A numerical model of debris flows with the Voellmy model over a real terrain

Abstract

In this paper, a depth-averaged numerical model is proposed to simulate the propagation of debris flows. To understand the dynamical features of debris flows, the granular-liquid mixture model is employed, where the resistance of liquid constituent consisting of water and fine particles is described by Herschel-Bulkley rheology, and the resistance of granular constituent involving the coarse particles is formulated by the Voellmy model. An Eulerian leapfrog finite difference scheme is employed to numerically solve the governing equations of debris flows. Several idealized dam break problems are numerically implemented to test the accuracy and stability of the numerical scheme, and good agreement with the analytical solutions is found. Furthermore, the experiment carried out by Iverson (2010) is numerically implemented in this study to validate the proposed model, and the results are in good agreement. Then, a typical debris flow in Tianmo gully in Tibet, China, is numerically analyzed based on the proposed model. The results show that several details of the motion of debris flows over a real terrain can be accurately predicted by the proposed model. The comparisons between the basal Coulomb friction model and the Voellmy model show that the mobility of debris flows can be overestimated when only the Coulomb friction model is employed. An analysis based on the unsteady motion of a uniform mass indicates that Voellmy resistance can rescue the issue in regard to the absence of resistance of the granular component due to fully liquefied states. Finally, the influence of the rheology of liquid slurry on the motion of debris flows is analyzed, and the results demonstrate that the basal resistances caused by the liquid slurry were very small relative to the basal resistances caused by granular constituent.

Appendix

Numerical scheme

In this study, the Eulerian leapfrog finite difference scheme, which has been widely applied to simulate the evolution of flood (Cho et al. 1998; Lin et al. 2011; Sun et al. 2011; Zhang et al. 2016), was employed to numerically solve the governing Eqs. (1)–(3). The numerical scheme in this study has second-order accuracy for the temporal step and spacial grid (Cho 1995; Saiduzzaman and Ray 2013; Zhang et al. 2016). First, the equation of mass balance is discretized as follows:

$$\frac{{h}_{i,j}^{n+1/2}-{h}_{i,j}^{n-1/2}}{\Delta t}+\frac{{p}_{i+1/2,j}^{n}-{p}_{i-1/2,j}^{n}}{\Delta {x}_{i}}+\frac{{q}_{i,j+1/2}^{n}-{q}_{i,j-1/2}^{n}}{\Delta {y}_{j}}=0$$

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where the superscript n represents the time level; the subscript (i, j) denotes the spatial node; Δt represents the time increment; Δx_i and Δy_j represent the grid cell sizes along the x- and y-directions, respectively; and h^n+1/2 = (hⁿ + hⁿ⁺¹)/2 represents the depth of flows at the n + 1/2-th time step in the center of the grid cells.

Second, the equations of momentum balance are numerically solved as formulated in Eqs. (A2) and (A3) at the n + 1-th time step in the right face and the upper face of the grid cell along the x- and y-directions, respectively.

$$\begin{array}{c}\frac{{p}_{i+1/2,j}^{n+1}-{p}_{i+1/2,j}^{n}}{\Delta t}+{\left[\frac{\partial }{\partial x}\left(\frac{{p}^{2}}{h}\right)+\frac{\partial }{\partial y}\left(\frac{pq}{h}\right)\right]}_{i+1/2,j}^{n}\\ =-g{h}_{i,j}^{n+1/2}\frac{(k{h}_{i+1,j}^{n+1/2}\text{+}{z}_{b}{}_{i+1,j}^{n+1/2})-(k{h}_{i,j}^{n+1/2}\text{+}{z}_{b}{}_{i,j}^{n+1/2})}{\Delta {x}_{i+1/2}}-{\left(\frac{{\tau }_{bx}}{\rho }\right)}_{i+1/2,j}^{n}\end{array}$$

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$$\begin{array}{c}\frac{{q}_{i,j+1/2}^{n+1}-{q}_{i,j+1/2}^{n}}{\Delta t}+{\left[\frac{\partial }{\partial x}\left(\frac{pq}{h}\right)+\frac{\partial }{\partial y}\left(\frac{{q}^{2}}{h}\right)\right]}_{i,j+1/2}^{n}\\ =-g{h}_{i,j}^{n+1/2}\frac{(k{h}_{i,j+1}^{n+1/2}\text{+}{z}_{b}{}_{i,j+1}^{n+1/2})-(k{h}_{i,j}^{n+1/2}\text{+}{z}_{b}{}_{i,j}^{n+1/2})}{\Delta {y}_{i+1/2}}-{\left(\frac{{\tau }_{by}}{\rho }\right)}_{i,j+1/2}^{n}\end{array}$$

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where the flux of momentums in this scheme is discretized by the upwind scheme as:

$${\left[\frac{\partial }{\partial x}\left(\frac{{p}^{2}}{h}\right)\right]}_{i+1/2,j}^{n}=\left\{\begin{array}{c}\frac{1}{\Delta {x}_{i}}\left[\frac{{({p}_{i+1/2,j}^{n})}^{2}}{{h}_{i+1/2,j}^{n}}-\frac{{({p}_{i-1/2,j}^{n})}^{2}}{{h}_{i-1/2,j}^{n}}\right]\qquad if\qquad{\ p}_{i+1/2,j}^{n}\ge 0\\ \frac{1}{\Delta {x}_{i}}\left[\frac{{({p}_{i+3/2,j}^{n})}^{2}}{{h}_{i+3/2,j}^{n}}-\frac{{({p}_{i+1/2,j}^{n})}^{2}}{{h}_{i+1/2,j}^{n}}\right]\qquad if\qquad {\ p}_{i+1/2,j}^{n}<0\end{array}\right.$$

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$${\left[\frac{\partial }{\partial y}\left(\frac{pq}{h}\right)\right]}_{i+1/2,j}^{n}=\left\{\begin{array}{c}\frac{1}{\Delta {y}_{i-1/2}}\left[\frac{{(pq)}_{i+1/2,j}^{n}}{{h}_{i+1/2,j}^{n}}-\frac{{(pq)}_{i+1/2,j-1}^{n}}{{h}_{i+1/2,j-1}^{n}}\right]\qquad if\qquad {\ q}_{i+1/2,j}^{n}\ge 0\\ \frac{1}{\Delta {y}_{i+1/2}}\left[\frac{{(pq)}_{i+1/2,j+1}^{n}}{{h}_{i+1/2,j+1}^{n}}-\frac{{(pq)}_{i+1/2,j}^{n}}{{h}_{i+1/2,j}^{n}}\right]\qquad if\qquad {\ q}_{i+1/2,j}^{n}<0\end{array}\right.$$

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$${\left[\frac{\partial }{\partial y}\left(\frac{{q}^{2}}{h}\right)\right]}_{i+1/2,j}^{n}=\left\{\begin{array}{c}\frac{1}{\Delta {y}_{j}}\left[\frac{{({q}_{i,j+1/2}^{n})}^{2}}{{h}_{i,j+1/2}^{n}}-\frac{{({q}_{i,j-1/2}^{n})}^{2}}{{h}_{i,j-1/2}^{n}}\right]\qquad if\qquad {\ q}_{i,j+1/2}^{n}\ge 0\\ \frac{1}{\Delta {y}_{j}}\left[\frac{{({q}_{i,j+3/2}^{n})}^{2}}{{h}_{i,j+3/2}^{n}}-\frac{{({q}_{i,j+1/2}^{n})}^{2}}{{h}_{i,j+1/2}^{n}}\right]\qquad if\qquad {\ q}_{i+1/2,j}^{n}<0\end{array}\right.$$

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$${\left[\frac{\partial }{\partial x}\left(\frac{pq}{h}\right)\right]}_{i+1/2,j}^{n}=\left\{\begin{array}{c}\frac{1}{\Delta {x}_{i-1/2}}\left[\frac{{(pq)}_{i+1/2,j}^{n}}{{h}_{i+1/2,j}^{n}}-\frac{{(pq)}_{i-1/2,j-1}^{n}}{{h}_{i+1/2,j-1}^{n}}\right]\qquad if\qquad {\ p}_{i+1/2,j}^{n}\ge 0\\ \frac{1}{\Delta {x}_{i+1/2}}\left[\frac{{(pq)}_{i+1/2,j+1}^{n}}{{h}_{i+1/2,j+1}^{n}}-\frac{{(pq)}_{i+1/2,j}^{n}}{{h}_{i+1/2,j}^{n}}\right]\qquad if\qquad {\ p}_{i+1/2,j}^{n}\ge 0\end{array}\right.$$

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The resistance term on the right-hand side is calculated by the semi-implicit scheme as follows:

$$\begin{array}{c}{\left({\tau }_{bx}\right)}_{i+1/2,j}^{n}={p}_{i+1/2,j}^{n+1}\{{\frac{\rho g{h}_{i+1/2,j}^{n}\mathrm{cos}{\theta }_{x}(1-\lambda )\mathrm{tan}\varphi }{\sqrt{{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}}}+\frac{{n}_{s}{\rho }_{s}g\sqrt{{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}}}{{({h}_{i+1/2,j}^{n})}^{2}{C}_{z}^{2}}} \\ +\frac{{n}_{f}{\tau }_{y}}{\sqrt{{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}}}+{n}_{f}{\eta }_{m}{(\frac{2m+1}{m})}^m\frac{{[{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}]}^{(m-1)/2}}{{({h}_{i+1/2,j}^{n})}^{2m}}\}\end{array}$$

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$$\begin{array}{c}{\left({\tau }_{by}\right)}_{i,j+1/2}^{n}={q}_{i,j+1/2}^{n+1}\{{\frac{\rho g{h}_{i+1/2,j}^{n}\mathrm{cos}{\theta }_{x}(1-\lambda )\mathrm{tan}\varphi }{\sqrt{{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}}}+\frac{{n}_{s}{\rho }_{s}g\sqrt{{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}}}{{({h}_{i+1/2,j}^{n})}^{2}{C}_{z}^{2}}}\\+\frac{{n}_{f}{\tau }_{y}}{\sqrt{{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}}}+{n}_{f}{\eta }_{m}{(\frac{2m+1}{m})}^m\frac{{[{({p}_{i+1/2,j}^{n})}^{2}+{({q}_{i+1/2,j}^{n})}^{2}]}^{(m-1)/2}}{{({h}_{i+1/2,j}^{n})}^{2m}}\}\end{array}$$

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Finally, the moving boundary treatment proposed by Cho (1995) was employed to deal with the moving boundary where the fluid feeds into or recedes from a dry land.