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Distributed Microstructure Models of Porous Media

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Flow in Porous Media

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 114))

Abstract

Laminar flow through fissured or otherwise highly inhomogeneous media leads to very singular initial-boundary-value problems for equations with rapidly oscillating coefficients. The limiting case (by homogenization) is a continuous distribution of model cells which represent a valid approximation of the finite (singular) case, and we survey some recent results on the theory of such systems. This is developed as an application of continuous direct sums of Banach spaces which arise rather naturally as the energy or state spaces for the corresponding (stationary) variational or (temporal) dynamic problems. We discuss the basic models for a totally fissured medium, the extension to include secondary flux in partially fissured media, and the classical model systems which are realized as limiting cases of the microstructure models.

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References

  1. Anzelius A. Über Erwärmung vermittels durchströmender Medien. Zeit. Ang. Math. Mech., 6:291–294, 1926.

    Article  Google Scholar 

  2. Arbogast T. Gravitational forces in dual-porosity models of single phase flow. to appear.

    Google Scholar 

  3. Arbogast T. The double porosity model for single phase flow in naturally fractured reservoirs. In Numerical Simulation in Oil Recovery, The IMA Volumes in Mathematics and its Applications 11, pages 23–45. Springer-Verlag, Berlin and New York, 1988. M. F. Wheeler, ed.

    Chapter  Google Scholar 

  4. Arbogast T. Analysis of the simulation of single phase flow through a naturally fractured reservoir. SIAM J. Numer. Anal., 26:12–29, 1989.

    Article  Google Scholar 

  5. Arbogast T., Douglas J., Hornung U. Modeling of naturally fractured petroleum reservoirs by formal homogenization techniques. In Frontiers in Pure and Applied Mathematics, pages 1–19. Elsevier, Amsterdam, 1991. R. Dautray, ed.

    Google Scholar 

  6. Arbogast T., Douglas J. Jr., Hornung U. Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. of Math. Anal, 21:823–836, 1990.

    Article  Google Scholar 

  7. Barenblatt G. I., Zheltov I. P., Kochina I. N. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math, and Mech., 24:1286–1303, 1960.

    Article  Google Scholar 

  8. Barker J. A. Block-geometry functions characterizing transport in densely fissured media. J. of Hydrology, 77:263–279, 1985.

    Article  Google Scholar 

  9. Charlaix E., Hulin J. P., Plona T. J. Experimental study of tracer dispersion in sintered glass porous materials of variable compation. Phys. Fluids, 30:1690–1698, 1987.

    Article  Google Scholar 

  10. Clark G. W., Showalter R. E. Fluid flow in a layered medium. Quarterly Appl. Math. to appear.

    Google Scholar 

  11. Coats K. H., Smith B. D. Dead-end pore volume and dispersion in porous media. Trans. Soc. Petr. Eng., 231:73–84, 1964.

    Google Scholar 

  12. Cook J. D., Showalter R. E. Microstructure diffusion models with secondary flux. Preprint.

    Google Scholar 

  13. Deans H. A. A mathematical model for dispersion in the direction of flow in porous media. Trans. Soc. Petr. Eng., 228:49–52, 1963.

    Google Scholar 

  14. DiBenedetto E., Showalter R. E. A free-boundary problem for a degenerate parabolic system. Jour. Diff. Eqn., 50:1–19, 1983.

    Article  Google Scholar 

  15. Diesler P. F. Jr., Wilhelm R. H. Diffusion in beds of porous solids: measurement by frequency response techniques. Ind. Eng. Chem., 45:1219–1227, 1953.

    Article  Google Scholar 

  16. Douglas J., Paes Leme P. J., Arbogast T., Schmitt T. Simulation of flow in naturally fractured reservoirs. In Proceedings, Ninth SPE Symposium on Reservoir Simulation, pages 271–279, Dallas, Texas, 1987. Society of Petroleum Engineers. Paper SPE 16019.

    Google Scholar 

  17. Friedman A., Knabner P. A transport model with micro-and macro-structure. Preprint.

    Google Scholar 

  18. Friedman A., Tzavaras A. A quasilinear parabolic system arising in modelling of catalytic reactors. Jour. Diff. Eqns., 70:167–196, 1987.

    Article  Google Scholar 

  19. Hornung U. Applications of the homogenization method to flow and transport through porous media. In Flow and Transport in Porous Media, Singapore. Summer School, Beijing 1988, World Scientific. Xiao Shutie, ed., to appear.

    Google Scholar 

  20. Hornung U. Miscible displacement in porous media influenced by mobile and immobile water. In Nonlinear Partial Differential Equations. Springer, New York, 1988. P. Fife and P. Bates, eds.

    Google Scholar 

  21. Hornung U. Homogenization of Miscible Displacement in Unsaturated Aggregated Soils. In Progress in Nonlinear Differential Equations and Their Applications, pages 143–153, Boston, 1991. Composite Media and Homogenization Theory, ICTP, Triest 1990, Birkhäuser. G. Dal Mase and G. F. Dell’Antonio, eds.

    Google Scholar 

  22. Hornung U. Miscible displacement in porous media influenced by mobile and immobile water. Rocky Mtn. Jour. Math., 21:645–669, 1991. Corr. pages 1153–1158.

    Article  Google Scholar 

  23. Hornung U. Modellierung von Stofftransport in aggregierten und geklüftten Böden. Wasserwirtschaft, 1, 1992. to appear.

    Google Scholar 

  24. Hornung IL, Jäger, W. Homogenization of reactive transport through porous media. In EQUADIFF 1991, Singapore. World Scientific Publishing. C. Perellö, ed., submitted 1992.

    Google Scholar 

  25. Hornung U., Jäger W. A model for chemical reactions in porous media. In Complex Chemical Reaction Systems. Mathematical Modeling and Simulation, volume 47 of Chemical Physics, pages 318–334. Springer, Berlin, 1987. J. Warnatz and W. Jäger, eds.

    Google Scholar 

  26. Hornung U., Jäger W. Diffusion, convection, adsorption, and reaction of chemicals in porous media. J. Diff. Equations, 92:199–225, 1991.

    Article  Google Scholar 

  27. Hornung U., Jäger W., Mikelić A. Reactive transport through an array of cells with semi-permeable membranes. In preparation.

    Google Scholar 

  28. Hornung U., Showalter R. E. Diffusion models for fractured media. Jour. Math. Anal. Appl, 147:69–80, 1990.

    Article  Google Scholar 

  29. Knabner P. Mathematishche modelle für den transport gelöster Stoffe in sorbierenden porösen medien. Technical Report 121, Univ. Augsburg, 1989.

    Google Scholar 

  30. Lindstrom F. T., Narasimham M. N. L. Mathematical theory of a kinetic model for dispersion of previously distributed chemicals in a sorbing porous medium. SIAM J. Appl. Math., 24:496–510, 1973.

    Article  Google Scholar 

  31. Lowan A. N. On the problem of the heat recuperator. Phil. Mag., 17:914–933, 1934.

    Google Scholar 

  32. Miller R. K. An integrodifferential equation for rigid heat conductors with memory. J. Math. Anal. Appl, 66:313–332, 1978.

    Article  Google Scholar 

  33. Nunziato J. W. On heat conduction in materials with memory. Quarterly Appl. Math., 29:187–204, 1971.

    Google Scholar 

  34. Packer L., Showalter R. E. Distributed capacitance microstructure in conductors. To appear in Applicable Analysis.

    Google Scholar 

  35. Peszenska M. Mathematical analysis and numerical approach to flow through fissured media. PhD thesis, Univ. Augsburg, May 1992.

    Google Scholar 

  36. Rosen J. B. Kinetics of a fixed bed system for solid diffusion into spherical particles. J. Chem. Phys., 20:387–394, 1952.

    Article  Google Scholar 

  37. Rosen J. B., Winshe W. E. The admittance concept in the kinetics of chromatography. J. Chem. Phys., 18:1587–1592, 1950.

    Article  Google Scholar 

  38. Rubinstein L.I. On the problem of the process of propagation of heat in heterogeneous media. Izv. Akad. Nauk SSSR, Ser. Geogr., 1, 1948.

    Google Scholar 

  39. Rulla J., Showalter R. E. Diffusion in partially fissured media and implicit evolution equations. In preparation.

    Google Scholar 

  40. Showalter R. E. Implicit evolution equations. In Differential Equations and Applications, Vol. II, pages 404–411, Columbus, OH, 1989. International Conf. on Theory and Applications of Differential Equations, University Press, Athens.

    Google Scholar 

  41. Showalter R. E. Diffusion models with micro-structure. Transport in Porous Media, 6:567–580, 1991.

    Article  Google Scholar 

  42. Showalter R. E. Diffusion in a Fissured Medium with Micro-Structure. In Free Boundary Problems in Fluid Flow with Applications, volume 282 of Pitman Research Notes in Mathematics, pages 136–141. Longman, 1993. J. M. Chadam and H. Rasmussen, eds.

    Google Scholar 

  43. Showalter R. E., Walkington N. J. A diffusion system for fluid in fractured media. Differential and Integral Eqns., 3:219–236, 1990.

    Google Scholar 

  44. Showalter R. E., Walkington N. J. Diffusion of fluid in a fissured medium with micro-structure. SI AM Jour. Math. Anal, 22:1702–1722, 1991.

    Article  Google Scholar 

  45. Showalter R. E., Walkington N. J. Micro-structure models of diffusion in fissured media. Jour. Math. Anal. Appl, 155:1–20, 1991.

    Article  Google Scholar 

  46. Showalter R. E., Walkington N. J. Elliptic systems for a medium with micro-structure. In Geometric Inequalities and Convex Bodies, pages 91–104. Marcel Dekker, New York, 1993. I. J. Bakelman, ed.

    Google Scholar 

  47. van Duijn C. J., Knabner P. Solute transport through porous media with slow adsorption. to appear.

    Google Scholar 

  48. van Genuchten M. Th., Wierenga R J. Mass transfer studies in sorbing porous media I. Analytical Solutions. Soil Sei. Soc. Amer. J., 40:473–480, 1976.

    Article  Google Scholar 

  49. Vogt Ch. A homogenization theorem leading to a Volterra integro-differential equation for permeation chromotography. Preprint #155, Sonderfachbereich 123, 1982.

    Google Scholar 

  50. Warren J. E., Root P. J. The behavior of naturally fractured reservoirs. Soc. Petr. Eng. J., 3:245–255, 1963.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, Texas, 78712-1082, USA

    Ralph E. Showalter

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  1. Ralph E. Showalter
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Editors and Affiliations

  1. Dept. of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Jim Douglas Jr.

  2. Fakultät für Informatik, Uni BwM, D-85577, Neubiberg, Germany

    Ulrich Hornung

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© 1993 Springer Basel AG

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Showalter, R.E. (1993). Distributed Microstructure Models of Porous Media. In: Douglas, J., Hornung, U. (eds) Flow in Porous Media. ISNM International Series of Numerical Mathematics, vol 114. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8564-5_14

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  • DOI: https://doi.org/10.1007/978-3-0348-8564-5_14

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9682-5

  • Online ISBN: 978-3-0348-8564-5

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