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

^{1}

^{2}

^{*}

## 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

- Nakamura, Y.; Kato, S. Testimony of the 3.11 Disaster—The Tsunami in Tagajo City. Kahoku-Shinpo, 13 May 2011; A106X0, K20110513A106X0050. (In Japanese) [Google Scholar]
- Oshiba, K. Tsunami that I Saw—A Sense of Despair Resulted from Heavy, Fast and Irresistible Tsunami. Kahoku-Shinpo, 15 June 2011; M206X0, K20110615M206X0020. (In Japanese) [Google Scholar]
- Rueben, M.; Holman, R.; Cox, D.; Shin, S.; Killian, J.; Stanley, J. Optical measurements of tsunami inundation through an urban waterfront modeled in a large-scale laboratory basin. Coast. Eng.
**2011**, 58, 229–238. [Google Scholar] [CrossRef] - Suppasri, A.; Latcharote, P.; Bricker, J.D.; Leelawat, N.; Hayashi, A.; Yamashita, K.; Makinoshima, F.; Roeber, V.; Imamura, F. Improvement of tsunami countermeasures based on lessons from the 2011 Great East Japan earthquake and tsunami—Situation after five years. Coast. Eng. J.
**2016**, 58. [Google Scholar] [CrossRef] - Arikawa, T.; Tomita, T. Development of High Resolution Tsunami Runup Calculation Method Based on a Multi Scale Simulation; Report of the Port and Airport Research Institute; Port and Airport Research Institute: Yokosuka, Japan, 2014; Volume 53, pp. 3–18, (In Japanese with English abstract).
- Arikawa, T.; Tomita, T. Development of high precision tsunami runup calculation method based on a hierarchical simulation. J. Disaster Res.
**2016**, 11, 639–646. [Google Scholar] [CrossRef] - Tomita, T.; Kakinuma, T. Storm Surge and Tsunami Simulator in Oceans and Coastal Areas (STOC); Report of the Port and Airport Research Institute; Port and Airport Research Institute: Yokosuka, Japan, 2005; Volume 44, pp. 83–98, (In Japanese with English abstract).
- Arikawa, T.; Yamada, F.; Akiyama, M. Study of applicability of tsunami wave force in a three-dimensional numerical wave flume. Proc. Coast. Eng. JSCE
**2005**, 52, 46–50. [Google Scholar] - Suwa, T.; Kazama, M.; Imamura, F.; Sugawara, D.; Yamashita, K. Resolution Dependency of Tsunami Simulation by an SPH Method; Proceedings of the Conference on Computational Engineering and Science; Japan Society for Computational Engineering and Science: Tokyo, Japan, 2016; Volume 21, (In Japanese with English abstract). [Google Scholar]
- Imamura, F.; Yalciner, A.; Ozyurt, G. Tsunami Modelling Manual; UNESCO Tsunami Modelling Course (UNESCO); UNESCO: Paris, France, 2006. [Google Scholar]
- UNESCO/IOC. IUGG/IOC Time Project: Numerical Method of Tsunami Simulation with the Leap-Frog Scheme; Manuals and Guides No. 30; UNESCO: Paris, France, 1997. [Google Scholar]
- Titov, V.V.; Gonzalez, F.I. Implementation and Testing of the Method of Splitting Tsunami (MOST) Model; NOAA Technical Memorandum ERL PMEL-112; NOAA: Silver Spring, MD, USA, 1997; 11p.
- Liu, P.L.F.; Cho, Y.S.; Yoon, S.B.; Seo, S.N. Numerical Simulations of the 1960 Chilean Tsunami Propagation and Inundation at Hilo, Hawaii, in Recent Developments in Tsunami Research; El-Sabh, M.I., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994; pp. 99–115. [Google Scholar]
- Liu, P.L.F.; Cho, Y.S.; Briggs, M.J.; Synolakis, C.E.; Kanoglu, U. Run-up of solitary waves on a circular island. J. Fluid Mech.
**1995**, 302, 259–285. [Google Scholar] [CrossRef] - Yalciner, A.C.; Pelinovsky, E.; Zaytsev, A.; Kurkin, A.; Ozer, C.; Karakus, H. NAMI DANCE Manual; Ocean Engineering Research Center, Civil Engineering Department, Middle East Technical University: Ankara, Turkey, 2006. [Google Scholar]
- Harig, S.; Chaeroni, C.; Behrens, J.; Schroeter, J. Tsunami Simulations with unstructured grids (TsunAWI) and a comparison to simulations with nested grids (Tsunami-N3). In Proceedings of the 6th International Workshop on Unstructured Mesh Numerical Modeling of Coastal, Shelf and Ocean Flows, London, UK, 19–21 September 2007. [Google Scholar]
- Nielsen, O.; Roberts, S.; Gray, D.; McPherson, A.; Hitchman, A. Hydrodynamic modeling of coastal Inundation. In Proceedings of the MODSIM 2005 International Congress on Modeling and Simulation, Melbourne, Australia, 12–15 December 2005; Zerger, A., Argent, R.M., Eds.; Modeling and Simulation Society of Australia and New Zealand: Melbourne, Australia, 2005; pp. 518–523. [Google Scholar]
- Taubenböck, H.; Goeseberg, N.; Setiadi, N.; Lämmel, G.; Moder, F.; Oczipka, M.; Klüpfel, H.; Wahl, R.; Schlurmann, T.; Strunz, G.; et al. “LastMile” preparation for a potential disaster—Interdisciplinary approach towards tsunami early warning and an evacuation information system for the coastal city of Padang, Indonesia. Nat. Hazards Earth Syst. Sci.
**2009**, 9, 1509–1528. [Google Scholar] [CrossRef] [Green Version] - Muhari, A.; Imamura, F.; Koshimura, S.; Post, J. Examination of three practical run-up models for assessing tsunami impact on highly populated areas. Nat. Hazards Earth Syst. Sci.
**2011**, 11, 3107–3123. [Google Scholar] [CrossRef] [Green Version] - Taubenböck, H.; Goeseberg, N.; Lämmel, G.; Setiadi, N.; Schlurmann, T.; Nagel, K.; Siegert, F.; Birkmann, J.; Traub, K.P.; Dech, S.; et al. Risk reduction at the “Last-Mile”: An attempt to turn science into action by the example of Padang, Indonesia. Nat. Hazards
**2013**, 65, 915–945. [Google Scholar] [CrossRef] - Oishi, Y.; Imamura, F.; Sugawara, D. Near-field tsunami inundation forecast using the parallel TUNAMI-N2 model: Application to the 2011 Tohoku-Oki earthquake combined with source inversions. Geophys. Res. Lett.
**2015**, 42, 1083–1091. [Google Scholar] [CrossRef] - Oishi, Y.; Imamura, F.; Sugawara, D.; Furumura, T. Investigation of reliable tsunami inundation model in urban areas using a supercomputer. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2016**, 72, I_409–I_414, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Adriano, B.; Hayashi, S.; Gokon, H.; Mas, E.; Koshimura, S. Understanding the extreme tsunami inundation in Onagawa town by the 2011 Tohoku earthquake, its effects in urban structures and coastal facilities. Coast. Eng. J.
**2016**, 58. [Google Scholar] [CrossRef] - Latcharote, P.; Suppasri, A.; Yamashita, A.; Adriano, B.; Koshimura, S.; Kai, Y.; Imamura, F. Possible failure mechanism of buildings overturned during the 2011 Great East Japan Tsunami in the Town of Onagawa. Front. Built Environ.
**2017**, 3, 1–18. [Google Scholar] [CrossRef] - Imamura, F.; Suppasri, A.; Sato, S.; Yamashita, K. The role of tsunami engineering in building resilient communities and issues be improved after the GEJE. In The 2011 Japan Earthquake and Tsunami: Reconstruction and Restoration; Springer: New York, USA, 2018; pp. 435–448. [Google Scholar]
- Tsudaka, R.; Shigihara, Y.; Fujima, K. Verification of tsunami inundation simulation on basic experiments. J. Jpn. Soc. Civ. Eng. Ser. B1 (Hydraul. Eng.)
**2012**, 68, I_1537–I_1542, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Kotani, M.; Imamura, F.; Shuto, N. Tsunami run-up simulation and damage estimation by using geographical information system. Proc. Coast. Eng. JSCE
**1998**, 45, 356–360. (In Japanese) [Google Scholar] - Aburaya, T.; Imamura, F. Proposal of a tsunami run-up simulation using combined equivalent roughness. Proc. Coast. Eng. JSCE
**2002**, 49, 276–280. (In Japanese) [Google Scholar] - Imai, K.; Imamura, F.; Iwama, S. Advanced tsunami computation for urban regions. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2013**, 69, I_311–I_315, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Baba, T.; Takahashi, N.; Kaneda, Y.; Inazawa, Y.; Kikkojin, M. Tsunami inundation modeling of the 2011 Tohoku Earthquake using three-dimensional building data for Sendai, Miyagi Prefecture, Japan. Tsunami Events and Lessons Learned. Adv. Nat. Technol. Hazards Res.
**2014**, 35, 89–98. [Google Scholar] - Matsutomi, H.; Onuma, K.; Imai, K. Basic equations for a flow in a vegetated area and a similarity law for a trunk. Proc. Coast. Eng. JSCE
**2004**, 51, 301–305. (In Japanese) [Google Scholar] - Guinot, V.; Soares-Frazão, S. Flux and source term discretization in two dimensional shallow water models with porosity on unstructured grids. Int. J. Numer. Methods Fluids
**2006**, 50, 309–345. [Google Scholar] [CrossRef] - Sanders, B.F.; Schubert, J.E.; Gallegos, H.A. Integral formulation of shallow water equations with anisotropic porosity for urban flood modeling. J. Hydrol.
**2008**, 362, 19–38. [Google Scholar] [CrossRef] - Miura, S.; Kawamura, I.; Kimura, I.; Miura, A. Study on inundation flow analysis method in densely populated urban area on alluvial fan. J. Jpn. Soc. Civ. Eng. Ser. B1 (Hydraul. Eng.)
**2011**, 67, I_979–I_984, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Ministry of Land, Infrastructure, Transport and Tourism, 2015. Available online: https://www.mlit.go.jp/river/shishin_guideline/pdf/manual_kouzuishinsui_1507.pdf (accessed on 4 February 2016). (In Japanese)
- Kim, B.; Sanders, B.F.; Schubert, J.E.; Famiglietti, J.S. Mesh type tradeoffs in 2D hydrodynamic modeling of flooding with a Godunov-based flow solver. Adv. Water Resour.
**2014**, 68, 42–61. [Google Scholar] [CrossRef] - Kim, B.; Sanders, B.F.; Famiglietti, J.S.; Guinot, V. Urban flood modeling with porous shallow-water equations: A case study of model errors in the presence of anisotropic porosity. J. Hydrol.
**2015**, 523, 680–692. [Google Scholar] [CrossRef] - Sha, W.T.; Domanus, H.M.; Schmitt, R.C.; Oras, J.J.; Lin, E.I.H. COMMIX-1: A Three Dimensional Transient Single-Phase Component Computer Program for Thermal-Hydraulic Analysis; NUREG/CR-0785, ANL-77-96; Argonne National Laboratory: Lemont, IL, USA, 1978. [Google Scholar]
- Sakakiyama, T.; Kajima, R. Numerical simulation of nonlinear wave interacting with permeable breakwater. In Proceedings of the 23rd International Conference on Coastal Engineering, Venice, Italy, 4–9 October 1992; pp. 1517–1530. [Google Scholar]
- The 2011 Tohoku Earthquake Tsunami Joint Survey Group. Nationwide field survey of the 2011 off the Pacific coast of Tohoku earthquake tsunami. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2011**, 67, 63–66. [Google Scholar] - Mori, N.; Takahashi, T.; Takahashi, T. The 2011 Tohoku Earthquake Tsunami Joint Survey Group. Nationwide post event survey and analysis of the 2011 Tohoku earthquake tsunami. Coast. Eng. J.
**2012**, 54. [Google Scholar] [CrossRef] - Mikami, T.; Shibayama, T.; Esteban, M. Field survey of the 2011 Tohoku earthquake and tsunami in Miyagi and Fukushima prefectures. Coast. Eng. J.
**2012**, 54, 54. [Google Scholar] [CrossRef] - Suppasri, A.; Koshimura, S.; Imai, K.; Mas, E.; Gokon, H.; Muhari, A.; Imamura, F. Damage characteristic and field survey of the 2011 Great East Japan tsunami in Miyagi prefecture. Coast. Eng. J.
**2012**, 54. [Google Scholar] [CrossRef] - Yagi, H.; Suginami, K.; Nakayama, A.; Nishi, T.; Okuno, M.; Koike, T.; Hayashi, K.; Igarashi, Y. A study of damage mechanism on port facilities at Onagawa Fishery port due to Tohoku Earthquake Tsunami. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2012**, 68, I_1346–I_1350, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Koshimura, S.; Hayashi, S.; Gokon, H. The impact of the 2011 Tohoku earthquake tsunami disaster and implications to the reconstruction. Soils Found.
**2014**, 54, 560–572. [Google Scholar] [CrossRef] - Hayashi, S.; Adriano, B.; Mas, E.; Koshimura, S. Improving tsunami numerical simulation with the time-dependent building destruction model. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2014**, 70, I_346–I_350, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Kozono, Y.; Takahashi, T.; Sakuraba, M.; Nojima, K. Application of tsunami numerical model considering collapsed buildings and disaster debris for the Nankai Trough. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2017**, 73, I_403–I_408, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Tajima, Y.; Kirigaya, N.; Sakurazawa, T. Impact of fluid-debris interactions on nearshore inundation characteristics. J. Jpn. Soc. Civ. Eng. Ser. B3 (Ocean Eng.)
**2016**, 72, I_205–I_210, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Sakurazawa, T.; Tajima, Y. Experimental study on horizontal movement of numerous wreckage under the inundating flow. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2016**, 72, I_1153–I_1158, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Sugawara, D.; Yamashita, K.; Takahashi, T.; Imamura, F. Role of sediment transport model to improve the tsunami numerical simulation. In Proceedings of the AGU 2015 Fall Meeting, San Francisco, CA, USA, 14–18 December 2015. [Google Scholar]
- Yamashita, K.; Sugawara, D.; Takahashi, T.; Imamura, F.; Saito, Y.; Imato, Y.; Kai, T.; Uehara, H.; Kato, T.; Nakata, K.; et al. Numerical simulations of large-scale sediment transport caused by the 2011 Tohoku Earthquake Tsunami in Hirota Bay, Southern Sanriku Coast. Coast. Eng. J.
**2016**, 58. [Google Scholar] [CrossRef] - Yamashita, K.; Shigihara, Y.; Sugawara, D.; Arikawa, T.; Takahashi, T.; Imamura, F. Effect of sediment transport on tsunami hazard and building damage—An integrated simulation of tsunami inudation, sediment transport and drifting vessels in Kesennuma City, Miyagi Prefecture during the Great East Japan Earthquake. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2017**, 73, I_355–I_360, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Yamashita, K.; Imamura, F.; Iwama, S.; Sugawara, D.; Takahashi, T. Effect of tsunami-indeced sediment transport and offshore tsunami waveform on enlargement of return flow. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2017**, 73, I_361–I_366, (In Japanese with English abstract). [Google Scholar] - Bear, J. Dynamics of Fluids in Porous Media; Elsevier: New York, NY, USA, 1972. [Google Scholar]
- Suga, Y.; Koshimura, S.; Kobayashi, E. Risk evaluation of drifting ship by tsunami. J. Disaster Res.
**2013**, 8, 573–583. [Google Scholar] [CrossRef] - Satake, K.; Fujii, Y.; Harada, T.; Namegaya, Y. Time and space distribution of coseismic slip of the 2011 Tohoku earthquake as inferred from tsunami waveform data. Bull. Seismol. Soc. Am.
**2013**, 103, 1473–1492. [Google Scholar] [CrossRef] - Haraguchi, T.; Iwamatsu, A. Detailed Maps of the Impacts of the 2011 Japan Tsunami. Aomori, Iwate and Miyagi prefectures, 1; Kokon-Shoin Publishers: Tokyo, Japan, 2011. (In Japanese) [Google Scholar]
- Takagawa, T.; Tomita, T. Tsunami source inversion with time evolution and real-time estimation of permanent deformation at observation point. J. Jpn. Soc. Civ. Eng. Ser. B2 (Coast. Eng.)
**2012**, 68, I_311–I_315, (In Japanese with English abstract). [Google Scholar] [CrossRef] - Cabinet Office, Government of Japan. Massive Earthquake Model Review Meeting of the Nankai Trough, 2012. Available online: http://www.bousai.go.jp/jishin/nankai/model/12/index.html (accessed on 21 July 2017).

**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 |

© 2018 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**

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