*2.2. Computational Parameters*

After vertical integration, the controlling equations can be reduced to a 2D format [28], which is described by the Navier-Stokes equations. The simplified controlling equations are shown below.

The continuity equation:

$$\frac{\partial \mathbf{h}}{\partial t} + \frac{\partial (\mathbf{u}h)}{\partial \mathbf{x}} + \frac{\partial (\nu h)}{\partial y} = I \tag{1}$$

The equations of motion:

$$S\_{fx} = S\_{\text{ox}} - \frac{\partial \mathbf{h}}{\partial \mathbf{x}} - \frac{\partial \mathbf{u}}{g \partial \mathbf{t}} - \mathbf{u} \frac{\partial \mathbf{u}}{\partial \mathbf{x}} - \upsilon \frac{\partial \mathbf{y}}{g \partial \mathbf{y}} \tag{2}$$

$$S\_{fy} = S\_{\upsilon y} - \frac{\partial h}{\partial y} - \frac{\partial \upsilon}{g \partial x} - \iota \frac{\partial \upsilon}{\partial x} - \upsilon \frac{\partial \upsilon}{g \partial y} \tag{3}$$

where *h* denotes the draining tailing flow depth (m), *I* is the drop in the water surface per unit distance within the simulated range and is called the hydraulic gradient (%), *u* and *v* refer to the flow rate in the horizontal and vertical directions (m/s), respectively; *Sf x* and *Sf y* are the differences in the unit distance of the frictional resistance in the *x* and *y* directions, respectively, and are called the frictional slope (%), *S*o*<sup>x</sup>* and *Soy* are the differences in elevation within a unit distance in the *x* and *y* directions, respectively, and are called the riverbed slope (%).

When the flood or tailings flow is simulated by the FLO-2D, it can be performed in the dynamic wave mode or the diffused wave mode. According to the similarity criterion, Equation (1) is a mass conservation equation, while Equation (2) and Equation (3) are the momentum conservation equations.

Additionally, the evolution process of leaked tailings flow can be described by the rheological equation for high sediment concentration, which is proposed by O'Brien [29], as shown below.

$$S\_f = S\_y + S\_\nu + S\_{td} = \frac{\tau\_y}{\gamma\_m h} + \frac{K \eta \mu}{8 \gamma\_m h^2} + \frac{n^2 u^2}{h^{4/3}} \tag{4}$$

where *Sf*, *Sy*, *Sv* and *Std* represent frictional slope, yield slope, viscous slope and turbulence-distribution slope, respectively. τ*<sup>y</sup>* denotes the yield stress of the fluid during the flow, γ*<sup>m</sup>* denotes the fluid specific gravity, η is the viscous coefficient of fluid, *K* is the laminar flow resistance coefficient, and *n* refers to the Manning coefficient representing the roughness of ground surface.

Yield stress in the current study refers to the Bingham yield stress, which is reflected primarily as the internal stress pattern of viscous tailings flow and present in the form of viscous force [30–32]. The viscous force is a resistance produced by the interaction between shear and tensile stresses of fluid [33,34]. As a result, the Bingham viscosity coefficient and viscous force are closely correlated in the present work, and the increase in fluid volume concentration leads to exponential increases in the Bingham yield stress and Bingham viscosity coefficient [35,36].

The relationship between the yield stress and the volume concentration is shown in Equation (5).

$$
\pi\_{\mathcal{Y}} = \alpha\_1 \mathbf{e}^{\beta\_1 \mathcal{C}\_{\mathcal{V}}} \tag{5}
$$

and the relational expression between the Bingham viscous coefficient and the volume concentration is presented in Equation (6).

$$
\eta = a\_2 \mathbf{e}^{\beta\_2 C\_V} \tag{6}
$$

where volume concentration (*CV*) indicates the percentage of soil and debris like aggregate and gravel in the leaked tailings flow over the entire tailings flow volume. α1, β1, α<sup>2</sup> and β<sup>2</sup> are the empirical coefficients of the yield stress and viscous force, which can be derived experimentally. On this basis, Table 1 provides the relevant parameters used in the present numerical simulation.


**Table 1.** The FLO-2D simulation parameters.

#### **3. Results and Discussion**

## *3.1. Various Downstream Riverbed Slope Conditions*
