This document presents an analytical solution for modeling seepage under an overflow spillway founded on anisotropic soil formations. The spillway is equipped with a downstream blanket and two end sheet piles. Variables investigated include the blanket length, permeability in horizontal and vertical directions, sheet pile depths, spillway length, head losses, discharge, and exit gradient. Fragment analysis is used to develop equations for three regions: upstream, under, and downstream of the spillway. Results show that increasing the blanket length significantly decreases maximum pressures on the sheet piles but only slightly increases uplift pressure under the spillway floor. The discharge and exit gradient are also affected. Charts presented allow simple design of the optimal blanket length while accounting for anisotropic soil conditions

1. Journal of Applied Sciences Research, 2(12): 1217-1227, 2006
© 2006, INSInet Publication
Corresponding Author: A.K. Shehata , Researcher, Department of Civil Engineering, National Research Center.
1217
Design of Downstream Blanket for Overflow Spillway Founded on Complex Formations
A.K. Shehata
Researcher, Department of Civil Engineering, National Research Center.
Abstract: The paper presents an analytical solution for the seepage problem under overflow spillway and
founded on complex formations. The spillway was furnished with downstream blanket and two end sheetpiles
to avoid the uplift and piping effects. All investigated models were founded on anisotropic foundation where
the coefficient of permeability in horizontal direction more than that of vertical direction. The analytical
solution was based on the fragment methodology. The study led to a criterion for designing the blanket and
investigating the optimal length of downstream blanket. Also, the effect of the anisotropic soil foundation was
indicated. The simple and practical design charts are presented.
Key words: Over flow Spillway-Analytical solution- Downstream blanket- Sheetpiles.
INTRODUCTION
The overflow spillway is subjected mostly to uplift
and piping effects. One method to maintain the safety of
the spillway structure against these effects is to extend the
length of creep. This safety is established by constructing
sheetpiles or extending the floor length (using
downstream blanket) or sheetpiles-blanket system. The
last system was considered the most practical one. The
amount of seepage through anisotropic foundation soil
under the spillway floor equipped with these systems
always has been more important in design considerations
of the Geotechnical engineers. Moreover, when hydraulic
gradients in the soil foundation are significant, knowledge
of their magnitude is essential to guard against heave or
piping phenomena.
One of the wide methods exit for solving the seepage
problem is the analytical method (conformal mapping and
complex variables)[6]
. A simple although analytical
approach for solving problem of confined flow is called
method of fragment[4,7]
. The accuracy of the fragment
method was assessed using finite element analysis by
Griffiths[6]
and electrical model analogy by Ghada[3]
and
a good agreement was achieved.
A solution for the seepage problem under a control
structure without cutoffs and provided with horizontal
upstream blanket founded on isotropic soil was presented
by El-Salawy[2]
who using F.E.M. Another case in which
the hydraulic structure was furnished with upstreamcutoff
and an impervious blanket located upstream the floor of
a gravity hydraulic structure was also investigated by
using Hele-Shaw model[1]
.
In this paper, an analytical solution for the seepage
problem under overflow spillway with two end sheetpiles
furnished with downstream blanket and founded on
complex formations was obtained. The analytical solution
was presented to investigate the proper length of a
downstream blanket. The effects of the downstream
blanket on seepage under the spillway were obtained.
Also, the effect of the anisotropic soil foundation on
seepage flow under the spillway was indicated. Finally a
set of design curves have been reproduced and their use
was simple and practical.
MATERIALS AND METHODS
2.1 The method of fragments: The method of fragment
forms basis of equations produced in this study. The main
assumptions of this method states that equipotential lines
at certain points within soil mass are vertical. The zones
between these vertical equipotential lines are called
fragments. Solutions for individual fragments can be
found and superposed to give overall solution.
Figure (1) shows how the region under spillway with
two end sheetpiles and downstream blanket could be
fragmented by assuming that equipotential lines are
vertical, wherever significant changes in boundary
geometry occur, the problem is reduced to superposition
of three fragments.
According to fragment method, discharge of seepage
flow is given in form [Harr]:
n
i
i 1
KH
Q
=
=
φ
∑ (1)
Where:
K = coefficient of permeability;
H = total head loss,
øi = form factor for ith fragment;
n = is number of fragments.

2. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
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Fig. 1: "Definition Sketch of the Fragments under Spillway with Two End Sheetpiles & Blanket and Founded on
Anisotropic Soil"
Also, the head loss through each fragment is
calculated from following equation[6]
:
i
Li n
i
i 1
H
h
=
φ
=
φ
∑ (2)
Where:
= the summation of the form factors.
n
i
i 1
=
φ
∑
2.2 Statement of the problem: In the present study the
case of spillway with two end sheetpiles and provided
with downstream blanket built on anisotropic soil is put
under investigation.
It is contemplated that correlations of the blanket
length (Lb) with parameters implying anisotropic soil
thickness (T), permeability of soil in horizontal direction
(Kx), permeability of soil in vertical direction (Ky),
upstreamsheetpile depth(d1),downstreamsheetpile depth
(d2), spillway length (L), the head loss upstream the
spillway (hU), the head loss downstream the spillway (hD),
the discharge of seepage flow (Q) and exit gradient down
stream the blanket (Ie) would be concluded.
Where:
H = head on spillway,
T = anisotropic soil thickness,
Kx = permeability of soil in horizontal direction,
Ky = permeability of soil in vertical direction,
L = spillway length,
d1 = upstream cutoff depth,
d2 = downstream cutoff depth,
Lb = the blanket length,
2.3 Analytical solution: In order to solve problem of
seepage through anisotropic layer, the profile should be
fragmented into three parts as shown in figure (1). The
fragments are as follows:
C Semi-infinite domain A (upstream the spillway)
C Rectangular domain B (under the spillway)
C Semi-infinite domain C (downstream the spillway)
In this study permeability coefficient of soil differs in
horizontal and vertical directions. So, effect of the
anisotropy degree should be consideredbeforedeveloping
equations of fragments using degree of anisotropy (R), as:
(3)
y x
R K K
=
Where:
Kx = coefficient of permeability of soil in horizontal
direction,
Ky = is coefficient of permeability of soil in vertical
direction.
2.3.1 Treatment of fragments:
C Dimensions:
The dimensions of fragments should be modified to
deal with anisotropy effect. All horizontal dimensions that
appear in dimensionless groupings are first subdivided by
anisotropy degree R before entering the equations.
C Permeability coefficients:
The permeability coefficients in both horizontal and
vertical directions will be modified to using the
K
following relation:
(4)
x y
K K K
=

3. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1219
Fig. 2: "The Fragment Types: [A], [B] & [C]"
The value of Q in this condition will be modified as
following:
y
m n
i
i 1
HK
Q
R
=
=
φ
∑ (5)
Now, equations of the fragments could be developed
for solving this problem using the last treatments.
2.3.2 Development of the fragments:
C Upstream spillway (fragment A)
This fragment represents a semi-infinite domain of
height (T) with impermeable boundaries at its base and a
sheetpile of depth (d1) at right side, as shown figure (2).
The form factor of fragment [A] calculated by:
( )
( )
2
2 2
0 1
2
2 2 2
0 1
d
1 a sin
1
d
cos a sin
π
π
θ
− θ
φ =
θ
θ + θ
∫
∫ (6)
Where:
1
1
d
a sin
2T
π
=
C Under the spillway (fragment B):
For rectangular domain, there are two cases
depending on the ratio of the spillway length (L) and the
summation of the sheetpiles depths as shown in figure (2).
-Case one (LR—G d1 + d2)
According to the fragments method formula
(dφ = dx/y), the form factor (φ2) can be obtained as
follow:
( ) ( )
2
2
a LR
2
0 a
1 2
dx dx
T d x T d LR x
φ = +
− + − + −
∫ ∫ (7)
Where:
1 2
2
LR d d
a
2
+ −
=
Thisintegration can be simplified using mathematical
program Matlab7 to:
( )
2
4
2
3
2 a
ln
4a
⎡ ⎤
+
φ = ⎢ ⎥
⎢ ⎥
⎣ ⎦ (8)
Where:
1 2 1 2
3 4
LR
d d d d
a (1 )(1 ) and a
T T T
− −
= − − =
-Case two (LR $ GD1 + d2)
The fragments method formula (dφ = dx/y) obtained
the form factor (φ2) as follow:
( ) ( )
1 2
1 2
d LR d LR
2
0 d LR d
1 2
dx dx dx
T
T d x T d LR x
−
−
φ = + +
− + − + −
∫ ∫ ∫
(9)
Thisintegrationcanbesimplifiedusing mathematical
program Matlab7 to:
(10)
[ ]
2 4 3
a lna
φ = −
Where:
1 2 1 2
3 4
LR
d d d d
a (1 )(1 ) and a
T T T
− −
= − − =
C Downstream spillway (fragment C):
This fragment represents a semi-infinite domain of
height (T), with impermeable boundaries at its base and in

4. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1220
the top corner, as shown figure (1). The form factor of
fragment [C] calculated by:
( )
( )
2
2 2
0 5
3 2
2 2 2
0 5
d
1 a sin
d
cos a sin
π
π
θ
− θ
=
φ
θ
θ + θ
∫
∫ (11)
Where:
b
2 2
2 2
5
R
d d
L
a cos tanh tan
2T 2T 2T
π π π
⎛ ⎞ ⎛ ⎞
= +
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
The summation of the form factors of the three
fragments and the head loss through each fragment can be
then obtained as follow:
-Case one (LR — G d1 = d2)
( )
( )
( )
2
4
1 5
2 2
3
1 5
2
a a
a
ln
4a
1 a 1 a
⎡ ⎤
+
+ +
⎢ ⎥
⎢ ⎥
− −
⎣ ⎦ (12)
( )
( )
( )
2
4
2
3
1 1
lA lB
3 3
i i
i 1 i 1
2
5 5
lC 3
i
i 1
2 a
ln
a 1 a 4a
h H ,h H
a 1 a
and h H
= =
=
⎡ ⎤
+
⎢ ⎥
− ⎢ ⎥
⎣ ⎦
= =
φ φ
−
=
φ
∑ ∑
∑ (13)
-Case two (LR $G d1 + d2)
( )
[ ]
( )
3
1 5
i 4 3
2 2
i 1
1 5
a a
a lna
1 a 1 a
=
φ = + − +
− −
∑
(14)
( ) [ ]
( )
2
1 1 4 3
lA lB
3 3
i i
i 1 i 1
2
5 5
lC 3
i
i 1
a 1 a a lna
h H ,h H
a 1 a
and h H
= =
=
− −
= =
φ φ
−
=
φ
∑ ∑
∑
(15)
2.3.2 Seepage characteristics: Once the head loss
through each fragment has been determined, the pressure
distribution under the spillway and the seepage discharge
as well as the exit gradient can be determined as
summarized below.
The pressure: Assuming a uniform rate of head loss
through each fragment, then the most important values of
pressure in designconsiderations (the maximumpressure)
can be determined as:
1. Max.net pressure on heel sheetpile (P1) = h lA
2. Max. Uplift pressure under spillway floor (P) = 1- hlA
3. Max.net pressure on toe sheetpile (P2) =
hIB[d2/(LR+d1+d2)]+hIC[d2/(LbR+d2)]
4. Max. Uplift pressure under downstream blanket (Pb) =
hIC[LbR/(LbR+d2)]
Exit gradient: The value of the exit gradient (Ie) depends
on the head loss through fragment [C] and the soil
thickness, It is expressed the relative exit gradient (Iv) by
the form[6]
:
( )
e lc
2
o
5
2 2
0 5
I Lh
I
I d
2a T
1 a sin
ν π
π
= =
θ
− θ
∫ (16)
Where:
Io= H/L
Performing the integration using Math Program
Matlab7, then value obtained as:
(17)
2
lC 5
e
o 5
h L 1 a
I
I
I a T
ν
−
= =
Seepagedischarge:Therelativedischarge underneaththe
spillway can be calculated as:
v y n
i
i 1
1
Q Q HK
R
=
= =
φ
∑ (18)
RESULTS AND DISCUSSIONS
Parameters of the Problem: The parameters of the
problem are written in the dimensionless form together
with there considered values as follows:
C The relative toe sheetpiles depth d2/L=0.1
C The relative ratio of sheetpile depths
dv=d2/d1=[0.5 to 2]
C The relative length of the spillway floor Lv=L/T=0.1
C The relative length of the downstream blanket
Lb/L =[0.0 to 1.0]
C The degree of anisotropy (R) ranges from [0.2 to 1.0]

5. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1221
Fig. 3: "Effect of Relative Blanket Length on Max. Net Pressure"
Fig. 4: "Effect of Relative Blanket Length on Max. Uplift"
Effect of the blanket length:
The pressure: Figure (3) indicates that the increase of the
value Lb/L creates significant decrease in maximum net
pressure at heel (P1) and toe (P2) sheetpiles. For most
values of (Lb/L), the value of (P1) decreases slightly with
increasing of value of (Lb/L). The rate of decreasing (P2)
is significant (see the dotted curves). Moreover, the trend
of the curves in case of isotropy soil (R=1) is slightly than
anisotropy soil (R=0.2) which means that the increasing
of Lb/L is more effective in case of anisotropy.
It is seen from figure (4) that the increase in the value
of (Lb/L) causes insignificant increase in maximum uplift

6. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1222
Fig. 5: "Effect of Relative Blanket Length on Relative Discharge and Relative Exit Gradient"
Fig. 6: "Effect of Anisotropy Degree on Max. Net Pressure"
on spillway floor (until Lb/L=0.6 the increase of (P) is less
than 10%). Also, the rate of change to (P) is
approximately the same for isotropy and anisotropy
media. On other hand, from the dotted curves it is clear
that the value of (Pb) increases with great extent as the
value of (Lb/L) increases especially in case of isotropy
foundation.
Seepage discharge: Figure (5) reveals that the relative
dischargeunderneaththe spillway decreasesslightlywhen
the relative length of the downstream blanket increases at
(R=1), while at (R=0.2) the rate of change for the relative
discharge increases steeply when Lb/L>0.3 which mean
that the increase of the downstream blanket length is

7. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1223
Fig. 7: "Effect of Anisotropy Degree on Max. Uplift Pressure"
Fig. 8: "Effect of Anisotropy Degree on Relative Discharge and Relative Exit Gradient"
important in case of anisotropy foundation to avoid piping
effects.
Exit gradient: Figure (5) also shows that the relative
blanket length (Lb/L) has remarkable effect on the relative
exit gradient (the dotted curves). The relative exit gradient
decreases with approximately constant rate at R=1.0 and
R=0.2 due to increase of (Lb/L) which means more safety
against piping in case of isotropy and anisotropy soils.
5.3. Effect of the anisotropy degree:
The pressure: Figure (6) shows that the increase of
the degree of anisotropy (R) creates significant
decrease in maximum net pressure at heel (P1) and
toe (P2) sheetpiles. The rate of decreasing values of
P1 (solid curves) and P2 (dotted curves) are
decreased significantly as (R) increases especially
when (R>0.4). Moreover, it is indicated that for
anisotropy soil (R=0.2) the maximum net pressure

8. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1224
Fig. 9: "Max. Net Pressure on Heel and Toe Sheetpiles At: D2/l=0.1"
increases approximately three times that of isotropy
soil (R=1).
Figure (7) is obtained that the maximum uplift
pressure under the spillwayfloor increases when the value
of (R) increases but the rate of increasing decreases when
(R<0.4). Also, the dotted curves in figure (7) are obtained
that the maximum uplift pressure under the downstream
blanket (Pb) increases as the value of (R) increases.
Seepage discharge: Figure (8) shows the effects of the
change in the anisotropy degree on the relative discharge
(Qv) under the spillway. It is seen that the value of (Qv)
was decreasing steeply until R=0.4 then the rate of change
becomes slightly.
Exit gradient: Figure (8) also shows that the anisotropy
degree has strongly effect on the relative exit gradient

9. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1225
Fig. 10: "Max. Uplift Pressure under Spillway Floor and Downstream Blanket At: D2/l=0.1"
(the dotted curves). The relative exit gradient decreases
with increasing in the anisotropy degree.
Design charts: The charts that representing each
of seepage characteristics as a function of all

10. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1226
Fig. 11: "Relative Discharge and Relative Exit Gradient At: D2/l=0.1"
the independent variables on the graph is very
helpful for the design purposes. The graphical
correlation method and the computer program
Matlab7 are used to construct such graph. A
sample of these charts is obtained in figures (9)
through (11).
Conclusions: In the present study, the case of overflow
spillway with two end sheetpiles and provided with
downstream blanket was put under investigation, the
following conclusions are reached:
C For given ratios of dv, Lv and d2/L the
maximum net pressure on heel and toe sheetpiles
decreases with increasing of blanket length
which mean that more safety for sheetpiles
sections in case of using blanket downstream the
spillway.

11. J. Appl. Sci. Res., 2(12): 1217-1227, 2006
1227
C The uplift under the spillway floor increases with
increasing of blanket length that means the extended
of blanket length more than the optimum causes a lot
of stresses on section of spillway floor (may be reach
to unsafe case).
C Both of discharge and exit gradient decreases
significant with increasing of blanket length. It is
mean that the safety against piping can be improved
by increasing the length of the blanket.
C The optimum length of the blanket is approximately
0.6 length of the spillway floor, where the blanket
efficiency reaches to 60% and the increase of uplift
pressure under the spillway floor is less than 10%
and even more economic.
C The increase of anisotropy degree causes strongly
decreasing of maximum net pressure on heel and toe
sheetpiles. It is very important conclusion and
indicates that the design concepts which deal with the
anisotropy media as isotropy gives unsafe sections of
sheetpiles which may be less than 30% of the safe
section (see figure 6)
C The maximum uplift under the spillway floor
increases with increasing of anisotropy degree that
means the assumption of consider the anisotropy soil
as isotropy soil is suitable in this case and achieves
safe section but uneconomic.
C The increase of the anisotropy degree causes
remarkable decreasing of both exit gradient and
discharge. It is clear that the anisotropy factor very
important and must be considered in design the
protection downstream overflow spillway, specially
the downstream blanket.
C The derived equations have been used to compute the
seepage characteristics. To facilitate the use of these
equations, simple and available design charts have
been obtained.
REFRENCES
1. Abou-Rahim, M., 1994. “Fragment solution of
seepage problem beneath dams on pervious
foundations” Magazine of civil engineering research,
Al-Azhar University, Vol. 16.
2. El-Salawy, M., 1999. “Creep ratio, a new concept for
better design of aprons for control structures
equipped with horizontal upstream blankets ” CMER
Vol. 21.
3. Ezeizah, G., 2000. “Seepage under hydraulic
structures founded on complex formations”
Ph.D. Thesis, Ain Shams University,
6. Harr, M. 1962. “Ground water and seepage” Mc
Graw-Hill, New York, N.Y.
5. Hathoot, H., 1980. “Seepage discharge through
homogeneous soil beneath a horizontal base of a
concrete dam”, ICID Bul., Vol. 29, No. 1.
6. Hathoot, H., 1986. “Seepage beneath a concrete dam
with a downstream filter” J. Applied Mathematical
Modelling, Vol. 10.
7. Hathoot, H., 1993. “Design of downstream filter for
a concrete dam with an end sheetpile” J. Alex. Eng.,
Vol. 32.
8. Shehata, A., 1994. “An advanced study for seepage
under hydraulic structures with sheetpiles” M. Sc.
Thesis, Ain Shams University, April,