The Mathematics of Fluid Flow Through Porous Media

ISBN-13: 9781119663843 / Angielski / Twarda / 2021 / 224 str.

Myron B. Allen

Spis treści

Preface xiii1 Introduction 11.1 Historical setting 11.2 Partial differential equations (PDEs) 21.3 Dimensions and units 51.4 Limitations in scope 62 Mechanics 92.1 Kinematics of simple continua 92.1.1 Referential and spatial coordinates 92.1.2 Velocity and the material derivative 122.2 Balance laws for simple continua 132.2.1 Mass balance 142.2.2 Momentum balance 162.3 Constitutive relationships 202.3.1 Body force 212.3.2 Stress in uids 222.3.3 The Navier-Stokes equation 232.4 Two classic problems in uid mechanics 252.4.1 Hagen-Poiseuille ow 262.4.2 The Stokes problem 282.5 Multiconstituent continua 302.5.1 Constituents 302.5.2 Densities and volume fractions 322.5.3 Multiconstituent mass balance 352.5.4 Multiconstituent momentum balance 373 Single-Fluid Flow Equations 393.1 Darcy's law 393.1.1 Fluid momentum balance 413.1.2 Constitutive laws for the uid 41viiviii CONTENTS3.1.3 Filtration velocity 443.1.4 Permeability 453.2 Non-Darcy ows 463.2.1 The Brinkman law 463.2.2 The Forchheimer equation 483.2.3 The Klinkenberg effect 493.3 The single-uid ow equation 503.3.1 Fluid compressibility and storage 513.3.2 Combining Darcy's law and the mass balance 523.4 Potential form of the ow equation 523.4.1 Conditions for the existence of a potential 533.4.2 Calculating the scalar potential 543.4.3 Piezometric head 563.4.4 Head-based ow equation 573.4.5 Auxiliary conditions for the ow equation 593.5 Areal ow equation 613.5.1 Vertically averaged mass balance 623.5.2 Vertically averaged Darcy's law 653.6 Variational forms for steady ow 663.6.1 Standard variational form 673.6.2 Mixed variational form 683.7 Flow in anisotropic porous media 703.7.1 The permeability tensor 703.7.2 Matrix representations of the permeability tensor 713.7.3 Isotropy and homgeneity 743.7.4 Properties of the permeability tensor 743.7.5 Is permeability symmetric? 774 Single-Fluid Flow Problems 814.1 Steady areal ows with wells 814.1.1 The Dupuit-Thiem model 814.1.2 Dirac _ models 854.1.3 Areal ow in an infinite aquifer with one well 884.2 The Theis model for transient ows 914.2.1 Model formulation 914.2.2 Dimensional analysis of the Theis model 924.2.3 The Theis drawdown solution 954.2.4 Solving the Theis model via similarity methods 974.3 Boussinesq and porous medium equations 1024.3.1 Derivation of the Boussinesq equation 104CONTENTS ix4.3.2 The porous medium equation 1074.3.3 A model problem with a self-similar solution 1085 Solute Transport 1155.1 The transport equation 1155.1.1 Mass balance of miscible species 1165.1.2 Hydrodynamic dispersion 1175.2 One-dimensional advection 1215.2.1 Pure advection and the method of characteristics 1225.2.2 Auxiliary conditions for first-order PDEs 1255.2.3 Weak solutions 1265.3 The advection-diffusion equation 1285.3.1 The moving plume problem 1295.3.2 The moving front problem 1315.4 Transport with adsorption 1355.4.1 Mass balance for adsorbate 1365.4.2 Linear isotherms and retardation 1375.4.3 Concave-down isotherms and front sharpening 1385.4.4 The Rankine-Hugoniot condition 1416 Multiuid Flows 1476.1 Capillarity 1486.1.1 Physics of curved interfaces 1486.1.2 Wettability 1526.1.3 Capillarity at the macroscale 1546.2 Variably saturated ow 1576.2.1 Pressure head and moisture content 1576.2.2 The Richards equation 1596.2.3 Alternative forms of the Richards equation 1616.2.4 Wetting fronts 1636.3 Two-uid ows 1646.3.1 The Muskat-Meres model 1646.3.2 Two-uid ow equations 1666.3.3 Classification of simplified ow equations 1676.4 The Buckley-Leverett problem 1706.4.1 The saturation equation 1706.4.2 Welge tangent construction 1736.4.3 Conservation form 1786.4.4 Analysis of oil recovery 1796.5 Viscous _ngering 182x CONTENTS6.5.1 The displacement front and its perturbation 1846.5.2 Dynamics of the displacement front 1876.5.3 Stability of the displacement front 1886.6 Three-uid ows 1906.6.1 Flow equations 1916.6.2 Rock-uid properties 1936.7 Three-uid fractional ow analysis 1956.7.1 A simplified three-uid system 1956.7.2 Classification of the three-uid system 1976.7.3 Saturation velocities and saturation paths 1996.7.4 An example of three-uid displacement 2027 Flows With Mass Exchange 2077.1 General compositional equations 2087.1.1 Constituents, species, and phases 2087.1.2 Mass balance equations 2107.1.3 Species ow equations 2117.2 Black-oil models 2137.2.1 Reservoir and stock-tank conditions 2137.2.2 The black-oil equations 2147.3 Compositional ows in porous media 2177.3.1 A simplified compositional formulation 2177.3.2 Conversion to molar variables 2187.4 Fluid-phase thermodynamics 2207.4.1 Flash calculations 2217.4.2 Equation-of-state methods 222Appendices 225A Dedicated Symbols 227B Useful Curvilinear Coordinates 229B.1 Polar coordinates 229B.2 Cylindrical coordinates 230B.3 Spherical coordinates 233C The Buckingham Pi theorem 235C.1 Physical dimensions and units 235C.2 The Buckingham theorem 236CONTENTS xiD Surface Integrals 239D.1 Definition of a surface integral 239D.2 The Stokes theorem 241D.3 A corollary to the Stokes theorem 241Bibliography 244Index 259

O autorze

Myron B. Allen, is Professor Emeritus of Mathematics at the University of Wyoming in Laramie, Wyoming, USA. He is the author of Continuum Mechanics: The Birthplace of Mathematical Models and co-author of the first and second editions of Numerical Analysis for Applied Science.