×

Finite element solvers for Biot’s poroelasticity equations in porous media. (English) Zbl 1451.76070

Math. Geosci. 52, No. 8, 977-1015 (2020); correction ibid. 53, No. 5, 1095 (2021).
Summary: We study and compare five different combinations of finite element spaces for approximating the coupled flow and solid deformation system, so-called Biot’s equations. The permeability and porosity fields are heterogeneous and depend on solid displacement and fluid pressure. We provide detailed comparisons among the continuous Galerkin, discontinuous Galerkin, enriched Galerkin, and two types of mixed finite element methods. Several advantages and disadvantages for each of the above techniques are investigated by comparing local mass conservation properties, the accuracy of the flux approximation, number of degrees of freedom (DOF), and wall and CPU times. Three-field formulation methods with fluid velocity as an additional primary variable generally require a larger number of DOF, longer wall and CPU times, and a greater number of iterations in the linear solver in order to converge. The two-field formulation, a combination of continuous and enriched Galerkin function space, requires the fewest DOF among the methods that conserve local mass. Moreover, our results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included. These comparisons of the key performance indicators of different combinations of finite element methods can be utilized to choose the preferred method based on the required accuracy and the available computational resources.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abou-Kassem, J.; Islam, M.; Farouq-Ali, S., Petroleum reservoir simulations (2013), London: Elsevier, London
[2] Adler J, Gaspar F, Hu X, Ohm P, Rodrigo C, Zikatanov L (2019) Robust preconditioners for a new stabilized discretization of the poroelastic equations. Preprint arXiv:1905.10353 · Zbl 1448.65145
[3] Almani, T.; Kumar, K.; Dogru, A.; Singh, G.; Wheeler, M., Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics, Comput Methods Appl Mech Eng, 311, 180-207 (2016) · Zbl 1439.74183
[4] Alnaes, M.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M.; Wells, G., The FEniCS project version 1.5, Arch Numer Softw, 3, 100, 12 (2015)
[5] Arnold, D.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344 (1984) · Zbl 0593.76039
[6] Ayuso B, Georgiev I, Kraus J, Zikatanov L (2009) Preconditioning techniques for discontinuous Galerkin methods discretizations for linear elasticity equations · Zbl 1311.74108
[7] Balay S, Abhyankar S, Adams M, Brown J, Brune P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp W, Kaushik D, Knepley M, May D, McInnes L, Mills R, Munson T, Rupp K, Sanan P, Smith B, Zampini S, Zhang H, Zhang H (2018) PETSc users manual. Technical report ANL-95/11—revision 3.10, Argonne National Laboratory
[8] Ballarin F, Rozza G (2019) Multiphenics-easy prototyping of multiphysics problems in FEniCS. Version: 0.2.0. https://gitlab.com/multiphenics/multiphenics
[9] Bazilevs, Y.; Hughes, T., Weak imposition of Dirichlet boundary conditions in fluid mechanics, Comput Fluids, 36, 1, 12-26 (2007) · Zbl 1115.76040
[10] Biot, M., General theory of three-dimensional consolidation, J Appl Phys, 12, 2, 155-164 (1941) · JFM 67.0837.01
[11] Biot, M.; Willis, D., The elastic coefficients of the theory of consolidation, J Appl Mech, 15, 594-601 (1957)
[12] Bisdom, K.; Bertotti, G.; Nick, H., A geometrically based method for predicting stress-induced fracture aperture and flow in discrete fracture networks, AAPG Bull, 100, 7, 1075-1097 (2016)
[13] Bouklas, N.; Landis, C.; Huang, R., Effect of solvent diffusion on crack-tip fields and driving force for fracture of hydrogels, J Appl Mech, 82, 8, 081007 (2015)
[14] Bouklas, N.; Landis, C.; Huang, R., A nonlinear, transient finite element method for coupled solvent diffusion and large deformation of hydrogels, J Mech Phys Solids, 79, 21-43 (2015) · Zbl 1349.74130
[15] Cao, T.; Sanavia, L.; Schrefler, B., A thermo-hydro-mechanical model for multiphase geomaterials in dynamics with application to strain localization simulation, Int J Numer Methods Eng, 107, 4, 312-337 (2016) · Zbl 1352.74180
[16] Castelletto, N.; White, J.; 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)
[17] Castelletto, N.; White, J.; Ferronato, M., Scalable algorithms for three-field mixed finite element coupled poromechanics, J Comput Phys, 327, 894-918 (2016) · Zbl 1373.76312
[18] Chen, Z., Reservoir simulation: mathematical techniques in oil recovery (2007), New York: SIAM, New York · Zbl 1167.86001
[19] Choo, J., Large deformation poromechanics with local mass conservation: an enriched Galerkin finite element framework, Int J Numer Methods Eng, 116, 1, 66-90 (2018)
[20] Choo, J., Stabilized mixed continuous/enriched Galerkin formulations for locally mass conservative poromechanics, Comput Methods Appl Mech Eng, 357, 112568 (2019) · Zbl 1442.65254
[21] Choo, J.; Borja, R., Stabilized mixed finite elements for deformable porous media with double porosity, Comput Methods Appl Mech Eng, 293, 131-154 (2015) · Zbl 1423.74265
[22] Choo, J.; Lee, S., Enriched Galerkin finite elements for coupled poromechanics with local mass conservation, Comput Methods Appl Mech Eng, 341, 311-332 (2018) · Zbl 1440.74120
[23] Coussy, O., Poromechanics (2004), New York: Wiley, New York
[24] Deng, Q.; Ginting, V.; McCaskill, B.; Torsu, P., A locally conservative stabilized continuous Galerkin finite element method for two-phase flow in poroelastic subsurfaces, J Comput Phys, 347, 78-98 (2017) · Zbl 1380.65256
[25] Du, J.; Wong, R., Application of strain-induced permeability model in a coupled geomechanics-reservoir simulator, J Can Pet Technol, 46, 12, 55-61 (2007)
[26] Ern, A.; Stephansen, A., A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods, J Comput Math, 2008, 488-510 (2008) · Zbl 1174.65034
[27] Ern, A.; Stephansen, A.; Zunino, P., A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J Numer Anal, 29, 2, 235-256 (2009) · Zbl 1165.65074
[28] 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
[29] Ferronato, M.; Franceschini, A.; Janna, C.; Castelletto, N.; Tchelepi, H., A general preconditioning framework for coupled multiphysics problems with application to contact-and poro-mechanics, J Comput Phys, 398, 108887 (2019) · Zbl 1453.65065
[30] Frigo, M.; Castelletto, N.; Ferronato, M., A relaxed physical factorization preconditioner for mixed finite element coupled poromechanics, SIAM J Sci Comput, 41, 4, B694-B720 (2019) · Zbl 1420.65032
[31] Girault, V.; Wheeler, M.; Almani, T.; Dana, S., A priori error estimates for a discretized poro-elastic-elastic system solved by a fixed-stress algorithm, Oil Gas Sci Technol Rev IFP Energ Nouvelles, 74, 24 (2019)
[32] Haga, J.; Osnes, H.; Langtangen, H., On the causes of pressure oscillations in low permeable and low compressible porous media, Int J Numer Anal Methods Geomech, 36, 12, 1507-1522 (2012)
[33] Hansbo, P., Nitsche’s method for interface problems in computational mechanics, GAMM Mitteilungen, 28, 2, 183-206 (2005) · Zbl 1179.65147
[34] Hong Q, Kraus J (2017) Parameter-robust stability of classical three-field formulation of biot’s consolidation model. Preprint arXiv:1706.00724 · Zbl 1398.65046
[35] Honorio, H.; Maliska, C.; Ferronato, M.; Janna, C., A stabilized element-based finite volume method for poroelastic problems, J Comput Phys, 364, 49-72 (2018) · Zbl 1392.74090
[36] Jaeger, J.; Cook, NG; Zimmerman, R., Fundamentals of rock mechanics (2009), Berlin: Wiley, Berlin
[37] Juanes, R.; Jha, B.; Hager, B.; Shaw, J.; Plesch, A.; Astiz, L.; Dieterich, J.; Frohlich, C., Were the May 2012 Emilia-Romagna earthquakes induced? A coupled flow-geomechanics modeling assessment, Geophys Res Lett, 43, 13, 6891-6897 (2016)
[38] Kadeethum T, Nick H, Lee S, Richardson C, Salimzadeh S, Ballarin F (2019a) A novel enriched Galerkin method for modelling coupled flow and mechanical deformation in heterogeneous porous media. In 53rd US rock mechanics/geomechanics symposium, New York, NY, USA. American Rock Mechanics Association
[39] Kadeethum, T.; Salimzadeh, S.; Nick, H., An investigation of hydromechanical effect on well productivity in fractured porous media using full factorial experimental design, J Petrol Sci Eng, 181, 106233 (2019)
[40] Kadeethum T, Jørgensen T, Nick H (2020a) Physics-informed neural networks for solving inverse problems of nonlinear Biot’s equations: batch training. In 54th US rock mechanics/geomechanics symposium, Golden, CO, USA. American Rock Mechanics Association
[41] Kadeethum, T.; Jørgensen, T.; Nick, H., Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations, PLoS ONE, 15, 5, e0232683 (2020)
[42] Kadeethum, T.; Nick, H.; Lee, S.; Ballarin, F., Flow in porous media with low dimensional fractures by employing Enriched Galerkin method, Adv Water Resour, 142, 103620 (2020)
[43] Kadeethum, T.; Salimzadeh, S.; Nick, H., Well productivity evaluation in deformable single-fracture media, Geothermics, 87, 101839 (2020)
[44] Kim, J.; Tchelepi, H.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits, Comput Methods Appl Mech Eng, 200, 13-16, 1591-1606 (2011) · Zbl 1228.74101
[45] Kumar, S.; Oyarzua, R.; Ruiz-Baier, R.; Sandilya, R., Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity, ESAIM Math Model Numer Anal, 54, 1, 273-299 (2020) · Zbl 1511.65114
[46] Lee, S.; Wheeler, M., Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization, J Comput Phys, 331, 19-37 (2017) · Zbl 1378.76048
[47] Lee, S.; Wheeler, M., Enriched Galerkin methods for two-phase flow in porous media with capillary pressure, J Comput Phys, 367, 65-86 (2018) · Zbl 1415.76456
[48] Lee, S.; Woocheol, C., Optimal error estimate of elliptic problems with Dirac sources for discontinuous and enriched Galerkin methods, Appl Numer Math, 150, 76-104 (2019) · Zbl 1434.65242
[49] Lee, S.; Wheeler, M.; Wick, T., Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model, Comput Methods Appl Mech Eng, 305, 111-132 (2016) · Zbl 1425.74419
[50] Lee, J.; Mardal, K.; Winther, R., Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J Sci Comput, 39, 1, A1-A24 (2017) · Zbl 1381.76183
[51] Lewis, R.; Schrefler, B., The finite element method in the static and dynamic deformation and consolidation of porous media (1998), Berlin: Wiley, Berlin · Zbl 0935.74004
[52] Liu, R.; Wheeler, M.; Dawson, C.; Dean, R., Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method, Comput Methods Appl Mech Eng, 198, 9-12, 912-919 (2009) · Zbl 1229.76053
[53] Liu, J.; Tavener, S.; Wang, Z., Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes, SIAM J Sci Comput, 40, 5, B1229-B1252 (2018) · Zbl 1404.65232
[54] Macminn, C.; Dufresne, E.; Wettlaufer, J., Large deformations of a soft porous material, Phys Rev Appl, 5, 4, 1-30 (2016)
[55] Murad, M.; Loula, A., On stability and convergence of finite element approximations of Biot’s consolidation problem, Int J Numer Methods Eng, 37, 4, 645-667 (1994) · Zbl 0791.76047
[56] Murad, M.; Borges, M.; Obregon, J.; Correa, M., A new locally conservative numerical method for two-phase flow in heterogeneous poroelastic media, Comput Geotech, 48, 192-207 (2013)
[57] Nick, H.; Raoof, A.; Centler, F.; Thullner, M.; Regnier, P., Reactive dispersive contaminant transport in coastal aquifers: numerical simulation of a reactive Henry problem, J Contam Hydrol, 145, 90-104 (2013)
[58] Nitsche, J., Uber ein variationsprinzip zur losung von dirichlet-problemen bei verwendung von teilraumen, die keinen randbedingungen unterworfen sind, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, 36, 1, 9-15 (1971) · Zbl 0229.65079
[59] Nordbotten, J., Cell-centered finite volume discretizations for deformable porous media, Int J Numer Methods Eng, 100, 6, 399-418 (2014) · Zbl 1352.76072
[60] Pain, C.; Saunders, J.; Worthington, M.; Singer, J.; Stuart-Bruges, W.; Mason, G.; Goddard, A., A mixed finite-element method for solving the poroelastic Biot equations with electrokinetic coupling, Geophys J Int, 160, 2, 592-608 (2005)
[61] Peaceman, D., Fundamentals of numerical reservoir simulation (2000), Berlin: Elsevier, Berlin
[62] Phillips, P.; Wheeler, M., A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case, Comput Geosci, 11, 2, 131 (2007) · Zbl 1117.74015
[63] Phillips, P.; Wheeler, M., A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case, Comput Geosci, 11, 2, 145-158 (2007) · Zbl 1117.74016
[64] Riviere, B.; Tan, J.; Thompson, T., Error analysis of primal discontinuous Galerkin methods for a mixed formulation of the Biot equations, Comput Math Appl, 73, 4, 666-683 (2017) · Zbl 1368.65195
[65] Rodrigo, C.; Hu, X.; Ohm, P.; Adler, J.; Gaspar, F.; Zikatanov, L., New stabilized discretizations for poroelasticity and the Stokes’ equations, Comput Methods Appl Mech Eng, 341, 467-484 (2018) · Zbl 1440.76027
[66] Salimzadeh, S.; Nick, H., A coupled model for reactive flow through deformable fractures in enhanced geothermal systems, Geothermics, 81, 88-100 (2019)
[67] Salimzadeh, S.; Nick, H.; Zimmerman, R., Thermoporoelastic effects during heat extraction from low-permeability reservoirs, Energy, 142, 546-558 (2018)
[68] Scovazzi, G.; Wheeler, M.; Mikelic, A.; Lee, S., Analytical and variational numerical methods for unstable miscible displacement flows in porous media, J Comput Phys, 335, 444-496 (2017) · Zbl 1375.76188
[69] Sokolova, I.; Bastisya, M.; Hajibeygi, H., Multiscale finite volume method for finite-volume-based simulation of poroelasticity, J Comput Phys, 379, 309-324 (2019) · Zbl 07581574
[70] Terzaghi, K., Theoretical soil mechanics (1951), London: Chapman And Hall, London
[71] Vermeer, P.; Verruijt, A., An accuracy condition for consolidation by finite elements, Int J Numer Anal Methods Geomech, 5, 1, 1-14 (1981) · Zbl 0456.73060
[72] Vik, H.; Salimzadeh, S.; Nick, H., Heat recovery from multiple-fracture enhanced geothermal systems: the effect of thermoelastic fracture interactions, Renew Energy, 121, 606-622 (2018)
[73] Vinje, V.; Brucker, J.; Rognes, M.; Mardal, K.; Haughton, V., Fluid dynamics in syringomyelia cavities: Effects of heart ate, CSF velocity, CSF velocity waveform and craniovertebral decompression, Neuroradiol J, 31, 1971400918795482 (2018)
[74] Virtanen, P.; Gommers, R.; Oliphant, TE; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; van der Walt, SJ; Brett, M.; Wilson, J.; Millman, KJ; Mayorov, N.; Nelson, ARJ; Jones, E.; Kern, R.; Larson, E.; Carey, CJ; Polat, İ.; Feng, Y.; Moore, EW; VanderPlas, J.; Laxalde, D.; Perktold, J.; Cimrman, R.; Henriksen, I.; Quintero, EA; Harris, CR; Archibald, AM; Ribeiro, AH; Pedregosa, F.; van Mulbregt, P., SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nat Methods, 17, 261-272 (2020) · doi:10.1038/s41592-019-0686-2
[75] Wang, H., Theory of linear poroelasticity with applications to geomechanics and hydrogeology (2017), Princeton: Princeton University Press, Princeton
[76] Wheeler, M.; Xue, G.; Yotov, I., Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity, Comput Geosci, 18, 1, 57-75 (2014) · Zbl 1395.65093
[77] White, J.; Borja, R., Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients, Comput Methods Appl Mech Eng, 197, 49-50, 4353-4366 (2008) · Zbl 1194.74480
[78] White, J.; Borja, R., Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics, Comput Geosci, 15, 4, 647 (2011) · Zbl 1367.76034
[79] White, J.; Castelletto, N.; Tchelepi, H., Block-partitioned solvers for coupled poromechanics: a unified framework, Comput Methods Appl Mech Eng, 303, 55-74 (2016) · Zbl 1425.74497
[80] Zdunek, A.; Rachowicz, W.; Eriksson, T., A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity, Comput Math Appl, 72, 1, 25-47 (2016) · Zbl 1443.74151
[81] Zhao, Y.; Choo, J., Stabilized material point methods for coupled large deformation and fluid flow in porous materials, Comput Methods Appl Mech Eng, 362, 112742 (2020) · Zbl 1439.76163
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.