Transport Equations for Fractured Porous Media

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.

[1]  Jacob Bear,et al.  Transport Phenomena in Porous Media — Basic Equations , 1984 .

[2]  J. V. Tracy,et al.  Flow through fractures , 1981 .

[3]  H. Tyrrell,et al.  Diffusion in liquids , 1984 .

[4]  J. Nunziato,et al.  On ideal multiphase mixtures with chemical reactions and diffusion , 1980 .

[5]  W. J. McGuire,et al.  The Effect of Vertical Fractures on Well Productivity , 1960 .

[6]  T. D. Streltsova.  Hydrodynamics of groundwater flow in a fractured formation , 1976 .

[7]  J. Rubin,et al.  Transport of chloride ion in a water-unsaturated soil exhibiting anion exclusion , 1986 .

[8]  Hossein Kazemi,et al.  Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution , 1969 .

[9]  Donald A. Drew,et al.  Averaged Field Equations for Two‐Phase Media , 1971 .

[10]  E. A. Sudicky,et al.  Contaminant transport in fractured porous media: Analytical solution for a single fracture , 1981 .

[11]  J. Andersson,et al.  Steady state fluid response in fractured rock: A boundary element solution for a coupled, discrete fracture continuum model , 1983 .

[12]  T. N. Narasimhan,et al.  Recent trends in hydrogeology , 1982 .

[13]  C. Truesdell,et al.  The Classical Field Theories , 1960 .

[14]  A. C. Gringarten,et al.  Flow-Test Evaluation of Fractured Reservoirs , 1982 .

[15]  John D. Ingram,et al.  A continuum theory of chemically reacting media—II Constitutive equations of reacting fluid mixtures☆ , 1967 .

[16]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[17]  I. Müller.  A thermodynamic theory of mixtures of fluids , 1968 .

[18]  R. M. Bowen.  Part I – Theory of Mixtures , 1976 .

[19]  Lincoln F. Elkins,et al.  Reservoir Performance and Well Spacing, Spraberry Trend Area Field of West Texas , 1953 .

[20]  Anders Rasmuson,et al.  An approach to modelling radionuclide migration in a medium with strongly varying velocity and block sizes along the flow path , 1984 .

[21]  J. E. Warren,et al.  The Effect of Horizontal Hydraulic Fracturing On Well Performance , 1961 .

[22]  J. Bear.  Hydraulics of Groundwater , 1979 .

[23]  H. J. Ramey,et al.  Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture , 1974 .

[24]  William G. Gray,et al.  General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. , 1980 .

[25]  J. A. Cherry,et al.  Hydrologic Characteristics and Response of Fractured Till and Clay Confining a Shallow Aquifer , 1975 .

[26]  Allen M. Shapiro,et al.  Simulation of steady-state flow in three-dimensional fracture networks using the boundary-element method , 1985 .

[27]  D. K. Babu,et al.  Flow to a Partially Penetrating Well in a Double‐Porosity Reservoir , 1984 .

[28]  Franklin W. Schwartz,et al.  A Stochastic Analysis of Macroscopic Dispersion in Fractured Media , 1983 .

[29]  Arnold Verruijt,et al.  Flow and transport in porous media , 1981 .

[30]  P. Witherspoon,et al.  Porous media equivalents for networks of discontinuous fractures , 1982 .

[31]  Lee A. Segel,et al.  Averaged Equations for Two-Phase Flows , 1971 .

[32]  T. D. Streltsova,et al.  UNSTEADY FLOW TO A PUMPED WELL IN A FISSURED WATER-BEARING FORMATION , 1977 .

[33]  Emil O. Frind,et al.  Contaminant Transport in Fractured Porous Media: Analytical Solution for a Two‐Member Decay Chain in a Single Fracture , 1984 .

[34]  J. E. Warren,et al.  The Behavior of Naturally Fractured Reservoirs , 1963 .

[35]  R. Scholten.  Stratigraphic Oil and Gas Fields. Robert E. King , 1975 .

[36]  H. J. Ramey,et al.  Unsteady-State Pressure Distributions Created by a Well With a Single Horizontal Fracture, Partial Penetration, or Restricted Entry , 1974 .

[37]  G. I. Barenblatt,et al.  Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata] , 1960 .

[38]  W. G. Gray,et al.  Finite elements in water resources. , 1982 .

[39]  William G. Gray,et al.  On the theorems for local volume averaging of multiphase systems , 1977 .

[40]  James O. Duguid,et al.  Flow in fractured porous media , 1977 .

[41]  D. T. Snow,et al.  Anisotropie Permeability of Fractured Media , 1969 .

[42]  Emil O. Frind,et al.  Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures , 1982 .

[43]  J. Oden,et al.  Finite Element Methods for Flow Problems , 2003 .

[44]  James W. Mercer,et al.  An efficient finite element technique for modeling transport in fractured porous media: 2. Nuclide decay chain transport , 1983 .

[45]  William G. Gray,et al.  General conservation equations for multi-phase systems: 1. Averaging procedure , 1979 .

[46]  Allen M. Shapiro,et al.  A Stochastic Model of a Fractured Rock Conditioned by Measured Information , 1984 .

[47]  Paul A. Witherspoon,et al.  Validity of cubic law for fluid flow in a deformable rock fracture. Technical information report No. 23 , 1979 .

[48]  John D. Ingram,et al.  A Continuum theory of chemically reacting media—I , 1965 .

[49]  D. G. Russell,et al.  Transient Pressure Behavior in Vertically Fractured Reservoirs , 1964 .

[50]  J. W. Mercer,et al.  An efficient finite element technique for modeling transport in fractured porous media: 1. Single species transport , 1983 .

[51]  R. E. Collins,et al.  Estimated Effect of Vertical Fractures on Secondary Recovery , 1954 .

[52]  P. B. Crawford,et al.  Effect of Isolated Vertical Fractures Existing in the Reservoir On Fluid Displacement Response , 1966 .

[53]  G. I. Barenblast,et al.  Fundamental Equations of Filtration of Homogeneous Liquids in Fissured Rocks , 1960 .

[54]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[55]  John F. Pickens,et al.  Solute transport through fractured media: 1. The effect of matrix diffusion , 1980 .

[56]  H. Kazemi,et al.  NUMERICAL SIMULATION OF WATER-OIL FLOW IN NATURALLY FRACTURED RESERVOIRS , 1976 .

[57]  F. Schwartz,et al.  An Analysis of the Influence of Fracture Geometry on Mass Transport in Fractured Media , 1984 .

[58]  T. D. Cook.  Underground Waste Management and Environmental Implications , 1972 .

[59]  T. N. Narasimhan,et al.  Multidimensional numerical simulation of fluid flow in fractured porous media , 1982 .

[60]  W. A. Beetem,et al.  Porosity and Dispersion Constant Calculations for a Fractured Carbonate Aquifer Using the Two Well Tracer Method , 1971 .

[61]  J. Noorishad,et al.  An upstream finite element method for solution of transient transport equation in fractured porous media , 1982 .

[62]  S. Whitaker.  Diffusion and dispersion in porous media , 1967 .

[63]  Jacob Bear,et al.  Flow through porous media , 1969 .

[64]  A. Cemal Eringen,et al.  Mechanics of continua , 1967 .

[65]  I. Liu,et al.  Thermodynamics of Mixtures of Fluids , 1984 .

[66]  M. King Hubbert,et al.  Mechanics of Hydraulic Fracturing , 1972 .

[67]  Paul A. Witherspoon,et al.  Steady state flow in rigid networks of fractures , 1974 .

[68]  Anders Rasmuson,et al.  Chemical transport in a fissured rock: Verification of a numerical model , 1982 .

[69]  Hossein Kazemi,et al.  SPE-002156-B - The Interpretation of Interference Tests in Naturally Fractured Reservoirs with Uniform Fracture Distribution , 1969 .

[70]  A. Bedford,et al.  A thermomechanical theory for reacting immiscible mixtures , 1980 .

[71]  R. Alverson,et al.  Mechanics Of Solids And Fluids , 1961 .

[72]  T. Narasimhan,et al.  A Purely Numerical Approach For Analyzing Fluid Flow To A Well Intercepting A Vertical Fracture , 1979 .

[73]  H. J. Ramey,et al.  Applied Pressure Analysis for Fractured Wells , 1975 .

[74]  William G. Gray,et al.  General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations , 1979 .

[75]  L. Király.  Groundwater flow in heterogeneous, anisotropic fractured media: A simple two-dimensional electric analog , 1971 .