Stereological Interpretation of Rock Fracture Traces on

Borehole Walls and Other Cylindrical Surfaces

Xiaohai Wang

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Civil Engineering

Mauldon, Matthew, Chair

Dove, Joseph E.

Dunne, William M.

Gutierrez, Marte S.

Westman, Erik C.

September 16, 2005

Blacksburg, Virginia

Keywords: fractures, cylindrical sampling, borehole, stereology, Monte Carlo method, intensity measures, conversion factors, mean fracture length and width

Copyright 2005, Xiaohai Wang

Stereological Interpretation of Rock Fracture Traces on Borehole Walls and Other Cylindrical Surfaces

Xiaohai Wang

Abstract

Fracture systems or networks always control the stability, deformability, fluid and gas

storage capacity and permeability, and other mechanical and hydraulic behavior of rock

masses. The characterization of fracture systems is of great significance for

understanding and analyzing the impact of fractures to rock mass behavior. Fracture

trace data have long been used by engineers and geologists to character fracture system.

For subsurface fractures, however, boreholes, wells, tunnels and other cylindrical

samplings of fractures often provide high quality fracture trace data and have not been

sufficiently utilized. The research work presented herein is intended to interpret fracture

traces on borehole walls and other cylindrical surfaces by using stereology. The

relationships between the three-dimension fracture intensity measure, P32, and the lower

dimension fracture intensity measures are studied. The analytical results show that the

conversion factor between the three-dimension fracture intensity measure and the two-

dimension intensity measure on borehole surface is not dependent on fracture size, shape

or circular cylinder radius, but is related to the orientation of the cylinder and the

orientation distribution of fractures weight by area. The conversion factor between the

two intensity measures is determined to be in the range of [1.0, /2]. The conversion

factors are also discussed when sampling in constant sized or unbounded fractures with

orientation of Fisher distribution. At last, the author proposed estimators for mean

fracture size (length and width) with borehole/shaft samplings in sedimentary rocks based

on a probabilistic model. The estimators and the intensity conversion factors are tested

and have got satisfactory results by Monte Carlo simulations.

iii

Acknowledgments

I am indebted to the assistance of my dissertation committee: Dr. Matthew Mauldon, Dr.

Joseph E. Dove, Dr. William M. Dunne, Dr. Marte S. Gutierrez, and Dr. Erik C.

Westman. From my proposal to the final form of this dissertation, they have given great

amount of valuable suggestions and made the study in this Ph.D. program priceless

experience to me.

My advisor, Matthew Mauldon, whom I met two weeks after I arrived at this country,

generously provided the support for me to enroll as a Ph.D. student. In the passed four

years, he and his insights had showed me many times the lights of the way and lead me

out of the darkness of confusion and uncertainty. Though, what I have learned from him

is far beyond what I can put in words. I Thank Matthew, his wife Amy and their

daughters for their kindness and support.

Special thanks to Dr. Dunne and his student Chris Heiny in the University of Tennessee.

The collaborations with them on fracture size estimators pushed the dissertation to a new

level. Their work and suggestions as geologists have made the estimators more practical

and useful.

I also owed thanks to Jeremy Decker of Virginia Tech, who helped me testing my

program and carrying out numerous simulations. I always regret that I can not include in

my dissertation the great figures he worked out in Matlab.

I am grateful to have my friends around me in the years in Ozawa library and Rm19,

Patton Hall. My colleagues consideration and thoughtfulness makes the days and nights

in the office wonderful memory.

Last, but not least, I am beholden to my wife Hui Cheng, her family and my family in

China. Without their great love, this dissertation is impossible.

iv

Contents

Acknowledgments .........................................................................................iii

1 Introduction..................................................................................................1

2 Multi-dimensional intensity measures for Fisher-distributed fractures ......3 2.1 Introduction ............................................................................................................3

2.2 General form of conversions ..................................................................................4

2.3 Linear and planar sampling of fisher-distributed fractures.....................................7

2.4 Sampling on a cylindrical surface ........................................................................14

2.5 Example: 3-d fracture intensity inferred from scanline data ................................19

2.6 Discussion and Conclusions .................................................................................22

Acknowledgments ......................................................................................................22

Appendix 2.A Probability density function (pdf) )(f of angle ........................23 Appendix 2.B Numerical approach for obtaining F() and F() ...........................27 References...................................................................................................................28

3 Estimating fracture intensity from traces on cylindrical exposures ..........31 3.1 Introduction ..........................................................................................................32

3.2 Basic assumptions.................................................................................................35

3.3 General form of the relationship between areal intensity P21,C and volumetric

intensity P32 for right circular cylinders ..................................................................36

3.4 General case of cylindrical sampling....................................................................38

3.5 Special case: Sampling fractures of constant orientation .....................................42

3.6 Special case: fractures with uniform orientation distribution...............................45

3.7 Cycloidal Scanline Technique ..............................................................................47

3.7.1 Unbiased sampling criterion........................................................................47

3.7.2 Cycloidal scanlines......................................................................................50

3.8 Monte Carlo Simulations......................................................................................50

v

3.9 Discussion & Conclusions....................................................................................54

Acknowledgements.....................................................................................................55

Appendix 3.A Determine |cos | .................................................................................56 References...................................................................................................................59

4 Estimating length and width of rectangular fractures from traces on

cylindrical exposures ............................................................................62 4.1 Introduction ..........................................................................................................63

4.2 Assumptions .........................................................................................................67

4.4 Probabilistic model for occurrence of intersection types .....................................71

4.4.1 w > D ..........................................................................................................77 4.4.2 w D ..........................................................................................................79 4.4.3 Summary of fracture length and width estimators ......................................84

4.5 Examples ..............................................................................................................86

4.6 Monte Carlo simulations ......................................................................................90

4.7 Discussion & Conclusions....................................................................................96

References...................................................................................................................98

5 Conclusions and discussions....................................................................103

6 Appendix: Programs used in the dissertation ..........................................106 A. FISHER - Simulate the Fisher distribution.........................................................106

B. TRACE - Simulate fracture population sampled by a borehole ..........................107

7 Vita...........................................................................................................111

vi

List of Figures

Fig. 2.1. Geometry of linear and planar sampling of a fracture.......................................... 5

Fig. 2.2. For an isotropic fracture orientation distribution, the distributions of and are

proportional to the sin and sin , respectively. ................................................. 7

Fig. 2.3. Spherical triangle formed by n, m, and s, where n is fracture normal, m is

Fisher mean pole, and s is the sampling line....................................................... 9

Fig. 2.4. Fisher-distributed fracture normals in relation to sampling line (+). ................. 10

Fig. 2.5. pdfs of Fisher distribution with = 20, 40, and 100......................................... 10 Fig. 2.6. Coefficients a, b and c for conversion factor [1/C13 ] as functions of Fisher

constant . ......................................................................................................... 12 Fig. 2.7. Coefficients a, b and c for conversion factor [1/C23 ] as functions of Fisher

constant . ......................................................................................................... 13 Fig. 2.8. Cylindrical sampling of Fisher-distributed fractures with mean pole m. The

shaded area is a slice of the cylinder surface with normal c. ............................ 15

Fig. 2.9. Cylinder axis (z), Fisher mean pole (m), and normal (c) of a slice on the

cylinder surface.................................................................................................. 16

Fig. 2.10. Coefficients a, b and c for conversion factor [1/C23,C] as functions of Fisher

constant . ......................................................................................................... 18 Fig. 2.11. Fracture normals () and mean pole () in lower hemisphere projection....... 21 Fig. 2.A-1. Coordinate system for spherical triangle formed by m, s and n. .................. 24

Fig. 2.A-2 The figure shows the range, R , of , as a function of , and . Angle (between m and s) is a constant. ....................................................................... 26

Fig. 3.1. Borehole or shaft sampling of fractures in a rock mass. .................................... 33

Fig. 3.2. Fracture traces on a cylindrical shaft. Intersections between fractures and the

shaft are traces (curved line segments) on the shaft surface.............................. 38

Fig. 3.3. A thin slice of the shell sampling in fractures. The total trace length on its

surface is dl. ....................................................................................................... 39

vii

Fig. 3.4. A cylindrical shell (axis Z, height = H) intersects a set of fractures with constant orientation (normal n)........................................................................................ 43

Fig. 3.5. For cylindrical sampling in fractures with constant orientation, the correction

factor C23,C between areal intensity P21,C and volumetric intensity P32 is a

function of angle 0 between the cylinder axis and fracture normal. ................ 45 Fig. 3.6. Illustration of linear (vector) IUR sampling in 3-d space................................... 49

Fig. 3.7. The cycloid (heavy curve) is the path of a point on the circle of radius r0 as the

circle rolls from left to right along the x-axis. .................................................. 51 Fig. 3.8. The computer program is used to generate rectangular fractures intersecting with

a borehole........................................................................................................... 52

Fig. 3.9. Illustration (to the scale) of the five cases studied. Shaded rectangles are

simulated fractures, and circles are sampling cylinders..................................... 53

Fig. 3.10. Simulation results of the conversion factor 1/ C23,C, compared with the

calculated curve by Eq.(3.19). ........................................................................... 54

Fig. 3.A-1. Unit vectors S, T, n, and nr in Cartesian coordinate system, where Z is parallel to the borehole axis. The coordinates of unit vectors S and n are given

based on the geometry. ...................................................................................... 57

Fig. 4.1. Joints on limestone bed at Llantwit Major, Wales (photo provided by Matthew

Mauldon). Cross joints terminate at primary systematic joints. ........................ 65

Fig. 4.2. Schematic drawing of dipping sedimentary beds, with primary joints either

terminating on bedding planes or cutting across several layers......................... 65

Fig. 4.3. Borehole/shaft and rectangular fractures and their projections on the axis-normal

plane. Note true width w and apparent width w. .............................................. 66 Fig. 4.4. A vertical borehole of diameter D intersects rectangular fractures in six ways.

The unrolled trace map is developed from the borehole wall by cutting along

fracture dip direction. Intersection types are marked beside the corresponding

traces. ................................................................................................................. 70

Fig. 4.5 Six types of intersection between projected fractures (shaded) and

boreholes/shafts (dashed circles) are shown on the axis-normal plane. ............ 71

viii

Fig. 4.6. The locus for borehole/shaft-projected fracture intersection on the axis-normal

plane is the region inside by the dashed line. ................................................... 73

Fig. 4.7. Each intersection type has a corresponding locus on the projected fracture (bold

rectangle) for the center of the borehole. In this case, w > D. .......................... 73 Fig. 4.8. Each intersection type has a corresponding locus on the projected fracture (bold

rectangle) for the center of the borehole. In this case, D/2 < w D. ................ 74 Fig. 4.9. The corresponding locus for the center of the borehole/shaft for each intersection

type around the projected fracture (bold rectangle) on the axis-normal plane for

case w D/2...................................................................................................... 74 Fig. 4.10. Flowchart of choosing estimators to estimate mean fracture length and width.

........................................................................................................................................... 85

Fig. 4.11. A computer program was developed to generate a population of rectangular

fractures intersected by a borehole/shaft............................................................ 92

Fig. 4.12. Comparison of computed fracture length and width vs. actual fracture length

and width for scenario 1..................................................................................... 94

Fig. 4.13. Percent error and coefficient of variation of estimators for (a) fracture length

and (b) fracture width, in comparison with observed counts of B3-type

borehole/shaft-fracture intersections.................................................................. 95

Fig. App-1. The geometry of fracture, sampling cylinder, and three different shapes of

generation region. ............................................................................................ 108

ix

List of Tables

Table 2.1. Factor 1/C13 vs. and ................................................................................... 12 Table 2.2. Factor 1/C23 with different values of and . ................................................. 13 Table 2.3. 1/C23,C, the conversion factor between P21 and P32 when sampling with

cylinder surface............................................................................................... 18

Table 2.4. Orientation data for a set of fractures on the Huckleberry Trail...................... 20

Table 3.1. Simulation parameters and results. .................................................................. 53

Table 4.1. Six borehole/shaft-fracture intersection types ................................................. 69

Table 4.2. Defined symbols .............................................................................................. 72

Table 4.3. Areas of regions corresponding to each borehole/shaft-fracture intersection

type from geometry......................................................................................... 76

Table 4.4. Borehole-fracture intersection counts from a borehole sampling.................... 86

Table 4.5. Borehole-fracture intersection counts from borehole sampling ...................... 89

Table 4.6. Parameters and results of Monte Carlo simulations ........................................ 93

Table App-1. Inputs for generating the Fisher distribution. ........................................... 106

Table App-2. Parameters for simulating fractures sampled by a borehole..................... 107

Table App-3. Minimum dimension of different generation regions............................... 109

x

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1

Chapter 1

1 Introduction

Characterization of rock fractures is essential in engineering geology, civil engineering,

mining engineering and oil and gas industry. Geometric and mechanical parameters of

fractures are widely used for estimating fractured rock mass strength, deformability,

permeability, and fluid storage capacity. Currently geological investigations have

provided a great amount of fracture data from boreholes, tunnels, shafts as well as other

cylindrical sampling surfaces. Therefore, the study of fracture characterization based on

cylindrical sampling of fractured rock mass is of great significance. In this dissertation,

the author intends to study the stereological relationships in cylindrical samplings,

unbiased scanline techniques and their applications, and estimation of fracture size in

sedimentary rocks. These studies are demonstrated in the following three chapters.

Chapter 2 discusses the conversions (linear fracture intensity measure P10, planar fracture

intensity measure P21 and cylindrical fracture intensity measure P21,C, to the volumetric

fracture intensity measure P32) appropriate for constant size or unbounded fractures with

a Fisher distribution of orientation. The corresponding paper is submitted to

Mathematical Geology. Chapter 3 discusses the estimating of fracture intensity, more

specifically, fracture volumetric intensity P32, from fracture trace data in cylindrical

(borehole, tunnel or shaft) samplings. The conversion factor between the cylindrical

fracture intensity measure P21,C and the fracture volumetric intensity P32, is presented in a

general form and some special cases are also discussed. The corresponding paper is for

submission to International Journal of Rock Mechanics & Mining Sciences. In Chapter 4

the author intends to develop a general model for estimating mean rectangular fracture

length and width from traces on cylinder walls. The corresponding paper is for

submission to Rock Mechanics & Rock Engineering.

2

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3

Chapter 2

2 Multi-dimensional intensity measures for Fisher-distributed

fractures

Abstract: Fracture intensity is fundamentally a three dimensional concept, relating the

total area (m2) or volume (m3) of fractures to the volume of the rock mass studied.

However, field measurements of fracture intensity in rock masses are usually either one

dimensional - along sampling lines or boreholes, or two dimensional - on tunnel walls or

trace planes. In this paper, conversions between these one and two dimensional intensity

measures, and the three dimensional intensity measure P32, are developed for constant

size or unbounded Fisher-distributed fractures, for three types of sampling domain: lines,

planes and cylindrical surfaces. Conversion factors for each of these sampling domains

are derived semi-analytically, and then computed, graphed and tabulated for a wide range

of cases. The practical significance of this work is that it enables rock engineers and

geologists to deduce 3-d fracture intensity from 1-d or 2-d field measurements.

2.1 Introduction

The Fisher distribution (Fisher, 1953) is the most commonly assumed distribution for

natural fracture orientations (Cheeney, 1983). This is largely due to its relatively simple

form, as compared to other distributions for spherical data (N. Fisher et al., 1987). The

Fisher distribution also has the advantage that it is the theoretical analogue of the normal

distribution, for spherical data. Because of these advantages, the Fisher distribution is

widely used for hydrological and geomechanical modeling in fractured rock (Cheeney,

1983; Priest, 1993).

One dimensional (1-d) and two dimensional (2-d) fracture intensity measures P10 and P21

are defined, respectively, as the number of fractures per unit length and the number of

fractures per unit area (Dershowitz & Herda, 1992; Mauldon, 1994) in the rock mass.

4

These measures are directionally dependent and are strongly affected by the relative

orientation of the fractures and the sampling domain, e.g., scanline, or planar surface. In

contrast, the three dimensional (3-d), or volumetric, fracture intensity measure P32,

defined as area of fractures per unit volume, is not directionally dependent (Dershowitz &

Herda, 1992; Mauldon, 1994). Measures P10 and P21 are easy to measure in the field, but

they cannot be used as general parameters to characterize fracture intensity because of

their directional dependence. For these reasons, the ability to convert linear intensity P10

or areal intensity P21 to the volumetric intensity P32, which is difficult to measure in the

field but directional independent, will be very useful.

Previous work (Dershowitz & Herda, 1992; Mauldon, 1994, Mauldon & Mauldon, 1997)

has developed some of the theoretical background for fracture intensity measures. In the

present paper, the authors derive conversions between field measures of fracture

intensity, P10 and P21, and the three dimensional volumetric fracture intensity measure P32

for fracture sets with the Fisher orientation distribution. This study focuses on fracture

orientation instead of fracture size; we assume fractures are either of constant size or are

unbounded. Based on this assumption, factors to convert measured 1-d or 2-d fracture

intensity for Fisher-distributed fractures to volumetric intensity are obtained semi-

analytically for sampling domains on lines, planes and cylinders.

2.2 General form of conversions

Conversions between 1-d intensity measure P10 and 3-d intensity measure P32, or between

2-d intensity measure P21 and 3-d intensity measure P32, require consideration of the

sampling bias that arises from the relative orientation of the sampling domain and the

fracture. This bias was first described by R. Terzaghi (1964), and later explored by Yow

(1987), Priest (1993), Martel (1999), and Mauldon and Mauldon (1997), among others.

In the general case, for linear or planar sampling of constant size or unbounded fractures,

P10 and P21 are related to P32 in the following ways (Dershowitz, 1992; Mauldon, 1994):

5

( ) dfPP = |cos|0

3210 and (2.1)

( ) dfPP =0

3221 sin , (2.2)

where is the angle between the sampling line and the fracture normal (Fig. 2.1a); is the angle between the sampling plane normal and fracture normal (Fig. 2.1b); and

f() and f() are the probability density functions (pdfs) of and , respectively. In the following, we assume a statistically homogeneous sampling domain, and it is to be

understood that the given relationships refer to expected values of the intensity measures.

Here, for simplicity, the integrals in Eqs. (2.1) and (2.2) are each functions of a single

variable. The angles or are themselves functions of conventional geologic fracture orientation parameters such as dip and dip-direction, and orientation of the sampling line

or sampling plane, and can be calculated from orientations of sampling line or plane and

fracture normal.

Fig. 2.1. Geometry of linear and planar sampling of a fracture.

(a) (b)

Sampling line

Fracture normal

Fracture

Fracture normal

Sampling plane normal

FractureSampling plane

6

Define conversion factors C13 and C23 by

( )1

013 |cos|

=

dfC and (2.3)

( )1

023 sin

=

dfC , (2.4)

so that

321013 PPC = and (2.5)322123 PPC = . (2.6)

The integrals in Eqs. (2.3) and (2.4) are on [0, 1], so the ranges of the conversion factors

C13 and C23 are from 1 to .

As an example, for the isotropic case of a uniform fracture orientation distribution,

sin)( 21=f and sin)( 21=f (Fig. 2.2), for and in the range [0, ]. Introducing these pdfs into (3) and (4), respectively, we have

1

021

)( 13 |cos|sin

= dC isotropic and (2.7)

( )1

0

1

0

221

)( 23 2cos141sin

=

= ddC isotropic , (2.8)which, combining with Eqs. (2.5) and (2.6), yield (Dershowitz, 1985)

32)( 10 21 PP isotropic

= and (2.9)

32)( 21 4PP isotropic

= . (2.10)

7

Fig. 2.2. For an isotropic fracture orientation distribution, the distributions of and are

proportional to the sin and sin , respectively.

Eqs. (2.9) and (2.10) imply that for uniformly distributed fractures (the isotropic case),

P32 is twice the average scanline frequency and 1.27 times the mean areal trace length

intensity. In the following, we determine the conversion factors C13 and C23 for the case

of Fisher-distributed fractures.

2.3 Linear and planar sampling of fisher-distributed fractures

The probability density function of the Fisher distribution is given as (N. Fisher et al,

1987)

(a) (b)

Fracture

Sampling line

Fracture normal

length sin

Samplingplane

Fracture

Fracture normal

Sampling plane normal

length

8

)0( )(

= eeSinef

Cos

, (2.11)

where is the angle between a fracture normal and the Fisher mean pole (Fig. 2.3); f() is the probability density function of ; and is the Fisher constant related to the amount of dispersion ( has high values for low dispersion and low values for high dispersion). Because of the radial symmetry of the Fisher distribution about its mean pole, we express

its probability density function as a function only of for a given dispersion constant. The local azimuth of the Fisher mean pole is uniform on [0, 2] and is independent of .

Fig. 2.4 shows a set of fracture normals following the Fisher distribution, in upper

hemisphere projection. Fisher mean pole m corresponds to a plane with dip 80 and dip-

direction 45. The Fisher dispersion constant in this case is equal to 60.

The theoretical range of is from 0 to , with low values indicating a high degree of dispersion. As approaches 0, the fractures approach a uniform orientation distribution. Typical graphs of the pdf of the Fisher distribution are shown in Fig. 2.5.

In order to obtain the conversion factor between 1-d intensity measure P10 and 3-d

intensity measure P32, we need to know f() , the probability density function of angle between the sampling line and the fracture normal.

Based on the geometry of the spherical triangle formed by the fracture normal n, the

Fisher mean pole m and the sampling line s (Fig. 2.3), the theoretical probability density

function f() is given by (see Appendix 2.A):

d

eeSinef

Cos

R

= 222 )coscos(cossinsin sin 1)( (2.12)

for in the range + || , where the range of integration Rd is given by: ],-[ +=R , if , or

]2 ,0[ =R , if > . (2.13)

9

The integral in Eq. (2.12) cannot, however, be expressed in closed form. We use

numerical simulation to find the set of values of the conversion factor, following the

procedure described in Appendix 2.B.

Fig. 2.3. Spherical triangle formed by n, m, and s, where n is fracture normal, m is

Fisher mean pole, and s is the sampling line. The spherical angles , , and are,

respectively, the angles between n & s, m & n, and m & s.

10

Fig. 2.4. Fisher-distributed fracture normals in relation to sampling line (+).

Fig. 2.5. pdfs of Fisher distribution with = 20, 40, and 100.

01234567

0 10 20 30 40 50 60 70 80 90Angular deviation (deg.) from Fisher mean pole

= 100

= 40

= 20f()

North

20

Fisher Mean Pole m(dip 80, dip -direction 45)

Upper Hemisphere Equal Area

Sampling line s (trend 225, plunge 45)

Small circle with = 20

Fracture normal ni

11

Tabulated values of the factor 1/C13 (= P10/P32) are shown in Table 2.1 as a function of

the Fisher constant and angle . The reciprocal of C13, rather than C13, is tabulated in order that values range between 0 and 1. When is relatively small ( < 1), indicating that fracture orientations have close to a uniform distribution, the factor 1/C13 is close to

0.50, which agrees with Eq. (2.9). The factor 1/C13 can be fitted to the family of curves

given by cbaC += )cos(/1 13 (Fig. 2.6). Regression coefficients a, b and c can be computed for 1, according to the logarithmic expression given in Fig. 2.6. For < 1, it is recommended to treat the distribution as uniform and to use the conversion factor

given by Eq. (2.9).

Following the procedure described in Appendix B, the conversion factor 1/C23 (= P21/P32)

is also computed numerically. The values of the factor 1/C23 are tabulated in Table 2.2 as

a function of and . As with the case of linear sampling, for a given value of , the conversion factor 1/C23 is relatively insensitive to changes in for > 50. When is relatively small ( < 1), the factor is close to 0.79, which agrees with Eq. (2.10). The conversion factor 1/C23 can be fitted to the family of curves given by

cdbaC += )2/sin(/1 23 (Fig. 2.7). Regression coefficients a, b, c and d can be computed for 1, according to the logarithmic expression given in Fig. 2.7. For < 1, it is recommended to treat the distribution as uniform and to use the conversion factor

given by Eq. (2.10).

12

Table 2.1. Factor 1/C13 vs. and 0.1 1 2 5 10 50 100 200 500

0 0.50 0.53 0.62 0.79 0.90 0.98 0.99 0.99 1.00 5 0.50 0.53 0.62 0.79 0.89 0.97 0.98 0.99 0.99

10 0.50 0.53 0.61 0.78 0.88 0.96 0.97 0.98 0.98 20 0.50 0.53 0.59 0.75 0.84 0.91 0.92 0.93 0.93 30 0.50 0.52 0.56 0.68 0.77 0.84 0.85 0.85 0.85 40 0.50 0.51 0.54 0.62 0.67 0.74 0.75 0.75 0.75 50 0.50 0.51 0.51 0.54 0.57 0.62 0.62 0.63 0.63 60 0.50 0.49 0.48 0.47 0.45 0.47 0.48 0.48 0.48 70 0.50 0.48 0.45 0.39 0.34 0.32 0.32 0.32 0.32 80 0.50 0.48 0.44 0.34 0.26 0.18 0.16 0.16 0.16 90 0.50 0.48 0.44 0.33 0.24 0.11 0.08 0.06 0.04

Cos

Fig. 2.6. Coefficients a, b and c for conversion factor [1/C13 ] as functions of Fisher

constant . The equations for a, b and c shown in the figure are for > 1.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.1 1 10 100 1000

Coe

ffici

ent

a b

c

cbaC += )cos(/1 13 2.0951

+ )ln( 0.1655

- =

b

0.0507 + )ln( 0.1247 =

a

0.5988 + )ln( 0.0551-

= c

13

Table 2.2. Factor 1/C23 with different values of and . 0.1 1 2 5 10 50 100 200 500

0 0.79 0.77 0.70 0.53 0.39 0.19 0.14 0.11 0.07 5 0.79 0.77 0.70 0.54 0.40 0.21 0.17 0.14 0.12

10 0.79 0.77 0.70 0.54 0.43 0.25 0.22 0.20 0.20 20 0.79 0.77 0.72 0.58 0.49 0.38 0.37 0.36 0.36 30 0.79 0.78 0.74 0.64 0.58 0.53 0.52 0.52 0.52 40 0.79 0.78 0.76 0.71 0.68 0.66 0.66 0.66 0.66 50 0.79 0.78 0.78 0.77 0.77 0.77 0.77 0.78 0.78 60 0.79 0.79 0.79 0.82 0.85 0.87 0.87 0.87 0.87 70 0.79 0.80 0.82 0.87 0.90 0.94 0.94 0.94 0.94 80 0.79 0.80 0.83 0.90 0.94 0.98 0.98 0.99 0.99 90 0.79 0.80 0.83 0.91 0.95 0.99 0.99 1.00 1.00

Sin

Fig. 2.7. Coefficients a, b and c for conversion factor [1/C23 ] as functions of Fisher

constant . The equations for a, b, c and d shown in the figure are for > 1.

0.0

0.5

1.0

1.5

2.0

2.5

0.1 1 10 100 1000

Coe

ffici

ent

b

a

c d

cdbaC += )2/sin(/1 23

0.0351 + )ln( 0.1064=

a

0.8112 + )ln( -0.0771

= c

2.1745 + )

ln( -0.1376

=

b

1.0783 + )ln(

-0.1297=

d

14

2.4 Sampling on a cylindrical surface

In this section we discuss the conversion factor between the areal intensity measure

obtained by sampling rock fractures on the surface of a cylinder, and the volumetric

intensity measure P32. Again, constant size or unbounded Fisher-distributed fractures are

assumed. The practical significance of this case arises, on one hand, from the availability

of fracture trace data obtained from borehole image or FMI and FMS logs (Dershowitz et

al., 2000), and on the other hand, from fracture trace maps obtained from circular tunnel

walls (Mauldon and Wang, 2003).

Let P21,C denote fracture trace length per unit area on the cylinder surface. For constant

size or unbounded Fisher-distributed fractures, the relationship between P21,C and

volumetric measure P32 is a function only of the angle between the Fisher mean pole m

and the cylinder axis z (Fig. 2.8) because of the circular symmetry of the sampling

surface with respect to the cylinder axis. Define C23,C as the conversion factor between

P21,C and P32 for cylinder sampling, with

32,21,23 PPC CC = . (2.14)

For a slice of the cylinder surface, such as the shaded area in Fig. 2.8, the normal c of the

surface element makes an angle with the mean fracture pole m. Let 213223 / PPC = denote the conversion factor between P21 and P32 for a sampling plane which has the

same normal as the slice (e.g., vector c in Fig. 2.8), then

( ) = maxmin

)(/1/1 23,23

dfCC C , (2.15)

where f() is the pdf of and the integration is carried out over the full range of . It should be noted that C23 in Eq. (2.15) is a function of and refers to a specific slice (such

as the shaded strip in Fig. 2.8).

15

We adopt the Cartesian coordinate system shown in Fig. 2.9, where the xy plane is

perpendicular to the cylinder axis z, and for convenience, the x axis is selected to be

perpendicular to the zm plane. If m and c are unit vectors, then

cossincos == cm , (2.16)from which

=

sincoscos 1 . (2.17)

where is the angle between the y-axis and c. Note that is uniformly distributed on [0,

2] because of the radial symmetry of the cylinder (Fig. 2.9). The pdf )(f of is given by

21)( =f . (2.18)

Fig. 2.8. Cylindrical sampling of Fisher-distributed fractures with mean pole m. The

shaded area is a slice of the cylinder surface with normal c.

zm

co

16

Fig. 2.9. Cylinder axis (z), Fisher mean pole (m), and normal (c) of a slice on the

cylinder surface. The xyz cylinder coordinate system is also shown.

And the cdf )(F of is =

17

( ) 2/122 sin/cos1sin sin1)()( == Ff , ( ) ( ) + 2/2/ . (2.22)Substituting Eq. (2.22) into Eq. (2.15), we obtain

( ) ( ) =max

min

2/12223,23 sin/cos1sin sin/11/1

dCC C , (2.23)

where ( ) ( ) +== 2/2/ maxmin and where (1/C23) is already given numerically in Table 2.2.

With Table 2.2 and Eq. (2.24), the conversion factor 1/C23,C is computed numerically, and

is tabulated in Table 2.3 as a function of and . It is interesting to note that when angle (between the Fisher mean pole and the cylinder axis) is around 60, the conversion

factor 1/C23,C is relatively insensitive to and has the value 0.79, which is also the value obtained (/4) on any sampling surface for a uniform fracture orientation distribution, as is implied by Eq. (2.10).

The conversion factor 1/C23,C increases to 1.0 if the cylinder axis is close to the Fisher

mean pole, and decreases to 0.64 (2/, the theoretical solution for ) if the cylinder axis is perpendicular to the mean pole. The factor 1/C23,C can be fitted to a family of

curves given by cbaC C += )cos(/1 ,23 (Fig. 2.10). Regression coefficients a, b and c can be computed for 1 according to the logarithmic expression given in Fig. 2.10. For < 1, it is recommended to treat the distribution as uniform and the conversion factor1/C23,C

0.79.

18

Table 2.3. 1/C23,C, the conversion factor between P21 and P32 when sampling with

cylinder surface.

0.1 1 2 5 10 50 100 500 5 0.79 0.81 0.83 0.91 0.95 0.99 0.99 1.00

10 0.79 0.81 0.83 0.91 0.95 0.99 0.99 1.00 20 0.79 0.81 0.83 0.90 0.93 0.97 0.97 0.98 30 0.79 0.80 0.82 0.88 0.91 0.95 0.95 0.95 40 0.79 0.80 0.81 0.85 0.88 0.91 0.91 0.92 50 0.79 0.79 0.80 0.83 0.84 0.86 0.86 0.87 70 0.79 0.79 0.78 0.77 0.77 0.77 0.76 0.77 90 0.79 0.79 0.77 0.75 0.73 0.70 0.70 0.70

Fig. 2.10. Coefficients a, b and c for conversion factor [1/C23,C] as functions of Fisher

constant . The equations for a, b and c shown in the figure are for > 1.

0.0

0.5

1.0

1.5

2.0

2.5

0.1 1 10 100 1000

Coe

ffici

ents

a

b

c

cbaC C += )cos(/1 ,23

0.0359 + )ln( 0.0274= a

2.1971 + )ln( -0.0944= b

0.8045 + )ln( 0.0026= c

19

2.5 Example: 3-d fracture intensity inferred from scanline data

Table 2.4 contains the orientation data for a set for 19 subparallel fractures. Fracture

orientation data were collected along a 400 ft straight scanline (trend 180, plunge 0) on

a rock slope along a former railroad alignment, on the Huckleberry Trail near

Blacksburg, Virginia. Fig. 2.11 shows the fracture normals in lower hemisphere

projection. The fractures are thought to be of approximately the same size, and

orientations to follow the Fisher distribution. Since all fractures belong to a well-defined

set, the Terzaghi bias associated with sampling along a straight scanline (Priest, 1993)

is approximately the same for all fractures, and is therefore neglected here.

Let the y-axis be directed horizontally to the north, the x-axis horizontally to the east, and

the z-axis vertically upward. The Fisher mean pole and dispersion constant can be

estimated as follows (Cheeney, 1983).

(1) The arithmetic means of the direction cosines are calculated from

NncNmcNlc iiil === nm , (2.24)where here the total number of fractures N = 19.

(2) The length of the mean vector R is calculated from:

222nml cccR ++= , (2.25)

(3) Direction cosines of the estimated Fisher mean pole are computed from:

RcnRcmRcl nml === , (2.26)

20

Table 2.4. Orientation data for a set of fractures on the Huckleberry Trail.

Direction cosines of

normals Number Dip

Dip-Direction l m n

1 80 44 0.684 0.708 0.174 2 84 76 0.965 0.241 0.105 3 80 44 0.684 0.708 0.174 4 80 270 0.985 0.000 -0.174 5 88 260 0.984 0.174 -0.035 6 82 48 0.736 0.663 0.139 7 88 227 0.731 0.682 -0.035 8 86 244 0.897 0.437 -0.070 9 86 238 0.846 0.529 -0.070

10 89 75 0.966 0.259 0.017 11 86 256 0.968 0.241 -0.070 12 76 43 0.662 0.710 0.242 13 75 58 0.819 0.512 0.259 14 74 50 0.736 0.618 0.276 15 70 42 0.629 0.698 0.342 16 90 248 0.927 0.375 0.000 17 90 66 0.914 0.407 0.000 18 86 252 0.949 0.308 -0.070 19 84 240 0.861 0.497 -0.105

totals 15.942 8.766 1.100 arithmetic means 0.839 0.461 0.058

(4) If R has magnitude greater than about 0.65, the Fisher constant can be approximated by:

)1(1 R= , (2.27)In this example, the direction cosines of the mean pole are estimated to be:

060.0 481.0 875.0 === nml , (2.28)which gives a mean plane with dip-direction 61.2 and dip 86.5. The angle between the Fisher mean pole and the scanline is calculated to be about 61.0. The mean resultant

length R is 0.96, from Eq. (2.25), and the Fisher constant is estimated by Eq. (2.27),

which gives = 24.6. A similar procedure for computing the Fisher parameters is given by Goodman (1989), who takes R to be the resultant vector rather than the mean.

21

Fig. 2.11. Fracture normals () and mean pole () in lower hemisphere projection.

We can calculate the (direction dependent) fracture frequency P10 along the scanline by

dividing the total number of fractures N by scanline length L: 1

10 048.040019=== ftLNP . (2.29)

By using Table 2.1, Fig. 2.6, or the curves defined by the coefficients in Fig. 2.6, we can

interpolate the value of 1/C13. In this example, 61.0 and 24.6, so 1/C13 0.46 and C13 2.17. The volumetric intensity measure P32 (fracture area per unit rock mass volume) can be determined for this fracture set by multiplying C13 and P10, giving

11101332 10.0) 048.0)(17.2(

== ftftPCP (2.30)

N

Mean

lower hemisphere

equal area +

Scanline

22

2.6 Discussion and Conclusions

Fracture intensity is a key input for computer models that deal with flow through a

fractured rock mass. Fracture intensity (P32) is inherently three-dimensional, but is

usually approximated via measurements on 1-d or 2-d sampling domains. Conversions

from 1-d or 2-d intensity, however, necessarily depend on the orientation (or orientation

distribution) of the sampling domain with respect to the fracture orientation distribution

of the rock mass. In this paper, conversion factors between 1-d and 2-d fracture intensity

measures (P10 and P21) and the 3-d intensity measure (P32) are discussed for the cases of

constant size or unbounded Fisher-distributed fractures. The needed conversion factors

for linear, planar and cylindrical sampling domains are computed semi-analytically, with

the aid of Monte Carlo simulation. For linear sampling and planar sampling, the

conversion factors C13 and C23 are determined to be in the range of [1.0, ]. For

cylindrical surface sampling of constant size or unbounded Fisher-distributed fractures,

the conversion factor C23,C is determined to be in the range of [1.0, /2]. These

conversion factors are graphed and tabulated for a wide variety of cases.

In practice, straight scanlines run on a rock mass exposure, as well as straight small-

diameter boreholes, can be considered linear sampling. Rock exposures such as rock

slopes, or mine drift walls, are typical examples of planar sampling of fractures. Tunnel,

shaft or borehole walls give rise to cylindrical surface sampling of fractures. After

collecting fracture data on a sampling domain, e.g., a scanline, a planar rock slope, or a

borehole, engineers and geologists can estimate the volumetric intensity measure P32 by

using the conversion factors presented in this paper.

Acknowledgments

Support from the National Science Foundation, Grant Number CMS-0085093, is

gratefully acknowledged.

23

Appendix 2.A Probability density function (pdf) )(f of angle

Angles , , and between Fisher mean pole m and sampling line s, sampling line s and fracture normal n, and fracture normal n and Fisher mean pole m, respectively, are

shown in Fig. 2.3 and Fig. 2.A-1.

To simplify determination of the pdf of , we define a coordinate system as shown in Fig.

2.A-1, in which m is perpendicular to the xy plane, and s is in the zy plane. Vectors s'

and n' are the projections of s and n on xy plane, respectively. The angle between s' and

n' is defined as . For the spherical triangle formed by m, n, and s, the following relationship holds (Ayres, 1954):

coscossinsincoscos += , (2.A-1)so that,

=

sinsin

coscoscoscos 1

. (2.A-2)

The probability density function of angle depends on and ( being kept constant in the derivation). For the Fisher distribution, the joint pdf of angle and is given by (N. Fisher et al., 1987)

)0( )(2

),(,

= ee

SinefCos

. (2.A-3)

24

Fig. 2.A-1. Coordinate system for spherical triangle formed by m, s and n.

Angle is uniformly distributed in the range [0, 2] and is independent of angle . Therefore the pdf of is

2/1)( =f , (2.A-4)from which,

)0( )(

= eeSinef

Cos

. (2.A-5)

The pdf of angle can be derived through its cumulative distribution function (cdf).

Given , the cdf of is

o

Xy

z

n

ms

n s

25

== R dfProbF )()|()|(| , (2.A-6)where R is the range of when . Given , increases from the minimum of |-| to the maximum of + when increases from 0 to . Fig. 2.A-2 shows the relationship among angles , , , and also the range of . Note that R is symmetric about the y-axis. Below is the determination of R with different range of , , and .

: lse

/2 )( if /2],,-[ /2 )( if ],,-[

0

],0[ max

eR

+++

=

,

:

=R

+

26

Then, the pdf of is

deeSinedfff

Cos

RR

== 222| )coscos(cossinsin sin 1)()|()(

+ || , (2.A-12)where Rd is given by

],-[ +=R , if , or ]2 ,0[ =R , if > . (2.A-13)

Fig. 2.A-2 The figure shows the range, R , of , as a function of , and . Angle (between m and s) is a constant. Angles and are the semi-apical angles of small circles about z and y, respectively. R delimits the intersection of the above-mentioned

small circles, projected into the xy plane.

R

27

Appendix 2.B Numerical approach for obtaining F() and F()

Rewrite Eqs. (2.3) and (2.4) as

( )1

013 cos

=

dFC , and (2.B-1)

( )1

023 sin

=

dFC , (2.B-2)

where F() and F() are the cdfs of and , respectively. Numerical evaluation of the finite integrals in the equations gives the conversion factors C13 and C23, using the

procedure described below for F(). F() can be obtained through a similar procedure. 1. Set the mean pole and the Fisher constant for the Fisher distribution. 2. Generate a set of Fisher-distributed fracture normals by using the cdf of the Fisher

distribution, given by (Dershowitz, 1985)

)0( 1

)(

= eeF

Cos

, (2.B-3)

Fig. 2.4 shows a simulated population of 3000 fracture normals with the Fisher

mean pole corresponding to a plane with dip 80 and dip-direction 45, and = 60. for a detailed description of the simulation procedure, see Priest (1993).

3. For a given sampling orientation, draw small circles (Fig. 2.4) with values of at

fixed increments.

4. The cdf F() of , is calculated empirically by the number of fracture normals falling inside small circles divided by the total number of fracture normals. For

instance, in Fig. 2.4, 104 out of 3000 fracture normals are inside the small circle

with = 20. Therefore the cdf F() of , evaluated at = 20, is

0.035 3000104)20( === F . (2.B-4)

28

References

Ayres, F. Jr. (1954) Schaums Outline Series of Theory and Problems of Plane & Spherical Trigonometry. McGraw-Hill

Cheeney, R. F. (1983) Statistical methods in geology for field and lab decisions, Allen & Unwin Ltd. London. UK

Dershowitz, W.S. (1985) Rock Joint System Ph.D. Dissertation, MIT, Cambridge, Mass.

Dershowitz, W.S. and H.H. Einstein (1988) Characterizing rock joint geometry with joint system models Rock Mechanics and Rock Engineering 21: 2151

Dershowitz, W. S. and Herda, H. H. (1992) Interpretation of fracture spacing and intensity Proceedings of the 33rd U.S. Symposium on Rock Mechanics, eds. Tillerson, J. R., and Wawersik, W. R., Rotterdam, Balkema. 757-766.

Dershowitz, W., J. Hermanson & S. Follin, M. Mauldon (2000) Fracture intensity measures in 1-D, 2-D, and 3-D at Aspo, Sweden, Proceedings of Pacific Rocks 2000, eds. Girard, Liebman, Breeds & Doe

Einstein, H. H. and Baecher, G. B. (1983) Probabilistic and statistical methods in engineering geology Rock Mechanics and Rock Engineering 16: 39-72.

Fisher, N. I., T., Lewis, B.J.J. Embleton (1987) Statistical analysis of spherical data. Cambridge University Press, Cambirdge UK

Fisher, R. A. (1953) Dispersion on a sphere Proc. Roy. Soc. London, Ser. A, 217: 295-305

Goodman, R. E. (1989) Introduction to Rock Mechanics. John Wiley & Sons, New York.

Martel, S.J. (1999) Analysis of fracture orientation data from boreholes. Environmental and Engineering Geoscience. 5: 213-233.

Mauldon, M. (1994) Intersection probabilities of impersistent joints, International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 31(2): 107-115.

Mauldon, M., J. G. Mauldon. (1997) Fracture sampling on a cylinder: from scanlines to boreholes and tunnels. Rock Mechanics and Rock Engineering. 30: 129-144.

Mauldon, M., M.B. Rohrbaugh, W.M. Dunne, W. Lawdermilk (1999) Fracture intensity estimates using circular scanlines. In Proceedings of the 37th US Rock Mechanics Symposium, eds. R.L. Krantz, G.A. Scott, P.H. Smeallie, Balkema, Rotterdam. 777-784.

29

Mauldon M., W. M. Dunne and M. B. Rohrbaugh, Jr. (2001) Circular scanlines and circular windows: new tools for characterizing the geometry of fracture traces. Journal of Structural Geology, 23(3): 247-258

Mauldon M. and X. Wang (2003) Measuring Fracture Intensity in Tunnels Using Cycloidal Scanlines Proceedings of the 12th Panamerican Conference on Soil Mechanics and Geotechnical Engineering and the 39th U.S. Rock Mechanics Symposium.

Owens, J.K., Miller, S.M., and DeHoff, R.T. (1994) Stereological Sampling and Analysis for Characterizing Discontinuous Rock Masses. Proceedings of 13th Conference on Ground Control in Mining. 269-276.

Priest, S.D. (1993) Discontinuity Analysis for Rock Engineering. Chapman and Hall, London.

Russ, J. C., DeHoff, R. T. (2000) Practical Stereology Kluwer Academic/Plenum Publishers, New York

Terzaghi, R.D. (1965) Sources of errors in joint surveys. Geotechnique. 15: 287-304.

Yow, J.L. (1987) Blind zones in the acquisition of discontinuity orientation data. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts. Technical Note. 24: 5, 317-318.

30

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31

Chapter 3

3 Estimating fracture intensity from traces on cylindrical

exposures Abstract

Fracture intensity is a fundamental parameter when characterizing fractures. In the field,

a great amount of fracture data is collected along boreholes, circular tunnel or shaft walls.

The data reveal some characteristics of fractures in rock masses; however, it has not been

sufficiently interpreted. In this paper, we discuss estimating of fracture intensity, more

specifically, fracture volumetric intensity P32, from fracture trace data in cylindrical

(borehole, tunnel or shaft) samplings. We built up the relationships between the 2-d

fracture intensity measure and the 3-d fracture intensity measure theoretically.

Stereological analyses show that the conversion factor between the two intensity

measures is not dependent on fracture size, shape or circular cylinder radius, but is related

to the orientation of the cylinder and the orientation distribution of fracture area. It is also

found that the fracture volumetric intensity measure P32 is always 1.0 to 1.57 times of

fracture trace length per unit borehole surface area (P21,C). The technique of using

cycloidal scanlines to estimate the fracture volumetric intensity is also discussed. A

computer program is developed to generate synthetic fractures sampled by a circular

cylinder and the derived conversion factor between the two intensity measures is tested

by Monte Carlo simulations.

Key words: cylindrical sampling, fracture networks, stereology, rock mass, intensity

measures, conversion factors

32

3.1 Introduction

Natural rock masses are commonly dissected by discontinuities such as fractures, faults

and bedding planes, which influence or even control the behavior of rock masses

(Goodman, 1989; Priest, 1993). Therefore, characterization of the fracture system in a

rock mass, including properties such as fracture orientation, shape, size, aperture, and

intensity (ISRM, 1978), is necessary for many engineering applications. Examples of

such applications include hydrocarbon extraction, control of contaminants in landfills,

tunneling, and rock slope engineering.

Fracture intensity, which represents the amount of fractures in the rock mass, is one of

the fundamental parameters for characterizing fracture systems. Fracture intensity can be

interpreted in several ways, corresponding to a set of fracture abundance measures,

depending on the dimension of the sampling domain. (Dershowitz, 1984, 1992; Mauldon

1994). The most commonly used measure is the frequency of fractures, defined as

number of fractures per unit length. Frequency, which is also referred to as the one-

dimensional (1-d, linear) intensity, P10, is often measured along a scanline (Fig. 3.1(a)) of

fixed orientation on a planar exposure, or along the length of a borehole. The sampling

bias (R. Terzaghi 1965) induced by scanline or borehole measurements of fracture

frequency, or P10, remains a problem with scanline measurements. The major difficulty

with implementing frequency data as a fracture intensity measure has to do with the so-

called blind zone (Terzaghi, 1965; Yow, 1987), which refers to fracture orientations

that are not seen or under-sampled by a borehole or scanline. The geometric

(Terzaghi) correction factor for fractures in the blind zone can lead to gross distortion

of the data (Yow, 1987). A review of scanline sampling is presented by Priest (1993,

2004).

On cylindrical exposures such as borehole walls, circular tunnel or shaft walls, the

fracture system is revealed in a two-dimensional (2-d) form. Besides features of fractures

such as orientation, aperture, or infilling that can be measured directly on cylindrical

33

exposures, the intensity, pattern, and termination relationships of fracture traces on the

cylindrical exposure surfaces provide much more information about fracture networks

than a one-dimensional exposure (scanline) does.

Fig. 3.1. Borehole or shaft sampling of fractures in a rock mass. (a) Vertical shaft

intersects several fractures, which yield traces on the cylinder surface and on the face of

the rock mass; horizontal scanline on the rock face intersects three fracture traces. (b)

Unrolled trace map developed from the borehole or shaft wall.

Rock mass

Scanline

Borehole

Fractures

Fracture trace on the slope

Fracture traces total length = l

Unrolled (developed) trace map (total area A)

(a) (b)

34

To explore the relationships between fracture traces on a cylindrical surface and the 3-d

fracture system, we introduce the following notation. Let P21,C denote the two-

dimensional (2-d, areal) fracture intensity on the circular sampling cylinder surface,

defined as trace length per unit sampling surface area. The subscript C denotes the

cylindrical sampling domain. P21,C is determined as the sum of trace length on tunnel or

borehole walls divided by the total surface area of tunnel or borehole walls. In Fig. 3.1(b),

for instance, assume the total trace length on the unrolled trace map is l and the total area

of the unrolled trace map is A. Then the areal fracture intensity is simply P21,C = l / A.

For a fractured rock mass, this measure is a function of tunnel or borehole size and

orientation, as well as the fracture orientation distribution (weighted by fracture size).

Therefore it is also a directionally biased measure, as is as the linear intensity measure

P10.

Let P32 denote the three-dimensional (3-d, or volumetric) fracture intensity, defined as

fracture area per unit volume of rock mass. P32 is independent of the sampling process

and is an unbiased measure of fracture intensity (Dershowitz, 1992; Mauldon 1994).

Interpreted as an expected value, P32 is also scale independent. P32 is a crucial parameter

for numerical analyses in models such as the discrete fracture flow and transport model

(Dershowitz et al., 1998). However, P32 is impossible to measure directly in an opaque

rock mass.

This paper proposes approaches to utilize fracture trace data collected on the cylindrical

exposures of rock mass, such as borehole walls, tunnel or shaft walls, to estimate

volumetric fracture intensity of the rock mass. This determination is based on the derived

relationship (conversion factor) between the fracture areal intensity on a cylindrical

surface (P21,C) and the fracture volumetric intensity measure (P32).

Following stereological principles (Russ and DeHoff, 2000) we first discuss the general

form of the conversion factor between the areal intensity P21,C on circular cylinder

surface and fracture volumetric intensity measures P32. Theoretical solutions for the

conversion factor between the two measures are derived in the case of cylindrical

35

sampling of constant orientated fractures, and also sampling of fractures with a uniform

distribution. The conversion factor can be calculated analytically if the fracture

orientation distribution with respect to its area is known. Secondly, another approach to

estimate fracture volumetric intensity, based on the cycloidal scanline technique, is also

discussed. By counting the intersections between cycloidal scanlines and fracture traces

on the circular cylinder surface, the fracture volumetric intensity can be estimated

without knowing the orientation of fractures. Finally Monte Carlo simulations are carried

out to verify the derived correction factors.

3.2 Basic assumptions

In this paper, we study a fractured rock mass sampled by a borehole or tunnel/shaft by

using stereology. For convenience, we make the following assumptions with respect to

the geometry of the sampling domain, e.g., the surface of the tunnel/shaft or borehole;

and of fractures in the rock mass.

a) The surface of the sampling domain is a right circular cylinder, long in relation to

its diameter. Borehole, tunnel or shaft ends are not included in the sampling

domain.

b) Fractures are planar features with negligible thickness. No assumptions are made

regarding the spatial distribution of fractures, or fracture shape. In particular, it is

not necessary that fracture centers follow a Poisson process, or that fractures have

the shape of circular or elliptical discs.

c) No prior assumptions are made about fracture size, or orientation distribution;

however, for the first method discussed below, the fracture orientation distribution

in terms of area must be known.

d) The sampling domain is independent of the rock mass fracture network to be

characterized. What this means in practical terms is the borehole/shaft or tunnel is

emplaced without consideration of fracture locations.

36

The above assumptions are fairly standard in engineering analysis of fractured rock

masses (Priest & Hudson, 1976; Warburton, 1980; Cheeney, 1983; Dershowitz, 1984;

Priest 1993; Mauldon & Mauldon, 1997). Furthermore, these assumptions are applicable

in most rock engineering situations either because of the lack of knowledge of

underground fracture networks before boreholes are excavated or, because the location of

a tunnel or shaft is predetermined, based on external factors.

In accordance with principles of stereology, the 1-d, 2-d and 3-d fracture intensities

discussed in this paper refer to expected values, if not specified otherwise. The acronym

IUR - isotropic, uniform, random denotes, in general, desirable properties of

stereological samples (Russ and DeHoff, 2000; Mauton 2002). In the present situation,

isotropy is ensured in the plane perpendicular to the borehole/shaft/tunnel axis by the

circular symmetry of the cylinder; the directional relationship between the cylinder axis

and the fracture system, however, is not in general, one of isotropy, except in the special

case of a uniform fracture orientation distribution. One of the primary tasks of this paper

is to account for the directional relationship between cylinder and fractures, with respect

to the determination of fracture intensity.

3.3 General form of the relationship between areal intensity P21,C and

volumetric intensity P32 for right circular cylinders

In this section, we relate the volumetric fracture intensity measure P32 (fracture area per

unit rock mass volume) to the areal fracture intensity measure P21,C as measured on a

cylinder (fracture trace length per unit sampling surface area). The relationship is

presented here in a general form.

We define a geometric correction factor, C23,C by

CC PCP ,21,2332 = , (3.1)

37

where the subscript 23 denotes conversion from a two-dimensional to a three-

dimensional measure, and the subscript C denotes a cylindrical surface sampling domain.

The conversion factor C23,C is a function of cylinder orientation and the fracture

orientation distribution; it does not depend on cylinder radius, as demonstrated in next

section.

The geometric meaning of this conversion factor can be illustrated using a simple model

of a cylindrical surface sample (Fig. 3.2), in which five fractures are sampled by a

vertical shaft of radius r and height H. Let l denote the total summed trace length on the

shaft surface. Given a population of fractures, l is a function of cylinder orientation,

radius r, and height H; and the area-weighted fracture orientation distribution.

Consider a thin cylindrical shell (Fig. 3.2) with radius r. The shell thickness dr is taken to

be infinitesimal, so that the area of fractures contained inside the shell, Afractures, can be

approximated as

drlCA Cfractures = ,23 . (3.2)where C23,C is the geometric correction factor. If the fractures are perpendicular to the

circular cylinder surface at the intersections, this correction factor is 1.0 (and the

expression is exact). Otherwise, it is greater than 1.0.

The volumetric fracture intensity measure P32, fracture area per unit volume, for the shell

can be expressed as

drHrA

P fractures= 232 (3.3)

Substituting Eq. (3.2) into (3.3), we obtain

CCC PCHrlCP ,21,23,2332 2

== . (3.4)

38

Fig. 3.2. Fracture traces on a cylindrical shaft. Intersections between fractures and the

shaft are traces (curved line segments) on the shaft surface

This is the general form of the relationship between the 2-d intensity measure for trace

length and the 3-d intensity measure for fracture area in a rock mass. In the following we

derive the correction factor C23,C for the general case of fractures that are distributed

according to a known probability density function for fracture orientation with respect to

fracture area. Then we discuss two special cases: fractures of constant orientation and

fracture orientations uniformly distributed in the rock mass.

3.4 General case of cylindrical sampling As discussed in section 2, the sampling cylinders radius, orientation and location is

assumed independent of the rock mass and fracture geometry. Let f(,) denote the probability density function (pdf) of fracture orientation weighted by area, where is the acute angle between the sampling cylinder axis (Z-axis in Fig. 3.3(a)) and the normal n to

a fracture; and is the angle between the Y-axis and the projection of the fracture normal n onto the XY plane (Fig. 3.3(a)).

Shaft with radius r, shell thickness dr

H Fracture traces, total

39

Fig. 3.3. A thin slice of the shell sampling in fractures. The total trace length on its

surface is dl. (a) A cylindrical shell (axis Z) intersects a set of fractures with orientation

distributed as f(,). For a fracture with unit normal n, is the angle between Y-axis and the projection of n on the XY plane; is the angle between n and Z. (b) A portion (unit height) of a slice from the shell is taken out for study. The ith fracture intersected with the

portion has a unit normal ni and the trace of this fracture on the circular cylinder surface

is represented by a unit vector Ti. The figure above shows the vectors in a lower

hemisphere projection.

Y

Z

Y

Z

X

S

n

d dr

Y

Z

rd

1 i ninpi

S

Ti

(a)

Y

S

Z

npi

Ti

ni

i

i LH

(b)

40

Consider a thin, narrow slice of unit length, width = rd and thickness = dr, taken out from the shell (Fig. 3.3(b)). Let dli denote the length of the trace of fracture i on the

outside surface of the slice; let Ti be the unit vector representing the direction of the corresponding fracture trace on the slice surface; and let npi denote the unit normal to a

plane passing through the trace, and perpendicular to the slice surface (npi is the

normalized vector of the cross product STi). Finally, let i be the angle between npi and the normal ni of fracture i. Then the infinitesimal area dAi of fracture i inside the slice is

i

ii

drdldA cos

= . (3.5)

Notice that i varies for different fractures intersecting the same slice, and for the same fracture intersecting by different slices from the cylindrical shell.

The expected area dAi of fracture i inside the unit length slice can also be expressed in

terms of P32 and the probability density function f(,), drrdfPdA iii = 1),(32 , (3.6)

where i and i are the angles representing the orientation of the normal to fracture i in the coordinate system shown in Fig. 3.3(a). Equating Eqs. (3.5) and (3.6), the expected

trace length dli of fracture i on the unit slice surface is found to be,

iiii rdfPdl cos),(32= . (3.7)

The expected total length dl of fracture trace segments on the outer cylindrical surface

contained within the slice of height H is the integration of trace lengths of all fractures

intersecting the slice, with respect to fracture orientation:

ddfHrdPdl cos),(,

32 = . (3.8)

41

where is the angle between the normal n to a fracture and the normal np to the plane passing through the trace of the fracture and perpendicular to the slice surface. Note that

is a function of , and (Appendix 3.A) and that, in this context, and d are constant.

Denote the integral in Eq. (3.8) as

ddfIo cos),(,= , (3.9)

where Io is a function of f(, ) and the orientation of the cylinder axis. For this general case, cos is determined in Appendix 3.A as

( ) += 222 sinsincoscos , (3.10)so that

oIHrdPdl = 32 . (3.11)

The expected total trace length l on the cylindrical sampling surface is obtained by

integrating dl over all values of ,

==

2

032 dIHrPdll o .

(3.12)

The fracture areal intensity on the cylinder surface can be expressed as

===

2

0

322

0

32,21 222

dIPdIrHHrP

rHlP ooC

(3.13)

Then the conversion factor C23,C relating areal intensity on a cylinder to volumetric

intensity (c.f. Eq. (3.1)) is given by

42

12

0,23 2

=

dIC oC . (3.14)

For this general case, Eq. (3.14) shows that the conversion factor C23,C is dependent

neither on the size of the circular cylinder surface, nor on fracture shape. It is a function

of the orientation of cylinder axis and the area-weighted fracture orientation pdf f(,). The range of the conversion factor will be discussed in the next section.

3.5 Special case: Sampling fractures of constant orientation

When a cylindrical surface samples a set of fractures with constant orientation, we can

always choose the Y-direction so that cylinder axis Z, fracture normal n, and the Y-axis

are coplanar (Fig. 3.4). In this coordinate system, angle between Y and the projection of n on the XY plane is 0. Let 0 denote the acute angle between n and Z (Fig. 3.4). It is a constant in this context.

For fractures with constant orientation, cos is determined to be (Appendix 3.A)

202 cossin1cos = . (3.15)

Note that cos is not a function of either or . Then the integral in Eq.(3.10) is

ddfddfIo ==,,

),(coscos),(

202 cossin1cos == . (3.16)

43

Fig. 3.4. A cylindrical shell (axis Z, height = H) intersects a set of fractures with constant

orientation (normal n). 0 is the angle between the fracture normal and the cylinder axis.

From Eq. (3.14),

12

0

20

212

0,23 cossin122

=

=

ddIC oC (3.17)

Evaluating the above integral using Mathematica (Wolfram Research, Inc, 2004), we

obtain

Y

Z

0 n

H

X

44

)(sincossin1 02

2

0

20

241

EllipticEd = , (3.18)

where )(sin 02 EllipticE is a complete elliptic integral of the second kind.

Combining Eqs. (3.18) and (3.14), the conversion factor C23,C relating areal intensity on

a cylinder to volumetric intensity is

[ ] 102,23 )(sin2 = EllipticEC C . (3.19)

The conversion factor C23,C takes on values ranging from 1 to /2 for 0 ranging from 0 to 90, respectively (Fig. 3.5). Note in particular that fracture volumetric intensity P32 is

equal to fracture areal intensity P21,C on the cylinder surface if fractures are perpendicular

to the sampling cylinder(C23,C = 1); and P32 is 1.57 times fracture areal intensity P21,C on

the cylinder surface if fractures are parallel to the cylinder axis (C23,C = /2).

It should be noted that the case above of constant fracture orientation is the least isotropic

of all orientation distributions and that the above orientations of the fractures relevant to

the cylinder i.e. parallel and perpendicular to the cylinder axis, also represent extreme

cases. Therefore, for a general case of fracture orientation distribution, the conversion

factor C23,C is in the range [1, /2] as well. This result is very important to rock engineering practitioners, especially when there is not much information about the

fracture orientation distribution with respect to area. Since the range of the conversion

factor C23,C is fairly small (1.0 to 1.57), it will be convenient and will not cause major

errors to approximate the fracture volumetric intensity P32 by using Eq. (3.19) or Fig. 3.5,

where 0 is estimated as the average acute angle between fractures and the sampling cylinder axis.

Finally, for the special case of constant fracture orientation, Eq. (3.19) shows clearly that

the conversion factor C23,C is only a function of the angle between the cylinder axis and

45

the fracture normal. It is independent of the radius of the sampling cylinder, as well as of

fracture shape and size.

Fig. 3.5. For cylindrical sampling in fractures with constant orientation, the correction

factor C23,C between areal intensity P21,C and volumetric intensity P32 is a function of

angle 0 between the cylinder axis and fracture normal. The elliptic integral required to obtain the curve was evaluated using Mathematica.

3.6 Special case: fractures with uniform orientation distribution

We apply the general result of the conversion factor C23,C to the isotropic case, in which

fracture orientations are uniformly distributed with respect to area.

In this case,

0 1 2 3 4 5 6 7 8 90 (degree)

Cor

rect

ion

fact

or C

23,C

/2

1.

1.1

1.2

1.3

1.4

1.5

1.6

46

Therefore, the integral in Eq.(3.9) is

( )

dd

ddfIo

=

=

=

=

+=

=2/

0

2

0

222

,

sinsincos2

sin

cos),(

(3.22)

And from Eq. (3.14),

( )1

2

0

2

0

2

0

222

12

0,23

sinsincos2

sin2

2

=

=

=

=

=

=

+=

=

ddd

dIC oC

(3.23)

The definite integral in Eq. (3.23) was evaluated in Mathematica (Wolfram Research, Inc,

2004), which gives

( )2

9348.4sinsincos2

sin 22

0

2

0

2

0

222

=+ =

=

=

=

=

=ddd

(3.24)

So,

2sin),( =f , and from Appendix 3.A (3.20)

( ) += 222 sinsincoscos . (3.21)

47

( )[ ]

412

12

0

2

0

2

0

222,23

2/2

sinsincos2

sin2

=

+=

=

=

=

=

=

= dddC C

(3.25)

This result can be compared with the results for plane sampling of isotropically

distributed fractures (Dershowitz, 1984), namely

32)( 21 4PP isotropic

= , (3.26)

where P21(isotropic) is the trace length per unit area of sampling plane.

3.7 Cycloidal Scanline Technique

In this section we discuss a sampling technique that uses a special curved scanline based

on a cycloid, which automatically takes care of the directional bias described by Terzaghi

(1965). By correctly deploying cycloidal scanlines on the cylindrical surface, we can

make an unbiased estimate of fracture volumetric intensity with no need to know the

orientation of fractures (either ahead of time or at the time of sampling).

3.7.1 Unbiased sampling criterion

As mentioned earlier, a basic strategy in stereology involves the use of IUR (Isotropic-

Uniform-Random) sampling (Russ and Dehoff, 2000). A perfectly isotropic 2-d

sampling surface is a sphere, on which the surface area is distributed uniformly with

respect to direction. Similarly, on a plane, a circular scanline is a perfectly isotropic 1-D

48

sampling domain, with length segments uniformly distributed in every direction

(Mauldon et al., 2001). IUR scanlines produce unbiased samples automatically, and thus

obviate the need for any bias correction.

Fig. 3.6(a) shows uniformly distributed unit vectors (directed line segments) on a

hemisphere. Let denote the angle (colatitude) between a unit vector and axis Z. If the unit vectors have a uniform orientation distribution, the probability p( ) of choosing a line segment of unit length and along a vector with colatitude must be proportional to l() = 2 sin ( ) (Fig. 3.6(a)). Choosing a normalizing constant such that p( ) d has the value unity when integrated over all values of (0 to ) for vectors uniformly distributed in all orientations), we have

sin)( 21=p (3.27)

As an alternative to selecting scanline orientations from a probability distribution, it is

possible to specify a curved scanline (Fig. 3.6(b)) that utilizes all values of (0 ) with differential scanline arc lengths dL() proportional in all cases to sin , or

sin)( dL . (3.28)

One form of scanline that has this property is the cycloid, which we discuss in the next

section.

49

Fig. 3.6. Illustration of linear (vector) IUR sampling in 3-d space. (a) Uniformly oriented

unit vectors on a hemisphere. (b) Length-scaled vectors on the cylinder surface.

From stereological principles, (Russ and DeHoff, 2000; Dershowitz, 1984; Mauldon and

Wang 2003), linear fracture intensity P10(unbiased) measured on the cylindrical surface by

such unbiased sampling probes (scanlines) has the following relationship with the

volumetric fracture intensity P32.

3221

)(10 PP unbiased = . (3.29)

(a)

l = 2sin sin

Z

Z

Cycloidal curve

(b)

50

3.7.2 Cycloidal scanlines

Mathematically, a cycloid is the locus of a point on the rim of a circle rolling along a

straight line, as shown in Fig. 3.7. For a generating circle of radius r0, the coordinates of a

point on the cycloid are given by:

==

)cos1()sin('

0

0

rzrx

(3.30)

where is the angle of rotation of the circle.

One of the properties of a cycloid is that for any point on the cycloid with angle between Z and the tangent to the cycloid, the incremental arc length dl is proportional to sin ( drld sin4 0= ). Therefore the cycloid as a sampling probe satisfies Eq. (3.28) and can be used as a directionally unbiased (IUR) sampling probe for measuring fracture

intensity on the walls of a borehole or tunnel/shaft. In other words, cycloids can be used

as scanlines on (right-circular) cylindrical surfaces without the need to correct for

directional sampling bias, and without the need to know fracture orientation. In practice,

cycloidal scanlines can be modified in various ways for more efficient deployment (Russ

and DeHoff, 2000; Mauldon and Wang 2003), as long as the correct relationship between

arc length and orientation is maintained. The fracture volumetric intensity P32 can then

be estimated by Eq. (3.29) - which in terms of expected values is an exact expression.

3.8 Monte Carlo Simulations

A computer program was developed in Visual C++ and used to generate a population of

synthetic fractures, of rectangular shape, intersecting a cylindrical surface such as the

wall of a borehole, tunnel or shaft (Fig. 3.8). The fracture traces are computed and shown

on the unwrapped cylindrical surface (right-hand window in Fig. 3.8). For each

simulation, the total number of generated fractures, the area of each fracture, as well as

the size of the generation region were recorded, in order to calculate the volumetric

51

fracture intensity P32. Total fracture trace length on the circular cylinder surface was also

recorded to calculate P21,C, the 2-d intensity on the sampling circular cylinder, by

dividing by total cylinder surface area.

Fig. 3.7. The cycloid (heavy curve) is the path of a point on the circle of radius r0 as the

circle rolls from left to right along the x-axis.

Five cases are chosen, to represent different fracture sizes and shapes intersecting a

cylinder of constant size (Fig. 3.9). In each case, the angle 0 between fracture normal and cylinder axis, is set to be 0, 30, 60, and 90, respectively. Ten simulations were run

for each fracture orientation. The parameters for each case and the results of the

simulations are listed in Table 3.1. For comparison, the conversion factor C23,C calculated

by Eq. (3.19) for each 0 is also listed in Table 3.1. In all the simulations, fracture volumetric intensity P32 was set constant, P32 = 1.0.

Z

Generating circle

Cycloid

d

X

r0

dl

drld sin4 0=

52

The simulation results are plotted in Fig. 3.10, where they are compared with the curve of

C23,C computed by Eq. (3.19). The simulations show that for fractures with constant

orientation, the areal fracture intensity measure on a cylindrical surface P21,C (trace length

per unit cylinder surface area), is related to the volumetric fracture intensity P32 (fracture

area per unit volume), only by angle 0 between the fracture normal and the cylinder axis. The conversion factor is independent of the cylinder radius, as well as of the size or shape

of fractures. The derived conversion factor, expressed by Eq. (3.19), is also verified from

the simulations.

Fig. 3.8. The computer program is used to generate rectangular fractures intersecting with

a borehole. Fracture orientation can be set to constant or vary according to given

parameters.

Fractures

Borehole Trace map

53

Fig. 3.9. Illustration (to the scale) of the five cases studied. Shaded rectangles are

simulated fractures, and circles are sampling cylinders (radius is constant 10 for all

simulations).

Table 3.1. Simulation parameters and results.

Average C23,C for each case

Fracture

length l

Fracture

width w

Aspect

ratio l /w 0 = 0 0 = 30 0 = 60 0 = 90 Case 1 100 100 1.0 1.00 1.04 1.31 1.53

Case 2 10 10 1.0 1.01 1.07 1.29 1.56

Case 3 20 20 1.0 1.01 1.06 1.28 1.62

Case 4 100 20 5.0 1.04 1.09 1.32 1.56

Case 5 20 4 5.0 0.99 1.06 1.30 1.57

Average C23,C for each angle 0 1.01 1.07 1.30 1.57 C23,C calculated by Eq. (3.19) 1.00 1.07 1.30 1.57

Case # 1 2 3 4 5

Cylinder radius 10 10 10 10 10

Fracture length l 100 10 20 100 20

Fracture width w 100 10 20 20 4

Aspect ratio l /w

1 1 1 5 5

54

Fig. 3.10. Simulation results of the conversion factor 1/ C23,C, compared with the

calculated curve by Eq.(3.19).

3.9 Discussion & Conclusions

In this paper, we used stereological principles to study the conversion factor between the

2-d fracture intensity measure on a cylinder surface and the 3-d fracture volumetric

intensity measure. The derived conversion factor between the two intensity measures is

not dependent on fracture size, shape or circular cylinder radius, but is related to the

orientation of the cylinder and the distribution of fracture area with respect to its

orientation.

0 1 2 3 4 5 6 7 8 9

Case 1 Case 2 Case 3 Case 4 Case 5

1.6

1.5

1.4

1.3

1.2

1.1

1.0

C23,C calculated by Eq.(3.19)

C23,C from simulations

0 (degree)

Con

vers

ion

fact

or C

23,C

55

By studying a special case of cylindrical sampling of fractures with constant orientation,

it is found that the fracture volumetric intensity measure P32 is always 1.0 to 1.57 times of

fracture trace length per unit borehole surface area (P21,C). The two values are also the

minimum and maximum limit of the conversion factor between the two measures in a

general case of cylindrical sampling of fractures, which provides a very practical means

in the field to estimate fracture volumetric intensity.

Based on Isotropic-Uniform-Random principle of stereology, cycloidal scanlines, as

directional unbiased probes to estimate the fracture volumetric intensity, is also

introduced in this paper.

A computer program simulating synthetic fractures sampled by a circular cylinder was

developed and the derived conversion factor between the two intensity measures is

confirmed by Monte Carlo simulations.

Acknowledgements

Partial support from the National Science Foundation, Grant Number CMS-0085093, is

gratefully acknowledged.

56

Appendix 3.A Determine |cos |

Fig. 3.A-1 shows the unit vectors S, T, n, nr in a Cartesian coordinate system.

In this coordinate system, Z represents the borehole or sampling circular cylinder axis. n

is the normal to a fracture and it makes an acute angle with Z-axis. is the angle between Y-axis and the projection of n on XY. S is a unit normal to a small slice of th