River Flood Modelling under Limited Data Acquisition using PWRI Hydrologic Model
50 points, even though the value is increased, a change in the discharge is not observed. The user needs to know the inflection points of all of these variables.
It requires adding a calibration module with a sequence of parameters so that the user can determine the optimal values using a numerical analysis method to determine those values.
5. Integrated Input of DEMs in another format, there are other sources of DEMS additional to the three integrated options, where other options can be added.
6. Minimum size cell: The minimum accepted cell size is 100 m, which is related to the resolution of the DEM included within the IFAS, such as the cases of the three integrated DEMs, Global Map, GTOPO30, HYDROK, which have a resolution of 30 arcsec. It should be considered that other digital models of elevation, free access have a lower resolution LiDAR from 1m to 30m, USGS from 5m to 30m, ASTERGDEM 30m, and others. The cell size determines the resolution of the DEM, which influences the delineation of the basin and the river.
7. Even are non-linear relationships, and it is not a complete Black Box, the parameters have not limited; the values of them can change the concept of the model.
8. There is some issue related to the storage function, the reproducibility in the different event depends on the parameter, there is a challenge to determine how many cases are necessary to study
9. The solution using time integral function is an approximation, that means to solve the differential equation; it does not use the convergence calculation process.
51 2.2. Some cells, called river cells, define the mainline of the drainage network or river.
There are four types of cells: cell type 0, cell type 1, cell type 2, and cell type 3, been cell type 3, those with greater drainage area, whereas cells type 0 are cells with the minimum drainage area. In cells type 1, 2, and 3, we find the four tanks (surface, subsurface/unsaturated, aquifer and river tank) while in type 0 cell, we only find the surface, the subsurface/unsaturated and the aquifer tank.
Figure 2.1. PWRI-DH cells classification schema.
The calculations are made to estimate the flow of each cell individually as well as the flow between cells. The discharge is the result of the flow originating from different cells and the flow originating from river routing. When solving the flood problem, the PWRI-DH model takes the nonlinear form and uses approximation functions to solve the differential equation for solving the time integral equation. In this way, the numerical calculation is performed smoothly, allowing real-time operation.
Figure 2.2 Configuration according to the cell classification
1
1 1
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
1
0
0 0
0 0 3 3
3 3
3
3 3
1 1
2 1
2 2
0
2
2
Flow from cell type 0 Flow from cell type 2
River Flood Modelling under Limited Data Acquisition using PWRI Hydrologic Model
52 As is showed in the drainage course diagram, the rainwater is routed by flow direction indicate, where the groundwater flow, intermediate flow, surface flow, and river channel flow follow the same direction. For the calculation, the flow direction (drainage course) is required for every cell. The outlet is only one and is identified as a river cell. IFAS use slope to determine the flow directions with the eight neighborhood cell.
PWRI-DH model works with the parameters of the surface, subsurface, aquifer, and river tanks in order to perform the runoff analysis. The parameters set contain, among others, the maximum storage height and runoff coefficient for each tank. It also includes parameters whose primary function is presented as follows.
In order to introduce the parameters required, it is necessary to study the configuration of the tanks used in the model. The model is composed of four tanks: the surface tank, the subsurface/unsaturated tank, the aquifer tank, and the river tank. Each tank has its group of parameters as it is explained in Equation 2.1 to 2.19. Table 2.1 to 2.4 shows the parameter definitions and nomenclature; also, the values of each parameter recommended as baseline parameters by IFAS. Note that in most of the cases, parameters are defined using two types of nomenclature: the existing in the literature and the one used by IFAS.
The reason we place the two nomenclature is to avoid confusion in IFAS users.
Figure 2.3 illustrates the surface tank, which is used to simulate the flow on the surface ground, the flow portion that streams to subsurface runoff, and the percolation to the groundwater tank. Equations 2.1 to 2.3 show the relationship governing the outflows obtained from this tank.
Figure 2.3. Representation of the surface tank 𝑄𝑜𝑓 = 𝐿
𝑁(ℎ𝑎− ℎ𝑓2)
5
3√𝑖 (2.1)
𝑄𝑜𝑓
𝑄𝑟𝑠𝑓
ℎ𝑓1 ℎ𝑓2
ℎ𝑓0 𝑄𝑔𝑖
Rainfall
ℎ𝑎
53 𝑄𝑟𝑠𝑓 = 𝛼𝐴𝐼0 (ℎ𝑎−ℎ𝑓1)
(ℎ𝑓2−ℎ𝑓1) (2.2)
𝑄𝑔𝑖 = 𝐴𝐼0 (ℎ𝑎−ℎ𝑓0)
(ℎ𝑓2−ℎ𝑓0) (2.3)
Where Qof is the saturation excess overland flow, Qrsf is the rapid subsurface stormflow, Qgi is the infiltration flow to the aquifer, L is the cell length, A is the cell area, i is the spatial increment, N is the surface roughness coefficient, ha is the water level, hf1 is the height where from which rapid subsurface stormflow occurs, α is a coefficient that regulates the rapid intermediate, I0 is the final infiltration capacity, hf2 is the maximum value of water height where saturated excess overland flow occurs, and hf0 is the height where the ground infiltration occurs. The surface roughness coefficient (N) depends on the land use classification.
Figure 2.4 shows the subsurface/unsaturated tank. This tank works as an intermediate zone and represents subsurface storage. The outflows from this tank are the infiltration to the aquifer, the subsurface runoff, and the low intermediate. The equations governing this model are given by Equations 2.4 to 2.5.
Figure 2.4. Representation of the subsurface/unsaturated tank 𝑄2𝑠𝑠𝑓 = 𝐶𝑥ℎ2𝑚𝑎𝑥 𝑖 (2.4)
𝑄2𝑔𝑖 = 𝐶𝑧 𝐴 (2.5)
𝑄2𝑠𝑠𝑓 ℎ2𝑚𝑎𝑥
𝑄2𝑔𝑖 𝑄𝑔𝑖
ℎ2
River Flood Modelling under Limited Data Acquisition using PWRI Hydrologic Model
54 Where Qgi is the flow originating from the surface tank, Q2ssf is the slow subsurface stormflow, Q2gi is the infiltration flow to the aquifer tank, Cx is the horizontal hydraulic conductivity at θ, Cz is the vertical hydraulic conductivity at θ, h2max is the maximum water height, A is the cell area, and i is the spatial increment.
𝐶𝑥 and 𝐶𝑧 are given by Equations 2.6 and 2.7, respectively, as follows:
𝐶𝑥 =𝐶𝑆𝑋
100∗ exp(𝑏𝜃)−exp (𝑏𝜃𝑤)
exp(𝑏𝜃𝑠)−exp (𝑏𝜃𝑤) (2.6)
𝐶𝑧= 𝐶𝑆𝑍∗ exp(𝑏𝜃)−exp (𝑏𝜃𝑤)
exp(𝑏𝜃𝑠)−exp (𝑏𝜃𝑤) (2.7) where θ is given by Equation 8 𝜃 = ℎ2
ℎ2𝑚𝑎𝑥 (2.8)
Where h2 is the water height, b is a constant that depends on the soil total porosity, θ is the soil moisture content, θs is the soil moisture at saturation, θw is the soil moisture at the wilting point, Csx is the horizontal saturated hydraulic conductivity at θs, and Csz is the vertical saturated hydraulic conductivity at θs.
Figure 2.5 represents the aquifer tank, which is used to simulate the outflow from the aquifer and the aquifer losses. Equations 2.9 and 2.10 show the relationship used in this tank. Inflow infiltration mode (Qif ) is the result of flow that originates from the ground infiltration of the surface tank and the subsurface/unsaturated tank
Figure 2.5. Representation of the aquifer tank
𝑄3𝑢𝑎𝑓
ℎ3
ℎ3𝑚𝑎𝑥
𝑄𝑢𝑤𝑙 𝑄𝑖𝑓
𝑄𝑎𝑏𝑓
55 𝑄3𝑢𝑎𝑓 = 𝐴𝑢2(ℎ3− ℎ3𝑚𝑎𝑥)2𝐴 (2.9)
𝑄𝑎𝑏𝑓 = 𝐴𝑔ℎ3𝐴 (2.10)
Where Q3uaf is the unconfined aquifer outflow, Qabf is the aquifer base flow, Qif is the inflow infiltration mode, Quwl is the unaccountable aquifer losses, h3max is the height from which unconfined aquifer outflow occurs, h3 is the water height, and Au and Ag are the unconfined and confined aquifer outflows coefficients, respectively.
Figure 2.6 represents the river tank. Here Qin is the flow resulting from the other three tanks; hR is the water level in the river tank, and 𝑄 is the river discharge.
Figure 2.6. Representation of the river tank (tank D).
The solution of Q in the river tank depends on the cell type classification.
In type 1 and 2 cells, the Manning equation, given by Equation 2.11, is used to calculate the discharge (Q). Herein the transversal section of the flow in these two cells is assumed to be square.
𝑄 = 1
𝑛((𝐵ℎ𝑅)53)𝑆12 (2.11) In which
𝐵 = 𝐶1𝐴𝑐𝐶2 (2.12)
Where n is the Manning roughness coefficient, B is the river width, S is the friction slope, Ac is the basin area, C2 is a constant that is assumed to be 0.5 and C1 is a constant related
𝑄𝑖𝑛
𝑄 ℎ𝑅
River Flood Modelling under Limited Data Acquisition using PWRI Hydrologic Model
56 to the characteristics of the river that is adjusted according to the relation between the width of the river and the discharge that can take values between 3.5 and 7.
In type 3 cells, the kinematic wave model is applied. Equation 2.13 shows the differential form of the kinematic wave model, assuming an idealized surface flow, considering a wide and shallow channel. The flow regime is considered entirely turbulent. At this point, the transversal section assumed is the combined geometric form shown in Figure 2.7.
𝑄𝑖+1𝑘+1= (
1 2∆𝑡+ 𝐶
2∆𝑥)𝑄𝑖𝑘+(1 2∆𝑡− 𝐶
2∆𝑥)𝑄𝑖+1𝑘 +( 𝐶 2∆𝑥− 1
2∆𝑡)𝑄𝑖𝑘+1 1
2∆𝑡+2∆𝑥𝐶 (2.13)
where the spatial increment is the cell size, ∆𝑥 = 𝐿.
Figure 2.7. Transversal section in cell type 3.
The constant C takes on different values, depending on the water height of the river tank (hR), as follows:
If ℎ𝑅 ≤ 𝑑2, then 𝐶 =5
3𝑄25𝑛−(
3 5)
𝑆103𝐵−(
2
5) = 𝐸 (2.14)
If 𝑑3< ℎ𝑅 < 𝑑2, then
𝐶 = 𝐵
𝐵+2( (𝑄 𝑛
𝐵 𝑆 1 2 )
3 5
−𝑑3 ) 𝐶6
𝐸 (2.15)
If ℎ𝑅 ≥ 𝑑2, then
𝑑1 𝑑1
𝑑3 𝑑2
𝐶4
B ℎ𝑅
57 𝐶 = 1
1+2𝐶5 𝐸 (2.16) where
𝑑1 = 𝐵𝐶3 (2.17) 𝑑2 = 𝐶4(𝐴𝑐)𝐶5 (2.18) 𝑑3 = 𝐶4(𝐴𝑐)𝐶5 + 𝐵𝐶3𝐶6 (2.19)
Where C3, C4, C5, and C6 are coefficients for maintaining the relationship between the transversal section shape and the discharge calculation, Δt is the time step, and k is the time increment.
It is essential to mention that the accuracy of the PWRI – DH model depends on two main aspects: the cell definition (cell size) and on the tank’s parameters. In the following sections of the thesis, these are explained in detail.