Skip to main content
SpringerLink
Log in

Transport Equations for Fractured Porous Media

  • Chapter
Advances in Transport Phenomena in Porous Media

Part of the book series: NATO ASI Series ((NSSE,volume 128))

Abstract

With the advent of analyzing geological settings as possible sites for hazardous waste isolation, the modeling of transport phenomena in fractured rock has been a topic of increasing interest. In studies to date, the means by which transport phenomena in fractured rock have been mathematically visualized has taken two distinct routes. The need for different conceptualizations of fractured rock has arisen due to the diverse nature of fracturing in rock formations. Usually, the length scale of a given transport problem, in relation to the intensity of fracturing, varies from one rock formation to the next. In some instances, there may exist only a few significant fractures (of a given fracture family) over the length scale of the transport problem. In other situations, the length scale of the transport problem may encompass large numbers of interconnected fractures. These observations have led to conceptualizations of fractured rock as either a system of individual and possibly interconnected fractures in a permeable or impermeable host rock, or as one or more overlapping fluid continua, in a manner similar to the mathematical treatment of granular porous materials. The assumptions implicit in the use of each of these conceptualizations are discussed in this chapter along with a selective review of the recent literature. A detailed analysis of the discrete fracture and continuum conceptualizations of fractured rock is provided by developing the appropriate equations of mass, momentum and energy transport for each conceptualization.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 429.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 549.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersson, J., Shapiro, A.M., and Bear, J., “A Stochastic Model of a Fractured Rock Conditioned by Measured Information,” Water Resources Research, Vol. 20, 1984, pp. 79–88.

    Article  Google Scholar 

  2. Barenblatt, G.I., and Zheltov, Yu.P., “Fundamental Equations of Filtration of Homogeneous Liquids in Fissured Rocks,” Soviet Physics-Doklady, Vol. 5, 1960, pp. 522–525.

    MATH  Google Scholar 

  3. Barenblatt, G.I., Zheltov, Yu.P., and Kochina, I.N., “Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks,” Journal of Applied Mathematics and Mechanics, Vol. 24, 1960, pp. 1286–1303.

    Article  MATH  Google Scholar 

  4. Bear, J., Dynamics of Fluids in Porous Media, Elsevier, New York, N.Y., 1972.

    MATH  Google Scholar 

  5. Bear, J., Hydraulics of Groundwater, McGraw-Hill, New York, N.Y., 1979.

    Google Scholar 

  6. Bear, J., and Bachmat, Y., “Transport Phenomena in Porous Media-Basic Equations,” Fundamental of Transport Phenomena in Porous Media, J. Bear and M.Y.Corapcioglu, eds., Martinus Nijhoff Publishers, Dordrecht, 1984, pp. 3–61.

    Google Scholar 

  7. Bear, J., and Braester, C., “On the Flow of Two Immiscible Fluid in Fractured Porous Media,” Proceedings of the Second Symposium on Fundamentals of Transport Phenomena in Porous Media, University of Guelph, 1972, pp. 177–202.

    Google Scholar 

  8. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley and Sons, Inc., New York, N.Y., 1960.

    Google Scholar 

  9. Bokserman, A.A., Zheltov, Yu.P., and A.A. Kocheshkov, “Motion of Immiscible Liquids in a Cracked Porous Medium,” Soviet Physics-Doklady, Vol. 9, 1964, pp. 285–287.

    Google Scholar 

  10. Boulton, A.S., and Streltsova, T.D., “Unsteady Flow to a Pumped Well in a Fissured Water Bearing Formation,” Journal of Hydrology, Vol. 35, 1977, pp. 257–269.

    Article  Google Scholar 

  11. Bowen, R.M., “Theory of Mixtures,” Continuum Physics, A.C. Eringen, ed., Vol. III, Academic Press Inc., New York, N.Y., 1976, pp. 2–127.

    Google Scholar 

  12. Braester, C., “A Shock Wave in Immiscible Displacement in a Fissured Porous Medium,” Israel Journal of Technology, Vol. 9, 1971, pp. 433–438.

    Google Scholar 

  13. Brown, D.M., “Stochastic Analysis of Flow and Solute Transport in a Variable-Aperture Rock Fracture,” thesis presented to the Massachusetts Institute of Technology, Cambridge, Massachusetts, in 1984, in partial fulfillment of the requirements for the degree of Master of Science.

    Google Scholar 

  14. Chu, Y., and Gelhar, L.W., “Turbulent Pipe Flow With Granular Permeable Boundaries,” Report No. 148, Dept. of Civil Engineering, Massachusetts Institute of Technology, 1972.

    Google Scholar 

  15. Coleman, B.D., and Noll, W., “Thermodynamics of Elastic Materials With Heat Conduction and Viscosity,” Archive for Rational Mechanics and Analysis, Vol. 13, 1963, pp. 167–178.

    Article  MathSciNet  MATH  Google Scholar 

  16. Crawford, P.B., and Collins, R.E., “Estimated Effect of Vertical Fractures on Secondary Recovery,” Transactions, American Institute of Mining Engineers, Vol. 201, 1954, pp. 192–196.

    Google Scholar 

  17. Davis, S.N., “Porosity and Permeability of Natural Materials,” Flow Through Porous Media, R.J.M. DeWiest, ed., Academic Press, New York, N.Y., 1969, pp. 54–89.

    Google Scholar 

  18. de Swann O.A., “Analytical Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing,” Transactions, American Institute of Mining Engineers, Vol. 261, 1976, pp. 117–122.

    Google Scholar 

  19. Dougherty, D.E., “On Equivalent Porous Media Modeling of Transport in Fractured Porous Reservoirs,” thesis presented to Princeton University, Princeton, N.J., in 1985, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

    Google Scholar 

  20. Dougherty, D.E., and Babu, D.K., “Flow to a Partially Penetrating Well in a Double-Porosity Reservoir,” Water Resources Research, Vol. 20, 1984, pp. 1116–1122.

    Article  Google Scholar 

  21. Drew, D.A., “Averaged Field Equations for Two-Phase Media,” Studies in Applied Mathematics, Vol. 50, 1971, pp. 133–166.

    MATH  Google Scholar 

  22. Drew, D.A., and Segel, L.A., “Averaged Equations for Two-Phase Flows,” Studies in Applied Mathematics, Vol. 50, 1971, pp. 205–231.

    MATH  Google Scholar 

  23. Drumheller, D.S., and Bedford, A., “A Thermomechanical Theory for Reacting Immiscible Mixtures,” Archive for Rational Mechanics and Analysis, Vol. 73, 1980, pp. 257–284.

    Article  MathSciNet  MATH  Google Scholar 

  24. Duguid, J.O., and Lee, P.C.Y., “Flow in Fractured Porous Media,” Water Resources Research, Vol. 13, 1977, pp. 558–566.

    Article  Google Scholar 

  25. Elkins, L.F., “Reservoir Performance and Well Spacing, Spraberry Trend Area Field of West Texas,” Transactions, American Institute of Mining Engineers, Vol. 198, 1953, pp. 177–196.

    Google Scholar 

  26. Elkins, L.F., and Skov, A.M., “Determination of Fracture Orientation From Pressure Interference,” Transactions, American Institute of Mining Engineers, Vol. 219, 1960, pp. 301–304.

    Google Scholar 

  27. Eringen, A.C., Mechanics of Continua, John Wiley and Sons, Inc., New York, N.Y., 1967.

    MATH  Google Scholar 

  28. Gale, J.E., Taylor, R.L., Witherspoon, P.A., and Ayatollahi, M.S., “Flow in Rocks with Deformable Fractures,” Finite Element Methods in Flow Problems, J.T. Oden, O.C. Zienkiewicz, R.H. Gallagher, C. Taylor, eds., University of Alabama Press, Huntsville, Alabama, 1974, pp. 583–598.

    Google Scholar 

  29. Givens, J.W., and Crawford, P.B., “Effect of Isolated Vertical Fractures Existing in the Reservoir on Fluid Displacement Response,” Transactions, American Institute of Mining Engineers, Vol. 237, 1966, pp. 81–86.

    Google Scholar 

  30. Gray, W.G., and Lee, P.C.Y., “On the Theorems for Local Volume Averaging of Multi-Phase Systems,” International Journal of Multi-Phase Flow, Vol. 3, 1977, pp. 333–340.

    Article  MATH  Google Scholar 

  31. Gringarten, A.C., “Flow Test Evaluation of Fractured Reservoirs,” Recent Trends in Hydrogeology, Geological Society of America Special Paper 189, T.N. Narasimhan, ed., 1982, pp. 237–263.

    Google Scholar 

  32. Gringarten, A.C., and Ramey, H.J., “Unsteady-State Pressure Distribution Created by a Well With a Single Horizontal Fracture, Partial Penetration or Restricted Entry,” Transactions, American Institute of Mining Engineers, Vol. 257, 1974, pp. 413–426.

    Google Scholar 

  33. Gringarten, A.C., Ramey, H.J., and Raghavan, R., “Unsteady-State Pressure Distribution Created by a Well With a Single Infinite-Conductivity Vertical Fracture,” Transactions, American Institute of Mining Engineers, Vol. 257, 1974, pp. 347–360.

    Google Scholar 

  34. Gringarten, A.C., Ramey, H.J., and Raghavan, R., “Applied Pressure Analysis for Fractured Wells,” Transactions, American Institute of Mining Engineers, Vol. 259, 1975, pp. 887–892.

    Google Scholar 

  35. Grisak, G.E., and Cherry, J.A., “Hydrologic Characteristics and Response of Fractured Till and Clay Confining a Shallow Aquifer,” Canadian Geotechnical Journal, Vol. 12, 1975, pp. 23–43.

    Article  Google Scholar 

  36. Grisak, G.E., and Pickens, J.F., “Solute Transport Through Fractured Media: 1. The Effect of Matrix Diffusion,” Water Resources Research, Vol. 16, 1980, pp. 719–730.

    Article  Google Scholar 

  37. Grove, D.B., and Beetem, W.A., “Porosity and Dispersion Constant Calculations for a Fractured Carbonate Aquifer Using Two Well Tracer Method,” Water Resources Research, Vol. 7, 1971, pp. 128–134.

    Article  Google Scholar 

  38. Hartsock, J.H., and Warren, J.E., “The Effect of Horizontal Hydraulic Fracturing on Well Performance,” Journal of Petroleum Technology, Vol. 13, 1961, pp. 1050–1056.

    Article  Google Scholar 

  39. Hassanizadeh, M., “Macroscopic Description of Multi-Phase Systems-A Thermodynamic Theory of Flow in Porous Media,” thesis presented to Princeton University, Princeton, N.J., in 1980, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

    Google Scholar 

  40. Hassanizadeh M., and Gray, W.G., “General Conservation Equations for Multi-Phase Systems 1. Averaging Procedure,” Advances in Water Resources, Vol. 2, 1979, pp. 131–144.

    Article  Google Scholar 

  41. Hassanizadeh, M., and Gray, W.G., “General Conservation Equations for Multi-Phase Systems 2. Mass, Momenta, Energy and Entrophy Equations,” Advances in Water Resources, Vol. 2, 1979, pp. 191–203.

    Article  Google Scholar 

  42. Hassanizadeh, M., and Gray, W.G., “General Conservation Equations for Multi-Phase Systems 3. Constitutive Theory for Porous Media Flow,” Advances in Water Resources, Vol.3, 1980, pp. 25–40.

    Article  Google Scholar 

  43. Hubbert, M.K., and Willis, D.G., “Mechanics of Hydraulic Fracturing,” Underground Waste Management and Environmental Implications, American Association of Petroleum Geologists Memoir 18, 1972, pp. 239–257.

    Google Scholar 

  44. Huyakorn, P.S., Lester, B.H., and Mercer, J.W., “An Efficient Finite Element Technique for Modeling Transport in Fractured Porous Media: 1. Single Species Transport,” Water Resources Research, Vol. 19, 1983, pp. 841–854.

    Article  Google Scholar 

  45. Huyakorn, P.S., Lester, B.H., and Mercer, J.W., “An Efficient Finite Element Technique for Modeling Transport in Fractured Porous Media: 2. Nuclide Decay Chain Transport,” Water Resources Research, Vol. 19, 1983, pp. 1286–1296.

    Article  Google Scholar 

  46. Ingram, J.D., and Eringen, A.C., “A Continuum Theory of Chemically Reacting Media-II. Constitutive Equations of Reacting Fluid Mixtures,” International Journal of Engineering Science, Vol. 5, 1967, pp. 289–322.

    Article  Google Scholar 

  47. James, R.V., and Rubin, J., “Transport of Chloride in a Water-Unsaturated Soil Exhibiting Anion Exclusion,” presented at the December 3–7, 1984, American Geophysical Union Fall Meeting, San Francisco, Califormia.

    Google Scholar 

  48. Kazemi, H., “Pressure Transient Analysis of Naturally Fractured Reservoirs with Uniform Fracture Distribution,” Society of Petroleum Engineers Journal, Vol. 9, 1969, pp. 451–462.

    Google Scholar 

  49. Kazemi, H., Seth, M.S., and Thomas, G.W., “The Interpretation of Interference Tests in Naturally Fractured Reservoirs with Uniform Fracture Distribution,” Society of Petroleum Engineers Journal, Vol. 9, 1969, pp. 463–472.

    Google Scholar 

  50. Kazemi, H., Merrill, L.S., Potterfield, K.L., and Zelman, P.R., “Numerical Simulation of Water-Oil Flow in Naturally Fractured Reservoirs,” Transactions, American Institute of Mining Engineers, Vol. 261, 1976, pp. 317–326.

    Google Scholar 

  51. Kiraly, L., “Groundwater Flow in Heterogeneous, Anisotropic Fractured Media: A Simple Two-Dimensional Electric Analog,” Journal of Hydrology, 1971, pp. 255–261.

    Google Scholar 

  52. Lewis, D.C., and Burgy, R.H., “Hydraulic Characteristics of Fractured and Jointed Rock,” Groundwater, Vol. 2, 1964, pp. 4–9.

    Google Scholar 

  53. Lightfoot, E.N., and Cussler, E.L., “Diffusion in Liquids,” Selected Topics in Transport Phenomena, Chemical Engineering Progress Symposium Series, American Institute of Chemical Engineers, No. 58, Vol. 61, 1965.

    Google Scholar 

  54. Long, R.R., Mechanics of Solids and Fluids, Prentice Hall, Englewood Cliffs, N.J., 1961.

    MATH  Google Scholar 

  55. Long, J.C.S., Remer, J.S., Wilson, C.R., and Witherspoon, P.A., “Porous Media Equivalents for Networks of Discontinuous Fractures,” Water Resources Research, Vol. 18, 1982, pp. 645–658.

    Article  Google Scholar 

  56. McGuire, W.J., and Sikora, V.J., “The Effect of Vertical Fractures on Well Productivity,” Transactions, American Institute of Mining Engineers, Vol. 219, 1960, pp. 401–403.

    Google Scholar 

  57. Muller, I., “A Thermodynamic Theory of Mixtures of Fluids,” Archive for Rational Mechanics and Analysis, Vol. 28, 1968, pp. 1–39.

    Article  MathSciNet  Google Scholar 

  58. Muller, I., “Thermodynamics of Mixtures of Fluids,” Journal de Mécanique, Vol. 14, 1975, pp. 267–303.

    Google Scholar 

  59. Munoz Görna, R.J., and Gelhar, L.W., “Turbulent Pipe Flow With Rough Porous Walls,” Report No. 109, Dept. of Civil Engineering, Massachusetts Institute of Technology, 1968.

    Google Scholar 

  60. Narasimhan, T.N., “Multidimensional Numerical Simulation of Fluid Flow in Fractured Porous Media,” Water Resources Research, Vol. 18, 1982, pp. 1235–1247.

    Article  Google Scholar 

  61. Narasimhan, T.N., and Palen, W.A., “A Purely Numerical Approach for Analyzing Fluid Flow to a Well Intercepting a Vertical Fracture,” presented at the 1979, AIME California Regional Meeting of the Society of Petroleum Engineers, held at Ventura, California.

    Google Scholar 

  62. Neretnieks, I. and Rasmuson, A., “An Approach to Modeling Radionuclide Migration in a Medium with Strongly Varying Velocity and Block Sizes Along the Flow Path,” Water Resources Research, Vol. 20, 1984, pp. 1823–1836.

    Article  Google Scholar 

  63. Neuzil, C.E., and Tracy, J.V., “Flow in Fractures,” Water Resources Research, Vol. 17, 1981, pp. 191–199.

    Article  Google Scholar 

  64. Noorishad, J., and Mehran, M., “An Upstream Finite Element Method for Solution of Transient Transport Equation in Fractured Porous Media,” Water Resources Research, Vol. 18, 1982, pp. 588–596.

    Article  Google Scholar 

  65. Nunziato, J.W., and Walsh, E.K., “On Ideal Multi-Phase Mixtures With Chemical Reactions and Diffusion,” Archive For Rational Mechanics and Analysis, Vol. 73, 1980, pp. 285–311.

    Article  MathSciNet  MATH  Google Scholar 

  66. O’Neill, K., “The Transient Three-Dimensional Transport of Liquid and Heat in Fractured Porous Media,” thesis presented to Princeton University, Princeton, N.J., in 1977, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

    Google Scholar 

  67. Rasmuson, A., Narasimhan, T.N., and Neretnieks, I., “Chemical Transport in Fissured Rock; Verification of a Numerical Model,” Water Resources Research, Vol. 18, 1982, pp. 1479–1492.

    Article  Google Scholar 

  68. Rüssel, D.G., and Truitt, N.E., “Transient Pressure Behavior in Vertically Fractured Reservoirs,” Transactions, American Institute of Mining Engineers, Vol. 231, 1964, pp. 1159–1170.

    Google Scholar 

  69. Schwartz, F.W., Smith, L., and Crowe, A.S., “A Stochastic Analysis of Macroscopic Dispersion in Fractured Media,” Water Resources Research, Vol. 19, 1983, pp. 1253–1265.

    Article  Google Scholar 

  70. Shapiro, A.M., “Fractured Porous Media: Equation Development and Parameter Identification,” thesis presented to Princeton University, Princeton, N.J., in 1981, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

    Google Scholar 

  71. Shapiro, A.M., “An Alternative Formulation for Hydrodynamic Dispersion in Porous Media,” Flow and Transport in Porous Media, A. Verruijt and F.B.J. Barends, eds., A.A. Balkema, Rotterdam, 1981, pp. 203–207.

    Google Scholar 

  72. Shapiro, A.M., and Andersson, J., “Finite Element Simulation of Contaminant Transport in Fractured Rock Near Karlshamn, Sweden,” Finite Elements in Water Resources. K.P. Holz, U. Meissner, W. Zielke, C.A. Brebbia, G. Pinder, W. Gray, eds., Springer-Verlag, Berlin, 1982, pp. 12.11–12.20.

    Google Scholar 

  73. Shapiro, A.M., and Andersson, J., “Steady-State Fluid Response in Fractured Rock: A Boundary Element Solution for a Coupled Discrete Fracture-Continuum Model,” Water Resources Research, Vol. 19, 1983, pp. 959–969.

    Article  Google Scholar 

  74. Shapiro, A.M., and Andersson, J., “Simulation of Steady-State Flow in Three-Dimensional Fracture Networks Using the Boundary Element Method,” Finite Elements in Water Resources, J.P. Laible, CA. Brebbia, W. Gray, G. Pinder, eds., Springer-Verlag, 1984, pp. 713–722.

    Google Scholar 

  75. Shapiro, A.M., and Bear, J., “Evaluating the Hydraulic Conductivity of Fractured Rock From Information on Fracture Geometry,” presented at the January 7–10, 1985, International Association of Hydrogeologists 17th International Congress on Hydrogeology of Rocks of Low Permeability, held at Tucson, Arizona.

    Google Scholar 

  76. Smith, L. and Schwartz, F.W., “An Analysis of the Influence of Fracture Geometry on Mass Transport in Fractured Media,” Water Resources Research, Vol. 20, 1984, pp. 1241–1252.

    Article  Google Scholar 

  77. Snow, D.T., “Anisotropic Permeability of Fractured Media,” Water Resources Research, Vol. 5, 1969, pp. 1273–1289.

    Article  Google Scholar 

  78. Stearns, D.W., and Friedman, M., “Reservoirs in Fractured Rock,” Stratigraphic Oil and Gas Fields, American Association of Petroleum Geologists Memoir 16, 1972, pp. 82–106.

    Google Scholar 

  79. Streltsova, T.D., “Hydrodynamics of Groundwater Flow in a Fractured Formation,” Water Resources Research, Vol. 12, 1976, pp. 405–414.

    Article  Google Scholar 

  80. Sudicky, E.A., and Frind, E.O., “Contaminant Transport in Fractured Porous Media: Analytical Solutions for a System of Parallel Fractures,” Water Resources Research, Vol. 18, 1982, pp. 1634–1642.

    Article  Google Scholar 

  81. Sudicky, E.A., and Frind, E.O., “Contaminant Transport in Fractured Porous Media: Analytical Solutions for a Two-Member Decay Chain in a Single Fracture,” Water Resources Research, Vol. 20, 1984, pp. 1021–1029.

    Article  Google Scholar 

  82. Tang, D.H., Frind, E.O., and Sudicky, E.A., “Contaminant Transport in Fractured Porous Media: Analytical Solution for a Single Fracture,” Water Resources Research, Vol. 17, 1981, pp. 555–564.

    Article  Google Scholar 

  83. Truesdell, C., and Toupin, R.A., “The Classical Field Theories,” Handbuch der Physik, Vol. III/l, Principles of Classical Mechanics and Field Theory, S. Flugge, ed., Springer-Verlag, Berlin, 1960, pp. 226–793.

    Google Scholar 

  84. Uldrich, D.O., and Ershaghi, I., “A Method for Estimating the Interporosity Flow Parameter in Naturally Fractured Reservoirs,” Society of Petroleum Engineers Journal, Vol. 19, 1979, pp. 324–332.

    Article  Google Scholar 

  85. Verma, A.P., “Imbibition in Flow of Two Immiscible Liquids Through a Cracked Porous Medium with Small Viscosity Difference, Proceedings of the Second Symposium on Fundamentals of Transport Phenomena in Porous Media, University of Guelph, 1972, pp. 393–402.

    Google Scholar 

  86. Wang, J.S.Y., Narasimhan, T.N., Tsang, G.F., and Witherspoon, P.A., “Transient Flow in Tight Fractures,” Report No. LBL-7026, Lawrence Berkeley Laboratory, Berkeley, California, 1977.

    Google Scholar 

  87. Warren, J.E., and Root, P.J., “The Behavior of Naturally Fractured Reservoirs,” Transactions, American Institute of Mining Engineers, Vol. 228, 1963, pp. 245–255.

    Google Scholar 

  88. Whitaker, S., “Diffusion and Dispersion in Porous Media,” American Institute of Chemical Engineers Journal, Vol. 13, 1967, pp. 420–427.

    Google Scholar 

  89. Whitehead, R.E., and Davis, R.T., “Surface Conditions in Slip Flow with Mass Transfer,” College of Engineering Report, Virginia Polytechnic Institute, 1969.

    Google Scholar 

  90. Wilson, CR., and Witherspoon, P.A., “Steady-State Flow in Rigid Networks of Fractures,” Water Resources Research, Vol. 10, 1974, pp. 328–335.

    Article  Google Scholar 

  91. Witherspoon, P.A., Wang, J.S.Y., Iwai, K., and Gale, J., “Validity of the Cubic Law for Fluid Flow in a Deformable Rock Fracture,” Technical Report 23, Lawrence Berkeley Laboratory, Berkeley, California, 1979.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Water Resources Division, U.S. Geological Survey, Reston, Virginia, 22092, USA

    Allen M. Shapiro

Authors
  1. Allen M. Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Civil Engineering, Technion-Israel Institute of Technology, Technion City, Haifa, 32000, Israel

    Jacob Bear

  2. Department of Civil Engineering, City College of New York, New York, NY, 10031, USA

    M. Yavuz Corapcioglu

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Martinus Nijhoff Publishers, Dordrecht

About this chapter

Cite this chapter

Shapiro, A.M. (1987). Transport Equations for Fractured Porous Media. In: Bear, J., Corapcioglu, M.Y. (eds) Advances in Transport Phenomena in Porous Media. NATO ASI Series, vol 128. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3625-6_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-3625-6_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8121-4

  • Online ISBN: 978-94-009-3625-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation