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Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media. (English) Zbl 1432.76181

Summary: Our goal of this paper is to develop a new upscaling method for multicontinua flow problems in fractured porous media. We consider a system of equations that describes flow phenomena with multiple flow variables defined on both matrix and fractures. To construct our upscaled model, we will apply the nonlocal multicontinua (NLMC) upscaling technique. The upscaled coefficients are obtained by using some multiscale basis functions, which are solutions of local problems defined on oversampled regions. For each continuum within a target coarse element, we will solve a local problem defined on an oversampling region obtained by extending the target element by few coarse grid layers, with a set of constraints which enforce the local solution to have mean value one on the chosen continuum and zero mean otherwise. The resulting multiscale basis functions have been shown to have good approximation properties. To illustrate the idea of our approach, we will consider a dual continua background model consisting of discrete fractures in two space dimensions, that is, we consider a system with three continua. We will present several numerical examples, and they show that our method is able to capture the interaction between matrix continua and discrete fractures on the coarse grid efficiently.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
76T99 Multiphase and multicomponent flows

Software:

FEniCS; SyFi
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References:

[1] Akkutlu, I. Yucel; Fathi, Ebrahim, Multiscale gas transport in shales with local kerogen heterogeneities, SPE J., 17, 04, 1-002 (2012)
[2] Akkutlu, I. Yucel; Efendiev, Yalchin; Vasilyeva, Maria; Wang, Yuhe, Multiscale model reduction for shale gas transport in a coupled discrete fracture and dual-continuum porous media, J. Nat. Gas Sci. Eng. (2017) · Zbl 1380.76032
[3] Chung, Eric T.; Efendiev, Yalchin; Leung, Tat; Vasilyeva, Maria, Coupling of multiscale and multi-continuum approaches, GEM-Int. J. Geomath., 8, 1, 9-41 (2017) · Zbl 1456.65107
[4] Li, Qiuqi; Wang, Yuhe; Vasilyeva, Maria, Multiscale model reduction for fluid infiltration simulation through dual-continuum porous media with localized uncertainties, J. Comput. Appl. Math., 336, 127-146 (2018) · Zbl 1524.65568
[5] Wu, Yu-Shu; Di, Yuan; Kang, Zhijiang; Fakcharoenphol, Perapon, A multiple-continuum model for simulating single-phase and multiphase flow in naturally fractured vuggy reservoirs, J. Pet. Sci. Eng., 78, 1, 13-22 (2011)
[6] Wu, Yu-Shu; Ehlig-Economides, Christine; Qin, Guan; Kang, Zhijang; Zhang, Wangming; Ajayi, Babatunde; Tao, Qingfeng, A Triple-Continuum Pressure-Transient Model for a Naturally Fractured Vuggy Reservoir (2007)
[7] Yao, Jun; Huang, Zhaoqin; Li, Yajun; Wang, Chenchen; Lv, Xinrui, Discrete fracture-vug network model for modeling fluid flow in fractured vuggy porous media, International Oil and Gas Conference and Exhibition in China (2010), Society of Petroleum Engineers · Zbl 1193.76144
[8] Hou, T.; 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
[9] Efendiev, Y.; Hou, T., (Multiscale Finite Element Methods: Theory and Applications. Multiscale Finite Element Methods: Theory and Applications, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4 (2009), Springer: Springer New York) · Zbl 1163.65080
[10] Efendiev, Y.; Galvis, J.; Hou, T., Generalized multiscale finite element methods, J. Comput. Phys., 251, 116-135 (2013) · Zbl 1349.65617
[11] Lunati, Ivan; Jenny, Patrick, Multiscale finite-volume method for compressible multiphase flow in porous media, J. Comput. Phys., 216, 2, 616-636 (2006) · Zbl 1220.76049
[12] Jenny, Patrick; Lee, Seong H.; Tchelepi, Hamdi A., Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale Model. Simul., 3, 1, 50-64 (2005) · Zbl 1160.76372
[13] Sánchez-Palencia, Enrique, Non-homogeneous media and vibration theory, Non-Homogeneous Media and Vibration Theory, Vol. 127 (1980) · Zbl 0432.70002
[14] Talonov, Alexey; Vasilyeva, Maria, On numerical homogenization of shale gas transport, J. Comput. Appl. Math., 301, 44-52 (2016) · Zbl 1382.76177
[15] Douglas Jr, Jim; Arbogast, T., Dual porosity models for flow in naturally fractured reservoirs, Dyn. Fluids Hierarchical Porous Media, 177-221 (1990)
[16] Xu, Tianfu; Pruess, Karsten, Modeling multiphase non-isothermal fluid flow and reactive geochemical transport in variably saturated fractured rocks: 1. Methodology, Amer. J. Sci., 301, 1, 16-33 (2001)
[17] Warren, JE; Root, P. Jj, The behavior of naturally fractured reservoirs, Soc. Pet. Eng. J., 3, 03, 245-255 (1963)
[18] Barenblatt, GI; Zheltov, Iu P.; Kochina, IN, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], J. Appl. Math. Mech., 24, 5, 1286-1303 (1960) · Zbl 0104.21702
[19] Praditia, Timothy; Helmig, Rainer; Hajibeygi, Hadi, Multiscale formulation for coupled flow-heat equations arising from single-phase flow in fractured geothermal reservoirs, Comput. Geosci., 1-18 (2018) · Zbl 1406.86020
[20] Ţene, Matei; Al Kobaisi, Mohammed Saad; Hajibeygi, Hadi, Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (f-ams), J. Comput. Phys., 321, 819-845 (2016) · Zbl 1349.76394
[21] Karimi-Fard, Mohammad; Durlofsky, Luis J.; Aziz, Khalid, An efficient discrete fracture model applicable for general purpose reservoir simulators, SPE Reservoir Simulation Symposium (2003), Society of Petroleum Engineers
[22] Karimi-Fard, Mohammad; Firoozabadi, Abbas, Numerical simulation of water injection in 2d fractured media using discrete-fracture model, SPE Annual Technical Conference and Exhibition (2001), Society of Petroleum Engineers
[23] Hajibeygi, H.; Kavounis, D.; Jenny, P., A hierarchical fracture model for the iterative multiscale finite volume method, J. Comput. Phys., 230, 24, 8729-8743 (2011) · Zbl 1370.76095
[24] Bosma, Sebastian; Hajibeygi, Hadi; Tene, Matei; Tchelepi, Hamdi A., Multiscale finite volume method for discrete fracture modeling on unstructured grids (MS-dfm), J. Comput. Phys. (2017) · Zbl 1375.76097
[25] Ţene, Matei; Bosma, Sebastian BM; Al Kobaisi, Mohammed Saad; Hajibeygi, Hadi, Projection-based embedded discrete fracture model (pedfm), Adv. Water Resour., 105, 205-216 (2017)
[26] Akkutlu, IY; Efendiev, Yalchin; Vasilyeva, Maria, Multiscale model reduction for shale gas transport in fractured media, Comput. Geosci., 1-21 (2015)
[27] Akkutlu, I. Yucel; Efendiev, Yalchin; Vasilyeva, Maria; Wang, Yuhe, Multiscale model reduction for shale gas transport in poroelastic fractured media, J. Comput. Phys., 353, 356-376 (2018) · Zbl 1380.76032
[28] Chung, Eric; Efendiev, Yalchin; Hou, Thomas Y., Adaptive multiscale model reduction with generalized multiscale finite element methods, J. Comput. Phys., 320, 69-95 (2016) · Zbl 1349.76191
[29] Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat; Vasilyeva, Maria; Wang, Yating, Non-local multi-continua upscaling for flows in heterogeneous fractured media, J. Comput. Phys. (2018) · Zbl 1432.76086
[30] Maria Vasilyeva, Eric T Chung, Wing Tat Leung, Valentin Alekseev, Nonlocal multicontinuum (NLMC) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models, 2018. arXiv preprint arXiv:1805.09407; Maria Vasilyeva, Eric T Chung, Wing Tat Leung, Valentin Alekseev, Nonlocal multicontinuum (NLMC) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models, 2018. arXiv preprint arXiv:1805.09407
[31] Maria Vasilyeva, Eric T Chung, Yalchin Efendiev, Jihoon Kim, Constrained energy minimization based upscaling for coupled flow and mechanics, 2018. arXiv preprint arXiv:1805.09382; Maria Vasilyeva, Eric T Chung, Yalchin Efendiev, Jihoon Kim, Constrained energy minimization based upscaling for coupled flow and mechanics, 2018. arXiv preprint arXiv:1805.09382
[32] Maria Vasilyeva, Eric T Chung, Wing Tat Leung, Yating Wang, Denis Spiridonov, Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using Non-Local Multi-Continuum method (NLMC), 2018. arXiv preprint arXiv:1805.09420; Maria Vasilyeva, Eric T Chung, Wing Tat Leung, Yating Wang, Denis Spiridonov, Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using Non-Local Multi-Continuum method (NLMC), 2018. arXiv preprint arXiv:1805.09420
[33] Logg, Anders, Efficient representation of computational meshes, Int. J. Comput. Sci. Eng., 4, 4, 283-295 (2009)
[34] Logg, Anders; Mardal, Kent-Andre; Wells, Garth, Automated Solution of Differential Equations by the Finite Element Method: The FeniCS Book, Vol. 84 (2012), Springer Science & Business Media · Zbl 1247.65105
[35] Arbogast, Todd; Douglas, Jim; Hornung, Ulrich, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21, 4, 823-836 (1990) · Zbl 0698.76106
[36] Kazemi, H.; Merrill Jr, LS; Porterfield, KL; Zeman, PR, Numerical simulation of water-oil flow in naturally fractured reservoirs, Soc. Pet. Eng. J., 16, 06, 317-326 (1976)
[37] Martin, Vincent; Jaffré, Jérôme; Roberts, Jean E., Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., 26, 5, 1667-1691 (2005) · Zbl 1083.76058
[38] Formaggia, Luca; Fumagalli, Alessio; Scotti, Anna; Ruffo, Paolo, A reduced model for darcys problem in networks of fractures, ESAIM Math. Model. Numer. Anal., 48, 4, 1089-1116 (2014) · Zbl 1299.76254
[39] D’Angelo, Carlo; Scotti, Anna, A mixed finite element method for darcy flow in fractured porous media with non-matching grids, ESAIM Math. Model. Numer. Anal., 46, 2, 465-489 (2012) · Zbl 1271.76322
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