Effects of porosity and inertia on the apparent permeability tensor in fibrous media

To cite this version :

Luminari, Nicola

and Airiau, Christophe

and

Bottaro, Alessandro Effects of porosity and inertia on the apparent

permeability tensor in fibrous media. (2018) International Journal of

Multiphase Flow, vol.106. pp. 60-74. ISSN 0301-9322

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Effects

of

porosity

and

inertia

on

the

apparent

permeability

tensor

in

fibrous

media

Nicola

Luminari

a

_,

_Christophe

_Airiau

a

_,

_Alessandro

_Bottaro

a,b,∗

a Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France b DICCA, Università di Genova, 1 via Montallegro, Genova 16145, Italy

a

b

s

t

r

a

c

t

Theflowinthree-dimensionalfibrousporousmediaisstudiedintheinertialregimebyfirstsimulating forthemotioninunit,periodiccells,andthensolvingsuccessiveclosureproblemsleading – after ap-plyingan intrinsicaveragingprocedure– tothe componentsoftheapparent permeabilitytensor.The parametersvariedincludetheorientationofthedrivingpressuregradient,itsmagnitude(whichpermits todefineamicroscopicReynoldsnumber),andtheporosityofthemedium.Allcasestestedreferto sit-uationsforwhichthemicroscopicflowissteady.Whenthedrivingforceisorientedinadirectionwhich liesontheplaneperpendiculartothefibers’axis,theresultsfoundagreewiththoseavailablethe litera-ture.Thefactthatthemediumiscomposedbybundlesofparallelfibersfavoursadeviationofthemean flowtowardsthefibers’ axiswhenthedrivingpressuregradienthasevenasmallcomponent alongit, andthisisenhancedbyadecreasingporosity;thisphenomenoniswellquantifiedbytheknowledgeof thecomponentsofthepermeability.Contrarytoourinitialexpectations,fortheoveronehundredcases whichwehavesimulated,theapparentpermeabilitytensorremains,toaverygoodapproximation, di-agonal,afact mainlyrelatedtothetransverselyisotropicnatureofthemedium.Toobtainacomplete, albeitapproximate,databaseofthediagonalcomponentsoftheapparent permeabilitytensorwehave developedametamodel,basedonkriginginterpolation,andcarefullycalibratedit.Theresultingresponse surfacescanbeinvaluableindeterminingtheforcecausedbythepresenceofinclusionsinmacroscopic simulationsoftheflowthroughbundlesoffiberswhoseorientationsanddimensionscanvaryinspace and/ortime.

1. Introduction

The flow through porous media is a problem of importance forseveralnaturaland technological applications.Since the orig-inal formulation by Darcy (1856), which relates the flow rate throughaporousbedtothepressuredropacrossthebed’ssides, manycorrectionshavebeenmadetoaccount,forexample,for vis-cous effects (Brinkman, 1949) or forthe consequences of inertia (Forchheimer, 1901). All of the cited works are of empirical na-ture, but a volume averaging approach has beenable to recover alloftheseformulationsrigorouslystartingfromtheNavier-Stokes equations(Whitaker,2013).

Thetheoryrequirestheknowledgeofanumberofterms,most notably,inthecaseofan isotropicporousbed, apermeability co-efficient and a Forchheimer coefficient. Initial efforts in defining theseterms were based on a combination of physical reasoning

∗ _{Corresponding author.}

E-mail address: alessandro.bottaro@unige.it (A. Bottaro).

andmeasurements,leadingtoexpressions knownastheKozeny– Carman (Kozeny, 1927;Carman, 1937) and the Ergun (Ergun and Orning, 1949) correlations. The first coefficient provides the per-meability for the laminar flow of a single-phase fluid through a packedbedofsandgrains,asfunctionoftheporosityandthe di-ameterofthegrains, whilethesecond extendsDarcy’s lawto let the pressuredrop dependon two terms,one proportional to the velocity and the second to its square, thus accounting for iner-tia.Theseapproachesdonotconsidermicrostructuralor geometri-calfeatures oftheporousbed,whichcanrenderthepermeability a tensorial quantity, and are often restricted to simple unidirec-tionalflows. Inthepresentwork weare concerned witha trans-versely isotropic material composed by parallel fibers of circular cross-section,withoneaxisofsymmetry,(O, x 3);insuchmaterials the permeabilityisa diagonal tensorwith thecomponent inthe directionparalleltothefibersgreaterthan thosealongthe trans-verseaxes.Forsuchanarrangementwewillinvestigatetheeffects ofboth thedirectionoftheforcing pressure gradientandinertia. Whenthelattereffectispresent,embodiedbyaReynoldsnumber

Re d,basedonthemeanintrinsicvelocitythroughthemediumand

thefibers’diameter,exceedinganorderonethreshold,the perme-abilityisnomoresimplydefinedupongeometricalproperties.This newpermeability,whicharisesfromawell-defined closure prob-lem,isthencalled apparent permeability .

Theinfluenceofthegeometryofthesolidinclusionshasbeen addressed previously by Yazdchi et al.(2011) for arraysof cylin-dersinbothsquareandhexagonal(orstaggered)patterns,withthe cylinders’sectionwhichcanvaryinshape.Theresults,inthe two-dimensionalandlowReynoldsnumberlimits,demonstratethe de-pendence ofthepermeabilitycomponentalong theflow direction toboththeporosityandthedirectionofthemacroscopicpressure gradient. The directionof the pressure gradient isfound tohave a weak effect for beds of medium-high porosity (

ε

> 0.7) and a stronger dependenceappears upon the geometryof thesolid in-clusions.

The influence of the Reynolds number on the permeability andontheForchheimercorrectionhasbeenpresentedina num-ber of papers. One of the contributions most relevant here is due to Edwards et al. (1990). These authors show that, for ar-rays of fibers, the apparent permeability decreases with the in-creaseof theReynoldsnumber, andtherateofthisdecrease de-pends on the geometry of the array; also, the Reynolds num-ber is found to have a stronger influence on the apparent per-meability when the medium ishighly porous. The results ofthe workbyEdwardsetal.(1990)agreewiththosebyZampognaand Bottaro (2016) and with our own work (as shown later), all for the case of cylindrical fibers, although some issues remain on the persistence of steady solutions in the simulations by

Edwardsetal.(1990)incasesforwhichalimitcycleshould have set in. A fully three-dimensional porous medium, more complex than thosediscussedso far,hasbeenconsidered bySoulaine and Quintard (2014),confirmingthe decreasing trendofthe apparent permeabilitywiththeReynoldsnumber.

Another contribution which deserves mention is that by

Lasseux et al. (2011); they have computed the permeability ten-sorforvariousReynoldsnumbers,inatwo-dimensionalgeometry withcylinders ofsquarecross-section. Forcing theflow alongthe main symmetricdirectionsofthefiber,Lasseuxetal.(2011)have identifieddifferentregimes:

• a creeping flow regime for 0 < Re d< 10−3, without

Forch-heimerterms;

• aweakinertiaregimefor10−3_< Re _{d <}1_,withtheForchheimer correctionquadraticin Re d;

• astronginertiaregimefor1<Re d<10,wheretheForchheimer

correctionislinearwiththeReynoldsnumber;

• aturbulent regime, for Re d> 10, withtheForchheimer

correc-tionagainquadraticwiththeReynoldsnumber.

The boundaries betweenthe different regimes are specific to the geometricalarrangements andto theporositiesbeing consid-ered;astepforwardinrendering(someof)theseboundaries rigor-ousandindependentofthearrangementofthepores,throughthe definition ofa Reynoldsnumber which accountsfor a ”topologi-cal” coefficient,hasbeenrecentlymadebyPauthenetetal.(2017). For the purposes of the present paper, we must retain that

Lasseux et al. (2011) have parametrized the Forchheimer correc-tion withthe Reynoldsnumber, andhave found that theinertial correctionisordersofmagnitudesmallerthantheDarcy’sterm,at least before the turbulent regime sets in. Moreover, Lasseux and co-workers have studied how a Forchheimer tensor, F, depends upon the direction ofthe macroscopic forcing term withrespect totheorientationofthesquare cross-sectionofthefibers,for Re d

upto30.Itisconcludedthatadeviationangle,

γ

,existsbetween the directionof thepressuregradient andthatofthe meanflow, becauseofthefibers’geometry.Theinertialcorrectionisstrongly

Fig. 1. Illustration of the REV concept.

influencedbytheorientationofthedrivingpressuregradient,and thetensorFisnotsymmetric(infacttheoff-diagonalcomponents arefound tobe inverselyproportionalto thediagonalterms,and symmetricwithrespecttorotationsaboutthediagonalaxisofthe square,i.e.thedirectionat45° inthe x 1− x2plane,cf.Fig.1).

Theeffectofvariationsintheforcingangle,withrestrictionsto anglesinthe x 1− x2plane,isalsoexaminedbySoulaineand

Quin-tard(2014)withconclusionsinqualitativeagreementwiththoseof boththe contributionjustcitedandour resultsdescribedfurther below. In all cases, the off-diagonalcomponents of the apparent permeabilitytensoraresmallandthediagonalcomponentsdisplay butasmallvariation uponrotation ofthedrivingpressure gradi-ent.

As already anticipated, this work investigates how the direc-tion of the macroscopic pressure gradient, the porosity and the Reynolds number can modify the Darcy and Forchheimer clo-sures arising from a volume-averaged model of a fibrous porous medium. We will consider a three-dimensional unit cell for the microscopicmodel(suchaunitcellissometimesdenotedREV,for RepresentativeElementary Volume),witha genericforcing whose direction is defined by two Euler angles. Given the formidable space ofparameters, some representative results are first shown anddiscussed. Response surfaces in the spaceof parameters are thenidentifiedbytheuseofametamodelbasedonkriging inter-polation.Forthesakeofspace, onlythefirstdiagonalcomponent of the apparent permeability tensor is discussed in detail in the paper;however,allcomponentshavebeencomputed.They repre-sentanextremely usefuldatabasewhichwe arenowinthe pro-cessofusinginmacroscopicsimulationsofflowsthroughbundles offibersofvaryingorientationanddensity.

2. Thevolume-averagedNavier–Stokes(VANS)method 2.1. A brief description of the method

The system under investigation consists of an incompressible Newtonian fluid which flows through a rigid porousmedium. In thefollowing, thesubscript

β

isused toindicate the fluidphase while

σ

is adoptedfor the solid phase. The governing equations validatthemicroscalefortheflowthroughtheporesare:

∂

v_β

∂

t +vβ·

∇

vβ =− 1

ρ

β

∇

pβ+

ν

β

∇

2_v β+f, (1)

∇

· v_β =0, (2) wherev_β, p _β,

ρ

_β and

ν

_β stand, respectively,forthevelocity,the pressure,thedensityandthekinematicviscosityofthefluid.The right-handsideterm,f,isaforce(perunitmass)whichdrivesthe fluidmotion andcan be interpreted asthe macroscopicpressure gradientactingonthesystem.

The concept ofREVofthe porousmedium isclassically intro-duced in the framework of the VANS approach. An example of REV is depictedon Fig. 1,together with relevant notations (vol-umeshape and size, indicationof the fact that the normal unit vectoris directedfromthe fluidto thesolid phase,centroid xof theREV). The REV representsthe domain overwhich the micro-scopicproblemissolved;itssizeisdefinedsoastocaptureallthe microscopicfeaturesoftheflow.Asaruleofthumb,theREVisthe smallestfluiddomainoverwhichperiodicboundaryconditionscan beapplied.

In thecomputational domain, anyflowvariable

ψ

canbe de-composedintoan intrinsicaveragepart <

ψ

>β plusa perturba-tion

ψ

˜,as:

ψ

=



ψ



β+

_ψ

˜_.

The intrinsic average is defined with an integration carried out onlyonthefluidphase(Whitaker,2013):



ψ

β



β=_V1 β  V_β

ψ

β

(

x

)

dV_β. (3)

Applying such an operator to Eqs. (1) and (2), and following

Whitaker(1996)wehave:

∂



v_β



β

∂

t +



v_β



β·

∇



v_β



β=−

_ρ

1 β

∇



p_β



β+

ν

β

∇

2



vβ



β+ f+ + 1 V_β  A_βσ



−

_ρ

p˜β βI+

ν

β

∇

v˜β



· nβσdA, (4)

∇

·



v_β



β =0, (5)

upon neglecting in Eq. (4) the sub-REV-scale dispersion term (linked to



v˜_βv˜_β



β) which is often small in porous media flows (Breugem,2005).

The surface integral term in Eq. (4) represents the drag (per unitmass)duetosurfaceforcesatthe fluid-solidinterface ofthe medium. It iscalled the Darcy-Forchheimer microscaleforce, Fm_.

Theequations are howeveroftento be solved atthe macroscale, sothatamacroscaleforcemodel,FM_{, mustbeusedtoreplaceF}m

inthegoverningequation.Such amodelisoftenbasedon a per-meabilitytensor,K,andaForchheimertensor,F,andreads: FM₌₋

_ν

β

ε

K−1

(

I+F

)

<vβ>β, (6)

sothatthesystemisclosedbyimposing

Fm_=FM_{. (7)}

ThedragforceFm_{computedbydirectnumericalsimulations(DNS)}

withaccountofallindividual poreswillbelatercomparedto the modelbasedonthepermeabilityandForchheimertensors(whose equationsaregivenbelow).Thisisausefulexercisetodemonstrate consistencyoftheapproach andaccuracy ofthenumerical simu-lations; it doesnothing else since, as briefly described below,to derivetheForchheimertensorthemicroscopic velocityfieldmust be known anyhow. Nonetheless, knowledge of the behaviour of thesetensors(or, equivalently,ofthe related apparent permeabil-ity)might proveboth useful andinstructive,in particularshould

onewishtoextendtherangeofapplicabilityofthemodeltocases forwhichthemicroscopicsolutionisnotavailable.

ThecoreoftheVANSapproachconsistsintheidentificationof thepermeability andForchheimertensors.This problem,referred to as the closure problem, is discussed at length by Whitaker (1986,1996).He derivestwopartial differentialequationsystems, the firstvalidin thezeroReynolds numberlimit (system(8) be-low),whilethesecond applieswhen inertialtermsarenot negli-gible(system(10)).

In the first system of equations a three-component vector d

anda 3× 3 tensor D are introduced. Thissystem can be divided intothreeseparateindependentproblemswhichresembleaforced Stokes problemwhereeach componentofd andthe correspond-ingrowofDplay,respectively,theroleofapressureandavelocity field.Togetherwiththeperiodicboundaryconditions,theproblem reads:

⎧

⎪

⎨

⎪

⎩

0=−

∇

d+

∇

2_D+I,

∇

· D=0, D=0 on A_βσ, d

(

x+i

)

=d

(

x

)

, D

(

x+i

)

=D

(

x

)

i=1,2,3. (8)

Thepermeabilitytensorisfoundbyapplyingtheintrinsicaverage onthe Dtensorandmultiplying bythe porosity

ε

₌V β

V ,i.e.K=

ε

<D>β.IntheStokesregime,itis

FM=−

ν

β

ε

K−1<v_β>β. (9) The second closureproblemdiffersfromthe firstonly forthe presenceofalinearisedconvectiveterminwhichthemicroscopic velocity obtained from the DNS, v_β, is used as an input. This of course implies knowledge ofthe microscopic velocity field. A Oseen-like approximation which relaxes thisconstraint has been proposedbyZampognaandBottaro(2016).

The new unknowns are a vector anda tensor called, respec-tively,mandM,withthesamemeanings ofd andD.The system reads:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

1

ν

βvβ·

∇

M=−

∇

m+

∇

2_M+I,

∇

· M=0, M=0 on A_βσ, m

(

x+i

)

=m

(

x

)

, M

(

x+i

)

=M

(

x

)

i=1,2,3. (10) The average of the tensor M multiplied by the porosity is the apparent permeability , H=

ε



M



β. When inertia is important

Eq.(6)canbewrittenas FM₌₋

_ν

β

ε

H−1<vβ>β, (11)

asshownbyWhitaker(1996).

Tworemarksare inorder atthispoint.First, theequationsin the closure problem (10) are time-independent because the mi-croscopicvelocity v_β isa solution ofa stationaryDNS. Thus, the Reynoldsnumbershouldbesufficientlysmallforunsteadyeffects nottobepresent.Shouldthewakebehindasolidinclusiondisplay regular orirregular temporaloscillations,theequationsofsystem

(10) maybe used,as an approximation, by replacing the instan-taneous velocity in the REV with its time-averaged distribution. Thiscaseishowevernotofpresentconcern.Secondly,theclosure problemsreflect thestructure ofthesolution ofthetwo systems

(8)and(10).In particular,the solution of(8)depends only on the geometryoftheporousmediumsothatthepermeabilitytensorK

is symmetric.Thisis not thecase forH,because ofthe effectof themicroscopicvelocityamplitudeanddirection.Clearly,the solu-tion ofsystem(8)tends tothat of(10) when Re d=



v_β



βd

Fig. 2. REV for the fiber geometry investigated.

withv_β theamplitudeoftheintrinsicvelocityand d thediameter ofthefiber.

3. Validationandsetup

Inthissectionthenumericalmethodology,theparameters,the setupandthevalidationforsomereferencecasesaregiven.

3.1. Computational domain

ThegeometryusedforthebaseREVisshowninFig.2:a cylin-dricalinclusionispresentatthecentreoftheREVandfour quar-tersof cylindersare situatedatthecorners. Thelateral lengthof the cubic envelop is ,which isused aslength scale forthe mi-croscopicproblem;thediameter d ofthecylindersisadaptedasa functionofthedesiredporosity

ε

,ratiobetweenthefluidvolume over thetotal REV volume( 3_{).The forcing termfof theDNSis} a vectorwhosedirectionisdefinedbytwo Eulerangles,with ro-tationsoftheform:

θ

e3+

φ

e2I (cf.Fig.2).Itsamplitudeisseta

prioriandisconnectedtotheReynoldsnumber, Re dwhichresults

fromthecalculationsoncethemeanvelocityisevaluated.

3.2. Numerical setup

ThenumericalsimulationsofthefullNavier–StokesEqs.(1)–(2)

intheREVhavebeencarriedoutwiththeopen-sourcecode Open-FOAM (Weller et al., 1998), based on a finite volume discretiza-tionwithacolocatedarrangementfortheunknowns.Thestandard solver icoFoam (incompressible Navier-Stokes)has beenmodified in ordertoinclude aconstant pressure gradientactingasa forc-ing termfin Eq.(1).The couplingbetween thevelocity andthe pressureequationsisbasedonthepressureimplicitsplitoperator referred toasthePISOalgorithm.Thetimederivativetermis dis-cretizedusingthesecondorderbackwardEulerschemeandallthe spatialtermsuseasecond-ordercentraldifferencestencilbasedon Gaussfinitevolumeapproach.Thevelocitysystemissolvedwitha preconditioned bi-conjugategradientiterativesolverwiththe tol-eranceonthevelocityresidualssetto10−8_,associatedtoa diago-nalincompletelowerupperpre-conditioner.Thepressureequation is solved with a geometric-algebraic multigrid algorithm associ-atedtoaGauss-Seidelsmootherandthetoleranceonthepressure residuals istakenequalto10−6.Periodicboundaryconditionsare appliedtoallfields(velocityandpressure) onall fluidboundaries alongthethreedirections,andtheno-slipconditionisimposedon the surfaceofthe solid inclusions. The time step

t is automat-ically determined to ensure that the maximum Courant number,

Fig. 3. Permeability versus porosity for a square arrangement of cylinders. The scal- ing of the permeability is 2 and is explicitely indicated in the vertical axis.

Table 1

Relatives errors in the effective permeability validation.

Regular REV Staggered REV ε erK11 erK33 erK11

0.4 2.34% 3.22% 4.13% 0.6 1.58% 0.38% 2.03% 0.8 4.10% 2.03% 2.17%

Co ,respects thecondition: Co =

v

_β

t/

s



1/ 2, inwhich v _β isthe modulusofthelocalvelocityontheREVand

s isthelocalgrid spacinginthedirectionofv_β. Co isbasicallytheratiobetweenthe fluidspeedandthevelocitytopropagateinformationthroughthe meshandthecondition Co <1/2isfoundtobesufficienttohavea stablesolver.

The convergence of the solutions is assessed on the basis of a Richardson extrapolation which employs the results obtained on successively refined grids (with up to 1.5 million cells in the finer grid used). The mesh convergence analysis is described in

AppendixA.

3.3. Validation on two different configurations

The results published in the literature by Zampogna and Bottaro(2016) andYazdchietal.(2011) arenowused tovalidate both the methodology and our choice of the computational pa-rameters.In the cited papers,three-dimensional computations of thepermeabilitycomponentsindifferentcellsgeometriesare pre-sented.

Fig.3displaysthecomparisonforacellwithasquare arrange-ment of the fibers; herethe permeability is evaluated along the two principal directions, x 1 and x 3. To quantify the accuracy of theresultsweintroducearelativeerroronthepermeability com-ponents, defined as e rK_ii =

|

K iipresent− Kiiliterature

|

K iiliterature

and reported in

Table1.Goodagreementisfound withthepublishedresults: the relativeserrors in all thecases tested isalways below5%. Fig. 4

shows a similar comparison for a staggered arrangement of the inclusionsin the unit cell. In this casethe section of the cell is rectangular. The agreement forthe only permeability component available in the literature is again satisfactory. Finally, to check the correct implementation of the closure model (10) it is im-portant to verify the equality (7) between the amplitude F M _of

the macroscopic force and its microscopic counterpart obtained through an integration of the DNS fields over the solid bound-ariesof theinclusionsinthe REV.Fig.5 showsaplot ofthe rel-ativeerrorbetweenthesetwo forces,i.e.

||

F

M_{− F}m

_||

Fig. 4. Permeability versus porosity for a staggered arrangement of cylinders. Note that the permeability component is here scaled with d 2 .

Fig. 5. Relative error between the microscopically computed forces along the x 1

direction and those arising from the Darcy-Forchheimer model; ε = 0 . 8 for the REV in the staggered arrangement of Yazdchi et al. (2011) .

ofthe Reynolds number. We consider the successful comparison displayedinFig.5astheconclusivedemonstration ofthevalidity oftheapproach describedhere.We havenonetheless carriedout thesame verification displayedin Fig.5 for mostof the simula-tionsdescribedinthefollowing,tooursatisfaction.

3.4. Tests with larger REV’s

Since theREV istheunit cellwithin theporous mediumover whichaverage quantitiesof theVANS are computed, itis impor-tanttochoose itsdimensionsappropriately intheinertialregime for,iftheREVistoosmall,itmightbeeasytomisscrucialfeatures ofthewakes.Forexample,topredictthecriticalReynoldsnumber,

Re c,of the first Hopfbifurcation, a REV containing atleast three

solidinclusionsin thedirectionof themeanpressure gradient is necessaryintheconfigurationusedbyAgnaouetal.(2016).Among theresultsreported,itisfound that,fora fixedREV size,the er-ror committedin the evaluation ofthe critical Reynolds number increases with the porosity. This same error is considerably re-ducedwhen the mean pressure gradient angle is

θ

=45◦. Thus, thechoiceofthenumberofinclusionsinaREVisatasknottobe overlooked,andthefinalchoicemustaccountfortheporosity,the directionof the pressure gradient and the microscopic Reynolds number.

Here, the influence of thenumbers of inclusionspresent ina REVisassessedbyfocussingonlyonthevelocitycomponentsafter averagingovertheREV.Theunitcubiccellofsideisusedas ref-erence:startingfromthis,twoadditionalREV’sarebuilt,asshown inFig.6.Thefirstoneisdoubledinboththe x 1 and x 2 directions andthe casetested numerically ischaracterised by

θ

=0,

φ

=0

Fig. 6. REV configurations. Left: 2 × 2 × 1 arrangement; centre: 1 × 1 × 1 arrange- ment (reference); right 1 × 1 × 3 arrangement.

(i.e. the forcing pressure gradient is directed along x 1), porosity

ε

=0. 6 and Re _d=50. The second REV configuration is a compo-sitionof3referenceREVsontopofoneanotheralong x 3,withthe parameterssetto

θ

=45◦,

φ

=45◦,

ε

=0.6and Re d=100.

Forboththesetestcases,noappreciabledifferences,neitherin the meanvelocity norin the forceson thefibers, havebeen ob-served, withrelative errorson themeanvelocity withrespect to the referencecase which remain below2%. We take thisas suf-ficientevidence touse,inthe following,onlythe referencecubic REVofsideequalto,withtheunderstandingthatonlycaseswith amicroscopic Reynoldsnumberup toaround 100can be consid-ered.

4. Microscopicsolutions

Inthissection,somelocalmicroscopicfieldscomputedwith di-rect numericalsimulations are shown, together withcomponents oftheintermediatetensorMcomingfromthenumericalsolution oftheclosureequations(10).

In Fig. 7 (top row) the local x 1 velocity component is drawn for the two-dimensional flow when

ε

=0.6, for three Reynolds numbers, to cover the transition from the Stokes to the inertial regime.Inallplots,thevelocitiesarerenderednon-dimensionalby thecorrespondingvalueof H 11

ν

β

||

f

||

, where H 11 isthefirst

compo-nentofthetensorH.Wheninertiaisabsent,theflowhasacentral symmetry;byincreasingtheReynoldsnumber,onlythesymmetry withrespecttothe x 1axisismaintained(x 1isthedirectionofthe forcingpressuregradient),withthewake’slengthwhichincreases with Re _d.When Re _disoforder100thewakespreadstothe down-streamboundaryoftheREV,re-entering,becauseofperiodicity,at theupstream side. This Re d represents theupperlimit ofvalidity

forthecubicunitcellofside;largervaluesof Re dcouldonlybe

investigatedwithlonger/larger/thickerREV’s.

The non-dimensional microscopic M 11 fields forthe same pa-rameters are displayedin Fig.7 (mid row). Allvalues in the fig-uresarise from scalingM with 2_{. Visually,these localfields are} stronglycorrelatedto thelocalstreamwisevelocitycomponentin thewhole Re d range.Thisisnotunexpectedsincethelocal

veloc-itydrives theconvective termofsystem(10).Thecentral symme-tryofall componentsofMintheStokesregime iscoupledtothe rotationalinvariance of theapparent permeability tensorin two-dimensionalflows.

The effect of varying the porosity is shown in Fig. 7(bottom row) where

ε

is takenequal to 0.4. Evenatsuch a low porosity the stretching of the wakecan be noticed,and it increaseswith

Re d.Interestingly,thiseffectismilderwhentheforcingisinclined

Fig. 7. Top row: plane view of the dimensionless x 1 component of the local velocity field v βfor the case θ= 0 , φ= 0 , ε = 0 . 6 and for three Reynolds numbers Re d = 0, 10,

50, from left to right. Mid row: microscopic M 11 fields corresponding to the images in the top row. Bottom row: M 11 fields for the same Euler angles and Reynolds number

as in the top two rows, and smaller porosity ( ε = 0 . 4 ).

astrongdeviationofthemeanflowalongtheaxisofthefiber.In thiscase, M 11 and M 22 behaveverysimilarlytothecase

φ

=90◦.

AnotherinterestingpointemergesbyinspectionofFig.8where two off-diagonal components of M are shown for two porosity values; the first image (left frame) represents a plane flow in the Stokes regime while the second isthe plane cut of a three-dimensional solutionin theinertialregime. Positive andnegative values ofthe microscopic fields can be seen inboth images but, once averaging is applied over the REV, the resulting permeabil-itycomponentisveryclosetozero(in fact,exactlyequalto zero in theStokes case).Thissame featuresoccurs foralloff-diagonal terms inall casesexamined, so that,within the currentrangeof Reynoldsnumbers,theapparent permeabilitytensoris,toagood approximation,diagonal1

Athree-dimensionalcaseisshowninFig.9,whereallthe non-zerotermsoftheMtensorareplottedforaporousstructurewith

1 In fact, there are always at least two orders of magnitude differences between

the diagonal and the off-diagonal components. While the latter should not, in

prin-ε

=0.6.The componentsshownare M 11, M 22, M 33, M 12 and M 21, while M _i3and M _3jarenotplottedbecausetheyareidenticallyzero tomachineaccuracy.Distinctfeaturesarevisibleineachimage;in particular,in the last frame the M 33 microscopiccomponent dis-playsalowwavelengthstructurealongthecylinder’saxis. Increas-ingthedimensionsoftheREValong x 3doesnotaltersucha struc-ture, i.e.the 3 _{domainchosen withitsperiodic boundary} condi-tionsdoesnotfilteroutsignificanthighwave-numbersoftheflow. We furthernote that the tensor Mis not symmetricin thiscase sinceeach off-diagonal componentrepresentsthe solutionof the closureprobleminaspecificdirection(firstindexofthefield)and theforcingtermactsorthogonallytoit(secondindexofthefield). OnceaveragedovertheREVitisfoundthatboth H 12and H 21 are veryclosetozero.

ciple, be ignored, we will focus attention here only on the dominant terms of the permeability tensor.

Fig. 8. Right: Non-dimensional M 21 field for θ= 0 , φ= 0 , Re d = 10 , ε = 0 . 8 , left:

Non-dimensional M 12 field for θ= 22 . 5 ◦, φ= 45 ◦, Re d = 50 , ε = 0 . 4 .

Table 2

Directions of the forcing tested and property of the solutions.

Index θ φ Field properties 1 0 ° 0 ° 2D symmetric 2 22.5 ° 0 ° 2D non-symmetric 3 0 ° 45 ° 3D symmetric 4 22.5 ° 45 ° 3D non-symmetric 5 - 90 ° 3D symmetric

5. Theapparentpermeabilitytensor

Inthissectionthevariationsofthediagonalcomponentsofthe permeabilitytensorHarediscussedasfunctionofthedirectionof themeanforcing,theReynoldsnumberandtheporosity.Asstated previously, the Reynolds number Re d ranges from 0 to

approxi-mately100inordertocapturephenomenaassociatedwithinertia; thecasesconsidered neverlead tounsteady signals.The porosity parameter

ε

issettoeither0.4(lowporosity),0.6(medium)or0.8 (high).

The forcing direction is defined by the Euler angles and all the configurations considered in this section are summarized in

Table2; the choicehas beenmade toexplore a reasonablylarge rangeofparameters,withbothtwo-dimensionalandthree- dimen-sionalflowscharacterizedbysymmetricandasymmetricpatterns. Let us briefly recall the methodology. First, a DNS is carried out to compute the microscopic flow. Then the closure problem issolved forthe tensorM.Finally, eachcomponentofthe appar-entpermeabilityHis obtainedby averaging(Eq.(3)). The results arecollected inFigs. 10,11 and12,showing thevariation of the diagonalcomponentsofH.

In the left column of each figure we focus on the low-Re d

regime (0<Re d<2), while in the right column the effect of

in-ertia canbe assessed. As expected, when Re d is smallthe

appar-entpermeabilityis quasi-Reynolds-number-independent(and can beapproximated well by thetrue permeability).As the Reynolds numberincreasesaboveafewunits,inertialeffectsgrowin impor-tanceyieldingtypicallyamonotonicdecreaseofallcomponentsof

H,asidefromcaseindexed5

(

φ

=90◦

)

forwhichtheflowremains alignedwiththecylinder’saxis.Incase5themicroscopicflow so-lutionis invariant with x 3 and doesnot change with Re d in the

rangeconsidered,sothatHisaconstanttensor.Whenthe poros-ityislarge all components show a similar behaviour irrespective oftheforcingangle(except,clearly,case5).

Differences startappearingat

ε

=0.6; thetwocaseswith

φ

= 0◦ (index 1and2) behavesimilarly, andso dothe two cases in-dexed3and4(with

φ

=45◦).Thisseems tosuggestaweaker ef-fectof

θ

onthepermeabilitycomponents.Forevensmaller

poros-Fig. 9. Non-dimensional M components fields for the case θ= 22 . 5 ◦_{, φ=}

45 ◦_{, Re}

d = 50 , ε = 0 . 6 .

ity(

ε

=0.4),theblockageeffectcausedbytheinclusionsproduces the unexpectedbehaviour displayedinFig. 12. When theflow is purelytwo-dimensional(cases1and2),variationsintheReynolds numberaffectHsignificantly;whenapressuregradientalong x 3is presentthestrongpackingofthefibersconstrainthefluidtoflow prevalentlyalongthefibers’axis,andtheapparentpermeabilityis almost Re d-independent.When assessingvariationsin H jj forthis

case,attentionshouldalsobepaidtothefactthatthepermeability isnowatleastoneorderofmagnitudesmallerthaninthe previ-ouscasessothatvariationsofthediagonalcomponentsshownin

Fig.12aretinyinabsoluteterms.Thisisrelatedtothefactthatthe inverseofthe permeabilityplays therole of adragcoefficient in themacroscopicexpressionoftheforce(cf.equation(6)).Inother

Fig. 10. Diagonal elements of the apparent permeability H as function of the Reynolds number for porosity ε = 0 . 8 . The forcing direction is represented through the couple

of Euler angles ( θ,φ) (cf. Table 2 for the case index). Left column: low- Re d regime; right column: inertial regime.

Fig. 12. Same as Fig. 10 with porosity ε = 0 . 4 .

words,materials withlowerporosity offerlargerresistancetothe motionofthefluid.

Applyingtheintrinsicaverageoperatortotheoff-diagonal com-ponentsofthe tensorM resultsintermsthat are negligiblewith respectto their diagonal counterparts, and theseresults are true foralltheparameters considered.Thismeansthatthereisavery weak couplingbetween the principal directions ofthe fiber. The directionaldecoupling andthe diagonalproperty ofthe apparent permeabilitytensorhavealsobeencomputationallydemonstrated on a completely different REV geometry by Soulaine and Quin-tard (2014). Conversely, Lasseux et al. (2011) have carried out a two-dimensionalstudywithfibersofsquarecross-section,finding thattheoff-diagonaltermsarenon-negligibleandonlyaboutone order of magnitude smaller than the diagonal components. This resultis a consequence of the non-rotationally-invariant geome-try considered. The presentwork andthe two articles justcited suggestthatthediagonalpropertyofthetensorHismoreclosely associatedtothegeometryoftheporousmaterialthantotheflow regime,atleastintherangeof Re dconsidered.

6. AmetamodelforH

Sections 4 and 5 have shownhow the apparent permeability dependson the two Euler angles, the Reynolds number andthe porosity. The space of parameters is formidable and the results found so far are not sufficient to treat, for example, cases char-acterizedbymultipleinclusions’sizesandorientationsindifferent regions of the domain, or cases involving a poroelasticmedium, withtemporallyandspatially varyingporosity,flowdirectionand localReynoldsnumber.Thecompletesolutionoftheclosure prob-lem for a single set of parameters takes approximately 4 CPU hoursonourtwo-processor Intel(r)IVYBRIDGE2.8Ghz,each with

10coresand64GBofRAM, sothata completeparametricstudy is,tosaytheleast,unpractical.Inviewofthis,theconstructionof ametamodelcapabletoprovideafull characterisationofthe per-meabilityasafunctionofallparametersisaworthyendeavor.We havetestedseveralsurrogatemodels,beforeeventuallysettlingon thekrigingapproach(Kleijnen,2017)describedinAppendixB.

Themetamodelprovidesascalarfunction(foreachtermofthe

Htensor) definedinafour-dimensionalspace.Ineach ofthe fol-lowingfigurestwo parametersare fixedandtheresponsesurface isdisplayedasfunctionoftheremainingtwo,focussingonthe H 11 component. The other diagonal components ofthe apparent per-meabilitytensorbehaveinasimilarfashionandwillnotbeshown forbrevity.Alltheresultsofthemetamodelare,however,available on:https://github.com/appanacca/porous_solvers_OF.git.

InFig.13theangle

φ

isfixedto zero,andtheisolinesdisplay

H 11 as functionof the angle

θ

and ofthe Reynoldsnumber, Re d,

forthreevaluesoftheporosity.Thewhitesquaresymbolsindicate thesamplesusedtobuildthemetamodel.Themaximumvalueof each surfaceis always found for Re d equalto zero and H 11 typi-callydecreaseswith Re d,whentheporosityissufficientlylarge.As

seen previously, fora porosity approximatelygreater or equal to 0.6thevariationoftheapparent permeabilitywiththeangle

θ

is weakinthistwo-dimensionalconfiguration.Forthelowest poros-itystudied(leftframe)thepermeabilityhasverysmallvaluesand theisolinesdisplay an irregularbehaviour; thisisa feature com-mon toall plots relative tothe smaller value of

ε

,signaling that it isprobably necessary, inthis specific case, to insertadditional samplepointsinbuildingtheresponsesurfaces.

InFig.14theparameter

θ

issetto0° andtheresponsesurface isdisplayedinthe Re d−

φ

plane.Asalreadyindicated,theresults

confirmthatanincreaseoftheReynoldsnumberisgenerally asso-ciatedtoadecreaseofthefirstdiagonalcomponentofthe

appar-Fig. 13. Response surfaces of H 11 with φ= 0 ◦for porosity ε = 0 . 4 , 0 . 6 , 0 . 8 , from left to right.

Fig. 14. Response surfaces of H 11 with θ= 0 ◦for porosity ε = 0 . 4 , 0 . 6 , 0 . 8 , from left to right.

Fig. 16. Response surface of H 11 ; in the left frame φ= θ= 0 ◦, in the centre frame φ= 90 ◦, θ= 0 ◦and on the right φ= 45 ◦, θ= 22 . 5 ◦.

entpermeabilitytensor.However, the H 11 variationswithrespect to

φ

aremorepronouncedthanthosefoundwithrespectto

θ

and areduetoarealthree-dimensionalizationoftheflow.This conclu-sionremains tobeverifiedinthelowerporosity case(leftframe) wherethevariationsareverytinyandmoreirregular.

InFig.15theReynoldsnumberissettotheinertialrangevalue of 40 and the response surface is displayed in the

θ

−

φ

plane. Forthetwo highestporosity values,0.6and0.8, the results con-firm that H 11 hasa much strongerdependence on

φ

than on

θ

, suggestingthat therealtestofpermeabilitymodels mustinclude three-dimensionaleffects.Asseenearlier,thebehaviourofthe per-meabilitywhentheporosityislow(leftframeinthefigure)isnot intuitive,witha significanteffectof theangle

φ

anda minor in-fluenceof

θ

.Againthisoccursfromtheconstraintprovidedtothe flow by the inclusions, andfrom the occurrenceof a large devi-ation

γ

in thesecases (see Appendix B forthe definition of the deviationangle

γ

).

Theresponsesurfaceisshowninthe Re _d−

ε

planeofFig.16for threesets of

θ

−

φ

angles. Here a significanteffect ofthe poros-ity with respect to the Reynolds number is observable. In fact thesurfacegradientisalmostalignedwiththeporositydirection, i.e.a quasi-Reynoldsindependenceis demonstratedinthisplane, andtheapparentpermeabilitycanchangebyoneorderof magni-tudein therange ofthe analysed porosity.Some relatively small Reynoldsnumbereffects are visiblefor

ε

equal to 0.8, whenthe wakeoftheflowhasmorespacetodevelopintheinertialregime. Inthecentral figurethe flowisalignedwiththe directionof the fibersand, asexpected, it showspractically no dependence with respecttotheReynoldsnumber.

The response surface analysis has confirmed the qualitative trendswhich hadbeen reachedearlieron the basis ofa few se-lected flowcases, yielding atthesame time muchmore detailed informationonthebehavioroftheapparentpermeabilitywiththe parameters of the problem. The data base which has been built will be used in future work which will focus, via the VANS ap-proach,on configurations for whichneither the porosity northe localReynoldsnumberareconstantinspaceortime.

7. Concludingremarks

Thecomponentsofthepermeabilitytensorareessential ingre-dientsforanysolution offlowthroughanisotropic porousmedia. Whentheflowthroughtheporesresentsofsignificantacceleration effects,thepermeabilitymustbemodified(itisthencalled appar-

ent )bythepresenceofasecond tensor,theForchheimertensorF, definedby

F=KH−1− I.

Thepermeability,K,andtheapparentpermeability,H,canbe for-mallydeduced by two closureproblems which havebeenbriefly recalledinSection2.Therealobstacletothesolutionofthe prob-lem for H is the need to know the microscopic velocity fields through the pores. We have solved for such fields in a unit cell (the REV), varying the forcing amplitude and direction, treating overonehundreddifferentcasesofflowsthrougharrangementsof parallelfibers.Fromthis,wehavethusbeenabletosolvethe lin-earsystem(10)foralltheunknown elementsoftheintermediate tensorM,fromwhich,throughaveraging,weobtainthe apparent permeability.Suchatensorisindispensabletoevaluateaccurately thedragforce causedby the presenceofthe fibers,fora macro-scopic solutionoftheflow onthe basis ofEqs.(4)and(5)when inertialeffectsarepresent.

It has been found that the apparent permeability tensor is strongly diagonally dominant for whatever forcing direction and porosity,provided the microscopic Reynoldsnumberremains be-low avalue approximatelyequal to100;thisresults– which isa directconsequenceofthetransverseisotropyofthematerialwhich hasbeenconsideredhere– canbeusedtocomputeHrapidly, ap-proximatingitasadiagonaltensor.

Finally,ametamodelhasbeenusedtoproduceresultssoasto coverthewholespaceofparameters,andthishasallowedthe con-structionofa completedata base.Theelementsofthe database providearobustapproximation tothe”true” apparent permeabil-ity values asdiscussed in Appendix B (see in particular B.2 and

B.3 ).Thisdatabaseisnowbeingusedinsimulationsofporoelastic mediabasedontheVANSapproach.

Acknowledgment

TheauthorsacknowledgetheIDEXFoundationoftheUniversity of Toulouse for the financial support granted to the last author undertheprojectAttractivityChairs.The computationshavebeen conducted at the CALMIP center, Grant no. P1540. The complete database is available at the link: https://github.com/appanacca/ porous_solvers_OF.git.

AppendixA. Meshconvergenceanalysis

The mesh has been computed using the internal OpenFOAM meshernamed snappyHexMesh .Thefinalgridismainlycomposed by hexahedral cellswith a refined regular grid in the boundary layerregionsnexttothesolidsurfaces.Threedifferentmeshsizes, with0.65_{× 10}6_,106_{and1.5× 10}6_{elements,havebeentestedin} or-dertodemonstratespatialconvergence.Thishasbeenassessed us-ingtheGridConvergenceIndex(GCI )introducedbyRoache(1998). Images of the coarsest mesh used are shownin Fig. A.17.On therightframeacloseupofthegridintheneighbourhoodofthe fiber’sboundaryisdisplayed:twentypointsareusedinthe struc-turedportionofthemeshalongthewall-normaldirection.

The GCI methodisbaseduponagridrefinementerrorestimator derivedfromthetheoryofgeneralizedRichardsonextrapolation.It measurestheratiobetweenthecomputedvalueofaquantityover theasymptoticnumericalvalue,thus indicatinghowfarthe solu-tionisfromtheasymptotic(”exact”)value.Theprocedureissimple andprovidesamethodtoestimatetheorderofthespatial conver-gence,based ontwo orthree differentgrid sizes. Firstof all,the grids mustbe generatedwiththe samealgorithm andthey must havethe samefinalquality.Ineach simulationthe intrinsic aver-agevelocityintheporousmediumissampled.Themethodfollows thefollowingfoursteps:

1. Estimatetheorderofconvergenceoftheprocedure,definedas p=ln



f3− f2 f2− f1



/lnr,

where r is the grid refinement ratio betweeneach grid (it is computedastheratiobetweenthenumberofelementsoftwo consecutivegrids;the approachimposes that r should remain constantbetweenanycoupleofconsecutivegridsandbelarger than 1.1), and f i represents the amplitude ofthe intrinsic

av-eragevelocity ineachgrid (1= coarse,2 =medium and3= fine).

2. Computetherelativeerrorbetweengrid i and j :

|

|

i j=

f j− fi

f i ,

for(i, j )∈{(1,2),(2,3)}. 3. Compute GCI _{i j=} F s

|

|

i j

r p_{− 1},with F sasafetyfactorequalto1.25if

the grids are three, and equal to 3 if the grids are only two (Roache,1998).

4. Checkwhether each gridlevel yields a solutionthat is inthe asymptoticrangeofconvergence;thismeansthat thequotient

AC =GCI 23 GCI 12

1

r p should be as close as possible to one.

Table A.3

Convergence analysis. Left: average velocity within the REV normalized with K11

νβ|| f || , with K 11 the first component of

the tensor K . Right: grid convergence metrics. The REV has ε = 0 . 6 , the motion is along x 1 , i.e. θ= φ= 0 and Re d = 180 .

Mesh Mesh Average REV Metric Value index identifier velocity

3 fine 1.11 GCI23 0.366%

2 medium 1.07 GCI12 1.11%

1 coarse 1.09 AC 1.006

TheresultsaresummarizedinTableA.3.

Fromthetableitcanbeseenthattheintrinsicvelocitydifference isvery smallfromone grid to thenext andthe coarse grid pro-videsresultsclosetotheexpectedasymptoticvalue.Thisistaken asa sufficientlyconvincingargumentto carryout all the compu-tationsinthepaperwithagriddensityequaltothatofgrid1.

AppendixB. Buildingthemetamodel B1. DACE sampling

Thefirststeptobuildametamodelisthecollectionofrelevant samples.The qualityof thefinal metamodelstrongly dependson thesamplescollectedandtheirnumberanddistributionisof pri-maryimportance.

Theapparentpermeabilitytensor,H,dependsonfour indepen-dentvariables;thesampleshavebeengeneratedstartingfromthe setofparametersgiveninTableB.4.

Oneofthebestoptionstogeneratetherelevantdatabasewould betousea fullfactorial designapproachinwhichall the combi-nationsofthefourvariablesfromTableB.4arecomputed.Because ofthe large numberofcomputations required,thisapproach has not been retained.We have resorted to the methodology known asDACE(Design andAnalysis ofComputer Experiments),a tech-niquetofillinthebestpossiblewaythespaceoftheparameters oftheproblem. The Dakotalibrary(Adams etal., 2014) hasbeen selectedforthepurposeandtheMonte-Carloincrementalrandom samplingalgorithm(Giuntaetal.,2003)hasbeenchosen,inorder to make efficient use ofthe cases already computed. This incre-mentalapproachselectsina quasi-randomwaythenewsamples togenerate,startingfromtheexistingones. Intheend,thesetof samplescomprises118cases.

Inthe scatter plot ofFig. B.18 thethree diagonal components ofthepermeability tensorare shownasfunction ofone another. Thethreeporositiesareseparatelyconsideredineachoftheabove

Table B.4 Sampling parameters. parameter values θ 0 ° 22.5 ° 45 ° φ 0 ° 22.5 ° 45 ° 67.5 ° 90 ° Red 0 10 50 100 ε 0.4 0.6 0.8

plot,andthepermeabilitypointsarerepresentedwiththeirlinear regressionon top.Thiskindofplotiscommoninstatistical anal-ysisto determineifcorrelationsinthedataare present.The per-meabilitycomponentsdisplayapositivecorrelation,withthedata pointswhichliereasonablyclosetoastraightline.

Thisresulthasaphysicalimplication.Remembering the diago-naldominanceofthepermeabilitytensor,wehaveinthelow Re d

limit:



u_β



β,



v

_β



β,



w_β



β



∼



H11

∂

p

∂

x1, H22

∂

p

∂

x2, H33

∂

p

∂

x3



. (B.1)

Itisthenpossibletocomputetheanglebetweentheforcingterm,

∇

p ,andtheaveragevelocityvector,representedinFig.B.19forthe two-dimensionalcase,

φ

₌0.This isachievedby takingthe ratio betweenthe first two componentsof Darcy’s equation, calling

γ

theangular flowdeviation withrespect tothe meanforcing. We thushave:

tan

(

θ

+

γ

)

=H22

H11

tan

θ

. (B.2)

Fig. B.19. Explanatory sketch for the relation between mean pressure gradient and mean velocity field.

Iftheratiobetweenthetwopermeabilitycomponentsisequalto one, the angle

γ

vanishes. The correlation between H 11 and H 22 controls the deviation of the flow in the (x 1, x 2) plane, and the argumentcaneasilybeextendedto H 11/H 33and H 22/H 33 for devi-ationanglesinthree-dimensions.

Usinga linearcorrelation suchasthat showninTableB.5and

Fig.B.18,itisobservedthatinthelowporositycase

(

ε

=0.4

)

the ratio can become very large indicating a strong deviation of the flow from the forcing direction, because ofthe strong constraint providedbytheinclusions.Astheporosityincreases,theratiodoes not differ much from unity, which means that the deviation re-mainslimited. Itissimpletoseethat thedeviationangle,for

Table B.5

Permeability components ratio for three values of the porosity. The permeability ratios here are given by the angular co- efficients of the linear correlations dis- played in Fig. B.18 .

ε H11 / H 22 H11 / H 33 H22 / H 33

0.4 1.57 11.06 96.03 0.6 1.50 1.62 0.99 0.8 1.20 0.82 0.66

ampleinthe(x 1, x 2)plane,satisfiestheapproximaterelation

tan

γ

=



1−H11 H22



tan

θ

H11 H22 +tan2

_θ

, sothatfor H 11 H 22

equalto,say,1.5,thelargestdeviationremains al-waysbelow12° forany

θ

.Itshouldhoweverbekeptinmindthat trendsbasedon theseratiosare validonlyaslongasDarcy’slaw andlinearcorrelationsareacceptable.Casesexistsforwhichsuch trends areviolated; forexample,aflow with

θ

=45◦ and

φ

=0◦ hasdeviationangle

γ

equaltozero,forwhateverporosity.Inthis case H 11/H 22 isequaltooneandsucha pointisan outlierinthe regressionplotsofFig.B.18.

B2. Kriging interpolation method

The kriging approach is a linear interpolation/extrapolation methodthat aimstobuild apredictorfield basedona setof ob-servations(xi,y(xi)),for i =1,...,n .

Thepredictor f ˆ

(

x

)

isasumofatrendfunction t (x)anda Gaus-sianprocesserrormodel e (x):

ˆ

f

(

x

)

=t

(

x

)

+e

(

x

)

. (B.3)

The aimofthe errormodelisto makeadjustments onthetrend functionsothat,foranypointofthesamplingthepredictoris ex-actly equalto the sample, i.e. f ˆ

(

xi

)

=y

(

xi

)

. Thisproperty

repre-sentsoneofthemainqualitiesofthisapproach.Inaddition,when themodelparametersareconvenientlyset,thetrendfunctionand thecovariancemodelcantakeintoaccountbothsmoothandsteep variationsinthedataset.

The trend function defined here is based on a second order least-square regression,with the coefficientsfound from the so-lution of the associated linear system. The Gaussian process er-ror modelhaszero-meananditscovariancebetweentwo generic data-points, x _iand x _j,iswrittenas

Cov

(

y

(

xi

)

,y

(

xj

))

=

σ

2r

(

xi,xj

)

.

Thecoefficient

σ

isanamplitudeparameterand r (x i_{, x} j_)isa

corre-lationfunction,basedontheMatérncovariancemodel,thatreads: r

(

xi,xj

)

= 21−ν

(

ν

)



√ 2

ν|

xi− xj

|

|

λ|



ν K_ν



√ 2

ν|

xi− xj

|

|

λ|



, (B.4)

where K _ν(.)is amodified Besselfunction and



(.)is thegamma function. The parameters that can be used to tune the meta-model are the amplitude parameter

σ

, the exponent

ν

and the scale vector

λ

. The kriging metamodel outputs can show differ-ent behavioursfordifferentselections oftheabove three param-eters and their setting is thus crucial. The amplitude parameter

σ

is chosen to be equal to 1;larger value lead to steeper gradi-ents and undesirable local extrema around the data points. The vector

λ

=

(

λ

θ,

λ

φ,

λ

Re_d,

λ

ε

)

is a scaling parameter for the

dis-tance

|

xi− xj

|

. Inthis study,through systematicvariationsofthe

Fig. B.20. Relative mean error computed using the k -fold approach presented against the number of folds k used to divide the dataset.

parameters itis foundthat the choice

λ

=

(

1.2,1,1,1

)

yields ac-ceptableresults; inparticular,theweightalong

θ

ismildly larger than in the other directions in order to obtain smoother meta-modelsurfacesinthisdirection. The exponent

ν

controlsthe co-variance function and more particularly its gradients. When

ν

= 1_/2thecovariancecanbeapproximatedbyanegativeexponential, exp

(

−

α

x

)

andwhen

ν

goestoinfinityitbehavesasexp

(

−

α

x 2

₎

_.In thepresentstudy,thebest(i.e.smoother)resultsareobtainedfor

ν

equal to 1.9. The above parameters havebeen chosen in order toavoidunphysicalorunrealisticbehavioroftheapparent perme-abilitysuchas,forinstance,negativevaluesorsteep,spuriouslocal maxima/minima.ThemethodaboveisimplementedinOpenTURNS (Baudinetal.,2016).

B3. Robustness of the metamodel

A procedure called k -fold, belonging to the class of cross-validationmethods,hasbeenusedinordertoprovetherobustness ofthemetamodel.The k -foldmethodstartswiththefulldatabase

S n=

(

xi,y

(

xi

))

,for i =1,...,n,splitintotwocomplementarysets

ofsize n 1 and n 2,suchthat S n= S n1∪S n2.Then,anewmetamodel

isbuiltusingonlythepointspresentintheset S n₁.Forthesakeof

clarity,themetamodelbuiltwithonlythesubset S n₁ willbecalled

fromnowon f ˆn_{1,andthe metamodelbuiltwithall thedatabase}

willbeindicatedas f ˆn_{.Theideanowistousethepointsintheset}

S n₂ astest,sincetheyareessentially”new” forthemetamodel f ˆn1.

The division ofthe subset is performed picking pointsin a ran-domway,andisrepeated k timesinordertoruleoutanypossible ”lucky” combination.

Thus,themetricused fortheerrorcomputationisthe follow-ing:

ξ

cv=_k1_n 2 k i=1 n2 j=1



ˆ fn i

(

xj

)

− ˆfin2

(

xj

)



2 ,

quantifying the quadratic error between the original metamodel andthe one built each time witha differentset that belongs to differentfolds.

Themetricis alsoaveragedover allthe testpoints n 2 present inallthe k folds.Therelativemeanerrorcanbecomputedas:

Ecv%=100



ξ

cv mean

(

|

fˆn i

|

)

.

Inourcasethenumberofpointsusedtotestthemodel n 2isequal to√N ≈ 12asrecommendedforkrigingmetamodelsbyWangand Shan (2007). The number offolds has beenvaried from5 to 25

andinallthecasestestedthe E cv%hasbeenfoundtodecrease be-low6%whenweuseatleast16folds(whichmeansleavingout7 to8pointsfromthemetamodelconstruction),whichismorethan acceptableto prove that ourkriging methodis a robust approxi-mation(Fig.B.20).

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