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Multiscale two-stage solver for Biot’s poroelasticity equations in subsurface media

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Abstract

We propose a two-stage preconditioner for accelerating the iterative solution by a Krylov subspace method of Biot’s poroelasticity equations based on a displacement-pressure formulation. The spatial discretization combines a finite element method for mechanics and a finite volume approach for flow. The fully implicit backward Euler scheme is used for time integration. The result is a 2 × 2 block linear system for each timestep. The preconditioning operator is obtained by applying a two-stage scheme. The first stage is a global preconditioner that employs multiscale basis functions to construct coarse-scale coupled systems using a Galerkin projection. This global stage is effective at damping low-frequency error modes associated with long-range coupling of the unknowns. The second stage is a local block-triangular smoothing preconditioner, which is aimed at high-frequency error modes associated with short-range coupling of the variables. Various numerical experiments are used to demonstrate the robustness of the proposed solver.

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References

  1. Adler, J.H., Gaspar, F.J., Hu, X., Rodrigo, C., Zikatanov, L.T.: Robust block preconditioners for Biot’s model. In: Bjøstad, P.E. , Brenner, S., Halpern, L., Kornhuber, R., Kim, H.H., Rahman, T., Widlund, O.B. (eds.) Domain Decomposition Methods in Science and Engineering XXIV. Springer International Publishing. https://doi.org/10.1007/978-3-319-93873-8 (2018)

    Google Scholar 

  2. Akkutlu, I.Y., Efendiev, Y., Vasilyeva, M., Wang, Y.: Multiscale model reduction for shale gas transport in poroelastic fractured media. J. Comput. Phys. 353, 356–376 (2018). https://doi.org/10.1016/j.jcp.2017.10.023

    Article  Google Scholar 

  3. Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Elsevier applied science publishers, London (1979)

    Google Scholar 

  4. Bause, M., Radu, F.A., Köcher, U.: Space–time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Meth. Appl. Mech. Eng. 320, 745–768 (2017). https://doi.org/10.1016/j.cma.2017.03.017

    Article  Google Scholar 

  5. Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182(2), 418–477 (2002). https://doi.org/10.1006/jcph.2002.7176

    Article  Google Scholar 

  6. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005). https://doi.org/10.1017/S0962492904000212

    Article  Google Scholar 

  7. Bergamaschi, L., Ferronato, M., Gambolati, G.: Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations. Comput. Meth. Appl. Mech. Eng. 196(25–28), 2647–2656 (2007). https://doi.org/10.1016/j.cma.2007.01.013

    Article  Google Scholar 

  8. Bergamaschi, L., Martínez, Á.: RMCP: relaxed mixed constraint preconditioners for saddle point linear systems arising in geomechanics. Comput. Meth. Appl. Mech. Eng. 221-222, 54–62 (2012). https://doi.org/10.1016/j.cma.2012.02.004

    Article  Google Scholar 

  9. Both, J.W., Borregales, M., Nordbotten, J.M., Kumar, K., Radu, F.A.: Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017). https://doi.org/10.1016/j.aml.2016.12.019

    Article  Google Scholar 

  10. Both, J.W., Kumar, K., Nordbotten, J.M., Radu, F.A.: Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media. Comput. Math. Appl. (2018). https://doi.org/10.1016/j.camwa.2018.07.033

    Article  Google Scholar 

  11. Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comput. 50(181), 1–17 (1988). https://doi.org/10.1090/S0025-5718-1988-0917816-8

    Article  Google Scholar 

  12. Brown, D.L., Vasilyeva, M.: A generalized multiscale finite element method for poroelasticity problems I: Linear problems. J. Comput. Appl. Math. 294, 372–388 (2016). https://doi.org/10.1016/j.cam.2015.08.007

    Article  Google Scholar 

  13. Brown, D.L., Vasilyeva, M.: A generalized multiscale finite element method for poroelasticity problems II: Nonlinear coupling. J. Comput. Appl. Math. 297, 132–146 (2016). https://doi.org/10.1016/j.cam.2015.11.007

    Article  Google Scholar 

  14. Buck, M., Iliev, O., Andrä, H.: Multiscale finite element coarse spaces for the application to linear elasticity. Cent. Eur. J. Math. 11(4), 680–701 (2013). https://doi.org/10.2478/s11533-012-0166-8

    Article  Google Scholar 

  15. Buck, M., Iliev, O., Andrä, H.: Domain Decomposition Methods in Science and Engineering XXI, pp. 237–245. Springer International Publishing. In: Erhel, J. , Gander, J.M. , Halpern, L. , Pichot, G. , Sassi, T. , Widlund, O. (eds.) . https://doi.org/10.1007/978-3-319-05789-7_20 (2014)

    Google Scholar 

  16. Cao, H., Tchelepi, H.A., Wallis, J., Yardumian, H.: Parallel scalable unstructured CPR-type linear solver for reservoir simulation. In: Proceedings - SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. https://doi.org/10.2118/173226-MS (2005)

  17. Castelletto, N., Ferronato, M., Gambolati, G.: Thermo-hydro-mechanical modeling of fluid geological storage by Godunov-mixed methods. Int. J. Numer. Meth. Eng. 90(8), 988–1009 (2012). https://doi.org/10.1002/nme.3352

    Article  Google Scholar 

  18. Castelletto, N., Hajibeygi, H., Tchelepi, H.: Hybrid multiscale formulation for coupled flow and geomechanics. In: Proceedings of the 15th European Conference on the Mathematics of Oil Recovery (ECMOR XV). EAGE. https://doi.org/10.3997/2214-4609.201601888 (2016)

  19. Castelletto, N., Hajibeygi, H., Tchelepi, H.: Multiscale finite-element method for linear elastic geomechanics. J. Comput. Phys. 331, 337–356 (2017). https://doi.org/10.1016/j.jcp.2016.11.044

    Article  Google Scholar 

  20. Castelletto, N., White, J.A., Ferronato, M.: Scalable algorithms for three-field mixed finite element coupled poromechanics. J. Comput. Phys. 327, 894–918 (2016). https://doi.org/10.1016/j.jcp.2016.09.063

    Article  Google Scholar 

  21. 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

    Article  Google Scholar 

  22. Christie, M., Blunt, M.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4(4), 308–316 (2001). https://doi.org/10.2118/72469-PA

    Article  Google Scholar 

  23. Coussy, O.: Poromechanics. Wiley, Chichester (2004)

    Google Scholar 

  24. Cusini, M., Fryer, B., van Kruijsdijk, C., Hajibeygi, H.: Algebraic dynamic multilevel method for compositional flow in heterogeneous porous media. J. Comput. Phys. 354, 593–612 (2018). https://doi.org/10.1016/j.jcp.2017.10.052

    Article  Google Scholar 

  25. Cusini, M., Lukyanov, A.A., Natvig, J., Hajibeygi, H.: Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous media. J. Comput. Phys. 299, 472–486 (2015). https://doi.org/10.1016/j.jcp.2015.07.019

    Article  Google Scholar 

  26. Dana, S., Ganis, B., Wheeler, M.F.: A multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs. J. Comput. Phys. 352, 1–22 (2018). https://doi.org/10.1016/j.jcp.2017.09.049

    Article  Google Scholar 

  27. Di Pietro, D.A., Eymard, R., Lemaire, S., Masson, R.: Hybrid finite volume discretization of linear elasticity models on general meshes. In: Fořt, J., et al. (eds.) Finite Volumes for Complex Applications VI – Problems & Perspectives, pp 331–339. Springer, Berlin (2011)

  28. Droniou, J.: Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Meth. Appl. Sci. 24(8), 1575–1619 (2014). https://doi.org/10.1142/S0218202514400041

    Article  Google Scholar 

  29. Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods: Theory and Applications. Springer, New York (2009). https://doi.org/10.1007/978-0-387-09496-0

    Google Scholar 

  30. Eymard, R., Gallouët, T., Guichard, C., Herbin, R., Masson, R.: TP Or not TP, that is the question. Comput. Geosci. 18(3), 285–296 (2014). https://doi.org/10.1007/s10596-013-9392-9

    Article  Google Scholar 

  31. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G. , Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 7, pp. 713–1018. Elsevier. https://doi.org/10.1016/S1570-8659(00)07005-8 (2000)

    Google Scholar 

  32. 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). https://doi.org/10.1016/j.jcp.2010.03.018

    Article  Google Scholar 

  33. Frijns, A.J.H.: A Four-Component Mixture Theory Applied to Cartilaginous Tissues: Numerical Modelling and Experiments. Phd Thesis. Technische Universiteit Eindhoven, The Netherlands (2000)

    Google Scholar 

  34. Gai, X., Sun, S., Wheeler, M.F., Klie, H.: A time-stepping scheme for coupled reservoir flow and geomechanics. In: Proceedings - SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. https://doi.org/10.2118/97054-MS (2005)

  35. Gaspar, F.J., Lisbona, F.J., Oosterlee, C.W., Wienands, R.: A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity system. Numer. Linear Algebr. Appl. 11(2-3), 93–113 (2004). https://doi.org/10.1002/nla.372

    Article  Google Scholar 

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

    Article  Google Scholar 

  37. Haga, J.B., Osnes, H., Langtangen, H.P.: A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters. Comput. Geosci. 16(3), 723–734 (2012). https://doi.org/10.1007/s10596-012-9284-4

    Article  Google Scholar 

  38. 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

    Article  Google Scholar 

  39. Hajibeygi, H., Bonfigli, G., Hesse, M.A., Jenny, P.: Iterative multiscale finite-volume method. J. Comput. Phys. 227(19), 8604–8621 (2008). https://doi.org/10.1016/j.jcp.2008.06.013

    Article  Google Scholar 

  40. Helmig, R., Niessner, J., Flemisch, B., Wolff, M., Fritz, J.: Efficient modeling of flow and transport in porous media using multiphysics and multiscale approaches. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp 417–457. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-01546-5_15

    Chapter  Google Scholar 

  41. Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997). https://doi.org/10.1006/jcph.1997.5682

    Article  Google Scholar 

  42. 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/10.1016/j.cam.2016.06.003

    Article  Google Scholar 

  43. Hughes, T.J.R.: The finite element method: linear static and dynamic finite element analysis dover publications (2000)

  44. Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187(1), 47–67 (2003). https://doi.org/10.1016/S0021-9991(03)00075-5

    Article  Google Scholar 

  45. Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. J. Comput. Phys. 217(2), 627–641 (2006). https://doi.org/10.1016/j.jcp.2006.01.028

    Article  Google Scholar 

  46. 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

    Article  Google Scholar 

  47. Jha, B., Juanes, R.: Coupled multiphase flow and poromechanics: a computational model of pore pressure effects on fault slip and earthquake triggering. Water Resour. Res. 5, 3776–3808 (2014). https://doi.org/10.1002/2013WR015175

    Article  Google Scholar 

  48. Kalchev, D.Z., Lee, C.S., Villa, U., Efendiev, Y., Vassilevski, P.S.: Upscaling of mixed finite element discretization problems by the spectral AMGe method. SIAM J. Sci. Comput. 38(5), A2912–A2933 (2016). https://doi.org/10.1137/15M1036683

    Article  Google Scholar 

  49. Keilegavlen, E., Nordbotten, J.M.: Finite volume methods for elasticity with weak symmetry. Int. J. Numer. Meth. Eng. 112, 939–962 (2017). https://doi.org/10.1002/nme.5538

    Article  Google Scholar 

  50. Kim, J., Tchelepi, H.A., Juanes, R.: Stability, accuracy and efficiency of sequential methods for coupled flow and geomechanics. SPE J. 16(2), 249–262 (2011). https://doi.org/10.2118/119084-PA

    Article  Google Scholar 

  51. 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. Meth. Appl. Mech. Eng. 200(13), 1591–1606 (2011). https://doi.org/10.1016/j.cma.2010.12.022

    Article  Google Scholar 

  52. Klevtsov, S., Castelletto, N., White, J., Tchelepi, H.: Block-preconditioned krylov methods for coupled multiphase reservoir flow and geomechanics. In: Proceedings of the 15th European Conference on the Mathematics of Oil Recovery (ECMOR XV). EAGE. https://doi.org/10.3997/2214-4609.201601900 (2016)

  53. Kozlova, A., Li, Z., Natvig, J.R., Watanabe, S., Zhou, Y., Bratvedt, K., Lee, S.H.: A real-field multiscale black-oil reservoir simulator. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers. https://doi.org/10.2118/173226-MS (2015)

  54. Lee, J.J., Mardal, K.A., Winther, R.: Parameter-robust discretization and preconditioning of Biot’s consolidation model. SIAM J. Sci. Comput. 39(1), A1–A24 (2010). https://doi.org/10.1137/15M1029473

    Article  Google Scholar 

  55. Lewis, R.W., Schrefler, B.A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2nd edn. Wiley, Chichester (1998)

    Google Scholar 

  56. Lie, K., Møyner, O., Natvig, J.R.: Use of multiple multiscale operators to accelerate simulation of complex geomodels. SPE J. 22(6), 1929–1945 (2017). https://doi.org/10.2118/182701-PA

    Article  Google Scholar 

  57. Lie, K., Møyner, O., Natvig, J.R., Kozlova, A., Bratvedt, K., Watanabe, S., Li, Z.: Successful application of multiscale methods in a real reservoir simulator environment. In: Proceedings of the 15th European Conference on the Mathematics of Oil Recovery (ECMOR XV). EAGE. https://doi.org/10.3997/2214-4609.201601893 (2016)

  58. Lipnikov, K.: Numerical Methods for the Biot Model in Poroelasticity. Phd Thesis. University of Houston, USA (2002)

  59. Lunati, I., Tyagi, M., Lee, S.H.: An iterative multiscale finite volume algorithm converging to the exact solution. J. Comput. Phys. 230(5), 1849–1864 (2011). https://doi.org/10.1016/j.jcp.2013.11.024

    Article  Google Scholar 

  60. Luo, P., Rodrigo, C., Gaspar, F.J., Oosterlee, C.W.: Multigrid method for nonlinear poroelasticity equations. Comput. Visual Sci. 17(5), 255–265 (2015). https://doi.org/10.1007/s00791-016-0260-8

    Article  Google Scholar 

  61. Luo, P., Rodrigo, C., Gaspar, F.J., Oosterlee, C.W.: On an Uzawa smoother in multigrid for poroelasticity equations. Numer. Linear Algebr. Appl. 24(1), e2074 (2017). https://doi.org/10.1002/nla.2074

    Article  Google Scholar 

  62. Meijerink, J.A., van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31(137), 148–162 (1977). https://doi.org/10.1090/S0025-5718-1977-0438681-4

    Article  Google Scholar 

  63. Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17(3), 455–461 (2013). https://doi.org/10.1007/s10596-012-9318-y

    Article  Google Scholar 

  64. Møyner, O., Lie, K.: A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids. J. Comput. Phys. 304, 46–71 (2016). https://doi.org/10.1016/j.jcp.2015.10.010

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  67. Nordbotten, J.M.: Stable cell-centered finite volume discretization for Biot equations. SIAM J. Numer. Anal. 54(6), 942–968 (2016). https://doi.org/10.1137/15M1014280

    Article  Google Scholar 

  68. Nordbotten, J.M., Bjøstad, P.E.: On the relationship between the multiscale finit volume method and domain decomposition preconditioners. Comput. Geosci. 13(3), 367–376 (2008). https://doi.org/10.1007/s10596-007-9066-6

    Article  Google Scholar 

  69. Pasetto, D., Ferronato, M., Putti, M.: A reduced order model-based preconditioner for the efficient solution of transient diffusion equations. Int. J. Numer. Meth. Eng. 109(8), 1159–1179 (2017). https://doi.org/10.1002/nme.5320

    Article  Google Scholar 

  70. Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: The continuous in time case. Comput. Geosci. 11(2), 131–144 (2007). https://doi.org/10.1007/s10596-007-9045-y

    Article  Google Scholar 

  71. 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

    Article  Google Scholar 

  72. Prevost, 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. Meth. Eng. 98(2), 612–624 (2014). https://doi.org/10.1002/nme.4657

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  75. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003). https://doi.org/10.1137/1.9780898718003

    Book  Google Scholar 

  76. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986). https://doi.org/10.1137/0907058

    Article  Google Scholar 

  77. Schneider, M., Flemisch, B., Helmig, R., Terekhov, K., Tchelepi, H.A.: Monotone nonlinear finite-volume method for challenging grids. Comput Geosci. 22(2), 565–586 (2018). https://doi.org/10.1007/s10596-017-9710-8

    Article  Google Scholar 

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

    Article  Google Scholar 

  79. Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2014). https://doi.org/10.1007/s00211-013-0576-y

    Article  Google Scholar 

  80. Ţene, M., Al Kobaisi, M.S., Hajibeygi, H.: Algebraic multiscale method for fractured porous media. J. Comput. Phys. 321, 819–845 (2016). https://doi.org/10.1016/j.jcp.2016.06.012

    Article  Google Scholar 

  81. Ţene, M., Wang, Y., Hajibeygi, H.: Algebraic multiscale method for fractured porous media. J. Comput. Phys. 300, 679–694 (2015). https://doi.org/10.1016/j.jcp.2015.08.009

    Article  Google Scholar 

  82. 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). https://doi.org/10.1016/j.jcp.2016.11.010

    Article  Google Scholar 

  83. Turan, E., Arbenz, P.: Large scale micro finite element analysis of 3D bone poroelasticity. Parallel Comput. 40(7), 239–250 (2014). https://doi.org/10.1016/j.parco.2013.09.002

    Article  Google Scholar 

  84. Wang, H.F.: Theory of Linear Poroelasticity. Princeton University Press, Princeton (2000)

    Google Scholar 

  85. Wang, Y., Hajibeygi, H., Tchelepi, H.A.: Algebraic multiscale solver for flow in heterogeneous porous media. J. Comput. Phys. 259, 284–303 (2014). https://doi.org/10.1016/j.jcp.2013.11.024

    Article  Google Scholar 

  86. Wang, Y., Hajibeygi, H., Tchelepi, H.A.: Monotone multiscale finite volume method. Comput. Geosci. 20, 509–524 (2016). https://doi.org/10.1007/s10596-015-9506-7

    Article  Google Scholar 

  87. White, J.A., Borja, R.: Block-preconditioned Newton–Krylov solvers for fully coupled flow and geomechanics. Comput. Geosci. 15(4), 647–659 (2011). https://doi.org/10.1007/s10596-011-9233-7

    Article  Google Scholar 

  88. 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. Meth. Appl. Mech. Eng. 197(49–50), 4353–4366 (2008). https://doi.org/10.1016/j.cma.2008.05.015

    Article  Google Scholar 

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

    Article  Google Scholar 

  90. Yi, S.Y.: Convergence analysis of a new mixed finite element method for Biot’s consolidation model. Numer. Meth. Part. Differ. Equ. 30(4), 1189–1210 (2014). https://doi.org/10.1002/num.21865

    Article  Google Scholar 

  91. Zhang, H.W., Fu, Z.D., Wu, J.K.: Coupling multiscale finite element method for consolidation analysis of heterogeneous saturated porous media. Adv. Water Resour. 32 (2), 268–279 (2009). https://doi.org/10.1016/j.advwatres.2008.11.002

    Article  Google Scholar 

  92. Zhou, H., Tchelepi, H.A.: Two-stage algebraic multiscale linear solver for highly heterogeneous reservoir models. SPE J. 17(2), 523–539 (2012). https://doi.org/10.2118/141473-PA

    Article  Google Scholar 

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Acknowledgements

The authors thank Andrea Franceschini for his insightful suggestions. 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.

Funding

N. Castelletto, S. Klevtsov and H.A. Tchelepi gratefully acknowledge the financial support provided by the Reservoir Simulation Industrial Affiliates Consortium at Stanford University (SUPRI-B) and Total S.A. through the Stanford Total Enhanced Modeling of Source rock (STEMS) project. H. Hajibeygi was sponsored by Schlumberger Petroleum Services CV, The Netherlands.

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Authors and Affiliations

  1. Energy Resources Engineering, Stanford University, Stanford, CA, USA

    Nicola Castelletto, Sergey Klevtsov & Hamdi A. Tchelepi

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

    Nicola Castelletto

  3. Department of Geoscience and Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

    Hadi Hajibeygi

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  1. Nicola Castelletto
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  2. Sergey Klevtsov
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  3. Hadi Hajibeygi
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  4. Hamdi A. Tchelepi
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Correspondence to Hamdi A. Tchelepi.

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Castelletto, N., Klevtsov, S., Hajibeygi, H. et al. Multiscale two-stage solver for Biot’s poroelasticity equations in subsurface media. Comput Geosci 23, 207–224 (2019). https://doi.org/10.1007/s10596-018-9791-z

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  • DOI: https://doi.org/10.1007/s10596-018-9791-z

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