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.
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 14-01-00785
Funding statement: This work was supported by the Russian Foundation for Basic Research (project 14-01-00785) and by the Ministry of Education of the Russian Federation (Agreement No. 02.a03.21.0008 of June 24, 2016)
References
[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.Search in Google Scholar
[2] M. S. Alnæs, J. Blechta, J. Hake, et al, The FEniCS Project Version 1.5. Arch. Numer. Software3 (2015), No. 100, 9–23.Search in Google Scholar
[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, 737–766.10.1002/nme.1620350408Search in Google Scholar
[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, 1621–1633.10.1029/92WR02746Search in Google Scholar
[5] S. Balay, S. Abhyankar, M. F. Adams, et al, PETSc Web page. URL: http://www.mcs.anl.gov/petsc, (2015).Search in Google Scholar
[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. Mechanics24 (1960), No. 5, 1286–1303.10.1016/0021-8928(60)90107-6Search in Google Scholar
[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, 1697–1716.10.1016/0020-7225(86)90076-5Search in Google Scholar
[8] M. A. Biot, General theory of three dimensional consolidation. J. Appl. Phys. 12 (1941), No. 2, 155–164.10.1063/1.1712886Search in Google Scholar
[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, 1090–1102.10.1016/j.cam.2011.07.032Search in Google Scholar
[10] N. Boal, Finite-difference analysis for the linear thermoporoelasticity problem and its numerical resolution by multigrid methods. Math. Modelling Analysis17 (2012), No. 2, 227–244.10.3846/13926292.2012.662177Search in Google Scholar
[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, 138–154.10.1002/num.20612Search in Google Scholar
[12] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York, 2008.10.1007/978-0-387-75934-0Search in Google Scholar
[13] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, New York, 1991.10.1007/978-1-4612-3172-1Search in Google Scholar
[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), 138–170.10.1016/j.cma.2014.10.047Search in Google Scholar
[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, 1054–1100.10.1002/num.21936Search in Google Scholar
[16] Yu. V. Bychenkov and E. V. Chizhonkov, Iterative Methods for Solving Saddle Problems. Binom, Moscow, 2010 (in Russian).Search in Google Scholar
[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, 1447–1457.10.1016/j.cma.2006.03.020Search in Google Scholar
[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, 60–78.10.1002/num.20242Search in Google Scholar
[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.Search in Google Scholar
[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, 1309–1331.10.1002/nme.2579Search in Google Scholar
[21] V. Hernandez, J. E. Roman, and V. Vidal, SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software31 (2005), No. 3, 351–362.10.1145/1089014.1089019Search in Google Scholar
[22] B. Jha and R. Juanes, A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotechnica2 (2007), No. 3, 139–153.10.1007/s11440-007-0033-0Search in Google Scholar
[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. Geomechanics8 (1984), No. 2, 101–123.10.1002/nag.1610080202Search in Google Scholar
[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), 2094–2116.10.1016/j.cma.2011.02.011Search in Google Scholar
[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), 1591–1606.10.1016/j.cma.2010.12.022Search in Google Scholar
[26] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer Verlag, New York, 2003.Search in Google Scholar
[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, 2185–2198.10.1007/978-3-319-20239-6_25Search in Google Scholar
[28] A. Logg and G. N. Wells, DOLFIN: automated finite element computing. ACM Trans. Math. Software37 (2010), No. 2, Article 20.10.1145/1731022.1731030Search in Google Scholar
[29] A. Logg, K.-R. Mardal, G. N. Wells, et al, Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin, 2012.10.1007/978-3-642-23099-8Search in Google Scholar
[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, 197–462.10.1016/S1570-8659(05)80035-3Search in Google Scholar
[31] A. Mikelić and M. F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosciences17 (2012), No. 3, 455–461.10.1007/s10596-012-9318-ySearch in Google Scholar
[32] S. G. Mikhlin, The spectrum of a family of operators in the theory of elasticity. Russ. Math. Surveys28 (1973), No. 3, 45–88.10.1070/RM1973v028n03ABEH001563Search in Google Scholar
[33] A. A. Samarskii, The Theory of Difference Schemes. Marcel Dekker, New York, 2001.10.1201/9780203908518Search in Google Scholar
[34] A. A. Samarskii. P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors. Kluwer, 2002.10.1007/978-94-015-9874-3Search in Google Scholar
[35] P. J. Phillips and M. F. Wheeler, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosciences13 (2008), No. 1, 5–12.10.1007/s10596-008-9114-xSearch in Google Scholar
[36] P. N. Vabishchevich, Additive Operator-Difference Schemes: Splitting Schemes. de Gruyter, Berlin, 2014.10.1515/9783110321463Search in Google Scholar
[37] P. N. Vabishchevich, M. V. Vasil’eva, and A. E. Kolesov, Splitting scheme for poroelasticity and thermoelasticity problems. Comput. Math. Math. Physics54 (2014), No. 8, 1305–1315.10.1134/S0965542514080132Search in Google Scholar
[38] M. F. Wheeler and X. Gai, Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity. Numer. Meth. Partial Differ. Equations23 (2007), No. 4, 785–797.10.1002/num.20258Search in Google Scholar
[39] M. F. Wheeler, G. Xue, and I. Yotov, Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity, Comput. Geosciences18 (2013), No. 1, 57–75.10.1007/s10596-013-9382-ySearch in Google Scholar
[40] R. N. Wilson and E. C. Aifantis, On the theory of consolidation with double porosity. Int. J. Engrg. Sci. 20 (1982), No. 9, 1009–1035.10.1016/0020-7225(82)90036-2Search in Google Scholar
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