Flow in porous media with low dimensional fractures by employing enriched Galerkin method

ContentslistsavailableatScienceDirect

Advances

in

Water

Resources

journalhomepage:www.elsevier.com/locate/advwatres

Flow

in

porous

media

with

low

dimensional

fractures

by

employing

enriched

Galerkin

method

T.

Kadeethum

_,

_H.M.

_Nick

_,

_S.

_Lee

_,

_F.

_Ballarin

a Technical University of Denmark, Denmark b Delft University of Technology, the Netherlands c Florida State University, Florida, USA d mathLab, Mathematics Area, SISSA, Italy

a

r

t

i

c

l

e

i

n

f

o

Fractured porous media Mixed-dimensional Enriched Galerkin Finite element method Heterogeneity Local mass conservative

a

b

s

t

r

a

c

t

This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the het- erogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same ac- curacy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures.

1. Introduction

Modelingof fluidflow in fractured porousmedia is essential for awidevarietyof applicationsincludingwaterresourcemanagement (Glaser etal., 2017;Penget al.,2017),geothermal energy(Willems andNick,2019;Salimzadehetal.,2019a;SalimzadehandNick,2019), oil andgas (Wheeler etal., 2019; Kadeethum et al., 2019c; Andri-anovand Nick,2019; Kadeethum et al., 2020c),induced seismicity (Rinaldi andRutqvist, 2019), CO₂ sequestration (Salimzadeh et al., 2018),andbiomedicalengineering(Vinjeetal.,2018;RuizBaieretal., 2019;Kadeethum etal., 2020a).Afractured porousmedium canbe decomposedintothebulkmatrixandfracturedomains,whichare gen-erallyanisotropic,heterogeneous,andhavesubstantiallydiscontinuous materialpropertiesthatcanspanseveralordersofmagnitude(Matthai andNick,2009;Flemischetal.,2018;Jiaetal.,2017;Bisdometal., 2016).These discontinuitiescancriticallyenhanceorhindertheflux withinandbetweenthebulkmatrixandfracturedomains.Accurately

∗ _{corresponding author at: Technical University of Denmark, Lyngby, Denmark, Delft University of Technology, The Netherlands.}

E-mailaddresses:teekad@dtu.dk(T. Kadeethum), h.m.nick@tudelft.nl(H.M. Nick), lee@math.fsu.edu(S. Lee), francesco.ballarin@sissa.it(F. Ballarin). capturingtheflowbehaviorcontrolledbythesediscontinuitiesin com-plex mediais stillchallenging(NickandMatthai,2011a;De Dreuzy etal.,2013;Flemischetal.,2016;HoteitandFiroozabadi,2008;Zhang etal.,2016).

Therearetwomainapproachestorepresentthefluidflowbetween thematrixandfracturedomain(NickandMatthai,2011b;Flemisch etal.,2018;Juanesetal.,2002).Thefirstmodel,anequi-dimensional model,discretizesthematrixandfracturedomainwithsame dimension-ality(Salinasetal.,2018).Thisapproachisstraightforwardto imple-ment,andnocouplingconditionisrequired.Thisapproachisutilized tomodel,forexample,coupledhydromechanicaloffracturedrocks us-ingthefinite-discreteelementmethod(Lathametal.,2013;2018),and canalsocapturefracturepropagationusinganimmersed-bodymethod (Obeysekaraetal.,2018)orphase-fieldapproach(Santillanetal.,2018; Leeetal., 2016b;2018).Thesecond model,whichwecalla mixed-dimensional modelhereafter,reducesthefracturedomaintoalower dimensionalitybyassumingthefracturethicknessismuchsmaller

com-https://doi.org/10.1016/j.advwatres.2020.103620

Received 21 December 2019; Received in revised form 6 May 2020; Accepted 10 May 2020 Available online 31 May 2020

Fig.1. Comparison of degrees of freedom for linear polynomial case among ( a) continuous Galerkin (CG), ( b) discontinuous Galerkin (DG), and ( c) enriched Galerkin (EG) function spaces.

paredtothesizeofmatrixdomain(Boonetal.,2018;Martinetal., 2005; Berroneetal., 2018).The secondapproach hasseveral bene-fits;forexample,itreducesthedegreesoffreedom(DOF)(Nickand Matthai,2011a)asthefracturedomainisrepresentedastheinterface, whichispartofthematrixdomain,andsubsequently,thisapproachcan improvemeshquality(reducethemeshskewness)(Matthaietal.,2010). Sincethefracturesareinterfaces,onecanusealargermeshsize,which satisfiesCourant-Friedrichs-Lewy(CFL)conditionmoreeasily(Juanes etal.,2002;NickandMatthai,2011b).

Inthepastdecades,manyapproacheshavebeenproposedtomodel thefracturedporousmediausingthemixed-dimensionalapproach;(1) two-point flux approximationin unstructuredcontrol volume finite-differencetechnique(Karimi-Fardetal.,2004),(2)multi-pointflux ap-proximationusingmixedfiniteelementmethodongeneralquadrilateral andhexahedralgrids(Wheeleretal.,2012),(3)eXtendedfiniteelement combinedwithmixedfiniteelementformulation(D’AngeloandScotti, 2012;PrevostandSukumar,2016;SanbornandPrevost,2011),(4) em-beddeddiscretefracture-matrix(DFM)modelingwithnon-conforming mesh(Hajibeygietal.,2011;Odsaeteretal.,2019),(5)mixed approx-imationsuchasmimeticfinitedifference(FlemischandHelmig,2008; Formaggia etal.,2018),(6)two-fieldformulationusingmixed finite element(MFE)(Martinetal.,2005;Fumagallietal.,2019),and(7) dis-coninuousGalerkin(DG)method(Rivieetal.,2000;Hoteitand Firooz-abadi,2008;Antoniettietal.,2019;Arnoldetal.,2002).

WefocusonthefiniteelementbaseddiscretizationsuchastheDG and MFEmethodsthat ensurethe localmass conservative property. Moreover, they areflexibleenough todiscretize complexsubsurface geometriessuch asintersectionsof fracturesor irregular-shaped ma-trix blocks.Additionally,theaforementioned methodsarecapable of mimickingthefracturepropagationinporoelasticmediausingeither cohesivezonemethodorlinearelasticfracturemechanicsframework (SalimzadehandKhalili,2015;Salimzadehetal., 2019b;Secchiand Schrefler, 2012;SeguraandCarol,2008).However,theMFEmethod requiresanadditionalprimaryvariable(fluidvelocity)whichmay re-quiremorecomputationalresources,especiallyinathree-dimensional

Fig.2. Comparison of ratio of degrees of freedom for 1 st ,2 nd , 3 rd ,4 th , and 5 th poly- nomial degree cases on triangular element among CG, EG, and DG discretizations: ( a) ratio of EG over CG DOF, ( b) ratio of DG over CG DOF, and ( c) ratio of DG over EG DOF.

Fig.3. Illustration of ( a) equi-dimensional and ( b) mixed-dimensional settings. Note that these illustra- tions are the graphical representations; in the numeri- cal model 𝜕Ω_p, 𝜕Ω_q, 𝜕Γ_p, or 𝜕Γ_qhave to be imposed on all boundary faces of each domain.

Fig.4. Illustration of EG elements ( a) without fracture interface and ( b) with fracture interface.

domainKadeethumetal.(2019a).Meshadaptivityisalsonot straight-forwardtoimplementLeeandWheeler(2017),anditrequiresthe inver-sionofthepermeabilitytensor,whichmayleadtoanill-posedproblem (ChooandLee,2018).TheDGmethodalsocanbeconsideredasa com-putationallyexpensivemethodasitrequiresalargenumberofDOF(Sun andLiu,2009;Leeetal.,2016a).

Toresolvesomeoftheshortcomingsmentionedabove,wepropose anenrichedGalerkin(EG)discretization(Leeetal., 2016a;Zi etal., 2004;Khoeietal.,2018)tomodelfluidflowinfracturedporousmedia usingthemixed-dimensionalapproach.TheEGmethod,utilizedinthis study,composesoftheCGfunctionspaceaugmentedbya piecewise-constantfunctionatthecenterofeachelement.Thismethodhasthe sameinteriorpenaltytypebilinearformastheDGmethod(SunandLiu, 2009;Leeetal.,2016a).TheEGmethod,however,onlyrequirestohave discontinuousconstantsasillustratedinFig.1,soithasfewerDOFthan theDGmethod.Fig.2presentsthecomparisonoftheDOFratioamong CG,EG,andDGmethods,anditshowsthattheEGmethodrequires halfoftheDOFneededbytheDGmethod(triangularelementswiththe firstpolynomialdegreeapproximation).Notethatthisratiodecreases asthepolynomialdegreeapproximationincrease.TheEGmethodhas beendevelopedtosolvegeneralellipticandparabolicproblemswith dy-namicmeshadaptivity(LeeandWheeler,2017;2018;Leeetal.,2018) andextendedtoaddressthemultiphasefluidflowproblems(Leeand Wheeler,2018).Recently,theEGmethodhasbeenalsoappliedtosolve thenon-linearporoelasticproblem(ChooandLee,2018; Kadeethum etal.,2019b;2020b),andcompareditsperformancewithother two-andthree-fieldformulationmethods(Kadeethumetal.,2019a).Tothe bestofourknowledge,thisisthefirstattempttoapplytheEG discretiza-tioninthemixed-dimensionalsetting.

Therestofthepaperisorganizedasfollows.Themethodology sec-tionincludesmodeldescription,mathematicalequations,andtheir dis-cretizationsfortheEGandDGmethods.Subsequently,theblock struc-ture usedtocomposetheEG functionspaceandthecouplingterms betweenmatrixandfracturedomainsisillustrated.Thenumerical ex-amplessectionpresentsfiveexamples,andtheconclusionisfinally pro-vided.

2. Methodology 2.1. Governingequations

Wefirstbrieflyintroducetheequi-dimensionalmodel,whichisused toderivethemixed-dimensionalmodel.Weareinterestedinsolving steady-stateandtime-dependent singlephase fluidflow in fractured porousmediaonΩ,whichcomposesofmatrixandfracturedomains, Ω_m andΩ_f,respectively. Let Ω⊂ ℝ𝑑 be thedomainof interestind -dimensionalspacewhere𝑑=1,2,or3boundedbyboundary,𝜕Ω.𝜕Ω canbedecomposedtopressureandfluxboundaries,𝜕Ωpand𝜕Ωq,

re-spectively.Thetimedomainisdenotedby𝕋 =(0,𝜏],where𝜏 >0isthe finaltime.

Theillustrationoftheequi-dimensionalmodelisshowninFig.3a. Thismodelcomposesoftwomatrixsubdomains,Ω_miwhere𝑖=1,2,and onefracturesubdomain,Ωf.Notethat,forthesakeofsimplicity,this

setup isusedtoillustratethegoverningequationwithonlytwo ma-trixsubdomains,butinageneralcase,thedomainmaycomposeofnm

subdomains,i.e.𝑖=1,2,…,𝑛_𝑚.Moreover,thedomainmaycontainup tonffractures,wherenmandnfarenumberofmatrixsubdomainand

fracture,respectively. Forsimplicity,in thissectionwe willconsider 𝑛𝑓=1.Thefracturesmaynotcutthroughthematrixdomain,whichwe

callimmersedfracturesetting.ThistopicwillbediscussedinSection3. Thegoverningsystemwithinitialandboundaryconditionsofthe equi-dimensionalmodelassumingaslightlycompressiblefluidforthematrix domainispresentedbelow:

𝑐𝜙𝑚𝑖_𝜕𝑡𝜕(𝑝𝑚𝑖)−∇⋅𝒌_𝜇𝒎𝑖(∇𝑝𝑚𝑖−𝜌𝐠)=𝑔𝑚𝑖inΩ𝑚𝑖×𝕋, for𝑖=1,2, (1)

𝑝𝑚=𝑝𝑚𝐷on𝜕Ω𝑝×𝕋, (2)

−𝒌𝒎𝑖

𝜇 (∇𝑝𝑚𝑖−𝜌𝐠)⋅ 𝒏=𝑞𝑚𝑁on𝜕Ω𝑞×𝕋, (3)

Fig.5. ( a) geometry used for the error convergence analysis, ( b) illustration of the exact solution, error convergence plot between EG and DG methods with polynomial approximation degree 1 and 2 of ( c) matrix and ( d) fracture domains. Note that for ( c) the slopes of the best fitted line are 2.235 (EG), 2.022 (DG) for the polynomial approximation degree 1, and 3.137 (EG), and 3.410 (DG) for the polynomial approximation degree 2. The slopes of the best fitted line for ( d) are 2.289 (EG), 2.313 (DG) for the polynomial approximation degree 1, and 3.194 (EG), and 2.948 (DG) for the polynomial approximation degree 2.

andforthefracturedomainis:

𝑐𝜙𝑓_𝜕𝑡𝜕(𝑝𝑓)−∇⋅𝒌_𝜇𝒇(∇𝑝𝑓−𝜌𝐠)=𝑔𝑓inΩ𝑓×𝕋, (5)

𝑝𝑓=𝑝𝑓𝐷on𝜕Ω𝑝×𝕋, (6)

−𝒌𝒇

𝜇 (∇𝑝𝑓−𝜌𝐠)⋅ 𝒏=𝑞𝑓𝑁on𝜕Ω𝑞×𝕋, (7)

𝑝𝑓=𝑝0𝑓 inΩ𝑓×𝕋 at𝑡=0, (8)

where(· )_irepresentsanindex,cisthefluidcompressibility,𝜙mand𝜙f

arethematrixandfractureporosity(𝜙fisassumedtobeone),kmandkf arethematrixandfracturepermeabilitytensor,respectively,𝜇 isfluid viscosity,p_mandp_farematrixandfracturepressure,respectively,𝜌 is

fluiddensity,gisthegravitationalvector,gmandgfaresink/sourcefor

matrixandfracturedomains,respectively,nisanormalunitvectorto anysurfaces,p_mDandp_fDareprescribedpressureformatrixandfracture domains,respectively,qmN andqfN areprescribedfluxformatrixand

fracturedomains,respectively,and𝑝_𝑚0

𝑖 and𝑝0𝑓 areprescribedpressure

formatrixandfracturedomainsat𝑡=0,respectively.

Toformulatethemixed-dimensionalsettingaspresentedinFig.3b, we integrate along the normal direction to the fracture plane Martinetal.(2005).AsaresultΩ_fisreducedtoaninterface,Γ.Note thatthegoverningequationofthemixed-dimensionalsettingusedin this studyis proposedbyMartinetal.(2005) inthemixedfinite el-ement formulation, which uses fluid pressure and fluid velocity as the primaryvariables. Themixed-dimensionalsetting has beenused in themixedformulation(Keilegavlenetal.,2017) oradapted to fi-nitevolumediscretization(Glaseretal.,2017;Stefanssonetal.,2018;

Fig.6. ( a) geometry and boundary conditions used for the quarter five-spot problem and ( b) mesh that has ℎ= 1 .2 × 10 −1 .

Fig.7. Pressure solution with ℎ= 7 .2 × 10 −2 ,pm, in the matrix of quarter five-spot example: ( a) permeable fracture, ( b) impermeable fracture cases, pmplot along

𝑥= 𝑦line of ( c) permeable fracture, and ( d) impermeable fracture cases. Note that we digitize the results of the reference solution from Antoniettietal.(2019). Note that the results obtained with the EG and DG methods overlap.

Fig. 8. Immersed fracture geometry and boundary conditions of ( a) permeable fracture, ( b) partially permeable fracture, ( c) imper- meable fracture cases, and ( d) mesh that has 𝑛𝑒= 272 . A’ is fracture tip.

Glaser et al., 2019) and DG discretization on polytopic grids (Antoniettietal.,2019).Themixed-dimensionstrongformulationand itsboundaryconditionsforthematrixdomainaresimilartothe equi-dimensionalmodel,(1)to(3),butthestrongformulationandits bound-aryconditionsofthefracturedomainare:

𝑐𝑎𝑓_𝜕𝑡𝜕(𝑝𝑓)−∇𝑇⋅ 𝑎𝑓 𝒌𝑻 𝒇 𝜇 ( ∇𝑇𝑝_𝑓−𝜌𝐠)=𝑔_𝑓+−𝒌𝒎 𝜇 ( ∇𝑝_𝑚−𝜌𝐠)inΓ ×𝕋, (9) 𝑝𝑓=𝑝𝑓𝐷on𝜕Γ𝑝×𝕋, (10) − 𝒌𝑻 𝒇 𝜇 (∇𝑝𝑓−𝜌𝐠)⋅ 𝒏=𝑞𝑓𝑁on𝜕Γ𝑞×𝕋, (11) 𝑝𝑓=𝑝0𝑓 inΓ ×𝕋at𝑡=0, (12)

where𝜕Γpand𝜕Γqrepresentpressureandfluxboundariesofthe

frac-ture domain, respectively, 𝒌𝑻

𝒇 is the tangentialfracture permeability tensor,∇Tand∇T· arethetangentialgradientanddivergence opera-tors,whicharedefinedas∇𝑇(⋅)=∇(⋅)−𝒏[𝒏⋅ ∇(⋅)]andtr(∇𝑇(⋅)), respec-tively,tr(⋅)istraceoperator,afisafractureaperture,−𝒌_𝜇𝒎

(

∇𝑝_𝑚−𝜌𝐠) representsthefluidmasstransferbetweenmatrixandfracturedomains Martinetal.(2005),⋅isjumpoperator,whichwillbediscussedlater inthediscretizationpart,andpfDandqfNarespecifiedpressureandflux

forthefracturedomain,respectively.

Inthisstudy,if𝑑=3,k_masafulltensorisdefinedas:

𝒌𝒎∶= ⎡ ⎢ ⎢ ⎣ 𝑘𝑥𝑥 𝑚 𝑘𝑥𝑦𝑚 𝑘𝑥𝑧𝑚 𝑘𝑦𝑥𝑚 𝑘𝑦𝑦𝑚 𝑘𝑦𝑧𝑚 𝑘𝑧𝑥 𝑚 𝑘𝑧𝑦𝑚 𝑘𝑧𝑧𝑚 ⎤ ⎥ ⎥ ⎦, (13)

wherealltensorcomponentscharacterizethetransformationofthe com-ponentsofthegradientoffluidpressureintothecomponentsofthe ve-locityvector.The𝑘𝑥𝑥

𝑚,𝑘𝑦𝑦𝑚,and𝑘𝑧𝑧𝑚 representthematrixpermeabilityin

x-,y-,andz-direction,respectively.kf,ontheotherhand,composesof twocomponents: 𝒌𝑓∶= [ 𝒌𝑻 𝒇 0 0 𝑘𝑛_𝑓 ] , (14) where𝑘𝑛

𝑓isthenormalfracturepermeability.Notethatwepresenthere

ageneralformof𝒌𝑻

𝒇,tobespecific,𝒌𝑻𝒇 isascalarif𝑑=2andtensorif 𝑑=3.Torepresentthefluidmasstransferbetweenmatrixandfracture domains,followingMartinetal.(2005), Antoniettietal.(2019),we definethecouplingconditionsbetweenthematrixandfracturedomain as: ⟨−𝒌_𝜇𝒎(∇𝑝_𝑚−𝜌𝐠)⟩ ⋅ 𝒏= 𝛼𝑓 2 ( 𝑝𝑚1 −𝑝𝑚2 ) onΓ, (15) −𝒌𝒎 𝜇 ( ∇𝑝_𝑚−𝜌𝐠)= 2𝛼𝑓 2𝜉 −1 ( ⟨𝑝𝑚⟩ −𝑝𝑓)onΓ, (16)

where⟨ · ⟩ isanaverageoperator,whichwillbepresentedlaterinthe discretizationpart,𝛼frepresentsaresistantfactorofthemasstransfer

Fig.9. Pressure solution, pm, in the matrix of

immersed fracture example using 𝑛𝑒= 6 ,698 :

( a) permeable fracture, ( b) partially permeable fracture, and ( c) impermeable fracture cases.

betweenthefractureandmatrixdomainsdefinedas: 𝛼𝑓=

2𝑘𝑛_𝑓

𝑎𝑓 , (17)

and𝜉 ∈ (0.5,1.0].Inthispaper,weset𝜉 =1.0forthesakeofsimplicity. Ingeneralcase,𝜉 isusedtorepresentafamilyofthemixed-dimensional model, and more details can be found in Martin et al. (2005), Antoniettietal.(2019).

2.2. Numericaldiscretization

Inthissection,thediscretizationofthemixed-dimensionalmodelis illustrated.ThedomainΩispartitionedintoneelements,ℎ,whichis

thefamilyofelementsT(trianglesin2D,tetrahedronsin3D).Wewill furtherdenotebyeafaceofTasillustratedinFig.4.Wedenoteh_Tasthe diameterofTanddefineh≔ max(hT),whichwemayreferasmeshsize.

Letℎdenotesthesetofallfacets,forthematrixdomain,ℎ0theinternal

facets,_ℎ𝐷theDirichletboundaryfaces,and_ℎ𝑁denotestheNeumann boundaryfaces.FollowingLeeetal.(2016a) forany𝑒∈0

ℎlet𝑇+and

𝑇− _{betwoneighboringelementssuchthat𝑒=𝜕𝑇}+_∩𝜕𝑇−_.Let𝒏+_and

𝒏−_{betheoutwardnormalunit vectorsto𝜕𝑇}+ _and𝜕𝑇−_{,respectively} (seeFig.4a).ℎ=ℎ0 ∪ ℎ𝐷 ∪ ℎ𝑁,ℎ0 ∩ ℎ𝐷=∅,ℎ0 ∩ ℎ𝑁=∅,and

𝐷

ℎ ∩ℎ𝑁=∅.Wefurtherdefineℎ1∶=ℎ0∪ℎ𝐷.

Thefracturedomainisconformingwith_ℎ,anditisnamedΓh

(in-tervalsin2D,trianglesin3Ddomain)hereafteraspresentedinFig.4b.

Wewillfurtherdenotebye_fafaceofΓ_h.LetΛ_hdenotesthesetofall facets,forthefracturedomain,Λ0

ℎ theinternalfacets,Λ𝐷ℎ the

Dirich-letboundaryfaces,andΛ𝑁_ℎ denotestheNeumannboundaryfaces,and Λ𝐷_ℎ∩ Λ𝑁_ℎ =∅.LetnbetheoutwardnormalunitvectortoΓ_h,whichis coincidedwith𝒏+_{ofthe𝜕𝑇}+_.

Next,wedefinethejumpoperatorofthescalarandvectorvalues as:

𝑋⋅ 𝒏∶=𝑋+𝒏++𝑋−𝒏−and

𝑿⋅ 𝒏∶=𝑿+⋅ 𝒏++𝑿−⋅ 𝒏−, (18)

respectively,Moreover,byassumingthatthenormalvectornisoriented from𝑇+_to𝑇−_,weobtain

𝑋∶=𝑋+−𝑋− (19)

FollowingLeeetal.(2016a),Scovazzietal.(2017),theweighted aver-ageisdefinedas:

⟨𝑋⟩𝛿𝑒=𝛿𝑒𝑋++(1−𝛿𝑒)𝑋−, (20) where 𝛿𝑒∶= 𝑘𝑚 − 𝑒 𝑘𝑚+𝑒 +𝑘𝑚−𝑒, (21) and 𝑘𝑚+𝑒 ∶=(𝒏+)𝑇⋅ 𝒌𝒎+⋅ 𝒏+,

Fig.10. pmplot along 𝑦= 0 .75 line of immersed fracture example: ( a) permeable fracture, ( b) partially permeable fracture, ( c) impermeable fracture cases. Note

that we digitize the reference results, Angot et al. 2009, from Angotetal.(2009). All the results obtained from the EG and DG methods with different mesh sizes overlap.

𝑘𝑚−𝑒 ∶=(𝒏−)𝑇⋅ 𝒌𝒎−⋅ 𝒏−, (22)

andaharmonicaverageof𝑘𝑚𝑒+and𝑘𝑚−𝑒 isdefinedas:

𝑘𝑚𝑒∶=

2𝑘_𝑚+_𝑒𝑘_𝑚−_𝑒 (

𝑘𝑚+𝑒 +𝑘𝑚−𝑒

) . (23)

Thearithmeticaverage,𝛿𝑒=0.5,issimplydenotedby⟨ · ⟩.Notethat,

ingeneralcase,𝛿e isalsodefinedforthekf;however,forthesakeof simplicity,weassumethematerialpropertiesofthefracturedomainare homogeneous.Hence,weperformtheseoperatorsinthematrixdomain only.

Remark1. Thenumericaldiscretizationdiscussedin thisstudyonly considersthecaseof aconformingmesh,i.e.thefracturedomain,Γ, elementiscoincidentwithasetoffacesofthematrixdomain,Ω,as illustratedinFig.4b.

Thisstudyfocusesontwofunctionspaces,arisingfromEG,andDG discretizations,respectively.WebeginwithdefiningtheCGfunction

spaceforthematrixpressure,pmas

CG_𝑘 ℎ ( ℎ)∶={𝜓𝑚∈ℂ0(Ω)∶ 𝜓𝑚||𝑇 ∈ℙ𝑘(𝑇),∀𝑇∈ℎ}, (24) whereCG_𝑘 ℎ (

ℎ)isthespacefortheCGapproximationwithkthdegree

polynomialsforthe pm unknown,ℂ0(Ω)denotesthespaceof

scalar-valuedpiecewisecontinuouspolynomials,ℙ𝑘(𝑇)isthespaceof polyno-mialsofdegreeatmostkovereachelementT,and𝜓mdenotesageneric

functionofCG_𝑘

ℎ

(

ℎ).Furthermore,wedefinethefollowingDGfunction

spaceforthematrixpressure,pm:

DG_𝑘 ℎ ( ℎ)∶={𝜓𝑚∈𝐿2(Ω)∶𝜓𝑚||𝑇∈ℙ𝑘(𝑇),∀𝑇∈ℎ}, (25) whereDG_𝑘 ℎ (

ℎ)isthespacefortheDGapproximationwithkthdegree

polynomialsforthepmspaceandL2(Ω)isthespaceofsquareintegrable

functions.Finally,wedefinetheEGfunctionspaceforpmas:

EG_𝑘 ℎ ( ℎ)∶=ℎCG𝑘 ( ℎ)+ℎDG0 ( ℎ), (26) whereEG_𝑘 ℎ (

ℎ)isthespacefortheEGapproximationwithkthdegree

polynomialsforthepspaceandDG₀

ℎ

(

approx-Fig.11. Regular fracture network example: ( a) geometry and boundary conditions (fractures are shown in red), pressure solution, pm, in the

matrix using 𝑛𝑒= 2046 of ( b) permeable frac-

ture, ( c) impermeable fracture cases, and ( d) mesh that has 𝑛𝑒= 184 . (For interpretation of

the references to colour in this figure legend, the reader is referred to the web version of this article.)

imationwith0thdegreepolynomials,inotherwords,apiecewise con-stantapproximation.NotethattheEGdiscretizationisexpectedtobe beneficialforanaccurateapproximationofthepmdiscontinuityacross

interfaceswherehighpermeabilitycontrast,eitherbetween𝒌+_𝒎and𝒌−_𝒎 ork_mandk_f,isobserved.Ontheotherhand,thematerialpropertiesof thefracturedomainareassumedtobehomogeneous,whichleadstono discontinuitywithinthefracturedomain.Therefore,theCG discretiza-tionofthep_funknownwillsufficeinthefollowing.Thep_ffunctionspace isdefinedas: CG_𝑘 ℎ ( Γ_ℎ)∶={𝜓_𝑓∈ℂ0(Γ)∶𝜓_𝑓|| |𝑒∈ℙ𝑘(𝑒),∀𝑒∈ Γℎ } , (27) whereCG_𝑘 ℎ (

Γ_ℎ)isthespacefor theCGapproximationwithkth de-greepolynomialsforthepfunknown,ℂ0(Γ)denotesthespaceof

scalar-valuedpiecewisecontinuouspolynomials,ℙ𝑘(𝑒)isthespaceof

polyno-mialsofdegreeatmostkovereachfacete,and𝜓fdenotesageneric

functionof_ℎCG𝑘(Γ_ℎ).

Remark2. Inthisstudy,weonlyfocusonthemixedfunctionspace be-tweenthematrixandfracturedomainsarisingfromeitherEG_𝑘

ℎ ( ℎ)× CG_𝑘 ℎ ( Γ_ℎ) orDG_𝑘 ℎ (

ℎ)×_ℎCG𝑘(Γ_ℎ). The fracturedomain can be dis-cretizedbyeitherEG_𝑘 ℎ ( Γ_ℎ)orDG_𝑘 ℎ (

Γ_ℎ)ifthereareanydiscontinuities insidethefracturemedium.

Thetimedomain,𝕋=(0,𝜏],ispartitionedintoNtopensubintervals

suchthat, 0=∶𝑡0<𝑡1<⋯<𝑡𝑁𝑡∶=𝜏. Thelength of the subinterval, Δtn_{,is definedasΔ𝑡}𝑛_=𝑡𝑛_−𝑡𝑛−1 _{wheren representsthecurrenttime}

step.Inthisstudy,implicitfirst-ordertimediscretizationisutilizedfor atimedomainof(1) and(5)asshownbelowforboth𝑝𝑛

𝑚,ℎand𝑝𝑛𝑓,ℎ: 𝜕𝑝𝑚,ℎ 𝜕𝑡 ≈ 𝑝𝑛 𝑚,ℎ−𝑝𝑛𝑚,ℎ−1 Δ𝑡𝑛 , and 𝜕𝑝𝑓,ℎ 𝜕𝑡 ≈ 𝑝𝑛 𝑓,ℎ−𝑝𝑛𝑓,ℎ−1 Δ𝑡𝑛 . (28)

WedenotethatthetemporalapproximationofthefunctionΦ(· ,tn_)by

Φn.

Withgiven𝑝𝑛−1

𝑚,ℎ and𝑝𝑛𝑓,ℎ−1,wenowseektheapproximatedsolutions

𝑝𝑛 𝑚,ℎ∈ EG_𝑘 ℎ ( ℎ)and𝑝𝑛𝑓,ℎ∈ CG_𝑘 ℎ ( Γ_ℎ)ofp_m(· ,tn_)andp f(· ,tn), respec-tively,satisfying ((𝑝𝑛 𝑚,ℎ,𝑝𝑛𝑓,ℎ;𝑝𝑛𝑚,ℎ−1,𝑝𝑛𝑓,ℎ−1 ) ,(𝜓𝑚,𝜓𝑓)) + (( 𝑝𝑛 𝑚,ℎ,𝑝𝑛𝑓,ℎ ) ,(𝜓𝑚,𝜓𝑓))−(𝜓_𝑚,𝜓_𝑓)=0, ∀𝜓_𝑚∈_ℎEG𝑘(_ℎ)and∀𝜓_𝑓∈_ℎCG𝑘(Γ_ℎ). (29) First,thetemporaldiscretizationpartisdefinedas

((𝑝𝑛 𝑚,ℎ,𝑝𝑛𝑓,ℎ;𝑝𝑛𝑚,ℎ−1,𝑝𝑛𝑓,ℎ−1 ) ,(𝜓𝑚,𝜓𝑓))∶=𝑚𝑚 ( 𝑝𝑛 𝑚,ℎ;𝑝𝑛𝑚,ℎ−1,𝜓𝑚 ) +𝑚_𝑓(𝑝𝑛_𝑓,ℎ;𝑝𝑛−1 𝑓,ℎ,𝜓𝑓 ) , (30) where 𝑚𝑚 ( 𝑝𝑛 𝑚,ℎ;𝑝𝑛𝑚,ℎ−1,𝜓𝑚 ) ∶=∑_{𝑇 ∈} ℎ∫𝑇𝑐𝜙𝑚 𝑝𝑛 𝑚,ℎ−𝑝𝑛𝑚,ℎ−1 Δ𝑡𝑛 𝜓𝑚𝑑𝑉, (31) and 𝑚𝑓 ( 𝑝𝑛 𝑓,ℎ;𝑝𝑛𝑓,ℎ−1,𝜓𝑓 ) ∶=∑_𝑒∈Γ_ℎ∫_𝑒𝑐𝑎𝑓 𝑝𝑛 𝑓,ℎ−𝑝𝑛𝑓,ℎ−1 Δ𝑡𝑛 𝜓𝑓𝑑𝑆. (32)

Fig.12. pmplot of regular fracture network example: ( a) along 𝑦= 0 .7 line of permeable fracture, ( b) along 𝑥= 0 .5 line of permeable fracture, ( c) along ( 𝑥= 0 .0 ,𝑦=

0 .1) to ( 𝑥= 0 .9 ,𝑦= 1 .0) line of impermeable fracture cases. All of the results obtained from the EG and DG methods of permeable fractures with different mesh sizes overlap. The results of impermeable fractures case with different mesh sizes, however, illustrate some differences.

Here,∫T· dVand∫e· dSrefertovolumeandsurfaceintegrals,

respec-tively. Next,wedefine((𝑝𝑛 𝑚,ℎ,𝑝𝑛𝑓,ℎ ) ,(𝜓𝑚,𝜓𝑓))as: ((𝑝𝑛 𝑚,ℎ,𝑝𝑛𝑓,ℎ ) ,(𝜓𝑚,𝜓𝑓))∶=𝑎 ( 𝑝𝑛 𝑚,ℎ,𝜓𝑚 ) +𝑏 ( 𝑝𝑛 𝑓,ℎ,𝜓𝑚 ) +𝑐(𝑝𝑛_𝑚,ℎ,𝜓_𝑓)+𝑑(𝑝𝑛_𝑓,ℎ,𝜓_𝑓), (33) where 𝑎(𝑝𝑛 𝑚,ℎ,𝜓𝑚 ) ∶= ∑ 𝑇 ∈ℎ∫𝑇 𝒌𝒎 𝜇 ( ∇𝑝𝑛_𝑚,ℎ−𝜌𝐠 ) ⋅ ∇𝜓𝑚𝑑𝑉 − ∑ 𝑒∈1 _ℎ⧵Γℎ∫𝑒 ⟨𝒌_𝜇𝒎(∇𝑝𝑛_𝑚,ℎ−𝜌𝐠)⟩_𝛿 𝑒⋅ 𝜓𝑚𝑑𝑆 +𝜃 ∑ 𝑒∈_ℎ1 ⧵Γℎ∫𝑒 ⟨𝒌𝒎 𝜇 ∇𝜓𝑚⟩𝛿𝑒⋅ 𝑝𝑛𝑚,ℎ𝑑𝑆 + ∑ 𝑒∈_ℎ1 ⧵Γℎ∫𝑒 𝛽 ℎ𝑒 𝑘𝑚𝑒 𝜇 𝑝𝑛𝑚,ℎ⋅ 𝜓𝑚𝑑𝑆 + ∑ 𝑒∈Γ_ℎ∫𝑒 𝛼𝑓 2 𝑝 𝑛 𝑚,ℎ⋅ 𝜓𝑚𝑑𝑆 + ∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1⟨𝑝 𝑛 𝑚,ℎ⟩⟨𝜓𝑚⟩ 𝑑𝑆, (34) 𝑏(𝑝𝑛 𝑓,ℎ,𝜓𝑚 ) ∶=−∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1𝑝 𝑛 𝑓,ℎ⟨𝜓𝑚⟩ 𝑑𝑆, (35) 𝑐(𝑝𝑛 𝑚,ℎ,𝜓𝑓 ) ∶=−∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1⟨𝑝 𝑛 𝑚,ℎ⟩𝜓𝑓𝑑𝑆, (36)

Fig.13. flux at surface A’ comparison for both permeable and impermeable cases between the EG and DG methods. All of the results obtained from the EG and DG methods with different mesh sizes overlap.

and 𝑑(𝑝𝑛 𝑓,ℎ,𝜓𝑓 ) ∶= ∑ 𝑒∈Γ_ℎ∫𝑒 𝑎𝑓 𝒌𝑻 𝒇 𝜇 ( ∇𝑝𝑛_𝑓,ℎ−𝜌𝐠 ) ⋅ ∇𝜓𝑓𝑑𝑆 +∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1𝑝 𝑛 𝑓,ℎ𝜓𝑓𝑑𝑆, (37)

Wenotethatthecouplingconditions,(15) and(16),areembeddedin theabovediscretizedequations.Inparticular,theconditions(15) and (16) arediscretizedas 1 ( 𝑝𝑛 𝑚,ℎ,𝜓𝑚 ) ∶= ∑ 𝑒∈Γ_ℎ∫𝑒 𝛼𝑓 2𝑝 𝑛 𝑚,ℎ⋅ 𝜓𝑚𝑑𝑆, ∀𝜓𝑚∈ℎEG𝑘 ( ℎ), (38) and 2 (( 𝑝𝑛 𝑚,ℎ,𝑝𝑛𝑓,ℎ ) ,(𝜓𝑚,𝜓𝑓))∶= ∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1⟨𝑝 𝑛 𝑚,ℎ⟩⟨𝜓𝑚⟩ 𝑑𝑆 − ∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1⟨𝑝 𝑛 𝑚,ℎ⟩𝜓𝑓𝑑𝑆− ∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1𝑝 𝑛 𝑓,ℎ⟨𝜓𝑚⟩ 𝑑𝑆 + ∑ 𝑒∈Γ_ℎ∫𝑒 2𝛼_𝑓 2𝜉 −1𝑝 𝑛 𝑓,ℎ𝜓𝑓𝑑𝑆, ∀𝜓𝑚∈ℎEG𝑘 ( ℎ)and∀𝜓𝑓∈ℎCG𝑘 ( Γ_ℎ), (39) respectively.Finally,wedefine(𝜓𝑚,𝜓𝑓)as:

(𝜓𝑚,𝜓𝑓)∶=𝓁𝑚(𝜓𝑚)+𝓁𝑓(𝜓𝑓) (40) where 𝓁𝑚(𝜓𝑚)∶= ∑ 𝑇 ∈ℎ∫𝑇 𝑔𝑚𝜓𝑚𝑑𝑉+ ∑ 𝑒∈𝑁 ℎ ∫𝑒𝑞𝑚𝑁𝜓𝑚𝑑𝑆 +𝜃 ∑ 𝑒∈𝐷 ℎ ∫_𝑒𝒌𝜇𝒎∇𝜓𝑚⋅ 𝑝𝑚𝐷𝒏𝑑𝑆+ ∑ 𝑒∈𝐷 ℎ ∫_𝑒ℎ𝛽𝑒 𝑘𝑚𝑒 𝜇 𝜓𝑝⋅ 𝑝𝑚𝐷𝒏𝑑𝑆 (41) and 𝓁𝑓(𝜓𝑓)∶=∑𝑒∈Γ_ℎ∫𝑒𝑎𝑓𝑔𝑓𝜓𝑓𝑑𝑆+∑𝑒𝑓∈Λ_ℎ𝑁∫𝑒𝑞𝑓𝑁𝜓𝑓𝑑𝑆. (42) Here,thechoicesoftheinteriorpenaltymethodisprovidedby𝜃.The discretizationbecomesthesymmetricinteriorpenaltyGalerkinmethod (SIPG),when𝜃 =−1,theincompleteinteriorpenaltyGalerkinmethod

(IIPG), when𝜃 =0, andthenon-symmetricinteriorpenalty Galerkin method(NIPG)when𝜃 =1Riviere(2008).Inthisstudy,weset𝜃 =−1 forthesimplicity.Theinteriorpenaltyparameter, 𝛽,isafunctionof thepolynomialdegree,k,andthecharacteristicmeshsize,he,whichis

definedas: ℎ𝑒∶=

meas(𝑇+)_+meas(𝑇−₎

2meas(𝑒) , (43)

wheremeas(· )representsameasurementoperator,measuringlength, area,orvolume.Somestudiesfortheoptimalchoiceof𝛽 isprovidedin Leeetal.(2019),Riviere(2008).

Remark3. TheNeumannboundaryconditionisnaturallyappliedon theboundaryfacesthatbelongtotheNeumannboundarydomainfor boththematrixandfracturedomains,𝑒∈_ℎ𝑁and𝑒_𝑓∈ Λ𝑁_ℎ.The Dirich-letboundarycondition,ontheotherhand,isweaklyenforcedonthe Dirichletboundaryfaces,𝑒∈_ℎ𝐷,forthematrixdomain,butstrongly enforcedontheDirichletbondaryfacesofthefracturedomain,𝑒𝑓∈ Λ𝐷_ℎ.

Let{𝜓_𝑚,𝜋(𝑖1)_}𝑝𝑚,𝜋

𝑖1 =1 denotethesetofbasisfunctionsof

𝜋𝑘 ℎ ( ℎ),i.e. 𝜋𝑘 ℎ ( ℎ)=span{𝜓_𝑚(𝑖₁ )_}𝑝𝑚,𝜋

𝑖1 =1 ,havingdenotedby𝑝𝑚,𝜋 thenumberof

DOFforthe𝜋 scalar-valuedspace,where𝜋 canmeaneitherEGorDG.In asimilarway,let{𝜓_𝑓,(𝑖3 _CG) }_𝑖𝑝𝑓,CG

3 =1 bethesetofbasisfunctionsforthespace CG_𝑘

ℎ

(

Γ_ℎ)._𝑝𝑓,CG thenumberofDOFfortheCGscalar-valuedspace. Hence,twomixedfunctionspaces,(1)EG_k×CG_kand(2)DG_k×CG_k wherekrepresentsthedegreeofpolynomialapproximation,are possi-ble,andwillbecomparedinthenumericalexamplesinthenextsection. Thematrixcorrespondingtotheleft-handsideof(29)isassembled com-posingthefollowingblocks:

[ 𝜋𝑘×𝜋𝑘 𝑚𝑚 ] 𝑖𝑖𝑖2 ∶=𝑚_𝑚(𝜓_𝑚(𝑖2 ),𝜓_𝑚(𝑖1 ))+𝑎(𝜓_𝑚(𝑖2 ),𝜓_𝑚(𝑖1 )),𝑖1 =1,…,_{𝑝𝑚,𝜋},𝑖2=1,…,𝑝𝑚,𝜋, [ 𝜋𝑘×CG_𝑘 𝑚𝑓 ] 𝑖𝑖𝑖3 ∶=𝑏 ( 𝜓(𝑖₃ ) 𝑓,CG,𝜓𝑚,𝜋 (𝑖1 ))_,𝑖 1=1,…,𝑝𝑚,𝜋,𝑖3=1,…,𝑝𝑓,CG , [ CG_𝑘×𝜋_𝑘 𝑓𝑚 ] 𝑖4 𝑖2 ∶=𝑐 ( 𝜓𝑚,𝜋(𝑖2 ),𝜓𝑝(𝑖𝑓4 ,)CG ) ,𝑖4=1,…,𝑝𝑓,CG,𝑖2=1,…,𝑝𝑚,𝜋, [ CG_𝑘×CG_𝑘 𝑓𝑓 ] 𝑖4 𝑖3 ∶=𝑚𝑓 ( 𝜓(𝑖3 ) 𝑓,CG,𝜓 (𝑖4 ) 𝑓,CG ) +𝑑(𝜓_𝑓,(𝑖3 _CG) ,𝜓_𝑓,(𝑖4 _CG) )𝑖4 =1,…,_𝑝 𝑓,CG,𝑖3=1,…,𝑝𝑓,CG. (44) Inasimilarway,theright-handsideof(29)givesrisetoablockvector ofcomponents [ 𝓁𝜋𝑘 𝑚]𝑖1 ∶=𝓁𝑚 ( 𝜓𝑚,𝜋(𝑖1 )), 𝑖1=1,…,𝑝𝑚,𝜋, [ 𝓁CG_𝑘 𝑓 ] 𝑖3 ∶=𝓁𝑓 ( 𝜓(𝑖3 ) 𝑓,CG ) 𝑖3=1,…,𝑓,CG. (45)

Theresultingblockstructureisthus [ 𝜋𝑘×𝜋_𝑘 𝑚𝑚 𝑚𝑓𝜋𝑘×CG𝑘 CG_𝑘×𝜋𝑘 𝑓𝑚  CG_𝑘×CG_𝑘 𝑓𝑓 ]{ 𝑝𝜋𝑘 𝑚,ℎ 𝑝CG_𝑘 𝑓,ℎ } = { 𝓁𝜋𝑘 𝑚 𝓁CG_𝑘 𝑓 } . (46) where𝑝𝜋𝑘 𝑚,ℎand𝑝 CG_𝑘

𝑓,ℎ collectthedegreesoffreedomformatrixand

frac-turepressure,respectively.Finally,weremarkthat(owingto(26))the case𝜋 =EGcanbeequivalentlydecomposedintoa(CG_k×DG₀) ×CG_k mixedfunctionspace,resultingin:

⎡ ⎢ ⎢ ⎢ ⎣ CG_𝑘×CG_𝑘 𝑚𝑚 𝑚𝑚CG𝑘×DG0 𝑚𝑓CG𝑘×CG𝑘 DG₀ ×CG_𝑘 𝑚𝑚 𝑚𝑚DG0 ×DG0 𝑚𝑓DG0 ×CG𝑘 CG_𝑘×CG_𝑘 𝑓𝑚  CG_𝑘×DG₀ 𝑓𝑚  CG_𝑘×CG_𝑘 𝑓𝑓 ⎤ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪ ⎨ ⎪ ⎩ 𝑝CG_𝑘 𝑚,ℎ 𝑝DG₀ 𝑚,ℎ 𝑝CG_𝑘 𝑓,ℎ ⎫ ⎪ ⎬ ⎪ ⎭ = ⎧ ⎪ ⎨ ⎪ ⎩ 𝓁CG_𝑘 𝑚 𝓁DG₀ 𝑚 𝓁CG_𝑘 𝑓 ⎫ ⎪ ⎬ ⎪ ⎭ . (47)

ThisformulationmakestheEGmethodologyeasilyimplementable inanyexistingDGcodes.MatricesandvectorsarebuiltbyFEniCSform compiler(Alnaesetal.,2015).Theblockstructureissetupusing multi-phenicstoolbox(BallarinandRozza,2019).Randomfieldof permeabil-ity(km)ispopulatedusingSciPypackage(Jonesetal.,2001).𝛽,penalty parameter,issetat1.1and1.0forDGandEGmethods,respectively.

Fig.14. Low k_m case: kmdistribution of: ( a) quarter five-spot ( 𝑛𝑒= 6 ,568 ), ( b) regular fracture network ( 𝑛𝑒= 26 ,952 ) examples, histogram of km of ( c) quarter five-spot, and ( d) regular fracture network examples.

Remark4. WenotethatCG,DG,andEGmethodsarebasedonGalerkin method,whichcouldbeextendedtoconsideradaptivemeshesthat con-tainhangingnodes.Inaddition,there arevariousadvanced develop-mentforeachmethodstoenhancetheefficiency,includingvariable ap-proximationordertechniques.Especially,forEGmethod,anadaptive enrichment,i.e.,thepiecewise-constantfunctionsonlyaddedtothe el-ementswherethesharpmaterialdiscontinuities(e.g.,betweenmatrix andfracturedomains)areobserved,canbedeveloped.However,inour followingnumericalexamples,wefocusontheclassicalformofeach methodsforthecomparisonbysimulatingtheproposedmixed dimen-sionalapproachformodelingfractures.

3. Numericalexamples

WeillustratethecapabilityoftheEGmethodusingsevennumerical examples.Webeginwithananalysisoftheerrorconvergencerate be-tweentheEGandDGmethodstoverifythedevelopedblockstructurein themixed-dimensionalsetting.WealsoinvestigatetheEGperformance inmodelingthequarterfive-spotpatternandhandlingthefracturetip intheimmersedfracturegeometry.Next,wetesttheEGmethodina regularfracturenetworkwithandwithoutaheterogeneityinmatrix permeabilityinput.Lastly,weapplytime-dependentproblemsfortwo three-dimensionalgeometries;thefirstonerepresentsthecasewhere

fracturesareorthogonaltotheaxes,andanotherrepresentsgeometry wherefracturesaregivenwitharbitraryorientationswiththeir interac-tionsinathree-dimensionaldomain.

3.1. Errorconvergenceanalysis

Toverifytheimplementationoftheproposedblockstructure uti-lizedtosolvethemixed-dimensionalmodelusingtheEGmethod,we illustratetheerrorconvergencerateoftheEGmethodandcomparethis valuewiththeDGmethod.Theexampleusedinthisanalysisisadapted fromAntoniettietal.(2019).WetakeΩ =[0,1]2,andchoosetheexact solutioninthematrix,Ω,andfracture,Γ ={(𝑥,𝑦)∈ Ω ∶𝑥+𝑦=1},as: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑝𝑚=𝑒𝑥+𝑦 inΩ1, 𝑝𝑚= 𝑒 𝑥+𝑦 2 + ⎛ ⎜ ⎜ ⎝ 1 2+ 3𝑎_𝑓 𝑘𝑛 𝑓 √ 2 ⎞ ⎟ ⎟ ⎠ 𝑒 inΩ2, 𝑝𝑓=𝑒 ( 1+√2𝑎𝑓 𝑘𝑛 𝑓 ) inΓ. (48)

Bychoosing𝒌_𝒎=𝑰,(48)satisfiesthesystemofequations,(1),(9),(15), and(16),presentedinthemethodologysectionwithsink/sourceterms,

Fig.15. High k_m case: kmdistribution of: ( a) quarter five-spot ( 𝑛𝑒= 6 ,568 ), ( b) regular fracture network ( 𝑛𝑒= 26 ,952 ) examples, histogram of km of ( c) quarter five-spot, and ( d) regular fracture network examples.

g,asfollows: ⎧ ⎪ ⎨ ⎪ ⎩ 𝑔𝑚=−2𝑒𝑥+𝑦 inΩ1, 𝑔𝑚=−𝑒𝑥+𝑦 inΩ2, 𝑔𝑓 =√𝑒 2 inΓ, (49)

All other physical parametersare setto one, andthe homogeneous boundaryconditionsareappliedtoallboundaries.Thegeometryused inthisanalysisandtheillustrationoftheexactsolutionarepresented inFig.5a-b,respectively.

WecalculateL2_{normofthedifferencebetweentheexactsolution,p,} andapproximatedsolution,ph,andtheresultsarepresentedinFig.5

c-dformatrixandfracturedomains,respectively.Forbothmatrixand fracturedomains,theEGandDGmethodsprovidetheexpected conver-gencerateoftwoandthreeforpolynomialdegreeapproximation,k,of oneandtwo,respectively(Antoniettietal.,2019;Babuska,1973). 3.2. Quarterfive-spotexample

ThisnumericalexampleteststheEGmethodperformanceinan in-jection/productionsetting usingfive-spotpattern,andweadoptthis examplefromChaveetal.(2018),Antonietti etal.(2019). The five-spot pattern,whereone injection well is located in the middleand

fourproducersarelocatedateachcornerofthesquare,iscommonly usedinundergroundenergyextractionChenetal.(2006).Duetothe symmetryofthisgeometry,onlyaquarterofthedomain(Ω =[0,1]2) is simulated. Theinjection well is located at (0,0), andthe produc-tion well islocated at (1,1).We placethefracturewith 𝑎𝑓=0.0005

atΓ ={(𝑥,𝑦)∈ Ω ∶𝑥+𝑦=1}.Thegeometry,boundaryconditions,and meshwithℎ=1.2× 10−1_{appliedinthisanalysisareshowninFig.6.}

Thefollowingsourcetermisapplied totheentirematrixdomain includingtheinjectionandproductionwells:

𝑔𝑚(𝑥,𝑦)=10.1tan ( 200 ( 0.2−√𝑥2_+𝑦2)) −10.1tanh ( 200 ( 0.2−√(𝑥−1)2_+(𝑦−1)2))_{. (50)} Toinvestigatetheeffectoffractureconductivity,weperformtwo sim-ulations using different fracture conductivity inputs. (i) We choose,

𝒌𝒎=𝑰,𝑘𝑛𝑓=1,and𝒌𝑻𝒇 =100𝑰forthepermeablefracturecase.(ii)We assumethefractureisimpermeableandset𝒌𝒎=𝑰,𝑘𝑛_𝑓=1× 10−2_,and

𝒌𝑻

𝒇 =𝑰.Alloftheremainingphysicalparametersaresettoone. ResultsoftwocasesarepresentedinFig.7a-bforthepressurevalue andFig.7c-dforthepressureprofilealong𝑥=𝑦line.Thepressure pro-fileofthepermeablefracturecaseissmoothlydecreasingfromthe in-jectionwelltowardstheproductionwell.Thepressureprofileofthe

im-Fig.16. The pmsolution of heterogeneous quarter five-spot example (low km ): ( a) permeable fracture, ( b) impermeable fracture cases, pmplot along 𝑥= 𝑦line of

( c) permeable fracture, and ( d) impermeable fracture cases. All of the results obtained from the EG and DG methods approximately overlap.

permeablefracturecase,ontheotherhand,illustratesajumpacrossthe fractureinterface.Thesefindingscomplywiththeresultsofthe previ-ousstudies(Chaveetal.,2018;Antoniettietal.,2019).Ourresults con-vergetothereferencesolutionasthehisreducedfromℎ=1.2× 10−1to ℎ=7.2× 10−2_{.Notethat(Antoniettietal.,2019)performthese} numer-icalexperimentsusingthesecond-orderDGmethodwithℎ=7.5× 10−2 onpolytopicgrids(Antoniettietal.,2019).Thereisnosignificant dif-ferencebetweentheEGandDGresultsforbothhvalues(Fig.7c-d). 3.3. Immersedfractureexample

The numerical examples discussedso far contain a fracture that cut through the matrix domain. To testthe EG discretization capa-bilityin the immersedfracturesetting, weadopt this example from Angotetal.(2009).Sinceweassumethatthefracturetipis substan-tiallysmall,thereisnofluidmasstransferbetweenthematrixand frac-turedomainsacrossthefracturetip,seeA’inFig.8.Hence,thefracture boundary,Λ𝑁_ℎ,thatintersectswith thebulkmatrixmaterialinternal boundary,0

ℎ,isenforcedwithno-flowboundarycondition,𝑞𝑓𝑁=0.

Inthisexample,wetakethebulkmatrix,Ω =[0,1]2,andthe frac-turewith𝑎𝑓=0.01,Γ ={(𝑥,𝑦)∈ Ω ∶𝑥=0.5,𝑦≥0.5}.Weperformthree

simulationsusingdifferentfractureconductivityinputs;(i)weassume

𝒌𝒎=𝑰,𝑘𝑛𝑓=1× 102,and𝒌𝑻𝒇=1× 106𝑰forthepermeablefracturecase, anditsgeometryandboundaryconditionsarepresentedinFig.8a.(ii) Thepartiallypermeablefracturecaseutilizes𝒌𝒎=𝑰,𝑘𝑛𝑓 =1× 102,and 𝒌𝑻

𝒇 =1× 102𝑰.Thiscase geometryandboundaryconditionsare illus-trated inFig.8b.(iii)Impermeable fracture,geometryandboundary conditionsareshowninFig.8c,andituses𝒌𝒎=𝑰,𝑘𝑛𝑓=1× 10−7,and 𝒌𝑻

𝒇 =1× 10−7𝑰.Allotherphysicalparametersareequaltoone.Example ofmeshthatcontains𝑛𝑒=272isshowninFig.8d.

InFig.9,thevaluesofpmwith𝑛𝑒=1,741forallcasesarepresented.

Thepressureplotalong𝑦=0.75ispresentedinFig.10.Thepermeable andpartiallypermeable fracturecasesillustratethecontinuityof the pressurewhile theimpermeablefracturecase showsthejumpacross thefractureinterface.Ourresultsconverge(𝑛𝑒=272,𝑛𝑒=1,741,and

𝑛𝑒=6,698)tothesolutionsprovidedbyAngotetal.(2009).Moreover,

theEGandDGmethodsprovidesimilarresults.Notethatthereference solutionsareperformedonthefinitevolumemethodwith65,536 con-trolvolumes.

Fig.17. The pmsolution of heterogeneous quarter five-spot example (high km ): ( a) permeable fracture, ( b) impermeable fracture cases, pmplot along 𝑥= 𝑦 line of

( c) permeable fracture, and ( d) impermeable fracture cases. All of the results obtained from the EG and DG methods approximately overlap.

3.4. Regularfracturenetworkexample

Weincreasethecomplexityoftheproblembyincreasingthenumber offracturesasshowninFlemischetal.(2018).Thisexample,however, iscalledregularfracturenetworksinceallfracturesareorthogonalto theaxes(xory).Thegeometryusedinthisexamplewasutilizedby Geigeretal.(2013)foranalyzingmulti-ratedual-porositymodel.Details formodelgeometryandboundaryconditionsareshown inFig.11a. Weset𝒌𝒎=𝑰forallthematrixdomain,Ω =[0,1]2,and𝑎𝑓=1× 10−4

forallfractures,Γ. Twofractureconductivityinputsareused;(i)we choose𝑘𝑛_𝑓=1× 104 _and𝒌𝑻

𝒇 =×104𝑰 forthepermeablefracturecase. (ii)Fortheimpermeablefracturecase,weassume𝑘𝑛

𝑓=1× 10−4,and 𝒌𝑻

𝒇=1× 10−4𝑰.Alloftheremainingphysicalparametersaresettoone. Exampleofmeshthatcontains𝑛𝑒=184isshowninFig.11d.

Thepmresultsofthepermeableandimpermeablefracturesare

pre-sentedinFig.11b-c,respectively.Thepermeablefracturecaseshows thesmoothpmprofilewhiletheimpermeablefracturecaseclearly

illus-tratesthejumpofpmacrossthefractureinterface.Theseresultsarethe

sameasthereferencesolutionsprovidedbyFlemischetal.(2018). Figs.12a-bpresentthepressureplotsalongthelines𝑦=0.7and𝑥= 0.5ofthepermeablefracturecase.Thepressureplotalong(𝑥=0.0,𝑦=

0.1)to(𝑥=0.9,𝑦=1.0)lineofimpermeablefracturecaseisshownin Fig.12c.Ourresultsusing𝑛𝑒=184,𝑛𝑒=382,𝑛𝑒=2,046,and𝑛𝑒=26,952

convergetothereferencesolutionsprovidedbyFlemischetal.(2018). The reference solution is simulatedbased on finite volume method with equi-dimensional setting, and it contains 1,175,056 elements Flemischetal.(2018).

Besidestheevaluationofpressureresults,wealsoinvestigatethe fluxcalculatedatfaceA’(seeFig.11a)asfollows:

flux=−∑

𝑒∈A′∫𝑒

𝒌𝑚

𝜇 (∇𝑝𝑚−𝜌𝐠)⋅ 𝒏𝑑𝑆onA′, (51)

AspresentedinFig.13,thedifferencebetweeneachnecaseis

insignif-icant,i.e.thedifferentbetween𝑛𝑒=26,952and𝑛𝑒=184casesisless

than1%.Furthermore,thereisnodifferencebetweentheEGandDG methods.TheDOFcomparisonbetweenEGandDGmethodsisshown in Table1. TheEGmethod requirestheDOF(inthematrixdomain) approximatelyhalfofthatoftheDGmethod.NotethattheDOFinthe fracturedomainisthesamebecausewediscretizethefracturedomain usingtheCGmethod.

Fig.18. The p_msolution of heterogeneous regular fracture network example (low k_m ): ( a) permeable fracture, ( b) impermeable fracture cases, p_mplot along ( c) 𝑦= 0 .7 and 𝑥= 0 .5 lines of permeable fracture case, and ( d) ( 𝑥= 0 .0 ,𝑦= 0 .1) to ( 𝑥= 0 .9 ,𝑦= 1 .0) line of impermeable fracture case.

Table1

Degrees of freedom (DOF) comparison between EG and DG methods of regular fracture network example.

n e EG DG

matrix domain fracture domain matrix domain fracture domain

26,952 40,629 13,677 80,856 13,677

2,046 3,123 1,077 6,138 1,077

382 595 213 1,146 213

184 293 109 552 109

3.5. Theheterogeneousinbulkmatrixpermeabilityexample

Thenumericalexamplespresentedsofaronlyconsideran homoge-neousmatrixpermeabilityvalue.Inthissection,weexaminethe capa-bilityoftheEGmethodinhandlingthediscontinuitynotonlybetween thefractureandmatrixdomainsbutalsowithinthematrixdomain(e.g. NickandMatthai,2011b)byemployingtheheterogeneouspermeability inthebulkmatrix.Weadaptthequarterfive-spotasinSection3.2and regularfracturenetworkasinSection3.4.Wechoosethefinestmesh frombothexamplestotesttheEGmethodcapabilitycomparedtothat

oftheDGmethodinhandlingthesharpdiscontinuitybetweenthe max-imumnumberofinterfaces.Thegeometries,boundaryconditions,and inputparametersareutilizedasSections3.2and3.4exceptforthek_m

value.

Inthis study,𝒌𝒎=𝑘_𝑚𝑰,km valueisrandomlyprovided valuefor

eachcell.Wewilldistinguishinparticulartwodifferentcases,named lowk_mcaseandhighk_mcaseinthefollowing.Thelowk_mcaseis char-acterizedbyalog-normaldistributionwithaverage𝑘̄𝑚=1.0,variance

var(𝑘_𝑚)=40,limitedtominimum value𝑘_𝑚min=1.0× 10−2,and maxi-mumvalue𝑘𝑚max=1.0× 102.Thehighkmcaseuses𝑘̄𝑚=30.0,var(𝑘𝑚)=

90,𝑘_𝑚min=1.0× 10−1,and𝑘𝑚max=1.5× 102.Thisheterogeneousfields forbothexamplesarepopulatedusingSciPypackage(Jonesetal.,2001) asshowninFigs.14and15forlowk_mandhighk_mcases,respectively.

3.5.1. Quarterfive-spotexample

Theresultsforthelowkm casewithpermeable(𝑘𝑛_𝑓=1and𝒌𝑻_𝒇= 100𝑰)andimpermeable(𝑘𝑛

𝑓=1× 10−2and𝒌𝑻𝒇=𝑰)fracturesare illus-tratedinFig.16.Ingeneral,thepmprofilefromthetwocasesaremore

dispersethanthehomogeneouskmsetting.TheEGandDGmethods pro-videsimilarresults,with𝑛𝑒=6,568andℎ=7.2× 10−2.Thediscussions

Fig.19. The pmsolution of heterogeneous regular fracture network example (high km ): ( a) permeable fracture, ( b) impermeable fracture cases, pmplot along ( c)

𝑦= 0 .7 and 𝑥= 0 .5 lines of permeable fracture case, and ( d) ( 𝑥= 0 .0 ,𝑦= 0 .1) to ( 𝑥= 0 .9 ,𝑦= 1 .0) line of impermeable fracture case.

Fig.21. The pmsolution of heterogeneous and anisotropy km in regular fracture network example: ( a) permeable fracture, ( b) impermeable fracture cases, the pm

plot along ( c) 𝑦= 0 .7 and 𝑥= 0 .5 lines of permeable fracture case, and ( d) ( 𝑥= 0 .0 ,𝑦= 0 .1) to ( 𝑥= 0 .9 ,𝑦= 1 .0) line of impermeable fracture case.

regardingpermeableandimpermablefracturesettingsareprovidedas follows:

1. Lowkmwithpermeablefractureresultillustratestheapproximately smoothpmsolutionbecausethe𝑘̄𝑚=1.0isequalto𝑘𝑛𝑓.Theplot

along𝑥=𝑦line,asexpected,showspmgraduallydecreasesfrom

theinjectionwelltotheproductionwell.Thisresultcomplies withthatofthehomogeneouskmsetting.

2. Lowkm withimpermeablefractureresultandtheplotalong𝑥= 𝑦lineexhibitalittlejumpofp_m across thefractureinterface. Thisbehaviorisdifferentfromthehomogeneouskmsettingsince 𝑘𝑚min=𝑘𝑛_𝑓,whichleadtothelesspermeabilitycontrastbetween thefractureandmatrixdomains.

Theresultsforthehighk_m casewithpermeableandimpermeable fracturesarepresentedinFig.17.Using𝑛𝑒=6,568,theEGandDG

re-sultsareapproximatelythesame.Theobservationsconcerningthe frac-turepermeabilityarepresentedbelow:

1. Highkmwithpermeablefracturedisplaysajumpacrossthefracture domainbecausek_m aroundthefractureontheplottinglineis higherthan𝑘𝑛

𝑓;asaresult,thefractureinterfaceactslikeaflow

barrier.Theplotalong𝑥=𝑦linesupportsthisobservationaspm

jumpsacrossthefractureinterface.

2. Highkmwithimpermeablefractureillustratesahugejumpacross thefractureinterface,ascanbeobservedfromthepmplotalong

𝑥=𝑦line,sincekmismuchhigherthan𝑘𝑛_𝑓.Thepmvariationis

lesspronouncedcomparedtothelowkm one,seeFigs.7b (ho-mogeneous)and16b(heterogeneous),becausethefluidflowis blockedbythefractureinterface(sharpmaterialdiscontinuity). 3.5.2. Regularfracturenetworkexample

The pressure results of the low km case between the permeable (𝑘𝑛

𝑓 =1× 104 and 𝒌𝑻𝒇 =×104𝑰) and impermeable (𝑘𝑛𝑓 =1× 10−4 and 𝒌𝑻

𝒇 =1× 10−4𝑰)fracturesareillustratedinFig.18.Similartothe five-spotexample,theEGandDGresultsaresimilarwith𝑛_𝑒=26,952,andpm

Fig.22. Three-dimensional regular fracture network. The illustrated surfaces with edges (blue in a and red in b and c) indicate the fractures. ( a) geometry and initial/boundary conditions are presented. pmis enforced as zero on two corner points shown in red and no-flow condition is applied to all boundaries. Initial

conditions (ICs) for pmand pfare set as one. ( b) presents pressure solution on plane C, pm, matrix velocity, vm , and fracture velocity, vf , with the permeable fracture

(case i), ( c) presents pmon plane D, vm , and vf with impermeable fracture cases (case ii). Note that the matrix pressure is only shown in the cross section between

the far two edges of the model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig.19a-b.Thediscussionsregardingpermeableandimpermable

frac-turesettingsareprovidedasfollows:

1. Lowkm withpermeablefractureillustratesthefracturedominate flow regime,even though 𝑘̄𝑚=1.0, kmmin is setto 1.0× 10−2, whichismuchlessthan𝑘𝑛

𝑓.Thissettingreducestheimpactof

thematrixdomain.pmplotalong𝑦=0.7and𝑥=0.5lines

sup-portthisobservationbyshowingthehighpressuregradientin thematrixdomain,butpmbecomesmuchlessvariedwhen

en-teringthefracturedomain.

2. Lowk_mwithimpermeablefracturealsopresentsthefracture dom-inate flow regime because 𝑘𝑛

𝑓=1× 10−4, which is less than

𝑘𝑚min=1.0× 10−2.Therefore,theflowinthematrixisblocked bythefracturedomain.Theplotalong(𝑥=0.0,𝑦=0.1)to(𝑥= 0.9,𝑦=1.0) line illustrates jumps across the fracture domain, whichissupportingourobservation.

Theresultsforthehighkm casewithpermeableandimpermeable fracturesarepresentedinFig.19.With𝑛𝑒=26,952,theEGandDG

re-sultsareapproximatelythesame.Theobservationsconcerningthe frac-turepermeabilityarepresentedbelow:

1. High km withpermeable fracture showsthatthematrixdomain gainsmoremomentumcomparingtothelowk_m case.p_m plot along𝑦=0.7and𝑥=0.5linesillustratesalsoapproximately lin-earreductionalongthematrixdomain,whilepressureisalmost constantinthefracturedomain.

2. High km withimpermeable fractureclearlypresents thefracture domaindominatetheflowbecause𝑘𝑛

𝑓ismuchlessthanthekm. Hence,theflowisblockedbythefractures.Theplotalong(𝑥= 0.0,𝑦=0.1)to(𝑥=0.9,𝑦=1.0)linesupportthisobservationby illustratingmultiplejumpsacrossthefracturedomainandnopm

variationinsideeachmatrixblock.

3.6. Theheterogeneousandanisotropyinbulkmatrixpermeabilityexample Thissectionillustratesthecomparisonbetweenthepmsolutionsof

con-Fig.23. Comparison of ( a) the average pmin the whole domain and blocks A and B, and ( b) the pmprofiles for t= 25 and t= 75 along ( 𝑥 = 0 .0 ,𝑦= 0 .0 ,𝑧= 0 .0) to

( 𝑥= 1 .0 ,𝑦= 1 .0 ,𝑧= 1 .0) line.

trastwiththepreviousexample,weutilizethecoarsestmesh(𝑛𝑒=184)

fromthe regularfracturenetworkexample.The matrixpermeability heterogeneity,km,isgeneratedwiththesamespecificationasthelow k_mcaseintheprevioussection.However,inthisstudy,theoff-diagonal termsof (13) arenot zero,andthekm ofeach elementisdefinedas follows: 𝒌𝒎∶= [ 𝑘𝑚 0.1𝑘𝑚 0.1𝑘_𝑚 𝑘_𝑚 ] , (52)

ThegeneratedheterogeneousfieldisshowninFig.20a-bforboth diag-onalandoff diagonalterms.

The pressure results of the low km case between the permeable (𝑘𝑛

𝑓=1× 104 and 𝒌𝑻𝒇=×104𝑰) and impermeable (𝑘𝑛𝑓=1× 10−4 and 𝒌𝑻

𝒇 =1× 10−4𝑰)fracturesarepresentedinFig.21.TheresultsofEGand DGmethodsareapproximatelysimilar forbothfracturepermeability settings.These resultsillustratethattheEGmethodcapturesthe dis-continuitiesandallowusingacoarsemeshtomaintainaccuracy. 3.7. Thetime-dependentproblems

Fromthissection,weconsidertime-dependentproblemwherecand 𝜙miarenonzero(1)–(8).Besides,weextendthespatialdomainto

three-dimensionalspacetofurtherillustratetheapplicabilityoftheproposed EGmethod.Here,thefirstexampleconsidersthegeometrycontaining onlytheorthogonalfracturestotheaxes(x,y,orz),andthesecond ex-ampleassumesthegeometrywitharbitraryorientatednaturalfractures. Notethatwe,here,presentonlytheresultsoftheEGmethod.Theresults oftheDGmethodarecomparabletothoseoftheEGmethod.

3.7.1. Three-dimensionalregularfracturenetworkexample.

Inthisexample,weconsiderathree-dimensionaldomain,whichis ananalogofthetwo-dimensionalcasepresentedinSection3.5.2.This domaincontainsasetofwell-interconnectedfracturesandmeshedwith tetrahedralandtriangularelements(forfractures)with𝑛𝑒=9,544.The

fracturegeometryisbasedontheexampleinBerreetal.(2020).The detailsofgeometrywithinitialandboundaryconditionsillustratedin Fig.22a.

Here,weconsidertwodifferent scenarios:casei)permeable frac-turecasewith𝑘𝑛

𝑓=1× 106and𝒌𝑻𝒇 =1× 106𝑰;andcaseii)impermeable fracturecasewith𝑘𝑛

𝑓=1× 10−12,and𝒌𝒇𝑻 =1× 10−12𝑰.Forbothcases,

weemployapermeableporousmediumbysetting𝒌𝒎=[1.0,1.0,0.1]𝑰, andalloftheremainingphysicalparametersaresettooneforthe sim-plicity.Thetemporaldomainisgivenas𝕋=[0,100]whereanuniform timestepsizeΔ𝑡𝑛=1.

InFig.22b-c,thenumerical results ofthepm,vm,andvf forthe permeable(casei)andimpermeablefracturecases(caseii)at𝑡=100 arepresented.Amerevisualexaminationoftheseresultsalreadyshows thatthefracturepermeabilitycontrolstheflowfield.Forthepermeable fracturecase,thevelocityatthecornerofblockBishigherthanthat ontheoppositecornerinblockAasthefracturesareclosertotheopen cornerpointinblockB.Fortheimpermeablefracturecasethevelocity atthecornerofblockBislowerthanthatontheoppositecornerin blockAsinceblockAislargerthanblockB,anditcansupportflowfor alongertime.

SimilarbehaviorsofthepressurevaluesareobservedinFig.23a, wheretheaveragepressurevaluesofthefulldomainandblockAand B areplottedfor𝕋 =[0,100]. Itis clearthat theaveragepressurein blockBdropsfasterthaninblockAfortheimpermeablecase.Moreover, Fig.23billustratesthevalueofpmalong(𝑥=0.0,𝑦=0.0,𝑧=0.0)to(𝑥=

1.0,𝑦=1.0,𝑧=1.0)lineat𝑡=25and75. 3.7.2. Algrøynaoutcropexample

The final example is a three-dimensional fracture network built basedonanoutcropmapinAlgrøyna,Norway(Fumagallietal.,2019). Themodelhasasizeof850 × 1400 × 600mandcontains52 inter-sectingfractures(themodelisdescribedindetailinBerreetal.,2020). Thefiniteelementmeshisdiscretizedbytetrahedral(rockmatrix)with 𝑛𝑒=163,575andtriangularelements(fractures)with𝑛𝑒=329,080.See

Fig.24aformoredetails.AsshowninthisfigureDirichletboundary conditionsareappliedontwoedgesofthemodeltorepresentan in-jectionandaproductionwell.Allotherboundariesareconsideredas no-flow.Therockmatrixandthefracturesareconsideredpermeable: 𝑘𝑛

𝑓=1× 102,m2𝒌𝒇𝑻 =1× 102𝑰 m2,and𝒌𝒎=[1.0,1.0,0.1]𝑰 m2.Allof theremainingphysical parametersaresettoone,and𝕋=[0,1× 106_] secusinganuniformΔtn_of1×105_sec.

Fig.24bandcshowthesimulationresultsincludingthepressure field,thepressureiso-surfaces,andvelocityvectorsintherockmatrix at𝑡=500,000sec.Thepressureprofilesalongalinebetweentwo op-positecornersofthemodelatdifferenttimesareplottedinFig.24d. ThisexampleillustratestheapplicabilityofthepresentedEGmethod

Fig.24. Algrøyna outcrop example: ( a) geometry and initial/boundary conditions (no-flow condition is applied to all boundaries (excepts the production and injection wells). Initial conditions (ICs) for pmand pfare set as one.), b) pressure solution, pm, matrix velocity (in black arrow), vm , and fractures mesh, ( c) pmand

its iso-surfaces shown in grey, and ( d) pressure profiles along a line from the top of injection well to the bottom of production well.

foracomplexthree-dimensionalfracturenetworkwitharbitrary orien-tations.

4. Conclusion

ThisstudypresentstheEGdiscretizationforsolvingasingle-phase fluidflowinthefracturedporousmediausingthemixed-dimensional approach.Ourproposedmethodhasbeentested againstseveral

pub-lishedbenchmarksandsubsequentlyassesseditsperformanceinthetest caseswiththeheterogeneousandanisotropicmatrixpermeability.Our resultsillustratethatthepressuresolutionsresultedfromtheEGandDG method,withthesamemeshsize,areapproximatelysimilar. Further-more,theEGmethodenjoysthesamebenefitsastheDGmethod;for instance,preserveslocalandglobalconservationforfluxes,canhandle discontinuitywithinandbetweenthesubdomains,andhastheoptimal