European Union’s Horizon 2020 research and innovation programme

Marie Sklodowska-Curie grant agreement NO 705402, POROSOS

OVERVIEW:

Over the last 150 years, the global temperature has risen at an accelerating pace. Human activity through emissions of greenhouse gases are considered the main cause of this relentless warming. Carbon dioxide (CO2) is one of the major greenhouse gases, which is most challenging to deal with, due to its prevalence as a by-product from industrial processes and electricity generation. To reduce greenhouse gas emissions into the atmosphere, CO2 capture and storage is considered as a highly relevant technology, and its study is therefore under active consideration. Due to their ubiquitous presence, deep saline aquifers provide the most substantial carbon dioxide storage capacity. The injection of CO2 into deep saline aquifers typically results in elevated pressure in the vicinity of the injection well. Due to the high injection pressure, the stress distribution in the reservoir region can change significantly, and therefore deformation of the porous medium must be considered to guarantee safety assessments of the injection process. This can result into uplift, fracture formation, and activation of existing faults. Therefore, there are potential risks to humans and ecosystems that arise from the leakage of CO2, or the displacement of saltwater from the saline to the fresh-water aquifers. The aim here is to advance in the applied mathematics techniques needed in this context. We will study modern numerical techniques and novel concepts, to be able to make a significant step forward in the numerical simulation of CO2 storage research. This involves detailed analysis to ensure accuracy of the numerical solutions, development of highly efficient multilevel solution methods for complicated governing nonlinear systems of partial differential equations (PDEs), but also uncertainty quantification (UQ) and the corresponding solution techniques.

In this project we have worked in the following topics:

· Stable discretizations for poroelasticity (joint work with C. Rodrigo, L. Zikatanov, X. Hu, J. Adler, P. Ohm)

· Fast solvers for multiphysic problems (joint work with K. Oosterlee, P. Luo, C. Rodrigo)

· Multilevel Monte Carlo methods for uncertainity quantification for porous media flows. (joint work with P. Kumar, C. Oosterlee, C. Rodrigo).

· Fast solvers for poroelasticity problems (joint work with C. Rodrigo, A. Radu, K. Kumar, M. Borregales)

· Fast solvers for the time fractional heat equation (joint work with C. Rodrigo, X. Hu)

· Fast solvers for fractured porous media (joint work with C. Rodrigo, A. Arraras, L. Portero)

· Fast solvers for isogeometric analysis (joint work with A. Pe de la Riva, C. Rodrigo).

PUBLICATIONS:

P1. “Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System”, Peiyao Luo, Carmen Rodrigo, Francisco J. Gaspar, Cornelis W. Oosterlee, SIAM Journal on Scientific Computing, S633-S661 (2017).

P2. “Multigrid Waveform Relaxation for the Time-Fractional Heat Equation”. Francisco J. Gaspar, Carmen Rodrigo. SIAM Journal on Scientific Computing 39, A1201-A1224 (2017).

P3. “On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics”. Francisco J. Gaspar, Carmen Rodrigo. Computer Methods in Applied Mechanics and Engineering 326, 526-540 (2017).

P4. “Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system”. P. Luo, C. Rodrigo, F.J. Gaspar, C.W. Oosterlee. Journal of Computational Physics 353, 148-168, (2018).

P5. “A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system”. Prashant Kumar, Peiyao Luo, Francisco J. Gaspar, Cornelis W. Oosterlee. Journal of Computational Physics 371, 382-408 (2018).

P6. “On cell-centered multigrid methods and Local Fourier Analysis for PDEs with random coefficients”. Kumar, Prashant; Rodrigo, Carmen; Gaspar, Francisco J.; Oosterlee, Cornelis W. Submitted. ArXiv: http://arxiv.org/abs/1803.08864.

P7. “Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation”. Hu, Xiaozhe; Rodrigo, Carmen; Gaspar, Francisco J. Submitted. ArXiv: http://arxiv.org/abs/1706.07632.

P8. “New stabilized discretizations for poroelasticity and the Stokes’ equations”. C. Rodrigo, X. Hu, P. Ohm, J.H. Adler, F.J. Gaspar, L.T. Zikatanov. Computer Methods in Applied Mechanics and Engineering 341. 467-484 (2018)

P9. “A parallel-in-time fixed-stress splitting method for Biot's consolidation model”. Borregales, Manuel; Kumar, Kundan; Radu, Florin Adrian; Rodrigo, Carmen; Gaspar, Francisco José. Computers & Mathematics with Applications. Accepted (2018). ArXiv: http://arxiv.org/abs/1802.00949.

P10. “On an Uzawa smoother in multigrid for poroelasticity equations”. P. Luo, C. Rodrigo, F. J. Gaspar, C. W. Oosterlee. Numerical Linear Algebra with Applications 24 (2017).

P11. “On The Validity of the Local Fourier Analysis”. Carmen Rodrigo, Francisco J. Gaspar, Ludmil T. Zikatanov. Journal of Computational Mathematics 37, 340-348 (2018).

P12. “An efficient multigrid solver for isogeometric analysis”. Pé de la Riva, Alvaro, Rodrigo, Carmen, Gaspar, Francisco. Submitted. https://arxiv.org/abs/1806.05848.

P13. “Monotone difference schemes for weakly coupled elliptic and parabolic systems”. P. Matus, F. J. Gaspar, V.T.K. Tuyen. Computational Methods in Applied Mathematics 15, 287-298 (2017).

P14. “Geometric multigrid methods for Darcy-Forchheimer flow in fractured porous media”, A. Arrarás, F.J. Gaspar, L. Portero, C. Rodrigo. In preparation.

P15. “Mixed-dimensional multi-grid methods for fractured porous media”, A. Arrarás, F.J. Gaspar, L. Portero, C. Rodrigo. In preparation.

CONFERENCES:

Plenary talks

C1. Ninth International Conference on Numerical Methods and Applications (NM\&A'18), August 20-24, 2018, Borovets, Bulgaria. “ Stabilized finite element discretizations for poroelasticity”.

C2. Valencia Numérica 2017, July 17-20, 2017, Valencia, Spain. “Multigrid waveform relaxation. Application to the time-fractional heat equation.”

Invited talks

C3. International Conference on Domain Decomposition Methods (DD XXIV), February 6-10, 2017, Svalbard, Longyearbyen, Norway. “A segregated Uzawa smoother in multigrid for poroelastic problems. “

C4. SIAM Conference on Computational Science and Engineering, February 27 - March 3, 2017, Atlanta, Georgia, USA. “About the Uzawa smoother for poroelastic problems.”

C5. 11th International Conference on Large-Scale Scientific Computations, June 5-9, 2017, Sozopol, Bulgaria. “Multigrid Treatment of Poroelasticity System.”

C6. SIAM Conference on Mathematical and Computational Issues in the Geosciences (SIAM GS 2017), September 11-14, 2017, Erlangen, Germany. “A new stabilized discretization for poroelasticity.”

C7. 7th International Conference on Advanced Computational Methods in Engineering (ACOMEN2017), September 18-22, 2017, Ghent, Belgium. “A new iterative algorithm based on the fixed-stress split scheme for solving the Biot's problem.”

C8. International Workshop on Flow in Deformable Porous Media: Numerics and Benchmarks, December 4-6, 2017, Hamburg, Germany. “Stable discretizations and fast solvers based on multigrid methods on semi-structured grids. “

C9. Finse workshop on efficient solvers for fractured porous media, January 9-11, 2018, Finse, Norway. “Multigrid methods on semi-structured grids.”

C10. 8th International Conference Computational Methods in Applied Mathematics (CMAM-8), July 2-6, 2018, Minsk, Belarus. “Stabilization techniques for finite element discretizations in poroelasticity.”

Contributed talks

C11. 18th Copper Mountain Conference on Multigrid Methods, March 26-30, 2017, Copper Mountain, Colorado, USA. “Seggregated smoothers in the multigrid treatment of poroelasticity system.

Mini-symposium organization

· European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2017), September 25-29, 2017, Voss, Norway. “Discretizations and solvers for multi-physics problems.

Workshop organization

The Computational Mathematics Aspects of Porous Media, and Fluid Flow. (2018).

In addition to the scientific dissemination of the results, I have recently given an invited talk for a general audience about my experience with the MSCA grant:

F.J. Gaspar, “Attractive opportunities for young researchers in European projects”, Virtual Physiological Human Conference VPH2018, Zaragoza, 5-7 September 2018.