A modified finite difference model for the modeling of flowslides

Abstract

In this paper, a modified finite difference model is proposed to simulate the propagation of flowslides. Modifications of the new model are conducted by calculating the lateral pressure coefficient k in the sliding mass and the entrainment and centrifugal effect during the transport process. The strength parameters are modified based on the size of the entrainment to consider the change in the landslide strength due to material mixing. Two dam break problems are simulated to test the accuracy and stability of the numerical scheme, and the results show good agreement with the analytical solutions and the measured data. Then, the model is used to analyze a typical flowslide: the Dagou landslide in Gansu Province, China. The model can accurately predict the details of the motion of the landslide, especially behaviors such as turning along the meandering gully and thrusting on the gully slopes due to centrifugal force.

Appendix A. Derivation of the supporting force

The unit normal vector of the sliding surface is $\mathbf{n}=\left(\mathrm{tan}\upalpha \mathbf{i}+\mathrm{tan}\upbeta \mathbf{j}+\mathbf{k}\right)/\sqrt{G}$, in which i, j, and k are the unit vectors in the x, y, and z directions, respectively. The static supporting force N_s is given by

$$ {N}_s=\left(\mathrm{d}{P}_x\mathbf{i}+\mathrm{d}{P}_y\mathbf{j}+W\mathbf{k}\right)\bullet \mathbf{n} $$

(A1)

Dividing each side of Eq. (A1) by mass m and rearranging the equation gives the following expression:

$$ \frac{N_s}{m}=\frac{\mathrm{d}{P}_x\tan \alpha +\mathrm{d}{P}_y\tan \beta +W}{m\sqrt{G}} $$

(A2)

Substituting Eq. (3) into Eq. (A2) and rearranging Eq. (A2) gives the following expression:

$$ \frac{N_s}{m}=\left(1+\left(k\frac{\partial h}{\partial x}+\frac{h}{2}\frac{\partial k}{\partial x}\right)\tan \alpha +\left(k\frac{\partial h}{\partial y}+\frac{h}{2}\frac{\partial k}{\partial y}\right)\tan \beta \right)\frac{g}{\sqrt{G}} $$

(A3)

Projecting N_s in the x direction gives the following expression:

$$ \frac{N_{sx}}{m}=\frac{N_s}{m}\mathbf{n}\bullet \mathbf{i}=\left(1+\left(k\frac{\partial h}{\partial x}+\frac{h}{2}\frac{\partial k}{\partial x}\right)\tan \alpha +\left(k\frac{\partial h}{\partial y}+\frac{h}{2}\frac{\partial k}{\partial y}\right)\tan \beta \right)\frac{g\tan \alpha }{G} $$

(A4)

The velocity vector is v = v_xi + v_yj + v_zk and is assumed to be parallel to the sliding surface. Therefore, the velocity vector can be divided into two components, v_xz and v_yz, as shown in Fig. 19. The two components are the projections of v onto the XOZ and YOZ planes, respectively, and the scalars of these projections are given by

Fig. 19

Velocity vector and centrifugal supporting force on the sliding surface

$$ {v}_{xz}=\frac{v_x}{\cos \alpha } $$

(A5)

$$ {v}_{yz}=\frac{v_y}{\cos \beta } $$

(A6)

According to Eq. (9), the centrifugal supporting forces parallel to the XOZ and YOZ plane are given by

$$ {N}_{xz}={mC}_x{\left(\frac{v_x}{\cos \alpha}\right)}^2 $$

(A7)

$$ {N}_{yz}={mC}_y{\left(\frac{v_y}{\cos \beta}\right)}^2 $$

(A8)

Projecting N_xz and N_yz in the direction normal to the sliding surface, the centrifugal supporting force N_c is given by

$$ {N}_c={N}_{xz}\frac{\left(\tan \alpha \mathbf{i}+\mathbf{k}\right)\bullet \mathbf{n}}{\sqrt{\tan^2\alpha +1}}+{N}_{yz}\frac{\left(\tan \beta \mathbf{j}+\mathbf{k}\right)\bullet \mathbf{n}}{\sqrt{\tan^2\beta +1}} $$

(A9)

Substituting n and Eqs. (A7) and (A8) into Eq. (A9) and rearranging the equation gives the following expression:

$$ {N}_c=\left(\frac{C_x}{\cos \alpha }{\left(\frac{v_x}{\cos \alpha}\right)}^2+\frac{C_y}{\cos \beta }{\left(\frac{v_y}{\cos \beta}\right)}^2\right)\frac{m}{\sqrt{G}} $$

(A10)

Dividing each side of Eq. (A10) by m and projecting N_c in the x direction gives the following expression:

$$ \frac{N_{cx}}{m}=\frac{N_c}{m}\mathbf{n}\bullet \mathbf{i}=\left(\frac{C_x}{\cos \alpha }{\left(\frac{v_x}{\cos \alpha}\right)}^2+\frac{C_y}{\cos \beta }{\left(\frac{v_y}{\cos \beta}\right)}^2\right)\frac{\tan \alpha }{G} $$

(A11)

Eq. (10) is the combination of Eqs. (A4) and (A11).