# The FLO Diffusive 1D-2D Model for Simulation of River Flooding

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. 1D Model Equations

_{b}is the topographic channel bottom level (the topographic elevation of the deepest point in the channel cross-section), H is the water level (or potential or piezometric head), the sum of the maximum water depth h

_{max}in the 1D cross-section and of z

_{b}, H = h

_{max}+ z

_{b}, p is the source term (i.e., the inlet flow rate per unit length) and J is the friction slope. Discharge Q is related to J through the relationship:

_{sp}is a function of h

_{max}, of the bed roughness and of the section geometry, called in the following conveyance and computed as a function of h

_{max}according to [19]. Equation (1) is the mass conservation equation, and Equation (2) is the momentum conservation equation where terms not proportional to acceleration due to gravity have been neglected according to the diffusive hypothesis. Equations (2) and (3) can be merged in Equation (1), to get:

## 3. 2D Model Equations

_{0}the horizontal plane, π

_{b}the bottom plane and π

_{v}the vertical plane orthogonal to flow velocity vector (see Figure 2a). Call

**n**and

**t**the unit vectors normal to flow velocity and parallel respectively to π

_{0}(

**n**) and to π

_{b}(

**t**) (see Figure 2a). Let α be the angle, in the vertical plane, between

**t**and

**n**(see Figure 2a). The energy slope is, according to the Manning formula:

_{n(t)}, m

_{n(t)}and r

_{n(t)}are the directional parameters of

**n**(

**t**), respectively equal to:

_{a}, pa

_{b}and pa

_{c}are the director cosines of the vector orthogonal to the bottom plane. If a triangular mesh with peace-wise linear variation of H and z inside each element is used, the parameters at the l.h.s. of Equation (11b) will be constant inside each element.

## 4. Initial and Boundary Conditions

_{D}U Γ

_{N}is the boundary of Ω, Γ

_{D}and Γ

_{N}are the portions of Γ where Dirichlet and Neumann boundary conditions hold, respectively, H

_{D}and h

_{D}are the assigned Dirichlet values for H and h, g

_{N}is the assigned Neumann flux,

**q**(

**x**, t) is the boundary flow rate vector,

**n**is the unit outward normal to the boundary,

**x**= (x, y) and the subscript 0 marks the initial state in the domain.

## 5. The Fractional Time Step Procedure

**F**(U) is the flux vector and

**B**(U) is a source term. Applying a fractional time step procedure, we compute the solution of Equation (14) at the end of each time step by the solution of the following systems [14]:

^{k+1/2}and U

^{k+1}are the unknown variables computed respectively at the end of the prediction and the correction phases. Indices k, k + 1/2, k + 1 respectively mark the beginning of the time step (time level t

^{k}), the end of the prediction step (time level t

^{k+1/2}) and the correction step (time level t

^{k+1}). Integrals ${\overline{F}}^{p}\mathsf{\Delta}t$ and ${\overline{B}}^{p}\mathsf{\Delta}t$ will be estimated “a posteriori” after the solution of the prediction problem, according to the procedure explained in [14]. In the present case, we have:

^{km}is a water depth value in the computational cell obtained by local mass balance, computed as explained in [14], as well as the ${\overline{Q}}_{sp}$ coefficient in Equation (20). The initial condition is η = 0, and L(h

^{km}) in Equation (20) is the water surface width corresponding to h

^{km}.

**x**, t) point [14,18]. The prediction PDE system is equivalent to a single non-linear convection equation, the function of the gradient of the piezometric head at time level t

^{k}, while the correction system has the functional characteristics of a pure diffusive problem.

## 6. Computational Mesh Properties and Computational Cells

_{h,2D}a polygonal approximation of Ω

_{2D}and T

_{h}an unstructured Delaunay-type triangulation of Ω

_{h,2D}[14]. The triangulation T

_{h}is called the basic mesh, and the triangle k

_{T}$\in $ T

_{h}is called the primary element. Let P

_{h}= $\{{P}_{i}i=1,\dots N\}$ be the set of all vertices (or nodes) of all k

_{T}$\in $ T

_{h}and N the total number of nodes. The dual finite volume e

_{i}associated with vertex P

_{i}is the closed polygon given by the union of sub-triangles resulting from the subdivision of each triangle of T

_{h}connected to node P

_{i}by means of its axes (see Figure 3). In the following part of the paper, the dual volumes e are called also (computational) cells.

_{i}, previously defined, is called the Voronoi cell or the Thiessen polygon [20]. The vertices of the Voronoi cells are the circumcenters of the Delaunay triangulation. Area A

_{V,i}of the Voronoi cell i associated with node P

_{i}is computed as:

_{t}triangles sharing node P

_{i}. Storage capacity is assumed concentrated in the cells (nodes) in the measure of 1/3 of the area of all of the triangles sharing each node. Let H

_{i}be the water level in node P

_{i}and ${h}_{{P}_{i}}$ the corresponding water depth, defined as:

_{i}of cell i is computed as:

_{i}= 1 if ${h}_{{P}_{i}}>0$, and δ

_{i}= 0 otherwise.

_{m}

_{,1 }and P

_{m}

_{,2}the points at the ends of section m and d

_{m,r}the average distance between sections m and m − 1 (r = 1) or between sections m and m + 1 (r = 2). Computational cell i associated with section m in the 1D domain is given by the two halves of downstream and upstream 1D elements next to section m, if existing. We assume P

_{m}

_{,1}and P

_{m}

_{,2}to have the same water level H

_{i}corresponding to the computational cell i associated with section m. Let ${h}_{max,m}^{1}$ and ${h}_{max,m}^{2}$ be the two maximum water depths upstream and downstream of the discontinuity along the 1D section, computed as:

_{m,1}and σ

_{m,2}and the two water surface widths L

_{m,1}and L

_{m,2}corresponding to ${h}_{max,m}^{1}$ and ${h}_{max,m}^{2}$. Horizontal area A

_{i}of computational cell i associated with section m is given by:

_{m,r}= 0 if m is a domain boundary section or a 1D section shared with the 2D domain, and δ

_{m,r}= 1 otherwise.

_{m}the number of 2D nodes P

_{m,s}shared by 1D section m and the 2D domain and ${z}_{{P}_{m,s}}$ their topographic elevation (s = 1, ..., N

_{m}). Let ${h}_{{P}_{m,s}}$ be the water depth at node P

_{m,s}, defined as:

_{m,s}is the Voronoi cell area associated with node P

_{m,s}in the 2D domain (see the previous Equations (23)–(25)), δ

_{m,s}= 1 if node P

_{m,s}is on the trace of section m and ${h}_{{P}_{m,s}}>0$, δ

_{m,s}= 0 otherwise, and the other symbols have been specified before.

_{c}the total number of computational cells.

## 7. Solution of the Prediction Step

_{i}and water level H

_{i}has been previously defined in Equations (24)–(29). The outgoing flux $F{l}_{i,j}^{out}$ is a function of H

_{i}since it depends on water depth in cell i, as explained in the following Equations (32) and (33).

_{c}equations by approximating the r. h. s. with its mean value along the given time step, that is by setting:

_{m}the number of nodes of sections m (s = 1, ..., N

_{m}), shared with the 2D domain, and call P

_{m,s}any of these nodes.

_{s}the node associated with cell i.

_{n}nodes (q = 1, ..., N

_{n}) shared with the 2D domain, and let P

_{n,q}be one of these nodes. Call P

_{q}any node in the 2D domain, not belonging to any 1D section, with P

_{q}≠ P

_{s}.

_{s}or P

_{m,s}, depending on whether cell i is a 2D or a mixed cell; q marks node P

_{q}or P

_{n,q}corresponding to cell j. ${\mathsf{\delta}}_{s,q}^{2D}$ is equal to one if nodes P

_{s}(or P

_{m,s}), P

_{q}(or P

_{n,q}) have a common side and ${H}_{i}^{k}>{H}_{j}^{k}$, equal to zero otherwise. ${\mathsf{\delta}}_{m,n}^{1D}$ is equal to one if both i and j are mixed cells, m and n have a common 1D element and ${H}_{i}^{k}>{H}_{j}^{k}$, equal to zero otherwise. Flux is the flux leaving from cell i to any cell j corresponding to node P

_{q}(or P

_{n,q}) sharing a side with node P

_{s}(or P

_{m,s}), $f{l}_{m,n}^{1D}$ is the flux in the 1D channel leaving from cell i to any j cell corresponding to a section n sharing a 1D element with section m.

_{s}is the computed water depth in P

_{m,s}, in Case 1 (${h}_{s}={h}_{{P}_{m,s}}$ as defined in Equation (28)), or in node P

_{s}in Case 2 (${h}_{s}={h}_{{P}_{s}}$ as defined in Equation (24)). Flux coefficient ${K}_{s,q}^{k}$ is computed as explained in [14].

_{m,r}is a geometric distance (see Figure 7) and r = 1 if n = m − 1, r = 2 if n = m + 1. If m is a measured section; ${Q}_{m}^{r}\left({h}_{max,m}^{r}\right)$ is given by the Q

_{sp}(h) function assigned to the same section. If m is an interpolated section, ${Q}_{m}^{r}\left({h}_{max,m}^{r}\right)$ is obtained by linear interpolation of the two functions Q

_{sp}(h) available in the two upstream and downstream measured sections. If both i and j cells are in the 2D domain, flux $f{l}_{i,j}^{1D}$ is zero.

## 8. Solution of the Correction Step

_{m,s}(or P

_{s}) and ${\overline{fl}}_{m,n}^{1D}$ the mean in time leaving flux in the 1D channel from cell i to any linked mixed cell j. These mean fluxes are computed respectively as:

_{m,s}(or P

_{s}) given by:

_{sq}in Equation (40) have been already defined for the prediction step; g is the number of triangles (one or two) sharing side $\overrightarrow{{P}_{m,s}{P}_{n,q}}$ (or $\overrightarrow{{P}_{m,s}{P}_{q}}$); δ

_{g}= 1 if the g-th triangle does exist, and δ

_{g}= 0 otherwise.

^{k}

^{+1}= H

^{k}

^{+1/2}+ η

## 9. Stability of the Solution at Discontinuous Nodes

_{j}is the set of the mixed type cells with a discontinuous node connected to cell i and r is the index of any cell connected to cell i. The equations of the mixed type cells j become:

_{i}is the set of the 2D cells connected to cell j with a discontinuous node.

_{s}is the sum of half lengths of the channels sharing section s and the r.h.s. is the discharge corresponding to the critical condition of the lateral overflow. Coefficient ${K}_{s,q}^{k}$ has been already defined for Equation (33). Equation (47) can be approximated with:

_{s}< 3 m.

## 10. Test 1: Comparison with the Results of Morales et al. [10]

^{1/3}in the floodplain and 0.015 s/m

^{1/3}in the channel, and sediment transport is not considered. The domain has been discretized with a 1D-2D mesh with lateral coupling, with 6960 triangles, 3600 nodes and 131 1D channel sections (six measured sections). A second unstructured fully 2D mesh has been used, with 29,857 triangles and 15,820 nodes. The triangle number increases in the 2D mesh because of the slope of the trapezoidal cross-section, and a fine mesh is required to represent the topography faithfully. Three scenarios have been selected for the upstream boundary condition: (1) a steady and (2) an unsteady flow in the most upstream channel section; (3) constant net rain, falling homogeneously above all of the floodplain.

^{3}/s is given as the upstream boundary condition at the channel inlet, and the model is run until convergence to the steady state. The time step is 40 seconds. The proposed coupling procedure computes water depths very similar to the ones obtained over the fully-2D mesh. Due to the different resistance law in the 1D channel and in the floodplain, water depths computed over the coupled 1D-2D mesh are slightly higher than the ones over the 2D mesh, with the highest difference less than 0.01 m. On the opposite side, the results provided by Morales et al. by their coupled procedure show large differences with respect to the ones computed by their 2D model. Morales et al. [10] justify these differences with the choice of the Manning coefficient in the 1D channel and its adjustment due to coupling strategies. See the results of the two models, for both mixed and 2D meshes, shown in Table 1.

^{3}/s, is assigned in the upstream boundary section. Zero water depth is the Dirichlet downstream boundary condition. The Manning coefficients in the channel and in the floodplain are 0.01605 s/m

^{1/3}and 0.03 s/m

^{1/3}. For this scenario, four simulations were carried out with a time step size equal to 20, 40, 100 and 400 seconds, in order to evaluate the stability of the model. The CFL number (CFL = VΔt/Δx, where V is the vertical averaged velocity and Δx is the cell length) in the 2D triangle l is computed for the FLO model as:

^{3}/s. The test was carried out with the same time step size used for Scenario 2. The relative mass balance error in the correction step due to the existence of discontinuous nodes is reported for each time step in Figure 12. It has been computed as the ratio between the total discharge neglected in the correction step and the total entering rain volume per unit time. The error obtained with a time step of 40 s (7.10 × 10

^{−3}) is negligible for practical applications. For higher time steps, FLO returns a higher error, although the outlet discharge hydrograph remains always stable (see Figure 11 and Figure 12). Observe that the hypothesis of instantaneous shift from zero to a finite net rain intensity is quite severe in real applications, and a time step of 40 s corresponds to a maximum CFL = 295 (Table 1).

## 11. Test 2: The Toce River (Italy) Test Case

^{3}/s (50,000 m

^{3}/s in the real scale).

^{3}/s (corresponding to 20,000 m

^{3}/s at the real scale), while peak flow for HY2 was 0.35 m

^{3}/s (35,000 m

^{3}/s at the real scale).

^{1/3}. This value has been taken from the literature (e.g., [24,25] and the cited references) and has not been previously calibrated.

## 12. The Severn River (U.K.) Test Case

^{2}and 42 1D sections profiles have been provided, with sections labelled from M013–M054 [27,28]. Uniform roughness has been assumed inside the channel and floodplains, equal respectively to n = 0.028 s/m

^{1/3}and n = 0.04 s/m

^{1/3}[27,28]. According to the topography of the studied area [27,28], the modelled flood is not expected to inundate roads and built-up areas to any significant extent. Therefore, a uniform roughness value has been used. Any effect of buildings has been neglected, as well as any head losses due to the river axis curvature [27,28].

## 13. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 5.**Water depths ${h}_{max,m}^{r}$ in section m of the 1D channel and at nodes shared with the 2D domain, lateral coupling.

**Figure 6.**Horizontal area of the computational cell associated with section m, lateral coupling. (

**a**) Case 1, (

**b**) Case 2 and (

**c**) Case 3 in Figure 5. (

**d**) Horizontal area of the computational cell associated with section m, frontal coupling.

**Figure 7.**Nomenclature of the geometrical variables involved for the computation of the leaving flux in the prediction step.

**Figure 8.**Test 1. Geometry of the domain. (

**a**) Plane view and gauges position; (

**b**) Channel section (from Morales et al. [10]).

**Figure 9.**Test 1. Unsteady flow. FLO computed water depths. (

**a**) Gauges 1–5. (

**b**) Gauges 6–10. n = 0.03 s/m

^{1/3}in the floodplain; n = 0.01605 s/m

^{1/3}in the 1D channel.

**Figure 10.**Test 1. Unsteady flow. Computed water depths by Morales et al. [10] (

**a**) Gauges 1–5. (

**b**) Gauges 6–10. n = 0.03 s/m

^{1/3}in the floodplain; n = 0.01605 s/m

^{1/3}in the 1D channel.

**Figure 11.**Test 1. Unsteady flow. Discharge leaving the domain computed by FLO for different time step size values (values around the peak).

**Figure 12.**Test 1. Relative mass balance error computed in the correction step due to the existence of discontinuous nodes.

**Figure 15.**(

**a**) Test 2. Measured and computed water level Hydrograph 1 (HY1) at Gauges P1, P5, P13, P21, P26 (dynamic wave and measured by [25]). (

**b**) Test 2. Measured and computed water level HY1 at Gauges S4, P4, P8, P18, P23 (measured by [24,26]). (

**c**) Test 2. Measured and computed water level HY2 at Gauges P1, P5, P13, P21, P26 (dynamic wave and measured by [25]). (

**d**) Test 2. Measured and computed water level HY2 at Gauges S4, P4, P8, P18, P23 (measured by [24,26]).

**Figure 18.**Test 3. (

**a**) Map of Floodplain 1. (

**b**) Map of Floodplain 2. (

**c**) Map of Floodplain 3. Purple dots are the gauges for water level registrations [27].

Water Depth (m) | ||||
---|---|---|---|---|

FLO model | Morales et al., 2013 | |||

coupled 1D-2D | fully 2D | coupled 1D-2D | fully 2D | |

Probe 1 | 0.075 | 0.062 | 0.000 | 0.035 |

Probe 2 | 0.277 | 0.278 | 0.171 | 0.279 |

Probe 3 | 0.501 | 0.495 | 0.394 | 0.513 |

Probe 4 | 0.871 | 0.861 | 0.870 | 0.991 |

Probe 5 | 0.993 | 0.983 | 0.878 | 0.992 |

Probe 6 | 0.020 | 0.012 | 0.000 | 0.000 |

Probe 7 | 0.239 | 0.229 | 0.000 | 0.021 |

Probe 8 | 0.493 | 0.475 | 0.136 | 0.270 |

Probe 9 | 0.747 | 0.731 | 0.381 | 0.510 |

Probe 10 | 0.998 | 0.979 | 0.660 | 0.791 |

CFL | ||
---|---|---|

1D domain | 2D domain | |

ΔT = 20 s | 147.22 | 39.73 |

ΔT = 40 s | 295.08 | 79.41 |

ΔT = 100 s | 742.54 | 197.96 |

ΔT = 400 s | 869.96 | 2992.40 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aricò, C.; Filianoti, P.; Sinagra, M.; Tucciarelli, T.
The FLO Diffusive 1D-2D Model for Simulation of River Flooding. *Water* **2016**, *8*, 200.
https://doi.org/10.3390/w8050200

**AMA Style**

Aricò C, Filianoti P, Sinagra M, Tucciarelli T.
The FLO Diffusive 1D-2D Model for Simulation of River Flooding. *Water*. 2016; 8(5):200.
https://doi.org/10.3390/w8050200

**Chicago/Turabian Style**

Aricò, Costanza, Pasquale Filianoti, Marco Sinagra, and Tullio Tucciarelli.
2016. "The FLO Diffusive 1D-2D Model for Simulation of River Flooding" *Water* 8, no. 5: 200.
https://doi.org/10.3390/w8050200