京都大学防災研究所年報 第 58 号 B 平成 27 年 6 月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 58 B, 2015

Numerical Simulation for the Erosion and Stability of Riverbanks Subjected to
Different Flood Hydrographs

Ahmed ALY EL-DIEN, Hiroshi TAKEBAYASHI, and Masaharu FUJITA
Synopsis
Stability of riverbank under the conditions of five different flood hydrographs and of
three bed conditions is discussed using the results of numerical simulation. Threemodels of hydraulic fluvial erosion, seepage flow, and slope stability are coupled to
discuss the effect of the seepage flow and river bed deformation on riverbank stability.
The fluvial erosion model predicts the distribution of boundary shear stress along theriver cross section and then predicts the erosion progress. The seepage model predicts
the spatial and temporal variations in the groundwater table and the pore-water pressure
inside the riverbank. Stability model calculates the factor of safety of bank material andestimates the possible failure plane. The three models are based on the finite element
method with moving boundaries. The response of riverbank to the oscillated water levelin the river and the consequent groundwater table is analyzed. The trend of factor ofsafety through time is presented. The inﬂuence of relevant geometrical, internal andexternal forces are also discussed.
Keywords: Riverbank stability, flood, seepage, fluvial erosion, hydrograph.

1.

Introduction

Flood characteristics such as magnitude,frequency, duration, peak discharge, and variability
govern some aspects of the river as they erode,transport and deposit sediments in the river reach.
Stability of riverbank depends on its geometry,its material properties, and the forces to which it issubjected. These forces include the effects of water
both internally, in the form of pore-water pressure
and seepage forces and externally in terms ofconfining water pressure and hydrodynamic forces(Lane and Griffiths, 2000). The movement of water
through voids generates a hydraulic drag in the
direction of flow which is known as seepage force.
Seepage force is generated when hydraulic headgradient exists between two points within thesaturated zone. The seepage force acts to separate
the soil particles and reduces the effective stress.

Many banks are frequently subjected to suddenchanges of water level (rise or fall), like theriverbanks during flood events, reservoir banks
during dam operation, and irrigation channels thatare operated by rotation system. In a two-turnrotation irrigation system, the agricultural land isdivided into two parts. The branch channels that
serving the first part are "working" and allowed toconvey water with full capacity while the otherchannels that serving the second part are "closed"with no flow. The two-turn rotation is reversed after
8 days (4 working + 4 closing) and so on, suchrotation irrigation system is commonly used in
Egypt. In this system there is a periodic filling andemptying for the channels which affect the stabilityof its banks. Such fluctuation in water level alters
the hydraulic conditions inside the bank and finallyleads to bank failure. Bank retreat occurs by a
combination of two processes: (1) hydraulic fluvial― 365 ―

erosion by the erosive stress of the flowing waterand it acts mainly on the lower portion of the bank,(2) mass failure under gravity which occurs on theupper part of the bank.
Fluvial erosion is caused by an imbalance
between hydraulic shear stress on the bank surface
and the resistance of a bank material. When theshear stress exceeds the critical shear strength of thebank material, hydraulic erosion processes are
initiated. Mass failure occurs if there is a change inthe bank geometry or a change in the applied loads.
During flood event, a great amount of sediment isremoved from the bank toe and bed (change in
geometry), the bank becomes too steep and, maybe,
an overhang is formed. Such alteration in thegeometry changes the distribution of normal andshear stresses inside the riverbank. Also, any
increase in the bank weight due to saturation or
external loads will also change the stress
distribution. Mass failure takes place when thegenerated stresses reach certain failure criterion that
depends on cohesion c ′ and angle of friction ϕ′
for riverbank material. It have been found that mass
failure takes place as a second step after thehydraulic erosion forms a deep undercut at the banktoe (Thorne, 1982; Darby et al., 2010).
Beside the erosive shear stress, field studies
have found other flow properties to be morestrongly correlated to hydraulic erosion during
flood events. For example, (Wolman, 1959) found a
relation between the storm duration and thehydraulic erosion. (Knighton, 1973) found that flow
variability (number of discharge peaks) also affects
bank erosion. (Hooke, 1979) showed that at sites
where hydraulic processes are dominant over
sub-aerial processes, the variable with the strongest
correlation to bank erosion is event peak discharge.
(Julian and Torres, 2006) found that event peak ofexcess shear stress is the best predictor of bankerosion for moderately cohesive banks andvariability of excess shear stress is the best
predictor for minimally cohesive banks. They
concluded that hydraulic bank erosion is dictated byflood peak intensities.
The flood event changes the internal forces
inside the riverbank. For example, during the highwater surface level (WSL) resulted from the flood,the banks are saturated and its weight becomeheavier than before, and also internal seepage flow
generates seepage forces in addition to thegeneration of positive pore water pressure.
The objective of this paper is to (1) investigate
the response of riverbanks under different flood
events and determine the most critical flood pattern,(2) achieve a better understanding for the factors
and mechanisms determining bank stability; and (3)to investigate interactions between fluvial erosion
processes and mechanisms of mass failure incontrolling bank morphology.
This research tries to investigate to what extent
a flood event can affect the riverbank on two main
aspects: (1) the change in bank geometry, (2) thechange in forces. To assess the change in geometry,fluvial erosion model is presented. To evaluate theinternal pore water pressure, the seepage model ispresented. Finally, results from both the erosion andseepage models are incorporated into a stability
model to calculate the factor of safety (FOS) andpredict the expected failure plane. Stability of
hypothetical riverbank subjected to five different
flood hydrographs and three bed conditions isnumerically simulated. First we briefly describe theformulations for fluvial erosion, seepage flow andriverbank stability models. Subsequently, theeffects of flood hydrograph characteristics and theresults from simulation cases are discussed. Finally,
we conclude our findings and outline the limitations
in our approach to be considered in the future
research.
2. Models
2.1 Fluvial Erosion model
The rate of fluvial bank erosion can bequantified using an excess shear stress formula(Osman and Thorne, 1988; Langendoen and Simon,
2008):
ω = k(τo − τcr )βω = 0.0

τo > τcr(1-a)

if τo ≤ τcr

(1-b)

if

where 𝜔 is the eroded distance normal to the banksurface per unit time (m/sec), τo is the shear stressfrom flow (N/m2), k is the erodibility coefficient(m3/N·s); τcr is the critical shear stress forentrainment (N/m2); andβis an empirically derived― 366 ―

exponent (dimensionless), often assumed to have avalue of 1.0 in bank erosion studies, (Rinaldi andDarby, 2007). Erosion distance perpendicular to thebank surface, E, can be calculated from:E = ω × ∆t

(2)

Where E in (m), and Δt is the time step (second).
The fluvial erosion model requires thedetermination of the three parameters τo , k, τcr .
The acting boundary shear stressτo , is estimatedusing the flow net method described by (Guoliang
Yu., and Soon-Keat, 2007). The distribution ofboundary shear stress τo is calculated each time
step according to the new updated geometry andflow conditions. Erodibility coefficients k, τcr
depends on the type of bank material and can bemeasured by different devices such as the erosion
function apparatus, the submerged jet test device,and by means of straight or annular flume tests,(Briaud et al., 2001). Erosion progress through thebank is predicted by using two conjugate adaptive
finite element meshes. The fluvial erosion modelthat simulates the temporal and spatial changes in
the profile of cohesive riverbanks is explained indetails by (Aly El-dien et al., 2013). Erosion modelis based on the following two assumptions:(1) The
river is straight which means there is no secondary
flow, (2) The bank material is cohesive andrelatively homogeneous, which means that erosion
advancement is synchronized in both bed and banks
according Eq.(1), (2) above. Deposition process is
not incorporated in the model. Accordingly,
aggradation/degradation model based on solving thesediment continuity equation, sediment transport
equation, should be applied to estimate the amountof bed erosion.
2.2 Seepage model
The finite element seepage model introduced by(Gottardi, and Venutelli, 2001) is employed tosimulate 2-D transient saturated-unsaturated
seepage flow through the riverbanks. The governing
equation for water flow through saturated andunsaturated soil can be obtained by introducing
Darcy’s law into the mass continuity equation. The
general governing differential equation for
two-dimensional seepage is given by Richard’s
equation as:

∂
∂ψ
∂
∂ψ
∂ψ
(K (ψ) ) + (K z (ψ)( + 1)) = C(ψ)
∂x x
∂x
∂z
∂z
∂t

(3)

where ψ is the pressure head, [L]; K x (ψ),
K z (ψ) are the unsaturated hydraulic conductivity inx, z directions respectively, [LT -1]; C(ψ) =∂θ/ ∂ψ is the specific water capacity, [L -1]; θ isthe volumetric water content [L 3L-3] , θ = nS ; n is
the soil porosity; S is the degree of saturation (Sranges from 0.0 in dry soil to 1.0 in fully saturatedsoil); x, z are the horizontal and the verticalcoordinates, respectively, [L]; and t is time, [T].
In an unsaturated soil, both the volume of water
stored within the voids, and hydraulic conductivity
will vary depending on the negative pore-waterpressure. A conductivity function ( K − θ curve)
and a storage function ( ψ − θ curve) should bewell defined to complete the seepage model. These
two functions could be obtained, for every soil type,by direct measurement in the laboratory and theninserted into the seepage model to get K(ψ) andC(ψ). Since it can sometimes be difficult or time
consuming to obtain the storage and conductivity
functions by laboratory measurements, thefunctions could be estimated by using closed-form
solutions as proposed by (Van Genuchten, 1980) or(Fredlund and Xing, 1994). Essential assumption in
the analysis of seepage model is that the initial porewater is assumed to vary hydrostatically with
distance above or below the groundwater table.
That is, pore-water pressure is negative anddecreases linearly above the groundwater table, andconversely below the groundwater table pore-waterpressure is positive and increases linearly, the same
approach as (Rinaldi and Casagli, 1999).
2.3 Stability Model
Riverbank stability can be performed using
either the traditional limit equilibrium methods suchas the Bishop's modified method, Janbu's
generalized procedure of slices, and Spencer's
method (Duncan, 1996) or by using the numericaldeformation finite element methods such as thegravity increase method (Swan and Seo, 1999) andthe strength reduction method (Matsui and San,
1992; Griffiths and Lane, 1999).
The stability model in the current study is a
finite element elastic-plastic deformation model― 367 ―

that uses the strength reduction technique. Shearstrength of bank material is proportional tocohesion c ′ and angle of friction ϕ′ and thestability of the banks increases with an increase inc ′ andϕ′ . In the strength reduction technique theoriginal shear strength parameters c ′ , ϕ′ arevirtually reduced by dividing by a strength
reduction factor (SRF) in order to bring the slope tothe point of failure (Smith and Griffiths, 2013). The
factored shear strength parameters cf′ , and ϕ′f are
therefore given by:flowing water; third the case of “Movable-bed”
where both banks and bed erodes simultaneously.

cf′ = c ′ /SRF
(4)
ϕ′f = arctan( tan ϕ′ /SRF)
The factor of safety for the slope is equal to thestrength reduction factor (FOS=SRF) when failure
takes place. Stability model consists of threesub-models:(1) plane strain model to calculate theshear and normal stress distribution inside theriverbank, (2) Mohr-Coulomb failure model todetermine which nodes have been yielded andoverstressed, and (3) elastic-plastic algorithm toredistribute the stresses of yielding nodes throughthe mesh. Detailed explanations are given by(Smith and Griffiths, 2013).
3.

Numerical Simulation

Fig.1 Patterns of flood hydrograph
3.2 Hypothetical data, Initial and boundary
conditions

3.1 Hydraulic Conditions
Five flood hydrographs having the same runoff
volume but with different patterns are assumed topass through the river cross section as shown in
Fig.1. Base time of Hyd-1is double that of Hyd-2
and it has constant peak discharge equals half thevalue of Hyd-2. Hyd-3 simulates the early peakflood event (rapid increase and gradual decrease)
while conversely Hyd-4 simulates the late peak
one .Hyd-5 is a multiple peak hydrograph. The
maximum peak discharge is 600 m3 /sec and thebase time lasts for 10 days except for Hyd-1 andHyd-2 to satisfy the constant runoff volume
criterion. In order to study the effect of fluvial
hydraulic erosion, three bank-bed conditions are
assumed; first the case of “No-erosion”; second thecase of “Fixed-bed” where the bed material is so
rigid that the stream cannot pick it up during floodand hence the banks only are attacked by the

Typical riverbank with dimensions asshown by Fig.2 is considered. Shearstrength parameters of bank material canbe determined by results of direct andtri-axial shear tests, but we use
hypothetical data with similar range tothose published before.c ′ = 12 KN/m2 , ϕ′ =20° , Young’s modulus E = 1 × 105 KN/m2
and Poisson’s ratio ν = 0.3 , dilation
angle, ψ∗ = 0° , and the saturated
permeabilityk x = k y = 1.389−5 m/sec,
porosity= 0.43, Ss (specific storage) = 10-5
m-1, γ soil(under GWT) = 20 KN/m3 ,γ soil(above GWT) = 18 KN/m3 , τcr =1.2 KN/m3 , k = 3.6 × 10−7 m3 /N. sec , Sb(bedslope) = 0.0005 , n (manning coefficient) =0.015. Initial GWT is assumed horizontal at thesame level of WSL at the beginning of thehydrograph. Pore-water pressure (PWP) under the― 368 ―

GWT is initially assumed as hydrostatic pressure asshown by Fig. 2 . The conductivity function( K − θ curve)
and a storage function ( ψ −θ curve) for the riverbank silt soil are drawn fromthe data obtained by (Gottardi and Venutelli, 2001)
as shown by Fig. 3.

Fig.2 Dimensions and boundary conditions of
seepage model(a)(b)
Fig.3(a)Storage function ( ψ − θ curve) ,Conductivity function (K − θ curve)(b)water pressure (PWP) for step i is the resulting
PWP from step i-1. (3) specified pressure head.
WSL is very important to define the specified headboundary condition for seepage model, and theforce vector in the stability model. In the stability
model, the horizontal displacement is zero for thetwo vertical sides, but both the vertical andhorizontal displacements are zero for the lower
boundary. The force vector represents theequivalent nodal gravity weight, and confining
water pressure. Pore water pressure is interpolated
from the seepage model and found at each gausspoint then subtracted from the total stress to get theeffective stress in the stability model. Mesh andboundary conditions for plane strain stability model
are shown in Fig. 4

Fig.4 Mesh and boundary conditions of stability
model.
4. Results and Discussion

3.3 Treatment of hydrograph and the dynamic
boundary conditions
Ten simulation days are divided into sixty time
periods each of 4-hours period. Flood discharge isconsidered constant during each time period and thecorresponding WSL is calculated by iterative
procedure using Manning equation. For every time
step and for the newly updated geometry of theriverbank, the seepage model is run and the GWT is
obtained, finally the factor of safety (FOS) is
calculated and the expected plane of failure isdetermined by the stability model. The above
mentioned procedure is repeated each time step
until the required simulation time is reached.
Each new time period requires three boundary
conditions from the previous step: (1) the newly
updated bank geometry, and (2) the initial poreIn the following paragraphs we will discuss how
floods can affect the geometry and both external
and internal forces. Three factors influence thestability of the river bank: (1) Change in thegeometry. (2) Change with the WSL (external
force). (3) Change of the GWT (internal force). It
should be considered that the factor of safety (FOS)
is calculated even it is less than 1.0 in order toinvestigate the trend of FOS. In the following, we
discuss each of them in details.
4.1 Effect of flood on the riverbank geometry(width, depth, slope):
Flow characteristics are governed by theexisting river dimensions, shape, slope and bed
roughness but these may themselves be altered by― 369 ―

the flow, especially during floods. As long as
discharge remains below the competent thresholdthe cross-section shape formed by the last flood
remain unchanged and governs the flow
characteristics.
A change in the geometry of the bank may
occur when the streambed lowers or degrades dueto the instability of the stream system. Generally,
fluvial erosion caused by floods (1) makes the rivercross section to become wider and deeper which
increases the bank height, (2) steepen the bankslopes,(3) helps the formation of overhangs.
(1) Formation of overhangs
Geometry of the riverbank subjected to thedifferent flood cases is shown by Fig.5. Bycomparing the “fixed-bed” cases to “movable-bed”
cases, an overhang or deep cut near the bank toe ismost likely to be formed in the “fixed-bed” cases.
One reason is that the flow energy is resisted by therigid bed and the other part strongly attacks thebanks, so the final result is erosion in the bank toe.
When the WSL remains constant for a period oftime, the flow erodes the same wetted perimeter andremoves particles which finally form an overhang.
Overhang is also formed in the “movable-bed”
case of Hyd-4, because the WSL changes very
slowly or almost constant. Bed condition andexistence of armor layers controls the stability ofbanks. Formation of overhang is a function of therigid bed and the time variation of WSL.
Overhanging conditions can only occur in cohesive
riverbanks, and they are inherently unstable and canfail with only slight changes in the bank conditions.
(2) Change in depth and width
Although the discharge of Hyd-2 is double thedischarge of Hyd-1, the base time of Hyd-2 is half
the base time of Hyd-1. The volume of water
passing through the river cross section is equal
whether from Hyd-1 or from Hyd-2. Fig.6 shows
how the hydrograph pattern affects the geometricshape of the river cross section. The river crosssection is wider and shallower in case of Hyd-2.

(a) Hyd-1(b) Hyd-2(c) Hyd-3(d) Hyd-4(e) Hyd-5
Fig. 5 Bank profile for different flood events.
Fixed bed case;Movable bed case― 370 ―

Fig.6 Effect of flood on width and depth of riverx-sec. Results after simulation time of 10 days
Since the discharge of Hyd-2 is double the valueof Hyd-1, the water depth and hydraulic radius ofcase Hyd-2 are greater than those for case Hyd-1.
As a result, a long portion of the wetted perimeteron the bank side is subjected to friction and shearedby the flow, this leads to the lateral erosion of thebank and widen the channel. But in case Hyd-1, thewater depth is small and the wetted perimeter of thebank is short, so the widening effect is not
significant. The boundary shear stress produced by
Hyd-2 is greater than that produced by Hyd-1 asshown by Fig.7, so it is expected that the erosion
depth accompany Hyd-2 is much deeper than theerosion depth from Hyd-1, which holds true if bothhydrographs have the same base time. But the basetime of Hyd-2 is only half that of Hyd-1 and thechannel widening makes the shear stress smaller
which explain why the erosion depth of Hyd-1 isdeeper than the erosion depth of Hyd-2.

Fig.7 Boundary shear stresses at the end of Hyd-1
and Hyd-2.
(3) Cumulative erosion volume
Fig.8 shows temporal change of cumulativeerosion volume. There is no significant differenceof the final cumulative erosion volume except for
Hyd-2. It is thought that the reason for being wider
and shallower mentioned above makes thisdifference. Erosion volume per unit length of riverincreases uniformly for cases of constant discharges(Hyd-1 and Hyd-2) and increases with an increasingrate during the rising stage of the hydrograph, andinversed at the peak discharge to increases with adecreasing rate during the recession hydrographstages.

Fig.8 Erosion volume at the end of eachhydrograph(4) Change in water surface level
Fig. 9 shows the change in water surface levelin the river for the five flood hydrographs. The
WSL increases during the rising stage of thehydrograph and decrease with its recession stage
which matches the logic sense. But in case of
movable bed of Hyd-4, the WSL remains constants
although during the rising stage because of thesynchronized increase in the discharge and thedegradation in bed area as shown by Fig.9-e.
From the above discussion, it can be concluded
that long-term changes in discharge magnitude or
duration have important implications for channel
form and process. River width and depth depend ondischarge value and material characteristics. Width
increases faster than depth for high discharges. The
flood events with small discharge and long duration
cause greater erosion volumes than floods with highdischarge and short duration if they are of the same
runoff volume.
Field observations by (Wolman, 1959) foundthat maximum bank erosion occurred during wintermonths. Winter storms were longer in duration andof less intensity than summer storms. Wolman’sobservations match reasonably our results.

― 371 ―

(a) Hyd-1(b) Hyd-2(c) Hyd-3(d) Hyd-4(e) Hyd-5(Knighton, 1973) found similar results as
Wolman’s, but added that flow variability (numberof discharge peaks) also affects bank erosion.
Knighton inferred that in addition to their longer
durations, winter storm-flow events caused a muchgreater amount of erosion because of their multiple
peaks as compared to summer storm-flow events
which had single peaks which also agrees well with
our results.
4.2 External factors affecting the riverbank
stability:
Both the confining pressure produced by thewater in the river and the soil mass at the toe of thebank are denoted as external factors that affect theriverbank stability in this section. The trend offactor of safety through time is shown by Fig.10.
During the rising stage of flood events, the WSL
increases and the factor of safety increases as aconsequence of the stabilizing confining pressure of
the water in the river.
During the rescission stage, the WSL drops
down in the river and the confining pressure in theriver decreases making the factor of safety falls tolower values than those experienced before. It isclear from any of the five cases in Fig.10 that at a
certain time, the FOS for the case of “No-erosion”
is higher than the case of “Fixed-bed” which is alsomuch more stable than the case of “Movable-bed”.
For example, at time 72 hour in Hyd-3 the FOSchanges from 1.62 for the No-erosion case to 1.46
for the “Fixed-bed” case to 1.22 for the“Movable-bed” case. The removed material and theeroded toe of the bank reduce the gravity forces
resisting failure so that the computed factor ofsafety reduces progressively. It should be noticed
that the FOS decreases rapidly at the start of thecalculation in Hyd-1 and Hyd-2 because the GWTinside the bank is initially assumed below the WSL
in the river, and with the passing time the GWTrises while the WSL remains almost constant or
slowly drops down as shown by Fig.9. This action
accelerates the instability condition inside the bank.
However, in the other cases, Hyd-3 to Hyd-5, therising discharge is accompanied by a slight rise in
the WSL and produces an outer confining pressure
great and fast enough to compensate the generated
inner pore pressure.

Fig.9 Variation of WSL for different flood events― 372 ―

The earliest failure is noticed to occur during
flood Hyd-5 (after 156 hr for “Fixed-bed” case andafter 72 hr for “Movable-bed” case). While thelatest bank failure occurs under flood Hyd-4 (after
224 hr for “Fixed-bed” case and after 164 hr for“Movable-bed” case). In the “No-erosion” case, thebank is stable under all hydrograph patterns. The
slow erosion rate (deformation) and slow variationin the WSL are thought to be the cause of delayed
failure under the effect of Hyd-4, while the very
rapid variation in WSL (and consequently the
GWT) under the effect of Hyd-5 accelerates thefailure.
In the multiple peak hydrograph, Fig.10-e, sharp
peaks in FOS values reveal that the peaks of flood
events have crucial effects on bank stability. During
the rising stage of each flow event, FOS increased
abruptly and during the drawdown it falls, reaching
lower values than those exhibited prior to the event.

Fig.10 Variation of FOS for different flood events

4.3 Internal factors affecting the riverbank
stability:
Pore water pressure (PWP) and the mechanicalproperties of bank material are denoted as theinternal factors that affect the riverbank stability inthis section.
(1) Change of the mechanical properties of bankmaterial
When the GWT rises up, a significant positive
pore water pressures is generated and the bank is
gradually saturated. Such saturation reduces thecontact between soil particles and then the internal
friction angle ϕ′ is reduced. Besides, saturation
affects the chemical bonds between particles andreduces the apparent cohesion c ′ . Moreover,saturation increases the unit weight of bankmaterial. Finally, saturation reduces the effective
stress. All the above four negative effects ofsaturation are against the stability of the bank andmay cause the factor of safety to become less than
1.0. However, the stabilizing effect of the confiningpressure from the water in the river may
counterbalance the destabilizing effects ofsaturation.
During the rising hydrograph stage, there are
two opposite actions on the riverbank. First, theconfining pressure acting on the riverbank facestabilizes the riverbank and increase the FOS.

― 373 ―

Second, the raising-up free surface or (GWT)
increases the field of positive pore water pressure
inside the bank which reduces the FOS. During therescission hydrograph stage these two actions areconversely changed.
(2) Effect of peak time of the flood
In most cases in Fig.10, failure occurs during
recession stage or some hours after the flood peak.
But this is not a governing rule because as we see inthe case of “Movable-bed”, the failure in Hyd-3
occurs during the recession stage while the failure
in Hyd-4 occurs during the rising stage. Although
the discharge continues to increase during the rising
stage of Hyd-4, the WSL remains almost constantbecause the bed and banks continues to degrade.
This situation changes the bank geometry (increaseheight, steepen its slope). The main reason of
failure is only the geometry change in this casebecause there is no significant variation in GWT.
The fluvial erosion is very active while theconfining water pressure remains constant while thesever degradation continuous to progress.
(3) Effect of recession rate on bank stability
Comparing the case of “Movable-bed” in both
Hyd-3 and Hyd-5, the water surface leveldrawdown rate in Hyd-3 is about 5 cm/day while in
Hyd-5 is about 15 cm/day as shown from Fig.9.
After 72 hours the riverbank collapses under theeffect of Hyd-5, while the FOS is still 1.22 underthe effect of Hyd-3. On the other hand, at the failure
time the volume of eroded material under the effectof Hyd-3 is about 24.4 m3 while under the effect of
Hyd-5 is about 20.36 m3 . In this case, it isintelligible that the failure doesn’t occur bygeometric change.
The stability of riverbank during recession stageof the hydrograph is greatly influenced by how fast
its pore water drains. If rapid recession rate took
place while the riverbank remains saturated, at least
for a while, this condition induces a reduction in theconfining pressure outside the riverbank but the
GWT is still high inside it. During high recessionrate, WSL drops rapidly while the GWT falls
slowly (depending on the hydraulic conductivity ofthe bank material) so that a pressure difference isgenerated between outside and inside the riverbank

which is supposed to be the cause of failure. Rapid
drawdown causes this difference to be high. Slow
drawdown causes the pressure difference to besmall.
In the multiple peak hydrograph, during
recession stages, the GWT drops down at a slower
rate than WSL drops. The FOS is expected toincrease again in two conditions 1) the rise-up of
the WSL, 2) the dropdown of the GWT and thebank is expected to restore its stability after totallydraining the inside water. Generally, its severity is afunction of the time that the GWT remains high.
These results are consistent with (Rinaldi et al.
2004)(4) Seepage flow field and directions
The direction of seepage flow depends mainlyon the relative difference in water level between
inside and outside the riverbank. Generally, thedirection seepage flow changes from an inward
direction during rising stage to an outward direction
during recession stage. Seepage force (hydraulicdrag which results from the movement of water
through pores) acts in the same direction of seepage
flow. This force is expected to (1) cause local scourand piping (for non-cohesive material) or
detachment (for cohesive material), (2) affect theboundary shear stress induced by the main river
flow (Yan Lu . et al. 2008). These two effects madeby seepage force are not considered in the current
study.
Fig.11 shows the distribution of pore waterpressure and the free surface line and seepage
velocity vectors at failure for the case of“Movable-bed” under the five flood hydrographpatterns. It is clear that the highest values of
seepage velocities take place at the exit point near
the toe of riverbank (as indicated by arrow size). In
all patterns the seepage velocity vectors moves
outward the bank except for Hyd-4 the seepage
flow moves inward direction which emphasis theresult obtained above that the failure occurs
because of the geometry change, and not because of
the pwp variation. Also, seepage flow is divided
into two directions under the highest point on of
GWT of Hyd-5 (point “m” Fig.11-e). This indicatesthat the recession rate is so fast that the GWT
couldn’t fall down uniformly along the whole bank

― 374 ―

region and falls down near the bank making a humpshape.
(a) Hyd-1m(b) Hyd-2mm(c) Hyd-3(d) Hyd-4

Initial toe point

Toe point at failure(e) Hyd-5m

Exit point

provided as follows:
1- By involving the three simulation models(erosion + seepage + stability), factor of safety,
expected failure plane, collapse time, riverbank
geometry, and pore water pressure could be
successfully calculated for different floodhydrograph patterns.
2- Innovative idea of incorporating the lateralerosion in the stability analysis is applied.
Here we add this model and it is very important
to consider the eroded parts (and overhangs).
The dynamic response is involved at every time
step during the flood considering moving
boundaries.
3- The process of lateral erosion increases the bed
width of the channel and results in steepeningof the bank, which reduces its stability. Bedlowering increases the bank height until acritical height is reached and the bank collapse.
4- The riverbank can restore back its stability as
long as the water inside it is drained out.
5- The increase of water level in the river sustains
the bank stability because the exertedhydrostatic pressures ensure a stabilizing
factor.
6- The high pore water pressure is an adverse
factor against riverbank stability.
7- The rate of change in the free-surface inside thebank controls its stability. The more drainablebank material, the more stable it is.
8- The simulated results show that riverbank
failure is triggered particularly when during thefalling stage, which has been pointed out byother researchers as well.
Acknowledgements

Fig.11 Free surface and seepage velocity vectors at
failure for the case of “ Movable-bed”

Authors would like to acknowledge thefinancial support from the Ministry of Education,
Culture, Sports, Science and Technology (MEXT)
of Japan.

5. Conclusions
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(Received June 11, 2015)