dynamics of f luids in porous media

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1 dynamics of f luids in porous media Jacob Bear Department of Civil Engineering Technion Israel Institute of Technology, Haifa DOVER PUBLICATIONS, INC. New York

2 Contents Preface xvii CHAPTER 1 Introduction Aquifers, Ground Water and Oil Reservoirs Definitions The Moisture Distribution in a Vertical Profile Classification of Aquifers Properties of Aquifers The Oil Reservoir The Porous Medium The Continuum Approach to Porous Media The Molecular and Microscopic Levels Porosity and Representative Elementary Volume Areal and Linear Porosities Velocity and Specific Discharge Concluding Remarks 24 CHAPTER 2 Fluids and Porous Matrix Properties Fluid Density Definitions Mixture of Fluids Measurement of Density Fluid Viscosity Definition Non-Newtonian Fluids Units Effect of Pressure and Temperature Measurement of Viscosity Fluid Compressibility Statistical Description of Porous Media Particle-Size Distribution Pore-Size Distribution Other Statistical Descriptions 42

2.5 Porosity 43 2.5.1 Porosity and Effective Porosity 43 2.5.2 Porosity, Structure and Packing 45 2.5.3 Porosity Measurement 47 2.6 Specific Surface 50 2.6.1 Definitions 50 2.6.2 Measurement of Specific Surface 51 2.

3 2.5 Porosity Porosity and Effective Porosity Porosity, Structure and Packing Porosity Measurement Specific Surface Definitions Measurement of Specific Surface Matrix and Medium Compressibility 52 CHAPTER 3 Pressure and Piezometric Head Stress at a Point Hydrostatic Pressure Distribution Piezometric Head 63 CHAPTER 4 The Fundamental Fluid Transport Equations in Porous Media Particles, Velocities and Fluxes in a Fluid Continuum 65 ' Definitions of Particles and Velocities Diffusive Velocities and Fluxes The Eulerian and Lagrangian Points of View The Substantial Derivative The General Conservation Principle Equations of Mass, Momentum and Energy Conservation in a Fluid Continuum Mass Conservation of a Species Mass Conservation of a Fluid System Conservation of Linear Momentum of a Species <x Conservation of Linear Momentum of a Fluid System Constitutive Assumptions and Coupled Processes General Considerations Principles to be Used in Forming Constitutive Equations Coupled Processes A Porous Medium Model The Conceptual Model Approach A Model of Flow Through a Porous Medium Frames of Reference An Averaging Procedure Equations of Volume and Mass Conservation Equation of Volume Conservation Equation of Mass Conservation of a Species in Solution Equation of Mass Conservation Equation of Motion 104

4.8 Tortuosity and Permeability 106 4.8.1 Relationship Between Tortuosity and Permeability 106 4.8.2 Tortuosity and Other Transport Coefficients 112 4.8.3 Formation Factor and Resistivity Index (Amyx 1960) in Reservoir Engineering 113 CHAPTER 5 The Equation of Motion of a Homogeneous Fluid 119 5.

4 4.8 Tortuosity and Permeability Relationship Between Tortuosity and Permeability Tortuosity and Other Transport Coefficients Formation Factor and Resistivity Index (Amyx 1960) in Reservoir Engineering 113 CHAPTER 5 The Equation of Motion of a Homogeneous Fluid The Experimental Law of Darcy Generalization of Darcy's Law Isotropie Medium Anisotropie Medium Deviations from Darcy's Law The Upper Limit The Lower Limit The Slip Phenomenon Rotational and Irrotational Motion The Potential and Pseudopotential Irrotational Flow Hydraulic Conductivity of Isotropie Media Hydraulic Conductivity and Permeability Units and Examples Anisotropie Permeability The Principal Directions Directional Permeability Measurement of Hydraulic Conductivity General The Constant Head Permeameter The Falling Head Permeameter Determining Anisotropie Hydraulic Conductivity Layered Porous Media Flow Normal and Parallel to the Medium Layers Equivalent Hydraulic Conductivity of Arbitrarily Directed Flow A Layered Medium as an Equivalent Anisotropie Medium Girinskii's Potential Compressible Fluids Derivations of Darcy's Law Capillary Tube Models Fissure Models Hydraulic Radius Models Resistance to Flow Models Statistical Models Averaging the Navier-Stokes Equations Ferrandon's Model 175

5.11 Flow At Large Reynolds Numbers 176 5.11.1 The Phenomenon 176 5.11.2 Turbulence, Inertial Forces and Separation 177 5.11.3 Some Examples of Proposed Nonlinear Motion Equations.... 182 5.

5 5.11 Flow At Large Reynolds Numbers The Phenomenon Turbulence, Inertial Forces and Separation Some Examples of Proposed Nonlinear Motion Equations Seepage Forces and Stresses The Forces Piping and Quicksand 186 CHAPTER6 Continuity and Conservation Equations for a Homogeneous Fluid The Control Volume Mass Conservation in a Nondeformable Porous Matrix The Basic Continuity Equation Continuity Equation for an Incompressible Fluid Continuity Equation for a Compressible Fluid Mass Conservation in a Consolidating Medium Vertical Compressibility Only Extension to Three Phases and to Three-Dimensional Consolidation Barometric Efficiency of Aquifers Continuity Equations for Flow in Confined and Leaky Aquifers The Horizontal Flow Approximation Flow in a Confined Aquifer Flow in a Leaky Aquifer Averaging the Exact Equations over a Vertical Line The Boltzmann Transformation Stream Functions Pathlines, Streamlines, Streaklines and Fronts The Stream Function in Two-Dimensional Flow The Stream Functions in Three-Dimensional Flow The Partial Differential Equations for the Lagrange and Stokes Stream Functions The Relationships between the Potential and the Stream Functions Solving Problems in the q>-<p Plane Flow Nets and Ground Water Contour Maps The cp-ifi Flow Net The Ground Water Contour Map The Partial Differential Equations Describing Flow of an Inhomogeneous Incompressible Fluid in Terms of W Two-Dimensional Flow Axisymmetric Flow 243 CHAPTER 7 Solving Boundary and Initial Value Problems Initial and Boundary Conditions 248

7.1.1 Boundary of Prescribed Potential 250 7.1.2 Boundary of Prescribed Flux 251 7.1.3 The Steady Free (or Phreatic) Surface without Accretion 252 7.1.4 The Unsteady Free (or Phreatic) Surface without Accretion.

6 7.1.1 Boundary of Prescribed Potential Boundary of Prescribed Flux The Steady Free (or Phreatic) Surface without Accretion The Unsteady Free (or Phreatic) Surface without Accretion The Steady Free (or Phreatic) Surface with Accretion The Unsteady Free (or Phreatic) Surface with Accretion Boundary of Saturated Zone (or of Capillary Fringe) The Seepage Face Capillary Exposed Faces Discontinuity in Permeability A Note on Anisotropie Media Boundary Conditions in Terms of Pressure or Density A Well Posed Problem Description of Boundaries in the Hodograph Plane The Hodograph Plane Boundaries in the Hodograph Plane Examples of Hodograph Representation of Boundaries Intersection of Boundaries of Different Types The Relations between Solutions of Flow Problems in Isotropie and Anisotropie Media The Flow Equations Relationships among Parameters in the Two Systems Examples Superposition and Duhamel's Principles Superposition Unsteady Flow with Boundary Conditions Independent of Time Unsteady Flow with Time-Dependent Boundary Conditions Direct Integration in One-Dimensional Problems Solution of the One-Dimensional Continuity Equation Advance of a Wetting Front The Method of Images Principles Examples Methods Based on the Theory of Functions Complex Variables and Analytic Functions The Complex Potential and the Complex Specific Discharge Sources and Sinks Conformal Mapping The Schwarz-Christoffel Transformation Fictitious Flow in the eö-plane Numerical Methods Method of Finite Differences The Method of Finite Elements Relaxation Methods 348

7.9.4 Schmidt's Graphic Method 350 7.10 Flow Nets by Graphic Methods 351 CHAPTER 8 Unconfined Flow and the Dupuit Approximation 361 8.1 The Dupuit Approximation 361 8.1.1 The Dupuit Assumptions 361 8.

7 7.9.4 Schmidt's Graphic Method Flow Nets by Graphic Methods 351 CHAPTER 8 Unconfined Flow and the Dupuit Approximation The Dupuit Approximation The Dupuit Assumptions Examples of Application to Hydraulic Steady Flows in Homogeneous Media Unconfined Flow in an Aquifer with Horizontal Stratification Unconfined Flow in an Aquifer with Vertical Strata Unconfined Flow in a Two-Dimensional Inhomogeneous Medium Continuity Equations Based on the Dupuit Approximation The Continuity Equation Boundary and Initial Conditions Some Solutions of Forchheimer's Equation Some Solutions of Boussinesq's Equation The Hodograph Method The Functions w and w The Hodograph Method Examples without a Seepage Face Hamel's Mapping Function Zhukovski's and Other Mapping Functions A Graphic Solution of the Hodograph Plane Linearization Techniques and Solutions First Method of Linearization of the Boussinesq Equation The Second Method of Linearization of the Boussinesq Equation The Third Method of Linearization of the Boussinesq Equation The Method of Successive Steady States The Method of Small Perturbations The Shallow Flow Approximation 430 CHAPTER 9 Flow of Immiscible Fluids Introduction Types of Two-Fluid Flows The Abrupt Interface Approximation Occurrence Interfacial Tension and Capillary Pressure Saturation and Fluid Content Interfacial Tension and Wettability Capillary Pressure Drainage and Imbibition 449

9.2.5 Saturation Discontinuity at a Medium Discontinuity 452 9.2.6 Laboratory Measurement of Capillary Pressure 453 9.3 Simultaneous Flow of Two Immiscible Fluids 457 9.3.1 The Basic Motion Equations 457 9.

8 9.2.5 Saturation Discontinuity at a Medium Discontinuity Laboratory Measurement of Capillary Pressure Simultaneous Flow of Two Immiscible Fluids The Basic Motion Equations Relative Permeability Mass Conservation in Multiphase Flow Statement of the Multiphase Flow Problem The Buckley-Leverett Equations Simultaneous Flow of a Liquid and a Gas Laboratory Determination of Relative Permeability Unsaturated Flow Capillary Pressure and Retention Curve The Capillary Fringe Field Capacity and Specific Yield The Motion Equation Relative Permeability of Unsaturated Soils The Continuity Equation Methods of Solution and Examples Additional Comments on Infiltration and Redistribution of Moisture Comments on Vapor Movement in Unsaturated Flow Immiscible Displacement with an Abrupt Interface The Abrupt Interface Approximation Piezometric Heads and Dynamic EquilibriumConditions at a Stationary Interface The Boundary Conditions along an Interface Horizontal Interface Displacement Interface Displacement in the Vertical Plane Numerical and Graphic Methods Approximate Solutions based on Linearization Interface Stability Determining the Steady Interface by the Hodograph Method Boundary Conditions Description of Boundaries in the Hodograph Plane Examples The Interface in a Coastal Aquifer Occurrence The Ghyben-Herzberg Approximation Determining the Shape of a Stationary Interface by the Dupuit- Ghyben-Herzberg Approximation Approximate Solution for the Moving Interface Interface Upconing The Dupuit-Ghyben-Herzberg Approximation for an Unsteady Interface in a Thick Aquifer 573

CHAPTER 10 Hydrodynamic Dispersion 579 10.1 Definition of Hydrodynamic Dispersion 579 10.2 Occurrence of Dispersion Phenomena 582 10.3 Review of Some Hydrodynamic Dispersion Theories 582 10.3.1 Capillary Tube and Cell Models 583 10.

9 CHAPTER 10 Hydrodynamic Dispersion Definition of Hydrodynamic Dispersion Occurrence of Dispersion Phenomena Review of Some Hydrodynamic Dispersion Theories Capillary Tube and Cell Models Statistical Models Spatial Averaging Parameters of Dispersion The Coefficients of Mechanical Dispersion and Hydrodynamic Dispersion The Medium's Dispersivity Dispersivity-Permeability Relationship The Governing Equations and Boundary Conditions The Partial Differential Equation in Cartesian Coordinates The Partial Differential Equation in Curvilinear Coordinates Initial and Boundary Conditions Solving the Boundary Value Problems The Use of Nondimensional Variables Some Solved Problems One-dimensional Flow Uniform Flow in a Plane Plane Radial Flow Heat and Mass Transfer Modes of Heat Transfer in a Porous Medium Formulation of the Problem of Heat and Mass Transfer in a Fluid Continuum Formulation of the Problem of Heat and Mass Transfer in a Porous Medium Comments on Some Heat and Mass Transfer Coefficients Simplifying the Macroscopic Heat and Mass Transfer Equations Convective Currents and Instability Some Similitude Considerations 660 CHAPTER 11 Models and Analogs General Scaling Principles and Procedure The Two Systems Geometrie Similarity Kinematic Similarity Dynamic Similarity Dimensional Analysis 671

11.2.6 Inspectional Analysis 673 11.2.7 Modified Inspectional Analysis 676 11.3 The Sand Box Model 678 11.3.1 Description 678 11.3.2 Scales 680 11.4 The Viscous Flow Analogs 687 11.4.1 General 687 11.

10 Inspectional Analysis Modified Inspectional Analysis The Sand Box Model Description Scales The Viscous Flow Analogs General Description of the Vertical Hele-Shaw Analog Establishing the Analogy between Analog and Prototype Scales for the Vertical Analog Recommended Applications of Vertical Analog The Liquids The Horizontal Hele-Shaw Analog Description and Scales ^ Simulation of an Infinite Horizontal Aquifer Electric Analogs Description of the Electrolytic Tank and the Conducting Paper Analogs Scales for the Electrolytic Tank Analog The Resistance Network Analog for Steady Flow The Resistance-Capacitance Network for Unsteady Flow The Ion Motion Analog The Membrane Analog Summary 725 Answers to Exercises 729 Bibliography 733 Index 757

Table of Contents. Preface... xiii

Table of Contents. Preface... xiii Preface... xiii PART I. ELEMENTS IN FLUID MECHANICS... 1 Chapter 1. Local Equations of Fluid Mechanics... 3 1.1. Forces, stress tensor, and pressure... 4 1.2. Navier Stokes equations in Cartesian coordinates...