Математика в общейфизике
The movement of fluids (water/oil/gas) in various media (channels, pipelines, fractured and fractured-porous subsurface formations) is described mathematically in this book. The writers used the unified systematic approach based on the continuity and conservation laws of continuum mechanics. The topics covered include: (1) hydrostatics, (2) Bernoulli’s equation, (3) permanence rules in integral and differential forms, (4) flow of compressible and incompressible fluids, (5) flow of ideal and viscous fluids, (6) flow in fractured and fractured-porous reservoir rocs, (7) turbulent flow, and (8) two-phase flow. The book can be used both as a textbook and reference book by engineers and geologists, both students and professionals.
PREFACE PART I. FUNDAMENTALS OF THE MECHANICS OF CONTINUA CHAPTER I. BASIC CONCEPTS OF THE MECHANICS OF CONTINUA Introduction 1. Continuity hypothesis 2. Movement of continuous medium: description techniques 3. Local and substantive derivative 4. Scalar and vector fields 0 5. Forces and stresses in the continuous medium. Stress tensor CHAPTER II. CONSERVATION LAWS. INTEGRAL AND DIFFERENTIAL EQUATIONS OF CONTINUOUS MEDIUM 1. Integral parameters of a continuous medium and the conservation laws 2. Time differentiation of the integral taken over a movable volume 3. Continuity equation (law of mass conservation) 4. Motion equation under stress 5. Law of variation of kinetic momentum. Law of pairing of tangential stresses 6. The law of conservation of energy 7. Theorem of variation of kinetic energy 8. Heat flow equation 9. Continuous medium motion equations CHAPTER III. CONTINUOUS MEDIUM DEFORMATION RATE 1. Small particle deformation rate. Helmholtz theorem 2. Tensor of the deformation velocity 3. Physical meaning of the deformation velocity tensor components 4. Tensor surface of a symmetric second rank tensor 5. Velocity circulation. Potential motion of the liquid CHAPTER IV. LIQUIDS 1. Mathematical model of ideal fluid 2. Mathematical model of ideal incompressible fluid 3. Viscous fluid. Stress tensor in viscous fluid 4. Motion equations of viscous fluids 5. Mathematical model of a viscous incompressible fluid 6. The work of internal forces. Equation of the heat inflow CHAPTER V. BASICS OF THE DIMENSIONALITY AND CONFORMITY THEORY 1. Systems of units. Dimensionality 2. Dimensionality formula 3. Values with independent dimensionalities 4. Π-theorem 5. Conformity of physical phenomena, modeling 6. Parameters determining the class of phenomena 7. Examples of application of the Π-theorem 8. Contraction of equations to dimensionless format PART II. HYDROMECHANICS CHAPTER VI. HYDROSTATICS 1. Liquids and gas equilibrium equations 2. Equilibrium of a liquid in the gravitational field 3. Relative quiescence of fluid 4. Static pressure of liquid on firm surfaces 5. Elements of buoyancy theory CHAPTER VII. FLOW OF IDEAL FLUID 1. Euler’s equations in the Gromeko-Lamb format 2. Bernoulli integral 3. Particular forms of Bernoulli’s integral 4. Simple applications of Bernoulli’s integral 5. Cauchy-Lagrange’s integra 6. Thomson’s theorem 7. Helmholtz equation 8. Potential flow of a incompressible fluid 9. Flow around the sphere 10. Applications of the of momentum law CHAPTER VIII. PARALLEL-PLANE FLOWS OF IDEAL INCOMPRESSIBLE FLUID 1. Complex-valued potential of flow 2. Examples of parallel-plane potential flows 3. Conformous reflection of flows 4. Zhukovsky’s transform 5. Flow-around an arbitrary profile 6. Forces acting on a profile under the stationary flow CHAPTER IX. FLOW OF VISCOUS INCOMPRESSIBLE FLUID IN PRISMATIC TUBES 1. Equations descring straight-line motion of a viscous incompressible fluid in prismatic tubes 2. Straight-line flow between two parallel walls 3. Straight-line flow within axisymmetric tubes 4. Equation of transient-free circular motion of a viscous fluid 5. Flow between two revolving cylinders CHAPTER X. TURBULENT FLOW OF FLUIDS IN TUBES 1. Reynolds’ experiments 2. Averaging the parameters of turbulent flow 3. Reynolds’ equations 4. Semi-empiric turbulency theory by L. Prandtl 5. Application of the dimensionality theory to the construction of semi-empirical turbulence theories 6. Logarithmic law of velocity distribution 7. Experimental studies of hydraulic resistivity CHAPTER XI. HYDRAULIC CALCULATION FOR PIPELINES 1. Bernoulli’s equation for a viscous fluid flow 2. Types of head loss 3. Designing simple pipelines 4. Designing complex pipelines 5. Pipelines performing under vacuum CHAPTER XII. FLUID’S OUTFLOW FROM ORIFICES AND NOZZLES 1. Outflow from a small orifice 3. Outflow through nozzles 3. Outflow of fluid at variable level CHAPTER XIII. NON-STATIONARY FLOW OF VISCOUS FLUID IN TUBES 1. Equations of the non-stationary fluid flow in tubes 2. Equation of non-stationary flow for slightly-compressible fluid in tubes 3. Equations of non-stationary gas flow in tubes at low subsonic velocities 4. Integrating equations of non-stationary fluid and gas flow using the characteristics technique 5. Integrating linearized equations of non-stationary flow using Laplace transformation 6. Examples of computing non-stationary flow in tubes 7. Hydraulic shock 8. Effect of flow instability on force of friction CHAPTER XIV. LAMINAR BOUNDARY LAYER 1. Equations of the boundary layer 2. Blasius problem 3. Detachment of the boundary layer CHAPTER XV. UNIDIMENSIONAL GAS FLOWS 1. Sound velocity 2. Energy conservation law 3. Mach number. Velocity factor 4. Linkage between the flow tube’s cross-section area and flow velocity 5. Gas outflow through a convergent nozzle 6. De Laval’s nozzle 7. Gas-dynamic functions 8. Shock waves 9. Computation of gas ejector 10. Transient-free gas flow in tubes 11. Shukhov’s equation CHAPTER XVI. LAMINAR FLOW OF NON-NEWTONIAN FLUIDS 1. Simple shear 2. Classification of non-Newtonian fluids 3. Viscosimetry 4. Fluid flow in an infinitely-long round tube 5. Rotational fluid flow within a ring gap 6. Integral technique in viscosimetry 7. Hydraulic resistance factor 8. Additional remarks to the calculation of non-Newtonian fluids flow in tubes CHAPTER XVII. TWO-PHASE FLOW IN PIPES 1. Equations of the conservation laws 2. Equations of two-phase mixture flow in tubes 3. Transformation of equations of two-phase flow in pipes 4. Flow regimes 5. Absolute open flow of a gas-condensate well PART III. OIL AND GAS SUBSURFACE HYDROMECHANICS CHAPTER XVIII. MAIN DEFINITIONS AND CONCEPTS OF FLUID AND GAS FLOW. DARCY’S LAW AND EXPERIMENT 1. Specifics of fluid flow in natural reservoirs 2. Basic model concepts of the subsurface liquid and gas hydrodynamics 3. Reservoir properties of porous bodies. Porosity, specific surface area 4. Darcy’s experiment and Darcy’s law. Permeability. The conceptof «true» average flow velocity and flow velocity 5. Applicability limits of Darcy’s law. Analysis and interpretation of experimental data 6. Nonlinear laws of filtration 7. Structural model of porous media 8. Darcy’s law for anisotropic media CHAPTER XIX. MATHEMATICAL MODELS OF UNIPHASE FILTRATION 1. Introductory notes. The concept of the mathematical model of a physical process 2. Mass conservation laws in a porous medium 3. Differential equation of fluid flow 4. Closing equations. Mathematical models of isothermal filtration 5. Filtration model of incompressible viscous fluid under Darcy’s law in a non-deformable reservoir 6. Gas filtration model under Darcy’s law. Leibensohn’s function 7. Uniphase filtration models in non-deformable reservoir under nonlinear filtration laws 8. Correlation between fluid parameters and porous medium parameters with pressure CHAPTER XX. UNIDIMENSIONAL TRANSIENT-FREE FILTRATION OF NONCOMPRESSIBLE FLUID AND GAS IN A UNIFORM POROUS MEDIUM 1. Schematics of unidimensional filtration flows 2. Rectiliner-parallel filtration of incompressible fluid 3. Radial-plane filtration of incompressible fluid 4. Radial-spherical filtration of incompressible fluid 5. Filtration similarity between incompressible liquid and gas 6. Unidimensional filtration flow of ideal gas 7. Parallel-plane filtration flow of real gas under Darcy’s law 8. Radial-plane filtration flow of incompressible liquid and gas under binomial filtration law 9. Radial-plane filtration flow on incompressible liquid and gas under the exponential filtration law CHAPTER XXI. UNIDIMENSIONAL FILTRATION OF INCOMPRESSIBLE LIQUID AND GAS IN A NONUNIFORM RESERVOIRS UNDER DARCY’S LAW 1. Major types of reservoir nonuniformities 2. Rectilinear-parallel flow within nonuniformly-laminated reservoir 3. Rectilinear-parallel flow in zonally-nonuniform bed 4. On the calculation of continuously-nonuniform reservoirs 5. Radial-plane flow in a nonuniformly stratified reservoir 6. Rectilineal-parallel flow in a nonuniformly stratified reservoir CHAPTER XXII. FLAT TRANSIENT-FREE FILTRATION FLOWS 1. Major definitions and concepts 2. Potential of a point source and sink on an isotropic plane. Superposition method 3. Liquid flow to a group of wells with the remote charge contour 4. Liquid inflow to a well in the reservoir with a rectilinear charge contour 5. Liquid inflow to a well in the reservoir near the impermeable boundary 6. Liquid inflow to a well positioned eccentrically in a round reservoir 7. The use of superposition technique at the gas flow 8. Fluids inflow to infinite well lines and ring well rows CHAPTER XXIII. NON-STATIONARY FLOW OF AN ELASTIC FLUID IN AN ELASTIC RESERVOIR 1. Elastic reservoir drive and its specifics 2. Calculation of elastic fluid reserves of a reservoir 3. Mathematical model of the elastic fluid non-stationary filtration in an elastic porous medium 4. Derivation of the differential equation of the elastic fluid filtration in an elastic porous medium under Darcy’s law 5. Unidimensional filtration flows of an elastic fluid. Point-solutions of the piezo-conductivity equation. Main equation of the elastic drive theory 5.1. Rectilinear-parallel filtration flow of an elastic fluid 5.2. Rectilinear-parallel filtration flow of an elastic fluid. The main equation of the elastic filtration regime theory 6. Approximate solution techniques of the elastic drive problems 6.1. Method of sequential change of stationary states 6.2. Pirverdian’s technique 6.3. Integral relationships technique 6.4. «Averaging» technique 7. Elastic fluid flow to an aggregate well CHAPTER XXIV. NON-STATIONARY FLOW OF GAS IN A POROUS MEDIUM 1. Mathematical model of non-stationary gas filtration 2. Linearization of Leibensohn’s equation and the main solution of linearized equation 3. Point solution of an automodel problem on axisymmetric gas flow to a well with a constant flow-rate 4. Solution of the problem of gas flow to a well using sequential change of stationary states technique 5. Solution of the gas flow to well problem using averaging technique 6. Application of superposition principle to problems of non-stationary gas filtration 7. Approximate solution of gas production from closed reservoir problems using the material balance eqvation CHAPTER XXV. FILTRATION OF NON-NEWTONIAN LIQUID 1. Viscoplastic liquid: filtration law and mathematical model 2. Rectilinear-parallel filtration flow for the viscoplastic liquid 3. Rectilinear-parallel filtration flow of viscoplastic liquid in a nonuniformly-laminated reservoir 4. Radial-plane filtration flow of viscoplastic liquid 5. Non-stationary filtration flow of viscoplastic liquid 7. Formation of bypass zones in the process oil-with-water displacement 8. Specifics of viscoplastic liquid filtration in anisotropic porous media CHAPTER XXVI. LIQUID AND GAS FLOW IN FRACTURED AND FRACTURED-POROUS MEDIA 1. Specifics of filtration in fractured and fractured-porous media 2. Filtration laws in fractured media 3. Permeability vs. pressure in fractured and fractured-porous media 4. On the fluid crossflow in fractured-porous media and gas flow within the fractured and fractured-porous media 5. Derivation of differential equations for liquids and gas flow within the fractured and fractured-porous media 6. Stationary unidimensional liquids and gas filtration in a fractured and fractured-porous reservoir 7. Non-stationary liquid and gas flow in fractured and fractured-porous reservoirs APPENDIX A REFERENCES SUBJECT INDEX