Coupled thermohaline groundwater flow and single-species reactive solute transport in fractured porous media

$Page 1: Coupled thermohaline groundwater flow and single-species reactive solute transport in fractured porous media$
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Advances in Water Resources 30 (2007) 742–771

Coupled thermohaline groundwater flow and single-species reactivesolute transport in fractured porous media

Thomas Graf *, Rene Therrien

Departement de Geologie et Genie Geologique, Universite Laval, Ste-Foy, Que., Canada G1K 7P4

Received 15 December 2005; received in revised form 26 June 2006; accepted 6 July 2006Available online 17 August 2006

Abstract

A 3D numerical model has been developed to solve coupled fluid flow, heat and single-species reactive mass transport with variablefluid density and viscosity. We focus on a single reaction between quartz and its aqueous form silica. The fluid density and viscosity andthe dissolution rate constant, equilibrium constant and activity coefficient are calculated as a function of the concentrations of major ionsand temperature. Reaction and flow parameters, such as mineral surface area and permeability, are updated at the end of each time stepwith explicitly calculated reaction rates. Adaptive time stepping is used to increase or decrease the time step size according to the rate oftemporal variation of the solution to prevent physically unrealistic results. The time step size depends on maximum changes in matrixporosity and/or fracture aperture. The model is verified against existing analytical solutions of heat transfer and reactive transport infractured porous media. The complexity of the model formulation allows studying chemical reactions and variable-density flow in a morerealistic way than done previously.

The newly developed model has been used to simulate illustrative examples of coupled thermohaline flow and reactive transport infractured porous media. Simulations indicate that thermohaline (double-diffusive) transport impacts both buoyancy-driven flow andchemical reactions. Hot zones correspond to upwelling and to quartz dissolution while in cooler zones, the plume sinks and silica pre-cipitates. The silica concentration is inversely proportional to salinity in high-salinity regions and proportional to temperature in low-salinity regions. Density contrasts are generally small and fractures do not act like preferential pathways but contribute to transversedispersion of the plume. Results of a long-term (100 years) simulation indicate that the coexistence of dissolution and precipitation leadsto self-sealing of fractures. Salt mass fluxes through fractures decrease significantly due to major fracture aperture reduction in the pre-cipitation zone. The system is the most sensitive to temperature because it impacts both the dissolution kinetics (Arrhenius equation) andthe quartz solubility. The system is least sensitive to quartz surface area in the fracture because the volumetric fraction of a fracture issmall compared to the volumetric fraction of the porous matrix.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Numerical modeling; Nuclear waste; Fracture; Quartz; Reactive transport; Density; Heat; Thermohaline

1. Introduction

1.1. Problem definition

Many countries generate electrical power using nuclearfuel. Altogether, 436 nuclear power plants around theworld operate in 31 countries. Nuclear energy production

0309-1708/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2006.07.001

* Corresponding author. Tel.: +1 418 6562131; fax: +1 418 6567339.E-mail address: [email protected] (T. Graf).

eventually creates waste in the form of spent nuclear fuel.Deep burial of radioactive waste in low-permeability geo-logical formations is a disposal option considered world-wide [11]. Studies show that, with increasing depth,groundwater and ambient rock temperature increaseaccording to geothermal gradient and water compositioncan reach that of hot saline Na–Ca–Cl brines [17,71].Because of these temperature and composition changes,deep-fluid properties such as viscosity and density can nolonger be assumed constant. For deep waste disposal,

mailto:[email protected]

Nomenclature

The use of symbols for main variables is consistentthroughout the entire textAll symbols represent scalar variables denoted in normalitalic lettersLatin letters

(2b) [L] fracture apertureAqz [M�1 L2] specific surface area in the matrixAfr

qz ½M�1 L2� specific surface area in the fractureAs [L2] active surface areaB [MOL�1 L3] coefficient in the Jones–Dole equation~c ½L2 T�2 #�1� specific heatC [M L�3] solute concentration, expressed as volumetric

massD [–] Marshall–Chen coefficientDd [L2 T�1] free-solution diffusion coefficientDij [L2 T�1] hydrodynamic matrix dispersion tensorDfr

ij ½L2 T�1� hydrodynamic fracture dispersion coefficientEa [MOL�1 M L2 T�2] activation energyg [L T�2] acceleration due to gravityh0 [L] equivalent freshwater headI+, I� [–] fracture–matrix interfacek [M L T�3 #�1] thermal conductivityk0þ ½MOL L�2 T�1� dissolution reaction constant in

deionized waterkcorrþ ½MOL L�2 T�1� dissolution reaction constant in

saltwaterKad [MOL�1 M] equilibrium adsorption coefficientKd [M�1 L3] equilibrium distribution coefficientK fr

d ½L� fracture–surface distribution coefficientKeq [MOL M�1] equilibrium constantK0

ij ½LT�1� coefficients of hydraulic conductivity tensorof freshwater

K fr0 ½LT�1� hydraulic freshwater conductivity of the

fracture‘v [L] geometry of the model domain; v = x,y,z

Lv [L] geometry of a block element; v = x,y,z

m [MOL M�1] molal concentrationM [MOL L�3] molar concentrationMw [M] mass of waterP [M L�1 T�2] dynamic pressure of the fluidPPM [–] mass parts per millionqi [L T�1] Darcy fluxr� [MOL M�1 T�1] precipitation (backward) reaction

rater+ [MOL M�1 T�1] dissolution (forward) reaction raternet [MOL M�1 T�1] net molal production raterM [MOL L�3 T�1] net molar production rateR [–] retardation factorR* [MOL�1 M L2 T�2 #�1] universal gas constant

R = 8.3144 mol�1 kg m2 s�2 K�1

Rfr [–] fracture retardation factorSS [L�1] specific storage of the porous matrixSfr

S ½L�1� specific storage of an open fracture

t [T] timeT [#] absolute temperature in KelvinTC [#] relative temperature in centigradevi [L T�1] linear flow velocityVqz [MOL�1 L3] molar volume of quartz

Greek letters

afl [M�1 L T2] coefficient of the compressibility of thefluid due to fluid pressure or hydraulic head vari-ations

al [L] matrix longitudinal dispersivityafr

l ½L� longitudinal fracture dispersivityam [M�1 L T2] coefficient of the compressibility of the

porous medium due to fluid pressure or hydrau-lic head variations

at [L] matrix transverse dispersivityafr

t ½L� transverse fracture dispersivityasalt [M�1 L3] solutal expansion coefficientcr [–] activity coefficient of species rCm variable mass sources and sinksdij [–] Kronecker delta functiongj [–] indicator for flow directionhr [–] fraction of sites occupied by cation rjij [L2] coefficients of the intrinsic permeability tensork [T�1] decay constantK [M T�3] convective–dispersive–conductive loss or

gain of heatl [M L�1 T�1] dynamic viscosity of the fluidq [M L�3] densityqr [–] relative fluid densityq~c ½M L�1 T�2 #�1� heat capacitys [–] factor of tortuosity/ [–] porosity of the rock matrix/qz [–] quartz volume fractionu [1�] fracture inclinex [–] fracture roughness coefficientX [M M�1 T�1] advective–dispersive–diffusive loss or

gain of solute mass

Sub- and superscripts

0 [–] reference fluidb [–] bulkfr [–] fracturei, j [–] spatial indicesinit [–] initial time levell [–] liquid phaseL [–] time leveln [–] normal directions [–] solid phaser [–] species

Special symbols

o [–] partial differential operatorn [–] difference

T. Graf, R. Therrien / Advances in Water Resources 30 (2007) 742–771 743

744 T. Graf, R. Therrien / Advances in Water Resources 30 (2007) 742–771

critical safety questions arise due to the presence of frac-tures in low-permeability rock formations. Fractures havea great impact on mass transport because they representpreferential pathways where accidentally released contam-inants migrate at velocities that are several orders of mag-nitude larger than within the rock matrix itself.

Significant increases in temperature can promote rock-fluid interactions such as mineral dissolution and precipita-tion. Physical properties, such as matrix permeability andfracture aperture are modified if chemical rock-fluid inter-actions occur. These modifications in physical propertiescan be significant because the Cubic Law states that, forexample, an increase of the fracture aperture by 26% dou-bles the discharge through this fracture. The high numberof conceivable feedback scenarios between variable-densityflow and reactive solute transport demonstrates that thetwo processes are strongly coupled. This is especially thecase in fractured media where high groundwater flowvelocities enable rapid transport of reactive species to thelocation of the chemical reaction and away from it. Clearly,the ability to predict the transport behavior of hazardouschemicals leaked to the geosphere is essential.

Processes that affect the fluid properties on one handand induce reactions on the other hand are elevated tem-peratures and high salt contents. Numerical models arevery helpful tools for studying the behavior and predictinglong-term effects in such complex systems. Models thatcouple variable-density flow with reactive transport are rel-atively new and ‘‘the development of these codes has onlyjust begun’’ [50]. Already available models vary greatly intheir coupling method and in model sophistication [50].

1.2. Prior studies

Reactive transport models are typically limited to thechemical system being investigated. Several modellingstudies considered multiple mobile species undergoing reac-tions [25,56,85,65,81,88,58,6,68,67,59,26,51,28,55,27,21,41,22].Some models consider reactions with aqueous silica(H4SiO4) as the only mobile reactive component[34,66,83]. In this case, the reactive transport equationremains linear and iterative solvers for nonlinear equations,such as those described by Steefel and MacQuarrie [68], donot need to be applied.

Several authors examined reactive solute transport inporous media assuming constant water density (e.g.[68,69]). Bolton et al. [6] and Freedman and Ibaraki [21]investigated the impact of density-driven flow on chemicalreactions. Bolton et al. [6] studied coupled thermal convec-tion and quartz dissolution/precipitation for large spatialand temporal scales. They found that long-term changesof porosity and permeability can either increase the flowvelocities and the degree of subsaturation (in regions of dis-solution) or decrease flow rates and the degree of supersat-uration (in regions of precipitation). However, Bolton et al.[6] did not account for the salinity dependency of thekinetic rate law nor for the salinity effect on water density

and viscosity. Freedman and Ibaraki [21] numerically sim-ulated the horizontal migration of a dense plume in anunfractured porous medium where density varies withsalinity but not with temperature. The results were com-pared with simulations where chemical reactions areignored. The most important outcome of the Freedmanand Ibaraki [21] study was that chemical reactions do notsignificantly impact density-driven flow in porous media.However, this finding may not be always valid becausethe type of chemical system studied will influence the out-come. In Freedman and Ibaraki [21], the solubility of thesolid phase (calcite) is fairly low and coupling between flowand reactive transport is therefore weak. In addition, [21]focused on a small temporal scale and did not studylong-term effects. They also ignored the influence of tem-perature and salinity on both solute solubility and reactionkinetics.

Simulations of reactive transport in fractured systemshave previously been carried out by a number of authors[65,67,28,27]. Not all of the studies addressed the questionof how dissolution/precipitation reactions will alter frac-ture aperture and matrix permeability and, thus, impactthe flow field. Modifications of flow parameters were eithernot considered [28] or only applied to the permeability ofthe porous matrix [27]. However, other investigationsshowed that chemical reactions within open fractures trig-ger complex reaction-flow feedback scenarios [65,67] andthat fracture aperture cannot be assumed constant.

Steefel and Lichtner [67] studied the infiltration of ahyper-alkaline fluid along a discrete fracture. Theyobserved that within tens of years, the permeability feed-back between reaction and transport is significant. Theyalso found that fluid flow through the fracture is likely tobe restricted, or even stopped, due to self-sealing if reactionrates in the fracture are only one order of magnitude largerthan in the adjacent matrix. On the other hand, if the ratesare of the same order of magnitude, the porous matrix willbe cemented first.

While most studies ignored density variations wheninvestigating reactive transport in fractured media[67,28,27,65,49] fully accounted for thermal density-drivenflow. According to Steefel and Lasaga [65], geothermalconvection cells in reactive fractured media are never stablebecause upwelling fluids cool and the resulting precipita-tion of minerals significantly reduces permeability leadingto highly dispersive plumes. On the other hand, if fluidsmove downward to a zone of higher temperature, dissolu-tion reactions locally increase permeability leading to chan-nelling of flow. However, Steefel and Lasaga [65] did notaccount for the impact of salinity on both reaction kineticsand fluid properties.

Being designed for simulating flow in salt domes, theTOUGH2 model accounts for thermohaline impact onfluid density and viscosity [51]. Oldenburg and Pruess [49]used this model to simulate thermohaline convection inporous media. In addition, TOUGH2 can simulate flowand transport in fractured media using the dual permeabil-


ity approach. TOUGH2 also simulates reactive transportof salt (NaCl), including temperature-dependent solubilityand dissolution/precipitation of salt. However, Pruesset al. [51] did not account for the impact of salinity on reac-tion kinetics. Nevertheless, other studies suggested that therate-enhancing effect of salt is significant [13] and that fluidsalinity also impacts the reactive species solubility [40,36].

Table 1 shows a selection of previous studies on coupledvariable-density flow and reactive transport. It highlightssubtle differences between model assumptions made by var-ious authors. The studies are not listed chronologically butaccording to increasing model complexity.

This paper presents the development and application ofa numerical model that simulates dense plume migration ina chemically reactive fractured geologic material, for non-isothermal conditions. This new model is based on theexisting FRAC3DVS model, which solves variable-satu-rated and multi-component transport in discretely frac-tured porous media [76]. The model can be used tosimulate the long-term behavior of coupled thermohalineflow and reactive solute transport in fractured media. Themodel developed in the present study continues the seriesof increasing model complexity and provides simulationcapabilities previously lacking (Table 1).

Table 1A selection of previous studies on reactive solute transport in porous and frac

Authors Simulated processes

Reactive transport Heat transfer

in PMa in FMb in PM in FM

Steefel and MacQuarrie [68]pe – – –

Johnson et al. [34]p/,A – – –

Steefel and Yabusaki [69]p/,A – – –

Freedman and Ibaraki [21]p/,j,A – – –

White and Mroczek [83]p/,j,A – – –

Bolton et al. [6]p/,j,A –

p–

Ghogomu and Therrien [28]pe pe – –

Geiger et al. [27]pj pe – –

Steefel and Lichtner [67]p/,A p(2b),A – –

Steefel and Lasaga [65]p/,A p(2b),A p p

Pruess et al. [51]p/,j p/,j,DP p p

Present studyp/,j,A p(2b),A p p

The models are listed in order of increasing complexity. If density is a functiondensity flow.DP Dual permeability approach used.

a Porous media.b Fractured media.c Salinity.d Temperature.e No change of simulation parameters considered.f Multi-species reactive transport.

qz Single-species reactive transport (silica) and nonreactive transport (electroly(2b) Change of fracture aperture considered.

j Change of matrix permeability considered./ Change of matrix porosity considered.A Change of specific mineral surface area considered.

2. Physicochemical system

2.1. Chemical system

We have chosen the quartz-water chemical systembecause silicate minerals are the most abundant mineralsin the earth’s crust, accounting for 90% of the total mass[35]. The focus will be on a-quartz, the most commonSiO2 polymorph in the upper crust. Several studies under-lined the importance to examine the reactive nature ofquartz-rich rock. Fournier [18,20] demonstrated thatquartz precipitation may significantly decrease permeabil-ity and thus lead to self-sealing of the rock matrix. Theessential role of quartz dissolution has been studied since1884 [9] and the high number of more modern studies[7,14,75,15,83,13] indicates that quartz dissolution is alsoa recent subject of intensive research.

It is assumed that aqueous silica is the only species thatundergoes chemical interactions (dissolution/precipitation)with the rock matrix and it is thus termed ‘‘reactive spe-cies’’. Other species that undergo sorption reactions butnot dissolution/precipitation will not be termed ‘‘reactive’’.Therefore, the model presented here is a single-species reac-tive transport model.

tured porous media

Densityfrom

Viscosityfrom

Reactionkineticsfrom

Reactivespeciessolubilityfrom

Cc Td C T C T C T

– – – – – –pf –

– – – – –p

–qz p

– – – – –p pf p

p–

p– – –

pf –– – – –

p p pqz pp p

–p

–p pf p

– – – – – –pf –

– – – – – –pf –

– – – – – –pf –

–p

–p

–p pf p

p p p p–

p pf pp p p p p p pqz p

of salinity and/or temperature, the model couples reactions with variable-

tes).


Rimstidt and Barnes [53] experimentally studied thequartz-water system, described by the reaction

SiO2ðsÞ þ 2H2OðaqÞ �rþ

r�H4SiO4ðaqÞ ð1Þ

where r+ and r� [both MOL M�1 T�1] are the dissolutionand precipitation rates of quartz (SiO2), respectively, andwhere H4SiO4 is aqueous silica. Upon applying the lawof mass action to reaction (1), the net rate of silica produc-tion in the porous matrix, rnet, can be written as [53,15,13]

rnet ¼ /qzkcorrþ Aqz 1�

cH4SiO4

Keq

mH4SiO4

� �ð2Þ

where /qz [–] is the volume fraction of quartz,kcorrþ ½MOL L�2 T�1� is the dissolution rate constant, cor-

rected for salt water, Aqz [M�1 L2] is the specific quartz sur-face area, cH4SiO4

½–� is the activity coefficient of silica, Keq

[MOL M�1] is the quartz solubility or equilibrium constantof reaction (1) and mH4SiO4

½MOL M�1� is the molal con-centration of silica. It is assumed here that pure solidsand pure liquids, for example SiO2 and H2O, have activitiesequal to unity [35].

For a discrete fracture, the net rate of silica productionis given by

rfrnet ¼ /qzk

corrþ Afr

qz 1�cH4SiO4

Keq

mfrH4SiO4

� �ð3Þ

where superscript ‘‘fr’’ refers to fracture and where othervariables are defined similarly to those used for the porousmedium.

This chemical model is based on transition state theoryand simulates a zeroth/first order chemical kinetic reaction.The model is similar to that used by Rimstidt and Barnes[53] and White and Mroczek [83], except that the rate lawin (2) and (3) is also a function of the quartz volume frac-tion, /qz, as proposed by Johnson et al. [34]. The followingparagraphs present in more detail parameters kcorr

þ , cH4SiO4

and Keq.

2.1.1. Corrected dissolution rate constant, kcorrþ

The corrected dissolution rate constant, kcorrþ , is obtained

from the dissolution rate constant in deionized water, k0þ,

which is given by the Arrhenius equation as (e.g. [37,69])

k0þ ¼ k0

25 exp�Ea

R�1

T� 1

298:15

� �� ð4Þ

where k025 ½MOL L�2 T�1� is the known dissolution rate

constant in deionized water at 25 �C, Ea [MOL�1 M L2

T�2] is the activation energy necessary to overcome the po-tential energy maximum of the transition state and T [#] isabsolute temperature. Values of the universal gas constant,R* [MOL�1 M L2 T�2 #�1], and constant k0

25 are given inRimstidt and Barnes [53] as 8.3144 mol�1 kg m2 s�2 K�1

and 4.3 · 10�14 mol m�2 s�1, respectively. Values of Ea

for quartz dissolution have been reported in the range be-tween 36 and 96 kJ mol�1, and a value 75.0 kJ mol�1 is

assumed here as proposed by Rimstidt and Barnes [53]and used in Steefel and Lasaga [65] as well as in the soft-ware packages OS3D and GIMRT [69].

Eq. (4) holds for deionized water but [14] have shownthat the presence of electrolytes in the fluid can increasethe reaction rate by 1.5 orders of magnitude. The increaseis caused by adsorbed cations that change the structure ofthe mineral surface and make the surface more vulnerableto water dipole attacks. According to Dove and Nix [15],the concentration of the bivalent (IIA) cations (Mg2+,Ca2+) have the greatest impact on dissolution rate whilethe effect of monovalent (IA) cations (Na+, K+) is minordue to their less effective adsorption.

Dove [13] demonstrated that the fraction of adsorptionsites occupied by species Na+, Mg2+ and Ca2+ can beexpressed by a Langmuir model for equilibrium adsorption[54,5,14] as

hNaþ ¼KNaþ

ad mNaþ

1þ KNaþ

ad mNaþ þ KMg2þ

ad mMg2þ þ KCa2þ

ad mCa2þ

ð5Þ

hMg2þ ¼KMg2þ

ad mMg2þ

1þ KNaþ

ad mNaþ þ KMg2þ

ad mMg2þ þ KCa2þ

ad mCa2þ

ð6Þ

hCa2þ ¼ KCa2þ

ad mCa2þ

1þ KNaþ

ad mNaþ þ KMg2þ

ad mMg2þ þ KCa2þ

ad mCa2þ

ð7Þ

where hr [–] is the fraction of sites occupied by cation r, mr

[MOL M�1] and Krad ½MOL�1 M� are the molal concentra-

tion and the equilibrium adsorption coefficient of cation r,respectively. Because sorption constants, Kr

ad, for cationson quartz at hydrothermal temperatures are unknown[15], we used Kr

ad values at 20 �C. These values are equalto 101.78, 103.7 and 103.35 mol�1 kg for sodium, magnesiumand calcium, respectively [13].

The constant krþ is computed from a fit to experimen-

tal data published by Dove [13] who measured thedependence of quartz dissolution rates on different elec-trolyte concentrations of sodium chloride, magnesiumchloride and calcium chloride at 200 �C. The logarithmof the dissolution rate constant for Na+, Mg2+ andCa2+ can be written as

log kNaþ

200 ¼ �2:8� 10�4

mNaþ� 6:35 ð8Þ

log kMg2þ

200 ¼ � 2:2� 10�4

mMg2þ� 6:80 ð9Þ

log kCa2þ

200 ¼ �1:3� 10�6

ðmCa2þÞ2� 6:35 ð10Þ

where log denotes the decadic logarithm log10. With thehelp of the Arrhenius equation (4) and with Eqs. (8)–(10), the dissolution rate constant of species r at any con-centration and temperature can be formulated as

krþ ¼ kr

200 exp�Ea

R�1

T� 1

473:15

� �� ð11Þ


The corrected dissolution rate constant, kcorrþ , accounts for

the presence of Na+, Mg2+ and Ca2+ in the solution and isdefined here as

kcorrþ ¼ kNaþ

þ hNaþ þ kMg2þ

þ hMg2þ þ kCa2þ

þ hCa2þ

þ k0þ 1� hNaþ þ hMg2þ þ hCa2þ

� �h ið12Þ

where Dove’s [13] idea of competitive adsorption isadapted in order to account for protons ‘‘adsorbed’’ onthe remaining sites, expressed by the last term in Eq.(12). This last term does not occur in Dove’s [13] originalformulation of the dissolution rate constant in a mixedelectrolyte solution but it is necessary to obtain a correctrate constant in water of very low salinity. In that case,the values of mr and hr approach zero and kcorr

þ becomesapproximately equal to k0

þ.Note that competitive adsorption of cations is only used

in a qualitative sense to assess the effect of salinity on dis-solution rates. This is because the model does not explicitlysimulate competitive adsorption but sorption of cations issimulated with a linear isotherm (Eqs. (39) and (42) in Sec-tion 3.2).

Electrolyte concentration and fluid temperature are themain factors that affect quartz dissolution rates. BetweenpH 8 and pH 12, quartz dissolution is also a function ofwater acidity [7]. However, the expected range of pH valuesof water in the quartz-rich rock considered here is below 7.This pH is controlled by silica dissolution and subsequentbuffering by the silicic acid buffer [47]. For pH values below7, changes of dissolution rates are ‘‘small [. . .] and difficultto interpret’’ [3]. Therefore, the pH dependency of quartzdissolution rates is neglected in this study.

2.1.2. Activity coefficient, cH4SiO4

In an electrolyte solution, the solubility of a neutral spe-cies, such as H4SiO4, is a function of the amount of dis-solved salt and temperature. Marshall and Chen [40]proposed a modified form of the Setchenow equation tocalculate the activity coefficient of H4SiO4 in a mixed elec-trolyte solution at any given temperature:

log cH4SiO4¼X

r

Drmr ð13Þ

where Dr is the dimensionless, temperature dependentMarshall–Chen coefficient of ion r and mr is the molal con-centration [MOL M�1] of r. Marshall and Chen [40] pro-vide temperature-dependent values of Dr for species Na+,Mg2+, Cl� and SO2�

4 in the range 25–300 �C. Due to thephysicochemical similarity of Mg2+ and Ca2+, their Mar-shall–Chen coefficients are assumed to be identical, suchthat DCa2þ ¼ DMg2þ . The further assumption is made thatDr can be extrapolated beyond the 25–300 �C temperaturerange down to 0 �C.

2.1.3. Equilibrium constant, Keq

In this model, the equilibrium constant, Keq, is expressedas a function of the absolute temperature, T, over the range0–300 �C [52] such that

log Keq ¼ �1107:12

T� 0:0254 ð14Þ

Other models use an apparent solubility, which also ac-counts for the impact of salinity [19,80,62,46]. However,the present model takes ion activity into consideration bycalculating nonzero Marshall–Chen coefficients in Eq.(13), thus using silica activity coefficients that are greaterthan or equal to one. In the calculation of the quartz solu-bility, it is further assumed that below pH 9, there is noinfluence of pH on quartz solubility [35,72] and that thepressure effect on quartz solubility is not significant withinthe temperature range considered in the scope of this study[83].

2.2. Physical system

In thermohaline convective systems, the fluid densityand viscosity depend on temperature and salinity, whilethe effect of pressure on fluid properties can be ignored[6]. In the model, both fluid quantities are first calculatedas a function of temperature alone q0

T ; l0T

� and then cor-

rected according to salinity.

2.2.1. Fluid density, qUnder isobaric conditions, the fluid density is calculated

as a function of temperature for different temperatureranges [32]:

q0T ¼

1000 � ð1� ð½T C � 3:98�2=503; 570Þ�ð½T C þ 283�=½T C þ 67:26�ÞÞfor 0 �C < T C 6 20 �C

996:9 � ð1� 3:17� 10�4½T � 298:15��2:56� 10�6½T � 298:15�2Þ

for 20 �C < T C 6 175 �C

1758:4þ 1000 � T ð�4:8434� 10�3

þT ð1:0907� 10�5 � T � 9:8467� 10�9ÞÞfor 175 �C < T C 6 300 �C

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð15Þwhere TC [#] and T [#] are the temperatures in centigradeand Kelvin, respectively. In a second step, the fluid densityat any given salinity and temperature is evaluated using thefreshwater density at temperature, q0

T , and from the sum ofall species concentrations using the following empiricalrelation:

q ¼ q0T þ asalt �

Xr

Cr ð16Þ

where asalt [M�1 L3] is the solutal expansion coefficient.Due to the low solubility of quartz, the impact of dissolvedsilica on fluid density is not significant [19,46] and, there-fore, ignored in Eq. (16). The model calculates density fromthe concentration of eight major ions found in naturalwaters: Na+, K+, Ca2+, Mg2+, Cl�, SO2�

4 , CO2�3 and

HCO�3 . The Pitzer ion interaction model is used [44,45],


where the fluid density is derived from the partial electro-lyte volumes. The Monnin model is used to derive anempirical expression for asalt as a function of the ground-water chemistry in the Canadian Shield given by Farvoldenet al. [17] in the form

asalt ¼ �0:0829 � lnX

r

Cr

!þ 1:1415 ð17Þ

where the unit of Cr [M L�3] is mg l�1.

2.2.2. Fluid viscosity, lAlthough fluid viscosity is assumed constant in some

thermohaline transport studies [70,78,79,16,8,86], it isrecommended to relate viscosity to both temperature [6]and salinity [21] because it can increase by a factor oftwo between pure water and a dense brine [49]. Relationsto calculate fluid viscosity for different temperature rangesare

l0T ¼

1:787� 10�3 � expðð�0:03288þ 1:962� 10�4 � T CÞ � T CÞfor 0 �C < T C 6 40 �C

10�3 � ð1þ 0:015512 � ½T C � 20�Þ�1:572

for 40 �C < T C 6 100 �C

0:2414 � 10^ð247:8=½T C þ 133:15�Þfor 100 �C < T C 6 300 �C

8>>>>>>>><>>>>>>>>:

ð18Þ

The relation for temperatures below 40 �C has been usedby Molson et al. [43] and the other two relations are givenby Holzbecher [32]. Viscosity can be expressed as a func-tion of salinity and temperature by substituting the temper-ature-dependent freshwater viscosity, l0

T , in the Jones–Doleequation:

l ¼ l0T � 1þ

Xr

BrMr

!ð19Þ

where Mr is the molar concentration of species r. Marcus[39] gives values of the B-coefficients [L3 MOL�1] for allspecies considered here.

2.3. Solid phase properties

Chemical reactions have a significant impact on a num-ber of physical flow and transport properties. In the model,the individual quartz volume fraction, /qz [–], is recalcu-lated using [69,65]:

o/qz

ot¼ �V qzrLþ1

M ð20Þ

where Vqz [MOL�1 L3] is the molar volume of quartz andrLþ1

M ½MOL L�3 T�1� is the molar reaction rate at time levelL+1. In finite difference form, this equation becomes

/Lþ1qz ¼ /L

qz � DtV qzrLþ1M ð21Þ

where Dt = tL+1 � tL [T] is the time step size. The molarvolume of a mineral is the ratio of its molecular weight

to its density [36]. The model updates porosity from thesum of all mineral volume fractions:

/Lþ1 ¼ 1�X

r

/Lþ1r ð22Þ

where it is assumed that quartz is the only reactive solidspecies and that /r is constant for any given mineral r ex-cept quartz. The specific surface area in the porous matrixis updated by means of the two-thirds power relation givenby Steefel and Yabusaki [69] as

ALþ1qz ¼ Ainit

qz

ð/Lþ1=/initÞ � /Lþ1qz =/init

qz

� �h i2=3

dissolution of quartz

ð/Lþ1=/initÞ2=3

precipitation of quartz

8>>>>><>>>>>:

ð23Þ

where Ainitqz ½L2 M�1� is the initial specific surface area in

the matrix and where /init [–] and /initqz ½–� are the initial

matrix porosity and quartz fraction, respectively. Thematrix permeability, jij [L2], is calculated from porosityfor the special case of dissolution and precipitation ofquartz [82]:

jLþ1ij ¼ jinit

ij � 1� 1� /Lþ1 � /c

/init � /c

!1:5824

35

0:468<:

9=; ð24Þ

where jinitij ½L

2� is the initial permeability and /c [–] is thecritical porosity at which jij = 0. This relation is obtainedfrom theoretical considerations of deposition and dissolu-tion of quartz grains, arranged in a rhombohedral arrayof uniform spheres [82].

Similar to the matrix porosity, fracture apertures arerecalculated from [66,67]:

ð2bÞLþ1 ¼ ð2bÞL � 1þ DtV qzrLþ1M

� ð25Þ

Finally, the specific surface area in the fracture is updatedusing [66]:

Afr;Lþ1qz ¼ Afr;init

qz � ð2bÞLþ1

ð2bÞinit

!ð26Þ

where Afr;initqz ½L2 M�1� is the initial specific surface area in

the fracture and where (2b)init [L] is the initial fracture aper-ture. The initial surface area in a 2D rectangular fractureelement is the ratio between active surface area, given byAs = 2xLxLz, and the mass of water stored in the 2D ele-ment, Mw = q Æ (2b)LxLz, where x [–] is the fracture rough-ness coefficient (Fig. 1).

Thus, a fluid moving through a large fracture willencounter less mineral surface area per unit fluid mass thana fluid moving through a narrow fracture. That relation-ship is expressed by the following equation for the initialspecific surface area:

Afr;initqz ¼ x

qbð27Þ

Fig. 1. Fracture roughness coefficient for rough-walled (left) and smoothfractures.


3. Governing equations

3.1. Variable-density, variable-viscosity flow

The model uses the equivalent freshwater head h0 [L],defined by Frind [24] as

h0 ¼P

q0gþ z ð28Þ

where P [M L�1 T�2] is the dynamic fluid pressure, q0 [ML�3] is the reference fluid density, g [L T�2] is the gravita-tional acceleration and z [L] is the elevation above datum.

The 3D variable-density, variable-viscosity Darcy flux inporous media can be completely expressed in terms offreshwater properties [24,30]:

qi ¼ �K0ij

l0

loh0

oxjþ qrgj

� �; i; j ¼ 1; 2; 3 ð29Þ

where the assumption of a horizontal datum (i.e., oz/ oz =1) is made and where gj [–] represents the direction of flowwith gj = 0 in the horizontal directions and gj = 1 in thevertical direction [24]. The relative density, qr [–] is givenby qr = (q/ q0) � 1. Using the ratio between the referencefluid viscosity, l0, and the fluid viscosity, l [both M L�1

T�1], the hydraulic conductivity of the porous mediumfor freshwater, K0

ij ½L T�1�, is [1]

K0ij ¼

jijq0gl0

ð30Þ

The 2D Darcy flux in differently oriented 2D fracture facesis calculated using a form of the Darcy equation similar tothat presented by Graf and Therrien [30]:

qfri ¼ �K fr

0

l0

lfr

ohfr0

oxjþ qfr

r gj cos u

� �; i; j ¼ 1; 2 ð31Þ

where gj is 0 in the horizontal direction and 1 along thefracture incline. The incline of a fracture face is given byu with u = 0� for a vertical face and u = 90� for a horizon-tal face. In the case of flow within fractures, a local 2DCartesian coordinate system is assumed. The freshwaterhydraulic conductivity of the fracture, K fr

0 ½L T�1�, is de-rived from the parallel plate model as

K fr0 ¼ð2bÞ2q0g

12l0

ð32Þ

where (2b) [L] is the fracture aperture. The application ofDarcy’s law in fractures (31) requires that the Reynoldsnumber be smaller than 1 [2].

The equation that governs variable-density, variable-vis-cosity flow in porous media has the following 3D form[24,30]:

o

oxiK0

ij

l0

loh0

oxjþ qrgj

� �� ¼ SS

oh0

ot; i; j ¼ 1; 2; 3 ð33Þ

The specific storage, SS [L�1], accounts for both matrix andfluid compressibility and is defined as [1]

SS ¼ q0gðam þ /aflÞ ð34Þwhere am and afl [both M�1 L T2] are the matrix and fluidcompressibility, respectively.

Flow in an open discrete fracture is assumed 2D here.Therefore, the corresponding governing equation is definedin a local 2D coordinate system. The governing flow equa-tion in fractured media is similar to that presented by sev-eral authors [4,73,64,77,30]:

ð2bÞ o

oxiK fr

0

l0

lfr

ohfr0

oxjþ qfr

r gj cos u

� �� Sfr

S

ohfr0

ot

�þ qnjIþ

� qnjI� ¼ 0; i; j ¼ 1; 2 ð35Þ

where the last two terms represent normal components offluid flux across the boundary interfaces (I+ and I�) thatseparate the fracture and the porous matrix. In the concep-tual model, fractures are idealized as 2D parallel plates.Therefore, both the total head, hfr

0 , and the relative density,qfr

r , are uniform across the fracture width. The specific stor-age in an open fracture, Sfr

S ½L�1�, can be derived from (34)by assuming that the fracture is essentially incompressible,such that am = 0, and by setting its porosity to 1:

SfrS ¼ q0gafl ð36Þ

3.2. Reactive solute transport

The governing reactive transport equation in porousmedia has the 3D form [1]

o

oxi/Dij

oCoxj� qiC

� �þ Cm ¼

oð/RCÞot

; i; j ¼ 1; 2; 3

ð37Þ

where C [M L�3] is solute concentration. In this form of thetransport equation, the assumptions of fluid incompress-ibility and constant fluid density are made. The coefficientsof the hydrodynamic dispersion tensor, Dij [L2 T�1], are gi-ven by Bear [1] as

/Dij ¼ ðal � atÞqiqj

jqj þ atjqjdij þ /sDddij; i; j ¼ 1; 2; 3

ð38Þ


where al [L] and at [L] are the longitudinal and transversedispersivity, respectively, s [–] is matrix tortuosity, Dd [L2

T�1] is the free-solution diffusion coefficient and dij [–] isthe Kronecker delta function. The dimensionless retarda-tion factor, R, is given by Freeze and Cherry [23] as

R ¼ 1þ qb

/Kd ð39Þ

where qb [M L�3] is the bulk density of the porous mediumand Kd [M�1 L3] is the equilibrium distribution coefficientdescribing a linear Freundlich isotherm.

The source/sink term, Cm [M L�3 T�1], is �/kRC forradioactive components with decay constant k [T�1]. Forchemically reactive species such as silica, the governingtransport equation is obtained from (37) by replacing theconcentration, C, by the silica molality, m, and by settingthe source/sink term, Cm [now MOL M�1 T�1], equal tothe net reaction rate, rnet, given by (2). Yeh and Tripathi[85] argue that precipitation/dissolution reactions andsorption cannot be simulated simultaneously if the aqueouscomponent is the primary dependent species. Thus, the dis-tribution coefficient of silica must be set to zero in Eq. (39).

Therrien and Sudicky [76] give the equation thatdescribes 2D solute transport in a discrete fracture as

ð2bÞ o

oxiDfr

ij

oCfr

oxj� qfr

i Cfr

� �þ Cfr

m � Rfr oCfr

ot

�þ XnjIþ

� XnjI� ¼ 0; i; j ¼ 1; 2 ð40Þ

where Dfrij ½L

2 T�1� is the hydrodynamic dispersion coeffi-cient of the fracture, calculated as

Dfrij ¼ ðafr

l � afrt Þ

qfri qfr

j

jqfrj þ afrt jqfrjdij þ Dddij; i; j ¼ 1; 2

ð41Þwhere afr

l and afrt [both L] are the longitudinal and trans-

verse fracture dispersivity, respectively. The dimensionlessfracture retardation factor, Rfr, is given by [23]

Rfr ¼ 1þ 2K frd

ð2bÞ ð42Þ

where K frd ½L� is the fracture–surface distribution coeffi-

cient. The source/sink term in Eq. (40) is Cfrm ¼ �kRfrCfr

for radioactive chemicals and Cfrm ¼ rfr

net for the silica spe-cies. The last two terms in (40) represent advective–disper-sive–diffusive loss or gain of solute mass across thefracture–matrix interfaces I+ and I� [73]. Sorption reac-tions of silica must be neglected, thus K fr

d ¼ 0 for silica [85].Note that Eqs. (39) and (42) simulate a linear sorption

isotherm but calculating corrected dissolution rate con-stants (Eq. (12)) assumes competitive adsorption of cat-ions. Although assumed in Section 2.1, competitiveadsorption is not actually simulated but only used in aqualitative sense to quantify the rate-enhancing effect ofsalt.

The reactive source/sink term always consists of a firstorder reaction term representing precipitation and a con-

stant term of zeroth order describing dissolution. Thus,both solute transport equations in porous and fracturedmedia are linear, allowing a one-step solution. Therefore,neither an iterative operator splitting, two-step schemenor a computationally demanding fully-coupled, one-stepapproach are required. Performing a Newton Iterationand formulating Jacobian matrix entries would have highlycomplicated the model development.

3.3. Heat transfer

The convective–dispersive–conductive heat transferequation in porous media can be written in a form similarto that given by Molson et al. [43] as

o

oxikb þ /Dijql~cl

� oToxj� qiql~clT

� �¼ qb~cb

oTot; i; j ¼ 1; 2; 3

ð43Þ

where kb [M L T�3#�1] is the bulk thermal conductivity, q[M L�3] is density and ~c ½L2 T�2#�1� is specific heat. Theabsolute temperature, T [#], is the average temperature be-tween the solid and the liquid phase [12]. The subscripts ‘‘l’’and ‘‘b’’ refer to the liquid and the bulk phase, respectively.It is assumed that the gaseous phase is absent and thatexternal heat sinks and sources due to chemical reactions(dissolution/precipitation) are negligibly small. Bulk prop-erties qb~cb ½M L�1 T�2#�1� and kb can be quantified con-sidering the volume fractions of the solid and the liquidphases according to Bolton et al. [6]

qb~cb ¼ ð1� /Þqs~cs þ /ql~cl ð44Þkb ¼ ð1� /Þks þ /kl ð45Þ

where subscript ‘‘s’’ refers to the solid phase.Heat transport in an open discrete fracture can be

described with a 2D equation similar to Eqs. (40) and(43) in the form

ð2bÞ o

oxikl þ Dfr

ijql~cl

� � oT fr

oxj� qfr

i ql~clT fr

� �� ql~cl

oT fr

ot

�þ KnjIþ � KnjI� ¼ 0; i; j ¼ 1; 2 ð46Þ

The last two terms represent convective–dispersive–con-ductive loss or gain of thermal energy across the frac-ture–matrix interfaces I+ and I�. The temperature isuniform across the fracture width. Furthermore, it is as-sumed that, along the fracture–matrix interface, the tem-perature in the fracture and the adjoining matrix areidentical.

4. Numerical Formulation

4.1. The FRAC3DVS model

FRAC3DVS is a 3D variable-density saturated-unsatu-rated numerical groundwater flow and multi-componentsolute transport model. A detailed description of the modelcan be found elsewhere [76,77,30,31] and is not repeated


here. The FRAC3DVS model was modified to simulate thechemical system described in Section 2.1. The extendedmodel couples reactive transport with variable-density, var-iable-viscosity flow and with changes of solid phaseproperties.

4.2. Coupling flow and reactive transport

The processes of variable-density flow and reactive sol-ute transport are naturally coupled. Density variationscause weak nonlinearities in the flow equation. In thenumerical model, they are treated by means of a sequentialiterative approach (SIA), also called Picard iteration, whichlinks the two governing equations for flow and transport.This method alternately solves the two governing equationsduring each time step until convergence is attained.

Mineral dissolution/precipitation has a direct impact ona variety of physicochemical and material properties duringthe simulation. A change of porosity and fracture apertureaffects the active surface area, which, in turn, changes thenet rate of reaction (1). The change of such parameters is

FLOW

first time step

q

h0

DARCY

HE

ρ, μ

Niterations += 1

Niterations = 0

initial Cσ, ,T h0 next time step

update fluidproperties

Picard Iteration

thexc

rrepe

no

Fig. 2. Flow chart of the Picard Iteration with chemistry loop to couple vachemical reactions and parameter updates.

naturally fully coupled with flow, heat transfer and solutetransport. However, mineral volume fractions changemuch more slowly than do the solute concentrations inthe fluid [56,65,69,55]. Therefore, in the present model, likein other common geochemical models, fluid properties areupdated after each iteration of the Picard loop whereasmaterial properties are updated after each time step ratherthan after each iteration (Fig. 2). This procedure of recalcu-lating material parameters at the end of each time step iscalled the quasi-stationary state approximation and hasfirst been introduced by Lichtner [38]. Using the reactionrate at time level L+1 (implicit time weighting scheme) toupdate all model parameters ensures numerical stability[65].

The decoupled, two step approach to update materialproperties works well for relatively small time step sizes.However, if nonuniform time step sizes are used to acceler-ate the simulation, the time increment may become toolarge. As a consequence, high reaction rates may lead tounrealistically great changes in quartz volume fraction dur-ing a single time step. In this case, variable-density, vari-

convergenceAND Niterations > 1

noyes

T

TRANSPORTAT

Cσ

reactiveTRANSPORT

radioactiveTRANSPORT

update materialproperties

Loop

resholdeeded ?

educe Δtat time step

yes

riable-density, variable-viscosity flow and solute transport with external


able-viscosity flow and reactive transport are not satisfac-torily coupled with parameter changes because the timeincrement is too large. As a solution, the model dynami-cally reduces the time step size. This adaptive time steppingscheme eliminates unphysically large changes of quartzfraction in order to stabilize the simulation. With adaptivetime stepping, the time step size depends on the absolutechange of porosity according to

ðDtÞLþ1 ¼ /�

max j/Lþ1 � /LjðDtÞL ð47Þ

where /* is the maximum absolute change in porosity al-lowed during a single time step. Because quartz is consid-ered as the only reactive mineral, the threshold /* appliesto absolute changes of both porosity and quartz fraction.The model includes a verification of newly calculatedquartz fractions to ensure that they are not negative. Ifthe maximum change in porosity is greater than the al-lowed threshold, the fraction in Eq. (47) is less than 1and the updated new time step size is smaller than the pre-vious one. In this case, the old time step is repeated usingthe new reduced time increment, (Dt)L :¼ (Dt)L+1, withoutupdating the material properties. In fractured systems,the adaptive time stepping can also be based on absolutechanges in fracture aperture by using an expression similarto (47). If both time step size controllers (porosity controland aperture control) are used, the new time step size is cal-culated from the material whose time step multiplier issmaller.

Therrien and Sudicky [76] previously used adaptive timestepping in the simulation of variably-saturated flow. How-ever, the time stepping presented above is different fromthat used by Therrien and Sudicky [76] where time stepsare not repeated and where new time step sizes alwaysapply to the following time step.

5. Model verification

Simulations are presented here to verify the model for-mulation for reactive solute transport and heat transfer.The verification problems shown here for a single fractureare identical to the fracture–matrix geometry used by Tanget al. [74].

5.1. Reactive solute transport

This verification problem examines 2D advective–reac-tive transport in a single fracture, embedded in a porousmatrix. It is assumed that solutes migrate due to advectionin the fracture and due to molecular diffusion in the porousmatrix. Chemical reactions take place in the fracture and inthe matrix. Molecular diffusion and mechanical dispersionin the fracture are neglected, allowing an easier formula-tion of the analytical solution of Tang et al. [74] with noneed to numerically integrate. Groundwater in the fracturemigrates at a constant velocity. It is assumed that ground-

water is free of dissolved electrolytes, thus kcorrþ ¼ k0

þ andcH4SiO4

¼ 1. For simplicity, the molal concentration of silicawill be written as m. Heat transfer is not considered hereand a constant background temperature is imposed. It isfurther assumed that the material properties (i.e., matrixporosity, hydraulic conductivity, fracture aperture, mineralsurface area) are constant in time. Different mineral surfaceareas in the porous matrix and in the fracture are used,resulting in two different net reaction rates. This assump-tion does not correspond to Tang et al. [74] where theradioactive decay rates in fracture and matrix are identical.

Initially, the entire domain is in thermodynamic equilib-rium. Silica-free freshwater enters the fracture at a constantrate during the entire simulation, diluting the silica-satu-rated fluid in the fracture. All boundaries, except the frac-ture inlet and outlet, are impermeable for flow and areassigned zero-dispersive transport rates. The resulting dropof silica molality creates a thermodynamic disequilibriumand initiates quartz dissolution. Eventually, the systemreaches equilibrium between dilution and dissolution.

With m = m 0 + Keq and mfr ¼ mfr0 þ Keq, the governingequations of this problem using the new variables, m 0

and mfr0 , are given by Steefel and Lichtner [66] in the form:

om0

ot� Dd

o2m0

ox2þ

/qzk0þAqz

/Keq

m0 ¼ 0; b 6 x 61 ð48Þ

and

omfr0

otþ vfr omfr0

ozþ

/qzk0þAfr

qz

Keq

mfr0 � /Dd

bomfr0

ox

��x¼b

¼ 0; 0 6 z 61 ð49Þ

for reactive transport in the porous matrix and in the dis-crete fracture, respectively. The groundwater velocity inthe fracture is given by vfr [L T�1]. Using the new governingequations (48) and (49), both initial and boundary condi-tions are identical to those used in Tang et al. [74]. Theyare formulated mathematically by Steefel and Lichtner[66] who presented the steady state as well as the transientanalytical solutions.

In the numerical simulation, the finite element domain issimilar to the fracture–matrix system used by Tang et al.[74]. It is spatially discretized in the x-direction using agradually increasing Dx with factor 1.1 fromDx = 0.005 cm near the fracture to Dx = 0.1 cm at thedomain boundary. In the flow direction, the Dz increaseswith factor 1.25 from Dz = 0.001 cm near the source toDz = 0.1 cm at the domain boundary. All model parame-ters are summarized in Table 2.

Fig. 3 shows the concentration profile versus distancealong the fracture for both the analytical and the numericalsolution. Because initial thermodynamic equilibrium hasbeen assumed, silica concentrations are initially high andclose to the equilibrium constant (Fig. 3a). Inflowing fresh-water dilutes the silica-saturated water and decreases silicaconcentration (Fig. 3b and c). This leads to subsaturated

Table 2Model parameters used in the verification example for 2D reactive silicatransport in fractured porous media

Parameter Value

Constant backgroundtemperaturea (TC)

239 �C

Matrix porosity (/) 0.35Quartz volume fractionb (/qz) 0.65Specific surface area in

the matrixa (Aqz)54.2 m2 kg�1

Specific surface area inthe fracturec Afr

qz

� � 6.15 m2 kg�1

Free-solution diffusioncoefficientd (Dd)

1.0 · 10�10 m2 s�1

Fracture apertured (2b) 200 lmGroundwater velocity in

the fractured (vfr)1.9727 · 10�5 m s�1

Dissolution rate constante k0þ

� 1.3298 · 10�8 mol m�2 s�1

Equilibrium constante (Keq) 6.4996 · 10�3 m kg�1

Domain sizea (‘x,‘z) 2.0 cm, 3.1 cmLocation of cross-sections (z1,z2) 0.1‘z, 0.5‘z

Output times (t1, t2, t3, t4) 500 s, 1000 s, 2000 s and steady state

a Johnson et al. [34].b 1 � /.c From Eq. (27) with x = 1.0.d Steefel and Lichtner [66].e Computed by this model for deionized water at TC = 239 �C.

0

1

2

3

4

5

6

7

0 1 2 3distance along fracture [cm]

Si[m

mol

kg-1

]

0

50

100

150

Si[PPM

]

Analytical

Numerical

vfr

porous matrix

fracture m = Kfr eq0

m = K eq0

m =1fr 0

(a) (b)(c)

(d)

Fig. 3. Concentration profiles of 1D reactive transport of silica indiscretely fractured porous media. Shown are the silica molalities in thefracture at (a) 500, (b) 1000 and (c) 2000 s and at (d) steady state.

vfr

porous matrix

fracture m = Kfr eq0

m = Keq0

m =1fr 0

location of cross-sections

(a) (b)

0

1

2

3

4

5

6

7

0 0.1 0.2 0.3

distance into matrix [cm]

Si [m

mol

kg-1

]

0

50

100

150

Si [PPM]

Analytical

Numerical

(a)

(b)

Fig. 4. Concentration profiles of 1D reactive transport of silica indiscretely fractured porous media. Shown are the silica molalities in thematrix at steady state at the distances (a) 0.31 cm and (b) 1.55 cm from thesolute source.


conditions promoting dissolution of quartz. Because theambient temperature is high (239 �C), silica concentrationchanges within minutes [34]. Eventually, the system reachesa steady-state condition where dilution and dissolution arein equilibrium (Fig. 3d).

The discrepancy between the analytical and numericalsolution at early times was previously described by Steefeland Lichtner [66], who interpreted this as numerical disper-

sion. However, as the simulation proceeds in time, thisinconsistency diminishes and eventually vanishes after aninfinitely long period of time. Perfect match between theanalytical solution and the results from this model areobtained with the molal concentrations in the matrix asshown in Fig. 4.

5.2. Heat transfer

This test case verifies 2D heat transfer in a single frac-ture embedded in a porous matrix. This example is basedon analytical results presented by Meyer [42] for advectiveheat transfer in a fracture coupled with heat conduction ina surrounding porous matrix. Mechanical heat dispersionas well as conduction within the fracture are not consid-ered, making numerical integration unnecessary. Thegroundwater flow velocity in the fracture is constant.Under these assumptions, the governing equations for heattransport in the matrix and in the discrete fracture simplifyfrom (43) and (46) to

qb~cb

oTot� kb

o2Tox2¼ 0; b 6 x 61 ð50Þ

and

ql~cloT fr

otþ ql~clvfr oT fr

oz� kb

boT fr

ox

��x¼b

¼ 0; 0 6 z 61 ð51Þ

The last term in (51) expresses conductive loss of heat fromthe fracture into the matrix on the fracture–matrix inter-face. Initially, the entire system has a uniform temperatureequal to T0. The fluid entering the fracture has the constanttemperature equal to T1. All boundaries, except the fracture

11

12

13

14

15

tem

pera

ture

[°C

]

Analytical

Numerical(a)(b)

v fr

porous matrix

fracture T =fr 10 Co0

T = 10 Co0

T =1fr 15 Co

location of cross-sections

(a) (b)


inlet and outlet, are impermeable for groundwater flow andfor heat exchange. According to Meyer [42], the transientsolution along the fracture is

T fr � T 0

T 1 � T 0

¼ erfczffiffiffiffiffiffiffiffiffiffiffiffiffiffikbqb~cb

p2vfrql~clb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt � z=vfrÞ

p !

ð52Þ

Using the analytical results presented by Tang et al. [74], itcan be shown that the transient solution along a cross-section from the fracture into the porous matrix is given by

T fr � T 0

T 1 � T 0

¼ erfczffiffiffiffiffiffiffiffiffiffiffiffiffiffikbqb~cb

p2vfrql~clb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt � z=vfrÞ

p þffiffiffiffiffiffiffiffiffiqb~cb

pðx� bÞ

2ffiffiffiffiffikb

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt � z=vfrÞ

p !

ð53ÞThe fracture–matrix geometry is identical to that used byTang et al. [74]. The finite element domain was spatially

10

11

12

13

14

15

0 1 2 3 4 5 6 7 8 9 10distance along fracture [m]

tem

pera

ture

[°C

]

Analytical

Numerical

vfr

porous matrix

fracture T =fr 10 Co0

T = 10 Co0

T =1fr 15 Co

(a)

(b)

Fig. 5. Temperature profiles of 1D heat transfer in discretely fracturedporous media. Shown are the temperatures in the fracture at (a) 5000 and(b) 10,000 s.

Table 3Model parameters used in the verification example for 2D heat transfer ina single fracture embedded in a porous matrix

Parameter Value

Bulk thermal conductivity (kb) 3.4 kg m s�3 K�1

Heat capacity of solid ð~csÞ 908 m2 s�2 K�1

Solid density (qs) 2550 kg m�3

Heat capacity of water ð~clÞ 4192 m2 s�2 K�1

Fluid density (ql) 997 kg m�3

Matrix porosity (/) 0.2Groundwater flow velocity in the fracture (vfr) 0.05 m s�1

Initial temperature (T0) 10 �CBoundary temperature (T1) 15 �CDomain size (‘x,‘z) 2 m, 10 mLocation of cross-sections (z1,z2) 0.1 m, 0.61 mOutput times (t1, t2) 5000 s and 10,000 s

All parameters are identical to those used by Meyer [42].

100 0.1 0.2 0.3 0.4 0.5

distance into matrix [m]

Fig. 6. Temperature profiles of 1D heat transfer in discretely fracturedporous media. Shown are the temperatures in the matrix at 10,000 ssimulation time at the distances (a) 0.1 m and (b) 0.61 m from the heatsource.

discretized in the x-direction by gradually increasing Dx

with constant factor 1.1 from Dx = 0.01 m near the frac-ture to D x = 0.1 m at the domain boundary. In the flowdirection, D z also increases gradually from Dz = 0.1 mnear the elevated temperature to Dz = 0.5 m at the domainboundary. All other parameters are presented in Table 3and the simulation results are exhibited in Figs. 5 and 6.

6. Illustrative examples

To simulate heat transfer and chemical reactions in afracture network, existing studies of reactive transport inporous media must first be expanded to include heat trans-fer. Section 6.2 presents simulations for an unfracturedporous medium, pm, that focus on (i) reactive transport,(ii) thermohaline variable-density, variable-viscosity flow,and (iii) coupled thermohaline reactive transport. Theseprocesses will in turn be considered for a fractured med-ium, fm, in Section 6.3. In Section 6.4, the long-term effectof chemical reactions on density-driven mass flux throughfractured porous media, fm_long, will be demonstrated.The description of problems pm and fm is given in Section6.1 while problem fm_long will be described in Section 6.4.

6.1. Problem description

The domain for simulations pm and fm is the verticalcross-section shown in Fig. 7. A similar domain has beenused for simulations presented by Schincariol et al. [61],Ibaraki [33] and Freedman and Ibaraki [21] and it corre-sponds to the laboratory tank used in the experimental

0.25 m0.03 m

0.085 m

0.135 m

source of constantconcentration and temperature

1.0 m

flow

Fig. 7. Model domain and location of the solute source for numerical simulations of reactive silica transport and variable-density thermohaline flow. Thedomain is used for simulations in unfractured and discretely fractured porous media.


work of Schincariol and Schwartz [60]. This domain waschosen to highlight the coupling between variable-densityflow and reactive solute transport as done previously byFreedman and Ibaraki [21].

The 2D simulation domain has dimensions of 1.0 m ·0.25 m with a unit thickness. The domain has been spatiallydiscretized using 63,000 rectangular finite elements, whichare smaller at the left boundary (Dx = 0.5 mm, Dz =2.0 mm), and whose size increase towards the right (Dx =2.0 mm, Dz = 2.0 mm). Fractures are shown in Fig. 7 butthe first simulations presented here are for an unfracturedporous medium. Simulations that incorporate fracturesare presented later.

We consider that the domain has a uniform initial tem-perature of 239 �C [34] and that the only dissolved speciesinitially in the fluid is aqueous silica (H4SiO4). The fluid isinitially in thermodynamic equilibrium (rnet = 0), wheremH4SiO4

¼ Keq=cH4SiO4¼ 6:4996 mmol kg�1 is the initial sil-

ica molality at Cr = 0.0 mg l�1 and TC = 239 �C.A horizontal flow field is imposed by assigning constant

fluid fluxes (q = 1.045 · 10�6 m s�1) along the left and rightboundaries, with top and bottom boundaries being imper-meable to fluid flow. Also shown in Fig. 7 is the location ofa source of fluid entering the domain at a constant concen-tration and temperature. It is assumed that this fluid con-tains the four common ions Na+, Ca2+, Mg2+ and Cl�.This choice of fluid composition is based on average con-centrations of major ions in groundwater of the Canadian

0

500

1000

1500

Br SO4 HCO3 Cl TDS

1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06

concentration [mg/l]

dept

h [m

]

Fig. 8. Average ion concentrations in grou

Shield at different depths (Fig. 8). In that environment,water below 1000 m is a Ca–Na–Cl brine with dissolvedsolids exceeding 100,000 mg l�1. Stober and Bucher [71]have stated that all deep waters below 500 m in the conti-nental crystalline crust are brines of the same chemicalcomposition. Although the deep water is depleted inMg2+, this cation was used as the fourth mobile speciesbecause Dove [13] has shown that even at low concentra-tions, Mg2+ can significantly enhance silica dissolutionrates. Fig. 8 also illustrates that water at depth 500 m is richin SO2�

4 . According to Marshall and Chen [40], SO2�4 forms

a sulphate–silicic-acid complex at temperatures above150 �C, which increases silica solubility. The significanceof sulphate was verified by calculating the silica activitycoefficient with typical Na+, Ca2+, Mg2+, Cl� and SO2�

4

concentrations at 500 m depth and at the temperature max-imum (300 �C), where the solubility increasing effect of sul-phate is highest. This verification indicated that the activitycoefficient is approximately equal to 1 and the presence ofSO2�

4 is therefore assumed not to influence silica dissolutionand it is neglected here.

In the numerical simulations, first-type boundary condi-tions are imposed at the source for solute and heat trans-port, with a constant concentration equal to 1000 mg l�1

assigned to the four species Na+, Ca2+, Mg2+ and Cl�anda constant temperature equal to TC = 247 �C, higher thanthe initial temperature. First-type boundary conditionsare assigned to the remainder of the left boundary of the

Sr K Mg Na Ca TDS

1E+00 1E+01 1E+02 1E+03 1E+04 1E+05 1E+06

concentration [mg/l]

ndwater of the Canadian Shield [17].

Table 4Model parameters used in reactive transport studies

Parameter Value

Domain sizea (‘x,‘z) 1.0 m, 0.25 mSpatial discretizationd (Dx,Dz) 0.5 mm, . . . , 2.0 mm, 2.0 mmTemporal discretizatione (Dt) 1 min, . . . , 2 hLongitudinal dispersivitya,b,c (al) 3.0 · 10�4 mTransverse dispersivitya,b,c (at) 0.0 mTortuosityb,c (s) 0.35Average Darcy fluxa,b,c (q) 1.045 · 10�6 m s�1

Free-solution diffusion coefficientb,c (Dd) 1.6 · 10�9 m2 s�1

Distribution coefficient (Kd) [Na+] 3.0 · 10�6 kg�1 m3

[Ca2+] 5.0 · 10�5 kg�1 m3

[Mg2+] 1.0 · 10�4 kg�1 m3

[Cl�] 0.0 kg�1 m3

[H4SiO4]f 0.0 kg�1 m3

Reference fluid densityg (q0) 815.969 kg m�3

Reference fluid dynamic viscosityg (l0) 1.1184 · 10�4 kg m�1 s�1

Fluid compressibilityh (afl) 4.4 · 10�10 kg�1 m s2

Matrix compressibilityh (am) 1.0 · 10�8 kg�1 m s2

Initial porosityb,c (/init) 0.38Initial hydraulic freshwater

conductivityb K0;initij

� � 5.6 · 10�4 m s�1

Initial specific surface area inthe matrixi Ainit

qz

� � 54.2 m2 kg�1

Solid phase densityj (qs) 2650 kg m�3

Specific heat of solidj ð~csÞ 738 J kg�1 K�1

Specific heat of liquidj ð~clÞ 4186 J kg�1 K�1

Thermal conductivity of solidj (ks) 5.0 W m�1 K�1

Thermal conductivity of liquidj (kl) 0.6 W m�1 K�1

a Freedman and Ibaraki [21].b Schincariol et al. [61].c Ibaraki [33].d To fulfill the Peclet criterion, Pe < 2.3, used by b and c.e To fulfill the Courant criterion, Cr 6 1.0, used by b.f Yeh and Tripathi [85].g Computed by this model for deionized water at TC = 239 �C.h Shikaze et al. [64].i Johnson et al. [34].j Bolton et al. [6].

1 For interpretation of colour in this figure, the reader is referred to theweb version of this article.


domain, outside the source, with zero concentration and aconstant temperature equal to 239 �C, to ensure that theonly solute input in the domain is through the source.The top, bottom, and right boundaries are zero-conductiveheat transfer and zero-dispersive solute flux boundaries.Thus, heat and solutes cannot cross the top and bottomboundaries but they are able to cross the right boundaryby convection and advection, respectively.

As opposed to the four source species, the aqueous silicamolality at the source is not constant but recalculated ateach time step. All boundaries are zero-dispersive fluxboundaries for aqueous silica. Initially, the equilibrium sil-ica molality at the source is mH4SiO4

¼ Keq=cH4SiO4¼

6:9361 mmol kg�1 at Cr = 1000.0 mg l�1 and TC = 247�C. The choice of initial thermodynamic equilibrium makesit easy to identify every deviation from the silica equilib-rium as the result of a chemical reaction (dissolution orprecipitation). Because silica is the only species whose con-centration will not be influenced by water density, we willgive silica concentrations in the density-independent unitmolality (mol kg�1).

The simulations cover a time of 3 days with increasingtime step sizes. Thermal deformations of the rock are notconsidered. The spatial and temporal discretization as wellas all simulation parameters are summarized in Table 4.

6.2. Coupled thermohaline and reactive transport in porous

media

The first simulation, entitled pm_reac, is for reactivetransport but assumes that the fluid has constant densityand viscosity, equal to those of the ambient groundwater.In pm_reac, the time step size changes dynamically, basedon porosity changes. The maximum permitted change inporosity per time step, /*, was set to 10�3 (0.1%). The ini-tial and maximum time step sizes chosen were 1 min and2 h, respectively. Fig. 9 shows simulation results forpm_reac after 3 days. Thermal energy is predominantlytransferred by conduction in both the longitudinal andthe transverse direction (Fig. 9a). Because buoyancy forcesare not considered in pm_reac, the plume is mainly trans-ported by advection and migrates laterally across thedomain. Molecular diffusion and transverse dispersionslightly increase the plume extension in the vertical direc-tion. The chloride concentration (Fig. 9b; no retardation)and the magnesium concentration (Fig. 9c; highest retarda-tion) illustrate this transport behavior. Concentration con-tours for the nonreactive and nonsorptive chloride indicatethe position of the advective front.

Fig. 9d shows the molal concentration of silica andreveals an interesting simulation result for pm_reac. Nearthe source, temperatures are relatively high such thatquartz dissolves. However, further away from the source,the temperatures are close to the background temperature,239 �C and solute concentrations are high, which decreasesthe solubility of silica. Therefore, salinity controls the silicaconcentration further away from the source. Conversely,

the silica concentration follows the isotherms near thesource as well as in regions of low salinity above and belowthe plume. Fig. 10 is a vertical cross-section located at adistance of x = 0.12 m from the source. The figure showsthat the silica concentration is proportional to temperaturein low-salinity zones and inversely proportional to salinityin high-salinity zones. Clearly, these observations demon-strate the solubility-lowering effect of salt and the solubil-ity-increasing effect of temperature as discussed by Dove[13]. Fig. 9e1 finally shows the distribution of the hydraulicfreshwater conductivity. As expected, the area around thesource became more conductive because of quartz dissolu-tion. However, the elongated bluish fields located to theright of the source indicate a conductivity value smallerthan the initial one. Apparently, dissolved silica is trans-ported by advection to the right. Silica is assumed to benonsorptive and its transport rate is, therefore, comparableto the chloride transport rate. If silica molecules are

Fig. 9. Results of reactive transport simulations in an unfractured porous medium with constant fluid density and viscosity (pm_reac). Shown are (a)temperature, (b) chloride and (c) magnesium ion concentration, (d) molal concentration of aqueous silica and (e) freshwater hydraulic conductivity at 3days.


transported laterally to regions of lower temperature andhigh salinity, the system becomes locally supersaturatedand some of the previously dissolved silica precipitates,resulting in lower hydraulic conductivity.

The second simulation, called pm_dens, is for variable-density flow in an unfractured porous medium but ignoreschemical reactions. Water density and viscosity are calcu-lated from temperature and salinity. Unlike the previoussimulation, time step sizes are prescribed and graduallyincrease from 1 min to 2 h.

The results show that the magnitude of buoyancy is con-trolled by water density, which is a function of both tem-perature (Fig. 11a) and salinity (Fig. 11b and c). Fig. 11bdemonstrates that density effects cause vertical fluid move-ment. The figure shows concentration profiles of Cl� at 3

days, highlighting the mixed convective flow character.Forced convection (advection) remains the main lateraltransport mechanism whereas buoyancy-induced free con-vection controls the shape of the plume in the verticaldirection. Different diffusivities explain the completely dif-ferent transport behavior of thermal energy and solutes,which is known as double diffusive transport [48]. In thepm_dens simulation, heat transfer is practically indepen-dent of groundwater flow, while water flow dominates sol-ute transport. This difference results in an interestingdensity distribution (Fig. 11d). In the near field of thesource, temperature appears to control water density, whilein the far field, the salt concentration has the greatest influ-ence on water density. The inflowing hot saline water has adensity of 808.443 kg m�3, lower than the reference density

0

0.05

0.1

0.15

0.2

0.25

zele

vatio

n [m

]

silica

chloride

temperature

chloride [mg l ]-1

0 200 400 600 800 1000

temperature [°C]239.8 239.9 240.0

silica [mmol kg ]-1

6.48 6.5 6.52 6.566.54

239.7

Fig. 10. Vertical cross-section at x = 0.12 m from the source for the simulation pm_reac at 3 days.


(815.969 kg m�3). In this case, the inflowing water is lessdense than the ambient water. As a consequence, the rela-tive density, qr, is negative (�9.223 · 10�3), resulting in apositive buoyancy effect near the source. However, furtheraway from the source, the influence of the solutes on den-sity dominates because advective solute transport is moreefficient than conductive heat transfer. Therefore, the waterdensity exceeds its reference value and the density contrastis positive (in the range of qr � 10�3), which results in anoticeable sinking of the plume.

The last simulation for an unfractured porous medium,pm_reac_dens, couples the effect of density with chemicalreactions. Similar to the pm_reac simulation, hydraulicconductivity and matrix porosity change with time as aresult of reactions. Time step sizes adapt to porositychanges with the maximum permitted porosity change,/*, set to 10�3 (0.1%). This contrasts with pm_dens, whereconductivity and porosity remain constant over time.Fig. 12 shows that, as before, the temperature (Fig. 12a)is the important factor in the near field of the source whereit controls quartz solubility (Fig. 12d) and water density(Fig. 12e). In the far field, however, the salt content(Fig. 12b and c) dictates variable-density flow and chemicalreactions.

Although conductivity and permeability vary with timein pm_reac_dens and are constant in pm_reac, the resultsof the two simulations are not significantly different. Thisobservation is in agreement with findings by Freedman

and Ibaraki [21], who simulated the chemistry of calcite,coupled with density-driven flow. They concluded thattime scales of a few days are too short to perceive amajor impact of the reactions. In addition, the solidphase of both the calcite system studied by Freedmanand Ibaraki [21] and the quartz-water system studied herehave a fairly low solubility. Therefore, coupling betweenflow and reactive transport is weak. Nevertheless, thethree simulations presented here illustrate the couplingbetween variable-density flow, heat transfer and reactivetransport in porous media. They also show that adaptivetime stepping is a useful tool and certainly competitivecompared with the conventional use of predefined timestep sizes (Table 5).

6.3. Coupled thermohaline and reactive transport in fractured

media

A second series of three simulations assumes the pres-ence of fractures (fm) oriented transversely to the ambientflow direction as shown in Fig. 7. Using the random frac-ture generator developed by Graf and Therrien [31], a totalof 60 random fractures are generated following the twomain orientations 60� and 120� with the standard deviationof the Gaussian distribution, sigma = 1�. All fractures are0.1 m in length and have a uniform aperture equal to100 lm. Initial and boundary conditions of the fm simula-tions are identical to those used in the previous example in

Fig. 11. Results of density dependent nonreactive transport simulations in porous media (pm_dens) at 3 days. Shown are (a) temperature, (b) chloride and(c) magnesium ion concentration and (d) water density.


porous media. Table 4 presents the simulation parameterswhile Table 6 shows additional parameters for the simula-tions including fractures.

The first simulation, fm_reac, ignores density effects butsimulates reactive transport. The time step sizes adapt tochanges in matrix porosity and/or fracture aperture. Max-imum permitted changes of porosity and aperture are cho-sen as /* = 10�3 (=0.1%) and (2b)* = 0.1 lm, respectively.Fig. 13 shows the results after 3 days. The fractures have asubstantial impact on fluid migration because their hydrau-lic conductivity is more than 100 times greater than that ofthe porous matrix between the fractures. Because the valueof matrix permeability has originally been used for simula-tions without fractures, the permeability ratio of 100 is nottypical for fractured rocks. Other flow studies in fracturedrocks [64,30,31] assume a ratio that is more than threeorders of magnitude greater than that observed here. Inthese cases, fractures have a most significant impact onflow. Nevertheless, the fractures do increase the transversedispersion of the plume. This results in a larger verticalextension of the plume and reduced lateral migration(Fig. 13b and c), compared with results in porous media(Fig. 9b and c). Fig. 13d exemplifies the thermohaline influ-

ence on silica solubility. The plume is now more dispersedand solute concentrations in the far field are low, causingless silica precipitation. Therefore, silica precipitation doesnot significantly lower hydraulic conductivity (Fig. 13e).

The second simulation, fm_dens, is for variable-densityflow but ignores chemical reactions. Time step sizes are pre-scribed and gradually increase from 1 min to 2 h. Resultsfor this simulation are shown in Fig. 14. Because heat con-duction is generally the principal heat transfer mechanism,fractures do not affect the temperature distribution(Fig. 14a). The simulated concentration profile of the non-reactive, nonsorptive chloride ion at 3 days (Fig. 14b) indi-cate that, in the present case, fractures do not act likepreferential pathways as in simulations by Shikaze et al.[64] and Graf and Therrien [30,31]. However, because theplume is more dispersed in the fm_dens simulation(Fig. 14b) than in the pm_dens simulation (Fig. 11b), waterdensity in the far field is smaller in fractured media(�816 kg m�3) than in porous media (�817 kg m�3). As aconsequence, density contrasts in the far field of the frac-tured medium are generally small and in the range ofqr � 10�4. Therefore, the buoyancy effect in the fracturesis minor. This contrasts with simulations carried out by

Fig. 12. Results of density dependent reactive transport simulations in porous media (pm_reac_dens) at 3 days. Shown are (a) temperature, (b) chlorideand (c) magnesium ion concentration, (d) molal concentration of aqueous silica and (e) freshwater hydraulic conductivity.

Table 5Simulations and CPU times in porous media (pm)

Simulation Chemicalreactions

Densityvariations

Timestepping

CPU timea

pm_reacp

– Adaptive 32 minpm_dens –

pPrescribed 1 h 24 min

pm_reac_densp p

Adaptive 1 h 32 min

a Computed on a Pentium 4, 2.6 GHz, 500 MB RAM.

Table 6Additional model parameters used in reactive transport studies infractured media

Parameter Value

Fracture dispersivitya,b (afr) 0.1 mInitial fracture apertureb,c (2b)init 100 lmInitial specific surface area in the fractured Afr;init

qz

� �49.021 m2 kg�1

Fracture roughness coefficiente (x) 1.0

a Therrien and Sudicky [76].b Tang et al. [74].c Sudicky and Frind [24].d Eq. (27).e Steefel and Lasaga [65].


Shikaze et al. [63] and Graf and Therrien [30,31] where rel-ative density has values up to 0.2, more than three orders ofmagnitude greater than those encountered here.

The last fm_reac_dens simulation is for variable-densityflow and reactive transport using adaptive time stepping asin fm_reac. The dissolved ions are now distributed over alarger cross-sectional area with smaller salt concentrations

(Fig. 15b and c) than before (Fig. 12b and c). The silicaconcentrations in the far field are smaller than in thepm_reac_dens example in unfractured porous media

Fig. 13. Results of nondensity dependent reactive transport simulations in fractured media (fm_reac) at 3 days. Shown are (a) temperature, (b) chlorideand (c) magnesium ion concentration, (d) molal concentration of aqueous silica and (e) freshwater hydraulic conductivity.


(Fig. 15d), which is a result of the silica solubility-loweringeffect of dissolved salt. Because silica precipitation isreduced in fractured media, the hydraulic conductivity ofthe porous matrix does not change significantly(Fig. 15e). Aperture changes are insignificant for the timescale of this simulation and are not shown.

The CPU times are typically greater than in the pm sim-ulations because fractures are present (Table 7). It is againshown that adaptive time stepping is a helpful means toaccelerate and control the simulation process.

6.4. Long-term simulation of coupled thermohaline flow and

reactive transport in fractured porous media

Long-term simulations in a fractured medium (fm_long)have been carried out to demonstrate the significant effect

of chemical reactions on thermohaline flow. The simulationdomain is similar to that used by Shikaze et al. [64] andshown in Fig. 16. The domain of dimension 10 m · 10 mis spatially discretized into rectangular elements of sizeDx = Dz = 0.1 m. Mobile species are Na+, Cl�, H4SiO4

and temperature. Initially, the system is free of salt, silicaconcentration corresponds to thermodynamic equilibrium,ensuring reaction rates equal to zero in the entire domainand temperature is uniform equal to 10 �C.

Boundary conditions are also shown in Fig. 16. Lateralboundaries are impermeable for flow, while both top andbottom boundaries are assigned constant hydraulic headsequal to zero. The domain top is assigned a constant tem-perature equal to 10 �C and a constant concentration of0.0 mg l�1 for both Na+ and Cl�, corresponding to fresh-water at air temperature. The bottom of the domain is

Fig. 14. Results of density dependent nonreactive transport simulations in fractured media (fm_dens) at 3 days. Shown are (a) temperature, (b) chlorideand (c) magnesium ion concentration and (d) water density.


assigned a constant temperature of 239 �C and a constantconcentration of 1000 mg l�1 for both Na+ and Cl�, corre-sponding to salty hot groundwater. Unlike salt concentra-tions and temperature, the silica molality at bottom andtop is not imposed but recalculated at each time step. Atemperature of 239 �C has been chosen because (i) it hasbeen used by Johnson et al. [34] to experimentally studyquartz chemistry, and (ii) temperature around a radioactivewaste facility is in the range of 239 �C [10,84]. With concen-trations of 1000 mg l�1, water density at the bottom of thedomain is 818.137 kg m�3, giving a relative density value of�0.19, which is similar to the value of 0.2 used by Shikazeet al. [64]. Flow and transport parameters are identical tothose used by Shikaze et al. [64]; Heat transfer and reactionparameters are identical to those used in the previous twosections and given in Table 4. The maximum permittedchanges (per time step) of porosity, /*, and fracture aper-ture, (2b)*, are 0.005 (0.5%) and 1.0 lm, respectively. Sim-ulation time is 100 years.

The 2D vertical slice shown in Fig. 16 represents the typ-ical situation of cold freshwater above hot saltwater foundin many aquifers. The block is assumed to be heated frombelow due to nuclear fuel waste heat generation. The spa-tial dimension of 10 m · 10 m is atypical for a geological

formation above a deep radioactive waste facility. How-ever, this dimension has been chosen in order to createan illustrative scenario of unstable density-driven flow thatis similar to that studied by Shikaze et al. [64].

Fig. 17 displays simulated chloride concentrations at dif-ferent times. Flow in the entire domain is upwards becausedenser fluid (cold freshwater) overlies less dense fluid (hotsaltwater). Thus, chloride is transported by upwards den-sity-driven advection and a steady-state is reached at about30 years where the chloride concentration in nearly theentire domain is close to 1000 mg l�1. Inspection of thevelocity field indicated that convection does not occurbecause isotherms are horizontal, leading to negative rela-tive densities throughout the domain.

Fig. 18 shows results of the fm_long simulation after 100years. The Fig. illustrates thermal and chemical profilesalong the fracture at x = 3.5 m. Because heat conductionis the most important heat transfer mechanism, thesteady-state temperature distribution is linear between239 �C (bottom) and 10 �C (top) and reached after about2 years. The hot zone near the bottom corresponds toquartz dissolution. Therefore, silica molality in the dissolu-tion zone is high (about 5 mmol kg�1), reaction rates arepositive () and the fracture widens up to 247 lm. Dis-

Fig. 15. Results of density dependent reactive transport simulations in fractured media (fm_reac_dens) at 3 days. Shown are (a) temperature, (b) chlorideand (c) magnesium ion concentration, (d) molal concentration of aqueous silica and (e) freshwater hydraulic conductivity.

Table 7Overview of the simulation trials in fractured porous media (fm)

Simulation Chemicalreactions

Densityvariations

Timestepping

CPU timea

fm_reacp

– Adaptive 36 minfm_dens –

pPrescribed 1 h 32 min

fm_reac_densp p

Adaptive 1 h 48 min

a Computed on a Pentium 4, 2.6 GHz, 500 MB RAM.


solved silica is transported by upwards density-drivenadvection through the porous matrix and the fractures.At z P 0.5 m, the combination of low temperature andupward migrating silica creates local supersaturation, caus-ing silica precipitation with negative reaction rates () andreduced fracture apertures down to 39.7 lm. The major

aperture reduction is the result of a self-sealing process offractures, which is due to the coexistence of a dissolutionzone (where silica is produced) and a precipitation zone(where imported silica precipitates). As shown for the sim-ulations presented in Sections 6.2 and 6.3, the modeldomain is divided into a near field of quartz dissolutionand a far field of silica precipitation.

A second simulation was run where chemical reactionswere neglected. The chloride mass flux through a cross-sec-tion at z = 5 m (shown in Fig. 16) was compared for thesimulations without and with reactions and the comparisonis shown in Fig. 19. For the case without reactions, massfluxes through fractures, porous matrix and total mass fluxincrease at early times and attain a plateau after 30 years.This corresponds to the time when both temperature and

0 2 6x -distance [m]

1084

10

z-d

ista

nce

[m]

0

2

4

6

8

h0

σ

= 0

C = 0

T = 10 Co

h0

σ

= 0

C = 1000 mg l-1

T = 239 Co

horizontalcross-section

impermeablelateralboundaries

discretefractures

Fig. 16. Simulation geometry and boundary conditions for the long-term simulation of coupled thermohaline flow and reactive transport (fm_long).


salinity have reached steady-state and when density-drivenupwelling is therefore uniform in the entire domain. How-ever, in the case with reactions, mass fluxes are significantlylower after 10 years. Ultimately, at 100 years, mass fluxesthrough the fractures of the reactive domain are about25% lower compared to the nonreactive case. The reasonis that the minimum fracture aperture of 39.7 lm is the lim-iting factor that controls mass flux through the fracture.With time, the ongoing dissolution and precipitation willfurther enhance fracture self-sealing such that fracturemass fluxes decrease even more. Fig. 19 demonstrates thatthe impact of reactions and flow parameter changes ondensity-driven flow is a long-term effect that can only beobserved after decades.

7. Sensitivity analysis

Additional simulations were performed to assess theimpact of parameter uncertainties on reactive solute trans-port. The verification example from Section 5.1 was used asa base case with original input parameter values. Startingwith the base case, two simulations were run for everyparameter tested, using a lower and a higher parametervalue compared to the base case. We conducted two sensi-tivity analyses: (i) a mathematical analysis and (ii) an anal-ysis for visualization purpose.

All simulations of the mathematical sensitivity test werecharacterized with a dependent variable that appropriatelydescribes the simulation. This quantity is named n and theterms nlow, norg and nhigh denote the results when simulatingwith a low, unmodified and high value of parameter p: plow,porg and phigh, respectively. A dimensionless mathematicalsensitivity of the model parameter p, Xp, is evaluated usingan equation presented by Zheng and Bennett [87]:

X p ¼on=norg

op=porgð54Þ

According to Zheng and Bennett [87], the partial derivativeof the dependent variable, n, with respect to the inputparameter, p, is normalized by the original value of the var-iable, norg, and the parameter, porg.

The steady state silica concentration at the fracture out-let (z = 3.1 cm) was chosen as the dependent variable, n.The simulation of the base case yields the characteristicnumber norg = 3.5887 · 10�3 mol kg�1. The reactive trans-port simulation was run with modified values of fourparameters: the specific quartz surface area in the fractureand matrix, quartz volume fraction and temperature.

The choice of the range over which the input parameteris varied, is subjective. However, if parameter changes (i.e.,perturbations) are too small, computer round-off errorsmay conceal differences of the dependent variable. On theother hand, perturbations which are too large may yieldinaccurate sensitivities, especially if the relation betweendependent variable and parameter is nonlinear. In the pres-ent mathematical sensitivity analysis, a uniform perturba-tion size of 5% is applied as suggested by Zheng andBennett [87].

The results of the mathematical sensitivity test areshown first. Table 8 presents the original value of eachinput parameter, which was lowered and increased uni-formly by 5%. Fig. 20 shows the calculated dimensionlesssensitivity for each input parameter.

In addition to the mathematical sensitivity analysis, fur-ther simulations were carried out with much wider rangesof the input parameters (Table 8), in order to visualizeparameter sensitivity. The perturbations are not identicalfor all parameters and are not used in a mathematical

Fig. 17. Chloride concentration at (a) 0.5 years, (b) 5 years and (c) 30years. Concentration contours are 200 mg l�1 (thin lines) and 600 mg l�1

(bold lines).


sense. The result of this visual sensitivity analysis is shownin Fig. 21.

Both sets of simulations show that the uncertainties ofthe fracture surface area have a negligible impact on theresults (Fig. 21a), expressed by the low sensitivity of 0.01.However, the fracture surface area is about one order ofmagnitude smaller than in the matrix. Consequently, thefracture reaction rate is also one order of magnitude slower.Therefore, the fast reaction in the matrix is dominant andsuppresses unprecise fracture reaction rates. However, thesensitivity might be greater than 0.01 if more than one frac-ture were present or if fracture surface area was in the sameorder of magnitude than matrix surface area.

Both the surface area in the matrix and the quartz vol-ume fraction are directly proportional to the reaction rate(Eq. (2)). Thus, the sensitivity of the two parameters is sim-ilar (Fig. 21b and c). However, uncertainties of the quartzvolume fraction also impact the fracture reaction rate,which may cause the slightly higher sensitivity (0.34) thanthat of the matrix surface area (0.32).

Temperature variations have the most significant influ-ence on results (Fig. 21d). If, for example, the ambient tem-perature increases, the dissolution reactions proceed faster(Arrhenius equation) and, in addition, more quartz dis-solves because quartz solubility increases. Both geochemi-cal processes considerably increase the net reaction rate,hence the high sensitivity of temperature with a value of4.28.

8. Summary and conclusion

Underground disposal of nuclear waste has been deter-mined by the international scientific community as the bestoption for permanently isolating high-level radioactivewaste from the biosphere. Countries like Finland, Swedenand Japan regard low-permeability fractured crystallinerock as a potential host rock for spent fuel. In the presentstudy, a new numerical model has been presented that canbe used to assess potential sites for underground disposalof nuclear waste in these countries. The model is free soft-ware when used for research purpose. Interested readersare welcome to contact one of the authors to get a DOSor LINUX version of the model along with input files ofverification problems and illustrative examples.

The new model simulates coupled variable-density, var-iable-viscosity flow and kinetically controlled reactivetransport in nonisothermal fractured porous media. Wefocused on the chemistry of the common quartz-water sys-tem with aqueous silica as the only mobile reactive species.The flow equation is linked with the heat transfer and sol-ute transport equations through an iterative Picardapproach. After each iteration, the fluid properties densityand viscosity are updated from the individual ion concen-trations and temperature as primary variables. Chemicalreactions are calculated outside the Picard Iterationbecause the reactive species silica does not significantlyimpact the fluid properties. According to the quasi-station-ary state approximation [38], flow and reactive transportparameters are also updated at the end of a time step.

The model calculates water density from temperatureusing different equations for different temperature ranges.The dissolved solid effect on density is accounted for withan empirical relationship calibrated by means of densitycalculations with the VOPO model [44,45]. Fluid viscosityis also evaluated differently for various temperature ranges.The developed model applies the Jones–Dole equation torepresent the salt impact on viscosity. The new model alsoaccounts for the reaction rate enhancing effect of dissolvedsolutes and elevated temperature using a modified form ofthe Arrhenius equation. The equilibrium constant of the

0

1

2

3

4

5

zel

evat

ion

[m]

T

H SiO4 4

(2b)

rnetfr

100 250

temperature [ C]o

150 200 -1.75 1.5

net rate [10 mol kg sec ]-8 -1 -1

-1 -0.5 0.5 10

1 5

H SiO [mmol kg ]4 4-1

2 3 4 40 250

aperture [μm]100 150 200

dissolution zone

precipitation zone

flow

Fig. 18. Profiles of temperature, silica molality, fracture reaction rate and fracture aperture up to elevation z = 5 m in the fracture located at x = 3.5 mand for t = 100 years.

0

0.05

0.1

0.15

0.2

0 25 50 75 100time [yr]

Cl-1

mas

sflu

x[g

sec-1

]

total

fractures

porous matrix

without reactions

with reactions

Fig. 19. Chloride mass flux at z = 5 m through the porous matrix, fractures and total mass flux versus time for simulations without (bold lines) and with(thin lines) chemical reactions.

Table 8Model parameter modifications used for visualization only in thesensitivity analysis of reactive solute transport

Parameter Low value Original value High value

Specific quartzsurface area inthe fracture ðAfr

qzÞ

1.15 m2 kg�1 6.15 m2 kg�1 11.15 m2 kg�1

Specific quartzsurface area inthe matrix (Aqz)

34.2 m2 kg�1 54.2 m2 kg�1 74.2 m2 kg�1

Quartz volumefraction (/qz)

0.6 1.0 1.0

Temperature (TC) 209 �C 239 �C 269 �C

4.28

0.34

0.32

0.01

0 1 2 3 4 5

Temperature

Quartz volume fraction

Specific quartz surface area inthe matrix

Specific quartz surface area inthe fracture

Dimensionless sensitivity

Fig. 20. Mathematical dimensionless sensitivity of model parameters inreactive solute transport simulations in order from least (top) to most(bottom) sensitive.


chemical system is calculated from the local temperaturewith a common equation presented by Rimstidt [52]. Weadapted the method presented by Marshall and Chen [40]to calculate activity coefficients as a function of salinity,accounting for the haline influence on quartz solubility.Adaptive time stepping is used to calculate the time stepsizes in the numerical simulations. New time incrementsdepend on maximum changes in matrix porosity and/orfracture aperture. Adaptive time stepping eliminates

unphysically great changes of quartz fraction and stabilizesthe simulation.

The new model was used to simulate coupled thermoha-line groundwater flow and kinetically-controlled reactivetransport in porous and fractured porous media. Resultsof short-term simulations (3 days) demonstrate the effectsof temperature and salinity on silica concentrations. The

0

1

2

3

4

5

6

0 1 2 3distance along fracture [cm] distance along fracture [cm]

Si[m

mol

kg-1

]

high

original

low

0

1

2

3

4

5

6

0 1 2 3

Si[m

mol

kg-1

]

high

original

low

0

1

2

3

4

5

6


Si[m

mol

kg-1

] original

low

0

1

2

3

4

5

6


Si[m

mol

kg-1

]

high temperature

original temperature

low temperature

Aqz Aqzfr

Aqz

Aqz

Aqzfr

Aqzfr

qz

qz

(a) (b)

(c) (d)

φ

φ

Fig. 21. Visual sensitivity input parameters at steady state. Shown is the steady state quartz concentration in the fracture if the following parameters areuncertain: (a) specific quartz surface area in the fracture and (b) in the matrix, (c) quartz volume fraction and (d) temperature.


results indicate that silica concentration is inversely pro-portional to salinity in high-salinity regions and propor-tional to temperature in low-salinity regions. However,the results also show that chemical reactions do not signif-icantly impact density-driven flow and that salt mass fluxesremain unchanged. As in Freedman and Ibaraki [21], thesolubility of the solid phase (here quartz) is fairly lowand, therefore, coupling between flow and reactive trans-port is weak when simulating over 3 days.

However, the model is designed to also simulate on alarger temporal scale and an additional long-term simula-tion (100 years) in a fractured porous medium was con-ducted. The results demonstrate that, after 100 years,fracture apertures increase fivefold in the hot dissolutionzone and decrease by 20% in the less hot precipitation zone.This self-sealing effect of fractures significantly reduces saltmass fluxes after about 10 years, which cannot be observedon a small temporal scale. Although fracture aperturesincrease substantially, reduction of fracture mass flux isattributed to the aperture minimum, which represents thebottleneck for upwelling fluid. Ultimately, at 100 years,fracture–sealing diminishes the fracture mass flux by 25%relative to the case where reactions and aperture changesare neglected.

In a sensitivity analysis, parameter uncertainties wereinvestigated. We used the simulation of reactive transport

in fractured porous media presented by Steefel and Licht-ner [66] as the base case. It was found that the fracture sur-face area has a negligible impact on the result because thereaction rate in the matrix is about one order of magnitudefaster than in the fracture and therefore dominates thereaction. The matrix surface area and the quartz volumefraction are almost equally sensitive. Both are directly pro-portional to the reaction rate. Variations of temperaturehave the greatest influence on the results because tempera-ture impacts both the reaction kinetics and the quartz sol-ubility. Thus, field measurements of temperature must bevery accurate.

The presented results of thermohaline flow and reactivetransport simulations are numerically stable and obtainedfrom the developed and fully verified model. However, themodel has not been validated because appropriate field dataare currently lacking. The used model input is mostly hypo-thetical although input parameters are representative of thetype of geological material considered, and it contains someuncertainties. For example, exact fracture locations arehard to measure in the field but were shown to have a cru-cial impact on plume migration. Another uncertainty is themineral surface area in the matrix and especially in the frac-ture. The fracture roughness is difficult to determine, suchthat prior studies have commonly assumed perfectlysmooth fracture surfaces for simplicity. Moreover, this

Table A.1Water chemistry at different depths in the Canadian Shield; all concen-trations are in mg l�1 [17]

Solute Freshwater Brackish water Saltwater Dense brine0 m �500 m �1000 m �1500 m

Na+ 9 360 3550 34,000Mg2+ 2 90 95 25Ca2+ 15 630 7600 60,000Cl� 30 730 24,000 150,000


model assumes that matrix permeability is only a functionof absolute porosity (Eq. (24)) and ignores the effects ofpore structure and connectivity. However, the latter effectsmay be controlled in various ways by dissolution/precipita-tion reactions [57]. As a consequence of uncertain modelinput, the 2D results presented thus far only allow an anal-ysis and interpretation in a conceptual way.

The necessary step from obtaining conceptual 2D resultsto making reliable 3D long-term predictions will involve aniterative cycle of further model development – sensitivityanalysis – data gathering – numerical modeling – modeldevelopment as proposed by Glynn and Plummer [29].Completing this cycle, however, is highly challengingbecause ‘‘there are relatively few studies that have used 3-D geochemical transport codes’’ [29]. Prior research thatwould help complete the cycle described above is rare

600

700

800

900

1000

1100

1200

0 100 200 300

fluid

den

sity

[kg

m-3

]

Pitzer’smodel

temperature [ C]

temperature [ C]

o

o

0

0.5

1

1.5

2

2.5

3

0 100 200 300

activ

ity c

oeffi

cien

t [--]

temperature [ C]o

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

0 100 200 300

log

Keq

[mol

kg-1

]

all fluids

Fig. A.1. All physicochemical parameters calculated by F

and it remains greatly demanding to simulate a long-term3D thermohaline flow – reactive transport feedback sce-nario in a numerically stable fashion.

The complexity of nature and the need to find securedeep repositories for hazardous waste are the reasonswhy exploring the coupled system of thermohaline flow

temperature [ C]o

dense brine

brackish water

freshwater

saltwater

-15-14-13-12-11-10-9-8-7-6-5

0.001 0.002 0.003 0.004

1/T [K-1]

log

k +[m

olm

-2se

c-1]

0

1

2

3

0 100 200 300

visc

osity

[10-3

kgm

-1 se

c-1]

RAC3DVS are functions of temperature and salinity.


and reactive transport ‘‘will be an area of ongoingresearch’’ [50]. The model developed here was shown tobe a very useful tool that can advance this future research.

Acknowledgements

We thank the Canadian Water Network (CWN) as wellas the Natural Sciences and Engineering Research Councilof Canada (NSERC) for financial support of this project.We are grateful to P.M. Dove for providing valuable infor-mation on quartz dissolution kinetics, C.J. Neville for pro-viding details of the Meyer [42] solution and ChristopheMonnin for providing the subroutine for calculating den-sity in this code. We also thank eight anonymous AWRreviewers whose constructive comments have helped im-prove the manuscript.

Appendix A. Parameter dependency on temperature and

salinity

The model computes every physicochemical systemparameter over the low-temperature range 0–300 �C anda wide range of salinity, shown in Table A.1. Fig. A.1 illus-trates the corresponding model parameters.

References

[1] Bear J. Dynamics of fluids in porous media. New York: Elsevier;1988. 764 pp.

[2] Bear J, Verruijt A. Modeling groundwater flow and pollu-tion. Dordrecht: Reidel Publishing Company; 1987. 414 pp.

[3] Bennett PC. Quartz dissolution in organic-rich aqueous systems.Geochim Cosmochim Acta 1991;55:1781–97.

[4] Berkowitz B, Bear J, Braester C. Continuum models for contaminanttransport in fractured porous formations. Water Resour Res1988;24(8):1225–36.

[5] Blum A, Lasaga AC. Role of surface speciation in the low-temperature dissolution of minerals. Nature 1988;331:431–3.

[6] Bolton EW, Lasaga AC, Rye DM. A model for the kinetic control ofquartz dissolution and precipitation in porous media flow withspatially variable permeability: formulation and examples of thermalconvection. J Geophys Res 1996;101(B10):22157–87.

[7] Brady PV, Walther JV. Controls on silicate dissolution rates inneutral and basic pH solutions at 25 �C. Geochim Cosmochim Acta1989;53:2823–30.

[8] Brandt A, Fernando HJS. Double-diffusive convection. Geophysicalmonograph, vol. 94. Washington (DC): American GeophysicalUnion; 1995. 334 pp.

[9] Dana JD. On the decay of quartzyte, and the formation of sand,kaolin and crystallized quartz. Am J Sci 1884;28:448–52.

[10] Davison CC, Chan T, Brown A. The disposal of Canada’s nuclearfuel waste: site screening and site evaluation technology. AtomicEnergy of Canada Limited research (AECL-10719). Pinawa(MB): Whiteshell Laboratories; 1994. 255 pp.

[11] Davison CC, Chan T, Brown A. The disposal of Canada’s nuclearfuel waste: the geosphere model for postclosure assessment. AtomicEnergy of Canada Limited research (AECL-10713). Pinawa(MB): Whiteshell Laboratories; 1994. 497 pp.

[12] Domenico PA, Schwartz FW. Physical and chemical hydrogeol-ogy. New York: John Wiley; 1998. 506 pp.

[13] Dove PM. The dissolution kinetics of quartz in aqueous mixed cationsolutions. Geochim Cosmochim Acta 1999;63(22):3715–27.

[14] Dove PM, Crerar DA. Kinetics of quartz dissolution in electrolytesolutions using a hydrothermal mixed flow reactor. GeochimCosmochim Acta 1990;54:955–69.

[15] Dove PM, Nix CJ. The influence of the alkaline earth cations,magnesium, calcium, and barium on the dissolution kinetics ofquartz. Geochim Cosmochim Acta 1997;61(16):3329–40.

[16] Evans GE, Nunn JA. Free thermohaline convection in sedimentssurrounding a salt column. J Geophys Res 1989;94:2707–16.

[17] Farvolden RN, Pfannkuch O, Pearson R, Fritz P. The Precambrianshield. In: The Geological Society of America, editor. The geology ofNorth America. Hydrogeology 1988;O-2:101–14 [chapter 15].

[18] Fournier RO. Self-sealing and brecciation resulting from quartzdeposition within hydrothermal systems. In: Proceedings of thefourth international symposium on water rock interaction, Misasa,Japan, 1983, p. 137–40.

[19] Fournier RO. A method of calculating quartz solubilities in aqueoussodium chloride solutions. Geochim Cosmochim Acta1983;47:579–86.

[20] Fournier RO. The behaviour of silica in hydrothermal solutions. In:Gerger BR, Bethke PM, editors. Geology and Geochemistry ofEpithermal Systems. Rev Eco Geol 1985;2:45–61.

[21] Freedman V, Ibaraki M. Effects of chemical reactions on density-dependent fluid flow: on the numerical formulation and the develop-ment of instabilities. Adv Water Resour 2002;25(4):439–53.

[22] Freedman V, Ibaraki M. Coupled reactive mass transport and fluidflow: issues in model verification. Adv Water Resour 2003;26(1):117–27.

[23] Freeze RA, Cherry JA. Groundwater. Englewood Cliffs (NJ): Pren-tice Hall; 1979. 604 pp.

[24] Frind EO. Simulation of long-term transient density-dependenttransport in groundwater. Adv Water Resour 1982;5(2):73–88.

[25] Garven G, Freeze RA. Theoretical analysis of the role of groundwa-ter flow in the genesis of stratabound ore deposits 1. Mathematicaland numerical model 2. Quantitative results. Am J Sci 1984;284(12):1085–174.

[26] Garven G, Appold MS, Toptygina VI, Hazlett TJ. Hydrogeologicmodeling of the genesis of carbonate-hosted lead–zinc ores. Hydro-geol J 1999;7:108–26.

[27] Geiger S, Haggerty R, Dilles JH, Reed MH, Matthai SK. Newinsights from reactive transport modelling: the formation of thesericitic vein envelopes during early hydrothermal alteration at Butte,Montana. Geofluids 2002;2:185–201.

[28] Ghogomu NF, Therrien R. Reactive mass transport modeling indiscretely-fractured porous media. In: Bentley LR et al., editors.Computational methods in water resources, vol. XIII. Nether-lands: Rotterdam; 2000, ISBN 90-5809-123-6. p. 285–92.

[29] Glynn PD, Plummer LN. Geochemistry and the understanding ofground-water systems. Hydrogeol J 2005;13:263–87.

[30] Graf T, Therrien R. Variable-density groundwater flow and solutetransport in porous media containing nonuniform discrete fractures.Adv Water Resour 2005;28(12):1351–67.

[31] Graf T, Therrien R. Variable-density groundwater flow and solutetransport in irregular 2D fracture networks. Adv Water Resour 2006.doi:10.1016/j.advwatres.2006.05.003.

[32] Holzbecher EO. Modeling density-driven flow in porous media. Ber-lin: Springer; 1998. 286 pp.

[33] Ibaraki M. A robust and efficient numerical model for analyses ofdensity-dependent flow in porous media. J Contaminant Hydrol1998;34(10):235–46.

[34] Johnson JW, Knauss KG, Glassley WE, DeLoach LD, TompsonAFB. Reactive transport modeling of plug-flow reactor experiments:quartz and tuff dissolution at 240 �C. J Hydrol 1998;209(10):81–111.

[35] Krauskopf KB, Bird DK. Introduction to geochemistry. NewYork: McGraw-Hill; 1995. 647 pp.

[36] Langmuir D. Aqueous environmental geochemistry. Upper SaddleRiver (NJ): Prentice Hall; 1997. 600 pp.

[37] Lasaga AC. Chemical kinetics of water–rock interactions. J GeophysRes 1984;89(B6):4009–25.

http://dx.doi.org/10.1016/j.advwatres.2006.05.003


[38] Lichtner PC. The quasi-stationary state approximation to coupledmass transport and fluid–rock interaction in a porous medium.Geochim Cosmochim Acta 1988;52:143–65.

[39] Marcus Y. Ion solvation. Chichester: John Wiley; 1985. 306 pp.[40] Marshall WL, Chen TAC. Amorphous silica solubilities V. Predic-

tions of solubility behavior in aqueous mixed electrolyte solutions to300 �C. Geochim Cosmochim Acta 1982;46(2):289–91.

[41] Mayer KU, Frind EO, Blowes DW. Multicomponent reactivetransport modeling in variably saturated porous media using ageneralized formulation for kinetically controlled reactions. WaterResour Res 2002;38(9):1174. doi:10.1029/2001WR000862.

[42] Meyer JR. Development of a heat transport analytical model for asingle fracture in a porous media. Unpublished project report of thecourse Earth 661. In: Sudicky EA, Neville CJ, editors. Analyticalsolutions in hydrogeology. Waterloo Center for GroundwaterResearch. 2004. 46 pp.

[43] Molson JWH, Frind EO, Palmer C. Thermal energy storage in anunconfined aquifer 2: model development, validation and application.Water Resour Res 1992;28(10):2857–67.

[44] Monnin C. An ion interaction model for the volumetric properties ofnatural waters: density of the solution and partial molal volumes ofelectrolytes to high concentrations at 25 �C. Geochim CosmochimActa 1989;53:1177–88.

[45] Monnin C. Density calculation and concentration scale conversionsfor natural waters. Comput Geosci 1994;20(10):1435–45.

[46] Mroczek EK, Christenson B. Solubility of quartz in hypersaline brine– implication for fracture permeability at the brittle–ductile transi-tion. In: Proceedings world geothermal congress, Kyushu-Tohoku,Japan. 2000. p. 1459–62.

[47] Myers J, Ulmer GC, Grandstaff DE, Brozdowski R, Danielson MJ,Koski OH. Developments in the monitoring and control of Eh andpH conditions in hydrothermal experiments. In: Barney GS, NavratilJD, Schulz WW, editors. Geochemical behavior of disposed radio-active waste. Washington (DC): American Chemical Society; 1984.p. 97–216.

[48] Nield DA, Bejan A. Convection in porous media. New York: -Springer; 1999. 546 pp.

[49] Oldenburg CM, Pruess K. Layered thermohaline convection inhypersaline geothermal systems. Transport Porous Media1998;33:29–63.

[50] Post VEA. Fresh and saline groundwater interaction in coastalaquifers: is our technology ready for the problems ahead? HydrogeolJ 2005;13:120–3.

[51] Pruess K, Oldenburg C, Moridis G. TOUGH2 user’s guide version2.0. Report LBNL-43134. Berkeley (CA), USA: Lawrence BerkeleyNational Laboratory; 1999. 197 pp.

[52] Rimstidt JD. Quartz solubility at low temperatures. GeochimCosmochim Acta 1997;61(13):2553–8.

[53] Rimstidt JD, Barnes HL. The kinetics of silica–water reactions.Geochim Cosmochim Acta 1980;44:1683–99.

[54] Rimstidt JD, Dove JD. Mineral/solution reaction rates in a mixedflow reactor: Wollastonite hydrolysis. Geochim Cosmochim Acta1986;50:2509–16.

[55] Saaltink MW, Carrera J, Ayora C. On the behavior of approaches tosimulate reactive transport. J Contaminant Hydrol 2001;48(4):213–35.

[56] Sanford WE, Konikow LF. Simulation of calcite dissolution andporosity changes in saltwater mixing zones in coastal aquifers. WaterResour Res 1989;25(4):655–67.

[57] Saripalli KP, Meyer PD, Bacon DH, Freedman VL. Changes inhydrologic properties of aquifer media due to chemical reactions: areview. Crit Rev Environmental Sci Technol 2001;31(4):311–49.

[58] Schafer W, Therrien R. Simulating transport and removal of xyleneduring remediation of a sandy aquifer. J Contaminant Hydrol1995;19(9):205–36.

[59] Schafer D, Schafer W, Kinzelbach W. Simulating of reactiveprocesses related to biodegradation in aquifers 1. Structure of thethree-dimensional reactive transport model. J Contaminant Hydrol1998;31(5):167–86.

[60] Schincariol RA, Schwartz FW. An experimental investigation ofvariable-density flow and mixing in homogeneous and heterogeneousmedia. Water Resour Res 1990;26(10):2317–29.

[61] Schincariol RA, Schwartz FW, Mendoza CA. On the generation ofinstabilities in variable-density flows. Water Resour Res1994;30(4):913–27.

[62] Shibue Y. An empirical equation for quartz solubility in NaClsolution. J Mineral Petrol Eco Geol 1994;89:203–12.

[63] Shikaze SG, Sudicky EA, Mendoza CA. Simulations of dense vapourmigration in discretely-fractured geologic media. Water Resour Res1994;30(7):1993–2009.

[64] Shikaze SG, Sudicky EA, Schwartz FW. Density-dependent solutetransport in discretely-fractured geologic media: is prediction possi-ble? J Contaminant Hydrol 1998;34(10):273–91.

[65] Steefel CI, Lasaga AC. A coupled model for transport of multiplechemical species and kinetic precipitation/dissolution reactions withapplication to reactive flow in single phase hydrothermal systems. AmJ Sci 1994;294(5):529–92.

[66] Steefel CI, Lichtner PC. Multicomponent reactive transport indiscrete fractures: I. Controls on reaction front geometry. J Hydrol1998;209:186–99.

[67] Steefel CI, Lichtner PC. Multicomponent reactive transport indiscrete fractures: II. Infiltration of hyperalkaline groundwater atMaqarin, Jordan, a natural analogue site. J Hydrol 1998;209:200–24.

[68] Steefel CI, MacQuarrie KTB. Approaches to modeling of reactivetransport in porous media. In: Lichtner PC, Steefel CI, Oelkers EH,editors. Reviews in mineralogy, vol. 34. Washington (DC): Miner-alogical Society of America; 1996. p. 3–129 [chapter 2].

[69] Steefel CI, Yabusaki SB. OS3D/GIMRT: Software for modelingmulticomponent-multidimensional reactive transport; user manualand programmer’s guide. Technical Report PNL-1116. Richland(WA), USA: Pacific Northwest National Laboratory; 1996. 56 pp.

[70] Stern ME. The ‘‘salt-fountain’’ and thermohaline convection. Tellus1960;12(1):172–5.

[71] Stober I, Bucher K. Deep fluids: neptune meets pluto. Hydrogeol J2005;13:112–5.

[72] Stumm W, Morgan JJ. Aquatic chemistry. New York: John Wiley;1996.

[73] Sudicky EA, McLaren RG. The Laplace transform Galerkintechnique for large-scale simulation of mass transport in discretely-fractured porous formations. Water Resour Res 1992;28(2):499–514.

[74] Tang DH, Frind EO, Sudicky EA. Contaminant transport infractured porous media: analytical solution for a single fracture.Water Resour Res 1981;17(3):555–64.

[75] Tester JW, Worley WG, Robinson BA, Grigsbay CO, Feerer JL.Correlating quartz dissolution kinetics in pure water from 25 to625 �C. Geochim Cosmochim Acta 1994;58(11):2407–20.

[76] Therrien R, Sudicky EA. Three-dimensional analysis of variablysaturated flow and solute transport in discretely-fractured porousmedia. J Contaminant Hydrol 1996;23(6):1–44.

[77] Therrien R, McLaren RG, Sudicky EA, Panday SM. HYDRO-SPHERE – a three-dimensional numerical model describing fully-integrated subsurface and surface flow and solute transport. Univer-site Laval, University of Waterloo; 2004. 275 pp.

[78] Turner JS. Buoyancy effects in fluids. Cambridge: Cambridge Uni-versity Press; 1979. 368 pp.

[79] Tyvand PA. Thermohaline instability in anisotropic porous media.Water Resour Res 1980;16:325–30.

[80] von Damm KL, Bischoff JL, Rosenbauer RJ. Quartz solubility inhydrothermal seawater: an experimental study and equation describ-ing quartz solubility for up to 0.5 m NaCl solutions. Am J Sci1991;291(12):977–1007.

[81] Walter AL, Frind EO, Blowes DW, Ptacek CJ, Molson JWH.Modelling of multicomponent reactive transport in groundwater: 2.Metal mobility in aquifers impacted by acidic mine tailings discharge.Water Resour Res 1994;30:3149–58.

[82] Weir GJ, White SP. Surface deposition from fluid flow in a porousmedium. Transport Por Media 1996;25:79–96.

http://dx.doi.org/10.1029/2001WR000862


[83] White SP, Mroczek EK. Permeability changes during the evolution ofa geothermal field due to the dissolution and precipitation of quartz.Transport Por Media 1998;33:81–101.

[84] Yang J, Edwards RN. Predicted groundwater circulation in fracturedand unfractured anisotropic porous media driven by nuclear fuelwaste heat generation. Can J Earth Sci 2000;37:1301–8.

[85] Yeh GT, Tripathi VS. A critical evaluation of recent developments inhydrogeochemical transport models of reactive multichemical com-ponents. Water Resour Res 1989;25(1):93–108.

[86] Yoshida J, Nagashima H, Nagasaka M. Numerical experiment ondouble diffusive currents. In: Brandt A, Fernando HJS, editors.Double-diffusive convection. Washington (DC): American Geophys-ical Union; 1995. p. 69–79.

[87] Zheng C, Bennett GD. Applied contaminant transport model-ing. New York: John Wiley; 2002. 621 pp.

[88] Zysset A, Stauffer F, Dracos T. Modeling of chemicallyreactive groundwater transport. Water Resour Res 1994;30(7):2217–28.

Documents

Coupled thermohaline groundwater flow and single-species reactive solute transport in fractured porous media