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Multi-physics flux coupling for hydraulic fracturing modelling within INMOST platform. (English) Zbl 1450.65097
The author presents a collocated finite volume method for a coupled system of equations in multiple domains. Each domain is characterized by the properties of heterogeneous media and features a distinctive multi-physics model. The coupled systems of equations, corresponding to multiple unknowns, results in a vector flux. The finite volume method requires continuity of intradomain and interdomain vector fluxes. The continuous flux is derived using an extension of the harmonic averaging point concept. The collocated coupling of the equations results in a saddle-point problem which leads to inf-sup stability problems. These problems are handled thanks to the eigen-splitting of indefinite matrix coefficients involved in the flux expression. The application of the techniques implemented within INMOST (integrated numerical modelling and object-oriented supercomputing technologies) platform to hydraulic fracturing problem is shown.
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S10 Finite volume methods applied to problems in solid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
Full Text: DOI
[1] I. Ambartsumyan, E. Khattatov, T. Nguyen, and I. Yotov, Flow and transport in fractured poroelastic media. GEM - Int. J. Geomathematics10 (2019), No. 1, 11. · Zbl 1419.76602
[2] K. Aziz and A. Settari, Petroleum Reservoir Simulation. Applied Science Publishers, 1979.
[3] R. L. Berge, I. Berre, E. Keilegavlen, J. M. Nordbotten, and B. Wohlmuth, Finite volume discretization for poroelastic media with fractures modeled by contact mechanics. Int. J. Numer. Meth. Engrg. 121 (2020), No. 4, 644-663.
[4] M. A. Biot, General theory of three-dimensional consolidation. J. Applied Physics12 (1941), No. 2, 155-164. · JFM 67.0837.01
[5] J. M. Carcione, Wave propagation in anisotropic, saturated porous media: Plane-wave theory and numerical simulation. J. Acoustical Soc. America99 (1996), No. 5, 2655-2666.
[6] N. Castelletto, J. A. White, and M. Ferronato, Scalable algorithms for three-field mixed finite element coupled poromechanics. J. Comput. Physics327 (2016), 894-918. · Zbl 1373.76312
[7] J. Choo and S. Lee, Enriched Galerkin finite elements for coupled poromechanics with local mass conservation. Comput. Meth. Appl. Mech. Engrg. 341 (2018), 311-332. · Zbl 1440.74120
[8] O. Coussy, Poromechanics. John Wiley & Sons, 2004.
[9] R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods. Handbook of Numerical Analysis7 (2000), 713-1018. · Zbl 0981.65095
[10] C. A. Felippa and E. Oñate, Stress, strain and energy splittings for anisotropic elastic solids under volumetric constraints. Computers & Structures81 (2003), No. 13, 1343-1357.
[11] B. Flemisch, K. M. Flornes, K. Lie, and A. Rasmussen, OPM: The Open Porous Media Initiative, AGU Fall Meeting Abstracts, 2011.
[12] B. Ganis, R. Liu, B. Wang, M. F. Wheeler, and I. Yotov, Multiscale modeling of flow and geomechanics. Radon Series on Computational and Applied Mathematics (2013), 165-204. · Zbl 1302.76174
[13] D. Garagash and E. Detournay, The tip region of a fluid-driven fracture in an elastic medium. J. Appl. Mech. 67 (2000), No. 1, 183-192. · Zbl 1110.74448
[14] T. T. Garipov, M. Karimi-Fard, and H. A. Tchelepi, Discrete fracture model for coupled flow and geomechanics. Computational Geosciences20 (2016), No. 1, 149-160. · Zbl 1392.76079
[15] F. J. Gaspar, F. J. Lisbona, and P. N. Vabishchevich, Staggered grid discretizations for the quasi-static Biot’s consolidation problem. Appl. Numer. Math. 56 (2006), No. 6, 888-898. · Zbl 1091.76047
[16] H. T. Honório, C. R. Maliska, M. Ferronato, and C. Janna, A stabilized element-based finite volume method for poroelastic problems. J. Comput. Physics364 (2018), 49-72. · Zbl 1392.74090
[17] M. Karimi-Fard, L. J. Durlofsky, and K. Aziz, An efficient discrete fracture model applicable for general purpose reservoir simulators. In: SPE Reservoir Simulation Symposium, 2003, Society of Petroleum Engineers.
[18] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach, New York, 1969. · Zbl 0184.52603
[19] A. Léo, E. Robert, and H. Raphaèl, A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. In: Comptes Rendus Mathematique. Elsevier France, Paris, 2009.
[20] R. Liu, M. F. Wheeler, C. N. Dawson, and R. Dean, Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method. Comput. Meth. Appl. Mech. Engrg. 198 (2009), No. 9-12, 912-919. · Zbl 1229.76053
[21] P. Luo, C. Rodrigo, F. J. Gaspar, and C. W. Oosterlee, Multigrid method for nonlinear poroelasticity equations. Computing and Visualization in Science17 (2015), No. 5, 255-265. · Zbl 1388.74013
[22] A. Naumovich, On finite volume discretization of the three-dimensional Biot poroelasticity system in multilayer domains. Comput. Meth. Appl. Math. 6 (2006), No. 3, 306-325. · Zbl 1100.74060
[23] J. M. Nordbotten, Stable cell-centered finite volume discretization for Biot equations. SIAM J. Numer. Analysis54 (2016), No. 2, 942-968. · Zbl 1382.76187
[24] D. W. Peaceman, Interpretation of well-block pressures in numerical reservoir simulation (includes associated paper 6988). Society of Petroleum Engineers J. 18 (1978), No. 3, 183-194.
[25] M. Preisig and J. H. Prévost, Stabilization procedures in coupled poromechanics problems: A critical assessment. Int. J. Numerical Analytical Methods in Geomechanics35 (2011), No. 11, 1207-1225.
[26] C. Rodrigo, F. J. Gaspar, X. Hu, and L. T. Zikatanov, Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Meth. Appl. Mech. Engrg. 298 (2016), 183-204. · Zbl 1425.74164
[27] A. Settari and K. Aziz, Use of irregular grid in reservoir simulation. Society of Petroleum Engineers Journal, 12 (1972), No. 2, 103-114.
[28] S. A. Shapiro, C. Dinske, and J. Kummerow, Probability of a given-magnitude earthquake induced by a fluid injection. Geophysical Research Letters34 (2007), No. 22.
[29] I. V. Sokolova, M. G. Bastisya, and H. Hajibeygi, Multiscale finite volume method for finite volume-based simulation of poroelasticity. J. Comput. Phys. 379 (2019), 309-324.
[30] I. V. Sokolova and H. Hajibeygi, Multiscale finite volume method for finite volume-based poromechanics simulations. In: ECMOR XVI - 16th European Conference on the Mathematics of Oil Recovery, 2018, pp. 1-13.
[31] M. Ţene, S. B. M. Bosma, M. S. Al Kobaisi, and H. Hajibeygi, Projection-based embedded discrete fracture model (pEDFM). Advances in Water Resources105 (2017), 205-216.
[32] K. M. Terekhov, B. T. Mallison, and H. A. Tchelepi, Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem. J. Comput. Physics330 (2017), 245-267. · Zbl 1380.65335
[33] K. M. Terekhov and H. A. Tchelepi, Cell-centered finite-volxume method for elastic deformation of heterogeneous media with full-tensor properties. J. Comput. Applied Math. 364 (2020), 112331. · Zbl 1457.74186
[34] K. M. Terekhov and Yu. V. Vassilevski, INMOST parallel platform for mathematical modeling and applications. In: Russian Supercomputing Days, 2018, 230-241.
[35] K. M. Terekhov and Yu. V. Vassilevski, Finite volume method for coupled subsurface flow problems, I: Darcy problem. J. Comput. Physics395 (2019), 298-306.
[36] K. M. Terekhov and Yu. V. Vassilevski, Mesh modification and adaptation within INMOST programming platform. In: Numerical Geometry, Grid Generation and Scientific Computing. Springer, 2019. · Zbl 07215542
[37] K. M. Terekhov, Cell-centered finite-volume method for heterogeneous anisotropic poromechanics problem. J. Comput. Appl. Math. 365 (2020), 112357. · Zbl 1423.76308
[38] Yu. Vassilevski, I. Konshin, G. Kopytov, and K. Terekhov, INMOST — a software platform and a graphical environment for development of parallel numerical models on general meshes. Moscow State Univ. Publ., Moscow, 2013 (in Russian).
[39] Yu. Vassilevski, K. Terekhov, K. Nikitin, and I. Kapyrin, Parallel Finite Volume Computation on General Meshes. Springer, Cham, 2020. URL: https://www.springer.com/gp/book/9783030472313. · Zbl 1437.65003
[40] M. Wheeler, G. Xue, and I. Yotov, A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numerische Mathematik121 (2012), No. 1, 165-204. · Zbl 1277.65100
[41] H. C. Yoon and J. Kim, Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity. Int. J. Numer. Meth. Engrg. 114 (2018), No. 7, 694-718.
[42] O. C. Zienkiewicz, The Finite Element Method. McGraw-Hill, London, 1977. · Zbl 0435.73072
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