dynamic of fluids in porous media

8/13/2019 Dynamic of Fluids in Porous Media

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dyn micsoffluidsinporous mediJacob ear

Department of Civil EngineeringTechnion-Israel Institute of Technology, Haifa

Technische Hochschule Darmstadt

F a c h b e r e i c h M e c h a n i kB i b j i p t h e kInv. Nr.

a m e r ic a n e lsevie rp ub lish ing c o m p a ny inc.NEW YORK LOND ON AM ST ERD AM


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ContentsPreface xviiCHAPTER 1Introduction 11.1 Aquifers, Groun d W ate r and Oil Reservo irs 1

1.1.1 Definitions 11.1.2 The Moisture D istributio n in a Vertical Profile 21.1.3 Classification of Aquifers 41.1.4 Prop erties of Aquifers 71.1.5. The Oil Reservoir 8

1.2 The Porous Medium 131.3 The Continuum Approach to Porous Media 15

1.3.1 The Molecular and Microscopic Levels 151.3.2 Porosity and Rep resentative Elem entary Volume 191.3.3 Areal and Linear Porosities 211.3.4 Veloc ity an d Specific Disc harge 221.3.5 Conclud ing Re m ark s 24

CHAPTER 2Fluids and Porous M at rix Properties 272.1 Fluid De nsity 27

2.1.1 De finitions 272.1.2 M ixture of Flu ids . . . 302.1.3 M easure me nt of D en sity . . . .' 31

2.2 Fluid Viscosity 322.2.1 Definition 322.2.2 No n-Ne wton ian Fluids 332.2.3 U nits 342.2.4 Effect of Pressure and Te m pe ratu re 342.2.5 M easu rem ent of Viscosity . . ' 35

2.3 Fluid Com pressibility 372.4 Statistica l Desc ription of Porou s Media 382.4.1 Particle-Size Dis tributio n 392.4.2 Pore-Size D istributio n 412.4.3 Othe r Statistic al Desc riptions 42


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2.5 Porosity 432.5.1 Por osity and Effective Po rosity 432.5.2 Po rosi ty /Stru ctur e and Packing 452.5.3 Poros ity Measurement 47

2.6 Specific Surface 502.6.1 De finitions 502.6.2 M easu rem ent of Specific Surface 51

2.7 M atrix and Medium Com pressibility 52CHAPTER 3Pressure and Piezom etric He ad 593.1 Stress at a Po int 593.2 H ydr ostatic Pressure Distribution 623.3 Piezometric He ad 63CHAPTER 4The Fundam ental Fluid Tran spo rt Equations in Porous Me dia 654.1 Pa rticles, Velocities and Flux es in a Fluid Co ntinuum 65

4.1.1 Definitions of Particle s and Velocities 654.1.2 Diffusive Veloc ities an d Flux es 684.1.3 The Eulerian and Lagran gian Poin ts of View 704.1.4 The Sub stantial De rivative 714.2 Th e General Conserva tion Principle 74

4.3 Eq uatio ns of Mass, Mo mentum and Ene rgy Conservation in a FluidContinuum 774.3.1 Mass Con serva tion of a. Species 774.3.2 Mass Conse rvation of a Fluid System 784.3.3 Conserva tion of Line ar M om entum of a Species a 794.3.4 Conse rvation of Line ar M om entum of a Fluid System 80

4.4 Co nstitutive Assu mp tions an d Coupled Processes 824.4.1 Gene ral Considerations 824.4.2 Principles to be Used in Form ing Con stitutive Eq uatio ns 844.4.3 Coupled Processes 85

4.5 A Poro us Medium Model 904.5.1 The Conc eptual Model Ap proa ch 904.5.2 A Model of Flow Th rou gh a Poro us Medium 924.5.3 Fra m es of Reference 934.5.4 An Averaging Procedure 95

4.6 Eq ua tion s of Volume and Mass Conse rvation 984.6.1 Eq ua tion of Volume Con servation 984.6.2 E qu atio n of Mass Conse rvation of a Species in Solution 1004.6.3 E qu atio n of Mass Conse rvation 102

4.7 Eq ua tion of Motion 104


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4.8 Tortuosity and Permeability 1064.8.1 Relationship Between Tortuosity and Permeability 1064.8.2 Tortuosity and Other Transport Coefficients 1124.8.3 Formation Factor and Resistivity Index (Amyx 1960) in Reservoir

Engineering 113C H A P T E R ST h e E qu a t io n o f M o t io n o f a H o m o g e ne o u s F l u i d 1195.1 The Experimental Law of Darcy 1195.2 Generalization of Darcy's Law 122

5.2.1 Isotropic Medium 1225.2.2 Anisotropic Medium 123

5.3 De viations from Da rcy 's Law 1255.3.1 The Upper Limit 1255.3.2 Th e Low er Lim it 1275.3:3 The Slip Pheno me non . ' 128

5.4 Ro tational and Irrotation al Motion 1295.4.1 The Potential and Pseudopotential 1295.4.2 Irrotational Flow 131

5.5 Hydraulic Conductivity of Isotropic Media 1325.5.1 Hydraulic Conductivity and Permeability 1325.5.2 Units and Examples 135

5.6 Anisotropic Permeability 1365.6.1 The Principal Directions 1375.6.2 Directional Permeability 143

5.7 Measurement of Hydraulic Conductivity 1485.7.1 General ; 1485.7.2 The Constant He ad Perm eam eter 1495.7.3 The Falling Head Permeameter 1505.7.4 Determining Anisotropic Hydraulic Conductivity 150

5.8 Layered Porous Media 1515.8.1 Flow Normal arid Parallel to the Medium Layers 1515.8.2 E quiv alen t Hy drau lic Con duc tivity of Ar bitrarily Direc ted Flow . . 1555.8.3 A Layered Medium as an Equivalent Anisotropic Medium 1565.8.4 Girinskii's Po te nt ia l. 157

5.9 Compressible Fluids 1595.10 Derivations of Darcy's Law 161

5.10.1 Capillary Tube Models 1625.10.2 Fissure Models 1645.10.3 H ydr aulic Ra dius Models . . . . 1655.10.4 Resistance to Flow Models 1675.10.5 Sta tistica l Models 1715.10.6 Averaging the Navier-Stokes Equations 173 15.10.7 Ferrandon's Model 175


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5.11 Flow At Large Rey nolds Nu m bers 1765.11.1 The Phenom enon 1765.11.2 Turbulence , Iner tial Forces and Separation 1775.11.3 Some Exam ples of Proposed Nonlinear Motion Eq uation s . . . . 1 8 2

5.12 Seepage Forc es an d Stresses . . . '. 1845.12.1 The Forces 1845.12.2 Piping and Quicksand 186

CHAPTER 6Co ntinu ity and Conservation Equations for a Homogeneous Fluid 1956.1 The Control Volume 1956.2 Mass Conserv ation in a Non deforma ble Porou s M atrix 196

6.2.1 The Basic Con tinuity Eq uatio n 1966.2.2 Co ntinu ity Eq ua tion for an Incom pressible Fluid 1986.2:3 Co ntinu ity Eq ua tion for a Compressible Fluid 200

6.3 Mass Conserv ation in a Consolidating Medium 2026 .3 .1 Vert ical Compres sib ility On ly . . . . . ' 2036.3.2 Extension to Three Phases and to Three:Dimensional Consolidation . 2086.3.3 Ba rom etric Efficiency of Aqu ifers 211

6.4 Co ntinu ity Eq ua tion s for Flow in Confined and Lea ky Aquifers 2136.4.1 The Horizon tal Flow Ap proxim ation 2136.4.2 Flow in a Confined Aqu ifer 2146.4.3 Flow in a Le aky Aquifer 2166.4.4 Averaging the Ex ac t Eq uatio ns over a Vertical Line 2186.4.5 The Boltzm ann Transform ation 221

6.5 Stream Func tions 2226.5.1 Pathlines, Streamlines, Streaklines and Fro nts 2236.5.2 The Stream Func tion in Two-Dim ensional Flow 2256.5.3 The Stream Fun ctions in Three-Dimensional Flow 2266.5.4 The Par tial Differential Eq uation s for the Lagrange and Stokes Stream

Functions 2306.5.5 The Relationships between the Potential and the Stream Functions 2336.5.6 Solving Problem s in the


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7.1.1 Bo und ary of Prescribed Pote ntial 2507.1.2 Bo und ary of Prescribed Flux 2517.1.3 The Stea dy Free (or Phre atic) Surface with out Accretion 2527.1.4 The Unsteady Free (or Phreatic) Surface without Accretion . . . . 2547.1.5 The Stea dy Free (or Phre atic) Surface with Accretion 2567.1.6 The U nste ady Free (or Phre atic) Surface w ith Accretion 2587.1.7 Bo un da ry of Sa tura ted Zone (or of Capillary Fringe) 2597.1.8 Th e Seepage Fac e 2607.1.9 Capillary Exp osed Faces 2627.1.10 Disco ntinuity in Perm eability . 2637.1.11 A No te on An isotropic Media 2697.1.12 Bo un da ry Conditions in Ter ms of Pressure or De nsity 270

7.2 A Well Posed Problem 2707.3 Description of Bou ndaries in the Ho dog raph Plan e 272

7.3.1 The Hod ograph Plane 2727.3.2 Bound aries in the Hod ograph Plane 2747.3.3 Exam ples of Hod ograph Represe ntation of Boundaries 2807.3.4 Inters ection of Bou ndaries of Different Typ es 284

7.4 The Relations between Solutions of Flow Problems in Isotropic and Aniso-tropic Media 2897.4.1 The Flow Eq uation s 2907.4.2 Re la tio ns hip s a mo ng P ar am e te rs in th e T wo S yste ms . . . . . . . 2917.4.3 Ex am ples 296

7.5 Superp osition and Du ham el's Principles 2977.5.1 Supe rposition 2977.5.2 U nstead y Flow with Bou ndary Conditions Indep ende nt of Time . . 2997.5 .3 Uns teady F low wi th Time-Dependent Boundary Condi tions . . . . 300

7.6 Direc t Inte gra tion in One-D imensional Problem s '. . 3017.6.1 Solution of the One-D imensional Co ntinu ity Eq ua tion 3017.6.2 Adva nce of a W etting Fro nt - 303

7.7 Th e M ethod of Im age s .' 3047.7.1 Princ iples . 3047.7.2 Ex am ple s' 3067.8 Methods Based on the The ory of Fun ctions 3127.8.1 Complex Variab les and An alytic Fun ction s 313

7.8.2 The Complex Po ten tial and the Complex Specific Discharge 3167.8.3 Sources an d Sinks 3167.8.4 Conformal M app ing 3247.8.5 The Schwarz-Christoffel Tra nsform ation 3337.8.6 Fic titiou s Flow in th e co-Plane 337

7.9 Nu merical M ethods 3387.9.1 M ethod of Fin ite Differences 3387.9.2 The Me thod of Fin ite Ele me nts 3467.9.3 Re laxa tion Me thods 348


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7.9.4 Schmidt 's Graphic Method. . f . 3507.10 Flow Netsby Graphic Methods 351CHAPTER 8Unconfined Flow and the Dupuit Approximation 3618.1 TheDup uit A pproximation 361

8.1.1 The Dupuit Assumptions 3618.1.2 ExamplesofApplication to Hydraulic Steady Flowsin Homogeneous

Media . 36 68.1.3 Unconfined Flow in anAquifer with Horiz ontal Stratification . . . 3698.1.4 Unconfined Flow in an Aquifer with Vertical Strata 3728.1.5 Unconfined Flow in aTwo-Dimensional Inhomogeneous Medium . . 373

8.2 Continuity Equations Basedon the Dupuit Approximation 3748.2.1 The Continuity Equation 3748.2.2 Boundary and Initial Conditions 3788.2.3 Some Solutionsof Forchheimer's Equation 3798.2.4 Some Solutionsof Boussinesq's Equation 381

8.3 TheHodograph Method. 3888.3.1 TheFun ctions coanda> 3888.3.2 The Hodograph Method 3898.3.3 Examples without a Seepage Face 3918.3.4 Hamel's Mapping Function 3988.3.5 Zhukovski's andOther Mapping Func tions 4038.3.6 A Graphic Solutionof the Hodograph Plane 406

8.4 Linearization Techniques andSolutions 4088.4.1 First MethodofL inearizationof theBoussinesq E qua tion 4088.4.2 The Second Methodof Linearizationof theBoussinesq E quatio n . . 4178.4.3 TheThird MethodofL inearizationof theBoussinesq E qua tion . . . 4198.4.4 TheMethodof Successive Steady States -. . 4208.4.5 TheMethodof Small Perturbations 4228.4.6 TheShallow. Flow Ap proxim ation 430

CHAPTER 9Flow of Immiscible Fluids 4399.1 Introduction 439

9.1.1 Typesof Two-Fluid Flows 4399.1.2 TheAb rupt Interface Approxim ation 4399.1.3 Occurrence 440

9.2 Interfacial Tension and Capillary Pressure 441.9 .2 .1 Saturat ion and Fluid Content r 4419.2.2 Interfacial Tension andW ettabi l i ty 4419.2.3 Capillary Pressure 4449.2.4 DrainageandImbibit ion 449


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9.2.5 Satu ration Discon tinuity at a Medium Disco ntinuity 4529.2.6 La bo rato ry M easurem ent of Capillary Pressure 453

9.3 Simu ltaneous Flow of Two Imm iscible Fluids 4579.3.1 The Basic Motion Eq ua tion s 4579.3.2 Relative Perm eability 4599.3.3 Mass Conserv ation in M ultiphase Flow 4669.3.4 State me nt of the M ultiphase Flow Problem 4669.3.5 The Buck ley-Leve rett Eq uatio ns 4689.3.6 Sim ultaneo us Flow of a Liqu id and a Gas 4729.3.7 La bora tory De term ination of Relative Perm eability 473

9.4 U nsa tura ted Flow 4749.4.1 Capillary Pressure and Re tentio n Curve 4759.4.2 The Cap illary Fringe , 4809.4.3 Field Ca pa city -and Specific Yield 4839.4.4 The Motion Eq ua tion 4879.4.5 Relative Perm eability of U nsa tura ted Soils 4919.4.6 The Con tinuity Eq uatio n 4959.4.7 M ethods of Solution an d Ex am ples 5039.4.8 Add itional Com men ts on Infiltra tion and Redistrib ution of Mo isture 5139.4.9 Comm ents on Vapo r Movem ent in U nsa tura ted Flow 515

9.5 Imm iscible Displacement with an Ab rupt Interface 5199.5.1 The Ab rupt Interface App roxima tion 5199.5.2 Piezometric Heads and Dynam ic Equ ilibrium Conditions at a Station aryInterface 5219.5.3 The Bo und ary Conditions along an Interface 5249.5.4 Ho rizontal Interface Displacement 5269.5.5 Interface Displac em ent in the Ve rtical Plan e 5339.5.6 Num erical and Graphic Methods 5369.5.7 Ap prox imate Solutions based on Linearization 5389.5.8 Interface Stabi l i ty . 5 4 4

9.6 Determ ining the Stead y Interface by the Ho dograp h Method 5479.6.1 Bo un da ry Cond itions 5489.6.2 Description of Bound aries in the Hod ograph Plane 5499.6.3 Exa mp les 549

9.7 The Interfac e in a Coastal Aquifer 5579.7.1 Occu rrence 5579.7.2 The Ghy ben-Herzb erg Ap proxim ation 5599.7.3 Determ ining the Shap e of a Stationar y Interface by the D upu it-

Ghyben-Herzberg Approximation 5619.7.4 A pp rox im ate Solution for the Moving Interfa ce 5639.7.5 Interfac e Up coning 5699.7.6 The Dupuit-Ghyb en-Herzberg Approxim ation for an Unsteady

Interfa ce in a Thic k Aquifer 573


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CHAPTER 10Hydrodynamic Dispersion 57910.1 Definition of Hydrodynamic Dispersion 57910.2 Occurrenceof Dispersion Phenomena 58210.3 Reviewof Some Hydrodynamic Dispersion Theories 582

10.3.1 Capillary Tubeand Cell Models 58310.3.2 Statistical Models 58710.3.3 Spatial Averaging 603

10.4 Parameters of Dispersion 60510.4.1 The Coefficients of Mechanical Dispersion and Hydrodynamic

Dispersion 60510.4.2 The Medium's Dispersivity 61110.4.3 Dispersivity-Permeability Relationship 615

10.5 The Governing Equations and Boundary Conditions 61710.5.1 ThePar tial Differential Eq uation in Cartesian Coordinates . . . . 61710.5.2 ThePa rtial Differential Eq uation in Curvilinear Coordinates . . . 61910.5.3 Initialand Boundary Conditions 62210.5.4 Solvingthe Boundary Value Problems 62410.5.5 The Use ofNondimensional Variables 626

10.6 Some Solved Problems 62610.6.1 One-dimensional Flow 62710.6.2 Uniform Flowin aPlane 63310.6.3 Plane Radial Flow 63410.7 Heat andMass Transfer . . . . . . . . : 64110.7.1 Modesof Heat Transfer in a Porous Medium 64110.7.2 Formulation of theProblemof Hea t and Mass Transfer in a Fluid

Con t inuum. . . / 64310.7.3 Formulationof theProblemofHea t andMass Transfer in a Porous

Medium 64410.7.4 Commentson Some Heat and Mass Transfer Coefficients 64710.7.5 Simplifying the Macroscopic Heat andMass Transfer Equ ations . 65110.7.6 Convective CurrentsandIn stabi l i ty 65310.7.7 Some Similitude Considerations 660

CHAPTER 11Models and Analogs . 66511.1 General 66511.2 Scaling PrinciplesandProcedure 668

11.2.1 The TwoSystems 66811.2.2 Geometric Similarity 66911.2.3 Kinematic Similarity ~ 67011.2.4 Dynamic Similarity 67011.2.5 Dimensional Analysis 671


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11.2.6 Inspe ctiona l Analysis 67311.2.7 Modified Insp ectio nal An alysis 676

11.3 Th e Sand Box Model 67811.3.1 De scription 67811.3.2 Scales 68011.4 Th e Viscous Flow Analogs 68711.4.1 Ge neral 68711.4.2 De scription of the Ve rtical Hele-Sh aw Ana log 68711.4.3 Establishing the Analogy between Analog and Pro totyp e . . . . 6 9 011.4.4 Scales for the Ve rtical An alog 69311.4.5 Rec om me nded Ap plications of Vertical Analog 69611.4.6 Th e Liquids' 69711.4.7 The Ho rizon tal He le-Shaw Analog Description and Scales . . . 69711.4.8 Simulation of an Infinite Ho rizon tal Aquifer 70111.5 Elec tric Analogs 70211.5.1 Description of the Electrolytic Ta nk and th e Conducting Pap er

Ana logs . . 70211.5.2 Scales for the Elec trolytic Ta nk Ana log 70811.5.3 The Res istance Ne twork Analog for Stea dy Flow 71011.5.4 The Resistance-Capacitance Netwo rk for Un steady Flow 71611.5.5 The Ion Motion Analog 719

11.6 The M emb rane Analog 72211.7 Sum mary 725Answers to Exercises 729Bibliography 733Index 757

dynamic of fluids in porous media

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