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, π/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) )(αΑf of angle α ........................23

Appendix 2.B Numerical approach for obtaining FΑ(α) and FΒ(β) ...........................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 γ| .................................................................................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 α and β are

proportional to the sin α and sin β, 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. pdf’s of Fisher distribution with κ = 20, 40, and 100......................................... 10

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

constant κ . ......................................................................................................... 12

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

constant κ . ......................................................................................................... 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 κ . ......................................................................................................... 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, Rθ , of θ , as a function of δ, α and ρ. Angle ρ

(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 β0 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. κ and ρ ................................................................................... 12

Table 2.2. Factor 1/C23 with different values of κ and ρ. ................................................. 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

This page intentionally left blank.

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

This page intentionally left blank.

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

( ) αααπ

dfPP Α∫= |cos|0

3210 and (2.1)

( ) βββπ

dfPP Β∫=0

3221 sin , (2.2)

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

the angle between the sampling plane normal and fracture normal (Fig. 2.1b); and

fΑ(α) and fΒ(β) are the probability density functions (pdf’s) of α and β, 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 α or β 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

α

Fracture

β

Fracture normal

Sampling plane normal

FractureSampling plane

6

Define conversion factors C13 and C23 by

( )1

013 |cos|

−

Α ⎥⎦

⎤⎢⎣

⎡= ∫ ααα

π

dfC and (2.3)

( )1

023 sin

−

Β ⎥⎦

⎤⎢⎣

⎡= ∫ βββ

π

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,

αα sin)( 21=Αf and ββ sin)( 2

1=Βf (Fig. 2.2), for α and β in the range [0, π].

Introducing these pdf’s into (3) and (4), respectively, we have 1

021

)( 13 |cos|sin−

⎥⎦

⎤⎢⎣

⎡= ∫

π

ααα dC isotropic and (2.7)

( )1

0

1

0

221

)( 23 2cos141sin

−−

⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡= ∫∫ ββββ

ππ

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 ⎟

⎠⎞

⎜⎝⎛=

π. (2.10)

7

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

proportional to the sin α and sin β, 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

α

length ∝ sinα

Samplingplane

Fracture

Fracture normal

Sampling plane normal

βlength ∝

8

)0( )( πδδκδ κκ

δκ

≤≤−

= −∆ eeSinef

Cos

, (2.11)

where δ is the angle between a fracture normal and the Fisher mean pole (Fig. 2.3); f∆(δ)

is the probability density function of δ; and κ is the Fisher constant related to the amount

of dispersion (κ 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 δ for a given dispersion constant.

The local azimuth of the Fisher mean pole is uniform on [0, 2π] and is independent of δ.

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 80º and dip-

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

The theoretical range of κ is from 0 to ∞, with low values indicating a high degree of

dispersion. As κ 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 fΑ(α) , the probability density function of angle α

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 fΑ(α) is given by (see Appendix 2.A):

δδκ

ρδαρδ

απ

α κκ

δκ

δ

deeSinef

Cos

R−Α −−−

= ∫ 222 )coscos(cossinsin

sin 1)( (2.12)

for α in the range ρδαρδ +≤≤− || , where the range of integration Rd is given by:

],-[ αραρδ +=R , if ρα ≤ , or

]2 ,0[ ραδ −=R , if ρα > . (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 α, δ, and ρ 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. pdf’s of Fisher distribution with κ = 20, 40, and 100.

01234567

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

κ = 100

κ = 40

κ = 20

f(δ)

North

20º

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

Upper Hemisphere Equal Area

Sampling line s (trend 225º, plunge 45º)

Small circle with α = 20º

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 κ and angle ρ. The reciprocal of C13, rather than C13, is tabulated in

order that values range between 0 and 1. When κ is relatively small (κ < 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 ρ (Fig. 2.6). Regression coefficients a, b and c can be

computed for κ ≥ 1, according to the logarithmic expression given in Fig. 2.6. For κ < 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 κ and ρ. As with the case of linear sampling, for a given value of ρ, the

conversion factor 1/C23 is relatively insensitive to changes in κ for κ > 50. When κ is

relatively small (κ < 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 πρ (Fig. 2.7). Regression coefficients a, b, c and d can be

computed for κ ≥ 1, according to the logarithmic expression given in Fig. 2.7. For κ < 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. κ and ρ

ρ κ 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 ρ

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

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

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.1 1 10 100 1000κ

Coe

ffici

ent

a b

c

cbaC += )cos(/1 13 ρ

2.0951 + )ln( 0.1655

- = κ

b

0.0507 + )ln( 0.1247 = κ

a

0.5988 + )ln( 0.0551- = κ

c

13

Table 2.2. Factor 1/C23 with different values of κ and ρ.

ρ κ 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 ρ

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

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

0.0

0.5

1.0

1.5

2.0

2.5

0.1 1 10 100 1000κ

Coe

ffici

ent

b

a

c d

cdbaC +−= )2/sin(/1 23 πρ

0.0351 + )ln( 0.1064=

κa

0.8112 + )ln( -0.0771= κ

c

2.1745 + )

ln( -0.1376

=

κ

b

1.0783 + )ln( -0.1297

= κ

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 ψ 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 δ 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

( )∫ ∆=max

min

)(/1/1 23,23

δ

δ

δδ dfCC C , (2.15)

where f∆(δ) is the pdf of δ and the integration is carried out over the full range of δ. It

should be noted that C23 in Eq. (2.15) is a function of δ 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

θψδ cossincos =•= cm , (2.16)

from which

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

ψδθ

sincoscos 1

. (2.17)

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

2π] because of the radial symmetry of the cylinder (Fig. 2.9). The pdf )(θΘf of θ is

given by

πθ

21)( =Θf . (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

coδ

ψ

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 )(δ∆F of δ is

∫ Θ∆ =<∆=θ

θθδδR

dfProbF

)()( )( , (2.19)

where Rθ is the range of θ corresponding to ∆ < δ, with limits determined by Eq. (2.17)

for specified ψ and δ. Noting the symmetry of the range of θ with respect to y-axis, and

utilizing Eq. (2.17),

∫⎟⎠⎞⎜

⎝⎛

∆

−

=ψδ

θπ

δsincos1cos

0 212)( dF , (2.20)

from which

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

∆ ψδ

πδ

sincoscos1)( 1F , ,20 πθ ≤≤ ( ) ( )ψπδψπ +≤≤− 2/2/ . (2.21)

The pdf )(δ∆f of δ is found by differentiation as

xc

y δ

z

m

ψ

θ o

(0, sin ψ, cos ψ)

(sin θ, cos θ, 0)

1

17

( ) 2/122 sin/cos1sinsin1)()(

ψδψ

δπ

δδ−

=′= ∆∆ Ff , ( ) ( )ψπδψπ +≤≤− 2/2/ . (2.22)

Substituting Eq. (2.22) into Eq. (2.15), we obtain

( ) ( )∫−

=max

min

2/12223,23sin/cos1sin

sin/11/1δ

δ ψδψ

δδπ

dCC C , (2.23)

where ( ) ( )ψπδδψπδ +=≤≤−= 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 κ and ψ. It is interesting to note that when angle

ψ (between the Fisher mean pole and the cylinder axis) is around 60º, the conversion

factor 1/C23,C is relatively insensitive to κ and has the value ≈ 0.79, which is also the

value obtained (π/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/π, the theoretical solution for κ → ∞) 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 ψ (Fig. 2.10). Regression coefficients a, b and c can

be computed for κ ≥ 1 according to the logarithmic expression given in Fig. 2.10. For κ <

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.

ψ κ

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

0.0

0.5

1.0

1.5

2.0

2.5

0.1 1 10 100 1000κ

Coe

ffici

ents

a

b

c

cbaC C += )cos(/1 ,23 ψ

0.0359 + )ln( 0.0274= κa

2.1971 + )ln( -0.0944= κb

0.8045 + )ln( 0.0026= κc

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 180º, plunge 0º) 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 κ can be

approximated by:

)1(1 R−=κ , (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 61.2º and dip 86.5º. The angle ρ between the

Fisher mean pole and the scanline is calculated to be about 61.0º. The mean resultant

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

which gives κ = 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, ρ ≈ 61.0º and κ ≈ 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 11

101332 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, π/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) )(αΑf of angle α

Angles ρ, α, and δ 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 α, 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 θ. For the spherical triangle formed by m, n, and s, the following

relationship holds (Ayres, 1954):

ρδρδθα coscossinsincoscos += , (2.A-1)

so that,

⎟⎟⎠

⎞⎜⎜⎝

⎛ −= −

ρδρδα

θsinsin

coscoscoscos 1

. (2.A-2)

The probability density function of angle α depends on δ and θ (ρ being kept constant in

the derivation). For the Fisher distribution, the joint pdf of angle δ and θ is given by (N.

Fisher et al., 1987)

)0( )(2

),(, πδπ

δκθδ κκ

δκ

≤≤−

= −Θ∆ eeSinef

Cos

. (2.A-3)

24

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

Angle θ is uniformly distributed in the range [0, 2π] and is independent of angle δ.

Therefore the pdf of θ is

πθ 2/1)( =Θf , (2.A-4)

from which,

)0( )( πδδκδ κκ

δκ

≤≤−

= −∆ eeSinef

Cos

. (2.A-5)

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

Given δ, the cdf of α is

o

Xy

z

n

ms

n′s′θ

αδρ

25

∫ ΘΑ =≤Α=θ

θθδαδαδ RdfProbF )()|()|(| , (2.A-6)

where Rθ is the range of θ when Α ≤ α. Given δ, α increases from the minimum of |δ-ρ|

to the maximum of δ+ρ when θ increases from 0 to π. Fig. 2.A-2 shows the relationship

among angles α, δ, ρ, and also the range of θ. Note that Rθ is symmetric about the y-axis.

Below is the determination of Rθ with different range of α, δ, and ρ.

:ρα ≤

lse

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

0

],0[ max

eR ⎩

⎨⎧

≥+≥++

∈

⎪⎩

⎪⎨⎧

= παρπαρπαραραρ

δθθ ,

:ρα ≥

=θR⎪⎪⎩

⎪⎪⎨

⎧⎩⎨⎧

≥+<++−

∈/2 )( if /2],,-[ /2 )( if ],,[

],0[ max παρπαρπαρραρα

δθ

]- [0, 2 αρδπ ∈

/2] ),[ 0 παρδ +∈

(2.A-7)

where θmax is the upper limit of θ, which is determined by Eq.(2.A-2), from which

⎟⎟⎠

⎞⎜⎜⎝

⎛ −= −

ρδρδαθ

sinsincoscoscoscos 1

max . (2.A-8)

Then

θπ

δα ρδρδα

δ dF ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −

Α

−

=≤Α sinsincoscoscoscos

0|

1

212)|( , ρδαρδ +≤≤− || , (2.A-9)

from which,

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=≤Α −

Α ρδρδα

πδαδ sinsin

coscoscoscos1)|( 1|F , ρδαρδ +≤≤− || , and (2.A-10)

222||)coscos(cossinsin

sin)|()|(ρδαρδπ

αδαδα δδ−−

=≤Α′= ΑΑ Ff ,

ρδαρδ +≤≤− || . (2.A-11)

26

Then, the pdf of α is

δδκ

ρδαρδ

απ

δδδαα κκ

δκ

δδδ

deeSinedfff

Cos

RR−∆ΑΑ −−−

== ∫∫ 222|)coscos(cossinsin

sin 1)()|()(

ρδαρδ +≤≤− || , (2.A-12)

where Rd is given by

],-[ αραρδ +=R , if ρα ≤ , or

]2 ,0[ ραδ −=R , if ρα > . (2.A-13)

Fig. 2.A-2 The figure shows the range, Rθ , of θ , as a function of δ, α and ρ. Angle ρ

(between m and s) is a constant. Angles δ and α are the semi-apical angles of small

circles about z and y, respectively. Rθ delimits the intersection of the above-mentioned

small circles, projected into the xy plane.

Rθ

27

Appendix 2.B Numerical approach for obtaining FΑ(α) and FΒ(β)

Rewrite Eqs. (2.3) and (2.4) as

( )1

013 cos

−

⎥⎦

⎤⎢⎣

⎡= ∫ αα α

π

dFC , and (2.B-1)

( )1

023 sin

−

⎥⎦

⎤⎢⎣

⎡= ∫ ββ β

π

dFC , (2.B-2)

where FΑ(α) and FΒ(β) are the cdf’s of α and β, respectively. Numerical evaluation of the

finite integrals in the equations gives the conversion factors C13 and C23, using the

procedure described below for FΑ(α). FΒ(β) can be obtained through a similar procedure.

1. Set the mean pole and the Fisher constant κ 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

)( πδκδ κ

δκ

≤≤−

=∆ 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 80º and dip-direction 45º, and κ = 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 α at

fixed increments.

4. The cdf FΑ(α) of α, 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 α = 20º. Therefore the cdf FΑ(α) of α, evaluated at α = 20º, is

0.035 3000104)20( ==°=Α αF . (2.B-4)

28

References

Ayres, F. Jr. (1954) “Schaum’s 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: 21–51

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

This page intentionally left blank.

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

⋅⋅=

π232 (3.3)

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

CCC PCHr

lCP ,21,23,2332 2=

⋅=

π . (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 cylinder’s radius, orientation and location is

assumed independent of the rock mass and fracture geometry. Let f(α,β) denote the

probability density function (pdf) of fracture orientation weighted by area, where β is the

acute angle between the sampling cylinder axis (Z-axis in Fig. 3.3(a)) and the normal n to

a fracture; and α 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(α,β). For a fracture with unit normal n, α is the angle between Y-axis and

the projection of n on the XY plane; β 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

β

X α

S

n

dθ

dr

θ

Y

Z

rdθ

1γi

ni

npi S

Ti

(a)

Y

S

Z

npi

Ti θ

ni

βi

γi LH

(b)

40

Consider a thin, narrow slice of unit length, width = rdθ 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 S×Ti). Finally, let γi 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

γcos⋅

= . (3.5)

Notice that γi 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(α,β),

drrdfPdA iii ⋅⋅⋅⋅= θβα 1),(32 , (3.6)

where αi and βi 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 γθβα 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:

βαγβαθβα

ddfHrdPdl cos),(,

32 ∫∫= . (3.8)

41

where γ 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

γ is a function of θ, α and β (Appendix 3.A) and that, in this context, θ and dθ are

constant.

Denote the integral in Eq. (3.8) as

βαγβαβα

ddfIo cos),(,∫∫= , (3.9)

where Io is a function of f(α, β) and the orientation of the cylinder axis. For this general

case, γcos is determined in Appendix 3.A as

( )αθββγ −+= 222 sinsincoscos , (3.10)

so that

oIHrdPdl ⋅= θ32 . (3.11)

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

integrating dl over all values of θ,

∫∫ ==π

θ2

032 dIHrPdll o .

(3.12)

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

∫∫ ===ππ

θπ

θππ

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

−

⎥⎦

⎤⎢⎣

⎡= ∫

π

θπ 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(α,β).

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 α between Y and the projection

of n on the XY plane is 0. Let β0 denote the acute angle between n and Z (Fig. 3.4). It is

a constant in this context.

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

θβγ 20

2 cossin1cos −= . (3.15)

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

βαβαγβαγβαβαβα

ddfddfIo ∫∫∫∫ ==,,

),(coscos),(

θβγ 20

2 cossin1cos −== . (3.16)

43

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

orientation (normal n). β0 is the angle between the fracture normal and the cylinder axis.

From Eq. (3.14),

12

0

20

212

0,23 cossin122

−−

⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡= ∫∫

ππ

θθβπθπ ddIC oC (3.17)

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

obtain

Y

Z

β0n

H

X

44

)(sincossin1 02

2

0

20

241 βθθβ

π

EllipticEd =−∫ , (3.18)

where )(sin 02 βEllipticE 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

[ ] 10

2,23 )(sin

2−

= βπ EllipticEC C . (3.19)

The conversion factor C23,C takes on values ranging from 1 to π/2 for β0 ranging from 0º

to 90º, 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 = π/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, π/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 β0 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 β0 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 9β0 (degree)

Cor

rect

ion

fact

or C

23,C

π /2

1.

1.1

1.2

1.3

1.4

1.5

1.6

46

Therefore, the integral in Eq.(3.9) is

( ) βααθββπβ

βαγβα

πβ

β

πα

α

βα

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

−=

=

=

=

=

=

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+=

⎥⎦

⎤⎢⎣

⎡=

∫ ∫ ∫

∫πθ

θ

πβ

β

πα

α

π

θβααθββπβπ

θπ

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 πθβααθββπβπθ

θ

πβ

β

πα

α

≈=−+∫ ∫ ∫=

=

=

=

=

=

ddd (3.24)

So,

πββα

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

( )αθββγ −+= 222 sinsincoscos . (3.21)

47

( )

[ ] π

πθ

θ

πβ

β

πα

α

ππ

θβααθββπβπ

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 ⎟

⎠⎞

⎜⎝⎛=

π, (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 ψ denote the angle (colatitude) between a unit vector and axis Z. If the

unit vectors have a uniform orientation distribution, the probability p(ψ ) of choosing a

line segment of unit length and along a vector with colatitude ψ ′ must be proportional to

l(ψ) = 2π sin (ψ ) (Fig. 3.6(a)). Choosing a normalizing constant such that ∫p(ψ ) dψ

has the value unity when integrated over all values of ψ (0 to π) for vectors uniformly

distributed in all orientations), we have

ψψ 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 ψ (0 ≤ ψ ≤ π)

with differential scanline arc lengths dL(ψ) proportional in all cases to sinψ , or

ψψ 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 = 2πsin ψsin ψ ψ

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

0φφφ

rzrx

(3.30)

where φ 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 ψ

between Z and the tangent to the cycloid, the incremental arc length dl′ is proportional to

sinψ ( ψψdrld 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 β0 between fracture normal

and cylinder axis, is set to be 0º, 30º, 60º, and 90º, 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 β0 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

ψdψ

φ

X′

r0

dl′

ψψdrld 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 β0 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 β0 = 0º β0 = 30º β0 = 60º β0 = 90º

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 β0 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

β0 (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 γ|

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 β with Z-axis. α is the angle

between Y-axis and the projection of n on XY. S is a unit normal to a small slice of the

circular cylinder surface, which is parallel to the circular cylinder axis, and it makes an

angle θ with Y-axis. T is a unit vector parallel to the intersection of two planes whose

normals are S and n respectively (Fig. 3.A-1). nr is the unit normal to a plane containing

both T and S.

Let vector T′ be the cross product of n and S.

( )[ ]zyx

zyxSnT

ˆsinsinˆsincosˆcoscos0cossin

coscossinsinsinˆˆˆ

αθβθβθβθθ

βαβαβ

−−+−=

=

×=′

(3.A-1)

Then unit vector T will be the normalized T′.

( )[ ] TzyxT ′−−+−= ˆsinsinˆsincosˆcoscos αθβθβθβ (3.A-2)

where

( )( )αθββ

αθβθβθβ

−+=

−++=′222

222222

sinsincos

sinsinsincoscoscosT.

(3.A-3)

57

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.

Let vector n′r be the cross product of S and T, which gives a unit vector.

( )

( ) ( ) zT

yT

xT

TTT

zyx

TSnr

ˆcosˆsinsinsinˆcossinsin

sinsinsincoscoscos0cossinˆˆˆ

′+

′−

+′−−

=

′−−

′′−

=

×=′

βθαθβθαθβ

αθβθβθβθθ

(3.A-4)

Then nr is the same as n′r.

Z

YX S T

n r

n

(sinθ, cosθ, 0)

(sinβ sinα, sinβ cosα, cosβ)

58

|cos γ| is given by the dot product of nr and n.

( ) ( )

( )

( )( )αθββ

θαθββ

θαθββββ

βθαθβθαθβ

γ

−+

−+=

′−+

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

′′−

′−−

=

⋅=

222

22

22

sinsincos

sinsinsincos

sinsinsincoscossin

0cossinsinsincossinsin

cos

T

TTT

nnr

(3.A-5)

In the special case that fractures are of constant orientation, we can always rotate the

coordinate system around Z-axis and make n inside ZY plane. Then angle α, the angle

between Y-axis and the projection of n on XY, turns to be zero. Let β0 denote the acute

angle between n and Z, which is a constant in this special case.

Therefore, |cos γ| given by Eq. (3.A-5) will be simplified as follows.

θβ

θββ

θββθββγ

20

2

20

20

2

20

20

2

20

20

2

cossin1

sinsincos

sinsincossinsincoscos

−=

+=

+

+=

(3.A-6)

59

References

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

Dershowitz, W.S. (1984) “Rock joint systems”. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts.

Dershowitz, W.S. and H.H. Einstein, (1988) “Characterizing rock joint geometry with joint system models” Rock Mechanics and Rock Engineering 21: 21–51

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.S., Lee, G., Geier, J., Foxford, T., LaPointe, P., and Thomas, A. (1998) “FracMan, Interactive discrete feature data analysis, geometric modeling, and exploration simulation”, User documentation, version 2.6, Seattle, Washington: Golder Associates Inc.

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

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

ISRM, Commission on Standardization of Laboratory and Field Tests. (1978) “Suggested methods for the quantitative description of discontinuities in rock masses”. International Journal of Rock Mechanics and Mining Science, 15: 319-368

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., 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

60

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.

Mauton, Peter R. (2002) “Principles and practices of unbiased stereology: an introduction for bioscientists”. Johns Hopkins University Press.

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. & Hudson, J. (1976) “Discontinuity spacing in rock”. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 13: 135-148

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.

Warburton, P. M. (1980) “A stereological interpretation of joint trace data”. Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 17: 181-190

Wolfram Research, Inc. (2004). Mathematica, Version 5.1, Champaign, IL.

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.

61

This page intentionally left blank.

62

Chapter 4

4 Estimating length and width of rectangular fractures from

traces on cylindrical exposures Abstract

This study focuses on estimating length and width of subsurface fractures in sedimentary

rocks. Fractures in sedimentary rock are typically elongated along their strikes and their

shapes can be considered rectangles. The study shows how information about length and

width of rectangular fractures can be discerned from study of borehole/shaft-fracture (or

core-fracture) intersections. Based on the possible geometric relations between a fracture

and a sampling cylinder, six types of intersection: transection, long-edge, short-edge,

corner, single piercing, and double piercing, are defined. The probabilities of occurrence

of these intersection types are related to the length and width of the fractures and

borehole/shaft diameter. The mean length and width of the fractures are estimated

directly from the observed counts of different types of intersection in a borehole/shaft or

rock core. A computer program is developed to generate synthetic fractures sampled by a

circular cylinder and the derived estimators are tested by Monte Carlo simulations, which

show satisfactory results.

Key words: cylindrical rock exposures, fracture networks, fracture length and width,

rectangular fractures

63

4.1 Introduction

Rock engineers, geologists, and hydrologists have long made use of fracture trace data

from planar rock exposures to extract characterization of rock fractures and fracture

systems, and procedures for inferring the three-dimensional (3-d) fracture geometry from

traces have been the subject of considerable research (Priest & Hudson, 1976; Cruden,

1977; Baecher et al., 1977; ISRM, 1978; Warburton 1980; Cheeney, 1983; Einstein &

Baecher, 1983; Kulatilake & Wu, 1984; LaPointe & Hudson 1985; Dershowitz &

Einstein, 1988; Dershowitz & Herda, 1992; Priest 1993, 2004; Mauldon et al. 1994;

Zhang & Einstein, 1998; Mauldon et al. 2001; Zhang et al., 2002). For subsurface rock

masses, large planar exposures are, however, rare. Direct measures from cylindrical

exposures, such as circular tunnel and shaft walls, borehole images (Dershowitz et al.,

2000), rock bores, as well as geophysical surveys, often provide the main sources of

subsurface fracture data, for characterization of fracture systems.

Common practice for borehole sampling of fractures (here used as a generic term for

discontinuities of all types) is to treat the borehole as a one-dimensional (1-d) sampling

domain, equivalent to a scanline. Fracture frequency is then taken to be inversely

proportional to the probability of the observed fractures being intersected by the 1-d

sampling line (Terzaghi, 1965; Priest & Hudson, 1976; Dershowitz & Einstein, 1988;

Priest, 1993). If the ratio of sampling cylinder (i.e. borehole) diameter to the average size

of fractures is, however, greater than about 20%, the sampling domain effectively takes

on a higher dimension – either 2-d or 3-d – depending on whether fractures are sampled

using the cylinder surface only, or using the cylinder volume (Mauldon & Mauldon,

1997). Yet another form of 2-d sampling has been described, making use of virtual 2-d

boreholes applied to a subsurface cross-section. (Narr, 1993; Pascal et al., 1997; Fouché

& Diebolt, 2004). Methods have been proposed to use borehole data to determine

fracture orientation distribution (Martel, 1999), average spacing (Narr 1996), 3-d fracture

intensity (Owens et al., 1994; Mauldon & Wang, 2003), fracture surface roughness

(Thapa et al., 1996) and fracture size (Stone, 1984; Mauldon, 2000; Zhang & Einstein,

64

2000; Özkaya, 2003; Wang et al., 2004, 2005). The present paper addresses inference of

fracture size, and also shape.

A wide variety of fracture geometry (or fracture system) models have been proposed in

the literature (e.g. Ruhland, 1973; LaPointe & Hudson, 1985; Dershowitz et al., 1998),

and most of these make assumptions about fracture shape and size distribution. The

Baecher model (Baecher et al. 1977; Dershowitz & Einstein, 1988), for example, assumes

circular disks with lognormally distributed radii. Orthogonal fracture models may

comprise either unbounded (e.g. Snow, 1965) or bounded (e.g. Müller, 1963; Gross, 1993)

joints. Field observations and mechanical consideration lend support to fracture models

for layered sedimentary rocks in which termination and propagation relationships (and

hence fracture size) are governed by elastic properties of the layers, boundary conditions

during fracturing and other mechanical and geometric factors (Engelder, 1993; Gross et

al., 1995). In particular, late forming fractures are likely to be normal to and terminate at

the primary fractures (or mechanical layer boundaries, Fig. 4.1), which may be either

bedding planes or systematic joint sets (Price, 1966; Helgeson & Aydin 1991; Gross,

1993; Engelder & Gross, 1993; Gross et al., 1995; Engelder & Fischer, 1996; Ruf et al.,

1998; Bai & Pollard 2000; Cooke & Underwood, 2001). One of the commonly observed

fracture patterns is that in which the late-forming cross joints propagate between and

orthogonal to preexisting primary joints (e.g. Fig. 4.1) in a “ladder” pattern (Gross, 1993;

Engelder & Gross, 1993). Field observations have also confirmed that fractures in

sedimentary rock are commonly perpendicular to bedding and elongated in one direction

(typically along strike, as shown schematically in Fig. 4.2) (Price, 1966; Suppe, 1985;

Priest, 1993). In all such cases, the shapes of fractures can be approximated as rectangles.

Estimates of (and models for) fracture size are usually predicated on assumed fracture

shape, such as circular disks (Baecher et al. 1977; Mauldon, 2000; Özkaya, 2003),

elliptical disks (Zhang & Einstein, 2002), or rectangles (Narr 1996; Wang et al. 2004,

2005).

65

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

Mauldon). Cross joints terminate at primary systematic joints.

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

terminating on bedding planes or cutting across several layers.

Rock mass

Bedding planes Fractures (joints)

66

In this paper we focus on fractures in sedimentary rocks, in which fracture shape is

assumed to be rectangular; and we introduce methods to estimate the mean length and

width of fractures by using borehole/shaft-fracture intersection (trace) data. For

convenience, the derivations are based on the model of a vertical borehole/shaft sampling

a layered rock mass that contains strike-elongated fractures. The results can also be

applied, however, to a general orientation of the borehole/shaft, as long as the

assumptions discussed in Section 2 are applicable.

A simple model for borehole/shaft sampling of fractures in sedimentary rock is shown in

Fig. 4.3, in which fracture long axes align in the direction perpendicular to the borehole

axis. Consider a fracture of length l and width w (Fig. 4.3). The apparent width w′ is

defined as the width of the fracture when projected onto a plane normal to borehole axis

(the axis-normal plane in Fig. 4.3), and is related to fracture true width by

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

plane. Note true width w and apparent width w′. (a) vertical borehole/shaft; (b) general

case of a skew borehole/shaft

(a)

l

w′

w

Axis-normal plane

w′

w

(b)

l

Axis-normal plane

Ground surface Ground surface

BoreholeBorehole ϕ

ϕ

67

ϕcosww =′ . (4.1)

where angle ϕ is the minimum angle between the fracture and the axis-normal plane (Fig.

4.3), i.e. the true dip in the case of a vertical borehole/shaft.

4.2 Assumptions

We make the following assumptions regarding fracture geometry, the borehole/shaft

sampling domain and the interrelationship between sampling domain and fracture system.

a) Fractures are planar rectangular objects with negligible thickness.

b) A shaft or a borehole is considered to be a right circular cylinder of diameter D,

oriented normal to the fracture elongation direction (or to strike when the

borehole/shaft is vertical).

c) The sampling domain refers to the cylindrical surface of the borehole/shaft.

d) The shaft/borehole is assumed to be long compared to its diameter. The end (or

ends) is not included in the cylindrical surface sampling domain.

e) Fracture length is greater than borehole/shaft diameter and also greater than the

apparent width w′of the fracture. Note that if the latter condition is not the case,

the length and width can be interchanged.

f) The location of the borehole/shaft is independent of the locations of fractures in

the rock mass to be explored. This is the case when we have little knowledge of

fracture networks before the excavation of a borehole or a shaft. Statistically, this

assumption ensures that the portion of the rock mass intersected by the cylindrical

surface of the borehole/shaft corresponds to a uniformly distributed, random

68

sample. Isotropy is achieved automatically with respect to directions

perpendicular to the borehole/shaft axis.

4.3 Borehole/shaft-fracture intersection types

Six types of intersection: transection, long-edge, short-edge, corner, single-piercing, and

double-piercing, are defined, based on the possible geometric relations between a

rectangular fracture and a borehole/shaft (Table 4.1). Corresponding to each intersection

type are characteristic types of fracture trace on the unrolled borehole surface (Fig. 4.4).

It may be observed that type A intersections can occur only if fracture apparent width w′

is greater than borehole/shaft diameter D. Type C1 and C2 intersections can occur only if

w′ is less than borehole/shaft diameter D. Note that Wang et al. (2004) used a slightly

different terminology, referring to transactions as complete intersections. A simple

illustration of all six intersection types is shown in Fig. 4.5, in which rectangular fractures

and boreholes/shafts are projected onto the axis-normal plane, on which boreholes/shafts

project as circles.

For the model discussed in this paper – rectangular fracture elongated along strike – we

can identify each of the six intersection types from fracture traces on the borehole/shaft

surface or unrolled trace map (Table 4.1, Fig. 4.4). Note that in Table 4.1, the

characteristics of each intersection type are based on knowing fracture dip-direction (cut

line along fracture dip-direction). If fractures are perpendicular to the borehole/shaft, this

direction can not be determined. In this case, only A-type (Transection) and C2-type

(double piercing) intersections can be easily identified; and it will be difficult to apply the

estimators discussed in this paper.

69

Table 4.1. Six borehole/shaft-fracture intersection types

Intersection

type

Symbol Example

in Fig. 4.4

Trace (or traces) on

borehole/shaft surface

Trace (or traces) on the

unrolled trace map with cut

line along dip-direction

Transection A 1 Full ellipse Full sine curve

Long-edge B1 2 Partial ellipse,

symmetric with respect

to dip-direction or anti-

dip-direction

Partial segments of sine curve,

symmetric with respect to cut

line or anti-dip-direction

Short-edge B2 3 Partial ellipse, centered

with respect to strike

Partial segments of sine curve,

centered along strike

Corner B3 4 Partial ellipse, not

symmetric with respect

to any direction

Partial segments of sine curve,

not symmetric with respect to

any direction

Single

piercing

C1 5 Single partial ellipse,

similar to one of the

paired C2 traces

Single partial segments of sine

curve, similar to one of the

paired C2 traces

Double

piercing

C2 6 Paired partial ellipse,

symmetric with respect

to dip-direction or anti-

dip-direction

Paired partial segments of sine

curve, symmetric with respect

to dip-direction or anti-dip-

direction

70

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. (a) borehole and

fractures; (b) Unrolled trace map. Coordinate axis θ is defined with θ = -π/2 at the cut

line. If the cut line were taken along strike, the angular coordinate θ would be from 0 to

2π.

(a) (b)

Borehole axis direction

Cut line Cut line

θπ/2 π 3π/2 0 -π/2

D

Cut line

1A

1

6 6hC2 C26

3B23

22B1 B1

2Dip direction

hC1

55

4B34

71

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

boreholes/shafts (dashed circles) are shown on the axis-normal plane. (a) D < w′; (b) D ≥

w′

4.4 Probabilistic model for occurrence of intersection types

We define symbols in Table 4.2.

In this section, we discuss the probabilities of occurrence of each borehole/shaft-fracture

intersection type. The key to the probabilistic model is that, on the axis-normal plane, the

center of a borehole/shaft must be inside a specific region around the projected fracture in

order for an intersection to occur (Fig. 4.6). Each intersection type, therefore, has a

B3

A B2

B1

Boreholes (diameter D < w′)

l

w′

Projected Fracture

(a)

C2 C1

(b)

l

w′

Boreholes (diameter D ≥ w′)

72

corresponding locus with respect to the projected fracture on the axis-normal plane (Figs.

4.7–4.9).

Table 4.2. Defined symbols

Symbol Definition

α Aspect ratio of a fracture: wl /=α .

N~ Number of occurrences of borehole/shaft-fracture intersections. N~ with a subscript (e.g. B1, B2…) indicates the number of occurrences for a specific intersection type (or several types).

H Length of the borehole/shaft.

λ′ Expected frequency of borehole/shaft-fracture intersections: HN /~=λ . λ′ with a subscript (e.g. B1, B2…) indicates the expected frequency of intersections for a specific intersection type (or several intersection types).

λ′(l, w′) Expected frequency of borehole/shaft-fracture intersections when sampling in fractures of constant size (the projected fracture has the dimension of l × w′ on the axis-normal plane).

∆ Area of a region on the axis-normal plane corresponding to an intersection type (or several intersection types). ∆ with a subscript (e.g. B1, B2…) indicates a specific intersection type (or several intersection types).

fL,W′(l,w′) Joint probability density function (pdf) of fracture length and fracture apparent width.

µl and µw′ Expected values of fracture length and apparent width, respectively. For constant fracture orientation, ϕµµ secww ′= , where µw is the mean fracture width.

73

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

plane is the region inside by the dashed line. If the center of borehole/shaft is inside the

region, an intersection occurs.

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.

w′

l

D/2

B1

B1

B2B2

B3 B3

B3B3

A

D/2 Projected fracture

D/2

D/2

l

w′

Region of intersection

Projected Fracture

Borehole/shaft location

74

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.

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.

Projected fracture

C1

w′C1

C2

B1

l

B2, B3 B2, B3

B1

C2D/2

D/2 D/2

l Projected fracture

B2, B3B2, B3

w′

B1

C2 C1C1

B1D/2

D/2 D/2

75

The regions, separated by dashed lines in Figs. 4.7–4.9, each define the possible locus of

the center of a borehole/shaft, corresponding to each intersection type (e.g. A-type, B1-

type, B2-type, B3-type, C1-type and C2-type). For instance, when the center of the

borehole/shaft falls into the shaded region marked as A, a transection intersection will

occur and a full cosine trace will be induced on the trace map. Consider a fracture of

dimension l × w projected on the axis-normal plane so that its projection has size of l × w′.

Then the area of the region for each intersection type as well as the area of all intersection

regions can be determined from simple geometry (Table 4.3).

If the last assumption in Section 4.2 holds, the location of a borehole/shaft is independent

of the location of fractures. If we were to introduce a Cartesian coordinate system on the

axis-normal plane, with origin at the borehole/shaft center, the locations of projected

fractures would be uniform on that plane (this holds even when projected fractures

overlay). In other words, projected fractures on the axis-normal plane have the same

probability to be at any point on that plane. Therefore, for a rectangular fracture, the

frequency λ of any type of intersection is proportional to the area of the corresponding

region on the axis-normal plane (Figs. 4.7-4.9). For fractures of constant orientation and

size, this can be expressed as the equations below.

∆=ηλ (4.2)

where λ is a frequency (Table 4.2), ∆ is an area (Tables 4.2 and 4.3), and η is identical to

the 3-d fracture density P30 (Dershowitz, 1992), i.e., number of fractures per unit volume

of the rock mass. P30 is assumed to be constant.

There are three cases to consider for the probabilistic model, depending on the relative

magnitudes of borehole/shaft diameter D and fracture apparent width w′. In each case,

fractures are assumed to be of constant orientation.

76

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

type from geometry

Area Case 4.4.1: w′ > D Case 4.4.2.1: D ≥ w′ > D/2 Case 4.4.2.2: w′ ≤ D/2

∆A 2DwDDllw +′−−′ NA NA

∆B1 42

22 DDDl π

−− 4

222

2 DDwDlw π−+′−′ * =∆ + 21 CB 4

2DwDDllw π−′−+′

∆B2 42

22 DDwD π

−−′ ⎟⎠⎞

⎜⎝⎛ ′

+−′ −

DwDDwD 12

2cos

4π

22 wDw ′−′−

∆B3 43 2

2 DD π+ ⎟

⎠⎞

⎜⎝⎛ ′

−+′ −

DwDDwD 12

2cos2

43π

222 wDw ′−′+

*2

22

32DwDBB

π+′=∆ +

2212 cos wDwDwD ′−′+⎟

⎠⎞

⎜⎝⎛ ′

− −

∆C1 NA 2212 cos wDwDwD ′−′−⎟

⎠⎞

⎜⎝⎛ ′− 2212 cos wDw

DwD ′−′−⎟

⎠⎞

⎜⎝⎛ ′−

∆C2 NA 2DDwDllw −′++′− * =∆ + 21 CB 4

2DwDDllw π−′−+′

∆total 4

2DwDDllw π+′++′

* Area of combined regions

77

4.4.1 w′ > D

In this case, the apparent widths of all fractures are greater than the borehole/ shaft

diameter D. When these relatively large fractures intersect a borehole/shaft, a transection

intersection (type A) may occur. Piercing intersection (C types), on the other hand, are

impossible.

For fractures of constant orientation and size with projected size of l × w′ on the axis-

normal plane, if the fracture density is η, from Eq. (4.2) and Table 4.3, the frequencies,

here interpreted as expected values (Owens et al., 1994), of B1, B2 and B3 intersection are

given by,

( ) ( )4/12, 211 πηηηλ +−=∆=′ DDlwl BB (4.3)

( ) ( )4/12 , 222 πηηηλ +−′=∆=′ DwDwl BB (4.4)

and

( ) ( )4/31 , 233 πηηλ +=∆=′ Dwl BB . (4.5)

For a set of fractures with constant orientation but varied size, let fL,W′(l, w′) denote the

joint pdf of fracture length and apparent width. The expected value of frequency λB3 is

obtained by integrating the right side of Eq. (4.5) over all values of l and w′ (in this case,

w′ ≤ l < ∞; D < w′ < ∞), noticing that λB3(l, w′) in Eq. (4.5) is expressed as a function

independent of fracture size.

The constant η can then be expressed as

( ) ( )

( )4/31

,4/31

2

,,

23

πη

πηλ

+=

′′+= ∫∫′

′

D

wdldwlfDwl

WLB (4.6)

78

Similarly, the expected value of frequencies λB1 and λB2 can be obtained by integrating

the right side of Eqs. (4.3) and (4.4), respectively, over all values of l and w′ (in this case,

w′ ≤ l < ∞; D < w′ < ∞). Noticing that the right side of Eq. (4.3) is not a function of w′

and the right side of Eq. (4.4) is not a function of l.

where µl and µw′ are the mean fracture length and the mean fracture apparent width,

respectively.

Substituting Eq.(4.7) into Eqs. (4.8) and (4.9), the mean fracture length µl and the mean

fracture apparent width µw′ can be determined as:

( ) ⎥⎦

⎤⎢⎣

⎡+++= ππ

λλµ 434

8 3

1

B

Bl

D,

and

(4.10)

( )4/3123

πλη+

=D

B . (4.7)

( )

( ) ( )

( )4/12

,4/1

,2

2

,,

2

,,1

πηµη

πη

ηλ

+−=

′′+−

′′=

∫∫

∫∫

′′

′′

DD

wdldwlfD

wdldwlDlf

l

wlWL

wlWLB

and

(4.8)

( )

( ) ( )

( )4/12

,4/1

,2

2

,,

2

,,2

πηµη

πη

ηλ

+−=

′′+−

′′′=

′

′′

′′

∫∫

∫∫

DD

wdldwlfD

wdldwlfwD

w

wlWL

wlWLB

(4.9)

79

( ) ⎥⎦

⎤⎢⎣

⎡+++=′ ππ

λλµ 434

8 3

2

B

Bw

D. (4.11)

Note that the expected values of fracture length and apparent width are proportional to

borehole diameter D, and are linear functions of the ratios of expected frequency of B1-

type and B2-type intersections over B3-type intersections, respectively.

4.4.2 w′ ≤ D

When fractures are narrow, or sufficiently steep that their apparent widths are smaller

than borehole/shaft diameter, piercing intersections (C types) may occur, whereas a

transection intersection (type A) is impossible. These narrow fractures are called piercing

fractures. Piercing fractures can pierce a borehole/shaft in either of two ways: singly or

doubly, as fracture #5 (doubly piercing) and fracture #6 (singly piercing) show in Fig.

4.4(a). Both single piercing and double piercing fractures intersect the borehole with two

long edges and leave similar traces on borehole/shaft walls, except that double piercing

fractures have paired traces (Fig. 4.4(b)). Double piercing fractures are easily identified

on unrolled trace maps derived from shaft surface or borehole imagery; and the amplitude

h of the traces (Fig. 4.4(b)) can be used to determine the apparent width of fractures by

ϕtan/hw =′ . (4.12)

80

4.4.2.1 D/2 < w′ ≤ D

The procedure to determine the length of piercing fractures with apparent widths greater

than the radius and less than the diameter of the sampling borehole/shaft is as follows.

For fractures of constant orientation and size with projected size of l × w′ on the axis-

normal plane, if the fracture density is η, from Eq. (4.2) and Table 4.3, consider the

following combinations of frequencies, as functions of fracture size.

( ) ( ) ( )[ ]

4/22

,,,

2

21

21

DwD

wlwlwl

CBtotal

CBtotal

πηη

η

λλλ

+′=

∆−∆−∆=

′−′−′

(4.13)

and

( ) ( ) [ ]( )4/12

2,2,2

2121

πηη

ηλ

+−=

∆+∆=′∆+′

DDl

wlwl CBCB

(4.14)

For a set of fractures with constant orientation but varied size, the expected value of the

linear frequency combination (λtotal - λB1 - λC2) can be obtained by integrating right side

of Eq. (4.13) over all values of l and w′(in this case, w′ ≤ l < ∞; D/2 < w′ ≤ D), noticing

that the combination, given fracture size, is not a function of l.

( )

( ) ( )

2/2

,4/2

,2

2

,,

2

,,21

DD

wdldwlfD

wdldwlfwD

w

wlWL

wlWLCBtotal

πµη

πη

ηλλλ

+=

′′+

′′′=−−

′

′′

′′

∫∫

∫∫

(4.15)

Note that the mean apparent width µw′ can be estimated directly by averaging all values

of w′ determined by Eq.(4.12). Then the constant η can be estimated by

81

2/2 221

DD w

CBtotal

πµλλλη

+−−

=′

. (4.16)

The expected value of the linear frequency combination (λB1 + 2λC2) can be obtained by

integrating right side of Eq. (4.14) over all values of l and w′(w′ ≤ l < ∞; D/2 < w′ ≤ D),

noticing that the combination, given fracture size, is not a function of w′.

( )

( ) ( )

( )4/12

,4/1

,22

2

,,

2

,,21

πηµη

πη

ηλλ

+−=

′′+−

′′=+

∫∫

∫∫

′′

′′

DD

wdldwlfD

wdldwllfD

l

wlWL

wlWLCB

(4.17)

Substituting Eq. (4.16) into Eq. (4.17) yields,

( )2121 2/2)4/1(22 CBtotal

w

lCB D

D λλλπµ

πµλλ −−+

+−=+

′ . (4.18)

Finally, the mean fracture length l can be determined as

( ) ( )ππµλλλ

λλµ +++−−

+= ′ 4

84/2

21

21 DDwCBtotal

CBl .

(4.19)

4.4.2.2 w′ ≤ D/2

The fractures in this case are very narrow piercing fractures or very steep fractures (angle

ϕ close to 90°) whose apparent widths are smaller than borehole/shaft radius. This

scenario is rare in borehole samplings, but may occur for shafts or tunnels. The regions

separated by the dashed lines in Fig. 4.9 show the locus of the center of a borehole/shaft

82

corresponding to each intersection type (e.g. C1-type, C2-type, B1-type, B2 and B3-type)

for this case.

For fractures of constant orientation and size with projected size of l × w′ on the axis-

normal plane, if the fracture density is η, consider the total borehole/shaft-fracture

intersection frequency and the following linear combination of frequencies from Eq. (4.2)

and substitute for areas from Table 4.3, we have

( ) ( )4/ , 2DwDDllwwl totaltotal πηηλ +′++′=∆=′ (4.20)

( ) ( ) ( )2/2

,,2

2121

DwD

wlwl CBtotalCBtotal

πηη

ηλλ

+′=

∆−∆=′−′ ++

(4.21)

For a set of fractures with constant orientation but varied size, the expected value of the

frequency combination (λtotal - λB1+C2) can be obtained by integrating right side of Eq.

(4.21) over all values of l and w′(in this case, w′ ≤ l < ∞; 0 ≤ w′ ≤ D/2), noticing that the

combination, given fracture size, is not a function of l.

( )

( ) ( )

2/2

,2/

,2

2

,,

2

,,21

DD

wdldwlfD

wdldwlfwD

w

wlWL

wlWLCBtotal

πηµη

πη

ηλλ

+=

′′+

′′′=−

′

′′

′′+

∫∫

∫∫

.

(4.22)

Again, the mean apparent width µw′ can be estimated by averaging all w′ determined by

Eq.(4.12). Then constant η can be estimated by

2/2 221

DDw

CBtotal

πµλλη

+−

=′

+ . (4.23)

83

The expected value of the total frequency λtotal is expressed as

( ) ( )∫∫′

′ ′′+′++′=wl

WLtotal wdldwlfDwDDllw,

,2 ,4/πηλ .

(4.24)

The integral in Eq.(4.24) is difficult to evaluate unless we know or could assume the joint

pdf of fracture length and apparent width. For instance, if we assume:

a) Fracture length is constant (and equal to l*). Then the integral in Eq.(4.24) is

simplified as follows.

( ) ( )

( ) ( )

( ) ( )

( ) ( )4/

,4/

,

,4/

**

,,

*

,,

*

,,

2

DlDDl

wdldwlfwDlD

wdldwlfwDl

wdldwlfDwDDllw

w

wlWL

wlWL

wlWLtotal

πηµη

πη

η

πηλ

+++=

′′′++

′′′+=

′′+′++′=

′

′′

′′

′′

∫∫

∫∫

∫∫

(4.25)

Substituting Eq. (4.23) into Eq. (4.25), l* can be determined.

w

w

CBtotal

total

DDDl

′

′

+ ++

⎥⎦

⎤⎢⎣

⎡−

−=

µπµ

λλλ 4/12 2

21

* . (4.26)

b) Fracture aspect ratio is constant (α* = l / w′ = µl / µw′).

Then Eq.(4.24) is expressed as

84

where E(w′2) is the second moment of fracture apparent width probability density

function, which can be estimated by averaging all w′2 determined by Eq.(4.12).

Substituting Eq. (4.23) into Eq. (4.27), α* can be solved.

( ) w

w

CBtotal

total

DwEDD

′

′

+ +′+

⎥⎦

⎤⎢⎣

⎡−

−=

µπµ

λλλα 2

2

21

* 4/12

(4.28)

Finally, the mean fracture length can be estimated as:

w

ww

CBtotal

totalwl DwE

DD′

′′

+′

+′+

⎥⎦

⎤⎢⎣

⎡−

−==

µµπµ

λλλµαµ

)(4/12

2

22

21

* (4.29)

4.4.3 Summary of fracture length and width estimators

The estimators discussed in this section (Eqs. (4.10, 4.11, 4.19, 4.26, 4.29)) are

categorized in three cases, depending on the relative magnitudes of borehole/shaft

diameter D and fracture apparent width w′. Judgment should be made to determine

which estimator(s) will be applied to estimate fracture length and apparent width. The

flowchart in Fig. 4.10 shows how this procedure is carried out. Note that if there are no

C-type intersections, it is much likely that fracture apparent width is greater than

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( ) 4/1

,4/

,1

,

,4/

2*2*

,,

2

,,

*

,,

2*

,,

2*2*

DDwE

wdldwlfD

wdldwlfwD

wdldwlfw

wdldwlfDwDwDw

w

wlWL

wlWL

wlWL

wlWLtotal

πηµαηηα

π

α

ηα

πααηλ

+++′=

′′+

′′′++

′′′=

′′+′+′+′=

′

′′

′′

′′

′′

∫∫

∫∫

∫∫

∫∫

(4.27)

85

borehole/shaft diameter. Estimators for the case 4.4.1 (Eqs. (4.10, 4.11)), therefore, can

be used to estimate fracture length and apparent width. If there are no A-type

intersections, on the other hand, it is much likely that fracture apparent width is less than

borehole/shaft diameter. Estimators for the case 4.4.2 (Eqs. (4.19, 4.26, 4.29)), can be

used, depending on whether the measured fracture apparent width is greater than

borehole/shaft radius or not.

In practice, when using the estimators, the ratios of expected frequencies can be replaced

by the ratios of the corresponding observed intersection counts. This is demonstrated in

the following example.

Fig. 4.10. Flowchart of choosing estimators to estimate mean fracture length and width.

Unrolled borehole/shaft trace map

Identify intersection types and their counts

No C-type intersections

Case 4.4.1: w′ > D

Both A and C intersections

Use judgment

Mean fracture length: Eq. (4.10) Mean fracture width: Eq. (4.11)

No A-type intersections

Case 4.4.2: w′ ≤ D

Determine mean fracture apparent width µw′ by averaging all w′ calculated by Eq. (4.12)

µw′ > D/2 ? Case 4.4.2.1

Case 4.4.2.2

Yes

NoMean fracture length: Eq. (4.19)

Mean fracture length: Eq. (4.26) or Eq. (4.29)

86

4.5 Examples

Example 1: suppose a 600.0 feet long, 6.0 inch in diameter vertical borehole was drilled

in a sedimentary rock mass. From borehole image, three sets of fractures were observed,

with an average dip of 35.0°, 60.0°, and 80.0°, respectively. The borehole-fracture

intersection types were identified by unrolled borehole images, and the counts of

intersections for each intersection type are listed in Table 4.4.

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

Intersection counts Intersection type

Set 1 Set 2 Set 3

A 18 0 0

B1 105 233 58

B2 38 78 34

B3 57 63 62

C1 0 0 0

C2 0 55 21

Total number of intersections 218 429 175

Average dip 35.0 60.0 80.0

Borehole length (ft.) 600.0

Borehole diameter (in.) 6.0

Estimated mean fracture length l (ft) 2.0 2.4 1.0

Estimated mean fracture apparent width w′ (ft) 1.0 0.4 0.1

Estimated mean fracture width w (ft) 1.2 0.8 0.6

Estimated mean squared fracture width w′2 (ft2) 0.015

Estimated Aspect Ratio (l/w) 1.6 3.0 1.7

87

For fracture set 1, no C-type intersections were found. Therefore, we use estimators Eqs.

(4.10) and (4.11) for the case 4.4.1 (w′ > D) to estimate mean fracture length and width.

The ratios of expected intersection frequencies are replaced by the ratios of intersection

counts.

( )

( )

( )

ftin

NND

D

B

B

B

Bl

0.2.9.23

4342570

80.6

434~~

8

4348

3

1

3

1

==

⎥⎦⎤

⎢⎣⎡ +++=

⎥⎦

⎤⎢⎣

⎡+++≈

⎥⎦

⎤⎢⎣

⎡+++=

ππ

ππ

ππλλµ

(4.30)

and

( )

( )

( )

ftin

NND

D

B

B

B

Bw

0.1.1.12

4342518

80.6

434~~

8

4348

3

2

3

2

==

⎥⎦⎤

⎢⎣⎡ +++=

⎥⎦

⎤⎢⎣

⎡+++≈

⎥⎦

⎤⎢⎣

⎡+++=′

ππ

ππ

ππλλµ

(4.31)

Assume that fractures in set 1 are of constant orientation (dip = 35°), fracture true width

is estimated by

ftftww 2.1)35cos(/0.1cos/ =°== ′ ϕµµ . (4.32)

For fracture set 2, no A-type intersections were identified and the average fracture

apparent width is µw′ = 0.4 ft by averaging all values of fracture apparent width

88

determined by Eq. (4.12). We use estimator Eq. (4.19) for case 4.4.2.1 (D/2 < w′ ≤ D) to

estimate the mean fracture length.

( ) ( )

( ) ( )

( ) ( )

ftin

DDNNN

NN

DD

wCBtotal

CB

wCBtotal

CBl

4.2.5.28

480.64/0.68.0

55110204552110

48

4/~~~~2~

48

4/2

21

21

21

21

==

+++−−

×+=

+++−−

+≈

+++−−

+=

′

′

ππ

ππµ

ππµλλλ

λλµ

(4.33)

Assume that fractures in set 2 are of constant orientation (dip = 60°), fracture true width

is estimated by

ftftww 8.0)60cos(/4.0cos/ =°== ′ ϕµµ . (4.34)

For fracture set 3, no A-type intersections were identified and the average fracture

apparent width is µw′ = 0.1 ft by averaging all values of fracture apparent width

determined by Eq. (4.12) and E(w′2) = 0.015 ft2. By assuming that fractures are of

constant aspect ratio, we use estimator Eq. (4.29) for case 4.4.2.2 (w′ ≤ D/2) to estimate

the mean fracture length.

( )

( )

( )

ft

DwEDD

NNN

DwEDD

w

ww

CBtotal

total

w

ww

CBtotal

totall

0.1

1.05.0015.04/1.05.05.01.01

21581751752

)(4/1~~

~2

)(4/12

22

2

22

21

2

22

21

=

×+×+×

⎥⎦⎤

⎢⎣⎡ −

−−×

=

+′+

⎥⎦

⎤⎢⎣

⎡−

−≈

+′+

⎥⎦

⎤⎢⎣

⎡−

−=

′

′′

+

′

′′

+

π

µµπµ

µµπµ

λλλµ

(4.35)

89

Assume that fractures in set 3 are of constant orientation (dip = 80°), fracture true width

is estimated by

ftftww 6.0)80cos(/1.0cos/ =°== ′ ϕµµ . (4.36)

Example 2: a 10.4 inch in diameter borehole was drilled in a sedimentary rock mass.

From borehole FMI image, a set of fractures was observed, with an average dip of 82.3°.

The borehole-fracture intersection types were identified and the counts of intersections

for each intersection type are listed in Table 4.5.

Table 4.5. Borehole-fracture intersection counts from borehole sampling

Intersection type Intersection counts

A 17

B1 103

B2 21

B3 23

C1 0

C2 98

Total number of intersections 262

Average dip 82.3°

Borehole diameter (in) 10.4

Estimated mean fracture length l (in) 59.0 ~ 65.9

Estimated mean fracture apparent width w′ (in) 3.1

Estimated mean fracture width w (in) 23.1

Estimated mean squared fracture width w′2 (in2) 14.5

Estimated Aspect Ratio (l/w) 2.5 ~ 2.8

90

Although both A-type and C-type intersections were identified for this fracture set, C-

type intersection counts (98) are much higher than A-type intersection counts (17).

Therefore, w′ ≤ D (case 4.4.2) is assumed to be more suitable for this case. The average

fracture apparent width is µw′ = 3.1 in by averaging all values of fracture apparent width

determined by Eq. (4.12) and E(w′2) = 14.5 in2. Since µw′ is less than borehole radius (5.2

in), assuming that fractures are of constant aspect ratio, we use estimators Eq. (4.29) for

case 4.4.2.2 (w′ ≤ D/2) to estimate the mean fracture length.

( )

( )

( )

in

DwEDD

NNN

DwEDD

w

ww

CBtotal

total

w

ww

CBtotal

totall

0.591.34.105.14

4/1.34.104.101.3198103262

2622

)(4/1~~

~2

)(4/12

22

2

22

21

2

22

21

=×+

×+×⎥⎦⎤

⎢⎣⎡ −

−−×

=

+′+

⎥⎦

⎤⎢⎣

⎡−

−≈

+′+

⎥⎦

⎤⎢⎣

⎡−

−=

′

′′

+

′

′′

+

π

µµπµ

µµπµ

λλλ

µ

(4.37)

Assume that fractures are of constant orientation (dip = 82.3°), fracture true width is

estimated by

ininww 1.23)3.82cos(/1.3cos/ =°== ′ ϕµµ . (4.38)

4.6 Monte Carlo simulations

Monte Carlo simulations were carried out using a computer program developed in Visual

C++. The program generates a population of synthetic rectangular fractures intersected

by a borehole/shaft (Fig. 4.11). Fracture traces are computed (right-hand side window in

Fig. 4.11) and used to determine the occurrence of each borehole/shaft-fracture

91

intersection type. For each simulation, the size (length and width) of fractures and the

dimensions of borehole (length and diameter) are constant; and the total number of

generated fractures, the area of each fracture, as well as the size of the generation region,

were recorded. The observed counts of intersection types are used to estimate fracture

length and width by using Eqs. (4.10) and (4.11), or Eqs. (4.12) and (4.19), depending on

the relative size of borehole/shaft diameter D and fracture apparent width w′.

Twenty-seven scenarios (Table 4.6) were simulated by systematically changing fracture

dip ϕ (0º, 30º, and 60 º), length l (2, 10, and 20) and aspect ratio α (1, 2, and 10). For all

cases, the cylinder (borehole/shaft) diameter was held constant (D = 0.2), and the fracture

intensity P32 was held constant at 1.0 [L-1]. Depending on the mean size of the fracture,

the number of fractures varied from one scenario to the next. For each scenario, fifty

simulations were run; and the average estimated fracture length and width, and average

percent error, variance and coefficient of variation are listed in Table 4.6.

An example of simulation results for scenario 1 is shown in Fig. 4.12. In this scenario,

the length and width of the generated fractures are both set to be 2.0, i.e., the fractures are

square. In 50 simulations, there are an average of 354 B1-type intersections, 353 B2-type

intersections and 65 B3-type intersections observed. The estimated fracture length and

width are 2.04 and 2.02, by using Eqs. (4.10) and (4.11), respectively.

Overall, the simulation results (Table 4.6) show that the derived equations produce

reasonable, good estimates of fracture length and width (absolute percent error is less

than 15% of the actual fracture size). The largest errors occur for scenarios 4, 7, 13, 16,

22, and 25, in which the data are distorted by very low or even no occurrence of B3-type

intersections observed in some simulations. The comparison of percent error and

coefficient of variation versus observed B3-type intersection counts (Fig. 4.13) shows a

big increase of both error and variation of the estimators when the observed B3-type

intersection counts drops from 26 to 7. This suggests that cares should be taken when

92

applying these estimators in practice, especially when fractures are much larger than

borehole/shaft diameter and the counts of B3-type intersections is very low.

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

fractures intersected by a borehole/shaft. In the simulation shown above, number of

fractures generated = 4796; fracture length = 2; aspect ratio = 1; dip/dip direction = 30

deg/north; borehole length = 10; borehole diameter = 0.2; fracture area per unit rock mass

volume = 5.0. The trace analysis produced 36 A-type intersections; 6 B1-type

intersections; 6 B2-type intersections; 1 B3-type intersections; 0 C1-type intersections; 0

C2-type intersections.

Borehole

Trace map

Fractures

123

4

5

6

7

8

9

1011

93

Table 4.6. Parameters and results of Monte Carlo simulations

Ave

rage

d nu

mbe

r of

inte

rsec

tions

Est

imat

ed fr

actu

re

leng

th [L

]

Est

imat

ed fr

actu

re

wid

th [L

]

Sce

nario

Dip

Frac

ture

leng

th [L

]

Frac

ture

wid

th [L

]

Asp

ect R

atio

App

aren

t wid

th [L

]

Num

ber o

f fra

ctur

es

gen

erat

ed

Frac

ture

den

sity

(P30

) [L-3

]

A

B1

B2

B3

C2

Tota

l

Ave

rage

Abs

olut

e pe

rcen

t er

ror

Var

ianc

e

Coe

ffici

ent o

f va

riatio

n

Ave

rage

Abs

olut

e pe

rcen

t er

ror

Var

ianc

e

Coe

ffici

ent o

f va

riatio

n

1 0° 2 2 1 2 3673 0.25 1569 354 353 65 0 2341 2.0 2% 0.08 14% 2.0 1% 0.08 14%

2 0° 2 1 2 1 4754 0.50 1443 735 329 134 0 2640 2.0 2% 0.03 9% 1.0 1% 0.01 8%

3 0° 2 0.2 10 0.2 19562 2.50 0 3660 45 665 0 4369 2.0 1% 0.01 5% 0.2 1% 0.00 2%

4 0° 10 10 1 10 3295 0.01 1917 77 78 3 0 2075 12.2 22% 55.56 61% 12.3 23% 56.94 62%

5 0° 10 5 2 5 4150 0.02 1884 157 79 6 0 2125 11.7 17% 60.40 66% 6.0 21% 21.01 76%

6 0° 10 1 10 1 16832 0.10 1566 794 66 26 0 2451 10.9 9% 4.76 20% 1.1 6% 0.04 19%

7 0° 20 20 1 20 3249 0.00 1955 40 41 1 0 2036 11.6 42% 12.00 30% 11.5 43% 10.77 29%

8 0° 20 10 2 10 4077 0.01 1939 79 40 1 0 2058 20.3 2% 62.12 39% 10.3 3% 16.46 39%

9 0° 20 2 10 2 16505 0.03 1783 400 37 7 0 2227 24.4 22% 137.81 48% 2.4 21% 1.18 45%

10 30° 2 2 1 1.73 3673 0.25 1374 366 316 66 0 2121 2.1 4% 0.07 13% 2.1 5% 0.07 12%

11 30° 2 1 2 0.87 4754 0.50 1198 730 276 134 0 2338 2.0 1% 0.03 9% 1.0 1% 0.01 8%

12 30° 2 0 10 0.17 19562 2.50 0 3162 27 610 239 4038 2.0 0% 0.01 4% 0.2 0% 0.00 0%

13 30° 10 10 1 8.66 3295 0 1654 81 70 2 0 1807 13.4 34% 59.92 58% 13.4 34% 61.76 58%

14 30° 10 5 2 4.33 4150 0.02 1623 156 65 6 0 1850 10.5 5% 44.11 63% 5.1 2% 12.34 69%

15 30° 10 1 10 0.87 16832 0.10 1308 793 55 28 0 2184 10.1 1% 3.67 19% 1.0 1% 0.03 18%

16 30° 20 20 1 17.3 3249 0.00 1692 39 34 1 0 1766 10.5 47% 14.20 36% 10.8 46% 16.40 38%

17 30° 20 10 2 8.66 4077 0.01 1677 81 34 1 0 1794 19.3 3% 70.41 43% 9.7 3% 21.64 48%

18 30° 20 2 10 1.73 16505 0.03 1513 396 32 7 0 1947 22.4 12% 101.61 45% 2.3 13% 1.09 46%

19 60° 2 2 1 1 3673 0.25 717 365 163 64 0 1309 2.1 5% 0.06 12% 2.1 4% 0.06 12%

20 60° 2 1 2 0.5 4754 0.50 539 725 128 133 0 1525 2.0 1% 0.02 7% 1.0 1% 0.00 7%

21 60° 2 0.2 10 0.1 19562 2.50 0 1847 6 381 896 3130 2.0 1% 0.01 4% 0.2 0% 0.00 0%

22 60° 10 10 1 5 3295 0.01 943 79 37 2 0 1062 14.1 41% 63.84 57% 13.2 32% 55.09 56%

23 60° 10 5 2 2.5 4150 0.02 901 156 40 6 0 1103 11.3 13% 64.42 71% 5.9 19% 13.07 61%

24 60° 10 1 10 0.5 16832 0.10 590 788 26 27 0 1432 10.2 2% 4.14 20% 1.0 2% 0.05 21%

25 60° 20 20 1 10 3249 0.00 971 41 20 1 0 1033 11.8 41% 17.73 36% 11.5 43% 20.33 39%

26 60° 20 10 2 5 4077 0.01 949 81 20 1 0 1051 21.7 9% 51.16 33% 11.3 13% 25.64 45%

27 60° 20 2 10 1 16505 0.03 800 398 17 7 0 1223 22.9 14% 123.09 48% 2.3 14% 0.92 42%

Diameter of borehole/shaft [L] 0.2

Fracture area per unit rock mass volume (P32) [L-1] 1

Simulations per scenario 50

94

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

and width for scenario 1. Fracture dip = 0°; length = 2.0; aspect ratio = 1.0.

0

1

2

3

0 10 20 30 40 50

Simulations

Leng

th

Estimated LengthAverage Estimated LengthActual Length

0

1

2

3

0 10 20 30 40 50

Simulations

Wid

th

Computed WidthAverage Estimated WidthActual Width

95

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.

(a)

(b)

0

100

200

300

400

500

600

700

0%

10%

20%

30%

40%

50%

60%

70%

80%

# of B3 intersectionsAbsolute percent errorCoefficient of variation

Scenarios

B3

inte

rsec

tion

coun

ts

Per

cent

err

or o

r co

effic

ient

ofva

riatio

n

26 B3 intersection

7 B3 intersection

0

100

200

300

400

500

600

700

0%

10%

20%

30%

40%

50%

60%

70%

80%

# of B3 intersectionsAbsolute percent errorCoefficient of variation

Scenarios

B3

inte

rsec

tion

coun

ts

Per

cent

err

or o

r co

effic

ient

ofva

riatio

n

26 B3 intersection

7 B3 intersection

96

The fracture apparent widths in Scenario 12 and 21 are 0.17 and 0.10, respectively, which

are less than borehole diameter (0.2). The estimators (Eqs. (4.12) and (4.19)) for case D

≥ w′ > D/2 give very good estimates for fracture width and length (Table 4.6). In

addition, estimator (Eq. (4.10)) for case w′ > D were also used to estimate fracture length

and resulted an estimated length of 1.9 and 1.8 for scenario 12 and 21, respectively. The

percent errors are 4% and 10%, respectively, which implies that it will not cause major

errors by using Eq. (4.10) to estimate fracture length even if fracture apparent width is

smaller than borehole/shaft diameter.

4.7 Discussion & Conclusions

The goal of this study was to develop a general model for estimating mean rectangular

fracture length and width from traces on cylinder walls. Fractures in sedimentary rocks

are commonly elongated along strike and terminated on bedding planes or primary joint

sets, therefore assumed rectangular in shapes. From the geometric relations between a

fracture and a borehole/shaft, six types of intersection are defined. The features of each

intersection type described in the paper can be used to identify the six intersection types

from unrolled borehole/shaft trace maps. The occurrences of the intersection types are

related to fracture size and borehole/shaft diameter, assuming independence between

locations of borehole/shaft and the fractures.

Three cases regarding relations between fracture apparent width and the borehole/shaft

diameter are discussed. For each case, estimators are derived to estimate mean fracture

length and width based on probabilistic models. The estimators are confirmed by Monte

Carlo simulations, which gave satisfactory results. It is also pointed out in the paper that

caution should be used when applying the estimator to the cases that the size of the

sampled fractures is much larger than the diameter of boreholes/shafts.

97

Acknowledgements

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

gratefully acknowledged. Also should be acknowledged are Chris Heiny from University

of Tennessee and Jeramy Decker from Virginia Tech, who helped carrying out

simulations.

98

References

Baecher, G.B., Einstein, H.H., and Lanney, N.A. (1977). “Statistical description of rock properties and sampling”, In: Proc. 18th U.S. Symp. Rock Mech., Colorado School of Mines. 5C: 1-8.

Bai, Taixu, Pollard, D.D. (2000). “Closely spaced fractures in layered rocks: initiation mechanism and propagation kinematics”, Journal of Structural Geology, 22: 1409-1425

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

Cooke, M.L. and Underwood, C.A. (2001). “Fracture termination and step-over at bedding interfaces due to frictional slip and interface opening”, Journal of Structural Geology, 23: 223-238.

Cruden, D.M. (1977). “Describing the size of discontinuities”, International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 14: 133-137

Dershowitz, W.S. (1984). “Rock joint systems”, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts.

Dershowitz, W.S. and Einstein, H.H. (1988). “Characterizing rock joint geometry with joint system models”, Rock Mechanics and Rock Engineering, 21: 21–51

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. pp. 757-766.

Dershowitz, W.S., Lee, G., Geier, J., Foxford, T., LaPointe, P., and Thomas, A. (1998). “FracMan, Interactive discrete feature data analysis, geometric modeling, and exploration simulation”, User documentation, version 2.6, Seattle, Washington: Golder Associates Inc.

Dershowitz, W., Hermanson, J., Follin, S., and Mauldon, M. (2000). “Fracture intensity measures in 1-D, 2-D, and 3-D at Aspo, Sweden”, Proceedings of the North American Rock Mechanics Symposium (Pacific rocks 2000), v.4: 849-853.

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

Engelder, T. (1993). “Stress regimes in the lithosphere”, Princeton University Press, Princeton, NJ.

99

Engelder, T. and Gross, M.R. (1993). “Curving cross joints and the lithospheric stress field in eastern North America”, Geology, 21: 817-820.

Engelder, T. and Fischer, M.P. (1996). “Loading configurations and driving mechanisms for joints based on the Griffith energy-balance concept”, Tectonophysics, 256: 253-277.

Fouché, O. and Diebolt, J. (2004). “Describing the Geometry of 3D Fracture Systems by Correcting for Linear Sampling Bias”, Mathematical Geology, 36(1): 33-63.

Gross, M.R. (1993). “The origin and spacing of cross joints: examples from the Monterey Formation, Santa Barbara Coastline, California”, Journal of Structural Geology, 15(6): 737-751

Gross, M.R., Fischer, M.P., Engelder, T., and Greenfield, R.J. (1995). “Factors controlling joint spacing in interbedded sedimentary rocks: integrating numerical models with field observations from the Monterey Formation, USA”, In: Ameen, M.S. (ed.) Fractography: Fracture topography as a tool in fracture mechanics and strain analysis. Geological Society, London, Special Publications, 92: 215-233.

Helgeson, D.E. & Aydin, A. (1991). “Characteristics of joint propagation across layer interfaces in sedimentary rocks”, Journal of Structural Geology, 13(8): 897-911.

ISRM, Commission on Standardization of Laboratory and Field Tests. (1978). “Suggested methods for the quantitative description of discontinuities in rock masses”, International Journal of Rock Mechanics and Mining Science, 15: 319-368

Kulatilake, P.H.S.W., Wu, T.H. (1984). “The density of discontinuity traces in sampling widows”, International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 21(6): 345-347.

LaPointe, P.R. and Hudson, J.A. (1985). “Characterization and Interpretation of Rock Mass Joint Patterns”, Special Paper 199, Geological Society of America Book Series.

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

Mauldon, M. (2000). “Borehole estimates of fracture size”, In Proceedings of the 4th North America Rock Mechanics Symposium. Seattle, pp. 715-721

100

Mauldon, M., Dunne, W.M. and Rohrbaugh, M.B.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 Wang, X. (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.

Müller-Salzburg, L. (1963). “Der Felsbau”, Bd.I, Theoretischer Teil, Felsbau über Tage, 1. Teil. Enke, Stuttgart 1-624

Narr, W. (1996). “Estimating average fracture spacing in subsurface rock”, AAPG Bulletin, 80(10): 1565-1586.

Ozkaya, S.I. (2003). “Fracture length estimation from borehole image logs”, Mathematical Geology, 35(6): 737-753.

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. pp. 269-276.

Pascal, C., Angelier, J., Cacas, M.-C., and Hancock P.L. (1997). “Distribution of joints: probabilistic modelling and case study near Cardiff (Wales, U.K.)”, Journal of Structural Geology, 19(10): 1273-1284

Price, N.J. (1966). “Fault and joint development in brittle and semi-brittle rock”, Pergamon Press, New York.

Priest, S.D. and Hudson, J. (1976). “Discontinuity spacing in rock”, International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 13: 135-148

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

Priest, S.D. (2004). “Determination of Discontinuity Size Distributions from Scanline Data”, Rock Mechanics and Rock Engineering, 37(5): 347-368

Ruf, J.C., Rust, K.A., Engelder, T. (1998). “Investigating the effect of mechanical discontinuities on joint spacing”, Tectonophysics 295: 245-257

Ruhland, M. (1973). “Méthode d'étude de la fracturation naturelle des roches associée a divers modeles structuraux”, Sciences Géologiques Bulletin, 26(2-3): 91-113

Snow D.T. (1969). “Anisotropic permeability of fractured media”, Water Resources Research, 5(6): 1273-1289.

Suppe, J. (1985). “Principles of structural geology”, Prentice-Hall, Inc., Englewood Cliffs, New jersey.

101

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

Thapa, B.B., Ke, T.C., Goodman, R.E., Tanimoto, C., and Kishida, K. (1996). “Numerically simulated direct shear testing of in situ joint roughness profiles”, International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 33(1): 75-82

Wang, X., Mauldon, M., Dunne, W., Heiny C. (2004). “Using Borehole Data to Estimate Size and Aspect Ratio of Subsurface Fractures”, In: Proceedings of the 6th North American Rock Mechanics Symposium (NARMS), Houston, Texas

Wang, X., Mauldon, M., Dunne, W., Heiny C. (2005). “Extracting fracture characteristics from piercing-type intersections on borehole walls”, In: Proceedings of the 40th U.S. Symposium on Rock Mechanics (USRMS), Anchorage, Alaska

Warburton, P.M. (1980). “A stereological interpretation of joint trace data”, International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 17(4): 181-190

Zhang, L. and Einstein, H.H. (1998). “Estimating the Mean Trace Length of Rock Discontinuities”, Rock Mechanics and Rock Engineering, 31(4): 217-235

Zhang, L. and Einstein, H.H. (2000). “Estimating the intensity of rock discontinuities”, International Journal of Rock Mechanics and Mining Sciences, 37(5): 819-837

Zhang, L., Einstein, H.H. and Dershowitz, W.S. (2002). “Stereological relationship between trace length and size distribution of elliptical discontinuities”, Geotechnique, 52(6): 419–433

102

This page intentionally left blank.

103

Chapter 5

5 Conclusions and discussions

Based on the stereological analyses and numerical simulation results of sampling

fractures by a line, a plane, and most important, cylindrical surfaces, the following

conclusions are drawn:

1. For linear sampling in constant sized or unbounded fractures with orientation

given by the Fisher distribution, the conversion factor C13 [1.0, ∞] between the

fracture linear intensity and the volumetric intensity is a function of the angle

between the sampling line and the Fisher mean pole, and the Fisher constant κ.

2. For planar sampling in constant sized or unbounded fractures with orientation

given by the Fisher distribution, the conversion factor C23 [1.0, ∞] between the

fracture areal (planar) intensity and the volumetric intensity is a function of the

angle between the normal of the sampling plane and the fracture Fisher mean pole,

and the Fisher constant κ.

3. For cylindrical surface sampling in constant sized or unbounded fractures with

orientation given by the Fisher distribution, the conversion factor C23,C [1.0, π/2]

between the fracture areal (cylindrical surface) intensity and the volumetric

intensity is a function of the angle between the axis of the sampling cylinder

(borehole) and the fracture Fisher mean pole, and the Fisher constant κ.

4. For a general case of cylindrical surface sampling of fractures, the conversion

factor C23,C [1.0, π/2] between the fracture areal (cylindrical surface) intensity and

the volumetric intensity is only a function of orientation of cylinder axis relative

to the fracture system and the pdf of fracture orientation weighted by area. It is

independent of fracture size or shape, or the sampling cylinder size.

104

5. Cycloidal scanlines, when deployed on the cylinder surface in a certain pattern,

give directional unbiased estimates of fracture volumetric intensity. Fracture

orientation information is not required by using this technique.

6. Fractures in sedimentary rocks can be approximated rectangular in shapes and the

estimators for their mean length and width are derived for three cases. The

estimators are independent of fracture size distributions.

The author also recommends the following work to be done in the future.

1. Interpretation of fracture traces to estimate other fracture properties, such as

roughness, connectivity, and so on, by means of stereology.

2. Study on fracture trace length distribution on borehole walls. It may be another

way to make estimates of fracture size, based on the assumptions of fracture

shapes.

3. Apply the conversion factors of fracture intensities and the estimators of fracture

size to real fracture trace data. Find ways to verify the obtained results.

105

This page intentionally left blank.

106

6 Appendix: Programs used in the dissertation

Two programs were used in this dissertation to carry out Monte Carlo simulations.

A. FISHER - Simulate the Fisher distribution

This program was developed by using Microsoft Excel. The inputs for this program are

listed in the table below.

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

Input Range

Fisher constant, κ [0.1, 700]

Number of fracture poles, N [0, 3000]

Fisher mean pole dip [0, 90]

Fisher mean pole dip-direction [0, 360]

To simulate a fracture normal given by the Fisher distribution, first we rotate the Fisher

mean pole to be upward (Fig. 2.A-1). A random number (between 0 and 1) is generated,

and by using the cdf of the Fisher distribution (Eq. (2.B-3)), angle δ, the angle between a

fracture normal and the Fisher mean pole, is calculated. This angle and another generated

random number between 0 and 360 define a unique orientation in the coordinate system

shown in Fig. 2.A-1. The dip and dip-direction of the simulated fracture are then

calculated from the paired angles by rotating the upward axis back to the Fisher mean

pole. An example of the simulated Fisher distributed fracture normals is shown in Fig.

2.4. This program was used to study linear and planar samplings of the Fisher distributed

fractures.

107

B. TRACE - Simulate fracture population sampled by a borehole

The program was developed in Visual C++. OpenGL was used to visualize the simulated

fracture population and the borehole in three-dimensional graphics (Figs. 3.8 and 4.11).

In this program, fractures are rectangular in shape and borehole is considered as a

cylinder. The parameters user may change are listed in Table App-2.

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

Parameter Range

Fracture length, f_l > 0

Fracture aspect ratio, α > 1.0

Fracture width, f_w f_w = α × f_l

Fracture Dip, or dip of the Fisher mean pole [0, 90]

Fracture dip-direction, or dip-direction of the Fisher

mean pole [0, 360]

Fisher constant, κ > 0

Generation region shape Box, Cylinder, Ball

Fracture volumetric intensity, P32 > 0

Number of fractures, N > 0

Borehole (sampling cylinder) length, c_l > 0

Borehole (sampling cylinder) radius, c_r > 0

Borehole plunge [0, 90]

Borehole trend [0, 360]

User may fix the number of fractures to be generated or fix fracture volumetric intensity

and let the program calculate the number of fractures. The generation region, i.e., a box,

108

a cylinder, or a ball (Fig. App-1), is a region in which the centers of generated fractures

are located. The sampling region refers to the region that the sampling cylinder (borehole)

is within. The relationship between sampling region and generation region is showed in

Fig. App-1. Note that the maximum fracture dimension is: max_f_l = 22 __ lfwf + ;

the maximum sampling cylinder dimension is max_c_l = 22 _4_ rclc + .

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

generation region.

l_Box l_Cylinder

R_Cylinder

R_Ball

Generation CylinderGeneration Box Generation Ball

2 c_r

c_l f_w

f_l

max_f_l

Largest fracture Sampling cylinder

max_c_l

109

Given dimensions of the largest fracture and sampling cylinder, the minimum size of

generation region varies with the shape of the region. The dimensions for different

shaped generation regions are listed in Table App-3.

Table App-3. Minimum dimension of different generation regions

Region shape Length Radius

Box lflc _max__max_ + -

Cylinder lclflclc _max_/)_max__(max__ + lfRc _max__ +

Ball - 2/)_max__(max_ lflc +

In three shapes of generation region, generation box is the simplest. The algorithm of

calculating fractures truncated by the region boundaries is also simple. Generation

cylinder is for the case that no rotation of the sampling cylinder is involved, while

generation ball allows rotation of the sampling cylinder.

After generating a set of synthetic fractures, the program computes the intersections

between the sampling cylinder and the rectangular fractures. Fracture traces are shown

on an unrolled trace map (Figs. 3.8 and 4.11). The outputs (in a text file) of the program

includes: fracture volumetric intensity (either set by the user, or calculated by the

program), fracture areal intensity on the borehole surface (calculated by the program by

dividing the total trace length by the cylinder area), and count of each intersection type.

110

This page intentionally left blank.

111

7 Xiaohai Wang

EDUCATION

Ph.D., Civil Engineering, Virginia Polytechnic Institute and State University, December 2005

Ph.D., Rock Mechanics & Rock Engineering, Institute of Soil & Rock Mechanics, Chinese Academy of Sciences, September 1999

M.S., Mining Engineering, Taiyuan University of Technology, 1996 B.S., Mining Engineering, Taiyuan University of Technology, 1993

RESEARCH AND WORK EXPERIENCE

Research

Research assistant, Department of Civil & Environmental Engineering, Virginia Tech, Blacksburg, VA August 2001 – September 2005 • Characterization of rock fractures based on cylindrical samples, supported by National

Science Foundation • Scanline bias estimate and techniques to minimize the directional bias in cylindrical

sampling • Techniques to estimate fracture size and aspect ratio in sedimentary rocks • Computer simulation of rock mass fractures with three-dimensional visualization • Computer program to deploy unbiased scanlines on fracture trace maps and characterize

fractures

Research Fellow, Institute of Soil & Rock Mechanics, Chinese Academy of Sciences, Wuhan, China August 1998 – May 2001 • Integration of Three-dimensional Strata Information System (3DSIS), supported by

Chinese Academy of Sciences • Strata geological structure analysis and modeling Research Assistant, Department of Mining Engineering, Shanxi Mining Institute, Taiyuan, Shanxi, China August 1994 – July 1996 • Fractals of distribution features of rock mass fissures, supported by Shanxi Natural

Science Foundation

Teach

Teaching Assistant, Department of Civil & Environmental Engineering, Virginia Tech, Blacksburg, VA May 2004 – July 2004 • Assist teaching the Intensive Summer Course in Geology Engineering & Rock

Mechanics for the US Army Corps of Engineers Work

112

Programmer, Institute of Soil & Rock Mechanics, Chinese Academy of Sciences, Wuhan, China August 1999 – May 2001 • Develop the visualization module for program Three-dimensional Limit Equilibrium

Method in Slope Stability Analysis • Design and develop the program Supporting System in Deep Foundation Excavation

PUBLICATIONS

Wang, Xiaohai, M. Mauldon, W. Dunne. Estimating size and aspect ratio of rectangular fractures from traces on cylindrical rock exposures. (for submission to Rock Mechanics & Rock Engineering)

Mauldon, Matthew, X. Wang. Estimating fracture intensity from traces on cylindrical exposures. (for submission to International Journal of Rock Mechanics & Mining Sciences)

Wang, Xiaohai, M. Mauldon, and W. S. Dershowitz. Multi-dimensional intensity measures for Fisher-distributed fractures. (submitted to Mathematical Geology, May 2005)

Wang, Xiaohai, M. Mauldon, W. Dunne, C. Heiny. 2005. Extracting fracture characteristics from piercing-type intersections on borehole walls. In: Proceedings of the 40th U.S. Symposium on Rock Mechanics (USRMS) (2005), Anchorage, Alaska.

Wang, X., M. Mauldon, W. Dunne, C. Heiny. 2004. Using Borehole Data to Estimate Size and Aspect Ratio of Subsurface Fractures. In: Proceedings of the 6th North American Rock Mechanics Symposium (NARMS), Houston, Texas

Mauldon, M. and X. Wang. 2003. Measuring Fracture Intensity in Tunnels Using Cycloidal Scanlines. In: Proceedings of the 12th Pan-Am. Conf on Soil Mech & Geotech Eng. & 39th U.S. Rock Mech Symp. Soil & Rock America 2003, Culligan, Einstein & Whittle, eds., Boston: Vol. 1, 123-128

Jiang, Q., X. Wang, D. Feng, S. Feng. 2003. Three Dimensional Limit Equilibrium Analysis System Software 3D_SLOPE for Slope Stability and its Application. Chinese Journal of Rock Mechanics and Engineering. 22 (7): 1121-1125

Mauldon, M., X. Wang, D. Peacock. 2002. Fracture frequency predictions using double-corrected data. In: Proc. of the 5th North American Rock Mechanics Symp. And the 17th Tunnelling Association of Canada Conference: NARMS-TAC 2002, Hammah, R. et al. ed., Toronto, Canada: 27-34

Jiang, Q., M.R. Yeung, X. Wang, D. Feng. 2002. Development of the interactive visualization system for three dimensional slope stability analysis. In: Proc. of the 9th Congress of the International Association of Engineering Geology and the Environment, Durban, September 16-20, 244-252

Zhao, Y., X. Wang, K. Duan, D. Yang. 2002. Unsymmetry of scale transformation of rock mass anisotropy, Chinese Journal of Rock Mechanics and Engineering, Vol. 21. No. 11: 1594-1597

Zhang, Y., X. Wang, J. Chen, S. Bai. 2000. Application of 3D Volume Visualization in Geology, Journal of Rock Mechanics & Engineering (in Chinese), Vol. 20. No. 5.

Wang, X., S. Bai. 1999. 3D Topological Grid Data Structure for Modeling Subsurface In: Proc. of International Symposium on Spatial Data Quality (ISSDQ 1999), Hong Kong.

113

Wang, X., S. Bai, Z. Gu. 1998. The Problems in the Applications of GIS in Rock and Soil Projects. Research and Practice in Rock and Soil Mechanics. Zhengzhou.

Wang, X., S. Bai. 1998. An Easily Integrated Three-dimensional Data Structure in Strata Modeling. In: Proceedings of International Conference on Modeling Geographical and Environmental Systems with Geographic Information Systems. Hong Kong

Zhao, Yangsheng, X. Wang, K. Duan. 1997. The Scale-invariability of the Distribution of Rock Mass Fissures. Modern Mechanics and Technology Progressing. Beijing.

AWARDS / AFFILIATION

Graduate Research Development Project Grant, Virginia Polytechnic Institute & State University, 2003-2004

Outstanding poster presenter, 20th Annual Graduate Student Assembly Research Symposium & Exposition, Virginia Polytechnic Institute & State University, 2004

Best poster (tied), 6th North American Rock Mechanics Symposium (NARMS), June 6-10, Houston, TX, 2004

Member of American Rock Mechanics Association, 2004, 2005