Integrating a compressible multicomponent two-phase flow into an existing reactive transport simulator

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flow into an existing reactive transport simulator

Irina Sin, Vincent Lagneau, Jérôme Corvisier

To cite this version:

Irina Sin, Vincent Lagneau, Jérôme Corvisier. Integrating a compressible multicomponent two-phase

flow into an existing reactive transport simulator. Advances in Water Resources, Elsevier, 2017, 100,

pp.62-77. �10.1016/j.advwatres.2016.11.014�. �hal-01516810�

ContentslistsavailableatScienceDirect

Advances in Water Resources

journalhomepage:www.elsevier.com/locate/advwatres

Integrating a compressible multicomponent two-phase flow into an existing reactive transport simulator

Irina Sin

^∗

, Vincent Lagneau, Jérôme Corvisier

MINES ParisTech, PSL Research University, Centre for Geosciences and Geoengineering, 35 rue Saint-Honoré, F-77305 Fontainebleau Cedex, France

a r t i c l e i n f o

Article history:

Received 15 March 2016 Revised 23 November 2016 Accepted 25 November 2016 Available online 27 November 2016 Keywords:

Compressible two-phase flow Reactive transport

Sequential iterative coupling Operator splitting HYTEC Equation of state

a b s t r a c t

Thiswork aimsto incorporate compressible multiphase flowinto theconventional reactive transport frameworkusinganoperatorsplittingapproach.Thisnewapproachwouldallowustoretainthegeneral paradigmoftheflowmoduleindependentofthe geochemicalprocessesand tomodel complexmulti- phasechemicalsystems,conservingtheversatilestructureofconventionalreactivetransport.Thephase flowformulationisemployedto minimizethenumber ofmass conservationnonlinearequations aris- ingfromtheflowmodule.Applyingappropriateequationsofstatefacilitatedprecisedescriptionsofthe compressiblemulticomponentphases,theirthermodynamicpropertiesandrelevantfluxes.

The proposed flowcoupling method was implemented in the reactive transport software HYTEC.

The entire framework preserves itsflexibility for further numerical developments. The verification of thecouplingwasachievedbymodelingaproblemwithaself-similarsolution.Thesimulationofa2D CO2-injectionproblemdemonstratesthepertinentphysical resultsandcomputational efficiencyofthis method.Thecouplingmethodwasemployedformodelinginjectionofacidgasmixtureincarbonated reservoir.

© 2016TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

1.1.Background/motivation

Human activity in the subsurface has been expanding and diversifying(wastedisposal,miningexcavationandhigh-frequency storage of energy), and the public and regulatory expectations havebeenincreasing.Theassessmentofeachstepofunderground operations, including environmental impact evaluation, relies on elaborate simulators and leads to an urgent need to develop multiphysicsmodeling.Reactivetransport,ageochemicalresearch and engineering tool, is used in multicomponent systems and sophisticated chemical processes (activity andfugacity correction accordingto different models, mineral dissolution and precipita- tion,cation exchange, oxidation and reduction reactions, isotopic fractionation and filiation), in addition to gas evaporation and dissolution(vanderLeeetal.,2003;Mayeretal.,2012;Parkhurst andAppelo,1999;Steefel,2009;Yehetal.,2004).Multiphaseflow is based on broad experiences in reservoir engineering research, includingthethermodynamic modeling ofcomplex phase behav- ior. In particular, the equations of state were used to simulate

∗ Corresponding author.

E-mail address: [email protected] (I. Sin).

and study interfacial tension, gas, steam and alkaline injection inoil reservoirs andenhanced oilrecovery (Delshadetal., 2000;

Farajzadehetal.,2012;Nghiemetal.,2004;Wangetal.,1997)..

Thisworkaims toincorporateacompressiblemultiphase flow moduleintoan existingreactivetransportsimulator.Ourcoupling methodshouldthereforemeetthefollowingrequirements:

1. The new approach should handle the different complex mul- tiphasechemical modelsandretain thegeneralparadigmofa multiphase flow moduleindependent of thegeochemical sys- temandconservetheconventionalreactivetransportstructure;

2. The numberof mass conservationnonlinear equations arising fromtheflow module should beminimized such thatthe re- duced flow system preserves the matrix structure and mini- mizesthecomputationalintensity;

3. The entireframework should preserve its flexibility for possi- blenon-isothermal,geomechanical anddomaindecomposition developmentsinthefuture.

Reactivetransport methodshavebeen extensivelyinvestigated overthe pasttwo decades (seebelow). Thiswork focusesonthe coupling between multicomponent multiphase flow (MMF¹) and

1The nomenclature is provided in Table 1 . The abbreviations are detailed in Table 2 .

http://dx.doi.org/10.1016/j.advwatres.2016.11.014

0309-1708/© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

Table 1 Nomenclature.

Latin symbols

a k activity of potential catalyzing or inhibiting species A s specific surface area, [ m ²/ m ³solution ] or [ m ²/ kg mineral ] c l,k total liquid mobile concentration of basis species k , [ mol / kg w ] c s,k immobile concentration of basis species k , [ mol / kg w ] c g,m gas concentration of basis species m , [ mol / m ³] C i concentration of primary species i in chemical module d dissolution parameter of transport model

dt time step

D α molecular diffusion coefficient of phase α, [ m ²/ s ] D ^e_α effective diffusion coefficient of phase α, [ m ²/ s ] D α diffusion-dispersion tensor of fluid phase α e evaporation parameter of transport model

f _i^α fugacity of species i in phase α F residual function

g gravitational acceleration vector, [ m / s ²]

J Jacobian

k number of iterations in flow coupling k kin kinetic rate constant in , [ mol / m ²/ s ]

k max^{f l} maximum number of iterations in flow coupling

k ^rtmax maximum number of iterations in reactive transport coupling k rα relative permeability of phase α

K intrinsic permeability, [ m ²] K intrinsic permeability tensor, [ m ²] K i K-value/equilibrium ratio K j equilibrium constant of reaction j

K s equilibrium solubility constant of solid phase K _i^h Henry’s law constant

M _· molar mass of species ·, [ kg / mol ] n α quantity of matter in phase α

n normal vector

N c number of primary species in chemistry module N f number of fluid phases α

N g number of gas species N kin number of kinetic reactions N Nit number of Newton iterations N Pit number of Picard iterations N p number of phases

N r number of independent chemical reactions N s number of species in chemistry module p α liquid/gas pressure, [ Pa ]

P pressure, [ Pa ]

P c set of the critical pressures q α mass source term of phase α, [ kg / s ]

q α,k mass source term of species k in phase α, [ kg / s ] q g,m source term of basis species m in the gas phase, [ mol / m ³] q l,k source term of basis species k in the liquid phase, [ mol / kg w ] Q s ion activity product

r αβ reaction term of phases αand βin αtransport operator R gas constant, [ J / K / mol ]

R α reaction term of phase α, [ kg / s ]

R α,k reaction term of species k in phase α, [ kg / s ]

R g,m reaction term of basis species m in the gas phase, [ mol / m ³] R l,k reaction term of basis species k in the liquid phase, [ mol / kg w ] R geochemical reaction operator

S j concentration of species j in chemical module S α saturation of phase α

t calculation time of entire system per time step t fl calculation time of flow operator per iteration t flc calculation time of flow coupling per iteration t gt calculation time of gas transport operator per iteration t rtc calculation time of reactive transport coupling per iteration T temperature, °C and K

T i total concentration in chemistry module T c set of the critical temperatures T α transport operator of phase α u α Darcy’s velocity of phase α v molar volume, [ m ³/ mol ]

V α volume of porous space occupied by phase α, [ m ³] V tot total volume, [ m ³]

x i mole fraction of basis species i in the liquid phase X α,k mass fraction of basis species k in phase α x vector of primary variables of the flow system y i mole fraction of basis species i in the gas phase

Table 1 ( continued ) Latin symbols

Z compressibility factor

Z c set of the critical compressibility factors Greek symbols

α⁼ {^{l, g}} liquid/gas phase αij stoichiometric coefficient γj activity coefficient

matrix of binary interaction coefficients of PR EOS εg gas quantity tolerance in reactive transport coupling εNf residual function tolerance in flow coupling εqss quasi-stationary state tolerance in flow coupling ε^rt tolerance in reactive transport coupling μα viscosity of phase α, [ Pa ·s ]

ρα mass density of phase α, [ kg / m ³]

ρα^g density of phase αin the gravity term, [ kg / m ³] τα tortuosity of phase α

φ porosity

ϕi^α fugacity coefficient of species i in phase α ψα volumetric rate of phase α, [ m ³/ s ]

acentric factor set

T α mathematical transport operator of phase α

·∞ infinity norm

1 R>0(·) indicator function of the set of strictly positive real numbers

Table 2 Abbreviations.

AIM adaptive implicit method

CFL Courant-Friedrichs-Lewy number

DAE differential algebraic equations based method

DSA direct substitution approach

EOS equation of state

FIM fully implicit method

FVM finite volume method

GIA global implicit approach

IMPEC implicit pressure/explicit concentration

MMF multiphase multicomponent flow

MMRF multiphase multicomponent reactive flow

ODE ordinary differential equations based method

PDE partial differential equation

OSA operator splitting approach

RT reactive transport

SI saturation index

SIA sequential iterative approach

SNIA sequential non-iterative approach

TPFA two-point flux approximation

reactivetransport(RT)modules andstartsbysurveyingtheexist- ingapproaches

1.2.Areviewofmultiphasemulticomponentflowandreactive transportcodes

1.2.1. OperatorsplittingalgorithmsbetweenMMFandRT

Thestrengthoftheoperator-splittingapproach (OSA)(sequen- tial iterative(SIA) or sequential non-iterative(SNIA)) arises from the framework flexibility, which allows each model to be devel- opedandverified independently.Theseare importantreasonsfor selecting the OSA for the coupling between MMF and RT, par- ticularly when extending a hydrogeochemical code from single- to two-phase flow. The following codes apply the OSA: Code- Bright(Olivella etal., 1996) (coupling withthe reactivetransport code RETRASO (Saaltink et al., 2004)), DuMu^X (based on DUNE) (Ahusbordeetal.,2015;Vostrikov,2014),DUNE(Hronetal.,2015), HYDROGEOCHEM(unsaturated)(Yehetal.,2004,2012),iCP(Nardi etal.,2014),IPARS(PeszynskaandSun,2002;Wheeleretal.,2012), MIN3P(thebubblemodel)(Mayeretal.,2012;MolinsandMayer, 2007),MoReS(Farajzadehetal.,2012;Wei,2012),NUFT(Haoetal., 2012), PFLOTRAN (Lichtner et al., 2015; Lu and Lichtner, 2007),

STOMP(Whiteetal.,2012;WhiteandOostrom,2006),TOUGHRE- ACT(XuandPruess,1998;Xuetal.,2012),andUTCHEM(Delshad etal.,2000);seealso(Steefeletal.,2014).

Thesimulatorsareprimarilybasedonthefinitevolumemethod (FVM)becauseofitsconservativeproperties.Thegeneraltendency incouplingistofirst(non-)iterativelysolvetheflowtoobtainthe velocities.Thecompositionalformulationistypicallychosen,oral- ternatively,theconservationequationscanbesolvedforthedom- inantcomponents(e.g.,water/air).Whenthe two-phaseflowsys- teminvolvesonly two components (e.g.,H₂OandCO₂),the final setsofequationsforthecompositional anddominantcomponent formulationsareidenticalandreducetothemassconservationfor eachcomponent.

Once the flow is established, the RT part can be solved us- ing the OSA (SIA, SNIA or predictor-corrector); a global implicit approach (GIA), such as the ordinary differential equation-based method(ODE, thechemistry moduleis usedas ablack box), the directsubstitutionapproach(DSA)(Saaltinketal.,2001)orthedif- ferential algebraic equation-based method (DAE) (de Dieuleveult, 2008; de Dieuleveult et al., 2009). The OSA and GIA applied to RTare compared in Carrayrou etal.(2010); de Dieuleveult etal.

(2009); Saaltink etal., (2001); Steefel andMacQuarrie (1996). In 2001,Saaltink etal.(2001) reportedthat SIA wasmorefavorable thanDSAincasesof largegrids andlow kineticratesbecause of thehigh computer storage requirements and slowlinear solvers.

Adecadelater,theMoMaSbenchmarkconfirmedthereliabilityof bothapproachesandtheenhancedcomputational potentialofthe DSAwiththereductiontechnique(KräutleandKnabner,2007).

1.2.2. FromglobalimplicitalgorithminRTtoglobalimplicit algorithminMMF&RT

The advantageofGIA isits accuracy,although itcomes atthe costofcomputational resources.However, thismethodis becom- ing more competitive because of increases in computer capabil- ities and advances in the methods that allow reducing the sys- temofequations.Additionally,reservoirsimulatorstypicallyutilize theglobal formulationfor the MMF problemcombined withthe fullyimplicitmethod(FIM)ortheadaptiveimplicitmethod(AIM), whichunifies theFIMandimplicitpressure/explicitconcentration (IMPEC).TheycanthereforebenaturallyextendedtotheRTusing GIA,e.g., COORES², GEM-GHG (Nghiem etal., 2004), GPAS(Pope etal.,2005),andGPRS(Cao,2002;Fanetal.,2012).

For the global implicit solution of the RT, several techniques havebeen proposedtoreduce theinitial massbalancesystemby linearcombinationandeliminate the reactionterms inthe equi- libriumreactions(KräutleandKnabner,2007;Molinsetal.,2004;

Saaltink etal., 1998). Such modifications of the DSA forRT lead tomathematicaldecouplingoftheentiresystemandconsequently makeit interestingforthe globalimplicitcouplingofmultiphase flowandreactive transport, asrecentlydemonstratedby Saaltink etal.(2013) andFanetal.(2012).Saaltink etal.(2013)proposed amethodforintroducing thechemistrycalculationsto aconven- tionalmultiphasesimulatortominimizethenumberofmasscon- servationequationsbytakingthefluidphasepressuresandporos- ity as primary variables and expressing all secondary variables (suchascomponentconcentrations,fugacity,pH,andsalinity)asa polynomialfunctionofgaspressure,whichrequirespre-processing priorto eachapplication. Fanetal.(2012) employed theelement balanceformulation(MichelsenandMollerup,2007)viareduction techniques(Kräutle and Knabner,2007; Molins et al., 2004) and the decoupled linearized system on the primary and secondary equations. Saaltink et al. (2013) demonstrated that the GIA and OSAprovided similar results for CO₂ storage andconcluded that thefullcouplingwasnotnecessaryforMMF&RTmodelingfroma

2A. Michel and T. Faney, personal communication , 2015

physicalperspective,althoughtheOSAcanbe morecomputation- allyexpensive.

Moreover,Gamazoetal.(2012) highlightedthe significantim- pact of geochemical reactions on the phase fluxes by modeling gypsumdehydration when thenon-isothermal flow is chemically restrained,i.e.,theanhydrite-gypsumparagenesiscontrolsthewa- ter activityandhencetheevaporationprocess. Theimportanceof DSA,whichimplicitlyconnectstheequilibriumheterogeneousre- actions and the phase flow, was emphasized compared with the decoupled formulation of the flow and the RT: the decoupled formulation overestimatesthe evaporation, butits computational timewasreducedby22.5%(comparedtoDSA).However,thedef- initionsof wateractivityandliquiddensity weredifferent inthe GIA’s andOSA’smodels.Furthermore,the decoupled flowsystem contained the conservation equations for dominant components (water and air), but the coupling between flow and RT was not specified. Given the nature of the formulations describedin this work,itislikelythatusingthestrongOSA(theSIA-basedconnec- tionbetweenflow andRTwiththeprecisereactiontermsinflow equations)wouldyieldresultssimilar tothoseofDSAatthecost ofadditionaliterations.

1.3. PreambletoanewMMF&RTcouplingapproach

Toreduce thenumberofnonlinearequationsarising fromthe flow system, the formulation of the dominant components was considered.Thisapproachisefficientwhentheimpactoftheother speciesis negligible.Ingeochemicalproblems,the speciation can vary largely over time and space. Ideally, the dominant compo- nentsshouldbeadaptedlocallytopreserve theaccuracy,butthis wouldrequirespecifictreatmentoftheglobalflowsolution.

Iftheflowsystemexpandssuch thatmanyspeciesplaysignif- icant roles in the thermodynamic state and phase displacement, thenthenumberofnonlinearequationsofcompositionalflowin- creaseswiththenumberofcomponents,whichiscomputationally expensive.

Despitethereductiontechniques,theprimarynonlinearsystem mustbecalculated.Ingeochemicalmodeling,whichincludescom- plex homogeneousandheterogeneousreactions,the primary(ba- sis)speciestypicallychangesspatially andtemporally.Thesystem of equations is therefore redefined,which requires modifications oftheJacobianstructurewithinNewton’smethod.Thesolutionof thetransportlinearsystemcanbeseveralorevenahundredtimes fasterthanthatofmultiphaseflow.

Tominimize thesize ofthenonlinearsystem, itis possibleto replace the compositional formulationby the phase formulation.

ThephaseflowcoupledtotheRTwasemployedbyPeszynskaand Sun(2002).Giventheirproblemconditions(slightlycompressible fluidsand density-independent flow), theauthors divided the RT intothreestages(advection,reaction,anddiffusion)withinthein- ternaltimestepsbyinterpolatingthefluxesandsaturations.Later, Hronetal.(2015)simulatedEscherichiacoli growthandtransport underaerobicandanaerobicconditionsviathesequentialcoupling of phase flow and the SNIA-based RT. The liquid fluid was as- sumedtobeincompressible,whereasthegasdensitywascompo- sitiondependent.Intheadvective-dominantregime,thetransport wassolvedexplicitlyinatimethat ledtoatime steprestriction:

CFL =0.4; thus, the split transport time step was applied. Note that the same issuearises fromthe IMPECscheme inwhich the componentequationsaresolvedexplicitly.

1.4. Ouralternativemethod

Thisworkproposesanewapproachforincorporatingthecom- pressibletwo-phaseflowinaconventionalreactivetransportsim- ulator.Itisbasedonthephaseflowformulationandpreservesall

the sustainability and facilities of the reactive part.This method builds on theunconditional stabilityof thefully implicitscheme andcanmodeladvective-dominantanddensity-drivenregimes.

The following simplifications are considered in this work:

isothermalflowandnogeomechanics.Themethodwasappliedto the(SIA)reactivetransportsimulatorHYTEC(Lagneauandvander Lee, 2010b;van der Leeet al., 2003). The HYTEC code hasbeen widely evaluated in several benchmark studies (Carrayrou et al., 2010; Lagneau and van der Lee, 2010a; Trotignon et al., 2005;

deWindtetal.,2003)andnumerousapplications,suchascement degradation(deWindtandDevillers,2010),radioactivewastedis- posal(Debureetal.,2013;deWindtetal.,2014),geologicalstorage ofacidgases(Corvisieretal.,2013;Jacquemetetal.,2012;Lagneau et al., 2005), and uranium in situ recovery processes (Regnault etal.,2014).

InSection2,aconcisedescriptionofthegoverningequationsof multicomponent multiphase flow andreactive transport is given.

Next, the proposed coupling methods are detailed in Section 3. Section 4 presents the method’s applicability and computational performance, firstusinga benchmarkproblemwitha self-similar solution and then for modeling 2D CO₂ injection. Section 4 also illustrates aproblemofacidgasinjectionincarbonatedreservoir.

TheconclusionsarepresentedinSection5.

2. Governingprocessesandmathematicalformulation

Thestandard entireisothermal MMRFproblemiscomposedof componentc_α,kmolconservation,massbalance,mass-actionlaws and other constitutive relations. The total number of phases is Np=3,wherethesolidphaseisimmobileandtheN_ffluidphases

α

^{are mobile (liquid andgas). The chemical system ofNs chem-}

ical speciesandNr linearlyindependent chemical reactionsrelies onMorel’smethod(MorelandHering,1993)ofN_cprimaryspecies thatformabasisforallNsspecies:Nc=Ns−Nr.

2.1. Massconservationforeachphase

Wefirstintroducethestandard multiphasecompositionalflow problem and the physical parameters, and then we present the phasemassconservationformulationanditsadvantages.Letusbe- ginwiththegeneralmassconservationintermsofthemassfrac- tionX_α,kofspecieskinfluidphase

α

^thatformsNcN_fequations:

∂ ( φ

^Sα

ρ

αX_α,k

)

∂

^{t =−}

∇

^·

( ρ

αX_α,ku_α−

ρ

αD_α

∇

^Xα,k

)

+R_α,k+q_α,k, (1) where

φ

^{istheporosity,S}α isthesaturationoffluidphase

α

^,

ρ

αis themass densityoffluidphase

α

^,uα is theDarcy-Muskatveloc- ityoffluid phase

α

^,Dα isthediffusion-dispersion tensoroffluid phase

α

^{. The classical Fick’s law is used forthe diffusion coeffi-}

cients D_α that are then scalar and independent of composition;

foreach phase

α

^{, thediffusioncoefficient should beidentical for}

all species to preserve the consistency of system (Hoteit, 2011).

In thiswork, the dispersion term isneglected. Consequently, the diffusion-dispersion tensorD_α transformsinto theeffectivediffu- sionD^e_α=

φ

^SαD_α.R_α,kisthereactionterm, andq_α,k istheexter- nal sourcetermof speciesk influid phase

α

^{. The} Darcy–Muskat law(Muskatetal.,1937)statesthat

u_α=−krα

μ

αK

( ∇

^pα−

ρ

αg

)

^{, (2)}

wherekrα,

μ

α andp_α are therelativepermeability, viscosityand pressureoffluidphase

α

^,respectively.Kistheintrinsicpermeabil- itytensor,andg isthegravitationalaccelerationvector. Thepres- suresareconnectedbyN_f−1capillarypressurerelations,andthe saturationsandmassfractionssumto1:

α

S_α =1, (3)

k

X_α,k=1. (4)

Thus, an additional 2N_f constitutive relations are generated. The conventional PDE systemofisothermal compositional multiphase flow can be expressed using (1) by deriving the conservation of each species k in all phases

α. These Nc nonlinear equations, 2N_f constitutive relations andNc

(

^Nf−1

)

^phase equilibrium rela- tions(Section2.3) yield2N_f+N_cN_f equations for2N_f+N_cN_f un- knowns p_α, S_α andX_α,k. According to the Gibbs phase rule, the number of intensive properties is Nc−Np+2. If the number of phases is locally known, then N_p=2 for fixed temperature, and at least Nc nonlinear equations (e.g., pressure and composition (Jindrová and Mikyška,2015)) canbe solved to establish the ther- modynamicstate.Notethatthegeochemicalsystemcanbeabun- dantinspecies,implyingthatconsiderablylargenonlinearsystems mustbesolved.

Inthiswork, we propose handlinga phasemass conservation system of N_f PDE whose size is independent of the number of chemicalspeciesNs.Weapply

kto(1)toobtain

∂ ( φ

^Sα

ρ

α

)

∂

^{t =−}

∇

^·

( ρ

αu_α

)

+R_α+q_α, (5) takingintoaccountthatthesumofdiffusivefluxesforeachphase

α

^{isequalto0becausethediffusioncoefficientsD}α areequalover all components in each phase. The external source term q_α can alsobepresentedas

q_α=

ρ

α

ψ

_α^{, (6)}

where

ψ

α isthe volumetricrate. Thesystem (5)ofN_f equations, N_f−1capillarypressurerelationsand(3)isassembledfor2N_fun- knowns,p_αandS_α,whicharenaturalvariablesofmultiphaseflow problems.

When one of the phases disappears, the corresponding equa- tion of system (5) degenerates, and the natural variables are no longerappropriatefordescribingthesystem.Inthiscase,wemove tothesingle-flowproblem.Numerousformalismsarededicatedto thetwo-component, two-phase flow and associatedphase disap- pearance/appearance(AbadpourandPanfilov,2009;Angelinietal., 2011;Bourgeatetal.,2013;Lauseretal.,2011;Massonetal.,2014;

Pruessetal., 1999). In thiswork, the liquidphase isassumed to be present throughout the system, even if it is present in small amounts. The liquid pressure and gas saturation are chosen as primary variables. Therefore, solving the system of N_f nonlinear Eq.(5)canbe beneficial when Nc > N_f.Althougha semi-implicit methodIMPEC is usedto reduce thenumber ofnonlinear equa- tions to be solved simultaneously (Hoteit and Firoozabadi,2006;

Mikyška and Firoozabadi, 2010), time stepping is constrained by theCFL condition. Alternatively,an implicit methodofdecoupled systemhasbeen proposed inZidane andFiroozabadi (2015). The authorssolveimplicitlythespeciestransportcoupledwiththeto- talfluxcalculatedbythemixedfiniteelementmethod.

2.2.Moleconservationforeachgascomponentandprimaryspecies

TheliquidandgasphasesconsistofN_c primaryspeciesandN_g gas species, respectively, Ng < Nc, for which the transport must besolved.Wechosethetransportformulationintermsofconcen- trationsc_α,k∝

ρ

αX_α,k/M_k,where M_k is themolarmass ofspecies k. Deriving the transport equations in mole/massfractions, as in Eq.(1),thedensitydeviation

∇ρ

α/

ρ

α thatarisesfromthediffusive partoffluxisneglected. Basedonthe primaryspeciesformalism (vanderLee, 2009;Lichtner, 1996;SteefelandMacQuarrie,1996;

YehandTripathi,1991),theliquidtransportisdefinedforthetotal

liquidmobileconcentration c_l,kof primaryspecies kandthe gas transport forthe gas concentration cg,m ofgas species m, which yieldsNc+Ngtransport(linear)equations:

∂φ

^Slc_l,k

∂

^{t =−Tl}

c_l,k

+R_l,k

c_l,k,c_s,k

+q_l,k, (7)

∂φ

^Sgcg,m

∂

^{t =−Tg}

⁽

^cg,m

⁾

^+Rg,m

⁽

^cg,m

⁾

^{+qg,m, (8)}

wherec_s,kistheimmobileconcentrationofspeciesk,operatorT_α includestheadvectivefluxpresentedbytheDarcy–Muskatlaw(2), andassuming non-Knudsendiffusion, Fick’slawfordiffusive flux yields(Lichtner,1996;deMarsily,2004):

T_α

c_α,k

=

∇

·

(

^cα,ku_α−D^e_α

∇

^cα,k

)

, (9) whereD^e_α caninvolvethetortuositycoefficient

τ

α(Millingtonand Quirk,1961).

2.3.Molebalanceforeachprimarybasisspecies

The chemical equations arise from the mass action law and phase equilibrium relations. As mentioned above, the reactive transport code isbased on the primary speciesformulation, and thus,we denotetheconcentration ofspeciesS_j, j=1,...,Ns,that canbeexpressedasafunctionofbasisspeciesC_i,i=1,...,Nc: Sj

Nc

i=1

α

^{i jCi, (10)}

where

α

ijisthestoichiometriccoefficient.Atequilibrium,themass actionlawprovidesthereactionaffinity:

Sj=K_j

γ

^j

Nc

i=1

( γ

^iCi

)

^{αi j, (11)}

where

γ

j isthe activity coefficient andK_j isthe thermodynamic equilibriumconstant ofreactionj.All aqueousreactions are con- sideredtobeatequilibrium.Foreachbasisspecies,themolebal- ance can be written in terms of the total concentration T_i, i= 1,...,N_c,T_i=N_s

j=1

α

jiS_j, T_i−C_i−

Ns

j=1,j=i

α

ji

Kj

γ

j Nc

i=1

( γ

iC_i

)

^{αi j=0, (12)}

constitutingasystemofNcequationsonC_i,i=1,...,Nc.

2.3.1. Liquidmixtures

In non-ideal liquid mixtures, the activity coefficients

γ

i are not trivial and can be calculated using different models whose complexityincreaseswiththeconcentrationofsolution,fromless tomoreconcentratedsolution: truncatedDaviesformula(Colston etal.,1990),B-dot(Helgeson,1969),SIT(Grentheetal.,1997),and Pitzer(1991).

2.3.2. Gas-liquidequilibrium

Thechemicalpotentialsandcorrespondingfugacitiesofspecies i in mixture f_i^α are equal under equilibrium conditions. The fugacity-activity(

ϕ

−

γ

^{)approachyields}

Pyi

ϕ

_i^g=fi=K_i^h

γ

ixi, (13) where y_i is the mole fraction of species i in the gas phase,

ϕ

_i^g

is the fugacity coefficient of species i, K_i^h=K_i^h

(

^T,P

)

^{is the cor-} rectedHenry’sconstant ofspeciesi,

γ

iistheasymmetric activity coefficient ofspecies i (Michelsen andMollerup, 2007),and x_i is themolefractionofspeciesiintheliquidphase.K_i^hincludes the Poyntingfactor,whichcorrectsthereferencefugacitywithrespect to the pressure and is near unity at low to moderate pressures.

The fugacity coefficients can be calculated using the equation of state(EOS).WechosethePeng–RobinsonEOS(RobinsonandPeng, 1978),whichprecisely reflectsthe gasmixturepropertiesathigh pressures/temperatures. For a given P, the cubic equation should besolved forthecompressibilityfactorZ=P

v

/RT,wherevisthe molarvolume.Thefugacitycoefficientisthencalculatedasfollows (RobinsonandPeng,1978):

ϕ

_i^g=

ϕ

_i^g

(

^T,P,Z,T_c,P_c,Z_c,

,

)

,where Tc,Pc,andZcarethesetsofthecriticaltemperatures,criticalpres- suresandcompressibilityfactorsofthespeciesinthemixture,re- spectively.

^{istheacentricfactorset,and}

^{isthematrixofem-} piricalbinaryinteractionparametersforeachpairofspeciesinthe mixture.

TheEOSalsoprovidesthemassdensityforgasmixtures:

ρ

^g=

Ng i=1yiMi

v

⁼

M¯

v

^{, (14)}

and,byanalogy,forliquidmixtures:M¯=N_c i=1x_iM_i.

NotethatRaoult’slaw(Py_i=P^satx_i)isderivedfromEq.(13),as- suminglowpressureconditionsandidealsolutions.

2.3.3. Liquid-solidequilibriumandkineticrelations

For minerals under the thermodynamic equilibrium assump- tion,thesolubilityconstantK_s,jcanbederivedfromthemassac- tionlaw.The mineralactivityisunityby convention.Byomitting theindexj,Eq.(11)yieldstheexpressionfortheequilibriumsolu- bilityconstantforsolids

Ks=

Nc

i=1

( γ

iC_i

)

^{αi j}. (15)

Whenthereactions arekineticallycontrolled,theratelawcan be described, e.g., by the transitionstate theory (Lasaga, 1984). The model uses the ion activity product Qs, which by definition be- comesKsatequilibrium,andthesaturationindex(SI):

SI=log

_Qs

Ks

= <0 undersaturated→dissolution,

=0 saturated→equilibrium,

>0 oversaturated→precipitation.

(16)

Underthetransitionstatetheory,thekineticratelawisthen

dS dt =Ask_kin

sign

(

^SI

)

_Q

s

Ks

^b1

−1

b2

k

aⁿ_k^k (17)

where As is the specific surface area; k_kin is the kinetic dissolu- tion/precipitationrateconstant;b₁,b₂,andn_kare fittingparame- ters;anda_kistheactivityofthepotentialcatalyzingorinhibiting species.

Themolebalanceequationswiththemassactionlawsforthe speciesat equilibrium, theratelaws forkinetically limitedsolids and phase equilibrium relations constitute a complete system of algebro-differentialequationswhosesolutionyieldstheconcentra- tionsofbasisandsecondaryspecies.Otherchemicalreactionscan alsobehandledby theformalismofbasis species,e.g.,cationex- changeandsurfaceandorganiccomplexation.

3. Numericalsolution

Applying the OSA to subsurface environmental modeling en- ablestheindependentdevelopmentofseparatedmodulesofcode andtherigoroussolutionofeach.Consequently,themajorityofRT simulators rely on the OSA(Steefel etal., 2014). The assessment ofdifferentcouplingmethodswasstudiedfortheMoMaSbench- markofRT codes(Carrayrouetal., 2010;Carrayrou andLagneau, 2007),duringwhichthereliabilitiesofboththeSIAandGIAwere determined, and the detailed results of HYTEC were reported in LagneauandvanderLee(2010a).Followingthegeneralstrategyof integratingthe(un)saturatedflow inRTcodes byOSA,we solved

thecompressibletwo-phase flowblockfirst.Thisprocessinvolved theflowsystem,thegastransportequations,andtheEOSandfluid property models. Then, reactive transport coupling was applied.

WeproposeemployingSIAforeachmodule,andhence,twointer- nalSIA-basedcouplingsexist.Letusdescribetheappliedmethods forflowandtransportdiscretizationandthesubsequentcoupling methods.

3.1. Discretizationofflowandtransport

The discretization of mass phase conservation (5) and mole species transport (7),(8) isconstructed based on a Voronoi-type finitevolume method.The timeapproximation oftwo-phase flow (5) is fully implicit, and the fluxes are handled by TPFA. In the 1960s, thetransmissibilitydiscretization wasdemonstratedto af- fect the numerical stability and accuracy (Allen, 1984; Blair and Weinaug, 1969; Settari and Aziz, 1975); in this work, the inter- face coefficients of flow between adjacent cells were evaluated implicitly and upstream. Forthe relative permeability(krα)_ij, the upstream spaceapproximationiswidely usedbecauseits conver- gence wasconfirmedfortheBuckley–Leverett problem(Aziz and Settari,1979;Bastian,1999).Thedetailedcomparisonoftemporal discretizationpresentedinAzizandSettari(1979);BlairandWein- aug (1969) demonstrated the stability advantagesof the implicit upstream treatmentrelative toexplicitonesbutalsoreportedin- creasedtruncationerrors.Wewilldiscusstheimpactoftruncation errorsinSection4.1.Therelativepermeabilitykrα,intrinsicperme- abilityK,phasedensity

ρ

α,andphaseviscosity

μ

α oftheinterface coefficientsbetweenadjacentcellsi,j atiterationk+1arethere- foredefinedas

(

·

)

^k_{i j}⁺¹=

₍

·

)

^k_i ^ifuk_αⁿi j>0,

(

·

)

^k_j ^{else (18)}

wheren_ijisthenormalvectorfromitoj.

The discretizationofdensity

( ρ

^gα

)

i j in thegravityterm

ρ

αg is treateddifferentlyandweightedrelativetotheeffectivephasevol- ume. If thegas phase isabsent in one ofthe cells, thenthe up- streamtreatmentisapplied(Coats,1980):

( ρ

α^g

)

i j=

⎧ ⎨

⎩

ρ

α,i

σ

+

ρ

α,j

(

¹−

σ )

^if

(

^Sα,i>0

)

∧

(

^Sα,j>0

)

^:

σ

^=Vα,i/

(

^V_α,i+V_α,j

)

;

( ρ

α

)

i j else

(19)

whereV_α=

φ

^SαV_totandV_totisthevolumeofthecell.Theresulting nonlinear system can be solved using Newton’smethod with an analyticalJacobian,aspresentedinSection3.2.

The space discretization of transport operators (7) and (8) is achieved byupstream weighting for advectiveflux andharmonic weightingforeffectivediffusion.Thesemi-implicitmethodischo- senforthetimediscretization:theimplicitEulerschemeisapplied forthediffusiveflux,whereastheadvectivepartisdiscretizedus- ingtheCrank–Nicolsonmethod.

3.2. Coupling1:compressibletwo-phaseflow

Because the phase flow system (5) is nonlinear, the classic Newton’smethodisapplied.Wedenote thediscretizedEq.(5)as F

(

^x

)

=0,wherex=

(

^pl,S_g

)

^{.Whenthefluidsarehighlycompress-} ible(e.g.,thegasphase),thedensitypropertiesshouldbeprecisely evaluated ateach deviationoftheintensive variablesP, V, andn, where n is the quantity of matter. To ensure the implicit treat- ment ofinterfacecoefficients, the densityis updatedin Newton’s loop,similar totheother flowparameters.However, thegasden- sitymaybe stronglydependentonitscomposition. Therefore,the gas composition must be calculated by employing the gastrans- port(8)denotedbyTg

(

^cg;x

)

=0.Thisisparticularlyimportantfor

modelingthegasappearanceanddisappearance.Theflowcoupling algorithmfortimestepn+1ispresentedinAlgorithm1,andits parsingisgivenbelow.

Algorithm1 Newton’smethodforflow.

1:

ε

N f=1×10⁻⁶ 2:

ε

^qss=1×10⁻²⁴ 3: k_max^{f l} =9 4: k=0

5: while

^F

(

^xk

)

∞≥

ε

N f

^F

(

^x0

)

∞

∧

^F

(

^xk

)

∞≥

ε

qss

∧

k≤k_max^{f l}

do

6: find

δ

^xk+1:J

(

^xk

) δ

^xk+1=−F

(

^xk

)

7: x^k+1←x^k+

δ

^xk+1

8: findc^k_g⁺¹:Tg

(

^ckg⁺¹;x^k+1

)

=0 9: updateEOSandphysicalparameters 10: k←k+1

11: endwhile

The user-defined or default tolerance

ε

Nf (Section 3.4.3) and maximumnumberofNewtoniterationsk_max^{f l} fortheflowcoupling areinitializedfirst.Next,thelinearizedsystemissolvedforthein- crement

δ

^{xatline6.Allthepartial}derivativesinvolvedintheJa- cobianJofthediscretizedflowsystemareanalytical.Forexample, thederivative ofthegasdensityinterface coefficientis expressed as

∂ ( ρ

^g

)

i j

∂

^pg,i =

∂ ( ρ

^g

)

i j

∂ρ

g,i

∂ρ

g,i

∂

^pg,i,

∂ρ

g,i

∂

^pg,i =− M¯i

v

²_i

₍ ∂

^pg/

∂ v ₎

i

, (20)

where

∂

^pg/

∂

^{v is the analytical derivative arising fromthe corre-}

sponding EOS. Note that the density derivatives are composition dependent and proportional to the average molecular weight M¯ andthe densityfunction Eq.(14).Variousmethodsexistforsolv- ingmultiphasesystemsoflinearequations(Chenetal.,2006);we apply GMRES(Saad andSchultz, 1986),which isone ofthemost prevalent and efficient methods, with ILU(0) (van der Vorst and Meijerink,1981)asapreconditioner.

Usingthevelocitiesandsaturationsgivenbyx^k+1 fromstep7, thelineartransportsystemTg

(

^ckg⁺¹;x^k+1

)

=0issolvedforc^k_g⁺¹at line8withGMRESandILU(0).Becauseofthe modifiedcomposi- tion,theEOSparametersmustbeupdatedtoevaluateanewmolar volumevbysolvingtheEOSanalytically.Then, thephysicalprop- erties can be calculated at line 9. Thereafter, three stop criteria mustbe checkedatline 5:two forthe flow systemresidualand oneforthenumberofiterations.

TheproposedcouplinginAlgorithm1correspondstoNewton’s familyfortheflowsystemregardingx^k+1,whereasitcanbecon- sideredasPicard’s method(fixed-pointmethod)forthetransport equationsonc^k_g⁺¹.Giventhatthegasphaseissupposedtobecom- pressible and that significant differences in gas density may oc- curthroughoutthemodeleddomain,anadditionalcriterionforgas quantityngdeviationcanbeincluded:

max

n^k_g⁺¹−n^k_g

n^kg⁺¹

≤

ε

^g, (21)

wheremax isthemaximumvalueoverthemodeleddomain.After numeroustests, wededucethatthismaynotbeanecessarycon- ditionbutissufficienttofinishtheloopthatdependsonthecom- plexityofgasdynamics.Despiteneglecting thecriterion(21),the solution’s accuracyis not lacking. In addition, the reactive trans- portcouplingdescribedinSection3.3,whichfollowstheflowcou- plinginAlgorithm1,entailstheconvergence(stop)conditionsfor the gasand solid phasesand guarantees the conservationof the entiresystem.