International Journal of Engineering & Applied Sciences (IJEAS)

Vol.4, Issue 3(2012)49-56

ANALYSIS AND ESTIMATION OF SEEPAGE DISCHARGE IN DAMS

S. Baghalian 1, F. Nazari 1* and S.S.Malihi 2

Abstract

Analysis and estimation of seepage discharge in dams is presented in this paper. First of all the continuity Laplace
equation is solved for a dam and piezometric head are computed under the dam. Based on piezometric head data values
of seepage discharge of dam are obtained for different conditions. After that a procedure for estimation of seepage
discharge under a diversion dam using feed forward multi layer perceptron artificial neural network is presented. Neural
network are trained based on pizometric head data for seepage discharge estimation in different conditions and its test
results are compared with actual data. Finally it is observed that estimated seepage discharges are in good agreement
with actual results.

Keywords: Seepage Discharge, Dams, Analysis, Estimation, Laplace Equation

1. Introduction

Dams are barrier constructed to save and hold back water and raise its level, the resulting reservoir
being used in the generation of electricity or other beneficial things. Because of high level
importance of dams several studies have been performed on analysis and investigation of dams. Rezk
and Senoon [1] presented an analytical solution for the same problem and also comparisons between
two solutions. Effect of relative permeability of core on each relative seepage discharge and relative
drop of phreatic surface due to core is investigated. Rochon-Cyr and Léger [2] presented a review
study about Shake table sliding response of a gravity dam model consist of water uplift pressure.
They performed a series of shear tests and shake table sliding tests on a 1.5 m high concrete gravity
dam model with a smooth concrete–concrete frictional interface corresponding to a cold lift joint.
Javanmardi et al. [3] developed a theoretical model for transient water pressure variations along a
tensile seismic concrete crack with known crack wall motion history. They performed Experimental
tests to validate the proposed model. Then the proposed model was implemented in a nonlinear
discrete crack finite element program for seismic analysis of concrete dams. Wei et al. [4] in a study
used an anisotropic laminar layer element with thickness to simulate mechanical deformation
properties of weak-bed intercalations at a dam’s foundation as well as a contact friction interface
element without thickness to simulate joints and fissures of the rock mass at the dam’s foundation.
They used nonlinear finite element analysis to compute the resistance to sliding of a high-concrete
gravity dam at the dam’s foundation. Yan et al. [5] presented a systematic analysis on the factors that
may contribute to the uplift. They performed three dimensional numerical analysis and rock
mechanical model to confirm the uplift mechanism of the confined hot aquifer test. Plizzari [6]
studied uplift pressure effects in cracked concrete gravity dams. He investigated influence of uplift
pressure on stress intensity factors and crack-propagation angle. Liu et al. [7] used a coupled hydro-
mechanical model to study of the uplift mechanism of Tongjiezi dam. They used a numerical model
for appraise the representative elementary volume and to investigate related parameters to hydraulic
and mechanical properties of the rock mass. They found if hydro-geological conditions at the
Tongjiezi dam site are specific, hydro-mechanical coupling during and after the reservoir

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impoundment is the most important factor to make the uplift, and the rheological behavior of rock
masses cause the time-dependent deformation under seepage pressure.
Nowadays a new method that often used for investigation and design of different characteristics and
parameters of dams in recent decades is ANN. Wang and He [8] presented an article about numerical
simulation and the model experiment upon a hypothetical concrete arch dam in order to crack
detection using the reduction of natural frequencies and effect of crack characteristics on the dynamic
property of the arch dam was investigated. Mata [9] studied the differences between multiple linear
regression and ANN models for the characterization of dam behavior under environment loads. Then
they investigated the horizontal displacement recorded by a pendulum in a large Portuguese arch
dam. Hasebe and Y. Nagayama [10] presented an article about multipurpose dam with drainage area
relatively smaller compared with dam capacity. They made a comparison between reservoir
operation using the fuzzy and ANN systems and actual one by operator, by using examples of floods
during flood and non-flood seasons. Kim and Kim [11] developed an ANN model for the estimation
of relative crest settlement of concrete-faced rockfill dams. The settlement values that were predicted
using the ANN model were in good agreement with these field data.
In this paper analysis and estimation of seepage discharge in dams is presented. The continuity
Laplace equation is solved for a dam and piezometric head are computed under the dam. Using
piezometric head data values of seepage discharge of dam are obtained for different conditions. A
procedure for estimation of seepage discharge under a diversion dam using feed forward multi layer
perceptron artificial neural network is presented. Trained network used to seepage discharge
estimation in different conditions and compared with actual results.

2. Analysis
2.1. Derivation of the Laplace Equation

In reality, the flow of water through soil is not in one direction only, nor is it uniforms over the entire
area perpendicular to the flow. In such cases, the groundwater flow is generally calculated by the use
of graphs referred to as flow nets. The concept of the flow net is based on Laplace’s equation of
continuity, which governs the steady flow condition for a given point in the soil mass. Laplace
equation is the combination of the equation of continuity and Darcy’s law [12].
A small element with dimensions of dx , dy and dz is considered in order to study of flow in point A
(Fig.1).

Figure 1. Flow in point A [12]

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In Fig.1, vx , v y and vz are flow velocity component in main three directions, vx .dz.dy , vz .dx.dy and
 ∂vx   ∂v 
v y .dx.dz are inlet water discharge and  vx + dx  dz.dy ,  vz + z dz  dx.dy and
 ∂x   ∂z 
 ∂v z 
 vz + dz  dx.dy are outlet water discharges.
 ∂z 
By assuming water as an incompressible flow and considering that volume to be constant, total inlet
discharge flow is equal to total outlet discharge flow.
So:

 ∂vx   ∂vz  
 vx + ∂x dx  dz.dy +  vz + ∂z dz  dx.dy  − [ vx .dz.dy + vz .dx.dy ] = 0 (1)
    
∂vx ∂v y ∂v z
+ + =0 (2)
∂x ∂y ∂z

From darcy’s law, flow velocity can be written as:

∂h
v x = k x ix = k x
∂x
∂h
vy = k yiy = k y
∂y (3)
∂h
v z = k z iz = k z
∂z

That k x and k z are coefficient of permeability in horizontal and vertical directions. By replacing
Eq.e in to continuity equations (Equation (2)), the following equation is obtained.

∂ 2h ∂2h ∂ 2h
kx 2 + k y 2 + kz 2 = 0 (4)
∂x ∂y ∂z
If soil be isotropic, so:

kx = k y = kz (5)

Then the preceding continuity equation simplifies to:

∂2h ∂2h ∂ 2h
+ + =0 (6)
∂x 2 ∂y 2 ∂z 2

Equation (6) is called Laplace equation.

In an isotropic medium the continuity equation represents two orthogonal families of curves:
1. Flow lines: the line along which a water particle will travel from upstream to the downstream side
in the permeable soil medium;
2. Equipotential lines: the line along which the potential (pressure) head at all points is equal [12]
The combination of flow lines and equipotential lines is called flow nets (Fig.2).

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 Figure 2. Flow Net [12]

2.2. Solving Laplace Equation

In present study using PDE toolbox of MATLAB, Eq.6 has been solved in 2 dimensional. Firstly
geometry of diversion dam has been drawn. Characteristics of considered diversion dam have been
shown in Fig. 3.

Figure. 3. Characteristics of considered diversion dam

For solving Eq.6 boundary conditions must be defined. There are two types of boundary conditions
that are natural (Neumann) and essential (Dirichlet). Piezometric head values and passing flow flux
are specified using Dirichlet and Neumann boundaries, respectively. In this study, except of upstream
and downstream boundaries, all of boundary conditions are Neumann. In Neumann boundaries flow
passing is zero and piezometric head values are 40 m and 35 m in upstream and downstream
boundaries, respectively. After meshing of model, Laplace equation is solved.
By solving Laplace equation, piezometric head in several points of considered model was derived.
Then uplift pressure under the diversion dam is obtained using following equation:

Pu = γ (U - Y ) (7)

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where Pu is uplift pressure, U is piezometric head, γ is Specific gravity of water and Y is height of
base level.

2.3. Seepage Discharge

In the next step, using derived piezometric head in different nodes and following equation seepage
discharge per unit width is calculated.

q = kiA (8)

where i is hydraulic gradient and obtains from Equation (8):

∆u
i= (9)
∆x
also

v = ki (10)

that v = ki is velocity.

3. Estimation of Seepage Discharge

Multi-Layer Feed Forward (MLFF) is the most popular type of neural network. ANNs offer a
procedure to tackle complex problems, and are applied in different fields of engineering. A schematic
diagram of typical MLFF neural-network architecture was illustrated in Fig. 4. In MLFF neural
networks knowledge is stored as a set of connection weights. The process of modifying the
connection weights, in some orderly fashion, using a suitable learning method is call training [13]. In
this study, an ANN was trained based on the Back Error Propagation (BEP) technique for the
estimation of pizometric head in different points under the dam. The inputs of the mentioned ANN
were coordinates of different points under the dam, and target outputs were corresponding pizometric
head. Also the same ANN was trained based on the BEP technique. In this table, Wij is the weight of
link that relates neuron number i from input layer to neuron number j from hidden layer. Also Vik
is the weight of link that relates neuron number j from hidden layer to neuron number k from
output layer. B1 j is the weight of the link that relates bias of input layer to neuron number j of
hidden layer and B2k is the weight of the link that relates bias of hidden layer to neuron number j of
output layer. In each network, transfer functions for neurons of hidden and output layers are Tansig
and are defined as equation (11).

H (n) = 2 / [(1 + exp(-2n)) -1]. (11)

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 Figure 4. Diagram of typical MLFF neural network

The estimation procedure of the present study consists of three main stages. In the first stage, a
MLFF neural network was created with an input, a hidden and an output layer with 2, 10 and 1
neuron, respectively. The inputs of the ANN were coordinates of different points under the diversion
dam, and the target output was corresponding pizometric head. In the second stage, ANN training
was performed by data of 56 different points that were obtained by solving continuity Laplace
equation. For training ANN, BEP technique was applied and layer weights of ANN were obtained. In
the third stage, some data were not used in the training process were used to test the trained ANN.
For this propose, coordinates of these data were applied as inputs to trained ANN and corresponding
outputs were obtained. All the data were applied to ANN of this article in normalized form. Then, the
outputs of the trained ANN were compared with corresponding pizometric heads from analytical
data. In the training procedure, BEP iteration was assumed to be 1000 epochs.

4. Result

Piezometric head and then seepage discharge in several points of considered model was computed by
solving continuity equation, and using equation (9). Coordinates of 56 different points under
diversion dam and corresponding pizometric heads were applied as inputs and outputs to ANN,
respectively.
The BEP technique was used to training the ANN and the calculated weights using BEP method was
obtained that were tabulated in Table 1. The test results of ANN for 16 other points were obtained
which were tabulated in Table 1. As it can be seen, the average error between actual and predicted
data for the ANN that trained by BEP methods was 3.14% and therefore it can be concluded that
there is good agreement between predicted and actual data and for this problem.
The actual seepage discharge value in the considered condition was 0.00138 (m2/s) which was
estimated 0.00132 (m2/s). It means that the estimation error in this case was less than 4.5% that is
very good result for this problem. So the proposed procedure can be used for estimation and
investigation of discharge’s seepage under the diversion dams. The main advantage of the proposed
procedure is respond to the points that the prizometric head data aren’t available there, and therefore
ANN can be used to estimation of prizometric head and then discharge’s seepage with good
approximation. In the

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 Figure 5. Training procedure of ANN using MATLAB

Table 1. Comparison of actual and predicted pizometric head under the diversion dam

Number X (m) Y (m) Actual Head (m) Estimated Head (m) Error (%)
1 31 0 38.2293 38.95242 1.891538
2 31 5 38.2327 38.87403 1.677447
3 31 10 38.4857 39.12392 1.658338
4 31 15 38.7884 39.30735 1.337911
5 31 20 39.115 39.44971 0.855719
6 31 25 39.3803 39.57033 0.482551
7 31 30 39.6344 39.71655 0.20726
8 31 35 39.8166 39.88174 0.163601
9 36 0 37.7085 37.44185 0.707123
10 36 5 37.7574 37.56109 0.519934
11 36 10 37.986 36.78514 3.161328
12 36 15 38.6572 36.44846 5.713663
13 36 20 39.0678 36.28089 7.133516
14 36 25 39.3291 36.22683 7.887976
15 36 30 39.6517 36.24364 8.594999
16 36 35 39.6383 36.37492 8.232897

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5. Conclusion

In dams is presented in this paper Analysis and estimation of seepage discharge. First of all the
continuity Laplace equation is solved for a dam and piezometric head are computed under the dam.
Based on piezometric head data values of seepage discharge of dam are obtained for different
conditions. After that a procedure for estimation of seepage discharge under a diversion dam using
feed forward multi layer perceptron artificial neural network is presented. Neural network are trained
based on pizometric head data for seepage discharge estimation in different conditions and its test
results are compared with actual data. Finally it is observed that estimated seepage discharges are in
good agreement with actual results. All in all, it is shown that analysis and estimation of seepage
discharge which is very important parameters in dams using proposed procedure of this study is so
accurate and time consuming.

6. References

[1] Rezk, M.AE. and Senoon, A.E.A, Analytical solution of seepage through earth dam with an
internal core, Alex, Eng. J., 50 (1), 111-115, 2011.
[2] Rochon-Cyr, M. and Léger. P., Shake table sliding response of a gravity dam model including
water uplift pressure, Eng. Struct., 31 (8), 1625-1633, 2009.
[3] Javanmardi, F., Léger, p. and Tinawi, R., Seismic structural stability of concrete gravity dams
considering transient uplift pressures in cracks, Eng. Struct., 27 (4), 616-628, 2005.
[4] Wei, Z., Xiaolin, C., Chuangbing, Z. and Xinghong. L., Failure analysis of high-concrete gravity
dam based on strength reserve factor method, Comput. Geotech., 35 (4), 627-636, 2008.
[5] Yan, F., Xinbin, T. and Li, G., The uplift mechanism of the rock masses around the Jiangya dam
after reservoir inundation, China. Eng. Geol., 117 (1-2), 134-150, 2011.
[6] Plizzari, G.A., On the influence of uplift pressure in concrete gravity dams, Eng. Fract. Mech., 59
(3), 253-267, 1998.
[7] Liu, X., Wang, S. and Wang, E., A study on the uplift mechanism of Tongjiezi dam using a
coupled hydro-mechanical model, Eng. Geol., 117 (1-2), 134-150, 2011.
[8] Wang, B.S. and He, Z.C., Crack detection of arch dam using statistical neural network based on
the reductions of natural frequencies, J. Sound. Vib., 302 (4-5), 1037-1047, 2007.
[9] Mata, J., Interpretation of concrete dam behaviour with artificial neural network and multiple
linear regression models, Eng. Struct., 33 (3), 903-910, 2011.
[10] Hasebe, M. and Nagayama, Y., Reservoir operation using the neural network and fuzzy systems
for dam control and operation support, Adv. Eng. Softw., 33 (5), 245-260, 2002.
[11] Kim, Y.S. and Kim, B.T., Estimation of relative crest settlement of concrete-faced rockfill dams
analyzed using an artificial neural network model, Comput. Geotech., 35 (3), 303-322, 2008.
[12] Textbook: BrajaM. Das, "Principles of Geotechnical Engineering", 7thE. (Chapter 8)
[13] Rumelhart, D.E. and Mcclelland, J.L., Parallel distributed processing: explorations in the
microstructure of cognition, volume 1, MIT Press, 1986.

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