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Computational Geosciences

, Volume 22, Issue 4, pp 1039–1057 | Cite as

Unified thermo-compositional-mechanical framework for reservoir simulation

  • T. T. Garipov
  • P. Tomin
  • R. Rin
  • D. V. Voskov
  • H. A. Tchelepi
Original Paper

Abstract

We present a reservoir simulation framework for coupled thermal-compositional-mechanics processes. We use finite-volume methods to discretize the mass and energy conservation equations and finite-element methods for the mechanics problem. We use the first-order backward Euler for time. We solve the resulting set of nonlinear algebraic equations using fully implicit (FI) and sequential-implicit (SI) solution schemes. The FI approach is attractive for general-purpose simulation due to its unconditional stability. However, the FI method requires the development of a complex thermo-compositional-mechanics framework for the nonlinear problems of interest, and that includes the construction of the full Jacobian matrix for the coupled multi-physics discrete system of equations. On the other hand, SI-based solution schemes allow for relatively fast development because different simulation modules can be coupled more easily. The challenge with SI schemes is that the nonlinear convergence rate depends strongly on the coupling strength across the physical mechanisms and on the details of the sequential updating strategy across the different physics modules. The flexible automatic differentiation-based framework described here allows for detailed assessment of the robustness and computational efficiency of different coupling schemes for a wide range of multi-physics subsurface problems.

Keywords

Geomechanics Thermal-compositional-mechanics Reservoir simulation Multiphase flow Multi-physics coupling 

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Notes

Funding information

The authors 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.

References

  1. 1.
    Aboustit, B., Advani, S., Lee, J.: Variational principles and finite element simulations for thermo-elastic consolidation. Int. J. Numer. Anal. Methods Geomech. 9, 49–69 (1985).  https://doi.org/10.1002/nag.1610090105 CrossRefGoogle Scholar
  2. 2.
    AD-GPRS: Automatic differentiation general purpose research simulator. https://supri-b.stanford.edu/research-areas/ad-gprs (2017)
  3. 3.
    Armero, F., Simo, J.: A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems. Int. J. Numer. Methods Eng. 35, 737–766 (1992).  https://doi.org/10.1002/nme.1620350408 CrossRefGoogle Scholar
  4. 4.
    Aziz, K., Settari, A: Petroleum reservoir simulation. Applied Science Publishers (1979)Google Scholar
  5. 5.
    Bevillon, D., Masson, R: Stability and convergence analysis of partially coupled schemes for geomechanical reservoir simulations. In: The European Conference on the Mathematics of Oil Recovery. Baveno, Italy (2000)Google Scholar
  6. 6.
    Borja, I.: Plasticity: modeling and computation. Springer, Berlin (2013)CrossRefGoogle Scholar
  7. 7.
    Butler, R.: New approach to the modelling of steam-assisted gravity drainage. J. Can. Pet. Technol. 24, 42–51 (1985).  https://doi.org/10.2118/85-03-01 CrossRefGoogle Scholar
  8. 8.
    Cao, H., Tchelepi, H.A., Wallis, J.R., Yardumian, H.E.: Parallel scalable unstructured CPR-type linear solver for reservoir simulation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2005)Google Scholar
  9. 9.
    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, 1593–1618 (2015).  https://doi.org/10.1002/nag.2400 CrossRefGoogle Scholar
  10. 10.
    Christie, M., Blunt, M.: Tenth SPE comparative solution project: a comparison of upscaling techniques. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2001)Google Scholar
  11. 11.
    Coats, K.H.: An equation of state compositional model. Society of Petroleum Engineers (1980).  https://doi.org/10.2118/8284-PA
  12. 12.
    Coussy, O.: Poromechanics. Wiley, New York (2004)Google Scholar
  13. 13.
    Crisfield, M.: Non-linear finite element analysis of solids and structures. Wiley, New York (1996)Google Scholar
  14. 14.
    David, C., Wong, T.-F., Zhu, W., Zhang, J.: Laboratory measurement of compaction-induced permeability change in porous rocks: implications for the generation and maintenance of pore pressure excess in the crust. Pure Appl. Geophys. 143, 425–456 (1994).  https://doi.org/10.1007/BF00874337 CrossRefGoogle Scholar
  15. 15.
    Dean, R., Gai, X., Stone, C., Minkoff, S.: A comparison of techniques for coupling porous flow and geomechanics. Soc. Petrol. Eng. 11, 132–140 (2006).  https://doi.org/10.2118/79709-PA Google Scholar
  16. 16.
    Drucker, D., Prager, W.: Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10, 157–165 (1952)CrossRefGoogle Scholar
  17. 17.
    Ehlers, W., Ellsiepen, P.: Pandas: Ein fe-system zur simulation von sonderproblemen der bodenmechanik. Finite Elemente in der Baupraxis: Modellierung, Berechnung und Konstruktion. Beiträge zur Tagung FEM 98, 431–400 (1998)Google Scholar
  18. 18.
    Fedorenko, R.: A relaxation method for solving elliptic difference equations. USSR Comput. Math. Math. Phys. 1, 1092–1096 (1962)CrossRefGoogle Scholar
  19. 19.
    Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., Helmig, R.: DuMux: DUNE for multi-{phase,component,scale,physics,...} flow and transport in porous media. Adv. Water Resour. 34, 1102–1112 (2011).  https://doi.org/10.1016/j.advwatres.2011.03.007 CrossRefGoogle Scholar
  20. 20.
    Gai, X.: A coupled geomechanics and reservoir flow model on parallel computers. Ph.D. thesis University of Texas at Austin (2004)Google Scholar
  21. 21.
    Garipov, T.T., Voskov, D., Tchelepi, H.A.: Rigorous coupling of geomechanics and thermal-compositional flow for SAGD and ES-SAGD operations. In: SPE Canada Heavy Oil Technical Conference. Calgary, Canada (2015).  https://doi.org/10.2118/174508-MS
  22. 22.
    Garipov, T.T., White, J., Lapene, A., Tchelepi, H.A.: Thermo-hydro-mechanical model for source rock thermal maturation. In: 50th US Rock Mechanics Geomechanics Symposium 2016. Houston, USA (2016)Google Scholar
  23. 23.
    Hu, L., Winterfeld, P.H., Fakcharoenphol, P., Wu, Y.-S.: A novel fully-coupled flow and geomechanics model in enhanced geothermal reservoirs. J. Pet. Sci. Eng. 107, 1–11 (2013).  https://doi.org/10.1016/j.petrol.2013.04.005 CrossRefGoogle Scholar
  24. 24.
    Huang, J., Griffiths, D.: Return mapping algorithms and stress predictors for failure analysis in geomechanics. J. Eng. Mech. 135, 276–284 (2009).  https://doi.org/10.1061/(ASCE)0733-9399(2009)135:4(276) CrossRefGoogle Scholar
  25. 25.
    Hughes, T.: The finite element method: linear static and dynamic finite element analysis. Courier Dover Publications (2012)Google Scholar
  26. 26.
    Jha, B., Juanes, R.: A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotech. 2, 139–153 (2007).  https://doi.org/10.1007/s11440-007-0033-0 CrossRefGoogle Scholar
  27. 27.
    Keilegavlen, E., Nordbotten, J.M.: Finite volume methods for elasticity with weak symmetry. International Journal for Numerical Methods in Engineering (2017)Google Scholar
  28. 28.
    Kim, J.: Sequential methods for coupled geomechanics and multiphase flow. Ph.D. thesis Stanford University (2010)Google Scholar
  29. 29.
    Kim, J.: Unconditionally stable sequential schemes for thermoporomechanics: undrained-adiabatic and extended fixed-stress splits. In: SPE Reservoir Simulation Symposium. Houston, USA (2015).  https://doi.org/10.2118/173294-MS
  30. 30.
    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, 1591–1606 (2011).  https://doi.org/10.1016/j.cma.2010.12.022 CrossRefGoogle Scholar
  31. 31.
    Kim, J., Tchelepi, H.A., Juanes, R.: Rigorous coupling of geomechanics and multiphase flow with strong capillarity. SPE J 18, 1–123 (2013)CrossRefGoogle Scholar
  32. 32.
    Klevtsov, S., Castelletto, N., White, J., Tchelepi, H.: Block-preconditioned Krylov methods for coupled multiphase reservoir flow and geomechanics. In: ECMOR XIV-15th European Conference on the Mathematics of Oil Recovery (2016)Google Scholar
  33. 33.
    Kolditz, O., Bauer, S., Bilke, L., Bottcher, N., Delfs, J., Fischer, T., Gorke, U., Kalbacher, T., Kosakowski, G., Mcdermott, C., Park, C., Radu, F., Rink, K., Shao, H., Shao, H., Sun, F., Sun, Y., Singh, A., Taron, J., Walther, M., Wang, W., Watanabe, N., Wu, Y., Xie, M., Xu, W., Zehner, B.: Opengeosys: an open-source initiative for numerical simulation of thermo-hydro-mechanical/ chemical (THM/C) processes in porous media. Environ. Earth Sci. 67, 589–599 (2012).  https://doi.org/10.1007/s12665-012-1546-x CrossRefGoogle Scholar
  34. 34.
    Lewis, R.W., Schrefler, B.A.: The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley, New York (1998)Google Scholar
  35. 35.
    Li, P., Chalaturnyk, R.: Gas-over-bitumen geometry and its SAGD performance analysis with coupled reservoir geomechanical simulation. J. Can. Pet. Technol. 46, 42–49 (2007).  https://doi.org/10.2118/07-01-05 Google Scholar
  36. 36.
    Li, P., Chalaturnyk, R., et al.: History match of the UTF phase A project with coupled reservoir geomechanical simulation. In: Canadian International Petroleum Conference. Petroleum Society of Canada (2005)Google Scholar
  37. 37.
    Lie, K.-A.: An introduction to reservoir simulation using MATLAB: user guide for the Matlab Reservoir Simulation Toolbox (MRST). SINTEF ICT (2014)Google Scholar
  38. 38.
    Mainguy, M., Longuemare, P.: Coupling fluid flow and rock mechanics: formulations of the partial coupling between reservoir and geomechanical simulators. Oil Gas Sci. Technol. 57, 355–367 (2002)CrossRefGoogle Scholar
  39. 39.
    Mandel, J.: Consolidation des sols (etude mathématique)́. Geotechnique 3, 287–299 (1953)CrossRefGoogle Scholar
  40. 40.
    Markert, B., Heider, Y., Ehlers, W.: Comparison of monolithic and splitting solution schemes for dynamic porous media problems. Int. J. Numer. Methods Eng. 82, 1341–1383 (2010).  https://doi.org/10.1002/nme.2789 Google Scholar
  41. 41.
    Mikelic, A., Wheeler, M.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17, 455–461 (2013).  https://doi.org/10.1007/s10596-012-9318-y CrossRefGoogle Scholar
  42. 42.
    Minkoff, S., Stone, C., Bryant, S., Peszynska, M., Wheeler, M.: Coupled fluid flow and geomechanical deformation modeling. J. Pet. Sci. Eng. 38, 37–56 (2003a).  https://doi.org/10.1016/S0920-4105(03)00021-4 CrossRefGoogle Scholar
  43. 43.
    Minkoff, S.E., Stone, C., Bryant, S., Peszynska, M., Wheeler, M.F.: Coupled fluid flow and geomechanical deformation modeling. J. Pet. Sci. Eng. 38, 37–56 (2003b).  https://doi.org/10.1016/S0920-4105(03)00021-4 CrossRefGoogle Scholar
  44. 44.
    Nikolaevskij, V.N.: Mechanics of porous and fractured media volume 8. World Scientific (1990)Google Scholar
  45. 45.
    Noorishad, J., Tsang, C.F., Witherspoon, P.A.: Coupled thermal-hydraulic-mechanical phenomena in saturated fractured porous rocks: numerical approach. J. Geophys. Res. Solid Earth 89, 10365–10373 (1984).  https://doi.org/10.1029/JB089iB12p10365 CrossRefGoogle Scholar
  46. 46.
    Ottosen, N., Ristinmaa, M.: The Mechanics of Constitutive Modeling. Elsevier, Amsterdam (2005)Google Scholar
  47. 47.
    Park, K.: Stabilization of partitioned solution procedure for pore fluid-soil interaction analysis. Int. J. Numer. Methods Eng. 19, 1669–1673 (1983).  https://doi.org/10.1002/nme.1620191106 CrossRefGoogle Scholar
  48. 48.
    Peneloux, A., Rauzy, E., Freze, R.: A consistent correction for Redlich-Kwong-Soave volumes. Fluid Phase Equilib. 8, 7–23 (1982).  https://doi.org/10.1016/0378-3812(82)80002-2 CrossRefGoogle Scholar
  49. 49.
    Prevost, J.H.: Partitioned solution procedure for simultaneous integration of coupled-field problems. Commun. Numer. Methods Eng. 13, 239–247 (1997).  https://doi.org/10.1002/(SICI)1099-0887(199704)13:43.0.CO;2-2 CrossRefGoogle Scholar
  50. 50.
    Rahmati, E., Nouri, A., Fattahpour, V., et al.: Caprock integrity analysis during a sagd operation using an anisotropic elasto-plastic model. In: SPE Heavy Oil Conference-Canada. Society of Petroleum Engineers (2014)Google Scholar
  51. 51.
    Rin, R.: Implicit Coupling Framework for Multi-Physics Reservoir Simulation. Ph.D. thesis Stanford University (2017)Google Scholar
  52. 52.
    Rin, R., Tomin, P., Garipov, T., Voskov, D., Tchelepi, H.: General implicit coupling framework for multi-physics problems. In: SPE-182714-MS, SPE Reservoir Simulation Conference. Montgomery, USA (2017)Google Scholar
  53. 53.
    Rutqvist, J.: Status of the tough-flac simulator and recent applications related to coupled fluid flow and crustal deformations. Comput. Geosci. 37, 739–750 (2011).  https://doi.org/10.1016/j.cageo.2010.08.006.2009 Transport of Unsaturated Groundwater and Heat SymposiumCrossRefGoogle Scholar
  54. 54.
    Samier, P., Onaisi, A., de Gennaro, S.: A practical iterative scheme for coupling geomechanics with reservoir simulation. SPE Reserv. Eval. Eng. 11, 892–901 (2008).  https://doi.org/10.2118/107077-PA CrossRefGoogle Scholar
  55. 55.
    Settari, A., Walters, D.: Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. Soc. Petrol. Eng. 6, 14–17 (2001).  https://doi.org/10.2118/74142-PA Google Scholar
  56. 56.
    Simo, J.C., Hughes, T.J.: Computational Inelasticity volume 7. Springer Science Business Media (2006)Google Scholar
  57. 57.
    Stüben, K.: Algebraic multigrid (AMG): experiences and comparisons. Appl. Math. Comput. 13, 419–451 (1983).  https://doi.org/10.1016/0096-3003(83)90023-1 Google Scholar
  58. 58.
    Thomas, L., Chin, L., Pierson, R., Sylte, J.: Coupled geomechanics and reservoir simulation. Soc. Petrol. Eng. 8, 350–358 (2003).  https://doi.org/10.2118/87339-PA Google Scholar
  59. 59.
    Tran, D., Settari, A., Nghiem, L.: New iterative coupling between a reservoir simulator and a geomechanics module. Soc. Petrol. Eng. 9, 362–369 (2004).  https://doi.org/10.2118/88989-PA Google Scholar
  60. 60.
    Voskov, D., Zaydullin, R., Lucia, A.: Heavy oil recovery efficiency using SAGD, SAGD with propane co-injection and STRIP-SAGD. Comput. Chem. Eng. 88, 115–125 (2016).  https://doi.org/10.1016/j.compchemeng.2016.02.010 CrossRefGoogle Scholar
  61. 61.
    Voskov, D.V., Tchelepi, H.A.: Comparison of nonlinear formulations for two-phase multi-component EoS based simulation. J. Pet. Sci. Eng. 82–83, 101–111 (2012).  https://doi.org/10.1016/j.petrol.2011.10.012 CrossRefGoogle Scholar
  62. 62.
    Wallis, J.R.: Incomplete gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: 7th SPE Reservoir Simulation Symposium. San Francisco, USA (1983). https://doi.org/10.2118/12265-MS
  63. 63.
    Wheeler, M.F., Gai, X.: Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity. Num Methods Partial Differential Equations 23, 785–797 (2007).  https://doi.org/10.1002/num.20258 CrossRefGoogle Scholar
  64. 64.
    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 CrossRefGoogle Scholar
  65. 65.
    White, M., Oostrom, M.: Stomp: subsurface transport over multiple phases. version 4.0, user’s guide. Richland: Pacific Northwest National Laboratory (2006)Google Scholar
  66. 66.
    Yang, D., Moridis, G.J., Blasingame, T.A.: A fully coupled multiphase flow and geomechanics solver for highly heterogeneous porous media. J. Comput. Appl. Math. 270, 417–432 (2014).  https://doi.org/10.1016/j.cam.2013.12.029 CrossRefGoogle Scholar
  67. 67.
    Younis, R.M.: Modern advances in software and solution algorithms for reservoir simulation. Ph.D. thesis Stanford University (2011)Google Scholar
  68. 68.
    Zaydullin, R., Voskov, D., Tchelepi, H.: Comparison of eos-based and k-values-based methods for three-phase thermal simulation. Transport in Porous Media, (pp. 1–24) (2016a).  https://doi.org/10.1007/s11242-016-0795-7
  69. 69.
    Zaydullin, R., Voskov, D., Tchelepi, H.: Phase-state identification bypass method for three-phase thermal compositional simulation. Comput. Geosci. 20, 461–474 (2016b).  https://doi.org/10.1007/s10596-015-9510-y CrossRefGoogle Scholar
  70. 70.
    Zhou, Y., Jiang, Y., Tchelepi, H.: A scalable multistage linear solver for reservoir models with multisegment wells. Comput. Geosci. 17, 197–216 (2013a).  https://doi.org/10.1007/s10596-012-9324-0 CrossRefGoogle Scholar
  71. 71.
    Zhou, Y., Jiang, Y., Tchelepi, H.A.: A scalable multistage linear solver for reservoir models with multisegment wells. Comput. Geosci. 17, 197–216 (2013b)CrossRefGoogle Scholar
  72. 72.
    Zienkiewicz, O., Paul, D., Chan, A.: Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems. Int. J. Numer. Methods Eng. 26, 1039–1055 (1988).  https://doi.org/10.1002/nme.1620260504 CrossRefGoogle Scholar
  73. 73.
    Zienkiewicz, O., Taylor, R.: The finite element method for solid and structural mechanics. Elsevier, Amsterdam (2005)Google Scholar

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Authors and Affiliations

  1. 1.Energy Resources EngineeringStanford UniversityStanfordUSA
  2. 2.Delft University of TechnologyDelftThe Netherlands

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