Abstract
We shall consider diffusion or single-phase flow in a multiscale porous medium which represents an infinite set of self-similar double-porosity media. At each scale, the medium consists of a highly permeable network of connected channels and low-permeable blocks. The characteristic scale of heterogeneity is ε at the highest level of hierarchy, wherein ε is a small parameter. The ratio between the channel and block permeability at each scale is ε 2. The process analyzed is described using a diffusion equation with an oscillating multiscale diffusion parameter. The macroscale behavior is of interest. The transition to the macroscale is performed by means of the two-scale homogenization procedure. One step of averaging at each level of hierarchy leads to the appearance of the memory terms in the averaged equation. The successive averaging steps lead to progressive memory accumulation, so at each step of averaging, the macroscale model changes its type, and even the result of the second step is unknown a priori. The objective was to determine the macroscopic limit model for the infinite number of scales. By the method of induction, we obtained the macroscale model for an arbitrary number of scales and its limit for the infinite hierarchy. The limit model represents the system of two equations with memory terms. The kernel of the memory operator is the solution of a nonlinear integro-differential equation. Its solution is obtained through Laplace transform.
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References
Adler, P.M.: Porous Media: Geometry and Transports, N.Y. USA. Butterworth-Heinemann (1992)
Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenisation. Proc. Roy. Soc. Edinburgh, Sect. A 126, 297–342 (1996)
Arbogast, T., Douglas, J., Hornung U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21(4), 823–836 (1990)
Auriault, J.L.: Effective macroscopic description for heat conduction in periodic composites. Int. J. Heat Mass Transfer 26(6), 861–869 (1983)
ben-Avraham, D., Havlin, S.: Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press (2000)
Barenblatt, G.I., Zheltov, Iu.P., Kochina, I.N.: Basic concept in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)
Bakhvalov, N.S., Panasenko, G.P.: Homogenization: Averaging Processes in Periodic Media. Kluwer, Dordrecht/Boston/London (1989)
Cushman, J.H.: The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. Kluwer Academic Publishers, Dordrtecht (1997)
De Swaan, A.O.: Analytic solutions for determining naturally fractured reservoir properties by well testing. SPE J. 16(3), 117–122 (1976)
Douglas, J., Arbogast, T., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21, 823–836 (1990)
Glimm, J., Sharp, D.H.: Stochastic methods for the prediction of complex multiscale phenomena. Q. J. Appl. Math. 56, 741–765 (1997)
Goncharenko, M., Amaziane, B., Pankratov, L.: Homogenization of a degenerate triple porosity model with thin fissures. Eur. J. Appl. Math. 16(03), 335–359 (2005)
de Hoog, F.R., Knight, J.H., Stokes, A.N.: An improved method for numerical inversion of Laplace transforms. SIAM J. Sci. Stat. Comput. 3, 357–366 (1982)
Lim, K.T., Aziz, K.: Matrix-Fracture transfer functions for dual porosity simulators. J. Pet. Sci. Eng. 13(3–4), 169–178 (1995)
Lions, J.L.: Computational Methods in Mathematics, Geophysics, and Optimum Control. Notes on Some Computational Aspects of Homogenization in Composite Materials, pp. 5–19. Nauka, Novosibirsk (1978)
Ozkan, E., Dreier, J., Kazemi, H.: New Analytical Pressure-Transient Models to Detect and Characterize Reservoir with Multiple Fracture System. SPE 92039, SPE International Petroleum Conference in Mexico (2004)
Paes-leme, P.J., Douglas, J., Kischinhevsky, M., Spagnuolo, A.M.: A multiple-porosity model for a single-phase flow through naturally-fractured porous media. Comput. Appl. Math. 1(17), 19–48 (1998)
Panfilov, M.: Main mode of porous flow in highly inhomogeneous media. Sov. Phys. Doclady, 35(3), 225–227 (1990)
Panfilov, M.B.: Macroscale Models of Flow through Highly Heterogeneous Porous Media. Kluwer Academic Publishers (2000)
Peszynska, M.L., Showalter, R.E.: Multiscale elliptic–parabolic systems for flow and transport. Electr. J. Differ. Equ. 147, 1–30 (2007)
Salimi, H., Bruining, J.: Upscaling in partially fractured oil reservoirs using homogenization. SPE J. 16(2), 273–293 (2011)
Sanchez-Palensia, E.: Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 129. Springer Verlag, Berlin (1980)
Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. SPE J. 3(3), 245–255; Trans. AIME, 228; SPE-426-PA (1963)
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Panfilov, M., Rasoulzadeh, M. Appearance of the nonlinearity from the nonlocality in diffusion through multiscale fractured porous media. Comput Geosci 17, 269–286 (2013). https://doi.org/10.1007/s10596-012-9338-7
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DOI: https://doi.org/10.1007/s10596-012-9338-7