Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 32, No. 3, pp. 251261, 1995
Pergamon
01489062(94)000344
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 01489062/95 $9.50 + 0.00
Permeability Predictions for Jointed Rock Masses Z. Q. WEIr P. EGGERy F. DESCOEUDRESt
The following problems have been studied: the precolation threshold of jointed rock masses, the Representative Elementary Volume, the permeability of rock masses with impersistent rock joints and the permeability variation with depth. The proposed simple methods are verified against the results of numerical computation and the Stripa Project, Sweden.
1. INTRODUCTION The following questions have to be put forward for design in a jointed rock mass. When is it permeable? If it is permeable, what is the Representative Elementary Volume (REV)? How can one predict the permeability of rock masses with impersistent rock joints? When is it not necessary to know the fracture dimension to predict the permeability? How can one predict the permeability variation with depth? What scale should be used to obtain an equivalent permeability by large scale in situ tests? Can this permeability be applied to a given dimension of underground structures? Snow's methods [1, 2] have been used but infinite fracture lengths are assumed. Several approaches have been introduced to account for the interconnectivity and heterogeneity effects. Parsons [3], CaldweU [4, 5] and La Pointe and Hudson [6] used electrical analog models to study finite fractures. The drawback is that the electric current is proportional to the width of conductor while the flow in fractures is proportional to the aperture cubed. Rocha and Franciss [7] proposed a field method for finding an empirical correction factor. Sagar and Runchal [8] attempted to take into account the finite size of fractures and found that the equivalent permeability of fractured rock depends upon not only the number of fracture groups, number of fractures in each group, fracture dimensions and orientations but also the rock volume considered. He obtained a nonsymmetric and nontensorial permeability matrix by summing up the contribution of each fracture that intersects at least one face of the rock cube considered. The fractures that do not intersect any face are not taken into account. In order to apply their theory for a cube of given rock volume, it is important to know how many fractures intersect at least one face of the cube.
It seems that the best way is to use numerical fracture network models (Robinson [9], Cundall [10], Long [11], Schwartz et al. [12], Samaniego [13], Long and Witherspoon [14], Witherspoon and Long [15], Dershowitz and Schrauf [16]). These models require detailed deterministic or statistical information about the geometry of fractures. Modelling is laborious and expensive. Neuman [17] also argued that there is growing laboratory and field evidence that the manner in which such models translate the data about fracture geometry into hydraulic and transport properties of the rock is open to serious questions. And he proposed a stochastic continuum approach and used a unified framework to analyse the hydraulic test data obtained on different scales, generally on subREV scales. As far as the authors known, the questions at the beginning of this article have not received clear answers, so we will try to find solutions for the following in later sections: percolation threshold, the REV, permeability correction function and permeability variation with depth. 2. P E R C O L A T I O N
THRESHOLD
Whether a jointed rock mass is permeable can be studied by percolation theory (Broadbent and Hammersley [18], Dienes [19], Englman et al. [20], Charlaix et al. [21], Gueguen and Dienes [22]). Here a very simple approach is used and verified against numerical results. Figure 1 shows rock masses with two orthogonal sets of fractures. The reason why only the rock mass in diagram (b) is permeable is that the length (L) of fractures is equal to or greater than the spacing (So): Length Id = Spacing
L So
20L >/1
(1)
Where Id is so called discontinuity index and 2o is the discontinuity frequency in the direction normal to the
#Laboratoire de Mecanique des Roches, Departement de Genie Civil, Ecole Polytechnique, CH1015 Lausanne, Switzerland. 251
WEI et al.: ROCK MASS PERMEABILITYPREDICTIONS
252 Discontinuity length
~ I
I
I
I I
for a permeable fracture network. Table 1 shows the predictions. It can be seen that only 3 out of 18 predictions (3B, 3C and 4A) do not agree with interconnected numerical networks. This difference results from the fact that equation (2) is for two orthogonal sets of identical fractures while the orientation and length of fractures in the numerical simulations are random variables, in which case the percolation threshold is about 2, as will be seen later. The predictions in parentheses in Table 1, are based on the percolation threshold of two and agree with the numerical networks. In consequence, the predictions are very good.
I I
a: Imoermeable
3. THE REPRESENTATIVE ELEMENTARY VOLUME
Length
> 1
Spacing b: Permeable
Fig. 1. Permeability threshold of jointed rock masses.
discontinuity set. In two and threedimensions, they are, respectively (Hudson and Priest [23]): •rnax
20=~
'~max
20=~.
Therefore in twodimensions, equation (1) becomes L Id = ~ = ~maxL ~
(2)
and in threedimensions, L
Id = ~min = '~'maxL ~
x//~,
(3)
where Smi, and 2m~xare the minimum and maximum total discontinuity spacing and frequency. Samaniego [13] undertook twodimensional numerical simulations of water flow in fracture networks. Fracture orientation follows a normal distribution, fracture length a negative exponential distribution and the aperture is constant. Figure 2 shows the generated and interconnected meshes. The prediction of percolation threshold can be made according to equation (2) as follows. Samaniego and Priest [24] gave the following relation between the discontinuity frequency (2), area density (Pa) and length (L) for random fractures: 2 = _2Pa L.
(4)
Substitution into equation (2) leads to:
Pa~> 2L 2 '
(5)
Only three area fracture densities were used in the numerical simulations, which are, respectively, 7.5, 10 and 12.5 fractures m 2. For a given fracture length, equation (5) gives the minimum fracture density required
The REV is the jointed rock mass volume beyond which there is no substantial variation in permeability. Samaniego [13] and Long and Witherspoon [14] conducted twodimensional numerical studies. The main difference between their models is that the fracture length is constant in Long and Witherspoon's model while it follows a negative exponential distribution in Samaniego's model. Their results will be used to obtain a rough REV dimension. Figures 3 and 4 show the permeability variation with simulated dimension. The vertical axis is the permeability ratio of rock masses with finite and infinite fracture lengths. The horizontal axis is the ratio of the simulated dimension (D) to the total fracture spacing (St = 1/2t), i.e.: D ID = fftt= 2t D.
(6)
Since Samaniego did not provide the permeability for infinite fractures, it is estimated as: k0 = ~'°e3 12
(7)
where e is the average hydraulic fracture aperture. The discontinuity frequency, 20, is estimated from equation (4). It can be seen that the permeability ratio tends to converge when the simulated dimension is larger than l0 times the total discontinuity spacing (St). Although the permeability still decreases, its variation is very small when the simulated dimension is larger than 50 times St. Therefore, the REV dimension (DREv) can be defined as: DaEv = (10 ,,~ 50)St Even if it is technically possible, it is not necessary to carry out in situ tests with dimensions larger than 50St since they cannot significantly increase the accuracy. When the fracture length is ( 1 0  5 0 ) S t , it may not be necessary to know the fracture dimension to predict the permeability and the theories based on infinite fracture length may not overestimate the permeability too much. In fact, as shown in the next section, it only overestimates 20% when the fracture length is greater than 15St. If the total discontinuity spacing is replaced by the average discontinuity spacing (Sa0) of different sets, the
WEI et al.:
predicted REV size is always larger than that predicted above since S~0 is always larger than St. Real fractures are not distributed randomly. As a result for practical application, the REV dimension can be assumed to be 1020 times the average spacing, i.e.: DRE v =
(10
253
R O C K MASS P E R M E A B I L I T Y P R E D I C T I O N S
For the Stripa site, Sweden, Olsson [25] obtained a REV of 7 m for the Hzone and 12 m for the average rock. Since the average fracture spacing, as given later, is 0.65 m, the predicted D~Ev = (1020)S~0 = (6.51.3 m). Thus, the two approaches are very close. Rats and Chernyashov [26], Maini [27] and Neuman [17] adopted a continuum approach if a sample
~ 20)Sao
q'...'_,::'4, "

~"':~J
t :'/% • q , ,.~,.x..,~ ,¢.~, ¢;;,~ .: :.1 i~'..'.~ .~." _.'.~,,., ~',~ =,
1"~".~
~.
,'~ ~'" ",%...:,.~._;r
~,. ,'2.,',~... ~. ~,,: :, ~'..~q ..'Y~. ;~,, "¢,~ . ~ " ~ ~._'~"4
~ " "/4~" "."~ .~ ,~'~..'~l, I , , "," ~ : ,.~",¢.,:", ~ ;.~ ~
13" .....~ "4: L~H:"~; .'."~~,
• ...,, I , .
Test 1A
Test IB
, \ 
~
•
.,'....~./.~ .,~)
Test IC
~//
../ /
f
f
Test 2A
Test 2B Fig. 2continued overleaf
RMMS 3213~E
Test 2C
254
WEI et al.: ,x
,(
ROCK MASS PERMEABILITY PREDICTIONS
N×
r_
N
e~ g~/"
.j
,
A
/x//
.,,,p°
c~v~
~
.,1
_( V
V Test 3A
,IL v
Test 4A
.
.
.
.
Test 3B
Test 3C
Test 4B
T e s t 4C
',¢.,."
Fig. 2continued opposite
I
¢,
WEI et
al.:
ROCKMASSPERMEABILITYPREDICTIONS
/
^
A
255
><"
v i
} >
V
~" 7  ~ , , . , ' ~ Test
5A
Test
5B
Test
5C
% +
¢
i,~L
< /"
'
• ..~
"¢
/
/
~,. i Z~'_. ~
,~ /
j.. Test 6A
Fig. 2. Generated
Test 6B
Test
6C
and interconnected meshes for study of the influence of discontinuity length and density on permeability (Samaniego
contains 10 fractures. This coincides with the above conclusion. Wei et al. [28] showed that an equivalent permeability is valid if the tunnel diameter intersects five continuous joints so that a continuum approach can be adopted if the smallest dimension of an underground structure intersects 510 fractures.
[13]). 4.
PERMEABILITYCORRECTION FUNCTION
Snow [1, 2] obtained a permeability tensor based on infinite fractures using an equivalent flow rate while Wei and Hudson [29] used a volume average flow velocity to indirectly take into account the effects of finite fractures. As stated in the Introduction, discrete numerical simu
256
WEI et aL:
ROCK MASS PERMEABILITY PREDICTIONS
Table 1. Predicted permeable and impermeable cases Discontinuity length, L(m)
Density P, ( l / m s)
(1) 0.2
(2) 0.4
(3) 0.5
(4) 0.6
(5) 0.8
(6) 1.0
(A) 7.5 (B) 10 (C) 12.5
No No No
No No No
No Yes? (No) Yes? (No)
Yes? (No) Yes Yes
Yes Yes Yes
Yes Yes Yes
k ko
1.0
1.0i~ o
°'8]i
k ko
II
o.e #1
D
0.8
"\
"4.
",~.,...,.___,
~0.6  0.4
._o
•
r
.0 .0
l
~
/
®
a.
2:21 0
I).2 [] ~ I
I 10
I 20
I 30
I 40
50
0.0
0
I
2
Flow domain dimension/spacing Fig. 3. Permeability variation with the ratio of the flow domain dimension to discontinuity spacing (data from Long and Witherspoon [14]).
I
I
I
4 6 8 10 Index = trace length/spacing
I
12
14
Fig. 5. Permeability variation with discontinuity index (data from Long and Witherspoon [14]).
for Fig. 5, lation seems the best way to obtain the permeability tensor; however, it is impractical and not necessary to carry out the simulation for each project, so we try to find an alternative by studying how the permeability varies with the fracture length and spacing based on the numerical results of Samaniego [13] and Long and Witherspoon [14]. Figures 5 and 6 show the permeability ratio varies with the discontinuity index (Id). The following hyperbolic function was fitted to the data:
(8)
k I~I~ ~ = 3 + (Id I~)
k
Ia1
[d1
k0
3+(Io1)
2+Id
for Fig. 6, k k0
Id  2.4 3+(I d2.4)
Ia  2.4 0.6+Id
where Io is the discontinuity index of percolation threshold. For Long and Witherspoon's networks, the I¢ is smaller than that of Samaniego. This is understandable as the fracture length of the former is constant while that of the latter follows a negative exponential distribution. From ideal fracture patterns, Section 2 has shown that the I¢ is lx/3; while, from the numerical
k 0.8 ko
k
•
1.0"~
"~ 0.4 I /
._>, "~ 0.6
i'~ui nmiil_l_ll_m_a
0.2
I /
o.o '~ I 0 10
I 20
I I I I I I 30 40 50 60 70 80 Flow domain dimension/spacing
I 90 100
Fig. 4. Permeability variation with the ratio of the flow domain dimension to discontinuity spacing (data from Samaniego [13]).
o.oI,~,,','~'/ 0 2
/.\ Numerical /
I I I I 4 6 8 10 Index = trace length/spacing
I 12
14
Fig. 6. Permeability variation with discontinuity index (data from Samaniego [13]).
WEI et al.:
R O C K MASS P E R M E A B I L I T Y P R E D I C T I O N S
257
for the maximum total discontinuity frequency.
X ko 1.0,
2 m= x/a 2 + b 2 + c 2
._o 0.8
N
a = ~ 2p sin ctpcos P=I
~ 0.6. e,J
N
o
b = ~ ,~ cos ap cos/~
~ 0.4. II)
P=I
0.2
N
c = ~ ,,],psin tip
0
. 0
5
0 10
. ~ 15 20 25 30 Index = trace length/spacing
35
40
Fig. 7. Permeabilityvariation with discontinuityindex. results in Figs 5 and 6, Ic is 13. As shown in Fig. 7, the choice of any value between 1 and 3 does not make much difference to predicted permeabilities since Id generally lies between 5 and 15. In Samaniego's networks, there may be too many shorter fractures. The three dimensional effects will make the Ic smaller than that predicted from two dimensions. Consequently, it can be assumed that the Ic = 2 and we have the following permeability correction function for practical application: k k0
Id2 1+I d
L Id = 2t L = St"
(1 O)
P=l
(9)
Section 3 has shown that the REV dimension is 1050 times the total fracture spacing so that, if the fracture length is larger than 10 times the spacing, the permeability is approaching that for infinite fractures. Figure 7 shows that the permeability is more than 70% of that for infinite fractures if the fracture length is larger than 10 times the total spacing. This proves that the proposed permeability correction function is correct. The success of the permeability correction function results from the successful choice of the discontinuity index Id, which was proposed based on the simple fracture patterns in Section 2 and the conclusions of numerical studies of Samaniego [13] and Long and Witherspoon [14]: the fracture length has a stronger influence on permeability than the fracture density; networks with shorter fractures and higher density will have lower permeability than those with longer fractures and lower density. Id is the product of fracture length and frequency, while the latter is a measure of the product of fracture density size. Hudson and Priest [23] gave the following formulae
where 2p, ap and tip respectively are the discontinuity frequency in the normal direction, the trend and plunge of the p th set of fractures. Although equation (9) was obtained from random fractures, it may be applied to anisotropic finite fractures to correct the following anisotropic permeability tensor obtained from infinite fracture assumption:
k ~ j = l ~ 2pe3(f~j_n,nj) or in term of hydraulic conductivity as:
Kij = g k~j = ? where ep, n~ and nj are the hydraulic aperture and directional cosines of the p th set of fractures, 6ij the Kronecker delta, g the gravity acceleration, v and # the kinematic and dynamic fluid viscosities, ? the specific weigh, V = pv and p is the density. There are three ways to calculate an equivalent isotropic permeability k0. The first is to take the average of principal anisotropic permeabilities of equation (11):
kll + k22 +
k0 =
k33
3
(m)
Mean spacing
(m)
Normal trend Normal plunge
1 e3
k0 = ~ 2a0e3 = 6Sa0"
~z r2e 3
k0

30 $30
Set 3
Set 4
2.16
0.83
1.51
1.03
1.38
0.51
0.65
0.36 262.8 0.6
(14)
where Sa0 is the average (not total) discontinuity spacing of different sets since it was assumed that the disc number per unit volume equals 230 . e and r are the hydraulic aperture and disc radius. Robertson [30] gave
Set 2
0.93
(13)
The last is to assume disctype fractures (Gueguen and Dienes [22]):
Set 1
216.3 20.8
(12)
The second is to assume three orthogonal identical sets of joints:
Table 2. Fracture geometric data at Stripa Site, Sweden
Mean trace length
(11)
P=I
0.79 149.4 62.5
294.2 62.9
Average

258
WEI et al.: ROCK MASS PERMEABILITYPREDICTIONS
the following relation between the trace length (L) and radius: 2 r =L. 7z
(15)
As an example of application, we try to predict the permeability of Stripa site, Sweden. The geometric data of fractures are given in Table 2 (Rouleau and Gale [31]; Bursey et al. [32]). Gale et al. [33] obtained a mean hydraulic aperture of 8.3 pm. First equations (10) and (11) are used to calculate the maximum discontinuity frequency and the permeability tensor of infinite fractures. Axes 1, 2 and 3, respectively, are in the east, north and downwards vertical. Then, equation (9) is applied to calculate the discontinuity index Id (=7.2) and the permeability correction factor (=0.64). After the permeability tensor of infinite fractures is multiplied by the factor, we have the following predicted permeability tensor for the Stripa site:
1.09  0 1 6 kij=
0.16 0.21
1.87 0.12
0.21\ 0.12~ x 1016(m2). 1.33]
In the final report of the Stripa Site Characterization and Validation, Olsson [25] gave the permeability tensor for the HZone as: 0.7 k~= 0.2 \0.2
0.2 2.7 0.3
0.2 ) 0.3 X 1016(m2) 2.2
where axes 1, 2 and 3 respectively are in the east, north and upwards vertical. It is clear that the two permeability tensors above are in the same order. With equations (12)(15) and the same correction procedure as above, the equivalent isotropic permeability was obtained. For the assumption of average principal permeability,
5. P E R M E A B I L I T Y
WITH DEPTH
Stresses increase with depth while the discontinuity frequency, aperture and permeability decrease with depth. From 5532 injection tests carried out at dam sites, Snow [36] found that the aperture decrease is more responsible for the permeability with depth so that the hyperbolic function of rock joint normal behaviour can be applied to describe the variations of the hydraulic aperture and permeability with depth (Z) as below (Wei and Hudson [29]). e ei

Z
1
k~
(16)
A + BZ
1
A +)~Z
(17)
1 3 ki = g2a0ei
(18)
where e i and k i a r e the initial hydraulic aperture and permeability at the ground surface or at zero normal stress on rock joints, and A and B are constants. Snow's in situ test results include eight rock types. He found that the variations of hydraulic aperture and permeability follow the same trends regardless of rock types, which means that the two constants A and B in equations (16) and (17) can be fixed and applied to any rock masses to predict the hydraulic aperture and permeability variations with depth. The fitted curves in Figs 8 and 9 have the fixed forms of equations (16) and (17) as: e ei
Z 58.0 + 1.02Z
(19) (20)
  =
k~
1
58.0 ~ 1.02Z
where depth Z is in metres. For Figs 8 and 9, ei 120 #m and ki = 2 x 1013 m:. As a comparison, we discuss the initial hydraulic aperture used by Barton et al. [37]. Their e~ is 100 #m. The Stripa test site is 385 m below the ground surface =
k = kl q k2 F k 3  1.43 × 1016(m2). 3 From the three orthogonal sets model,
VARIATION
Hydraulic aperture (p.m)
O,
0
1O0 200 300 • ,=l :;',.;;V I 01 1
qq ~
,o
• •
•
•
k = 0.93 x 10 16(m2). From the disctype model, k = 1.07 x 1016(m2). The ouflow and injection tests carried out in the Buffer Mass Test Drift gave a permeability, k = 1.41 x 1016 m 2 (Gale and Rouleau [34], Herbert et al. [35]). The three predictions are also in the same order as that from in situ tests, so that they are all applicable. Finally, it must be pointed out that counting too many very small fractures and measuring their orientations may lead to unrealistic discontinuity frequency and set orientation and calculated permeability.
g 60
r "
OT'!iI •
,°°l. I. 1201 t,
140/ Fig. 8. Hydraulic aperture variation with depth for eight rocks (data
from Snow [36]).
WEI et al.:
R O C K MASS P E R M E A B I L I T Y P R E D I C T I O N S
Permeability ( m 2) 1015 1013 1011 109 0
i f
For the empirical estimation of the initial hydraulic aperture, Barton et al. [40] gave
~,,, "~.'ST"
2o i+ "
e i  JRC2.5
¢o~ • •
40
g 60. t
a
!
80
259
p
,
where e~ and E0 (mechanical aperture) are in microns. Wei et al. [41] gave the following alternative method. ei k
!o
I Vm
and Vm is determined according to the method of Barton et al. [40]:
100"
Vm = A + B ( J R C ) +
cFJCS1 v [_ E 0 J
(mm)
120'
JRC / E0=T~0.2
140 Fig. 9. Permeability variation with depth for eight rocks (data from Bianchi and Snow [38]).
(Olsson [25]), with e i = 100 #m, the hydraulic aperture at this depth is calculated as 14.6 # m from equation (19), which is about two times that (8.3/am) given in the last section so they had a very good estimation in the light of the difficulties in obtaining a realistic hydraulic aperture. For different rock types, it remains to determine the initial hydraulic aperture and permeability (ei and ki). Both in situ tests and empirical methods can be used but for important projects the in situ tests should be carried out at a certain depth and calculate k~ from equation (20) or ei and ki using equations (18) and (19). Attention must be paid to what Snow [36] found: although they are important for the very near surface structures such as dam foundation, the e~ and k~ measured at the ground surface are either unrealistic or too large to be used to interpret or predict the hydraulic aperture and permeability below the ground surface. Permeability ( m 2) 1019 10 18 1017 10161015 10141013 .,ilil
I/~1
500.
g
1000 
t
~ 1500a
f
2000 
J
2500 
B
~Predicted
!
Measured
3000 Fig. 10. Permeability variation with depth at Stripa, Sweden (data from Carlsson and Olsson [39]).
j ac~ _ 0.1
A = 0.2960 C = 2.241
)
(ram)
B = 0.0056
D = 0.2450.
Here E0 is in mm. The parameters, JRC and JCS, are determined on the laboratory sample scales, kl is a constant with a range of 0.41.1. Hardin et al.'s [42] 8 m 2 in situ block test has a kl of about 0.56, Makurat et al.'s [43] 1 x 1 × 2 m in situ block test has a kl of about 0.7 and their two laboratory tests have kl values of 0.48 and 0.57, respectively. These tests were well carried out and the results are reliable. Since the rock joint type and roughness have already been taken into account in the maximum joint closure (Vm), a k~ value of 0.60.7 may be good enough for practical application. As an example of application, we consider the permeability variation with depth at Stripa, Sweden, as shown in Fig. 10 (Carlsson and Olsson [39]). For prediction, the ki in equation (20) is set to 1014m 2. It can be seen that the prediction matches the borehole data very well. If the in situ measured hydraulic aperture of 8.3 p m at the test site depth of 385 m is used to calculate the e~ ( = 56.94/am) from equation (19), equation (18) will give k+ = 4.73 x 10 ~+m 2. If the in situ measured permeability of 1.41 × 10 16 m 2 and the test site depth of 385 m are used, equation (20) will lead to ki = 4.55 x 10 14 m 2. The two calculated initial permeabilities are all in the same order a s 10 14 m 2. This shows that the proposed method for p, ediction of permeability variation with depth is applicable. However, it must be noticed that the data in Fig. 10 are the average permeabilities measured in a large number of in situ tests at different depths and that the scattering near the ground surface is between 1018 and 10 13 m 2 (Herbert et al. [35]).
6. D I S C U S S I O N
AND CONCLUSIONS
We have discussed when a jointed rock mass is permeable and how to determine its permeability. A discontinuity index was introduced, which is the product of the discontinuity frequency and length and mades the
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analysis quantified. T h e m a i n p o i n t s are s u m m a r i z e d as follows. (1) T h e p e r c o l a t i o n t h r e s h o l d o f j o i n t e d r o c k masses is two, i.e. it is p e r m e a b l e when the a v e r a g e fracture length is larger t h a n two times the total spacing (reciprocal o f the t o t a l d i s c o n t i n u i t y frequency) o r larger t h a n the average spacing o f different sets. (2) T h e R E V d i m e n s i o n is 1050 times the t o t a l spacing for r a n d o m fractures a n d 1020 times the average spacing for n o n  r a n d o m fractures (practical application). In o r d e r to o b t a i n an equivalent perm e a b i l i t y b y large scale in s i t u tests, the test section s h o u l d intersect 10 fractures. I f the smallest d i m e n sion o f a n u n d e r g r o u n d structure intersects 5  1 0 fractures, a c o n t i n u u m t h e o r y can be a d o p t e d for p e r m e a b i l i t y analysis. (3) T h e p e r m e a b i l i t y o f r o c k masses with finite fractures can be p r e d i c t e d f r o m t h a t o f infinite fractures c o r r e c t e d by a h y p e r b o l i c function. Since the fracture length generally is 5  1 5 times the total spacing, the p e r m e a b i l i t y r e d u c t i o n b y c o r r e c t i o n is a b o u t 2050%. (4) T h e v a r i a t i o n s o f the h y d r a u l i c a p e r t u r e a n d perm e a b i l i t y with d e p t h generally follow a fixed h y p e r bolic function. T h e initial h y d r a u l i c a p e r t u r e a n d p e r m e a b i l i t y can be e s t i m a t e d o r d e t e r m i n e d by in s i t u tests at a certain depth. Since the initial hyd r a u l i c a p e r t u r e a n d p e r m e a b i l i t y m e a s u r e d at the g r o u n d surface are unrealistic o r t o o large to be used, in s i t u tests s h o u l d be carried o u t for i m p o r t a n t projects. (5) It is n o t necessary to use the p e r m e a b i l i t y c o r r e c t i o n for o r d i n a r y projects o r if the fracture is longer t h a n 15 times the t o t a l spacing because (a) the perm e a b i l i t y c o r r e c t i o n requires fracture length d a t a , which are n o t easily o b t a i n e d ; (b) the p e r m e a b i l i t y scattering is generally at least two o r d e r s while the c o r r e c t e d p e r m e a b i l i t y r e m a i n s o f the same o r d e r o f m a g n i t u d e as the uncorrected; (c) the p e r m e a b i l i t y c o r r e c t i o n can be estimated, which is 20500/0 • o f the p e r m e a b i l i t y o f infinite fractures. (6) A l t h o u g h there are three m e t h o d s to calculate the a v e r a g e p e r m e a b i l i t y , the three o r t h o g o n a l identical set m o d e l is preferred because it does n o t need the d a t a for fracture o r i e n t a t i o n a n d length a n d it only needs the h y d r a u l i c a p e r t u r e a n d the average spacing o f different sets. T h e effects o f fracture length is p a r t l y t a k e n into a c c o u n t b y the d i s c o n t i n u i t y frequency. T h e h y d r a u l i c a p e r t u r e can be e s t i m a t e d o r d e t e r m i n e d by in s i t u tests. T h e average spacing can be e s t i m a t e d f r o m x / 3 times the total d i s c o n t i n u i t y frequency o b t a i n e d b y simply c o u n t i n g the fractures a l o n g a straight line o r b o r e h o l e , which should be generally in three o r t h o g o n a l directions. Accepted for publication 1 October 1994.
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