Evaluation of one-dimensionalin situ leaching processes

*Correspondence to: Jishan Liu, Department of Civil Engineering, The University of Western Australia, Nedlands, Perth,WA 6097, Australia. E-mail: [email protected]

Contract/grant sponsor: Australian Research Council; Contract/grant number: A89600730

CCC 0363}9061/99/151857}16$17.50 Received 14 October 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 27 April 1998 and 3 August 1998

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 23, 1857}1872 (1999)

EVALUATION OF ONE-DIMENSIONAL IN SI¹;LEACHING PROCESSES

J. LIU1,* AND B. H. BRADY2

1 Department of Civil Engineering, The University of Western Australia, Nedlands, Perth 6907, Australia2 Faculty of Engineering and Mathematical Sciences, The University of Western Australia, Nedlands WA 6907, Australia

SUMMARY

A methodology to characterizing processes of in situ leaching is developed to investigate various parametersthat may a!ect the recovery of a valuable mineral and assure a successful application of the in situ leachingtechnology. The governing equations of in situ leaching processes for the one-dimensional case are solvedboth analytically and numerically for both the consumption rate of a lixiviant and the production rate ofa target mineral. A numerical simulator, which is developed to evaluate coupled e!ects among leachingkinetics, solution #ow and transport of the dissolved mineral species in saturated ore deposits, is validatedagainst the steady-state solutions and applied to investigate the transient e!ects of various parameters on themineral recovery. Results from the evaluation indicate that there exists an optimal #ow velocity range ofleach solution for the e!ective leaching of a particular ore deposit. The determination of this optimalleaching velocity may become a key to the design of a real in situ leach mine. Results of the parametersensitivity study illustrate the relative importance of other parameters such as porosity, ore grade, andreaction rate. Copyright ( 1999 John Wiley & Sons, Ltd.

KEY WORDS: In situ leaching; analytical solution; numerical model; #ow; transport

1. INTRODUCTION

Leaching is de"ned as removal of minerals by dissolving them from the solid matrix. In situleaching may be de"ned as recovery of mineral values by dissolving them directly from in-placeore deposits. An inexpensive lixiviant is injected into a mineralized rock mass and moves throughthe ore body. The reagent is selected in part to maximize the leaching rate of a speci"c mineral ina particular orebody, ordinarily the rate constant would be only one of a number of chemicalfactors that would be considered.1}3 The signi"cance of this mining technology is illustrated byproblems which are becoming apparent with current practice of mineral resource engineering.Decreasing grade of near-surface deposits (i.e. within 500 m of the ground surface) will in futureresult in mining, transport and processing of much larger volumes of rock per unit productrecovered. In addition to continued exploitation of low-grade and near-surface deposits, thetechnology in in situ leaching may be used to extract higher grade and deeper deposits. Assumingexploration technology advances su$ciently to permit economic location and delineation of deep

orebodies, current practice has already confronted limitations in mining technology for large-scale and deep-level mining. In particular, it is now recognized that virtually all mining operationsin hard-rock settings at depth will be subject to induced seismicity and rockbursts, particularly ifthe orebody is mined at an extraction ratio consistent with optimum economic use of the totalmineral resource. Rockbursts cause severe damage to underground installations, endanger thelives and safety of mine sta! and introduce a high degree of unpredictability in deep miningoperations. Induced seismicity and rockbursts are recognized as a pervasive problem in deep-level hard-rock mining.4 Results of the research on the in situ leaching processes can also beapplied to solve a wide range of other problems such as waste rock drainage,5,6 industrial wastedisposal,7 rock weathering,8 and concrete degradation by dissolving soluble constituents due torainwater.9 Although in situ leaching is increasingly becoming an attractive mining method forextraction of mineral values from near-surface low-grade ore deposits or higher grade and deeperdeposits,1}3 this method may not be used in mining practice until the governing sciences are fullyunderstood.

1.1. Coupling processes

In situ leaching is a process of mass transfer between immobile (mineral aggregate) and mobile(lixiviant) zones. The transfer of mass in an ore deposit is a consequence of simultaneousprocesses. Once injected into an orebody, the lixiviant will react with minerals that make up therock. Major hydrological processes of the interaction include di!usion, dispersion and advec-tion.8,10}12 Major chemical processes of the interaction may involve aqueous complexation,redox reaction, acid}base reaction, adsorption via surface reactions (complexation and ionexchange),10,13}17 dissolution and precipitation.15,17,18 All of these processes will contribute tothe distribution and redistribution of chemicals including the lixiviant and the dissolved mineralin the leach solution. Three basic requirements19 for an e!ective in situ leaching have to be met: (1)leach solution must be able to contact ore minerals; (2) target minerals must be selectivelysolubilized by a leaching agent (lixiviant); (3) metallic species must be transported to a recoverywell. As a result of leaching, some mechanical properties of an ore deposit such as porosity willalso change.20 In order to meet these three requirements, the e!ects of coupling between the #owof leach solution through an ore deposit, the chemistry of leaching in the ore deposit, and thetransport of chemicals through the ore deposit have to be fully understood. One of the moste!ective means to better understand these coupled e!ects is computational simulation.

1.2. Previous researches

In situ leaching, as a well-recognized potential mining method for low ore grade deposits, hasbeen reviewed by several publications.21}23 An improved understanding of the leaching processeshas been achieved through laboratory tests19,24}33 and mathematical modelling.34}43 Mathemat-ical models can be classi"ed as chemical models, #ow models and transport models. Chemicalmodels are developed to characterize the dynamics of leaching. Shrinking-core models10,27 havebeen proven to be very useful in describing the leaching behaviour of many ores.25,27,28,31}33However, whether these chemical models can be applied to represent the dynamics of in situleaching remains undetermined. Flow models are developed to characterize the #ow of leachsolution in an ore deposit. It is shown in the study44 that fractures and their distribution in anorebody control the #ow characteristic of a lixiviant. Transport models are developed to

1858 J. LIU AND B. H. BRADY

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 1857}1872 (1999)

characterize the e!ects of couplings between leaching dynamics, solution #ow, and chemicalstransport in an orebody. It is obvious that only transport models are able to address the questionof how the lixiviant in leach solution is consumed and how the valuable mineral is produced.Some limitations of these models are lack of the capability to deal with the complex solution #owin an ore deposit,15}17,40,41 and others are lack of the capability to handle the dynamics of in situleaching.42,44 The most signi"cant limitation of these transport models is that they are notveri"ed against analytical solutions. A long-term e!ort is being made through this research todevelop a computer simulator which can simultaneously handle both the solution #ow ina complex ore deposit and the dynamics of in situ leaching. Both analytical and numerical resultsfor the one-dimensional in situ leaching case are presented in this paper.

1.3. Scientixc rationale

In this research a methodology to characterize processes of in situ leaching is developed tostudy various parameters that may a!ect the recovery of a valuable mineral and to assurea successful application of the in situ leaching technology. A leaching kinetics model is "rstderived based on the concept of representative elementary volume (REV). Every parameter in themodel is clearly de"ned and may be easy to be obtained in practice. Then the governing equationsare obtained for transport of both the lixiviant and the dissolved mineral. These equations aresolved both analytically and numerically. A numerical simulator (SWIFT II with some modi"ca-tions) is validated against the analytical solutions and used to investigate the sensitivity ofparameters such as #ow velocity, ore grade, reaction rate, and lixiviant concentrations.

2. GOVERNING EQUATIONS

Governing equations for processes of in situ leaching are derived based on the concept ofrepresentative elementary volume (REV).45 Consider a mathematical point x in a #ow region, withco-ordinates (x

1, x

2, x

3), in a three-dimensional coordinate system. A small volume, which can be

either spherical or cubic with its centre at x, is de"ned as a particle of the ore deposit. On the onehand, the volume must be large enough to cover a su$cient number of solid particles and pores sothat the stable mean values of certain physical properties associated with it can be obtained overthe volume. On the other, the volume must also be small enough in comparison with the wholeregion so that it can be treated as a point. The particle thus de"ned is called a REV. If everymathematical point in an ore deposit is associated with such a particle, the ore deposit, which isconstructed with solid matrices and pores, can be considered as a continuum fully "lled withthose particles. In the following derivations of governing equations for the one-dimensional in situleaching, a cubic REV is adopted to derive the #ow and transport equations, and a spherical REVfor the in situ leaching kinetics, as illustrated in Figure 1.

2.1. Transport equations

The e!ects of in situ leaching on solute transport are incorporated into the advection}disper-sion equation through a chemical source/sink term. A particle may be either represented asa cubic REV in a #ow domain or a spherical REV in a chemical domain, as illustrated in Figure 1.The particle may also be considered as an in situ reactor. Several processes such as advection,dispersion, consumption of a lixiviant contained in the leach solution, and production of a target

EVALUATION OF ONE-DIMENSIONAL IN SI¹; LEACHING PROCESSES 1859


Figure 1. Conceptual representation of an ore deposit. A particle is represented by a cubic REV for the #ow and transportmodel and a spherical REV for the leaching model. q is the Darcy's #ux and o the density of leach solution

mineral will simultaneously proceed within the reactor. Transport equations of a component(lixiviant or dissolved mineral) can be de"ned based on the principle of mass conservation as

LC1

Lt"D

L2C1

Lx2!<

LC1

Lx!R

1(1)

LC2

Lt"D

L2C2

Lx2!<

LC2

Lx#R

2(2)

and seepage velocity of the leach solution in an ore deposit, <, is de"ned as

<"q

/"!

K)

/

Lh

Lx(3)

where C1

and C2

are concentrations of the lixiviant and the dissolved mineral, respectively, D thehydrodynamic dispersion coe$cient, x the co-ordinate, t the time, q the Darcy's #ux, o the densityof the leach solution, K

)the hydraulic conductivity, h the total head, / the e!ective porosity and

R1

and R2

are the consumption rate of the lixiviant and production rate of the target mineral,respectively. R

1and R

2can be evaluated based on the kinetics of in situ leaching. The

hydrodynamic dispersion coe$cient, D, is de"ned as46

D"aL<#D

$¹ (4)

where aL

is the longitudinal dispersivity, D$

the di!usion coe$cient in the solution and ¹ thetortuosity of the porous medium. D

$is a function of the composition and the temperature of

the solution. Experimental results show that aLis mainly dependent on the mean particle size, the

uniformity coe$cient46 and the scale.47 Therefore, dispersion coe$cient in equations (1) and (2)may be assumed as equal.

2.2. Kinetics of in situ leaching

Shrinking-core models have been applied to describe the leaching behaviour of many oresalthough they are used as empirical tools rather than in any predictive way. Generally, attemptsare made to "t di!erent shrinking-core equations with batch leaching data, and then the-best-"t



equation is chosen to characterize the leaching kinetics for a speci"c type of ore. These columnleaching tests are carried out on the ground ores, and the lixiviant concentration is maintainedconstant. These two conditions signi"cantly deviate from the in situ situation. Because of thesedeviations, whether these shrinking-core equations can be applied to represent the situation insitu leaching remains undetermined. In the following derivations, the overall dissolution processis considered to consist of (1) transport of reactants (lixiviant) from the bulk solution into the oreparticles. (2) chemical reactions at the solid surface, (3) desorption of soluble products of thereactions and (4) transport of soluble products back to the bulk solution. The chemical reaction ofthe in situ leaching process may be described as

lAA (aqueous)#l

BB (solid)Pl

MM (aqueous)#l

SS (aqueous) (5)

where A represents a lixiviant, B represents a solid reactant (target mineral), M representsa dissolved product (recoverable precious metal), S represents non-mineral reaction product, andlA, l

B, l

M, and l

Sare stoichiometric numbers. A conceptual model of characterizing the overall

leaching process is presented in Figure 1, where a is the radius of the spherical REV for an oreparticle, r is radius of the unreacted particle core at time t, and d is the thickness of the di!usionlater. Assuming the mass of the dissolved mineral is directly proportional to time, t, and theconcentration of the lixiviant, C

i, in the di!usion layer. As illustrated in the leaching model in

Figure 1, the concentration of the dissolved mineral is de"ned as the ratio of the mass of thedissolved mineral and the mass of the solution in the di!usion layer. It can be expressed as

C2"

4nr2o3Gl

M4nr2do/l

A

kCit"

o3Gl

Mo/l

A

kCit (6)

Therefore, the rate of change in the concentration of the dissolved mineral is de"ned as

dC2

dt"

o3Gkl

Mo/l

A

Ci

(7)

where C2

is the concentration of the dissolved mineral, / the porosity of ore rock, k the rateconstant, o the density of the solution, o

3the density of the rock, G the ore grade, and C

ithe

concentration of the lixiviant in the di!usion layer. Under steady-state conditions, the rate oftransport of the lixiviant into the di!usion layer will be equal to the rate within the di!usion layer.Di!usion through pores to the di!usion layer can be expressed according to the Fick's law as

dC2

dt"4nr2D

lM

lA

dC

dr(8)

Substituting equation (8) into (7) gives

4nr2DlM

lA

dC

dr"

orGklM

o/lA

Ci

(9)

Integrating equation (9) by letting r vary from r to a and C from Cito C

1(bulk concentration of

the lixiviant) yields

Ci"

C1

1#orGk

4nDo/

da2

(10)



The surface chemical analyses indicate that mineral dissolution is a process controlled bysurface reactions.8,48,49 The depth of this &surface region' is not extensive (less than 5 nm) and inacid solutions depends on the interplay between the exchange rate of hydrogen ions for cationsand the subsequent decomposition of the remaining structure.8 It is obvious that the latter is fastenough so that the cation}proton exchange occurs only in the top several unit cells and noleached layer develops. Therefore, it may be reasonable to assume the radius r at time t is equal tothe original radius a. This is particularly true for the fracture-hosted ore deposit because leachsolution predominantly #ows in fractures and minerals deposit at the fracture surface. Thereforeit may be reasonable to assume d/a2"0 and equation (10) is simpli"ed as

Ci"C

1(11)

The "nal rate of dissolution of the mineral is obtained by substituting equation (11 into (7):

dC2

dt"

o3Gkl

Mo/l

A

C1"

lM

lA

KC1

(12)

and the parameter K is de"ned as

K"

o3Gk

o/(13)

Consequently, the consumption rate of the lixiviant, R1, and the dissolution rate of the target

mineral, R2, can be de"ned as

R1"KC

1"

o3Gk

o/C

1(14)

R2"

lM

lA

KC1"

o3Gkl

Mo/l

A

C1

(15)

respectively.Equation (12) indicates that the linear leaching kinetics may be used to represent the processes

of in situ leaching. The conclusion is also supported by several studies.8,24,26,48}51 It is found thatthe most signi"cant deviation of batch leaching tests away from the in situ situation stems fromsample preparation, and linear leaching kinetics were obtained if the e!ects of grinding thesamples on the structure of the surface and the size and the shape of the various small grain wereremoved.52,53 Leaching results on partially removed "ne grains indicated that non-linear kineticswere initially observed, and essentially linear kinetics were produced thereafter.24 The initialnon-linearity was believed due to the leaching of remaining "ne particles or super"cial oxidecoatings. Linear kinetics was also produced by a real in situ leaching test conducted by the USBMin a copper deposit at the Emerald Isle Mine in 1977, and a linear concentration versus leachingduration curve can be obtained by use of the in situ leaching data.26 Linear kinetics, as expressedin equation (12), is also consistent with the result of Meng and Han.51 It is also demonstrated inReference 51 that linear kinetics may be applied to characterize the in situ leaching dynamics ofcobalt, nickel, copper and gold in acid, cyanide or ammoniacal solutions.



2.3. Initial and boundary conditions

Equations (1) and (2) are solved under the following boundary and initial conditions:

C1"C0

1, C

2"0, x"0, t*0

C1"0, C

2"0, x'0, t"0

(16)

where C01

is the constant concentration, maintained at the top of a leach column. The physicalmeaning of th initial and boundary conditions corresponds to a situation where a lixiviant iscontinually supplied on to the top of a leach column which does not contain both the lixiviantand the dissolved mineral initially.

2.4. Governing equations

Substituting equations (14) and (15) into (1) and (2), respectively, gives the complete set ofgoverning equations:

LC1

Lt"D

L2C1

Lx2!<

LC1

Lx!KC

1(17)

LC2

Lt"D

L2C2

Lx2!<

LC2

Lx#

lM

lA

KC1

(18)

It is apparent in equations (17) and (18) are coupled by chemical reaction terms.

3. ANALYTICAL SOLUTIONS

The complete set of governing equations (17) and (18), with boundary and initial condition asshown in equation (16), are solved analytically under the simpli"cation of l

M/l

A"1 and

R1"R

2"KC

1. Solutions for concentrations of both a lixiviant and a target mineral can be

obtained based on the research results on convective transport of ammonium.54,55 Assuming thatthe value of DK is very small as compared with<2, steady state (tP#Rand x is "nite) solutionscan be approximated to

C1"C0

1e~Kx@V (19)

C2"C0

1[1!e~Kx@V]. (20)

It is shown in equation (20) that breakthrough curves are controlled by two parameters, K and<. The breakthrough curves of a dissolved mineral as a function of the #ow velocity of a leachsolution are plotted with di!erent lumped rate constants in Figure 2. It can be concluded that thee!ectiveness of in situ leaching is determined by the #ow velocity for a speci"c lumped rateconstant, and that the higher #ow velocity should be applied to e!ectively leach a valuablemineral with a higher lumped rate constant. As an example, #ow velocities must not exceed 0)01,0)1, 1, and 10 m/day for the lumped rate constants are equal to 0)001, 0)1, 1, and 10 day~1,respectively, if the minimum recovery concentration of the mineral is maintained not less than0)62, as indicated by dash lines in Figure 2. Breakthrough curves of a dissolved mineral are alsoplotted as a function of dimensionless leaching parameter Kx/< in Figure 3. The relationshipbetween Kx/< and C

2/C0

1may be used to optimally design a leach column test and to



Figure 2. Breakthrough curves of a dissolved mineral as a function of the #ow velocity of a leach solution with di!erentlumped rate constants K

Figure 3. Dimensionless breakthrough curves of a dissolved mineral as a function of dimensionless leachingparameter Kx/<

experimentally determine the lumped rate constants. As an example, if the dimensionless steady-state recovery concentration C

2/C0

1"0)80, x"1)0 m, and <"0)1 m/day, then the dimension-

less leaching parameter Kx/<"1)609, as illustrated by dash lines in Figure 3, and the lumpedrate constant K is calculated as 1)29 day~1.

Equation (12) alone may be applied to evaluate laboratory leaching tests in which theconcentration of the lixiviant is maintained constant. As shown in equation (12), the lumped rate



Figure 4. Relation between dimensionless leaching rate of the target mineral with porosity for a leaching test witha constant lixiviant concentration

constant K is directly proportional to the ore grade and the rate constant, and inverselyproportional to the porosity. The dimensionless form of equation (12) is written as

1

/"

R2

orGklM/ol

A

C1

(21)

The relation between dimensionless leaching rate and ore porosity, as de"ned in equation (21), isshown in Figure 4 under di!erent lixiviant concentrations. It is apparent that both the lixiviantconcentration and the porosity have signi"cant e!ects on the mineral recovery rate. The smallerthe porosity the higher the mineral recovery rate. This may be explained by the fact that a higherinitial porosity corresponds to a smaller initial volume for the mineral phases.

4. NUMERICAL MODELLING

The computer code used in this application is Sandia Waste-Isolation Flow and TransportModel (SWIFT II), a fully three-dimensional and transient "nite-di!erence program.56}58 Thecode was developed under rigorous quality assurance59 and has been applied to a variety ofgeohydrological problems.58,60,61 The computer program was developed primarily for use in theanalysis of deep geologic nuclear waste disposal facilities. Because of the similarities of governingequations for the radionuclide transport and the in situ leaching process in fractured porousmedia, the computer program has been applied to the analysis of the in situ leaching processes,governed by equations (1)}(3), (16) and (12). In this analysis, the problem of 1D in situ leaching istreated a two-component nuclide transport in fractured porous media. The lixiviant, treated asthe mother radionuclide component, which derives from a source of constant concentration at thetop of a leach column, is advected, dispersed and di!used through the ore deposit column ina constant velocity "eld. The target mineral, treated as the daughter radionuclide, is dissoved into



Figure 5. Validation of the computer simulator. Numerical data, represented by legends, are compared with analyticalsolutions, represented by solid lines

the leach solution, and transported to the other end. Because the #ow is not directly speci"ed inthe computer program, the #ow within the leach column is maintained using injection andproduction wells at each end of the column. In order to demonstrate the validity of the computerprogram for the in situ leaching processes, the numerical results for the lumped rate constantK"0)24 d~1 with di!erent velocities <"0)001, 0)01 and D"0)1 m/day are compared withanalytical solutions for the steady state, as shown in Figure 5. It can be concluded that numericaldata are well-matched analytical ones. The validated computer program is applied to investigatevarious parameters that may a!ect the mineral recovery. These results are reported in thefollowing section.

5. SENSITIVITY STUDIES

The relation between the dimensionless concentration of a dissolved mineral and the leachingduration is determined by three basic parameters, the lumped rate constant K, the #ow velocity<, and the dispersion coe$cient D, as de"ned in equation (4). K is a function of ore grade G,chemical reaction rate constant k, and ore porosity /, as shown in equation (12).< is a function ofDarcy's #ux q and porosity /, as shown in equation (3). These controlling factors may besite-speci"c and vary widely. Therefore, it is necessary to know how the mineral recovery isa!ected by change of these parameters within reasonable regions.

The numerical results of breakthrough curves of a dissolved valuable mineral with di!erentlumped rate constants, #ow velocities, and dispersion coe$cients are plotted in Figures 6}8. Eachof the three parameters, K, <, and D, signi"cantly a!ects the recovery concentration of a dis-solved valuable mineral. However, the three parameters may have quite di!erent e!ects on themineral recovery.



Figure 6. Numerical results of the breakthrough curves of a dissolved valuable mineral with di!erent lumped rateconstants

Figure 7. Numerical results of the breakthrough curves of a dissolved valuable mineral with di!erent #ow velocitiesof the leach solution

5.1. Ewect of lumped rate constant

Numerical simulations are performed for the lumped rate constants K"1, 2 and 10 day~1.The injection #ux (Darcy's #ux, volume/unit area), porosity, and the dispersion coe$cient are



Figure 8. Numerical results of the breakthrough curves of a dissolved valuable mineral with di!erent dispersioncoe$cients

chosen as 0)1 m/day, 0)10 and 0)01 m2/day, respectively. It s apparent that the three curves aresimilar in shape, but the steady-state recovery concentrations are quite di!erent, as illustrated inFigure 6. The higher the lumped rate constant, the higher the steady-state recovery concentration.The steady-state recovery concentrations for the lumped rate constants K"1, 2, and 10 day~1

are 0)6, 0)8, and 1)0, respectively, as indicated in Figure 6.

5.2. Ewect of leach yow velocity

Numerical simulations are performed for the injection #ux q"0)01, 0)1 and 0)2 m/day. Thelumped rate constant, porosity, and the dispersion coe$cient are chosen as 2 day~1, 0)10 and0)01 m2/day, respectively. It is apparent that the leach solution #ow velocity a!ects not only theshape of the breakthrough curves but also the steady-state recovery concentrations of a mineral,as illustrated in Figure 7. The steady-state recovery concentrations of a mineral for the injection#uxes q"0)01, 0)1 and q"0)2 m/day are 0)6, 0)8, and 1)0, respectively, as indicated in Figure 7.

5.3. Ewect of dispersion coezcient

Numerical simulations are performed for the dispersion coe$cient D"0)002, 0)01 and0)05 m2/day. The injection #ux, porosity, and the lumped rate constant are chosen as 0)1 m/day,0)10 and 2day~1, respectively. It is concluded that the dispersion coe$cient a!ects both the shapeof breakthrough curves of a mineral and the steady-state mineral recovery concentrations, asillustrated in Figure 8. The steady-state mineral recovery concentrations of a mineral for thedispersion coe$cients D"0)002, 0)01 and 0)05 m2/day are 0)68, 0)82, and 0)86, respectively, asindicated in Figure 8.



6. CONCLUSIONS

Governing equations of in situ leaching processes are developed based on the concept ofrepresentative elementary volume (REV). A cubic REV is adopted to derive the #ow andtransport equations,. and a spherical REV for the in situ leaching kinetics. Analytical solutions ofthe governing equations provide one-dimensional solutions for both the consumption process ofa lixiviant and the production process of a targeted mineral. The adopted computer simulatorSWIFT II is applied to investigate the e!ects of lumped rate constant, #ow velocity and dispersioncoe$cient on the breakthrough curves of a dissolved mineral. Each of the three parameterssigni"cantly a!ects the recovery concentration of a dissolved valuable mineral. However, the threeparameters may have quite di!erent e!ects on the mineral recovery. The lumped rate constant doesnot a!ect the shape of the concentration pro"le of a valuable mineral, but signi"cantly a!ects thesteady-state recovery concentrations. The higher the lumped rate constant, the higher the steady-state recovery concentration. The #ow velocity a!ects not only the shape of the breakthroughcurves but also the steady-state recovery concentrations of a mineral. An optimal #ow velocityrange of leach solution exists for the e!ective leaching of a particular ore deposit. The dispersioncoe$cient also a!ects both the shape of breakthrough curves of a mineral and the steady-statemineral recovery concentrations, but this in#uence is signi"cantly less than the #ow velocity.

The research results may have signi"cant values for the mining industry. The lumped rateconstant can be easily obtained through laboratory leaching tests. The optimal #ow velocity ofleach solution, obtained based on the magnitude of the lumped rate constant, may be the key tothe design of a real in situ leaching mine because it determines whether the permeability of an oredeposit needs to be enhanced for e!ectively leaching the ore deposit. However, cautions should bemade when applying these conclusions, drawn from the one-dimensional leaching with an axial#ow of the leach solution, to the in situ leaching situation where the leach solution #ows radiallyand the transport is greatly dependent on the velocity.

ACKNOWLEDGEMENTS

The work reported in this paper was supported by the Australian Research Council under LargeGrant No. A89600730.

APPENDIX

NomenclatureA reagentB target mineralC

1concentration of the reagent (kg/kg)

C01

concentration of lixiviant at the top of a leach column (kg/kg)C

iconcentration of lixiviant in the di!usion layer (kg/kg)

C2

concentration of the mineral (kg/kg)D

$molecular di!usion coe$cient (m2/s)

D dispersion coe$cient (m2/s)G ore grade (kg/kg)K lumped rate constant (l/s)K

)hydraulic conductivity (m/s)



M dissolved mineralP #uid pressure (kg/m2)Q injected (!) or withdrawn (#) rate of the leach solution (kg/m3s)Qm ion activity productR dimensionless recovery rateR

1consumption rate of the reagent (l/s)

R2

dissolution rate of the mineral (l/s)S nonmineral reaction product¹ tortuosity of porous media (m/m)< pore #ow velocity (m/s)a radium of the spherical representative elementary volume (m)h water head (m)k rate constant (l/s)q Darcy #ux (m/s)t time (s)xi(i"1, 2, 3) co-ordinates (m)

aL

longitudinal dispersivity (m)aT

transverse dispersivity (m)d thickness of the di!usion layer (m)dij

Kronecker functionk dynamic viscosity (ms)lA, l

B, l

M, and l

Sstoichiometric numbers

o density of the leach solution (kg/m3)o3

density of the rock mass (kg/m3)/ e!ective porosity of the ore deposit (m3/m3)

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Evaluation of one-dimensionalin situ leaching processes

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