831

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

AbstractThis 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 thisthe 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 gradientand the character of earth will influence it in

uniform porous medium. The normal form of Darcy Law is as flowing:

dHu k

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:

832

0yx zuu u

x y z

+ + =

2

Form the equation (1), the projection velocities in three directions can be expressed as:

x x

dHu k

dx=

y ydH

u kdy

= 3

z z

dHu k

dz=

Then substitute (2) with (3):

2 2 2

2 2 20x y z

H H Hk k k

x y z

+ + =

4

When in isotropy seepage area, the seepage coefficient is the same for all direction, the equation (4) can

be transformed into:

2 2 2

2 2 20

H H H

x 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)(i-1,j)

(i,j+1)

(i+1,j)

(i,j-1) Figure1. Two Dimensional Difference Grids

Equation (5) can be expressed in difference scheme:

( )

( )

( 1, ) ( , ) ( , ) ( 1, )

( 1, ) ( , ) ( , ) ( 1, )

( 1, ) ( 1, ) 2

( , 1) ( , ) ( , ) ( , 1)

( , 1) ( , ) ( , ) ( , 1)0

( , 1) ( , 1) 2

H i j H i j H i j H i j

x i j x i j x i j x i j

x i j x i j

H i j H i j H i j H i j

y i j y i j y i j y i j

y i j y i j

+ +

+

+ + + =

+

6

833

If the grid size is the same, this equation is simplified to:

2 2

( 1, ) ( 1, ) ( , 1) ( , 1)0

H i j H i j H i j H i j

x y

+ + + =

7

For the universality of equation(6), the geometric parameter of every node is given, the unknown

parameters for each algebraic formula are H(i1j)H(ij)H(i+1j)H(ij-1)H(ij+1)which form the following

matrix to solve the seepage field.

(1,1) (1,2) (1, 2) (1, 1)

(2,1) (2,2) (2, 2) (2, 2)

( ,1) ( 1, ) ( , 1) ( , ) (

... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ...

... ... ... ...

... ... ... ...

... ... ... ...

... ... ... ...

... ... ... ...

... ...

n n

n n

i i j i j i j

K K K K

K K K K

K K K K K

(1,1)

(1,2)

( 1, )

( , 1)

( , )

, 1) ( 1, ) ( , 1) ( , 1)

(

( 1,1) ( 1, 1)

...

...

... ... ...

... ... ...

... ...

... ...

... ...

... ... ... ... ... ... ... ... ... ...

i j

i j

i j

i j i j i n i j

i

n n n

H

H

H

H

H

K K H

H

K K

+ + +

+

1, )

0

...

( 1, 2)

( 1, 1)

j

H n n

H n n

=

8

In this matrix, the K(i1j)K (ij)K (i+1j)K (ij-1)K (ij+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.

150.0

158.5

1:2.0

160.0

1:2.0

152.0

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

90

80

70

60

50

40

30

20

10

150.0

158.5

1:2.0

160.0

1:2.0

152.0

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 fielda 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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