Pressure-stabilized fixed-stress iterative solutions of compositional poromechanics. (English) Zbl 07867410
Summary: We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological \(\mathrm{CO}_2\) sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.
References:
| [1] | Benson, S.; Cook, P., IPCC Special Report on Carbon dioxide Capture and Storage, 2005, Cambridge University Press |
| [2] | Rutqvist, J., The geomechanics of CO2 storage in deep sedimentary formations, Geotech. Geol. Eng., 30, 525-551, 2012 |
| [3] | Zienkiewicz, O. C.; Chan, A.; Pastor, M.; Paul, D.; Shiomi, T., Static and dynamic behaviour of soils: a rational approach to quantitative solutions. I. Fully saturated problems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 429, 1877, 285-309, 1990 · Zbl 0725.73074 |
| [4] | Truty, A.; Zimmermann, T., Stabilized mixed finite element formulations for materially nonlinear partially saturated two-phase media, Comput. Methods Appl. Mech. Engrg., 195, 13-16, 1517-1546, 2006 · Zbl 1116.74067 |
| [5] | Aguilar, G.; Gaspar, F.; Lisbona, F.; Rodrigo, C., Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation, Internat. J. Numer. Methods Engrg., 75, 11, 1282-1300, 2008 · Zbl 1158.74473 |
| [6] | Preisig, M.; Prévost, J. H., Stabilization procedures in coupled poromechanics problems: A critical assessment, Int. J. Numer. Anal. Methods Geomech., 35, 11, 1207-1225, 2011 |
| [7] | Wan, J., Stabilized Finite Element Methods for Coupled Geomechanics and Multiphase Flow, 2003, Stanford University |
| [8] | White, J. A.; Borja, R. I., Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput. Methods Appl. Mech. Engrg., 197, 49-50, 4353-4366, 2008 · Zbl 1194.74480 |
| [9] | Berger, L.; Bordas, R.; Kay, D.; Tavener, S., Stabilized lowest-order finite element approximation for linear three-field poroelasticity, SIAM J. Sci. Comput., 37, 5, A2222-A2245, 2015 · Zbl 1326.76054 |
| [10] | Camargo, J. T.; White, J. A.; Borja, R. I., A macroelement stabilization for mixed finite element/finite volume discretizations of multiphase poromechanics, Comput. Geosci., 25, 775-792, 2021 · Zbl 1460.65118 |
| [11] | Frigo, M.; Castelletto, N.; Ferronato, M.; White, J. A., Efficient solvers for hybridized three-field mixed finite element coupled poromechanics, Comput. Math. Appl., 91, 36-52, 2021 · Zbl 1524.76451 |
| [12] | Garipov, T. T.; Tomin, P.; Rin, R.; Voskov, D. V.; Tchelepi, H. A., Unified thermo-compositional-mechanical framework for reservoir simulation, Comput. Geosci., 22, 1039-1057, 2018 · Zbl 1401.86012 |
| [13] | Bui, Q. M.; Osei-Kuffuor, D.; Castelletto, N.; White, J. A., A scalable multigrid reduction framework for multiphase poromechanics of heterogeneous media, SIAM J. Sci. Comput., 42, 2, B379-B396, 2020 · Zbl 1435.65153 |
| [14] | Bui, Q. M.; Hamon, F. P.; Castelletto, N.; Osei-Kuffuor, D.; Settgast, R. R.; White, J. A., Multigrid reduction preconditioning framework for coupled processes in porous and fractured media, Comput. Methods Appl. Mech. Engrg., 387, Article 114111 pp., 2021 · Zbl 1507.74375 |
| [15] | Dean, R. H.; Gai, X.; Stone, C. M.; Minkoff, S. E., A comparison of techniques for coupling porous flow and geomechanics, SPE J., 11, 01, 132-140, 2006 |
| [16] | Park, K., Stabilization of partitioned solution procedure for pore fluid-soil interaction analysis, Internat. J. Numer. Methods Engrg., 19, 11, 1669-1673, 1983 · Zbl 0519.76095 |
| [17] | Settari, A.; Walters, D. A., Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction, SPE J., 6, 03, 334-342, 2001 |
| [18] | Mikelić, A.; Wheeler, M. F., Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 455-461, 2013 · Zbl 1392.35235 |
| [19] | Settari, A.; Mounts, F., A coupled reservoir and geomechanical simulation system, SPE J., 3, 03, 219-226, 1998 |
| [20] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits, Comput. Methods Appl. Mech. Engrg., 200, 13-16, 1591-1606, 2011 · Zbl 1228.74101 |
| [21] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability, accuracy, and efficiency of sequential methods for coupled flow and geomechanics, SPE J., 16, 02, 249-262, 2011 |
| [22] | Castelletto, N.; White, J. A.; Tchelepi, H., Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics, Int. J. Numer. Anal. Methods Geomech., 39, 14, 1593-1618, 2015 |
| [23] | Kim, J.; Tchelepi, H. A.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits, Comput. Methods Appl. Mech. Engrg., 200, 23-24, 2094-2116, 2011 · Zbl 1228.74106 |
| [24] | Janenko, N. N., The Method of Fractional Steps, 1971, Springer · Zbl 0209.47103 |
| [25] | Yoon, H. C.; Kim, J., Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity, Internat. J. Numer. Methods Engrg., 114, 7, 694-718, 2018 · Zbl 07878380 |
| [26] | Storvik, E.; Both, J. W.; Kumar, K.; Nordbotten, J. M.; Radu, F. A., On the optimization of the fixed-stress splitting for Biot’s equations, Internat. J. Numer. Methods Engrg., 120, 2, 179-194, 2019 · Zbl 07859732 |
| [27] | Aronson, R.; Hamon, F.; Castelletto, N.; White, J.; Tchelepi, H., Pressure jump stabilization for compositional poromechanics on unstructured meshes, (SPE Reservoir Simulation Conference, 2023, SPE), Article D011S003R003 pp. |
| [28] | Storvik, E.; Both, J. W.; Kumar, K.; Nordbotten, J. M.; Radu, F. A., On the optimization of the fixed-stress splitting for Biot’s equations, 2018, arXiv preprint arXiv:1811.06242 |
| [29] | Biot, M. A., General theory of three-dimensional consolidation, J. Appl. Phys., 12, 2, 155-164, 1941 · JFM 67.0837.01 |
| [30] | Coussy, O., Poromechanics, 2004, John Wiley & Sons |
| [31] | Wang, H., Theory of linear poroelasticity with applications to geomechanics and hydrogeology, 2000, Princeton University Press |
| [32] | Murad, M. A.; Loula, A. F., Improved accuracy in finite element analysis of Biot’s consolidation problem, Comput. Methods Appl. Mech. Engrg., 95, 3, 359-382, 1992 · Zbl 0760.73068 |
| [33] | Murad, M. A.; Loula, A. F., On stability and convergence of finite element approximations of Biot’s consolidation problem, Internat. J. Numer. Methods Engrg., 37, 4, 645-667, 1994 · Zbl 0791.76047 |
| [34] | Ferronato, M.; Castelletto, N.; Gambolati, G., A fully coupled 3-D mixed finite element model of Biot consolidation, J. Comput. Phys., 229, 12, 4813-4830, 2010 · Zbl 1305.76055 |
| [35] | Nordbotten, J. M., Cell-centered finite volume discretizations for deformable porous media, Internat. J. Numer. Methods Engrg., 100, 6, 399-418, 2014 · Zbl 1352.76072 |
| [36] | Liu, R., Discontinuous Galerkin Finite Element Solution for Poromechanics, 2004, The University of Texas at Austin |
| [37] | Choo, J.; Lee, S., Enriched Galerkin finite elements for coupled poromechanics with local mass conservation, Comput. Methods Appl. Mech. Engrg., 341, 311-332, 2018 · Zbl 1440.74120 |
| [38] | Voskov, D. V.; Tchelepi, H. A., Comparison of nonlinear formulations for two-phase multi-component EoS based simulation, J. Pet. Sci. Eng., 82, 101-111, 2012 |
| [39] | Aziz, K.; Settari, A., (Petroleum Reservoir Simulation, 1979, Elsevier Applied Science Publishers: Elsevier Applied Science Publishers London, UK) |
| [40] | White, J. A.; Castelletto, N.; Klevtsov, S.; Bui, Q. M.; Osei-Kuffuor, D.; Tchelepi, H. A., A two-stage preconditioner for multiphase poromechanics in reservoir simulation, Comput. Methods Appl. Mech. Engrg., 357, Article 112575 pp., 2019 · Zbl 1442.76118 |
| [41] | Terekhov, K. M.; Mallison, B. T.; Tchelepi, H. A., Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem, J. Comput. Phys., 330, 245-267, 2017 · Zbl 1380.65335 |
| [42] | Both, J. W.; Köcher, U., Numerical investigation on the fixed-stress splitting scheme for Biot’s equations: Optimality of the tuning parameter, (Numerical Mathematics and Advanced Applications ENUMATH 2017, 2019, Springer), 789-797 · Zbl 1425.76132 |
| [43] | Guermond, J.-L.; Quartapelle, L., On stability and convergence of projection methods based on pressure Poisson equation, Internat. J. Numer. Methods Fluids, 26, 9, 1039-1053, 1998 · Zbl 0912.76054 |
| [44] | Guermond, J.-L.; Minev, P.; Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195, 44-47, 6011-6045, 2006 · Zbl 1122.76072 |
| [45] | Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comp., 22, 104, 745-762, 1968 · Zbl 0198.50103 |
| [46] | Chorin, A. J., On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp., 23, 106, 341-353, 1969 · Zbl 0184.20103 |
| [47] | Badia, S.; Codina, R., Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition, Numer. Math., 107, 533-557, 2007 · Zbl 1132.76029 |
| [48] | Goda, K., A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows, J. Comput. Phys., 30, 1, 76-95, 1979 · Zbl 0405.76017 |
| [49] | Barry, S.; Mercer, G., Exact solutions for two-dimensional time-dependent flow and deformation within a poroelastic medium, J. Appl. Mech., 66, 2, 536-540, 1999 |
| [50] | Hughes, T. J.; Franca, L. P., A new finite element formulation for computational fluid dynamics: VII. the Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg., 65, 1, 85-96, 1987 · Zbl 0635.76067 |
| [51] | Silvester, D. J.; Kechkar, N., Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem, Comput. Methods Appl. Mech. Engrg., 79, 1, 71-86, 1990 · Zbl 0706.76075 |
| [52] | Silvester, D., Optimal low order finite element methods for incompressible flow, Comput. Methods Appl. Mech. Engrg., 111, 3-4, 357-368, 1994 · Zbl 0844.76059 |
| [53] | Borio, A.; Hamon, F. P.; Castelletto, N.; White, J. A.; Settgast, R. R., Hybrid mimetic finite-difference and virtual element formulation for coupled poromechanics, Comput. Methods Appl. Mech. Engrg., 383, Article 113917 pp., 2021 · Zbl 1506.74390 |
| [54] | Berger, L.; Bordas, R.; Kay, D.; Tavener, S., A stabilized finite element method for finite-strain three-field poroelasticity, Comput. Mech., 60, 51-68, 2017 · Zbl 1386.74134 |
| [55] | Dohrmann, C. R.; Bochev, P. B., A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Internat. J. Numer. Methods Fluids, 46, 2, 183-201, 2004 · Zbl 1060.76569 |
| [56] | Camargo, J. T.; Hamon, F.; Mazuyer, A.; Meckel, T.; Castelletto, N.; White, J. A., Deformation monitoring feasibility for offshore carbon storage in the Gulf-of-Mexico, (Proceedings of the 16th Greenhouse Gas Control Technologies Conference (GHGT-16), 2022), 23-24 |
| [57] | Geos, J. T., Next-gen simulation for geologic carbon storage, 2023, http://www.geos.dev |
| [58] | Duan, Z.; Sun, R., An improved model calculating CO2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar, Chem. Geol., 193, 3-4, 257-271, 2003 |
| [59] | Span, R.; Wagner, W., A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa, J. Phys. Chem. Ref. Data, 25, 6, 1509-1596, 1996 |
| [60] | Fenghour, A.; Wakeham, W. A.; Vesovic, V., The viscosity of carbon dioxide, J. Phys. Chem. Ref. Data, 27, 1, 31-44, 1998 |
| [61] | Phillips, S., A technical databook for geothermal energy utilization, 1981 |
| [62] | Ruiz, I., Characterization of the High Island 24 L Field For Modeling and Estimating CO2 Storage Capacity in the Offshore Texas State Waters, Gulf of Mexico, 2019, (Ph.D. thesis) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
