RADIOACTIVE WASTEMANAGEMENTAND DISPOSAL

KEYWORDS: Yucca Mountain,Pitzer model, dryout

MODELING REACTIVE MULTIPHASEFLOW AND TRANSPORT OFCONCENTRATED SOLUTIONSGUOXIANG ZHANG,* NICOLAS SPYCHER, ERIC SONNENTHAL,CARL STEEFEL, and TIANFU XU Lawrence Berkeley National Laboratory1 Cyclotron Road, MS 90-1116, Berkeley, California 94720

Received November 29, 2006Accepted for Publication January 8, 2008

A Pitzer ion-interaction model for concentrated aque-ous solutions was added to the reactive multiphase flowand transport code TOUGHREACT. The model is de-scribed and verified against published experimental dataand the geochemical code EQ3/6. The model is used tosimulate water-rock-gas interactions caused by boilingand evaporation within and around nuclear waste em-placement tunnels at the proposed high-level waste re-pository at Yucca Mountain, Nevada. The coupled thermal,hydrological, and chemical processes considered consistof water and air/vapor flow, evaporation, boiling, con-densation, solute and gas transport, formation of highlyconcentrated brines, precipitation of deliquescent salts,generation of acid gases, and vapor-pressure loweringcaused by the high salinity of the concentrated brine.

I. INTRODUCTION

Aqueous solutions are generally defined as concen-trated when their ionic strength ~I ! is greater than 1 molal~mol0kg H2O!. Such solutions may result from many nat-ural and artificial processes, such as evaporation,1 boil-ing,2 seawater intrusion,1,3,4 leakage of toxic solutionsand electrolytic fluids from storage tanks,5–7 and acidmine drainage.8 Concentrated aqueous solutions are sig-nificantly different from diluted solutions in terms oftheir geochemical behavior because the thermodynamicactivities of both water and solutes deviate from idealbehavior.1,3,9,10 In this paper, we discuss the implemen-tation of the Pitzer ion-interaction model1,3,9,10 into theTOUGHREACT existing reactive transport simulator11

for the purpose of modeling the geochemical reactivetransport of concentrated solutions and the relevant gasesat the proposed high-level nuclear waste repository atYucca Mountain, Nevada.

TOUGHREACT ~Ref. 11! was originally developedby integrating geochemical transport into the frameworkof the TOUGH2 multiphase flow code.12 It can simulatenonisothermal multiphase groundwater flow, diffusiveand advective transport of gases ~including vapor! andsolutes, aqueous speciation, and mineral dissolution andprecipitation under both equilibrium and kinetic con-straints. These processes can be simulated in complexflow and transport systems, such as in variably saturatedenvironments and multiple porosity0permeability media~e.g., rock matrix-fracture systems!. TOUGHREACT hasbeen used to simulate a number of reactive geochemicaltransport processes at various scales and over a widerange of geochemical conditions.11 Prior to the presentstudy, this code could only handle diluted to moderatelyconcentrated solutions ~I, 1 molal! and moderately con-centrated NaCl-dominant solutions ~typically I, 2 molal!by using an extended Debye-Hückel ionic activity model~i.e., Helgeson-Kirkham-Flowers, or HKF, model!.13 Theimplementation of the Pitzer ion-interaction model nowallows the application of this simulator to problems in-volving more concentrated aqueous solutions, such asthose involving geochemical processes in and aroundhigh-level nuclear waste repositories where fluid evapo-ration and0or boiling is expected to occur.

The Pitzer ion-interaction model, which we definehere as the Pitzer virial approach and associated ion-interaction parameters,9,10 has been applied successfullyto study nonideal concentrated aqueous solutions.4–7

Pitzer’s original formulation and parameters were re-arranged by Harvie, Moller, and Weare1,3 to model sea-water systems ~Na-K-Mg-Ca-Cl-SO4-H2O!, leading to amore practical formulation ~see the Appendix!. This re-arranged formulation ~here referred to as “HMW”! andits parameters differ from Pitzer’s original model in terms*E-mail: gxzhang@lbl.gov

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of mathematical expressions and parameter values, butnevertheless express the same theory ~and results! asPitzer’s original work. It should be noted that for com-plex systems, ion-interaction parameters are often morereadily available for the HMW formulation than forPitzer’s original formulation, although parameters for ei-ther formulation can be converted from one to the other.14

Pitzer’s original formulation was implemented in theEQ306 geochemical reaction code15,16; however, this codereads interaction parameters for the HMW formulation,which are then internally mapped into Pitzer’s originalformulation. The HMW formulation has been imple-mented in several geochemical and reactive transportcodes, including PHRQPITZ ~Ref. 17!, GMIN ~Ref. 18!,UNSATCHEM-2D ~Ref. 19!, and BIO-CORE2D©

~Refs. 7, 20, 21, and 22!, using various sources of ion-interaction parameters. In this paper, we present the im-plementation and verification of the HMW formulationinto TOUGHREACT, using a database of ion-interactionparameters developed by others14,23–25 for specific ap-plications to nuclear waste–related problems. We alsodemonstrate the code’s capability to handle reactive trans-port processes of concentrated aqueous solutions ~brines!involving dissolution0precipitation of minerals and salts~including deliquescent salts!, generation of acid gasesfrom the brines at varying temperatures, dryout causedby boiling and evaporation, and vapor-pressure loweringcaused by high salinity. This modeling capability is use-ful for the safety assessment of underground high-levelradioactive waste repositories.

II. ION-INTERACTION PARAMETERS

In this paper, we make use of the EQ306 databasedata0.ypf ~Refs. 14, 23, 24, and 25! for both Pitzer ion-interaction parameters and thermodynamic equilibriumconstants. A dependence on temperature is considered, asdescribed below, whereas the effect of pressure is notbecause it is much less significant, and because the pres-sure range considered here is near atmospheric.

The interpolation and extrapolation equations for var-ious thermodynamic properties of aqueous solutions as afunction of temperature, for binary and ternary systems,and for multiple-component mixtures within the Pitzerformulation have been reported in many papers,3,7,9,10,26–32

covering various ranges of temperatures depending onthe system considered. In the present paper, we make useof the temperature-dependence formulation implementedin the EQ306 Pitzer database14,23–25:

P~T ! � a1 � a2� 1

T�

1

T0�� a3 ln� T

T0�

� a4~T � T0 ! , ~1!

where P~T ! represents Pitzer parameters ~see the Appen-dix! b~0!, b~1!, b~2!, a, F, C, and CMX

F at temperature T~absolute temperature! and T0 is the reference tempera-ture ~298.15 K!. These data provide good results up toat least 908C and ionic strengths in the 20 to 40 molalrange for chloride-dominant waters in the context ofevaporation0boiling studies at Yucca Mountain.25 Moredetails on this database can be found in Refs. 23, 24,and 25.

Because the effects of ion pairing and aqueous com-plexation are generally taken into account by the ion-interaction parameters, much care must be taken to avoid“double counting” these effects by including only thoseion pairs and complexes that were considered in derivingthe ion-interaction parameters.

III. TESTS

The code was verified using experimental data andbenchmarked against EQ306 ~Ref. 16!. The followingare three selected test cases that show how the code re-produces experimental data, how it matches the resultscalculated by EQ306, and how it captures the water-vapor-pressure–lowering effects of saline solutions resultingfrom high salinity.

The first test calculates the mean activity coeffi-cients of NaCl and the osmotic coefficient of NaCl solu-tions up to 6 molal, at 0, 25, 50, 80, 100, and 1108C,respectively. The calculated mean activity coefficientsare compared with measured data in Fig. 1 ~Ref. 33!.This test validates the temperature dependence of theactivity coefficients calculated in the model. Note thatcomparisons of mean activity coefficients, rather thanmean activities, are appropriate here, because there areno Na and Cl species other than the free ions included inthe simulations ~i.e., the effect of NaCl ion pairing isaccounted for by the activity coefficients alone, and notby a separate NaCl secondary species!. The mean activ-ity coefficient of NaCl, gNaCl, is defined as

ln~gNaCl! �ln~gCl!� ln~gNa!

2. ~2!

Another test case involved the calculation of the meanactivity of CaCl2, osmotic coefficient, water activity, andwater-vapor pressure of CaCl2 solutions at ionic strengthup to 27 molal ~9 molal of CaCl2!. The calculated os-motic coefficient at 608C and mean activity of CaCl2were compared with literature data34 in Fig. 2 and showreasonably good agreement. The osmotic coefficient iscalculated, according to Eq. ~A.1!, as

f � �1000 ln~aw !

Ww(i

mi

. ~3!

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The differences between the simulation resultsand the data at extremely high concentration ~close tosaturation of the salt! could be caused by the peculiarbehavior of this salt at high concentrations,35 possibleuncertainties in the measurements ~possible precipita-tion of solid phases when the solution is concentrated upto saturation with respect to the salt!, and0or uncertain-ties in ionic interaction parameters.

Vapor-pressure lowering caused by dissolved saltswas implemented directly through the water activity com-puted with the Pitzer ion-interaction model.36 For equi-librium between water and H2O vapor ~i.e., for the reactionH2O~l ! ? H2O~g! !, equating the chemical potentials ofboth phases yields

mv0 � mw

0 � RT ln~ fv 0fv0!� RT ln~ fw 0fw

0!

� RT ln~ fv 0aw !� RT ln~K ! , ~4!

where

w � liquid water

v � H2O gas

m0 � reference chemical potential

f � fugacity

a � activity ~defined as f0f 0, with f 0 being thefugacity in the reference state!

K � thermodynamic equilibrium constant

R � gas constant

T � temperature.

The reference ~standard! state of H2O gas is taken as theunit fugacity of the pure gas at pressure of 1 bar and alltemperatures, whereas that of liquid water is taken as the

Fig. 1. Comparison of TOUGHREACT-calculated ~solid lines! and measured33 ~symbols! mean activity coefficients for NaClsolutions at ~a! 258C and ~b! 1108C.

Fig. 2. Comparison of TOUGHREACT-calculated and measured ~a! osmotic coefficient34 of CaCl2 solutions and ~b! meanactivity coefficient34 of CaCl2 ~right!, at 608C and concentrations up to 9 molal of CaCl2 salt ~I � 27 molal!.

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unit activity of pure water at all temperatures and pres-sures. In our case, at low pressure ~atmospheric!, fugac-ity is approximated by pressure, and fv and thus K inEq. ~4! are respectively approximated by the actual vaporpressure Pv and the vapor pressure of pure water Psat

0 suchthat Pv0aw � Psat

0 ~thus, for the pure system, aw � 1 andPv � Psat

0 !. Accordingly, the vapor pressure of the solu-tion is computed as

Pv � aw Psat0 . ~5!

Equation ~5! is used in the coupling of chemistry andflow calculations, which arises primarily through the ef-fect of salt concentrations on vapor pressure in the multi-phase flow computations. From Eq. ~5!, it is also apparentthat if relative humidity Rh is defined as the ratio of theactual vapor pressure to that of pure water, then

Rh � aw . ~6!

The implementations of Eqs. ~5! and ~6! were veri-fied by taking the vapor pressure of pure water from theNational Institute of Standards and Technology steamtables37 and then calculating the vapor pressure of thesolution using these equations and the water activity cal-culated by the Pitzer ion-interaction model. Simulatedvapor pressure and relative humidity values for solutionsup to 9 molal CaCl2 agree well with the values calculatedfrom the steam tables ~Fig. 3!.

The third test case involves calculating the activitycoefficients of Ca�2, Na�, and Cl� in mixtures of CaCl2and NaCl solutions up to 6 molal CaCl2 and 6 molalNaCl, yielding ionic strengths up to 24 molal at 258C. Wecompare the TOUGHREACT results against EQ306 re-sults. Both codes use the same chemical setup, sameinput data, and same thermodynamic databases; a com-parison of individual activity coefficients, rather thanactivities, is appropriate because the same database is

Fig. 3. ~a! Comparison between hand-calculated and simulated vapor pressures, and ~b! comparison between calculated relativehumidities and water activities of CaCl2 solutions up to 9 molal of CaCl2 salt at 258C, 1 bar.

Fig. 4. Comparisons between TOUGHREACT-calculated ~solid lines! and EQ306-calculated ~symbols! individual activity coef-ficients at 258C @~a! Ca�2 and ~b! Na�# in aqueous mixtures of NaCl and CaCl2 with ionic strengths up to 24 molal.

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used in both cases, and specific ~separate! ion pairs arenot included in the calculations. Figure 4 shows that theresults match well.

IV. EXAMPLE SIMULATIONS OF BOILING AND

EVAPORATIVE CONCENTRATION IN AND AROUND

NUCLEAR WASTE EMPLACEMENT TUNNELS

The proposed U.S. high-level nuclear waste reposi-tory at Yucca Mountain, Nevada, is located in unsatu-rated volcanic tuffs several hundred meters above thewater table. The radioactive decay from the spent fuelwill heat the rock around waste emplacement tunnels anddepending on the areal loading of spent fuel, and venti-lation design, the temperature could rise above the boil-ing point of pure water for several hundred years.38,39

Under such conditions, water moisture in the surround-ing rock ~i.e., porewater in the unsaturated volcanic tuff !will boil within several meters of the walls of emplace-ment tunnels. Accordingly, a dryout zone will developbetween the boiling front and tunnel walls during theheating period.38,39 In the dryout zone, a tiny amount ofconcentrated aqueous solutions ~brines!may remain as aresult of vapor-pressure lowering, and some salts mayprecipitate. Upon evaporative concentration of the rockmoisture, some acid gases could be volatilized from the

residual brines. These and other coupled thermal, hydro-logical, and chemical processes ~Fig. 5! may affect theintegrity of waste packages and therefore need to be care-fully evaluated as to their impact on the safety of theproposed repository.

The TOUGHREACT-Pitzer reactive transport simu-lator was applied to investigate some of these processes,as shown below with two example simulations: ~a! theboiling and evaporative concentration of porewater ~rockmoisture! in contact with an open atmosphere and with-out buffering effects from rock minerals and ~b! a two-dimensional reactive transport simulation depicting thethermal, chemical, and hydrological evolution of the dry-out zone around a typical waste emplacement tunnel.

IV.A. Evaporation/Boiling Without

Water-Rock Interactions

Here, a dynamic model is developed to simulate theevaporation of a Yucca Mountain tuff porewater open tothe atmosphere. The simulation is set up as if the waterwas heated in an open beaker, thus allowing for exchangeof gases between the solution and the atmosphere. Thebeaker is represented by a partially liquid saturated modelgrid block of a given volume, and is connected to a sec-ond air-filled infinitely large grid block representing theatmosphere. The grid block representing the beaker ismaintained at a constant temperature of 958C, the boiling

Fig. 5. Schematic illustration of geochemical processes expected to take place in and around a typical nuclear waste emplacementtunnel at the proposed Yucca Mountain repository.

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temperature of pure water at the elevation of the pro-posed repository. The vapor and gases generated in thebeaker grid block are allowed to advect and diffuse intothe grid block representing the atmosphere ~at ;1 bar!,which is set to a fixed atmospheric CO2 partial pressureof 10�3.5 bar. This setup captures the chemical processesthat would accompany the evaporation of seepage in thehot, open emplacement tunnels without contacting hostrock ~Fig. 5!. Note that this setup cannot capture precisegradients at the liquid-atmosphere boundary, and is onlyintended to illustrate chemical effects in the beaker gridblock during rapid evaporation under conditions whereinitial CO2 volatilization from the solution is not imme-diately replenished by diffusion of CO2 from the atmo-

sphere. This setup, therefore, differs from the work ofothers25,40 who consider a constant CO2 partial pressurein the solution.

The water used in this simulation ~Table I! repre-sents porewater extracted from rock core samples col-lected from borehole ESF-PERM-3 ~Ref. 2! near thelocation of the Drift Scale Test ~prior to the test! in theEngineered System Facility ~ESF! at Yucca Mountain.This porewater is characterized by a higher calcium andsulfate content than is found in porewaters sampled atother locations in the repository host rocks and producesa slightly acidic brine when it is extremely concentrated,in agreement with observations by others.25

The code’s capability allows simulating evaporationuntil near dryness at concentration factors . 106 andwater activity values essentially nil. In the real system,however, evaporation would stop when the brine wateractivity reaches the relative humidity of the prevailing airin the tunnel @Eqs. ~4!, ~5!, and ~6!# . Also, at the proposedYucca Mountain repository, in-drift relative humiditiesare expected to remain mostly above ;0.2 ~Ref. 41!. Inaddition, ion-interaction parameters input in the simula-tions are valid only up to a certain range of ionic strength~in our case to maximum ionic strength values in therange of 20 to 40 molal!. For these reasons, the computedchemical evolution of dissolved species, solids, and gasesare shown ~Figs. 6 through 9! only down to water activ-ities around 0.2 ~at a concentration factor;105 and ionicstrength within the validity range of the thermodynamicdata!.

The remaining brine at a water activity near 0.2 ismildly acidic ~pH of 5.5, Fig. 6!, with acid gas fugacitiesreaching 10�8 bar for HF and HNO3 and 10�7 bar forHCl ~Fig. 9!. Predicted major solid precipitates ~volumefraction .1%! are halite ~NaCl!, calcite ~CaCO3!, an-hydrite ~CaSO4!, amorphous silica ~SiO2!, sepiolite~Mg4Si6O15~OH!2:6H2O!, and sylvite ~KCl! ~Fig. 8!. Thepredicted major mineral phases are in relatively good

TABLE I

Chemical Composition of the Solution Used inSimulations of Evaporative Concentration*

Components Molality

Ca�2 2.5 � 10�3

Mg�2 6.9 � 10�4

Na� 2.6 � 10�3

Cl� 3.3 � 10�3

SiO2~aq! 1.1 � 10�3

HCO3� 3.2 � 10�3

SO4�2 1.2 � 10�3

K� 2.0 � 10�4

Al~OH!4� 6.2 � 10�10

F� 4.5 � 10�5

HFeO2 1.2 � 10�12

NO3� 1.0 � 10�4

Ionic strength 0.013pH 8.3

*See Sec. IV.A.

Fig. 6. Computed evolution of ~a! water activity and ~b! brine pH.

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agreement with experimental and modeling results fromothers25 using a similar water composition. Note that inthis simulation, sepiolite ~a magnesium silicate! and amor-phous silica are reacting under kinetic constraints, whereas

other minerals react at equilibrium. Sellaite ~MgF2! even-tually forms because the precipitation of sepiolite is ki-netically retarded relative to amorphous silica. However,in the real system, an amorphous magnesium silicate

Fig. 7. Evolution of ~a! cation and ~b! anion concentrations in solution ~as total dissolved concentrations!, and total amounts of~c! cations and ~d! anions per kilogram of initial water, as a function of the concentration factor ~see Sec. IV.A!.

Fig. 8. Evolution of precipitated solids ~mol0m3 medium!.Halite, calcite, and anhydrite dominate the saltassemblage.

Fig. 9. Evolution of gas partial pressures. Acid gas pressuresincrease as the solution is concentrated and pH de-creases ~see Sec. IV.A!.

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phase would likely precipitate in place of, and fasterthan, sepiolite42 and deplete magnesium from the solu-tion so that other magnesium salts may not form.

Evaporative concentration and the precipitation ofmajor solid phases control the concentration profiles ofdissolved species. As a result, the concentration profilesof dissolved species ~Fig. 7! show breaks in slope whenminerals start to precipitate ~Fig. 8!. It should be notedthat the initial solution ~Table I! is supersaturated withrespect to calcite, which forces calcite precipitation priorto the start of evaporation ~at concentration factor � 1!,driving the pH down ~Fig. 6! and, accordingly, drivingthe partial pressures of CO2 and other acid gases up~Fig. 9!. The sharp calcium concentration drop ~Fig. 7! atthe start of the simulation ~at concentration factor �1! isalso caused by this initial precipitation of calcite. Onceevaporation starts, however, pH is driven up by CO2volatilization ~e.g., HCO3

� � H� ] CO2~g!F � H2O!,which further induces calcite precipitation. Eventually,the effects of evaporative concentration take over theeffect of CO2 volatilization and the pH trend reverses~Fig. 6!. As the brine is concentrated, calcium concen-trations rise until anhydrite precipitation becomes themajor process removing calcium from the brine. At thispoint, the rate of increase in calcium concentration sharplydrops, although the calcium concentration still slowlyincreases as calcite dissolves owing to the decreasingpH. Further dissolution of calcite and precipitation ofanhydrite ~Fig. 8! cause the sulfate concentrations to dropas calcium continues to increase ~Fig. 7!.

As mentioned above, the simulation is set up as ifwater was heated in a beaker in direct contact with theatmosphere. The atmosphere is simulated as a gas reser-voir of constant pressure ~;1 bar! with a constant CO2partial pressure of 10�3.5 bar. Initially, the CO2 partialpressure ~Fig. 9! and total aqueous carbonate concentra-tions in the water ~expressed as HCO3

� , Fig. 7! decrease,and the pH increases ~Fig. 6! owing to the volatilizationand transport of CO2 from the water to the atmosphereprimarily by advection. As long as this advection is sig-nificant, equilibration of the water at an atmospheric CO2partial pressure cannot occur, because the back diffusionof CO2 gas into the beaker is too slow relative to theadvective transport of CO2 out of the beaker. It should benoted that in this simulation, the atmosphere does notinitially contain HCl, HF, and HNO3 gases. These acidgases are generated as a direct result of the evaporativeconcentration of dissolved chloride, fluoride, and nitrate,although their partial pressures remain fairly small. Thepartial pressure trends of these gases ~Fig. 9! are consis-tent with the pH trend ~Fig. 6!, with breaks correspond-ing to the precipitation of chloride and fluoride minerals~Fig. 8!. Note that because the initial fluoride concentra-tion in solution is much smaller than the nitrate and chlo-ride concentrations, the HF partial pressure eventuallybecomes limited by the precipitation of fluorite ~CaF2!and sellaite ~MgF2!.

IV.B. Reactive Transport Simulation of Near-Field

Thermal, Chemical, and Hydrological Evolution

As a demonstration, we also present results of a two-dimensional simulation, using the same numerical gridand general setup as described in earlier studies of drift-scale processes at the proposed repository at YuccaMountain.2,43– 45 This model consists of a vertical crosssection through a waste emplacement tunnel ~drift!. Themodel extends from the ground surface to the ground-water table, with the modeled drift located;360 m belowthe surface and several hundred meters above the watertable. Relying on the principle of symmetry, only half adrift is simulated, with a total model width ~40.5 m!representing half the horizontal distance between drifts~81 m, as currently planned at the proposed Yucca Moun-tain repository!. A heterogeneous fracture permeabilityfield is specified based on field measurements.43,44

In this simulation, we conservatively assume that themodel domain receives 60 mm0yr infiltration, about tentimes higher than the average infiltration rate in that area,so as to maximize seepage and possible flow-channelingeffects. A dual-continuum model is used to representrock fractures and matrix, respectively. Main processesconsidered in this model have been detailed else-where2,38,39,43,44 and include infiltration through groundsurface; multiphase flow ~water, air, and vapor! in theunsaturated zone; heating by the waste package ~up to;1508C!; boiling in the near-field rock fractures andmatrix; condensation0reflux in cooler areas away fromthe drift; reactive transport of solutes and gases; geo-chemical reactions including aqueous complexation, min-eral ~salt! precipitation, and dissolution; and permeabilitychanges resulting from mineral ~salt! precipitation anddissolution. Differences from earlier works consist in thetreatment of evaporative concentration and salt precipi-tation using a full Pitzer approach and consideration ofvapor-pressure lowering resulting from the increase insalinity.36 Also, the generation and reaction of acid gases~HCl, HF, and HNO3! is considered in addition to thereactive transport of CO2. Details regarding the domainspatial discretization, model boundary conditions, rockthermal properties, and rock hydrological properties arereported elsewhere.2,44

The simulation discussed in this paper covers a timeperiod of 0 to 600 yr, including a 50-yr ventilation periodfollowed by an unventilated post-closure period. As thedrift heats up, the temperature of the waste package reaches;518C during the first 50 yr, while 86% of the heat isremoved by ventilation. After 50 yr, the temperature ofthe waste package sharply increases to above 1508C andthe temperature of the drift wall reaches;1278C at;75 yr~Fig. 10, top left plot!. The porewater ~moisture! in thehost rock is quickly heated up to the boiling temperature~;958C at the repository elevation!, then vaporized, lead-ing to the evaporative concentration of residual solutionsand eventual precipitation of small amounts of minerals

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Fig. 10. Simulated distributions of temperature, liquid water saturation, pH, chloride concentration, CO2 gas partial pressure, andHCl gas partial pressure in rock fractures at 75 yr after the emplacement of the waste package ~grid blocks with zero liquidsaturation are left blank on plots of dissolved species!. Note that the outline of the dry areas, the distribution of pH, Clconcentration, CO2 gas, and HCl gas are all affected by the fracture permeability heterogeneity.

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and salts, as well as generation of small quantities of acidgases.

The chemical evolution of porewater in the near-field rock upon evaporation ~Fig. 10! is somewhat sim-ilar to the results discussed previously for “batch”evaporation without reaction with the host rock ~Figs. 6through 9!. However, in the present case, the processestaking place in the near-field rock are more complex. Theboiling point of residual solutions is elevated because ofthe vapor-pressure–lowering effects. When these solu-tions are concentrated, the water activity decreases andvaporization declines owing to vapor-pressure lowering.When the water activity reaches the prevailing ambientair relative humidity, vaporization ceases and very smallamounts of residual solution remain.

In this dryout zone, precipitated salts are left behindthe boiling front ~Fig. 11!. Generated acid gases diffuseinto the drift throughout the dryout zone and beyond,although the flux of acid gases is not sufficient to pro-duce acid condensate ~the partial pressures of the acidgases are lower than 10�7 bar; see Fig. 10, bottom right!.The vapor generated at the boiling front moves outwardand condenses in the cooler zone a few meters from theboiling front. In the condensation zone, the liquid satu-ration increases, and the porewater is diluted.

After 75 yr, the temperature starts to decrease grad-ually as heat dissipates in the rock and the overall heatload from radioactive decay decreases owing to the de-pletion of short-lived radionuclides. The boiling front infractures around the drift starts to collapse and retractsback about halfway to the drift wall at ;600 yr. Thepreviously precipitated salts dissolve again, and gener-ated acid gases gradually dissipate. The effect of thesegases on pH in the rewetting front ~Fig. 12! is not sig-nificant. The salt redissolution causes the dissolved chlo-ride concentrations in drainage areas around the modeleddrift to increase relative to background ~Fig. 12!, al-though these concentrations remain at low to moderatelevels ~typically ,0.1 molal!.

V. CONCLUSIONS

The existing geochemical reactive transport numer-ical simulator TOUGHREACT ~Ref. 11! was extendedby adding a Pitzer ion-interaction model using the HMWformulation and then was used to simulate highly con-centrated solutions at varying temperatures and evapo-ration to near-dry conditions. Simulation examplesrelevant to the geologic storage of nuclear waste atYucca Mountain are presented, including dissolution0precipitation of minerals and salts ~including deliques-cent salts!, generation and transport of acid gases fromthe brines at varying temperatures, and dryout while ac-counting for vapor-pressure–lowering effects due to highsalinity. This modeling capability is a useful tool for

assessing water-gas-rock interactions in the context ofhigh-level radioactive waste storage and other hydrogeo-logic systems. However, one challenge remains in thatthese simulations are computationally much more inten-sive than simulations using simpler ion-interaction mod-els ~such as Debye-Hückel, which cannot be used withhighly concentrated solutions! because of the added com-putations of specific ion-interaction effects. Therefore,the large-scale application of such models remains lim-ited. Efforts are currently underway to increase the effi-ciency of algorithms as well as parallelize the numericalmodel for improving computational efficiency.

APPENDIX

FORMULATION OF THE PITZER IONIC ACTIVITY MODEL

The Pitzer ion-interaction model evaluates the ionicactivities of a solution as a function of the solution’sionic strength ~long-distance interaction!, interaction terms~short-distance interaction!, temperature, and pressure.This model contains a Debye-Hückel term to account forthe effects of long-distance interactions and several virialequations to account for the effects of short-distance in-teractions. These virial equations are sometimes called“specific interaction equations,” “Pitzer equations,” or“phenomenological equations.” These equations ade-quately describe the thermodynamic properties of con-centrated solutions over a wide range of concentrationsand temperatures.23 The Pitzer model, based on a virialexpansion,9,10 is reduced to a modified form of the Debye-Hückel formula at low ionic strength.9 This virial expan-sion involves summations over all possible binary andternary short-range interactions, including mixing terms.A generally accepted form of the Pitzer model is theformulation rearranged by Harvie, Moller, and Weare1,3

~HMW formulation!. This formulation has been imple-mented in TOUGHREACT.

In the HMW formulation, water activity is ex-pressed as

ln~aH2O! � �mw

1000�(

i�1

N

mi�f , ~A.1!

where

aH2O � water activity

mi � molality of species i

mw � molecular weight of water

N � number of species in the system

f � osmotic coefficient.

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Fig. 11. Simulated distributions of salts ~minerals, in volume fraction change! precipitated in rock fractures at 75 yr after theemplacement of the waste package. Note that the distributions of the minerals ~salts! are affected by the heterogeneity ofthe fracture permeability.

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Fig. 12. Simulated distributions of temperature, chloride concentration, and pH at 300 and 600 yr ~grid blocks with zero liquidsaturation are left blank on plots of dissolved species!.

Zhang et al. MODELING REACTIVE MULTIPHASE FLOW AND TRANSPORT OF CONCENTRATED SOLUTIONS

NUCLEAR TECHNOLOGY VOL. 164 NOV. 2008 191

The osmotic coefficient f is defined as

(i�1

N

mi ~f� 1!

� 2��AFI 302

1 � 1.2MI�

� (c�1

Nc

(a�1

Na

mc ma~BcaF � ZCca !

�(c(

c '�c�1

mc mc '�Fcc 'f

� (a�1

maccc 'a��(

a(

a '�a�1

ma ma '�Faa 'f

� (c�1

mccaa 'c�� (

n�1

Nn

(c�1

Nc

mn mclnc � (n�1

Nn

(a�1

Na

mn malna

� (n�1

Nn

(c�1

Nc

(a�1

Na

mn mc maznca , ~A.2!

where

I � ionic strength, defined as I � 12_(k�1

N zk2 mk

zk � electrical charge of species k

M,C, c � cations

X, A, a � anions.

The activity coefficients of cations ~gM !, anions ~gX !,and neutral species ~gN ! are respectively calculated as

ln gM � ZM2 F � (

a�1

Na

ma~2BMa � ZCMa !

� (c�1

mc�2FMc � (a�1

macMca��(

a(

a '�a�1

ma ma 'Caa 'M

� 6ZM 6(c�1

Nc

(a�1

Na

mc ma Cca

� 2 (n�1

Nn

mnlnM , ~A.3!

ln gX � ZX2 F � (

c�1

Nc

mc~2BcX � ZCcX !

� (a�1

ma�2FXa � (c�1

mccXac��(

c(

c '�c�1

mc mc 'Ccc 'X

� 6Zx 6(c�1

Nc

(a�1

Na

mc ma Cca

� 2 (n�1

Nn

mnlnX , ~A.4!

and

ln gN � (a�1

Na

ma~2lna !� (c�1

Nc

mc~2lnc !

� (c�1

Nc

(a�1

Na

mc mazNca , ~A.5!

where

F � �AF� MI

1 � 1.2MI�

2

1.2ln~1 � 1.2MI !�

� (c�1(

c '�c�1

mc mc 'Fcc '' � (

a�1(

a '�a�1

ma ma 'Faa ''

� (c�1

Nc

(a�1

Na

mc ma Bca' , ~A.6!

CMX �CMXF

2M6zM zX 6, ~A.7!

and

Z � (k�1

N

6zk 6mk . ~A.8!

All Pitzer virial coefficients ~BMXF , BMX , BMX

' , aMX ,CMXF , lNC , and lNA! in Eqs. ~A.2! through ~A.7! are

defined as follows:

1. BMXF is used to calculate the osmotic coefficient

and water activity according to Eq. ~A.9!:

BMXF � bMX

~0! � bMX~1! e�aMXMI � bMX

~2! e�aMX' MI , ~A.9!

where bMX~0! , bMX

~1! , bMX~2! , and aMX are temperature-

dependent ion-interaction parameters.

Zhang et al. MODELING REACTIVE MULTIPHASE FLOW AND TRANSPORT OF CONCENTRATED SOLUTIONS

192 NUCLEAR TECHNOLOGY VOL. 164 NOV. 2008

2. BMX is used to calculate the activity coefficient ofions; this coefficient is calculated as

BMX � bMX~0! � bMX

~1! g~aMXMI !� bMX~2! g~aMX

' MI ! ,

~A.10!

with the function g~x! defined as

g~x! � 2~1 � ~1 � x!e�x !0x 2 , ~A.11!

with x denoting aMXMI or aMX' MI , respectively.

3. BMX' , used to calculate the modified Debye-

Hückel term, is formulated as

BMX' �

]BMX

]I

� bMX~1! g '~aMXMI !

I� bMX

~2! g '~aMX' MI !

I, ~A.12!

with the function g '~x! defined as

g '~x! � �2�1 � �1 � x �x 2

2 �e�x��x 2 , ~A.13!

where x denotes aMXMI or aMX' MI , respectively, for any

salt containing a monovalent ion, aMX � 2, and aMX' �12;

for 2-2 electrolytes, aMX �1.4 and aMX' �12; for 3-2, 4-2,

and higher valence electrolytes, aMX � 2.0 and aMX' � 50.

4. CMXF is a temperature-dependent interaction pa-

rameter of cation-anion pair.

5. Fccf , Faa

f , Fcc, Faa, Fcc' , and Faa

' are interactionparameters for like-sign ionic pairs ~mixing terms!; theyare temperature and ionic strength dependent:

Fijf � uij � Euij ~I !� I Euij

' ~I ! , ~A.14!

Fij � uij � Euij ~I ! , ~A.15!

and

Fij' � Euij

' ~I ! , ~A.16!

where Euij~I ! and Euij' ~I ! are functions of the ionic

charges between the pair and I. These functions are de-fined in Ref. 10 and can normally be omitted in moder-ately concentrated solutions of I , 10 molal @for alllike-sign pairs, Euij~I !� 0 and Euij

' ~I !� 0# . Also, uij aretemperature-dependent fitting parameters, with Euij~I !and Euij

' ~I ! calculated according to Ref. 10.

6. Ccca and Ccaa are the temperature-dependent in-teraction coefficients of ternary terms.

7. znca is the temperature-dependent interaction co-efficient of neutral-cation-anion terms; normally, this termis ignored ~for all neutral-cation-anion triplets, z� 0!.

8. lNC and lNA are the temperature-dependent in-teraction coefficients of neutral-cation pairs and neutral-anion pairs, respectively.

ACKNOWLEDGMENTS

We thank S. Mukhopadhyay, C. Bryan, M. Zhu, and twoanonymous reviewers for their valuable comments and sugges-tions, as well as D. Hawkes for his technical editing support.This work was supported by the Office of the Chief Scientist,Office of Civilian Radioactive Waste Management, providedto Lawrence Berkeley National Laboratory through the U.S.Department of Energy contract DE-AC02-05CH11231.

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