Splitting schemes with respect to physical processes for double-porosity poroelasticity problems

Alexander E. Kolesov 1  and Petr N. Vabishchevich 2 , 3
  • 1 North-Eastern Federal University, Belinskogo 58, Yakutsk 677000, Russia
  • 2 Nuclear Safety Institute of the RAS, B. Tulskaya 52, Moscow 113191, Russia
  • 3 RUDN University, Moscow 117198, Russia
Alexander E. Kolesov and Petr N. Vabishchevich

Abstract

We consider unsteady poroelasticity problem in fractured porous medium within the classical Barenblatt double-porosity model. For numerical solution of double-porosity poroelasticity problems we construct splitting schemes with respect to physical processes, where transition to a new time level is associated with solving separate problem for the displacements and fluid pressures in pores and fractures. The stability of schemes is achieved by switching to three-level explicit-implicit difference scheme with some of the terms in the system of equations taken from the lower time level and by choosing a weight parameter used as a regularization parameter. The computational algorithm is based on the finite element approximation in space. The investigation of stability of splitting schemes is based on the general stability (well-posedness) theory of operator-difference schemes. A priori estimates for proposed splitting schemes and the standard two-level scheme are provided. The accuracy and stability of considered schemes are demonstrated by numerical experiments.

  • [1]

    O. Alexsson, R. Blaheta, and P. Byczanski, Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices. Comput. Vissual. Sci. 15 (2013), No. 15, 191–2007.

  • [2]

    M. S. Alnæs, J. Blechta, J. Hake, et al, The FEniCS Project Version 1.5. Arch. Numer. Software 3 (2015), No. 100, 9–23.

  • [3]

    F. Armero and J. C. Simo, A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems. Int. J. Numer. Meth. Engrg. 35 (1992), No. 4, 737–766.

  • [4]

    M. Bai, D. Elsworth, and J. C. Roegiers, Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs. Water Resources Research (1993), No. 29, 1621–1633.

  • [5]

    S. Balay, S. Abhyankar, M. F. Adams, et al, PETSc Web page. URL: http://www.mcs.anl.gov/petsc, (2015).

  • [6]

    G. I. Barenblatt, Iu. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata). J. Appl. Math. Mechanics 24 (1960), No. 5, 1286–1303.

  • [7]

    D. E. Beskos and E. C. Aifantis, On the theory of consolidation with double porosity, II. Int. J. Engrg. Sci. 24 (1986), No. 11, 1697–1716.

  • [8]

    M. A. Biot, General theory of three dimensional consolidation. J. Appl. Phys. 12 (1941), No. 2, 155–164.

  • [9]

    N. Boal, F. J. Gaspar, F. J. Lisbona, and P. N. Vabishchevich, Finite-difference analysis of fully dynamic problems for saturated porous media. J. Comput. Appl. Math. 236 (2011), No. 6, 1090–1102.

  • [10]

    N. Boal, Finite-difference analysis for the linear thermoporoelasticity problem and its numerical resolution by multigrid methods. Math. Modelling Analysis 17 (2012), No. 2, 227–244.

  • [11]

    N. Boal, F. J. Gaspar, F. J. Lisbona, and P. N. Vabishchevich, Finite difference analysis of a double-porosity consolidation model. Numer. Meth. Partial Differ. Equ. 28 (2012), No. 1, 138–154.

  • [12]

    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York, 2008.

  • [13]

    F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, New York, 1991.

  • [14]

    M. Bukač, I. Yotov, R. Zakerzadeh, and P. Zunino, Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Comput. Meth. Appl. Mechanics Engrg. 292 (2015), 138–170.

  • [15]

    M. Bukač, I. Yotov, R. Zakerzadeh, and P. Zunino, An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure. Numer. Meth. Partial Differ. Equ. 31 (2015), No. 4, 1054–1100.

  • [16]

    Yu. V. Bychenkov and E. V. Chizhonkov, Iterative Methods for Solving Saddle Problems. Binom, Moscow, 2010 (in Russian).

  • [17]

    F. J. Gaspar and F. J. Lisbona, An efficient multigrid solver for a reformulated version of the poroelasticity system. Comput. Meth. Appl. Mech. Engrg. 196 (2007), No. 8, 1447–1457.

  • [18]

    F. J. Gaspar, J. L. Gracia, F. J. Lisbona, and P. N. Vabishchevich, A stabilized method for a secondary consolidation Biot’s model. Numer. Meth. Partial Differ. Equ. 24 (2008), No. 1, 60–78.

  • [19]

    F. J. Gaspar, A. V. Grigoriev, and P. N. Vabishchevich, Explicit-implicit splitting schemes for some systems of evolutionary equations. Int. J. Numer. Anal. Model. 11 (2014), No. 2, 346–357.

  • [20]

    C. Geuzaine and J.-F. Remacle, Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Engrg. 79 (2009), No. 11, 1309–1331.

  • [21]

    V. Hernandez, J. E. Roman, and V. Vidal, SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software 31 (2005), No. 3, 351–362.

  • [22]

    B. Jha and R. Juanes, A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotechnica 2 (2007), No. 3, 139–153.

  • [23]

    M. Y. Khaled, D. E. Beskos, and E. C. Aifantis, On the theory of consolidation with double porosity, III. A finite element formulation. Int. J. Numer. Anal. Meth. Geomechanics 8 (1984), No. 2, 101–123.

  • [24]

    J. Kim, H. A. Tchelepi, and R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits. Comput. Meth. Appl. Mech. Engrg. 200 (2011), 2094–2116.

  • [25]

    J. Kim, H. A. Tchelepi, and R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Meth. Appl. Mech. Engrg. 200 (2011), 1591–1606.

  • [26]

    P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer Verlag, New York, 2003.

  • [27]

    A. E. Kolesov, P. N. Vabishchevich, and M. V. Vasil’eva, Splitting schemes for poroelasticity and thermoelasticity problems. Computers & Math. Appl. 67 (2014), No. 12, 2185–2198.

  • [28]

    A. Logg and G. N. Wells, DOLFIN: automated finite element computing. ACM Trans. Math. Software 37 (2010), No. 2, Article 20.

  • [29]

    A. Logg, K.-R. Mardal, G. N. Wells, et al, Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin, 2012.

  • [30]

    G. I. Marchuk, Splitting and alternating direction methods. In: Handbook of Numerical Analysis (Eds. P. G. Ciarlet and J.-L. Lions). North-Holland, 1990, 197–462.

  • [31]

    A. Mikelić and M. F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosciences 17 (2012), No. 3, 455–461.

  • [32]

    S. G. Mikhlin, The spectrum of a family of operators in the theory of elasticity. Russ. Math. Surveys 28 (1973), No. 3, 45–88.

  • [33]

    A. A. Samarskii, The Theory of Difference Schemes. Marcel Dekker, New York, 2001.

  • [34]

    A. A. Samarskii. P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors. Kluwer, 2002.

  • [35]

    P. J. Phillips and M. F. Wheeler, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosciences 13 (2008), No. 1, 5–12.

  • [36]

    P. N. Vabishchevich, Additive Operator-Difference Schemes: Splitting Schemes. de Gruyter, Berlin, 2014.

  • [37]

    P. N. Vabishchevich, M. V. Vasil’eva, and A. E. Kolesov, Splitting scheme for poroelasticity and thermoelasticity problems. Comput. Math. Math. Physics 54 (2014), No. 8, 1305–1315.

  • [38]

    M. F. Wheeler and X. Gai, Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity. Numer. Meth. Partial Differ. Equations 23 (2007), No. 4, 785–797.

  • [39]

    M. F. Wheeler, G. Xue, and I. Yotov, Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity, Comput. Geosciences 18 (2013), No. 1, 57–75.

  • [40]

    R. N. Wilson and E. C. Aifantis, On the theory of consolidation with double porosity. Int. J. Engrg. Sci. 20 (1982), No. 9, 1009–1035.

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