porous media transport phenomena (civan/transport phenomena) || index

Porous Media Transport Phenomena, First Edition. Faruk Civan.© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

455

INDEX

Absolute permeability, 296Adsorbed molecule layer, 154Adsorbed phase diffusion, 152, 158Apparent gas permeability, 162, 163Apparent heat capacity formulation for heat

transfer with phase change, 318Apparent thermal conductivity, 313Apparent volumetric heat capacity, 313Asymptotic analytical solutions, 405Attraction number, 362Averaging

applications, 68bulk volume, 63divergence of a property, 66generalized volume-averaged transport

equations, 76gradient of a property, 66interface surface area, 68mass-weighted volume, 67phase volume, 63product of three space properties, 66product of two space properties, 65shape-averaged formulations, 305

cross-sectional area-averaged formulation, 306

thickness-averaged formulation, 305surface area, 68volume, 59volumetric average density, 314time derivative of a property, 66

Berea sandstone, 27Beskok and Karniadakis equation, 161Biot’s effective stress coeffi cient, 100Black oil model of a nonvolatile oil system,

295Blackwell et al. dispersion coeffi cient

correlation, 286

Blick and Civan equation, 133Body force, 128Boundary conditions of reservoir, 182Brinkman equation, 133Brooks and Corey equations, 263Buckley and Leverett equation, 193Bulk volume average, 63Bulk volume average velocity (superfi cial

fl uid velocity), 128Bundle of capillary tube models, 1, 10

permeability of porous media, 10power-law fl ow unit equation, 15

Capillarycondensation fl ow, 153, 159orifi ce model, 126pressure, 191, 231

dynamic, 250temperature effect, 232

tube fl uid velocity, 129Cake fi ltration over an effective fi lter,

370Cement exclusion factor, 15, 16Chapman–Enskog viscosity, 150Characteristic length, 87, 150Characteristic time, 87Characterization of parameters for fl uid

transfer, 5Chemical potential, 119Coeffi cient

Biot’s effective stress, 100density variation by species

concentration, 395dispersion, 286grain volume expansion, 364fractal, 17hydrodynamic dispersion, 284hydraulic resistance, 14

bindex.indd 455bindex.indd 455 5/27/2011 12:23:52 PM5/27/2011 12:23:52 PM

456 INDEX

isothermal fl uid compressibility, 339isothermal rock compressibility, 340Joule–Thomson throttling, 341longitudinal dispersivity, 284pore compressibility, 395rarefaction, 161thermal fl uid expansion, 340transverse dispersivity, 284wetting, 232

Collision parameter, dimensionless, 160Compositional nonisothermal mixture

model, 288Condensate and gas containing

noncondensable gas in the near-wellbore region, 301

Conductive heat transferapparent heat capacity formulation,

318enthalpy formulation, 322phase change, 307

Confi guration of pore structure, 7Confi ning pressure, 99Contact angle, 232Control volume analysis

capillary orifi ce model, 126generalized Darcy’s law, 124macroscopic equation of motion, 102macroscopic transport equations, 74

Coordination number, 7, 21Coupling

single-phase mass and momentum equations, 178

two-phase mass and momentum equations, immiscible displacement, 186

Cross-sectional areaaveraged formulation, 306collision, 149

Darcy’s lawgeneralized, 124modifi cation, 99

non-Newtonian fl uids, 135threshold pressure gradient, 105

Darcy–Brinkman–Forchheimer equation, 133

Darcy–Forchheimer equation, 133Darcy number, 109Darcy velocity, 147, 171

Data analysis and correlationdispersivity and dispersion, 286kinetics of freezing/thawing, 311modifi ed Kozeny–Carman equation,

51power-law fl ow unit model, 23scale deposition, 33, 39Vogel–Tammann–Fulcher equation, 51

Decompositiondouble decomposition for turbulent

processes, 70gas hydrates, 328spatial, space, 64temporal, time, 70

Deep-bed fi ltration under nonisothermal conditions, 355

Demarcation criteria, fl ow, 115de Swaan single-phase transient pressure

model for naturally fractured reservoirs, 392

DiameterLeverett pore size factor, 14mean pore, 15

Diffusionmolecular, 282, 283Soret effect, 282

Dimensional analysis method, 80, 81macroscopic equation of motion, 102

Dimensionless collision parameter, 160Dispersive transport, 282

Blackwell et al. dispersion coeffi cient correlation, 286

Civan dispersion coeffi cient correlation, 287

correlation of dispersivity and dispersion, 286, 287

Hiby dispersion coeffi cient correlation, 286

hydraulic, 282, 283longitudinal dispersivity, 284transverse dispersivity, 284

Dissociationgas hydrates, 328

Dissolution, 35, 40gas hydrates, 328

Dolomite, parametric relationships, 26Double decomposition for turbulent

processes, 70Drag force, interfacial, 239Dupuit relationship, 148, 286

Coeffi cient (cont’d)


INDEX 457

Effective permeability, 296Effective pressure, 127Electric double-layer force parameter, 362Electrokinetic parameter, fi rst and second,

362Empirical correlation methods, 2

dimensionless, 2End-point mobility ratio, 404Energy transport, 281

jump balance conditions, 294Enthalpy formulation, conduction heat

transfer, phase change, 322Entropy generation, production, 117, 120Equation

apparent gas permeability, 162Beskok and Karniadakis, 161Brooks and Corey, 263Buckley and Leverett, 193continuity, 173, 186coupling single-phase mass and

momentum equations, 178Dupuit relationship, 148, 286energy transport, 281Gibbs, 119Klinkenberg, 168, 173macroscopic equation of motion, 102

Beskok and Karniadakis equation, 161

Blick and Civan equation, 133Brinkman equation, 133capillary orifi ce model, 126Darcy’s law, 99Darcy–Forchheimer equation, 133Darcy–Brinkman–Forchheimer

equation, 133Ergun equation, 115Forchheimer equation, 108, 120linear momentum balance, 126non-Newtonian fl uids, 134simplifi ed equations, 132

Mass transport, 281Momentum transport, 281Permeability, 13Stokes–Einstein, 286

apparent, gas, 162derivation, 16

Equilibriumthermal conductivity, 313saturation ion activity product, 35

Ergun equation, 115

Extended Klinkenberg equation, 168External open fl ow surfaces, 125Extrapolated limit concept, 127

Filter coeffi cient, 361Finite difference, 197, 380, 415Fixed cross-section hard-sphere model

(FHS), 149Fissured/fractured porous media, 388Flow demarcation criteria, 115Flow functions and parameters, 238

direct interpretation methodsneglecting the capillary end effect for

constant fl uid properties, 242Tóth et al. formulae, 251

constant pressure conditions, 256constant rate conditions, 257variable pressure and rate

conditions, 253unsteady-state core tests, 238

indirect interpretation methodssteady-state core tests, 260unsteady-state core tests, 261

drainage tests, 267imbibition tests, 269

Flow patternssource fl ow, 385translation fl ow, 385

Flow potential, 98extended pseudofl ow potential function,

131Hubbert’s pseudofl ow potential function,

99true fl ow potential function, 99

Flow rate, 118capillary condensation, 153mass, 118, 147mole, 147, 171volumetric, 118, 147

Flowregimes, 116, 146, 152units, 21zone indicator, 14

Fluidcoupling fl uid mass and motion, 5characterization of parameters, 5mobility, 106motion in porous media, 4transfer, 5volume average velocity, 128


458 INDEX

Fluxmass, 357molar, 148volume, 109

Forchheimer equation, 108modifi ed for non-Newtonian fl uids, 137

Formation volume factor, 340Fractal coeffi cient, 17Fractal dimension, 17Fractional fl ow formulation, 192Fracture

fl ow equation, 402skin, 389

Friction factor, 109, 135Front tracking, 205Fully compositional nonisothermal mixture

model, general multiphase, 288

Gasfl ow regimes, 146, 152hydrates, 328Klinkenberg slippage factor, 163real gas density, 108transport in tight porous media, 4

isothermal, 152multicomponent, 166nonisothermal, 159pore size distribution effect, 170single-component, 165

Gibbs equation, 119Glass bead

parametric relationships, 31Grain

consolidation, 7grain volume expansion coeffi cient, 364packing fraction or solidity, 14surface area, 14

Gravity number, 362

Heat capacity, apparent volumetric, 313Heat transfer

apparent heat capacity formulation with phase change, 318

conductive, 307enthalpy formulation with phase change,

322phase change, 307

Heterogeneous porous media, 6modeling of transport, 6

Hiby dispersion coeffi cient, 286

Hybrid models, 2Hydraulic dispersion, 282, 283Hydraulic hydrodynamic dispersion

coeffi cient, 284Hydraulic resistance, 14Hydraulic coeffi cient, 14Hydraulic tube, 14

cross-sectional area, 18diameter, 118, 386effective mean, 14fl ow, 118, 146perimeter, 17, 18, 119surface area, 17volume, 17

Hydrates, gas, 328

Immiscible fl uid displacementasymptotic analytical solutions, 405capillary pressure effect, 191convenient formulation, 194coupling two-phase mass and momentum

equations, 186fractional fl ow formulation, 192fracture fl ow equation, 402linear displacement, one-dimensional,

190matrix–fracture interchange, 388naturally fractured porous media,

396scaling, 89unit end-point mobility ratio formulation,

195work of displacement, 234

Immobile fl uids, 63Interfacial drag force, 239Inertial fl ow coeffi cient, 111, 130Inspectional analysis method, 80, 82Interconnectivity parameter, 15

parametric relationships, 26valve effect, 15

Internal fl ow structure, 125Interstitial fl uid velocity, 128, 147, 286Isothermal black oil model of a nonvolatile

oil system, 295Isothermal limited compositional model of

a volatile oil system, 298Isothermal fl uid compressibility coeffi cient,

339Isothermal rock compressibility coeffi cient,

340


INDEX 459

Joule–Thomson throttling coeffi cient, 341

Jump balance conditions, 294

Klinkenberg equationbasic, 108, 173extended, 168gas slippage factor, 163

Knudsen number, 150Knudsen regime, 154Kozeny–Carman equation, 9

linear fl ow unit equation, 9modifi ed, 46

Leaky tank reservoir model, 179Leaky tube model, 12Leverett J-function, 232Leverett pore size (or diameter) factor, 14Linear displacement, immiscible, one-

dimensional, 190Linear fl ow unit equation, 9, 14Linear Kozeny–Carman equation, 9, 14Linear momentum balance, 126Liquid viscous fl ow, 159London force parameter, 363

Mach number, 151Macroscopic transport equations, 4, 57

control volume analysis, 74equation of motion, 102

Massfl ow rate, 118jump balance conditions, 294transport, 281

Matrix–fractureinterchange transfer functions, 388oil transfer, 397

Mean free path, 149, 150Mean hydraulic tube, 14, 17Mean pore diameter, 15Mean value theorem, 108Method of images

basic, 202expanded, 205

Microscopic fi eld equations, 2averaging of, 2

Mixture properties, 341Mobility, 106

end-point mobility ratio, 404unit end-point mobility ratio, 195

Modelingcondensate and gas containing

noncondensable gas in the near-wellbore region, 301

de Swaan single-phase transient pressure model for naturally fractured reservoirs, 392

gas and vaporizing water phases in the near-wellbore region, 299

gas hydrates, 328isothermal black oil model of a

nonvolatile oil system, 295isothermal limited compositional model

of a volatile oil system, 298leaky-tube model, 12matrix–fracture interchange transfer

functions, 388multiphase fully compositional

nonisothermal mixture model, 288

nonisothermal hydrocarbon fl uid fl ow considering Joule–Thomson effect, 339

particulate transport in porous media, 6sugar cube model of naturally fractured

porous media, 386transport

fi ssured/fractured porous media, 388heterogeneous porous media, 6porous media, 5transient-state, advection, dispersion,

and source/sink, 380Warren–Root lump-parameter model,

390Modifi ed Darcy’s law

non-Newtonian fl uids, 135Molecular diffusion, 282, 283

Stokes–Einstein equation, 286Momentum transport, 281

jump balance conditions, 294Multicomponent gas fl ow, 166Multiphase models

condensate and gas containing noncondensable gas in the near-wellbore region, 301

gas and vaporizing water phases in the near-wellbore region, 299

fracture fl ow equation, 402fully compositional nonisothermal

mixture model, 288


460 INDEX

isothermal black oil model of a nonvolatile oil system, 295

isothermal limited compositional model of a volatile oil system, 298

nonisothermal hydrocarbon fl uid fl ow considering Joule–Thomson effect, 339

Naturally fractured porous mediaasymptotic analytical solutions, 405de Swaan single-phase transient pressure

model for naturally fractured reservoirs, 392

fracturefl ow equation, 402skin, 389

immiscible displacement, 396mass rate of transfer from matrix to

fracture, 396matrix-to-fracture oil transfer, 397species transport in fractured porous

media, 394sugar cube model, 386transient steady-state pressure over

matrix block, 391Warren–Root lump-parameter model,

390Net pressure, 127Non-Darcy effect, 179Nonisothermal hydrocarbon fl uid fl ow

considering Joule–Thomson effect, 339

Non-Newtonian fl uidsequation of motion for, 134frictional drag, 134modifi ed Darcy’s law, 135

Normalized variables, 86saturation, 263

Numerical solutionsfi nite analytic, 211fi nite difference, 197, 380, 415quadrature solution, 410, 413weighted sum, 410

One-dimensional linear displacement, 190Orifi ce-drag force, 129

Pack of solid grain models, 2Packing fraction or solidity, 14

grain, 14

Parametric relationshipsfl uid transfer, 227interconnectivity, 26particle retention dependency, 364simulated annealing method, 265specifi c surface, 26temperature dependency, 364tortuosity, 26

Particle retention dependency factor, 364

Particle-to-grain diameter ratio, 362Particulate suspensions, 353Peclet number, 286, 363Permeability of porous media, 10

absolute permeability, 296alteration by scale deposition, 33, 36apparent gas permeability, 162, 163bundle of capillary tube models, 1,

10derivation, 16

effective permeability, 296equation, 13, 16extremely low-permeability porous rock,

145power-law fl ow unit equation, 15relative permeability, 296temperature effect, 44Vogel–Tammann–Fulcher (VTF)

equation, 49Phase

change, 307conductive heat transfer, 307

apparent heat capacity formulation, 318

enthalpy formulation, 322gas hydrates, 328kinetics of freezing/thawing and

correlation, 311volume average, 63

Porecompressibility, 395connectivity function, 20, 23effective or net pressure, 127fl uid pressure, 99fl uid velocity, 128friction factor, 135geometry function, 15, 16mean diameter, 15parametric function, 20, 23Reynolds number, 135pressure, 153

Multiphase models (cont’d)


INDEX 461

size distribution, 170structure, 11surface area, 14surface friction force, 128throat-drag force, 129volume, 17

Power-law fl ow unit equation, 15data analysis and correlation, 23

Pressurebulk-versus-fl uid volume average, 99capillary, 191, 231confi ning pressure, 99effective or net pressure, 127pore fl uid pressure, 99pseudopressure function, 110

Pressure and saturation formulation, 188

Pluggingpore–throat, 12surface, 12

Porous media, 3contact angle, 232coupling fl uid mass and motion, 5entropy generation, 117fi nite analytic representation of wells,

211fl uid motion, 4fi ssured/fractured porous media, 388gas transport in tight porous media, 4heterogeneous, 383leaky-tube model, 12modeling

transport, 5particulate transport, 6transport in heterogeneous porous

media, 6parameters of fl uid transfer, 227particulate suspensions, 353potential fl ow, 200scaling and correlation, 4sugar cube model of naturally fractured

porous media, 386tortuosity, 14transport properties, 3, 7viscous dissipation, 123

Porosity, 62function, 15, 17noncontributing, 7variation by

scale, 35temperature, 364

Potential fl ow, 200Precipitation, 35Principle

superposition, 200imaging, 202

Pressurepore, 153

Pseudopressure function, 110

Quadrature solution, 410, 413

Rarefaction coeffi cient, 161Real gas

density, 108viscosity, 149

Relative permeability, 296direct interpretation methods

determination, 238neglecting the capillary end effect

for constant fl uid properties, 242

Tóth et al. formulae, 251constant pressure conditions, 256constant rate conditions, 257variable pressure and rate

conditions, 253unsteady-state core tests, 238

indirect interpretation methods determination, 260

steady-state core tests, 260unsteady-state core tests, 261

drainage tests, 267imbibition tests, 269

Representative elementarycore length, 112volume (REV), 58, 384

Retardation factor, 395Reynolds number, 109, 135

Sand packs, parametric relationships, 31Saturation

formulation, 189ratio, 35

Scale deposition, 33, 39Scaling and correlation of transport, 4, 79,

84Scaling criteria and options, 87Shape-averaged formulations, 305Silty soil, 33Simulated annealing method, 265Single-component gas fl ow, 165


462 INDEX

Skin friction force, 128Slip regime, 156Solidity, 14

grain packing fraction, 14Soret effect, 282Source/sink, 292Source fl ow, 385Spatial, space decomposition, 64Speed of sound, 151Specifi c heat capacity at constant pressure,

341Species

advective/convective fl ux, 285dispersive transport of, 282hydraulic dispersion, 282molecular diffusion, 282, 283retardation factor, 395transport in fractured porous media, 394Soret effect, 282

Steam saturation, 237Stokes–Einstein equation, 286Streamline formulation, 205

basic, 206fi nite analytic representation of wells,

211immiscible displacement

confi ned reservoirs, 214unconfi ned reservoirs, 213

stream tube formulation, 205basic, 206fi nite analytic representation of wells,

211immiscible displacement

confi ned reservoirs, 214unconfi ned reservoirs, 213

Sugar cube model of naturally fractured porous media, 386

Superfi cial velocity, 147, 171Surface area, 14

external open fl ow surfaces, 125grain (or pore), 14, 17internal fl ow structure, 125parametric relationships, 26

Surfacediffusion, 152friction force, 128

Suspensionsparticulate, 353cake fi ltration over an effective fi lter,

370

deep-bed fi ltration under nonisothermal conditions, 355

Syntheticporous media parametric relationships,

29spheres parametric relationships, 26

Temperature effectpermeability, 44, 364Vogel–Tammann–Fulcher (VTF)

equation, 49Temporal decomposition, 70Thermal conductivity

apparent, 313equilibrium, 313

Thermal regimesfreezing or thawing, 308kinetics, 311single temperature, 308

Thickness-averaged formulation, 305Threshold pressure gradient, 105, 127Tight porous media, 4

gas transport, 4Time

averaging, 70decomposition, 70

Tortuosity, 14, 73parametric relationships, 26porous media, 14, 17

Tóth et al. formulae, 251, 253, 256, 257constant pressure conditions, 256constant rate conditions, 257variable pressure and rate conditions,

253Transition regime, 156Translation fl ow, 385Transfer functions, matrix–fracture

interchange, 388Transport equations

generalized volume-averaged, 76heterogeneous porous media, 383source/sink, 292transient-state, advection, dispersion, and

source/sink, 380Transport processes

external interaction, 293internal or bulk volume, 293interface, 293jump balance conditions, 294transport units, 385


INDEX 463

Transport propertiesporous media, 3, 7tortuosity, 14transport units, 385

Transport units, 385True—fl ow potential, 99Turbulent processes, 70

Unfrozen water content, 236, 315gradual temperature phase change, 311,

326instantaneous fi xed temperature phase

change, 317Unifi ed Hagen–Poiseuille equation, 160Unit end-point mobility ratio formulation,

195asymptotic analytical solutions, 405exact analytical solution, 404method of weighted sum (quadrature)

numerical solutions, 410

Valve effect, 15van der Waals number, 363Variable cross-section hard-sphere model

(VHS), 149Velocity

Darcy, 147interstitial, 128, 147pore velocity, 128superfi cial, 147

Viscous dissipation, 123

Viscous fl ow, liquid, 159Viscous regime, 157Viscosity

Chapman–Enskog viscosity, 150real gas, 149

Vogel–Tammann–Fulcher (VTF) equation, 49, 366

Volumeaveraging rules, 59fl ow rate, 118pore, 17

Wall shear stress, 128, 147Warren–Root lump-parameter model, 390Waterfl ooding

asymptotic analytical solutions, 405fracture fl ow equation, 402method of weighted sum (quadrature)

numerical solutions, 410naturally fractured porous media, 396

Wellsfi nite analytic representation, 211hydraulically fractured, 224

Wettability, 230index, 230temperature effect, 235

steam saturation, 237unfrozen water content, 236

Wetting coeffi cient, 232Work of fl uid displacement, 234Wormhole development, 42


porous media transport phenomena (civan/transport phenomena) || index

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