modeling reactive multiphase flow and transport of concentrated ...

MODELING REACTIVE MULTIPHASE FLOW AND TRANSPORT OF CONCENTRATED SOLUTIONS

RADIOACTIVE WASTE MANAGEMENT AND DISPOSAL KEYWORDS: Yucca Mountain, Pitzer model, dryout

GUOXIANG ZHANG,* NICOLAS SPYCHER, ERIC SONNENTHAL, CARL STEEFEL, and TIANFU XU Lawrence Berkeley National Laboratory 1 Cyclotron Road, MS 90-1116, Berkeley, California 94720

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

A Pitzer ion-interaction model for concentrated aqueous solutions was added to the reactive multiphase flow and transport code TOUGHREACT. The model is described and verified against published experimental data and the geochemical code EQ3/6. The model is used to simulate water-rock-gas interactions caused by boiling and evaporation within and around nuclear waste emplacement tunnels at the proposed high-level waste repository at Yucca Mountain, Nevada. The coupled thermal, hydrological, and chemical processes considered consist of water and air/vapor flow, evaporation, boiling, condensation, solute and gas transport, formation of highly concentrated brines, precipitation of deliquescent salts, generation of acid gases, and vapor-pressure lowering caused by the high salinity of the concentrated brine.

I. INTRODUCTION Aqueous solutions are generally defined as concentrated when their ionic strength ~I ! is greater than 1 molal ~mol0kg H 2O!. Such solutions may result from many natural and artificial processes, such as evaporation,1 boiling,2 seawater intrusion,1,3,4 leakage of toxic solutions and electrolytic fluids from storage tanks,5–7 and acid mine drainage.8 Concentrated aqueous solutions are significantly different from diluted solutions in terms of their geochemical behavior because the thermodynamic activities of both water and solutes deviate from ideal behavior.1,3,9,10 In this paper, we discuss the implementation of the Pitzer ion-interaction model 1,3,9,10 into the TOUGHREACT existing reactive transport simulator 11 *E-mail: [email protected] 180

for the purpose of modeling the geochemical reactive transport of concentrated solutions and the relevant gases at the proposed high-level nuclear waste repository at Yucca Mountain, Nevada. TOUGHREACT ~Ref. 11! was originally developed by integrating geochemical transport into the framework of the TOUGH2 multiphase flow code.12 It can simulate nonisothermal multiphase groundwater flow, diffusive and advective transport of gases ~including vapor! and solutes, aqueous speciation, and mineral dissolution and precipitation under both equilibrium and kinetic constraints. These processes can be simulated in complex flow and transport systems, such as in variably saturated environments and multiple porosity0permeability media ~e.g., rock matrix-fracture systems!. TOUGHREACT has been used to simulate a number of reactive geochemical transport processes at various scales and over a wide range of geochemical conditions.11 Prior to the present study, this code could only handle diluted to moderately concentrated solutions ~I , 1 molal! and moderately concentrated 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 The implementation of the Pitzer ion-interaction model now allows the application of this simulator to problems involving more concentrated aqueous solutions, such as those involving geochemical processes in and around high-level nuclear waste repositories where fluid evaporation and0or boiling is expected to occur. The Pitzer ion-interaction model, which we define here as the Pitzer virial approach and associated ioninteraction parameters,9,10 has been applied successfully to study nonideal concentrated aqueous solutions.4 –7 Pitzer’s original formulation and parameters were rearranged by Harvie, Moller, and Weare 1,3 to model seawater systems ~Na-K-Mg-Ca-Cl-SO 4-H 2O!, leading to a more practical formulation ~see the Appendix!. This rearranged formulation ~here referred to as “HMW”! and its parameters differ from Pitzer’s original model in terms NUCLEAR TECHNOLOGY

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of mathematical expressions and parameter values, but nevertheless express the same theory ~and results! as Pitzer’s original work. It should be noted that for complex systems, ion-interaction parameters are often more readily available for the HMW formulation than for Pitzer’s original formulation, although parameters for either formulation can be converted from one to the other.14 Pitzer’s original formulation was implemented in the EQ306 geochemical reaction code 15,16 ; however, this code reads interaction parameters for the HMW formulation, which are then internally mapped into Pitzer’s original formulation. The HMW formulation has been implemented in several geochemical and reactive transport codes, including PHRQPITZ ~Ref. 17!, GMIN ~Ref. 18!, UNSATCHEM-2D ~Ref. 19!, and BIO-CORE 2D© ~Refs. 7, 20, 21, and 22!, using various sources of ioninteraction parameters. In this paper, we present the implementation and verification of the HMW formulation into TOUGHREACT, using a database of ion-interaction parameters developed by others 14,23–25 for specific applications to nuclear waste–related problems. We also demonstrate the code’s capability to handle reactive transport processes of concentrated aqueous solutions ~brines! involving dissolution0precipitation of minerals and salts ~including deliquescent salts!, generation of acid gases from the brines at varying temperatures, dryout caused by boiling and evaporation, and vapor-pressure lowering caused by high salinity. This modeling capability is useful for the safety assessment of underground high-level radioactive waste repositories.

II. ION-INTERACTION PARAMETERS In this paper, we make use of the EQ306 database data0.ypf ~Refs. 14, 23, 24, and 25! for both Pitzer ioninteraction parameters and thermodynamic equilibrium constants. A dependence on temperature is considered, as described below, whereas the effect of pressure is not because it is much less significant, and because the pressure range considered here is near atmospheric. The interpolation and extrapolation equations for various thermodynamic properties of aqueous solutions as a function of temperature, for binary and ternary systems, and for multiple-component mixtures within the Pitzer formulation have been reported in many papers,3,7,9,10,26–32 covering various ranges of temperatures depending on the system considered. In the present paper, we make use of the temperature-dependence formulation implemented in the EQ306 Pitzer database 14,23–25 : P~T ! ⫽ a 1 ⫹ a 2

冉

1 T

⫺

1 T0

冊

⫹ a 3 ln

冉冊 T

T0

⫹ a 4 ~T ⫺ T0 ! , NUCLEAR TECHNOLOGY

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III. TESTS The code was verified using experimental data and benchmarked against EQ306 ~Ref. 16!. The following are three selected test cases that show how the code reproduces experimental data, how it matches the results calculated by EQ306, and how it captures the water-vaporpressure–lowering effects of saline solutions resulting from high salinity. The first test calculates the mean activity coefficients of NaCl and the osmotic coefficient of NaCl solutions up to 6 molal, at 0, 25, 50, 80, 100, and 1108C, respectively. The calculated mean activity coefficients are compared with measured data in Fig. 1 ~Ref. 33!. This test validates the temperature dependence of the activity coefficients calculated in the model. Note that comparisons of mean activity coefficients, rather than mean activities, are appropriate here, because there are no Na and Cl species other than the free ions included in the simulations ~i.e., the effect of NaCl ion pairing is accounted for by the activity coefficients alone, and not by a separate NaCl secondary species!. The mean activity coefficient of NaCl, gNaCl , is defined as ln~gNaCl ! ⫽

ln~gCl ! ⫹ ln~gNa ! 2

.

~2!

Another test case involved the calculation of the mean activity of CaCl2 , osmotic coefficient, water activity, and water-vapor pressure of CaCl2 solutions at ionic strength up to 27 molal ~9 molal of CaCl2 !. The calculated osmotic coefficient at 608C and mean activity of CaCl2 were compared with literature data 34 in Fig. 2 and show reasonably good agreement. The osmotic coefficient is calculated, according to Eq. ~A.1!, as f⫽⫺

~1! NOV. 2008

where P~T ! represents Pitzer parameters ~see the AppenF dix! b ~0!, b ~1!, b ~2!, a, F, C, and CMX at temperature T ~absolute temperature! and T0 is the reference temperature ~298.15 K!. These data provide good results up to at least 908C and ionic strengths in the 20 to 40 molal range for chloride-dominant waters in the context of evaporation0boiling studies at Yucca Mountain.25 More details on this database can be found in Refs. 23, 24, and 25. Because the effects of ion pairing and aqueous complexation are generally taken into account by the ioninteraction parameters, much care must be taken to avoid “double counting” these effects by including only those ion pairs and complexes that were considered in deriving the ion-interaction parameters.

1000 ln~a w ! Ww ( m i

.

~3!

i

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Fig. 1. Comparison of TOUGHREACT-calculated ~solid lines! and measured 33 ~symbols! mean activity coefficients for NaCl solutions at ~a! 258C and ~b! 1108C.

Fig. 2. Comparison of TOUGHREACT-calculated and measured ~a! osmotic coefficient 34 of CaCl2 solutions and ~b! mean activity coefficient 34 of CaCl2 ~right!, at 608C and concentrations up to 9 molal of CaCl2 salt ~I ⫽ 27 molal!.

The differences between the simulation results and the data at extremely high concentration ~close to saturation of the salt! could be caused by the peculiar behavior of this salt at high concentrations,35 possible uncertainties in the measurements ~possible precipitation of solid phases when the solution is concentrated up to saturation with respect to the salt!, and0or uncertainties in ionic interaction parameters. Vapor-pressure lowering caused by dissolved salts was implemented directly through the water activity computed with the Pitzer ion-interaction model.36 For equilibrium between water and H2O vapor ~i.e., for the reaction H 2O ~l ! ? H 2O ~g! !, equating the chemical potentials of both phases yields m 0v

⫺

m w0

⫽ RT ln~

fv 0fv0 !

⫺ RT ln~

fw 0fw0 !

⫽ RT ln~ fv 0a w ! ⫽ RT ln~K ! , 182

~4!

where w ⫽ liquid water v ⫽ H 2O gas m 0 ⫽ reference chemical potential f ⫽ fugacity a ⫽ activity ~defined as f0f 0 , with f 0 being the fugacity in the reference state! K ⫽ thermodynamic equilibrium constant R ⫽ gas constant T ⫽ temperature. The reference ~standard! state of H 2O gas is taken as the unit fugacity of the pure gas at pressure of 1 bar and all temperatures, whereas that of liquid water is taken as the NUCLEAR TECHNOLOGY

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

unit activity of pure water at all temperatures and pressures. In our case, at low pressure ~atmospheric!, fugacity is approximated by pressure, and fv and thus K in Eq. ~4! are respectively approximated by the actual vapor 0 pressure Pv and the vapor pressure of pure water Psat such 0 that Pv 0a w ⬵ Psat ~thus, for the pure system, a w ⫽ 1 and 0 Pv ⫽ Psat !. Accordingly, the vapor pressure of the solution is computed as 0 Pv ⫽ a w Psat .

~5!

Equation ~5! is used in the coupling of chemistry and flow calculations, which arises primarily through the effect of salt concentrations on vapor pressure in the multiphase flow computations. From Eq. ~5!, it is also apparent that if relative humidity R h is defined as the ratio of the actual vapor pressure to that of pure water, then Rh ⫽ aw .

~6!

The implementations of Eqs. ~5! and ~6! were verified by taking the vapor pressure of pure water from the National Institute of Standards and Technology steam tables 37 and then calculating the vapor pressure of the solution using these equations and the water activity calculated by the Pitzer ion-interaction model. Simulated vapor pressure and relative humidity values for solutions up to 9 molal CaCl2 agree well with the values calculated from the steam tables ~Fig. 3!. The third test case involves calculating the activity coefficients of Ca⫹2, Na⫹, and Cl⫺ in mixtures of CaCl2 and NaCl solutions up to 6 molal CaCl2 and 6 molal NaCl, yielding ionic strengths up to 24 molal at 258C. We compare the TOUGHREACT results against EQ306 results. Both codes use the same chemical setup, same input data, and same thermodynamic databases; a comparison of individual activity coefficients, rather than activities, is appropriate because the same database is

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

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used in both cases, and specific ~separate! ion pairs are not included in the calculations. Figure 4 shows that the results 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 repository at Yucca Mountain, Nevada, is located in unsaturated volcanic tuffs several hundred meters above the water table. The radioactive decay from the spent fuel will heat the rock around waste emplacement tunnels and depending on the areal loading of spent fuel, and ventilation design, the temperature could rise above the boiling point of pure water for several hundred years.38,39 Under such conditions, water moisture in the surrounding rock ~i.e., porewater in the unsaturated volcanic tuff ! will boil within several meters of the walls of emplacement tunnels. Accordingly, a dryout zone will develop between the boiling front and tunnel walls during the heating period.38,39 In the dryout zone, a tiny amount of concentrated aqueous solutions ~brines! may remain as a result of vapor-pressure lowering, and some salts may precipitate. Upon evaporative concentration of the rock moisture, some acid gases could be volatilized from the

residual brines. These and other coupled thermal, hydrological, and chemical processes ~Fig. 5! may affect the integrity of waste packages and therefore need to be carefully evaluated as to their impact on the safety of the proposed repository. The TOUGHREACT-Pitzer reactive transport simulator was applied to investigate some of these processes, as shown below with two example simulations: ~a! the boiling and evaporative concentration of porewater ~rock moisture! in contact with an open atmosphere and without buffering effects from rock minerals and ~b! a twodimensional reactive transport simulation depicting the thermal, chemical, and hydrological evolution of the dryout zone around a typical waste emplacement tunnel. IV.A. Evaporation/Boiling Without Water-Rock Interactions Here, a dynamic model is developed to simulate the evaporation of a Yucca Mountain tuff porewater open to the atmosphere. The simulation is set up as if the water was heated in an open beaker, thus allowing for exchange of gases between the solution and the atmosphere. The beaker is represented by a partially liquid saturated model grid block of a given volume, and is connected to a second air-filled infinitely large grid block representing the atmosphere. The grid block representing the beaker is maintained 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 emplacement tunnel at the proposed Yucca Mountain repository. 184

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TABLE I Chemical Composition of the Solution Used in Simulations of Evaporative Concentration* Components

Molality

Ca⫹2 Mg⫹2 Na⫹ Cl⫺ SiO 2~aq! HCO⫺ 3 SO⫺2 4 K⫹ Al~OH!⫺ 4 F⫺ HFeO 2 NO⫺ 3 Ionic strength pH

2.5 ⫻ 10⫺3 6.9 ⫻ 10⫺4 2.6 ⫻ 10⫺3 3.3 ⫻ 10⫺3 1.1 ⫻ 10⫺3 3.2 ⫻ 10⫺3 1.2 ⫻ 10⫺3 2.0 ⫻ 10⫺4 6.2 ⫻ 10⫺10 4.5 ⫻ 10⫺5 1.2 ⫻ 10⫺12 1.0 ⫻ 10⫺4 0.013 8.3

*See Sec. IV.A.

temperature of pure water at the elevation of the proposed repository. The vapor and gases generated in the beaker grid block are allowed to advect and diffuse into the grid block representing the atmosphere ~at ;1 bar!, which is set to a fixed atmospheric CO 2 partial pressure of 10⫺3.5 bar. This setup captures the chemical processes that would accompany the evaporation of seepage in the hot, open emplacement tunnels without contacting host rock ~Fig. 5!. Note that this setup cannot capture precise gradients at the liquid-atmosphere boundary, and is only intended to illustrate chemical effects in the beaker grid block during rapid evaporation under conditions where initial CO 2 volatilization from the solution is not immediately replenished by diffusion of CO 2 from the atmo-

sphere. This setup, therefore, differs from the work of others 25,40 who consider a constant CO 2 partial pressure in the solution. The water used in this simulation ~Table I! represents porewater extracted from rock core samples collected from borehole ESF-PERM-3 ~Ref. 2! near the location of the Drift Scale Test ~prior to the test! in the Engineered System Facility ~ESF! at Yucca Mountain. This porewater is characterized by a higher calcium and sulfate content than is found in porewaters sampled at other locations in the repository host rocks and produces a slightly acidic brine when it is extremely concentrated, in agreement with observations by others.25 The code’s capability allows simulating evaporation until near dryness at concentration factors . 10 6 and water activity values essentially nil. In the real system, however, evaporation would stop when the brine water activity reaches the relative humidity of the prevailing air in the tunnel @Eqs. ~4!, ~5!, and ~6!#. Also, at the proposed Yucca Mountain repository, in-drift relative humidities are expected to remain mostly above ;0.2 ~Ref. 41!. In addition, ion-interaction parameters input in the simulations are valid only up to a certain range of ionic strength ~in our case to maximum ionic strength values in the range of 20 to 40 molal!. For these reasons, the computed chemical evolution of dissolved species, solids, and gases are shown ~Figs. 6 through 9! only down to water activities around 0.2 ~at a concentration factor ;10 5 and ionic strength within the validity range of the thermodynamic data!. The remaining brine at a water activity near 0.2 is mildly acidic ~pH of 5.5, Fig. 6!, with acid gas fugacities reaching 10⫺8 bar for HF and HNO 3 and 10⫺7 bar for HCl ~Fig. 9!. Predicted major solid precipitates ~volume fraction .1%! are halite ~NaCl!, calcite ~CaCO 3 !, anhydrite ~CaSO 4 !, amorphous silica ~SiO 2 !, sepiolite ~Mg 4Si6O 15 ~OH!2 :6H 2O!, and sylvite ~KCl! ~Fig. 8!. The predicted major mineral phases are in relatively good

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

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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 ~mol0m 3 medium!. Halite, calcite, and anhydrite dominate the salt assemblage.

Fig. 9. Evolution of gas partial pressures. Acid gas pressures increase as the solution is concentrated and pH decreases ~see Sec. IV.A!.

agreement with experimental and modeling results from others 25 using a similar water composition. Note that in this simulation, sepiolite ~a magnesium silicate! and amorphous silica are reacting under kinetic constraints, whereas

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

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phase would likely precipitate in place of, and faster than, sepiolite 42 and deplete magnesium from the solution so that other magnesium salts may not form. Evaporative concentration and the precipitation of major solid phases control the concentration profiles of dissolved species. As a result, the concentration profiles of dissolved species ~Fig. 7! show breaks in slope when minerals start to precipitate ~Fig. 8!. It should be noted that the initial solution ~Table I! is supersaturated with respect to calcite, which forces calcite precipitation prior to the start of evaporation ~at concentration factor ⫽ 1!, driving the pH down ~Fig. 6! and, accordingly, driving the partial pressures of CO 2 and other acid gases up ~Fig. 9!. The sharp calcium concentration drop ~Fig. 7! at the start of the simulation ~at concentration factor ⫽ 1! is also caused by this initial precipitation of calcite. Once evaporation starts, however, pH is driven up by CO 2 ⫹ volatilization ~e.g., HCO⫺ ] CO 2~g!F ⫹ H 2O!, 3 ⫹ H which further induces calcite precipitation. Eventually, the effects of evaporative concentration take over the effect of CO 2 volatilization and the pH trend reverses ~Fig. 6!. As the brine is concentrated, calcium concentrations rise until anhydrite precipitation becomes the major process removing calcium from the brine. At this point, the rate of increase in calcium concentration sharply drops, although the calcium concentration still slowly increases as calcite dissolves owing to the decreasing pH. Further dissolution of calcite and precipitation of anhydrite ~Fig. 8! cause the sulfate concentrations to drop as calcium continues to increase ~Fig. 7!. As mentioned above, the simulation is set up as if water was heated in a beaker in direct contact with the atmosphere. The atmosphere is simulated as a gas reservoir of constant pressure ~;1 bar! with a constant CO 2 partial pressure of 10⫺3.5 bar. Initially, the CO 2 partial pressure ~Fig. 9! and total aqueous carbonate concentrations in the water ~expressed as HCO⫺ 3 , Fig. 7! decrease, and the pH increases ~Fig. 6! owing to the volatilization and transport of CO 2 from the water to the atmosphere primarily by advection. As long as this advection is significant, equilibration of the water at an atmospheric CO 2 partial pressure cannot occur, because the back diffusion of CO 2 gas into the beaker is too slow relative to the advective transport of CO 2 out of the beaker. It should be noted that in this simulation, the atmosphere does not initially contain HCl, HF, and HNO 3 gases. These acid gases are generated as a direct result of the evaporative concentration of dissolved chloride, fluoride, and nitrate, although their partial pressures remain fairly small. The partial pressure trends of these gases ~Fig. 9! are consistent with the pH trend ~Fig. 6!, with breaks corresponding to the precipitation of chloride and fluoride minerals ~Fig. 8!. Note that because the initial fluoride concentration in solution is much smaller than the nitrate and chloride concentrations, the HF partial pressure eventually becomes limited by the precipitation of fluorite ~CaF2 ! and sellaite ~MgF2 !. NUCLEAR TECHNOLOGY

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IV.B. Reactive Transport Simulation of Near-Field Thermal, Chemical, and Hydrological Evolution As a demonstration, we also present results of a twodimensional simulation, using the same numerical grid and general setup as described in earlier studies of driftscale processes at the proposed repository at Yucca Mountain.2,43– 45 This model consists of a vertical cross section through a waste emplacement tunnel ~drift!. The model extends from the ground surface to the groundwater table, with the modeled drift located ;360 m below the surface and several hundred meters above the water table. Relying on the principle of symmetry, only half a drift 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 Mountain repository!. A heterogeneous fracture permeability field is specified based on field measurements.43,44 In this simulation, we conservatively assume that the model domain receives 60 mm0yr infiltration, about ten times higher than the average infiltration rate in that area, so as to maximize seepage and possible flow-channeling effects. A dual-continuum model is used to represent rock fractures and matrix, respectively. Main processes considered in this model have been detailed elsewhere 2,38,39,43,44 and include infiltration through ground surface; multiphase flow ~water, air, and vapor! in the unsaturated zone; heating by the waste package ~up to ;1508C!; boiling in the near-field rock fractures and matrix; condensation0reflux in cooler areas away from the drift; reactive transport of solutes and gases; geochemical reactions including aqueous complexation, mineral ~salt! precipitation, and dissolution; and permeability changes resulting from mineral ~salt! precipitation and dissolution. Differences from earlier works consist in the treatment of evaporative concentration and salt precipitation using a full Pitzer approach and consideration of vapor-pressure lowering resulting from the increase in salinity.36 Also, the generation and reaction of acid gases ~HCl, HF, and HNO 3 ! is considered in addition to the reactive transport of CO 2 . Details regarding the domain spatial discretization, model boundary conditions, rock thermal properties, and rock hydrological properties are reported elsewhere.2,44 The simulation discussed in this paper covers a time period of 0 to 600 yr, including a 50-yr ventilation period followed by an unventilated post-closure period. As the drift heats up, the temperature of the waste package reaches ;518C during the first 50 yr, while 86% of the heat is removed by ventilation. After 50 yr, the temperature of the waste package sharply increases to above 1508C and the temperature of the drift wall reaches ;1278C at ;75 yr ~Fig. 10, top left plot!. The porewater ~moisture! in the host rock is quickly heated up to the boiling temperature ~;958C at the repository elevation!, then vaporized, leading to the evaporative concentration of residual solutions and eventual precipitation of small amounts of minerals 187

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Fig. 10. Simulated distributions of temperature, liquid water saturation, pH, chloride concentration, CO 2 gas partial pressure, and HCl gas partial pressure in rock fractures at 75 yr after the emplacement of the waste package ~grid blocks with zero liquid saturation are left blank on plots of dissolved species!. Note that the outline of the dry areas, the distribution of pH, Cl concentration, CO 2 gas, and HCl gas are all affected by the fracture permeability heterogeneity. 188

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and salts, as well as generation of small quantities of acid gases. The chemical evolution of porewater in the nearfield rock upon evaporation ~Fig. 10! is somewhat similar to the results discussed previously for “batch” evaporation without reaction with the host rock ~Figs. 6 through 9!. However, in the present case, the processes taking place in the near-field rock are more complex. The boiling point of residual solutions is elevated because of the vapor-pressure–lowering effects. When these solutions are concentrated, the water activity decreases and vaporization declines owing to vapor-pressure lowering. When the water activity reaches the prevailing ambient air relative humidity, vaporization ceases and very small amounts of residual solution remain. In this dryout zone, precipitated salts are left behind the boiling front ~Fig. 11!. Generated acid gases diffuse into the drift throughout the dryout zone and beyond, although the flux of acid gases is not sufficient to produce acid condensate ~the partial pressures of the acid gases are lower than 10⫺7 bar; see Fig. 10, bottom right!. The vapor generated at the boiling front moves outward and condenses in the cooler zone a few meters from the boiling front. In the condensation zone, the liquid saturation increases, and the porewater is diluted. After 75 yr, the temperature starts to decrease gradually as heat dissipates in the rock and the overall heat load from radioactive decay decreases owing to the depletion of short-lived radionuclides. The boiling front in fractures around the drift starts to collapse and retracts back about halfway to the drift wall at ;600 yr. The previously precipitated salts dissolve again, and generated acid gases gradually dissipate. The effect of these gases on pH in the rewetting front ~Fig. 12! is not significant. The salt redissolution causes the dissolved chloride concentrations in drainage areas around the modeled drift to increase relative to background ~Fig. 12!, although these concentrations remain at low to moderate levels ~typically ,0.1 molal!.

V. CONCLUSIONS The existing geochemical reactive transport numerical simulator TOUGHREACT ~Ref. 11! was extended by adding a Pitzer ion-interaction model using the HMW formulation and then was used to simulate highly concentrated solutions at varying temperatures and evaporation to near-dry conditions. Simulation examples relevant to the geologic storage of nuclear waste at Yucca Mountain are presented, including dissolution0 precipitation of minerals and salts ~including deliquescent salts!, generation and transport of acid gases from the brines at varying temperatures, and dryout while accounting for vapor-pressure–lowering effects due to high salinity. This modeling capability is a useful tool for NUCLEAR TECHNOLOGY

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assessing water-gas-rock interactions in the context of high-level radioactive waste storage and other hydrogeologic systems. However, one challenge remains in that these simulations are computationally much more intensive than simulations using simpler ion-interaction models ~such as Debye-Hückel, which cannot be used with highly concentrated solutions! because of the added computations of specific ion-interaction effects. Therefore, the large-scale application of such models remains limited. Efforts are currently underway to increase the efficiency of algorithms as well as parallelize the numerical model for improving computational efficiency.

APPENDIX FORMULATION OF THE PITZER IONIC ACTIVITY MODEL The Pitzer ion-interaction model evaluates the ionic activities of a solution as a function of the solution’s ionic strength ~long-distance interaction!, interaction terms ~short-distance interaction!, temperature, and pressure. This model contains a Debye-Hückel term to account for the effects of long-distance interactions and several virial equations to account for the effects of short-distance interactions. These virial equations are sometimes called “specific interaction equations,” “ Pitzer equations,” or “phenomenological equations.” These equations adequately describe the thermodynamic properties of concentrated solutions over a wide range of concentrations and temperatures.23 The Pitzer model, based on a virial expansion,9,10 is reduced to a modified form of the DebyeHückel formula at low ionic strength.9 This virial expansion involves summations over all possible binary and ternary short-range interactions, including mixing terms. A generally accepted form of the Pitzer model is the formulation rearranged by Harvie, Moller, and Weare 1,3 ~HMW formulation!. This formulation has been implemented in TOUGHREACT. In the HMW formulation, water activity is expressed as ln~a H2O ! ⫽ ⫺

mw 1000

冉( 冊 N

mi f ,

~A.1!

i⫽1

where a H2O ⫽ water activity m i ⫽ molality of species i m w ⫽ molecular weight of water N ⫽ number of species in the system f ⫽ osmotic coefficient. 189

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Fig. 11. Simulated distributions of salts ~minerals, in volume fraction change! precipitated in rock fractures at 75 yr after the emplacement of the waste package. Note that the distributions of the minerals ~salts! are affected by the heterogeneity of the fracture permeability. 190

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

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The osmotic coefficient f is defined as

Nc

ln gX ⫽ Z X2 F ⫹

N

( m c ~2BcX ⫹ ZCcX !

c⫽1

( m i ~f ⫺ 1!

i⫽1

冉

⫽2 ⫺

A F I 302 1 ⫹ 1.2M I

Nc

⫹

⫹

冊

⫹(

Na

⫹(

冉冉

c c ⫽c⫹1

a a ⫽a⫹1

( m a ccc a '

f

Nc

Nn

⫹ 6 Zx 6

冊冊

⫹

冊

m c m c ' Ccc ' X

( ( m c m a Cca

( m n l nX

,

~A.4!

and

( m c caa c c⫽1 '

Na

n⫽1

Na

ln gN ⫽

Na

Nc

( m a ~2l na ! ⫹ c⫽1 ( m c ~2l nc !

a⫽1

Nc

⫹

Na

Na

( ( m c m a zNca

,

~A.5!

c⫽1 a⫽1

( ( ( m n m c m a znca

,

~A.2!

n⫽1 c⫽1 a⫽1

where I ⫽ ionic strength, defined as I ⫽ _12 ( Nk⫽1 z k2 m k

where F ⫽ ⫺A F

z k ⫽ electrical charge of species k

⫹

M, C, c ⫽ cations

1 ⫹ 1.2M I

c⫽1 c ⫽c⫹1 Nc

Na

⫹

⫹

2 1.2

冉

2FMc ⫹

( m a cMca

a⫽1

ln~1 ⫹ 1.2M I !

' m c m c ' Fcc ' ⫹

( (

a⫽1 a ⫽a⫹1

冊

' m a m a ' Faa '

'

Na

( ( m c m a Bca'

,

~A.6!

c⫽1 a⫽1

CMX ⫽

( m a ~2BMa ⫹ ZCMa ! a⫽1

( mc

c⫽1

MI

'

The activity coefficients of cations ~gM !, anions ~gX !, and neutral species ~gN ! are respectively calculated as 2 F⫹ ln gM ⫽ Z M

冉

( (

X, A, a ⫽ anions.

⫹

( m c cXac

c⫽1

Nn

⫹2

( ( m n m c l nc ⫹ n⫽1 ( a⫽1 ( m n m a l na Nc

2FXa ⫹

c⫽1 a⫽1

n⫽1 c⫽1 Nn

( Nc

a⫽1

m a m a ' Faa ' ⫹

(

'

⫹

f

m c m c ' Fcc ' ⫹

(

'

⫹(

冉

c c ' ⫽c⫹1

F m c m a ~Bca ⫹ ZCca ! ( ( c⫽1 a⫽1

Nn

( ma

a⫽1

F CMX

,

~A.7!

2 6z M z X 6

M

冊

and N

Z⫽

( 6z k 6 m k

.

~A.8!

k⫽1

⫹(

(

F ' , BMX , BMX , aMX , All Pitzer virial coefficients ~BMX l NC , and l NA ! in Eqs. ~A.2! through ~A.7! are defined as follows:

m a m a ' Caa ' M

F CMX ,

a a ' ⫽a⫹1 Nc

⫹ 6 ZM 6

F 1. BMX is used to calculate the osmotic coefficient and water activity according to Eq. ~A.9!:

Na

((

m c m a Cca

c⫽1 a⫽1 F ⫽ bMX ⫹ bMX e ⫺aMX M I ⫹ bMX e ⫺aMXM I , BMX ~0!

~1!

~2!

'

~A.9!

Nn

⫹2 192

( m n l nM n⫽1

,

~A.3!

~0!

~1!

~2!

where bMX , bMX , bMX , and aMX are temperaturedependent ion-interaction parameters. NUCLEAR TECHNOLOGY

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2. BMX is used to calculate the activity coefficient of ions; this coefficient is calculated as ' BMX ⫽ bMX ⫹ bMX g~aMX M I ! ⫹ bMX g~aMX MI ! , ~0!

~1!

~2!

~A.10! with the function g~ x! defined as g~ x! ⫽ 2~1 ⫺ ~1 ⫹ x!e

⫺x

!0x

2

,

~A.11!

' M with x denoting aMX M I or aMX I , respectively. ' , used to calculate the modified Debye3. BMX Hückel term, is formulated as ' BMX ⫽

8. l NC and l NA are the temperature-dependent interaction coefficients of neutral-cation pairs and neutralanion pairs, respectively.

ACKNOWLEDGMENTS We thank S. Mukhopadhyay, C. Bryan, M. Zhu, and two anonymous reviewers for their valuable comments and suggestions, 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, provided to Lawrence Berkeley National Laboratory through the U.S. Department of Energy contract DE-AC02-05CH11231.

]BMX ]I

⫽ bMX ~1!

REFERENCES g ' ~aMX M I ! I

' g ' ~aMX MI !

⫹ bMX ~2!

I

, ~A.12!

with the function g ' ~ x! defined as

冉冉

g ' ~ x! ⫽ ⫺2 1 ⫺ 1 ⫹ x ⫹

x2 2

冊冊冒 e ⫺x

x2 ,

~A.13!

' M I , respectively, for any where x denotes aMX M I or aMX ' 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. F is a temperature-dependent interaction pa4. CMX rameter of cation-anion pair. f

f

' ' 5. Fcc , Faa , Fcc , Faa , Fcc , and Faa are interaction parameters for like-sign ionic pairs ~mixing terms!; they are temperature and ionic strength dependent: f Fij

⫽ uij ⫹ E uij ~I ! ⫹ I E uij' ~I ! ,

Fij ⫽ uij ⫹ E uij ~I ! ,

~A.14! ~A.15!

and Fij' ⫽ E uij' ~I ! ,

~A.16!

where E uij ~I ! and E uij' ~I ! are functions of the ionic charges between the pair and I. These functions are defined in Ref. 10 and can normally be omitted in moderately concentrated solutions of I , 10 molal @for all like-sign pairs, E uij ~I ! ⫽ 0 and E uij' ~I ! ⫽ 0# . Also, uij are temperature-dependent fitting parameters, with E uij ~I ! and E uij' ~I ! calculated according to Ref. 10.

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