Engineering Geology, 33 (1992) 11-30 l 1 Elsevier Science Publishers B.V., Amsterdam

Evaluation of connectivity characteristics of naturally jointed rock masses

Xing Zhang a, Richard M. Harkness a and Nigel C. Last b

"Department of Civil Engineering, The University of Southampton, Southampton, UK bGeomechanics Section, Sunbury Research Centre, B.P. Int. Ltd, Sunbury, UK

(Received February 28, 1992; revised version accepted June 3, 1992)

ABSTRACT

Zhang, X., Harkness, R.M. and Last, N.C., 1992. Evaluation of connectivity characteristics of naturally jointed rock masses. Eng. Geol., 33: 11-30.

A method is presented to predict the connectivity characteristics of two-dimensional joint patterns in jointed rock masses. The statistical results obtained from in-situ data show that the average joint length, L, joint density, D, and an intersection parameter of joint sets, 0, within a joint pattern, each affects the connectivity ratio, C, of the joint pattern. In addition, there is evidence that a connectivity index, no, incorporating the three joint parameters L, D and 0, can be used to characterise their combined effect on the connectivity ratio. Some additional geometric parameters, termed the k-order connectivity ratios and the maximum network extent, can be used to define more accurately the connectivity characteristics of joint patterns. Alterna- tively, a parameter based upon joint density and joint length can be employed to quantify connectivity. Furthermore, simulated joint patterns are used to study the connectivity ratio of joints so that the effect on large domains of jointed rock can be investigated. There is a significant effect of the sampling area on the connectivity ratio, where the sample dimension is rather small relative to the mean joint length in the joint pattern. Most importantly, several connectivity measures are compared to calculated fluid flow through simulated jointed rock masses.

Introduction

Rock discont inui t ies (hereaf ter referred to as jo in ts) have a significant effect on the proper t ies

o f rock structures. I t is i m p o r t a n t to different iate

between the character is t ics o f ind iv idua l jo in t s in

a rock mass and those o f a jo in t pa t t e rn or a jo in t system. This d i f ferent ia t ion is necessary, because

the s tabi l i ty and permeabi l i ty o f a rock s t ructure

are not only affected by the proper t ies o f the indiv idual jo in ts , but also, and perhaps domi-

nant ly, by the geometr ic character is t ics o f the jo in t system. In par t icu la r , the connect iv i ty proper t ies

o f jo in t systems p lay a d o m i n a n t role in the effective pe rmeab i l i ty o f some rock structures.

Measu remen t and pred ic t ion o f the connect iv i ty o f na tu ra l jo in t pa t t e rns is, therefore, cons idered

Correspondence to: Xing Zhang, Department of Civil Engineer- ing, The University of Southampton, SO9 5NH, UK.

to be par t i cu la r ly impor t an t for some rock engi- neering designs. Examples include the u n d e r g r o u n d s torage o f nuclear or toxic waste in jo in ted rock masses, and the p roduc t ion o f hydro- ca rbons f rom na tura l ly f rac tured reservoirs. In these activities, the mis judgment o f permeabi l i ty or s tabi l i ty can lead to cons iderab le financial loss or social risk.

Dur ing the pas t two decades much effort has been devoted to s tudying the size, o r ien ta t ion and densi ty o f jo in ts in rock masses. F o r the or ien ta t ion o f joints , var ious me thods have been deve loped to measure and record str ike and dip, pole di rect ion cosines, az imuth and d ip (Terzaghi, 1965; Rober t - son, 1970; Hergi t , 1978; M a h t a b and Yegulalp, 1982). M u c h more effort has been devoted to es tabl ishing a me thod for de te rmin ing unbiased jo in t size (trace length, areas, radii) and expla ining how any bias can be overcome (Baecher and Lanney, 1978; Beyer and Rolofs , 1981; Priest and

0013 7952/92/$05.00 (~3 1992- Elsevier Science Publishers B.V. All rights reserved.

12 XING Z H A N G ET AL.

Hudson, 1981; Kulatilake and Wu, 1984; Panek, 1985). The density of joints is usually described with joint spacing, joint numbers per unit of rock volume, or numbers per unit of outcrop area (Priest and Hudson, 1976; Beyer, 1982; Priest and Samaniego, 1983; Grossmann, 1988).

However, the study of all individual joints in a rock mass, as well as joint patterns and the collec- tive effects on bulk rock properties represent a relatively recent development in the characteriza- tion of jointed rock masses. For example, Ders- howitz and Einstein (1988) summarized and emphasized the importance of joint system geome- try and its rock mechanics implications, and Zhang (1989) proposed a two-dimensional mechanical model of an en-echelon jointed rock mass, which highlights the effect of geometry on bulk strength.

Among the possible geometric parameters for joint patterns, it seems likely that connectivity is one of the most important. However, the predic- tion of connectivity remains an elusive goal. Rou- leau and Gale (1987) proposed a composite index that describes the inter-connectivity of a fracture set. La Pointe (1988) stated that his proposed block density formulation may provide an index of fracture permeability, but this has not been tested.

In this paper, the connectivity ratio of a joint pattern, C, is defined, and a method for calculating C is presented. A set of additional indices, called k-order connectivity ratios, is used to describe the connectivity characteristics of a joint pattern in greater detail. Furthermore, another parameter, termed the maximum network extent of a joint pattern, is used to indicate the maximum connected area of a joint network of a joint pattern within a rock region. In addition, based on the field data recorded on the outcrops of the slope benches of Ekou Open Pit Mine (P.R. China), the effects of average joint length, L, joint density, D, and an intersection parameter of joint sets 0 on the connec- tivity ratio are discussed. In order to show the common effect of these three separate parameters, a connectivity index, no, is developed from them, and its correlation with C is discussed. By using a random simulation approach, the connectivity ratio of large sizes of joint patterns is calculated, so that the effects of sample size on the connectivity

ratio can be studied. Then, a theoretical demon- stration of intersections and connectivity for two joint sets is presented. Finally, an example is given to illustrate how to apply the connectivity ratio to rock engineering problems, and the effect of con- nectivity ratio on the permeability of an excavation in jointed rock is discussed. The results of some numerical simulations support the conclusions.

Two-dimensional connectivity characteristics of joint patterns in rock

Consider an outcrop in jointed rock where a two-dimensional joint pattern, consisting of indivi- dual joints and joint networks can be observed. For example, in the joint pattern of Fig. 1, there are two individual joints and a joint network consisting of several joints. The joint pattern can be considered to consist of nodes and branches. A node is an intersection point between two or more joints. For branches, there are two main classes, connected and non-connected. An individual joint in a joint pattern is called a non-connected branch, whereas a segment of a joint attached to a node, or between two nodes, is called a connected branch. Following this definition, Fig. 1 shows two nodes indicated by the encircled numbers 1 and 2, and eight branches indicated with 1-8. Branches 1 and 2 are non-connected and all the others are con- nected.

In natural joint patterns, it is sometimes difficult

I 1

O

Fig. 1. Nodes and branches in a joint pattern.

CONNECTIVITY CHARACTERISTICS OF NATURALLY JOINTED ROCKS 13

to distinguish between a through joint and, say, two intersecting joints, especially if three or more branches connect together at a node. For example, in Fig. 2, it is difficult to determine if branch 1 and branch 2, or if branch 1 and branch 3 belong to the same through joint or, indeed, they are all separate joints. Therefore, it is convenient to indi- cate the connectivity of a joint pattern by charac- terising the branches rather than the joints. The connectivity ratio, C, of a joint pattern is defined as:

c = Bc/(B° + Bo) (I)

where B c is the total number of connected branches

2

1

Fig. 2. Representation of three branches connected at a node.

and Bo is the total number of non-connected branches in the joint pattern.

It is evident that when Bc is zero, the connectivity ratio of the joint pattern will be zero. This extreme case indicates that the joint pattern has no connec- tivity, because all the joints of the joint pattern are separate. By contrast, in the opposite extreme case, where C is unity, all joints will be connected together. For all other cases, C lies between zero and unity. Thus the connectivity ratio, C, of a joint pattern can be determined from the number of connected and non-connected branches.

Unfortunately, C is not unique as different joint patterns may lead to the same connectivity ratio. In Figs. 3(a) and 3(b), for example, the two joint patterns have the same connectivity ratio, but their connectivity characteristics are rather different: in Fig. 3(a), there are separate joint networks, each of which consists of one node with four branches, whilst in Fig. 3(b), all the branches connect together in a single network.

Therefore, it is necessary to determine a set of additional ratios, called k-order connectivity ratios, C k (k= 0, 1, 2, 3 ..... s), to more fully characterise the connectivity of a joint pattern. Here, a joint network with one node is called a first-order connected joint network because if the node is removed from the joint network, all the connected branches will become non-connected, as shown in Fig. 4. Thus a joint network with k nodes is called a k-order connected network. Usually there are

©

4

ii

2 @

13 u l l

3

I Y 6 I

0

14 f

12 8

7

9

1 3 I

i0

2

6

ii

:0

13

O 7

12

5 U--

14

Fig. 3. Difference of two joint patterns with the same connectivity ratio. (a) Three separate joint networks; (b) entire joint network.

14 XING ZHANG ET AL,

• 4

Fig. 4. Representation of one-order connected joint network.

some non-connected branches and several con- nected networks of different order in a joint pattern. I f the number of branches of a kth-order network in a joint pattern is denoted by Bk, the kth-order connectivity ratio of the joint pattern, Ck, can be calculated from:

Ck = EBk/(Bo + Be) (2)

where EBR is the sum of the connected branches of all kth-order connected networks in a joint pattern; 5:Bo= Bo. The highest-order connectivity ratio, C s, is given by:

cs= Y Bs/(ao + Bo) (3)

Similarly, there may be more than one connected network with the highest order s, so EBs is the number of the connected branches of all networks with the highest order. Thus, the connectivity characteristics of a joint pattern can be more accurately described by the connectivity ratio C together with the k-order connectivity ratios Ck.

From Eq. 1 and 2, C can be expressed as:

C = Z Ck (4) k = 0

One further geometric parameter is proposed. It is clear that the permeability of jointed rock masses will be dominated by those joints which provide the main conduits of fluid flow. In particular, the larger, connected, joint networks play a key role. An additional parameter, termed the joint network extent of a joint pattern, Ce, is introduced in an attempt to quantify this geometric characteristic.

C e = (Bs)m.x/(B o + Be) (5)

where , (Bs)ma x is the number of connected branches

in a single joint network of order s having the highest number of connected branches.

Ce is different from Cs when there are two or more joint networks with the highest-order connec- tivity in a given joint pattern; where there is a single joint network with the highest-order connec- tivity, Co and Cs are equivalent. For example, in Fig. 3(a), Co = 0.286, but C~ = 0.857. This is because there are three joint networks with the highest- order connectivity. In contrast, in Fig. 3(b), Ce = Cs---0.857, because there is only one joint network with the highest-order connectivity in the joint pattern. Thus, there is a basic constraint on Ce:

Ce ~< C~ (6)

Figure 5 shows a joint pattern recorded at an outcrop on the slope bench in the Ekou Open Pit

Fig. 5. Quartz-mica schist joint pattern with a sample area of 2 x 2 m. [] indicates joint network 1; A indicates joint network 2; G indicates joint network 3.

CONNECTIVITY CHARACTERISTICS OF NATURALLY JOINTED ROCKS 15

Mine, Shanxi Province (P.R. China). The connec- tivity ratio C and k-order connectivity ratios Ck of the joint pattern are shown in Table I. In the joint pattern, there are twelve non-connected branches and three joint networks whose orders are 1, 7 and 13, and which contain 4, 18 and 34 connected branches, respectively. Moreover, Cc = Cs = 0.5.

Effects of joint pattern parameters on connectivity

The volume of rock affected by an engineering project is usually much larger than that seen as local outcrops or accessible exposures of the rock

TABLE l

Connectivity ratio C and k-order connectivity ratio Ck of a joint pattern

Bo B1 Bv B,3 C C1 Cv Cla

12 4 18 34 0.824 0.059 0.265 0.5

mass. It is almost impossible to obtain joint data over the whole engineering region. However, it is possible to describe the statistical characteristics of connectivity within an engineering project region, by means of the information gained from several outcrops and/or exposures of the rock mass.

The senior author recorded the joint characteris- tics of 24 sample areas (each measuring 2 × 2 m) on the quartz-mica schist in Ekou Open Pit Mine (China). From the statistical results of these data, the average connectivity ratio C and the k-order connectivity ratios Ck of the joint patterns were calculated and are shown in Fig. 6. Here L is the average joint length (here, 'joint' is the conven- tional long, straight feature and may comprise several co-linear branches) and D is the joint density defined as:

D : (YL)/A (7)

where EL is the total length of joints in a sampling area A.

C k

0.06

0.05

0.04

0.03

0.02

0.01

C = 0.735

D = 2.45 m/m 2

L = 0.94 m

1 2 3 4 5 6 7 9 12

Fig. 6. Statistical results from k-order connectivity ratio of 24 joint patterns; each sample area is 2 x 2 m.

15

Order

(K)

16 XING Z H A N G ET AL.

The connectivity ratio of a joint pattern C is dependent not only on joint length and joint density, but also on the relative orientation between joint sets. To quantify this effect, an intersection parameter, 0, of two joint sets within a joint pattern is introduced. Consider a joint pattern containing n joints, and assume that two joint sets can be identified, each containing a number of sub-parallel joints, with nA in one set and na in the other set. Clearly an average angle, fl, between the two sets can be estimated, and this angle will characterise the relative orientation of one set with respect to the other. The probability that joints of the two sets intersect may be assumed to be directly proportional to the sine of angle fl when nA and nB are given. The probability is also directly proportional to the product of nA and nR when fl is given. An intersection parameter, 0, of two joint sets can be defined as:

0 = sin fl nAnB/n 2 (8)

where nA/n and nB/n are the relative proportion of the numbers of joints in the two sets.

From the data of the 24 sample areas, the regression results of the three parameters L, D and

0 against the connectivity ratios of the joint pat- terns are shown in Figs. 7, 8 and 9 respectively. All the statistical results show that there is a tendency for the connectivity ratio of these joint patterns to increase with increasing 0, L or D, but that each one of them is poorly correlated with the connectivity ratio. This is not surprising per- haps, because, intuitively, the likelihood of inter- sections will depend on a combination of relative orientation, density and length of joints within the system. For this reason, a connectivity index, no, consisting of the product of these three parameters, L, D and 0, is proposed as a first measure of connectivity for fracture patterns.

Hence:

no = OLD (9)

Figure 10 indicates the relation between this connectivity index and the connectivity ratio. The results show that the connectivity ratio increases with connectivity index and that there is a reason- ably good correlation. All the data used in the regression analysis are shown in Table 2. Figure 10 also includes results for simulated joint patterns which are discussed in the next section.

0.8

o

0.6

o.4 o

0.2

1.0 0 '

0

0 o D O ~

o 11 o

0 o

o o [ o

o

C = 0.17 + 0.55 L I' = 0.62

o l J .5 .7 .9 I.I 1.3

Mean length of joints L (m)

o

Fig. 7. Relation between average joint length, L, and connectivity ratio, C, for 24 joint patterns.

O

1.5

CONNECTIVITY CHARACTERISTICS OF N A T U R A L L Y JOINTED ROCKS 17

C 1.0

0 .8

.o

0.6

.4

c 0 . 4 e~

/ J

/

0 f~

0

f

iO 0

i o

o

C 0 (

o

/ o

I o.2 I I I

-- C = 0.18 + 0.22 D T = 0.5

0 I I I I I

1 1.6 2.2 2.8 3.4

Joint density D (m/m 2)

Fig. 8. Relation between average joint density, D, and connectivity ratio, C, for 24 joint patterns.

1.0 I I il O v O j o!~

- i / / I o o ~ o >'0.6 ( J

• / ~ o.~

/ I o 0 Oi o

0.2 . /

o

o 0.05 o.I o.15 0.2 0.25

Intersection parameter of joint set

Fig. 9. Relation between intersection parameter, 0, for joint sets and connectivity ratio, C, for 24 joint patterns.

Size effect of sample area on connectivity ratio

The size of sample area may have some effect on estimating the connectivity ratio because the average length of joints may increase as the sam-

piing area is increased (Priest and Hudson, 1981; Zhang, 1990). This may result in a change of connectivity ratio. To examine this, it is necessary to obtain data for larger sample sizes than our current field measurements provide. However, to

18 XING Z H A N G El" AL.

C 1

~ .8

o

.u

.6 b

.4

A A

f A f I

o I

f /

i i ~I/° A

f ° ° 0

/o- o.o>

J ° &

0 0 ~ , ~._...~- , -- ]

A

-- O O Natural joint patterns --

C = 0.99 + 0.37 In n

? = 0.82 / I

A------g~ Simulated joint patterns

C = 0.941 + 0.215 Inno

i/ o o

0 i t I I I I 0 .6 .8 1.0 .2 .4 1.2

Connectivity Index (empirical parameter)

1.4

n o

Fig. 10. Relation between connectivity index n o and connectivity ratio, C, for 24 natural joint patterns and 30 simulated joint patterns.

alleviate this difficulty, joint patterns can be simu- lated. Figure 11 shows a simulated joint pattern developed from the computer programme UDEC*. In this joint pattern, there are two joint sets: set A which has a mean dip of 60 °, a standard deviation of 30 °, a mean joint length 30 m and a joint density of 0.2 m/m2; set B with a mean dip of - 50 °, a standard deviation of 30 °, mean joint length of 40 m and a joint density of 0.2 m / m 2. For this joint pattern, the connectivity ratio is 0.983.

In the original UDEC code, joint length, joint spacing and joint dip are each uniformly distrib-

*UDEC (The Universal Distinct Element Code used for the numerical simulations) was written by P.A. Cundall and is marketed by ITASCA Consulting Group, Inc. of Minneapolis, Minnesota, USA. The source code has been made available to the authors by BP Research and has been modified so that appropriate randomisation of the three parameters L, D and 0 can be specified in the simulations.

uted, and joints are generated line by line. Joint density is thus determined by the joint mean spacing. However, the joint data of the 24 field samples used in this paper do not support uniform distributions for all parameters. Four probability- density functions (exponential, normal, uniform and log-normal) were tested against the measured distributions. The results of Z2 statistical tests (used to test the goodness-of-fit) show that joint lengths were satisfied best by the exponential model and joint orientation by the normal model. Hence, the original UDEC code was modified to generate joint lengths with an exponential distribution, joint dips with a normal distribution, and mid-point coordinates of joints with independent uniform distributions in the x- and y-directions.

In order to examine the similarity of simulated and natural joint patterns, the connectivity ratios of joint patterns which have been simulated using

CONNECTIVITY CHARACTERISTICS OF NATURALLY JOINTED ROCKS

TABLE 2

Data of joint parameters and statistic results of connectivity ratio

C L D 0 n o

I 0.255 0.525 1.400 0.218 0.160 2 0.283 0.780 2.320 0.12 0.217

3 0.289 0.520 2.500 0.19 0.247 4 0.318 1.22 2.45 0.15 0.448

5 0.556 0.780 2.85 0.15 0.333 6 0.604 1.090 1.95 0.162 0.344

7 0.639 1.030 2.08 0.149 0.319 8 0.688 0.78 3.05 0.08 0.191 9 0.734 0.916 2.57 0.245 0.577

10 0.776 1.44 2.46 0.223 0.788 I 1 0.803 0.913 2.00 0.235 0.429

12 0.830 1.175 2.32 0.205 0.559 13 0.850 1.128 2.550 0.203 0.58

14 0.861 1.040 3.380 0.193 0.678 15 0.871 1.254 1.990 0.206 0.514 16 0.891 1.173 2.430 0.215 0.613

17 0.913 1.430 2.870 0.231 0.948

18 0.923 1.125 3.720 0.222 0.93 19 0.938 1.29 3.520 0.216 0.978

20 0.950 1.210 2.450 0.248 0.735 21 0.969 1.32 3.31 0.245 1.07

22 0.975 1.28 3.1 0.23 0.911 23 0.990 0.810 2.89 0.24 0.562 24 0.995 1.2 2.540 0.24 0.706

the geometric parameters derived from natural joint patterns have been calculated and are plotted in Fig. 10. For both natural and simulated joint patterns, the range of mean joint length is 0.5 to 1.5 m, the range of joint density is 1 to 4 m / m e, and the range of the intersection parameters is zero to 0.25. The comparison clearly shows that the connectivity ratio of the simulated joint pat- terns is quite close to that of the corresponding natural joint patterns. This provides an approach for calculating the connectivity ratios of large joint patterns.

Figures 12, 13 and Table 3 show the effect on the connectivity ratio of different sizes of joint patterns with different mean joint lengths and different joint densities, respectively.

Clearly, sample areas which are rather small in relation to the mean length of joints have a major effect on the connectivity ratio. The connectivity ratio is a reliable indicator of joint geometry only to the extent that the data are statistically signifi-

19

cant. This means that joint data should be recorded in a relatively large sample area: the results in Figs. 12 and 13 indicate that where the side length of the sample area is about ten times the mean joint length, there is little effect on the connectivity ratio.

An alternative characterisation of connectivity

A more rigorous approach to characterising the connectivity of joint patterns is now developed.

Consider a representative area in the form of a parallelogram whose sides are of length L A and LB, parallel, respectively, with the directions of joint sets A and B (the definition of L g and LB is discussed later). Thus the acute angle, fl, of the parallelogram is the angle of intersection of the joint sets, as shown in Fig. 14. The probability that a joint exists at an arbitrary point along a joint line is just the length of the joint (or the sum of such lengths if more than one joint occurs along the line) divided by the total length of the line. For joint line 'i' of set A let this be Pi and correspondingly, pj for joint line, j, of set B. Then the probability of an intersection of joints on the two joint lines is just p~o~ or Pii. For all the joint lines in the parallelogram the probable total number of intersections become EiZplj.

The total length of joints in joint set A is ~,i(piLA), SO that the density of joints in set A is this total length divided by the area of the paral- lelogram or:

DA = Z~(P~LA)/(LALBsinfl) = Z~pi/(LRsin fl) (10)

Thus, using the same argument for the B set, we have:

DADa = EiE~pij /(LgLasin 2 fl) (! 1)

or, re-arranging, the total probable number of intersections per unit area, n, is given by:

n = DADRsin fl (l 2)

Now consider the concept of connectivity. Clearly, this should relate to the ability to move through a pattern of joints via the joint intersec- tions and may be distinguished from the permeabil- ity of the joint system since the latter depends also on joint apertures, etc. The problem can be

20

i

] ' I 0. 000 0. 200

' I ' I ' 1 ' I 0 . 4 0 0 0 . 6 0 0 0 . 8 0 0 l . O 0 0

( ~ i O w w 2 )

XING ZHANG ET AL.

' ~. .oo(

_ O . 8 0 C

_ 0 . 6 0 0

_ 0 . 4 0 0

. 0 . 2 0 0

• 0 . 000

Fig. 11. Represen ta t ion of a s imula ted jo in t pat tern .

approached by assuming that a given pattern of joints (with the number of joint intersections deter- mined by the pattern) will have the same measure of connectivity, irrespective of the scale of the pattern. Since there is, in general, no such pattern in nature, we need to provide an artificial length scale and this we do by determining (somewhat arbitrarily) a characteristic length (some sort of average length which we discuss shortly) for each of the joint sets. Then, by requiring the lengths LA and LB of our representative parallelogram to be these characteristic lengths, we may deduce that the total number of intersections in the parallelo-

gram is independent of scale and thus a possible measure of connectivity. Letting this measure by denoted by nc we have:

nc = (DADBSin fl)(LnLBsin fl) (13)

As they are to be used in this connectivity context, the characteristic lengths ought, perhaps, to be chosen so that long joints (which have a greater chance of making connections) have more weight than short ones. One possible way to do this might be to weight each joint in proportion to its length. Thus the weighted number of joints of length Li in band i of set A would be ni(Li/LA)

C O N N E C T I V I T Y C H A R A C T E R I S T I C S O F N A T U R A L L Y J O I N T E D R O C K S 21

C 1

.8

.6

.4

.2

I I I I I I I I I I

o ~ - 4 , . . . A - 0 ~ 0 0 0 0 0 0 0 0 0 0

0 0 0

0 0

Mean joint length = 0.6m

O = 0.196

G ..... Density of Joints = 3 m/m 2

A ..... Density of joints = 2.5 m/m 2

iOO •

O ..... Density of joints = 2 m/m 2

1 I I I I I I I I I 4 8 12 16 20 24 28 32 36 40

0

Side length of s a m p l e areas

Fig. 12. Relationship connectivity ratio of different sizes of sample areas with different joint densities.

rm)

and the weighted total length for all joints in the set would be Ei(ni(LI/LA)LI). We would then divide this by the total weighted number of joints, E~n~(LJ LA), to get the characteristic length. Thus:

L A = (~,n,L,2/LA)/(~iniLi/LA) = (~,n,L,2)/(~,n,L,) ( 1 4 )

and similarly for set B. The above definition of characteristic length

relates easily to a commonly accepted function for the length distribution in numerical studies and it also offers, in field studies, the possibility of obtain- ing a characteristic length value even though a small proportion of the joint lines were found to pass right through the sample area and were thus of indeterminate length. The function referred to is the exponential relation giving the propor- tion of joint lengths in a given band 6(L/2) as e-L/~6(L/2) where 2 is a scaling length equal to the arithmatic mean length. The weighted charac-

teristic length as given above would then be:

[ ~ L 2 e - L/ z ,5 (L /2 ) ] / [~Le - L /~ f (L /2)] = 2 F ( 3 ) / F ( 2 )

= 2 2 F ( 2 ) / F ( 2 )

=2 ,~ (15)

where F( ) is the Gamma function. That is, the characteristic length would be twice that normally assumed for the mean length of the exponentially- distributed length function.

In order to demonstrate the validity of the parameter, n c, the connectivity ratios of 42 simu- lated joint patterns have been calculated. All the joint patterns have been developed for a square of 24 x 24 m, having a range of joint densities from 1 to 4 m/m 2 and an intersection parameter, 0, from 0.177 to 0.25 (corresponding to angles between the joint sets ranging from 45 ° to 90°). The sides of the sample area are thus at least ten times that of the mean joint length so, as demon-

22 XING ZHANG ET AL.

.= o

=

1 . o ~ I

0

.4 n

.2 m

0 "0~ ' , 0

0

I I I I

=_. o---=~=___~ =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

O O

I I 36 40

Density of joints = i m/m 2

0 = 0.196

D ..... Mean joint length = 2.4 m

..... Mean joint length = 1.2 m

O ..... Mean joint length = 0.6 m

I I I I I I I i 4 8 12 16 20 24 28 32

Side length of sample areas

Fig. 13. Re l a t i onsh ip connec t iv i t y ra t io o f dif ferent sizes o f s amp le a reas wi th different m e a n j o in t s lengths.

(m)

strated earlier, the connectivity ratio of these joint patterns is independent of sample size.

The relationships of the connectivity ratio with no, and with the empirical parameter, no, are shown in Fig. 15 and Table 4. It is evident that both parameters, nc and no, can be an indirect measure of the joint connectivity ratio. If nc is used to indicate the connectivity ratio, the mean length of each joint set is needed for calculation; if no is used to indicate the connectivity ratio, only the mean length of all joints is needed.

It should be noted that the mean lengths used to characterise the scale of a pattern (LA and LB) can not be used if there is no scale to the pattern (e.g., the distribution of joint lengths is fractal). In such cases a pattern scale must be determined based on both (a) the smallest joint length to be included in the measurement set (based, perhaps, on observability, see Heifer and Bevan, 1990), and (b) the size of the largest joint not affected by the

boundaries of the measurement area. Numerical modelling could be used to establish a relation between any such new measure and that discussed above, but the point is not discussed further in this paper.

Example problem

This example is concerned with ground-water flow into an excavation in jointed rock. Joints are assumed to be hydrodynamically smooth with sub- parallel walls. In such a conduit, the volume flow rate of fluid per unit width is expressed by the cubic law (Rouleau and Gale, 1987):

q = W3~fAh/( I 2pAx) (16)

where W is the joint aperture, 7f is the weight density of fluid, p is the dynamic viscosity, Ax is the distance along the conduit and Ah is the hydraulic head.

C O N N E C T I V I T Y C H A R A C T E R I S T I C S O F N A T U R A L L Y J O I N T E D R O C K S

TABLE 3

Connectivity ratio of joint patterns with different sample areas

23

Sample areas

(m 2)

Length = 0 . 6 m Length = 1.2m Length = 2 . 4 m

Density = Density = Density = Density =

2 m/m z 2.5 m/m z 3m/m 2 2 m/m 2 Density = 2 m/m 2

1 × 1 0 0.667 0.875 0.8 1

1.5 × 1.5 0 0.7 0.611 0 1 2 x 2 0.731 0.733 0.639 0.875 0.8 3 x 3 0.568 0.683 0.793 0.731 0.875

4 × 4 0.733 0.667 0.705 0.755 0.912 6 x 6 0.614 0.665 0.666 0.786 0.828 8 x 8 0.562 0.579 0.613 0.737 0.888

10 x 10 0.519 0.583 0.614 0.753 0.871

12 × 12 0.523 0.573 0.622 0.735 0.867 14 x 14 0.479 0.548 0.62 0.694 0.878

16 × 16 0.48 0.572 0.64 0.72 0.883 18 × 18 0.519 0.585 0.64 0.718 0.883

20 × 20 0.528 0.612 0.671 0.707 0.837

22 x 22 0.535 0.619 0.687 0.704 0.85 24 x 24 0.552 0.628 0.681 0.706 0.852

26 × 26 0.558 0.623 0.675 0.724 0.864 28 × 28 0.557 0.622 0.671 0.721 0.867

30 x 30 0.561 0.629 0.681 0.723 0.859 32 × 32 0.555 0.618 0.663 0.73l 0.855

34 × 34 0.559 0.626 0.665 0.733 0.861 36 x 36 0.556 0.612 0.668 0.74 0.868

38 × 38 0.562 0.61 0.669 0.75 0.864 40 x 40 0.541 0.621 0.671 0.751 0.864

,. LA L/

Fig. 14. Diagram for calculating joint geometric parameter, no.

For the problem of ground-water flow into a drift excavated in jointed rock, consider the volume flow rate, Q, of fluid within an engineering region A × B, shown in Fig. 16. Then:

Q = Kg W 3 ~ f A h / ( 1 2 p ) (17)

in which Kg is a geometric parameter of the engineering region and the joints within it. If there is a single set of joints, which is parallel to A, the

fluid flows only through those key joints which extend from boundary to boundary of the region of interest, as shown in Fig. 16(b;part a). In this case:

Kg = B/(ASk) (18)

in which Sk is the mean spacing of those key joints in the region. Sk can be obtained from the following (see Zhang, 1990):

Sk = S~ exp ( - AlL) (19)

where S is the mean spacing and L the mean length of all joints in the system.

Alternatively, if there is another joint set in the rock mass and the two joint sets connect together completely (i.e., the connectivity ratio of the joint pattern, C, equals unity, as shown in Fig. 16(b; part b), then Kg can approximately be expressed by:

K, = B/(AS) (20)

24 XING Z H A N G ET A L

0 (: 1.0

o "~ ° 6

o

.4

== o

.2

Connectivity Index (theoretical parameter)

i0 20 30 40 50 60 70 80

~ o0~I 9 f

~9

At

n c

90 I00

o o C = 0.873 + 0.209 in n o

T = 0.984

C = 0.561 + 0.106 in n c

y = 0.981

0 0 .25 .5 .75 1 1.25 1.5 1.75 2 2.25

Connectivity Index (Empirical parameter)

Fig. 15. Relationship of connectivity ratio with empirical parameter, no, and theoretical parameter, n c, of joint patterns.

2 . 5

n o

Tunnel wall ~ . f

Fig. 16(a). Geometry of rock mass adjacent to the tunnel considered in the underground excavation example problem.

C O N N E C T I V I T Y C H A R A C T E R I S T I C S O F N A T U R A L L Y J O I N T E D R O C K S 25

runnel wall

A A Ke

(a)

~ Tunnel wall

~ I

? !

I .L______

==

Tunnel wall A

\

First joint set

Second joint set /

\ /

\

(c)

Fig. 16(b). Different connectivity cases of the second example problem.

This is because all the joints become conduits of fluid flow. However, in Eqs. 18 and 20, the relative volume flow rate is expressed from two extreme cases of network extent. For a normal case, where 0 < C < 1, Kg can be estimated by:

K, = B/[A(Sk(1 - C) + SC)] (21)

In the example of Fig. 16(b; part c), if L = 10 m and S=0 .2 m (for the first joint set), the variation of the relative volume flow rate of the excavation, Kg, against the connectivity ratio of the joint patterns, C, is shown in Fig. 17 and Table 5. Clearly the connectivity ratio of a joint system has a considerable effect on the volume flow rate

26 XING ZHANG ET AL.

K g 300

200

100

Q, 4) A = 10m

Q ., ~ A = 20m

O A = 30m

O O A = 40m

Q, Q A = 50m

-°

~ 0 6 I * B I O

. J

J j ,

j "

/

J t f

w

. / / ./

. . . . ~ ¢' . )

/ ,

0 0 .2 0 .4 0 .6 0 . 8 1 .0 C

Connectivity ratio

Fig. 17. Results of an example problem-relative volume flow rate of excavation, Kg, against connectivity ratio, C, within different rock region A.

Head /

gradie~dH/dL

j Head gradient vector

Fig. 18. Hydraulic boundary condition of jointed rock.

L X

Qy

Qx

Equivalent hydraulic boundary condition

of an excavation, particularly where A is rather long.

The effect o f the connectivity ratio on rock permeability was examined by using U D E C to

calculate the permeability o f simulated jointed rock masses. Using the principle o f superposition, gene- ral flow conditions such as those illustrated in Fig. 18 were analysed as two independent flow

C O N N E C T I V I T Y C H A R A C T E R I S T I C S O F N A T U R A L L Y J O I N T E D R O C K S

k,,

o

Qxy Trrrvr ~-~_

Q x x

~-- +

Fig. 19. Equivalents of hydraulic boundary condition in Fig. 20.

aff/Ox

I I I I I I

Qyy

IIIIII

Qyx

27

TABLE 4

Data of connectivity ratio C against geometric parameters no and n c

C r/o n c C n o n c

0.706 0,422 4.35 0.852 0.845 17.51 0.552 0,2[2 1.037 0.628 0.265 1.619 0.681 0,318 2.27 0.917 1.195 24.76 0.935 1.5 36.8 0.948 1.79 56.48 0.45 0.159 0.584 0.721 0.371 3.079 0.753 0,424 3.98 0.622 0.316 2.5 0.847 0.742 12.68 0.873 0.849 16.59 0.802 0,636 10.2 0.933 1.48 53.1 0.945 1.7 69.6 0.3 0.08 0.142 0.367 0.106 0.248 0.429 0.133 0.392 0.469 0.159 0.56 0.522 0.177 0.764 0.555 0.212 1.012 0.541 0.225 0.826 0.789 0.525 4.354 0.819 0.6 5.62 0.732 0.447 3.53 0.901 1.049 17.93 0.917 1.2 23.47 0.876 0.9 14.42 0.958 2.09 75.1 0.968 2.4 98.44 0.372 0. 112 0.2 0.442 0.15 0.35 0.51 0.187 0.555 0.547 0.225 0.793 0.6 0.262 1.081 0.635 0.3 1.431 0.783 0.6 6.15 0.842 0.75 9.53 0.917 1.2 23.47

TABLE 5

Results of the second example, problem-relative volume flow rate of excavation, Kg, against connectivity ratio, C, of joint patterns within different rock regions

C= 0 0.2 0.4 0.6 0.8 1.0

A = 10 m 110.3 1 2 6 . 3 1 4 7 . 8 177.5 223 300 = 2 0 m 20.3 24.6 31.0 42.2 65.8 150 = 3 0 m 5.0 6.1 8.0 11.6 20.7 100 = 4 0 m 1.4 1.7 2.3 3.3 6.4 75 = 50 m 0.4 0.5 0.7 1 2 6

B=600 m, L= 10 m, S=0.2 m.

p r o b l e m s , as s h o w n in Fig. 19. F i g u r e s 20(a) a n d

(b) s h o w the f low p a t h a n d f low d i r e c t i o n o f t he

j o i n t p a t t e r n in Fig. 11, u n d e r h e a d g r a d i e n t s in

t he x- a n d y - d i r e c t i o n , r espec t ive ly . T h u s , f o u r

v a l u e s o f f low r a t e c a n be m e a s u r e d a n d a m e a n

p e r m e a b i l i t y t e n s o r d e t e r m i n e d . H o w e v e r , fo r t he

p u r p o s e o f th i s d e m o n s t r a t i o n , t he f low r a t e u n d e r

u n i t g r a d i e n t in e a c h d i r e c t i o n h a s b e e n a v e r a g e d

to g ive a s ingle va lue , Q, fo r t he s i m u l a t i o n .

A v e r a g e f low r a t e s h a v e b e e n c a l c u l a t e d fo r t he

28 XING ZHANG ET AL.

Fig. 20(a). Flow path and flow direction of the joint pattern in Fig. 11 under hydraulic head gradient in the x-direction. (b). Flow path and flow direction of the joint pattern in Fig. 11 under hydraulic head gradient in the y-direction.

CONNECTIVITY CHARACTERISTICS OF NATURALLY JOINTED ROCKS

Q 50

29

40,

c~

o

30

20

0

I0

0 0 0 0 0 0

0 0 0

o~

0 0.2 0.4 0.6 0,8

Connectivity Ratio

Fig. 21. Relationship of the connectivity ratio against the flow rate of 42 simulated joint patterns.

0

o~

o

o : P °

O o

0

0 D °

O0

1 . 0

C

TABLE 6

Data of connectivity ratio, C, and flowrate, Q, of simulated joint panerns(l x 10-3m3/s)

c Q c Q

0.639 4.93 0.906 16.82 0.805 17.19 0.976 21.36 0.924 23.66 0.881 12.86 0.714 17.38 0.913 28.03 0.948 32.51 0.963 32.78 0.970 29.94 0.876 15.67 0.937 24.64 0.958 26.74 0.96 26.03 0.968 35.77 0.967 35.45 0.886 13.12 0.912 15.59 0.934 24.85 0.955 28.92 0.964 33.17 0.967 40.19 0.923 20.32 0.908 28. ! 4 0.94 27.26 0.88 28.56 0.861 25.56 0.801 17.74 0.983 37.77 0.579 2.54 0.488 2.38 0.326 2.02 0.528 5.29 0.642 6.16 0.259 2.61 0.774 5.16 0.763 6.86 0.84 6.96 0.778 6.54 0.6 4.30 0.66 4.29

42 jo inted rock masses, previously generated with

the modified U D E C code. For each one, the flows

were measured in a 100 x 100 m square. The results

of flow rate against connectivi ty ratio are shown

in Fig. 21 and Table 6. The results of the numerical

modell ing show that the connectivi ty of a jo in t

pat tern has a major effect on the flow rate through

jointed rock masses, and that the connectivi ty ratio

provides a measure of this effect (Fig. 21).

Conclusions

This study defined a connectivi ty ratio, C, of a

jo in t pat tern within a rock mass, and developed

an approach for recording and est imating the

connectivi ty ratio for natura l jo in t systems. Also,

a set of addi t ional indices, called k-order connectiv-

ity ratios, Ck, was used to describe in more detail

the connectivi ty characteristics of a jo in t system.

Addit ional ly, a parameter called the ma x i mum

network extent of a jo in t system, Ce, is used to

indicate the ma x i mum region of connected joints.

The statistical results obta ined from the in-situ data enabled the following conclusions to be made:

30 X I N G Z H A N G ET AL.

(1) The connectivi ty ratio of a jo in t pat tern

within a rock region, C, will increase with increase

of the average jo in t length, L, density, D, and

intersection parameter , 0, of the jo in t pattern,

respectively, but the correlat ion is poor.

(2) The connectivity ratio, C, of a joint pattern

within a rock region will increase with increasing

the connectivity index of the joint pattern, n o , and

there is a good correlation. Here, no=OLD. The

connectivity ratio, C, of a jo int pattern within a

rock region will increase with increasing the theoret-

ical parameter of the joint pattern, n c, and there is

a good correlation. Here, nc = (DaDRsin ~)(LAL~ sin

/~). This means that either of the parameters nc or

n o can be used as a measure of joint connectivity.

(3) The size of the sample area, which is rather

small in relation to the mean length of the joints,

has a major effect on the connectivity ratio. It seems

that, when the dimensions of the sampling area are

larger than ten times the mean joint length, there

is little effect on the connectivity ratio.

The results obtained from the example problem

show that the connectivi ty ratio of jo ints within a

jo in t pat tern, C, has a major effect on the volume

flow rate of fluid into an excavation, and that the

volume flow rate of fluid is strongly correlated to

the connectivi ty ratio. Al though this work represents only an initial

step, the proposed approach promises considerable

insight into a major problem in rock engineering.

Acknowledgements

The authors would like to thank both the Gov-

e rnment of China and BP Research for their part

of the financial suppor t for this project and the

Depar tment of Civil Engineering, Universi ty of

Southampton , U.K. for its support . X. Zhang

would also like to thank Professor Liao G u o h a

for his encouragement and guidance for work done

previously in the Depar tment of Min ing Engineer-

ing, Universi ty of Science and Technology Beijing,

and Mr. Yong Qing for his assistance in collecting

the data used in this paper.

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