4.0 Scaling Analysis Of Coupling Between Chemical Kinetics And Transport Phenomena

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

4.0 SCALING ANALYSIS OF COUPLING BETWEEN CHEMICAL KINETICS AND TRANSPORT PHENOMENA

Mass and energy transport can be modelled by nonlinear partial differential equations which describe the spatially nonhomogeneous distribution of oxygen, water, reaction products and temperature as a function of physical and mineralogical properties such as pile porosity, oxygen transport rate, thermal conductivity, etc. [ScD], [Sch], [Ar], [Ot1], [Ot2].

In this Chapter, multiple steady state solutions are calculated exactly and the bifurcation diagram is analyzed. The scaling analysis is performed in order to establish the relative importance of individual factors responsible for acid rock drainage. The relative importance of various physical and chemical factors responsible for acid rock drainage is described by means of a dimensionless scaling parameter δ which contains quantitative information about the different processes involved. The scaling model covers a broad range of chemical and physical parameters.

Long term predictions can be usually based on quantitative results obtained for steady state solutions of reaction-transport models. One has to remember, however, that rock texture undergoes changes during weathering processes, and parameters such as the reactive surface area change over time. To our knowledge, the only detailed study of temporal changes in the reactive surface area has been performed for tailings [Ni], [NiG], A similar long term study for waste rock would provide useful information for the present scaling analysis. Because of the lack of quantitative experimental data we do not attempt to model temporal changes in the reactive surface area, S.

4.1 Formulation of the Reaction-Transport Model

From a physico-chemical point of view a pile of waste rock reacting with oxygen and water is an open non-equilibrium system evolving towards an equilibrium state in which pyrite is transformed into ferrous/ferric iron and acid.

The overall process of pyrite oxidation in waste rock piles can be described by a set of two coupled nonlinear parabolic equations in two fields, oxygen concentration, Y and temperature, T:

Equations (4.1) and (4.2) are the energy and mass conservation equations, respectively. (T - temperature, Y - oxygen concentration in the gas phase, ε - porosity of the pile, ρ_a - air density, C_a - specific heat of air, ρ_r - rock density, C_r - specific heat of rock, λ - thermal conductivity of the pile (air+rock), h - heat of reaction of oxygen with pyrite, D - diffusion coefficient of oxygen in the gas phase, q - effective thermal exponent, p - effective order of chemical reaction of oxygen with pyrite.) 

The values of a, p and q are different for different pH values. For pH less than 4 p=0.6, q=2.1. For pH greater than 4, p=0.6, q=2.8. In order to simplify the scaling analysis we will use the value of p=1. We plan to properly calibrate our results when the results of thermokinetic tests at different values of oxygen partial pressure become available for rock samples.

We consider a pile characterized by the linear dimension L, with the oxygen concentration Y_b at the boundaries, and temperature T_b at the boundaries. For symmetric boundary conditions, for a steady state we have:

(For non symmetric boundary conditions

where C is a constant depending on the boundary conditions).

From equations (4.1) and (4.2) one obtains for steady state solutions:

where

The integration constant C is not a free parameter, but (for given values of parameters) is determined by T_b and is implicitly related to L. For different values of C we obtain different values of the maximum temperature, T_m in the pile. The linear size, L is given by the formula

Formula (4.6) defines implicitly the distribution of temperature T(x).

The maximum temperature in the pile cannot exceed the value:

At the point where the temperature T_m* is reached, the concentration of oxygen is equal to zero. Note that T_m* does not depend on the rate constant aS. Thus the state with zero oxygen concentration at the point of maximum temperature is controlled by oxygen transport.

In the next step we derive a dimensionless parameter δ which will define the scaling law involving all the parameters in equations (4.4) and (4.5).

We introduce the variables:

In terms of the new variables, the linear size can be expressed as an integral which depends only on the parameter r: 

The universal function I[U_m; U_b] is presented in Figs. 4.2 and 4.3 for q=2.1 corresponding to low pH values and two different values of U_b (see p. 27). 

4.2 Scaling Properties and Sensitivity Analysis of Acid Rock Drainage 

It is useful to define the dimensionless parameter:

which after substituting expressions (4.5) defining the parameters A and B takes the form:

The scaling parameter δ is a combination of parameters describing physico-chemical properties of pyritic rock and storage conditions. The scaling parameter δ can be used as a quantitative indicator of acid rock drainage. Values of δ define different regimes of acid rock drainage. The greater the value of δ the greater the rate of acid generation. δ shows a strong nonlinear dependence on parameters characterizing acid rock drainage. Some of these parameters, like pile porosity and pile size, can be controlled. Maintaining the smallest possible value of δ is important in order to minimize the environmental impact of acid rock drainage. The expected environmental impact increases with the value of δ. (On the other hand, one may consider a controlled preemptive leaching over a short time in order to reduce the long term environmental impact and to increase production [Go]. In this scenario the large values of δ may be required in order to intensify leaching.)

The scaling indicator δ provides important quantitative information about relative importance of individual factors contributing to ARD. Because δ is proportional to S^1/2 the pile design parameters become increasingly important for large values of pyrite concentration. In particular, the effectiveness of impermeable covers increases with the pyrite concentration. In Chapter 5 we demonstrate how to use the scaling indicator δ for the predictive analysis of ARD.

The scaling formula (4.12) indicates the required precision of individual laboratory tests in which parameter values are determined. This aspect is analyzed in Section 5.4.

Laboratory tests usually measure acid generation over time and provide information about the chemical activity of pyritic rock (factor aS). Acid/base tests should also determine a detailed temperature dependence of acid generation rates at temperatures between 0°C and 60°C. Experimental validation of the formula describing the dependence of reaction rates on oxygen concentration in the gas phase for different pH values also seems necessary. Such experimental tests would determine the values of parameters a, S and q which are crucial for reliable estimates based on the scaling parameter δ.

For given values of other parameters we obtain the condition:

which defines the optimum size of the pile as the function pile porosity, ε. The form of condition (4.13) indicates a very strong nonlinear dependence of acid rock drainage on ε. The porosity factor:

which enters the relation (4.14), affects very significantly the total rates of acid generation.

Fig. 4.1. Dependence of the porosity factor defined by eq. (4.14 on porosity values for high and low pH values (q=2.8 and 2.1 respectively).

Fig. 4.1 shows the plots of P(ε) for values of q characteristic for low and high pH values. Additional strongly nonlinear effects appear as the result of discontinuities and cross-over effects discussed in the next section.

4.3 Bifurcations and Thermodynamic Catastrophes in Acid Generation Rates

Numerical values of the integral I[U_m; U_b] show that the value of δ should be less than a certain critical value δ* for which a transition to accelerated rates of acid production is expected. The values of δ* depend on the scaling exponent, q, and are different for low and high pH values. The values of U_b and U_m in the integral define the temperature at the pile surface and the maximum temperature in the pile respectively (see eq. (4.8)). Figs. 4.2 and 4.3 present plots of the maximum value of U as the function of the scaling parameter δ for q=2.1 and two different values of U_b: U_b=0.0792 and U_b=0.13636 used later in Chapter 5 when various alternative scenarios are discussed. For δ=δ*=2.75 in Fig. 4.2 we observe a discontinuity in U_m. Because the maximum temperature in the pile is proportional to U_m, we call this discontinuity a thermodynamic catastrophe. The thermodynamic catastrophe is responsible for accelerated rates of acid generation.

For given values of a and S the values of pile porosity and pile size should be controlled in such a way that the value of δ* is not exceeded. The values of generation rates at δ>δ* can exceed by several times the acid generation rates at δ<δ*. The critical values, δ*(pH) can be determined from numerical results obtained for the integral I(U_m; U_b] which is a multivalued function of U_m and U_b.

The curve U_m(δ*) in Fig. 4.3 is continuous but also shows a dramatic increase in the vicinity of δ*=2.35. The critical value of the scaling parameter is defined in this case as the value of δ at which the maximum slope of the curve U_m(δ) occurs. The values of δ* depend strongly on the temperature at the pile boundary. The greater the ambient temperature, the smaller the value of δ that should be maintained in order to avoid the thermodynamic catastrophe.

Chapter 5 demonstrates the practical use of the scaling parameter δ as the practical indicator of acid rock drainage. The qualitative features of the integral I[U_m; U_b] for other values of the thermokinetic exponent q are the same as for the particular value of q=2.1 used in our examples. Thus the occurrence of thermodynamic catastrophes is a general feature of waste rock piles.

Fig. 4.2 Dependence of the maximum value of U (defined by eqs. (4.8)-(4.10)) on the values of the parameter δ; U_b=0.0792. At δ=δ*=2.75 a discontinuity in the maximum temperature and acid generation rates occurs. (Note: The top branch extends beyond the point P₂ - this feature is not visualised here and is responsible for the thermodynamic hysteresis not discussed in this report).

Fig. 4.3 Dependence of the maximum value of U (defined by eqs. (4.8)-(4.10)) on the values of the parameter δ; U_b=0.13636. In the vicinity of δ=δ*=2.35 a dramatic increase in the maximum temperature and acid generation rates occurs.

3.0 Effective Kinetic Equations For The Oxidation Of Pyrite

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

3.0 EFFECTIVE KINETIC EQUATIONS FOR THE OXIDATION OF PYRITE

This Chapter presents the results of our mathematical analysis of numerical results obtained for a set of kinetic equations discussed in Chapter 2. The purpose of our analysis is to derive closed form formulae for overall oxygen consumption rates and thermal energy generation rates. The effective formulae have a simple form suitable for the scaling analysis of coupled physical and chemical processes.

The scaling parameters discussed in this Chapter allow the estimation of the temperature dependent rates for processes contributing to acid rock drainage, without solving the differential equations.

In Sections 3.1 and 3.2 we analyze two aspects of pyrite oxidation:

i) Scaling relations determined by the quantitative analysis of the chemical reactions described in the previous section; and

ii) Scaling relations for the concentration of oxygen dissolved in water as a function of temperature and concentration of oxygen in the gas phase.

In Section 3.3 effective equations for oxygen consumption rates and energy generation rates are derived for different pH values.

3.1 Scaling Relations for Chemical Reactions

The scaling parameters should be useful for analyzing certain site-specific situations where values for material parameters and initial conditions are different than the particular values used in this report. The scaling analysis for chemical rates will be used subsequently in our bifurcation analysis of the reaction-transport scaling model which will include processes of mass and energy transport.

There is one obvious scaling parameter, S/V, which relates time scales of the surface reactions (R1) and (R3) to the active surface area and water volume. We can also define other scaling factors which relate quantities characterizing acidic drainage for different conditions labelled by i and j.

For the reaction (R1) we define the scaling factors: 

and 

The thermodynamic scaling factor f₁(T₀,ΔT) tells us that when we increase temperature by ΔT, the rate constant will change f₁(T₀,ΔT) times. The thermodynamic scaling factors satisfy a recursion relation:

It turns out that for activation energy values on the order of 10⁵J and temperatures between T_min=283 K and T_max=333 K the scaling factor defined by eq.(3.2), can be approximated by the factor f₁(ΔT) which does not depend on T₀. At first we find the ratio between the greatest and the smallest values of the rate constant k₁(T):

Then the thermodynamic scaling factor f₁(ΔT) is defined as:

which satisfies a recursion relation:

The accuracy of the approximation f₁(ΔT)≈f₁(T₀,ΔT) is better than 10%. Subscripts l and h label results obtained for low (at pH<4) and high pH values respectively. For the reaction (R1) at pH<4, when E_1l=57 kJ/mol, we obtain:

which means that if we increase the temperature by 10K, then the rate constant for the oxidation by dissolved oxygen increases 2.1 times. For high pH values, when E_1h=88 kJ/mol we obtain: 

We may combine factors r₁ and f₁ and define the scaling factor:

which takes into account the effective active surface area, the amount of water, oxygen concentration and temperature. We can use the scaling factor F₁ to transform the results obtained for one set of material parameters and initial conditions into results for different values of parameters and initial conditions. The conclusions are based on our analysis of experimental data and numerical results presented in [SyT] (See Appendix A and Appendix B). For example, by comparing Figs. A3(21) and A3(31) we see that Fig. A3(21) can be obtained from Fig. A3(31) after rescaling the time axis by the factor F₁(ΔT=20K) equal to 8.2 (calculated by using the eqs. (3.1), (3.8) and (3.9) and taking into account the decreasing concentration of dissolved oxygen - see Table 3.1).

In a similar way we may derive scaling factors f₂ and f₃ for the reactions (R2) and (R3). At low pH values we can use the average activation energy (E_1l+E_2l)/2 to define:

For the reaction (R3) (for which the activation energy E₃=90 kJ/mol [MaR2], [WiR]) we obtain:

For ΔT=20 K the thermodynamic scaling factor f₃(20K)=10.7. Fig. 3.1 presents numerical solutions to eqs. (D2) for large initial values of [Fe³⁺] at different temperatures and for different values of S/V.

Fig. 3.1 Numerical solutions to eqs. (D2) for the initial values [Fe²⁺(t₀)]=Fe³⁺(t₀)]=0.01M, [H⁺(t₀)]=10^-4M. Plots (a), (b) and (c) obtained for S/V=1m²/l and temperatures 283K, 303K and 323K respectively; (d) obtained for S/V=10m² and T=303 K. Vertical axis: molar concentrations; horizontal axis: time in seconds (30 000 s = 8 hours).

As we see, Figures 3.1 (a) and 3.1 (d) are almost identical. They are related by the scaling factor F₃=f₃(20K)S(d)/S(a)=1.07.  Figs. 3.1 (a), (b) and (c) are related by the scaling factor f₃(20K)=10.7.

More complex symmetries of differential equations and more complicated scaling relations will be used in Chapter 4 to analyze reaction-transport models defined by coupled partial differential equations [Ar], [BaZ]. In our further analysis we follow principles used for other known examples of reaction-transport problems for which the scaling relations allow the determination of a bifurcation structure and reduce computation time [Ar], [Ot], [SyT], The results of this report can be used as a starting point for analyzing the scaling properties of a waste rock model or a model for underwater disposal.

3.2 Scaling Formula for Concentration of Oxygen Dissolved in Water 

Table 3.1 presents data on the concentrations of oxygen dissolved in water, [O₂] (in units of mol per litre) as a function of the temperature and oxygen concentration in the gas phase, [O₂]_gas (in volume %). The temperature values correspond to those observed in waste rock piles. [O₂] increases slightly faster than linearly with [O₂]_gas at constant temperature. At 50°C and [O₂]_gas=21%, [O₂] is less than 50% of its value at 10°C. In this way we observe a competitive effect of temperature and dissolved oxygen concentration on the reaction rates.

The values in Table 3.1 are obtained for water without any dissolved iron or sulphate ions. Unfortunately we do not have any experimental data on the concentrations of dissolved oxygen in the presence of iron and sulphate ions at various pH values. We will simply assume that the maximum concentration of oxygen dissolved in water does not depend on pH. This assumption is supported by the fact that the order of reaction (R3) with respect to oxygen does not depend on pH, as discussed byMcKibben and Barnes [McK].

The solubility of oxygen in water can be described by the relation:

The last formula gives the values of concentration of dissolved oxygen in water in micromoles of oxygen per litre when concentration of oxygen in the gas phase is measured as a percentage of the total volume of air and temperature is measured in °C. This formula gives better than 10% average agreement with the experimental data presented in Table 3.1.

Eq. (3.12) introduces the temperature dependent factor which inhibits oxidation rates at high temperatures. To our knowledge, this effect has not been analyzed in previous models for acid rock drainage.

In Chapters 4 and 5 we will use the variable Y which will denote [O₂]_gas expressed in units of mol per cubic meter and appropriately transformed formula (3.12) will be used.

Table 3.1 Saturation concentrations of dissolved oxygen in water, [O₂], at atmospheric pressure (1013.25 hPa) for different oxygen concentrations in the gaseous phase [O₂]_gas. The solubilities of oxygen are defined in terms of the mass of oxygen dissolved in one litre of the water in equilibrium with an atmosphere saturated with water vapour at various temperatures. The data are calculated following Benson and Krause [BeK] and using a computer program published by Beer [Be].

3.3 Effective Kinetic Equations

Rates of oxygen consumption and energy release can be derived as effective rates taking into consideration the summary effect of all reactions involved. Our quantitative analysis of experimental data and numerical analysis of eqs. (D1) and (D2) issummarized below.

3.3.1 pH Between 4 and 7

1) The acid production rates are directly proportional to the coefficient S/V which measures the ratio between the active surface area of pyritic rock per volume of water.

2) Oxygen consumption rates increase at the rate of 3.1 times per 10K. This corresponds to the power law increase T^3.30 with temperature T measured in °C. This formula gives better than 5% agreement with both laboratory data and our previous numerical results over the temperature range from 0°C to 60°C typical for waste rock piles.

3) The total rate of iron production (as Fe²⁺ and Fe(OH)₃) is proportional to the rate of oxygen consumption with the molar proportionality coefficient (0.28±0.01) in the temperature range between 0°C and 60°C (see Figs. A1 and A2).

4) The rate of sulphate ion production follows the stoichiometry of the reactions (R1)-(R4) and is twice as high as that of the ferrous iron production. The molar coefficient with respect to oxygen is 0.56±02 (i.e. 0.56mol of sulphate produced per 1mol of oxygen consumed).

5) The solubility of oxygen in water can be described by the relation (μmol/1 of oxygen in water)=100 μmol/1 {[O₂]_gas/(10%)}^1.2(30°C/T)^0.5 which describes the dependence of dissolved oxygen as a function of temperature and oxygen concentration in the gas phase.

Based on the results summarized above the effective kinetic equations for oxygen consumption and energy release have the form:

where h is the heat of pyrite oxidation per mol of oxygen consumed.

3.3.2 pH Less Than 4

1) The acid production rates are directly proportional to the coefficient S/V which measures the ratio between the active surface area of pyritic rock per volume of water.

2) Oxygen consumption rates increase at the rate of 2.5 times per 10K. This corresponds to the power law increase T^2.6 with temperature T.

3) The rate of ferrous iron production is proportional to the rate of oxygen consumption with the molar proportionality coefficient (0.29±0.01) in the temperature range between 0°C and 60°C.

4) The rate of sulphate ion production follows the stoichiometry of the reactions (R1)-(R4) and is twice as high as that of the ferrous iron production. The molar coefficient with respect to oxygen is 0.58±02 (i.e. 0.58mol of sulphate produced per 1mol of oxygen consumed).

5) The solubility of oxygen in water can be described by the relation (μmol/1 of oxygen in water) = 100 μmol {[O₂]_gas/(10%)}^1.2(30°C/T)^0.5 which describes the dependence of dissolved oxygen as a function of temperature and oxygen concentration in the gas phase.

The molar coefficients are significantly affected by the bacterial oxidation. (See ref. [SyT] for details).

The kinetic equations have the form:

In Chapter 4 the rate equations (3.13)-(3.16) will be used in the reaction-transport scaling model. The effective kinetic equations derived in this Chapter are based on laboratory data obtained for small samples of pyrite in water. In our further analysis we replace the coefficient kS/V by the coefficient aS in which S is the reactive surface area of pyrite in one cubic meter of waste rock and a is the rate constant in the appropriate units (see Tables 5.1 and 5.2).

2.0 Oxidation Of Pyrite

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage 

2.0 OXIDATION OF PYRITE

In this Chapter, the complex kinetics of pyrite oxidation is briefly summarized. We refer the reader to [SyT] and references therein for more details and further references.

The overall stoichiometry of pyrite oxidation may be described by the following reactions: 

Reactions (S1) and (S2) summarize several elementary chemical reactions which take place at low pH values (reaction (S1)) or high pH values (reaction (S2)).

In order to describe the oxidation kinetics, the intermediate steps in this reaction must be considered. The following reactions characterizing the oxidation of pyrite have been proposed [StM], [SiS], [TaW]:

The chemical network responsible for pyrite oxidation consists of four reactions:

(R1) The oxidation of pyrite by molecular oxygen to Fe²⁺ and sulphate. The oxidation of iron sulphide (pyrite) to sulphate (eq. (R1)) releases dissolved ferrous ions and acidity into the water.

(R2) Subsequently, the dissolved ferrous ions undergo oxidation to ferric ions (eq. (R2)). This is a slow reaction and viewed as the rate-limiting step determining the overall rate of pyrite oxidation.

(R3) Sulphide is oxidized again by ferric ion and acidity is released along with additional ferrous ions which may re-enter the reaction cycle via reaction (R2). This is regarded as a fast step.

(R4) Ferric ions hydrolyse to form insoluble ferric hydroxide (eq. (R3)), releasing more acidity to the stream. This reaction takes place only at high pH values which can be attained when the mineral composition of the waste rock is such that self-buffering processes take place or when neutralizing minerals are added.

Reactions (R2) and (R3) provide a feedback loop discussed by Singer and Stumm (p. 471 of ref. [StM]). Ferrous iron produced in (R3) is utilized again in reaction (R2). The number of reacting species during acid formation is greater than two. This leads to strong nonlinearities in the kinetic equations.

2.1 Oxidation at pH Between 4 and 7

The set of coupled kinetic differential equations has been derived in our previous study and has the form:

In our previous study we have presented numerical solutions to the set of equations (D1) for the experimentally determined values of the temperature and pH dependent rate constants k(T) and reactive surface area, S per unit volume, V.

2.2 Oxidation at pH Less Than 4 

At low pH values ferric hydroxide does not precipitate if concentrations of ferric iron are sufficiently small. This means that the ferric iron, Fe³⁺ (which is formed by oxidation of ferrous iron) oxidizes pyrite according to the reaction (R3). The set of coupled kinetic differential equations has the form:

The use of the full set of the kinetic equations in a physicochemical model is not practical, however, because of the required long computer time when transport processes are included. The main factors affecting the rates of acid production are concentration of oxygen, temperature and pH value. Because in our previous study no oscillatory or chaotic behaviour of the coupled chemical equations (D1) and (D2) has been found, the detailed kinetic equations can be replaced by much simpler effective equations suitable for the physico-chemical model. The most significant information can be obtained by analyzing the temperature dependence of the overall rates of oxygen consumption and the dependence of acid production rates on temperature. We have analyzed our previous numerical results and on that basis it is possible to derive effective kinetic equations for oxygen. The effective kinetic equations are subsequently used as a starting point for the scaling analysis. Numerical solutions to the kinetic equations (D1) and (D2) are presented in Appendix A. The previous report [SyT] discusses the agreement of the results obtained for the kinetic equations with laboratory data.

1.0 Introduction

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage 

1.0 INTRODUCTION 

The purpose of this project is to provide a simple quantitative description of the interrelated nonlinear physical and chemical processes responsible for ARD. Accurate prediction and understanding of acid rock drainage may be based on the analysis of quantitative data from experiments and physico-chemical models, and potentially offers a cost-effective means of reducing the environmental impact of ARD.

It is believed that practical decisions should be based on the quantitative results of laboratory and field tests. Laboratory tests analyze various aspects of acid rock drainage by using a limited amount of waste rock, ranging from one to a thousand kilograms (meso-scale). Such tests concentrate on characterization of the chemical properties of waste rock. Because the processes involved are nonlinear, scaling up the meso-scale results to obtain macro-scale characteristics for millions of tons of waste rock is a difficult task which requires an extensive modelling effort. Tests for ARD prediction (e.g. humidity cells, column tests, small scale treatment systems) are used frequently to determine the optimum approach to minimizing the environmental impact and treatment cost. No researcher, however, has examined the size effects that may be significant for the prediction of macro-scale behaviour. Such efforts are often hindered by the long computation time required for parametric analysis of numerical models. Finding the appropriate scaling relations between physical, chemical and biological processes is one of the goals of this study. Knowledge of the scaling laws may allow one to improve interpretation of the existing experimental data, design better micro-scale and meso-scale experiments and reduce the cost of prediction techniques by using cheaper small scale experiments.

Prediction of long term environmental impact, based on the characterization of waste materials is not a simple task, however. Apart from chemical processes, ARD depends also on physical, biological and mineralogical factors. Chemical and biological aspects of acidic drainage were analyzed in our recent study [SyT]. It has been found that the interrelated elementary chemical and microbial reactions have to be described as coupled nonlinear processes which respond in a rather complex way to preventive measures like neutralization and oxygen depletion. Temperature is a very important factor. The rates of oxygen and water transport also are important factors to consider due to their effect on both oxidation and heat exchange with the surroundings.

Extensive computer models which have to be analyzed numerically, often require a very long computational time before one is able to gather quantitative information about the complexity of the coupled physical and chemical processes. Scaling relations based on the consideration of simplified reaction transport models often give good quantitative information about the relative importance of the factors involved, and allow one to translate laboratory data into large-scale estimates.

In this report we concentrate on the interplay between the chemical kinetics and transport of mass and energy. Mass and energy transport are physical factors which ultimately control reaction rates. The concentration of oxygen and the temperature distribution are non-homogeneous inside waste rock pile [PaR], [FeM], [BeR], [Ge], [NoD]. The quantitative analysis of the expected distribution of oxygen and temperature is important for the estimation of acid generation.

Previous waste rock models did not describe properly the process of pyrite oxidation in regions where oxygen concentration is low and temperature is high. In particular, to our knowledge, the temperature dependence of oxygen dissolved in water has not been included in previous models. [BeR], [BrC], [CdO], [Da], [DaR1], [DaR2], [DaR3], [FeE1], [Ge], [Res], [WhJ]. 

The report starts with an analysis of a kinetic chemical model which lays the ground for the quantitative physico-chemical model. In Chapter 2, we analyze the following coupled processes:

• pyrite oxidation by water and oxygen dissolved in water and the release ot Fe²⁺ (ferrous) ions and acid (reaction (Rl))

• oxidation of ferrous ions to ferric (Fe³⁺) ions by oxygen (reaction (R2))

• anaerobic oxidation of pyrite by ferric ions and water (reaction (R3))

• precipitation of ferric hydroxide, which eliminates ferric ions from the stream (reaction (R4))

• the neutralization process

Effective kinetic models for chemical reactions at high and low pH values are constructed and analyzed in Chapter 3. The temporal behaviour of the concentrations of chemical species is considered separately for pH values greater than four, and less than four. This distinction is necessary because of the dramatic changes in the nature of pyrite oxidation due to the precipitation of ferric hydroxide at pH values above four. Pyrite oxidation rates increase dramatically with temperature - for temperatures between 273K and 323K the rate of pyrite oxidation accelerates about ten times per 20K. Transport processes which control the supply of water and oxygen, are ultimately responsible for both the rate of acid generation and its release to the environment. The form of the effective kinetic equations derived in Chapter 3 is suitable for the scaling analysis of the reaction-diffusion equations in Chapter 4.

In Chapter 4 the mathematical scaling analysis of coupled chemical and physical processes is presented. We derive a dimensionless physico-chemical scaling parameter, δ, which describes the combined effects of chemical reactions and mass and energy transport. The dimensionless scaling parameter δ describes the relative effects of oxygen diffusion, thermal conductivity, pile porosity, ambient conditions, etc. The relative importance of the different factors involved in acid rock drainage is described quantitatively by the dimensionless scaling parameter which may become useful as a standard tool for predictive analysis of laboratory and field data. The strongly nonlinear properties of the combined physical and chemical processes are analyzed as a function of the parameters describing pile size, pile porosity, pyrite content and other quantities. It is found that there exist critical values of these parameters at which a discontinuous increase of chemical rates may take place together with a jump-wise increase of temperature and acid generation rates inside waste rock piles. We call this phenomenon a thermodynamic catastrophe. This aspect of ARD is very important for the prediction and control of acid rock drainage and to the best of our knowledge has not been discussed in previous studies.

In Chapter 5, different scenarios of acid generation for different sets of input data are analyzed in detail. The power of scaling analysis is demonstrated by showing how the results for one set of entry data can be obtained from results derived for a different set of entry data. Maximum allowed values of pile size and effective reactive surface area are calculated for two different realistic scenarios of ARD. Also, in Chapter 5, the required accuracy of experimental tests is analyzed.

The dimensionless scaling parameter, δ, presented in this report is more general and provides more information than the Thiele modulus typically used in literature [Ar], [DrA], The scaling analysis has been performed by analytical mathematical methods with a minimum use of numerical analysis. In the next step, the results presented in this report should be confronted with available experimental data on acid drainage. Data collected during field tests and laboratory thermokinetic experiments involving large samples of pyritic rocks should be used to calibrate the model.

Résumé

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage 

RÉSUMÉ 

La nature particulièrement non linéaire des équations cinétiques decrivant les processus géochimiques et physiques agissant ensemble dans l'oxydation de la pyrite a soulevé des questions importantes sur la prévisibilité des répercussions sur l'environnement du drainage des eaux acides provenant des stériles. En réponse à ces questions, le présent rapport décrit les résultats d'un projet de recherche qui a été amorcé dans le but d'analyser quantitativement les processus chimiques et physiques élémentaires combinés qui sont à l'origine de l'oxydation de la pyrite et de l'acidification des eaux de drainage des stériles. Ce projet fait partie de mesures prises pour concevoir des indicateurs pratiques qui combineraient les données quantitatives de laboratoire, obtenues de petits échantillons de stériles, dont les effets observés dans les haldes de stériles sont à grande échelle.

Le principal objectif du présent projet était de déterminer les simples lois d'échelle qui gouvernent le comportement géochimique et thermodynamique de la roche pyritique dans les haldes de stériles. L'indicateur d'échelle δ qui combine les informations sur les processus se produisant à différentes échelles, est un élément innovateur du projet.

Les équations cinétiques effectives des réactions chimiques couplées se produisant pendant l'oxydation de la pyrite ont été établies. La concentration d'oxygène dissous a été décrite par une simple formule qui corrobore quantitativement les données expérimentales pour tout l'intervalle des températures et des concentrations de l'oxygène dans la phase gazeuse, tel qu'exigé pour l'analyse du drainage des eaux acides provenant de stériles. La forte diminution de la concentration de l'oxygène dissous avec la température est incluse dans le modèle. Elie ne l'avait pas été dans les modèles antérieurs - ce qui a pu causer une surestimation du potentiel d'acidification.

L'énergie et le transport de l'oxygène sont décrits en utilisant un modèle de réaction- transport. La forte dépendance non linéaire des vitesses de réaction effectives sur les paramètres physiques, minéralogiques et chimiques a été décrite au moyen d'un paramètre d'échelle δ que l'on peut utiliser comme indicateur pratique de drainage d'eaux acides provenant de haldes de stériles. Le paramètre d'échelle sans dimension δ combine des informations sur la porosité de la halde, la faille de la halde, la superficie de réaction effective, l'effet de la température sur la vitesse d'oxydation de la pyrite, la diffusion de l'oxygène dans la phase gazeuse, la chaleur de réaction de l'oxydation de la pyrite, la conductivité thermique des stériles et la température ambiante.

L'analyse de la sensibilité foumit des informations sur la précision requise pour les essais expérimentaux et l'importance relative des paramètres agissant sur les différents processus physiques et géochimiques à l'origine des eaux acides provenant de haldes de stériles. En particulier, il est montré que l'efficacité des couvertures imperméables s'accroît avec la concentration de la pyrite.

L'analyse d'échelle indique que les processus géochimiques et de transport fonctionnent à l'échelle moyenne d'une façon fondamentalement différente de l'échelle réelle. Les valeurs critiques du paramètre d'échelle δ auxquelles ont lieu des bifurcations ou des catastrophes thermodynamiques aboutissant à une accélération de l'acidification, ont été déterminées dans différents scénarios. Le fait crucial que l'acidification des eaux drainant les haldes de stériles dépend de la porosité des haldes, de la taille des haldes et de leur surface de réaction est l'une des conclusions de l'analyse de bifurcation. Les résultats de l'analyse d'échelle offrent la possibilité d'une évaluation provisoire rapide et peu coûteuse des répercussions prévues sur l'environnement.

Tous les paramètres et toutes les variables du présent modèle peuvent être mesurés dans des expériences indépendantes. Le modèle produit des résultats réalistes en ce qui concerne les vitesses d'acidification sans l'introduction de paramètres ajustables utilisés par tous (connus de nous) les autres modèles de stériles. Plusieurs résultats de la présente étude different considérablement des conclusions et des hypothèses d'autres modèles.

Les résultats quantitatifs présentés dans l'étude devraient être confrontés aux données recueillies sur le terrain. Des essais thermocinétiques additionnels sur des échantillons de roches de stériles doivent être réalisés pour obtenir des données d'entrée fiables pour les modèles prédictifs futurs de stériles. Les effets du transport de l'oxygène par convection et du transport de l'eau sur les valeurs critiques du paramètre d'échelle sans dimension devraient être analysés dans une étude future.

Les essais habituels pour déterminer l'équilibre acide-base ne permettent pas de concevoir un modèle prédictif qui produirait des informations quantitatives sur les effluents. Pour obtenir les informations expérimentales nécessaires à un modèle prédictif des stériles, il est proposé de réaliser des essais thermocinétiques additionnels.

Nous espérons qu'après d'autres modifications et étalonnages, l'analyse d'échelle présentée facilitera la conception et la gestion des haldes de stériles.

Executive Summary

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage 

EXECUTIVE SUMMARY

The highly nonlinear nature of the kinetic equations describing the coupled geochemical and physical processes involved in pyrite oxidation has posed serious questions about the predictability of the environmental impact of acid rock drainage. In response to these questions this report describes the results of a research project which has been initiated with the purpose to provide a quantitative analysis of the interrelated elementary chemical and physical processes which are responsible for pyrite oxidation and acid rock drainage (ARD). This project is part of an effort to design practical indicators which would combine quantitative laboratory data, obtained for small samples of waste rock, with large-scale effects observed in waste rock piles.

The main objective of this project was to determine the simple scaling laws which govern the geochemical and thermodynamic behaviour of pyritic rock in waste rock piles. The scaling indicator δ which combines information about processes occurring at different scales, is an innovative feature of this project.

Effective kinetic equations for coupled chemical reactions involved in pyrite oxidation have been derived. The concentration of dissolved oxygen has been described by a simple formula which gives reasonable quantitative agreement with experimental data for the whole range of temperatures and oxygen concentrations in the gas phase, as required for ARD analysis. The strong decrease of dissolved oxygen concentration with temperature is included in the model. This property has not been included in previous models - the omittance of which could result in an overestimation of the acid generation potential.

Energy and oxygen transport are described by using a reaction-transport model. Strong nonlinear dependence of the effective reaction rates on the physical, mineralogical and chemical parameters has been described by means of a scaling parameter δ which can be used as a practical indicator of ARD for waste rock piles. The dimensionless scaling parameter δ combines information about pile porosity, pile size, effective reactive surface area, temperature dependence of the rate of pyrite oxidation, oxygen diffusion in the gas phase, heat of the pyrite oxidation reaction, thermal conductivity of waste rock, and ambient temperature.

The sensitivity analysis provides information about the required accuracy of experimental tests and the relative importance of parameters governing different physical and geochemical processes responsible for ARD. In particular, it is shown that the effectiveness of impermeable covers increases with the pyrite concentration.

The scaling analysis indicates that geochemical and transport processes operate at the meso-scale in a way fundamentally different from the full-scale. Critical values of the scaling parameter δ, at which bifurcation’s or thermodynamic catastrophes leading to accelerated acid generation rates, have been determined for different scenarios. The critical dependence of ARD on pile porosity, pile size and reactive surface area is one of the conclusions of the bifurcation analysis. The results of the scaling analysis offer the possibility of a cheap and fast preliminary assessment of the expected environmental impact.

All parameters and variables of the present model can be measured in independent experiments. The model produces realistic results for acid generation rates without introducing adjustable fitting parameters used by all (known to us) other waste rock models. Several results of this study are significantly different from conclusions and assumptions of other existing models.

Quantitative results presented in this study should be confronted with field data. Additional thermokinetic tests for waste rock samples are required in order to provide reliable entry data for future predictive waste rock models. The effects of convective oxygen transport and water transport on the critical values of the dimensionless scaling parameter should be analyzed in a future study.

Usual acid/base accounting tests do not provide data necessary for a predictive model which should generate quantitative information about the effluent. Additional thermokinetic tests are proposed in order to provide experimental information necessary for a predictive waste rock model.

We hope that after further modifications and calibration, the scaling analysis presented here may help to properly design and manage waste rock piles and dumps.

Acknowledgements

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage 

ACKNOWLEDGEMENTS 

The guidance by Benoit Godin, Section Head, Environmental Contaminants Division, Pacific and Yukon Region, Environment Canada, who provided several ideas and continued support for this project is gratefully acknowledged. This project benefited greatly from discussions and information provided by M. Blanchette, P. Gelinas, K. Ferguson, P. Grabinski, B. Krusche, R. Lefebvre, G.W. Luther III, R. McCandless, M.A. McKibben, K.A. Morin, R.V. Nicholson, W.A. Price, A. Robertson, I. Suzuki, G.A. Tremblay, J. Weres and W.W. White III. The reviewers of the draft report; Stephen Day, Benoit Godin, and Ronald V. Nicholson supplied helpful comments and suggestions.

Matthew Otwinowski: Scaling Analysis Of Acid Rock Drainage - Table Of Contents

Matthew Otwinowski
Synergetic Technology 
The University of Calgary,
Department of Physics and Astronomy, Calgary AB,
Canada T2N 1N4

Scaling Analysis Of Acid Rock Drainage 

MEND Project 1.19.2 

October 1995 

REPORT PREPARED FOR 

MINE ENVIRONMENT NEUTRAL DRAINAGE PROGRAM 

and 

ENVIRONMENT CANADA 

This work was done on behalf of MEND and sponsored by Environment Canada.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS
EXECUTIVE SUMMARY 
RÉSUMÉ
1.0 INTRODUCTION
2.0 OXIDATION OF PYRITE 
2.1 Oxidation at pH Between 4 and 7
2.2 Oxidation at pH Less Than 4 
3.0 EFFECTIVE KINETIC EQUATIONS FOR THE OXIDATION OF PYRITE
3.1 Scaling Relations for Chemical Reactions
3.2 Scaling Formula for Concentration of Oxygen Dissolved in Water
3.3 Effective Kinetic Equations
3.3.1 pH Between 4 and 7
3.3.2 pH Less Than 4
4.0 SCALING ANALYSIS OF COUPLING BETWEEN CHEMICAL KINETICS AND TRANSPORT PHENOMENA 
4.1 Formulation of the Reaction-Transport Model
4.2 Scaling Properties and Sensitivity Analysis of Acid Rock Drainage
4.3 Bifurcations and Thermodynamic Catastrophes in Acid Generation Rates
5.0 PRACTICAL IMPLICATIONS OF PHYSICO-CHEMICAL SCALING RELATIONS 
5.1 Alternative Scenarios of Acid Rock Drainage
5.2 Acid Generation Rates
5.3 Required Accuracy of Laboratory and Field Tests
5.4 Discussion of Results
6.0 CONCLUSIONS AND RECOMMENDATIONS
APPENDIX A
APPENDIX B
REFERENCES