Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media
Abstract
:1. Introduction
2. Problem Formulation and Approximation on a Fine Grid
- The sequential division of the intersection segments and the fracture boundaries by points into segments of a given size, which are subsequently used as mesh faces;
- Uniform filling of fracture regions by points with a restriction on the proximity of internal points to the boundary and points on intersection segments;
- Construction of triangulation for each fracture according to the given frameworks (points, segments).
3. Multiscale Method for a Coarse-Grid Approximation on Offline Space
- Coarse grid and local domain construction.
- The solution of the local problems with different boundary conditions to construct a snapshot space in each local domain.
- An offline space construction via the solution of the local spectral problems on the snapshot space.
- Generation of the projection matrix.
- Construction of the coarse grid system using the projection matrix.
- Solving the problem on the coarse grid at the current time step.
- Moving to the next time step.
4. Online Enrichment of Multiscale Space
- Construction of the coarse-grid system using the projection matrix.
- Solving the problem on the coarse grid at the current time step.
- Multiscale space enrichment by calculation of the online basis functions. We enrich the offline space and update the projection matrix using the obtained online basis functions. After that, we repeatedly solve the current time step problem on the coarse grid to update the solution.
- Moving to the next time step.
- Define projection matrix for the current time step n with for ( is the projection matrix from the previous time step).
- Construct and solve the coarse-scale problem at the current time step.
- –
- If we want to add/update online basis functions for the current time step, we solveFor the projection matrix, we have for andHere, online basis functions are calculated using the solution from the previous iteration .
- –
- Otherwise, we solve
- Move to the next time step.
5. Numerical Results
- Case 1. Poroelasticity problem in heterogeneous media with the following parameters: , . The calculation is performed by with 50 time steps ().
- Case 2. Poroelasticity problem in heterogeneous and fractured media with the following parameters: . The calculation is performed by with 50 time steps ().
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
References
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() | () | ||||||||
---|---|---|---|---|---|---|---|---|---|
363(0) | 1.012 | 22.64 | 2.713 | 14.82 | 363(0) | 1.012 | 22.64 | 2.713 | 14.82 |
726(1) | 0.128 | 4.080 | 0.212 | 3.289 | 726(1) | 0.127 | 4.078 | 0.212 | 3.288 |
1089(2) | 0.093 | 2.876 | 0.101 | 2.153 | 1089(2) | 0.092 | 2.869 | 0.101 | 2.152 |
726(0) | 0.605 | 15.28 | 1.110 | 10.32 | 726(0) | 0.605 | 15.28 | 1.110 | 10.32 |
1089(1) | 0.052 | 2.354 | 0.085 | 1.973 | 1089(1) | 0.052 | 2.35 | 0.084 | 1.973 |
1452(2) | 0.032 | 1.495 | 0.053 | 1.201 | 1452(2) | 0.032 | 1.496 | 0.053 | 1.201 |
1452(0) | 0.360 | 12.01 | 0.444 | 6.784 | 1452(0) | 0.360 | 12.01 | 0.444 | 6.784 |
1815(1) | 0.016 | 0.974 | 0.043 | 0.755 | 1815(1) | 0.016 | 0.974 | 0.043 | 0.755 |
2178(2) | 0.010 | 0.590 | 0.021 | 0.504 | 2178(2) | 0.010 | 0.591 | 0.021 | 0.504 |
2904(0) | 0.168 | 6.672 | 0.223 | 4.471 | 2904(0) | 0.168 | 6.672 | 0.223 | 4.471 |
3267(1) | 0.007 | 0.499 | 0.004 | 0.224 | 3267(1) | 0.007 | 0.499 | 0.004 | 0.224 |
3630(2) | 0.004 | 0.361 | 0.002 | 0.156 | 3630(2) | 0.004 | 0.361 | 0.002 | 0.156 |
4356(0) | 0.126 | 5.362 | 0.158 | 3.775 | 4356(0) | 0.126 | 5.362 | 0.158 | 3.775 |
4719(1) | 0.004 | 0.398 | 0.002 | 0.136 | 4719(1) | 0.004 | 0.397 | 0.002 | 0.136 |
5082(2) | 0.002 | 0.211 | 0.001 | 0.098 | 5082(2) | 0.002 | 0.210 | 0.001 | 0.098 |
() | () | ||||||||
---|---|---|---|---|---|---|---|---|---|
726(0) | 52.87 | >100 | >100 | 84.16 | 726(0) | 52.87 | >100 | >100 | 84.16 |
1089(1) | 67.55 | >100 | >100 | >100 | 1089(1) | 61.38 | >100 | 99.71 | >100 |
1452(2) | 27.90 | >100 | 44.57 | 51.76 | 1452(2) | 28.99 | >100 | 51.39 | 54.91 |
1452(0) | 3.297 | 22.77 | 9.126 | 11.98 | 1452(0) | 3.297 | 22.73 | 9.126 | 11.98 |
1815(1) | 1.910 | 16.95 | 4.173 | 8.675 | 1815(1) | 1.896 | 19.96 | 4.093 | 8.670 |
2178(2) | 1.598 | 15.31 | 3.06 | 8.082 | 2178(2) | 1.616 | 15.38 | 2.867 | 8.099 |
2904(0) | 0.719 | 9.834 | 1.402 | 5.645 | 2904(0) | 0.719 | 9.834 | 1.402 | 5.645 |
3267(1) | 0.458 | 7.457 | 0.617 | 4.858 | 3267(1) | 0.460 | 7.457 | 0.632 | 4.859 |
3630(2) | 0.396 | 6.646 | 0.562 | 4.581 | 3630(2) | 0.402 | 6.654 | 0.576 | 4.585 |
4356(0) | 0.301 | 5.697 | 0.618 | 4.245 | 4356(0) | 0.301 | 5.697 | 0.618 | 4.245 |
4719(1) | 0.205 | 4.506 | 0.380 | 3.851 | 4719(1) | 0.205 | 4.506 | 0.382 | 3.851 |
5082(2) | 0.173 | 4.012 | 0.357 | 3.685 | 5082(2) | 0.173 | 4.012 | 0.357 | 3.685 |
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Tyrylgin, A.; Vasilyeva, M.; Ammosov, D.; Chung, E.T.; Efendiev, Y. Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media. Fluids 2021, 6, 298. https://doi.org/10.3390/fluids6080298
Tyrylgin A, Vasilyeva M, Ammosov D, Chung ET, Efendiev Y. Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media. Fluids. 2021; 6(8):298. https://doi.org/10.3390/fluids6080298
Chicago/Turabian StyleTyrylgin, Aleksei, Maria Vasilyeva, Dmitry Ammosov, Eric T. Chung, and Yalchin Efendiev. 2021. "Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media" Fluids 6, no. 8: 298. https://doi.org/10.3390/fluids6080298