Characterization of solute transport parameters in leach ore: inverse modeling based on column experiments

Abstract

Heap leaching is essentially a process in which metals are extracted from mine ores with lixiant. For a better understanding and modeling of this process, solute transport parameters are required to characterize the solute transport system of the leach heap. For porous media like leach ores, which contain substantial gravelly particles and have a broad range of particle size distributions, traditional small-scale laboratory experimental apparatus is not appropriate. In this paper, a 2.44 m long, 0.3 m inner diameter column was used for tracer test with boron as the tracer. Tracer tests were conducted for 2 bulk densities (1.92 and 1.62 g/cm³) and 2 irrigation rates (2 and 5 L/ (m²·h^-1)). Inverse modeling with two-region transport model using computer code CXTFIT was conducted based on the measured breakthrough curves to estimate the transport parameters. Fitting was focused on three parameters: dispersion coefficient D, partition coefficient β, and mass transfer coefficient ω. The results turned out to fall within reasonable ranges. Sensitivity analysis was conducted for the three parameters and showed that the order of sensitivity is β>ω>D. In addition, scaling of these parameters was discussed and applied to a real scale heap leach to predict the tracer breakthrough.

Keywords

leach ore / tracer test / inverse modeling / parameter up-scaling

Introduction

Heap leaching has been developed to a broadly practiced approach for copper and other precious minerals production from low-grade ores since the 1970s. Heap leaching is a complicated process, in which microbial activities, chemical reactions, and physical processes function simultaneously to extract metal from the ores. Many researches have addressed the microbial behavior and effectiveness (Clark et al., 2006; Petersen and Dixon, 2006; Coram-Uliana et al., 2006; Rawlings, 2002) and chemical reactions in the leaching process (Watling, 2006; Miller et al., 2003; Wan and LeVier, 2003). There are also lots of researches on the modeling of these processes with focus on the unsaturated flow system and chemical kinetics, in which most of them applies the shrinking core model to account for the change of ore formation (McLaughlin and Agar, 1991; Sanchez-Chacon and Lapidus, 1997; Dixon, 2000; Bouffard and Dixon, 2001; Petersen and Dixon, 2002). Both the application of these models and the better understanding of the biochemical reactions and their effectiveness rely on a representative description of the unsaturated solute transport system.

The leaching heap is usually composed of run of mine or crushed ore or their combination, which includes substantial gravel and has broad grain particle size distribution. For this type of earth material, the spatial distribution of the gravel can create macro-pores and discontinuity in the pore size distribution (Milczarek et al., 2006; Al-Yahyai et al., 2006; Poulsen et al., 2002). This large-scale heterogeneity requires large-scale experimental apparatus and makes the traditional small-scale laboratory method unsuitable for this material. In this paper, a set of laboratory measured long column tracer test data is presented. Based on these data, the solute transport parameters are derived through inverse modeling. Sensitivity analysis and up-scaling of these parameters are also discussed.

Materials and methods

Boron was used as the tracer in this study in which the lixiant, raffinate and the leach ore contain many different types of metals and ions, which make the background concentration of these ions and cations very high. Therefore, traditional tracers, such as bromine, are not appropriate. Tracer tests were conducted in a column of 2.44 m length and 30 cm inner diameter. Leach ore sample collected from a copper mine in the southwest of USA was packed in the column in two different bulk densities (1.62 and 1.92 g/cm³) respectively. Three ECHO-5 sensors (Decagon Devices, WA, USA) were installed in the depths of 60, 120, and 180 cm to monitor the water contents. The tracer was mixed with the raffinate to 100 mg/L. An irrigation head connected to the raffinate and tracer solution reservoir was placed on top of the column to leach the ore. There were 12 needle sprinklers in the irrigation head to ensure even distribution. Two irrigation rates, 2 and 5 L/ (m²·h^-1), were applied for each of the two different density columns, respectively. The boron tracer solution was applied for approximately two pore volumes and then the column was eluted for a minimum of two pore volumes. A sample collector was set below the column to collect the effluent every 2 hours. The collected sample was analyzed in a commercial chemistry lab.

The boron tracer breakthrough was observed to be retarded, so ore/tracer batch tests were performed to determine the sorption coefficients of boron in raffinate to the leach ore. Inverse modeling was then conducted with computer code CXTFIT 2.1 (Toride et al., 1995) based on the experimental tracer breakthrough curves and the measured boron/ore partition coefficient. CXTFIT is a widely used computer model for transport simulation, which provides both standard advection-dispersion transport model and two-region non-equilibrium transport model.

Sensitivity analysis was conducted subsequently for three critical parameters. To make the modeling results useful for a real leach heap, which is 18 m high, the inverse modeling estimated parameters for the 2.44 m column were up-scaled to predict the tracer transport in the real scale leach heap.

Results and discussion

The batch test results are shown in Fig. 1. The sorption coefficient K_d was calculated as 0.0625 cm³/g (K_d = C_s/C_w, where C is the solute concentration, subscript s and w refer to solid and water phase, respectively). Figures. 2 and 3 present the long column tracer test results for the high and low density columns with two irrigation rates, respectively. Observation of the boron breakthrough curves shows that other than retardation, non-ideal transport occurred. In these cases, the degree of non-ideal transport is such that the standard advection-dispersion transport model cannot provide a satisfactory simulation of the observed breakthrough curve. Thus, the breakthrough curves obtained for the columns were analyzed with the two-region non-equilibrium transport model using CXTFIT.

The two-region model is also called the dual-porosity or mobile-immobile model where the liquid phase is assumed to partition into mobile and immobile regions. Solute mass transfer between the two regions is modeled as a first-order rate process. The dimensionless form of the model is given by

(1)\beta R{\displaystyle \frac{\partial C_m}{\partial T}}={\displaystyle \frac{\mathrm{1}}{P}}{\displaystyle \frac{{\partial }^{\mathrm{2}}{C}_m}{\partial Z^{\mathrm{2}}}}-{\displaystyle \frac{\partial C_m}{\partial Z}}-\omega (C_m-C_{im}),

(2)(\mathrm{1}-\beta )R{\displaystyle \frac{\partial C_{im}}{\partial T}}=\omega (C_m-C_{im}),

where C_m is relative concentration in the mobile phase, C_im is relative concentration in the immobile phase, T is dimensionless time, R is the retardation factor, which is calculated as (1+ρ_bK_d/θ) for this tracer where the retardation is caused by the sorption on the solid particles. P is the Peclet number (P=vL/D, where v is pore water velocity [L]/[T], L is the domain length, and D is dispersion coefficient [L]²/[T]). β is a partition coefficient, (β=(θ_m+fρ_bK_d)/(θ+ρ_bK_d)), that will equal θ_m/θ if the solute is non-reactive (θ_m, θ_im, is mobile and immobile water content respectively), ρ_b is bulk density; f represents the fraction of sorption sites that are in equilibrium with the mobile liquid phase), ω is the mass transfer coefficient where ω= α_mtL/q and α_mt is the dimensional mass transfer coefficient [T^-1], while q is the flux [L]/[T]. Table 1 lists the parameters in the two-region model determined through experiments. These parameters were used as initial input parameters for CXTFIT to estimate the other 3 transport parameters: D, β, and ω.

Inverse modeling results for high density and low density columns

Figure 2 shows the observed and fitted breakthrough curves for the high density column experiments. The fitted 3 transport parameters are included in Table 2. Dispersivity (α) is also included in Table 2, which is calculated as D/v. Inspection of Table 2 reveals that the values of the estimated parameters are within a reasonable range. For instance, dispersivity is usually 1–2 orders of magnitude smaller than the system scale (Gelhar and Rehfeldt, 1992). The values of dispersivity obtained here is about 1/30 – 1/40 of the column length. The dispersivity of low flux column is slightly larger than that of the high flux column. This can be attributed to the increased torturosity for a system with lower water content (Sato et al., 2003). The value β is in the range of 0.6 to 0.68. A larger β is predicted for the high flux column. This indicates a larger mobile phase fraction and/or a larger fraction of boron sorption sites that are in equilibrium with the mobile liquid phase for the high flux column. The mass transfer coefficient, ω can be interpreted as the ratio of the effective hydraulic residence time (L/q) to the characteristic time of mass transfer (1/α_mt). The estimated ωfor the low flux column is 1.73 times that of the high flux column (0.71/0.41). With the fact that the low flux column has a longer hydraulic residence time than the high flux column with a factor of 2.5 (=12/4.8), this indicates that the mass transfer characteristic time for the low flux column is longer than that for the high flux column with a factor of 1.44 (2.5/1.73). This is reasonable since for the low flux column, there is greater fraction of immobile water; therefore, the rate-limiting factor such as diffusion in the immobile phase will take longer and lead to a longer mass transfer time.

For the low density columns, the same fitting procedures as that for the high density column were performed. However, the best fit resulted in an unrealistic value of ω for the low flux column (ω =1E-7). Therefore, the fitting procedures were adjusted with more reasonable ω. The dimensional mass-transfer coefficient α_mt has been reported to have a log linear relationship with the residence time LR/v (Maraqa et al., 2001). Consequently, a Maraqa-form empirical equation was used to determine the α_mt and subsequently ω (ω= α_mtL/q) for the low density columns using (Maraqa et al., 2001):

(3){\alpha }_{mt}=a{(LR/v)}^b.

The parameters a and b are material specific. Therefore, the predicted ω from the high density column was used to estimate the coefficients a and b for the leach ore. The values are 0.04 and-0.44 for a and b, respectively. Subsequently, for the leach ore, Eq. (4) can be used to estimate ω:

(4)\begin{array}{l}{\alpha }_{mt}=\mathrm{0.04}{(LR/v)}^{-\mathrm{0.44}}\hfill \\ \omega ={\alpha }_{mt}L/q\hfill \end{array}.

Using this equation, ω for the low density columns can be estimated and then used as the input parameters with v to estimate D, R, and β. The best fitting results from this method are presented in Table 2 and Fig. 3.

Inspection of Fig. 3 indicates that the estimated values of ω result in a decent fit for the low density columns. Furthermore, the other estimated parameters (D and β) are within a reasonable range and are consistent with the results for the high density columns. This indicates the effectiveness of Eq. (4) in the estimation of α_mt and ω. The estimated R for the high flux column is the same as the experimental determined value, while that for the low flux column (1.90) is slightly smaller than the experimental determined value (2.05), which can be considered as reasonable.

Parameter sensitivity analysis

To evaluate the impact of changes in input parameters on the model output, a sensitivity analysis was performed for three critical parameters D, β, and ω, respectively. Since the dependence of model output should be similar for different columns, the sensitivity analysis was only performed for the high density, high flux column. Table 3 lists the variability in parameters used. Forward modeling was conducted with these parameters. Figure 4 (a)-(c) shows the results.

Inspection of Fig. 4(a) shows that a smaller β (fraction of mobile water) results in an earlier arrival limb and longer tailing, which indicates a more heterogeneous system. Smaller fraction of mobile water makes the effective path of the solute shorter, thus the arrival time is earlier. On the other hand, non-equilibrium mass transfer between the mobile and the larger fraction of the immobile part causes the longer tailing. Conversely, a greater fraction of mobile water generates more symmetric breakthrough curve because the impact of immobile water turns out to be smaller and the system behaves more like a homogeneous one. Figure 4(a) also indicates that a relatively small change of ±10% on β will cause a substantial change in the model output. This indicates that the model is very sensitive to the parameter β. Figure 4(b) shows the change in ω, which can be interpreted as the ratio of the effective hydraulic residence time (L/q) to the characteristic time of mass transfer (1/α_mt). ω represents the impact of large-scale physical heterogeneity (e.g., preferential flow, inter-region mass transfer) on transport with large ω values considered to be close to a homogeneous system. In Fig. 4(b), the+100% ω curve is more symmetric and will have less tailing than the-100% curve. The-100% ω curve has the earliest breakthrough and longest tailing.

The effect of D on the breakthrough curves is straightforward, as shown in Fig. 4(c). A small D causes less spreading of the breakthrough curve. It is worthwhile to note that the effect of changing D on the breakthrough curve does not become substantial until it is 100% larger or smaller. Overall, the order of sensitivity of the model output put to the parameters is: β>ω>D.

Prediction of 18–m heap

Parameters such as dispersivity (α) and mass transfer coefficient (ω) are known to be scale-dependent. Therefore, to make the results obtained from the laboratory experiment applicable to the real larger scale leach heap, up-scaling of the parameters is needed. The dispersivity has been reported to be 1-2 order smaller than the scale characteristic length (Gelhar and Rehfeldt, 1992). The dimensional mass-transfer coefficient has been reported to be related to the residence time or experimental time (Maraqa, 2001; Haggerty et al., 2004).

The fitted dispersivity for the 2.44 m column is approximately 1/30-1/40 of the column length. For an 18 m column, the same ratio of dispersivity to column length can be assumed and the scaled dispersivity values and corresponding dispersion coefficients are presented in Table 4.

The scaling of ω is embedded in Eq. (4). Using Eq. (4), the ω for the 18 m column can be estimated and is listed in Table 4. Parameter β is considered constant for different column lengths in that the influencing factors for β should be the porous media and the pore water velocity, which are the same for different column lengths.

The parameters in Table 4 were applied to predict the BTCs for the 18 m heap. An irrigation pulse of 1.79 pore volume was applied to all the four scenarios. Figure 5 (a) and (b) shows the results.

Observation from Fig. 5(a) reveals that all the BTCs are close to each other. This indicates that the larger scale somehow smoothes out the non-ideal behavior caused by the system’s heterogeneity. Figure 5(b) shows that for high flux scenarios, when the irrigation pulse is around 30 days (corresponding to the 1.79 pore volume), the full breakthrough of the solute (when the C/C₀ is less than 0.01) will occur at around 135 days; while for the low flux scenarios, when the irrigation pulse is approximately 60 days, the time required for the complete breakthrough is 270 days. These can be used as guide in heap leaching design and operation.

Conclusions

Boron was selected as tracer for the leach ore column experiments to identify the characteristics of solute transport parameters. Batch tests were conducted to obtain the sorption coefficient of boron/leach ore in raffinate, which was determined as 0.0625 cm³/g. 2.44 m long column tracer tests were conducted for two bulk densities (1.92 and 1.62 g/cm³) under two different irrigation rates (2 and 5L/ (m²·h^-1)), through which breakthrough curves were obtained. Inverse modeling was then conducted with the observed breakthrough curves to estimate the transport parameters D, β, and ω with other pre-determined parameters in the two-region non-equilibrium transport model using the computer code CXTFIT.

Two-region transport model is shown to be effective to simulate boron transport in the leach ore column. β was estimated to be within the range of 0.6 to 0.68. The Maraqa form Eq. (4) is provided to estimate the α_mt and subsequently ω for the leach ore. The estimated parameters fall in the reasonable range. Therefore, the fitted parameters are expected not only to be applied to the solute transport for the column experiments but also have the potential to be applied for scaling. The scaling is proceeded with dispersivity estimated as 1/30-1/40 of the scale length, and ω obtained from Eq. (4). Scaled parameters were applied to an 18 m leach heap to predict the breakthrough of the tracer, which can be used as guide in heap leaching design and operation.

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