Italian Journal of Engineering Geology and Environment  Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
101
DOI: 10.4408/IJEGE.201103.B012
NONDIMENSIONAL PARAMETERS CONTROLLING OCCURRENCE
AND CHARACTERISTIC OF LANDSLIDES THAT PROVIDE
SEDIMENT FOR DEBRIS FLOW DEVELOPMENT
F
UMITOSHI
IMAIZUMI
(*)
& K
UNIAKI
MIYAMOTO
(*)
(*)
Graduate School of Life and Environmental Sciences, University of Tsukuba, Japan
landslide slopes to immediately subsequent landslides,
as well as the cyclical nature of landslide processes.
K
EY
WORDS
: landslide, nondimensional parameters, inﬁ nite
slope model
INTRODUCTION
Landslides are one of the most important proc
esses supplying debris ﬂ ow materials into channels.
The ability to predict the timing, location, and vol
ume of landslides would improve the estimation of
debrisﬂ ow occurrence. Many physical models that
predict the occurrence of landslides have been pro
posed for the mitigation of landslide and debrisﬂ ow
disasters, as well as the estimation of sediment sup
ply rates into channel networks (e.g., B
URTON
et alii,
1998; D
YMOND
et alii, 1999; S
IDLE
& O
CHIAI
, 2006).
Several studies have used sensibility analysis and nu
merical simulations to determine that the occurrence of
landslides is affected by multiple parameters, includ
ing topographic factors, i.e., slope gradient and shape
geometry (O
KIMURA
AND
N
AKAGAWA
, 1988; O
HSAKA
et
alii, 1992; M
ONTGOMERY
& D
IETRICH
, 1994; S
ASAHARA
et alii, 1995), soil constants (S
AMMORI
et alii, 1993;
M
ONTGOMERY
& D
IETRICH
, 1994; W
O
& S
IDLE
, 1995),
and hydraulic conductivities (H
IRAMATSU
et alii, 1990;
S
AMMORI
et alii, 1993). Field studies have also empiri
cally documented the inﬂ uence of many parameters on
slope stabilities (e.g., D
UMAN
et alii, 2004). To improve
ABSTRACT
Landslides are one of the most important processes
supplying debris ﬂ ow materials into channels. We need
to predict the timing, location, and volume of landslides
for better estimation of the occurrence of debris ﬂ ows.
However, a number of soil parameters (e.g., the angle
of internal friction, cohesion, and porosity), which have
signiﬁ cant spatial variability, are needed to predict land
slide occurrence. Therefore, it is important to make clear
the contribution of these parameters to overall slope sta
bility and their relationships to one another. In this study,
we normalized the safety factor equation for inﬁ nite
slope model, and introduced multi soil layer structure
into the model. We also tried to clarify factors affect
ing the pore water pressure on the basis of the equations
for the vertical inﬁ ltration process (i.e., continuity equa
tion and Darcy’s law). Depth gradient of the porewater
pressure at the given soil layer is controlled by the ratio
of the water velocity in saturated zone to the hydraulic
conductivity at that layer. New, nondimensional repre
sentations for the effects of groundwater table and cohe
sion of soil were obtained by the normalization of the
safety factor equation. They are evaluated relative to
each other by comparison to the stability of dry nonco
hesive soil. The effect of the groundwater table and the
cohesion of soil on the slope stability are both affected
by depth of the sliding surface. We also found that the
effect of cohesion should be evaluated from the com
parison with the maximum effect of groundwater table
to the stability. This can explain the immunity of post
F. IMAIZUMI & K. MIYAMOTO
102
5th International Conference on DebrisFlow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy  1417 June 2011
this study, soil porosity is considered to be constant
throughout all soil layers to clarify the relationship be
tween the spatial distribution of hydraulic conductiv
ity and porewater pressure. As demonstrated by many
ﬁ eld surveys, hydraulic conductivity is generally lower
in deeper soil layers (e.g., H
IRAMATSU
et alii, 1993; I
RA

SAWA
et alii, 1997; H
IRAMATSU
& B
ITO
, 2001). Therefore,
we set a smaller hydraulic conductivity for deeper soil
layers. The continuity of vertical inﬁ ltration of incom
pressible water can be expressed as following equation:
where ρ is the mass density of water, n is the porosity of
soil, S is the degree of saturation, u is the vertical (zaxis)
water ﬂ ow velocity. As S equals 1 in the saturated zone,
equation (1) in the saturated zone can be rewritten as:
When we treat groundwater as the steady state, the
equation of motion is described by the following equation:
where дp / дz is the pressure gradient, and g is the
gravitational acceleration.
The pressure gradient дp / дz consists of the
porewaterpressure gradient дp
s
/ дz and the hydrau
lic gradient дp
d
/ дz:
The relationship between the hydraulic gradient
and water velocity is expressed by Darcy’s law:
where K is hydraulic conductivity. By substituting equa
tions (4) and (5) into equation (3), the porewaterpres
the prediction of debris ﬂ ows into channel networks,
we must be able to estimate the occurrence of land
slides in the entire area supplying debrisﬂ ow material.
However, signiﬁ cant spatial variability in these param
eters prevents us from accurately predicting landslides
at the catchment area scale. Therefore, the extraction
of control parameters for the occurrence of landslides
from numerous soil and topographic parameters is nec
essary for the effective prediction of landslides.
Some studies have revealed that landslide vol
ume and depth are affected by the magnitude and
pattern of rainfall (D
AI
& L
EE
, 2001; H
ATTANJI
, 2003).
Therefore, the depth proﬁ le of hydraulic conductiv
ity, which controls temporal changes in the depth
proﬁ le of porewater pressure, should be considered
in the estimation of landslide volume.
To extract the important parameters controlling the
occurrence and volume of landslides, their interrelation
ships and contributions to overall slope stability must be
clariﬁ ed. In this study, we discuss a simple but versatile
model for the prediction of landslides based on the sta
bility of an inﬁ nite slope (e.g., S
IDLE
& O
CHIAI
, 2006).
We assume the multilayer soil structure to reﬂ ect the
depth distribution of soil parameters for the slope sta
bility analysis. First, we identify important factors af
fecting the magnitude of porewater pressure that induce
shallow landslides during rainfall events. Because the
inﬁ nite model is employed in this study, the groundwa
ter table should be parallel to the ground surface. We
thus analyze the porewater pressure far from the top of
the slope, where the water table is parallel to the ground
surface. This allows us to consider only the vertical in
ﬁ ltration of groundwater in the analysis of pore water
pressure. Second, we normalize the safety factor equa
tion for an inﬁ nite slope to obtain the nondimensional
parameters controlling the occurrence of landslides.
Given the nature of the inﬁ nite slope, we consider the
spatial distribution of soil parameters only in the direc
tion of depth. To simplify the model, a constant is used
to describe the parameters of each soil layer.
BASIC EQUATIONS FOR THE VERTICAL
INFILTRATION PROCESS
VERTICAL INFILTRATION IN THE SATURATED
ZONE
To estimate the pore water pressure above a slid
ing surface, we must ﬁ rst develop basic equations for
groundwater in multilayer soil structure (Fig. 1). In
(1)
Fig. 1  Schematic diagram of vertical inﬁ ltration
(2)
(3)
(4)
(5)
NONDIMENSIONAL PARAMETERS CONTROLLING OCCURRENCE AND CHARACTERISTIC OF LANDSLIDES
THAT PROVIDE SEDIMENT FOR DEBRIS FLOW DEVELOPMENT
Italian Journal of Engineering Geology and Environment  Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
103
S in the direction of depth. The integrals in equations
(10) and (11) can be performed as follows:
where η
w
, S are:
By substituting equations (12) and (13) into equa
tion (9) and multiplying the equation by tanθ / tan
f,
the normalized safety factor equation is
where F
w
and F
c
are:
F
w
(groundwater term) and F
c
(cohesion term) are
the nondimensional parameters describing the ef
fects of groundwater and cohesion on the safety fac
tor, respectively. The effects of these terms on slope
stability are evaluated by their scale relative to the
ﬁ rst term on the right side of member “1” in equation
(16), and also as the linear sum of the ﬁ rst term. The
safety factor equation for the soil layers without the
sure gradient can be expressed by the following equation:
The porewater pressure at a boundary between soil
layers can be obtained by integrating equation (6) from
the groundwater table of the saturated zone with the soil
layer boundary. By setting the boundary condition p
s
=
0 at the groundwater table, the pore water pressure is:
where m is number of soil layers above the analyzed
boundary,
K
i
is the hydraulic conductivity, and D
i
is
the thickness of ith soil layer in the saturated zone.
Equation (7) indicates that porewater pressure is
controlled not only by the hydraulic conductivity of
an individual layer (K
i
), but also by the downward
velocity of water in the saturated zone (u). As dem
onstrated by equation (7), this downward velocity
is constant in the saturated zone. In cases where the
saturated zone is formed on impermeable bedrock,
the vertical water velocity is equal to 0. Therefore,
porewater pressure on impermeable bedrock is:
Equation (8) indicates that pore water pressure
agrees with hydrostatic pressure. As presented in
equation (8), the presence of another saturated zone
above the analyzed zone (Fig. 1) does not affect the
magnitude of porewater pressure on the analyzed
soil layer boundary.
NORMALIZATION OF THE SAFETY
FACTOR EQUATION
By introducing the coordinate system shown in
Fig. 2, the safety factor equation for a sliding surface
can be expressed by the following equations:
where F is the safety factor, τ
r
is the shear strength of
soil on the sliding surface, τ is the shear stress on the
sliding surface, σ is the mass density of soil particles,
f is the angle of internal friction, and c is the cohesion
of soil at the sliding surface.
As shown in equations (10) and (11), the safety
factor is affected by the spatial distribution of n and
(6)
(7)
(8)
Fig. 2  Schematic diagram of the inﬁ nite slope
(11)
(10)
(9)
(13)
(12)
(14)
(15)
(16)
(17)
(20)
(19)
(18)
F. IMAIZUMI & K. MIYAMOTO
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5th International Conference on DebrisFlow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy  1417 June 2011
EFFECT OF COHESION ON SLOPE STA
BILITY
The effect of cohesion on slope stability is rep
resented by the cohesion term (F
c
), the third term
on the right side of equation (16). The magnitude of
the numerator of the cohesion term (η
c
) depends on
soil depth (h  z
1
) and cohesion (c). Equations (18)
and (20) indicate that F
c
is larger for shallower slid
ing surfaces. In the case of shallow landslides, h  z
1
is generally 1 m, and c ranges from 0 to 9800 N/m
2
(e.g., I
RASAWA
et alii, 1997; H
IRAMATSU
& B
ITO
, 2001).
In this case, the range of η
c
is:
If θ˚ and tan
f = 0.7, the maximum value of the
η
c
is 2.0. Given that the denominator of the cohesion
term ranges from 1.3 to 1.8, the maximum value of
F
c
is similar to or higher than the basic part of slope
stability “1” on the right side of equation (16). Thus,
the cohesion term (F
c
) is an important factor affecting
the occurrence of shallow landslides. In contrast, the
maximum value of η
c
for a sliding surface at a depth of
10 m is 0.2 if the other parameters (i.e., θ˚,
f) remain
the same as above. Therefore, the inﬂ uence of F
c
on
slope stability is negligible for deepseated landslides.
Because the denominator of F
c
is the same as that
of F
w
, the numerators of these terms (η
c
and εη
w
) can
be simply compared. In cases where η
c
is much small
er than 1 (= maximum value of εη
w
), the inﬂ uence of
η
c
on slope stability is negligible. To demonstrate the
inﬂ uence of η
c
on slope stability, we assume that the
slope has a gradient θ that is similar to
f. This hy
pothesis is not unusual, as many ﬁ eld surveys (e.g.,
T
SUCHIYA
& K
OMOTO
, 1995; H
IRAMATSU
& B
ITO
, 2001)
have found similar relationships between θ and
f.
saturated zone and soil cohesion ((h
w
 z
l
) = 0, c = 0 )
is considered the basic part of equation (16):
Equation (21) indicates that neither porosity (n)
nor degree of saturation (S) affects the safety factor
in cases lacking a saturated layer and soil cohesion.
EFFECT OF GROUNDWATER ON SLOPE
STABILITY
The denominator of the second term on the right side
of equation (16) (hereafter called W), which is identical
to that of the third term, indicates the average speciﬁ c
gravity of the soil layer. The denominator is deformed as:
Comparison of the denominator of equation (17)
with equation (22) shows that η
w
and S in W are symmet
rical. Given that the range of both η
w
and S values is 0 to
1, W ranges from (1n)σ / ρ to (1n)σ / ρ + n. Thus, in cases
where n ≈ 0.5 and σ / ρ ≈ 2.6, W ranges from 1.3 to 1.8.
The numerator of the groundwater term (F
w
) is the
product of η
w
, the ratio of thickness of the saturated
zone to the depth of the sliding surface (equation 14),
and ε, the ratio of porewater pressure p
S
to hydro
static pressure. Given that both of these nondimen
sional parameters range from 0 to 1, the numerator
also ranges from 0 to 1. Consequently, changes in the
numerator εη
w
affect the safety factor more strongly
than do changes in the denominator W.
When the sliding surface is formed on imper
meable bedrock, ε is equal to 1 (see equations 19
and 8). In this case, F
w
can be obtained by substitut
ing ε = 1 into equation (17):
To clarify the degree of dependence on ground
water parameters (i.e. η
w
, and S), changes in F
w
with
increasing η
w
were investigated under conditions of
n = 0.5, σ / ρ = 2.6, and several S values (0, 0.3,
0.6, 0.9; Fig. 3). The inﬂ uence of S on F
w
is small if
porosity n is 0.5 (Fig. 3). In addition, F
w
is approxi
mately proportional to η
w
. As the maximum value
of F
w
is 1/1.8, F
w
decreases the safety factor by a
maximum of 50%.
(21)
(22)
(23)
Fig. 3  Degree of dependence of
F
w
on η
w
and
S
(24)
NONDIMENSIONAL PARAMETERS CONTROLLING OCCURRENCE AND CHARACTERISTIC OF LANDSLIDES
THAT PROVIDE SEDIMENT FOR DEBRIS FLOW DEVELOPMENT
Italian Journal of Engineering Geology and Environment  Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
105
We also assume that the sliding surface is located
on bedrock with a hydraulic conductivity of almost 0.
In this case, F
w
is expressed by equation (23). Therefore,
the safety factor (F) is equal to 1 when η
w
 η
c
=
0. Given
that η
w
ranges from 0 to 1, landslides should occur when
η
c
is less than 1. Based on equation (20), the relationship
between c and h  z
1
that satisﬁ es is expressed as:
The relationship between c and (h  z
1
) that satis
ﬁ es η
c
=1 is not clearly affected by the value of θ˚ (=
f;
Fig. 4). In cases where cohesion (c) ranges from 3000
to 9000 N/m
2
, the range of (h  z
1
) required for land
slides to occur is 0.4 to 1.5 m (Fig. 4). No landslide
should occur on the slopes plotted at the lower right
of ﬁ g. 4 because η
c
the of these slopes always exceeds
1. In contrast, the η
c
of slopes plotted at the upper left
of Fig. 4 is less than 1. Therefore, landslides occur on
these slopes when η
w
satisﬁ es 1 ≥ η
w
= η
c
.
Based on these discussions, the following im
portant characteristics of landslides may be deduced.
Once a landslide occurs on a slope, the soil depth (h
 z
1
) approaches zero or is decreased signiﬁ cantly.
Based on equation (20), η
c
is considered to exceed 1
when little or no regolith remains on the sliding sur
face. Thus, a landslide should never occur on these
slopes, even if surface ﬂ ow is generated. Thereafter,
soil depth may be recovered gradually by the weath
ering of bedrock and inﬁ lling from the surrounding
area. Landslides can occur again when the depth of
the soil layer on the sliding surface reaches a critical
level that satisﬁ es η
c
=1. This explains the immunity
of postlandslide slopes to immediately subsequent
landslides, as well as the cyclical nature of landslide
processes (e.g., I
IDA
, 2004).
SUMMARY AND CONCLUSIONS
In this study, we tried to identify the nondi
mensional parameters that control the occurrence of
landslides, which are one of the most important proc
esses by which debrisﬂ ow materials enter channel
networks. We also clariﬁ ed the contribution of these
parameters to overall slope stability and their relation
ships to one another. We used the inﬁ nite slope model,
which is a simple but versatile model for landslide
prediction. Multilayer soil structure was also as
sumed in order to reﬂ ect the depth distribution of soil
parameters in the analysis of slope stability.
On the basis of equations describing the verti
cal inﬁ ltration process (i.e., continuity equation
and Darcy’s law), we identiﬁ ed the factors affect
ing porewater pressure. Our study revealed that the
depth gradient of pore water pressure in an individu
al soil layer is affected by the ratio of vertical water
velocity in the saturated zone to hydraulic conductiv
ity in that layer. In cases where the saturated zone
is developing on impermeable bedrock, porewater
pressure agrees with hydrostatic pressure.
New nondimensional representations of the ef
fects of the groundwater table and soil cohesion were
obtained by normalizing the safety factor equation.
They were evaluated relative to the stability of dry
noncohesive soil. The effects of the groundwater ta
ble and soil cohesion on slope stability depend on the
depth of the sliding surface. We also found that the
effect of cohesion should be evaluated by comparison
with the maximum effect of the underground water ta
ble on stability. This comparison can explain a slope’s
immunity to the occurrence of landslide, as well as the
periodicity of landslide occurrence.
Our analysis is applicable to slopes with multi
ple soil layer boundaries and saturated zones (Figs. 1,
2). Because the parameters relevant to groundwater
and cohesion include soil depth, the vertical structure
of soil layers must be considered for the prediction
of landslides. Furthermore, depth of sliding surface
should be estimated in order to predict volume of
the landslide sediment supplied to mountainous tor
rents. Therefore, analysis of slopes with multilayer
soil structure is also effective for prediction of debris
ﬂ ows. To predict debris ﬂ ows in mountainous tor
rents, we need to predict the occurrence of landslides
in the entire area that supplies debrisﬂ ow material.
Therefore, methods for investigating the regional spa
(25)
Fig. 4  Relationship between c and hz
1
when η
c
=1
F. IMAIZUMI & K. MIYAMOTO
106
5th International Conference on DebrisFlow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy  1417 June 2011
ACKNOWLEDGEMENTS
We appreciate Dr. Hideji Maita for kindly giving
us useful advice on this study.
tial and temporal distribution of the nondimensional
parameters should be developed to apply our results to
the prediction of debris ﬂ ows.
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