Numerical study on the evolution process of a geohazards chain resulting from the Yigong landslide

Abstract

Geohazard chain processes in mountainous areas generally entail a landslide, followed by a dammed lake, a dam breach, and then outburst flooding. These chains have greater destructive power and a larger area of coverage than a single process, of which a representative event is the April 2000 Yigong landslide in Tibet, China. In this study, a two-part, numerical back-analysis of the entire chain process is carried out. Enhanced one-layer Savage-Hutter models, which incorporate a multiscale, empirical friction model (velocity-weakening) and appropriate erosion mechanics, are solved using a non-staggered central differencing scheme. A reasonable reproduction of the geohazard event chain was obtained. Results show that the use of the multiscale friction law is able to reproduce the dynamic process of the landslide with acceptable accuracy. In addition, the variation of soil shear resistance along the dam depth (against the water flow above) during the dam breach is considered in the study, in which the outburst flooding process is better modeled. The numerical results, validated by field measurements, provide reliable assessment and interpretation of the actual event.

Appendix

Central differencing schemes are often used to solve nonlinear dynamic equations since they are not linked to the specific eigenstructure of the problem and can thus be implemented in a more straightforward way. The first-order Lax-Friedrichs scheme is the forerunner for such central schemes. The central Nessyahu–Tadmor (NT) scheme offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. Godunov’s original scheme is also the forerunner of all upwind schemes. Though its higher-order and multi-dimensional generalizations were constructed, it requires characteristic information along the discontinuous interfaces of these spatial cells using approximate Riemann solvers, dimensional splitting, etc. which greatly complicates the upwind methods.

In this section, we first briefly introduce the high-resolution scheme on staggered grids over regular square grids- the NT scheme (Nessyahu and Tadmor 1990), which is regarded as a natural extension of the first-order LxF scheme. The cells in the NT scheme alternate every adjacent time step ∆t, The importance of staggering is due to the fact that cell interfaces are stable in neighborhoods around the smooth regular mid-cells of the previous time step (Jiang et al. 1999). The two-dimensional system of conservation laws Equations without source terms are proposed:

$$ \frac{\partial \boldsymbol{U}}{\partial t}+\frac{\partial f\left(\boldsymbol{U}\right)}{\partial x}+\frac{\partial g\left(\boldsymbol{U}\right)}{\partial y}=0 $$

(15)

Lax and Friedrich introduced the first-order stable central scheme, the general Lax-Friedrich scheme:

$$ {{\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}}^{n+1}={{\overline{U}}_{i,j}}^n-\lambda \left(f\left({{\overline{U}}_{i+1,j}}^n\left)-f\right({{\overline{U}}_{i,j}}^n\right)\right)-\eta \left(g\left({{\overline{U}}_{i,j+1}}^n\left)-g\right({{\overline{U}}_{i,j}}^n\right)\right) $$

(16)

Here,$\lambda =\frac{\Delta t}{\Delta x}$ and$\eta =\frac{\Delta t}{\Delta y}$denote the mesh ratios of fixed size.

As a first step, we use the piecewise constant solution of the form$\sum {{\overline{U}}_{ij}}^n{\chi}_{ij}\left(x,y\right)$, in which ${{\overline{U}}_{ij}}^n$is the approximate cell average at tⁿ, associated with the cell ${C}_{ij}=\left\{|x-{x}_i|\le \frac{\Delta x}{2},|y-{y}_j|\le \frac{\Delta y}{2}\right\}$ centered at (x_i, y_j), x_i = ∆x . i,y_j = ∆y . j. The function χ_ij(x, y) is the characteristic function of the cell C_ij, i.e., ${\chi}_{ij}\left(x,y\right)={1}_{C_{ij}}$.

We construct a piecewise-linear approximation of the form:

$$ U\left(x,y,{t}^n\right)=\sum \left({\overline{U}}_{ij}+{\overset{\acute{\mkern6mu}}{U}}_{ij}\left(\frac{x-{x}_i}{\Delta x}\right)+{\overset{`}{U}}_{ij}\left(\frac{y-{y}_j}{\Delta y}\right)\right){\chi}_{ij}\left(x,y\right) $$

(17)

where ${\overset{\acute{\mkern6mu}}{U}}_{ij}$ and ${\overset{`}{U}}_{ij}$ are discrete slopes in the x, y direction. To guarantee second-order accuracy, ${\overset{\acute{\mkern6mu}}{U}}_{ij}$ and ${\overset{`}{U}}_{ij}$ should satisfy:

$$ {\overset{`}{U}}_{ij}\sim \Delta x\cdotp {U}_x\left({x}_i,{y}_j,{t}^n\right)+\mathrm{o}{\left(\Delta x\right)}^2 $$

(18)

$$ {\overset{`}{U}}_{ij}\sim \Delta y\cdotp {U}_y\left({x}_i,{y}_j,{t}^n\right)+\mathrm{o}{\left(\varDelta y\right)}^2 $$

(19)

The second step is to replace the exact solution at next time step t^n + 1 by its averages over staggered cells: ${C}_{i+\frac{1}{2},j+\frac{1}{2}}={I}_{i+\frac{1}{2}}\times {J}_{j+\frac{1}{2}},{C}_{i+\frac{1}{2},j+\frac{1}{2}}=\left\{|x-{x}_{i+\frac{1}{2}}|\le \frac{\varDelta x}{2},|y-{y}_{j+\frac{1}{2}}|\le \frac{\varDelta y}{2}\right\}$, integrate the equation over the staggered control volume ${C}_{i+\frac{1}{2},j+\frac{1}{2}}\times \left[{t}^n,{t}^{n+1}\right)$ yield:

$$ {\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^{n+1}\right)={\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^n\right)-\lambda \left(\frac{1}{\mid \varDelta t\mid}\frac{1}{\mid {J}_{j+\frac{1}{2}}\mid }{\int}_{t^n}^{t^{n+1}}{\int}_{J_{j+\frac{1}{2}}}\Big(f\left(U\left({x}_{i+1},y,t\right)\Big)-f\left(U\left({x}_i,y,t\right)\right)\right) dydt\right)-\eta \left(\frac{1}{\mid \varDelta t\mid}\frac{1}{\mid {I}_{i+\frac{1}{2}}\mid }{\int}_{t^n}^{t^{n+1}}{\int}_{I_{i+\frac{1}{2}}}\Big(g\left(U\left(x,{y}_{j+1},t\right)\Big)-g\left(U\left(x,{y}_j,t\right)\right)\right) xdt\right) $$

(20)

Here${\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^{n+1}\right)=\frac{1}{\left|{C}_{i+\frac{1}{2},j+\frac{1}{2}}\right|}{\iint}_{C_{i+\frac{1}{2},j+\frac{1}{2}}}U\left(x,y,{t}^{n+1}\right) dxdy$, ${\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^n\right)=\frac{1}{\left|{C}_{i+\frac{1}{2},j+\frac{1}{2}}\right|}{\iint}_{C_{i+\frac{1}{2},j+\frac{1}{2}}}U\left(x,y,{t}^n\right) dxdy$ denote the cell averages over staggered grids.

The next step is to evaluate the staggered grid averages ${\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^n\right)$ in Eq. 20:

$$ {\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^n\right)=\frac{1}{\left|{C}_{i+\frac{1}{2},j+\frac{1}{2}}\right|}{\iint}_{C_{i+\frac{1}{2},j+\frac{1}{2}}}U\left(x,y,{t}^n\right) dxdy=\sum \limits_{i=1}^4\frac{1}{\mid {A}_i\mid }{\iint}_{A_i}U\left(x,y,{t}^n\right) dxdy $$

(21)

Here, the staggered cell ${C}_{i+\frac{1}{2},j+\frac{1}{2}}$ can be separated into $\sum \limits_{i=1}^4{A}_i$, where A₁, A₂, A₃, and A₄ represent the cell: $\left\{\left({x}_i,{x}_{i+\frac{1}{2}}\right)\times \left({y}_{j+\frac{1}{2}},{y}_{j+1}\right)\right\}$, $\left\{\left({x}_{i+\frac{1}{2}},{x}_{i+1}\right)\times \left({y}_{j+\frac{1}{2}},{y}_{j+1}\right)\right\}$, $\left\{\left({x}_{i+\frac{1}{2}},{x}_{i+1}\right)\times \left({y}_j,{y}_{j+\frac{1}{2}}\right)\right\}$, and $\left\{\left({x}_i,{x}_{i+\frac{1}{2}}\right)\times \left({y}_j,{y}_{j+\frac{1}{2}}\right)\right\}$, respectively.

Using the integration over A₁for example:

$$ \frac{1}{\mid {A}_1\mid }{\iint}_{A_i}U\left(x,y,{t}^n\right) dxdy=\frac{1}{\mid \varDelta x\mid}\frac{1}{\mid \varDelta y\mid }{\int}_{x_i}^{x_{i+\frac{1}{2}}}{\int}_{y_{j+\frac{1}{2}}}^{y_{j+1}}\left({\overline{U}}_{ij}+{\overset{\acute{\mkern6mu}}{U}}_{ij}\left(\frac{x-{x}_i}{\varDelta x}\right)+{\overset{`}{U}}_{ij}\left(\frac{y-{y}_j}{\varDelta y}\right)\right) dxdy $$

(22)

The other three cell averages can be treated the same way. By adding the four integrals, the value of the staggered average ${\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^n\right)$ can be expressed as:

$$ {\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^n\right)=\frac{1}{4}\left({{\overline{U}}_{i,j}}^n+{{\overline{U}}_{i+1,j}}^n+{{\overline{U}}_{i,j+1}}^n+{{\overline{U}}_{i+1,j+1}}^n\right)+\frac{1}{16}\left(\left({\overset{\acute{\mkern6mu}}{U}}_{i,j}+{\overset{\acute{\mkern6mu}}{U}}_{i,j+1}-{\overset{\acute{\mkern6mu}}{U}}_{i+1,j}-{\overset{\acute{\mkern6mu}}{U}}_{i+1,j+1}\right)+\left({\overset{`}{U}}_{i,j}+{\overset{`}{U}}_{i+1,j}-{\overset{`}{U}}_{i,j+1}-{\overset{`}{U}}_{i+1,j+1}\right)\right) $$

(23)

The next step is to approximate the fluxes in equation:

$$ \frac{1}{\mid \varDelta t\mid}\frac{1}{\mid {J}_{j+\frac{1}{2}}\mid }{\int}_{t^n}^{t^{n+1}}{\int}_{J_{j+\frac{1}{2}}}\Big(f\left(U\left({x}_{i+1},y,t\right)\Big)-f\left(U\left({x}_i,y,t\right)\right)\right) dydt $$

$$ \approx \frac{1}{\mid {J}_{j+\frac{1}{2}}\mid }{\int}_{J_{j+\frac{1}{2}}}\left(f\left(U\left({x}_{i+1},y,{t}^{n+\frac{1}{2}}\right)\right)-f\left(U\left({x}_i,y,{t}^{n+\frac{1}{2}}\right)\right)\right) dy $$

$$ \approx \frac{1}{2}\left\{\left(f\left({U_{i+1,j}}^{n+\frac{1}{2}}\right)+f\left({U_{i+1,j+1}}^{n+\frac{1}{2}}\right)\right)-\left(f\left({U_{i,j}}^{n+\frac{1}{2}}\right)+f\left({U_{i,j+1}}^{n+\frac{1}{2}}\right)\right)\right\} $$

$$ \frac{1}{\mid \varDelta t\mid}\frac{1}{\mid {I}_{i+\frac{1}{2}}\mid }{\int}_{t^n}^{t^{n+1}}{\int}_{I_{i+\frac{1}{2}}}\Big(g\left(U\left(x,{y}_{j+1},t\right)\Big)-g\left(U\left(x,{y}_j,t\right)\right)\right) dxdt $$

$$ \approx \frac{1}{\mid {I}_{i+\frac{1}{2}}\mid }{\int}_{I_{i+\frac{1}{2}}}\left(g\left(U\left(x,{y}_{j+1},{t}^{n+\frac{1}{2}}\right)\right)-g\left(U\left(x,{y}_j,{t}^{n+\frac{1}{2}}\right)\right)\right) dx $$

$$ \approx \frac{1}{2}\left\{\left(g\left({U_{i,j+1}}^{n+\frac{1}{2}}\right)+g\left({U_{i+1,j+1}}^{n+\frac{1}{2}}\right)\right)-\left(g\left({U_{i+1,j}}^{n+\frac{1}{2}}\right)+g\left({U_{i,j}}^{n+\frac{1}{2}}\right)\right)\right\} $$

(24)

The staggered form of our central differencing scheme can thus be expressed as:

$$ {\overline{U}}_{i+\frac{1}{2},j+\frac{1}{2}}\left({t}^n\right)=\frac{1}{4}\left({{\overline{U}}_{i,j}}^n+{{\overline{U}}_{i+1,j}}^n+{{\overline{U}}_{i,j+1}}^n+{{\overline{U}}_{i+1,j+1}}^n\right)+\frac{1}{16}\left(\left({\overset{\acute{\mkern6mu}}{U}}_{i,j}+{\overset{\acute{\mkern6mu}}{U}}_{i,j+1}-{\overset{\acute{\mkern6mu}}{U}}_{i+1,j}-{\overset{\acute{\mkern6mu}}{U}}_{i+1,j+1}\right)+\left({\overset{`}{U}}_{i,j}+{\overset{`}{U}}_{i+1,j}-{\overset{`}{U}}_{i,j+1}-{\overset{`}{U}}_{i+1,j+1}\right)\right)-\frac{\lambda }{2}\left\{\left(f\left({U_{i+1,j}}^{n+\frac{1}{2}}\right)-f\left({U_{i,j}}^{n+\frac{1}{2}}\right)\right)+\left(f\left({U_{i+1,j+1}}^{n+\frac{1}{2}}\right)-f\left({U_{i,j+1}}^{n+\frac{1}{2}}\right)\right)\right\} $$

$$ -\frac{\eta }{2}\left\{\left(g\left({U_{i,j+1}}^{n+\frac{1}{2}}\right)-g\left({U_{i,j}}^{n+\frac{1}{2}}\right)\right)+\left(g\left({U_{i+1,j+1}}^{n+\frac{1}{2}}\right)-f\left({U_{i+1,j}}^{n+\frac{1}{2}}\right)\right)\right\} $$

(25)

The values $f\left({U_{i,j}}^{n+\frac{1}{2}}\right)$ can be evaluated by Taylor’s expansion and conservation law Eq. 15:

$$ {U_{i,j}}^{n+\frac{1}{2}}={{\overline{U}}_{i,j}}^n+\frac{\varDelta t}{2}{U}_t\left({x}_i,{y}_j,{t}^n\right) $$

$$ ={{\overline{U}}_{i,j}}^n-\frac{\varDelta t}{2}{f}_x\left(U\left({x}_i,{y}_j,{t}^n\right)\right)-\frac{\varDelta t}{2}{g}_y\left(U\left({x}_i,{y}_j,{t}^n\right)\right) $$

(26)

Finally, the staggered form can be transformed into non-staggered form. The non-staggered cell averages ${{\overline{U}}_{i,j}}^{n+1}$ using a similar method:

$$ {{\overline{U}}_{i,j}}^{n+1}=\frac{1}{4\varDelta x\varDelta y}\left({\iint}_{C_{i+\frac{1}{2},j+\frac{1}{2}}}{U_{i+\frac{1}{2},j+\frac{1}{2}}}^{n+1} dxdy+{\iint}_{C_{i-\frac{1}{2},j+\frac{1}{2}}}{U_{i-\frac{1}{2},j+\frac{1}{2}}}^{n+1} dxdy+{\iint}_{C_{i+\frac{1}{2},j-\frac{1}{2}}}{U_{i+\frac{1}{2},j-\frac{1}{2}}}^{n+1} dxdy+{\iint}_{C_{i-\frac{1}{2},j-\frac{1}{2}}}{U_{i-\frac{1}{2},j-\frac{1}{2}}}^{n+1} dxdy\right) $$

(27)

The discrete derivatives${\overset{\acute{\mkern6mu}}{U}}_{i,j}$and staggered derivatives${\overset{\acute{\mkern6mu}}{U}}_{i\pm \frac{1}{2},j\pm \frac{1}{2}}$ are computed using a limiter function at timetⁿ:

$$ {\overset{\acute{\mkern6mu}}{U}}_{i,j}= MM\left\{\alpha \left({U}_{i+1.j}-{U}_{i.j}\right),\frac{1}{2}\left({U}_{i+1,j}-{U}_{i-1,j}\right),\alpha \left({U}_{i.j}-{U}_{i-1.j}\right)\right\}\left(1\le \alpha \le 2\right) $$

(28)

Here, MM is the min-mod limiter function:

$$ MM\left\{{a}_1,{a}_2,\dots, {a}_k,\dots \right\}=\left\{\begin{array}{c}\min \left({a}_k\right) if{a}_k>0\forall k\\ {}\max \left({a}_k\right) if{a}_k<0\forall k\\ {}0\kern4.5em otherwise\end{array}\right. $$

(29)

To prevent numerical instabilities, the time steps are computed according to the Courant criterion:

$$ Courant=\frac{1}{2\alpha}\left(\sqrt{4+4\alpha -{\alpha}^2}-4\right)\left(1\le \alpha \le 4\right) $$

$$ dt=\frac{Courant.\mathit{\min}\left(\varDelta x,\varDelta y\right)}{\rho (A)} $$

(30)

Here, ρ(A) is the spectral radius of A.