Peng S., Zhang J., Engineering Geology for Underground Rocks
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8.2 Concrete block tests on stress and fluid flow 205
0
1
2
3
4
5
012345
Flowrate (l/min)
)(MPa
yx
V
V
Fig. 8.7. Flowrate versus biaxial loading (
V
x
/
V
y
= 3) in the concrete block.
0
1
2
3
4
5
0 0.4 0.8 1.2 1.6 2
Aperture increment (mm)
Flowrate (l/min
)
Fig. 8.8. Flowrate versus changes of fracture aperture for biaxial loading (
V
x
/
V
y
=
3) in the concrete block.
206 8 Stress-dependent permeability
3
]704.01[27.0 bQ
z
'
(8.3)
where
'
b is the aperture change (ǻb = b b
0
); b
0
and b are the aperture be-
fore and after load application.
The experimental data of flowrate versus fracture aperture is shown
graphically in Fig. 8.8. This relationship verifies that the cubic law is still
valid for a medium with randomly distributed multiple fractures and with
confining stresses.
The flowrate and fracture aperture relationship from Eq. 8.3 can be
written as the following general form:
3
2
1
1
¸
¸
¹
·
¨
¨
©
§
'
C
b
CQ
(8.4)
where C
1
and C
2
are constants.
8.3 Stress-dependent permeability in fractured media
Researchers have developed the parallel model to describe fluid flow in a
single fracture. In the model a single fracture is conceptualized as a chan-
nel between two parallel plates, and the vertical separation between the
plates, or aperture, is constant throughout the length of the fracture. The
flow rate in this model is governed by the “cubic law”, i.e. the volumetric
flow rate of a fluid through a cross section of the fracture is proportional to
the applied pressure gradient and the cube of the vertical separation (Waite
et al. 1999):
dx
dpb
wQ
P
12
3
(8.5)
where Q is the volumetric flow rate, w is the width of the fracture perpen-
dicular to flow, b is the fracture aperture,
P
is the dynamic viscosity of the
fluid.
Using Dacy’s Law, the fracture permeability k
f
can be defined as:
12
2
b
k
f
(8.6)
The parallel plate model for a single fracture can be extended to mul-
tiple fracture systems by considering regular families of parallel fractures
8.3 Stress-dependent permeability in fractured media 207
(Bear et al. 1993). For a set of parallel fractures of equal aperture, oriented
parallel to the flow direction, the permeability through this set is given by:
s
b
k
f
12
3
(8.7)
where b is the fracture aperture; and s is the mean fracture spacing.
Since natural rock fractures are neither smooth nor parallel, investiga-
tors have questioned the accuracy of applying the cubic law to natural frac-
tures. Investigations show that the cubic law is valid when corrected by
considering the fracture tortuosity, correction factor or using effective ap-
erture (Witherspoon et al. 1980). Waite et al. (1999) examined experimen-
tally the parallel plate geometry and confirmed that the rate of fluid
movement through a parallel plate fracture can be predicted by the cubic
law with the traditional estimations of aperture and pressure gradient.
However, for the sinusoidal geometries (Fig. 8.9), the cubic law is valid
only when the fracture aperture is measured normal to the flow path and
averaged according to a harmonic averaging scheme. The normal aperture
can be calculated at any point in the cycle of the sine wave using the verti-
cal separation b and the local slope of the average flow path (dy/dx) as fol-
lows:
»
¼
º
«
¬
ª
¸
¹
·
¨
©
§
dx
dy
bb
n
1
tancos
(8.8)
x
y
b
b
n
Average flow path
Fig. 8.9. Schematic sinusoidal geometry of a natural fracture.
Along the fracture, an average normal aperture <b
n
> can be deter-
mined by averaging over the length of the fracture and used in the cubic
law, i.e.:
208 8 Stress-dependent permeability
3
1
1
3
»
¼
º
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
!
¦
k
i
ni
i
n
b
L
Lb
(8.9)
where L is the total length of the fracture; L
i
is the interval length; and, b
ni
is the normal aperture of the ith interval.
Iwai (1976) showed that the validity of the cubic laws is dependent on
the stress applied onto the fracture. After measuring water flowrates
through granite fractures under an applied normal stress, Raven and Gale
(1985) found an increasing deviation from the cubic law associated with
stress and larger samples. Many experimental and theoretical research of
stress-dependent permeability has been presented (such as; Louis 1974,
Bawden et al. 1980, Witherspoon et al. 1980, Barton et al. 1985, Bai et al.
1994, Zimmerman and Bodvarsson 1996, Zhang et al. 1999, Zhang and
Wang 2000, Zhang 2002).
Applying the cubic law, the permeability change in the z-direction
due to an aperture change for only one set of fractures can be expressed as
follows (Zhang et al. 2007):
3
0
0
1
¸
¸
¹
·
¨
¨
©
§
'
x
x
z
b
u
kk
(8.10)
where k
z
is the permeability change due to the aperture increment
'
u
x
;b
xo
is the average normal aperture of the original fracture in the x-direction as
defined by Eq. 8.9, and k
0
is the original permeability in the z-direction.
The permeability change due to the aperture changes for two sets of
fractures (Fig. 8.10) can be expressed as:
x
z
y
b
x
b
y
s
x
s
y
Fig. 8.10. Multiple fracture system for two sets of parallel fractures in the z-
direction.
8.3 Stress-dependent permeability in fractured media 209
»
»
¼
º
«
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
'
'
3
00
0
1
y
y
x
x
z
b
u
b
u
kk
(8.11)
where k
z
is the permeability change due to the aperture increment
'
u
x
and
'
u
y
.
In order to determine stress-dependent permeability in fractured me-
dia, both the fracture aperture variation and the rock matrix deformation
need to be considered. To derive permeability and stress relation, an ideal-
ized three-dimensional regularly spaced fracture-matrix system can be as-
sumed, as illustrated in Fig. 8.11.
matrix
K
n
Kn
s
x
s
x
s
x
b
x
b
x
'V
'V
x
x
'V
y
z
'V
x
y
z
Fig. 8.11. Coupled fracture-matrix system with 3-D stresses.
The change of total displacement along x-direction is, therefore, the
sum of the fracture displacement and the matrix displacement changes; i.e.:
rxfxtx
uuu '' '
(8.12)
where 'u
tx
, 'u
fx
, and 'u
rx
are the displacements of total fracture-matrix
system, fracture and rock matrix, respectively.
Therefore, the displacement across the fracture can be obtained by:
rxtxfx
uuu '' '
(8.13)
The above equation can be expressed as the strain form:
rxxtxxxfx
sbsu
H
H
'
' ' )(
(8.14)
where s
x
and b
x
are the fracture spacing and aperture along the x-direction,
respectively; '
H
tx
, '
H
rx
are the total and matrix strains, respectively.
210 8 Stress-dependent permeability
The total strain along the x-direction can be obtained according to
Hooke's law:
>@
)(
1
zyx
mx
tx
E
VVQVH
''' '
(8.15)
where E
mx
is the Young's modulus of the rockmass in the x-direction.
According to Hooke's law, the matrix strain along the x-direction may
be described by:
>@
)(
1
zyx
r
rx
E
VVQVH
''' '
(8.16)
where E
r
is the Young's modulus of the rock matrix; 'V
x
, 'V
y
, and 'V
z
are
the stress changes in the x-, y- and z-directions, respectively.
Substituting Eqs. 8.15 and 8.16 into Eq. 8.14, the change of the frac-
ture aperture due to the stress variation can be obtained; i.e.:
>@
)(
)(
zyx
r
x
mx
xx
fx
E
s
E
bs
u
VVQV
'''
¸
¸
¹
·
¨
¨
©
§
'
(8.17)
The Young's modulus of the rockmass (E
mx
) related to the properties
of the intact rock matrix and the fractures can be expressed as follows:
xnxrmx
sKEE
111
(8.18)
where K
nx
is the fracture normal stiffness in the x-direction.
Substituting Eq. 8.18 into Eq. 8.17, the fracture displacement in x-
direction can be obtained:
>@
)(
1
zyx
r
x
xnx
x
nx
fx
E
b
sK
b
K
u
VVQV
'''
¸
¸
¹
·
¨
¨
©
§
'
(8.19)
The fracture permeability in the direction perpendicular to the x-
direction can be calculated directly from the parallel plate analog defined
in Eq. 8.10. With reference to Eqs. 8.19 and 8.10, and considering the
compressive strain as negative, the change of fracture permeability in z-
direction can be expressed as:
>@
3
0
)(
111
1
¿
¾
½
¯
®
'''
¸
¸
¹
·
¨
¨
©
§
zyx
rxnxxnx
zz
EsKbK
kk
VVQV
(8.20)
8.4 Permeability-stress relation in a porous medium 211
where k
0z
and k
z
are the permeabilities along the z-direction before and
after stress change, respectively.
Hence, the generalized permeability-stress relation may be written ac-
cording to Eq. 8.20:
>@
3
0
)(
111
1
¿
¾
½
¯
®
'''
¸
¸
¹
·
¨
¨
©
§
kji
riniini
kk
EsKbK
kk
VVQV
(8.21)
where k
0k
and k
k
are the permeabilities along the k-direction before and
after stress changes; i = x, y, z; j = y, z, x; k = z, x, y; and, i
z
j
z
k.
Only one set of fractures is examined in the above analysis. When
three parallel fracture sets exist in the x-, y- and z-directions, as shown in
Fig. 8.10, permeability changes along the k-direction by referring to Eq.
8.11 may be written as:
>@
>@
3
0
)(
111
)(
111
1
°
¿
°
¾
½
'''
¸
¸
¹
·
¨
¨
©
§
¯
®
'''
¸
¸
¹
·
¨
¨
©
§
kij
rjnjjnj
kji
riniini
kk
EsKbK
EsKbK
kk
VVQV
VVQV
(8.22)
8.4 Permeability-stress relation in a porous medium
For porous media, fluid flow moves mainly through pore spaces. The
variation in grain sizes or pore spaces due to the effect of an applied stress
results in a change of permeability. For a simple cubical grain packing
structure, the permeability k can be expressed as (Bai and Elsworth 1994):
2
2
2
S
R
k
(8.23)
where R is the grain radius.
Under a three-dimensional effective stress condition, the change in
grain sizes within a cubical packing (Fig. 8.12) can be determined by ana-
lyzing the elastic contact of spheres. Applying the theory of Hertzian con-
tact of spherical grains, the change of grain radius due to the effective
compressive stresses changes of ǻı
x
, ǻı
y
, and ǻı
z
can be obtained by:
212 8 Stress-dependent permeability
°
°
°
°
°
¯
°
°
°
°
°
®
»
»
¼
º
«
«
¬
ª
¸
¹
·
¨
©
§
»
»
¼
º
«
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
»
»
¼
º
«
«
¬
ª
¸
¹
·
¨
©
§
3
1
2
2
0
3
1
2
2
0
3
1
2
2
0
2
19
2
2
19
2
2
19
2
E
R
R
E
R
R
E
R
R
z
z
y
y
x
x
ıǻʌ
Ȟ
ǻ
ıǻʌ
Ȟ
ǻ
ıǻʌ
Ȟ
ǻ
(8.24)
The grain radius after stress changes can be represented as R = R
0
+
'
R
x
+
'
R
y
+
'
R
z
, i.e.:
2
3
2
3
2
3
2
3
1
2
2
22
0
2
19
2
1
1
°
¿
°
¾
½
°
¯
°
®
¸
¹
·
¨
©
§
'''
»
»
¼
º
«
«
¬
ª
zyx
E
RR
VVV
QS
(8.25)
where R
0
is the original grain radius.
x
y
z
'V
x
'V
x
'V
z
'V
x
'V
x
'V
z
'V
z
'V
z
'V
y
'V
y
'V
y
'V
y
Fig. 8.12. Spherical contact of grains with 3-D stresses.
Substituting Eq. 8.25 into Eq. 8.23 and considering both effects of
compressional and dilatational stresses, the change in permeability in a po-
rous medium can be expressed as (Zhang and Wang 2006):
8.5 Case application in mining panel 213
2
3
2
3
2
3
2
3
1
2
2
22
0
2
19
2
1
1
°
¿
°
¾
½
°
¯
°
®
¸
¹
·
¨
©
§
'r'r'r
»
»
¼
º
«
«
¬
ª
zyx
E
kk
VVV
QS
(8.26)
where a negative sign refers to compressional loading and a positive sign
corresponds to dilatational loading; k
0
is the initial permeability.
8.5 Case application in mining panel
The solid deformation and fluid flow problems can be solved by using the
finite element method (FEM) using governing equations for a dual-
porosity poro-mechanical model found in Zhang (2002) and Zhang and
Roegiers (2005). Introducing stress-permeability into the finite element
model, permeability variations induced by stress changes can be obtained.
A case example is given below to examine permeability changes due to
mining.
When mining near aquifers, it is of critical importance to determine
the changes of permeability due to mining (Zhang and Shen 2004). The
coal mine considered here is located in the Yanzhou coalfield, Eastern
China. The average mining depth is 305 m and the extraction thickness of
the pertinent coal seam is 5 m. A water-bearing sand lies 75 m above the
coal horizon, and the thickness of the sand layer is 30 m (Zhang et al.
2001, Zhang and Wang 2006). Figure 8.13 shows the finite element model
and mesh of a section perpendicular to the mining direction. In this model,
a half of the mining panel is considered due to geometric symmetry. The
generalized plane strain model is adopted since the mining direction of the
panel is much longer than the direction of the panel width. The model is
laterally confined and impermeable. The bottom of the model is considered
as a rigid and impermeable boundary. The strata gravity with average unit
specific weight of
J
= 23 kPa/m (or 2.3 sg) is considered as the far-field
stress acting on the panel. The far-field stresses and pore pressure are re-
spectively:
V
v
= 7.0 MPa,
V
H
=
V
h
=3.8 MPa, p
w
= 3.0 MPa. The main pa-
rameters of these strata are listed in Table 8.2. In the table most rock pa-
rameters are based on laboratory experiments. However, the parameters for
the sand aquifer and the mined area are simply estimated. The fracture
spacing is assumed to be 1 m for all layers.
214 8 Stress-dependent permeability
Table 8.2. Input rock parameters for the FEM calculations
Layer
number in
Fig. 8.13
Strata
location
Layer
thickness
(m)
Elastic
modulus
(GPa)
Fracture
stiffness
(GPa)
Poisson’s
ratio
5 Sand aquifer 30 1 7 0.4
1 Roof 75 5 12 0.32
2 Immediate roof 25 2.4 10 0.38
3 Coal seam 5 1 10 0.38
4 Mined area 5 10
-4
10
-3
0.35
2 Immediate floor 20 2.4 10 0.38
1 Floor 50 5 12 0.32
0
20
40
60
80
100
120
140
160
180
0 20 40 60 80 100 120 140 160 180 200
D i s t a n c e f r o m p a n e l c e n t e r ( m )
Vertical distance from floor reference (m)
E
4
,
Q
4
E
5
,
Q
5
E
1
,
Q
1
E
3
,
Q
3
E
2
,
Q
2
E
2
,
Q
2
E
1
,
Q
1
Aquifer
50
50
20
25
30
5
305
to surface
Fig. 8.13. Partial finite element mesh in a long-wall mining panel.
Figure 8.14 shows the FEM calculated contours of the permeability
variations (permeability ratios of post- to pre-mining) in the vertical direc-
tion. In this figure, the mining width of the panel is 90 m, and the vertical
axis is the central line of the panel. It can be seen that permeability in-
creases in the strata around and beyond the mined area and decreases in
some areas near the unmined coal seam. The maximum magnitude of in-
creased permeability lies in the immediate roof and floor of the mined
seam of the panel center. However, the maximum height of increased per-
meability zone appears in the strata over the unmined coal pillar.