E3S Web of Conferences 415, 02020 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502020

Development of an open source Debris Flow Simulator for“Sabo” (DFSS)
Takeshi Shimizu1*, Yusuke Yamazaki1,2, and Naoki Imamori1
1Public Works Research Institute, Volcano and Debris flow Team, 1-6 Minamihara, Tsukuba City, Ibaraki Prefecture, Japan
2Nippon Koei Co., Ltd., Land Infrastructure Development Operations, 5-4 Kojimachi, Chiyoda-ku, Tokyo, 102-8539, Japan

Abstract. We report the development of an open source “Debris Flow Simulator for Sabo” (DFSS). DFSS
consists of three models corresponding to three geomorphological regions (mountain hillside slopes,mountain-to-plain river channels, and lower plains). A 2D rainfall-runoff (RR) model is adopted mainly forhillside slopes; a 1D debris flow runoff (DR) model is used for the river channel; and a 2D debris flow
flooding or inundation (DF) model is adopted for the lower plains. These models are weakly coupled suchthat one can modify the overall model to fit diverse problems. In this article, we provide a detailed
introduction to the models comprising DFSS, and demonstrate its utility.

1 Introduction
The purpose of this article is to introduce our recently
developed open-source “Debris Flow Simulator forSabo” (hereafter, DFSS; [1-2]). The Japanese word
“Sabo” is interpreted to mean not only Sabo works suchas check dams, but also an entire Sabo Project, including
structural and non-structural measures against sedimentand debris flow disasters [3]. Among the important nonstructural measures adopted is setting a debris flow
evacuation zone. A range of physical models are used topredict areas at the risk of hazard and runout areaassociated with debris flows. It is crucial to selectappropriate governing equations, geometric models, andspatial dimensions to adequately approximate specificproblems. Furthermore, the recent expanding use ofsatellite-based spatiotemporal high-resolution data hasenable the occurrence and magnitude of debris flows to
be determined more accurately. Given this background,the development of DFSS was conceptualized to use avariety of distributed data, such as rainfall and land use,
and to adopt a different spatial dimensional models toachieve a reasonable calculation time. DFSS reflects theheavy rainfall conditions and topography typical ofJapan, which has many mountains and few plains.
In this article, we provide a detailed introduction toDFSS and concisely present results using this approach,since [1] is currently only available in Japanese. It is
hoped that this will encourage other researchers to use
the open source code in [2]. Note that DFSS is currently
in the early development stage.

2 Models and Data
DFSS consists of three models corresponding
respectively to three regions (mountain hillside slopes,

mountain-to-plain river channels, and lower plains) as
shown in Fig. 1. A 2D rainfall-runoff (RR) model isadopted for hillside slopes; a 1D debris flow runoff (DR)model is used for the river channel; and a 2D debris flow
flooding or inundation (DF) model is used for the lowerplains. Standard
calculations are performed
independently in this order, and information regarding
the flow is weakly coupled, such that the time-series
output files of the previous model are used as input data
for the next model. These output data to the next model
could be regarded as the data such that real-world data
which are observed in discrete time intervals. Althoughless accurate in its calculations than a strongly coupledmodel, where data connects models at each calculation
time step (dt), the approach presented herein has theadvantage of a configuration and weak coupling modethat are able to be controlled to fit a diverse range ofproblems. Furthermore, the calculation times associatedwith DFSS are particularly short when a 1D model isused. For example, if the target area features only a steep
mountainous topography, the 1D DR model would
provide a more appropriate model than the 2D DF model interms of geometry and calculation time. Whendeveloping DFSS, we adopted the concept of easy
modification of the model to cover a wide range ofproblems. The governing equations and constitutive
equations of each model are based on previous research[4-8].
2.1 Configuration of topographic models
In the 2D region (mountain-side slopes and lower
plains), the region of interest is divided into square grids
according to the north–south and east–westdirections asshown in Fig. 1. When setting up a non- calculation grid,a binary mask of any shape may be used.

_________________________* Corresponding author: t-shimizu@pwri.go.jp© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 415, 02020 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502020■：distributed rainfall-runoff analysis―：debris flow runoff■：debris flow inundation

Slope①

seepage flow is steady and uniform, the equation ofmotion is the same as Darcy’s law as follows :𝑞𝑞𝑠𝑠 = 𝑘𝑘ℎ𝐼𝐼

Subsurface flow③
Sediment yieldstability analysis/erosion(3)

where 𝐼𝐼 is the steepest gradient of the water surface and𝑘𝑘 is saturated hydraulic conductivity.
In a surface flow, the following continuity
equation holds :②④𝜕𝜕ℎ 𝜕𝜕𝑞𝑞𝑠𝑠 𝜕𝜕𝑞𝑞𝑠𝑠
2𝑞𝑞𝑠𝑠𝑠𝑠 𝐿𝐿++= (𝑟𝑟 − 𝑓𝑓𝑠𝑠2 − 𝑒𝑒)cos𝜃𝜃 −
𝜕𝜕𝑥𝑥
𝜕𝜕𝑡𝑡
𝜕𝜕𝑦𝑦
∆𝑥𝑥∆𝑦𝑦Debris flow runoff
deposition / inundation(4)

where 𝑞𝑞𝑠𝑠 , 𝑞𝑞𝑠𝑠 are the 𝑥𝑥 and 𝑦𝑦 components of discharge
flux 𝑞𝑞, respectively, 𝑞𝑞𝑠𝑠𝑠𝑠 is the unit width discharge fromthe slopes to the river channel, and 𝐿𝐿 is river channel
length : which is given the same value as grid length.
Assuming a steady and uniform flow, the equation ofmotion using Manning’s law holds as follows :Debris flow runoff
overflow / inundation

Fig. 1. Geometrical and physical models in “DebrisFlow Simulator for Sabo” (DFSS)e𝑞𝑞 =1√𝐼𝐼ℎ5/3𝑁𝑁(5)where 𝑁𝑁 is the equivalent roughness coefficient of theslope. Discharge from the slopes to the river channel iscalculated using overflow formula as follows :𝑞𝑞𝑠𝑠𝑠𝑠 = 𝜇𝜇ℎ�2𝑔𝑔ℎFig. 2. Water exchange along slopes in the RR model(6)

where 𝜇𝜇 is constant value of 0.35 which value is basedon the design manual of river works of Japan(“Technical Criteria for River Works”) and 𝑔𝑔 isacceleration due to gravity. For weak coupling, the timeseries data of the discharge from the slope to the river
channel are stored in the nodesl, which output text files
passing the information to the next model. When the RR
model is used alone, runoff analysis in the river channel
without sediment is performed ; this concept is not
explained in detail herein.

The 1D region (river channel) is represented bylines connecting the center of the grid (node) with thecenter of the adjacent grids in eight directions with thesteepest downward gradients. In other words, the river
channel is portrayed as a graph consisting of nodes andlines. The nodes store information including time anddischarge. The upstream end of a river channel isdetermined by a grid with a catchment area that exceeds
a user-defined threshold. The width and depth of theriver channel are determined according to regime theory,and do not depend on the size of the square grid.

2.2.2 Debris flow runoff (DR) model
2.2 physical models

The DR model inputs only the output
information reflecing the RR flowing from the slope into
the river channel to the nodes at the appropriate time
during the DR calculation. In the DR model, the amountof sediement yield is calculated by analysis of slope
stability in the river channel. The driving force 𝐺𝐺 is abody force from gravity and the resistance force 𝑅𝑅
represents Coulomb friction and adhesive force asfollows:2.2.1 2D rainfall-runoff (RR) model
In the RR model, the amount of water suppliedby rainfall is divided into surface water and subsurface(ground) water. As shown in Fig. 2, the continuity
equation of water in topsoil layer holds as follows:𝜕𝜕(𝑝𝑝𝑤𝑤 𝐷𝐷)
𝜕𝜕𝜕𝜕= 𝑟𝑟 − 𝑒𝑒(1)𝜎𝜎ℎ𝑠𝑠ℎ𝑠𝑠ℎ
𝐺𝐺 = 𝜌𝜌𝑔𝑔𝐷𝐷sin𝜃𝜃 �� � 𝑐𝑐∗ + �1 − � 𝑝𝑝𝑤𝑤 + � � (1 − 𝑐𝑐∗) + �𝜌𝜌
𝐷𝐷𝐷𝐷𝐷𝐷

where 𝑝𝑝𝑤𝑤 is the volumetric water content of the topsoil,
𝑡𝑡 is time, 𝐷𝐷 is the depth of topsoil layer, 𝑟𝑟 is rainfall
intensity, and 𝑒𝑒 is evapotranspiration rate.
In the case of seepage flow, because the topsoil
layer is a solid–water–air phase mixture, the porosity oftopsoil layer 𝜆𝜆 can be integrated into the continuity
equation of seepage flow as follows:𝜕𝜕ℎ𝑠𝑠
𝜕𝜕𝑞𝑞𝑠𝑠𝑠𝑠 𝜕𝜕𝑞𝑞𝑠𝑠𝑠𝑠cos𝜃𝜃
1(𝑓𝑓 −𝑓𝑓 − 𝑒𝑒)++��=𝜕𝜕𝑡𝑡 𝜆𝜆 − 𝑝𝑝𝑤𝑤 𝜕𝜕𝑥𝑥𝜕𝜕𝑦𝑦
𝜆𝜆 − 𝑝𝑝𝑤𝑤 𝑠𝑠1 𝑠𝑠2ℎ𝑠𝑠ℎ𝑠𝑠𝜎𝜎
𝑅𝑅 = 𝜌𝜌𝑔𝑔𝐷𝐷cos𝜃𝜃 �� � 𝑐𝑐∗ + �1 − � 𝑝𝑝𝑤𝑤 − � � 𝑐𝑐∗� tan𝜙𝜙 + 𝑐𝑐
𝜌𝜌𝐷𝐷𝐷𝐷(7)(8)

where 𝜎𝜎 is soil particle density, 𝜌𝜌 is water density, 𝑐𝑐∗ isvolume concentration of the topsoil layer, 𝜙𝜙 is theinternal friction angle, and 𝑐𝑐 is the adhesive force at thebottom of the topsoil layer. If under state of limit
equilibrium, 𝐺𝐺 = 𝑅𝑅 and solving for 𝜃𝜃 yields the limit
equilibrium angle as follows:(2)

where 𝑥𝑥,𝑦𝑦 are the axis parallel to slope and the verticalaxis, ℎ𝑠𝑠 is water depth in the topsoil, 𝑞𝑞𝑠𝑠𝑠𝑠 , 𝑞𝑞𝑠𝑠𝑠𝑠 are the 𝑥𝑥
and 𝑦𝑦 components of seepage flux 𝑞𝑞𝑠𝑠 , respectively, 𝜃𝜃 isthe gradient of the slope, and 𝑓𝑓𝑠𝑠1 , 𝑓𝑓𝑠𝑠2 are the equilibrium
infiltration rate of the surface layer and the bottom partof the topsoil layer, respectively. Assuming that thetan𝜃𝜃𝑐𝑐 =𝜎𝜎 ℎℎ� − 𝑠𝑠 � 𝑐𝑐∗ + �1 − 𝑠𝑠 � 𝑝𝑝𝑤𝑤 + 𝑐𝑐 ′𝜌𝜌 𝐷𝐷𝐷𝐷tan𝜙𝜙(ℎ𝑠𝑠 + ℎ)
ℎ𝜎𝜎 ℎ𝑠𝑠+ �1 − 𝑠𝑠 � 𝑝𝑝𝑤𝑤
� − � 𝑐𝑐∗ +𝜌𝜌 𝐷𝐷
𝐷𝐷𝐷𝐷(9)where 𝜃𝜃𝑐𝑐 is the critical gradient for the occurrence ofcollapse, and 𝑐𝑐 ′ is the dimensionless adhesive force :𝑐𝑐 ′ = 𝑐𝑐/(𝜌𝜌𝑔𝑔𝐷𝐷 cos𝜃𝜃 tan𝜙𝜙) . The all collapsed material is2

E3S Web of Conferences 415, 02020 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502020

25ℎ −2
�𝑓𝑓𝑑𝑑 + 𝑓𝑓𝑓𝑓 � � �4𝑑𝑑𝜎𝜎2
𝑓𝑓𝑑𝑑 = 𝑘𝑘𝑑𝑑 � � (1 − 𝑒𝑒 )𝑐𝑐𝑐𝑐 1/3
𝜌𝜌𝑚𝑚regarded as river bed sediment which is contributed tothe debris flow through the source term.
In the DR model, the debris flow is a mixed
flow comprising coarse and fine sediment, as well aswater. Fine particles are treated as “particles dissolved
in interstitial fluid” and, as a result the bulk density ofthe interstitial fluid is greater than that of water.
Continuity equations of mixed flow as a whole, the flowof coarse sediment, and the flow of fine sediment holdas follows :𝜕𝜕ℎ 1 𝜕𝜕𝜕𝜕ℎ𝐵𝐵 𝐸𝐸+= + 𝑓𝑓𝑖𝑖𝑖𝑖 − 𝑓𝑓𝑜𝑜𝑜𝑜𝜕𝜕𝜕𝜕𝑡𝑡 𝐵𝐵 𝜕𝜕𝑥𝑥𝑐𝑐∗
𝜕𝜕𝑐𝑐𝑐𝑐 ℎ 1 𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 𝜕𝜕ℎ𝐵𝐵
𝑝𝑝 𝐸𝐸 − 𝑐𝑐𝑐𝑐 𝑓𝑓𝑜𝑜𝑜𝑜𝜕𝜕 (𝑒𝑒𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒)+= � 𝑐𝑐
𝐸𝐸 − 𝑐𝑐𝑐𝑐 𝑓𝑓𝑜𝑜𝑜𝑜𝜕𝜕 (𝑑𝑑𝑒𝑒𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒)
𝐵𝐵
𝜕𝜕𝑡𝑡
𝜕𝜕𝑥𝑥

(10)
(11)

static pressure and gravity acceleration is clarified.
2.2.3 Debris flow flooding (DF) model

The DF model calculates the inundation
overflowing from river channels to the plain. Places
prone to flooding along the river channel are determined
by DR calculations. The governing equations of the DF
model are similar to those of the DR model describedabove, with the exception of dimensions (1D and 2D).
From here onward, only continuity equations andequations of motion in 2D form are described.
Continuity equations of the mixed flow as a whole, aflow of coarse and a flow of fine sediment are,respectively, as follows:(13)𝜕𝜕ℎ 𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕ℎ 𝐸𝐸++= + 𝑓𝑓𝑑𝑑𝑓𝑓
𝜕𝜕𝑡𝑡
𝜕𝜕𝑥𝑥𝜕𝜕𝑦𝑦𝑐𝑐∗(14)𝜕𝜕𝑐𝑐𝑐𝑐 ℎ 𝜕𝜕𝜕𝜕𝑐𝑐𝑐𝑐 𝜕𝜕ℎ 𝜕𝜕𝑐𝑐𝑐𝑐 𝜕𝜕ℎ
𝑝𝑝 𝐸𝐸 + 𝑓𝑓𝑐𝑐𝑐𝑐 (𝑒𝑒𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒)++= � 𝑐𝑐
𝐸𝐸 + 𝑓𝑓𝑐𝑐𝑐𝑐 (𝑑𝑑𝑒𝑒𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒)
𝜕𝜕𝑡𝑡
𝜕𝜕𝑥𝑥
𝜕𝜕𝑦𝑦

(15)

𝜕𝜕𝑐𝑐𝑓𝑓 (1 − 𝑐𝑐𝑐𝑐 )ℎ 𝜕𝜕𝑐𝑐𝑓𝑓 (1 − 𝑐𝑐𝑐𝑐 )𝜕𝜕ℎ 𝜕𝜕𝑐𝑐𝑓𝑓 (1 − 𝑐𝑐𝑐𝑐 )𝜕𝜕ℎ++
𝜕𝜕𝑥𝑥
𝜕𝜕𝑦𝑦
𝜕𝜕𝑡𝑡
𝑝𝑝𝑓𝑓 𝐸𝐸 + 𝑓𝑓𝑐𝑐𝑓𝑓 (𝑒𝑒𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒)=�(1/𝑐𝑐∗ − 1)𝑐𝑐𝑓𝑓 𝐸𝐸 + 𝑓𝑓𝑐𝑐𝑓𝑓 (𝑑𝑑𝑒𝑒𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒)

(16)

(17)

where 𝜕𝜕 is the momentum correction factor, 𝜕𝜕 is thewater level of the debris flow expressed as 𝜕𝜕 = 𝑧𝑧 +ℎ cos 𝜃𝜃 , and 𝜏𝜏𝑏𝑏 is basal stress acting on the riverbed,
which is expressed as follows [7]:𝜏𝜏𝑏𝑏 = 𝜏𝜏𝑠𝑠 + 𝜌𝜌𝑚𝑚 𝑓𝑓𝑏𝑏 𝜕𝜕2𝜕𝜕𝜕𝜕 𝜏𝜏𝑏𝑏𝑠𝑠
𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ++= −𝑔𝑔ℎ−𝜕𝜕𝑥𝑥𝜕𝜕𝑦𝑦𝜕𝜕𝑥𝑥 𝜌𝜌𝑚𝑚
𝜕𝜕𝑡𝑡

(18)

where 𝜏𝜏𝑠𝑠 is yield stress and 𝑓𝑓𝑏𝑏 is the flow resistancecoefficient. These are expressed as follows:
𝑐𝑐𝑐𝑐 1/5
𝜏𝜏𝑠𝑠 = � � (𝜎𝜎 − 𝜌𝜌𝑚𝑚 )𝑐𝑐𝑐𝑐 𝑔𝑔ℎ cos 𝜃𝜃 tan 𝜙𝜙𝑐𝑐∗

(25)

(26)

(27)

where 𝜕𝜕 and 𝜕𝜕 are the depth averaged velocity of mixed
flow along the x and y directions, respectively, 𝑓𝑓𝑑𝑑𝑓𝑓 , 𝑓𝑓𝑐𝑐𝑐𝑐
and 𝑓𝑓𝑐𝑐𝑓𝑓 are the total, coarse, and fine sediment fluxes
from the river channel to the plain, respectively, which
are given as outputs of the DR model calculation.
Equations of motion of x and y direction are,respectively, as follows:

The equation of motion of the DR model is a depthaveraged one-fluid model as follows:𝜕𝜕𝜕𝜕 𝜏𝜏𝑏𝑏
𝜕𝜕𝜕𝜕ℎ 1 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ𝐵𝐵+= −𝑔𝑔ℎ−𝐵𝐵 𝜕𝜕𝑥𝑥
𝜕𝜕𝑥𝑥 𝜌𝜌𝑚𝑚
𝜕𝜕𝑡𝑡(23)𝜕𝜕𝜕𝜕𝜕𝜕ℎ(24)= −𝑔𝑔ℎ cos 𝜃𝜃− 𝑔𝑔ℎ sin 𝜃𝜃
𝜕𝜕𝑥𝑥
𝜕𝜕𝑥𝑥
where 𝜕𝜕𝑧𝑧/𝜕𝜕𝑥𝑥 = sin 𝜃𝜃 . In this expression, the term ofTo describe the change of a river bed due to 𝐸𝐸, thecontinuity equation of riverbed 𝑧𝑧 holds as follows:𝜕𝜕𝑧𝑧𝐸𝐸=−𝜕𝜕𝑡𝑡𝑐𝑐∗ cos 𝜃𝜃

1 1 ℎ −2+ ln �𝜅𝜅 𝜅𝜅 𝜅𝜅𝑠𝑠−𝑔𝑔ℎwhere 𝜌𝜌𝑚𝑚 is the bulk (quasi-homogenous) density of aninterstitial fluid with fine particle and water as follows:𝜌𝜌𝑚𝑚 = (𝜎𝜎 − 𝜌𝜌)𝑐𝑐𝑓𝑓 + 𝜌𝜌 .

(22)

(21)

where 𝐴𝐴𝑠𝑠 is constant (8.5), 𝜅𝜅 is the Karman constant(0.4), 𝜅𝜅𝑠𝑠 is the diameter of coarse sediment, 𝑑𝑑. These
constants are values usually used in the log law ofvelocity profiles on a movable bed. Note that the firstterm of right-hand side of eq. (17) could be expressed asfollows:

where 𝜃𝜃𝑒𝑒 is the equilibrium gradient of the debris flow
expressed as Takahashi’s equilibrium condition for adebris flow of non-adhesive material [6]:(𝜎𝜎/𝜌𝜌𝑚𝑚 − 1)𝑐𝑐𝑐𝑐
tan 𝜙𝜙(𝜎𝜎/𝜌𝜌𝑚𝑚 − 1)𝑐𝑐𝑐𝑐 + 1

𝑓𝑓𝑓𝑓 = 𝑘𝑘𝑓𝑓 (1 − 𝑐𝑐𝑐𝑐 )5/3 𝑐𝑐𝑐𝑐 −2/3

𝑓𝑓𝑏𝑏 = �𝐴𝐴𝑠𝑠 −where ℎ is the depth of the mixed flow, 𝐵𝐵 is the riverwidth, 𝜕𝜕 is the depth averaged velocity of flow, 𝐸𝐸 iserosion/deposition velocity (described in detail below),𝑐𝑐∗ is the volume concentration of the static riverbed, 𝑓𝑓𝑖𝑖𝑖𝑖
is the discharge from the slopes to the river channel as
RR output, 𝑓𝑓𝑜𝑜𝑜𝑜𝜕𝜕 is the overflow discharge from the river
channel to the plain (which is passed to the DF model),𝑐𝑐𝑐𝑐 and 𝑐𝑐f are the depth-averaged volume concentrationsof coarse particles in whole space and fine particles ininterstitial space, respectively, 𝜕𝜕 is the sediment flux
correction factor, 𝑝𝑝𝑐𝑐 and 𝑝𝑝𝑓𝑓 are the volume fractions ofcoarse and fine particles on the riverbed, respectively,
and 𝑐𝑐∗𝐷𝐷 is the volume concentration of coarse particles
on the static riverbed after deposition. 𝐸𝐸 [5] is given by :tan 𝜃𝜃𝑒𝑒 =(20)where 𝑑𝑑 is the representative diameter of a coarse
particle, 𝑘𝑘𝑓𝑓 and 𝑘𝑘𝑑𝑑 are constant parameters with valuesof 0.16 and 0.0828, respectively [5], 𝑒𝑒 is the restitution
coefficient between coarse particles, and 𝑓𝑓b is as follows[8]:𝜕𝜕𝑐𝑐𝑓𝑓 (1 − 𝑐𝑐𝑐𝑐 )ℎ 1 𝜕𝜕𝑐𝑐𝑓𝑓 (1 − 𝑐𝑐𝑐𝑐 )𝜕𝜕ℎ𝐵𝐵+𝜕𝜕𝑡𝑡
𝜕𝜕𝑥𝑥𝐵𝐵(12)𝑝𝑝𝑓𝑓 𝐸𝐸 − (1 − 𝑐𝑐𝑐𝑐 )𝑐𝑐𝑓𝑓 𝑓𝑓𝑜𝑜𝑜𝑜𝜕𝜕 (𝑒𝑒𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒)=�(1/𝑐𝑐∗ − 1)𝑐𝑐𝑓𝑓 𝐸𝐸 − (1 − 𝑐𝑐𝑐𝑐 )𝑐𝑐𝑓𝑓 𝑓𝑓𝑜𝑜𝑜𝑜𝜕𝜕 (𝑑𝑑𝑒𝑒𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑡𝑡𝑒𝑒𝑒𝑒𝑒𝑒)

𝐸𝐸= 𝑐𝑐∗ tan(𝜃𝜃 − 𝜃𝜃𝑒𝑒 )𝜕𝜕𝑓𝑓𝑏𝑏 =𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕ℎ𝜕𝜕𝜕𝜕 𝜏𝜏𝑏𝑏𝑠𝑠++= −𝑔𝑔ℎ−𝜕𝜕𝑡𝑡𝜕𝜕𝑥𝑥𝜕𝜕𝑦𝑦𝜕𝜕𝑦𝑦 𝜌𝜌𝑚𝑚

(19)

3(28)

(29)

E3S Web of Conferences 415, 02020 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502020

References
1.

2.

3.
4.

5.
Fig. 3. Location of Aso Caldera (top left) and thetargeting area shown as transparent black. The redpoint is the outlet point that divided 2D region of theRR and the DF model.

6.
7.

whereandare basal stresses (ref. eq. 18) along
the x and y directions, respectively. Note that therelationship of eq. (24) can be applied to first term of theright-hand side of these equations.

8.
9.

1.1 Data
On July 2nd, 1992, a large scale debris flow disasteroccurred at Aso-cho, Kyusyu, Japan (Fig. 3). Aso isfamous for its large caldera and the surrounding hillsides
feature large amount of unstable sediment.
Data such as soil parameters and deposition

Y. Yamazaki, T. Shimizu, Y. Ishii and K. Ishida,Public Works Research Institute (PWRI)Technical Note, Vol.4419 (2022) (in Japanese)
ibid, DFSS Ver.1.00, https://github.com/PWRIvolcano-debrisflow-team/DFSS, accessed 2023
Jan. 20thJ. Kanbara and N. Imamori, Landslides, 17, 25032513, DOI 10.1007/s10346-020-01554-9 (2020)
Y. Yamazaki, S. Egashira, Y. Iwami, J.
Soci, (J. Hydraul. Eng.), B1, 60, I 1327-I
1332 (2016) (in Japanese. with Englishabstract)
S. Egashira, N. Honda and T. Itoh, Phys. Chem. Earth,26, 9, 645-650 (2001)
T. Takahashi, Jour. Hydraulics Div. ASCE,
106, 381-396 (1980)
S. Egashira, K. Miyamoto and T. Itoh,
Proceedings of5th international conference.
Water Res. Eng. Div. ASCE (1997)
P. Y. Julien, and A. Paris, Jour. Hydraul Eng.,
136, 9, 676-679 (2010)
H. Hashimoto, M. Hirano, S. Hayashi and J.
Umemura,J. Jpn. Soc. Hydrol. Water Resour., 4, 25-32,
https://doi.org/10.3178/jjshwr.4.25 (1991) (inJapanese).

area from [9] were used for confirmation of theapplicability of DSFF to debris flow disasters involcanic area. Rainfall data from [9] was also used bythe Automated Meteorological Data Acquisition Systemof Japanese Meteorological Agency (see [1] for furtherdetail).

2 Results
The results of our calculations are shown in Fig.
4. The calculated area was got by the DF model using
the output of the RR and DR model as input files. As
shown in Fig. 4, comparison between the calculated areaof maximum depth and the area of deposition
determined from aerial photography [9], shown in thedotted area, indicates that the calculated results matchthe observed depositional area near the mountain, but do
not match the broader observed depositional area,especially in the lower region. The reason for thisinaccuracy is thought to be the topography of the river
channel in the mountain, which is confined by thesurrounding hillsides. As such, the results of DFSS,
especially in 2D calculation, were found to be sensitive
to digital elevation models.
In order to accurately estimate the inundation
area across the plains, it will be necessary to improve theDFSS by considering the presence of various detailed
features such as topography and Sabo structures, theseas elevation data or boundary conditions.

Max. depthof mixture

Fig. 4. Estimated maximum depth map for mixed
flows. Dotted area represents the area of deposition
based on [9].

4