Numerical modelling of fluids mixing, heat transfer and non-equilibrium redox chemical reactions in...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2006; 66:1061–1078Published online 12 December 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1581

Numerical modelling of fluids mixing, heat transfer andnon-equilibrium redox chemical reactions in

fluid-saturated porous rocks

Chongbin Zhao1,2,∗,†, B. E. Hobbs1, P. Hornby1, A. Ord1 and Shenglin Peng2

1CSIRO Division of Exploration and Mining, P.O. Box 1130, Bentley, WA 6102, Australia2Computational Geosciences Research Centre, Central South University, Changsha, China

SUMMARY

Non-equilibrium redox chemical reactions of high orders are ubiquitous in fluid-saturated porous rockswithin the crust of the Earth. The numerical modelling of such high-order chemical reactions becomesa challenging problem because these chemical reactions are not only produced strong non-linearsource/sink terms for reactive transport equations, but also often coupled with the fluids mixing,heat transfer and reactive mass transport processes. In order to solve this problem effectively andefficiently, it is desirable to reduce the total number of reactive transport equations with strong non-linear source/sink terms to a minimum in a computational model. For this purpose, the concept ofthe chemical reaction rate invariant is used to develop a numerical procedure for dealing with fluidsmixing, heat transfer and non-equilibrium redox chemical reactions in fluid-saturated porous rocks.Using the proposed concept and numerical procedure, only one reactive transport equation, which isused to describe the distribution of the chemical product and has a strong non-linear source/sink term,needs to be solved for each of the non-equilibrium redox chemical reactions. The original reactivetransport equations of the chemical reactants with strong non-linear source/sink terms are turnedinto the conventional mass transport equations of the chemical reaction rate invariants without anynon-linear source/sink terms. A testing example, for some aspects of which the analytical solutionsare available, is used to validate the proposed numerical procedure. The related numerical solutionshave demonstrated that (1) the proposed numerical procedure is useful and applicable for dealingwith the coupled problem between fluids mixing, heat transfer and non-equilibrium redox chemicalreactions of high orders in fluid-saturated porous rocks; (2) the interaction between the solute diffusion,solute advection and chemical kinetics is an important mechanism to control distribution patterns ofchemical products in an ore-forming process; and (3) if the pore-fluid pressure gradient is lithostatic,it is difficult for the chemical equilibrium to be attained within permeable cracks and geological faultswithin the crust of the Earth. Copyright � 2005 John Wiley & Sons, Ltd.

KEY WORDS: non-equilibrium; redox reactions; chemical kinetics; rate invariants; numericalprocedure; fluids mixing

∗Correspondence to: C. Zhao, CSIRO Division of Exploration and Mining, P.O. Box 1130, Bentely, WA 6120,Australia.

†E-mail: [email protected]

Received 31 May 2005Revised 9 October 2005

Copyright � 2005 John Wiley & Sons, Ltd. Accepted 10 October 2005

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1. INTRODUCTION

Non-equilibrium redox chemical reactions of high orders are ubiquitous in fluid-saturatedporous rocks within the crust of the Earth. They play a very important role in ore-bodyformation and mineralization closely associated with a mineralizing system. Since pore-fluid isa major carrier to transport chemical species from one part of the crust into another, the chem-ical process is coupled with the pore-fluid flow process in fluid-saturated porous rocks. On theother hand, if the rate of a chemical reaction is dependent on temperature, the chemical processis also coupled with the heat transfer process. When a pore-fluid carrying one type of chemicalspecies meets the one carrying another type of chemical species, these two types of pore-fluids can mix together to allow the related chemical reaction to take place due to the solutemolecular diffusion/dispersion and advection. For these reasons, the resulting patterns of min-eral dissolution, transportation, precipitation and rock alteration are a direct consequence ofcoupled processes between fluids mixing, heat transfer and chemical reactions in fluid-saturatedporous rocks.

Due to ever-increasing demands for minerals and possible exhaustion of the existing mineraldeposits in a foreseeable future, it becomes a very important research field to understandcontrolling mechanisms behind ore-body formation and mineralization within the upper crustof the Earth. There is no doubt that the better understanding of ore-forming processes cansignificantly promote mineral exploration for new ore deposits within the upper crust of theEarth. Although extensive studies have been carried out to understand the possible physical andchemical processes associated with ore-body formation and mineralization [1–19], the kineticsof a redox chemical reaction and its interaction with physical processes are often overlookedin the numerical modelling of ore-forming systems. In most chemical reactions associated withan ore-forming system, the chemical reaction rate is finite so that an interaction between thesolute molecular diffusion/dispersion, advection and chemical kinetics must be considered.

In terms of numerical modelling of coupled problems between fluids mixing, heat transferand chemical reactions in fluid-saturated porous rocks, it is possible to divide the coupledproblems into the following three categories [9]. In the first category of coupled problem, thetime scale of the advective flow is much smaller than that of the relevant chemical reactionin porous rock masses so that the rate of the chemical reaction can be essentially taken to bezero in the numerical analysis. For this reason, the first category of coupled problem is oftencalled the non-reactive mass transport problem. In contrast, for the second category of coupledproblem, the time scale of the advective flow is much larger than that of the relevant chemicalreaction in pore-fluid-saturated porous rocks so that the rate of the chemical reaction can beessentially taken to be infinite, at least from the mathematical point of view. This means that theequilibrium state of the chemical reaction involved is always attained in this category of coupledproblem. As a result, the second category of coupled problem is called the quasi-instantaneousequilibrium problem. The intermediate case between the first and the second category belongsto the third category of coupled problem, in which the rate of the relevant chemical reactionis a positive real number of finite value. Another significant characteristic of the third categoryof coupled problem is that the detailed chemical kinetics of chemical reactions must be takeninto account. It is the chemical kinetics of a chemical reaction that describes the reactionterm in a reactive species transport equation. If a redox chemical reaction is considered, boththe forward reaction rate and the backward one need to be included in the reaction term ofa reactive species transport equation. Although significant achievements have been made for

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2006; 66:1061–1078

NON-EQUILIBRIUM REDOX CHEMICAL REACTIONS 1063

the numerical modelling of non-reactive species and quasi-instantaneous equilibrium reactiontransport problems, research on the numerical modelling of the third category of coupledproblem with redox chemical reactions is rather limited. Considering this fact, we will developa numerical procedure to solve coupled problems between fluids mixing, heat transfer andredox chemical reactions in fluid-saturated porous rocks.

Large geological faults and cracks are favourable locations for fluids carrying different chem-ical species to focus and mix. For this reason, ore-body formation and mineralization are oftenassociated with geological faults and cracks. Since the permeability of a fault/crack is muchbigger than that of the surrounding rock, the pore-fluid flow in the fault/crack is much fasterthan that in the surrounding rock. This implies that an interaction between the solute diffusion,advection and chemical kinetics is very strong within and around a fault/crack. Although it iswell known that ore-body formation and mineralization is associated with geological faults andcracks, the major factors controlling the reaction patterns within and around large faults andcracks remain unclear. Thus, it becomes one of the main research purposes of this paper.

Keeping the above-mentioned considerations in mind, a numerical approach based on thefinite element method is used to solve coupled problems between fluids mixing, heat transferand redox chemical reactions in fluid-saturated porous rocks. In order to improve the efficiencyof numerical modelling, the concept of the chemical reaction rate invariant is used to convertthe conventional reactive transport equations with strong chemical reaction terms into the fol-lowing two different kinds of equations: one is the same as the first category of mass transportequation without any reaction term; while another remains the same as the third category ofreactive transport equation with a strong reaction term. Since the solution of a reactive transportequation with a chemical reaction term is computationally much more expensive than that of amass transport equation without a chemical reaction term, any reduction in the total number ofreactive transport equations can significantly save computer time in a numerical computation.Based on this idea, a new numerical procedure is proposed to solve coupled problems be-tween fluids mixing, heat transfer and redox chemical reactions in fluid-saturated porous rocks.This allows the interaction between the solute molecular diffusion/dispersion, advection andchemical kinetics to be investigated within and around faults and cracks in the upper crust ofthe Earth.

2. THEORETICAL AND NUMERICAL CONSIDERATIONS OF COUPLEDPROBLEMS BETWEEN FLUIDS MIXING, HEAT TRANSFER

AND REDOX CHEMICAL REACTIONS

For pore-fluid-saturated porous rocks, Darcy’s law can be used to describe pore-fluid flow andthe Boussinesq approximation is employed to describe a change in pore-fluid density due to achange in the pore-fluid temperature. Fourier’s law and Fick’s law can be used to describe theheat transfer and mass transport phenomena, respectively. If the pore-fluid is assumed to beincompressible, the governing equations of the coupled problem between fluids mixing, heattransfer and redox chemical reactions in fluid-saturated porous rocks can be expressed as

(�fuj ),j = 0 (1)

ui = 1

�k0(−p,i + �fgi) (2)


1064 C. ZHAO ET AL.

[��fcpf + (1 − �)�scps]�T

�t+ (�fcpf)ujT,j = (�e

0T,j ),j (3)

��Ck

�t+ ujCk,j = (De

0Ck,j ),j + �Rk (k = 1, 2, . . . , N) (4)

�f = �0[1 − �T (T − T0)] (5)

�e0 = ��0 + (1 − �)�s

0, De0 = �D0 (6)

where ui is the Darcy velocity component in the xi direction, p, T and Ck are pressure,temperature and the concentration of chemical species k, �0, T0 and Ck0 are the referencedensity of pore-fluid, reference temperature of the medium and reference concentration of thekth chemical species, � is the dynamic viscosity of the pore-fluid, cpf and cps are the specificheat of the pore-fluid and solid matrix, respectively, �0 and �s

0 are the thermal conductivitycoefficients for the pore-fluid and solid matrix in the porous medium, � and �T are theporosity of the medium and the thermal volume expansion coefficient of the pore-fluid, D0 isthe dispersion/diffusivity of the chemical species, �f and �s are the densities of the pore-fluidand solid matrix in the porous medium, N is the total number of all the chemical species tobe considered in the system, Rk is the source/sink term of chemical species k due to chemicalreactions within the systems, and k0 is the permeability of the medium and gi is the gravityacceleration component in the xi direction.

It is noted that if the aqueous mineral concentrations associated with ore-body formationand mineralization are relatively small, their contributions to the density of the pore-fluid arenegligible so that the mass transport process can be decoupled from the pore-fluid flow andheat transfer processes. This means that the entire coupled problem between fluids mixing,heat transfer and redox chemical reactions in fluid-saturated porous rocks can be divided intotwo new problems. The first is a coupled problem between the pore-fluid flow and the heattransfer process, while the second is a coupled problem between the mass transport processand the redox chemical reaction process. Since the first coupled problem, which is describedby Equations (1)–(3), can be solved using the existing finite element method [20, 21]. Themain purpose of this study is to develop a numerical algorithm to effectively and efficientlysolve the second coupled problem, which is described by Equation (4) and the related chemicalreaction equations.

If the reaction term in Equation (4) can be determined and is linearly dependent on thechemical species concentration, then the coupled problem defined between fluids mixing, heattransfer and redox chemical reactions in fluid-saturated porous rocks above is solvable usingthe numerical methods available [9, 19]. This requires the chemical reaction to be of the firstorder. Since many chemical reactions of different orders are associated with ore-body formationand mineralization in fluid-saturated porous rocks, both the second-order and the high-orderchemical reactions are very common in nature. Without loss of generality, the second-orderredox chemical reaction is considered to develop new concepts and numerical procedures inthis study. In principle, the developed new concepts and numerical procedures can be extendedto deal with the high-order redox chemical reactions. For this reason, a redox chemical reactionof the second order is considered as follows:

A + Bkf⇐⇒kb

AB (7)



where A and B are two chemical reactants, AB is the chemical product due to this redox chem-ical reaction, and kf and kb are the forward and backward reaction rates of this redox chemicalreaction. It needs to be pointed out that Equation (7) represents a class of redox chemical

reactions such as H+ + OH− kf⇐⇒kb

H2O, Na+ + Cl− kf⇐⇒kb

NaCl, Ca2+ + CO2−3

kf⇐⇒kb

CaCO3 and

so forth in geochemical systems. It is clear that in the first chemical reaction example, chemicalreactants A and B are H+ and OH−, while chemical product AB is H2O. In the second chem-ical reaction example, chemical reactants A and B are Na+ and Cl−, while chemical productAB is NaCl. Similarly, in the third chemical reaction example, chemical reactants A and B areCa2+ and CO2−

3 , while chemical product AB is CaCO3.From the chemical reaction point of view, the general chemical reaction source/sink terms

due to the redox chemical reaction expressed by Equation (7) can be written as follows:

RA = −kfrnf−1CACB + kbr

nb−1CAB (8)

RB = −kfrnf−1CACB + kbr

nb−1CAB (9)

RAB = kfrnf−1CACB − kbr

nb−1CAB (10)

where CA, CB and CAB are the concentrations of chemical species A, B and AB, RA, RBand RAB are the chemical reaction source/sink terms associated with chemical species A, Band AB, nf and nb are the orders of the forward and backward reactions, respectively, andr is a quantity of unity value to balance the unit of the reaction source/sink terms due todifferent orders of chemical reactions so that it has a reciprocal unit of the chemical speciesconcentration. For the redox reaction expressed by Equation (7), the forward reaction is of thesecond order, while the backward reaction is of the first order. Since a redox system allowschemical reactions to be proceeded towards both the product and the reactant directions, theorders of the forward reaction (i.e. the chemical reaction proceeds towards the product direction)and backward reaction (i.e. the chemical reaction proceeds towards the reactant direction) canbe determined from the related chemical kinetics.

It is noted that the accumulation or diffusion of chemical species in the rock matrix maylead to some change in porosity, which in turn affects permeability and fluid flow in the rockmatrix [14, 22]. This influence can be straightforwardly considered using variable permeabilitywithin the computational model. The permeability change induced by a chemical reaction canbe determined from the porosity variation induced by the chemical reaction. For example, theCarman–Kozeny law can be used to establish a relationship between the chemically inducedporosity change and the chemically induced permeability change in the rock matrix.

Substituting Equations (8)–(10) into Equation (4) yields the following equations:

��CA

�t+ ujCA, j = (De

0CA, j ),j + �RA (11)

��CB

�t+ ujCB, j = (De

0CB, j ),j + �RB (12)

��CAB

�t+ ujCAB, j = (De

0CAB, j ),j + �RAB (13)


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Since the total number of the linear-independent reaction rates is identical to the total numberof the chemical reactions involved, there is only one linear-independent reaction rate for thisredox chemical reaction. From this point of view, the total number of the reactive transportequations with source/sink terms due to chemical reactions can be reduced into one, for thisparticular redox chemical system.

Through some algebraic manipulations, Equations (11)–(13) can be rewritten as follows:

��(CA + CAB)

�t+ uj (CA + CAB),j − [De

0(CA + CAB),j ],j = 0 (14)

��(CB + CAB)

�t+ uj (CB + CAB),j − [De

0(CB + CAB),j ],j = 0 (15)

��CAB

�t+ ujCAB, j − (De

0CAB, j ),j = �kb

(kf

kbrnf−1CACB − rnb−1CAB

)(16)

It is obvious that Equations (14) and (15) are two conventional mass transport equations withoutany source/sink terms due to the redox chemical reaction so that they can be solved by the well-developed numerical methods available. Since the two new variables, namely CI = CA + CABand CII = CA+CAB, are independent of chemical reaction rates, they can be called the chemicalreaction rate invariants, which are the analogues of the stress and strain invariants in the fieldof solid mechanics.

If the redox chemical reaction is an equilibrium one, then both the forward and the backwardreaction rates are theoretically infinite so that the chemical reaction becomes predominant inthe reactive transport process. In this case, Equation (16) can be written as

Kernf−1CACB − rnb−1CAB = 0 (17)

where Ke = kf/kb is the chemical equilibrium constant.Inserting the two chemical reaction rate invariants, CI = CA +CAB and CII = CB +CAB, into

Equation (17) yields the following equation:

Ker(CI − CAB)(CII − CAB) − CAB = 0 (18)

It is noted that nf = 2 and nb = 1 are substituted into Equation (17) so as to obtain Equation(18). Clearly, Equation (18) has the following mathematical solution for the chemical productof the redox chemical reaction:

CAB = Ker(CI + CII) + 1

2Ker−

√[Ker(CI + CII) + 1]2 − 4K2

e r2CICII

4K2e r2 (19)

This indicates that for an equilibrium chemical reaction, we only need to solve the masstransport equations of chemical reaction rate invariants (i.e. CI = CA +CAB and CII = CB +CABin this particular example) using the conventional numerical methods. Once the distributionsof the chemical reaction rate invariants are obtained in a computational domain, the chemicalproduct distribution due to the chemical reaction can be calculated analytically. As a result,the distributions of the chemical reactants can be calculated using the distributions of both thechemical product and the chemical reaction rate invariants.



However, for non-equilibrium chemical reactions, the chemical reaction rates are finite sothat we need to solve at least one reactive transport equation with the source/sink term foreach of the chemical reactions in the geochemical system. This means that for the generalform of the redox chemical reaction considered in this study, we need to solve Equation (16)numerically if the reaction rates of this redox chemical reaction are finite. For this reason,Equation (16) can be rewritten in the following form:

��CAB

�t+ ujCAB, j − (De

0CAB, j ),j − �kb{KerC2AB − [Ker(CI + CII) + 1]CAB + KerCICII} = 0

(20)

Since Equation (20) is a strong non-linear equation with the non-linear term of �kbKerC2AB,

the Newton–Raphson algorithm is suitable for solving this equation.At this stage, it is possible to theoretically investigate the relationship between the con-

trolling processes associated with Equation (20) so that the overall picture of its solutioncan be theoretically understood, especially for several extreme situations. If the flow advec-tion and chemical kinetics are two controlling processes, a dimensionless parameter known asthe Damköhler number [23, 24] can be used to express the relative time scale of these twocontrolling processes:

Da = �kRl

V(21)

where Da is the Damköhler number, V is the controlling Darcy velocity of the system, l is thecharacteristic length of the controlling process in the system, and kR is the controlling reactionrate. For the redox chemical reaction considered in this study, kR = kb in the case of Ke�1,while kR = kbKe in the case of Ke>1. Since the Damköhler number is an indicator to expressthe ratio of the fluid advection time scale to the chemical kinetic time scale, it becomes unitywhen the two time scales are comparable. In this case, the chemical equilibrium length of thesystem can be expressed as follows:

lma = V

�kR(22)

where lma is the chemical equilibrium length associated with the fluid advection and chemicalreaction processes.

If the flow paths of two fluids carrying different reactive chemical species are parallel to eachother in a fluid-mixing system, then the solute diffusion/dispersion plays an important role inpromoting chemical reactions between different reactive chemical species. In this case, anotherdimensionless parameter needs to be defined for expressing the relative time scale between thesolute diffusion process and the chemical reaction process.

Z = kRl2

D(23)

where Z is a dimensionless parameter, D is the solute diffusion/dispersion coefficient, l isthe characteristic length of the controlling process in the system, and kR is the controllingreaction rate of the same definition as in Equation (21). Since this dimensionless number isan indicator to express the ratio of the solute diffusion time scale to the chemical kinetic time


1068 C. ZHAO ET AL.

scale, it becomes unity when the two time scales are comparable. In this situation, the chemicalequilibrium length of the system can be expressed as follows:

lmd =√

D

kR(24)

where lmd is the chemical equilibrium length associated with the solute diffusion and chemicalreaction processes.

The proposed numerical procedure for simulating the coupled problem between fluids mix-ing, heat transfer and redox chemical reactions in fluid-saturated porous rocks includes thefollowing five main steps: (1) For a given time step, the coupled problem described by Equa-tions (1)–(3) with the related boundary and initial conditions are solved using the conven-tional finite element method; (2) After the pore-fluid velocities are obtained from the firststep, mass transport equations of the chemical reaction rate invariants (i.e. Equations (14) and(15)) with the related boundary and initial conditions are then solved using the existing finiteelement method; (3) The chemical reaction source/sink terms involved in Equation (20) isdetermined from the related redox chemical reaction so that Equation (20) can be solved usingthe Newton–Raphson algorithm; (4) According to the definitions of the chemical reaction rateinvariants, CI = CA + CAB and CII = CB + CAB, the chemical reactant concentrations (i.e. CAand CB) can be determined from simple algebraic operations; (5) Steps (1)–(4) are repeateduntil the desired time step is reached. These solution steps have been programmed into ourresearch code.

3. NUMERICAL SIMULATION RESULTS OF COUPLED PROBLEMSBETWEEN FLUIDS MIXING, HEAT TRANSFER AND

REDOX CHEMICAL REACTIONS

The main and ultimate purpose of a numerical simulation is to provide numerical solutionsfor practical problems in a real world. These practical problems are impossible and imprac-ticable to be solved analytically and experimentally. Since numerical methods are the basicfoundation of a numerical simulation, only an approximate solution can be obtained from acomputational model, which is the discretized description of a continuum mathematical model.Due to inevitable round-off errors in computation and discretized errors in temporal and spatialvariables, it is necessary to validate, at least from the qualitative point of view, the proposednumerical procedure so that meaningful numerical results can be obtained from a discretizedcomputational model. For this reason, a testing coupled problem, for some aspects of whichthe analytical solutions are available, is considered in this study.

Figure 1 shows the geometry of the coupled problem between pore-fluids mixing, heat trans-fer and redox chemical reactions around a vertical geological fault within the crust of theEarth. For this problem, the pore-fluid pressure is assumed to be lithostatic, implying thatthere is an upward throughflow at the bottom of the computational model. The height andwidth of the computational model are 10 and 20 km, respectively. The length of the fault is5 km, with an aspect ratio of 20. The porosities of the fault and its surrounding rock are0.35 and 0.1. The permeability of the surrounding rock is assumed to have the permeability of10−16 m2, while the permeability of the fault is calculated using the Carman–Kozeny formula,



Figure 1. Geometry of a buried vertical fault in the crust.

which gives rise to a permeability of about 43 times that of the surrounding rock. The toptemperature is 25◦C and the initial geothermal gradient is 10◦C/km, meaning that the initialtemperature at all bottom is 125◦C. At the beginning of running the computational model,the temperature is increased to 325◦C and then is kept at a constant temperature boundary of325◦C. The two chemical reactants of a concentration of 1kmol/m3 are injected at the left-halfand right-half bottom boundaries, respectively, while the concentrations of both the reactantsand the product are assumed to be zero at the top boundary of the computational model.The dispersion/diffusivity of the chemical species is 3 × 10−10 m2/s. For the pore-fluid, dy-namic viscosity is 10−3 N s/m2; reference density is 1000 kg/m3; volumetric thermal expansioncoefficient is 2.1 × 10−4(1/◦C); specific heat is 4184 J/(kg ◦C); thermal conductivity coefficientis 0.59W/(m ◦C). For the porous matrix, thermal conductivity coefficient is 2.9 W/(m ◦C);specific heat is 878 J/(kg ◦C); reference rock density is 2600 kg/m3. In order to appropriatelysimulate the fluids focusing and chemical reactions within the fault, the fine mesh of smallelement sizes is used to simulate the fault zone, while the mesh gradation scheme is used tosimulate the surrounding rock by gradually changing the mesh size from the outline of thefault within the computational model. As a result, the whole computational domain is simulatedby 306 417 three-node triangle elements.

Figure 2 shows the streamline distribution of the system with the vertical fault in the com-putational model. Since the pore-fluids carrying two different chemical reactants are uniformlyand vertically injected into the computational model at the left and right parts of the bottom,the pore-fluid flow converges into the vertical fault at the inlet (i.e. the lower end) of the fault,but diverges out of the vertical fault at the outlet (i.e. the upper end) of the fault. This phe-nomenon can be clearly observed from Figure 3, where the velocity distributions are displayedat both the inlet and outlet of the vertical fault. For an elongated elliptic fault of large aspectratios in an infinite medium, the existing analytical solution indicates that the streamlines of thepore-fluid flow are parallel to each other within the inside of the fault [25]. As expected, thecomputed streamline distributes vertically within the vertical fault, indicating that the numericalresult obtained from the computational model has good agreement with the existing analytical


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Figure 2. Streamline distributions due to the vertical fault in the crust.

result. This demonstrates that the proposed numerical procedure can produce useful numericalsolutions for the fluids focusing and mixing, at least from the qualitative point of view. In orderto quantitatively validate the numerical solutions, the analytical solution for the flow-focusingfactor of an elongated elliptic fault of large aspect ratios can be employed. Since the elongated



Figure 3. Velocity distributions due to the vertical fault in the crust.


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fault within the computational model is basically of a rectangular shape, the analytical solutionfor its flow-focusing factor can be evaluated using the following modified formula:

� =(

4

�

)�(� + 1)

� + �(25)

where � is the pore-fluid flow-focusing factor of the rectangular fault of a large aspect ratio,� is the aspect ratio of the rectangular fault, and � is the permeability ratio of the fault to itssurrounding rock.

Since the vertical velocity of the injected fluids due to the lithostatic pore-fluid pressure isequal to 1.6 × 10−9 m/s and the numerical solution for the maximum vertical velocity withinthe fault is equal to 3.02 × 10−8 m/s, the corresponding flow-focusing factor of the fault isequal to the ratio of the maximum velocity within the fault to that of the injected fluids atthe bottom of the computational model. This results in a flow-focusing factor of 18.88 forthe rectangular fault within the computational model. The analytical value of the flow-focusingfactor for the rectangular fault can be calculated from Equation (25). Substituting the relatedparameters into Equation (25) yields the analytical flow-focusing factor of 18.25. Since therelative error of the flow-focusing factor from the numerical simulation is within 3.5%, itquantitatively demonstrates that the proposed numerical procedure used in the computationalmodel can produce accurate numerical solutions for the fluids focusing and mixing within thefault, which are very important for accurately simulating the chemical species transport andreaction within the fault of the computational model.

In order to validate the proposed numerical procedure for solving reactive transport equa-tions with strong non-linear reaction source/sink terms, the redox chemical reaction due toan equilibrium reaction is considered and solved using the proposed numerical procedure. Theequilibrium constant is assumed to be 10 and the time step used in the simulation is 100years. Figure 4 shows the concentration distributions of the two chemical reactants at threedifferent time instants, while Figure 5 shows the comparison of the numerical solutions (whichare obtained from the proposed numerical procedure) with the analytical solutions (which arederived mathematically and expressed by Equation (19) in Section 2) for the chemical product.It can be observed that with the increase of time, both chemical reactants are transported intothe computational domain from the left half and right half of the bottom. Due to the fluidflow focusing, both chemical reactants are transported much faster in the fault zone than in thesurrounding rock. As expected, these chemical reactants are divergent around the exit regionof the fault. The comparison of the numerical solutions with the analytical ones for the chem-ical product clearly demonstrates that the proposed numerical procedure can produce accuratenumerical solutions for simulating the equilibrium chemical reaction, in which the chemicalreaction rate approaches infinity. It is interesting to note that there is a strong interactionbetween the solute advection, diffusion and chemical reaction speed in the considered equi-librium chemical system. Although two reactants are well transported into the fault zone, themixing of the two fluids carrying them is controlled by the solute diffusion. In this particularcase, the chemical equilibrium length can be calculated from Equation (24). Since the chemicalreaction rate is infinite for the equilibrium reaction, the corresponding chemical equilibriumlength is identical to zero due to the solute diffusion. This implies that the chemical reactionspeed is too fast to allow both the reactants to diffuse across the common boundary betweenthem so that the fluids mixing cannot effectively take place within the fault zone. This is thereason why both chemical reactants are abundant but no chemical product is produced within



Figure 4. Concentration distributions of the chemical reactants at differenttime instants (equilibrium reaction).

the fault zone in the computational model. However, around the exit region of the fault zone,the flow of the fluids is slowed and divergent so that the fluids carrying two different reactantscan be mixed. Consequently, the chemical product of high concentration is produced aroundthe exit region of the fault zone.


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Figure 5. Comparison of numerical solutions with analytical ones for thechemical product (equilibrium reaction).

After the proposed numerical procedure is validated through the equilibrium reaction, it isused to investigate the effect of the reaction rate of a non-equilibrium redox chemical reaction onthe distribution pattern within and around the fault zone. Three different values of the chemicalreaction rates, namely kb = 10−8(1/s), kb = 10−10(1/s) and kb = 10−12(1/s), are considered in



Figure 6. Effects of chemical reaction rates on concentration distributions of thechemical product at different time instants.

the related numerical simulations. Figure 6 shows the effects of chemical reaction rates onconcentration distributions of the chemical product at different time instants. For these threechemical reaction rates, the theoretical chemical equilibrium lengths calculated from Equation(24) are 0.0548, 0.548 and 5.48m due to the control of the solute diffusion, while the theoretical


1076 C. ZHAO ET AL.

chemical equilibrium lengths calculated from Equation (22) are 0.863, 86.3 and 8630 m due tothe control of the solute advection within the fault zone. This indicates that in the cases ofkb = 10−8(1/s) and kb = 10−10(1/s), the chemical equilibrium length due to the solute advectionis smaller than that of fault zone so that the chemical product distribution pattern within thefault zone is controlled by the solute diffusion. However, in the case of kb = 10−12(1/s), thechemical equilibrium length due to the solute advection is greater than that of fault zone sothat the chemical product distribution pattern within the fault zone is controlled by the soluteadvection. Since the chemical equilibrium lengths due to the solute diffusion are relativelysmall in the cases of kb = 10−8(1/s) and kb = 10−10(1/s), the chemical equilibrium cannotbe reached within the fault zone, indicating that two fluids cannot be effectively mixed toproduce a considerable mixing region due to the slow diffusion of the solute within the faultzone. Because of the mixing and divergence of the fluids carrying two chemical reactants, thechemical equilibrium, which produces the maximum value of the chemical product, has beenattained around the exit region of the fault zone, as can be clearly seen in Figure 6. On theother hand, since the chemical equilibrium length due to the solute advection is greater than thetotal length of the fault plus its exit region (i.e. 5000+2500 m) in the case of kb = 10−12(1/s),the chemical equilibrium cannot be reached within the fault zone, indicating that two fluidscannot be well mixed to produce a considerable mixing region due to the fast advection ofthe fluids within the fault zone. Thus, it can be concluded that despite different values of thechemical reaction rate, chemical equilibrium cannot be attained within the fault zone for theredox chemical reaction considered in this study. This recognition has the following significantgeological implications. If the pore-fluid pressure gradient is lithostatic in the crust of the Earth,the chemical equilibrium cannot be reached in the large cracks and faults, implying that if aquartz vein exists in the crust, then the pore-fluid pressure gradient could not be lithostatic,unless other processes instead of the chemical equilibrium process are the driving forces tocause the formation of the quartz vein. From the ore-body formation and mineralization pointof view, the exit region of large faults and cracks are favourable location for the formation ofhigh-quality ore deposits because it is the most probable region for the maximum concentrationof the chemical product to be generated.

4. CONCLUSIONS

A new numerical procedure is proposed, in this paper, to solve non-equilibrium redox chemicalreactions of high orders, which are ubiquitous in fluid-saturated porous rocks within the crustof the Earth. The modelling of such high-order chemical reactions is a challenging problembecause they not only produced strong non-linear source/sink terms for reactive transport equa-tions, but also often coupled with the fluids mixing, heat transfer and reactive mass transportprocesses. In order to solve this problem effectively and efficiently, the total number of reactivetransport equations with strong non-linear source/sink terms needs to be reduced to a minimumin a computational model. For this purpose, the concept of the chemical reaction rate invari-ant is used to develop the numerical procedure for dealing with fluids mixing, heat transferand non-equilibrium redox chemical reactions in fluid-saturated porous rocks. Using the pro-posed concept and numerical procedure, only one reactive transport equation, which is used todescribe the distribution of the chemical product and has a strong non-linear source/sink term,needs to be solved for each of the non-equilibrium redox chemical reactions. The original



reactive transport equations of the chemical reactants with strong non-linear source/sink termsare turned into the conventional mass transport equations of the chemical reaction rate invariantswithout any non-linear source/sink terms. This certainly reduces the total number of reactivetransport equations with strong non-linear source/sink terms to a minimum in a computationalmodel.

Through the application of the proposed numerical procedure to the numerical solution ofa testing example, for some aspects of which the analytical solutions are available, it hasbeen demonstrated that: (1) the proposed numerical procedure is useful and applicable fordealing with the coupled problem between fluids mixing, heat transfer and non-equilibriumredox chemical reactions of high orders in fluid-saturated porous rocks; (2) the interactionbetween the solute diffusion, solute advection and chemical kinetics is an important mechanismto control distribution patterns of chemical products in an ore-forming process; (3) if the pore-fluid pressure gradient is lithostatic, it is difficult for the chemical equilibrium to be attainedwithin permeable cracks and geological faults within the crust of the Earth; and (4) the exitregion of large faults and cracks are favourable location for the formation of high-quality oredeposits because it is the most probable region for the maximum concentration of the chemicalproduct to be generated.

ACKNOWLEDGEMENTS

The authors express their thanks to the anonymous referees for their valuable comments, which ledto a significant improvement over an early version of the paper. The authors also thank Professor R.W. Lewis at University of Wales Swansea for his encouragement in revising the paper.

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