# zbMATH — the first resource for mathematics

A hierarchical fracture model for the iterative multiscale finite volume method. (English) Zbl 1370.76095
Summary: An iterative multiscale finite volume (i-MSFV) method is devised for the simulation of multiphase flow in fractured porous media in the context of a hierarchical fracture modeling framework. Motivated by the small pressure change inside highly conductive fractures, the fully coupled system is split into smaller systems, which are then sequentially solved. This splitting technique results in only one additional degree of freedom for each connected fracture network appearing in the matrix system. It can be interpreted as an agglomeration of highly connected cells; similar as in algebraic multigrid methods. For the solution of the resulting algebraic system, an i-MSFV method is introduced. In addition to the local basis and correction functions, which were previously developed in this framework, local fracture functions are introduced to accurately capture the fractures at the coarse scale. In this multiscale approach there exists one fracture function per network and local domain, and in the coarse scale problem there appears only one additional degree of freedom per connected fracture network. Numerical results are presented for validation and verification of this new iterative multiscale approach for fractured porous media, and to investigate its computational efficiency. Finally, it is demonstrated that the new method is an effective multiscale approach for simulations of realistic multiphase flows in fractured heterogeneous porous media.

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage
ML
Full Text:
##### References:
 [1] Barenblatt, G.I.; Zheltov, Iu.P.; Kochina, I.M., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, Pmm, 24, 5, 852-864, (1960) · Zbl 0104.21702 [2] Warren, J.; Root, P., The behavior of naturally fractured reservoirs, Spe j., 245-255, (1963) [3] Kazemi, H.; Gilman, J., Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution, Spe j., 451-462, (1969) [4] Kazemi, H.; Merill, L.S.; Porterfield, K.L.; Zeman, P.R., Numerical simulation of water-oil flow in naturally fractured reservoirs, Spe j., 16, 6, 317-326, (1976) [5] Noorishad, J.; Mehran, M., An upstream finite element method for solution of transient transport equation in fractured porous media, Water resour. res., 3, 18, 588-596, (1982) [6] Barenblatt, G.; Zheltov, Y.; Kochina, I., Basic concepts in the theory of seepage of homogeneous fluids in fissurized rocks, J. appl. math. mech., 5, 24, 1286-1303, (1983) · Zbl 0104.21702 [7] Thomas, L.K.; Dixon, T.N.; Pierson, R.G., Fractured reservoir simulation, Spe j., 23, 1, 42-54, (1983) [8] Baca, R.; Arnett, R.; Langford, D., Modeling fluid flow in fractured porous rock masses by finite element techniques, Int. J. numer. methods fluids, 4, 337-348, (1984) · Zbl 0579.76095 [9] T. Barkve, A. Firoozabadi, Analysis of reinfiltration in fractured porous media. SPE 24900, in: The 67th SPE Annual Technical Conference and Exhibition, Washington D.C., Oct. 4-7, 1992. [10] Lee, S.H.; Lough, M.F.; Jensen, C.L., Hierarchical modeling of flow in naturally fractured formations with multiple length scales, Water resour. res., 37, 3, 443-455, (2001) [11] Lee, S.H.; Jensen, C.L.; Lough, M.F., Efficient finite-difference model for flow in a reservoir with multiple length-scale fractures, Spe j., 3, 5, 268-275, (2000) [12] Li, L.; Lee, S.H., Efficient field-scale simulation of black oil in naturally fractured reservoir through discrete fracture networks and homogenized media, SPE reserv. evaluat. eng., 750-758, (2008) [13] Karimi-Fard, M.; Firoozabadi, A., Numerical simulation of water injection in 2d fractured media using discrete-fracture model, Spe ree j., 4, 117-126, (2003) [14] S. Matthai, A. Mezentsev, M. Belayneh, Control-volume finite-element two phase flow experiments with fractured rock represented by unstructured 3d hybrid meshes. SPE 93341-MS, 31 Jan.-2 Feb., The Woodlands, Texas, 2005. [15] J.R. Natvig, B. Skaflestad, F. Bratvedt, K. Bratvedt, K.-A. Lie, V. Laptev, S.K. Khataniar, Multiscale mimetic solvers for efficient streamline simulation of fractured reservoirs. SPE 119132-MS, 2-4 Feb., The Woodlands, Texas, 2009. [16] Gulbransen, A.F.; Hauge, V.L.; Lie, K.-A., A multiscale mixed finite element method for vuggy and naturally fractured reservoirs, Spe j., 15, 2, 395-403, (2010) [17] Hou, T.Y.; Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comput. phys., 134, 169-189, (1997) · Zbl 0880.73065 [18] Hughes, T.; Feijoo, G.; Mazzei, L.; Quincy, J., The variational multiscale method – a paradigm for computational mechanics, Comput. methods appl. mech. eng., 166, 3-24, (1998) · Zbl 1017.65525 [19] Hou, T.Y.; Wu, X.-H.; Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. comput., 68, 227, 913-943, (1999) · Zbl 0922.65071 [20] Efendiev, Y.; Ginting, V.; Hou, T.; Ewing, R., Convergence of a nonconforming multiscale finite element method, SIAM J. numer. anal., 37, 3, 888-910, (2000) · Zbl 0951.65105 [21] T. Arbogast, Numerical subgrid upscaling of two-phase flow in porous media, in: Numerical Treatment of Multiphase Flows in Porous Media. Lecture Notes in Physics, vol. 552, issue 3-4, 2000, pp. 35-49. · Zbl 1072.76560 [22] Arbogast, T., Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow, Comput. geosci., 6, 3-4, 453-481, (2002) · Zbl 1094.76532 [23] Chen, Z.; Hou, T.Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. comput., 72, 242, 541-576, (2002) · Zbl 1017.65088 [24] Jenny, P.; Lee, S.H.; Tchelepi, H.A., Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. comput. phys., 187, 47-67, (2003) · Zbl 1047.76538 [25] Jenny, P.; Lee, S.H.; Tchelepi, H.A., Adaptive multiscale finite volume method for multi-phase flow and transport, SIAM multiscale model. simul., 3, 1, 50-64, (2004) · Zbl 1160.76372 [26] Efendiev, Y.; Hou, T.; Ginting, V., Multiscale finite element methods for nonlinear problems and their applications, Commun. math. sci., 2, 4, 553-589, (2004) · Zbl 1083.65105 [27] Aarnes, J.E., On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation, Multiscale model. simul., 2, 3, 421-439, (2004) · Zbl 1181.76125 [28] Ginting, V., Analysis of two-scale finite volume element method for elliptic problem, J. numer. math., 12, 119-141, (2004) · Zbl 1067.65124 [29] Aarnes, J.E.; Kippe, V.; Lie, K.-A., Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels, Adv. water resour., 28, 3, 257-271, (2005) [30] Jenny, P.; Lee, S.H.; Tchelepi, H.A., Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media, J. comput. phys., 217, 627-641, (2006) · Zbl 1160.76373 [31] Juanes, R.; Dub, F.X., A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms, Comput. geosci., 12, 273-295, (2008) · Zbl 1259.76067 [32] H. Hajibeygi, I. Lunati, S.H. Lee, Error estimate and control in the msfv method for multiphase flow in porous media, in: Proceedings of XVIII International Conference on Computational Methods in Water Resources (CMWR XVIII), Barcelona, Spain, 2010. [33] H. Hajibeygi, I. Lunati, S.H. Lee, Accurate and efficient simulation of multiphase flow in a heterogeneous reservoir by using error estimate and control in the multiscale finite-volume framework. SPE 141954-PP, 21-23 Feb., The Woodlands, Texas, 2011. [34] Hajibeygi, H.; Bonfigli, G.; Hesse, M.A.; Jenny, P., Iterative multiscale finite-volume method, J. comput. phys., 227, 8604-8621, (2008) · Zbl 1151.65091 [35] Hajibeygi, H.; Jenny, P., Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media, J. comput. phys., 228, 5129-5147, (2009) · Zbl 1280.76019 [36] Bonfigli, G.; Jenny, P., An efficient multi-scale Poisson solver for the incompressible navier – stokes equations with immersed boundaries, J. comput. phys., 228, 4568-4587, (2009) · Zbl 1165.76030 [37] Lunati, I.; Tyagi, M.; Lee, S.H., An iterative multiscale finite volume algorithm converging to exact solution, J. comput. phys., (2010), available online doi:10.1016/j.jcp.2010.11.036 [38] G. Bonfigli, P. Jenny, Recent developments in the multi-scale-finite-volume procedure. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 5910 LNCS:124-131, 2010. · Zbl 1280.65091 [39] H. Zhou, H.A. Tchelepi, Two-stage algebraic multiscale linear solver for highly heterogeneous reservoir models. SPE 141473-PP, 21-23 Feb., The Woodlands, Texas, 2011. [40] Hajibeygi, H.; Jenny, P., Adaptive iterative multiscale finite volume method, J. comput. phys., 230, 3, 628-643, (2011) · Zbl 1283.76041 [41] Trottenberg, U.; Oosterlee, C.W.; Schueller, A., Multigrid, (2001), Elsevier Academic Press [42] Saad, Y., Iterative methods for sparse linear systems, (2003), SIAM Philadelphia, USA · Zbl 1002.65042 [43] D.W. Peaceman, Interpretation of well-block pressure in numerical reservoir simulation. Society of Petroleum Engineers of AIME (SPE 6893), presented at the PSE-AIME 52nd Annual Fall Technical Conference and Exhibition, Denver, 1978, pp. 183-194. [44] LeVeque, R.V., Numerical methods for conservation laws. lecture notes in mathematics - ETH zurich, (2006), Birkhaeuser Verlag Basel. Switzerland [45] Jenny, P.; Lunati, I., Modeling complex wells with the multi-scale finite volume method, J. comput. phys., 228, 687-702, (2009) · Zbl 1155.76040 [46] M.W. Gee, C.M. Siefert, J.J. Hu, R.S. Tuminaro, M.G. Sala, ML 5.0 smoothed aggregation user’s guide. Technical Report SAND2006-2649, Sandia National Laboratories, 2006. [47] Smith, B.; Bjorstad, P.; Gropp, W., Domain decomposition: parallel multilevel methods for elliptic partial differential equations, (1996), Cambridge University Press · Zbl 0857.65126 [48] Nordbotten, J.M.; Bjøstad, P.E., On the relationship between the multiscale finite volume method and domain decomposition preconditioners, Comput. geosci., 13, 3, 367-376, (2008) · Zbl 1155.76042
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.