DYNAMICS OF FLUIDS IN POROUS MEDIA,/ JACOB BEAR

Chapter 1 Introduction 1.1. Aquifers, ground water and oil reservoirs 1.1.1. Definitions 1.1.2. The moisture distribution in a vertical profile 1.1.3. Classification of aquifers 1.1.4. Properties of aquifers 1.1.5. The oil reservoir 1.2. The porous medium 1.3. The continuum approach to porous media 1.3.1. The molecular and microscopic levels 1.3.2. Porosity and representative elementary volume 1.3.3. Areal and linear porosities 1.3.4. Velocity and specific discharge 1.3.5. Concluding remarks. Chapter 2 Fluids and Porous Matrix Properties 2.1. Fluids density 2.1.1. Definitions 2.1.2. Mixture of fluids 2.1.3. Measurement of density 2.2. Fluid viscosity 2.2.1. Definition 2.2.2. Non-Newtonian fluids 2.2.3. Units 2.2.4. Effect of pressure and temperature 2.2.5. Measurement of viscosity 2.3. Fluid compressibility 2.4. Statistical description of porous media 2.4.1. Particle-size distribution 2.4.2. Pore-size distribution 2.4.3. Other statistical descriptions 2.5. Porosity 2.5.1. Porosity and effective porosity 2.5.2. Porosity, structure and packing 2.5.3. Porosity measurement 2.6. Specific surface 2.6.1. Definitions 2.6.2. Measurement of specific surface 2.7. Matrix and medium compressibility. Chapter 3 Pressure and Piezometric Head 3.1. Stress at a point 3.2. Hydrostatic pressure distribution 3.3. Piezometric head. Chapter 4 The Fundamental Fluid Transport Equations in Porous Media 4.1. Particles, velocities and fluxes in a fluid continuum 4.1.1. Definitions of particles and velocities 4.1.2. Diffusive velocities and fluxes 4.1.3. The Eularian and Langrangian points of view 4.1.4. The substantial derivative 4.2. The general conservation principle 4.3. Equations of mass, momentum and energy conservation in a fluid continuum 4.3.1. Mass conservation of a species 4.3.2. Mass conservation of a fluid system 4.3.3. Conservation of linear momentum of a species alpha 4.3.4. Conservation of linear momentum of a fluid system 4.4. Constitutive assumptions and coupled processes 4.4.1. General consideration 4.4.2. Principles to be used in forming constitutive equations 4.4.3. Coupled processes 4.5. A porous medium model 4.5.1. The conceptual model approach 4.5.2. A model of flow through a porous medium 4.5.3. Frames of reference 4.5.4. An averaging procedure 4.6. Equations of volume and mass conservation 4.6.1. Equation of volume conservation 4.6.2. Equation of mass conservation of a species in solution 4.6.3. Equation of mass conservation 4.7. Equation of motion 4.8. Tortuosity and permeability 4.8.1. Relationship between tortuosity and permeability 4.8.2.Tortuosity and other transport coefficients 4.8.3. Formation factor and resistivity index (Amyx 1960) in reservoir engineering. Chapter 5 The Equation of Motion of a Homogeneous Fluid 5.1. The experimental law of Darcy 5.2. Generalization of Darcy's law 5.2.1. Isotropic medium 5.2.2. Anistropic medium 5.3. Deviations from Darcy's law 5.3.1. The upper limit 5.3.2. The lower limit 5.3.3. The slip phenomenon 5.4. Rotational and irrotational motion 5.4.1.The potential and pseudopotential 5.4.2. Irrotational flow 5.5. Hydraulic conductivity of isotropic media 5.5.1. Hydraulic conductivity and permeability 5.5.2. Units and examples 5.6. Anistropic permeability 5.6.1.The principal directions 5.6.2. Directional permeability 5.7. Measurement of hydraulic conductivity 5.7.1. General 5.7.2. The constant head permeameter 5.7.3. The falling head permeameter 5.7.4. Determining anisotropic hydraulic conductivity 5.8. Layered porous media 5.8.1.Flow normal and parallel to the medium layers 5.8.2. Equivalent hydraulic conductivity of arbitrarily directed flow 5.8.3. A layered medium as an equivalent anisotropic medium 5.8.4. Girinskii's potential 5.9. Compressible fluids 5.10. Derivations of Darcy's law 5.10.1. Capillary tube models 5.10.2. Fissure models 5.10.3. Hydraulic radius models 5.10.4. Resistance to flow models 5.10.5. Statistical models 5.10.6. Averaging the Navier-Strokes equations 5.10.7. Ferrandon's model 5.11. Flow at large Reynolds numbers 5.11.1. The phenomenon 5.11.2. Turbulence, inertial forces and separation 5.11.3. Some examples of proposed nonlinear motion equations 5.12. Seepage forced and stresses 5.12.1. The forces 5.12.2. Piping and quicksand. Chapter 6 Continuity and Conservation Equations for a Homogeneous Fluid 6.1. The control volume 6.2 Mass conservation in a nondeformable porous matrix 6.2.1. The basic continuity equation 6.2.2. Continuity equation for an incompressible fluid 6.2.3. Continuity equation for a compressible fluid 6.3. Mass conservation in a consolidating medium 6.3.1. Vertical compressibility only 6.3.2. Extension to three phases and three-dimensional consolidation 6.3.3. Barometric efficiency of a aquifers 6.4. Continuity equations for flow in confined and leaky aquifers 6.4.1. The horizontal flow approximation 6.4.2. Flow in a confined aquifer 6.4.3. Flow in a leaky aquifer 6.4.4. Averaging the exact equation over a vertical line 6.4.5. The Boltzmann transformation 6.5. Stream functions 6.5.1. Pathline, streamlines, streaklines and fronts 6.5.2. The stream function in two-dimensional flow 6.5.3. The stream functions in three-dimensional flow 6.5.4. The partial differential equations for the lagrange and strokes stream functions 6.5.5. The relationships between potential and the stream functions 6.5.6. Solving problems in the @ plane 6.6. Flow nets and ground water contour map 6.6.1. The @ flow net 6.6.2. The ground water contour map 6.7. The partial differential equations describing flow of an inhomogeneous incompressible fluid in term of @ 6.7.1. Two-dimensional flow 6.7.2. Axisymmetric flow. Chapter 7 Solving Boundary and Initial Value Problems 7.1. Initial and boundary problem 7.1.1. Boundary of prescribed potential 7.1.2. Boundary of prescribed flux 7.1.3. The steady free (or Phreatic) surface without accretion 7.1.4. The unsteady free (or Phreatic) surface without accretion 7.1.5. The steady free (or Phreatic) surface with accretion 7.1.6. The unsteady free (or Phreatic) surface with accretion 7.1.7. Boundary of Saturated zone (or of capillary fringe) 7.1.8. The seepage face 7.1.9. Capillary exposed faces 7.1.10. Discontinuity in permeability 7.1.11. A note on a Anistropic media 7.1.12. Boundary conditions in terms of pressure or density 7.2. A well posed problem 7.3. Description of boundaries in the Hodograph plane 7.3.1. The Hodograph plane 7.3.2. Boundaries in the Hodograph plane 7.3.3. Example of Hodograph representation of boundaries 7.3.4. Intersection of boundaries of different types 7.4. The relations between solutions of flow problems in Isotropic and Anistropic media 7.4.1. The flow equations 7.4.2. Relationships among parameters in the two systems 7.4.3. Examples 7.5. Superposition and Duhamel's principles 7.5.1.Superposition 7.5.2. Unsteady flow with boundary conditions independent of time 7.5.3. Unsteady flow with time-dependent boundary conditions 7.6. Direct integration in one-dimensional problems 7.6.1. Solution of the one-dimensional continuity equation 7.6.2. Advance of a wetting front 7.7. The method of images 7.7.1. Principles 7.7.2. Examples 7.8. Method based on the theory of functions 7.8.1. Complex variables and Analytic functions 7.8.2. The complex potential and the complex specific discharge 7.8.3. Sources and sinks 7.8.4. Conformal mapping 7.8.5. The Schwarz-Christoffel transformation 7.8.6. Fictitious flow in the @-plane 7.9. Numrical methods 7.9.1. numerical method 7.9.2. Method of finite elements 7.9.3. Relaxation methods 7.9.4. Schmidt's graphic methods. Chapter 8 to hydraulic steady flows in homogeneous media 8.1.3. Unconfined Flow and the Dupuit Approximation 8.1. The dupuit approximation 8.1.1. The dupuit assumptions 8.1.2. Examples of application to hydraulic steady flows in homogeneous media 8.1.3. Unconfined flow in an aquifer with horizontal stratification 8.1.4. Unconfined flow in an aquifer with vertical strata 8.1.5. Unconfined flow in a two-dimensional inhomogeneous medium 8.2. Continuity equations based on the dupuit approximation 8.2.1. The continuity equation 8.2.2. Boundary and initial conditions 8.2.3. Some of Forchheimer's equation 8.2.4. Some solutions of Boussinesq's equation 8.3. The hodograph method 8.3.1. The functions @ and @ 8.3.2. The hodograph method 8.3.3. Examples without a seepage face 8.3.4. Hamels mapping function 8.3.5.Zhukovski's and other mapping functions 8.3.6.

Agraphic solution of the hodograph plane 8.4. Linearization techniques and solutions 8.4.1. First method of linearization of the Boussinesq equation 8.4.2.The second method of linearization of the Boussinesq equation 8.4.3. The third method of linearization of the Boussinesq equation 8.4.4.The method of successive steady states 8.4.5. The method of small perturbations 8.4.6. The swallow flow approximation. Chapter 9 Flow of Immiscible Fluids 9.1. Introduction 9.1.1. Types of two-fluid flows 9.1.2. The abrupt interface approximation 9.1.3. Occurrence 9.1.2. Interfacial tension and capillary 9.1.3. Capillary pressure 9.1.4. Drainage and imbibition 9.1.5. Saturation discontinuity at a medium discontinuity 9.2.6. Laboratory measurement of capillary pressure 9.3. Simultaneous flow of two immiscible fluids 9.3.1. The basic motion equations 9.3.2. Relative permeability 9.3.3. Mass conservation in multiphase flow 9.3.4. Statement of the multiphase flow problem 9.3.5. The Buckley-Leverett equations 9.3.6. Simultaneous flow of liquid and a gas 9.3.7. Laboratory determination of relative permeability 9.4. Unsaturated flow 9.4.1. Capillary pressure and retention pressure curve 9.4.2. The capillary fridge 9.4.3. Field capacity and specific yield 9.4.4. The motion equation 9.4.5. Relative permeability of unsaturated soils 9.4.6. The continuity Equation 9.4.7. Methods of solution and examples 9.4.8. Additional comments on infiltration and redistribution of moisture 9.4.9. Comments or Vapor movement in unsaturated flow 9.5. Immiscible displacement with an abrupt interface 9.5.1. The abrupt interface approximation 9.5.2. Piezometric heads and dynamic equilibrium conditions at a stationary interface 9.5.3. The boundary conditions along an interface 9.5.4. Horizontal interface displacement 9.5.5. Interface displacement in the vertical plane 9.5.6. Numerical and graphic methods 9.5.7. Approximate solution based on Linearization 9.5.8. Interface stability 9.6. Determining the steady interface by the hodograph method 9.6.1. boundary conditions 9.6.2. Description of boundaries in the hodograph plane 9.6.3. Examples 9.7. The interface in coastal aquifer 9.7.1. Occurrence 9.7.2. The Ghyben-Herzberg approximation 9.7.3. Determining the shape of a stationary interface by the Dupuit-Ghyben-Herzberg approximation 9.7.5. Approximation solution for the moving interface 9.7.5.Interface upconing 9.7.6. The Dupuit-Ghyben-Herzberg approximation for an unsteady interface in a thick aquifer. Chapter 10 Hydrodinamic Dispersion 10.1. Definition of hydrodynamic dispersion 10.2.Occurrence of dispersion phenomena 9.3.7. Laboratory determination of relative permeability 9.4. Unsaturated flow 9.4.1. Capillary pressure and retention pressure curve 9.4.2. The capillary fridge 9.4.3. Field capacity and specific yield 9.4.4. The motion equation 9.4.5. Relative permeability of unsaturated soils 9.4.6. The continuity Equation 9.4.7. Methods of solution and examples 9.4.8. Additional comments on infiltration and redistribution of moisture 9.4.9. Comments or Vapor movement in unsaturated flow 9.5. Immiscible displacement with an abrupt interface 9.5.1. The abrupt interface approximation 9.5.2. Piezometric heads and dynamic equilibrium conditions at a stationary interface 9.5.3. The boundary conditions along an interface 9.5.4. Horizontal interface displacement 9.5.5. Interface displacement in the vertical plane 9.5.6. Numerical and graphic methods 9.5.7. Approximate solution based on Linearization 9.5.8. Interface stability 9.6. Determining the steady interface by the hodograph method 9.6.1. boundary conditions 9.6.2. Description of boundaries in the hodograph plane 9.6.3. Examples 9.7. The interface in coastal aquifer 9.7.1. Occurrence 9.7.2. The Ghyben-Herzberg approximation 9.7.3. Determining the shape of a stationary interface by the Dupuit-Ghyben-Herzberg approximation 9.7.5. Approximation solution for the moving interface 9.7.5.Interface upconing 9.7.6. The Dupuit-Ghyben-Herzberg approximation for an unsteady interface in a thick aquifer. Chapter 10 Hydrodinamic Dispersion 10.1. Definition of hydrodynamic dispersion 10.2.Occurrence of dispersion phenomena 10.3. Review some hydrodynamic dispersion theories 10.3.1. Capillary tube and cell models 10.3.2.Statistical models 10.3.3. Spatial averaging 10.4. Parameters of dispersion 10.4.1. The coefficients of mechanical dispersion and hydrodynamic dispersion 10.4.2. The medium dispersivity 10.4.3. Dispersivity-permeability relationship 10.5. The governing equations and boundary conditions 10.5.1. The partial differential equation in Cartesian coordinates 10.5.2. The partial differential equation in curvilinear coordinates 10.5.3. Initial and boundary conditions 10.5.4. Solving the boundary value problems 10.5.5. The use of nondimensional variables 10.6. Some solved problems 10.6.1. One-dimensional flow 10.6.2. Uniform flow in a plane 10.6.3. Plane redial flow 10.7. Heat and mass transfer 10.7.1. Modes of heat transfer in porous medium 10.7.2. Formulation of the problem of heat and mass transfer in a fluid continuum 10.7.3.Formulation of the problem of heat and mass transfer in a porous medium 10.7.4. Comment on some heat and mass transfer coefficients 10.7.5. Simplifying the macroscopic heat and mass transfer equations 10.7.6. Convective currents and instability 10.7.7. Some similitude considerations. Chapter 11 Models and Analogs 11.1. General 11.2. Scaling principles and procedure 11.2.1. The two systems 11.2.2. Geometric similarity 11.2.3. Kinematic similarity 11.2.4. Dynamic similarity 11.2.5. Dimensional analysis 11.2.6. Inspectional analysis 11.2.7. Modified inspectional analysis 11.3. The sand box model 11.3.1. Description 11.3.2.Scales 11.4. The viscous flow analogs 11.4.1. General 11.4.2. Description of the vertical Hele-Shaw analog 11.4.3. Establishing the analogy between analog and prototype 11.4.4. Scales for the vertical analog 11.4.5. Recommended applications of vertical analog 11.4.6. The liquids 11.4.7. The horizontal Hele-Shaw analog - Description and scales 11.4.8. Simulation of an infinite horizontal aquifer 11.5. Electric analogs 11.5.1. Description of the electrolytic tank and the conducting paper analogs 11.5.2. Scales for the electrolytic tank analog 11.5.3. The resistance network analog for steady flow 11.5.4. The resistance-capacitance network for unsteady flow 11.5.5. The ion motion analog 11.6. The membrane analog 11.7. Summary

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