This Study Walsh et al. This Study ... solid phase concentrations are compared with Walsh et al.'s ...... 52658, 41 pp., Lawrence Livermore Lab., Livermore, Calif.,.
WATER RESOURCES
RESEARCH,
VOL. 25, NO. 5, PAGES 883-910, MAY 1989
Redox-Controlled Multiple-Species Reactive Chemical Transport 2. Verification and Application ellEN
WUING
LIU
AND T. N. NARASIMHAN
Department of Materials Science and Mineral Engineering, University of California, Berkeley A redox-controlled multiple-species, multidimensional reactive chemical transport model, DYNAMIX, has been presented in part 1 of this paper. In part 2 the model is checked against two other independentlydeveloped reactive chemical transport codes. Simulated results from DYNAMIX agree reasonably well with those obtained from the other two models. Two approachesfor simulatingredox reactions, an external approach (based on hypothetical electron activity) and an effective internal approach (based on conservation of electrons), have been examined. The results show that in the external approach the simulated redox front is smearedout by diffusion and dispersion, whereas in the effective internal approach the redox front is sharp and is controlled by the redox reactions. The evolution of solid phasesfrom the two approachesgive markedly different mineral assemblages.It is suggestedthat the external approach is of relevance in industrial processessuch as electrometallurgy, in which mineral dissolution is driven by externally supplied electric power. The effective internal approach is applicable to hydrogeochemical systems (groundwater contamination, diagenesis, ore formation, rock weathering, and soil genesis)in which redox potential is dictated by the states of the redox species. The results also suggest that in the presence of precipitation (and consequent retardation of the concentrationfront), spreadingdue to hydrodynamic dispersionmay be significantly inhibited. To demonstratethe applicability of DYNAMIX for realistic field problems, two field cases are simulated by this model. In the first case, the supergeneenrichment of copper at Butte, Montana, the simulated mineral assemblagesand their distributions agree reasonably with the mineral assemblages and the ore grade observed in the field. In the second case, involving two-dimensional contaminanttransport in a hypothetic aquifer, the simulatedresults suggestthat DYNAMIX is capable of handling a realistic multi-dimensional field problem with several hundred grid blocks and time scales of practical interest.
1.
INTRODUCTION
THCC [Carnahan, 1986] are both one-dimensional simula-
We presented the development of a redox-controlled multiple-species,multi-dimensionalreactive chemicaltransport model, DYNAMIX in part 1 of this two-part series[Liu and Narasimhan, this issue]. The model includes advection,
diffusionand dispersion,transport of oxygen, oxidation and reduction and their consequentreactions, and also acid-base reactions, aqueouscomplexation, precipitation and dissolution, and kinetic mineral dissolution.
In this paper, the secondof the two-part series,we present two examples illustrating the verification of the model and two field problems illustrating the applicability of the model to realistic
field conditions.
2.
MODEL
VERIFICATION
Since no closed-form solutionsare available for problems involving multiple-speciesreactive chemical transport, the approach used to verify the computational model DYNAMIX is to compare its performance with other independently developed hydrogeochemicalmodels available in the literature. To this end, two hypothetical cases involving multiple-speciesreactive chemicaltransportwere simulated. The first of these involves a dissolution reaction, and the second involves dissolution-precipitationand redox reactions. Although DYNAMIX can handle multidimensional systems, the simulated cases are both restricted to one dimension because the two computer codes used for comparison,namely, PHASEQL/FLOW [Walsh et al., 1984]and Copyright 1989 by the American Geophysical Union.
tors.
2.1.
A Four-Component Dissolution Problem
The test case is a hypothetical one-dimensional column with
unit cross section
area which
contains
four fictitious
aqueousspeciesA, B, C, and D and one solid phase AB. No aqueouscomplex is considered.The activity of the aqueous speciesis assumedequalto their concentrations.The system initially contains two aqueous species A, B and one solid phase AB. At time greater than zero, an incoming fluid containingaqueous speciesA, C, and D in known concentrations enters the upstream boundary of the system. The concentrations describing the initial and boundary conditions are shown schematically in Figure 1. The solid phase AB has a solubilityproduct of 1. The pore velocity is 1 m/yr and the column length is 1 m. Note that the initial solutionin the column is saturated with respect to solid phase AB, inasmuchas the product of C^ ßCB equals 1 as specifiedby the solubility product of solid AB. However, the incoming fluid is undersaturatedwith respect to solid AB, since the productof C^ ß CB is less than 1. Owing to this undersaturation, the incoming fluid will dissolve solid AB, and a dissolutionfront will migrate through the column. This case serves as a check on the dissolution
reactions
of the chem-
ical speciation algorithm under equilibrium conditions. It also serves as a check on the accuracy of the algorithm in relation to massconservationbetween the aqueousand solid phases. The nonreactive species C and D serve as pathfinders, indicating the disposition of the hydrodynamic advective
front.
The problem has been studied by Walsh et al. [1984], who developed a one-dimensional,fully advective (diffusion and
Paper number 89WR00152. 0043-1397/89/89WR-0015255.00 883
884
LIO AND NARASIMHAN: REDOX-CONTROLLED CHEMICALTRANSPORT MODEL, 2
Aqueous Species: A, B, C, D Solid
Phase:
comix•ition of incomingfluid
CA Ca==0 C½ ==2
AB solubility • CA X ½• •
l
Fig. 1. Schematicillustrationof the four-component dissolutionproblemandits initial andboundaryconditions[after Walsh et al., 1984].
dispersionneglected),reactive chemicaltransportsimulator Figure 2 shows the model results of concentration versus PHASEQL/FLOW. A two-stepmixing-cellapproachis used distance at the instant when 0.5 pore volume of fluid has to couple the chemical equilibrium and chemical transport been injected. Figure 2a is for the solid phase AB, and equations which is similar to the approach used in DYFigure 2b is for aqueous speciesA, B, C, and D. The solid NAMIX.
phase concentration profile shows that solid AB is com-
Concentration Profilesat 1/2 PoreVolume
j2
0
------
0
1
i':•.,e-e e-e *-e-e e-e• •
Legend ß SpeciesA
o :? o
0.,
'
I
o.2
o.3
• 0.•
• o.•
0.•
o.7
Distonce (m) Fig. 2. Four-componentdissolutionproblem: Concentrationprofiles versus distance after 0.5 pore volume of incomingfluid which has been injected. (a) Solid phaseconcentration.(b) Aqueousphaseconcentration.
LIU AND NARASIMHAN' REDOX-CONTROLLEDCHEMICAL TRANSPORTMODEL, 2 TABLE
1.
885
Four-Component Dissolution Problem: Comparison of This Study With Results of Walsh et al. [1984]
Region I
Region II
Region III
Concentration
This Study
Walsh et al.
This Study
Walsh et al.
This Study
Walsh et al.
CA CB Cc
0.5 0.0 2.0
0.5 0.0 2.0
1.283 0.780 2.0
1.281 0.781 2.0
1.0 1.0 0.0
1.0 1.0 0.0
C•o
2.0
2.0
2.0
2.0
0.0
0.0
CAB
0.0
0.0
2.0
2.0
2.0
2.0
MineralAB solubility= CAßCB -'-'1. pletely dissolvedfor 0 < x < 0.14 m and remains undissolved on the downstream
side for 0.14
-< x
106 s for the externalapproach.This showsthat no extra acid is added through interphasereactions in the effective internal approach. Indeed, coffinite is the only precipitating mineral phase in the entire simulation (see Figure 12) because the precipitation reaction involved no hydrogenion to decreasethe aqueouspH. Figure 10 shows the distribution of the redox potential. Instead of the smeared out redox profiles observed in the external approach(Figure 6), a sharp redox front is obtained from the effective internal approach. The migration of redox front is controlled by the continuousprecipitation of coffinite. Sharp redox fronts in the natural environment have
-1
-4
-5 -6 -7
o
1
2
•
Disfonce (m) Fig. 10. Uraniumtransportproblem:pE distributions from effectiveinternalredox approachat 5 x 105, 106, 1.5 x 106, and 2 x 106 s.
LIU AND NARASIMHAN.' REDOX-CONTROLLED CHEMICAL TRANSPORTMODEL, 2
891
4.0
3.0
2.0
1.0
Legend
0.0
0
1
2
3
t = (5x10'sec.} t = {10' sec.) _ •= (c.5•j_o's._ec..) t: (2x•o_' sec.)
Disfonce (m) Fig. 11. Uraniumtransport problem:ZUO2 2+distributions fromeffectiveinternalredoxapproach at $ x 105, 106, 1.5 x 106, and2 x 106s.
been reported by several investigators. Wilson et al. [1986] observed a sharp oxidation front migrating progressively downward into deep ocean sediments. Cherry et al. [1986] measured a sharp vertical distribution of oxygen in tailing water from an abandoned
mill. The effective
internal
redox
potential approach is therefore believed to be more appropriate than the external approach for the modeling of redox reactions in many natural hydrogeochemical environments.
Figure11showsthedistribution of ZUO2 2+. Comparing with Figure 7 of the external approach, a more steady but less
retardedZUO2 2+ front propagates to abouta third of the columnat 2.0 x 106 s. This sharpuraniumfront is consistent with the sharp front of redox potential (see Figure 10). The sharpuraniumfront also reflectsthe sharpprecipitationfront of coffinitc as shown in Figure 12. In the effective internal approach, the aqueous solution is at equilibrium only with
12.0
10.0
8.0
6.0
4.0
Legend
2.0
t: (5x•O'sec.) t = (10' sec.) __ 0.0
L= O.Sx•O'sec..) 0
0.5
1
1.5.
l= (2x•O_' sec.)
Distance (m) Fig. 12. Uraniumtransportproblem:coffinitedistributions from effectiveinternalredoxapproachat 5 x 105,106, 1.5 X 106,and 2 x 106s.
892
Liu AND NARASIMHAN:REDOX-CONTROLLED CHEMICALTRANSPORT MODEL, 2
gO
g- 4.o 12,0
• 10.0 •
8ø0
,•
t5.0
0
G,3 4.0 2.0
Legend no reaction
0.0
external
0
1
2
3
,
effective
internal
Dislance Fig. 13. Uraniumtransportproblem:comparison of •UO• + distributions amongthe external,effectiveinternal,and no chemicalreactionapproachesat 2 x 106 s.
coffinite, which allows the systemto developa stable, sharp the same dispersion process is occurring in the reactive front of redox potential. Calcium uranateremainsundersat- chemical transport, the actual "effect" of dispersion has urated even at 2.0 x 106 s.
been suppressedby the precipitation chemical reaction and
It is worthwhile to compare the migration of nonreactive specieswith the reactive chemical speciesjust to obtain an idea of the magnitude of the reaction effects on transport. Using the sameinitial and boundary conditionsbut removing all the chemicalreactions,Figure 13 illustratesa comparison
can not be seen from the distribution
of the reactive chemical
species.The fact that pronounced precipitation suppresses dispersive spreading strongly suggeststhat in the case of stronglyretarding systems,hydrodynamic dispersionis not a critical transport parameter.
of the concentration distributions of •UO• + amongthe external, effective internal and no chemical reaction ap-
3.
MODEL
APPLICATION
proachesat 2 x 106 s. The nonreactiveuraniumspecies migratestwice as fast as the uranium speciesin the effective internal approachand 5 times as fast as the uranium species in the external approach. The figure also shows that the nonreactive chemical front is spread out significantly by hydrodynamic dispersion,whereas in the reactive case, the spreadingof the chemicalfront is greatly subdued.Although
3.1.
Supergene Copper Enrichment
3.1.1. Nature of the field problem. Supergene sulfide enrichment is an important mineralization process in the formation of near-land surface ore deposits. Supergene sulfide enrichment is the near-surface process of mineral
Field Problem
Meteoric Water
Rain Water leached
saturation < 1 oxygen excess
VadoseZone
Oxidation andLeaching
ß:':':':':':':':':':" ..............
enrichment
::'
Water Table •
Mobilizing Pb, Cu, Feetc.
water table surface
stable
pratore
Reductionand Precipitation
Phreatic Zone
oxygendeficiency
Fixation Pb, Cu, Fe etc. Fig. 14. Schematicillustration of the supergeneenrichment process.
Fig. 15. Schematic diagram of leached, enriched, and protore zones in the supergene enrichment process [after Cunningham, 1984].
LIU AND NARASIMHAN' REDOX-CONTROLLED CHEMICAL TRANSPORTMODEL, 2
Copper
I
Leaching Zone
Z
Enrichment, Zone
Prot,ore Zone
Fig. 16. Supergeneenrichmentproblem: a typical copper grade distribution in the leached, enriched and protore zones (modified from Ransome [1919]).
depositionby which oxidation producesacidic solutionsthat leach metals into groundwater, which carries these metals downwardand reprecipitatethem further down the flow path [Bates and Jackson, 1984]. The overall processis controlled by oxidation-reduction chemical reactions and the movement of ground water, as is schematically illustrated in Figure 14. As sulfide minerals are exposed to the oxygenrich infiltrating meteoric water, many oxidizable chemical species are released from solid phase by the oxidation reactions, and at the same time hydrogen ions are dissociated from water, leading to the generation of strong acidic solutions.The dissolvedchemical speciesare carried by the acidic solutions which move downward into the phreatic zone. Upon reacting with the primary sulfide minerals, the dissolvedmetal speciesare precipitated from aqueoussolu-
893
tion as secondary enrichment minerals. Repeating the same depositionalprocess, a high-gradeore deposit may gradually form right below the ground water table from the cumulative vertical flux. The zone with high grade of ore is called the enrichment zone. The primary, disseminated, and less concentrated ore body underlying the enrichment zone is known as protore. The zone above the enrichment zone is called leached zone. Conceptually, then, three distinct zones exist in the supergene system: the leached (oxidized) zone, the enrichment zone, and the protore (or hypogene zone). These zones are shown schematically in Figure 15. The supergeneenrichment process is an effective natural mechanismfor concentratingdispersed metals, contributing to an increase in the ore grade by a factor of 2 or more. The economic value of many otherwise uneconomical disseminated copper deposits is dependent on this process. The distribution of copper grade in the leaching, enriched, and protore zones after concentrating by the supergeneenrichment process is sketched in Figure 16. The copper in the leached zone has been dissolved and transported into the enrichment zone, whereas the copper in the protore remains unchanged. Because water mainly moves downward in the vadose zone, the chemical fluxes are conserved in the vertical direction as shown in Figure 15. This assumptionhas been confirmed by Brimhall et al. [1985], who used the mass balance principle to analyze the supergeneore-forming processes at La Escondida, Chile, and at Butte, Montana, and concluded that the lateral copper fluxes can be neglected in practice. Although the supergeneprocesshas been recognized as an important ore-forming process for decades [Emmons, 1918], the understanding of the interactions between hydrological and geochemicalprocessesremained mainly qualitative until recently. The geochemical modeling of the supergene ore forming processwas long restricted to idealization as a static batch system. The leaching and enriched or protore zones were treated as separatesystems,neglectingthe fluid and the chemical interactions between them. Only recently, with the increased
use of numerical
models
for reactive
TERTIARY GRAVELS LEACHED CAPPING ENRICN#[NT BLANKET
VEIN-BEARING (MAINS?•E) PRaTORE DISSEMINATED PY-CP
(PREMAINSTAGE)•O•RE ---
Faults
Fig. 17. The cross section of the copper ore deposit, Butte, Montana [after Cunningham, 1984].
chemical
894
Liu AND NARASIMHAN.' REDOX-CONTROLLED CHEMICAL TRANSPORTMODEL, 2
TABLE 5.
SupergeneEnrichment Problem: Initial Aqueous Chemical Compositions
Concentration,mol/m3 Chemical Species
Le•chin8 zone
60m ($ nodes)
W&ter c&ble
In Situ Water
Rain Water
9 x 1029.9 x 10-4 9.9 x 10-4 9.7 x 10-7 9.9 x 10-3 6.2 0.71 1 x 10-1
1 x 10-1 1 x 10-3 1 x 10-3 1 x 10-6 1 X 10-4 6.2 12.0 1 x 10-1
Na K Ca A1 Si pH pE C1
SO4
1 x 10-3
1 x 10-3
Fe Cu
5 x 10-14 9.1 x 10-9
5 x 10-14 9 x 10-9
After Cunningham [1984].
40m (4 nodes)
Enrichment
zone
The simulated results agreed with the observed natural system.
Protore zone
Fig. 18. Supergeneenrichmentproblem: numericalmesh usedfor simulation.
Cunningham, in his thesis, has provided detailed geochemical and hydrological data for Butte, Montana, which is used as the basis for simulations using DYNAMIX. 3.1.2. Modeling problem. The Butte district is located in southwest Montana and covers an area of approximately
64 squaremiles (164 km2). The regionalsetting,lithology, transport, has the quantitative analysis of ore-forming systems become possible. For example, Garven and Freeze [1984a, b] considered the effect of gravitational flow of aqueousbrines through carbonate rocks forming Mississippi Valley type Pb-Zn deposits. Cunningham [1984] coupled a fluid flow model TRUST [Narasimhan and Witherspoon, 1977; Narasimhan et al., 1978] with a chemical speciation reaction path model EQ3/EQ6 [Wolery, 1979] to study the supergeneweathering process at Butte, Montana. A simple one-dimensional elements
vertical
to account
for
mesh was discretized the three
different
into three
zones
in the
supergeneenrichment process. Instead of solving the advective diffusive-dispersive transport equation, at each time step, Cunningham solved only the fluid flow equation to get the advective chemical fluxes and neglected diffusivedispersive transport. Ague [1987] extended Cunningham's work by adding a numerical procedure which increases the computationalefficiency during periods of quasi steady state geochemical and hydrogeological behavior to simulate the same Butte problem. The flow region was divided into five volume elements, and the total model time was 12,026 years. TABLE 4. Relation
SupergeneEnrichment Problem: Hypothetical for Saturation
Versus
Reduction
Factor
of
Groundwater Velocity Saturation 1.0 0.94 0.87 0.80 0.74 0.71 0.65 0.62 0.59 0.55
Reduction
Factor
1.0 0.95 0.90 0.75 0.55 0.525 0.52 0.515 0.5145 0.5135
mineralization, structure, and alteration have been extensively studied by Meyer [1965] and Brimhall [1977, 1979]. Premainstage hypogene mineralization and alteration occurred in this district about 63 million years ago. This mainstage ore consists primarily of quartz, pyrite, molybdenite, and low-grade disseminated chalcopyrite. Figure 17
[after Cunningham, 1984] illustrates a cross sectionthrough the Butte district which is largely composed of quartz monzonite. A layer of stream gravels blankets a 200foot-thick
leached zone. A 130-foot enriched
zone immedi-
ately underlies the leached zone. The stable protore zone, in turn, underlies the enriched blanket. The copper grades are approximately2 wt % and 0.5 wt % in the enrichedzone and in the protore, respectively. Because the supergeneprocess is mass-conservingin the vertical direction [Brimhall et al., 1985], a one-dimensional column can be used for numerical simulations. The vertical column designed for numerical study using DYNAMIX is 100 m tall with a 10 m x 10 m cross section.
This column
is discretized
into 10 nodes of
dimension 10 m x 10 m x 10 m (see Figure 18). The water table is assumed initially at a depth of 60 m. The upper six nodes are above the groundwater table, under partially saturated conditions, representing the leached zone. The next four nodes are below the groundwater table under fully saturated conditions to represent the enriched zone. The bottom node is assigned an arbitrarily large capacity to maintain its initial condition representing the stable protore. Rainfall is applied to the first node at the top of the surface at a constant flux rate 2.54 cm/yr. The advective flux is thus 2.541 cm/yr. The rate of rainfall is slightly less than the advective flux inside the flow region, so that as time progresses, the saturation decreases gradually and the groundwater table falls slowly. A reasonable, hypothetical relation between
the saturation
and the reduction
factor for
the groundwater velocity, used to correct the change of the saturation correspondingto the change of Darcy's velocity, is given in Table 4. The initial aqueous chemical composi-
LIU AND NARASIMHAN: REDOX-CONTROLLEDCHEMICAL TRANSPORTMODEL, 2
TABLE 6. Supergene Enrichment Problem: Intra-aqueous Reactions and Their Equilibrium Constantsat 25øC
895
TABLE 7. SupergeneEnrichment Problem: Mineral Solubility Reactions and Their Equilibrium Constants at 25øC Log of Equilibrium
Log of Equilibrium Chemical
Reaction
Chemical
Constant
H20 - H + = OH-
13.99
H + + CO32-= HCO•2H+ + CO32-= H2CO3 H+ + SO42-+ HSO] SO42-+ 8H+ + 8e- - 4H20 = S2SO42-= 9H+ + Be- - 4H20 = HSSO42-+ 10H+ + 8e- -4H20 = H2S Ca2+ q- H20 - H + = CaOH+ Ca2+ + CO32----CaCo3 Ca2+ + CO32-+ H += CaliCOSCa2+ + 8042-= CaSO4 Na+ + CO32-= NaCO•Na+ + CO32-+ H + = NaHCO3 Na+ + SO42-= NaSO•K + + SO42-= KSO•Fe2+ + H20- H + = FeOH+ Fe2+ + 2H20- 2H+ = Fe(OH)2 Fe2+ + 3H20 - 3H+ = Fe(OH)j Fe2+ + SO42-= FeS04 Fe2+ + 2SO42-+ 18H+ + 16e- - 3H2¸ = Fe(HS)2 Fe2+ + 3SO42-+ 27H+ + 24e- - 12H20 = Fe(HS)3
10.34 16.70
1.987 20.735
Constant
Reaction
KAISi308 = K + + AI3+ + 3Si(OH)4- 4H20 - 4H+ AI2Si2Os(OH) 4 -- 2Si(OH)4+ 2AI(OH)•- + H + 7H20 KAI3Si301o(OH)2 = K+ + 3AI(OH)•- + 3Si(OH)4 + 2H +- 12H20 SiO2 = Si(OH)4- 2H20
33.652
FeS2= Fe2+ + 2HS- - 2H+ - 2e-
40.644
FeS = Fe 2+ + HS-
-12.598 3.153 11.345 2.309 1.268 10.08 0.700 0.85
-9.50
0.875 -36.921 -49.102 -4.006
-18.48
- H+
-5.9476
Fe203= Fe3+ + 3H20- 6H+ FeOOH = Fe3+ + 2H20- 3H+ Fe(OH)3- Fe3+ + 3H20- 3H+ Fe304 -- Fe2+ + 2Fe3+ + 4H20 - 8H+ CuFeS2= Cu2+ + Fe2+ + 2HS--2H + Cu2S= 2Cu+ + HS- - H +
-4.008 0.50 4.891 3.8270 -35.27 -34.619
Cu = Cu + + e-
-8.76
CusFeS 4 = Fe2+ + 4Cu+ + 4HS- + Cu2+ -4H 4
-104.6916
Cu20 = 2Cu+ + HS- - H + AI(OH)3 = AI3+ + 3H20- 3H+
-1.55 -36.921
-20.57 -31.00
After Cunningham [1984].
2.25
76.25 111.937
Fe 2+ - e- = Fe 3+
-13.032
Fe2+ + H20 + H + - e- = Fe(OH)2+ Fe2+ + 2H20 - 2H+ - e- = Fe(OH)•Fe2+ + 3H20 - 3H+ - e- = Fe(OH)3 Fe2+ + 4H20 - 4H+ - e- = Fe(OH)•2Fe2+ + 2H20- 2H+ - 2e- = Fe2(OH)24+ 3Fe2+ + 4H20- 4H+ - 3e- = Fe3(OH)•+
-15.22
Fe 2+ + CI- - e- = FeCI 2+
-11.55
Fe2+ + 2C1- - e- = FeCI•Fe2+ + 3C1- - e- = FeCI3 Fe2+ + SO42-= e- = FeSO•Fe2+ + 28042-- e- = Fe(SO4)•AI3+ + H20 - H + = AI(OH)2+ AI3+ + 2H20 - 2H+ = AI(OH)•AI3+ + 3H20 - 3H+ = AI(OH)3 AI3+ + 4H20 - 4H+ = AI(OH)•A13++ SO42-= AI(804)+ AI3+ + 2SO42-=AI(804)• Si(OH)4- H + = H3SiO•Si(OH)4- 2H+ = H2SiO4 2Cu2+ q- 2C1- q- e- --CuCl•Cu2+ q- 3C1- q- e- = CuCl3 2-
- 10.90
-18.70 -26.63 -34.63 -29.01 -45.4
- 11.90 -9.11 -7.61 -4.99 -10.1 -16.0
-23.0 3.02 4.92 -9.929 -21.617 8.22
8.42
Cu 2+ = e- = Cu +
2.72
Cu2+ q- CO32-•_ CuCO3 Cu2+ + 2CO32-= Cu(CO3)2 2-
6.73
Cu 2+ + CI- = CuCI +
0.43
Cu2+ + 2C1- = CuCI+ Cu2+ + 3C1- = CuCI•Cu2+ + 4C1- = CuCI42Cu2+ + H20 - H + = CuOH+ Cu2+ + 2H20 - 2H+ - Cu(OH)2 Cu2+ + 3H20- 3H+ = Cu(OH)•Cu2+ + 4H20 - 4H+ = Cu(OH)422Cu2+ + 2H20- 2H+ + Cu2(OH)22+ Cu2+ q- 5042- -- CuSO4 Cu2+ + 3SO42-+ 27H+ + 24e-- 12H20 = Cu(HS)•-
0.16
9.83
-2.29 -4.59 -8.0
-13.68 - 26.90 - 39.60
tions are listed in Table 5 [Cunningham, 1984]. The aqueous complexes derived from the aqueous components and their equilibrium constants are listed in Table 6. The mineral solubility reactions of interest and their equilibrium constantsare given in Table 7. The volume fractions of the initial mineral phases in the system and the rate constant of the dissolution reactions are given in Table 8. The porosity is assumed to be 12% uniformly in the system. Becausethe mineral phasesin the leached zone are under weathering condition, the dissolution reaction is expected to be kinetically controlled. As a first approximation, the initial rate of dissolution reaction of silicates in the aqueous solution, suggestedby Wood and Walther [1983], can be described as a zero-order kinetic rate equation which is con-
trolledby the reactingsurfaceand a rate constantof 10-12 mol/m2. Their interpretationhas been questionedby other investigators [e.g., Murphy, 1985]. The calculation of reactant surfacearea has been discussedin part 1 of this two-part series.
The dissolution
mechanisms
of sulfide minerals
are much
more complicatedthan silicate dissolutionmechanisms.For example, the dissolution and the oxidation of pyrite are controlled by several mechanisms under different environmental conditions. Under acidic conditions (pH -< 3.0), the rate-determining step is [Nordstrom, 1982] TABLE 8.
SupergeneEnrichment Problem' Initial Mineral
Phases, Volume Percents and Reaction Rate Constants
Volume %
Rate Constant, mol/cm2/s
Orthoclase Kaolinite Muscovite
22.2 18.2 32.2
1.25 X 10-17' 2.0 X 10-17' 8.3 x 10-18'
Quartz Pyrite Chalcopyrite
21.4 4.0 2.0
Mineral
5.0 x 10-17' 2.7 X 10-147 2.7 X 10-147
-10.359 2.31
126.856
After Cunningham [1984] * Wood and Walther [1983]. ?The rate constant - 500 times as fast as the silicate rate constant.
896
Liu AND NARASIMHAN' REDOX-CONTROLLED CHEMICAL TRANSPORTMODEL, 2
Total Copper(in 104 moles) 3.5
4.
•..'
',,'• ' •
Leaching zone
pE
4.$
5.
-2
•
\• •
0
2
4
6
8
10
12
600yea, rs
/
80m (8 nodes)
i•,
•
/
2000 years
60m (6nodes) 40•
I
,nn.....
•//
i/
6O
•oox-• •
•
/
•
Enrichment zonll0ye•
z(m)t
-'
40m (4nodes) l
_/:
'11 t /
100
!/i
Protore zone t Fig. 19. Supergene enrichment problem: simulated solid phase copper distributions at 0, 100, 500, 1000, and 2000 years.
FeS2(s) + 02 + 2H+ = Fe2+ + S2+ H202
(1)
and at slightly acid to base solution (pH _> 4.0), the ratedetermined step is 3
Fig. 21. Supergeneenrichment problem: simulatedaqueousphase copper distributionsat 0, 100, 500, 1000, and 2000 years.
3
FeS2(s) + • 02q-• H20= Fe(OH)3(s) + S2
(2)
Thiobacillus bacteria usually exist in the aqueous solution, acting as a catalytic phase [Nordstrom, 1982]. The ratedetermining step of pyrite oxidation becomes
FeS2(s) + 14Fe 3+ + 8H20= 15Fe 2+ + 2SO42+ 16H+ (3) To•al Copper(in t0• moles) 3.5
4.
i
I
4.5
The determination
of sulfide dissolution
rate is thus difficult.
Cunningham [1984] used the rate equation proposed by Wiersma and Rimstidt [1984] for pyrite dissolution and obtained a rate constant approximately 400 times greater than
the
rate
constant
of
silicate
dissolution.
The
rate
equation for chalcopyrite is taken from Braithwaite [1976]. By assuming that dissolved oxygen is saturated in the aqueous phase, Cunningham [1984] inferred that the rate constant of chalcopyrite is approximately 800 times greater than that for silicate dissolution.
In the DYNAMIX
simula-
tion, the initial dissolutionrate constantsfor both pyrite and
,L
log[Cu2+]moleIm3
60m (6nodes) Le•chin$ zone !0 chllcopyrite 6
Enrichment zone 40m(4nodes)
(½u•$) corellite (½u$) •rnite (•u
Protore zone
chaicopyrite
(
Fig. 20. Supergene enrichment problem: simulated averaged solid phase copper distribution and copper mineral assemblagesin the leached, enriched, and protore zones after 2000 years.
Fig. 22.
Supergene enrichment problem: simulated pE distributions at 0, 100, 500, 1000, and 2000 years.
LIU AND NARASIMHAN' REDOX-CONTROLLED CHEMICAL TRANSPORTMODEL, 2
ANNUAL
897
INFILTRATION
TAILINGS
SAND AQUIFER
GROUND
WATER
FLOW
BED ROCK
Fig. 23. Two-dimensional reactivechemicaltransport:schematicdescriptionof the field problem.
empirical material properties. The diffusive oxygen flux is evaluated by Fick's law. The solubility of oxygen in the aqueous phase is 8 mg/L under standard state condition,
chalcopyrite are assumedto be 500 times greater than the rate constant of silicate dissolution. Table 8 gives the rate
constantof each mineral phasewhich is subjectedto kinetic dissolution
in the leached
zone.
Because
there
are few
studieson the reaction rate of precipitation, the precipitation of mineral phases in the simulations is assumed to be controlled by thermodynamic equilibrium. In modeling the leaching process, it is important to accurately describe the movement of oxygen in the partially saturatedporous medium, since oxygen is the key driving force of the redox reaction in the supergene enrichment process.Diffusion of oxygen throughair-filled pore spaceis considered the dominant process of oxygen interchange between the atmosphere and the porous medium. Oxygen moved by advection in responseto the gradient of the total pressure is not as important as that moved by diffusion becausethe total pressureof the gasphaseis nearly constant at 1 atm pressure in the unsaturated region [Tokunaga, 1986]. Troeh et al. [1982] proposed an equation for the effective gas diffusion coefficient in the partially saturated porous media of the form
which serves as a constraint on maximum oxygen concentration in the unsaturated region.
The problem was run for a total time of 2000 years. The results are discussed
3.1.3.
below.
Results and discussions. Figure 19 shows the
computedprofiles of total amount of copper in the solid phaseas a functionof time. Shownplottedare the amountof copper versus depth in the leached, enriched, and protore zones for the initial year, 100 years, 500 years, 1000 years, and 2000 years. The amount of copper at each nodal location is calculatedby summingup all copper in the mineral phases. Initially, all three zones have the same mineral composition, and the amount of copper contained in each zone is the
same.At 100years, only a trace amountof copperis leached from the weathering zone and deposited in the enrichment zone. As time reaches500 years, a small bell-shapedprofile of enrichedcopperis developedand graduallyincreaseswith time. A significantamount of enriched copper has accumulated in the enrichment zone at 2000 years. These calculations suggestthat total leachingof copper occurs on a time Deft = Do 1- u scale of 10,000 to 100,000 years. By plotting the average whereD Ois the diffusioncoefficientfor opengaseousflow, •b amount of copper distribution at 2000 years in the three is porosity,Sg is the gaseous phasesaturation andu, v are zones as is shown in Figure 20, one obtains a conceptual
[qb(1Sg)tt.] •' CONTAMINANT
(4)
CONSTANT
FLUX
SOURCE
2S METERS
z o
z o
NO
.•
1000
FLOW
METERS
Fig. 24. Two-dimensionalreactivechemicaltransport:flow field and boundaryconditions.
898
LIU AND NARASIMHAN' REDOX-CONTROLLED CHEMICAL TRANSPORTMODEL, 2
TABLE 9.
Initial Aqueous Chemical Composition in the
Simulation
of Two-Dimensional
Reactive
Contaminant
Transport
TABLE 11. Intra-aqueous Reactions and Their Equilibrium Constants at 25øC for Species in the Simulation of TwoDimensional Reactive Contaminant Transport
Concentration,mol/m3
Log of Equilibrium
Chemical
Component
Tailing Water
Groundwater
Rain Water
pH pE Ca
3.5 8.0
7.6 -2.0
5.0
1.5
6.0 8.0 5 X 10-3
Mg
100.0
1.0
1 X 10-3
Na K Fe Mn A1 Si C1 C S Se As
10.0 1 x 10-2 500.0 300.0 400.0 10.0 6.0 1.0 X 10-2 1000.0 2.78 x 10-3 3.523 x 10-3
5.0 1.0 1 x 10-2 1 x 10-2 1 X 10-3 0.5 2.5 4.0 2.0 0.0 0.0
1x 1 X 1X 1x 1X 1 X 1 x 1X 1X 0.0 0.0
10-2 10-3 10-4 10-s 10-3 10-3 10-2 10-4 10-3
copper distribution model in the supergene enrichment process. This distribution has been confirmed by several investigators [e.g., Brimhall et al., 1985; Kwong, 1981]. The simulated copper mineral assemblagesin the three zones are also shown in Figure 20. As copper is liberated from chalcopyrite in the leached zone and transported to the enriched blanket zone, the solution becomes supersaturated, and copper precipitates from solution as chalcocite, covellite, and bornite in the transition environment between the highly oxidized leaching environment and the reducing protore environment. The simulated precipitated copper minerals agree with the core samples taken from Butte district [Cunningham, 1984]. Figure 21 shows the total concentration distribution of aqueous copper ions. Copper ion concentration in the aqueous phase is increased as the oxidizing environment progressively migrates down. A redox barrier is developed at the interface
between
leached
and enriched
zones which delim-
its the migration of redox front. Similar redox barriers have also been reported at sediment-water interfaces and soil systems [Davison, 1985] The time-dependent evolution of redox profiles is illustrated in Figure 22. A sharp redox front is formed due to the movement of dissolved oxygen in the leached zone and the interphase equilibrium between the aqueous and the solid phases in the enrichment zone. The dissolved oxygen is continuously consumed by the oxidation by pyrite and chalcopyrite and is replenished from the atmospheric air to maintain the high oxidation potential in the leached zone. The equilibrium between the aqueous and the solid phases TABLE 10. Initial Mineral Phases and Their Volume Percents for Simulation of Two-Dimensional Reactive Contaminant
Transport Mineral Calcite
Gypsum
Volume 10.0
5.0
Dolomite Gibbsite Orthoclase
5.0 2.0 20.0
Quartz
58.0
%
Chemical
Reaction
Constant
H20 - H + = OH-
13.99
H + + CO32-= HCO; 2H+ + CO32-= H2CO3 H+ + SO•- = HSO] SO•- + 8H+ + 8e- - 4H20 = S2SO•- + 9H+ + 8e- - 4H20 = HSSO•- + 10H+ + 8e- - 4H20 = H20 = H2S Ca2+ + H20 - H + = CaOH+ Ca2+ + CO32-= CaCO3 Ca2+ q-CO32-q-H + = CaHCO• Ca2+q- SO•- = CaSO4 Mg2+ + H20- H + = MgOHMg2+ + CO32-= MgCO3 Mg2+ + CO32-+ H + = MgHCO3 Mg2+ + SO•- = MgSO4 Na+ + CO32-= NaCO• Na+ + CO32-+ H + = NaHCO3 Na+ + SO•- = NaSO• K + + SO•- = KSO• Fe2+ + H20- H + = FeOH+ Fe2+ + 2H20- 2H+ = Fe(OH)2 Fe2+ + 3H20- 3H+ = Fe(OH)• Fe2+ + SO•- = FeSO4 Fe2+ + 2SO•- + 18H+ + 16e--8H2 ¸ = Fe(HS)2 Fe2+ + 3SO•- + 27H+ + 24e- - 12H20 = Fe(HS)3
10.34 16.70 1.987 20.735 33.652 40.644 -12.598 3.153 11.345 2.309 -11.794 2.98 11.396 2.25 1.268 10.08 0.700 0.85 -9.50 -20.57 -31.00 2.25 76.25 111.937
Fe 2+ - e- = Fe 3+
-13.032
Fe2+ + H20 + H + - e- = Fe(OH)2+ Fe2+ + 2H20 - 2H+ - e- = Fe(OH)• Fe2+ + 3H20- 3H+ - e- = Fe(OH)3 Fe2+ + 4H20 - 4H+ - e- = Fe(OH)• 2Fe2+ + 2H20 - 2H+ - 2e- = Fe(OH)24+ 3Fe2+ + 4H20 - 4H+ - 3e- = Fe3(OH)•+
-15.22 -18.70 -26.63 -34.63 -29.01 -45.4
Fe 2+ + C1- - e- = FeC12+
-11.55
Fe2+ + 2C1- - e = FeC12 Fe2+ + 3C1- - e- = FeC13 Fe2+ + SO•- - e- = FeSO• Fe2+ + 2SO•- - e- = Fe(SO4) • Mn2+ + H20- H + = MnOH+ Mn2+ + 3H20- 3H+ = Mn(OH)• Mn 2+ + C1- = MnC1 +
Mn2+ + 2C1-= MnC12 Mn2+ + 3C1- = MnCI• Mn2+ + CO32-+ H + = MnHCO• Mn2+ +SO•- = MnSO4 Mn 2+ - e- = Mn 3+
Mn2+ + 4H20 - 8H+ - 4e- = MnO•Mn2+ + 4H20- 8H+ - 5e- = MnO; A13++ H20- H + = AI(OH)2+ A13++ 2H20- 2H+ = AI(OH)• AI3+ + 3H20 + 3H+ - AI(OH)3 A13++ 4H20 - 4H+ = AI(OH)g A13++ SO•- = AI(804)+ A13++ 2SO]- = Al(804)• Si(OH)4 - H + - H3SiO4• Si(OH)4- 2H+ = H2SiO•-
SeO32-+ H20-
2H+ + 2e- = SeO]-
+
-10.90 -11.90 -9.11 -7.61 -10.59 -34.08 0.607
0.041 -0.305 11.60 2.26 -25.0507
-118.44 -127.824 -4.99 -10.1 -16.0 -23.0 3.02 4.92 -9.929 -21.617 -29.07
LIU AND NARASIMHAN: REDOX-CONTROLLEDCHEMICAL TRANSPORTMODEL, 2 TABLE
11.
(continued)
Log of Equilibrium Chemical
Reaction
SeO]- + H20- H + + 2e- = HSeOg SeO]- + H += HSeO•SeO]- + 2H+ = H2SeO 3 SeO]- - 3H20 + 6H+ + 6e- = Se2SeO]- - 3H20 + 7H+ + 6e- = HSeSeO]- - 3H20 + 8H+ + 6e- = H2Se SeO]- + H20 - 2H+ - 2e- + Ca2+ = CaSeO 4 SeO]- + Ca2+ = CaSeO3 SeO•- + H20- 2H+ - 2e- + Mg2+ = MgSeO 4 SeO]- + Mg2+ = MgSeO3 H3AsO4 + H + + 2e- - H20 = H2AsO•H3AsO4 + 2e- - H20 = HAsO32H3AsO4 + e- - H20- H + = AsO33H3AsO4 + H3AsO4 H3AsO4 H3AsO4 H3AsO3+
3H+ + 2e- - H20 = H4•tsO3 H + = H2AsO•2H+ = HAsO423H+ = AsO43H20 - 2H + - 2e- = H3AsO•-
Constant -27.16 7.31
9.87 37.35 52.33 56.14 -26.4 3.17
-26.72
2.87 -10.216 - 1.886
-15.3 19.139
-2.243 -9.001
-20.597 -19.444
899
contaminant transport, most studies have been restricted to qualitative description [e.g., Baedecker and Back, 1979; Cherry et al., 1984]. In this section, simulation of two-dimensional contaminant transport is presented. Various chemical reactions are considered,including acid-base, aqueouscomplexation, redox, and dissolution-precipitation.Sixteen aqueous components, 26 minerals, and over 76 aqueous complexes are included in the simulation. The redox reactive trace elements selenium and arsenic are included because these two
elements are toxic and are of great concern to environmentalists. The toxicity of selenium in ground waters, when present in large concentration, is well known [Robberecht and Van Grieken, 1982]. Arsenic is generally associatedwith selenium and has been found in many mine tailings [e.g., Cherry et al., 1986] which have polluted the aquatic ecosystem and groundwater. Selenium and arsenic are also the major toxicants generated from coal-fired power plants, which produce 70 million tons of coal ash and 7 million tons of scrubber sludgeper year. As a result of the high concentrations of aqueousSe and As, fish populationsin the cooling reservoirs adjacentto coal-fired plants at three sites in North Carolina and Texas have decreased significantly [Shepard, 1987].
causes copper precipitation and a gradual increase of the redox potential in the enrichmentzone and releaseof oxygen leading to a gradual increase in the redox potential of the enrichment zone. The protore is stable throughout the simulation. The sharp redox front is a characteristic of the natural redox processes of supergeneenrichment that have been reported by Titley [1975] and Ney et al. [1976].
3.2. 3.2.1.
The simulation was run to 30 years, a time scale relevant to many engineering applications. Results of simulation of aqueous concentrations and the dissolution-precipitation fronts are presented in terms of contour plots at 10, 20, and TABLE 12. Mineral Solubility Reactions and Their Equilibrium Constants at 25øC for Minerals
In order to demonstrate
of Two-
Log of Equilibrium
Two-Dimensional Contaminant Transport Introduction.
in the Simulation
Dimensional Reactive Contaminant Transport
Chemical
Reaction
Constant
the multi-
dimensionalcapability of DYNAMIX and its applicabilityto realistic field problems, we apply DYNAMIX to the study of a hypothetical, large-scale, two-dimensional reactive chemical transport problem. This type of problem is frequently encounteredin the field of contaminanthydrology where one wants to determine and to predict the movement of the contaminated zone and the
effect of contaminationon groundwaterquality. Examples of such situations include municipal and industrial landfills, industriallagoons,mine and mill tailings, and piles of fly ash generated from coal-fired power plants. The assumed field conditionsfor the contaminanttransportproblem are shown schematically illustrated in Figure 23. As the leachate from contaminant source moves downward into the groundwater aquifer, it mixes with the regionalgroundwaterflow system. The zone of contamination and its chemical composition continuously spread and change because of advection, diffusion and dispersionof the solutes,and the complex chemical interactionsbetweenaqueousand solidphases.Because of the complexity of chemical interactions between the aqueousand solid phases,previous studieson modelingthis type of problem have been limited to either a single-species transport in two or three dimensions [e.g., Robson, 1981; Hoford, 1985;Gilliland and Nguyen, 1987]or a rather simple multiple-speciestransport in one dimension [e.g., Miller and Benson, 1983; Cederberg, 1985; Narasimhan et al., 1986]. To fully account for all the complex chemical reactions in
CaCO3 = Ca2+ + CO32-
-8.47
CaMg(CO3) 2 = Ca2+ + Mg2+ q-2CO•-
-17.02
CaSO4 ß2H20 = Ca2+ + SO•- + 2H20 AL(OH)3 = A13++ 3H20- 3H+ KalSi308- K + + A13++ 3Si(OH)4 + 4H20
-22.792
-4.848
0.875
_ 4H +
SiO2 = Si(OH)4- 2H20 Ko.bMgo.25AI2.3Si3.sH20•2 = 2H20 + 0.6K +
+ 0.25Mg2+2.3A13+ + 3.5Si(OH)4- 8H+ FeS2= Fe2+ + 2HS- - 2H+ - 2eFeS = Fe 2+ + HS-
- H+
FeO(OH) = Fe3+ + 2H20- 3H+ Fe(OH)3= Fe3+ + 3H20- 3H+ Fe203= Fe3+ + 3H20- 6H+ Se = Se2032+ + 2H+ - H20 - 0 2 CaSe = Ca 2+ + Se2+
MgSe = Mg2+ + Se2-
CaSeO3.2H20 = Ca2+ + SeO32-+ 2H20 CaSeO 4 ß2H20 = Ca2+ + SeO42-+ 2H20 As205 = 2H3AsO4 - 3H20
A1AsO 4 ß2H20 = A13+= AsO43-+ 2H20 Ca3(AsO4)2.4H20= 3Ca2+ + 2ASO43+ 4H20 FeAsO4 ß2H20 = Fe3+ + AsO43-+ H20 Mn3(AsO4)2.8H20= 3Mn2+ + 2ASO43+ + 8H20 As406 = 4H3AsO3 - 6H20
As2S3 = 2H3AsO3 + 3HS- + 3H+-6H2 ¸ AsS = H3AsO3 + Hs- + 2H + + e-- 3H20
-4.006
9.8046
-18.48
-5.9476 0.5
4.891 -4.008 26.14
10.58 9.95 -4.61 -3.1
6.99 -15.837 18.905 -20.249 -28.707 14.33 -60.971 -19.747
900
LIu ANDNARASIMHAN: REDOX-CONTROLLED CHEMICALTRANSPORT MODEL,2
LU
s.s /
-5
I-U.I -10
O
-15
z
•C
I-
-2o
i
-25
200
i
i
400
600
DISTANCE,
i
800
lOOO
IN METERS
Fig. 25a
LU
-5
I-LU -10
Z
O
--6.5
-lS
6.9
Z
•C
I-
-2o
i
-25
200
i
i
400
600
DISTANCE,
i
800
1000
IN METERS
Fig. 25b
LU
-5
I-' LU
Z
0
-10
-15
z
I-
-20
6.9 ••'--25
200
400
600
DISTANCE,
800
1000
IN METERS
Fig. 25c
Fig. 25. Two-dimensional reactivechemicaltransport:pH contourat (a) 10 years,(b) 20 years,and(c) 30 years. 30 years to show the physicochemical evolution of the contaminants along the groundwater aquifer, the effect of chemical reactions on the migration of contaminantsin the aqueous phase, and the development of dissolution and precipitation in the solid phases. 3.2.2. Problem definition. To model the contaminant transport in the ground water aquifer as shownin Figure 23,
a simplifiedtwo-dimensionaldomainas shownin Figure 24 is used. The area simulatedis 1000 m long by 25 m deep by 1 m thick. Constanthydraulicheadsare appliedto both the left and right edges.At the left edge the hydraulic head is 1.6 m higherthan at the right edge suchthat the groundwaterflow is from left to right. The annualprecipitationis approximated
througha constantnormalflux of 1.3 x 10-7 m3/senteringall
LIU AND NARASIMHAN:REDOX-CONTROLLED CHEMICALTRANSPORT MODEL, 2
901
o
W
-$ -8.$
w -lo
Z
-$.$
I
-25
0
i
400
200
CO0
DISTANCE,
I lOOO
800
IN METERS
Fig. 26a
W
-5
IuJ
Z -lo
-25
I o
200
i
I
i
400
600
800
DISTANCE,
1000
iN METERS
Fig. 26b
• -sk\ '-. y: _ -•o
',
•-
: , :
-15
.,o0
...... 200
400
600
DISTAHCE,
IH IdF..TER$
800
1000
Fig. 26c
Fig. 26. Two-dimensional reactivechemical transport: pE contour at (a) 10years,(b) 20 years,and(c) 30years. acrossthe top, representingan infiltrationof 0.17 m/yr. A Dispersivity no-flowboundaryconditionis prescribedalongthe bottom. Otherphysicalparametersare asfollows:hydraulicconduc-
tivity is 10-5 m/s,
a=lOm
Porosity
&=0.2 Diffusion coefficient of each component
Ddiff, i = 1.6x 10-]0m2/s
Saturation s=
1.0
902
LIO ANDNARASIMHAN' REDOX-CONTROLLED CHEMICAL TRANSPORT MODEL,2
-5
-10
-15
-20
I................
-25
•1 ,
200
I
400
000
DISTANCE.
I,
800
lOOO
IN METERS
Fig. 27a
i
,
200
i
i
i
400
600
800
DISTANCE.
lOOO
IN METERS
Fig. 27b 0
2:: -10 O
-15 -
2:
I..- -20
-
, I ........
-25
0
.....
200
I,I
I
400
000
DISTANCE,
800
1000
IN METERS
Fig. 27c
Fig. 27. Two-dimensional reactivechemicaltransport'contourof aqueousSe concentration, in molesper cubic meter, at (a) 10 years, (b) 20 years, and (c) 30 years.
Note that dispersionis only applied to the horizontal con- [Narasimhan et al., 1978]. The steady state fluxes were then nectionsneglectingtransversedispersivity.The dispersion used as an input for the DYNAMIX simulation of contamiin the vertical connections is assumed to be zero.
nant transport.
The computationaldomainwas dividedinto 40 x 10equal The initial aqueouscompositionsfor the tailings water, sizegrid with a total of 400boxes.Each box is 25 m longby groundwaterand rain water are given in Table 9. While both 2.5 m wide by 1 m thick. The steady state flow field was first the tailingswater and rain water are under oxidizingcondisolvedby usingTRUST, an integralfinitedifferenceprogram tions, the tailings water is assumed to be more acidic. The for transient fluid flow under variably saturated conditions
initial groundwater is under near-neutral reduced conditions.
LIU AND NARASIMHAN' REDOX-CONTROLLED CHEMICAL TRANSPORTMODEL, 2
crj
903
1)
