Electroosmotic Drainage Applied to Mining Waste

Abstract

One factor affecting the stability of mining stockpiles is the moisture defined mainly by copper solution trapped in the porous by capillary. This moisture is not easy to remove if conventional methods are applied which use pressure or gravity as driving force. In the case of saturated or partially saturated soils with water, containing a large fraction of fine material, electroosmosis not only allows to reduce the humidity but also changes the structure, giving a higher strength and stability to the soils. Since the movement of the water, due to the electric gradient, is from the anode toward the cathode, the soil water content will decrease at the anode and will increase at the cathode. Water accumulated at the cathode then can be discharged by providing a drainage system at the cathode. This chapter presents theoretical and experimental aspects on electroosmotic drainage technique, based in works realized by the authors of this chapter. To explain the water flow through a mining residue containing a certain fraction of fine material and that in addition presents a high humidity, a model for the fluid flow in porous media is described here, taking into consideration two driving forces, defined by hydraulic and electric potentials.

1. Introduction

Mine waste represents one of the critical aspects of mining activity due to the environmental and social impacts that could generated not only in the period of extraction and processing of mineral but also in the post-closure period of the mining activity. Whether extraction and processing are stages that generate large quantities of waste that are stored above ground where they may constitute a potential risk of groundwater contamination or they may present structural stabilization problems by its humidity.

In waste from copper leaching, the remaining humidity is due to the solution trapped in the capillary interstices of the solid matrix. To avoid structural stabilization problems, it is necessary to reduce the moisture of the waste. One possibility is to let the humidity reduce naturally, by letting the water drain gravitationally. The action of this technique is limited, since gravity has no action on capillary trapped water. Thus, to reduce the remaining moisture content of the solid material and accelerate the drainage process, electroosmotic drainage technique is proposed. Electroosmotic drainage consists of applying a low electric potential to dewater a porous medium, in this case the solid mining waste. This technique was employed in the 50 years to consolidated clay soils as a simple and efficient way to accelerate the dewatering process in soils with low hydraulic conductivity [1, 2]. Since then, electroosmotic drainage has been successfully applied mainly to the drying process of sediments, and remediation of contaminated soils with heavy metals [3, 4, 5, 6, 7, 8, 9, 10]. Applications of this technique in mining activities are still under development. In a previous work, through the realization of mining waste drainage experiments, at laboratory and bench scale, the conceptual validation of the technique was demonstrated [7, 10, 11]. Electroosmotic drainage is more efficient than conventional drainage techniques such as vacuum filtering, belt filter pressing and centrifuging in terms of collected solution, time of drained solution, and energy consumption [12]. Water in a porous material can be divided into four types: free water, interstitial or capillary water, surface or vicinal water and intracellular water [13]. While conventional drainage techniques, which are based on mechanical pressure, are effective at removing free water, electroosmotic drainage can be applied to remove free, interstitial, and vicinal water [14].

The establishment of an electric field, in a porous medium with high humidity, in addition to the removing of the capillary trapped water, causes the ionic species present in solution to migrate toward the oppositely charged electrode (anode or cathode), and this ion migration is called electromigration, while the mechanism that govern the movement of the water toward the positively charged electrode (cathode) is called electroosmosis. In this chapter, the theoretical basis, complemented with experimental data at laboratory and bench scale, is established, describing conceptual and mathematically; in addition to the gravitational drainage, the electroosmotic flow as a transport mechanism allows an additional drainage of solution in a process of heap pile copper leaching. The importance of the application of this mechanism is that it allows a larger volume of solution to be drained than that could be drained naturally, in which the gravity is the only driving force acting on the medium. Keeping in mind that with the application of an electric field to the medium, the electroosmotic mechanism acts mainly on that fraction of solution trapped capillary, and this occurs in the fine solid fraction.

2. Flow of fluid in porous media

Flow of water in porous media is a relevant subject in applications such as hydrogeology and in multiple branches of engineering. At large physical scales, Darcy’s law is used, which provides a macroscopic description of flow in porous media. At smaller physical scales, flow and transport are simulated using more detailed descriptions such as Hagen-Poiseuille or Navier-Stokes flow. In the past decades, the topic of multiphase flow in porous media has received widespread attention. There are numerous pore-scale simulation and experimental studies on single-phase and multiphase flow and transport in porous media. However, very few studies of flow and transport in charged porous media have been reported [15, 16, 17]. When the porous medium and the fluids present in the pore space are exposed to an electric field, a number of electrokinetic and electro-hydrodynamic effects or mechanisms can take place, such as electromigration, electrophoresis, and electroosmosis. These phenomena have been used in applications, such as electrophoretic migration of contaminants in soils [4], microfluidics [18], or enhanced oil recovery [19].

2.1 Conceptualization of the electroosmotic waterflow

Electroosmosis is the flow of water induced by the application of an electric field [20, 21]. This waterflow, named electroosmotic flow, is generated by the electrical interaction between the surface of the solid particles and the fluid, which leads to charge separation at a “double layer” interface. The volumetric flow of water is dependent of the double-layer properties, the chemical composition of the porous material, the type of fluid in the pores, the geometry and size of the pores, the saturated condition of the medium, and the intensity of the applied electric potential [22]. Fluid flow usually occurs in the same direction as the applied electric potential, from anode to cathode [23, 24].

One of the earliest and most widely used theoretical models of the electroosmosis process is based on a model introduced by Helmholtz in 1879 and refined by Smoluchowski in 1914 [25]. Thus, to estimate the electroosmotic flow of water per unit of area (q_eo) in a saturated porous media, the Helmholtz-Smoluchowski model (H-S model) is used [22]. This flux of water (q_eo), expressed in units of water volume per unit of area and unit of time is formulated as:

$$q_{\mathit{eo}}=-\frac{{\epsilon }_p{\epsilon }_w\zeta }{\eta }\frac{\Delta \varnothing }{\mathit{\Delta L}}$$E1

where ε_p is the soil porosity, ε_w is the electrical permittivity of the soil, ζ is the zeta-potential, η is the dynamic viscosity of the fluid, Δϕ is the applied electric voltage difference between electrodes, and ΔL is the separation distances between electrodes.

The direction of the electroosmotic flow, relative to the electric potential, is solely determined by whether the zeta-potential is positive or negative. Most soils and porous ceramics exhibit a negative zeta-potential causing the direction of the electroosmotic flux to be from the anode toward the cathode. As the zeta-potential (ζ) is a parameter dependent of the pH value, the movement of the water may be in one direction, toward the cathode, or toward the anode, [23, 24, 25]. The electroosmotic flow can be expressed in terms of the electric gradient and the electroosmotic permeability of the porous media (k_eo), which is a measure of the fluid flux per unit area of the porous media. The value of k_eo is assumed to be a function of the zeta-potential of the soil-pore fluid interface, the viscosity of the pore fluid, the porosity, and electrical permittivity of the soil:

$$k_{\mathit{eo}}=\frac{{\epsilon }_p{\epsilon }_w\zeta }{\eta }$$E2

The electroosmotic permeability is pH-dependent as the z-potential (ζ) do it. Taken as reference the Kaolinite, an increase of the pH from 0 to 14 makes the electroosmotic permeability increases in the rage of (3 to 8) × 10⁻⁹ (m²/V/s) [26].

2.2 Flux of water for the case of saturated medium

The fluid flux is the consequence of three gradients: hydraulic gradient ∇(−h) (Darcy’s law), electrical gradient ∇(−ϕ), and a chemical gradient. The latter, chemical gradient, is significant only in the presence of large molecular chains and in very active clay deposits. Assuming that the chemical gradient is not significant [27, 28], the porous fluid mass flux (J_w), expressed in mass flow of water per unit of area, is thus estimated by two contributions:

$$J_W=J_h+J_{\mathit{eo}}$$E3

where J_h, and J_eo (kg/m²/s) are the hydraulic and electroosmotic mass flux of water, respectively. Both fluxes, J_h y J_eo, are defined by a driven force times the conductivity coefficient (K) that describes the ease with which a fluid (water) can move through pore spaces. Thus, rewriting the mass flux in terms of the volumetric flux (q) gives:

$$J_W=J_h+J_{\mathit{eo}}={\rho }_w\left(q_h+q_{\mathit{eo}}\right)={\rho }_w\left[K_h\nabla \left(-H\right)+K_{\mathit{eo}}\nabla \left(-\varnothing \right)\right]$$E4

where ρ_w (kg/m³) is water density, q_h and q_eo (m³/m²/s) are hydraulic and electroosmotic flow velocity, respectively, H is hydraulic head (m), and ϕ is the applied electric potential (V). The K_h (m/s) and K_eo (m²/V/s) are the hydraulic and electroosmotic conductivity, respectively. The negative sign indicates that flow of water occurs from large to small hydraulic head, for the hydraulic flow, and from anode to cathode, opposite to the electric gradient, for the electroosmotic flow. The hydraulic head, H, in length units, at any point in the soil is defined as the summa of the elevation head, z, and the pressure head $\left(\mathrm{\Psi }\right)$:

$$H=\frac{p_c}{{\rho }_{w}g}+z=\mathrm{\Psi }+z$$E5

where p_c is the pore water pressure, ρ_w is the density of the fluid (solution), and g is the gravitational acceleration. Then, the equation for the mass flux, Eq. (4), may be rewritten as:

$$J_W=J_h+J_{\mathit{eo}}={\rho }_w\left(q_h+q_{\mathit{eo}}\right)=-{\rho }_w\left[K_h\nabla \left(\mathrm{\Psi }+z\right)+K_{\mathit{eo}}\nabla \left(\varnothing \right)\right]$$E6

The relative importance of the hydraulic and electroosmotic contributions, to the flow of water, is determined by the ratio of the electroosmotic to the hydraulic conductivity, expressed as K_eo/K_h. This ratio is affected by the soil type, microstructure, and pore fluid conditions. In coarse-grained soil, the ratio is very low, due to the almost nonexistent electroosmotic flow and relatively high hydraulic conductivity (>10⁻³ cm/s) of such soils. In soft, fine-grained soils, the ratio is significant as K_eo is usually on the order of 10⁻⁵ cm²V⁻¹s⁻¹, while K_h varies between 10⁻⁵ cm/s for fine sand and 10⁻⁹ cm/s for bentonite [27, 28, 29]. Therefore, for moving liquid through fine-grained soils, the electrical gradient is more effective than the hydraulic gradient.

The hydraulic conductivity value (K_h) is always positive, and therefore, the hydraulic gradient always determines the direction of the hydraulic flow. In contrast, the K_eo can be either positive or negative resulting in flow with or against the electric potential. On the other hand, the magnitude of the K_h is dependent of the pore size distribution. However, K_eo is independent of pore size (Figure 1). In low-permeability zones such as clay, particle size smaller than 0.002 mm in diameter, the electroosmotic flow is easier to achieve than hydraulic flow (pressure-driven) (see Figure 1).

Figure 1.

a) Electro osmotic (K_eo) and hydraulic (K_h) conductivity for selected sediments; b) Comparison of hydraulic (q_h) and electro osmotic (q_eo) flow rates [24].

When water is trapped between fine solid particles, the electroosmotic technique is more efficient than the technique driven by pressure [29], because electroosmotic flow is based on the surface and colloidal characteristics of particles in suspension and is independent of pore size, unlike conventional hydraulic flow, which falls off significantly with pore size [30]. For copper mining waste, as a reference, tailing material from mineral processing, finely ground waste with particle size range 0.06–0.002 mm (similar to fine sand/coarse silt), the hydraulic conductivity is 1.65 × 10⁻⁷ m/s [31]. Since pressure dewatering is a popular technique to reduce the water content of solid waste, some works have combined electroosmotic drainage and pressure dewatering [32, 33].

3. Humidity and fraction of fine material on the drained solution

The total fluid flowing in a porous material under the effects of an electrical potential and the gravitational force is mainly caused by two gradients: the hydraulic gradient (Darcy’s law) and the electrical gradient (electroosmosis). The contributions of each of the gradients are affected by soil type, soil microstructure, and pore fluid conditions [28]. For example, in coarse-grained soils, the ratio of electroosmotic to hydraulic flow is very low, and it means that the electroosmotic flow is almost nonexistent, while in fine-grained soils the ratio is high [34]. Therefore, a large electrical gradient is more effective than a hydraulic gradient for moving liquid through fine-grained soils.

Experimental tests have been carried out, at laboratory scale, to evaluate the effect of the initial moisture degree and the fraction of fine material contained in the mining residues, over the volume of solution drained out from the electroosmotic cell containing the residues [35]. All the tests were done in 20-liter electroosmotic cells as shown in Figure 2, using a linear configuration of stainless steel electrodes, 24 V of constant applied voltage between electrodes, and an operating time of 48 hours. The tests followed a 3² factorial design experiment with and without replication, considering two factors with three levels each. The factors were the fraction of fine material and the moisture of the sample, and the levels of them were 10, 15, and 20% for fine material and 10, 20, and 30% for the moisture. The volume of drained solution was defined as the response variable.

Figure 2.

(a) electroosmotic drainage cell; and (b) gravitational drainage cell [35].

The solid matrix is made up of two materials: coarse material made up of solid particles of different granulometric sizes (gravel, coarse, and fine sand) and fine material made up of clay material (Table 1).

Material	10% fine particle		15% fine particle		20% fine particle
	Weight (kg)	%	Weight (kg)	%	Weight (kg)	%
Gravel	0.28	4	0.2	3	0.13	2
Gross sand	1.86	27	1.65	25	1.46	23
Fine sand	4.07	59	3.77	57	3.49	55
Clay	0.69	10	0.99	15	1.27	20

Table 1.

Distribution of the particle size fraction of the solid matrix [35].

The highest percentage of fine material (FP) was that established by slope safety regulations, while the other two were chosen to study the influence of lower levels of fine material content. The initial moisture (IM) levels of 10, 20, and 30% were defined based on mining applications. The first value of 10% is typical in leaching residues, while 30% is the typical level in tailings.

Two series of experimental test were run, differencing them only in the applying of an electrical field. In both tests act the gravity as one of the forces responsible of the volume of solution drained. In the first one test (Figure 2a), a voltage of 24 [V] between stainless electrodes was applied, following a lineal configuration of electrodes with a separation distance of 20 cm. In the other one test (Figure 2b), no voltage was applied, acting only the gravity as the force responsible of the drained volume of solution. An operating time of 48 hours was determined for each of the tests carried out, whose results are summarized in Tables 2 and 3.

Test	% Fine particles (FP)	Initial Moisture	% Final moisture (FM)		Contribution*
		%, (IM)	FMGD	FMG&EOD	CEOD, (%)
1	10	10	10	10	0
2	10	20	17.53	18.01	3.15
3	10	30	18.27	19.49	6.26
4	15	10	10	10	0
5	15	20	16.65	18.03	7.65
6	15	30	16.81	21.1	20.33
7	20	10	10	10	0
8	20	20	15.89	18.11	12.26
9	20	30	14.8	21.2	30.19

Table 2.

Data series collected from carried out run with electroosmotic and gravitational drainage, following a 3²-factorial design [35].

^*

C_EOD: contribution of EOD to final moisture, $C_{\mathit{EOD}}=\left({\mathit{FM}}_{\mathit{G}\&\mathit{EOD}}-{\mathit{FM}}_{\mathit{GD}}\right)/{\mathit{FM}}_{\mathit{G}\&\mathit{EOD}}$

% Initial moisture	% Fine particles
	A1	A2	A3
B1	0	0	0
B1-replica	0	0	0
B2	12.33	16.76	20.54
B2-replica	11.80	16.95	19.98
B3	39.11	43.96	50.67
B3-replica	36.37	41.20	50.01

Table 3.

Data series collected from carried-out tests run with electroosmotic and gravitational drainage, following a 3² factorial design with replica [35].

For the data series in Table 2, an ANOVA was applied based on the proposed 3² factorial design test to identify the influence of the relevant factors and their interaction (Table 4). Subsequently, a nonlinear regression was applied to obtain a mathematical model.

Source of variation	Sum of square	Degrees of freedom	Quadratic mean	F-ratio	p-Value
A: % Fine material	4.35202	1	4.35202	311.83	0.0001
B: % Initial moisture	65.8691	1	65.8691	4719.68	0.0000
AB	3.01022	1	3.01022	215.69	0.0001
BB	22.8038	1	22.8038	1633.95	0.0000
Residual	0.055825	4	0.0139562
Total	96.0909	8
R^2	99.9419%

Table 4.

ANOVA, using the data series as shown in Table 2 and the software Statgraphics, for the experimental run applying electrical field [35].

The analysis of variance for the quadratic model, with a confidence interval of 95%, showed that four p-values are less than 0.05 (see Table 4), which means that the null hypothesis (zero contribution to the final moisture) is rejected by the four effects (A, B, AB, and BB), while the effect AA on the final moisture is not significant. The resulting statistic model is formulated by Eq. (7) and is graphically shown in Figure 3:

Figure 3.

Final moisture affected by the % of fine material and the % of initial moisture when an electric potential is applied [35].

$$\mathit{\text{Final Moisture}}\left(\%\right)=-6.093+0.177\left(A\right)+1.942\left(B\right)-0.017\left(A\ast B\right)-0.034{\left(B\right)}^2$$E7

If the contribution of electroosmotic drainage on the final moisture level is analyzed, the analysis of variance for the quadratic model, with a confidence of 95%, showed that the three p-values are less than 0.05, which means that three null hypothesis is rejected for the effects A, B, and AB. The effects AA and BB did not significantly affect the final moisture level. The model describing the contribution of electroosmotic drainage, C_eod, to the final moisture is:

$$C_{\mathit{eod}}=9.55-0.47\left(A\right)-1.56\left(B\right)+0.12\left(A\times B\right)$$E8

Figure 4 represents the contribution of electroosmotic drainage on the final moisture level of the sample. The results show that the higher the initial moisture, the greater the reduction of moisture when an electric field is induced. As can be seen, the tests with natural drainage were less efficient. This electroosmotic contribution is greater as the % fine particles of the solid matrix increases.

Figure 4.

Contribution of the electroosmotic drainage technique on final moisture [35].

3.1 Model describing the volume of drained solution from the solid matrix

A 3² factorial design test with replication was used to determine the effect of the factors (fraction of fine particles (FP) = B and initial moisture (IM) = A, notation in Table 2), and their interaction against the drained volume as response variable. Table 3 summarizes data series collected from the carried-out tests.

According to the realized ANOVA, four p-values were less than 0.05, which means that the null hypothesis (zero contribution to the final moisture) is rejected for the effects A, B, AB, and BB on the drained volume, while the effect AA results to be not Significative (see Table 4). The equation representing the electroosmotic model is:

$$V_{\mathit{\text{drained}}}=2.876-0.567\left(A\right)-0.921\left(B\right)+0.063\left(\mathit{AB}\right)+0.054{\left(B\right)}^2$$E9

where V_drained is the drained volume of the solution in cubic centimeter. Figure 5 graphically represents the electroosmotic drainage model.

Figure 5.

Representation of the model for the volume of drained solution from the sample by applying the electroosmotic technique [35].

3.2 Comparison in the drained solution by the two techniques

The dewatering experimental tests realized to compare the electroosmotic technique with the conventional technique (gravitational dewatering) were done in the 20-liter electroosmotic cells at laboratory scale as those shown in Figure 2. The results plotted in Figure 6 shows that the volume of collected solution using the electroosmotic technique may reach three times more than the collected by the gravitational technique (0.32 over 0.1 L, respectively). This difference of 0.22 L may be explained for the high fraction of fine particles contained in the sample material. The capacity of drained solution, defined as the volume of collected solution per mass of sample material, is estimated in 0.046 L/kg, value that varies depending of the fine solid fraction which is directly related to the high volume of water trapped by capillary. Thus, the larger the fraction of fine material in the mining waste, the larger the volume of drained solution by electroosmotic flow.

Figure 6.

Solution collected by electroosmotic and gravitational drainage technique in cells containing 7 kg of dried material with an induced humidity of 15% [35].

4. Flux of water for the case of unsaturated medium

For the case of a saturated medium, the conductivity (K), associated to the fluid properties, density and viscosity, and of the soil, intrinsic permeability (k), is constant. But when the medium is partially saturated it is not, because the permeability (k) varies with the saturation degree (S_e). In the case of an unsaturated medium, as the pore space is occupied by two fluids (in this study water and air), the conductivity coefficient (K) is redefined by an effective permeability coefficient (K_e) related to one fluid. The subscript “e” to the coefficient K indicates that the conductivity is not necessarily that of a medium occupied by only one fluid. Usually, the effective conductivity (K_e) will be smaller than the conductivity (K) when only one fluid occupies the totality of the porous medium cavities, at saturated conditions. Then, the ratio K_e/K_sat is defined as the relative permeability (k_r) which reaches its maximum value of 1 at saturation (the cavities or the porous of the soil are fully with water). Then, rewriting the mass flux Eq. (6) for the effective conductivity (K_e) in terms of the relative permeability (k_rel) and the conductivity at saturated conditions (K_sat) gives

$$J_w={\rho }_w\left(q_w^h+q_w^{\mathit{eo}}\right)=-{\rho }_w\left[K_{\mathit{sat}}^h{k}_{\mathit{rel}}^h\nabla \left(\Psi +z\right)+K_{\mathit{sat}}^{\mathit{eo}}{k}_{\mathit{rel}}^{\mathit{eo}}\nabla \left(\varnothing \right)\right]$$E10

where $K_{\mathit{sat}}^h\mathit{\text{and}}{k}_{\mathit{rel}}^h$ are the saturated conductivity and the relative hydraulic permeability of the soil, respectively, and $K_{\mathit{sat}}^{\mathit{eo}}\mathit{\text{an}}{k}_{\mathit{rel}}^{\mathit{eo}}$ are the electroosmotic saturated conductivity and the relative electroosmotic permeability of the soil, respectively. Then, both hydraulic and electroosmotic conductivities are defined as products of a permeability, depending on the microstructural properties of the porous medium, and a nondimensional relative permeability coefficient, which accounts for the effects of partial saturation on soil.

Relative permeabilities are functions of the saturation degree (S_e), and the estimates of them are especially difficult to obtain, partly and mainly because of its extensive variability in the field. Models have been used for calculating the unsaturated conductivity from the more easily measured soil-water retention curve, and analytical expressions have also been developed [36, 37]. To simplify the analysis of the present work, the model formulated by Eqs. (11) and (12), defined by the power law of the effective saturation, is adopted for the relative hydraulic and electroosmotic permeabilities, respectively [38]:

$$k_{\mathit{rel}}^h=a_h{\left(S_e\right)}^{b_h}$$E11

$$k_{\mathit{rel}}^{\mathit{eo}}=a_{\mathit{eo}}{\left(S_e\right)}^{b_{\mathit{eo}}}$$E12

where a_h, a_eo, b_h and b_eo are numerical curve fitting parameters. For a wide variety of soils, proposed values of b_h, for hydraulic flow, vary between 3 and 3.5 [34, 38, 39, 40]. For the case of compacted clay material, a value of b_h = 5 was suggested [41].

For the case of electroosmotic flow, the dependence of the saturation degree on the electroosmotic permeability was experimentally investigated in electrokinetic filtration tests [43]. The experimental results compare well with the results reported in Figure 7 [43].

Figure 7.

Relative electro-osmotic permeability vs. saturation. The data adjusts well to the model expressed by eq. (12), with a value of 1 for a_eo, and a value of 3.2 for b_eo [38, 42, 43].

Rewriting Eq. (10) for the total mass flux, by inserting Eqs. (11) and (12) gives

$$J_w=-{\rho }_w\left[K_{\mathit{sat}}^h{\left(S_e\right)}^{b_h}\nabla \left(\Psi +z\right)+K_{\mathit{sat}}^{\mathit{eo}}{\left(S_e\right)}^{b_{\mathit{eo}}}\nabla \left(\varnothing \right)\right]$$E13

4.1 Governing equations of electroosmotic flow in a porous media partially saturated

Unsaturated flow conditions in porous media correspond to a particular case of two-phase flow. Generally, the “non-wetting” phase, which in this case corresponds to the gas phase, is such that its pressure is constant, thus, this phase is considered as a stagnant or inactive phase.

4.1.1 Balance of the drained water mass flux

For the case of a porous media under the application of an electrical potential, the balance of the mass of water per unit volume in a porous medium is given by the expression:

$$\frac{\partial m_w}{\partial t}=\nabla J_w$$E14

where J_w is the total mass flux of water (kg m⁻² s⁻¹), defined by two contributions, hydraulic and electroosmotic flow. The mass of water, m_w, per unit of total volume (kg m⁻³) is defined as:

$$m_w={\epsilon }_p{S}_e{\rho }_w$$E15

where ε_p is the porosity of the soil, ρ_w is the water density, and S_e is the effective saturation. S_e is defined as the fraction of the total pore space occupied by the wetting phase, fraction dependent of the pressure head, and that can vary from point to point within the medium. The S_e is defined as:

$$S_e=\frac{\left(\theta -{\theta }_r\right)}{\left({\theta }_s-{\theta }_r\right)}$$E16

with θ as the volumetric water content, defined as the volume of water per unit volume of soil (m³/m³), θ_s as the water content at saturation, and θ_r as the residual water content. By inserting Eq. (13), for the total mass flux, and Eq. (15), for the mass of water in the porous, into Eq. (14), the following equation is obtained:

$$\frac{\partial \left({\epsilon }_p{S}_e\right)}{\partial t}+\nabla \left[K_{\mathit{sat}}^h{\left(S_e\right)}^{b_h}\nabla \left(\mathrm{\Psi }+z\right)+K_{\mathit{sat}}^{\mathit{eo}}{\left(S_e\right)}^{b_{\mathit{eo}}}\nabla \left(\varnothing \right)\right]=0$$E17

The conductivity coefficient, K_sat, is constant and determined at saturation conditions. The saturation degree, S_e, is dependent of the pressure head, ψ, and to solve this equation it is necessary also to solve the continuity equation for the current density (J_e) that is obtained by doing a charge electric balance.

4.1.2 Balance of electric charge

In the presence of an external electric field, the balance of electric charge equation is expressed as:

$$\nabla ·J_e+\frac{\partial Q_e}{\partial t}=0$$E18

where J_e is the electric current density and Q_e is the charge density per unit volume. Neglecting the soil electric capacitance, the source term in Eq. (18) vanishes and the equation reduces to:

$$\nabla ·J_e=0$$E19

Neglecting the contribution of streaming currents produced by liquid flow, the equation for the electric current density, J_e, is provided by the Ohm’s law for the porous medium:

$$J_e=-k_e\nabla \varnothing$$E20

where k_e is the effective electric conductivity of the unsaturated porous medium that can be expressed as the sum of two terms [44]:

$$k_e=k_{\mathit{es}}+k_{\mathit{\text{esat}}}{k}_{\mathit{\text{erel}}}$$E21

where k_es is the apparent electric conductivity of the solid and generally assumed negligible, k_esat is the apparent electric conductivity of bulk pore water under saturated conditions, and k_erel is the relative electrical conductivity, accounting for the effects of partial saturation. The following power law has been assumed in this work [44, 45]:

$$k_{\mathit{\text{erel}}}=a_e{\left(S_e\right)}^{b_e}$$E22

where a_e and b_e are fitting parameters, with a_e = 1 and b_e = 2, as suggested by literature experimental data [45, 46]. Then, by inserting Eqs. (20)–(22) into Eq. (19), the following equation is obtained:

$$\nabla ·k_{\mathit{\text{esat}}}{\left(S_e\right)}^{b_e}\nabla \varnothing =0$$E23

4.1.3 Approach of the effective Saturation

Then, to solve the system of Eqs. (17) and (23), it is necessary to find some mathematical model that correlates the dependence of S_e with the pressure head. In scientific literature, numerous models of soil-water retention curve and permeability have been reported, showing that dependence [36, 46]. Then the effective saturation, S_e, could be estimated using van Genuchten’s retention curve model formulated as:

$$S_e={\left[\frac{1}{1+{\left(\alpha ·\Psi \right)}^n}\right]}^m$$E24

where “α”, “n”, and “m” are fitting parameters, and “ψ” is the matric suction defined by the pressure difference at the interface between the gas (air) and liquid (water) and named as pressure head. Combining Eqs. (16) and (24), the following equation is obtained:

$$\theta ={\theta }_r+\frac{\left({\theta }_s-{\theta }_r\right)}{{\left[1+{\left(\alpha \mathrm{\Psi }\right)}^n\right]}^m}$$E25

where, the pressure head, ψ, is positive, and the parameters “m” and “n” are correlated by Mualem’s model [45] as:

$$m=1-\frac{1}{n}$$E26

Eq. (25) contains four independent parameters (θ_r, θ_s, α, and n), which must be estimated from observed soil-water retention data. A graphical methodology may be found in a paper written by Van Genuchten (1980) [45].

The functional relationship of S_e, Eq. (24), used to estimate the relative permeability of the unsaturated porous media, has not a unique value because it is affected by hysteresis. The value of k_rel is different if S_e is obtained by starting with a dry medium and increasing the liquid saturation than if S_e is obtained by removing liquid from an initially vacuum-saturated medium. Hereby, as this paper is concerned with the drainage of liquid, by the moment, the hysteresis problem will be not considered, only as a way of simplifying the estimate of k_rel.

5. Application of the electroosmotic flow model

Once the copper is extracted from the ore through a heap leaching process, an acid-wetted solid waste is left with a 15–18% moisture. At the beginning of the leaching cycle, the agglomerated mineral has a good permeability, approximately 10⁻⁴ cm/s, but at the end, because of the drag of fine material (mainly clay or some precipitate generated during the leaching cycle) by the leaching solution, the permeability at the bottom of the heap become lower, making not easy the drainage of solution by gravity. The application of the electroosmotic model will be done to a copper leaching heap in its last stage of the leaching process, in the drainage step before to dispose the waste material left after the copper extraction.

The drainage electroosmotic model formulated by Eqs. (14)–(26) describes the partial saturated heap with a volumetric water content, θ, varying with height of the heap, showing larger water content at the bottom than at the top of the heap, $\mathit{at}t=0,H=-\left(y+2\right)$, where (H) is the hydraulic head and (y) is the independent variable of position related to the y-axis. The hydraulic head H is related to the water volumetric content θ by developing Eqs. (5) and (25). The initial electric field is assumed to be zero everywhere: $\mathit{at}t=0,V=0$. The boundary conditions (BCs) are no flow across the boundary top and through the right vertical side of the heap pile. At the left wall, a symmetry boundary is assumed (Figure 8 shows the sketch of the 2D pile). At the bottom, the boundary condition is open and free draining. For the electric BC, the voltages are maintained at the top and bottom, whereas all other boundaries are assumed to be isolated to the electric current. The parameters required by the model are listed in Table 5.

Figure 8.

Volumetric water content, in cm³ of water per cm³ of soil, varying between 0 and 1, of the leaching pile in 2D, at (a) initial time; and (b) time of 480 minutes. (Software Comsol Multiphysics).

Parameter	Value	Description	Unit
ɛp	0.1–0.7	Porosity	—
θr	0.13	Residual volumetric water content	—
θs	0.70	Volumetric water content at saturation	—
αcr	1.70	Parameter of the water retention curve model	1/m
ncr	1.38	Parameter of the water retention curve model	—
Ksath	10^−6–10^−11	Hydraulic conductivity at saturation	m/s
Ksateo	10^−9–10^−8	Electroosmotic conductivity at saturation	m^2/V/s
Ksate	3.5 × 10^−4	Electric conductivity at saturation	S/m
sign	1 or − 1	Sign of the zeta-potential	—
bh	3.5	Parameter of hydraulic relative conductivity model	—
beo	3	Parameter of electroosmotic relative conductivity model	—
be	2	Parameter of electric relative conductivity model	—
ρb	1012	Bulk density of solution	Kg/m^3
σe	0.01–1.0	Electric conductivity coefficient of the soil	S/m
Vanode	200	Anode voltage	V
Vcathode	0	Cathode voltage	V

Table 5.

Parameters utilized in the simulations for the case of a heap leaching.

The results of the simulation, including hydraulic and electroosmotic flow, are represented in Figure 8 for the volumetric water content θ at initial time and after 480 minutes. These results are only demonstrative, because the data used in the simulation are only slightly approximated to the studied case. In addition to the volumetric water content, Figure 8 shows the schematic view of the 2D pile to be simulated (8 meters wide and 2 meters high). The anode and cathode bars, represented by circles, locate at the top and bottom, respectively.

The simulation results represented in Figure 8 show the dewatering evolution of the heap leaching in a situation where the mineral bed presents a very low hydraulic permeability and high electroosmotic conductivity (see Table 5). This means that the relevant driving force, responsible of the drainage process, is the electroosmotic force. This situation may be realistic only if during the leaching process, it exists a great drag of fine material toward the bottom of the pile that could difficult the drainage of solution from the pile. Due to the lack of field data, (permeability, porosity, volumetric water content, and soil-water characteristic curve (SWCC)) for the studied case, it is not possible to make a comparison with results obtained from an experimental pilot test of one cubic meter of acid mining waste, realized and documented previously [11].

6. Conclusion

The two-phase fluid flow in porous media may be used to explain the drainage from a residue with high humidity, taking into consideration two driving forces, defined by the hydraulic and electric potentials. Electroosmotic flow is significative only when water is trapped capillary, as in those samples where the fraction of fine material is considerable. In an electroosmotic process, the movement of the water is from the anode toward the cathode; thus, the soil water content will decrease at the anode and will increase at the cathode. Thus, to discharge the water, a drainage system has to be provided at the cathode proximity.

To apply the electroosmotic model in the dewatering process of mining waste, it is necessary to know beforehand some hydraulic and electrical parameters such as permeability, porosity, volumetric water content, and the soil-water characteristic curve.

Acknowledgments

The authors thank RMV Ltda. and the UCN (Universidad Católica del Norte) for supporting this work, funded by a CONICYT Fondecyt Initiation Grant No. 11180329—Phenomenological aspects of electroosmotic drainage technique: application to copper leaching.

Conflict of interest

The authors declare no conflict of interest.