# Development of a Tsunami Inundation Analysis Model for Urban Areas Using a Porous Body Model

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^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

- (Model 1) the roughness map model, which uses Manning’s roughness coefficient “n” according to the land-use type [27] (hereafter, Landuse-n model),

#### 1.2. Objectives

#### 1.2.1. Development of NSWW Theory and Numerical Model Based on a Porous Body Model

#### 1.2.2. Investigation of Characteristics of Tsunami Hazards Simulated by Numerical Models

## 2. Nonlinear Shallow Water Wave Equations Based on a Porous Body Model

#### 2.1. Governing Equations

_{V}and γ

_{i}are the volume porosity (γ

_{V}= fluid volume space/dxdydz) and components in the i-th direction of the surface permeability (e.g., γ

_{x}= fluid surface area/dydz) in the unit element, respectively. The variables t, x

_{i}, u

_{i}, p, ρ and g

_{i}denote the time, components in the i-th direction of the spatial coordinates (x, y, z), components in the i-th direction of the fluid velocity coordinates (u, v, w), pressure, fluid density and gravity (0, 0, −g), respectively. The parameters M

_{i}and R

_{i}are the components in the i-th direction of the fluid inertia force and drag force, respectively, τ

_{ij}is the stress tensor acting on the surface of the unit volume element, and u

_{i}and γ

_{V}u

_{i}(or γ

_{i}u

_{i}) are the pore velocity and the Darcy velocity in the porous medium, respectively.

_{0}and η are the atmospheric pressure and the water surface displacement in the porous medium, respectively.

#### 2.2. Kinematic and Dynamic Boundary Conditions

_{V}and γ

_{i}are defined using the integral values of γ

_{V}

_{or i}as follows:

_{0}= 0. The following derivation processes for the two-dimensional equations are explained using an x-z vertical plane for brevity.

#### 2.3. Integration of the Continuity Equation

_{x}(x, t) into Equation (7), we can obtain the general form of the continuity equation for the shallow water wave equations based on the porous body model as follows:

_{x}(x, t) (Q

_{x}= U

_{x}D, where D = η + h is flow depth), Equation (8) can be rewritten as:

#### 2.4. Integration of the Momentum Equation

_{i}, R

_{i}and τ

_{ij}), leading to the following:

_{i}and R

_{i}are omitted, whereas the bottom friction due to τ

_{ij}is considered, although the terms represent important forces to simulate the hydraulics in urban areas where the effect of buildings suddenly changes spatially. Assuming that the temporal variation in the water level is sufficiently small similar to a long wave and that buildings are impermeable/non-destructive, Equations (11), (16) and (17) are applied as the basic equations in this study: by eliminating $\partial {\overline{R}}_{V}/\partial t$, e.g., the equations would overestimate the water level and flow velocity if a tsunami rapidly overflowed the buildings, and tsunamis passing through and staying in the buildings are not considered. The equations are incorporated into TUNAMI-N2 [10,11] using the FDM, and the proposed model is hereafter labeled a PBM in the following sections.

## 3. Establishing the Porosity and Surface Permeability for a Group of Buildings

#### 3.1. Porosity of the Water Column

_{n}and the building height H

_{n}from the calculation cell mesh, where n is the building ID within the cell mesh. In the case when the number of buildings that exist in the mesh is N, n = 0, 1,…, N, the integral values of the occupancy area and the characteristic building height on the cell mesh are defined as:

_{c}(=ΔxΔy) is the area of the cell mesh and Δx and Δy are the lengths of the edge of the cell mesh in the x and y directions, respectively.

#### 3.2. Horizontal Surface Permeability of the Water Column

_{n′}and the building height H′

_{n′}within the edge of the cell mesh, where n′ is the building ID in the edge of the cell mesh. In the case where the number of buildings that exist in the edge of the cell mesh is N′, n′ = 0, 1,…, N′, the integral values of the occupancy length and the characteristic building height on the edge of the cell mesh are defined as:

_{c}> 0.95, we add the height H to the DEM to produce a DSM. Hence, the porous-type model is not applied to the cell meshes using DSM and is built by adding the porous medium to Landuse-n/Topography.

## 4. Differences between Conventional Porous-Type Models and the Proposed Model

_{V}= γ

_{i}. The surface permeability in the flood model is defined by using the value of porosity as shown in Equations (23) or (24), and building height is not considered, as shown in Equation (22). However, since the proposed PBM is based on adequate boundary conditions considering both the porosity and the surface permeability [54], the PBM can more accurately reflect the geometric effects of urban areas on the tsunami inundation simulations.

## 5. Numerical Simulations of Tsunami Inundation in Onagawa, Miyagi Prefecture, during the Great East Japan Earthquake

#### 5.1. Background of the Target Domain

#### 5.2. Initial Setup of the Tsunami Simulation

_{c}, which is used in the porous-type models (i.e., the coastal-forest, the flood models and PBM).

#### 5.3. Results and Discussions

## 6. Conclusions and Recommendations

- The proposed PBM exhibited as good accuracy for the inundation heights around building near the coastline as with conventional three-dimensional simulation with high resolution.
- In the case where the inundation area is restricted by the topography with a steep slope behind the urban area and the flow depth is much greater than the building height, the differences in the modeled building effects and computational resolution do not significantly affect the maximum water level near the coastline. Therefore, two-dimensional simulations with a practical resolution demonstrate good accuracy while estimating the inundation height near the coastline, although we must examine the applicability of these models to regions where the locality can become stronger in future work.
- A model that includes porosity can yield an increase in the water level, which makes it easier for the tsunami to spread out over the land, thereby decreasing the flow velocity.
- A model that includes the surface permeability restricts the progress of the tsunami according to the distribution of the buildings and reproduces a flow field concentrated along the straight road.
- By properly incorporating the porosity and the surface permeability into the theory and the numerical model, the model can adequately reproduce high flow velocities due to the increase in the gradient of the water surface and the concentration of the flow during the inundation process within the urban area. In addition, the water level at the time will increase due to the flow and geometric effects. Therefore, a numerical model that does not consider geometric effects could underestimate the local hydrodynamic force.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Unit volume element in the porous body model, where the black and white are solid and fluid, respectively.

**Figure 2.**Modeling process for buildings using the porous body model. (

**a**) Buildings on computational mesh; (

**b**) Buildings on cell mesh; (

**c**) Occupancy area A

_{n}on cell mesh and length L

_{n′}at the edge of cell mesh; and (

**d**) porosity of the water column on cell mesh and horizontal surface permeability at the edge of the cell mesh.

**Figure 3.**Accumulation of crustal deformation in the tsunami source model proposed by Satake et al. [24], where the circle indicates the location of Onagawa, Miyagi Prefecture, Japan, and the star is the epicenter of the earthquake. The units of distance are km.

**Figure 5.**Topography in Onagawa, Miyagi Prefecture, Japan (Region 6: Δx

_{6}= 5 m). (

**a**) The blue line is the inundation area measured by Haraguchi and Iwamatsu [57], and the black colors are the locations where the building height data are incorporated into the topographic data, i.e., it indicates A/A

_{c}> 0.95 in Equation (18). (

**b**) The ground elevation around Marine Pal Onagawa, which is the region in the red box shown in Figure 5a.

**Figure 6.**The occupancy area ratio of buildings around Marine Pal Onagawa in Figure 5b. The black colors indicate the locations where building height data are incorporated into the topographic data, i.e., it indicates A/A

_{c}> 0.95 in Equation (18).

**Figure 7.**Maximum water level obtained by the proposed PBM, where the green line is the field survey result for the inundation area [57].

**Figure 8.**Difference in the maximum water level between the porous-type models (Coastal-forest: coastal-forest model, Flood: flood model, PBM: proposed model based on the porous body model) and Landuse-n/Topography, where the green lines are the field survey result for the inundation area [57].

**Figure 9.**Maximum flow velocities obtained using the (

**a**) Landuse-n: roughness map model, (

**b**) Landuse-n/Topography: roughness map and topographic model, (

**c**) Coastal-forest: coastal forest model, (

**d**) Flood: flood model, and (

**e**) PBM: proposed model based on the porous body model, where the porous-type models (i.e., the coastal-forest model, the flood model and PBM) describe the Darcy velocity and the pink lines are the field survey result for the inundation area [57].

**Figure 10.**Difference in the maximum flow velocity between the porous-type models (i.e., the coastal-forest model, the flood model and PBM) and Landuse-n/Topography, where the green lines are the field survey result for the inundation area [57].

**Figure 11.**Time series of the flow depth at the point near Marine Pal Onagawa shown in Figure 5b.

**Figure 12.**Water level at 2400 s after the earthquake: (

**a**) Landuse-n (roughness map model); (

**b**) the proposed PBM based on the porous body model.

**Figure 13.**Flow velocity at 2400 s after the earthquake: (

**a**) Landuse-n (roughness map model); (

**b**) the proposed PBM based on the porous body model, where the pore velocity is described.

**Figure 14.**Time series of the inundation height and flow velocity at Point A in Figure 5b: (

**a**) inundation height; (

**b**) flow velocity, where the pore velocity is described (Landuse-n: roughness map model; Landuse-n/Topography: roughness map and topographic model; Coastal-forest: coastal forest model; Flood: flood model; PBM: proposed model based on porous body model).

Model | Geometric Effect | Mechanical Effect | Rough indication of Computational Cost | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Space Volume (Porosity) | Flow Path Area (Surface Permeability) | Building Height Effect | Bottom Friction | Drag Force | ||||||||

Wall Friction | Wake | Array Effect | ||||||||||

High Resolution | 3-D NS | VOF | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | >>10^{4} | ||

SPH | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | >>10^{4} | |||||

2-D NSWW | ✓ | ✓ | ✓ | ✓ | ✓ | 10^{3} | ||||||

Practical Resolution (2D) | Roughness-Type Model | With DEM | Constant-n | (✓) | 1 | |||||||

○ Landuse-n | ✓ | (✓) | (✓) | (✓) | 1 | |||||||

Equivalent-n | ✓ | ✓ | ✓ | (✓) | 1 | |||||||

With DSM | ○ Landuse-n/Topography | (✓) | (✓) | (✓) | ✓ | (✓) | (✓) | (✓) | 1 | |||

Equivalent-n/Topography | (✓) | (✓) | (✓) | ✓ | ✓ | ✓ | (✓) | 1 | ||||

Porous-Type Model with DSM | ○ Coastal-Forest | ✓ | (✓) | ✓ | ✓ | ✓ | ✓ | (✓) | 1 | |||

Flood | ○ FDM | ✓ | (✓) | (✓) | ✓ | ✓ | ✓ | (✓) | 1 | |||

FVM | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | (✓) | 1 | ||||

○ PBM (FDM)(This Study) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | (✓) | 1 |

**Table 2.**Inundation area and maximum flow velocities in Onagawa town as shown in Figure 5a (field survey [57]; Landuse-n: roughness map model; Landuse-n/Topography: roughness map and topographic model; Coastal-forest: coastal forest model; Flood: flood model; PBM: proposed model based on porous body model).

Field Survey | Roughness-Type Model | Porous-Type Model | ||||
---|---|---|---|---|---|---|

Landuse-n | Landuse-n/ Topography | Coastal-Forest | Flood | PBM | ||

Inundation Area (km^{2}) | 1.64 | 1.68 | 1.56 | 1.59 | 1.48 | 1.55 |

Ratio with Landuse-n | 0.98 | 1.00 | 0.93 | 0.95 | 0.88 | 0.93 |

Mean Value of 100% of Maximum Velocity (m/s) | - | 3.18 | 2.78 | 2.64 | 2.43 | 3.02 |

Mean Value of Top 30% of Maximum Velocity (m/s) | - | 5.34 | 4.69 | 4.50 | 4.21 | 5.42 |

Mean Value of Top 10% of Maximum Velocity (m/s) | - | 6.71 | 6.00 | 5.80 | 5.50 | 7.78 |

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**MDPI and ACS Style**

Yamashita, K.; Suppasri, A.; Oishi, Y.; Imamura, F. Development of a Tsunami Inundation Analysis Model for Urban Areas Using a Porous Body Model. *Geosciences* **2018**, *8*, 12.
https://doi.org/10.3390/geosciences8010012

**AMA Style**

Yamashita K, Suppasri A, Oishi Y, Imamura F. Development of a Tsunami Inundation Analysis Model for Urban Areas Using a Porous Body Model. *Geosciences*. 2018; 8(1):12.
https://doi.org/10.3390/geosciences8010012

**Chicago/Turabian Style**

Yamashita, Kei, Anawat Suppasri, Yusuke Oishi, and Fumihiko Imamura. 2018. "Development of a Tsunami Inundation Analysis Model for Urban Areas Using a Porous Body Model" *Geosciences* 8, no. 1: 12.
https://doi.org/10.3390/geosciences8010012