In this paper, the Mike21C model is used to simulate the water depth, flow discharge, suspended load and bed load concentration of the study river reaches in the flood and dry season. Mike21C is one of the most comprehensive and well-established tools for simulating river bed and channel planform development caused by changes in the hydraulic regime. Simulated processes include alluvial resistance, bank erosion, and scouring and shoaling caused by various activities, such as construction and dredging, and seasonal flow fluctuations (Khue 1985; Lai 1987; Jin and Steffler1993). This model is approximated by using FDM in curved coordinates (Ahmadi et al. 2009; Beck and Basson 2007; DHI 1995; Dang and Park 2016; Talmon 1992; Gulkac 2005; McGuirk and Rodi 1978). Structurally, Mike21C has three main modules: the flow module, sediment transport module, and river morphology module. Mike21C model is applied to simulate the water level fluctuation, flow discharge distribution, suspended load transport rate, and bed level variation in the river downstream.

### Hydrodynamic module

The flow module based on the three-dimensional (3D) hydrodynamic model is complex. Application of the 3D model for simulating long time scales (e.g., months, season, and years) elevation to river morphology is a complicated process. To overcome this obstacle, scientists have converted the main hydrodynamics module into 2D equations representing the conservation of momentum and mass horizontally (DHI 1995; Ye and McCorquodale 1997).

The hydrodynamic equations are expressed as follows:

$$\frac{{\partial {\text{p}}}}{{\partial {\text{t}}}} + \frac{\partial }{{\partial {\text{s}}}}\left( {\frac{{{\text{p}}^{ 2} }}{\text{h}}} \right) + \frac{\partial }{{\partial {\text{n}}}}\left( {\frac{\text{pq}}{\text{h}}} \right) - 2\frac{\text{pq}}{{{\text{hR}}_{\text{n}} }} + \frac{{{\text{p}}^{ 2} - {\text{q}}^{ 2} }}{{{\text{hR}}_{\text{s}} }} + {\text{gh}}\frac{{\partial \,{\text{H}}}}{{\partial {\text{s}}}} + \frac{\text{g}}{{{\text{C}}^{ 2} }}\frac{{{\text{p}}\sqrt {{\text{p}}^{ 2} + {\text{q}}^{ 2} } }}{{{\text{h}}^{ 2} }} = {\text{RHS}}$$

(1)

$$\frac{{\partial {\text{q}}}}{{\partial {\text{t}}}} + \frac{\partial }{{\partial {\text{s}}}}\left( {\frac{\text{pq}}{\text{h}}} \right) + \frac{\partial }{{\partial {\text{n}}}}\left( {\frac{{{\text{q}}^{ 2} }}{\text{h}}} \right) + 2\frac{\text{pq}}{{{\text{hR}}_{\text{s}} }} + \frac{{{\text{q}}^{ 2} - {\text{p}}^{ 2} }}{{{\text{hR}}_{\text{n}} }} + {\text{gh}}\frac{{\partial {\text{H}}}}{{\partial {\text{n}}}} + \frac{\text{g}}{{{\text{C}}^{ 2} }}\frac{{{\text{q}}\sqrt {{\text{p}}^{ 2} + {\text{q}}^{ 2} } }}{{{\text{h}}^{ 2} }} = {\text{RHS}}$$

(2)

$$\frac{{\partial {\text{H}}}}{{\partial {\text{t}}}} + \frac{{\partial {\text{p}}}}{{\partial {\text{s}}}} + \frac{{\partial {\text{q}}}}{{\partial {\text{n}}}} - \frac{\text{q}}{{{\text{R}}_{\text{s}} }} + \frac{\text{p}}{{{\text{R}}_{\text{n}} }} = 0$$

(3)

where s, n are the co-ordinates in the curvilinear co-ordinate system; h is the water depth; p and q are the mass fluxes in the s-and n directions, respectively. H is the water level; C is the Chezy roughness coefficient; g is the gravitational acceleration; R_{s} and R_{n} are the radius of curvatures of s- and n-line, respectively; RHS is the right hand side describing the Reynolds’ stresses.

### Sediment transport module

In the sediment transport module, the suspended load transport equations under the control of convection and diffusion are expressed as follows (Duc 2004; Meyer and Müller 1948; Galappatti 1983; Jia and Wang 2001):

$$\frac{{\partial {\text{c}}}}{{\partial {\text{t}}}} + {\text{u}}\frac{{\partial {\text{c}}}}{{\partial {\text{x}}}} + \nu \frac{{\partial {\text{c}}}}{{\partial {\text{y}}}} + {\text{w}}\frac{{\partial {\text{c}}}}{{\partial {\text{z}}}} = {\text{w}}_{\text{s}} \frac{{\partial {\text{c}}}}{{\partial {\text{z}}}} + \frac{\partial }{{\partial {\text{x}}}}\left( {\varepsilon \frac{{\partial {\text{c}}}}{{\partial {\text{x}}}}} \right) + \frac{\partial }{{\partial {\text{y}}}}\left( {\varepsilon \frac{{\partial {\text{c}}}}{{\partial {\text{y}}}}} \right) + \frac{\partial }{{\partial {\text{z}}}}\left( {\varepsilon \frac{{\partial {\text{c}}}}{{\partial {\text{z}}}}} \right)$$

(4)

where z is the vertical coordinate; W_{S} is the settling velocity of the sediment particles, c is the suspended load concentration; ε is the eddy viscosity coefficient; u, v, and w are the flow velocity components in the x-, y-, and z-directions, respectively.

Ignoring the limited diffusion outside of the vertical diffusion, (4) becomes:

$$\frac{{\partial {\text{c}}}}{{\partial {\text{t}}}} + {\text{u}}\frac{{\partial {\text{c}}}}{{\partial {\text{s}}}} + {\text{v}}\frac{{\partial {\text{c}}}}{{\partial {\text{n}}}} + {\text{w}}\frac{{\partial {\text{c}}}}{{\partial {\text{z}}}} = {\text{w}}_{\text{s}} \frac{{\partial {\text{c}}}}{{\partial {\text{z}}}} + \frac{\partial }{{\partial {\text{z}}}}\left( {\varepsilon \frac{{\partial {\text{c}}}}{{\partial {\text{z}}}}} \right)$$

(5)

Bed load (S_{bl}) is very closely related to the suspended load. Many formulas of the bed load transport are based on the calibration coefficients k_{b} and k_{s}. Engelund and Hansen (1967) had established the relationship k_{b} + k_{s} = 1, which is used in many models, including Mike21C (DHI 1995).

$${\text{S}}_{\text{bl}} \, = {\text{ k}}_{\text{b}} \cdot {\text{S}}_{\text{tl}}$$

(6)

$${\text{S}}_{\text{sl}} \, = {\text{ k}}_{\text{s}} \cdot {\text{S}}_{\text{tl}}$$

(7)

where k_{s} and k_{b} are the suspended load and the bed load coefficients, respectively

S_{tl} is the total volume of sediment transported determined according to the formula:

$${\text{S}}_{\text{tl}} = 0.05\frac{{{\text{C}}^{2} }}{\text{g}}\uptheta^{{\frac{5}{2}}} \sqrt {\left( {\rho_{\text{ds}} - 1} \right){\text{gd}}_{50}^{3} }$$

(8)

where ρ_{ds} is the relative proportion of sediment (relative density of the sediment); d_{50} is the diameter of the sediment particles; Shields parameter θ is determined by;

$$\uptheta = \frac{\tau }{{\rho {\text{g(}}\rho_{\text{ds}} - 1 ) {\text{d}}_{ 5 0} }}$$

(9)

where τ is the flow shear stress; ρ is the water density (density of water); Shear flow is divided into two types: form drag and friction, as estimated based on local flow velocity u and the local Chézy C.

### Morphological module

In the river morphology module, the hydrodynamic solution must first be obtained before solving the sediment transport equation. Next, the river bed and hydrodynamic model are applied (Vriend and Struiksma 1983; Koch 1980; Mosselman 1992; Odgaard 1983; Olesen 1987; DHI 1995).

The equation describing bank erosion is given as follows:

$${\text{E}}_{\text{b}} = - \alpha \frac{{\partial {\text{z}}}}{{\partial {\text{t}}}} + \upbeta \frac{\text{S}}{\text{h}} + \upgamma$$

(10)

where E_{b} is the bank erosion rate in m/s; S is the near bank sediment transport; z is the local bed level; α, β, and γ are the calibration coefficients specified in the model.

Calculation of the bed level variation is based on the sediment continuity equation in the Cartesian coordinate system (Vanoni 1975; Galappatti 1983; DHI 1995):

$$\left( {1 - {\rm p}} \right)\frac{\partial z}{\partial t} + \frac{{\partial {\rm S}_{x} }}{\partial x} + \frac{{\partial {\rm S}_{y} }}{\partial y} = \Delta {\rm S}_{e}$$

(11)

where S_{x}, S_{y} are the total volume of sediment transported along x and y, respectively; p is the porosity of the bed; ∆S_{e} is the excess sediment supply from erosion of the bed (Fig. 1).