Flood inundation simulation in river basin using a shallow water model: application to river Yamuna, Delhi region

Abstract

River Yamuna is one of the major rivers of India and the largest tributary of the Ganga. The river faces the hazards of annual floods, susceptibility to erosion and adverse impact of anthropogenic factors. The Delhi stretch of the river Yamuna is located between 28°24′17″ and 28°53′00″N and between 76°50′24″ and 77°20′37″E. The area is more vulnerable to annual floods and apart from this the stretch has wide sandy beds bordered by high banks which are subjected to annual inundation. To estimate the inundation in the river banks, our previously developed mathematical model (Senthil et al., Appl Math Comput 216:2544–2558, 2010), based on unsteady, depth averaged shallow water equations is applied to this particular case study of Yamuna river. The model uses a coordinate transformation which enables the deforming boundaries to be transformed to fixed boundaries in the computational plane. The model is based on the finite difference scheme which is conditionally stable. The 2001 bank lines data of the river is taken as the input along with the actual discharge data of 2001 to predict the change in the width of the river. The average increase of 3.19 % (simulated) of the entire stretch compares well with the observed result of 3.87 %. The simulated results are in conformity with the observed data, the maximum increase in the width being found between the Nizamuddin–Noida toll bridge regions. The characteristics such as (i) spatial extent of the flood on the floodplain, (ii) depth of the flood water, (iii) velocity of the flood water and (iv) the time it takes for the flood water to spread, can be produced with the help of the simulated results such as flooded area, inundation-depth, inundation-time and flow-field. In other words these results can be used as a tool to produce hydrologic data of the river banks as well as to evaluate the effects on the river banks due to flooding.

Appendix

1.1 Coordinate transformation

Equations (1)–(3), after the coordinate transformation, reduce to

$$ \frac{\partial}{\partial t}(Hb)+\frac{\partial}{\partial \xi}(HbU)+\frac{\partial}{\partial y}{(\widetilde{v})}=0 $$

(12)

$$ \frac{\partial \widetilde{u}}{\partial t}+\frac{\partial}{\partial \xi}{(U \widetilde{u})}+\frac{\partial}{\partial y}{(\widetilde{u}v)}=-gH \frac{\partial \zeta}{\partial \xi}-{ \frac{c_f \widetilde{u}}{H}}{(u^2 + v^2)}^{1/2} $$

(13)

$$ \frac{\partial \widetilde{v}}{\partial t}+\frac{\partial}{\partial \xi}{(U \widetilde{v})}+\frac{\partial}{\partial y}{(\widetilde{v}v)}=-gH \left[b\frac{\partial \zeta}{\partial y}-\left[\frac{\partial b_1}{\partial y}+\xi\frac{\partial b}{\partial y}\right]\frac{\partial \zeta}{\partial \xi}\right] - \frac{c_f \widetilde{v}}{H}{(u^2 + v^2)}^{1/2} $$

(14)

$$ bU=u-\left[\frac{\partial b_1}{\partial t}+\xi \frac{\partial b}{\partial t}\right]-v\left[\frac{\partial b_1}{\partial y}+\xi \frac{\partial b}{\partial y}\right], $$

(15)

$$ \widetilde{u}=Hbu, \widetilde{v}=Hbv $$

(16)

where $H=(\zeta+h)$ is the total depth of the basin. From Eqs. (7), (8) and (15) we see that the kinematical condition at the curvilinear boundaries is always satisfied provided U = 0 at ξ = 0 and ξ = 1. The open boundary conditions are as given in (9) and (10).

1.2 Numerical procedure

In the ξ–y plane we use a staggered Arakawa C grid where there are three distinct types of computational points. The grid lines are all parallel to the coordinate axis and form a uniform network with a rectangular mesh having sides of length $\Updelta \xi$ in the ξ-direction and $\Updelta y$ in the y-direction. We use forward difference in the t direction ($\Updelta_t G$) and central difference for ξ and y directions (δ_ξ G and δ _y G).

Any variable G, at a grid point (i,j) and time t may be represented as

$$ G(\xi_i, y_j, t_p)= G^{p}_{i, j}; \quad p=0,1{\ldots};\quad i=1,2{\ldots}m;\quad j=1,2{\ldots} n; $$

(17)

The finite difference equations are as follows:

$$ {\Updelta_t}(Hb)+\delta_{\xi}(\bar H{^\xi}bU)+{\delta_y}(\widetilde v)=0 $$

(18)

$$ {\Updelta_t}\widetilde u+\delta_{\xi}(\bar U^{\xi}{\bar {\widetilde u}^{\xi}})+{\delta_y}(\bar v^{\xi}{\bar {\widetilde u}^y})=-g E_t(\bar H^{\xi} \delta_{\xi}\zeta) -\frac{c_f[u^2+(\bar v^{\xi y})^2]^{\frac{1}{2}}E_t(\bar u)}{E_t(\bar H^{\xi})} $$

(19)

$$ \begin{aligned} {\Updelta_t}\widetilde v+\delta_{\xi}(\bar U^y{\bar {\widetilde v}^{\xi}})+{\delta_y}(\bar v^y {\bar {\widetilde v}^y})&=-g E_t\left[b\bar H^y {\delta_y}\zeta) -\left( \delta_y b_1+\xi \delta_y b \right) \bar H^y {\delta_{\xi}}\bar \zeta^{\xi y}\right]\\ &\quad- \frac{c_f[v^2+(\bar u^{\xi y})^2]^{\frac{1}{2}}E_t(\bar v)}{E_t(\bar H^y)} \end{aligned} $$

(20)

where E _t G = G ^p+1 _i,j . Bar over a variable corresponds to average over two neighbouring points in the ξ i.e., $(\overline{G}^\xi)$ or y i.e., $(\overline{G}^y)$ directions.

Equations (19) and (20) are used for updating $\widetilde u$ and $\widetilde v$ respectively and then the values of u and v are deduced from (16),

$$ u=\frac{\widetilde u}{b \bar H^{\xi}}, \quad v=\frac{\widetilde v}{b \bar H^y} $$

(21)

After updating u and v, the updated value of U be obtained by applying (15) in the discretized form,

$$ U=\frac{1}{b}\left[u-(\delta_t b_1+\xi \delta_t b)-(\delta_y b_1+\xi \delta_y b)\bar v^{\xi y}\right] $$

(22)

The computational stability subject only to the time-step being limited by the space increment and gravity wave speed. This is governed by the Courant–Friedrich–Lewy (CFL) condition which provides a guideline for choosing an appropriate $\Updelta t$ (i.e., a $\Updelta t$ for which the scheme is stable), i.e.,

$$ \sqrt{2gh_{\rm max}} \frac{\Updelta t}{\Updelta x}\leq 1 $$

(23)

where h _max is the maximum depth and $\Updelta x$ is the space increment. The details relating to the stability characteristics of this computational scheme are given in Senthil et al. [14].