Development of Precursory Non-Segregation Criteria for Hard Rock Mine Tailings Slurries: Integration of Flume Testing and Buckingham π Dimensional Analysis

Abstract

Natural lateral particle segregation commonly occurs during the hydraulic deposition of slurry and thickened tailings in surface tailings storage facilities (TSFs), producing spatial heterogeneity in physical, hydrogeotechnical, and mineralogical properties, as well as in the water table. In sulfide-rich tailings, such heterogeneity complicates the design of reclamation cover systems, which are themselves affected by it. This study investigates the impact of physical and rheological properties of hard-rock mine tailings slurries on their segregation under hydrodynamic conditions. It proposes a multiparametric equation for the segregation index (SI) based on Buckingham’s π theorem. For this purpose, six flume experiments were conducted using tailings with initial solid mass concentrations of 63%, 66%, and 69% at slopes of 0.5% and 1%. Results revealed strong exponential correlations (R² > 0.95) between SI and tailings’ physical properties (solid concentration, bulk density) as well as rheological parameters (Herschel–Bulkley yield stress and flow index, Cross infinite dynamic viscosity). The SI equation was developed using MATLAB R2025b nonlinear least-squares optimization with a trust-region reflective algorithm. Using an SI threshold of 0.05 to define non-segregating behavior, the proposed model can predict segregation tendencies as a function of tailings properties and slope conditions. Further laboratory and field investigations are needed to validate and generalize the criterion.

1. Introduction

Lateral particle segregation is a phenomenon by which certain particles with similar properties tend to preferentially collect in one physical zone or another of an assembly. In the case of mine tailings, lateral particle segregation can occur when hydraulically discharged as a slurry in a tailings storage facility (TSF), creating a spatial distribution of the grain size distribution (GSD) as a function of distance from the discharge point. Three zones can then be distinguished: a beach zone with coarser particles near the discharge point, a decanting zone with finer particles, and a transition zone between the last two zones [1,2]. In addition to GSD, segregation of slurry tailings can generate spatial heterogeneity in their physical properties (specific gravity G_s), GSD-controlled hydrogeotechnical properties (saturated hydraulic conductivity and the water retention capacity), and mineralogical properties.

In sulfide-rich tailings with high acid mine drainage (AMD) potential, this mineralogical lateral heterogeneity can also extend to the material’s reactivity and oxygen consumption capacity [3]. The design of efficient cover systems that serve as oxygen and water barriers for the TSF reclamation becomes complex because it must account for the heterogeneities mentioned above [3,4]. This design complexity could be alleviated by discharging non-segregating tailings, resulting in tailings properties that are relatively homogeneous throughout the TSF.

Achieving non-segregating tailings requires a comprehensive understanding of the primary factors governing particle size segregation during hydraulic deposition within TSFs. Generally, the mechanisms that govern the particle segregation process are still not well documented, owing to the complex nature of the phenomenon, where particle features (such as particle size and distribution, shape, and specific gravity), rheological behavior (non-Newtonian fluids), and hydrodynamic conditions (flow velocity) contribute to the complexity. Characterization of the rheological behavior of non-Newtonian fluids then becomes very important from an engineering perspective. Since the mid-1990s, Australian researchers have established relationships between solid mass concentration (mass of solids/total mass) and tailings rheology [5,6,7]. Zhang et al. [8] correlated particle-size distribution and solid mass concentration (C_w, 72–78%) with tailings rheology, demonstrating that the shear yield stress (τ_y) increases quadratically and the dynamic viscosity (η) increases linearly with C_w.

Flume tests are widely used for tailings beach slope prediction [9,10,11,12,13,14,15,16,17,18,19], segregation during tailings deposition [12,18,20,21,22,23,24], and yield stress estimation of non-Newtonian slurries [25]. A summary of the flume characteristics, tested material, and principal results from a few studies is provided in Appendix A. Tests were conducted with flumes of lengths 10, 12.2, and 20 m for 3, 1, and 1 channels, respectively; the remaining flumes were shorter than 10 m. The shortest length was 1.5 m.

The nearly one-dimensional segregation observed in short-length flume tests may not be representative of flow behavior in tailings storage facilities (TSFs). Unfortunately, few studies have evaluated the extent of in situ segregation of flowing tailings. Maqsoud et al. [26] investigated the rheological and hydrogeological behavior of flowing, thickened gold tailings (C_w-ini = 69%) using a field-scale experimental cell. The grain size distributions and rheological properties of the ten samples collected along the flow path were highly similar, indicating the absence of segregation and confirming homogeneous deposition. Rheological property analysis showed that the yield stress obtained using the Herschel–Bulkley model was approximately 8 Pa.

Despite these laboratory and field segregation studies, limited attention has been given to developing a predictive criterion for particle lateral segregation that considers both physical and rheological properties. However, in the case of vertical segregation simulated by static sedimentation tests, various researchers used a segregation index (SI) for this purpose [21,27]. Chalaturnyk and Scott [26] introduced the concepts of average solid mass concentration (C_w-avg) and a segregation index (SI) to characterize this segregation boundary. Mihiretu [21] proposed the segregation index SI, defined with Equation (1), as a quantitative method for assessing particle segregation in static sedimentation tests

$$SI=\sqrt{{\displaystyle \frac{\sum H_i{(C_{w-i}-C_{w-avg})}^2}{\sum H_i}}}$$

(1)

where $H_i$ = sample height, C_w-i = mass concentration of the sample section i, C_w-avg = average solid mass concentration of the samples.

Mihiretu [21] suggested that the above-mentioned SI approach and criterion could also be applied to dynamic segregation evaluation (i.e., flume tests). In the case of a flume test, Equation (1) can be rewritten as follows:

$$SI=\sqrt{{\displaystyle \frac{\sum L_i{(C_{w-i} -{ C}_{w-avg})}^2}{\sum L_i}}}$$

(2)

where $L_i$ = length of section i along the flume (m), C_wi = solid content at the end of section i, and C_w-avg = average solid content along the flume length.

Given that no unified model exists to predict particle segregation while accounting for all relevant physical and rheological factors, this paper develops preliminary uni- and multi-parametric non-segregation criteria for hard rock mine tailings slurries. Uniparametric criteria are based on the relationships between the segregation index (SI) and the initial properties of the tailings. The selected initial solid mass concentrations of 63%, 66%, and 69% were chosen to represent the typical concentration range commonly encountered in thickened hard rock mine tailings. Furthermore, results obtained from flume tests and dimensional analysis using Buckingham’s π theorem are used to develop the multi-parametric criterion that includes rheological properties (shear yield stress and viscosity), solid content, and the flume slope and length.

2. Materials and Methods

2.1. Laboratory Flume Setup

Figure 1 shows the schematic drawing of the laboratory flume setup constructed to examine the depositional behavior of hard rock mine tailings. The flume consists of six sections, fabricated from polytetrafluoroethylene (PTFE), designed by the authors and manufactured by Plastiques G+, Rouyn-Noranda, QC, Canada, also known commercially as Teflon. Each section has a length (L_i) of 2.44 m, resulting in a total flume length (L_tot) of 14.63 m. The internal dimensions of the channel are 0.52 m wide and 0.23 m deep.

The flume was assembled outside the laboratory, where the ground surface was not level. A metal support structure with height adjustment capability was used to stabilize the channel (Figure 2a). To achieve the target slope (0.5% or 1%) along the 14.63 m-long flume, the system first required precise leveling to a zero-slope using a Topcon RL-H4C laser level, Topcon Corporation, Tokyo, Japan. The slope of the flume was then adjusted to the desired inclinations (Figure 2b). Silicone sealant was applied along the joints between the flume panels to ensure watertightness and prevent leakage (Figure 2c), and leakage tests were performed by filling the flume with tap water. During the experiments, no noticeable particle accumulation or abnormal deposition behavior was observed at the joint locations.

2.2. Experimental Procedure and Program

The following steps were carried out before conducting the flume tests. The tailings received in 200 L barrels were homogenized using a pump-mixer system (Figure 2d), with water added to achieve target initial solid mass contents (C_w) of 63%, 66%, and 69%. For this purpose, supernatant water was first carefully removed from the top of each barrel and transferred into buckets. Then, the highly concentrated tailings (C_w = 80%) were poured into a mixing pump chamber with the aid of a shovel. Due to the high initial solid content, supernatant water was gradually added until the mixture reached the target solid content C_w.

The amount of water required to achieve the desired solid mass concentration for each flume test was calculated using the following relationship.

$$M_{w\_add}=M_{Ti}\times \left({\displaystyle \frac{C_{w-s}}{C_{w-f}}}-1\right),$$

(3)

where M_{w_add} is the mass of water to be added (kg), M_Ti is the initial total mass of tailings (kg), C_w-s is the starting solid mass concentration (%), and C_w-f is the final solid mass concentration (%), with C_w-s > C_w-f.

The homogenization process lasted approximately 3 h. The homogenization duration of approximately 3 h was selected based on preliminary mixing trials to ensure uniform slurry consistency before sampling and testing. Once homogenized, samples were taken to verify the target solid mass concentration. Then, the tailings were pumped from the mixing unit into a small upstream compartment and distributed across the full width of the flume (Figure 2e). For performing the flume test, four people were required to control the operating speed of the pumps, to hold the discharge tube to the flume (with an internal diameter of 38 mm), to take samples, and to keep a record of time for flow velocity (v) measurement, respectively. To calculate the flow velocity of the tailings along the flume, a time-of-arrival method was employed: the time at which the tailings reached the end of each flume section was recorded with a stopwatch. By calculating the time difference between the arrival of the tailings at consecutive sections, the average velocity (v_m) within each section was determined by dividing the section length (2.44 m) by the corresponding travel time. The temperature (°C) of the tailings was also continuously monitored along the flume using a thermocouple probe during the tests (see datalogger in Figure 2e). Tailings samples were collected at 2.44 m intervals along the flume (Figure 2f) for physical and rheological characterization. The number of samples collected depends on the distance the tailings travel. For example, with C_w-ini = 69%, the flow stopped at a distance L = 11.6 m. Figure 3 is an illustrative image of the flume test at the end of the flow test. After each test, the flume was cleaned thoroughly to prepare it for the next experimental run (Figure 2g).

Table 1. Experimental program.

Test #	Slope (%)	Cw-ini (%)	Mass of Tailings Prepared (kg)	Measured Density (g/cm3)
1	0.5	63	712	1.67
2		66	731	1.75
3		69	740	1.78
4	1.0	63	649	1.67
5		66	711	1.73
6		69	695	1.79

summarizes the experimental program and lists the properties of the tailing samples used for the six flume tests that were conducted at ambient air temperatures ranging between 21 and 23 °C.

2.3. Physical and Rheological Characterization of Tailings Samples Collected Along the Flume

The physical properties determined include the solid mass concentration, bulk density, specific gravity, and particle size distribution (PSD). The solid mass concentration, C_w, was derived from the water content (w) obtained after drying the samples in an oven at 70 °C until a constant dry mass was reached. All samples were maintained in the oven for approximately 72 h until a constant dry mass was achieved. The drying temperature of 70 °C is a standard practice in our laboratory, aiming to preserve the natural mineralogy of mine tailings. The bulk or total density (ρ), specific gravity (G_s), and grain-size distribution were obtained using a mud balance (Metal Mud Balance, OFI Testing Equipment, Inc., Houston, TX, USA), a helium pycnometer (Ultrapyc 1200e, Quantachrome Instruments, Westborough, MA, USA), and a laser particle size analyzer (Panalytical Mastersizer 3000, Malvern Panalytical Ltd., Malvern, UK), respectively. All these characterizations were realized in triplicate.

Rheological tests were conducted in triplicate using an AR2000 rheometer (TA Instruments, New Castle, DE, USA) with a vane geometry (rotor) measuring 28 mm in diameter and 42 mm in length. The vane was immersed in a cylindrical container (stator) (30 mm in internal diameter and 79 mm in height), with a lateral gap of 1 mm and a bottom gap of 4 mm (Figure 4). Before testing, instrument inertia was calibrated, and temperature was maintained at 21 °C with the help of a Peltier temperature control system. Sample density was used to prepare the required 28.72 mL of slurry. Controlled shear rate tests were performed by applying a steady-state flow step procedure over shear rates ($\dot{\gamma }$) ranging from 0.1 to 100 s⁻¹ (upward ramp) and from 100 to 0.1 s⁻¹ (downward ramp), and measuring the shear stress (τ).

Experimental data consisted of rheograms (or flow curves) τ($\dot{\gamma }$) and dynamic viscosity curves η($\dot{\gamma })$, where η = τ/$\dot{\gamma }$. These curves were subsequently fitted with rheological models available in the analysis software (Rheology Advantage, Rheology Data Analysis, V 5.8.2.0). To adjust the flow curves, the generalized Herschel–Bulkley flow model (Equation (4)) was applied.

$$\tau ={\tau }_{0-HB}+k{\dot{\gamma }}^n.$$

(4)

In this equation, τ is the shear stress (Pa), τ_0-HB is the shear yield stress (Pa), i.e., the minimum stress to be applied to initiate flow, $\dot{\gamma }$ is the shear rate (1/s), $n$ is the flow index, and k is the consistency index $\left({\mathrm{P}\mathrm{a}·\mathrm{s}}^n\right)$ (related to the viscosity of the fluid). Fluids exhibit a shear thinning behavior when n < 1, a shear thickening behavior when n > 1, and a Bingham behavior for n = 1. For analyzing the viscosity curves, the Cross rheological model was applied (Equation (5)):

$$\eta ={\eta }_{\mathrm{\infty }}+{\displaystyle \frac{{\eta }_0-{\eta }_{\mathrm{\infty }}}{1+(k_c\dot{\gamma }{)}^{n_c}}},$$

(5)

where $\eta$ is the dynamic viscosity (Pa·s), ${\eta }_{\infty }$ is the dynamic viscosity at infinite or sufficiently high shear rate (Pa·s), ${\eta }_0$ is the initial dynamic viscosity (Pa·s), $k_c$ is a constant, and $n_c$ is the flow index specific to the Cross model.

The goodness of fit between the experimental and predicted curves was evaluated using the standard error (SE) (Equation (6)), which was calculated by the data analysis software through the least-squares method. According to the TA instruments manual, the standard error is considered acceptable if it is less than 20‰.

$$SE(‰)={\displaystyle \frac{\sqrt{\frac{\sum (x_m-x_c{)}^2}{N-2}}}{\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}}}\times 1000,$$

(6)

where $x_m$ is the experimental value, $x_c$ is the rheological model value, $N$ is the total number of measured data points, and Range is the difference between the maximum and minimum measured values.

3. Results

In the following, distributions of physical and rheological properties of the tailings along the flume length are presented. It is worth noting that the average flow velocity was 0.05 m/s, 0.03 m/s, and 0.02 m/s for C_w-ini = 63%, 66%, and 69%, respectively, for the flume with a 0.5% slope. For the flume with a slope of 1%, the average velocity was 0.1 m/s, 0.06 m/s, and 0.03 m/s for C_w-ini = 63%, 66%, and 69%. Flow velocity is not constant along the flume and is influenced by the initial properties of the tailing and the slope of the flume. The average temperature of the tailings’ slurries ranged from 19 to 23 °C. In the case of a slope of 0.5%, the times of deposition were 270 s, 540 s, and 589 s for C_w-ini = 63%, 66%, and 69%, respectively. In the case of a slope 1%, the times of deposition were 193 s, 437 s, and 477 s for C_w-ini = 63%, 66%, and 69%, respectively.

3.1. Distribution of Particle Size Along the Flume

Particle size distribution (PSD) curves as a function of sampling distance (L) along the flume with a 1% slope are presented in Figure 5 for tailings with solid contents (C_w-ini) of 63%, 66%, and 69%. The sample collected at a distance of 0 corresponds to the original tested tailings. By comparing the PSD curves along the flume, a noticeable difference in segregation behavior with respect to the solid content can be observed. At the highest tested C_w-ini of 69% (Figure 5a), the PSD curves at various distances nearly overlap, indicating a homogeneous and non-segregating flow where particles remain uniformly distributed along the flume. When the C_w-ini decreases to 66% (Figure 5b), slight divergence appears among the PSD curves, suggesting moderate segregation with coarser particles tending to settle near the upstream section. At a C_w-ini of 63% (Figure 5c), the variation between the PSD curves along the flume length becomes more pronounced, showing clearer particle-size separation along the flow path, where finer particles are transported further downstream. It should be mentioned that the variations in GSD along the flume with a slope of 0.5% (PSD not provided), for a given solid content, were less pronounced than those observed for a flume slope of 1%.

Figure 6a shows the variation in median particle size (D₅₀, corresponding to 50% passing) along the flume length for different initial solid mass concentrations (C_w-ini = 63%, 66%, and 69%). In all cases, D₅₀ decreases progressively with distance, indicating downstream particle segregation during slurry flow. The reduction is more pronounced at a lower solid concentration (63%), whereas higher concentrations (69%) exhibit relatively smaller changes in D₅₀, suggesting more homogeneous particle transport at higher solid contents. Figure 6b illustrates the variation in the fine particle fraction (P₈₀ µm) along the flume length for different initial solid mass concentrations (C_w-ini = 63%, 66%, and 69%). In all tests, P₈₀ µm increases progressively with distance, indicating downstream enrichment of fine particles due to segregation during slurry transport. The increase is most significant at the lowest solid concentration (63%), while higher concentrations show smaller variations, suggesting reduced segregation intensity and more uniform particle distribution at higher solid contents.

3.2. Distribution of Solid Content and Bulk Density Along the Flume

Figure 7 presents the variations in the solid content (C_w) and bulk density (ρ) along the flume for slopes of 0.5% and 1% (marks) together with the fitted linear trend curves (lines). It should be mentioned that these two parameters are interrelated and can be expressed through a mathematical relationship. As shown in Figure 7a,c, the solid mass concentration decreases progressively along the flume for all tests, with a steeper decline at lower initial C_w-ini.

At C_w-ini of 69%, the variation is minimal (0.32 for 0.5% slope and 0.27 for 1% slope). In contrast, at C_w-ini of 63%, the sharp reduction downstream (1.14 for a 0.5% slope and 0.87 for a 1% slope) reflects stronger particle segregation and greater slurry dilution. This trend shows that lower C_w-ini promotes higher segregation along the flow path. Based on Figure 7b,d, the bulk density (ρ) decreases gradually along with the flume for all C_w-ini, with the reduction becoming more pronounced at lower C_w-ini values. At C_w-ini = 69%, the bulk density remains nearly constant (variation of 0.005 for 0.5% slope and 0.003 for 1% slope), indicating limited segregation. In contrast, at C_w-ini = 63%, a sharp downstream decline (0.017 for a 0.5% slope, and from 0.015 for a 1% slope) reflects significant particle separation and the transport of finer, less dense material. Regarding the influence of the slope, Figure 7 and the above given values for C_w and ρ at upstream and downstream of the flume indicate that the variation of C_W and ρ with respect to the sampling distance along the flume is more pronounced for a 1% slope than for a 0.5% slope.

3.3. Distribution of Rheological Properties Along the Flume

Figure 8a,b display typical flow and viscosity curves obtained from the rheological tests (marks), together with the fitted curves based on the Herschel–Bulkley and Cross models (lines) for the flume test conducted with tailings at C_w-ini = 63% and slope of 0.5%. As shown in Figure 8a, the measured shear stress ranges from approximately 0.5 to 5.8 Pa over a shear rate range of 10 to 100 s⁻¹. The results show that shear stress decreases progressively with the sampling distance along the flume for a given shear rate. Indeed, near the discharge point (distance L = 0 m), shear stress values are the highest (2.0 Pa at 10 s⁻¹ and 5.8 Pa at 100 s⁻¹). As the flow advances downstream, shear stress drops significantly to around 0.5 Pa at 10 s⁻¹ and 1.0 Pa at 100 s⁻¹ at L = 14.63 m, reflecting particle segregation and increased water content. Based on Figure 8b, for C_w-ini = 63% with a flume slope of 0.5%, the viscosity consistently decreases with both increasing shear rate and sampling distance along the flume. Near the discharge point (L = 0 m), the viscosity values range from approximately 0.20 Pa·s at ~10 s⁻¹ to about 0.05 Pa·s at 100 s⁻¹. Farther downstream (L = 14.63 m), viscosity drops to approximately 0.03 Pa·s at ~10 s⁻¹ to about 0.01 Pa·s at ~100 s, indicating a more diluted and less resistant flow due to particle segregation and water enrichment along the flume. The flow and viscosity curves were fitted using the Herschel–Bulkley and Cross rheological models, respectively (see the lines in Figure 8).

Figure 9 presents the boxplots of triplicate rheological test results along the sampling distance at a 0.5% slope, showing the minimum and maximum values, the 25th and 75th percentiles, with the median line splitting them, and the mean values of the shear yield stress from the Herschel–Bulkley model and the dynamic viscosity at an infinite (or sufficiently high) shear rate from the Cross model. For all cases, both shear yield stress and viscosity decrease exponentially with increasing sampling distance, indicating a clear reduction in rheological resistance downstream. At a higher solid concentration (C_w-ini = 69%), average shear yield stress values are the largest (around 9 Pa near L = 0 m, decreasing to ~2.5 Pa at L = 14.63 m) and the average viscosity η_∞ drops from ~0.14 Pa·s to ~0.05 Pa·s. Conversely, at lower solid concentration (C_w-ini = 63%), the average shear yield stress decreases gradually from ~2 Pa to ~0.25 Pa, and the average viscosity η_∞ drops from ~0.04 Pa·s to ~0.01 Pa·s.

Figure 10 presents the boxplot of triplicate rheological test results along the sampling distance for a 1% slope, showing the minimum, maximum, and mean values of the shear yield stress from the Herschel–Bulkley model, as well as the dynamic viscosity at an infinite (or sufficiently high) shear rate from the Cross model.

For the three solid concentration values, both shear yield stress and viscosity decrease exponentially with sampling distance. The reduction is most pronounced for C_w-ini = 63%, where average shear yield stress decreases from about 1.47 Pa near the discharge point to below 0.21 Pa at the end of the flume, and average viscosity falls from roughly 0.03 Pa·s to 0.01 Pa·s. In contrast, for C_w-ini = 69%, both parameters remain higher (the average shear yield stress decreases from 7.32 Pa to 1.97 Pa; the average viscosity decreases from 0.13 Pa·s to 0.03 Pa·s).

4. Development of Precursory Non-Segregation Criteria

Uni- and multi-parametric non-segregation criteria are proposed in the following. Uniparametric criteria represent the threshold values of each initial physical and rheological property of the tailings above which segregation does not occur. In contrast, the multi-parametric criterion combines all these properties into a single equation.

4.1. Uni-Iparametric Non-Segregation Threshold Limits

Figure 11 illustrates the relationship between the segregation index (SI) calculated using Equation (2) for the two flume slopes (0.5% and 1%) and various physical and rheological parameters of the tailings. In all cases, the segregation index (SI) decreases exponentially with increasing values of the studied parameters (the initial solid mass concentration (C_w-ini), bulk density (ρ_ini), Herschel–Bulkley shear yield stress (τ_0-HB-ini), infinite dynamic viscosity (η_∞-ini)). The exponential fits yielded coefficients of determination (R-squared) greater than 0.95 in all cases confirming the strong predictive correlation between SI and the examined physical and rheological parameters. The dashed line represents the approximate threshold between segregating and non-segregating behavior (SI = 0.05). Based on the results presented in this figure, segregation threshold limits can be determined for each parameter. These threshold values define the parameter ranges beyond which segregation no longer occurs. These limits may be identified graphically by locating the intersection between the curve corresponding to SI = 0.05 and the function SI = f (studied parameter). For the slope of 0.5%, the tailings tested would be non-segregating when the initial solid mass concentration (C_w-ini), bulk density (ρ_ini), Herschel–Bulkley shear yield stress (τ_0-HB-ini), infinite dynamic viscosity (η_∞-ini) is approximately higher than 63% (Figure 11a), 1.68 g/cm3 (Figure 11b), 2.0 Pa (Figure 11c), and 0.04 Pa·s (Figure 11d), respectively. For a slope of 1%, the tested tailings exhibited non-segregating behavior within the investigated solid concentration range of 63% to 69%. For a slope of 1%, the tailings tested would be non-segregating across all tested solid contents. Since segregation depends not only on the solid concentration but also on rheological properties and hydrodynamic conditions, single-parameter criteria are inherently specific to given tailings. They, therefore, cannot be considered universally applicable segregation criteria. A multi-parameter criterion, such as the one developed below, would provide a more robust and generalized framework for assessing segregation behavior.

4.2. Multiparametric Non-Segregation Threshold Limit

To develop the non-segregation criterion based on physical and rheological properties, dimensional analysis involving Buckingham’s π theorem was used. Different authors present applications of the π theorem to various fluid mechanics and heat/mass transfer problems. It provides a systematic method for reducing the number of variables in a problem by grouping them into dimensionless parameters [28,29,30,31,32,33,34].

In this study, an empirical model was developed using the Buckingham π theorem to evaluate the influence of both rheological and physical properties on the segregation index (SI). The model demonstrates multiparametric nonlinear behavior and was solved in MATLAB R2025b [35] using the trust-region reflective algorithm, a robust iterative optimization technique for nonlinear least-squares problems. In this framework, the objective is to minimize the following function:

$${\displaystyle {\sum }_{i=1}^N}(S{I}_i-f(X_i,P){)}^2,$$

(7)

where SI_i = observed segregation index, f(X_i,P) = model prediction for given input X_i and coefficients P = [α, β, γ, …] are unknown coefficients to be determined. The subscript i denotes the observation (data point) number in the dataset, and N is the total number of data points. The following six steps are considered to obtain f(X_i,P) based on Buckingham’s π theorem.

• Step 1: The independent variables involved in the problem and their dimensions are determined (see

  Table 2. Variables used in the SI model and their units.

  Parameter	Symbol	Unit	Dimensions
  Flow running Length	L	$\mathrm{m}$	$\mathrm{L}$
  Shear yield stress	τ0_HB	$\mathrm{P}\mathrm{a}=\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$	$\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}$
  Infinite dynamic Viscosity	η∞	$\mathrm{P}\mathrm{a}\cdot \mathrm{s}$	$\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}$
  Flow index	n	-	-
  Slope of flume	tan(θ)	-	-
  Bulk density	ρ	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	$\mathrm{M}{\mathrm{L}}^{-3}$
  Segregation index	SI	-	-

  ). Here, the total number of variables is n_t = 7. The solid content (C_w) is not considered an independent variable, as it is directly dependent on density (ρ), which is already included among the seven variables. It would also have been necessary to account for the initial flow velocity (at L = 0 m) of the tailings; however, this parameter was not measured, and it varies along the flume.
• Step 2: The fundamental dimensions governing the physical system are identified. The selected dimensions are mass (M), length (L), and time (T), which are sufficient to represent all physical quantities involved in the analysis. The number of fundamental dimensions k is 3.
• Step 3: The total number of dimensionless groups according to the Buckingham π theorem is determined as:

  “Number of “π”-groups” = n_t − k = 7 − 3 = 4
• Step 4: The repeating variables that must collectively represent all the fundamental dimensions (M, L, T) are selected. This selection is not necessarily unique, as different researchers may choose different sets of repeating variables. Consequently, the resulting π terms are also not unique. The methodology recommends selecting a suitable combination of variables, typically including one geometric variable, one kinematic variable, and one fluid property, as is common in fluid mechanics problems. In this study, the repeating variables chosen for each π group are ρ [ML⁻³], η_∞ [ML⁻¹T⁻¹], and L [L].
• Step 5: The π groups are constructed by combining the selected repeating variables with the remaining variables to form dimensionless groups, as illustrated in

  Table 3. π groups by combining the repeating variables with the remaining variables.

  Π Groups	Formula
  Π1: Shear yield stress (${\tau }_y$)	${\Pi }_1={\displaystyle \frac{\rho {\tau }_{0-HB}{L}^2}{{{\eta }_{\infty }}^2}}$
  Π2: Flow index ($n$)	${\Pi }_2=n$
  Π3: Slope of flume ($tan\left(\theta \right)$)	${\Pi }_3=\mathrm{t}\mathrm{a}\mathrm{n}\left(\theta \right)$
  Π4: Segregation index ($SI$)	${\Pi }_4=SI$

  . Each π group represents a unique combination that captures the relationship among the physical parameters governing the system. The group Π₁: Shear yield stress (${\tau }_y$), is similar to the well-known Hedström number (He), where L is the characteristic length, D (m) is replaced by the flow running length (L).
• Step 6: The final dimensionless relationship is established by combining the constructed π groups. The relationship can be expressed as follows:

  $$f\left({\displaystyle \frac{{\mathsf{\tau }}_{0-HB} \rho L^2}{{{\eta }_{\infty }}^2}},SI,n,tan\left(\theta \right)\right)=0,$$

  (8)

  or

  $$SI=\varphi \left({\displaystyle \frac{{\mathsf{\tau }}_{0-HB} \rho L^2}{{{\eta }_{\infty }}^2}},n,tan\left(\theta \right)\right).$$

  (9)

The proposed multi-parametric segregation model for predicting the segregation index (SI) based solely on the initial physical and rheological parameters of the mine tailings and the flume slope is given as follows:

$$SI={\displaystyle \frac{\beta \times (\frac{\rho {\tau }_{0-HB}{L}^2}{{\eta }_{\infty }^2}{)}^a}{tan{\left(\theta \right)}^c \times n_{HB}^b}}$$

(10)

For the mine tailings used in this paper, β = 7.77629 × 10⁻⁷ (physico-rheological weighting factor), a = 0.613127 (rheological factor), b = 8.77633 (flow index factor), and c = 0.38665 (slope factor).

Table 4. Comparison between experimental (Exp.) and predicted (Pred.) SI values.

Flume Test #	Cw-ini (%)	tan(θ) (-)	ρini (kg/m3)	τ0-HB-ini (Pa)	nini (-)	η∞-ini (Pa·s)	SIExp (-)	SIModel (-)
1	69	0.005	1780	9.04	1.67	0.14	0.007	0.001
2	66	0.005	1750	4.78	1.31	0.08	0.013	0.016
3	63	0.005	1670	2.05	1.18	0.04	0.061	0.060
4	69	0.01	1790	7.23	1.18	0.12	0.020	0.021
5	66	0.01	1730	3.51	1.23	0.06	0.031	0.027
6	63	0.01	1670	1.46	1.2	0.03	0.042	0.044

presents the physico-rheological properties of the tailings used to calculate the segregation index (SI) and compares the calculated and experimental SI values. Each row in the table represents a single flume test. The columns list the corresponding initial physical and rheological properties. Figure 12 illustrates the correlation between the experimental segregation index (SI_Exp) and the modeled segregation index (SI_Model). Although the relative difference for Test #1 appears significant given the very small magnitudes of the SI values involved (0.007 vs. 0.001), the model’s overall predictive capability should be evaluated based on global statistical performance indicators rather than a single data point. The low root mean square error (RMSE = 0.003) and determination coefficient (R² = 0.96) indicate a strong agreement between the two.

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{\left({SI}_{Exp,i}-{SI}_{Model,i}\right)}^2}{N}}},$$

(11)

where SI_Exp = segregation index obtained from experimental measurements, SI_Model = segregation index predicted by the model, N = number of data points, and i = flume test number (1, 2, …, 6). The authors are aware that this validation using dependent data (used to develop the model) is insufficient. Independent data obtained from other flume tests with different types of tailings will allow for validation of the model’s robustness.

4.3. Limitations of the Study

In this study, a channel fabricated from Teflon was used; its overall roughness may differ from field conditions. To obtain a roughness representative of the expected ground conditions in future studies, a layer of residue will be allowed to settle at the bottom of the channel, over which the fresh residue will subsequently flow.

This model, derived from a single tested material, remains preliminary. Indeed, it was expected that the segregation index (SI) would decrease with increasing yield stress; however, this trend is not observed according to Equation (10). Compared with previous models, classical models such as Mihiretu [21] and Nik [17] mainly focused on the influence of solid concentration on segregation behavior, whereas the proposed criterion incorporates both physical and rheological properties, providing a more comprehensive representation of the mechanisms governing segregation in hard rock mine tailings. The model will be improved using a larger dataset obtained from tailings with varying initial characteristics. It will then be extended to incorporate additional parameters such as grain-size distribution, specific gravity, and other relevant independent properties. The initial velocity of the residue should also be considered.

5. Conclusions

Flume experiments were conducted to investigate the depositional characteristics of hard-rock mine tailings slurries from a geotechnical perspective, focusing on how their physico-rheological properties evolved along a 14.63 m-long flow path. Tailings were sampled at 2.44 m intervals and tested using a laser diffraction particle size analyzer, a helium pycnometer, and an AR2000 rheometer to determine particle size distribution, solid grain density, and rheological properties. The solid content was obtained from the gravimetric water content.

The physical properties of samples collected along the flume showed a progressive decrease in solid concentration (C_w) and bulk density (ρ). The experimental results indicated that decreasing the initial solid concentration (C_w-ini) intensified segregation behavior along the flow path. Higher C_w-ini values (69%) produced relatively stable segregation indexes and bulk density profiles, suggesting more homogeneous slurry transport with limited particle separation. Conversely, lower C_w-ini values (63%) led to marked downstream reductions in both parameters, highlighting segregation, slurry dilution, and preferential migration of finer, lower-density particles. These observations confirm the critical influence of the initial solid concentration on segregation mechanisms during tailings slurry transport. Rheological parameters based on the Herschel–Bulkley and Cross models indicated a noticeable reduction in shear yield stress (τ_0-HB) and infinite-shear viscosity (η_∞), respectively. Higher initial solid concentration (C_w-ini = 69%) resulted in the highest rheological resistance, with the average shear yield stress decreasing from approximately 9 Pa near the discharge point to about 2.5 Pa downstream. In contrast, the average infinite viscosity (η_∞) decreased from ~0.14 Pa·s to ~0.05 Pa·s. In contrast, at lower solid concentration (C_w-ini = 63%), both rheological parameters were substantially lower, with shear yield stress decreasing from ~2 Pa to ~0.25 Pa and η_∞ declining from ~0.04 Pa·s to ~0.01 Pa·s along the flume. These results indicate that increasing C_w-ini enhances the rheological stability of the slurry and reduces the tendency for particle segregation during flow deposition. Overall, the segregation index (SI) was found to decrease exponentially with increasing initial C_w, ρ, τ_0-HB, and η_∞. Using a segregation index threshold of SI < 5% or 0.05 as a criterion for non-segregating conditions, the tested tailings displayed non-segregating behavior when C_w-ini, ρ_ini, τ_0-HB-ini, or η_∞-ini are approximately higher than 63%, 1.68 g/cm³, 2.0 Pa, or 0.04 Pa·s, respectively.

A fundamental dimensional analysis technique, the Buckingham π theorem, enabled the development of a multiparametric segregation index model that integrates physical and rheological tailings properties and the flume slope. The proposed model accurately predicts the segregation index (and, consequently, segregation behavior) using a SI threshold of 5%, achieving a high coefficient of determination (R² = 0.96) and a low RMSE (0.003) between experimental and modeled values. Further flume tests will be carried out soon with other types of tailings to validate the proposed model with independent data.