A combined method for modeling the triggering and propagation of debris flows

Abstract

This study describes a combined method for rainfall-induced debris flow initiations (landslides) and propagations including a series of numerical analyses. In this study, the debris is assumed as a mixture of soils and water which is physically non-Newtonian fluid. The emphasis is placed on applying the effect of the combination of rainfall-induced landslides and debris flows to the numerical analysis. An analysis of rainfall-induced landslides is conducted to identify the thickness and location of the initial volume of debris flow. The movement of debris flow is subsequently simulated considering entrainments affected by the initial wetting condition (wetting front) estimated from the rainfall-infiltration analysis. The proposed method can simulate a sequential process from the initiation of the debris flows to the deposition based on the prediction of slope failure by rainfall, fluid dynamics based on Navier–Stokes equation, and the analysis of entrainments by considering the effect of the weight of debris and the wetting condition of soil beds. Based on the numerical results of this study, the proposed method could be applied to the analysis of regional-scale landslides and debris flows.

Appendix

Here, the integration procedure of the continuity equation (Eq. 7) and Navier–Stokes momentum equations (Eqs. 8-10) is described. Equation (10) is simplified as Eq. (26) because the z-direction velocity is assumed to be nearly zero and notably smaller than the velocities of the x-direction and y-direction in this numerical geometry for the semi-3D finite volume method (u, v ≫ w ≈0).

$$ {\rho}_dg-\frac{dp}{dz}=0 $$

(26)

The integration from z = ζ_b to z = ζ is performed as follows:

$$ {\int}_{\zeta_b}^{\zeta}\left({\rho}_dg\right) dz-{\int}_{\zeta_b}^{\zeta}\left(\frac{dp}{dz}\right) dz=0 $$

(27)

$$ p={\rho}_dg\left(\zeta -{\zeta}_b\right) $$

(28)

To further simplify the continuity and momentum equations, depth averaging is applied to eliminate the explicit dependence on the z-direction. In this work, the depth average velocity and the flow flux of the x- and y-components from z = ζ_b to z = ζ are defined as follows:

$$ \overline{u}=\frac{1}{h}{\int}_{\zeta_b}^{\zeta }u\ dz $$

(29)

$$ \overline{v}=\frac{1}{h}{\int}_{\zeta_b}^{\zeta }v\ dz $$

(30)

$$ {\int}_{\zeta_b}^{\zeta }u\ dz=\overline{u}h={FL}_x $$

(31)

$$ {\int}_{\zeta_b}^{\zeta }v\ dz=\overline{v}h={FL}_y $$

(32)

where $\overline{u}$ and $\overline{v}$ are the average velocity components of the x- and y-directions, and FL_x and FL_y are the flow flux components of the x- and y-directions.

From Eqs. (7), (31), and (32), the integration of continuity equation is derived as follows:

$$ {\int}_{\zeta_b}^{\zeta}\left(\frac{du}{dx}+\frac{dv}{dy}+\frac{dw}{dz}\right) dz=0 $$

(33)

$$ \frac{d}{dx}{\int}_{\zeta_b}^{\zeta } udz+\frac{d}{dy}{\int}_{\zeta_b}^{\zeta } vdz+{\left[-u\frac{dh}{dx}-v\frac{dh}{dy}+w\right]}_{\zeta_b}^{\zeta }=0 $$

(34)

$$ -u\frac{dh}{dx}-v\frac{dh}{dy}+w=0\kern0.5em \left(z={\zeta}_b\right) $$

(35)

(Bottom boundary conditions)

$$ -\frac{dh}{dt}-u\frac{dh}{dx}-v\frac{dh}{dy}+w=0\kern0.5em \left(z=\zeta \right) $$

(36)

(Kinematic free surface boundary conditions)

$$ \frac{d}{dx}{\int}_{\zeta_b}^{\zeta } udz+\frac{d}{dy}{\int}_{\zeta_b}^{\zeta } vdz+\frac{dh}{dt}=0 $$

(37)

$$ \frac{d}{dx}\left({FL}_x\right)+\frac{d}{dy}\left({FL}_y\right)+\frac{dh}{dt}=0 $$

(38)

We can divide both sides of the x-direction momentum equation (Eq. 8) by ρ_d, as shown in Eq. (39), and the integration of the left-hand side of the x-direction momentum equation from z = ζ_b to z = ζ is written as follows:

$$ \left(\frac{du}{dt}+u\frac{du}{dx}+v\frac{du}{dy}+w\frac{du}{dz}\right)=\frac{1}{\rho_d}\left[-\frac{dp}{dx}+\mu \left(\frac{d^2u}{dx^2}+\frac{d^2u}{dy^2}+\frac{d^2u}{dz^2}\right)\right] $$

(39)

$$ {\displaystyle \begin{array}{cc}{\int}_{\zeta_b}^{\zeta}\left(\frac{du}{dt}+u\frac{du}{dx}+v\frac{du}{dy}+w\frac{du}{dz}\right)& ={\int}_{\zeta_b}^{\zeta}\left[\frac{du}{dt}+\frac{d}{dx}(uu)+\frac{d}{dy}(uv)+\frac{d}{dz}(uw)\right] dz\\ {}& =\frac{d}{dt}{\int}_{\zeta_b}^{\zeta } udz+\frac{d}{dx}{\int}_{\zeta_b}^{\zeta }{u}^2 dz+\frac{d}{dy}{\int}_{\zeta_b}^{\zeta } uvdz\end{array}} $$

(40)

$$ {\displaystyle \begin{array}{cc}\frac{d}{dt}{\int}_{\zeta_b}^{\zeta } udz+\frac{d}{dx}{\int}_{\zeta_b}^{\zeta }{u}^2 dz+\frac{d}{dy}{\int}_{\zeta_b}^{\zeta } uvdz& =\frac{d{FL}_x}{dt}+\alpha \frac{d\left(\overline{u}{FL}_x\right)}{dx}+\alpha \frac{d\left(\overline{v}{FL}_x\right)}{dy}\\ {}& =\frac{d{FL}_y}{dt}+\frac{\alpha }{h}\frac{d\left({FL_x}^2\right)}{dx}+\frac{\alpha }{h}\frac{d\left({FL}_x{FL}_y\right)}{dy}\end{array}} $$

(41)

with

$$ {\int}_{\zeta_b}^{\zeta }{u}^2 dz=\alpha h{\overline{u}}^2=\alpha \overline{u}{FL}_x $$

(42)

where α is a coefficient that defines the deviation of the vertical velocity profile from uniformity (Savage and Hutter 1989). Takahashi et al. (1992) suggested general values of the coefficient α. If the debris flow profile is reasonably blunt, then α = 1; for the parabolic velocity profile and debris flow with no basal sliding, α = 6/5; and for a stone-type debris flow on a rough inclined plane, α = 1.25.

Similar to Eq. (9), integration of the left-hand side of the y-direction momentum equation from z = ζ_b to z = ζ is written as follows:

$$ \left(\frac{dv}{dt}+u\frac{dv}{dx}+v\frac{dv}{dy}+w\frac{dv}{dz}\right)=\frac{1}{\rho_d}\left[-\frac{dp}{dx}+\mu \left(\frac{d^2v}{dx^2}+\frac{d^2v}{dy^2}+\frac{d^2v}{dz^2}\right)\right] $$

(43)

$$ {\displaystyle \begin{array}{cc}{\int}_{\zeta_b}^{\zeta}\left(\frac{dv}{dt}+u\frac{dv}{dx}+v\frac{dv}{dy}+w\frac{dv}{dz}\right)& ={\int}_{\zeta_b}^{\zeta}\left[\frac{dv}{dt}+\frac{d}{dx}(uv)+\frac{d}{dy}(vv)+\frac{d}{dz}(vw)\right] dz\\ {}& =\frac{d}{dt}{\int}_{\zeta_b}^{\zeta } vdz+\frac{d}{dx}{\int}_{\zeta_b}^{\zeta } uvdz+\frac{d}{dy}{\int}_{\zeta_b}^{\zeta }{v}^2 dz\end{array}} $$

(44)

$$ =\frac{d{FL}_y}{dt}+\frac{\alpha }{h}\frac{d\left({FL}_x{FL}_y\right)}{dx}+\frac{\alpha }{h}\frac{d\left({FL_y}^2\right)}{dy} $$

(45)

with

$$ {\int}_{\zeta_b}^{\zeta }{v}^2 dz=\alpha h{\overline{v}}^2=\alpha \overline{v}{FL}_y $$

(46)

From Eq. (28), we can integrate the pressure component of the x-direction momentum equation as follows:

$$ {\int}_{\zeta_b}^{\zeta}\frac{dp}{dx} dz=\frac{d}{dx}{\int}_{\zeta_b}^{\zeta }{\rho}_dg\left(\eta -{\eta}_b\right) dz={\rho}_d hg\frac{dH}{dx} $$

(47)

$$ H={\zeta}_b+h $$

(48)

$$ \frac{\mu }{\rho_d}{\int}_{\zeta_b}^{\zeta}\frac{d^2u}{d{x}^2} dz=\frac{\mu \beta}{\rho_d}\frac{d^2h\overline{u}}{d{x}^2}=\frac{d^2{FL}_x}{d{x}^2} $$

(49)

$$ \frac{\mu }{\rho_d}{\int}_{\zeta_b}^{\zeta}\frac{d^2u}{d{y}^2} dz=\frac{\mu \beta}{\rho_d}\frac{d^2h\overline{u}}{d{y}^2}=\frac{d^2{FL}_x}{d{y}^2} $$

(50)

$$ \frac{\mu }{\rho_d}{\int}_{\zeta_b}^{\zeta}\frac{d^2u}{d{z}^2} dz={\left.\frac{\mu }{\rho_d}\frac{du}{dz}\right|}_{z=\zeta }-{\left.\frac{\mu }{\rho_d}\frac{du}{dz}\right|}_{z={\zeta}_b} $$

(51)

$$ {\left.\frac{\mu }{\rho_d}\frac{du}{dz}\right|}_{z=\zeta }=\frac{\tau_{sx}}{\rho_d}=0 $$

(52)

$$ {\left.\frac{\mu }{\rho_d}\frac{du}{dz}\right|}_{z={\zeta}_b}=\frac{\tau_{bx}}{\rho_d} $$

(53)

where τ_sx (=0) is the friction at free surface of debris flow, and τ_bx is the total resistance of the flowing media between the debris and residual soil.

In this study, the total resistance between the debris and residual soil is defined as a combination of viscous-Coulomb friction flow resistance with normal stress and a turbulent drag resistance with the square of the velocity (Voellmy 1964). The x-component of total resistance (τ_bx) used in this simulation method is written as follows:

$$ {\tau}_{bx}={\rho}_d\mu \sqrt{gh}\mathit{\cos}{\delta}_x tan\varphi +\left(\frac{\rho_dg{u}^2}{\xi}\right) $$

(54)

where δ_x is the slope angle for the x-direction, φ is a basal friction angle, and ξ [ms⁻²] is a turbulent drag coefficient.

Therefore, we can obtain the integrated equation of the right-hand side of the momentum equation for the x-direction (Eq. 8).

$$ -\frac{dH}{dx} gh+\frac{\mu \beta}{\rho_d}\left(\frac{d^2{FL}_x}{dx^2}+\frac{d^2{FL}_x}{dy^2}\right)-\mu \sqrt{gh}\mathit{\cos}{\delta}_x tan\varphi -\left(\frac{g{\overline{u}}^2}{\xi}\right) $$

(55)

Similarly, from Eq. (9), the integration of the right-hand side of the y-direction momentum equation from z = ζ_b to z = ζ is given as follows:

$$ -\frac{dH}{dy} gh+\frac{\mu \beta}{\rho_d}\left(\frac{d^2{FL}_y}{dx^2}+\frac{d^2{FL}_y}{dy^2}\right)-\mu \sqrt{gh}\mathit{\cos}{\delta}_y tan\varphi -\left(\frac{g{\overline{v}}^2}{\xi}\right) $$

(56)

with

$$ {\tau}_{by}={\rho}_d\mu \sqrt{gh}\mathit{\cos}{\delta}_y tan\varphi +\left(\frac{\rho_dg{\overline{v}}^2}{\xi}\right) $$

(55)

Finally, the integration of the momentum equations for the x- and y-directions are derived as follows:

$$ \frac{d{FL}_x}{dt}+\frac{\alpha }{h}\frac{d\left({FL_x}^2\right)}{dx}+\frac{\alpha }{h}\frac{d\left({FL}_x{FL}_y\right)}{dy}=-\frac{dH}{dx} gh+\frac{\mu \beta}{\rho_d}\left(\frac{d^2{FL}_x}{dx^2}+\frac{d^2{FL}_x}{dy^2}\right)-\mu \sqrt{gh}\mathit{\cos}{\delta}_x tan\varphi -\left(\frac{g{FL_x}^2}{\xi {h}^2}\right) $$

(58)

$$ \frac{d{FL}_y}{dt}+\frac{\alpha }{h}\frac{d\left({FL}_x{FL}_y\right)}{dx}+\frac{\alpha }{h}\frac{d\left({FL_y}^2\right)}{dy}=-\frac{dH}{dy} gh+\frac{\mu \beta}{\rho_d}\left(\frac{d^2{FL}_y}{dx^2}+\frac{d^2{FL}_y}{dy^2}\right)-\mu \sqrt{gh}\mathit{\cos}{\delta}_y tan\varphi -\left(\frac{g{FL_y}^2}{\xi {h}^2}\right) $$

(59)