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
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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.
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
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31
Chapter 3
3 Estimating fracture intensity from traces on cylindrical
exposures Abstract
Fracture intensity is a fundamental parameter when characterizing fractures. In the field,
a great amount of fracture data is collected along boreholes, circular tunnel or shaft walls.
The data reveal some characteristics of fractures in rock masses; however, it has not been
sufficiently interpreted. In this paper, we discuss estimating of fracture intensity, more
specifically, fracture volumetric intensity P32, from fracture trace data in cylindrical
(borehole, tunnel or shaft) samplings. We built up the relationships between the 2-d
fracture intensity measure and the 3-d fracture intensity measure theoretically.
Stereological analyses show that the conversion factor between the two intensity
measures is not dependent on fracture size, shape or circular cylinder radius, but is related
to the orientation of the cylinder and the orientation distribution of fracture area. It is also
found that the fracture volumetric intensity measure P32 is always 1.0 to 1.57 times of
fracture trace length per unit borehole surface area (P21,C). The technique of using
cycloidal scanlines to estimate the fracture volumetric intensity is also discussed. A
computer program is developed to generate synthetic fractures sampled by a circular
cylinder and the derived conversion factor between the two intensity measures is tested
by Monte Carlo simulations.
Key words: cylindrical sampling, fracture networks, stereology, rock mass, intensity
measures, conversion factors
32
3.1 Introduction
Natural rock masses are commonly dissected by discontinuities such as fractures, faults
and bedding planes, which influence or even control the behavior of rock masses
(Goodman, 1989; Priest, 1993). Therefore, characterization of the fracture system in a
rock mass, including properties such as fracture orientation, shape, size, aperture, and
intensity (ISRM, 1978), is necessary for many engineering applications. Examples of
such applications include hydrocarbon extraction, control of contaminants in landfills,
tunneling, and rock slope engineering.
Fracture intensity, which represents the amount of fractures in the rock mass, is one of
the fundamental parameters for characterizing fracture systems. Fracture intensity can be
interpreted in several ways, corresponding to a set of fracture abundance measures,
depending on the dimension of the sampling domain. (Dershowitz, 1984, 1992; Mauldon
1994). The most commonly used measure is the frequency of fractures, defined as
number of fractures per unit length. Frequency, which is also referred to as the one-
dimensional (1-d, linear) intensity, P10, is often measured along a scanline (Fig. 3.1(a)) of
fixed orientation on a planar exposure, or along the length of a borehole. The sampling
bias (R. Terzaghi 1965) induced by scanline or borehole measurements of fracture
frequency, or P10, remains a problem with scanline measurements. The major difficulty
with implementing frequency data as a fracture intensity measure has to do with the so-
called “blind zone” (Terzaghi, 1965; Yow, 1987), which refers to fracture orientations
that are “not seen” or under-sampled by a borehole or scanline. The geometric
(“Terzaghi”) correction factor for fractures in the blind zone can lead to gross distortion
of the data (Yow, 1987). A review of scanline sampling is presented by Priest (1993,
2004).
On cylindrical exposures such as borehole walls, circular tunnel or shaft walls, the
fracture system is revealed in a two-dimensional (2-d) form. Besides features of fractures
such as orientation, aperture, or infilling that can be measured directly on cylindrical
33
exposures, the intensity, pattern, and termination relationships of fracture traces on the
cylindrical exposure surfaces provide much more information about fracture networks
than a one-dimensional exposure (scanline) does.
Fig. 3.1. Borehole or shaft sampling of fractures in a rock mass. (a) Vertical shaft
intersects several fractures, which yield traces on the cylinder surface and on the face of
the rock mass; horizontal scanline on the rock face intersects three fracture traces. (b)
Unrolled trace map developed from the borehole or shaft wall.
Rock mass
Scanline
Borehole
Fractures
Fracture trace on the slope
Fracture traces total length = l
Unrolled (developed) trace map (total area A)
(a) (b)
34
To explore the relationships between fracture traces on a cylindrical surface and the 3-d
fracture system, we introduce the following notation. Let P21,C denote the two-
dimensional (2-d, areal) fracture intensity on the circular sampling cylinder surface,
defined as trace length per unit sampling surface area. The subscript C denotes the
cylindrical sampling domain. P21,C is determined as the sum of trace length on tunnel or
borehole walls divided by the total surface area of tunnel or borehole walls. In Fig. 3.1(b),
for instance, assume the total trace length on the unrolled trace map is l and the total area
of the unrolled trace map is A. Then the areal fracture intensity is simply P21,C = l / A.
For a fractured rock mass, this measure is a function of tunnel or borehole size and
orientation, as well as the fracture orientation distribution (weighted by fracture size).
Therefore it is also a directionally biased measure, as is as the linear intensity measure
P10.
Let P32 denote the three-dimensional (3-d, or volumetric) fracture intensity, defined as
fracture area per unit volume of rock mass. P32 is independent of the sampling process
and is an unbiased measure of fracture intensity (Dershowitz, 1992; Mauldon 1994).
Interpreted as an expected value, P32 is also scale independent. P32 is a crucial parameter
for numerical analyses in models such as the discrete fracture flow and transport model
(Dershowitz et al., 1998). However, P32 is impossible to measure directly in an opaque
rock mass.
This paper proposes approaches to utilize fracture trace data collected on the cylindrical
exposures of rock mass, such as borehole walls, tunnel or shaft walls, to estimate
volumetric fracture intensity of the rock mass. This determination is based on the derived
relationship (conversion factor) between the fracture areal intensity on a cylindrical
surface (P21,C) and the fracture volumetric intensity measure (P32).
Following stereological principles (Russ and DeHoff, 2000) we first discuss the general
form of the conversion factor between the areal intensity P21,C on circular cylinder
surface and fracture volumetric intensity measures P32. Theoretical solutions for the
conversion factor between the two measures are derived in the case of cylindrical
35
sampling of constant orientated fractures, and also sampling of fractures with a uniform
distribution. The conversion factor can be calculated analytically if the fracture
orientation distribution with respect to its area is known. Secondly, another approach to
estimate fracture volumetric intensity, based on the cycloidal scanline technique, is also
discussed. By counting the intersections between cycloidal scanlines and fracture traces
on the circular cylinder surface, the fracture volumetric intensity can be estimated
without knowing the orientation of fractures. Finally Monte Carlo simulations are carried
out to verify the derived correction factors.
3.2 Basic assumptions
In this paper, we study a fractured rock mass sampled by a borehole or tunnel/shaft by
using stereology. For convenience, we make the following assumptions with respect to
the geometry of the sampling domain, e.g., the surface of the tunnel/shaft or borehole;
and of fractures in the rock mass.
a) The surface of the sampling domain is a right circular cylinder, long in relation to
its diameter. Borehole, tunnel or shaft ends are not included in the sampling
domain.
b) Fractures are planar features with negligible thickness. No assumptions are made
regarding the spatial distribution of fractures, or fracture shape. In particular, it is
not necessary that fracture centers follow a Poisson process, or that fractures have
the shape of circular or elliptical discs.
c) No prior assumptions are made about fracture size, or orientation distribution;
however, for the first method discussed below, the fracture orientation distribution
in terms of area must be known.
d) The sampling domain is independent of the rock mass fracture network to be
characterized. What this means in practical terms is the borehole/shaft or tunnel is
emplaced without consideration of fracture locations.
36
The above assumptions are fairly standard in engineering analysis of fractured rock
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
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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 &
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)
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
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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.
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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.
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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.
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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,
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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
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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
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.
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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
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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.
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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