×

Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method. (English) Zbl 1392.76078

Summary: We describe an algorithm for modeling saturated fractures in a poroelastic domain in which the reservoir simulator is coupled with a boundary element method. A fixed stress splitting is used on the underlying fractured Biot system to iteratively couple fluid and solid mechanics systems. The fluid system consists of Darcy’s law in the reservoir and is computed with a multipoint flux mixed finite element method, and a Reynolds’ lubrication equation in the fracture solved with a mimetic finite difference method. The mechanics system consists of linear elasticity in the reservoir and is computed with a continuous Galerkin method, and linear elasticity in the fracture is solved with a weakly singular symmetric Galerkin boundary element method. This algorithm is able to compute both unknown fracture width and unknown fluid leakage rate. An interesting numerical example is presented with an injection well inside of a circular fracture.

MSC:

76S05 Flows in porous media; filtration; seepage
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q86 PDEs in connection with geophysics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M10 Finite element methods applied to problems in fluid mechanics

Software:

IPARS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adachi, J; Siebrits, E; Peirce, A; Desroches, J, Computer simulation of hydraulic fractures, Int. J. Rock Mech. Min. Sci., 44, 739-757, (2007) · doi:10.1016/j.ijrmms.2006.11.006
[2] Al-Hinai, O., Singh, G., Almani, T., Pencheva, G., Wheeler, M.F.: Modeling multiphase flow with nonplanar fractures. In: 2013 SPE Reservoir Simulation Symposium. Texas, The Woodlands. SPE-163605 (2013)
[3] Bishop, J., Martinez, M., Newell, P.: A finite-element method for modeling fluid-Pressure induced discrete-fracture propagation using random meshes. In: American Rock Mechanics Association, Chicago. ARMA 12-190 (2012)
[4] Bourdin, B., Chukwudozie, C., Yoshioka, K.: A variational approach to the numerical simulation of hydraulic fracturing. SPE PP 146951 (2012) · Zbl 1193.74158
[5] Brezzi, F; Douglas, J; Marini, LD, Two families of mixed elements for second order elliptic problems, Numer. Math., 88, 217-235, (1985) · Zbl 0599.65072 · doi:10.1007/BF01389710
[6] Brezzi, F; Lipnikov, K; Simoncini, V, A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci., 15, 1533-1551, (2005) · Zbl 1083.65099 · doi:10.1142/S0218202505000832
[7] Carrier, B; Granet, S, Numerical modeling of hydraulic fracture problem in permeable medium using cohesive zone model, Eng. Fract. Mech., 79, 312-328, (2012) · doi:10.1016/j.engfracmech.2011.11.012
[8] Carter, RD, Derivation of the general equation for estimating the extent of the fractured area, (1957), New York
[9] Ciarlet, P.G.: The Finite Element Method For Elliptic Problems. North-Holland, Amsterdam (1987)
[10] Program Development Company. GridPro mesh generation software. http://www.gridpro.com (2013) Accessed 12 July 2013
[11] Dean, R; Schmidt, J, Hydraulic-fracture predictions with a fully coupled geomechanical reservoir simulator, SPE J., 14, 707-714, (2009) · doi:10.2118/116470-PA
[12] Ganis, B., Girault, V., Mear, M.E., Singh, G., Wheeler, M.F.: Modeling fractures in a poro-elastic medium. In: Technical Report ICES Report 13-09. The University of Texas, Austin (2013) · Zbl 1309.76194
[13] Ingram, R; Wheeler, M.F; Yotov, I, A multipoint flux mixed finite element method on hexahedra, SIAM J. Numer. Anal., 48, 1281-1312, (2010) · Zbl 1228.65225 · doi:10.1137/090766176
[14] Irzal, F; Remmers, J; Huyghe, JM; de Borst, R, A large deformation formulation for fluid flow in a progressively fracturing porous material, Comput. Methods Appl. Mech. Eng., 256, 29-37, (2013) · Zbl 1352.76113 · doi:10.1016/j.cma.2012.12.011
[15] Jaffré, J; Mnejja, M; Roberts, JE, A discrete fracture model for two-phase flow with matrix-fracture interaction, Procedia Comput. Sci., 4, 967-973, (2011) · doi:10.1016/j.procs.2011.04.102
[16] Ji, L; Settari, A; Sullivan, R, A novel hydraulic fracturing model fully coupled with geomechanics and reservoir simulation, SPE J., 14, 423-430, (2009) · doi:10.2118/110845-PA
[17] Johns, R., Sepehrnoori, K., Varavei, A., Moinfar, A.: Development of a coupled dual continuum and discrete fracture model for the simulation of unconventional reservoirs. In: 2013 SPE Reservoir Simulation Symposium (2013)
[18] Keat, WD; Annigeri, BS; Cleary, MP, Surface integral and finite element hybrid method for two-and three-dimensional fracture mechanics analysis, Int. J. Fract., 36, 35-53, (1988)
[19] Lacroix, S., Vassilevski, Y.V., Wheeler, M.F.: Iterative solvers of the implicit parallel accurate reservoir simulator (IPARS). I: Single processor case. In: Technical Report 00-28, TICAM. University of Texas, Austin (2000)
[20] Latham, J-P; Xiang, J; Belayneh, M; Nick, HM; Tsang, C-F; Blunt, MJ, Modelling stress-dependent permeability in fractured rock including effects of propagating and bending fractures, Int. J. Rock Mech. Min. Sci., 57, 100-112, (2012)
[21] Lecampion, B; Detournay, E, An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag, Comput. Methods Appl. Mech. Engrg., 196, 4863-4880, (2007) · Zbl 1173.74383 · doi:10.1016/j.cma.2007.06.011
[22] Li, S; Mear, ME, Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media, Int. J. Fract., 93, 87-114, (1998) · doi:10.1023/A:1007513307368
[23] Li, S; Mear, ME; Xiao, L, Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Methods Appl. Mech. Eng., 151, 435-459, (1998) · Zbl 0906.73074 · doi:10.1016/S0045-7825(97)00199-0
[24] Mikelić, A., Wheeler, M.F.: Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 1-7 (2012)
[25] Nikishkov, GP; Park, JH; Atluri, SN, SGBEM-FEM alternating method for analyzing 3D non-planar cracks and their growth in structural components, Comput. Model. Eng. Sci., 2, 401-422, (2001) · Zbl 1074.74641
[26] Olson, J.E.: Multi-fracture propagation modeling: applications to hydraulic fracturing in shales and tight gas sands. In: The 42nd US Rock Mechanics Symposium (USRMS) (2008)
[27] Rungamornrat, J; Mear, ME, SGBEM-FEM coupling for analysis of cracks in 3D anisotropic media, Int. J. Numer. Methods Eng., 86, 224-248, (2011) · Zbl 1235.74339 · doi:10.1002/nme.3055
[28] Rungamornrat, J; Wheeler, MF; Mear, ME, A numerical technique for simulating nonplanar evolution of hydraulic fractures, SPE, 96968, 2005, (2005)
[29] Schrefler, BA; Secchi, S; Simoni, L, On adaptive refinement techniques in multi-field problems including cohesive fracture, Comput. Methods Appl. Engrg., 195, 444-461, (2006) · Zbl 1193.74158 · doi:10.1016/j.cma.2004.10.014
[30] Vassilevski, Y.V.: Iterative Solvers for the Implicit Parallel Accurate Reservoir Simulator (IPARS). II: parallelization issues. In: Technical Report 00-33, TICAM. University of Texas, Austin (2000)
[31] Wheeler, M.F., Xue, G., Yotov, I.: Accurate cell-centered discretizations for modeling multiphase flow in porous media on general hexahedral and simplicial grids. In: SPE Reservoir Simulation Symposium. The Woodlands, Texas. SPE 141534 (2011)
[32] Youngquist, W; Duncan, RC, North American natural gas: data show supply problems, Nat. Resour. Res, 12, 229-240, (2003) · doi:10.1023/B:NARR.0000007803.89812.06
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.