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A macroelement stabilization for mixed finite element/finite volume discretizations of multiphase poromechanics

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Abstract

Strong coupling between geomechanical deformation and multiphase fluid flow appears in a variety of geoscience applications. A common discretization strategy for these problems is a continuous Galerkin finite element scheme for the momentum balance equation and a finite volume scheme for the mass balance equations. When applied within a fully implicit solution strategy, however, this discretization is not intrinsically stable. In the limit of small time steps or low permeabilities, spurious oscillations in the piecewise-constant pressure field, i.e., checkerboarding, may be observed. Further, eigenvalues associated with the spurious modes will control the conditioning of the matrices and can dramatically degrade the convergence rate of iterative linear solvers. Here, we propose a stabilization technique in which the mass balance equations are supplemented with stabilizing flux terms on a macroelement basis. The additional stabilization terms are dependent on a stabilization parameter. We identify an optimal value for this parameter using an analysis of the eigenvalue distribution of the macroelement Schur complement matrix. The resulting method is simple to implement and preserves the underlying sparsity pattern of the original discretization. Another appealing feature of the method is that mass is exactly conserved on macroelements, despite the addition of artificial fluxes. The efficacy of the proposed technique is demonstrated with several numerical examples.

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References

  1. Teatini, P., Ferronato, M., Gambolati, G., Gonella, M.: Groundwater pumping and land subsidence in the Emilia-Romagna coastland, Italy: modeling the past occurrence and the future trend. Water Resources Research 42(1). https://doi.org/10.1029/2005WR004242

  2. Borja, R.I., White, J.A.: Continuum deformation and stability analyses of a steep hillside slope under rainfall infiltration. Acta Geotech. 5(1), 1–14 (2010). https://doi.org/10.1007/s11440-009-0108-1

    Google Scholar 

  3. Borja, R.I., Liu, X., White, J.A.: Multiphysics hillslope processes triggering landslides. Acta Geotech. 7(4), 261–269 (2012). https://doi.org/10.1007/s11440-012-0175-6

    Google Scholar 

  4. Camargo, J., Velloso, R.Q., Vargas, E.A.: Numerical limit analysis of three-dimensional slope stability problems in catchment areas. Acta Geotech. 11(6), 1369–1383 (2016). https://doi.org/10.1007/s11440-016-0459-3

    Google Scholar 

  5. Cao, J., Jung, J., Song, X., Bate, B.: On the soil water characteristic curves of poorly graded granular materials in aqueous polymer solutions. Acta Geotech. 13 (1), 103–116 (2018). https://doi.org/10.1007/s11440-017-0568-7

    Google Scholar 

  6. Fávero Neto, A.H., Borja, R.I.: Continuum hydrodynamics of dry granular flows employing multiplicative elastoplasticity. Acta Geotech. 13(5), 1027–1040 (2018). https://doi.org/10.1007/s11440-018-0700-3

    Google Scholar 

  7. Ghaffaripour, O., Esgandani, G.A., Khoshghalb, A., Shahbodaghkhan, B.: Fully coupled elastoplastic hydro-mechanical analysis of unsaturated porous media using a meshfree method. International Journal for Numerical and Analytical Methods in Geomechanics. https://doi.org/10.1002/nag.2931

  8. Gholami Korzani, M., Galindo-Torres, S.A., Scheuermann, A., Williams, D.J.: SPH approach for simulating hydro-mechanical processes with large deformations and variable permeabilities. Acta Geotech. 13(2), 303–316 (2018). https://doi.org/10.1007/s11440-017-0610-9

    Google Scholar 

  9. Mikaeili, E., Schrefler, B.: XFEM, strong discontinuities and second-order work in shear band modeling of saturated porous media. Acta Geotech. 13(6), 1249–1264 (2018). https://doi.org/10.1007/s11440-018-0734-6

    Google Scholar 

  10. Navas, P., Sanavia, L., López-Querol, S., Yu, R.C.: Explicit meshfree solution for large deformation dynamic problems in saturated porous media. Acta Geotech. 13(2), 227–242 (2018). https://doi.org/10.1007/s11440-017-0612-7

    Google Scholar 

  11. Oka, F., Shahbodagh, B., Kimoto, S.: A computational model for dynamic strain localization in unsaturated elasto-viscoplastic soils. Int. J. Numer. Anal. Methods Geomech. 43(1), 138–165 (2019). https://doi.org/10.1002/nag.2857

    Google Scholar 

  12. Prassetyo, S.H., Gutierrez, M.: High-order ADE scheme for solving the fluid diffusion equation in non-uniform grids and its application in coupled hydro-mechanical simulation. Int. J. Numer. Anal. Methods Geomech. 42(16), 1976–2000 (2018). https://doi.org/10.1002/nag.2843

    Google Scholar 

  13. Song, X., Wang, K., Ye, M.: Localized failure in unsaturated soils under non-isothermal conditions. Acta Geotech. 13(1), 73–85 (2018). https://doi.org/10.1007/s11440-017-0534-4

    Google Scholar 

  14. Wang, M., Feng, Y., Pande, G., Zhao, T.: A coupled 3-dimensional bonded discrete element and lattice Boltzmann method for fluid-solid coupling in cohesive geomaterials. Int. J. Numer. Anal. Methods Geomech. 42(12), 1405–1424 (2018). https://doi.org/10.1002/nag.2799

    Google Scholar 

  15. Wang, W., Zhou, M., Zhang, B., Peng, C.: A dual mortar contact method for porous media and its application to clay-core rockfill dams. Int. J. Numer. Anal. Methods Geomech. 43(9), 1744–1769 (2019). https://doi.org/10.1002/nag.2930

    Google Scholar 

  16. Yan, C., Jiao, Y.-Y., Yang, S.: A 2D coupled hydro-thermal model for the combined finite-discrete element method. Acta Geotech. 14(2), 403–416 (2019). https://doi.org/10.1007/s11440-018-0653-6

    Google Scholar 

  17. Zhou, Z., Du, X., Wang, S., Cai, X., Chen, L.: Micromechanism of the diffusion of cement-based grouts in porous media under two hydraulic operating conditions: constant flow rate and constant pressure. Acta Geotech. 14(3), 825–841 (2019). https://doi.org/10.1007/s11440-018-0704-z

    Google Scholar 

  18. Borja, R.I., Choo, J.: Cam-Clay plasticity, Part VIII: a constitutive framework for porous materials with evolving internal structure. Comput. Methods Appl. Mech. Eng. 309, 653–679 (2016). https://doi.org/10.1016/j.cma.2016.06.016

    Google Scholar 

  19. Borja, R.I., Song, X., Wu, W.: Critical state plasticity. Part VII: triggering a shear band in variably saturated porous media. Comput. Methods Appl. Mech. Eng. 261-262, 66–82 (2013). https://doi.org/10.1016/j.cma.2013.03.008

    Google Scholar 

  20. Borja, R.I., Yin, Q., Zhao, Y.: Cam-Clay plasticity. Part IX: on the anisotropy, heterogeneity, and viscoplasticity of shale. Comput. Methods Appl. Mech. Eng. 360, 112695 (2020). https://doi.org/10.1016/j.cma.2019.112695

    Google Scholar 

  21. Zoback, M.D.: Reservoir Geomechanics. Cambridge University Press. https://doi.org/10.1017/CBO9780511586477 (2007)

  22. Feng, Y., Gray, K.E.: XFEM-based cohesive zone approach for modeling near-wellbore hydraulic fracture complexity. Acta Geotech. 14(2), 377–402 (2019). https://doi.org/10.1007/s11440-018-0645-6

    Google Scholar 

  23. Mashhadian, M., Abedi, S., Noshadravan, A.: Probabilistic multiscale characterization and modeling of organic-rich shale poroelastic properties. Acta Geotech. 13(4), 781–800 (2018). https://doi.org/10.1007/s11440-018-0652-7

    Google Scholar 

  24. Peng, S., Zhang, Z., Mou, J., Zhao, B., Liu, Z., Wang, J.: Fully coupled hydraulic fracture simulation using the improved element partition method. Int. J. Numer. Anal. Methods Geomech. 43(1), 441–460 (2019). https://doi.org/10.1002/nag.2870

    Google Scholar 

  25. Semnani, S.J., White, J.A., Borja, R.I.: Thermoplasticity and strain localization in transversely isotropic materials based on anisotropic critical state plasticity. Int. J. Numer. Anal. Methods Geomech. 40(18), 2423–2449 (2016). https://doi.org/10.1002/nag.2536

    Google Scholar 

  26. Shiozawa, S., Lee, S., Wheeler, M.F.: The effect of stress boundary conditions on fluid-driven fracture propagation in porous media using a phase-field modeling approach. Int. J. Numer. Anal. Methods Geomech. 43(6), 1316–1340 (2019). https://doi.org/10.1002/nag.2899

    Google Scholar 

  27. Yan, C., Jiao, Y.-Y.: FDEM-TH3D: a three-dimensional coupled hydrothermal model for fractured rock. Int. J. Numer. Anal. Methods Geomech. 43(1), 415–440 (2019). https://doi.org/10.1002/nag.2869

    Google Scholar 

  28. Zhang, Q., Choo, J., Borja, R.I.: On the preferential flow patterns induced by transverse isotropy and non-Darcy flow in double porosity media. Comput. Methods Appl. Mech. Eng. 353, 570–592 (2019). https://doi.org/10.1016/j.cma.2019.04.037

    Google Scholar 

  29. Zhao, Y., Semnani, S.J., Yin, Q., Borja, R.I.: On the strength of transversely isotropic rocks. Int. J. Numer. Anal. Methods Geomech. 42(16), 1917–1934 (2018). https://doi.org/10.1002/nag.2809

    Google Scholar 

  30. Zhao, Y., Borja, R.I.: Deformation and strength of transversely isotropic rocks. In: Wu, W. (ed.) Desiderata Geotechnica, Springer International Publishing, pp 237–241, Cham (2019). https://doi.org/10.1007/978-3-030-14987-1_28

  31. Verdon, J.P., Kendall, J.-M., Stork, A.L., Chadwick, R.A., White, D.J., Bissell, R.C.: Comparison of geomechanical deformation induced by megatonne-scale CO2 storage at Sleipner, Weyburn, and In Salah. Proc. Natl. Acad. Sci. 110(30), E2762–E2771 (2013). https://doi.org/10.1073/pnas.1302156110

    Google Scholar 

  32. White, J.A., Chiaramonte, L., Ezzedine, S., Foxall, W., Hao, Y., Ramirez, A., McNab, W.: Geomechanical behavior of the reservoir and caprock system at the In Salah CO2 storage project. Proc. Natl. Acad. Sci. 111(24), 8747–8752 (2014). https://doi.org/10.1073/pnas.1316465111

    Google Scholar 

  33. Jin, W., Arson, C.: Fluid-driven transition from damage to fracture in anisotropic porous media: a multi-scale XFEM approach, Acta Geotechnica. https://doi.org/10.1007/s11440-019-00813-x

  34. de Boer, R.: Theory of porous media: highlights in historical development and current state. Springer Science & Business Media. https://doi.org/10.1007/978-3-642-59637-7 (2012)

  35. Coussy, O.: Poromechanics. Wiley, New York. https://doi.org/10.1002/0470092718 (2005)

  36. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941). https://doi.org/10.1063/1.1712886

    Google Scholar 

  37. White, J.A., Castelletto, N., Tchelepi, H.A.: Block-partitioned solvers for coupled poromechanics: a unified framework. Comput. Methods Appl. Mech. Eng. 303, 55–74 (2016). https://doi.org/10.1016/j.cma.2016.01.008

    Google Scholar 

  38. 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. Eng. 357:112575. https://doi.org/10.1016/j.cma.2019.112575 (2019)

  39. Murad, M.A., Loula, A.F.D.: On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37(4), 645–667 (1994). https://doi.org/10.1002/nme.1620370407

    Google Scholar 

  40. Settari, A., Mourits, F.M.: A coupled reservoir and geomechanical simulation system. SPE J. 3(03), 219–226 (1998). https://doi.org/10.2118/50939-PA

    Google Scholar 

  41. Jha, B., Juanes, R.: A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotech. 2(3), 139–153 (2007). https://doi.org/10.1007/s11440-007-0033-0

    Google Scholar 

  42. Phillips, P.J., Wheeler, M.F.: 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). https://doi.org/10.1007/s10596-007-9044-z

    Google Scholar 

  43. 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. https://doi.org/10.1016/j.jcp.2010.03.018 (2010)

  44. 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). https://doi.org/10.2118/119084-PA

    Google Scholar 

  45. Haga, J.B., Osnes, H., Langtangen, H.P.: 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). https://doi.org/10.1002/nag.1062

    Google Scholar 

  46. Prévost, J.H.: Two-way coupling in reservoir geomechanical models: vertex-centered Galerkin geomechanical model cell-centered and vertex-centered finite volume reservoir models. Int. J. Numer. Methods Eng. 98(8), 612–624 (2014). https://doi.org/10.1002/nme.4657

    Google Scholar 

  47. Castelletto, N., White, J.A., Tchelepi, H.A.: 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). https://doi.org/10.1002/nag.2400

    Google Scholar 

  48. Nordbotten, J.M.: Cell-centered finite volume discretizations for deformable porous media. Int. J. Numer. Methods Eng. 100(6), 399–418 (2014). https://doi.org/10.1002/nme.4734

    Google Scholar 

  49. Nordbotten, J.M.: Stable cell-centered finite volume discretization for Biot equations. SIAM J. Numer. Anal. 54(2), 942–968 (2016). https://doi.org/10.1137/15M1014280

    Google Scholar 

  50. Rodrigo, C., Gaspar, F., Hu, X., Zikatanov, L.: Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 298, 183–204 (2016). https://doi.org/10.1016/j.cma.2015.09.019

    Google Scholar 

  51. Hu, X., Rodrigo, C., Gaspar, F.J., Zikatanov, L.T.: A nonconforming finite element method for the Biot’s consolidation model in poroelasticity. J. Comput. Appl. Math. 310, 143–154 (2017). https://doi.org/j.cam.2016.06.003

    Google Scholar 

  52. Honório, H.T., Maliska, C.R., Ferronato, M., Janna, C.: A stabilized element-based finite volume method for poroelastic problems. J. Comput. Phys. 364, 49–72. https://doi.org/10.1016/j.jcp.2018.03.010 (2018)

  53. Choo, J., Lee, S.: Enriched Galerkin finite elements for coupled poromechanics with local mass conservation. Comput. Methods Appl. Mech. Eng. 341, 311–332 (2018). https://doi.org/10.1016/j.cma.2018.06.022

    Google Scholar 

  54. Sokolova, I., Bastisya, M.G., Hajibeygi, H.: Multiscale finite volume method for finite-volume-based simulation of poroelasticity. J. Comput. Phys. 379:309–324. https://doi.org/10.1016/j.jcp.2018.11.039 (2019)

  55. Liu, F., Borja, R.I.: Stabilized low-order finite elements for frictional contact with the extended finite element method. Compute. Methods Appl. Mech. Eng. 199(37), 2456–2471. https://doi.org/10.1016/j.cma.2010.03.030 (2010)

  56. Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20(3), 179–192 (1973). https://doi.org/10.1007/BF01436561

    Google Scholar 

  57. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. R.A.I.R.O Anal. Numér. 8(R2), 129–151 (1974). https://doi.org/10.1051/m2an/197408R201291

    Google Scholar 

  58. Brezzi, F., Pitkäranta, J.: On the Stabilization of Finite Element Approximations of the Stokes Equations, Vieweg+Teubner Verlag, pp. 11–19. https://doi.org/10.1007/978-3-663-14169-3_2 (1984)

  59. Hughes, T.J., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59(1), 85–99. https://doi.org/10.1016/0045-7825(86)90025-3 (1986)

  60. Dohrmann, C.R., Bochev, P.B.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46(2), 183–201 (2004). https://doi.org/10.1002/fld.752

    Google Scholar 

  61. Fortin, M., Boivin, S.: Iterative stabilization of the bilinear velocity–constant pressure element. Int. J. Numer. Methods Fluids 10(2), 125–140 (1990). https://doi.org/10.1002/fld.1650100202

    Google Scholar 

  62. Bochev, P., Dohrmann, C., Gunzburger, M.: Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44(1), 82–101 (2006). https://doi.org/10.1137/S0036142905444482

    Google Scholar 

  63. Burman, E., Hansbo, P.: A unified stabilized method for Stokes and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51. https://doi.org/10.1016/j.cam.2005.11.022 (2007)

  64. 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. Eng. 65(1), 85–96. https://doi.org/10.1016/0045-7825(87)90184-8 (1987)

  65. Silvester, D., Kechkar, N.: Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem. Comput. Methods Appl. Mech. Eng. 79(1), 71–86. https://doi.org/10.1016/0045-7825(90)90095-4 (1990)

  66. Pitkäranta, J., Saarinen, T.: A multigrid version of a simple finite element method for the Stokes problem. Math. Comput. 45(171), 1–14 (1985)

    Google Scholar 

  67. Aguilar, G., Gaspar, F., Lisbona, F., Rodrigo, C.: Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int. J. Numer. Methods Eng. 75(11), 1282–1300 (2008). https://doi.org/10.1002/nme.2295

    Google Scholar 

  68. 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. Eng. 197(49), 4353–4366. https://doi.org/10.1016/j.cma.2008.05.015 (2008)

  69. 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). https://doi.org/10.1002/nag.951

    Google Scholar 

  70. Choo, J., Borja, R.I.: Stabilized mixed finite elements for deformable porous media with double porosity. Comput. Methods Appl. Mech. Eng. 29(3), 131–154 (2015). https://doi.org/10.1016/j.cma.2015.03.023

    Google Scholar 

  71. Sun, W., Ostien, J.T., Salinger, A.G.: A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain. Int. J. Numer. Anal. Methods Geomech. 37(16), 2755–2788 (2013). https://doi.org/10.1002/nag.2161

    Google Scholar 

  72. 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). https://doi.org/10.1137/15M1009822

    Google Scholar 

  73. Rodrigo, C., Hu, X., Ohm, P., Adler, J.H., Gaspar, F.J., Zikatanov, L.: New stabilized discretizations for poroelasticity and the Stokes’ equations. Comput. Methods Appl. Mech. Eng. 341, 467–484 (2018). https://doi.org/10.1016/j.cma.2018.07.003

    Google Scholar 

  74. Wan, J.: Stabilized Finite Element Methods for Coupled Geomechanics and Multiphase Flow. Ph.D. thesis, Stanford University (2002)

  75. Truty, A., Zimmermann, T.: Stabilized mixed finite element formulations for materially nonlinear partially saturated two-phase media. Comput. Methods Appl. Mech. Eng. 195(13), 1517–1546. https://doi.org/10.1016/j.cma.2005.05.044 (2006)

  76. Silvester, D.: Optimal low order finite element methods for incompressible flow. Comput. Methods Appl. Mech. Eng. 111(3-4), 357–368 (1994). https://doi.org/10.1016/0045-7825(94)90139-2

    Google Scholar 

  77. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press (2005)

  78. Tchonkova, M., Peters, J., Sture, S.: A new mixed finite element method for poro-elasticity. Int. J. Numer. Anal. Methods Geomech. 32(6), 579–606 (2008). https://doi.org/10.1002/nag.630

    Google Scholar 

  79. Gaspar, F.J., Rodrigo, C.: On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics. Comput. Methods Appl. Mech. Eng. 326, 526–540 (2017). https://doi.org/10.1016/j.cma.2017.08.025

    Google Scholar 

  80. Bui, Q.M., Osei-Kuffuor, D., Castelletto, N., White, J.A.: A scalable multigrid reduction framework for multiphase poromechanics of heterogeneous media, arXiv:1904.05960

  81. Ertekin, T., Abou-Kassem, J.H., King, G.R.: Basic Applied Reservoir Simulation. Society of Petroleum Engineers, Richardson (2001)

  82. Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation. Soc. Pet. Eng. AIME J. 18(3), 183–194 (1978). https://doi.org/10.2118/6893-PA

    Google Scholar 

  83. Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous. Society for Industrial and Applied Mathematics, Philadelphia (2006)

  84. Qin, J., Zhang, S.: On the selective local stabilization of the mixed Q1–P0 element. Int. J. Numer. Methods Fluids 55(12), 1121–1141 (2007). https://doi.org/10.1002/fld.1505

    Google Scholar 

  85. Griffiths, D., Silvester, D.: Unstable modes of the Q1–P0 element, Technical Report NA - 257

  86. Aziz, K.: Petroleum Reservoir Simulation, vol. 476. Applied Science Publishers, London (1979)

    Google Scholar 

  87. Boland, J.M., Nicolaides, R.A.: Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20(4), 722–731 (1983). https://doi.org/10.1137/0720048

    Google Scholar 

  88. Stenberg, R.: Analysis of mixed finite element methods for the Stokes problem: a unified approach. Math. Comput. 42(165), 9–23 (1984). https://doi.org/10.2307/2007557

    Google Scholar 

  89. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II – a General Purpose Object Oriented Finite Element Library, ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)

  90. 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). https://doi.org/10.1115/1.2791080

    Google Scholar 

  91. Fu, G.: A high-order HDG method for the Biot’s consolidation model. Comput. Math. Appl. 77(1), 237–252. https://doi.org/10.1016/j.camwa.2018.09.029 (2019)

  92. Boffi, D., Botti, M., Di Pietro, D.A.: A nonconforming high-order method for the Biot problem on general meshes. SIAM J. Sci. Comput. 38(3), A1508–A1537 (2016). https://doi.org/10.1137/15M1025505

    Google Scholar 

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Acknowledgments

Funding for JTC and JAW was provided by Total S.A. through the FC-MAELSTROM Project. JTC also acknowledges financial support provided by the Brazilian National Council for Scientific and Technological Development (CNPq) and the John A. Blume Earthquake Engineering Center. RIB was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Geosciences Research Program, under Award Number DE-FG02-03ER15454 and by the National Science Foundation under Award Numbers CMMI-1462231 and CMMI-1914780. The authors wish to thank Nicola Castelletto for helpful discussions. Portions of this work were performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07-NA27344.

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  1. Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, USA

    Julia T. Camargo & Ronaldo I. Borja

  2. Atmospheric, Earth and Energy Division, Lawrence Livermore National Laboratory, Livermore, CA, USA

    Joshua A. White

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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). https://doi.org/10.1007/s10596-020-09964-3

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