract
- We consider a reactive transport model in a fractured porous medium. The particularity appears in the conditions imposed at the interface separating the block and the fracture, which involves a nonlinear transmission condition. Assuming that the fracture has thickness epsilon, we analyze the resulting problem and prove the convergence toward a reduced model in the limit epsilon a dagger y0. The result is a model defined on an interface (the reduced fracture) and acting as a boundary condition for the equations defined in the block. Using both formal and rigorous arguments, we obtain the reduced models for different flow regimes, expressed through a moderate or a high P,clet number.
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| 30. |
- 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
-
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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| 31. |
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| 32. |
- 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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| 33. |
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| 34. |
- Salama, Amgad, et al.
(författare)
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Flow and transport in tight and shale formations : A review
- 2017
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Ingår i: Geofluids. - : Hindawi Publishing Corporation. - 1468-8115 .- 1468-8123. ; , s. 1-21
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Forskningsöversikt (refereegranskat)abstract
- A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.
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| 35. |
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| 36. |
- Storvik, Erlend, et al.
(författare)
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On the optimization of the fixed-stress splitting for Biot’s equations
- 2019
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Ingår i: International Journal for Numerical Methods in Engineering. - : John Wiley & Sons. - 0029-5981 .- 1097-0207. ; 120:2, s. 179-194
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we are interested in efficiently solving the quasi-static, linear Biot model for poroelasticity. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot’s equations. It is well known that the convergence properties of the method strongly depend on the applied stabilization/tuning parameter. We show theoretically that, in addition to depending on the mechanical properties of the porous medium and the coupling coefficient, they also depend on the fluid flow and spatial discretization properties. The type of analysis presented in this paper is not restricted to a particular spatial discretization, although it is required to be inf-sup stable with respect to the displacement-pressure formulation. Furthermore, we propose a way to optimize this parameter that relies on the mesh independence of the scheme’s optimal stabilization parameter. Illustrative numerical examples show that using the optimized stabilization parameter can significantly reduce the number of iterations.
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| 37. |
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| 38. |
- 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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