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Numerical Investigation into Coarse-Scale Models of Diffusion in Complex Heterogeneous Media

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Abstract

Computational modelling of diffusion in heterogeneous media is prohibitively expensive for problems with fine-scale heterogeneities. A common strategy for resolving this issue is to decompose the domain into a number of non-overlapping sub-domains and homogenize the spatially-dependent diffusivity within each sub-domain (homogenization cell). This process yields a coarse-scale model for approximating the solution behaviour of the original fine-scale model at a reduced computational cost. In this paper, we study coarse-scale diffusion models in block heterogeneous media and investigate, for the first time, the effect that various factors have on the accuracy of resulting coarse-scale solutions. We present new findings on the error associated with homogenization as well as confirm via numerical experimentation that periodic boundary conditions are the best choice for the homogenization cell and demonstrate that the smallest homogenization cell that is computationally feasible should be used in numerical simulations.

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References

  • Abdulle, A., Weinan, E.: Finite difference heterogeneous multi-scale method for homogenization problems. J. Comput. Phys. 191, 18–39 (2003)

    Article  Google Scholar 

  • Allaire, G., Brizzi, R.: A multiscale finite element method for numerical homogenization. Multisc. Model. Simul. 4(3), 790–812 (2005)

    Article  Google Scholar 

  • Arbogast, T.: Gravitational forces in dual-porosity systems: I. Model derivation by homogenization. Transp. Porous Med. 13(2), 179–203 (1993a)

    Article  Google Scholar 

  • Arbogast, T.: Gravitational forces in dual-porosity systems: II. Computational validation of the homogenized model. Transp. Porous Med. 13(2), 205–220 (1993b)

    Article  Google Scholar 

  • Asvestas, M., Sifalakis, A.G., Papadopoulou, E.P., Saridakis, Y.G.: Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity. J. Phys. Conf. Ser. 490, 012143 (2014)

    Article  Google Scholar 

  • Bensoussan, A., Lions, J.L., Papanicolau, G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978)

    Google Scholar 

  • Cardwell, W.T., Parsons, R.L.: Average permeabilities of heterogeneous oils sands. Trans. Am. Inst. Min. Met. Pet. Eng. 160(1), 34–42 (1945)

    Google Scholar 

  • Carr, E.J., March, N.G.: Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions. Appl. Math. Comput. 333, 286–303 (2018)

    Google Scholar 

  • Carr, E.J., Turner, I.W.: Two-scale computational modelling of water flow in unsaturated soils containing irregular-shaped inclusions. Int. J. Numer. Methods Eng. 98(3), 157–173 (2014)

    Article  Google Scholar 

  • Carr, E.J., Turner, I.W.: A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers. Appl. Math. Model. 40, 7034–7050 (2016)

    Article  Google Scholar 

  • Carr, E.J., Turner, I.W., Perré, P.: A dual-scale modelling approach for drying hygroscopic porous media. Multisc. Model. Simul. 11(1), 362–384 (2013a)

    Article  Google Scholar 

  • Carr, E.J., Turner, I.W., Perré, P.: A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media: application to wood drying. J. Comput. Phys. 233, 66–82 (2013b)

    Article  Google Scholar 

  • Carr, E.J., Perré, P., Turner, I.W.: The extended distributed microstructure model for gradient-driven transport: A two-scale model for bypassing effective parameters. J. Comput. Phys. 327, 810–829 (2016)

    Article  Google Scholar 

  • Carr, E.J., Turner, I.W., Perré, P.: Macroscale modelling of multilayer diffusion: using volume averaging to correct the boundary conditions. Appl. Math. Model. 47, 600–618 (2017)

    Article  Google Scholar 

  • Chen, F., Ren, L.: Application of the finite difference heterogeneous multiscale method to the Richards’ equation. Water Resour. Res. 44, W07413 (2008)

    Article  Google Scholar 

  • Davit, Y., Bell, C.G., Byrne, H.M., Chapman, L.A.C., Kimpton, L.S., Lang, G.E., Leonard, K.H.L., Oliver, J.M., Pearson, N.C., Shipley, R.J., Waters, S.L., Whiteley, J.P., Wood, B.D., Quintard, M.: Homogenization via formal multiscale asymptotics and volume averaging: how do the two techniques compare? Adv. Water Resour. 62, 178–206 (2013)

    Article  Google Scholar 

  • Ferguson, W.J., Turner, I.W.: A comparison of the finite element and control volume numerical solution techniques applied to timber drying problems below the boiling point. Int. J. Numer. Methods Eng. 38(3), 451–467 (1995)

    Article  Google Scholar 

  • Ferguson, W.J., Turner, I.W.: A control volume finite element numerical simulation of the drying of spruce. J. Comput. Phys. 125(1), 59–70 (1996)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Hickson, R.I., Barry, S.I., Mercer, G.N.: Critical times in multilayer diffusion. Part 1: Exact solutions. Int. J. Heat Mass Transf. 52, 5776–5783 (2009a)

    Article  Google Scholar 

  • Hickson, R.I., Barry, S.I., Mercer, G.N.: Critical times in multilayer diffusion. Part 2: Approximate solutions. Int. J. Heat Mass Transf. 52(25–26), 5784–5791 (2009b)

    Article  Google Scholar 

  • Hornung, U.: Homogenization and Porous Media. Springer, New York (1997)

    Book  Google Scholar 

  • Jayantha, P.A., Turner, I.W.: On the use of surface interpolation techniques in generalised finite volume strategies for simulating transport in highly anisotropic porous media. J. Comput. Appl. Math. 152(1), 199–216 (2003)

    Article  Google Scholar 

  • Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, New York (1994)

    Book  Google Scholar 

  • Kong, X., Wang, H., Wang, J.G., Gao, F., Wang, X.: A two-phase flowback model for multiscale diffusion and flow in fractured shale gas reservoirs. Geofluids 2018, 5910437 (2018)

    Google Scholar 

  • Kouznetsova, V., Brekelmans, W.A., Baaijens, F.P.: An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 27(1), 37–48 (2001)

    Article  Google Scholar 

  • Liu, C., Ball, W.P.: Analytical modeling of diffusion-limited contamination and decontamination in a two-layer porous medium. Adv. Water Resour. 21, 297–313 (1998)

    Article  Google Scholar 

  • Mantzavinos, D., Papadomanolaki, M.G., Saridakis, Y.G., Sifalakis, A.G.: Fokas transform method for a brain tumor invasion model with heterogeneous diffusion in 1+1 dimensions. Appl. Numer. Math. 104, 47–61 (2014)

    Article  Google Scholar 

  • March, N.G., Carr, E.J.: Finite volume schemes for multilayer diffusion. J. Comput. Math. 345, 206–223 (2019)

    Article  Google Scholar 

  • March, N.G., Carr, E.J., Turner, I.W.: A fast algorithm for semi-analytically solving the homogenization boundary value problem for block locally-isotropic heterogeneous media. Appl. Math. Model. 92, 23–43 (2021)

    Article  Google Scholar 

  • Matache, A., Babuška, I., Schwab, C.: Generalized p-FEM in homogenization. Numer. Math. 86(2), 319–375 (2000)

    Article  Google Scholar 

  • Moroney, T., Turner, I.: A finite volume method based on radial basis functions for two-dimensional nonlinear diffusion equations. Appl. Math. Model. 30(10), 1118–1133 (2006)

    Article  Google Scholar 

  • Moulinec, H., Suquet, P.: A FFT-based numerical method for computing the mechanical properties of composites from images of their microstructures. In: Pyrz, R. (ed.) Solid Mech. Appl., pp. 235–246. Springer, Dordrecht (1995)

    Google Scholar 

  • Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Eng. 157(1), 69–94 (1998)

    Article  Google Scholar 

  • Renard, P., de Marsily, G.: Calculating equivalent permeability: a review. Adv. Water Resour. 20(5), 253–278 (1997)

    Article  Google Scholar 

  • Sadrnejad, S., Ghasemzadeh, H., Amiri, S.A.G., Montazeri, G.H.: A control volume based finite element method for simulating incompressible two-phase flow in heterogeneous porous media and its application to reservoir engineering. Pet. Sci. 9(4), 485–497 (2012)

    Article  Google Scholar 

  • van der Sluis, O., Schreurs, P., Brekelmans, W., Meijer, H.: Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling. Mech. Mater. 32(8), 449–462 (2000)

    Article  Google Scholar 

  • Sviercoski, R.F., Winter, C.L., Warrick, A.: An analytical effective tensor and its approximation properties for upscaling flows through generalized composites. Adv. Water Resour. 33(7), 728–739 (2010)

    Article  Google Scholar 

  • Szymkiewicz, A.: Calculating effective conductivity of heterogeneous soils by homogenization. Arch. Hydro-Eng. Environ. Mech. 52(2), 111–130 (2005)

    Google Scholar 

  • Szymkiewicz, A.: Modelling Water Flow in Unsaturated Porous Media. Springer, Berlin (2013)

    Book  Google Scholar 

  • Talebi, H., Silani, M., Klusemann, B.: The scaled boundary finite element method for computational homogenization of heterogeneous media. Int. J. Numer. Methods Eng. 118(1), 1–17 (2019)

    Article  Google Scholar 

  • Terada, K., Hori, M., Kyoya, T., Kikuchi, N.: Simulation of the multi-scale convergence in computational homogenization approaches. Int. J. Solids Struct. 37(16), 2285–2311 (2000)

    Article  Google Scholar 

  • Wattier, M.L., Descamps, F., Vandycke, S., Tshibangu, J.P.: Chalk fractures geometry: a comprehensive description of fracture surfaces. In: Engineering in Chalk, pp 663–668 (2018)

  • Welch, M.J., Souque, C., Davies, R.K., Knipe, R.J.: Using mechanical models to investigate the controls on fracture geometry and distribution in chalk. In: Fundamental Controls on Fluid Flow in Carbonates: Current Workflows to Emerging Technologies. Geological Society of London (2015)

  • Yi, S., Xu, L., Cheng, G., Cai, Y.: FEM formulation of homogenization method for effective properties of periodic heterogeneous beam and size effect of basic cell in thickness direction. Comput. Struct. 156, 1–11 (2015)

    Article  Google Scholar 

  • Zhou, Z., Su, Y., Wang, W., Yan, Y.: Application of the fractal geometry theory on fracture network simulation. J. Petrol. Explor. Prod. Technol. 7(2), 487–496 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The second and third authors acknowledge funding from the Australian Research Council (DE150101137, DP150103675). All authors acknowledge the helpful comments of the anonymous reviewers and editors that helped improve the quality of the manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia

    Nathan G. March, Elliot J. Carr & Ian W. Turner

  2. ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), Queensland University of Technology (QUT), Brisbane, Australia

    Ian W. Turner

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  1. Nathan G. March
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  2. Elliot J. Carr
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  3. Ian W. Turner
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Correspondence to Nathan G. March.

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March, N.G., Carr, E.J. & Turner, I.W. Numerical Investigation into Coarse-Scale Models of Diffusion in Complex Heterogeneous Media. Transp Porous Med 139, 467–489 (2021). https://doi.org/10.1007/s11242-021-01665-4

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