Hydrothermal coupling in a self-affine rough fracture

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Amélie Neuville, Renaud Toussaint, Jean Schmittbuhl

To cite this version:

Amélie Neuville, Renaud Toussaint, Jean Schmittbuhl. Hydrothermal coupling in a self-affine rough

fracture. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical

Society, 2010, 82, pp.036317. �10.1103/PhysRevE.82.036317�. �hal-00553182�

Hydro-thermal oupling in a self-ane rough fra ture A. Neuville 1,2 ,

∗

R. Toussaint 1,2 , and J. S hmittbuhl 1,2 1 EOST, Université de Strasbourg, Fran e and

2 Institut de Physique du Globe de Strasbourg, UMR CNRS ULP 7516,

5 rue René Des artes, 67084, Strasbourg Cedex, Fran e

Theinuen eofthemulti-s alefra tureroughnessontheheatex hangewhena olduidenters a fra tured hot solidis studied numeri allyonthe basis of the Stokesequation and inthe limit of both hydro- and thermo-lubri ations. The geometri al omplexity of the fra ture aperture is modeledbysmallself-aneperturbationsaddedtoauniformapertureeld. Thermalandhydrauli propertiesare hara terizedviathedenitionofhydrauli andthermalaperturesbothatmi roand ma ros opi s ales and obtained by omparing the uxes to the ones of atfra tures. Statisti s over a large numberof fra ture ongurations provide an estimate of the average behavior and itsvariability. Weshowthat thelong range orrelations ofthe fra ture roughnessindu esstrong hannellingee tsthatsigni antlyinuen ethehydrauli andthermalproperties. Animportant parameter is the aspe t ratio (length over width) of the fra ture: we show for example that a downstreamelongatedroughfra tureismorelikelyto inhibitthehydrauli owand subsequently to enhan e the thermalex hange. Fra tureroughness might, intheopposite onguration, favor strong hanneling whi hinhibitsheating oftheuid. Thethermalbehaviorisingeneralshownto bemainlydependentonthehydrauli one,whi hisexpressedthroughasimplelaw.

PACSnumbers: 47.56.+r44.05.+e47.11.B 44.30.+v

I. INTRODUCTION

Amongsituationswhereheatex hangebetweena

pass-ing uid and a fra tured medium is of entral

impor-tan e,geothermyisanintensivelydevelopingeld. Deep

Enhan ed GeothermalSystems(EGS) arebasedon the

energyextra tion obtainedwhena olduid is inje ted

from the surfa einside a hotfra tured massifat depth

andextra tedafter ir ulationintheopenfra tures

pos-sibly arti ially reated from an hydrauli or hemi al

stimulation (e.g. the EGS pilot plant in

Soultz-sous-Forêts,Fran e[14℄).

The e ien y of the heat ex hange depends on the

balan e between ondu tiveand onve tiveheat uxes.

Theformerismainlydependentonthegeometryofea h

individualinterfa e,i.e. fa ingfra turesurfa es,butthe later is denitively related to the hydrauli properties

of the fra ture network whi h results from the network

onne tivityandthefra turepermeability.

Hydrauli ondu tivityoffra turedro kshavebeen in-tensivelystudiedfordierentmotivations. Forinstan e, thehydrauli propertiesofthe rystallineaquiferof Ploe-meur,Fran e,hasbeenstudiedbyLeBorgneetal(2004)

[5℄ to address water supply issues. Another example is

themitigationofradionu lidemigrationwhi h hasbeen

fore asted in the ase of the repository sitefor nu lear

wastes storage in Äspö, Sweden on the basis of a

dis- retefra turenetwork[6℄. Themodelingofthetransport properties of fra ture networks is a tually averya tive

∗

resear harea. A lassi alapproa his tomodel theow

pathsviaparallelintera tingatfra tures[7℄. More ad-van edstudiesaddresstheee tsofthe onne tivityand orrelationsofthefra tures(e.g. in Refs[8,9℄).

Inmanymodelsofhydrauli orhydro-thermalow

de-velopedso far,thegeometryofea hfra tureofthe net-workisnevertheless onsideredassimple,e.g. asparallel plateswithasimplegeometryoftheedgeasanellipseor

apolyhedron. Thisisthe asein mostfra turenetwork

modelsusedforgeothermal[10℄orforuidtransport ap-pli ations[8,9℄. Thenon-trivial hara terofthefra ture aperturegeometryishoweververylikelytoinuen ethe

fra ture ow given their omplex real geometry. Most

natural fra ture surfa es are indeed self-ane obje ts.

Surprisingly,the omplexityofthemultis aleproperties of the fra tures has some remarkablesimpli ity, in the

sensethat theirHurstexponentisveryrobustlyaround

0.8 [11℄. Ex eptions however exist like for fra tures in

sandstoneswhere the Hurst exponentis 0.5 [12, 13℄, or

in glassy erami s whi h showan exponent lose to 0.4

[14℄.

Theaperturebetweenfra turesurfa esissubsequently farfrom at in parti ular iffa ingfra ture surfa es are un orrelated, at least at small s ales[15, 16℄. A rough

self-aneapertureisindeeddened betweentwo

un or-relatedself-anefra turesurfa es,orbetweentwo iden-ti alself-ane fra tures translatedtangentiallyto their average planebyatranslation largerthanthes ale

un-der study. Self-ane apertures have been shown to be

responsible for tortuous ow path. The related

han-neling of the uid ow was experimentally observed in

22℄). The appli ability of su h an approximation has beenstudied, e.g. in Refs[2325℄. Extensionof this

sit-uation was onsidered for exampleby Plouraboué et al

[26℄wheretheReynoldsequationis oupledtothe hem-i al onve tion-diusionstudy. Moreadvan edhydrauli

simulations in luding the solving of the Navier-Stokes

equation have been proposed either onsidering

simpli-edgeometry[23, 27℄ormorere entlywithinarealisti

fra ture geometry [2832℄. However, these re ent

sim-ulationsrequireheavy omputations (although dierent

methods are available) and aretherefore notfully

om-patiblewithstatisti alapproa heswherealargenumber ofrealizationsisne essary.

Beyond the problem of mass owin roughfra tures,

dierent kinds of numeri al simulations have already

beenproposedtoa ountforhydro-thermal oupling. As

arstapproa h,analyti alsolutionshavebeenobtained

to ompute the heat ux along parallel ir ular ra ks

embedded ina3Dinnitemediumusingsimpliedheat

equations [33℄. Atlarges ale and forlongterm predi -tions, models like that of Batailléet al. [10℄ havebeen proposedtopredi ttheresponseofgeothermalreservoirs.

This type of nite-element model in ludes ondu tion,

freeand for ed onve tion, but redu es thegeometry of

thehydrauli networktoadoublepermeability

distribu-tion to a ount forbothmatrix andfra ture transport.

A variety of more omplexmodels have also been

pro-posedlikethemodelingofa3Dnetworkoffra tures orga-nizeda ordingtogeologi alobservationsand ompleted

with sto hasti fra tures for underdetermined parts of

themodel[34℄,orthatforSoultz-sous-Forêts,Fran e,by Ra hezetal [35℄orthat ofKolditzand Clauser[36℄ for

Rosemanowes,UK.

In the present study, we fo us on the hydro-thermal

oupling at the fra ture s ale where the hanneling

ef-fe t is expe ted to ae t not only the uid transport

properties,butalsotheheatuxproperties,assuggested

byKolditzand Clauser[36℄ who proposed that the

dis- repan y between lassi al heat model predi tions and

realobservations ouldbedue toow hanneling

result-ing from fra tureroughness. We aimat obtainingfrom

themi ros opi analysisoftheowats alesofthe fra -tureasperities,thema ros opi parameters(i.e. the hy-drauli transmissivityand the hara teristi

thermaliza-tion length) that governthe e ien y of theuid mass

and heat transport through the overall fra ture. This

willallowto oarsegrain thedes riptionoftheee t of

mi ros opi asperities, i.e. the fra ture roughness, on

the hydrauli and thermal behavior in large s ale

net-work modelsastheones mentionedabove. The

ups al-ing from the mi ros opi asperity s ale to the fra ture

s ale is donevia asystemati statisti al analysis of the ma ros opi owparameters,foralargesetofsto hasti syntheti fra turesurfa es,des ribedwithafewkey

pa-rametersforsu h apertures: average aperture,standard

deviation. The ma ros opi parameters obtained after

theups aling redu e to two: thehydrauli

transmissiv-ity, hara terizingtheuidmasstransport,andtheother

one hara terizingthee ien yoftheheatex hange

be-tweenthero kandtheuid. Thisex hangeisexpressed

viathe hara teristi length

R

in a ma ros opi law of

type

(q/ kqk) · ∇

2

T − (T − T

r

)/R = 0

(1)

with

T

a uid temperature,

T

r

the temperature in the

surrounding ro k,

q

the uid ux integrated over the

fra turethi kness,and

∇

2

thetwodimensionalgradient

operator along the fra ture plane. The hydro-thermal

modeling is performed as in [3℄. The present study is

in the framework of the lubri ation approximation [37℄

whi h implies that the Reynolds number is small and

thatthefra turewallsarelo allyatenoughtoprovide

amainly in-plane velo ity eld, with a negligible

om-ponentnormalto themean fra tureplane. We propose

toextend thelubri ation approximationto the thermal

uxes. Bybalan ingheat ondu tion andfor ed

onve -tion we obtain a tri-dimensional (3D) temperature law

whi h will then be redu ed to a2D temperature

equa-tionbyaveragingitalongthethi knessofthefra tureas

proposede.g. byTur otteandS hubert[38℄.

Se tion IIdes ribesourgeometri almodelofthe fra -tureaperturebasedonaself-anes alinginvarian e. In Se tionIII,using lubri ationapproximations,weobtain

thebidimensionalpressureandthermal equationswhen

a old uid is inje ted through a fra ture in a

station-aryregime. Asa rststep, thetemperaturewithin the

surroundingro kissupposedtobehotand onstant(in

timeandspa e),andthedensityoftheuidis onsidered as onstant. Weshowthatata oarsegraineds ale,the

twodimensional (2D) equation forheat ux is identi al

totheoneforparallelplates, Eq.(1),butwitha hara -teristi thermalizationlength asso iatedto an aperture

(namedthermalaperture),dierentfromthegeometri al

aperture(alsooftenlabeledastheme hani alaperture). Otherrelevantquantitiesaredened todes ribethe

hy-drauli and thermalbehaviorsat lo aland ma ros opi

s ales. The numeri al approa h is des ribed in details

in Se tion IV. Equations are dis retized using a nite

dieren es hemeandsolvedwithabi onjugategradient

method. The numeri al hydrauli and thermal results

are respe tively set out in IV and V. In ea h of these

se tions,werstdes ribetheresultsforagivenfra ture

morphology(lo allyandma ros opi ally),thenaveraged

trendsofma ros opi parametersthatareobserved

sta-tisti allyfromlargesets ofsyntheti fra tures.

II. DESCRIPTION OFTHEROUGHNESS OF

THEFRACTUREAPERTURE

Theroughnessofaself-anesurfa eisstatisti ally

in-variantupon anisotropi s aling within its meanplane

(x, y)

whileontheperpendi ulardire tion

z

,thes aling isanisotropi . Indeed, itis statisti ally invariantunder

thes aling transformation

x → λx, y → λy, ∆z → λ

ζ

_z

exponent. A self-ane geometri al model has been ex-perimentally shown to be a realisti des ription of nat-uralro ksurfa es [11, 16, 42, 43℄, withHurst exponent

equalat larges ale to

ζ ≃ 0.8

for manykinds of

natu-ral fra tures and material surfa es [4245℄ and

ζ ≃ 0.5

for sandstones [46, 47℄. It is important to note that a

self-ane surfa e having a roughness exponent smaller

than oneis asymptoti ally atat large s ales[48℄.

A - ordingly, aself-anetopography anbeseenasa

per-turbation of a at interfa e. On the other end of the

s ales, thelo al slope ofaself-ane surfa edivergesat

small s ales, and the maximum slopeof su h surfa e is

determinedbythelower utooftheself-anebehavior

- orrespondinge.g. togranulardiameterwhenpresent.

In prin iple, modeling a ow boundary ondition along

su h surfa erequires to he kthat the ma ros opi ally

obtainedresultdoesnotdependonsu hlower uto.

Theaperture isthespa ebetweenthe fa ingfra ture

surfa es. Ourstudyislimitedtothe asewheretwonon orrelatedfra turesurfa eswiththesameroughness ex-ponentsarefa ingea hother. Subsequentlytheaperture

a(x, y)

isalso aself-anefun tion with thesameHurst exponentwhi hfullls thefollowingproperty[39,40℄:

λ

ζ

Pr(λ

ζ

∆a, [λ∆x, λ∆y]) = Pr(∆a, [∆x, ∆y])

(2)

where

Pr(∆a, [∆x, ∆y])

istheprobabilitytogetan

aper-turedieren e

∆a

between twopoints separatedbythe

distan e

[∆x, ∆y]

,

λ

isanarbitrarys aling fa torand

ζ

theroughnessexponent.

The self-ane aperture eld is numeri ally obtained

byrstgeneratingawhitenoise

ǫ(x, y)

[49℄onagridof size

2·n

x

×2·n

y

withasquaremesh-size

d

. Thenthe sta-tisti alspatial orrelationsareintrodu edbymultiplying the2DFouriertransformofthewhitenoise

˜

ǫ (k

x

, k

y

)

by

kkk

(−1−ζ)

[50℄, where

k

is the wave ve tor. When

de-sired,alower utolengths ale

l

c

anbeintrodu edby ltering as: if

kkk ≥ π/l

c

,

a (k

˜

x

, k

y

) = 0

. Finally we perform the inverse Fourier transform of

˜

a (k

x

, k

y

)

and normalize it to geta syntheti aperture

a(x, y)

with an

average

A

andarootmeansquare(RMS)

σ

. Using

dier-entseedsoftherandomgeneratorofthewhitenoise,itis possibletogenerateindependentself-aneaperture

mor-phologies showing dierent patterns, even if they share

the same roughness exponent hosenequal to

ζ = 0.8

,

thesamemeanaperture

A

andsameRMS

σ

. Theupper

limitof

σ

isprovided by the onditionof positive aper-ture,i.e. weprevent onta tbetweenthe fra turefa es

tokeepa onstantsimpleboundarygeometryofthe

do-main where the equations are solved. In pra ti e

a

is

imposed to range between

10

−

4

and

10

, whi h leadsto

0.7 > σ/A > 10

−

3

. Thetypi algridsizesthat wereused are:

1024 × 1024

,

1024 × 2048

,

1024 × 512

. Themeshsize

d

hasbeenadjustedtogetasu ientnumeri alpre ision ofthetemperaturesolutioninthe aseofaparallelplate

ongurationwhereananalyti alsolutionisknown. The

numeri alstability ofthe solutionshasalso been tested

against slight shifts of the mesh position on an

over-sampledself-aneapertureeld:

2·n

x

×2·n

y

= 2

12

_×2

12

,

Figure 1: 2D sket h of the fra ture modelwith parameter denitions.

x−

axisisalongthemeanhydrauli ow,y-axisis alongthemeanfra tureplanebutperpendi ulartothemain hydrauli ow and

z−

axis denotes the out-ofmean fra ture planedire tion.

z = z

1

and

z = z

2

are theaverage positions ofthefa ingfra turesurfa es.

a(x, y)

isthefra tureaperture.

T

r

isthe temperature of the solid, supposed to be homoge-neousand onstant,

T

0

is theuidtemperatureattheinlet. Fluidpropertiesare:

ρ

,

c

,

χ

,and

η

respe tivelydensity,heat apa ity,thermaldiusivityanddynami vis osity.

and against the introdu tion of a lower uto

l

c

of the

self-ane perturbations varying between the mesh size

and10timesthemeshsize: thederminedowand

tem-perature elds were found to be independent of su h

smalls alemodi ations.

III. HYDRAULIC ANDTHERMALFLOW

EQUATIONS

A. Hydrauli ow

We onsider the steady ow of a Newtonian uid at

low Reynolds number, so that the vis ous term of the

Navier-Stokesequation dominatestheinertialone. The

Navier-Stokesequationisthereforeredu edtotheStokes equation[51,52℄:

∇

_{P = η∆v,}

(3)

where

η

is the dynami vis osity,

v

the velo ity of the

uidand

P

thepressuredeviation from thehydrostati

prole (i.e. the hydrauli head whi h is equal to the

pressure orre ted by the gravity ee t). To be in the

frameworkofthelubri ationapproximation[37℄,besides

smallReynoldsnumber, wealso onsider fra tureswith

at enough sides as mentioned above (i.e. with small

lo alslopes). Therefore, uidvelo ityve torsget negli-gible

z

- omponents(normaltothemeanfra tureplane),

and a ordingly the velo ity eld is dominated by

in-plane omponents. Theunitaryve tor

x

ˆ

isalignedwith

thema ros opi imposed pressuregradient(see Fig.1);

z

1

(x, y)

and

z

2

(x, y)

arethebottomandtopfra ture o-ordinates,with

z

2

−z

1

= a

. Undertheseapproximations, thepressuredependen e is

P (x, y)

andthevelo ity

v

is orientedalong the unitary ve tor

v

ˆ

(x, y)

. By

v

(z

1,2

) = 0

, we get a lo al paraboli law in

z

(Fig. 2) [25℄:

v(x, y, z) =

∇

2

P (x, y)

12η

(z − z

1

) (z − z

2

)

(4) where

∇

2

= ˆ

x

∂

∂x

+ ˆ

y

∂

∂y

isthein-planegradientoperator.

T

v

v=0

T=T

r

z=a/2

z

z=−a/2

,

Figure2:(Coloronline)Lo alvelo ityquadrati prole(short dashed line) and temperature quarti prole (long dashed line)insideafra ture(with oe ientsfromEqs.(9)and(4)); arbitraryabs issaunits.Alongthe onta twiththefra ture,

v

= 0

and

T = T

r

.

Integrating Eq. (4) along

z

leads to express the

hy-drauli owthroughthefra turethi kness

q

as:

q

_{= −}

a

3

12η

∇

2

P.

(5)

Furthermore, we assumethe uidto be in ompressible,

i.e.

∇

2

· q = 0

whi hleadstotheReynoldsequation[19℄:

∇

₂

_{· a}

3

∇

₂

_P

_{= 0.}

(6) Asboundary onditionsofthisequation(Fig.3),we im-posethe pressureatthe inletand outlet ofthefra ture (if

x = 0

,

P = P

0

andif

x = l

x

,

P = P

L

,with

P

0

> P

L

)

and onsiderimpermeablesides(nomassex hangewith

thero kmatrix)at

y = 0

and

y = l

y

.

Figure 3: Fra ture model with pressure and temperature

boundary onditions.

B. Thermalow

In this work, we negle t the natural onve tion that

happensinfra turedro kswhentheuiddensityis ther-mally sensitive, asstudied for instan e byBatailleet al

[10℄. Natural onve tionmighthappenwithinthe

thi k-nessofthefra ture(owingtothetemperaturedieren e

betweenthefra ture boundary andthe oreof theow

alongthegravitydire tion)and at larges alewhenthe fra tureisnonhorizontal. Forthesakeofsimpli ity,we

onsider that the for ed uid ow studied here is only

weaklyae tedbydensity hanges. Aquantitative rite-rionofthisassumptionisgivenbythe omparisonofthe

pressuredieren es

∆P

for ingtheowandthat

gener-atedbythetemperature hanges:

∆P ≫ gα

T

ρ∆T,

with

g

thegravity,

α

T

the uid oe ient ofthermal

expan-sion,

ρ

theuiddensity,and

∆T

thetemperature

dier-en es in the system. We also assume that the Prandtl

numberoftheuidissu ientlyhighfortheowto be

dominatedbyhydrodynami ee tsratherthanthermal

ee ts.

Sin eourfo usisto understandhowthema ros opi

massand heat owsare ae ted bythe fra ture

rough-nessinthestationarylimit,wedonot onsidertimeand spa evariationsofthetemperatureinthero k: the fra -turesidesareassumedtobepermanentlyhotatthexed temperature

T

r

. Thissimpli ationisvalidifwe onsider

either long time s ales i.e. when the ro k temperature

prolesstabilizes,ortimes alesshorterthanthat ofthe hostingro kevolution. Takingtheslowtemperature evo-lutionof thehostingro kintoa ountwouldrequireto

ombinethepresentstudywithanon-stationary

ondu -tiveheatsolverforthero kwhi hisbeyondthes opeof thismanus ript. Inprin iple,to model these

intermedi-atetime s ales, thema ros opi parameters ontrolling

theheat ex hange determined in this manus ript ould

beutilizedinahybridmodel, ouplingtheheat diusion-adve tionintheuidwiththeheatdiusioninthesolid.

Lo al energy onservation implies that the uid

tem-perature is ontrolled by the balan e between thermal

onve tion and ondu tion inside the uid whi h reads

as(heatsour eduetofri tionbetweenuidlayersbeing negle ted)[53℄:

v

_{· ∇T = χ∆T,}

(7)

where

χ

isthethermaldiusivityoftheuidand

T

the

uidtemperature. Weextendthelubri ation

approxima-tion( .f.I)by onsideringthattheslopesofthefra ture

morphology are small enough to provide a ondu tion

at thero k interfa e lo ally orientedalong

z

ˆ

. This

im-pliesthat theout-of-plane ondu tion termisdominant

in front of the in-plane ones. Otherwise

v

z

∂T/∂z

an

be negle ted in

v

· ∇T

sin eout-of-plane velo ity

v

z

is negligible. A ordingly theleadingtermsin Eq. (7)are the ondu tion along

z

ˆ

axisandthein-plane onve tion terms,andthisredu es to:

∂

2

T

∂z

2

=

v

x

χ

∂T

∂x

+

v

y

χ

∂T

∂y

,

(8)

Fortheboundary onditions,weassumethattheuid

temperatureis equalto the ro ktemperaturealongthe

fra -tureinlet:

T (x, y, z) −→

x→∞

T

r

. Thetemperatureof

inje -tionat theinlet is

T

0

sothat

T (0, y, z) = T

0

(forany

y

and

z

). Byassumingthat

β = q

x

∂T/∂x + q

y

∂T/∂y

isonly fun tion of

x

and

y

, the following quarti expression of

T

issolutionofEq. (8):

T (x, y, z) = T

r

−

β(x, y)

2 · a

3

_{· χ}

(z − z

1

) (z − z

2

)

·



z −

√

5z

1

 

z −

√

5z

2



(9)

Fortheparti ular aseofsymmetri aperturesaround anaverageplane,i.e. where

z

1

= −z

2

= a/2

,thisredu es to

T = −3 · β z

4

/6 − a

2

z

2

/4 + 5a

4

/96
/ a

3

· χ

+ T

r

.

By uniqueness of the solution for given boundary

on-ditions (the problem is well-posed), this quarti law is

the only solution of Eq. (7). The temperature prole

along

z

isillustratedtogetherwiththevelo ityprolein Fig.2).

The energy onservation equation (Eq. (7)) is

inte-gratedalongthe

z

-dire tion,throughthethi knessofthe

fra ture (as done for the hydrauli des ription), whi h

providesanin-plane des riptionofthethermalbalan e. First,weestimatetheadve tedenergyux. Forthis,we note

c

theuidspe i heat apa ityand

U

0

itsinternal energydensityat

T = T

0

,and writetheinternal energy density

U

as

U = U

0

+ ρc (T − T

0

)

. Integratingalongthe fra ture thi kness ( i.e. along the

z

-axis), leadsto the

internal energy ux per unit volume

f

(x, y) =

R U vdz

whi h anbeexpressedas:

f

(x, y) =

U

0

+ ρc T − T

0

 q(x, y)

(10)

where

T

isaweightedaveragetemperaturedened as:

T (x, y) =

R

a

v (x, y, z) · T (x, y, z) dz

R

a

v (x, y, z) dz

,

(11)

with

T (0, y) = T

0

= T

0

at the inlet. The heat sour e

oming theadve ted energy is then given by:

−∇

2

· f

.

Usingthe mass onservationequation,

∇

2

· q = 0

, leads to:

∇

₂

_{.f = ρcq.∇}

₂

_{T .}

(12)

Theadve tedenergyuxbalan esthe ondu tiveux

through the upper and lower fra ture walls. To

evalu-ate thethermal ondu tiveow orientedalong the

out-going normal to the fra ture walls

n

ˆ

, the lubri ation approximation ( .f. I), leads to

n

ˆ

≃ ±ˆ

z

. Let

ϕ

w

be

the proje tion of the ondu tive ow along

n

ˆ

,

evalu-ated along the walls, at

z

1,2

. The Fourierlaw provides

ϕ

w

= −χρc

∂T

_∂z





z=z

1,2

ˆ

z

· ˆ

n

. Eqs.(9) and(4)inserted in-sideEq.(11),leadto

∂T

∂z





z=z

1,2

= T − T

r

₇₀

17·a

z

ˆ

· ˆ

n

. The Nusseltnumber

Nu = −ϕ

w

/ϕ

ref

= 70/17

isusedto

har-a terizethee ien y ofthepresentheatex hange

om-paredto thereferen eheat ow

ϕ

ref

= χρc T

r

− T

/a

,

whi ho ursinsituations withonly ondu tion.

Theenergynetux:

∇

₂

_{· f + 2ϕ}

_w

_{= 0,}

(13)

annallybeexpressed as:

q

_{· ∇}

₂

T + 2

χ

a

Nu · T − T

r

= 0.

(14)

For the boundary onditions of the two-dimensional

eld

T

,weassumethattheuidisinje tedata onstant

temperature

T (0, y) = T

0

older than the ro k and we

onsiderthelengthof thefra tureto belongenoughto

gettheuidat thesametemperatureasthero katthe

end ofit:

T (l

x

, y) = T

r

. On the ontrary, temperature settingsalongtheboundaries

y = 0

and

y = l

y

are

with-outanyinuen e, sin e the hydrauli owis null there

(seeIIIA).

Let the referen e ase be a fra ture modeled with

two parallel plates separated by a onstant aperture

a

//

(i.e., no self-ane perturbation). In this ase, the gradient of pressure is onstant all along the fra ture,

as well as the hydrauli ow whi h is equal to

q

//

=

−∆P a

3

//

/ (12l

x

η) ˆ

x

, where the subs ript

//

denotes re-sultsvalidforparallelplatesand

∆P = P

L

− P

0

. Under these onditionsEq.(14)isinvariantalong

y

and anbe writtenas:

∂T

//

∂x

+

T

//

− T

r

R

//

= 0,

(15)

wherethethermal length

R

//

hara terizesthedistan e

at whi h the uid rea hes the temperature of the

sur-roundingro k:

R

//

=

a

//

·

q

//

2 · Nu

//

· χ

= −

∆P

l

x

·

a

4

//

24η · Nu

//

· χ

,

(16)

with

Nu

//

= 70/17 ≃ 4.12

. Then theanalyti alsolution ofEq. (15)forparallelplatesis:

T

//

− T

r

= (T

0

− T

r

) exp



−

_R

x

//



.

(17)

Forroughfra tures,weaimatusingEq.(17)asaproxy

oftheaveragetemperatureprole

T

alongthe owand

deninganee tivema ros opi thermallength

R

as:

T − T

r

= (T

0

− T

r

) exp



−

x

R



.

(18)

C. Denition ofmi ros opi and ma ros opi

apertures

Dierent types of fra ture apertures an be dened.

Themostobviousoneisthegeometri alaperturebut ef-fe tiveapertures likehydrauli orthermal aperture an

also beintrodu ed. Thelatter are dened on the basis ofaninversiononaspe i modelliketheparallelplate

model. For instan e, the hydrauli aperture isdedu ed

fromtheknowledgeoftheuidowthroughthefra ture

and it representsthe aperture of aparallel plate model

that reprodu esthe observeduid ow. Equivalently a

thermal aperture an be introdu ed as the aperture of

aparallel platemodel thatreprodu esasimilarthermal prole. Aspatial s alehastoatta hedto denethe hy-drauli orthermal equivalent behaviorin parti ular for

a multi-s ale geometry. Sin e weaim at understanding

theups alingofthefra tureproperties,wewillintrodu e twospe i s ales: thesmallestone,i.e. thegridsize of

the dis retization and the largest one, i.e. the system

size. Thesmallestwillbereferredasthemi ros opi or lo als aleandsmallletterswillbeusedfortheirlabeling

and thelargest, asthema ros opi s ale and des ribed

with apitalletters.

Wealreadyusethemi ros opi geometri alor

me han-i al aperture

a

and its spatial average,i.e. the ma ro-s opi geometri alaperture:

A = ha(x, y)i

x,y

.

Themi ros opi hydrauli apertureisdened asfrom

Eq.(5)[19, 54℄:

h =





kqk

12η







∆P

l

x











1

_/

3

.

(19)

It depends on the lo al hydrauli ow

q

, and an be

related to the lo al pressure gradient

k∇P k

and lo al

aperture

a

as:

h = a ·

k∇P k

∆P

l

x

!

1

/

3

.

(20)

If thelo al pressuregradient

k∇P k

is smallerthan the ma ros opi gradient

∆P/l

x

, then

h (x, y) < a (x, y)

,

whi h means that lo ally the hydrauli ondu tivity is

lowerthanexpe ted from itslo al me hani alaperture.

The ma ros opi hydrauli aperture

H

an also be

de-nedatthesystems alefromtheaveragehydrauli ow

Q

x

= hq · n

x

i

x,y

:

H =



Q

x



12η





∆P

l

x











1

_/

3

.

(21)

Ma ros opi and mi ros opi hydrauli aperture are

re-lated,sin e

H

isa tuallyproportionaltothe ubi root ofthethird ordermomentof

h

:

H = hh(x, y)

3

_i

1

_/

3

x,y

whi h

isproportionaltotherstordermomentofthehydrauli ux,topower

1/3

. If

H/A > 1

,thenthefra tureismore permeablethanparallelplatesseparatedby

a(x, y) = A

.

The ma ros opi thermal aperture is dened from a

1D temperature prole

T (x)

along the for ed pressure

gradientdire tion(seeEq.(18))wheretheaverage

tem-peratureisdenedas:

T (x) =

R

l

y

u

x

(x, y) · T (x, y) dy

R

l

y

u

x

(x, y) dy

.

(22)

It is an average of

T

along the width of the

fra -ture

l

y

, weighted by the lo al uid velo ity

u

x

(x, y) =

q

x

(x, y)/a(x, y)

whi histheratioofthe

x

- omponentof thelo alux overthelo al fra tureaperture. Then,by ttingtheparallelplatetemperaturesolution(Eq.(18))

to the average temperature prole

T (x)

, we get the

ma ros opi thermal length

R

. In pra ti e the t is

omputed from a least square minimization, for

ab-s issa from

x = 0

to the minimum

x

value so that





(T − T

r

)/ (T

0

− T

r

)





 < 2 · 10

−

6

. Thema ros opi

ther-malaperture

Γ

isthendenedbyanalogytotheparallel platesolution(Eq.(16)) as:

Γ =



−R · 24η · Nu · χ

_∆P

l

x



1

/

4

= A · (R

∗

)

1

/

4

,

(23) where

R

∗

= R/R

//

isthenormalizedthermallength. At

a oarse grained s ale, the rough fra ture is thermally

equivalent to parallel plates separated by the onstant

aperture

a(x, y) = Γ

. Indeed,bothwillexhibitthesame

thermallength

R

under thesame ma ros opi pressure

gradient

∆P /l

x

.

Themi ros opi thermalaperture

γ

analsobe

intro-du edafter dening alo al thermallength

r

. Similarly tothedenitionofami ros opi hydrauli aperturefrom thelo alpressuregradient,orlo alux,ratherthanthe

ma ros opi pressuredieren e, orma ros opi ux,we

estimatethelo al thermallengthfrom alo al

tempera-turegradientratherthanalarges alepressuredieren e. Eq.(14) anberewrittenas:

q

· ∇

2



ln

h

T

∗

i

+

kqk

r

= 0

(24) with

r =

a · kqk

2 · Nu · χ

,

(25)

whi hisanestimateofthegradientalong

s

ˆ

thelo al hy-drauli owdire tion. Finally,thelo althermalaperture

γ

anbedenedby( onsistentlywithEq. (16)):

γ =



−r · 24η · Nu · χ

_∆P

l

x



1

/

4

(26)

A link between ma ros opi and mi ros opi thermal

apertures analsobeshownasfollows: atrstorder, on-sideringthattheaverageof

kqk

isvery losetothe aver-ageof

q

x

thenthelo allengthofreferen ewouldbeequal to

r = −

˜



∂ ln



T

∗



/∂x



−

1

. On the other hand,

inte-grating equation (18) between

0

and signi ant length

L

, resultsin

R = −

hh

ln



T

∗

(L)



− ln



T

∗

(0)

i

/L

i

−

1

,

whi h shows the link between ma ros opi and

mi ro-s opi thermal apertures:

R = L



_R

_L

0

(−1/˜r)dx



−

1

, i.e.

R =

r

−

1



−

1

∝

D

(a kqk)

−

1

E

−

1

,a ordingtoEq. (25).

Forparallelplates, allmi ros opi aperturesareequal

and also equalto the ma ros opi ones:

h = a = γ =

H = Γ = A

. For rough fra tures, this is denitively not the ase sin e thelo al apertures vary spatially in-sidethefra ture. Wewillseeinthenextse tionhowall

these apertures are inuen ed by the roughness

ampli-tude ofthe fra ture aperture, forwhi h we will

empha-sizeontwomainparameters: thenormalizedroot mean

squaredeviation

σ/A

ofthegeometri alapertureandthe aspe t ratio of the fra ture

J = l

x

/l

y

, i.e. the ratio of thedownstream lengthof thefra ture

l

x

overitswidth

l

y

.

D. Dimensionless quantities

Dimensionlesspositions,apertures,pressure,

tempera-tureandhydrauli owaredened asfollow:

x

∗

=

x

d

,

y

∗

=

y

d

a

∗

=

a

A

,

H

∗

=

H

A

,

Γ

∗

=

Γ

A

P

∗

= −

(P − P

_2d

0

)

_∆P

l

x

,

(27)

T

∗

=

T − T

r

T

0

− T

r

,

q

∗

= −

12η · l

x

∆P · A

3

q.

where

d

is the mesh size of the aperture grid.

Other-wise,wenotethatinthedimensionlesstemperature,the

dieren ebetweentheinje tiontemperature

T

0

andthe

ro k temperature

T

r

intervenes onlyas afa tor of pro-portionality.

IV. HYDRAULIC FLOWSIMULATIONS

A. Des riptionofthe pressuresolver

The Reynolds and temperature equations (Eqs. (6)

and (14)) are numeri ally solved by using a nite

dif-feren e s heme. The pressure

P

, the hydrauli ow

q

and temperature

T

aredis retized on agridof

n

x

× n

y

pointswithameshsizeof

2d

i.e. halfoftheaperturegrid points. In thefollowing,whenindexes

(i, j)

arepositive integers,theyreferto nodepositionswhereanaperture, apressureandatemperaturearedened,onthe ontrary ofthenon-integernodeposition(

i ± 0.5

or

j ± 0.5

)where

onlyanapertureisdened.

The Reynolds equation (Eq. (6)) is dis retized and

solvedin thesamewayasbyMéheust andS hmittbuhl

[20℄: weusenite dieren es enteredonasquaremesh

oflatti estep-size

2d

,and thelinearequationsystemis

inverted usingan iterativebi onjugate gradientmethod

[49℄. The hosenpressuredropalongthefra turelength

is

∆P

∗

= P

∗

n

x

,j

− P

∗

1,j

= 1 − n

x

for

1 ≤ j ≤ n

y

. The hydrauli ow

q

∗

i,j

=



q

∗

i,j x

, q

∗

i,j y

, 0



is omputed from

thepressureeld,as:

q

∗

i,j x

= −

a

∗

3

i,j

2

P

∗

i+1,j

− P

∗

i−1,j

q

∗

i,j y

= −

a

∗

3

i,j

2

P

∗

i,j+1

− P

∗

i,j−1

Foraparallelplate onguration(i.e. modelingwithout self-ane perturbation),

q

∗

i,j x

= 1

and

q

∗

i,j y

= 0

every-whereinthefra ture.

B. Exampleof ami ros opi hydrauli aperture

eld

Anexampleofafra tureapertureisshowninFig.4a. It is generated asexplained in II on a

1024 × 512

grid,

andhasaRMSequalto

σ/A = 0.25

. Thehydrauli ow

omputedinside thismorphologyisshownin Fig.4b,as

wellasthemi ros opi hydrauli apertures(Fig. 4 ). In this ase,thehydrauli owexhibitsastrong hanneling aspreviouslydes ribedbyMéheustandS hmittbuhl[20℄.

Themi ros opi hydrauli apertures anbeobservednot

tobesimply orrelatedtotheapertureeld.

The link between mi ros opi me hani al apertures

a

and the mi ros opi hydrauli apertures

h

, is given

in Fig. 5, where the s ale shows the orresponding

o - urren e probability of ea h lo al onguration. We

seethat thenormalizedme hani alandhydrauli

aper-turevaluesaredistributedarounda hara teristi point:

(h/A, < a > /A) = (1, 1)

. Nevertheless, the

orrela-tionbetweenbothapertures isnotsimple. Some ofthe

highestdensity valuesare lo atedbelow and abovethe

straight line whi h represents

h = a

. A ordingly, the

permeability anlo allybelowerorhigherthanwhatis givenbyanaveragePoiseuillelaw. Thes atteringaround thestraightlineshowsthatatonepoint,thelo alowis

notdetermined bythelo alme hani alaperture,but is

inuen edbyallthesurroundingmi ros opi me hani al

apertures. From omputations with other

σ

, we noti e

that the lower the roughness amplitude, the loser to

(1, 1)

the loudis.

C. Variability ofthema ros opi hydrauli

aperture

The dimensionless ma ros opi hydrauli aperture is

measured for our fra ture example as

H/A = 0.94

(or-dinateof the ross in Fig.5).

H/A < 1

meansthat the

fra turepermeabilityisredu ed ompared totheone of

a.

b.

c.

q*

h/A

a/A

y/d

y/d

y/d

x/d

x/d

x/d

Figure4: (Color online) a.: Self ane aperture with

σ/A =

0.25

. b.: Dimensionless hydrauli ow norm omputed with the aperture of Fig. 4a., having for dimensionless hydrauli

aperture

H

∗

_{= 0.94}

. .: Mi ros opi hydrauli apertures, omputedfromthethirdrootofthehydrauli owshownin Fig.4b.

Figure 5: (Color online) 2D histogram of the link between themi ros opi hydrauli aperture and themi ros opi me- hani alaperturefor the fra tureshowninFig. 4(thes ale indi atestheprobabilityinper ents%);the rosshasfor o-ordinates

(H/A, < a > /A) = (0.94, 1)

. Thestraight line is

h = a

,whi histheequalitygivenbyalo alPoiseuille law.

0

0.2

0.4

0.6

0.4

0.5

0.6

0.7

0.8

0.9

1

σ

/A

H/A

Example

Dataset

Average

Figure6: (Coloronline)Ma ros opi hydrauli aperture

H/A

versus

σ/A

for fra tureswithaspe tratio

l

x

/l

y

= 2

;Crosses: Variationofthehydrauli aperture byin reasing the rough-nessamplitude

σ/A

for the aperture shown inFig. 4; Dots: loudof omputeddata(about

20 000

aperturerealizations); Squares:Averagehydrauli behaviorwithvariabilitybars. On average,

H/A < 1

: thepermeability issmallerthanexpe ted fromthePoiseuillelawinparallel plateapertures.

withoutanyself-ane perturbation. Forthe same

mor-phologypattern(Fig.4), weexaminehowtheroughness

amplitude inuen es the ma ros opi hydrauli

aper-ture by hanging

σ/A

( .f. II). In Fig. 6 we see that

the ma ros opi hydrauli aperture is lose to

1

when

σ/A = 0.05

, whi h orrespondsto a quasiat aperture.

When the roughness amplitude in reases,

H

de reases,

whi h means that this morphology pattern tends to

in-hibitthe hydrauli ow and makesthe fra ture

perme-abilityde rease.

Forvariousrealizationswiththesame

σ/A

value,

vari-oushydrauli behaviorsmayhappenowingtothe hannel

variability in the hydrauli ow. In Fig.6, we plotthe

dimensionlessma ros opi hydrauli apertures

H/A

ver-sus

σ/A

(for about

20 000

omputations with

1 700

dif-ferent fra ture aperture patterns). Here, ea h fra ture

hasthesamesize asthefra tureshownin Fig.4where

l

x

/l

y

= 2

. We omputethemeanhydrauli apertures in-sidewindowsofsize

0.025 σ/A

andea hplotted bar

rep-resents twi e the standard deviation of

H/A

inside the

orrespondingwindows. Weseethat formost ases,the

permeabilityis redu ed. For

σ/A < 0.25

, thehydrauli aperture is still quite lose to

A

and the dispersivity is

relativelysmallevenifsome ongurationsshowsaow

enhan ement owing to the fra ture roughness:

H > A

[20℄. Then,forhigherRMS,theaverageof

H/A

de reases signi antlyonaverage(up to

50%

)with

σ/A

,but with ahighervariabilityoftheresults.

D. Inuen eofthe fra tureaspe tratioonthe

hydrauli ow

Togeta ompletedes ription,wenowmodifyone

ad-ditional parameter: the aspe t ratio of thefra ture, by hangingtheratio ofthefra ture lengthoveritswidth,

J = l

x

/l

y

. Figure7showsthesamekindofaverageplots

of

H/A

as a fun tion of

σ/A

but for three dierent

as-pe t ratios:

J = 2

(square symbols) whi h is the one

presentedinFig.6,

J = 1

(triangle)and

J = 0.5

(disks). Sin elesssimulationsweredonefor

J = 1

and

J = 2

(see thelegendofFig.7),fewapertureshow

σ/A > 0.45

,and thereforenoaveragepointsisrepresentedinthese ases.

For square systems(

J = 1

) and downstream elongated

fra ture(

J ≥ 1

),

H/A

isonaveragesmallerthanone(i.e. inhibitinghydrauli ow omparetotheonethrough

par-allelplates separatedby thesame opening

A

), whereas

forsystemswiderperpendi ularlyto thepressure gradi-entdire tion,

H/A

isonaveragehigherthanone. A qual-itative explanation might be that, it is stati ally more likelyto getalarge s ale onne ting hannelfor awide and short fra ture (

J < 1

) rather than for a thin and long fra ture (

J > 1

). In other words, qualitatively, hannels are ratherin parallel in widefra tures, and in seriesin longones. Forsquare systemswhi h shouldbe

isotropi andprovidingasmanyperpendi ularand

par-allel hannels,weseethatwhentheroughnessamplitude in reases,thehydrauli aperture getonaverageslightly

smaller than

A

. We ansuspe t that it would exist an

aspe tratio

J

inv

sothatthehydrauli apertureison

av-erage independent ofthefra ture roughness magnitude:

H/A = 1

for any

σ/A

. Followingthemodel proposed be-lowinIVE,weget

J

inv

≃ 0.65 ± 0.05

. Forany

J

value, weseethatthehighertheratio

σ/A

,thehigherthe vari-abilityof thebehaviorsis, espe ially for squaresystems whi hexhibitbothhigh(

H > A

)andlow(

H < A

)

per-meabilityforthesameroughnessmagnitude.

E. Modelofthe averagema ros opi hydrauli

aperture

One of the main questions we want to address here,

is the relationship between the ma ros opi hydrauli

aperture

H

and the mi ros opi me hani al aperture

eld

a(x, y)

. The knowledge of the me hani al aper-ture eld

a(x, y)

provides us the following bounds for

H

:

ha

−

3

i

−

1

< H

3

_{< ha}

3

_i

 the lower ase

orrespond-ingtoasystemofapertureu tuationspurelyalignedin

series, and theupperoneto u tuationspurely aligned

in parallel [55℄. However,

a (x, y)

is rarely known and

subsequently

ha

−

3

i

−

1

and

ha

3

_i

aredi ulttoestimate.

FromFig.7,

σ/A

and

J

appeartobeimportant

param-eters ontrollingthe ma ros opi hydrauli aperture of

thefra ture

H

. Ref[20℄proposedarstmodelofthe

H

behavioras:

H/A = 1 + α

σ

A

κ

. Herewesimilarlymodel

the average hydrauli aperture urves orresponding to

ea haspe t ratio( ontinuous urvesin Fig.7) and nd

0

0.2

0.4

0.6

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

σ

/A

H/A

l

_x

/l

_y

=2, Average

l

_x

/l

_y

=1, Average

l

x

/l

y

=0.5 Average

Fit curves 1

Fit curves 2

Figure7: (Coloronline)Ma ros opi hydrauli aperture ver-sus

σ/A

,forthreeaspe tratios

J = l

x

/l

y

.Averages omputed from data are shown with symbols, with error bars, orre-sponding to plusor minusthe standard deviation (see how theaverage is omputed inIVC).

J = l

x

/l

y

= 2

shows an

enhan ed ow (same data as presented in Fig. 6);

J = 1

showsonaverageaslightlyinhibitedow,i.e.

H ≤ A

( om-puted from a loud of about

1 300

points); for

J = 0.5

, onaverage, higherpermeabilityis observed( omputedfrom

a loud of about

900

points). Continuous urves are

t-tingmodels (1)

H/A = 1 + α

`

σ

A

´

κ

, with parameters

(κ, α)

equalto

(2.05, −1.46)

,

(1.57, −0.31)

,

(2.69, 0.67)

respe tively for

J

equal to

2

,

1

and

0.5

. Dotted urves are obtained withttingmodels(2)

H/A = 1 − µ [log

2

(J) + δ]

`

_σ

A

´

κ

,with

(µ, δ, κ) = (0.98, 0.59, 2.16)

,forthethree urves.

(κ, α)

su essively equal to

(2.05, −1.46)

,

(1.57, −0.31)

,

(2.69, 0.67)

respe tivelyfor

J

equalto

2

,

1

,

0.5

. Depend-ingonthesignof

α

,wegeteitherapermeabilityloweror

higherthanthat expe ted withatplates. Thenwet

thesethreebehaviorsbyamoregeneralmodelwhi h

in- ludestheaspe t ratiovariation,with three parameters (

µ

,

δ

,

κ

)tobeoptimized:

H/A = 1−µ [log

2

(J) + δ]

σ

A

κ

. With

(µ, δ, κ) = (0.98, 0.59, 2.16)

, we get thethree dot-tedlines in Fig.7whi h are a eptable tsof the aver-agetrend. Howeverithastobehighlightedthatthereal hydrauli aperture of a spe i surfa eis possibly very dierent from this average value (see size of variability barsinFig.7),espe iallyathigh

σ/A

.

Othermodelsfornumeri alorexperimentalhydrauli

apertures have been proposed in the literature [19℄, as

(H/A)

3

= 1 − C

1

exp (−C

2

A/σ)

or

(H/A)

3

= 1/[1 +

C

3

(2A/σ)

1.5

],

where

C

1−3

are onstantsbut the shape ofthese fun tions doesnot twellour averagedpoints, andthesetsarenotrepresentedhere.

V. THERMAL FLOWSIMULATIONS

A. Des ription ofthetemperature solver

Thetemperatureequation(Eq.(14))isdis retizedas:

q

∗

i,j x



T

∗

i+1,j

− T

∗

i−1,j



+ q

∗

i,j y



T

∗

i,j+1

− T

∗

i,j−1



+

4d

R

//

·

T

∗

i,j

a

∗

i,j

= 0,

(28)

where

(i, j) ∈ [|2, n

x

− 1|] × [|2, n

y

− 1|]

and

R

//

is the

thermallengthexpe tedbynegle tingtheroughness

am-plitude(seeEq.16). Theboundary onditionsare:

1 ≤ j ≤ n

y

,

T

∗

1,j

= 1

and

T

∗

n

x

,j

= 0

2 ≤ i ≤ n

x

− 1, T

∗

n

x

,j

= 0

and

T

∗

n

x

,j

= 0

Thesystemissolvedin thesamewayasthepressure

system(IVA). Twolimitingnumeri alfa torsintervene

forthe e ien yof thedis retizations heme: themesh

step

d

hasto besu ientlysmallto apturewitha

suf- ient a ura y the relative variations of

T − T

r

over

a latti e step. In pra ti e, the mesh step used in this

manus riptis hosenas

d = R

//

/50

. We he kedthat di-vidingthismeshsizeby2didnot hangesigni antlythe

omputedtemperatureeld. These ondnumeri allimit

isthatthesystemsize

l

x

hastobelargerthan

20 · R

//

to

avoid a possible numeri al instability (mostly with the

aperture grid size

1024 × 2048

whi h is more likely to

exhibitalongerthermallength,asexplainedin VE). If not,theuidpassingthefra tureissoslowlywarmedup

that the ondition

T

∗

n

x

,j

= 0

at theoutlet badly repre-sentsthe onditionimposedinprin ipleatinnityinthe

hannel,andthisboundary onditionimposedata

phys-i allytooshortdistan efromtheinlet annotbefullled withoutnumeri alartifa t. Tofa e thisprobleminsu h rare situations, we dupli ate the aperture grid to get a

longer systemlength and impose thesame ma ros opi

pressuregradient,and thero ktemperatureat thenew

end:

T

∗

2·n

x

,j

= 0

.

B. Example ofalo almi ros opi temperature

eld

Foranearly onstantaperture(

σ/A = 0.05)

,we

numer-i allyobtain atemperature law lose to an exponential

downstreamprole(Fig.8),asweexpe tfromEq.(17).

The2DtemperatureeldshowninFig.9a(

σ/A = 0.25

)

is omputed from theaperture and itspreviously

om-puted hydrauli oweld, shown in Fig. 4b. It anbe

observedthattheuidisgettinginhomogeneouslywarm,

with hannelizedfeatures. The thermal hannel begins

inazonewherethehydrauli ow omingfromtheinlet

onverges(Fig.4b). Thelo alnormalizedthermal

aper-ture

γ/A

(mapshowninFig.9b)exhibitslesspronoun ed

−ln(T )

x/d

y/d

*

Figure8: (Coloronline)

−

ln

“

T

∗

”

,oppositeofthelogarithm

ofthetemperatureeld

T

∗

omputedfromtheaperture mor-phologypattern shown inFig. 4 with avery low roughness amplitude:

σ/A = 0.05

. Thehydrauli apertureofthis fra -tureis

H/A = 0.99

. The temperature eld exhibits a nor-malized thermal length equal to

R

∗

_{= 0.97}

and a thermal apertureof

Γ/A = 0.99

.

γ

/A

−ln(T )

*

a.

b.

x/d

x/d

y/d

y/d

Figure9: (Color online)a.:

−

ln

“

T

∗

”

, opposite ofthe loga-rithmofthe 2Dtemperatureeld, omputedfromthe aper-turesinFig. 4a(

σ/A = 0.25

). b.: Normalizedlo althermal aperture

γ/A

asso iated withthe temperatureeldshownin Fig.9a.

hannelee tthaninFig.9a. Figure10istheplotofthe lo almi ros opi thermalapertures

γ/A

versusthelo al

apertures

a/A

, using a shading showing the o uren e

densityin the

(γ/A, a/A)

spa e. Thedispersivityofthe

loudaroundtheline

γ = a

showsthat thereis no

sim-plelink betweenthelo alapertureandthethermalone. Asimilarplot(Fig.11)allowstoobservethe orrelation

betweenthelo almi ros opi thermalaperturesandthe

lo al mi ros opi hydrauli apertures. It shows a good

orrelation of the lo al thermal aperture and the lo al

hydrauli aperture(i.e. the loudis losetothestraight line

γ = h

). Note that it is more probable (

59%

) to

Figure 10: (Color online) 2D histogram in per ents of the fra ture shown inFig. 4 as a fun tion of the lo al thermal aperture

γ

and lo al aperture

a

(the shading indi ates the probability density). The straight line is

γ = a

. The dis-persivityofthe loudaroundtheline shows thatthereisno simplelinkbetweenthelo alapertureandthethermalone.

have

γ > h

, whi h orresponds to a heat ex hange lo- ally lesse ientthan what isexpe ted from aparallel platemodelwhi h isequivalentin permeability.

Figure 11: (Color online) 2D Histogram in per ents of the fra ture shown inFig. 4 as a fun tion of the lo al thermal aperture

γ

andlo alhydrauli aperture

h

(thes aleindi ates the probability inper ents%). Thestraight line is

γ = h

; the lo alization of the loud around the line shows a good orrelationbetween

γ

and

h

.

C. Variability ofthe ma ros opi thermal aperture

Theaveragetemperature

T

(seedenitioninEq.(22))

is a semi lo al property whi h shows how the thermal

behaviorevolvesonaveragealong thepressuregradient

dire tion. Theshapeof

T (x)

(Fig. 12)is losetoan ex-ponentiallaw, but withathermallength

R

slightly

dif-ferentfrom thefra ture withoutself-ane perturbation

(i.e. parallel plates). This thermallength is omputed

fromthe slopeofthe linearregressionof

ln



T (x)



(see inIIIC). Intheexampledisplayedin Fig.12,the ther-mallengthis

R

∗

//

= 1.09

, whi h resultsin anequivalent

thermalapertureof

Γ

∗

= 1.02

.

Figure12: (Coloronline)Continuous urve:

−

ln

“

T

∗

”

, oppo-siteofthelogarithmofthetemperatureeld omputedfrom thetemperatureeld

T

showninFig.9. Dash-dotted urve: Lineart of urve A (from

x/d = 0

to

x/d = 772

), whi h provides the thermal length:

−ln

“

T

∗

”

= x/1.09 + 0.6

, i.e.

R

∗

_{= 1.09}

. Dashed urve:

−

ln

“

T

//

∗

”

oppositeofthe loga-rithmof thetemperaturelaw forthesame fra turemodeled withoutself-anityperturbation(i.e. parallelplates),whi h hasforthermallength

R

∗

//

= 1

.

Ingure13,the rossesillustratetheroughness

ampli-tudeinuen eonthethermalapertureforthe

morphol-ogypatternshowninFig.4a,whosereliefisampliedby hanging

σ

value(seein II). Forthisexample,

Γ

vs

σ

is notmonotoni . Thedimensionlessthermallengthis lose to

1

when

σ/A = 0.05

,whi h orrespondsto aquasiat

aperture. Whentheroughness amplitudeis big enough

(

σ > 0.1

),

Γ

in reases with

σ

and is higher than one,

whi h means that this morphology pattern tends to

in-hibitthethermalex hange. InFig.14,the rossesshow

thethermalapertureversus

H/A

usingthesamedataas

fortheplotsshownby rossesin Figs.13and6.

D. Variability ofthe thermal behavior

Statisti al thermal results are omputed for

numer-ous ases (more than

20 000

) whose ma ros opi

hy-drauli apertures are presented in IVC for various

σ/A

values.Similarly,anormalizedaveragema ros opi ther-malaperture,

Γ/A

,anditsstandarddeviationisobtained asfun tionof

σ/A

. Theresulting

Γ/A

fortheaspe tratio

J = 2

isdisplayedinFig.13,withbarsrepresentingthe

doubleofthestandarddeviation. Forthesame

normal-izedroughnessamplitude

σ/A

,variousthermalbehaviors mayhappen,espe iallyfor

σ/A > 0.25

,with hannels

ap-pearingornotanddimensionless thermallengthshigher

orlowerthan one. At rstorder,boththema ros opi

aper-0

0.2

0.4

0.6

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

H/A and

Γ

/A

σ

/A

Example

Γ

/A

Dataset

Γ

/A

Average

Γ

/A

Average H/A

Figure13: (Color online) Crosses: Variationof the thermal aperture

Γ/A

byin reasing the roughnessamplitude

σ/A

for the aperture patternshownin Fig. 4; Dots: Cloudof om-puteddata(about

20 000

points)forfra tureswithaspe t ra-tio

l

x

/l

y

= 2

;Triangles: Averagethermalbehaviorwith vari-abilitybarsofthe loud;Squares: Averagehydrauli aperture

H/A

versus

σ/A

,re alledherefor omparison.

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

H/A

Γ

/A

Example Γ/A

Dataset Γ/A

Average Γ/A

Γ=H

Figure 14: (Color online) Normalized thermalaperture

Γ/A

versus

H/A

forfra tureswithaspe tratio

l

x

/l

y

= 2

. Crosses: Variation of the thermal aperture by in reasing the rough-ness amplitude for the aperture pattern shown in Fig. 4a

versus

H/A

; Dots: Cloud of omputed data (about

20 000

points); Squares: Average thermal behavior with variability bars. Continuous urve:

Γ/A = H/A

,whi hholdsforparallel platesseparatesby

a(x, y) = H

.

tures (Fig. 13, square symbols) are de reasing as

fun -tionsof

σ

. This trend is signi antly morepronoun ed

for

H

than for

Γ

. The thermal results are ompared

with systems equivalent in permeability (same

normal-izedhydrauli aperture) in Fig.14whi h representsthe

normalizedthermal aperture versusthehydrauli

aper-turewiththeaveragepoints omputedinsidewindowsof

size

0.05 H/A

. Themoststrikingresultisthatroughness

inhibitsthermalization: nearlyallthe loudisabovethe

ontinuous urve

Γ = H

,whi hmeansthatthe

thermal-izationoftheuid(thermalizationisobtainedwhenthe uidtemperaturerea hesthero kone)isinhibited

om-paredtowhatweexpe tfromthehydrauli behavior.In

thesame time, we note that, on average,

Γ/A < 1

, i.e. mostoftheaperturesexhibitanenhan edthermalization

omparedtowhatwouldbeexpe tedwithamodelofat

fra turesseparatedby

A

,i.e. havingthesamegeometri (orme hani al)aperture.

E. Inuen eofthe fra tureaspe tratioon the

thermalbehavior

0

0.2

0.4

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Γ

/A

σ/A

l

_x

/l

_y

=2

l

_x

/l

_y

=1

l

x

/l

y

=0.5

Figure15: (Coloronline)Averagesofthenormalizedthermal aperture

Γ/A

and theirdeviationbarsversus

σ/A

forvarious aspe tratios

J = l

x

/l

y

,as indi atedbythe labels. Seehow theaverageis omputedinIVC.

0.6

0.8

1

1.2

0.6

0.8

1

1.2

1.4

1.6

H/A

Γ

/A

l

_x

/l

_y

=2

l

_x

/l

_y

=1

l

_x

/l

_y

=0.5

Γ=H

Fit H<A

Fit H>A

Figure16: (Coloronline)Averagesofthenormalizedthermal aperture

Γ/A

andtheirdeviationbarsversus

H/A

forvarious aspe tratios

J = l

x

/l

y

,asindi atedbythelabels(seehowthe average is omputedinIVC). Modelslinesare

Γ = 0.9H +

0.2A

for

H < A

and

Γ = 3.5H −2.4A

for

H ≥ A

;no ontinuity onditionbetweenbothlinesisimposed.

We omplete our study by omputing the averaged

us-ing the hydrauli ows omputed in IVD. The aver-aged valuesof thethermalapertures, with the

variabil-ity bars (dened similarly to what is done in VD) for

J ∈ {0.5, 1, 2}

are plottedin gs. 15and16. When

Γ/A

is plotted as a fun tion of

σ/A

(Fig. 15), various

ther-mal behaviorsare observed,a ordingto the aspe t

ra-tio,with high variability, parti ularlywhen

σ/A > 0.25

.

On the ontraryto fra tures with aspe t ratio equalto

J = 2

(des ribedinVD)theones with

J

in

{0.5, 1}

are morelikelytoinhibitthethermalization omparedtoat fra tureswiththesameme hani alaperture(

Γ/A > 1

).

Figure16 showstheaverageof

Γ/A

versus

H/A

.

Con-trarilytowhat anbeobservedfor

Γ/A

vs

σ/A

(Fig.15),

theaverage urves

Γ/A

vs

H/A

areroughlyindependent

ontheaspe tratio. Thisshowsthatthehydrauli aper-tureisabetterparameterthantheroughness

σ/A

to as-sess thethermalproperties.Thethermalapertureis sys-temati allylargerthanthehydrauli aperture(

Γ > H

).

It means that on e the permeability known, e.g. by

pumpingtests,usingaparallelplatemodelseparatedby

H

forestimatingthethermalbehavioroverestimatesthe

e ien y oftheheatex hange: theuid needsindeeda

longerdistan etobethermalizedthanexpe tedfromat

fra tureswiththesamepermeability. Onaverage

Γ/A

vs

H/A

ismonotoni (Fig.16),i.e. thisaveragedependen e displaysasimplerbehaviorthanforaparti ular aseof morphologyofvaryingamplitude(e.g. Fig.14, rosses).

Going more into details, Fig. 16 also shows that for

H/A > 1

,theslopeof

Γ

vs

H

issteeperthanfor

H/A < 1

; bothpartsofthe urve anbemodelledwithstraightline ts(dottedanddot-dashed urves). This ouldbe inter-pretedasfollows: fra tureswithhighhydrauli apertures provide highvelo itiessothat uidparti les needto go

further to rea h the ro k temperature. Fra tures with

small hydrauli apertures

H/A < 1

mightbedominated

by small me hani al apertures (fen es) providing small

velo ities,whi hleadstothermalapertures losertothe line

Γ = H

.

VI. DISCUSSIONANDCONCLUSION

A. Modellimits andpossibleextension

Despitethehydrauli lubri ationhypothesiswhi h

im-plies notably a low Reynolds number, the uid

velo -ityshould notbe toosmall. Indeed, thevelo itydrives thein-planethermal onve tion,whi hissupposedtobe

large omparedtothein-planethermal ondu tion.This

an be quantied by the Pé let number (ratio between

the hara teristi time of diusion and adve tion): our

modelisvalidatlowin-planePé letnumber. Therefore, owingtoin-plane ondu tion,thethermal hanneling ef-fe tmightberedu edespe iallyin aseofhigh

temper-ature ontrastalongthe hannel andverylowhydrauli

ow. Thishomogenizationmightbereinfor ediftheuid temperatureisstillinhomogeneousbutvery losetothe

ro k temperature: in this ase the in-plane ondu tion

inside the uid mightbe ashigh as the ondu tion

be-tween ro kand uid. Free onve tion (temperature

de-penden eof

ρ

),whi hisnottakenintoa ounthere,may alsointervene,espe iallyforthi kfra tures [56℄.

Inpra ti e,some3Dee tsmighthappenasthe

lubri- ationapproximationis notne essarilyrespe tedowing

tothero kmorphology,(e.g. [23,24℄). Innatural ases, theroughnessamplitude

σ/A

oversalargerangea ross thenatural ases,fromsmalltolargevaluesa ordingto thetypeofro kandfra tures. Forinstan e,were ently measuredtheroughnessamplitudeofnaturalfra turesin

bla kmarl at borehole s ale,and weobtainedvaluesof

σ/A < 0.04

foroneand

σ/A = 0.3

for anotherone [22℄. Someother values, typi ally

σ/A > 0.4

, havealso been reported for instan e in graniti ro ks [57, 58℄. If the aseswithlargeroughnessamplitudesalso orrespondto

large lo al slopes (angle between the fra ture side and

theaverage plane), it is likelythat the Reynolds

equa-tionand2Dtemperatureequationdoesnotapplysowell

tothese ases,andthattheresultsreportedhereareonly

approximateforthose.

When the fra ture morphology is highly developed,

duetomoresurfa eex hange,thero kmightlo ally

pro-videbetterheatex hange. Theassumptionofaveraging

thermalphenomenain2Dhasbeenstudiede.g. byVolik

orSangareet al. [59, 60℄, who onsidered only

ondu -tion. The3Dsolvingof thefullNavier-Stokesand heat

adve tion-diusionequationsisalsopossible,forexample

witha oupledlatti e-Boltzmannmethod[61℄. However,

onsideringthe omplexityoffra ture morphologyfrom

verysmall s alesto largeones requires heavy

omputa-tions,whi h makesstatisti al resultsdi ult to obtain.

When onve tionalso a ts, 3Dee ts lead to zones

de- oupledfrom the main mass andheat ux,asthe uid

mightbeblo kedintoeddies(olubri ationregime)

pro-voked by sharp morphologies [23, 2932℄ (like Moatt

eddies[62℄). Ithasindeedto benoti edthat evenwhen lowpressuregradientisimposed,turbulentowmightbe

observedduetohighroughnessamplitude. Thisee tis

omplementary to observations made at high Reynolds

number[6366℄,when evenaverylowroughness

ampli-tudeofthewallindu es turbulentow.

Alltheresultsaboutthethermalaperturemayalsobe

inuen edbythethermal boundary onditions. In

par-ti ularwehaveassumedthat

T

r

is onstant. Spatial vari-ationsof

T

r

aneasilybetakenintoa ountby hanging

the boundary onditions of the thermal equation while

temporal variations require to model the ro k getting

olderinthesurrounding( onsequen esofthero k diu-sivity). Intime, thehypothesisof onstanttemperature

T

r

holdseitherforveryshortdurationswhentheregime istransitory,orforlongerdurations,atquasi-stationary

regime,when the ro k temperature evolvesveryslowly

andtheuidtemperatureadaptsfast. Thisisthe aseif thesolidismu hmorethermallydiusivethantheuid, whi h is quite true in our ase: for instan e, the ratio

of the granite thermal diusivity over the water one is