Unsteady Seepage Through Random Fracture Network in Rock Mass CHAI Junrui

Physical and Numerical Simulation of Geotechnical Engineering
1st Issue, Sep. 2010
Unsteady Seepage Through Random Fracture Network in
Rock Mass
CHAI Junrui1, 2, HE Yang2, LI Kanghong2, LIU Zhao2
1. College of Civil and Hydroelectric Engineering, China Three Gorges university, Yichang, Hubei Province, People’s Republic of
China, 443002
2. College of Hydroelectric Engineering, Xi’an University of Technology, Xi’ an, Shaanxi Province, People’s Republic of China,
710048
ABSTRACT：It is of great importance to study the unsteady fluid flow in the fracture network in rock
mass and to simulate the seepage field in the fractured rock mass. The random distribution of the
fracture network in rock mass is simulated by the Monte-Carlo method. The numerical program is
developed based on the mathematical model of unsteady seepage in the fracture network. A case study
is carried out by means of the program. It can be concluded that (1) The hysteresis of the hydraulic
head distribution in the unsteady seepage field occurs in the case when the boundary condition is
varying; (2) The hysteresis of the hydraulic head distribution in the unsteady seepage field is more
obvious on the condition of the randomly variable apertures than the uniform apertures; (3) The
seepage discharge is centralized in the larger-aperture channel; and (4) The hydraulic head distribution
in the unsteady seepage field can be effected greatly by the variable apertures when the boundary
hydraulic head conditions of the upstream and the downstream are varying greatly.
KEYWORDS: Random fracture network, Unsteady seepage, Monte-Carlo method, Variable
apertures
1
INTRODUCTION
Seepage analysis is very important in hydraulic
engineering, especially in high dam engineering. Seepage
stability has more and more obvious effect on the safety
and economy of the project [1-5]. The natural characteristics
of the structure surface of rock mass fractures in the space
can be shown by the geometry characteristics which
include the orientation, configuration, scale, aperture and so
on. These geometry characteristics have direct effect on the
permeability of fractured media, such as the quantity,
orientation and distribution. In engineering practice, we can
investigate the nature of rock mass in the project area, and
then take some appropriate measures to make sure the
project safety. In most cases, the permeability of rock
matrix is so weak that it can be ignored, and the fluid flows
only along the connected fracture network. There are
different types of fracture network, formed as the result of
different mechanical reasons and different scales. Because
the fractures resistant to water exist in the system, the
connecting characteristic of fractures is very poor, which
leads to the discontinuous distribution of fluid flows in
fracture net work. This system is called as the unconnected
system. For this reason, seepage in rock mass can be
considered as seepage through the connected fracture
network[6-9]. In this paper, the distribution of fractures in
rock mass is simulated by the statistical characteristics of
the geometric elements of fractures and the Monte-Carlo
method. Then analysis of unsteady seepage through random
fracture network in rock mass is carried out. Finally the
hydraulic head and seepage discharge distributions in
fractures in rock mass varying with time are obtained and
studied, from which the hysteresis of the hydraulic head
© ST. PLUM-BLOSSOM PRESS PTY LTD
distribution and the centralized channel flow effect are
observed obviously.
2 RANDOM
SIMULATED
METHOD
2.1
FRACTURE
NETWORK
BY
THE
MONTE-CARLO
Principle of the Monte-Carlo method
The Monte-Carlo method [10] for solving the
mathematical problem is from the contrary procedures. If
the probability distribution satisfies a certain mathematical
equation, the trials of random sample are made to produce
the random variables, and the average values of the results
are taken as the appropriate estimate solution of
mathematical equation [8, 9]. In the same way, we can use
the Monte-Carlo method to simulate the fracture network.
In the process of random simulation, it is the most
important thing to produce the random numbers which have
the regular distribution. At present, the linear congruent
method is an important and popular one to refine the
random numbers. Because it has a simple algorithm which
can be understood easily, the linear congruent method is
performed easily and has the good statistic law. The
iteration equation of the multiple congruent method is as
follows [11]:
(1)
xi  axi 1  cm o dm
In which a is the non-negative factor, c is the
non-negative increment, m is the module, mod m
is the remainder which get after division m . So we can
Unsteady Seepage Through Random Fracture Network in Rock Mass
DOI: 10. 5503/J. PNSGE. 2010. 01.005
call this method as the remainder expression of
the initial value
x0 ,the
m .Given
crosses with each other or with the boundary to form the
connected fracture network [9]. Because the fractures
produced by this method are strongly random, we directly
number the junctions and elements of the fracture network.
A case study of simulating fracture network:
On a dam site, there is a 20 m  20 m area with three
groups of fractures. The probability distributions of the
geometric parameters of fractures in rock mass can be
obtained from the joint statistics and category, which are
shown as Table 1. The fracture network simulated by the
Monte-Carlo method is shown as Figure 1. The simulated
results demonstrate clearly the relationships and
distribution of the fracture network in rock mass. Figure 2
is the connecting fracture network, which is got by deleting
the unconnecting fractures in Figure 1. Thus, we can use
these results to analyze effectively the unsteady seepage
through the fracture network in rock mass.
average random number can be
calculated by the iteration step.
In fact, the distribution of the fracture network in rock
mass is so complex that it is difficult to describe it by the
investigation method. The process to simulate the fracture
network is as follows: (1) The fractures of outcrop rock
should be investigated, including the spacing, aperture,
tendency, dip angle, trace length and so on; (2) The
probability distribution functions of fracture geometric
elements are analyzed and established by statistics; (3) The
fracture network is simulated from its probability
distribution functions by the Monte-Carlo method. The
practical statistical data is used to simulate the fracture
network, the unconnected fractures may occur. In this
situation, the independent fractures along which fluid can
not flow through must be deleted. Thus, every fracture
Table 1
The input data for Monte-Carlo simulation
Group of joints
Dip angle (°)
1
2
3
Average value
90
180
45
Standard deviation
5.7
5.8
6.4
Average value
3.4
4.1
4.3
Average value
2.3
3.5
2.1
Standard deviation
0.2
0.3
0.4
Average value
1.75
2.98
3.63
Average value
0.00025
0.00034
0.00041
Standard deviation
0.00002
0.00001
0.00003
Normal distribution
Trace(m)
Negative exponential distribution
Spacing distance(m)
Homogeneous distribution
Discontinous distance (m) Negative exponential distribution
Aperture(m)
Normal distribution
A
D
A
4
1
5
6
5
8
13
11
14
10 11
16 16
6
7
12
10
7
9
4
17
D
3
2
1
2
15
12 13
17
20
14
15 19
18
21
18
23
22
21
24
22
28
29
58
54
39
36
30
33
52
38
53
20
25
24
23
31
56
41 40
42 43
59
19
26
27
2534 26
32
37
35
38
35 34
48
36
55
37
46
2739 28
40
3141 29
33 32 44 30
44
61
50
45
51
77
62
65
47
63
48
46
49
66
64
54
68
55
70
90
69
88
60
71
72
69
62
C
B
Figure 1 The fracture network simulated by the Monte-Carlo method
3 NUMERICAL SIMULATION OF UNSTEADY
SEEPAGE THROUGH RANDOM FRACTURE
NETWORK
3.1 Numerical
program
model
and
86
59
79
83
58
87
89
63
70
90
91
93
64
84
85
66
92
B
95
56
61
5280 50
51
81
57
79
73
78
80
74
87
53
65
67
68
C
Figure 2 The connected fracture network
The network system method based on discontinuous
media should be used for seepage through fracture network
in rock mass [12]. The point where two fractures cross is
taken as a node and the fracture between two nodes is
regarded as a linear element. The volume of water flowing
into or out of the same node is equal to the variation of
water storage volume in each node. So the model for 2-D
unsteady seepage through fracture network can be obtained
computational
30
Physical and Numerical Simulation of Geotechnical Engineering
1st Issue, Sep. 2010
as follows[13, 14], which should be combined with the initial
and boundary conditions.
in which D is the storage matrix of fracture network， I
is the unit matrix. According to equation (3), the hydraulic
heads at t  t can be determined by the computational
hydraulic heads at t . Area is studied, and then the quantity
of flow on boundary can be obtained.
The computational program for unsteady seepage
through random fracture network in rock mass is based on
equations (2) and (3). From a simple example verification,
the maximal error between the numerical solution and the
theoretical one is 0.02%. Thus, the effectiveness of the
computational program has been verified.
N
  N'

dH i
  q j    w j  i Qi   d i
 j 1   j 1 
dt


i 
(2)
1,2,3,, N 
in which q j  j  1,2, , N  is the flow discharge
through node i along the line element j ; j is the
'
connecting linear element; w j  j  1,2, …， N
is
'

the quantity of supply on each linear element;
source (or sink) of node
node
i
;
S
di  i
2
i ; Hi
Qi
is the
3.2 A case study
is the hydraulic head at
N'
b l
j 1
j j
, where
Si
3.2.1 Fracture network with uniform apertures
Here we use the fracture network simulated by the
Monte-Carlo method in section 2.2 to carry out analysis of
unsteady seepage. The fracture network can be meshed into
68 nodes and 86 line elements shown as Figure 2. The
aperture of each line element is 0.001 m , the depth is 1 m .
is the elastic
storage coefficient of the fracture which is connected with
i ; b j , l j are the
element j , respectively.
node
aperture and length of line
5
The coefficient of water reserve is S = 6.9  10 . The
initial and boundary conditions are as follows for six cases.
If the vertical water supply was ignored, the equation (2)
can be rewritten as follows [15].
D 
D

t  t
t
G  t I H   t H 
Table 2
Case
(3)
The initial and boundary conditions for six cases
Initial condition
Boundary condition
1
H
AB
 100 m , H
CD
 20m , Q
AC
 Q BD  0
H
AB
descending with
1m / s
2
H
AB
 100 m , H
CD
 20m , Q
AC
 Q BD  0
H
AB
descending with
1m / h
3
H
AB
 100 m , H
CD
 20m , Q
AC
 Q BD  0
H
AB
descending with
1m / d
4
H
AB
 20m , H
CD
 20m , Q
AC
 Q BD  0
H
AB
ascending with
1m / s
5
H
AB
 20m , H
CD
 20m , Q
AC
 Q BD  0
H
AB
ascending with
1m / h
6
H
AB
 20m , H
CD
 20m , Q
AC
 Q BD  0
H
AB
ascending with
1m / d
The computational program for unsteady seepage
through random fracture network in rock mass is used to
calculate the seepage fields under the above six cases. The
typical hydraulic head contours of seepage fields at the
initial moment and each moment under case 3 are shown as
Figure 4 to Figure 7.
According to Figure 4 to Figure 7, the seepage field in
rock mass is varying with the boundary conditions and time.
From Figure 7 we can see clearly that the hydraulic heads
inside are higher than hydraulic heads nearby the boundary.
When the hydraulic head on the boundary descends
suddenly, the hydraulic heads inside need time to reach the
steady condition. So the hysteresis of the hydraulic head is
caused by the change of hydraulic head sometimes. The
boundary hydraulic heads vary so fast that the hydraulic
heads inside have no time to keep up with it, making the
hydraulic head inside higher than the boundaries. This is an
extremely disadvantage factor to the stability of rock mass.
31
Unsteady Seepage Through Random Fracture Network in Rock Mass
DOI: 10. 5503/J. PNSGE. 2010. 01.005
20
20
18
18
16
16
14
14
12
10
Y(m)
Y(m)
12
10
8
8
6
6
4
4
2
0
0
2
4
6
8
10
12
14
16
18
2
20
X(m)
0
0
2
4
6
8
10
12
14
16
18
20
X(m)
Figure 4 The contour of hydraulic head at initial time
Figure 5 The contour of hydraulic head after
falling for 20 days
20
20
18
18
16
16
14
14
12
Y(m)
Y(m)
12
10
10
8
8
6
6
4
4
2
2
0
0
2
4
6
8
10
12
14
16
18
0
0
20
2
4
X(m)
6
8
10
12
14
16
18
20
X(m)
Figure 6 The contour of hydraulic head after falling
Figure 7 The contour of hydraulic head after falling
for 60 days
for 80 days
3.2.2 Fracture network with randomly variable apertures
As the above example, the engineering condition, the
basic assumption and the geometry elements are also the
same. It is only that the apertures of fracture network are
randomly variable. The aperture of each fracture is also
simulated by the Monte-Carlo method. The apertures of
fractures accord with the normal distribution shown as
Table 1. Some typical apertures which is produced
randomly are shown as Table 3. The depth of fracture is
also 1 m . The coefficient of water reserve is also
S  6.9  10 5 .
The initial and boundary conditions
are also shown as Table 2 for six cases. The typical
hydraulic head contours of seepage fields at the initial
moment and each moment under case 3 are shown as
Figure 8 to Figure 11.
32
Physical and Numerical Simulation of Geotechnical Engineering
1st Issue, Sep. 2010
The typical random apertures
Group of joints
X1（m）
Y1 (m)
X2 (m)
Y2 (m)
Aperture（m）
1
16.637861
16.056965
16.784464
20.000000
0.000237
1
14.411842
0.000000
14.353835
0.877495
0.000260
1
14.632769
3.217984
14.655147
5.145730
0.000209
1
15.177883
10.160059
14.196506
20.000000
0.000245
2
18.126020
15.724996
3.532354
11.015261
0.000343
2
20.000000
12.260617
5.337903
13.779658
0.000340
2
3.158185
12.188874
0.000000
12.541636
0.000331
2
20.000000
11.872809
17.372505
11.721912
0.000340
3
17.957096
10.186799
19.529942
12.373300
0.000418
3
4.682361
0.000000
5.974410
1.592494
0.000487
3
7.727613
2.568667
20.000000
14.521752
0.000352
3
0.000000
0.940207
4.478462
5.853339
0.000413
20
20
18
18
16
16
14
14
12
12
Y(m)
Y(m)
Table 3
10
10
8
8
6
6
4
4
2
2
0
0
2
4
6
8
10
12
14
16
18
0
0
20
2
4
6
8
X(m)
Figure 8 The contour of hydraulic head at initial time
14
16
18
20
20
18
18
16
16
14
14
12
12
Y(m)
Y(m)
12
Figure 9 The contour of hydraulic head after falling for 20
days
20
10
10
8
8
6
6
4
4
2
2
0
0
10
X(m)
2
4
6
8
10
12
14
16
18
0
0
20
X(m)
2
4
6
8
10
12
14
16
18
20
X(m)
Figure 10 The contour of hydraulic head after falling
for 60 days
Figure 11 The contour of hydraulic head after
falling for 80 days
According to Figure 8 to Figure 11, the seepage field in
rock mass is varying with the boundary conditions and time.
It can be clearly shown from Figure 11 that the hydraulic
heads inside are higher than hydraulic head nearby the
33
Unsteady Seepage Through Random Fracture Network in Rock Mass
DOI: 10. 5503/J. PNSGE. 2010. 01.005
boundaries. This is the hysteresis of hydraulic head which
is just as the example in Section 3.2.1. Comparing Figure 7
with Figure 11, we can see that the hydraulic head inside on
the variable apertures is higher than that on the uniform
apertures. When the apertures are considered as random
distribution, the hysteresis of the hydraulic head is more
obvious, comparing with the uniform apertures.
Comparing Figure 4-7 with Figure 8-11, the maximal
relative difference of hydraulic head at the same node at the
same time is 18.17%, which is resulted only from the
randomly variable apertures. When the hydraulic head
difference between upstream and downstream is great, the
seepage field was effected greatly by random apertures;
when the hydraulic head difference is small, the seepage
field was little effected by random apertures.
In addition, the hydraulic heads and flow rates of typical
nodes and elements are shown as Table 4. From it, the
centralized channel flow effect can be seen clearly. If the
hydraulic gradient is the same, the flow rate is proportional
to the cube of the aperture. Thus, at the node of the fracture
network, the most of the flow rate flows along the fracture
with larger aperture. So, it can be shown that the fracture
network model can reflect the centralized channel flow
effect.
Table 4 The hydraulic heads and flow rates of typical nodes and elements (after falling for 20 days)
Flow rate ( 106 m3 s )
Aperture （m）
Hydraulic head (m)
Element
Node
18
39
49
4
32.29768
18
21
22
23
0.0002448
0.0004239
0.0004239
0.0002448
18.99125
24.18184
-6.81789
-1.77417
38.34599
29
53
54
55
0.0002448
0.0003446
0.0003446
0.0002448
7.71453
23.3983
-13.2056
-6.51999
21.38649
51
74
76
77
0.0003989
0.0004456
0.0003989
0.0004456
6.78837
-10.48982
-3.92423
13.37318
CONCLUSION
ACKNOWLEDGEMENTS
The fracture network model in rock mass hydraulics is
employed in this paper to simulate 2D unsteady seepage
field in the discontinuous and anisotropic rock mass. From
the case study, the conclusions can be followed as follows.
(1) The Monte-Carlo method is feasible to simulate the
fracture network in rock mass, which is based on the
fracture geometric element statistical law.
(2) It can be shown from the case study that analysis of
unsteady seepage reflects more effectively the seepage field
in practice. The hysteresis of the hydraulic head distribution
in the unsteady seepage field occurs in all the six cases
when the boundary condition is varying.
(3) The hysteresis of the hydraulic head distribution in
the unsteady seepage field is more obvious on the condition
of the randomly variable apertures than the uniform
apertures; and the hydraulic head distribution in the
unsteady seepage field can be effected greatly by the
variable apertures when the boundary hydraulic head
conditions of the upstream and the downstream are varying
greatly.
(4) The seepage discharge is centralized in the
larger-aperture channel (fracture), i.e., the centralized
channel flow effect can be shown in the case study.
The financial support from the Project 50579092
sponsored by National Natural Science Foundation of
China (NSFC), Research Fund 20096118110007 for the
Doctoral Program of Higher Education of China, the
Project 106-220331 sponsored by the Scientific Research
Foundation (SRF) for the Returned Overseas Chinese
Scholars (ROCS) by State Education Ministry (SEM), the
Research Project 03JK098 sponsored by Shaanxi Provincial
Education Department (SNED), the Project 2004ABB012
sponsored by Hubei Provincial Science and Technology
Department (HBSTD), the Project 603108, 603402
sponsored by China Three Gorges University (CTGU) and
the Scientific Innovation Project 106-210303, 220275
sponsored by Xi’an University of Technology (XAUT) is
gratefully acknowledged.
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