River Basin Management X 63

DERIVATION OF THE INDEX GOVERNING THE SEEPAGE
FLOW AND DYNAMIC SIMILARITY CONDITION OF
LEVEE FAILURES DUE TO SEEPAGE FLOWSHOJI FUKUOKA & KOSUKE TABATA
Research Development Initiative, Chuo University, Japan

ABSTRACT
It is necessary to establish an estimation method of levee weak points due to seepage flow during floods
to make appropriate levee reinforcements and risk management measures in a river basin, preparing forlarge-scale floods. The internal flow with large phreatic surface gradient in the body of the levee tends
to concentrate around the landside toe of the levee during a flood event, which may trigger the levee
failure. This study shows that the idea of the levee failure is effective for primary selection of dangerous
levee failure positions to the seepage flow. First, the only dynamic index governing the seepage flow
in soils, the Seepage Flow number (SFn), is derived by nondimensionalizing Richards’ governing
equation for a general seepage flow. Next, by applying SFn to the seepage flow in the levee, the levee
vulnerability index (t*) is derived in the expression of a dynamic dimensionless quantity. Also, thevulnerability index of the levee on the permeable foundation is derived by using the unsteady quasi
two-dimensional seepage flow analysis method. Finally, the vulnerability index (t*) is calculated forthe actual field levee and model levee experiment in which each failure mode such as leakage, sliding
or collapse and breach occurred. As a result, it is confirmed that t* shows approximately common
values in the field levee and model levee taking the same failure mode. Therefore, it is demonstrated
that t* gives a dynamic similarity condition on the levee failure phenomena, due to the seepage flow
and the judge of the degree of levee failures.
Keywords: seepage flow number, seepage line, levee failure, levee vulnerability index, dynamic
similarity law, unsteady quasi two-dimensional equation, prototype and model.

1 INTRODUCTION
It is necessary to establish an estimation method of levee weak points due to seepage flow
during a flood event to make appropriate levee reinforcements and risk management
measures in a river basin. A general evaluation method of the levee safety was to calculate
the danger of the slope sliding or collapse failure, for the steady seepage line assumed that acertain water level of the flood acts for a long time [1]. However, the internal flow with alarge phreatic surface gradient in the body of the levee develops through the soil toward thelandside toe during a flood event. The danger of levee failure will rise considerably if theseepage flow abnormally concentrates around the landside toe. Therefore, it will be possible
to estimate when, where and what kind of levee failures will occur if the degree ofconcentration of the seepage flow at the landside toe can be appropriately evaluated.
In this study, the seepage phenomenon in the levee is expressed by the continuity equation
and Darcy’s law as the equation of motion. First, the index governing the seepage flow in thesoil, the Seepage Flow number (SFn), is derived by nondimensionalizing Richards’ governing
equation [2] for a general seepage flow. Next, by applying the SFn to the seepage flow in the
body of the levee, the levee vulnerability index (t*) is derived in the expression of a dynamic
dimensionless quantity. Moreover, it is confirmed that t* gives a dynamic similar condition
to the model and prototype levee. Also, each damaged level of the leakage, slope sliding or
collapse and breach that t* can take is discussed. This paper does not target the piping failurecaused by the movement of water and the soil particle in a permeable foundation layer.

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64 River Basin Management X

2 DIMENSIONLESS SIMILARITY INDEX GOVERNING GENERAL SEEPAGE
FLOW AND LEVEE VULNERABILITY INDEX
2.1 Derivation of the similarity index governing the general seepage flow (SFn)First, we derive a general similarity index governing a seepage flow in soils, since thecomparison of levee failures are made between field and model levees in following sections.
The research on the similarity law of the seepage flow has been conducted in the
1950–1960s. Bear [3] considered the similarity law of the steady seepage flow by nondimensionalizing the flow continuity equation taking account into the seepage flow velocity
expressed by Darcy’s law. He proposed two conditions expressing the relationships among
the dimensionless scales x', y', z' and dimensionless coefficients of permeability kx', ky', kz'.
x '2 y '2 z '2==
kx ' k y ' kz '(1)

Corey et al. [4] suggested a similarity condition by nondimensionalization of Richards’
equation [2]. However, it was a study focusing on the vertical seepage flow under thegravitational drainage in the saturated ground, and the pressure head and seepage length were
given as one common length. Therefore, the suggested dimensionless value could not be asimilarity law of the general seepage flow. Although other several similarity relations [5], [6]in which were expressed by empirically combining the length, permeability and porosity
were suggested, the verifications were not enough performed. There is no similarity condition
for the seepage flow with generality as far as the authors know.
The governing equation of the seepage flow in the soil is expressed by Richards’ equation(eqn (2)),
1 S r   h    h    h 
−  k  −  k  −  k  = 0 , tx  x  y  y  z  z (2)where  is density of water,  is porosity of soil, Sr is degree of saturation, k is coefficient of
permeability, h is total head (=Ψ+z), Ψ is pressure head. It is necessary to hold the same
equation between the prototype and the model by nondimensionalizing the governing
equation for deriving the dynamic similarity law. The dimensionless value of each variableof eqn (2) is given as follows by using each representative value. Here, superscript ' andsubscript 0 mean dimensionless and representative quantity, respectively.

'=

S


h
t
k
x
y
z
, Sr ' = r ,  ' = , h ' =,t ' = , k ' = , x' = , y ' = , z ' =
0
Sr 0
0
H0
T0
k0
L0
L0
L0

(3)

When eqn (2) is arranged taking account into eqn (3), the following dimensionlessequation is finally obtained.
1  '  ' S r ' 1 k0 H 0T0    h '    h '    h '  
+
−
k '
−
k '
−
k '
 = 0
t 'S r 0 0 L0 2  x '  x '  y '  y '  z '  z '  

'

(4)

In this dimensionless eqn (4), the dimensionless value enclosed with a red frame ahead ofparentheses that settle the second, third and fourth terms in the left side appears. Thisdimensionless value is defined as SFn, which shows the dynamic similarity condition of theseepage flow. If the seepage flow in the field (prototype) and the model have the same valueof SFn, the seepage flow of both will be dynamically similar.

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River Basin Management X 65

2.2 Derivation of levee vulnerability index t* from SFn
The presented SFn is expressed in the ratio of the time of Sr0λ0L02/k0H0 to T0. The former
indicates the time required to move the representative horizontal distance L0 by the true
seepage flow velocity k0H0/Sr0λ0L0. The latter indicates the representative time of the seepage
flow.
The SFn can be rewritten in the index that relates the seepage flow in the levee to the levee
failure if each representative physical value that consists SFn is chosen properly so as todescribe the seepage phenomena in the levee.
By substituting H, b, t', k,  and Sr to H0, L0, T0, k0, 0 and Sr0, SFn is finally shown by eqn(5). Here, H is water level, b is horizontal length from water front to levee toe of landside, t'
is duration time from when the water level exceeds the ground level of the landside toe. k, 
and Sr are averaged permeability, porosity and degree of saturation of the levee (see Fig. 1).
Then, SFn is defined as levee vulnerability index t*.

H: water levelt* =1 kHt '
Sr b2(5)

landside

flood plaintt’: duration time from when the water levelexceeds the ground level of the landside toe

k: coefficient of permeability
: porosity
Sr: degree of saturation(Sr=0.4 as average value between
unsaturation and saturation)

b: horizontal length from water front
to levee toe of landside

Figure 1: Definition of variables of levee vulnerability index t*.
2.3 Derivation of t* from quasi two-dimensional unsteady seepage flow equation
Fukuoka and Tabata [7] assumed the seepage line in the levee on the permeable foundation
as the similar function form (eqn (6)).
  x m h ( x, t ) = H 1 −   ( t )   ,

 
 


(6)

where  is a front position of seepage line, m is a parameter which defines the shape of theseepage line.
In addition, a differential equation of quasi two-dimensional unsteady seepage flow (seeFig. 2) that can describe the movement of the seepage line was derived based on Darcy’s law
and continuity condition of water in the levee and foundation. The derived equation was
solved for the front position  of the seepage line and let T be the time required for the seepage
line to reach the horizontal distance b. Then, the general levee vulnerability index (eqn (7))
defined by t*m=t'/T was obtained.
tm* =2 ( m + 1) mk1 H − m ( 2m − 1) k 2 D Ht ' ,2m − 1mH + (m + 1) D

b 2

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66 River Basin Management X

where, k1 and k2 are the coefficients of permeability of the levee and foundation, D is the
thickness of the permeable layer and defined as the distance from the impermeable layer orground water level to the ground level under the levee. The parameter m in eqn (6) can bedecided by Table 1, based on the values of k2/k1 and D/H of the levee and foundation [7].
z

Levee

H
Water level

②
①

qin1

V

watersideq*

Permeable foundation① Seepage flow in levee on
permeable foundation:  x m h( x, t ) = H 1 −     (t )  
② Seepage flow in levee on
impermeable foundation:(Uchida’s equation)32  x  h( x, t ) = H 1 −     (t )  

Coefficient of permeability: k1
Porosity: λ

qin2

V’

ξ(t)

ξ(t)

qout

D

x

landside

Coefficient of permeability: k2
Porosity: λImpermeable layer (or ground water level)

Figure 2: Quasi two-dimensional unsteady seepage flow in the levee [7].
Table 1: Relationship of k2/k1, D/H and parameter m [7].
k2/k1
D/H

0

0.1

0.5

1

2

5

10

20

0

1.500

1.500

1.500

1.500

1.500

1.500

1.500

1.500

0.1

1.500

1.500

1.500

1.500

1.426

1.060

0.858

0.711

0.5

1.500

1.500

1.472

1.075

0.823

0.660

0.588

0.545

1

1.500

1.500

1.118

0.860

0.700

0.591

0.546

0.524

2

1.500

1.384

0.866

0.710

0.614

0.547

0.524

0.510

5

1.500

1.014

0.671

0.592

0.549

0.519

0.509

0.504

10

1.500

0.821

0.591

0.548

0.524

0.509

0.504

0.501

20

1.500

0.691

0.548

0.524

0.510

0.504

0.501

0.501

In the case of the levee on the impermeable foundation, D and k2 in eqn (7) are zero. Also,
the parameter m is chosen 3/2 in order to coincide with the one-dimensional unsteady seepage
line equation suggested by Uchida [8]. Then, 2(m+1)/(2m–1) in eqn (7) becomes the value of
5/2. As a result, the levee vulnerability index t* of the levee on the impermeable foundation
is given by eqn (8).
t* =5 Hk1t '2 b 2(8)This equation means that the true seepage flow velocity in the levee is described by
5/2Hk1/b when the water level in the river H acts on the unsaturated levee with width b,coefficient of permeability k1 and porosity . Also, it can be interpreted that the influence ofdegree of saturation Sr in eqn (5), which changes as the seepage line develops, is included in
the coefficient 5/2.
Uchida [8] expressed the averaged velocity of the seepage flow in the levee as Hk1/b
by solving the temporary change of the seepage line based on the graphical analysis method
and confirmed it by seepage flow experiments. Therefore, as the seepage line develops andthe effects of saturation and unsaturation in the soil are integrated, finally the moving speedof the seepage line is described by Hk1/b on average. Since the degree of saturation Sr ofsoil is about 0.3 in the completely gravity drained state and 1 in the saturated state, 1/Sr of t*

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River Basin Management X 67

(eqn (5)) derived from SFn takes about 1 to 3. Therefore, it is can be regarded that eqns (5)and (8) almost coincide. In general, it is known that the permeability coefficient varies
according to the degree of saturation, but this paper does not refer to how to incorporate thechange in saturation at unsteady seepage phenomena and the corresponding
permeability coefficient.
In the case of the levee on the permeable foundation, the time T required for the seepage
line to reach the landside toe will be larger compared with the case of the levee on theimpermeable one, because a part of water in the levee flows into the permeable foundationand it makes the seepage line lower. For these reasons, the levee vulnerability index t*m on
the permeable foundation is calculated smaller than t* on the impermeable one. Therefore,
the levee vulnerability index t* that assumes the impermeable foundation should be applied
as the index of a dangerous failure.
The levee vulnerability index t* on the impermeable foundation shown by eqn (8) is
described in the same formula form with index (eqn (5)) obtained by applying SFn to theseepage flown in the levee. This means that the levee vulnerability index t* is the only
dynamic similarity condition of the seepage flow in the levee.
3 APPLICATION OF LEVEE VULNERABILITY INDEX
3.1 Application of the levee vulnerability index t* to field levees
The levee vulnerability index t* of the field levees in the Kinu River 2015 flood, the Koyoshi
River 2013 flood, the Kakehashi River 2012 flood, the Yabe River 2012 flood and the Nagara
River 1976 flood is calculated. In these rivers, the levee damage occurred considerably due
to seepage flows during flood events (see Fig. 3(a) and (b)).

(a)
Figure 3:(b)

The levee failure due to the seepage flow observed in the field levee. (a) Sliding
at 8.4 k right bank in the Kakehashi river during the 2012 flood; (b) Levee breachat 33.8 k right bank in the Nagara river during the 1976 flood.

Fig. 4 shows the value of the levee vulnerability index of the respective field levee
calculated by using eqns (7) and (8). The coefficient of permeability is estimated based on
Hazen [9] or Creager et al.’s [10] method, by using D10 and D20 observed by the boring survey
data of the levees and larger one of Hazen and Creager’s method is given to the calculation.
The calculated t* are plotted with larger variation compared with the model levee
experiments stated in 3.2 since the influences of rainfall and slope protection are not taken
into consideration. However, the levee vulnerability index t* of the field levee in which the

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68 River Basin Management X

levee breach (red circles in Fig. 4) and sliding or collapse (yellow circles in Fig. 4) occurred
take approximately 0.1–1.0 and 0.01–0.1 respectively. In which the leakage from the levee(without circles in Fig. 4) occurred, the variation of the data is large, but most of the t* are
distributed in the range larger than 0.001.
Kinu RIver

1.0E+1

Yabe River

Nagara River

Kakehashi River
Koyoshi River

1.0E+01.0E-1t*

1.0E-2
1.0E-3Sliding or collapse

R8.4k

R10.8k

R37.4k

R35.6k

R33.8k

R32.4kt*m(breach)R31.6kt*m(sliding or collapse)t*(breach)L34kt*m(leakage)t*(sliding or collapse)

L28kt*(leakage)R11.8k

R9k

R6k

L16k

L17.2k

L11.6k

L11.4k

L5k

R41.8k

L35k

Levee breach

R28.5k

L26.5k

L22.25kL21k

1.0E-6

L18.5k

1.0E-5R11k

1.0E-4

No frame：leakege from levee

Figure 4: The values of the levee vulnerability index t* of the field levees.
3.2 Application of the levee vulnerability index t* to model levee experiments
Yoshikawa and Kodaka [11] conducted a series of model levee experiments for clarifying
the relationship between the soil structure and the levee failure mechanism due to the seepage
flow. Three kinds of models such as two-dimensional half cross section, two-dimensional
full cross section and three-dimensional full cross section were created, and a total of 22 casesof experiments whose thickness and permeability of the foundation layer were different werecarried out. Fig. 5 shows the specification of the two-dimensional full section model andexample of the levee failure due to the seepage flow observed in the model levee. The model
levee is 0.2–0.3 m in height and 0.7–0.9 m in width. The foundation is created by the different
permeable materials in the upper and lower layer. The outside water level was rapidly
increased from the initial water level condition while the lower layer was saturated, then thewater level was kept constant. The deformation process of the body of levees and its
occurrence time were observed.
400mm

Levee

Upper layerbLower layer(saturated)
1700mmFigure 5:

100mm

200mm

Thickness
D :20-50mm

H

Boundary
condition(Duration time t’)Rising rapidly

Occurring of Sliding

Seepageflow

Initialcondition

Specification of the two-dimensional full section model levee and example ofthe levee failure due to the seepage flow observed in the model levee.

By using these experimental data, the relationship between t* and the failure modes isinvestigated. Fig. 6 shows the values of the levee vulnerability index t* of the model. The
levee vulnerability index t* of the experiments in which the levee breached are almost all

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River Basin Management X 69

plotted in the range of 0.1–1.0. In which the sliding or collapse occurred, t* are concentrated
at the range of 0.01–0.1 and plotted 16 data at this range. In addition, most of t* calculated
for cases which the leakage from the levee are observed are approximately plotted in therange of 0.001 to 0.01. These results are the same as ones of the field levees in 3.1.
Levee breach

1.0E+1Sliding or collapse

No frame：leakege from levee

1.0E+01.0E-1t*

1.0E-2
1.0E-3
1.0E-4
1.0E-5
1.0E-6

1

2

3

4

5

2D half cross section

6

7

8

9

10

t*(leakage)t*m(leakage)t*(sliding or collapse)

t*m(sliding or collapse)t*(breach)t*m(breach)

11

12

13

14

15

16

17

18

19

20

21

2D full cross section

22

3D

Figure 6: The values of the levee vulnerability index t* of the model levees.
Consequently, t* can approximately elucidate the degree of the levee damage and thedanger due to the seepage flow. It is clear that t* is the dynamic similarity condition
established commonly for model levees and field levees since the range of t* taken byeach failure mode in the field levee and model levee experiment is almost in agreement.
4 CONCLUSION
Based on the idea that the internal flow with large phreatic surface gradient in the body oflevees concentrates around the landside toe of the levee during a flood event, which may
trigger the levee failure, the estimation method for the degree of the levee failure due to theseepage flow is presented. First, the only dynamic index governing the seepage flow in thesoil, SFn, is derived by nondimensionalizing Richards’ governing equation for a general
seepage flow. Next, by applying SFn to the seepage flow in the levee, levee vulnerability
index t* is derived in the expression of a dynamic dimensionless quantity. Also, thevulnerability index of the levee on the permeable foundation is derived by using unsteadyquasi two-dimensional seepage flow analysis method. Finally, the vulnerability index t* iscalculated for the actual field levee and model levee experiment in which each failure modesuch as leakage, sliding or collapse and breach occurred. As a result, it is confirmed that t*
shows approximately common values in the field levee and model levee experiment taking
the same failure mode. Therefore, it is demonstrated that t* gives a dynamic similarity
condition on the levee failure phenomena due to seepage flow. Also, it is shown that thedegree of levee failure can be judged by t* value.
ACKNOWLEDGEMENT
In this study, we used valuable data of the seepage flow in the model levee experimentconducted by the research group of Professor Takeshi Kodaka of Meijo University, Japan.
We would like to express our appreciation.

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70 River Basin Management X

REFERENCESCIRIA, The International Levee Handbook, C731, London, 2013.
Richards, L.A., Capillary conduction of liquids through porous mediums. Journal ofApplied Physics, 1, pp. 318–333, 1931.
[3] Bear, J., Dynamics of Fluids in Porous Media, American Elsevier, pp. 671–677, 1972.
[4] Corey, G.L., Corey, A.T. & Brooks, R.H., Similitude for Non-Steady Drainage ofPartially Saturated Soils, Hydrology Papers 9, Colorado State University, 1965.
[5] Kawaharada, R., On the Hydraulic Similarity of the Unsteady Seepage Flow, Facultyof Agriculture, Kagoshima University, vol. 5, pp. 115–117, 1956 (Japanese).
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[7] Fukuoka, S. & Tabata, K., Development of seepage line estimation method
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[8] Uchida, S., On the non-steady percolation with a free boundary. Japan Society of Civil
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[9] Hazen, A., Some physical properties of sands and gravels with special reference totheir use in filtration, Massachusetts State Board of Health, 24th Annual Report, pp.
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