×

Multiscale formulation of pore-scale compressible Darcy-Stokes flow. (English) Zbl 1453.65304

Summary: Direct numerical simulation (DNS) of fluid dynamics in digital images of porous materials is challenging due to the cut-off length issue where interstitial voids below the resolution of the imaging instrument cannot be resolved. Such subresolution microporosity can be critical for flow and transport because they could provide important flow pathways. A micro-continuum framework can be used to address this problem, which applies to the entire domain a single momentum equation, i.e., Darcy-Brinkman-Stokes (DBS) equation, that recovers Stokes equation in the resolved void space (i.e., macropores) and Darcy equation in the microporous regions. However, the DBS-based micro-continuum framework is computationally demanding. Here, we develop an efficient multiscale method for the compressible Darcy-Stokes flow arising from the micro-continuum approach. The method decomposes the domain into subdomains that either belong to the macropores or the microporous regions, on which Stokes or Darcy problems are solved locally, only once, to build basis functions. The nonlinearity from compressible flow is accounted for in a local correction problem on each subdomain. A global interface problem is solved to couple the local bases and correction functions to obtain an approximate global multiscale solution, which is in excellent agreement with the reference single-scale solution. The multiscale solution can be improved through an iterative strategy that guarantees convergence to the single-scale solution. The method is computationally efficient and well-suited for parallelization to simulate fluid dynamics in large high-resolution digital images of porous materials.

MSC:

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
76M12 Finite volume methods applied to problems in fluid mechanics
65Z05 Applications to the sciences

Software:

C-AMS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Blunt, M. J.; Bijeljic, B.; Dong, H.; Gharbi, O.; Iglauer, S.; Mostaghimi, P.; Paluszny, A.; Pentland, C., Pore-scale imaging and modelling, Adv. Water Resour., 51, 197-216 (2013)
[2] Wildenschild, D.; Sheppard, A. P., X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems, Adv. Water Resour., 51, 217-246 (2013)
[3] Blunt, M. J., Multiphase Flow in Permeable Media: A Pore-Scale Perspective (2017), Cambridge University Press
[4] Mehmani, A.; Prodanović, M., The effect of microporosity on transport properties in porous media, Adv. Water Resour., 63, 104-119 (2014)
[5] Bultreys, T.; Van Hoorebeke, L.; Cnudde, V., Multi-scale, micro-computed tomography-based pore network models to simulate drainage in heterogeneous rocks, Adv. Water Resour., 78, 36-49 (2015)
[6] Scheibe, T. D.; Perkins, W. A.; Richmond, M. C.; McKinley, M. I.; Romero-Gomez, P. D.; Oostrom, M.; Wietsma, T. W.; Serkowski, J. A.; Zachara, J. M., Pore-scale and multiscale numerical simulation of flow and transport in a laboratory-scale column, Water Resour. Res., 51, 1023-1035 (2015)
[7] Soulaine, C.; Gjetvaj, F.; Garing, C.; Roman, S.; Russian, A.; Gouze, P.; Tchelepi, H. A., The impact of sub-resolution porosity of x-ray microtomography images on the permeability, Transp. Porous Media, 113, 227-243 (2016)
[8] Lin, Q.; Al-Khulaifi, Y.; Blunt, M. J.; Bijeljic, B., Quantification of sub-resolution porosity in carbonate rocks by applying high-salinity contrast brine using x-ray microtomography differential imaging, Adv. Water Resour., 96, 306-322 (2016)
[9] Soulaine, C.; Tchelepi, H. A., Micro-continuum approach for pore-scale simulation of subsurface processes, Transp. Porous Media, 113, 431-456 (2016)
[10] Brinkman, H., A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Flow Turbul. Combust., 1, 27 (1949) · Zbl 0041.54204
[11] Chalmers, G. R.; Bustin, R. M.; Power, I. M., Characterization of gas shale pore systems by porosimetry, pycnometry, surface area, and field emission scanning electron microscopy/transmission electron microscopy image analyses: examples from the Barnett, Woodford, Haynesville, Marcellus, and Doig units, Am. Assoc. Pet. Geol. Bull., 96, 1099-1119 (2012)
[12] Ma, L.; Taylor, K. G.; Lee, P. D.; Dobson, K. J.; Dowey, P. J.; Courtois, L., Novel 3d centimetre-to nano-scale quantification of an organic-rich mudstone: the carboniferous bowland shale, northern england, Mar. Pet. Geol., 72, 193-205 (2016)
[13] Wu, T.; Li, X.; Zhao, J.; Zhang, D., Multiscale pore structure and its effect on gas transport in organic-rich shale, Water Resour. Res., 53, 5438-5450 (2017)
[14] Guo, B.; Ma, L.; Tchelepi, H., Image-based micro-continuum model for gas flow in organic-rich shale rock, Adv. Water Resour., 122, 70-84 (2018)
[15] Jenny, P.; Lee, S.; 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
[16] Jenny, P.; Lee, S. H.; Tchelepi, H. A., Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale Model. Simul., 3, 50-64 (2005) · Zbl 1160.76372
[17] Tomin, P.; Lunati, I., Hybrid multiscale finite volume method for two-phase flow in porous media, J. Comput. Phys., 250, 293-307 (2013)
[18] Khayrat, K.; Jenny, P., A multi-scale network method for two-phase flow in porous media, J. Comput. Phys., 342, 194-210 (2017) · Zbl 1376.76066
[19] Bernardi, C.; Maday, Y.; Patera, A. T., Domain decomposition by the mortar element method, (Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters (1993), Springer), 269-286 · Zbl 0799.65124
[20] Arbogast, T.; Cowsar, L. C.; Wheeler, M. F.; Yotov, I., Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal., 37, 1295-1315 (2000) · Zbl 1001.65126
[21] Arbogast, T.; Pencheva, G.; Wheeler, M. F.; Yotov, I., A multiscale mortar mixed finite element method, Multiscale Model. Simul., 6, 319-346 (2007) · Zbl 1322.76039
[22] Balhoff, M. T.; Thomas, S. G.; Wheeler, M. F., Mortar coupling and upscaling of pore-scale models, Comput. Geosci., 12, 15-27 (2008) · Zbl 1153.76421
[23] Mehmani, Y.; Sun, T.; Balhoff, M.; Eichhubl, P.; Bryant, S., Multiblock pore-scale modeling and upscaling of reactive transport: application to carbon sequestration, Transp. Porous Media, 95, 305-326 (2012)
[24] Mehmani, Y.; Balhoff, M. T., Bridging from pore to continuum: a hybrid mortar domain decomposition framework for subsurface flow and transport, Multiscale Model. Simul., 12, 667-693 (2014) · Zbl 1312.65155
[25] Mehmani, Y.; Tchelepi, H. A., Multiscale computation of pore-scale fluid dynamics: single-phase flow, J. Comput. Phys., 375, 1469-1487 (2018) · Zbl 1416.76219
[26] Mehmani, Y.; Tchelepi, H. A., Multiscale formulation of two-phase flow at the pore scale, J. Comput. Phys. (2019) · Zbl 1452.65218
[27] 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
[28] Golfier, F.; Zarecone, C.; Bazin, B.; Lenormand, R.; Lasseux, D.; Quintard, M., On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium, J. Fluid Mech., 457, 213-254 (2002) · Zbl 1016.76079
[29] Goyeau, B.; Lhuillier, D.; Gobin, D.; Velarde, M., Momentum transport at a fluid – porous interface, Int. J. Heat Mass Transf., 46, 4071-4081 (2003) · Zbl 1065.76606
[30] Van Doormaal, J.; Raithby, G., Enhancements of the simple method for predicting incompressible fluid flows, Numer. Heat Transf., 7, 147-163 (1984) · Zbl 0553.76005
[31] Ferziger, J. H.; Peric, M., Computational Methods for Fluid Dynamics (2012), Springer Science & Business Media · Zbl 0869.76003
[32] Beucher, S., Use of watersheds in contour detection, (Proceedings of the International Workshop on Image Processing. Proceedings of the International Workshop on Image Processing, CCETT (1979))
[33] Serra Jean, C., Image Analysis and Mathematical Morphology (1983), Academic Press, Inc
[34] Lunati, I.; Jenny, P., The multiscale finite volume method: a flexible tool to model physically complex flow in porous media, (Proceedings of European Conference of Mathematics of Oil Recovery X. Proceedings of European Conference of Mathematics of Oil Recovery X, Amsterdam, The Netherlands (2006))
[35] Hajibeygi, H.; Bonfigli, G.; Hesse, M. A.; Jenny, P., Iterative multiscale finite-volume method, J. Comput. Phys., 227, 8604-8621 (2008) · Zbl 1151.65091
[36] Dolean, V.; Jolivet, P.; d ric Nataf, F., An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation, vol. 144 (2015), SIAM · Zbl 1364.65277
[37] Ţene, M.; Wang, Y.; Hajibeygi, H., Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media, J. Comput. Phys., 300, 679-694 (2015) · Zbl 1349.76272
[38] Beavers, G. S.; Joseph, D. D., Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30, 197-207 (1967)
[39] Saffman, P. G., On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 50, 93-101 (1971) · Zbl 0271.76080
[40] Payne, L.; Straughan, B., Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions, J. Math. Pures Appl., 77, 317-354 (1998) · Zbl 0906.35067
[41] Mosthaf, K.; Baber, K.; Flemisch, B.; Helmig, R.; Leijnse, A.; Rybak, I.; Wohlmuth, B., A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow, Water Resour. Res., 47 (2011)
[42] Baber, K.; Mosthaf, K.; Flemisch, B.; Helmig, R.; Müthing, S.; Wohlmuth, B., Numerical scheme for coupling two-phase compositional porous-media flow and one-phase compositional free flow, IMA J. Appl. Math., 77, 887-909 (2012) · Zbl 1343.76024
[43] Neale, G.; Nader, W., Practical significance of Brinkman’s extension of Darcy’s law: coupled parallel flows within a channel and a bounding porous medium, Can. J. Chem. Eng., 52, 475-478 (1974)
[44] Zhang, S.; Zhao, X.; Bayyuk, S., Generalized formulations for the Rhie-Chow interpolation, J. Comput. Phys., 258, 880-914 (2014) · Zbl 1349.76562
[45] Nordlund, M.; Stanic, M.; Kuczaj, A. K.; Frederix, E. M.; Geurts, B. J., Improved PISO algorithms for modeling density varying flow in conjugate fluid – porous domains, J. Comput. Phys., 306, 199-215 (2016) · Zbl 1351.76120
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.