Thermo-hydro-mechanical study of deformable porous media with double porosity in local thermal non-equilibrium

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Thermo-hydro-mechanical study of deformable porous

media with double porosity in local thermal

non-equilibrium

Rachel Gelet

To cite this version:

Rachel Gelet. Thermo-hydro-mechanical study of deformable porous media with double porosity in

local thermal non-equilibrium. Materials. Université de Grenoble; University of New South Wales,

2011. English. �NNT : 2011GRENI079�. �tel-00712459�

THÈSE

Pour obtenir le grade de

DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE

Spécialité :

Matériaux, Mécanique, Génie civil, Electrochimie

Arrêté ministériel : 7 août 2006

Présentée par

Rachel M. Gelet

Thèse dirigée par

Benjamin Loret et

codirigée par

Nasser Khalili

préparée au sein du

Laboratoire Sols, Solides, Structures -

Risques

dans

l'École Doctorale : Ingénierie – Matériaux Mécanique

Energétique Environnement Procédé Production

Thermo-hydro-mécanique des

milieux poreux déformables

avec double porosité et

non-équilibre thermique local

Thèse soutenue publiquement le

23 Septembre 2011,

devant le jury composé de :

M. Claude Boutin

Chercheur HDR a l’ENTPE, (Président)

M. Alessandro Gajo

Professeur Associé à l’Université de Trento (Rapporteur)

M. Alain Millard

Professeur en mécanique des structures à l’ECP (Rapporteur)

M. Benjamin Loret

Professeur à Grenoble INP, (Directeur de thèse)

M. Nasser Khalili

(Co-A fully oupled onstitutive modelispresented for arigorous analysisofdeformation,

hy-drauli andheatowsinsaturateddualporositymediasubje ttothermo-hydro-me hani al

loadings in luding those able to ause lo al thermal non-equilibrium. The solid phase is

assumed to ontain two distin t avities: the porousblo ks and the ssure network. The

governingequationsarederivedbasedontheequationsof onservationofmass,momentum

and energy. Solution to the governing equations is obtained numeri ally using the nite

element approa h. The apabilities ofthe modeladdresstwo energyappli ations: the

sta-bilityofaboreholein athermallyenhan edoilre overy ontext andthe heatextra tionof

enhan ed geothermal systems. Substantial dieren es, parti ularly in the ee tive stress

response, highlight the major inuen e of the dualporositymodel and the importan e of

Un modèle onstitutif omplètement ouplé est présenté pour l'analyse rigoureuse de la

déformation, del'é oulement de uides et detransfert de haleur danslesmilieux poreux

saturés à double porosité soumis à des hargements thermo-hydro-mé aniques, y

om-pris eux induisant un non-équilibre thermique lo al. La phase solide ontient deux

av-ités distin tes: le blo poreux et le réseau des ssures. Les équations de hamps sont

obtenuesàpartir deséquationsde onservationdelamasse, dumouvementet del'énergie

et sontrésolues par uneappro he parélément nis. Le modèleestutilisé pour deuxtypes

d'appli ations: lastabilité d'un puits deforagestimuléethermiquement pourla

ré upéra-tion de pétrole et l'extra tion de haleur dans un réservoir géothermique fra turé. Les

diéren essubstantielles, parti ulièrement de la ontrainte ee tive,soulignent l'inuen e

majeuredeladoubleporositéetdunon-équilibrethermiquepourprédirele omportement

soonwill be otherwith;

only what lasts an bring

us to the truth.

Young one,don't put your trust

into the trials of ight,

into the hot and qui k.

All things alreadyrest:

darkness and morning light,

ower and book.

Theworkdes ribedinthisthesiswas arriedoutaspartofajoinedinternational resear h

partnership ( o- otutellePhD)between the Sols,Solides, Stru tures- Risques(3S-R)

lab-oratory at the Institut Polyte hnique de Grenoble, Fran e and the S hool of Civil and

Environmental Engineeringat the Universityof NewSouth Wales(UNSW), Sydney,

Aus-tralia. Finan ialassistan ein theform ofa3 yearPhDfellowshipprovided bythe Fren h

Ministryof Higher Edu ation is gratefullya knowledged, asis the 6 month nan ial

sup-port from UNSW through the guidan e of Prof. Khalili. In addition, the author would

like to thank the Asso iation Française desFemmes Diplomées desUniversités (AFFDU)

and the InstitutNationalPolyte hnique de Grenoble for their travelgrants.

Iwould liketo thank many members ofsta in the Sols,Solides, Stru tures - Risques

laboratory and in the S hool of Civil and Environmental Engineering. I am indebted

to the senior s ientists at Grenoble and to Professor David Waite (UNSW) for fostering

an outstanding resear h environment and for wel oming me in the laboratory and in the

S hool. Iamenormouslygratefulto mysupervisorsProfessor BenjaminLoret(3S-R) and

Professor NasserKhalili (UNSW)for sparking myinterest in soilme hani s andfor their

invaluable guidan e and en ouragement. In parti ular, Ithank Professor Benjamin Loret

for histireless eortsand qui kresponses to my allsfor assistan e.

Fortheirfriendshipandday-to-daysupport,IthankHa,An a,Christina, Elma,Daïki,

Ola, Bertrand, Xiangwei, Mar os, Luisa, Séverine, Mark, Jane, Lam, Jessi a, Brunella,

Wenji, Mi hal, Jérémy, Hana, Fabriozio, Matias and the boys of the E-building: Cédri ,

Jean-François, Florent, Jérme, Stéphane, Rémi, Barthélémy and Guilherme from

3S-R; and Aurélie, Adam, Andrew, Adele, Daniel, Juan Pablo, Yun, Arman, Mohammad,

Samaneh, YongJia,Ce ilia, Irene, Pattie and MaryfromUNSW.I apologise forthose left

out and Ithankthem too.

Furthermore,Iwouldliketothankmyparentsandmybrothersfortheiren ouragement

and their kindness. I am extremely grateful to my husband, Sylvain Blanvillain, who

monthspriorto ompletion. Thiswholeexperien e,in ludingthelongseparation emented

our bond far beyond expe ted. Sin e marriage has already taken pla e down-under, the

next stepis yet to ome...

Ra helM. Gelet Blanvillain

Sydney, Australia

Abstra t 1

Resumé 2

List of gures 29

List of tables 34

Introdu tion 35

1 Non-isothermal uid-saturated porous media: Literature review 40

1.1 Porous media with multi porosity . . . 40

1.1.1 Classi ation ofmulti porousmedia . . . 41

1.1.2 Denition ofdualporousmedia . . . 42

1.2 Poroelasti onstitutive framework . . . 45

1.2.1 Averaging models . . . 45

1.2.2 Re ommendation forthe mixture theory . . . 49

1.3 Poroelasti modelsfor dualporosity . . . 50

1.3.1 Coupling indual porous media: two approa hes . . . 54

1.3.2 Appli ations. . . 55

1.4 Thermo-hydro-me hani al oupled models . . . 57

1.4.1 Models with lo althermal equilibrium . . . 58

1.4.2 Models with lo althermal non-equilibrium. . . 59

1.4.3 Models with dualporosityand thermal ontribution . . . 61

1.4.4 Models with onve tion . . . 63

2 Field and Constitutive equations 66 2.1 General eldequations . . . 67

2.1.1 Denition . . . 68

2.1.2 General formof the balan eequation . . . 71

2.1.4 Balan eof momentum . . . 75

2.1.5 Balan eof energy . . . 79

2.1.6 Balan eof entropy . . . 85

2.1.7 Clausius-Duhem inequality . . . 90

2.2 Constitutive equations basedon athermodynami approa h . . . 96

2.2.1 Onsager'sre ipro ityprin iple versus `rational' thermodynami s . . 98

2.2.2 Thermo-me hani al onstitutive equations . . . 103

2.2.3 Diusion onstitutive equations . . . 144

2.2.4 Transfer onstitutive equations . . . 155

2.3 Comprehensive eldequations . . . 163

2.3.1 Balan eof momentumfor the mixture . . . 164

2.3.2 Balan eof massequationfor the uids . . . 166

2.3.3 The omprehensive energy equations . . . 169

2.4 Summary ofgoverning equations . . . 177

2.5 Parameters identi ation . . . 181

3 Constitutive parameters: interpretation and identi ation 184 3.1 Intensive parameters . . . 185

3.1.1 TheYoung's modulus, the Poisson'sratio andthe Lamé's onstants 185 3.1.2 Thedrained ompressibilities . . . 186

3.1.3 Theporosities. . . 189

3.1.4 Thepermeabilities . . . 190

3.1.5 Theaperture fa tor. . . 192

3.2 Extensive parameters . . . 193

3.2.1 Thethermal expansion oe ients . . . 194

3.2.2 Thedensities . . . 195

3.2.3 Thedynami vis osity . . . 197

3.2.4 Thespe i heat apa ities . . . 197

3.2.5 Thethermal ondu tivities . . . 198

3.3 Solid-uid extensive parameters . . . 199

3.3.1 The oe ient of thermo-osmosis . . . 199

3.3.2 The oe ients ofinter-phase heattransfer . . . 200

3.4 TheReynolds, Prandtl, Pé let, Nusselt,Sparrow numbers . . . 204

3.4.1 TheReynolds number . . . 204

3.4.2 ThePrandtl number . . . 205

3.4.3 ThePé let number . . . 206

3.4.5 TheSparrownumber . . . 207

4 Finite element method 209 4.1 Thenite element method . . . 209

4.2 Thesemi-dis rete equations . . . 213

4.2.1 Theweakformulation . . . 213

4.2.2 Denition ofthe nodaland globalunknownve tors . . . 215

4.2.3 The(Bubnov-)Galerkin method . . . 216

4.2.4 Thesemi-dis rete system . . . 216

4.3 Thetime integration methods for equationsolving . . . 226

4.3.1 Thegeneralised trapezoidalmethod . . . 227

4.3.2 Predi tor multi- orre tor algorithms with operator split . . . 228

5 Preliminary numeri al results 233 5.1 Validation oftransient ondu tion tests . . . 234

5.1.1 A one-dimensionalappli ation. . . 234

5.1.2 Constant heatuxloading . . . 235

5.1.3 Fixedtemperature loading . . . 238

5.2 Thermo-Hydro-Me hani al tests: thermalloading . . . 241

5.2.1 A partially oupled THMmodelin lo althermalequilibrium . . . . 241

5.2.2 A one-dimensionaltest . . . 242

5.2.3 Analyti al formulation . . . 247

5.2.4 Comparing analyti al andnumeri al results . . . 255

5.2.5 Theimportan e ofthe diusivityratio

R

. . . 265

5.2.6 Summary and on lusion . . . 269

5.3 Axi-symmetri boundary valueproblems . . . 270

5.3.1 Analyti al solutions . . . 270

5.3.2 Numeri al onsiderations . . . 274

6 A borehole stability analysis: fo us on diusion and mass transfer 279 6.1 Introdu tion . . . 280

6.2 Governing equations . . . 282

6.3 Finite element formulation . . . 285

6.3.1 Thesemi-dis rete equations . . . 285

6.3.2 Timeintegration . . . 287

6.4 Non-isothermalborehole stabilityanalysis . . . 289

6.4.1 Boundary onditions . . . 291

6.5 Thermal ee tson dualporousmedia . . . 293

6.5.1 Inuen eof temperature . . . 294

6.5.2 Inuen eof masstransfer . . . 295

6.5.3 Time proles . . . 298

6.5.4 Borehole stabilityanalysis . . . 299

6.6 Con lusion. . . 301

7 The streamline-upwind/Petrov-Galerkinmethod 303 7.1 Presentation ofthe SUPG method . . . 304

7.1.1 Introdu tionof the SUPG method . . . 305

7.1.2 Origin ofthe SUPG method: the arti ial diusion . . . 306

7.1.3 Standarddiusion- onve tion formulations. . . 307

7.1.4 Weightingthe modi ation : the stabilisationparameter

τ

. . . 309

7.2 Validationof the SUPG method. . . 311

7.2.1 One-dimensional steadystate diusion- onve tion problems . . . 312

7.2.2 One-dimensional transient diusion- onve tion problems . . . 316

7.2.3 Two-dimensional steadydiusion- onve tion problems . . . 325

7.2.4 Two-dimensional transient diusion- onve tion problems . . . 329

7.2.5 Stabilityrequirements . . . 336

7.3 Beyond SUPG . . . 337

7.3.1 Me hanismsof numeri al `noise'and spuriousos illations . . . 337

7.3.2 Thelimitations ofthe SUPGmethod. . . 338

7.3.3 TheSUPG method applied toporousmedia: ashort literaturereview340 7.3.4 Thedis ontinuity apturing method . . . 342

8 Simulation of heat extra tion in geothermal reservoirs 348 8.1 Thestabilisation pro essfor a THMmodel . . . 349

8.1.1 Stabilisation for a mixture inLTNE . . . 350

8.1.2 Stabilisation for a mixture inthermal equilibrium . . . 354

8.2 Preliminaryresults onfor ed onve tion in a oupled model . . . 357

8.2.1 Problemsetup at Soultz-sous-Forêts . . . 357

8.2.2 Inuen eof the thermo-hydro-me hani al ouplings. . . 362

8.2.3 Inuen eof the thermalboundary onditions . . . 365

8.2.4 Inuen eof lo althermal non-equilibrium . . . 370

8.3 Asingleporositymediuminlo althermalnon-equil ibrium . . . 374

8.3.1 Introdu tion. . . 375

8.3.3 Finite element dis retization. . . 380

8.3.4 HDRreservoiranalysis . . . 385

8.3.5 Thedouble-step pattern of thermaldepletion in LTNE . . . 389

8.3.6 Fenton HillHDRreservoir . . . 397

8.3.7 Con lusions . . . 403

8.4 A dualporosity mediumin lo althermal non-equilibrium . . . 404

8.4.1 Introdu tion. . . 405

8.4.2 Balan eequations for the three phasemixture . . . 407

8.4.3 Constitutive equations . . . 412

8.4.4 The oupled eldequations . . . 417

8.4.5 Finite element dis retization. . . 418

8.4.6 HDRreservoiranalysis . . . 421

8.4.7 Calibration with elddata . . . 425

8.4.8 Thermo-hydro-me hani al response . . . 429

8.4.9 Con lusions . . . 438

Summary and on lusions 440 A Ray M. Bowen (1970) onstitutive model 460 A.1 Bowen Multiple temperature theory . . . 460

A.2 Bowen's multiple temperature linear isotropi theory . . . 462

B Pe ker and Deresiewi z (1973) onstitutive model 471 B.1 TheClausius-Duhem inequality . . . 471

B.2 A partialstress/strainformulation . . . 473

B.3 Theequivalent totalstress/strain formulation . . . 474

B.4 Reexionon the physi al meaning of the terms

α

sf

and

α

f s

. . . 477

C M Tigue (1986) onstitutive model 480 D Appendi es of Chapter 2.3.3 482 E Appendi es of Chapter 4 491 E.1 Sub-matri es ofthe weak formulation . . . 491

E.2 Detailed elementary weak formulation . . . 493

E.3 Sub-matri es ofthe ee tive onve tion-diusion matrix . . . 496

F Appendi es of Chapter 6 498 F.1 Finite element sub-matri es . . . 498

F.2 Denition ofthe matri esKand D in eq. (6.24) . . . 500

G Appendi es of Se tion 8.3 501

G.1 Denition ofthe ve tors

F

grav

and

F

surf

. . . 501

G.2 Denition ofthe matri es

K

e

and

D

e

in eqn(8.77) . . . 501

G.3 Anexpression for the spe i surfa e . . . 503

G.4 Thedis ontinuity apturing method . . . 504

H Appendi es of Se tion 8.4 506

H.1 Redu tion ofthe dual porosity modelto asingle porositymodel . . . 506

H.2 Denition ofthe ve tors

F

grav

and

F

surf

. . . 507

H.3 Denition ofthe element matri es

K

e

and

D

e

1.1 Illustrationofthede ompositionofadualporousmediumintwooverlapping

singleporousmedia;thessurenetworkandtheporousblo k. Masstransfer

of uid, momentum transfer, energy transfer and entropytransfer between

the two sub-systemsispermitted. . . 44

2.1 Sket h of the ex hanges that are a ounted for in the multi-spe ies

multi-phaseopen system. For illustration, it isassumed that, at anygeometri al

point, the spe ies an be segregated in three phases,ea h of them

ontain-ing several spe ies. (1) Transport: Spe ies of the uid phases (the ssure

phase and the pore phase) are transported to, and from, the boundary by

(hydrauli and thermal) diusion and by for ed onve tion. In addition,

the ouplingofthe hydrauli andthermaldiusionindu esthermo-osmosis.

(2) Transfer: Phases an ex hange mass, momentum, energy and entropy.

Theseex hangesaretermed transfers. The hara teristi s ofea hex hange

dependonboththe on ernedphases(itsdensity, vis osity, hemi al

poten-tial)and on the nature of the `membranes' thatseparate the phases. Thus

lowpermeablephasesareasso iatedtoalargetransfertimeby ontrastwith

higher permeable phases whi h areasso iated to a small transfertime. An

innite transfer time implies impermeability. (3) Ex hanges at the

bound-ary: If in addition, the system is thermodynami ally open, supply and/or

removal form the surroundings, of mass, momentum, energy and entropy

2.2 Sket hofex hangesthatarea ounted forin theporousmediumwith

dou-bleporosity of a non-isothermal losed system. At ea h geometri al point,

porousmediawithdoubleporosityarepartitionedinthreephases;onesolid

phase and two uid phases: the uid of the porous blo k and the uid of

thessurenetwork. Withinea huidphase,hydrauli andthermal oupled

diusion o ur. A rossuid phases masstransfero urs due to a hemi al

potential dieren e. Inbetweenthethreephases,energytransfero ursdue

to temperature dieren es,represented bya dashedarrow. . . 97

2.3 Totalstress de omposition ofarepresentativessured porouselement(Nur

and Byerlee,1971). The stressequilibrium is segregated in two parts: (M)

a me hani al stress stateand (Th) a thermalstress state. . . 113

2.4 Me hani alstressde ompositionofarepresentativessuredporouselement.

The me hani al stress equilibrium is segregated in four stress states: (I)

long-term equilibrium, (II) amedium-term non-equilibrium,(III)adrained

isotropi onditionand (IV)a drained deviatori ondition. . . 114

2.5 S hemati representationofathreephasethermo-elasti porouselement,at

onstant pressuresandstresses,subje tedto onstituenttemperature

T

s

,

T

p

and

T

f

. . . 117

2.6 Coe ient of thermalexpansion

c

T

of i eand water at atmospheri pressure.128

2.7 Entropy of water for

K = 1/c

H

= 2.2 GPa,

C

p

= 4.18 KJ/kg/K, and the

referen e values

p

0

=0 Pa,

T

0

= 0

◦

C,

v

0

= 10

−3

m

3

/kg,

S

0

= 0 J/kg/K,

µ

0

=0J/kg/K. Theleftand enter ontourplotsdisplaythelinearisedand

non linear entropies respe tively for

K

= 2.2 GPa, and the right ontour

plotdisplays the nonlinearentropy for

K

=0.022 GPa. . . 129

2.8 Chemi al potential, enthalpy, free energy, and internal energy of water, as

a fun tionof temperature and pressure. Samedata asFigure 2.7. . . 130

2.9 Entropy and hemi al potential asafun tion of temperature and pressure.

Samedata asFigure2.7;

K

=0.022 GPa (involvingair bubbles) andusing

the non linearrelations. Comparing the inuen e of the pressurerange. . . 131

2.10 Entropy and hemi al potential asafun tion of temperature and pressure.

Same data as Figure 2.9 ex ept for the pressure range : -50 MPa

< p <

50 MPa;

K

=0.022 GPa (involving air bubbles) and using the non linear

relations. Comparing the inuen e of the initial temperature. . . 132

2.11 Two types of dire t ows through a soil mass: dire t uid ow and dire t

heatow.

A

representsthe rossse tionareanormalto thedire tion ofthe

2.12 To illustrate theisothermal heatowphenomenon, asimple aseis

onsid-ered: asystem ontainingonlypureuid isdividedintotwobyapermeable

membrane. The two sub-systems (a and b) are homogeneous with

them-selves, but pressure vary asbetween the ompartments. The experimental

arrangementsaresu hthatthe twosub-systemsaremaintainedatthesame

temperature (thatis,

∇T = 0

)bybeingin intimate onta twithasuitable

heat sour e or sinkand the volume of the entire systemremains onstant.

Mattermovesfromahigherpressuretoalower andeveryunitwhi hpasses

will onveyheatfrom one tothe other. Isothermal heat ow des ribesow

ofheat indu edbyapressure gradient. . . 151

5.1 Solidtemperature

T

s

historyat(left)

x = 0

mand(right)

x = 2

m.

Compar-ison between Carslaw's analyti al solution eq. (5.4) and the nite element

response. The olumn is omposedof an homogeneoussolid phase su h as

n

s

= 1

and

α

T,s

=1.25

×10

−6

m

2

/s. . . 237

5.2 Prole of the solid temperature in the slab for four dierent times. A

on-stant heatuxis appliedat

x = 0

m andthe initialtemperature isimposed

at

x = 3

m. The olumn is omposed ofa singlesolidphase su h as

n

s

= 1

and

α

T,s

=1.25

×10

−6

m

2

/s. . . 237

5.3 Contour representation at four dierent times of a olumn of solidunder a

onstantheatuxapplied at

x = 0

mandatinitialtemperatureat

x = 3

m.

The olumn is omposedof a single solidphase su h as

n

s

= 1

and

α

T,s

=

1.25

×10

−6

m

2

/s. . . 238

5.4 SameasFigure 5.1for the uid temperature

T

p

. . . 238

5.5 Solidtemperature

T

s

historyat(left)

x = 0

mand(right)

x = 2

m.

Compar-ison between Carslaw's analyti al solution eq. (5.7) and the nite element

response. The olumn is omposedof an homogeneoussolid phase su h as

n

s

= 1

and

α

T,s

=1.25

×10

−6

m

2

/s. . . 240

5.6 Sket hofthe stepfun tion

G

1

(t)

usedforthe onstant temperature loading

at the inje tionpoint

x = 0

m,sothat toavoidnumeri al wiggles. . . 245

5.7 (left)S hemati ofthemeshusedforthenumeri alappli ations. (right)

Ini-tialboundary onditions: Homogeneousthermal,pressureanddispla ement

equilibrium. . . 245

5.8 Illustrations of two types of boundary onditions. (Top) 1. Imposed

tem-perature and i. Drainedboundary. (Bottom) 2. Constant heatux and ii.

Undrainedboundary. Forbothtests,azeroverti aldispla ementisimposed

5.9 (a)Sket h ofthe meshandof theboundary onditionsapplied tothe

semi-innite olumn. (b) Temperature, ( ) pressure and (d)displa ement

histo-ries at

x = 3

m in the olumn, for a onstant temperature and a drained

boundary. The temperature eldrea hesa steadystate. Thepressure eld

displays a peak when the heated boundary is drained. This peak is due

to a pressure rise into the pores beforea signi ant dilatation o urs. The

pressure propagation is ahead of the thermal front be ause

R = 3.9

. The

displa ement eld showsa dilatation syn hronisedwith the thermal front. . 256

5.10 (e)Temperature,(f)pressureand(g)displa ementprolesinthe olumn,for

a onstant temperature and adrained boundary: (left)analyti al solution,

(right) numeri al response. The thermal front has penetrated up to 5m.

A onstant pressure peak propagates into the olumn. The pressure front

is ahead of the thermal penetration be ause

R = 3.9

. The dilatation has

rea hed up to10m due to pull-on indu ed bythe dilatation below 5m. . . . 257

5.11 SameasFigure5.9fora onstant temperature andanundrained boundary.

The temperature and the displa ement elds display a similar behaviour

to the drained boundary ondition. On the other hand, the pressure eld

ends by a plateau when the heated boundary is undrained. Lo ally the

uidistrappedintothepores: thepressure annot dissipateinspiteofthe

dilatation. Notethatsomenumeri aldisturban esariseintworegions: (i)at

earlytimes,thesedisturban esareindu edbythenumeri altime-dependent

loading;(ii) atlate times, bythe niteproperty ofthe implemented olumn. 258

5.12 SameasFigure5.10fora onstanttemperatureandanundrained boundary.

Theproleofthetemperatureremainssimilartothedrainedboundary ase.

Ontheotherhand,thepressurepropagatesintothe olumnandismaximum

and onstant at the undrained boundary. Furthermore, the displa ement

magnitudeis in reased ompared with the drainedboundary ase. . . 259

5.13 (a)Sket h ofthe meshandof theboundary onditionsapplied tothe

semi-innite olumn. (b) Temperature, ( ) pressure and (d) displa ement

his-tories at

x = 3

m in the olumn, for a onstant heat ux and a drained

boundary. The temperature rises exponentially, i.e. no steady state is

rea hed. The pressure eld, although lose to the drained boundary, is

ontinuously heated: therefore the pressure does not dissipate easily. The

pressureplateau representsthis time an equilibriumbetween hydrauli

dis-sipationandpressure riseindu edbythermal ex itation. Thedispla ement

5.14 (e)Temperature,(f)pressureand(g)displa ementprolesinthe olumn,for

a onstant heat ux and a undrained boundary: (left) analyti al solution,

(right) numeri al response. The thermal front has penetrated up to 5m.

The pressure propagatesinto the olumn. Thepressure peak isdue to the

drained boundary ondition. The magnitude of whi h is in reasing with

time due to the ontinuous heating. The dilatation hasrea hed up to 10m

due topull-on indu edbythe dilatation below5m. . . 262

5.15 Same as Figure 5.13 for a onstant heat ux and an undrained

bound-ary. The temperature andthe displa ement elds remainthe sameasfor a

drained boundary problem. On the other hand, the pressure eld an not

dissipatedue to the undrained boundary ondition. Therefore the pressure

risesexponentiallyindu ed by ontinuousthermal ex itation. . . 264

5.16 Same as Figure 5.14 for a onstant heat ux and an undrained boundary.

The prole of the temperature remains similar to the drained boundary

ase. On the other hand, the pressure propagates into the olumn with

a maximum lo ated at the undrained boundary. The magnitude of the

maximum pressure is in reasing with time due to the ontinuous heating.

Again, the displa ement magnitude is slightly larger ompared with the

drained boundary ase. . . 265

5.17 Pressurehistoryof (left)abyssalred layand(right)salt;at

x = 3

m in the

olumn,for a onstant temperature anda drained boundary. For the same

loading, the pressure peak in the salt mixture is signi antly larger than

the pressurein the laymixture. Thisisdue tothe fa tthatthe diusivity

ratio

R

ofthe salt is loserto one thanthe diusivity ratioof the lay. . . . 269

5.18 Hollow ylinder subje ted to internal and external pressure,

p

1

and

p

2

,

re-spe tively. . . 271

5.19 Prole of the error along the

r

-dire tion for the radial displa ement, the

strain and the stress. Note that with a non-homogeneous mesh, rened at

the inner radius

r

1

= 0.5

mm (on the left-hand-side), the maximum error

is not lo ated at

r/r

1

= 0

mm. For the displa ement, the maximum error

is lo ated at

r

=

r

2

. For the strain, the maximum error is lo ated around

r = 2

mm. Finally, forthe stress,themaximum errorislo atedintherange

1 < r < 1.5

mm. . . 278

6.1 S hemati diagramofa verti alborehole subje tedto in-situstresses(left).

2D representation of the problem with an axi-symmetri mesh in the

r

-

z

6.2 Sket h of asemi-permeable hydrauli boundary onditionwith zero uxat

the porous blo ksboundary. . . 292

6.3 Proles of pore pressure and ssure pressure, at time 80

s

and for three

imposedtemperature hanges

∆t

at theborehole. Theleakageparameteris

setequal to the averageleakage parameter

η

∗

_{= η}

∗

av

. . . 294

6.4 Prolesof radial ee tive stress and tangential ee tive stress, at time 80

s

andforthreeimposedtemperature hanges

∆t

attheborehole. Theleakage

parameter issetequal to the average leakage parameter

η

∗

_{= η}

∗

av

. . . 295

6.5 Proles of pore pressure and ssure pressure, at time 80

s

, for a thermal

loading equal to

T

w

− T

0

=50

◦

C. . . 296

6.6 Prolesofee tive radial stressandee tive tangential stress,at time80

s

,

a ountingfor apermeable boundary. Thethermalloading isequal to

T

w

−

T

0

=50

◦

C. . . 297

6.7 SameasFigure6.6 for asemi-permeable boundaryon the porous blo ks. . . 297

6.8 Proles of pore pressure and ssure pressure for two values of the leakage

parameter

η

∗

=0 and

η

∗

=

η

∗

av

. The results arepresented for three

repre-sentative times. . . 298

6.9 Proles of ee tive radial stress and ee tive tangential stress, for

η

∗

= 0 and

η

∗

=

η

∗

av

. Theresults arepresented for threerepresentative times. . . . 299

6.10 Ee tive stress path alongdimensionless radial dire tion

r/r

1

, in the Mean

shearstress

√

J

2

- Meanee tive stress

S

p

plane, for twodistin t boundary

onditions. Theresultsarepresentedattime6

s

andwithanaverageleakage

parameter

η

∗

=

η

∗

av

. . . 300

7.1 The optimal upwind fun tion is approximated by two simpli ations: the

doubly asymptoti approximation and the riti al approximation, Brooks

and Hughes(1982, p. 214). . . 314

7.2 Relativetemperatureprolealongthehorizontaldistan e

x

. Comparisonof

thevariousstabilisationfun tions

ξ

˜

withtheexa tsolutioneq. (7.29). While

the optimal rule a) gives a nodally exa t solution, the doubly asymptoti

approximation ) is under- onve tive and the riti al approximation d) is

over- onve tive. b) The omparison between nite element solutions with

and withoutthe SUPGmethod illustratesthe spuriousos illationsand the

7.3 Relative temperature proles along the horizontal dire tion

x

for four

dif-ferent timesandfor gridPé let numbers

Pe

g

rankingfrom0.34to 4.3. The

hara teristi length of the elements

h

is equal to 0.086m and the uid

velo ityis setequal to

v

=

2 Pe

g

α

T

/h

. The niteelement response

repro-du eswell the onve tion-dominated prole of the analyti al solution. The

a ura y of the solution is good when using the denition of the arti ial

diusivityexpressed in eq. (7.31). . . 318

7.4 Relative temperature prole along the horizontal dire tion

x

for

t = 100

s.

The hara teristi length oftheelements

h

isequalto

0.086

m andthe uid

velo ityis set equal to

v

=

2 Pe

g

α

T

/h

. Os illations ariseduring the early

periodduetothe sharptemperaturegradient appliedattheboundary:

x =

0

m. When a ounting for the SUPG method, these os illations are larger

than the non-stabilised response. This in rease of numeri al perturbation

may indu e serious stability and onvergen e issues in oupled problems.

The remedial a tion to this problem is to add a transient- ontribution in

the arti ial diusivityformulawhi his dis ussedin Se tion 7.2.2.2. . . 319

7.5 Relative temperature prole along the horizontal dire tion

x

. The

hara -teristi length of the elements

h

is equal to

0.086

m and the uid velo ity

is set equal to

v

=

2 Pe

g

α

T

/h

. The simulation responsefor diusion only

is ompared with that of diusion- onve tion with SUPG and

diusion- onve tion without SUPG. With no onve tion, the heat front propagates

slowly in the layer and all the urves remainon the left-hand-side. On the

otherhand,if onve tion isa ounted for,theheatpropagatesmorequi kly

in the layer. Spurious os illations arise when the heat front hits the hard

boundary

ϕ(x = 10 m)

= 0 and when the SUPG method is not a ounted

for. ThepopularSUPGmethodremovese iently theseunwanted

numeri- alwiggles. Withoutthe SUPGmethod, thehigherthegridPé letnumber,

the more numeri al wigglesarise. . . 320

7.6 Integration rule for optimal upwind s heme (Brooks and Hughes, 1982),

7.7 Relative temperature proles along the horizontal dire tion

x

for arti ial

diusivities

α

˜

T i

, for

i

= 1, 6 (top);

α

˜

T i

, for

i

= 7, 10 (bottom). The grid

Pé letnumber

Pe

g

isequalto2.15. The hara teristi lengthoftheelements

h

is equal to

0.086

m and the uid velo ityis set equal to

v

=

2 Pe

g

α

T

/h

.

Alongthe oldside(right),theparameter

α

˜

T 1

providesabetterstabilisation

than the other propositions. On the other hand, lose to the perturbation

(left), the parameters

α

˜

T 5

,

α

˜

T 6

and

α

˜

T 9

,

α

˜

T 10

provide a better transition

betweenthetransient-dominatedperiodandthe onve tion-dominatedperiod.323

7.8 Relative temperature prole along the horizontal dire tion

x

for

t = 100

s.

The hara teristi length oftheelements

h

isequalto

0.086

m andthe uid

velo ity is set equal to

v

=

2 Pe

g

α

T

/h

. The streamline upwind

diusiv-ity proposed by Tezduyar and Osawa (2000) in eq. (7.45) redu es

signi- antly the spurious os illations en ountered at early times with respe t to

thestandard streamlineupwind diusivity

α

˜

T 1

(7.39). The`transient/short

time wiggles' are now of the same magnitude as the response of

onve -tion without SUPG, that is the use of the SUPG method will not indu e

morespuriousos illationsin oupledtransientproblems omparedwith the

Galerkinmethod. . . 324

7.9 Four node parallelogram nite element geometry; denitions of element

lengths

h

x

and

h

y

used in eq. (7.56). For more general quadrilaterals, a

propositionis given in Se tion 7.2.4. . . 326

7.10 Relativetemperature prolesalong the verti aldire tion

y

∗

, at

x

∗

= 5and

fora onve tiveowskewtothemesh,

Pe

g

= 10

6

. Thetestsarerunforthree

valuesof

y

∗

c

whi h orrespondtothreeangles

θ

ofthe velo itywith thegrid

lines:

θ

= 0

◦

,

θ

= 21.8

◦

and

θ

= 45

◦

. The analyti al solution is ompared

withtheniteelementresponsewithandwithoutSUPG.TheSUPG

pro e-dure providessatisfa toryresultsand stabiliseswellthe response ompared

with the Galerkin method(without SUPG). Whenthe angle is

θ

=0

◦

, the

SUPGresponseisinperfe tagreementwith theanalyti alresponse. Asthe

angle

θ

in reasesfrom

0

to45

◦

,theSUPGresponseslightly omesawayfrom

thestepbutremainsquitesmoothin omparison withthenon-stabilised

re-sponse. The stabilisation ee t of the SUPG method is mainly visible for

θ

=45

◦

7.11 Relativetemperatureprolesintheplane(

x

∗

,

y

∗

),fora onve tiveowskew

to the mesh,

Pe

g

= 10

6

. E illustrates the exa t solution of a

onve tion-dominated problem, from eq. (7.59). SUPG stands for the nite element

response a ounting for the SUPG method whi h demonstrates the

stabil-isation ee t of the method. G represents the nite element response with

no stabilisation (Galerkin method): spurious wiggles disturb the solution.

The SUPG method signi antly improves upon the Galerkin method asit

e iently ures the spurious wiggles. In addition, a good agreement with

the exa t solutionis obtained for allthe proposedangles. . . 330

7.12 Relativetemperatureprolesintheplane(

x

∗

,

y

∗

),fora onve tiveowskew

to the mesh and a very high gridPé let number

Pe

g

= 10

6

. SUPG stands

fortheniteelementresponsea ountingfortheSUPGmethod; whereasG

representstheniteelementresponsewith nostabilisationonthe weighting

fun tion, that is the standard Galerkin method, whi h displays a heavily

disturbed response. The SUPG method using eq. (7.56) produ es better

results (atall times) than when a ounting for a transient ontribution eq.

(7.68). ThisisduetothehighgridPé letnumber

Pe

g

=10

6

,whi hindu es

the transient ontribution of (7.68) to be equal or smallerthan the

onve -tion ontribution. Therefore,forhighgridPé let numbers, thestabilisation

parameter with notransient ontribution (7.56) is moree ient. . . 334

7.13 SameasFigure 7.12for a smallergridPé let number:

Pe

g

=10

2

. At early

times(left),theSUPGmethodwitheq. (7.68)displaysasmoother

temper-atureprolethantheSUPGmethodwitheq. (7.56),thankstothetransient

ontribution. On the other, during the intermediate and the late periods,

the SUPG method with eq. (7.56) produ es smoother results than T

ezdu-yar and Osawa's proposition. The des ription of the earlyperiod isbetter

viewed byusingeq. (7.68);whereasthe des riptionofthe intermediateand

of the latter periods is better reprodu ed with eq. (7.56). No optimum

7.14 Relativetemperatureprolesfora onve tiveowskewtothe mesh. SUPG

stands for the nite element response a ounting for the SUPG method.

DC1 and DC2 stand for the nite element response with the dis ontinuity

apturing method (addedto SUPG) with

τ

2

= τ

k

and

τ

2

= max(0, τ

k

− τ)

,

respe tively. The signi ant improvements of the dis ontinuity apturing

methods over SUPG are manifest: the solutions do not exhibit the

over-shootsof theSUPG at the downwind boundary;in addition the overshoots

at the internal dis ontinuity are redu ed. The DC2 response exhibits less

arti ial diusion when

v

= v

k

andtherefore asmootherresponse. . . 346

7.15 Relativetemperatureprolesfora onve tiveowskewtothe mesh. SUPG

stands for the nite element response a ounting for the SUPG method.

EC1 and EC2 stand for the nite element response with the dis ontinuity

apturingmethod(addedtoSUPG)witheq. (7.81)and(7.84),respe tively.

The overshoots at the downwind boundary are ee tively removed with

both EC1and EC2methods. Minorimprovements areobserved alsoat the

internal boundary. . . 347

8.1 Fluid ir ulationin the geothermal reservoir atSoultz-sous-Forêts after

hy-drauli stimulation. . . 358

8.2 Con eptual model of the 2-D ir ulation test,at Soultz-sous-Forêts, in the

horizontal

x −y

planeofspa e. Notethattheabovesket hisenlargedalong

the

y

-axis. The boundary onditions illustrated onthis sket hare properly

listedin Table8.2. . . 360

8.3 S aled velo ity ve tors

v

p

at steady state for an inje tion pressure equal

to 1 MPa at GPK1 and a produ tion pressure equal to -1 MPa at GPK2.

The velo ityeld is non-homogeneous in spa e in the vi inityof the wells.

The horizontal omponent of the velo ity between the wells is equal to

v

p,x

= 1.44 × 10

−5

m/s in average. . . 361

8.4 (left)Relativetemperatureprolesand(left)pressureproles,onthe

Soultz-sous-Forêts site at

y = 0

m. Thethermal frontpropagates towardsthe

pro-du tion well at a highunrealisti rate due to the lo al thermalequilibrium

assumption. Thepressureeldisundisturbedbythe oolingfront.

Further-more,the inuen e ofthe me hani al ouplingson the temperature andon

the pressureelds arenegligible. . . 364

8.5 Horizontal displa ement prole; on the Soultz-sous-Forêts site at

y = 0

m.

The horizontal displa ement is ontrolled bythe propagationof the ooled

8.6 Ee tive stressesproles

σ

x

(left) and

σ

y

(right)on the Soultz-sous-Forêts site at

y = 0

m. Tensile stresses are ounted positive. Cooling indu es

thermal ontra tionofthesolidskeleton,whi hindu esanin reaseoftensile

ee tive stress. . . 365

8.7 S aledvelo ityve tors

v

p

ofa18m reservoirwith a36m pie eofformation

lo ated at (

x, 18 < y < 36

m). The velo ity eld is non-monotoni in the

vi inityof the wells and the x- omponent of the velo itybetween the wells

isequal to

v

p,x

= 1.44 × 10

−5

m/s in average. . . 368

8.8 Relative temperature proles(left)and horizontal displa ement proles

u

x

(right)onthe Soultz-sous-Forêts siteat

y

=0 m,with a pie e offormation

on the

y

-boundary of the reservoir. The hot formation does not modify

signi antly the temperature prole; whereas the horizontal displa ements

aresigni antly redu ed ompared with Figure 8.4. . . 368

8.9 Horizontal ee tive stressesproles

σ

x

(left) and

σ

y

(right) on the

Soultz-sous-Forêtssite at

y = 0

m, with a pie eof formationon the

y

-boundaryof

the reservoir. The hotformation redu es the thermallyindu ed in rease in

ee tive stress ompared with Figure 8.6. . . 369

8.10 Contour of the mixture temperature

T

[

◦

C℄, with a pie e of formation on

the

y

-boundary of the reservoir, at 3.81months. The buer zone brings a

sharpinternal layer thatrequiresthe useof the SUPGstabilisation method. 369

8.11 (left) Gringarten analyti al setup for a single ra k embedded in a

semi-inniteimpermeablero k. (right)A ontinuoussingleporousmediumsetup

for the nite element analysis. The reservoir is modeled with 30 elements

of size

10 × 100

m

2

forming a total volume of

300 × 100

m

2

.

n

p

is xed to

0.003. Theimposeduid velo ityalong

x

is equal to

v

p

= 5 × 10

−5

m/s. . 371

8.12 (Left) Relative outlet temperature versus dimensionless time. (Right)

Rel-ative temperature prole at times 3.7months and 32years. Gringarten

an-alyti al solution is presented in eq. (8.44). LTE stands for lo al thermal

equilibrium and LTNE for lo al thermal non-equilibrium. The LTNE

sim-ulation uses an inter-phase heat transfer

κ

sp

= 1.0 × 10

−2

W/m

3

.K found

bytrialanderror to`bestt'theanalyti al response. Thegapbetween the

analyti al solution and the LTNE response is attributed to the estimation

of the uid porosity

n

p

. In spite of the signi ant dieren e between the

urves,theLTNEresponseis losertotheanalyti alsolution omparedwith

8.13 Relativeoutlettemperaturesofthesolidandtheuid phasesversus

dimen-sionless time. The solid and the uid phase remain in LTNE until

t

d

=

20. . . 372

8.14 Con eptualrepresentationoftheFentonHillreservoir,inspiredfromZyvoloski

et al. (1981, Figure 3-2). The inje tion well is denoted EE-1 and the

pro-du tion well is denoted GT-2. The extent and the amount of fra tures

linkingthe two wellsarenotpre isely knownandareindi atedherefor the

illustration. . . 375

8.15 Representation ofageneri HDRreservoir(notat s ale). Thepermeability

k

f

ofthereservoirdependsontheaveragefra turespa ing

B

andonthe

av-eragefra ture aperture

2 b

. Thesimulationsassumeaplanestrainanalysis,

in the

x − z

plane, and symmetrywith respe tto

z

-axis. . . 386

8.16 Thermal,hydrauli and me hani al boundary onditions.. . . 388

8.17 Dimensionlesstemperature outlet

T

D

asafun tionof timefor three

porosi-ties

n

f

and three fra ture spa ings

B

. All results are for

Z

R

= 230

m,

v

∞

= 2 10

−4

m/s and thermal properties from Table 8.4. In LTNE, time

prolesdisplayadouble-step pattern,whereasinLTE,time prolesdisplay

a ontinuous pattern. The thresholds between LTE and LTNE are

asso i-ated with the fra ture spa ings

B

T

= 6

m (left),

B

T

= 2.5

m (middle) and

B

T

= 2

m (right). The orresponding dimensionless values of

η

D

are equal

respe tively to 11, 13 and 10. Therefore,LTE is asso iated with

η

D

larger

than, say13, whilevalues of

η

D

smallerthan 13requires a LTNE analysis.

Thelateovershootingos illationsarenumeri alartifa tsduetoanimperfe t

damping ofthe onve tive ontribution, seeRemark8.3. . . 390

8.18 Dimensionless temperature outlet

T

D

asa fun tion of time, for a porosity

n

f

= 0.005

, three owrates

v

∞

and three fra ture spa ings

B

. All results

arefor

Z

R

= 230

m and thermal properties fromTable8.4. In LTNE, time

proles display a double-step pattern, whereas, in LTE (

B = 0.5

m), time

proles display a ontinuous pattern. The thresholds between LTE and

LTNE express in terms of fra ture spa ing

B

T

= 7.5

m (left),

B

T

= 2.5

m

(middle),

B

T

= 1.5

m (right). The resulting dimensionless threshold

η

D

≃

8.19 Square of the fra ture spa ing threshold

B

2

T

as a fun tion of the average

uid velo ityat steady state

v

∞

, for a porosity

n

f

= 0.005

, a uniform ow

pathandaninsulatedreservoir. Allresults arefor

Z

R

= 230

mandthermal

propertiesfromTable8.4. Thehyperboli relationshipdened byeqn(8.99)

isrepresented by ablue line. It iswell aptured bythe nite element (FE)

simulation. . . 393

8.20 S aled uid velo ityve torsin the reservoir. Wells (thi khorizontal lines)

penetrate the reservoir either totally (left) or partially ( enter, right). The

reservoir is insulated from the ro k formation (left, enter) or ex hanges

heatwith the formation (right). . . 394

8.21 Same asFig.8.19 but for a non-uniform ow path and heat ex hange with

a formation of 30m width. The drawdown results orrespond to the tipof

the produ tion well, i.e.

x = X

W

= 60

m. A non-linear non-monotoni

response isobtained fromthe niteelement (FE) simulations inopposition

with the power response suggested by eqn (8.99). Heat ex hange between

the reservoir andthe ro k formationrequiresthe useofthe SUPGmethod,

Remark8.5. . . 395

8.22 Fluid temperature ontours for

v

∞

_{= 2.0 10}

−4

m/s, at

t = 3.18

years,

a - ounting for heat transfer with the ro k formation and for a non-uniform

ow eld. The Galerkin method (left) displays spuriousnumeri al wiggles,

whi h arepartly uredbythe SUPGmethod (right). . . 396

8.23 Relative temperature outlet

T

D

versus time

t

along the produ tion well at

x

=60m. Experimental datapertaintodierentdepths, namely

◦

2703m,

⋄

2673m,

×

2626m and

✷

in the asing 2660m. Colors are available on the

ele troni version. The experimental temperatures at day one result from

thespatialheterogeneityalongthe produ tionwell,seetext. (left)Uniform

ow eld,

k

f

= 8.0 10

−15

m

2

and

n

f

= 0.005

. Optimum spe i inter-phase

heat transfer oe ient

κ

sf

= 33.0

mW/m

3

.K. (right) Non-uniform ow

eld,

k

f

= 2.35 10

−14

m

2

and

n

f

= 0.005

. Optimum spe i inter-phase

heattransfer oe ient

κ

sf

= 30.0

mW/m

3

8.24 Proles, along the

z

-axis at

x = 60

m, of uid temperature (left) and

solid temperature (right), with

n

f

= 0.005

,

k

f

= 8.0 10

−15

m

2

,

κ

sf

=

33.0

mW/m

3

.K and a uniform ow eld. P.W. stands for produ tion well

and I.W. for inje tion well. The early wiggles near the inje tion well are

numeri al artifa ts due to an imperfe t damping of the onve tive

ontri-bution, see text. The thermal depletion of the uid phase is signi antly

ahead ofthe thermal depletionof the solidphase. . . 399

8.25 Proles,alongthe

z

-axisat

x = 60

m,ofuid pressure

p

f

(left)andrelative

uid pressure with respe t to the LTE response (right). Same parameters

as Fig.8.24. The uid pressure rea hes qui kly steady state and remains

undisturbed in spite of the thermal depletion of the uid phase and of the

thermal ontra tion ofthe solid phase. . . 399

8.26 Proles, along the

z

-axis at

x = 60

m, of verti al strain (left) and lateral

strain(right). Sameparameters asFig.8.24. Extension is ounted positive.

The simulations assume a plane strain analysis,

ǫ

y

=0. Contra tion of the

solid phase,

∆ǫ

z

< 0

and

∆ǫ

x

< 0

, is ontrolled by the solid temperature

response and,thus, develops in the late period. . . 400

8.27 Proles, along the

z

-axis at

x = 60

m, of the indu ed hange in verti al

ee tivestress(left),lateralee tivestress( enter)andoutofplaneee tive

stress (right). Same parameters as Fig.8.24. Tensile stresses are ounted

positive. The ontra tion of the solid phase indu es the ee tive lateral

stresstobetensileneartheinje tionwell(

z < 120

m)and ompressivenear

the produ tion well (

z > 120

m). . . 401

8.28 Same as Fig.8.27 but with a rened mesh in the vi inity of the inje tion

well. The prole at 31.0 years orresponds to the omplete ooling of the

reservoir.. . . 401

8.29 Representation ofa generi HDRreservoir. The exa t onve tive owpath

isunknown andonly the average fra ture spa ing

B

and

n

f

theporosityof

thefra turenetworkarerequiredtoobtaintheaveragefra tureaperture

2 b

,

eqn (8.165).

k

p

and

k

f

denote the permeabilities of the porous blo ks and

ofthefra turenetwork,respe tively. Thesimulations assumeaplanestrain

analysis, in the

x − z

plane. Symmetry with respe tto

z

-axis isassumed. . 422

8.30 Thermal, hydrauli and me hani al boundary onditions (BC). Symmetry

8.31 Relative temperature outlet

T

D

= (T

0

_{− T}

f

(z = z

R

))/(T

0

− T

inj

)

versus

time in days. LTE stands for lo althermal equilibriumand is obtained for

κ

sf

= 100

W/m

3

.K. Colors are available on the ele troni version. (left)

Fenton Hill hot dryro k reservoir with

k

f

= 8.0 10

−15

m

2

and

n

f

= 0.005

.

Field data pertain to

◦

2703m,

⋄

2673m,

×

2626m and

✷

in the asing

2660m (Zyvoloski etal.,1981). Theoptimumspe i solid-to-fra ture uid

heattransfer oe ient

κ

sf

is equal to

33

mW/m

3

.K.(right)Rosemanowes

hot dry ro k reservoir with

k

f

= 3.2 10

−14

m

2

,

n

f

= 0.005

. Field data

pertainto

◦

the asingshoeoftheprodu tionwell(

≈ 2125

mintrueverti al

depth)(Kolditzand Clauser,1998). Theoptimum spe i solid-to-fra ture

uid heattransfer oe ient

κ

sf

liesinthe range60to 120mW/m

3

.K.The

late overshooting os illations for the LTE solution are due to an imperfe t

damping ofthe onve tive ontribution (Se tion8.3). . . 428

8.32 Fenton Hill reservoir, late time (

t = 1.9

years) verti al proles of the

tem-peratures of solid and pore uid (top-left), the temperature of fra ture

uid (top-right), the pressure of pore uid (bottom-left), and the

pres-sure of fra ture uid (bottom-right) for

k

f

= 8.0 10

−15

m

2

,

n

f

= 0.005

,

κ

sf

= 33

mW/m

3

.K and

B = 13

m (Se t. 8.4.7.1). I.W. stands for

inje -tionwellandP.W.forprodu tionwell. Theresponsesofthevariousmodels

mat hforthetemperaturesandforthefra tureuidpressure. Ontheother

hand,theporepressureresponseofthe dualporositymodeldisplaysa

pres-sure drop near the inje tion point. The magnitude of the pressure drop is

ontrolled by the diusivity ratio

R

p

and is larger for smaller pore

perme-ability (

k

p

= 10

−21

m

2

). The single porosity model leaves out of a ount

the porepressureresponse. Regarding the porepressure, the dualporosity

response for

k

p

= 10

−18

m

2

is bounded by the dual porosity response for

smallerporepermeabilities andbythe fra ture uid pressure(1P), lose to

8.33 FentonHillreservoir,latetime(

t = 1.9

years)verti alprolesofthe hanges

in verti al ee tive stress (left), lateralee tive stress (middle) and

out-of-plane ee tive stress (right). Tensile stresses are ounted positive. Owing

tothe porepressure ontribution, thestress responsesdes ribedbythe

sin-gle porosity model (1P) are not equivalent to the responses des ribed by

the dual porosity model. The single porosity model predi ts a thermally

indu ed tensile stressin the vi inityof the inje tion well, whereasthe dual

porositymodelpredi ts a smallertensile stress (

∆¯

σ

x

and

∆¯

σ

y

) and an

in- reaseof ompressivestress(

∆¯

σ

z

). Theporepressuredrop ounterbalan es

the ontra tion indu ed by the solid temperature. As expe ted, the dual

porosityresponsewith

k

p

= 10

−18

m

2

isboundedbythesingleporosityand

by the dual porosity with

k

p

= 10

−21

m

2

responses, lose to the inje tion

well

z/z

R

< 0.3

. . . 432

8.34 Fenton Hill reservoir, early time (

t = 34.72

days) verti al proles of uid

temperatures(left),anduid pressures(right), andjumpins aled hemi al

potential at early and late times (middle). At early time, the temperature

of the fra ture uid de reases, whereas the temperature of the pore uid

remains high. This large dieren e asso iated with a negative dieren e

in s aled hemi al potentials

G

p

/T

p

− G

f

/T

f

< 0

indu es a large

trans-fer of mass from the fra ture network towards the porous blo ks, whi h is

hara terized by a signi ant pore pressure drop, while the fra ture uid

pressureremains undisturbed asin Figure8.32. This behaviormat hes the

observations ofeld experimentsMurphyet al. (1981). . . 433

8.35 Fenton Hill reservoir, late time (

t = 1.9

years) verti al proles of solidand

pore uid temperatures (top-left), fra ture uid temperature (top-right),

pore uid pressure (bottom-left)and fra ture uid pressure(bottom-right)

for

k

f

= 8.0 10

−15

m

2

,

k

p

= 10

−20

m

2

and

n

f

= 0.005

. A small fra ture

spa ing

B

represents a dense fra ture network that overlaps the response

of the single porosity model, whereas a large

B

represents a sparsely

fra -tured reservoir. The fra ture spa ing

B

ontrols the departure from both

hydrauli and thermal equilibria. Hydrauli equilibrium is not re overed

unless thermal equilibrium is attained, whi h takes pla e only for

B → 0

.

Solely, thefra tureuidpressureisnotinuen edbythefra turespa ing

B

.

Theovershootingos illationsforthe temperature solutions near

z/z

R

= 0.2

areagain due to animperfe t damping ofthe onve tive ontribution, Se .

8.3,whi h slightlypollutesthestress andthestrainresponses(Figures8.36

8.36 FentonHillreservoir, latetime(

t = 1.9

years) verti alprolesofthe hanges

in verti al ee tive stress (left), in lateral ee tive stress (middle) and in

out-of-plane ee tive stress (right) for

k

f

= 8.0 10

−15

m

2

,

k

p

= 10

−20

m

2

and

n

f

= 0.005

. For small fra ture spa ings

B → 0

, hydrauli andthermal

equilibria arerea hed andthe hangesin ee tive stressaretensile lose to

the inje tion area

z/z

R

< 0.3

. In addition, the single porosity response is

wellre overed. Thedualporositymodelrevealsthatlargefra turespa ings

B

redu etheporepressureandthereforetheee tivestress

σ

¯

= σ+ξ

p

p

p

I+

ξ

f

p

f

I

ismore ompressive. . . 435

8.37 FentonHillreservoir, latetime(

t = 1.9

years) verti alprolesofthe hanges

inverti alstrain(left)andinlateralstrain(right)withthesameparameters

asinFigure8.36. Forsmallfra turespa ings

B → 0

,hydrauli andthermal

equilibriaarerea hedandnegativestrains losetotheinje tionarea

z/z

R

<

0.3

hara terize a sharp thermally indu ed ontra tion. In addition, the

singleporosity response is well re overed. The dualporosity modelreveals

that largefra ture spa ings

B

redu ethe thermally indu ed ontra tion of

the ro k in the vi inity of the inje tion well and thereby the potential for

apertureenlargement of the mi ro-fra turesor pores. . . 435

8.38 S aled hemi al potential [J/kg.K.10

3

℄ as a fun tion of temperature and

pressure in a range appropriate to the present analysis. The material data

areissuedfromTables 8.6and 8.8. (left): theinje tion state ofthe

ir ula-tion test is taken asreferen e; (right): the triple point of water istaken as

areferen e. . . 436

8.39 Fenton Hill reservoir, late time (

t = 1.9

years) verti al proles of jump in

s aled hemi al potentials (left), pore uid pressure (middle), and hange

in verti al ee tive stress (right) for

k

f

= 8.0 10

−15

m

2

,

k

p

= 10

−20

m

2

and

n

f

= 0.005

. I.S. : the inje tion state of the ir ulation test is used as a

referen e. T.P.W. : the triple point of water is used as a referen e. Mass

transfer is larger for T.P.W. Although the pore pressure drop dissipates

more ee tively for larger mass transfer, the onsequen eson the reservoir

response are quite small. For the pore uid pressure and the hange in

8.40 Fenton Hillreservoir lose to the inje tionwell

z/z

R

≤ 0.1

, verti al proles

ofporepressure(left)and hangein verti alee tivestress(right)for

k

f

=

8.0 10

−15

m

2

,

k

p

= 10

−21

m

2

,

B = 13

m and

n

f

= 0.005

. The hydrauli

hara teristi timeis

t

H

≈ 40

days. Fortimes loseto

t

H

,the porepressure

remainshighand negative. However, for

t > t

H

the ex essiveporepressure

an dissipatein the fra ture network. . . 438

G.1 Contoursofuidtemperatureforadimensionlessux

v

∞

_{= 9.0 10}

−3

m/sat

time

t = 0.38

hour, a ounting for heattransfer with the surrounding, with

the SUPG stabilization (left) and with the SUPG and DC stabilizations

(right). The DC stabilization ee tively ures some overshootings that are

1.1 Summary ofporoelasti onstantsfor various porousmaterials, reprodu ed

from Cowin (2001, p. 23, Chapter 18). Data on bone orresponds to the

la unar- anali ular porosity (PLC). Data on granites represent the range

fortwo granites,and dataonsandstones indi atetherangea rosssix

sand-stones. DataarefromDetournayandCheng(1993)andZhangetal.(1998),

where details ofspe i ro ks aregiven. . . 41

1.2 Classi ation of porous media with respe tto void spa e and solid matrix

onne tivity, reprodu ed from Bear and Ba hmat (1991). PM = Porous

medium, MC=Single multiply- onne ted domain only, SC =Ensembleof

simply- onne ted domain only, MSC =Combination of MCand SC, IP =

Isolatedpores, FB=Fluidised bed.. . . 42

1.3 Fra tured media lassi ation based on the omparison of the storage

a-pa ityofthe twosub-systems(Bai etal.,1993). The porosityofthe porous

blo kisdenoted

n

p

and the porosityofthe ssure network isdenoted

n

f

. . 43

1.4 Illustration of the ee tive porous medium approa h and of the mixture

theorybasedontheaveragingpro ess. Theaveragingpro essisrepresented

by dashedlines. The illustrationof the ee tive porousmedium is derived

fromCowin (2001). . . 46

1.5 Illustrationof the homogenisation approa h and ofthe statisti al approa h

based on the averaging pro ess. The averaging pro ess is represented by

dashed lines. The illustration of the homogenisation approa h is derived

fromHornung(1997). Theillustrationofthestatisti alapproa hisa

three-dimensionalrepresentationofafra turenetworkwithinagraniteblo k

1.6 Classi ationofmultiporosity/multipermeabilitymodels and

re ommen-dations for pra ti al appli ationsfrom Baiet al.(1993). The porosities are

denoted as in Table 1.3. The subs ript

T

refers to a total uid quantity,

that is the sum over all the sub- avity systems. The permeability of the

porous blo k is denoted

k

p

and the permeability of the ssure network is

denoted

k

f

. Inthe last ase,thesubs ript

f 1

referstonatural fra turesand

f 2

refersto arti iallyenhan ed fra turesor fault gouges. . . 53

2.1 Thermodynami almeasuresofthestate. Notethatthein rementdenitions

are arbitrary; therefore the total derivative

d(·)

may be repla ed by the

gradient operator

∇(·)

or bythe partialderivative

∂(·)

.. . . 121

2.2 Some physi al properties of pure water, from http://www.engineeringtool

box. omand

∗

fromKestin (1968, p. 541). . . 128

2.3 Dire tand oupleddiusionphenomena from(Mit hell,1993,p. 230). The

phenomena a ounted for in this modelarehighlighted in boldletters. . . . 148

2.4 Set ofeld equations andlist ofprimary unknowns . . . 164

2.5 Set of onstitutive equations,

k

∗

=

p

,

f

,

k

=

s

,

p

,

f

and

l

=

p

,

f

,

s

.. . . 165

2.6 Origins ofthe oe ients usedin equation (2.309). . . 173

2.7 Origins ofthe oe ients usedin equation (2.312). . . 175

2.8 Origins ofthe oe ients usedin equation (2.313). . . 176

2.9 Coe ients ofthe model . . . 179

2.10 Sket h of the repartition of the oupling oe ients in the model. The

model displays non symmetry. The diagonal or dire t oe ients are not

represented here. . . 180

3.1 Range ofvalues ofvariousdrainedYoung's modulifromPhilipponnat etal.

(2003). . . 185

3.2 Bulk moduli of typi al soil minerals from Gebrande (1982) and Fran ois

(2008). . . 187

3.3 Range ofvalues of ompressibiliy ratiosfor dualporousmedia. . . 188

3.4 Bulkmoduli ofvarious uids, at atmospheri pressure and20

◦

C. . . 189

3.5 Range of values of the porous blo kporosity

n

p

and of the ssure network

porosity

n

f

from the literature. . . 190

3.6 Range of values ofthe porous blo kpermeability

k

p

and of the ssure

net-work permeability

k

f

from the literature.. . . 191

3.7 Linear thermal expansion oe ients, of typi al lay minerals, for a

tem-perature range between 25

◦

Cand 100

◦

3.8 Volumetri thermal expansion oe ient of i e and pure water at

atmo-spheri pressure(Kestin, 1968,p. 541). . . 195

3.9 Density oe ients ofvarious soils(Burger et al.,1985,p. 139). . . 196

3.10 (left) Density variation with temperature for pure water, at atmospheri

pressure, from www.engineeringtoolbox. om. (right) Spe i volume

(in-versedensity) ontourwithpressureandtemperatureasdenedinequation

(2.158). . . 196

3.11 Dynami vis osity ofpure water, from www.engineeringtoolbox. om . . . . 197

3.12 (left) Spe i heat apa ity of dierent soils, from www.engineeringtool

box. om. (right) Spe i heat apa ity of pure water at onstant

atmo-spheri pressure(Kestin, 1968,p. 541). . . 198

3.13 (left)Thermal ondu tivityofdierentsoils,fromwww.engineeringtoolbox. om.

(right)Thermal ondu tivityof pure water (Burgeret al.,1985, p137). . . 199

3.14 Thermo-osmosis oe ientsfor ompa t lays-water systemfrom the

liter-ature. . . 200

3.15 Spe i surfa e areavalues forvarious materials (de Marsily,1986, p. 22). . 201

4.1 Shape fun tions asso iated to a four node bilinear quadrilateral referen e

element. The nodal pointsare labeled in as endingorder orresponding to

the ounter lo kwisedire tion. . . 220

4.2 Typi al numbersof the generalisedtrapezoidal methods. . . 227

5.1 Loading andboundary onditions. . . 235

5.2 Solidanduidparameters,representativeofgraniteandwater,respe tively,

at 25

◦

C and atmospheri pressure. The thermal diusivitiesaredened in

eq. (5.3). The hara teristi diusion times are al ulated with eq. (5.2)

and

H = 3

m. . . 236

5.3 A ombination of four boundary onditions isproposedbyM Tigue(1986). 241

5.4 Loading values. 1: The heat ux is only applied when onsidering a

on-stant heat ux loading. 2: The imposed temperature is only applied when

onsidering a onstant temperature loading. . . 247

5.5 Materialpropertiesofanabyssalred lay(Illite), NorthPa i O ean

(M -Tigue, 1986,p. 9540).

∗

Denitions of these parameters are provided in eq.

(5.3) and(5.24), respe tively. . . 247

5.6 Comparisonbetween the hydrauli andthe thermal ontributions for a

dif-fusivity ratio

R = 1

.

δ

is of order of magnitude of one: O

(10

0

₎

and the