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Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs. (English) Zbl 1405.76056
Summary: In simulation of fluid injection in fractured geothermal reservoirs, the characteristics of the physical processes are severely affected by the local occurence of connected fractures. To resolve these structurally dominated processes, there is a need to develop discretization strategies that also limit computational effort. In this paper, we present an upscaling methodology for geothermal heat transport with fractures represented explicitly in the computational grid. The heat transport is modeled by an advection-conduction equation for the temperature, and solved on a highly irregular coarse grid that preserves the fracture heterogeneity. The upscaling is based on different strategies for the advective term and the conductive term. The coarse scale advective term is constructed from sums of fine scale fluxes, whereas the coarse scale conductive term is constructed based on numerically computed basis functions. The method naturally incorporates the coupling between solution variables in the matrix and in the fractures, respectively, via the discretization. In this way, explicit transfer terms that couple fracture and matrix solution variables are avoided. Numerical results show that the upscaling methodology performs well, in particular for large upscaling ratios, and that it is applicable also to highly complex fracture networks.

76S05 Flows in porous media; filtration; seepage
86-08 Computational methods for problems pertaining to geophysics
80A20 Heat and mass transfer, heat flow (MSC2010)
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[1] Aarnes, JE; Efendiev, Y, An adaptive multiscale method for simulation of fluid flow in heterogeneous porous media, Multiscale Model Simul., 5, 918-939, (2006) · Zbl 1205.76156
[2] Aarnes, JE; Hauge, VL; Efendiev, Y, Coarsening of three-dimensional structured and unstructured grids for subsurface flow, Adv. Water Resour., 30, 2177-2193, (2007)
[3] Aavatsmark, I, An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci., 6, 405-432, (2002) · Zbl 1094.76550
[4] Aziz, K., Settari, A.: Petroleum Reservoir Simulation, 476 p. Science Publishers Ltd, London (1979)
[5] Berkowitz, B, Characterizing flow and transport in fractured geological reservoirs, A Review, Adv. Water Resour., 25, 861-884, (2002)
[6] Berkowitz, B; Bear, J; Braester, C, Continuum models for contaminant transport in fractured porous formations, Water Resour. Res., 24, 1225-1236, (1988)
[7] Bonnet, E; Bour, O; Odling, NE; Davy, P; Main, I; Cowie, P; Berkowitz, B, Scaling of fracture systems in geological media, Rev. Geophys., 39, 347-383, (2001)
[8] Bruel, D., Cacas, M.C.: Numerical modeling technique: contribution to the Soultz HDR project. In: Geothermal Energy in Europe -The Soultz Hot Dry Rock Project, Gordon and Breach Science Publishers, pp 267-279 (1992)
[9] Buck, M; Iliev, O; Andrä, H, Multiscale finite element coarse spaces for the application to linear elasticity, Cent. Eur. J. Math., 11, 680-701, (2013) · Zbl 1352.74331
[10] Cheng, P, Heat transfer in geothermal systems, Adv. Heat Trans., 14, 1-105, (1979)
[11] Dietrich, P., Helmig, R., Sauter, M., Hötzl, H., Köngeter, J., Teutsch, G.: Flow and Transport in Fractured Porous Media, 447 p. Springer, Berlin (2005)
[12] Hajibeygi, H; Bonfigli, G; Hesse, MA; Jenny, P, Iterative multiscale finite-volume method, J. Comput. Phys., 227, 8604-8621, (2008) · Zbl 1151.65091
[13] Hauge, VL; Lie, K-A; Natvig, JR, Flow-based coarsening for multiscale simulation of transport in porous media, Comput. Geosci., 16, 391-408, (2012)
[14] Hayashi, K; Willis-Richards, J; Hopkirk, RJ; Niibori, Y, Numerical models of HDR geothermal reservoirs - a review of current thinking and progress, Geothermics, 28, 507-518, (1999)
[15] Hajibeygi, H; Karvounis, D; Jenny, P, A hierarchical fracture model for the iterative multiscale finite volume method, J. Comput. Phys., 230, 8729-8743, (2011) · Zbl 1370.76095
[16] Holm, R; Kaufmann, R; Heimsund, B-O; Øian, E; Espedal, MS, Meshing of domains with complex internal geometries, Numer. Linear Algebra Appl., 13, 717-731, (2006) · Zbl 1174.76363
[17] Iserles, A.: A first course in numerical analysis of differential equations. Cambridge University Press, Cambridge (1996). ISBN 978-0-521-55655-2
[18] Jenny, P; Lee, SH; Tchelepi, HA, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187, 47-67, (2003) · Zbl 1047.76538
[19] Karimi-Fard, M; Durlofsky, L; Aziz, K, An efficient discrete-fracture model applicable for general-purpose reservoir simulators, SPE J., 9, 227-236, (2004)
[20] Karimi-Fard, M; Durlofsky, LJ, A general gridding, discretization, and coarsening methodology for modeling flow in porous formations with discrete geological features, Adv. Water Resour., 96, 354-372, (2016)
[21] Keilegavlen, E., Nordbotten J.M.: Finite volume methods for elasticity with weak symmetry, Int. J. Numer. Methods Eng. https://doi.org/10.1002/nme.5538 (2017)
[22] Martin, V; Jaffré, J; Roberts, JE, Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., 26, 1667-1691, (2005) · Zbl 1083.76058
[23] Møyner, O; Lie, K-A, A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids, J. Comput. Phys., 304, 46-71, (2016) · Zbl 1349.76824
[24] Mustapha, H; Mustapha, K, A new approach to simulating flow in discrete fracture networks with an optimized mesh, SIAM J. Sci. Comput., 29, 1439-1459, (2007) · Zbl 1251.76056
[25] Nordbotten, JM; Bjørstad, PE, On the relationship between the multiscale finite volume method and domain decomposition preconditioners, Comput. Geosci., 12, 367-376, (2008) · Zbl 1155.76042
[26] Nordbotten, JM, Cell-centered finite volume methods for deformable porous media, Int. J. Numer. Methods Eng., 100, 399-418, (2014) · Zbl 1352.76072
[27] Nordbotten, JM, Stable cell-centered finite volume discretization for biot’s equations, SIAM J. Numer. Anal., 54, 942-968, (2016) · Zbl 1382.76187
[28] Pruess, K, Heat transfer in fractured geothermal reservoirs with boiling, Water Resour. Res., 19, 201-208, (1983)
[29] Pruess, K; Narasimhan, TN, A practical method for modeling fluid and heat flow in fractured porous media, Soc. Petrol. Eng. J., 25, 14-26, (1985)
[30] Reichenberger, V; Jakobs, H; Bastian, P; Helmig, R, A mixed-dimensional finite volume method for two-phase flow in fractured porous media, Adv. Water Resour., 29, 1020-1036, (2006)
[31] Sandve, TH; Berre, I; Nordbotten, JM, An efficient multi-point flux approximation method for discrete fracture-matrix simulations, J. Comput. Phys., 231, 3784-3800, (2012) · Zbl 1402.76131
[32] Sandve, TH; Keilegavlen, E; Nordbotten, JM, Physics-based preconditioners for flow in fractured porous media, Water Resour. Res., 50, 1357-1373, (2014)
[33] Shah, S; Møyner, O; Tene, M; Lie, K-A; Hajibeygi, H, The multiscale restriction smoothed basis method for fractured porous media (F-msrsb), J. Comput. Phys., 318, 36-57, (2016) · Zbl 1349.76385
[34] Silberhorn-Hemminger, A.: Modellierung von Kluftaquifersystemen: Geostatistische Analyse und deterministisch-stochastische Kluftgenerierung, Dissertation. Institut für Wasserbau, Universität Stuttgart (2002). ISBN: 3-933761-17-4
[35] Tene, M., Al Kobaisi, M.S., Hajibeygi, H.: Algebraic multiscale solver for flow in heterogeneous fractured porous media. In: Proceedings of SPE Reservoir Simulation Symposium (2015) · Zbl 1349.76394
[36] Toselli, A., Widlund, O.B.: Domain decomposition methods: algorithms and theory, 450 p. Springer, Berlin (2005) · Zbl 1069.65138
[37] Ucar, E., Berre, I., Keilegavlen, E., Nordbotten, J.M.: Finite volume method for deformation of fractured media, arXiv:1612.06594 (2016) · Zbl 1396.74099
[38] Lent, J; Scheichl, R; Graham, IG, Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDE, Numer. Linear Algebra Appl., 16, 775-799, (2009) · Zbl 1224.65292
[39] Vanek, P; Mandel, J; Brezina, M, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56, 179-196, (1996) · Zbl 0851.65087
[40] Watanabe, K; Takahashi, H, Fractal geometry characterization of geothermal fracture networks, J. Geophys. Res., 100, 521-528, (1995)
[41] Zhou, H; Tchelepi, HA, Operator-based multiscale method for compressible flow, Soc. Petrol. Eng. J., 13, 267-273, (2008)
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