Flood modeling by 2D shallow water equation

Last update: Nov 30, 2022

hydraulicmodel

Flood modeling by 2D shallow water equation. Refer to Hunter et al (2005), Bates et al. (2010).

Diffusive wave approximation

$\frac{\partial h ^{i, j}}{\partial t} = \frac{Q_x^{i-1, j} - Q_x^{i, j} Q_y^{i, j-1} - Q_y^{i, j}}{\Delta x \Delta y}$

$Q_x^{i, j} = \frac{h_{flow}^{5/3}}{n} \left( \frac{h^{i-1, j} - h^{i, j}}{\Delta x} \right) ^{1/2} \Delta y$

$Q_y^{i, j} = \frac{h_{flow}^{5/3}}{n} \left( \frac{h^{i-1, j} - h^{i, j}}{\Delta y} \right) ^{1/2} \Delta x$

Local inertail approaximation

$\frac{\partial h ^{i, j}}{\partial t} = \frac{Q_x^{i-1, j} - Q_x^{i, j} Q_y^{i, j-1} - Q_y^{i, j}}{\Delta x \Delta y}$

$q_{t \Delta t} = \frac{q_t - gh_{flow, t} \Delta t \frac{\partial h_t}{\partial x}} {1 g n^2 \Delta t q_t / h_{flow, t}^{7/3}}$

Time steps in two methods

Diffusive wave approximation

$\Delta t = \frac{\Delta x^2}{4} min \left( \frac{2n}{h_{flow}^{5/3}}|\frac{\partial h}{\partial x}|^{1/2}, \frac{2n}{h_{flow}^{5/3}}|\frac{\partial h}{\partial y}|^{1/2} \right)$

Local inertial approximation

$\Delta t = \alpha \frac{\Delta x}{\sqrt{g (h-z)_{max}}}$

Implementation of these two approximation for simulating the flood in Mobara city on October 25, 2019

Setting of the domain and boundary conditions

Results of the maximum inundation area calculated by the diffusive wave approximation, compared with the flood extent reported by Chiba Office

Results of the inundation range considering the flood recession by the local inertial approximation, compared with the flood extent estimated by Geospatial Information Authority of Japan (GSI).

Flood modeling by 2D shallow water equation

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Overview

hydraulicmodel

Diffusive wave approximation

Local inertail approaximation

Time steps in two methods

Implementation of these two approximation for simulating the flood in Mobara city on October 25, 2019

Evolution of flood volume

Owner

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