Modelling and Simulation in Engineering

Volume 2019, Article ID 7053131, 13 pages

https://doi.org/10.1155/2019/7053131

## Flood Simulation by a Well-Balanced Finite Volume Method in Tapi River Basin, Thailand, 2017

^{1}Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok, Thailand^{2}Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, Thailand

Correspondence should be addressed to Montri Maleewong; ht.ca.uk@m.irtnom

Received 30 August 2018; Revised 13 November 2018; Accepted 6 December 2018; Published 15 January 2019

Academic Editor: Mohammed Seaid

Copyright © 2019 Sutatip Vichiantong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Flood simulation of a region in southern Thailand during January 2017 is presented in this work. The study area covers the Tapi river, the longest river in southern Thailand. The simulation is performed by applying the two-dimensional shallow water model in the presence of strong source terms to the local bottom topography. The model is solved numerically by our finite volume method with well-balanced property and linear reconstruction technique. This technique is accurate and efficient at solving for complex flows in the wet/dry interface problem. Measurements of flows are collected from two gauging stations in the area. The initial conditions are prepared to match the simulated flow to the measurements recorded at the gauging stations. The accuracy of the numerical simulations is demonstrated by comparing the simulated flood area to satellite images from the same period. The results are in good agreement, indicating the suitability of the shallow water model and the presented numerical method for simulating floodplain inundation.

#### 1. Introduction

To simulate flooding over an affected area of terrain, the two-dimensional shallow water model is one of the most efficient models. Since the nonlinear shallow water model is complicated, an efficient and accurate numerical method is required to find approximate solutions in terms of water depth and velocity. The finite volume method is an accurate numerical method that can be developed to solve the problem (for more details and reviews, see [1, 2]). This scheme requires an accurate numerical flux scheme for approximating the flux at cell interfaces in the shallow water equations. One extensively used scheme is Harten-Lax-van Leer (HLL) [3–5]. The modified version for two-dimensional problems is HLLC [5]. To obtain a more accurate approximation, the weighted average flux (WAF) has been introduced [4, 5–10, 11]. The WAF approximation is widely applied to the finite volume method. It can solve various types of problems (see, for instance, [5, 7, 9, 11, 12]).

In this work, we will apply the finite volume method with the WAF approximation for simulating a flood. The accuracy of numerical scheme depends on the method for approximating the bottom slope as discussed in [5]. Here, we improve the accuracy of numerical results by applying a linear reconstruction as described in [13]. The numerical scheme is second-order accurate in space for smooth flows with smooth bottom [13]. Instead of approximating bottom slope as presented in [11], a well-balanced scheme with bottom slope approximation is developed. The structured rectangular meshes are used due to its simple structure to develop a well-balanced WAF finite volume scheme [7, 10, 14, 15]. In addition, this kind of discretization can be applied directly to simulate real flood using the digital elevation model (DEM) from [16]. In application, we have to solve the nonlinear model that interacts with the nonlinear source term from the bottom topography. In this case, we improve the accuracy of numerical results by applying a linear reconstruction [1–3] for both water depth and bottom profile. The developed scheme is second-order accurate in space not only for smooth flow problem but also high-gradient water depth flow (see numerical experiments in [13]). To ensure second-order accuracy in time integration, we apply the second-order Runge–Kutta (RK2) method. When dealing with another source term of friction slope, the splitting implicit method proposed by Kesserwani and Liang [17, 18] is applied in our scheme. By combining all of these techniques, our numerical scheme is accurate and efficient; this will be demonstrated by the numerical experiments described below before applying the scheme to simulate a natural flood event recorded in Thailand in January 2017. The study area is the Tapi river basin located in southern Thailand. We propose a method to prepare the initial conditions necessary to conduct the flood simulation. All data utilized in our work for demonstrating the capability and performance of our numerical scheme are provided on the internet: the bottom topology, the satellite imagery, and the flow data collected at two gauging stations.

The paper is organized as follows. We describe the finite volume method with the weighted average flux and linear reconstruction for two-dimensional shallow water equations in Sections 2.1–2.3. The well-balanced finite volume method is presented in Section 2.4. Numerical tests are shown in Sections 3.1–3.3. Flood simulations are shown in Section 4. The conclusions are finally given in Section 5.

#### 2. Numerical Scheme for Shallow Water Equations

Flood over a large area can be simulated by considering the two-dimensional shallow water equations. The governing equations are given bywhere is the water depth, and are the flow velocities in the *x*- and *y*-direction, respectively, is the acceleration due to gravity, and *z* is the bottom elevation. and are the friction terms in the in *x*- and *y*-direction, respectively. Here, with *n* denoting Manning’s Roughness coefficient.

The conservative form of equations (1)–(3) iswhere , , , and .

Next, we will apply our developed scheme to numerically approximate the water level and velocity at each location and time of our studied area.

##### 2.1. Finite Volume Method

A discretized form of (4) iswhere is the approximation of , over cell , given bywith and for and . is the source term approximation at cell . and are the numerical fluxes in the *x*- and *y*-direction, respectively. We will apply the weighted average flux (WAF) to approximate these terms. Details are provided in Section 2.2.

The discretization in time is performed by the second-order Runge-Kutta (RK2) method to ensure the second-order accuracy in time of our method.

##### 2.2. Weighted Average Flux (WAF)

We first consider the approximation of numerical flux in the *x*-direction at interface () denoted by as follows:where is the solution of the Riemann problem from constant data and .

The weighted average flux in two dimensions is proposed by [5, 11]. It is composed of three flux components as follows:where *p* denotes the component of the numerical flux vector at the interface and is the value of flux in the region *k* of the solution of the Riemann problem at component *p*. The first region , the third region , and the flux in the intermediate region are approximated by the Harten-Lax-van Leer (HLL) approach [11], where and are the solutions from the left and the right limits at the interface and are the velocities in *y*-direction from the left and the right limits at the interface. The weighted values are calculated from the wave speeds in the left, the right, and the intermediate regions, respectively. WAF approximation in the *y*-direction, and at other interfaces can be obtained similarly.

To avoid unexpected oscillations near a discontinuous water level profile, the WAF can be applied while enforcing the total variation diminishing (TVD); more details can be found in [13].

##### 2.3. Linear Reconstruction

Second-order accuracy in space from constant data can be obtained by applying linear reconstruction [1–3, 13]. For instance, in the *x*-direction, the unknown variables are reconstructed before calculating numerical fluxes bywhere is a that there are various forms. Here, we applied the minmod slope limiter given bywhere

By the same concept, the linear reconstruction in the *y*-direction can be obtained by applying equations (9)–(12).

##### 2.4. Well-Balanced Scheme

A well-balanced concept is included in our developing scheme for preserving the stationary solution at the steady state. The concept is obtained from considering just the one-dimensional flow at the steady state where the stationary solution must satisfy the following conditions:

Following Bermudez and Vazquez [19], a numerical scheme is called a well-balanced scheme if it satisfies the exact C-property, namely,

Similarly, for the two-dimensional flow problem, the conditions are

To obtain the well-balanced scheme, we follow the reconstruction approach proposed by Audusse et al. [20]. We reconstruct *h* at the interfaces in the *x*- and *y*-direction bywhere and .

The advantage of these reconstructions is that it can preserve the non-negativity of the water depth [20].

In this work, we propose a technique to modify the conservative variables to be

The finite volume scheme is then expressed bywhere the numerical flux in the *x*-direction and the bottom slope terms are modified to

Similarly, the numerical flux in the *y*-direction, and can be obtained by the decomposition of the bottom slopes in the third component. By applying these reconstructions, the finite volume method with the WAF approximation becomes a well-balanced scheme that preserves the exact C-property in two dimensions at the steady state. Some numerical tests are shown in the next section to confirm this property.

Moreover, to overcome the difficulties in calculating the source term for wet/dry problem, the friction term is approximated by the splitting implicit technique (see more details in [13, 17, 18]).

#### 3. Numerical Tests

In this work, the scheme without linear reconstruction is referred to as scheme I, and the scheme with linear reconstruction is referred to as scheme II. Scheme II is second-order accurate in space for a smooth flow over a smooth bottom (see numerical experiment in [13]). Before applying our numerical method to simulate the observed flood event in Thailand, the accuracy of numerical scheme is checked by performing three test cases: still water stationary state, convergence of flow to still water stationary state, and partial dam-break flow. The problems and simulation setting are given in the following sections.

##### 3.1. Still Water Stationary State

This experiment is performed to check the well-balanced property of the present scheme. The scheme is a well-balanced scheme if it satisfies the exact C-property; that is, the numerical solution should preserve the still water stationary solution at steady state. In this experiment, we consider a rectangular domain of 1500 m long with the discontinuous bottom defined by

The initial water depth is and velocity is zero in an entire domain. Simulation is run on uniform 1000 cells. The numerical result of water depth and velocity by scheme II at final time 100 s preserves still water stationary solution as shown in Figure 1.