Influence of runoff on debris flow propagation at a catchment scale: a case study

Abstract

Debris flow mobility can vary during propagation due to changes in flow volume and bulk flow behavior resulting from the absorption of water from runoff. This study aims to investigate the effect of runoff on debris flow propagation by presenting an integrated model that considers the processes of rainfall, vegetation interception, soil infiltration, runoff generation, and debris flow propagation. Specifically, the study adopts an elevation-based empirical formula to evaluate the spatial distribution of rainfall and introduces a parameter for water absorption rate into the depth-averaged two-layer model that is used for describing the dynamics of runoff and debris flow. Through alternative simulations of the 2020 debris flow in the Meilong catchment, the study illustrates the significant effects of water absorption on debris flow propagation. The results indicate that as the water absorption rate of the debris mass increases, debris flow mobility also increases, since more mass and energy are transferred from runoff to debris flow. In addition, the spatial and temporal patterns of rainfall intensity can modify the propagation velocity of debris flow by influencing runoff dynamics.

Appendix. Derivation of the two-layer model by considering mass transfer at intermediate interface

Basic equations

The fundamental laws for mass and momentum conservations for an incompressible continuum are expressed as follows:

$$\left.\begin{array}{c}\partial_t\gamma_m+\nabla\cdot\left(\gamma_m{\mathbf U}_m\right)=0\\\partial_t\left(\gamma_m{\mathbf U}_m\right)+\nabla\cdot\left(\gamma_m{\mathbf U}_m\otimes{\mathbf U}_m\right)=-\nabla\cdot{\mathbf T}_m\end{array}\right\}$$

(13)

where U_m = (u_1,2, v_1,2, w_1,2) is the medium velocity field, in which the subscript m = (1, 2) refers to the upper layer and lower layer, respectively; γ_m is the medium density keeping constant; t is time; and T_m is the Cauchy stress tensor.

Boundary kinematic conditions

To express the boundary variation, such as the rates of boundary elevation change and the rates of material flux through each boundary, kinematic and dynamic boundary conditions are applied at the top and bottom boundary of any layer.

Kinematic boundary conditions imposed at the top and bottom interface of the upper layer are expressed as follows:

$${z}_{m}(x,t)={z}_{t}(x,t)-{h}_{1}=0$$

(14)

$$\left.\begin{array}{l}w_1(z_t)=\frac{\partial z_t}{\partial t}+u{}_1\left(z_t\right)\frac{\partial z_t}{\partial x}+v_1\left(z_t\right)\frac{\partial z_t}{\partial y}\\w_1(z_m)=\frac{\partial z_m}{\partial t}+u{}_1\left(z_m\right)\frac{\partial z_m}{\partial x}+v_1\left(z_m\right)\frac{\partial z_m}{\partial y}-E_m\end{array}\right\}$$

(15)

where E_m is the mixture rate from runoff to debris flow as volumetric fluxes per unit boundary area normal to the bottom boundary z_m.

Kinematic boundary conditions imposed at the top and bottom interface of the lower layer are expressed as follows:

$${z}_{b}(x,t)={z}_{t}(x,t)-{h}_{1}-{h}_{2}=0$$

(16)

$$\left.\begin{array}{c}{w}_{2}({z}_{\text{m}})=\frac{\partial {z}_{m}}{\partial t}+u{}_{2}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial x}+{v}_{2}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial y}-{E}_{m}\\ {w}_{2}({z}_{b})=\frac{\partial {z}_{b}}{\partial t}+u{}_{2}\left({z}_{b}\right)\frac{\partial {z}_{b}}{\partial x}+{v}_{2}\left({z}_{b}\right)\frac{\partial {z}_{b}}{\partial y}+{E}_{b}\end{array}\right\}$$

(17)

where E_b is entrainment rate of sediments as volumetric fluxes per unit boundary area normal to the bottom boundary.

The free surface of the two-layer is stress-free condition:

$${\mathbf T}_m\cdot\mathbf n=0$$

(18)

where $\mathbf{n}$ is the exterior unit normal vector. Dynamic boundary conditions for the upper and lower layers are assumed to satisfy Manning friction law and combined friction law that couples Coulomb friction law and Manning friction law, respectively.

$$\left.\begin{array}{l}\mathbf{pn}-\mathbf n\left(\mathbf n\cdot\textbf{pn}\right)=-\gamma_1\mathbf gn_b\left({\mathbf u}_1-{\mathbf u}_2\right)\left|{\mathbf u}_1-{\mathbf u}_2\right|/h_1^{4/3}\\\mathbf{pn}-\mathbf n\left(\mathbf n\cdot\textbf{pn}\right)=-\left[\left({\mathbf u}_2/\left|{\mathbf u}_2\right|\right)c_{d\;}\text{tan }\varphi_{bed}\left(\mathbf n\cdot\textbf{pn}\right)+\left(1-c_d\right)\gamma_{2\;}\mathbf gn_b{\mathbf u}_2\left|{\mathbf u}_2\right|/h_2^{4/3}\right]\end{array}\right\}$$

(19)

where pn, n·pn and pn–n(n·pn) represent the negative traction vector, normal pressure, and negative shear traction, respectively; g = (g_x, g_y, g_z) is the gravity components.

Depth-integrated equations for upper layer

Before driving the equations by integration, some mean values are defined as follows:

$$\overline u=\frac1h\int\limits_{z_1}^{z_2}udz;\;\overline v=\frac1h\int\limits_{z_1}^{z_2}\overset{}{vdz;\;\overline w}=\frac1h\int\limits_{z_1}^{z_2}wdz;\;\overline{\boldsymbol\tau}=\frac1h\int\limits_{z_1}^{z_2}\tau dz$$

(20)

where z₁ and z₂ refer to the top-bottom boundaries of any layer with a thickness h and a stress tensor τ. The superscript “–” refers to the depth-averaged form of any value.

Using Leibniz’s formula to integrate the mass balance equation of the upper layer

$$\begin{array}{c}\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}\left[\frac{\partial {\gamma }_{1}}{\partial t}+\frac{\partial }{\partial x}\left({\gamma }_{1}{u}_{1}\right)+\frac{\partial }{\partial y}\left({\gamma }_{1}{v}_{1}\right)+\frac{\partial }{\partial z}\left({\gamma }_{1}{w}_{1}\right)\right]dz=\frac{\partial }{\partial t}\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}{\gamma }_{1}dz-{\gamma }_{1}\left({z}_{t}\right)\frac{\partial {z}_{t}}{\partial x}+{\gamma }_{1}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial x}\\ +\frac{\partial }{\partial x}\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}\left({\gamma }_{1}{u}_{1}\right)dz-{\gamma }_{1}\left({z}_{t}\right){u}_{1}\left({z}_{t}\right)\frac{\partial {z}_{t}}{\partial x}+{\gamma }_{1}\left({z}_{m}\right){u}_{1}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial x}\\ +\frac{\partial }{\partial y}\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}\left({\gamma }_{1}{v}_{1}\right)dz-{\gamma }_{1}\left({z}_{t}\right){v}_{1}\left({z}_{t}\right)\frac{\partial {z}_{t}}{\partial y}+{\gamma }_{1}\left({z}_{m}\right){v}_{1}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial y}\\ +{\gamma }_{1}\left({z}_{t}\right){w}_{1}\left({z}_{t}\right)-{\gamma }_{1}\left({z}_{m}\right){w}_{1}\left({z}_{m}\right)\end{array}$$

(21)

We assume that ∂h₁/∂t = ∂z_t/∂t–∂z_m/∂t and then couple Eqs. (15) and (21) to obtain

$$\frac{\partial }{\partial t}\left({\gamma }_{1}{h}_{1}\right)+\frac{\partial }{\partial x}\left({\gamma }_{1}{h}_{1}{\overline{u} }_{1}\right)+\frac{\partial }{\partial y}\left({\gamma }_{1}{h}_{1}{\overline{v} }_{1}\right)=-{\gamma }_{1}\left({z}_{m}\right){E}_{m}$$

(22)

Take the momentum equation in x direction as an example, its left-hand side is changed into

$$\begin{array}{c}\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}\left(LHS\right)dz\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}\left[\frac{\partial }{\partial t}\left({\gamma }_{1}{u}_{1}\right)+\frac{\partial }{\partial x}\left({\gamma }_{1}{u}_{1}^{2}\right)+\frac{\partial }{\partial y}\left({\gamma }_{1}{u}_{1}{v}_{1}\right)+\frac{\partial }{\partial z}\left({\gamma }_{1}{u}_{1}{w}_{1}\right)\right]dz=\frac{\partial }{\partial t}\left({\gamma }_{1}{h}_{1}{\overline{u} }_{1}\right)+\frac{\partial }{\partial x}\left({\gamma }_{1}{h}_{1}{\overline{u} }_{1}^{2}\right)+\frac{\partial }{\partial y}\left({\gamma }_{1}{h}_{1}\overline{u }{\overline{v} }_{1}\right)\\ +{\gamma }_{1}\left({z}_{t}\right){u}_{1}\left({z}_{t}\right)\left[{w}_{1}\left({z}_{t}\right)-\frac{\partial {z}_{t}}{\partial t}-{u}_{1}\left({z}_{t}\right)\frac{\partial {z}_{t}}{\partial x}-{v}_{1}\left({z}_{t}\right)\frac{\partial {z}_{t}}{\partial y}\right]\\ -{\gamma }_{1}\left({z}_{m}\right){u}_{1}\left({z}_{m}\right)\left[{w}_{1}\left({z}_{m}\right)-\frac{\partial {z}_{m}}{\partial t}-{u}_{1}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial x}-{v}_{1}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial y}\right]\end{array}$$

(23)

By coupling Eqs. (15) and (23) to obtain

$$\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}\left(LHS\right)dz=\frac{\partial }{\partial t}\left({\gamma }_{1}{h}_{1}{\overline{u} }_{1}\right)+\frac{\partial }{\partial x}\left({\gamma }_{1}{h}_{1}{\overline{u} }_{1}^{2}\right)+\frac{\partial }{\partial y}\left({\gamma }_{1}{h}_{1}{\overline{u} }_{1}{\overline{v} }_{1}\right)+{\gamma }_{1}\left({z}_{m}\right){u}_{1}\left({z}_{m}\right){E}_{m}$$

(24)

The right-hand side of the x-momentum equation for the upper layer yields

$$\begin{array}{c}\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}\left(RHS\right)dz=\underset{{z}_{m}}{\overset{{z}_{t}}{\int }}-\left(\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}+\frac{\partial {\tau }_{xz}}{\partial z}\right)dz\\ \text{\hspace{0.05em}}=-\frac{\partial }{\partial x}\left({h}_{1}{\overline{\tau }}_{xx}\right)-\frac{\partial }{\partial y}\left({h}_{1}{\overline{\tau }}_{xy}\right)+{\tau }_{xx}\left({z}_{t}\right)\frac{\partial {z}_{t}}{\partial x}-{\tau }_{xx}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial x}+{\tau }_{xy}\left({z}_{t}\right)\frac{\partial {z}_{t}}{\partial y}-{\tau }_{xy}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial y}-{\tau }_{xz}\left({z}_{t}\right)+{\tau }_{xz}\left({z}_{m}\right)\end{array}$$

(25)

Based on Eq. (18), kinematic and stress condition at the free surfaces are as follows:

$$\left.\begin{array}{c}{\tau }_{xy}\left({z}_{t}\right)={\tau }_{xx}\left({z}_{t}\right)={\tau }_{xz}\left({z}_{\text{t}}\right)={\tau }_{xy}\left({z}_{m}\right)=0\\ {\overline{\tau }}_{xx}=\frac{1}{2}\left[{\tau }_{xx}\left({z}_{t}\right)+{\tau }_{xx}\left({z}_{m}\right)\right]=\frac{1}{2}{\gamma }_{1}g{h}_{1}\end{array}\right\}$$

(26)

With Eq. (26), the right-hand side of the x-momentum equation is changed into

$$\underset{{Z}_{m}}{\overset{{Z}_{t}}{\int }}\left(RHS\right)dz=-\frac{\partial }{\partial x}\left(\frac{1}{2}{\gamma }_{1}g{h}_{1}^{2}\right)-{\gamma }_{1}g{h}_{1}\frac{\partial {z}_{m}}{\partial x}+{\tau }_{xz}\left({z}_{m}\right)$$

(27)

Thus, we obtain the depth-averaged x-momentum equation of the upper layer as

$$\begin{gathered} \frac{\partial }{\partial t}\left( {{\gamma_1}{h_1}{{\bar u}_1}} \right) + \frac{\partial }{\partial x}\left( {{\gamma_1}{h_1}\bar u_1^2 + \frac{1}{2}{\gamma_1}gh_1^2} \right) + \frac{\partial }{\partial y}\left( {{\gamma_1}{h_1}{{\bar u}_1}{{\bar v}_1}} \right) = - {\gamma_1}\left( {z_m} \right){u_1}\left( {z_m} \right){E_m} \hfill \\ - {\gamma_1}g{h_1}\frac{{\partial {z_m}}}{\partial x} - \frac{{{\gamma_1}{g_z}{n_b}}}{{h_1^{1/3}}}\left( {{{\bar u}_1} - {{\bar u}_2}} \right)\left| {{{{\mathbf{\bar u}}}_1} - {{{\mathbf{\bar u}}}_2}} \right| \hfill \\ \end{gathered}$$

(28)

Using similar procedure, the depth-averaged y-momentum component for the upper layer is obtained

$$\begin{gathered} \frac{\partial }{\partial t}\left( {{\gamma_1}{h_1}{{\bar v}_1}} \right) + \frac{\partial }{\partial x}\left( {{\gamma_1}{h_1}{{\bar u}_1}{{\bar v}_1}} \right) + \frac{\partial }{\partial y}\left( {{\gamma_1}{h_1}\bar v_1^2 + \frac{1}{2}{\gamma_1}gh_1^2} \right) = - {\gamma_1}\left( {z_m} \right){v_1}\left( {z_m} \right){E_m} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\gamma_1}g{h_1}\frac{{\partial {z_m}}}{\partial y} - \frac{{{\gamma_1}{g_z}{n_b}}}{{h_1^{1/3}}}\left( {{{\bar v}_1} - {{\bar v}_2}} \right)\left| {{{{\mathbf{\bar u}}}_1} - {{{\mathbf{\bar u}}}_2}} \right| \hfill \\ \end{gathered}$$

(29)

Depth-integrated equations for lower layer

Also using the Leibnitz rule to interchange the mass balance equation of the lower layer

$$\begin{gathered} \int\limits_{z_b}^{z_m} {\left[ {\frac{{\partial {\gamma_2}}}{\partial t} + \frac{\partial }{\partial x}\left( {{\gamma_2}{u_2}} \right) + \frac{\partial }{\partial y}\left( {{\gamma_2}{v_2}} \right) + \frac{\partial }{\partial z}\left( {{\gamma_2}{w_2}} \right)} \right]dz} = \frac{\partial }{\partial t}\int\limits_{z_b}^{z_m} {\gamma_2} dz - {\gamma_2}\left( {z_m} \right)\frac{{\partial {z_m}}}{\partial x} + {\gamma_2}\left( {z_b} \right)\frac{{\partial {z_b}}}{\partial x} \hfill \\ +\; \frac{\partial }{\partial x}\int\limits_{z_b}^{z_m} {\left( {{\gamma_2}{u_2}} \right)dz} - {\gamma_2}\left( {z_m} \right){u_2}\left( {z_m} \right)\frac{{\partial {z_m}}}{\partial x} + {\gamma_2}\left( {z_b} \right){u_2}\left( {z_b} \right)\frac{{\partial {z_b}}}{\partial x} \hfill \\ +\; \frac{\partial }{\partial y}\int\limits_{z_b}^{z_m} {\left( {{\gamma_2}{v_2}} \right)dz} - {\gamma_2}\left( {z_m} \right){v_2}\left( {z_m} \right)\frac{{\partial {z_m}}}{\partial y} + {\gamma_2}\left( {z_b} \right){v_2}\left( {z_b} \right)\frac{{\partial {z_b}}}{\partial y} \hfill \\ +\; {\gamma_2}\left( {z_m} \right){w_2}\left( {z_m} \right) - {\gamma_2}\left( {z_b} \right){w_2}\left( {z_b} \right) \hfill \\ \end{gathered}$$

(30)

Coupling Eqs. (17) and (30) to obtain

$$\frac{\partial }{\partial t}\left( {{\gamma_2}{h_2}} \right) + \frac{\partial }{\partial x}\left( {{\gamma_2}{h_2}{{\bar u}_2}} \right) + \frac{\partial }{\partial y}\left( {{\gamma_2}{h_2}{{\bar v}_2}} \right) = {\gamma_2}\left( {z_m} \right){E_m} + {\gamma_2}\left( {z_b} \right){E_b}$$

(31)

Integrating the momentum equation for this layer is similar with that of the upper layer. The left-hand side of the x-momentum equation of the lower layer is written as follows:

$$\begin{array}{c}\underset{{z}_{b}}{\overset{{z}_{m}}{\int }}\left(LHS\right)dz=\underset{{z}_{b}}{\overset{{z}_{m}}{\int }}\left[\frac{\partial }{\partial t}\left({\gamma }_{2}{u}_{2}\right)+\frac{\partial }{\partial x}\left({\gamma }_{2}{u}_{2}^{2}\right)+\frac{\partial }{\partial y}\left({\gamma }_{2}{u}_{2}{v}_{2}\right)+\frac{\partial }{\partial z}\left({\gamma }_{2}{u}_{2}{w}_{2}\right)\right]dz=\frac{\partial }{\partial t}\left({\gamma }_{2}{h}_{2}{\overline{u} }_{2}\right)+\frac{\partial }{\partial x}\left({\gamma }_{2}{h}_{2}{\overline{u} }_{2}^{2}\right)+\frac{\partial }{\partial y}\left({\gamma }_{2}{h}_{2}{\overline{u} }_{2}{\overline{v} }_{2}\right)\\ +{\gamma }_{2}\left({z}_{m}\right){u}_{2}\left({z}_{m}\right)\left[{w}_{2}\left({z}_{m}\right)-\frac{\partial {z}_{m}}{\partial t}-{u}_{2}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial x}-{v}_{2}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial y}\right]\\ -{\gamma }_{2}\left({z}_{b}\right){u}_{2}\left({z}_{b}\right)\left[{w}_{2}\left({z}_{b}\right)-\frac{\partial {z}_{b}}{\partial t}-{u}_{2}\left({z}_{b}\right)\frac{\partial {z}_{b}}{\partial x}-{v}_{2}\left({z}_{b}\right)\frac{\partial {z}_{b}}{\partial y}\right]\end{array}$$

(32)

By applying the kinematic boundary conditions (Eq. 17), the left-hand side of the x-momentum equation of the lower layer is changed into

$$\underset{{Z}_{b}}{\overset{{Z}_{m}}{\int }}\left(LHS\right)dz=\frac{\partial }{\partial t}\left({\gamma }_{2}{h}_{2}{\overline{u} }_{2}\right)+\frac{\partial }{\partial x}\left({\gamma }_{2}{h}_{2}{\overline{u} }_{2}^{2}\right)+\frac{\partial }{\partial y}\left({\gamma }_{2}{h}_{2}{\overline{u} }_{2}{\overline{v} }_{2}\right)-{\gamma }_{2}\left({z}_{m}\right){u}_{1}\left({z}_{m}\right){E}_{m}-{\gamma }_{2}\left({z}_{b}\right){u}_{2}\left({z}_{b}\right){E}_{b}$$

(33)

The right-hand side of the x-momentum equation for the lower layer yields

$$\begin{array}{c}\underset{{z}_{b}}{\overset{{z}_{m}}{\int }}\left(RHS\right)dz=\underset{{z}_{b}}{\overset{{z}_{m}}{\int }}-\left(\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}+\frac{\partial {\tau }_{xz}}{\partial z}\right)dz\\ \text{\hspace{0.05em}}=-\frac{\partial }{\partial x}\left({h}_{2}{\overline{\tau }}_{xx}\right)+\frac{\partial }{\partial y}\left({h}_{2}{\overline{\tau }}_{xy}\right)+{\tau }_{xx}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial x}-{\tau }_{xx}\left({z}_{b}\right)\frac{\partial {z}_{b}}{\partial x}+{\tau }_{xy}\left({z}_{m}\right)\frac{\partial {z}_{m}}{\partial y}-{\tau }_{xy}\left({z}_{b}\right)\frac{\partial {z}_{b}}{\partial y}-{\tau }_{xz}\left({z}_{m}\right)+{\tau }_{xz}\left({z}_{b}\right)\end{array}$$

(34)

We assume that the normal stresses in the z direction are hydrostatic and thus

$${\tau }_{zz}\left({z}_{b}\right)={\gamma }_{1}g{h}_{1}+{\gamma }_{2}g{h}_{2}$$

(35)

$${\overline{\tau }}_{zz}=\frac{{\tau }_{zz}\left({z}_{m}\right)+{\tau }_{zz}\left({z}_{b}\right)}{2}=\frac{1}{2}\left(2{\gamma }_{1}g{h}_{1}+{\gamma }_{2}g{h}_{2}\right)$$

(36)

The depth-averaged normal stresses are related to the normal stress based on the Mohr–Coulomb theory and so that

$${\overline{\tau }}_{xx}={\overline{\tau }}_{yy}={k}_{ap}{\overline{\tau }}_{zz}=\frac{1}{2}{k}_{ap}\left(2{\gamma }_{1}g{h}_{1}+{\gamma }_{2}g{h}_{2}\right)$$

(37)

where k_ap is the lateral stress coefficient. By neglecting the lateral shear stress terms in Eq. (34), the right side of the x-momentum equation for the lower layer is expressed as follows:

$$\underset{{z}_{b}}{\overset{{z}_{m}}{\int }}\left(RHS\right)dz=-\frac{\partial }{\partial x}\left[\frac{1}{2}{k}_{ap}\left(2{\gamma }_{1}g{h}_{1}{h}_{2}+{\gamma }_{2}g{h}_{2}^{2}\right)\right]+{\gamma }_{1}g{h}_{1}\frac{\partial {z}_{m}}{\partial x}-{k}_{ap}\left({\gamma }_{1}g{h}_{1}+{\gamma }_{2}g{h}_{2}\right)\frac{\partial {z}_{b}}{\partial x}-{\tau }_{xz}\left({z}_{m}\right)+{\tau }_{xz}\left({z}_{m}\right)$$

(38)

Thus, we obtain the depth-averaged x-momentum equation of the lower layer as follows:

$$\begin{gathered} \frac{\partial }{\partial t}\left( {{\gamma_2}{h_2}{{\bar u}_2}} \right) + \frac{\partial }{\partial x}\left( {{\gamma_2}{h_2}\bar u_2^2 + \frac{1}{2}{k_{ap}}{\gamma_2}gh_2^2 + {k_{ap}}{\gamma_1}g{h_1}{h_2}} \right) + \frac{\partial }{\partial y}\left( {{\gamma_2}{h_2}{{\bar u}_2}{{\bar v}_2}} \right) = {\gamma_2}\left( {z_m} \right){u_1}\left( {z_m} \right){E_m} + {\gamma_2}\left( {z_b} \right){u_2}\left( {z_b} \right){E_b} \hfill \\ {\kern 1pt} + {\gamma_1}g{h_1}\frac{{\partial {z_m}}}{\partial x} - {k_{ap}}\left( {{\gamma_{1}}g{h_{1}} + {\gamma_{2}}g{h_{2}}} \right)\frac{{\partial {z_b}}}{\partial x} + \frac{{{\gamma_1}{g_z}{n_b}}}{{h_1^{1/3}}}\left( {{{\bar u}_1} - {{\bar u}_2}} \right)\left| {{{{\mathbf{\bar u}}}_1} - {{{\mathbf{\bar u}}}_2}} \right| \hfill \\ - {\gamma_{2}}\left[ {\frac{{{{\bar u}_2}}}{{\left| {{{{\mathbf{\bar u}}}_2}} \right|}}{c_d}{g_z}{h_2}\tan {\varphi_{bed}} + \left( {1 - {c_d}} \right){g_z}{n_b}\frac{{{{\bar u}_2}\left| {{{{\mathbf{\bar u}}}_2}} \right|}}{{h_2^{1/3}}}} \right] \hfill \\ \end{gathered}$$

(39)

Using similar procedure, the depth-averaged y-momentum component for the lower layer is obtained

$$\begin{gathered} \frac{\partial }{\partial t}\left( {{\gamma_2}{h_2}{{\bar v}_2}} \right) + \frac{\partial }{\partial x}\left( {{\gamma_2}{h_2}{{\bar u}_2}{{\bar v}_2}} \right) + \frac{\partial }{\partial y}\left( {{\gamma_2}{h_2}\bar v_2^2 + \frac{1}{2}{k_{ap}}{\gamma_2}gh_2^2 + {k_{ap}}{\gamma_1}g{h_1}{h_2}} \right) = {\gamma_2}\left( {z_m} \right){v_1}\left( {z_m} \right){E_m} + {\gamma_2}\left( {z_b} \right){v_2}\left( {z_b} \right){E_b} \hfill \\ + {\gamma_1}g{h_1}\frac{{\partial {z_m}}}{\partial y} - {k_{ap}}\left( {{\gamma_{1}}g{h_{1}} + {\gamma_{2}}g{h_{2}}} \right)\frac{{\partial {z_b}}}{\partial y} + \frac{{{\gamma_1}{g_z}{n_b}}}{{h_1^{1/3}}}\left( {{{\bar v}_1} - {{\bar v}_2}} \right)\left| {{{{\mathbf{\bar u}}}_1} - {{{\mathbf{\bar u}}}_2}} \right| \hfill \\ - {\gamma_{2}}\left[ {\frac{{{{\bar v}_2}}}{{\left| {{{{\mathbf{\bar u}}}_2}} \right|}}{c_d}{g_z}{h_2}\tan {\varphi_{bed}} + \left( {1 - {c_d}} \right){g_z}{n_b}\frac{{{{\bar v}_2}\left| {{{{\mathbf{\bar u}}}_2}} \right|}}{{h_2^{1/3}}}} \right] \hfill \\ \end{gathered}$$

(40)

To address the variations of the sediment in the lower layer and the bed material, the mass conservation equations are integrated from the base to the interface using Leibniz’s rule to swap the order of differentiation and integration, and then simplified by using conditions (4) and (5), which gives

$$\frac{\partial }{\partial t}\left({\gamma }_{2}{h}_{2}{c}_{d}\right)+\frac{\partial }{\partial x}\left({\gamma }_{2}{h}_{2}{c}_{d}{\overline{u} }_{2}\right)+\frac{\partial }{\partial y}\left({\gamma }_{2}{h}_{2}{c}_{d}{\overline{v} }_{2}\right)={\gamma }_{2}\left({z}_{b}\right)\left(1-p\right){E}_{b}$$

(41)

where p is the sediment porosity. To describe the variation of bed terrain caused by debris flow erosion, the mass conservation equations for bed materials are needed as follows:

$$\frac{\partial {z}_{b}}{\partial t}=-{E}_{b}$$

(42)