2.3.2. Weir Flow Type

If the areas are divided by hydraulic or artificial structures, such as roadways, levees, field ridges, or banks, then the border may be treated as a broad-crested weir, and the weir flow formula can be used to obtain flow from one cell to the other. Such flow exchange between cells is regarded as the weir flow type. If *hk* > *hi*, then there are two possible situations, the free weir, and the submerged weir, as shown in Figure 3. When the flow condition is the free weir, the status of the flow will be critical, and when the flow condition is submerged weir, the status of the flow will be sub-critical. Below are the formulas for both flow conditions:

1. Free weir

$$\mu \left( h\_i - h\_w \right) < \frac{2}{3} (h\_k - h\_w)\_r \ Q\_{i,k} = \mu\_1 b \sqrt{2g} (h\_k - h\_w)^{\frac{3}{2}} \tag{8}$$

2. Submerged weir

$$(h\_i - h\_w) \ge \frac{2}{3}(h\_k - h\_w), \ Q\_{i,k} = \mu\_2 b \sqrt{2g} (h\_i - h\_w)(h\_k - h\_i)^{\frac{1}{2}} \tag{9}$$

where *hw* is the weir height, which is the roadway, levees or ground height; *b* is the effective width of the weir top, which is equivalent to the intersection length of two adjacent cells; *g* is gravitational constant; and *μ*<sup>1</sup> and *μ*<sup>2</sup> are the weir coefficients of the free and submerged weirs, respectively. *μ*1= 0.36–0.57. In this study, *μ*1= 0.4 and *μ*2= 2.6*μ*<sup>1</sup> [24] are used.

**Figure 3.** The free weir and the submerged weir flow.

#### 2.3.3. Pumping Station Type

If a pumping station is set up in the *i* cell, the water exchange between two adjacent cells is based on the operation principle of the pumping station. When the water level in the *i* cell exceeds the initial water-pumping level, the water exchange between the cells will be carried out according to the pumping capacity of the pump.

If the water level in *i* cell exceeds the initial water-pumping level:

$$h\_i \ge h\_{\mathcal{P}'} \ Q\_{i,k} = Q\_{\mathcal{P}} \cdot \Delta t \tag{10}$$

If the water level in *i* cell is below the initial water-pumping level:

$$
\hbar\_i < \hbar\_{p\prime} \ Q\_{i,k} = 0 \tag{11}
$$

where *hp* is the initial water-pumping level of the pumping station operation rules, Δ*t* is the time step, and *Qp* is the pumping rate during Δ*t*.

The PHD model is based on the basic equation of the quasi-2D discharge where the mathematical model is established by the explicit finite-difference method. After the discretization of Equation (1) via the use of the explicit finite difference method, we can arrive at Equation (12):

$$
\Delta h\_i = \left[ Pe\_i + \sum Q\_{i,k}(h\_\prime h\_k) \right] \Delta t / A s\_i \tag{12}
$$

Δ*hi* is the water level increment during time.
