An FDM Modeling to Compute the Seepage Field for Fill Dam
ZHAO Wanhua1,2
1 Civil Department, Wuhan Polytechnic University, Wuhan, P.R.China, 430023
2 Economics and Management School, Wuhan University, Wuhan, P.R.China, 430072
chuanchengzhang@126.com
：
Abstract This Paper derives the differential equation to compute the seepage field based on the Darcy
Law and continuity equation, then discrete the differential equation in finite difference method, establish
a calculation matrix by changing it into algebraic equation, combined with the boundary condition, the
seepage field can be computed. Besides, the seepage of a fill dam with homogeneous material is
computed in this modeling, by dividing the grids, considering the boundary condition, the seepage water
head is solved and witch can reflect the real engineering.
It shows the modeling can compute the similar seepage field to some precision extent.
Key words: seepage; numeral simulation; Finite Difference Method
As for the filled dam, seepage is another significant reason which causes dam failure, the study to
which is benefit to both theory and engineering application. The formal method used in analyzing the
seepage flow is hydraulic method, then electronic simulation method occurs, which is used together with
hydraulic method to verify each other, so that the more reasonable seepage computation result is
achieved. It is difficult to get a precise result because of the simple presume for hydraulic method. It
requires a precise result from reference book, in addition to this the seepage area and boundary
condition is complicated, therefore, the numeral computation method is necessary.
So far, the common methods are Finite Difference Method, Finite Element Method, Boundary
Element Method, None Element Method, Numeral Manifold Method, all of these are based on seepage
in continue medium theory. The FDM is widely applied in engineering computation as a discrete
method, this model is established on FDM.
，
1. Governing Equation
1.1 Darcy Law
Liquid flows among the porous medium, Darcy Law shows the direct proportion relationship
between the seepage velocity and the hydraulic gradient and the character of earth will influence it in
uniform porous medium. The normal form of Darcy Law is as flowing:
，
u=k
dH
ds
（）
1
Where: u is the seepage velocity; k is the seepage coefficient; H is the piezometric head on
corresponding point, which is the summation for pressure head and the location altitude. Because of the
linear relation , Darcy Law is adequate to the laminar flow with linear drag force, excepting the
turbulence seepage in big pore like rock filled dam, most seepage can be defined as laminar flow, that
why the Darcy Law is applied so widely.
When Darcy Law is used in conducting seepage computation, the seepage coefficient k for earth,
which could reflect the earth/s seepage character, the value is determined by the
shape ,size ,ununiformity coefficient and temperature of earth grain.
1.2 Basic differential equation of motion for seepage
If it is considered that the seepage in element is continue, the liquid can not be compressed, the earth
framework can not be deformed, the continuity equation for steady seepage can be derived like this:
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∂u x ∂u y ∂u z
+
+
=0
∂x
∂y
∂z
（ 2）
Form the equation (1), the projection velocities in three directions can be expressed as:
dH
dx
dH
uy = ky
dy
dH
uz = kz
dz
ux = k x
（3）
Then substitute (2) with (3):
kx
∂2 H
∂2 H
∂2H
k
k
+
+
=0
y
z
∂x 2
∂y 2
∂z 2
（4）
When in isotropy seepage area, the seepage coefficient is the same for all direction, the equation (4) can
be transformed into:
∂2 H ∂2 H ∂ 2 H
+ 2 + 2 =0
∂x 2
∂y
∂z
（5）
From the equation (5), for uncompressed steady seepage, the water head accords with the Laplace
equation, so, the seepage field can be deducted by this equation with boundary condition.
2. Governing equation discrete and solving
This model applies the FDM to separate governing equation (5), for 2-dimensional seepage flow
(Figure 1):
(i,j+1)
(i-1,j)
(i,j)
(i+1,j)
(i,j-1)
Figure1. Two Dimensional Difference Grids
Equation (5) can be expressed in difference scheme:
 H (i + 1, j ) − H (i, j ) H (i, j ) − H (i − 1, j ) 
−


x (i, j ) − x(i − 1, j ) 
 x(i + 1, j ) − x(i, j )
( x(i + 1, j ) − x(i − 1, j ) ) 2
 H (i, j + 1) − H (i, j ) H (i, j ) − H (i, j − 1) 
−


y (i, j + 1) − y (i, j )
y (i, j ) − y (i, j − 1) 

+
=0
( y (i, j + 1) − y(i, j − 1) ) 2
832
（6）
If the grid size is the same, this equation is simplified to:
H (i + 1, j ) − H (i − 1, j ) H (i, j + 1) − H (i, j − 1)
+
=0
∆x 2
∆y 2
（7）
For the universality of equation(6), the geometric parameter of every node is given, the unknown
parameters for each algebraic formula are H(i －1，j) H(i ，j) H(i+1，j) H(i ，j-1) H(i，j+1) which form the following
matrix to solve the seepage field.
， ，
 K(1,1)
K
 (2,1)
 ...

 ...
 ...

 ...
 ...

 K(i,1)
 ...

 ...

 ...
 ...

K(n−1,1)
， ，
K(1,2)
...
...
...
...
...
...
...
...
K(1,n−2)
K(2,2)
...
...
...
...
...
...
...
...
K(2,n−2)
...
...
...
...
...
...
...
...
...
...
K(i−1, j)
...
K(i, j−1)
K(i, j)
K(i, j+1)
...
K(i+1, j)
...
...
...
...
...
...
...
...
...
...
...
...
...
， ，
，
，
H

K(1,n−1)   (1,1)



K(2,n−2)  H(1,2)
...

...  



...  H (i−1, j)


...  ...



...  H(i, j−1)


...  H(i, j)
 =0


K(i,n−1)  H(i, j+1)



... ...



...  H

  (i+1, j)

... 
...


...  H(n −1, n −2)


K(n−1,n−1) 
H(n −1, n −1) 
（8）
，
In this matrix, the K(i －1，j) K (i，j) K (i+1，j) K (i，j-1) K (i，j+1) in every row is determined by the geometric
values of every nodes, others are 0, this matrix can be solved combined with the boundary condition for
the seepage area, the seepage head can be got in this way. The example is conducted as following.
3. Engineering Example
This example computes the seepage field of a homogenous earth dam, the dam has a height of 10m,
the gradient for both upstream and downstream slops are 1:2, the upstream water level and downstream
water level is 158.5m and 152.0m relatively, the configuration is showed in Figure2, the material filled
is homogenous sandy clay, the seepage coefficient k=0.25m/d, the grids are divided as Figure3, there
are 440 structural elements for the whole computation area. The boundary downstream and upstream
immerging in water is defined as dank boundary, the dam bottom is considered as waterproof boundary,
after obtaining the phreatic line, the modeling above can be applied to compute the steady seepage field
within the seepage area, the results are showed in Figure 4, the parameter is seepage water head, here
the unit used is total water height difference H, which is 6.5m in this example. It finds that the
computation results reflect the real seepage flow in earth dam on the theory side.
（正常蓄水位）
160.0
158.5
1:2
.0
.0
1:2
152.0
150.0
Figure2. The hydraulic condition and configuration for a homogeneous earth dam
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Figure3. The seepage computation grids for a homogeneous earth dam
160.0
158.5
1: 2
152.0
10
30
％ ％ ％％
％
20
150.0
％
.0
40
80
％
50
％
60
％
70
.0
90
1:2
Figure4. The seepage water head contours for a homogeneous earth dam (Unit: 6.5m)
4. Conclusion and Analyze
This modeling based on the Darcy Equation and Continuity Equation, derives
the differential equation to compute the seepage, the discrete is conducted by FDM, changing it into
algebraic equation and establishing a matrix, which combined with the boundary condition within the
seepage area, can get a computation seepage field.
This paper computes a engineering example of seepage field—a seepage computation in a
homogenous earth dam, dividing the computation grids, combining with the boundary condition, using
this model to compute , then get a seepage computation results which fit the real seepage field in theory,
it concludes that this modeling can compute the seepage in earth dam to some extent. Every coin has
two sides, the shortages for this model is the presume that the seepage coefficients in three directions are
the same, when deriving the governing equation, divide out them, that is different with the real dam,
which influents the precision of the seepage computation,
However, this model can meet the ordinary need for seepage computation; beside of this, the
engineering example is used here in two dimensions, the three dimension example is never computed, so
the rationality of the three dimensional modeling is still a developing area in this research.
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