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Multiscale methods for model order reduction of non-linear multiphase flow problems. (English) Zbl 1414.76088
Summary: Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

76T30 Three or more component flows
76S05 Flows in porous media; filtration; seepage
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[1] Aarnes, JE, 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, 421-439, (2004) · Zbl 1181.76125
[2] 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
[3] Allaire, G., Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, 1482-1518, (1992) · Zbl 0770.35005
[4] Amaziane, B.; Antontsev, S.; Pankratov, L.; Piatnitski, A., Homogenization of immiscible compressible two-phase flow in porous media: application to gas migration in a nuclear waste repository, Multiscale Model. Simul., 8, 2023-2047, (2010) · Zbl 1215.35024
[5] Amaziane, B.; Bourgeat, A.; Jurak, M., Effective macrodiffusion in solute transport through heterogeneous porous media, Multiscale Model. Simul., 5, 184-204, (2006) · Zbl 1187.76746
[6] Amaziane, B.; Pankratov, L.; Piatnitski, A., Homogenization of immiscible compressible two-phase flow in highly heterogeneous porous media with discontinuous capillary pressures, Math. Models Methods Appl. Sci., 24, 1421-1451, (2014) · Zbl 1293.35025
[7] Arbogast, T.; Pencheva, G.; Wheeler, MF; Yotov, I., A multiscale mortar mixed finite element method, Multiscale Model. Simul., 6, 319-346, (2007) · Zbl 1322.76039
[8] Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures, volume 5. North-Holland Publishing Company, Amsterdam (1978) · Zbl 0411.60078
[9] Bourgeat, A., Homogenized behavior of two-phase flows in naturally fractured reservoirs with uniform fractures distribution, Comput. Methods Appl. Mech. Eng., 47, 205-216, (1984) · Zbl 0545.76125
[10] Chan, HY; Chung, E.; Efendiev, Y., Adaptive mixed gmsfem for flows in heterogeneous media, Numer. Math. Theory Methods Appl., 9, 497-527, (2016) · Zbl 1399.65322
[11] Chen, Y.; Durlofsky, LJ; Gerritsen, M.; Wen, X-H, A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations, Adv. Water Resour., 26, 1041-1060, (2003)
[12] Chen, Z.; Hou, T., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comput., 72, 541-576, (2003) · Zbl 1017.65088
[13] Christie, MA; Blunt, MJ, Tenth spe comparative solution project: a comparison of upscaling techniques, SPE Reservoir Evaluation and Engineering, 4, 308-317, (2001)
[14] Chung, E.; Efendiev, Y.; Hou, TY, Adaptive multiscale model reduction with generalized multiscale finite element methods, J. Comput. Phys., 320, 69-95, (2016) · Zbl 1349.76191
[15] Chung, ET; Efendiev, Y.; Lee, CS, Mixed generalized multiscale finite element methods and applications, Multiscale Model. Simul., 13, 338-366, (2015) · Zbl 1317.65204
[16] Chung, ET; Leung, WT; Vasilyeva, M.; Wang, Y., Multiscale model reduction for transport and flow problems in perforated domains, J. Comput. Appl. Math., 330, 519-535, (2018) · Zbl 1432.76086
[17] Durlofsky, LJ, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resour. Res., 27, 699-708, (1991)
[18] Efendiev, Y.; Galvis, J.; Hou, TY, Generalized multiscale finite element methods (gmsfem), J. Comput. Phys., 251, 116-135, (2013) · Zbl 1349.65617
[19] Gong, B., Karimi-Fard, M., Durlofsky, L.J., et al.: An upscaling procedure for constructing generalized dual-porosity/dual-permeability models from discrete fracture characterizations. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2006)
[20] Hou, TY; 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
[21] 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
[22] Jikov, V.V., Kozlov, S.M., Oleinik, A.O.: Homogenization of differential operators and integral functionals. Springer, Berlin Heidelberg (2012)
[23] Kozlova, A.; Li, Z.; Watanabe, JR; Zhou, SY; Bratvedt, K.; Lee, SH, A real-field multiscale black-oil reservoir simulator, Society of Petroleum Engineers Journal, 21, 2049-2061, (2016)
[24] Lee, SH; Wolfsteiner, C.; Tchelepi, HA, Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravity, Comput. Geosci., 12, 351-366, (2008) · Zbl 1259.76049
[25] Li, H.; Durlofsky, LJ, Local-global upscaling for compositional subsurface flow simulation, Transp. Porous Media, 111, 701-730, (2016)
[26] Lie, K-A; Møyner, O.; Natvig, JR; Kozlova, A.; Bratvedt, K.; Watanabe, S.; Li, Z., Successful application of multiscale methods in a real reservoir simulator environment, Comput. Geosci., 21, 981-998, (2017)
[27] Mikelić, A.; Devigne, V.; Duijn, CJ, Rigorous upscaling of the reactive flow through a pore, under dominant peclet and damkohler numbers, SIAM J. Math. Anal., 38, 1262-1287, (2006) · Zbl 1120.35007
[28] Moyner, O.; Lie, KA, 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
[29] Moyner, O.; Lie, KA, A multiscale restriction-smoothed basis method for compressible black-oil model, Society of Petroleum Engineers Journal, 21, 2079-2096, (2016)
[30] Pal, M.; Lamine, S.; Lie, KA, Validation of the multiscale mixed finite-element method, International Journal for Numerical Methods in Fluids, 77, 206-223, (2015)
[31] Tavakoli, R.; Yoon, H.; Delshad, M.; ElSheikh, AH; Wheeler, MF; Arnold, BW, Comparison of ensemble filtering algorithms and null-space monte carlo for parameter estimation and uncertainty quantification using co2 sequestration data, Water Resour. Res., 49, 8108-8127, (2013)
[32] Thomas, SG; Wheeler, MF, Enhanced velocity mixed finite element methods for modeling coupled flow and transport on non-matching multiblock grids, Comput. Geosci., 15, 605-625, (2011) · Zbl 1348.76101
[33] Weinan, E.; Engquist, B.; Li, X.; Ren, W.; Vanden-Eijnden, E., Heterogeneous multiscale methods: a review, Commun. Comput. Phys, 2, 367-450, (2007) · Zbl 1164.65496
[34] Wheeler, JA; Wheeler, MF; Yotov, I., Enhanced velocity mixed finite element methods for flow in multiblock domains, Comput. Geosci., 6, 315-332, (2002) · Zbl 1023.76023
[35] Wu, X-H; Efendiev, Y.; Hou, TY, Analysis of upscaling absolute permeability, Discrete and Continuous Dynamical Systems Series B, 2, 185-204, (2002) · Zbl 1162.65327
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