A finite-volume discretization for deformation of fractured media.

*(English)*Zbl 1396.74099Summary: Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multi-point stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (face pairs in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation, considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate
our proposed approach using both problems, for which analytical solutions are available, and more complex benchmark problems, including comparison with a finite-element discretization.

##### MSC:

74S10 | Finite volume methods applied to problems in solid mechanics |

74R10 | Brittle fracture |

74L10 | Soil and rock mechanics |

##### References:

[1] | Jaeger, J.C., Cook, N.G., Zimmerman, R.: Fundamentals of Rock Mechanics. Wiley, New York (2009) |

[2] | Jing, L, A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering, Int. J. Rock Mech. Min., 40, 283-353, (2003) |

[3] | McClure, MW; Horne, RN, An investigation of stimulation mechanisms in enhanced geothermal systems, Int. J. Rock Mech. Min, 72, 242-260, (2014) |

[4] | Oldenburg, C; Pruess, K; Benson, SM, Process modeling of CO2 injection into natural gas reservoirs for carbon sequestration and enhanced gas recovery, Energ. Fuel, 15, 293-298, (2001) |

[5] | 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) |

[6] | Rutqvist, J; Stephansson, O, The role of hydromechanical coupling in fractured rock engineering, Hydrogeol. J., 11, 7-40, (2003) |

[7] | Anderson, T.L.: Fracture Mechanics: Fundamentals and Applications. CRC, Boca Raton (2005) |

[8] | Adachi, J; Siebrits, E; Peirce, A; Desroches, J, Computer simulation of hydraulic fractures, Int. J. Rock Mech. Min., 44, 739-757, (2007) |

[9] | Liu, YJ; Mukherjee, S; Nishimura, N; Schanz, M; Ye, W; Sutradhar, A; Pan, E; Dumont, NA; Frangi, A; Saez, A, Recent advances and emerging applications of the boundary element method, Appl. Mech. Rev., 64, 030802, (2012) |

[10] | McClure, MW; Horne, RN, Investigation of injection-induced seismicity using a coupled fluid flow and rate/state friction model, Geophysics, 76, wc181-wc198, (2011) |

[11] | Norbeck, JH; McClure, MW; Lo, JW; Horne, RN, An embedded fracture modeling framework for simulation of hydraulic fracturing and shear stimulation, Computat. Geosci., 20, 1-18, (2016) · Zbl 1392.86022 |

[12] | Crouch, S.L., Starfield, A.: Boundary Element Methods in Solid Mechanics: With Applications in Rock Mechanics and Geological Engineering. Allen & Unwin, London (1982) · Zbl 0528.73083 |

[13] | Shou, K; Crouch, S, A higher order displacement discontinuity method for analysis of crack problems, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 1, 49-55, (1995) |

[14] | Zhou, X; Ghassemi, A, Three-dimensional poroelastic analysis of a pressurized natural fracture, Int. J. Rock Mech. Min., 48, 527-534, (2011) |

[15] | Borja, R.I.: Plasticity. Springer, Berlin (2013) · Zbl 1279.74003 |

[16] | Aagaard, BT; Knepley, MG; Williams, CA, A domain decomposition approach to implementing fault slip in finite-element models of quasi-static and dynamic crustal deformation, J. Geophys. Res. Solid Earth, 118, 3059-3079, (2013) |

[17] | Kuna, M.: Finite elements in fracture mechanics. Springer, Berlin (2013) · Zbl 1277.74002 |

[18] | Garipov, TT; Karimi-Fard, M; Tchelepi, HA, Discrete fracture model for coupled flow and geomechanics, Computat. Geosci., 20, 149-160, (2016) · Zbl 1392.76079 |

[19] | Kim, J; Tchelepi, HA; Juanes, R, Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits, Comput. Method Appl. M, 200, 1591-1606, (2011) · Zbl 1228.74101 |

[20] | Kim, J., Tchelepi, H.A., Juanes, R.: Stability, accuracy and efficiency of sequential methods for coupled flow and geomechanics. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2009) · Zbl 1228.74106 |

[21] | Castelletto, N; White, JA; Tchelepi, HA, Accuracy and convergence properties of the fixed-stress iterative solution of two-way coupled poromechanics, Int. J. Numer. Anal. Methods, 39, 1593-1618, (2015) |

[22] | Nordbotten, JM, Cell-centered finite volume discretizations for deformable porous media, Int. J. Numer. Methods Eng., 100, 399-418, (2014) · Zbl 1352.76072 |

[23] | Aavatsmark, I, An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci., 6, 405-432, (2002) · Zbl 1094.76550 |

[24] | Keilegavlen, E; Nordbotten, JM, Finite volume methods for elasticity with weak symmetry., Int. J. Numer. Methods Eng., 112, 939-962, (2017) |

[25] | Nordbotten, JM, Convergence of a cell-centered finite volume discretization for linear elasticity, SIAM J. Numer. Anal., 53, 2605-2625, (2015) · Zbl 1330.74171 |

[26] | Nordbotten, JM, Stable cell-centered finite volume discretization for Biot equations, SIAM J. Numer. Anal., 54, 942-968, (2016) · Zbl 1382.76187 |

[27] | Ucar, E; Keilegavlen, E; Berre, I, Post-injection normal closure of fractures as a mechanism for induced seismicity., Geophys. Res. Lett., 44, 9598-9606, (2017) |

[28] | Andrews, D, Test of two methods for faulting in finite-difference calculations, Bull. Seismol. Soc. Am., 89, 931-937, (1999) |

[29] | Daub, EG; Carlson, JM, Friction, fracture, and earthquakes, Annu. Rev. Condens. Matter Phys., 1, 397-418, (2010) |

[30] | Ida, Y, Cohesive force across the tip of a longitudinal-shear crack and griffith’s specific surface energy, J. Geophys. Res., 77, 3796-3805, (1972) |

[31] | Andrews, D, Rupture models with dynamically determined breakdown displacement, Bull. Seismol. Soc. Am., 94, 769-775, (2004) |

[32] | Dieterich, JH, Modeling of rock friction: 1. experimental results and constitutive equations, J. Geophys. Res., 84, 2161, (1979) |

[33] | Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method, vol. 1. SIAM, Philadelphia (2003) · Zbl 1031.65069 |

[34] | Shewchuck, J.: Engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin MC, Manocha D (eds.) Applied computational geometry: towards geometric engineering. Lecture notes in computer science, vol. 1148. From the First ACM Workshop on Applied Computational Geometry, pp 203-222. Springer, New York (1996) |

[35] | Zehnder, A.T.: Fracture mechanics. (Lecture notes in applied and computational mechanics, vol. 62). Springer, Netherlands (2012) |

[36] | Lawn, B.: Fracture of Brittle Solids. Cambridge University Press, Cambridge (1993) |

[37] | Dugdale, DS, Yielding of steel sheets containing slits, J. Mech. Phys. Solids, 8, 100-104, (1960) |

[38] | Sneddon, I.N.: Fourier Transforms. McGraw Hill Book Co, Inc., New York (1951) |

[39] | Phan, AV; Napier, JAL; Gray, LJ; Kaplan, T, Symmetric-Galerkin BEM simulation of fracture with frictional contact, Int. J. Numer. Methods Eng., 57, 835-851, (2003) · Zbl 1062.74641 |

[40] | Lie, K.A.: An introduction to reservoir simulation using MATLAB user guide for the MATLAB reservoir simulation toolbox (MRST). SINTEF ICT, Department of Applied Mathematics (2014) · Zbl 1425.76001 |

[41] | Aagaard, B., Knepley, M., Williams, C., Strand, L., Kientz, S.: PyLith user manual, version 2.1.0. Computational Infrastructure for Geodynamics (CIG), University of California, Davis (2015) |

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