Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits

Computer Methods in Applied Mechanics and Engineering

Volumn 200, Issue 23-24, 2011, Pages 2094-2116

  Kim, J ^a,b   Tchelepi, H A ^a   Juanes, R ^c  

^a STANFORD UNIVERSITY   (United States)

^b LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

^c Massachusetts Institute of Technology   (United States)

Author keywords

Convergence analysis; Drained split; Geomechanics; Poromechanics; Stability analysis; Undrained split

Indexed keywords

CONVERGENCE ANALYSIS; DRAINED SPLIT; GEOMECHANICS; PORO-MECHANICS; STABILITY ANALYSIS; UNDRAINED;

APPROXIMATION THEORY; CONVERGENCE OF NUMERICAL METHODS; ESTIMATION; INCOMPRESSIBLE FLOW; ITERATIVE METHODS; MATRIX ALGEBRA; MECHANICS; NON NEWTONIAN FLOW; SPECTRUM ANALYSIS;

STABILITY;

EID: 79955446198     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2011.02.011     Document Type: Article

References (73)

1. Truesdell C., Toupin R.A. The classical field theories. Handbuch der Physik 1960, vol. III/1:226-793. Springer-Verlag, Berlin. S. Flügge (Ed.).
2. Waugh R., Evans E.A. Thermoelasticity of red blood-cell membrane. Biophys. J. 1979, 26(1):115-131.
3. Kaczmarek M., Subramaniam R.P., Neff S.R. The hydromechanics of hydrocephalus: steady-state solutions for cylindrical geometry. Bull. Math. Biol. 1997, 59(2):295-323.
4. Biot M.A. General theory of three-dimensional consolidation. J. Appl. Phys. 1941, 12:155-164.
5. Coussy O. Mechanics of Porous Media 1995, John Wiley and Sons, Chichester, England.
6. Wang H.F. Theory of Linear Poroelasticity 2000, Princeton University Press.
7. Barton C.A., Zoback M.D., Moos D. Fluid flow along potentially active faults in crystalline rock. Geology 1995, 23(8):683-686.
8. Wiprut D., Zoback M.D. Fault reactivation and fluid flow along a previously dormant normal fault in the northern North Sea. Geology 2000, 28(7):595-598.
9. 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. 2008, 197:4353-4366.
10. Jain A.K., Juanes R. Preferential mode of gas invasion in sediments: grain-scale mechanistic model of coupled multiphase fluid flow and sediment mechanics. J. Geophys. Res. 2009, 114:B08101. 10.1029/2008JB006002.
11. Holtzman R., Juanes R. Crossover from fingering to fracturing in deformable disordered media. Phys. Rev. E 2010, 82:046305. 10.1103/PhysRevE.82.046305.
12. Merle H.A., Kentie C.J.P., van Opstal G.H.C., Schneider G.M.G. The Bachaquero study - a composite analysis of the behavior of a compaction drive/solution gas drive reservoir. J. Petrol Technol. 1976, 1107-1114.
13. Kosloff D., Scott R.F., Scranton J. Finite element simulation of Wilmington oil field subsidence: I. Linear modelling. Tectonophysics 1980, 65:339-368.
14. Bruno M.S. Subsidence-induced well failure. SPE Drill. Eng. 1992, 148-152.
15. Fredrich J.T., Arguello J.G., Deitrick G.L., de Rouffignac E.P. Geomechanical modeling of reservoir compaction, surface subsidence, and casing damage at the Belridge diatomite field. SPE Reserv. Eval. Eng. 2000, 3(4):348-359.
16. Settari A., Mourits F.M. A coupled reservoir and geomechanical simulation system. Soc. Petrol Eng. J. 1998, 3(3):219-226.
17. Settari A., Walters D.A. Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. Soc. Petrol Eng. J. 2001, 6(3):334-342.
18. Mainguy M., Longuemare P. Coupling fluid flow and rock mechanics: formulations of the partial coupling between reservoir and geomechanics simulators. Oil Gas Sci. Technol. 2002, 57:355-367.
19. Minkoff S.E., Stone C.M., Bryant S., Peszynska M., Wheeler M.F. Coupled fluid flow and geomechanical deformation modeling. J. Petrol Sci. Eng. 2003, 38:37-56.
20. Thomas L.K., Chin L.Y., Pierson R.G., Sylte J.E. Coupled geomechanics and reservoir simulation. Soc. Petrol Eng. J. 2003, 8(4):350-358.
21. Tran D., Settari A., Nghiem L. New iterative coupling between a reservoir simulator and a geomechanics module. Soc. Petrol Eng. J. 2004, 9(3):362-369.
22. Dean R.H., Gai X., Stone C.M., Minkoff S.E. A comparison of techniques for coupling porous flow and geomechanics. Soc. Petrol Eng. J. 2006, 11(1):132-140.
23. Jha B., Juanes R. A locally conservative finite element framework for modeling coupled fluid flow and reservoir geomechanics. Acta Geotech. 2007, 2(3):139-153.
24. Lewis R.W., Sukirman Y. Finite-element modeling of 3-phase flow in deforming saturated oil-reservoirs. Int. J. Numer. Anal. Methods Geomech. 1993, 17(8):577-598.
25. Lewis R.W., Schrefler B.A. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media 1998, John Wiley and Sons, Chichester, UK. second ed.
26. Armero F. Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions. Comput. Methods Appl. Mech. Engrg. 1999, 171:205-241.
27. Jean L., Mainguy M., Masson R., Vidal-Gilbert S. Accelerating the convergence of coupled geomechanical-reservoir simulations. Int. J. Numer. Anal. Methods Geomech. 2007, 31:1163-1181.
28. Park K.C. Stabilization of partitioned solution procedure for pore fluid-soil interaction analysis. Int. J. Numer. Meth. Engrg. 1983, 19(11):1669-1673.
29. Zienkiewicz O.C., Paul D.K., Chan A.H.C. Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems. Int. J. Numer. Meth. Engrg. 1988, 26(5):1039-1055.
30. Armero F., Simo J.C. A new unconditionally stable fractional step method for coupled thermomechanical problems. Int. J. Numer. Meth. Engrg. 1992, 35:737-766.
31. Prevost J.H. Partitioned solution procedure for simultaneous integration of coupled-field problems. Commun. Numer. Methods Engrg. 1997, 13:239-247.
32. Schrefler B.A., Simoni L., Turska E. Standard staggered and staggered Newton schemes in thermo-hydro-mechanical problems. Comput. Methods Appl. Mech. Engrg. 1997, 144:93-109.
33. Huang M., Zienkiewicz O.C. New unconditionally stable staggered solution procedures for coupled soil-pore fluid dynamic problems. Int. J. Numer. Meth. Engrg. 1998, 43:1029-1052.
34. Felippa C.A., Park K.C. Staggered transient analysis procedures for coupled mechanical systems: formulation. Comput. Methods Appl. Mech. Engrg. 1980, 24:61-111.
35. Farhat C., Park K.C., Dubois-Pelerin Y. An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems. Comput. Methods Appl. Mech. Engrg. 1991, 85:349-365.
36. Armero F., Simo J.C. A prior stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity. Int. J. Plasticity 1993, 9:749-782.
37. Simo J.C. Nonlinear stability of the time-discrete variational problem of evolution in nonlinear heat conduction, plasticity and viscoplasticity. Comput. Methods Appl. Mech. Engrg. 1991, 88:111-131.
38. Araújo A. A note on B-stability of splitting methods. Comput. Visual. Sci. 2004, 6:53-57.
39. Hundsdorfer W.H., Splijer M.L. A note on B-stability of Runge-Kutta methods. Numer. Math. 1981, 36:319-331.
40. Romero I. Thermodynamically consistent time-stepping algorithms for non-linear thermomechanical systems. Int. J. Numer. Meth. Engrg. 2009, 79:706-732.
41. Richtmyer R.D., Morton K.W. Difference Methods for Initial Value Problems 1967, John Wiley and Sons. second ed.
42. Brenan K.E., Campbell S.L., Petzold L.R. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations 1996, SIAM, Philadelphia.
43. Chorin A., Hughes T.R., Mccraken M.F., Marsden J.E. Product formulas and numerical algorithms. Commun. Pure Appl. Math. 1978, 205-256.
44. Lapidus M.L. Generalization of the Trotter-Lie formula. Integr. Equat. Oper. Theor. 1981, 4/3:366-413.
45. Turska E., Wisniewski K., Schrefler B.A. Error propagation of staggered solution procedures for transient problems. Comput. Methods Appl. Mech. Engrg. 1994, 114:177-188.
46. Vijalapura P.K., Govindjee S. An adaptive hybrid time-stepping scheme for highly non-linear strongly coupled Problems. Int. J. Numer. Meth. Engrg. 2005, 64:819-848.
47. Vijalapura P.K., Strain J., Govindjee S. Fractional step methods for index-1 differential-algebraic equations. J. Comput. Phys. 2005, 203:305-320.
48. Simo J.C., Govindjee S. Nonlinear B-stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity. Int. J. Numer. Meth. Engrg. 1991, 31:151-176.
49. Schrefler B.A., Simoni L., Turska E. Standard staggered and staggered Newton schemes in thermo-hydro-mechanical problems. Comput. Methods Appl. Mech. Engrg. 1997, 144:93-109.
50. Vermeer P.A., Verruijt A. An accuracy condition for consolidation by finite elements. Int. J. Numer. Anal. Methods Geomech. 1981, 5:1-14.
51. Ženišek A. The existence and uniqueness theorem in Biot's consolidation theory. Aplik. Matem. 1984, 29:194-210.
52. J. Wan, L.J. Durlofsky, T.J.R. Hughes, K. Aziz, Stabilized finite element methods for coupled geomechanics-reservoir flow simulations, in: SPE Reservoir Simulation Symposium, Houston, TX (SPE 79694), 2003.
53. 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. 2007, 11:131-144.
54. 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. 2007, 11:145-158.
55. Geertsma J. The effect of fluid pressure decline on volumetric change of porous rocks. Trans. AIME 1957, 210:331-340.
56. Borja R.I. On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int. J. Solids Struct. 2006, 43(6):1764-1786.
57. Aziz K., Settari A. Petroleum Reservoir Simulation 1979, Elsevier, London.
58. Hughes T.J.R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis 1987, Prentice-Hall, Englewood Cliffs, NJ.
59. Kim J., Tchelepi H.A., Juanes R. Stability, accuracy and efficiency of sequential methods for coupled flow and geomechanics. Soc. Petrol Eng. J. 2011, 10.2118/119084-PA.
60. Aavatsmark I. An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 2002, 6:405-432.
61. Isaacson E., Keller H.B. Analysis of Numerical Methods 1966, John Wiley and Sons, New York.
62. Strikwerda J.C. Finite Difference Schemes and Partial Differential Equations 2004, SIAM, Philadelphia, PA.
63. Miga M.I., Paulsen K.D., Kennedy F.E. Von Neumann stability analysis of Biot's general two dimensional theory of consolidation. Int. J. Numer. Meth. Engrg. 1998, 43:955-974.
64. Marsden J.E., Hughes T.J.R. Mathematical Foundations of Elasticity 1983, Prentice-Hall, Englewood Cliffs, NJ, (Reprinted with corrections, Dover, New York, 1994).
65. Butcher J.C. A stability property of implicit Runge-Kutta methods. BIT 1975, 15:358-361.
66. Simo J.C., Hughes T.J.R. Computational Inelasticity 1998, Springer, New York.
67. Simo J.C., Taylor R.L. Consistent tangent operators for rate dependent elastoplasticity. Comput. Methods Appl. Mech. Engrg. 1985, 38:101-118.
68. Taylor R.L., Simo J.C. A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Meth. Engrg. 1986, 22:649-676.
69. Thomée V. Galerkin Finite Element Methods for Parabolic Problems 1997, Springer-Verlag, Berlin.
70. Turska E., Schrefler B.A. On convergence condition of partitioned solution procedures for consolidation problems. Comput. Methods Appl. Mech. Engrg. 1993, 106:51-63.
71. Strang G. Linear Algebra and its Applications 1988, Brooks/Cole. third ed.
72. Borja R.I., Lee S.R. Cam-clay plasticity, part I: implicit integration of elasto-plastic constitutive relations. Comput. Methods Appl. Mech. Engrg. 1990, 78:49-72.
73. 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. 2011, 200(13-16):1591-1606.