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8/19/2019 Estimating Fracture Trace Intensity Density and Mean Lenght Using Circular Scan Line and Windows
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AAPG Bulletin, v. 86, no. 12 (December 2002), pp. 2089–2104 2089
Estimating fracture traceintensity, density, and meanlength using circular scan lines
and windowsM. B. Rohrbaugh Jr., W. M. Dunne, and M. Mauldon
A B S T R A C T
Fracture characterization protocols that reduce sampling bias are
likely to yield higher quality input for exploration and development
decisions when dealing with naturally fractured reservoirs. A new
set of estimators for fracture density, intensity, and mean tracelength corrects for sampling biases and provides a useful integrated
description for bulk aspects of a fracture network. These estimators
are based on counts of intersections between fracture traces and
circular scan lines and of trace terminations in circular windows.
Application to synthetic fracture patterns with known parameters
validates the use of the new estimators, which are then applied to
natural fault trace maps from seismic volumes and joint trace maps
from rock pavements. The new estimators are distribution inde-
pendent and eliminate the effects of orientation, censoring, and
length biases, which limit the effectiveness of other sampling tech-
niques. Estimator accuracy improves as sample size increases, par-ticularly for larger circles that exceed a fracture-defined block size.
Estimator accuracy for mean trace length improves when the sam-
ple exceeds threshold count values for fracture terminations based
on guidance from the analysis of similar synthetic patterns. These
new estimators also provide both inputs and independent checks of
predictions for fracture-generator programs used to model fracture
populations in a rock volume.
I N T R O D U C T I O N
Quantification of fracture parameters such as density, size, and in-
tensity aids in the assessment of hydrocarbon flow and storage in
fractured reservoirs (Reiss, 1982; Nelson, 1985; Dershowitz and
LaPointe, 1994; Narr, 1996) but is complicated by the difficulty of
deciding on the best approach for incorporating fractures into a
reservoir model. Problems arise from the lack of consensus as to
Copyright2002. The American Association of Petroleum Geologists. All rights reserved.
Manuscript received January 24, 2000; revised manuscript received January 10, 2001; final acceptance
June 15, 2002.
A U T H O R S
M. B. Rohrbaugh Jr. Tennessee Department of Environment and Conservation, Division of Underground Storage Tanks, 540 McCallie Avenue, Suite
550, Chattanooga, Tennessee, 37402–2013
M. Bruce Rohrbaugh Jr. received his B.S.degree in geology from West VirginiaUniversity in 1997 and his M.S. degree instructural geology from the University of Tennessee, Knoxville in 2000. His researchinterests include hydrogeology, application of computers to solving geologic problems, andstructural geology. He is currently employedas a geologist with the Tennessee Departmentof Environment and Conservation.
W. M. Dunne Department of Geological Sciences, 306 G&G Building, University of Tennessee, Knoxville, Tennessee, 37996-1410; wdunne@utk.edu
William M. Dunne, although born in theUnited States, received his B.S. degree andPh.D. in geology from the University of Bristol,England. He joined the Department of Geological Sciences at the University of Tennessee in 1988 as an associate professorand is now is a professor and departmenthead. His research interests include fracture
characterization particularly in younger rocks,deformation in thrust belts from large to smallscale, and deformation analysis of sedimentary rocks.
M. Mauldon Department of Civil and Environmental Engineering, Virginia Tech, 200 Patton Hall, Mail Code 0105, Blacksburg,Virginia, 24061; mauldon@vt.edu
Matthew Mauldon, although born in England,has geology (B.A.) and civil engineering (M.S.)degrees and a Ph.D. in civil engineering from
the University of California at Berkeley. Hespent eight years on the faculty at theUniversity of Tennessee, where hecollaborated with Dunne and Rohrbaugh.Mauldon is now an associate professor in the Via Department of Civil and EnvironmentalEngineering at Virginia Tech, where he teaches and conducts research in the areas of rock mechanics, engineering geology, andgeotechnical engineering.
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2090 Fracture Characterization Using Circular Scanlines and Windows
which parameters to quantify, the difficulties in measuring the pa-
rameters, and intrinsic sampling biases. To address these problems,
we offer a set of estimators for fracture parameters based on the
use of circular scan lines and windows where a scan line is the pe-
rimeter and the window is the interior of a circle. These estimators
reduce sampling biases and provide a useful integrated description
of bulk aspects of a fracture network. Use of these estimators re-
quires only counts of the number (n) of fracture traces intersectingthe circumference and/or counts of the number (m) of fracture
traces terminating in the circle interior. Performance of the new
circular scan line/window estimators is evaluated for both synthetic
and natural fracture patterns to demonstrate that estimates con-
verge on true values for a fracture trace population, to demonstrate
that the new estimators outperform or match existing estimators,
and to discuss issues of estimator performance.
Fracture Parameters
A redundant, commonly mutually inconsistent vocabulary exists todescribe the amount of fracturing in a rock. Rather than reviewing
this terminology, we define three key parameters of a fracture pat-
tern: density, size, and intensity (Table 1).
Density
Fracture density is commonly treated as the number of observed
isolated fractures or fracture segments per unit length, area, or vol-
ume (Dershowitz and Herda, 1992; Ghosh and Daemen, 1993).
This is a scale-dependent quantity that we call “apparent density.”
Fracture density is defined in this article as the number of fractures
per unit length, area, or volume, enumerated in terms of uniquepoints, such as fracture centers (Mauldon, 1998; Mauldon and Der-
showitz, 2000). Apparent density overestimates density (Kulatilake
and Wu, 1984; Mauldon et al., 2001), and the magnitude of this
overestimation increases as sample size decreases (Figure 1B). For
example, the apparent density (number of visible traces divided by
circle area) of fractures of Set 1 in Figure 1A is 0.0014 per m 2 for
a circular window of radius 75 m and increases to 0.0020 per m2
for a smaller window of radius 25 m. If trace centers (dots in Figure
1A) are counted to estimate true density for Set 1, the estimates
(count divided by circle area) are 0.0010 per m2 for both the circle
of radius 25 m and the circle of radius 75 m. In practical applica-tions, one half the number of fracture trace terminations is used as
an unbiased estimate of the number of trace centers, because cen-
ters cannot be identified unless both ends of a trace are visible
within the window (Mauldon, 1998; Mauldon et al., 2001).
Size
Fracture size is defined in one, two, or three dimensions, as frac-
ture trace length, area, and volume, respectively. For composite
connected fractures, an investigator should decide whether to
characterize individual segments or the entire composite. Typically
A C K N O W L E D G E M E N T S
Acknowledgment is made to the donors of The Petroleum Research Fund, administeredby the American Chemical Society, for partialsupport of this research. Additionally, theGeological Society of America and the South-eastern Section of the Geological Society of America are thanked for their partial supportof the field work. We would also like to thank M. F. Schaeffer for permission to use the frac- ture trace map from Rocky Creek, South Caro-lina, and Camilo Montes, Yen-Yit Chan, You Li,and Jim Calcagno for programming assistance.Steve Laubach, John Lorenz, and Bill Der-showitz are thanked for their insightful andconstructive reviews.
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Table 1. Vocabulary for Fracture Parameters*
Parameter Definition Estimator
Density Linear Number of fractures per unit length m /2p r 2 q̂
Areal (q) Number of fractures per unit area
Volumetric Number of fractures per unit volume (in all cases count is conducted using
unique points such as fracture centers or half of the fracture terminations)
Size Linear (l) Mean fracture trace length (p r /2)( n / m) l̂
Areal Mean fracture area
Volumetric Mean fracture volume
Intensity Linear Number of fracture per unit length (L0 /L1 L1) Î n /4 r
Areal (I) Fracture length per unit area (L1 /L2 L1)
Volumetric Fracture area per unit volume (L2 /L3 L1)
*Where L is a dimension of length and r is radius. See Figure 3 for illustration of m and n.
Figure 1. Problems due to censoring and length bias whensampling fracture traces. (A) Fracture pattern with two sets sam-pled by three progressively larger circles (dots trace centers;
r radius). (B) Decreasing density overestimates for increasingsample (circle) size. (C) Decreasing mean trace-length under-estimates for increasing sample size. (D) Undersampling of joint trace lengths due to censoring as shown by the probability dis- tribution function (pdf). (E) Oversampling of longer traces due to length bias as shown by a pdf.
for fracture studies on exposed surfaces, trace lengthor aperture is measured because fracture areas and
volumes are commonly not directly measurable (Der-
showitz and Herda, 1992; Marrett et al., 1999; Ortegaand Marrett, 2000). For the two-dimensional fracture
patterns discussed in this article, we use mean trace
length as our size parameter (Table 1; Figure 1).
Intensity
Fracture intensity is a pattern characteristic that in-
corporates both density and size (Dershowitz and
Herda, 1992; Mauldon and Dershowitz, 2000). Inten-
sity is defined as number of fractures per unit sample
length, fracture length per unit surface area, or frac-
ture area per unit rock volume, in one, two, or threedimensions, respectively (Table 1). Consequently, in-
tensity has the same dimensions whether calculated
linearly, areally, or volumetrically. In this article, we
examine two-dimensional samples and, therefore, use
the areal intensity: fracture length per unit area.
Current Measurement Methods
Two common sampling methods are used for esti-
mating fracture parameters: straight scan lines and
areal sampling (Figure 2) (LaPointe and Hudson,1985; Priest, 1993; Wu and Pollard, 1995, Becker and
Gross, 1996; Marrett et al., 1999; Ortega and Marrett,
2000). Straight scan lines sample the fractures they
intersect and are used to systematically record fracture
characteristics, such as number, orientation, aperture,
and so on. (Priest and Hudson, 1981). Areal sampling
involves mapping the fracture trace pattern and re-
cording desired fracture characteristics at locations in
the map area (e.g., Priest, 1993; Wu and Pollard,
1995).
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Figure 2. Circular scan line/window (dotted circle), areal (ir-regular white window), and straight scan line (dotted line) sam-pling of a fracture trace population. Solid lines represent visiblefracture traces, and dashed lines represent covered fracture traces.
Sampling Biases
Orientation bias occurs where scan lines or the long
axes of inequant sampling areas are not perpendicular
to a fracture set. In both cases, intensity is underesti-
mated (Terzaghi, 1965; Priest, 1993; Mauldon and
Mauldon, 1997). Scan line estimates are corrected by
dividing by the cosine of the angle between the scan
line and the normal to the fracture set (Terzaghi, 1965;Peacock et al., in press). However, as this angle ap-
proaches 90, the cosine approaches zero, and cor-
rected estimates approach infinity, significantly over-
estimating intensity (Priest, 1993). No procedures are
currently available for correcting orientation bias from
inequant sampling areas.
Censored fracture traces extend beyond the ex-
posure or seismic coverage, so that one or both ends
are not visible (Figure 1A) (e.g., Cruden, 1977;
Baecher and Lanney, 1978; Einstein and Baecher,
1983; Kulatilake and Wu, 1984; LaPointe and Hudson,1985; Pickering et al., 1995; Mauldon, 1998; Marrett
et al., 1999). Such traces are referred to as singly or
doubly censored, respectively. Using censored traces to
estimate density and size directly, rather than using es-
timators such as those in Table 1, leads to overesti-
mates of density and underestimates of size (Figure 1).
For example, a count of all visible trace segments in a
circle of radius 25 m (4 segments vs. 2 centers) of Set
1 (Figure 1A) overestimates density by a factor of two
(0.002 vs. 0.001 per m2). Similarly, using the censored
lengths, the estimate of mean trace length for Set 1
(total visible trace length divided by number of traces)
is 21.3 m, in contrast to the true mean trace length of
40 m. In both cases, increasing the size of the sampling
circle relative to fracture size decreases censoring and,
consequently, error. For example, increasing circle ra-
dius from 25 to 75 m (Figure 1A) decreases apparent
density from 0.0020 to 0.0014 per m2 (true density0.001 per m2) and increases apparent mean trace
length from 21.3 to 27.4 m (true value 40 m).
Length bias occurs because longer fracture traces
have a greater probability of being sampled than
shorter traces (Baecher and Lanney, 1978; Einstein and
Baecher, 1983; LaPointe and Hudson, 1985; Mauldon
1998). Consequently, mean trace-length estimates and
trace-length distributions are skewed toward longer
fractures (Figure 1E). For example in Figure 1A, Sets
1 and 2 have the same fracture density, but Set 1 has
longer traces. As a result, the largest window (75 mradius) samples 24 of the longer Set 1 fractures vs. 17
of Set 2. If these data are taken at face value,
mean trace length for the whole pattern will be
overestimated.
Pattern heterogeneity refers to a change in fracture
parameters with a change in position. This sampling
bias can occur, for example, when sampling rock vol-
umes with localized fracture development near faults.
Heterogeneity is partly a function of scale and may be
especially pronounced for borehole sampling, which
provides limited samples of subsurface fracture net-works. Use of multiple subdomains with homogeneous
parameters minimizes the effects of heterogeneity
(Turner and Weiss, 1963; Whitten, 1966; LaPointe
and Hudson, 1985; Kulatilake et al., 1997). Alterna-
tively, geostatistical methods may be employed to de-
termine the magnitude and rate of change of a param-
eter as a function of distance and direction (LaPointe
and Hudson, 1985; LaPointe, 1993; Priest, 1993; Jian
et al., 1996).
Advantages/Disadvantages of Current Estimation Methods
Straight scan lines (Figure 2) provide rapid estimates
of fracture intensity (Priest and Hudson, 1981; La-
Pointe and Hudson, 1985; Becker and Gross, 1996).
Unprocessed straight scan line data, however, are sub-
ject to orientation bias, length bias, censoring, and pat-
tern heterogeneity (Terzaghi, 1965; Baecher and Lan-
ney, 1978; Priest and Hudson, 1981; LaPointe and
Hudson, 1985; Priest, 1993; Mauldon and Mauldon,
1997; Peacock et al., in press).
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Figure 3. Fracture trace pattern with sampling circle. (A) Soliddots are intersection points ( n) between fractures and circle. (B)Triangles are fracture endpoints ( m) in the circular windows.
Areal sampling (Figure 2) reduces censoring and
length bias as compared to scan line sampling but (1)
is subject to orientation bias in the plane for anything
except a circular sampling area, (2) is more time con-
suming than use of scan lines, (3) may disguise pattern
heterogeneities, and (4) may introduce censoring and/
or length biases for inequant or small sampling areas
(LaPointe and Hudson, 1985; Kulatilake et al., 1993; Wu and Pollard, 1995; Kulatilake et al., 1997).
N E W E S T I M A T O R S
Because neither areal sampling nor straight scan lines
are completely satisfactory in terms of efficiency, ac-
curacy, and lack of bias, this article examines the use
of circular scan lines and windows for characterizing
intensity, density, and mean trace length (Figure 2).
Data from circular scan lines and windows are appliedto estimators (Table 1) that do not require knowledge
of fracture spacing, trace length, or orientation distri-
butions and are, therefore, distribution independent
(Mauldon, 1998; Mauldon et al., 2001).
E S T I M A T O R P E R F O R M A N C E F O R S Y N T H E T I C F R A C T U R E P A T T E R N S
The ability of the new estimators to correct for sam-
pling biases was investigated by comparing estimatesof intensity, density, and mean trace length to known
values for synthetic fracture trace patterns. A new
computer program called JAWS (Joint Analysis using
Windows and Scanlines) (Rohrbaugh, 2000), was used
to generate and sample traces with uniformly distrib-
uted centers in a square region. These traces were sam-
pled with 100 circles of known radius placed randomly
and independently in a smaller square analysis region
centered on the generation region so as to avoid edge
effects (Gilmour et al., 1986). Trace-circle intersec-
tions and trace terminations inside circular windowswere counted (Figure 3), and counts were input into
estimators (Table 1) to yield comparison values.
For example, two synthetic fracture sets with ori-
entations of 60 30 and 150 30 were deployed
to produce orientation bias when sampled, with frac-
ture lengths of 50 10 L (L is an arbitrary unit of
length) and 100 10 L that exceed circle radius of 20
L by factors of 2.5 and 5, respectively, to create a cen-
soring bias, and with a difference in length between
sets of a factor of two to yield a length bias. Running
means for the estimates (Figure 4) converge on the in-
put values for intensity, density, and mean trace length
and correspond closely to them after about 40 samples,whereas individual estimates fluctuate about the mean.
These results show that the new estimators deal suc-
cessfully with orientation, censoring, and length biases.
Similar results were found for all other synthetic cases.
Comparison to Other Estimators
Having established that the new estimators overcome
sampling biases, their accuracy was compared to that
from straight scan lines and/or areal sampling of a syn-
thetic pattern. To achieve equivalent sampling com-parison, estimates were obtained using 100 circular
scan lines/windows of radius 10 L , 100 straight scan
lines of the same length (2p (10 L) 63 L), and
areal samples consisting of 100 circle interiors.
For example, a fracture set with orientation of 060
30, mean trace length of 40 20 L, intensity of
1.95 L/L2, and density of 0.049 per L2 was generated
(Figure 5A). Intensity estimates from both circular
scan lines and areal samples accurately estimate inten-
sity, but map construction for areal samples is likely to
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Figure 4. Estimates (jagged black lines) and running meansof estimates (small open circles) for intensity, density, and mean trace length of the synthetic pattern with dashed lines and num-bers for control values, showing that estimators successfully deal with sampling bias issues.
be much more time intensive than is deployment of
circular scan lines. In contrast, running means for the
straight scan lines underestimate intensity by a factor
of 1.7 with the Terzaghi correction applied to the
mean set orientation (Figure 5B).The running mean density estimate from the cir-
cular window estimator corresponds closely to the con-
trol value of 0.049 per L2, whereas the same size areal
samples overestimate density by a factor of 3.5 if trace
segments are counted (Figure 5C). Running means for
circular window estimates of mean trace length also
correspond to the control value, whereas the same size
areal samples tend to yield estimates of a little less than
4 L, which is an underestimate by a factor of 11 as
compared to the input value of 40 L (Figure 5D).
Based on these results, the use of the estimators based
on circular scan lines and windows is recommended.
E S T I M A T O R P E R F O R M A N C E W I T HN A T U R A L F R A C T U R E P A T T E R N S
Given the success of the new estimators with syntheticfracture patterns, we investigated their applicability to
natural patterns. This comparison used new and exist-
ing trace maps of fault and/or joint patterns that range
from single sets of parallel fractures to multiset pat-
terns to near polygonal patterns (Table 2; Figure 6).
This geometric variety was selected so as to provide a
thorough test of the estimator equations. Also, fault
trace maps from seismic reflection data (Figure 6C, D;
Table 2) were included to demonstrate that circular
scan lines may be used effectively with this common
industry data source. Trace maps were digitized andimported into JAWS for analysis. After specifying the
number and size of the sample circles, JAWS randomly
distributed them in the sample area and only retained
intersection and termination counts for circles that
were completely contained within a map (Stoyan et al.,
1995). The new circle-based estimates for the natural
data sets were compared to estimates from areal sam-
ples of the maps.
Circle-based intensity estimates mostly match
areal estimates for both fault and joint trace patterns
(e.g., Figures 6; 7C, D), as would be expected from thesimilar correspondence that was found during the anal-
ysis of synthetic fracture patterns (e.g., Figure 5B). The
circle-based density and mean trace-length estimates
mostly matched the apparent density and mean trace-
length estimates where fracture/fault traces are small
relative to window size, so that censoring is minimal
(e.g., Figures 6C, D; 7E, F). In contrast, at Llantwit
Major, 86% of the master joint traces are censored, and
direct density and mean trace-length estimates from
areal samples differ by a factor of 4 from the circle-
based estimates (density: 1.5 vs. 0.35 per m2
; meantrace length: 2.2 vs. 8.7 m)(Figures 6A; 7A, B). Having
already tested and demonstrated the performance of
the circle-based estimators using synthetic traces, we
interpret these circle-based estimates at Llantwit Ma-
jor as being representative of true population charac-
teristics. It follows that the direct areal estimates for
the characteristics of these greatly censored traces are
off by a factor of 4. Again, the circle-based estimators
yield superior results to previous estimators where the
potential for sampling biases exists.
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Figure 5. Estimator performances for a synthetic fracture trace pattern. (A) Part of a synthetic fracture pattern used to test relativeperformance of new and old estimators. Pattern characteristics are described in text. (B) Intensity estimates from circular scan lines(thick black line), circular areas (light gray line), and straight scan lines (thin line). Control value is 1.95 L/L 2. (C) Density estimatesfrom circular windows (thick black line) and circular areal sampling of apparent density (light gray line). Control value is 0.049 perL2. (D) Mean trace length estimates from circular windows (thick black line) and circular areal sampling of apparent trace length(light gray line). Control value is 40.0 L. Open circles in (B), (C), and (D) are the running means for appropriate estimators.
D I S C U S S I O N O F E S T I M A T O R P E R F O R M A N C E
Effects of Block Size on Estimates
Block size in two dimensions is the unfractured area
bounded by fracture traces from two or more sets
(LaPointe, 1988). Block size depends on the fracture
spacing or frequency and could be an issue for the new
estimators for circles smaller than block size. Block size
effects on the intensity estimator were investigated at
Llantwit Major and the light gray region at Amroth
(Figure 6; Table 2). Block size at Llantwit Major is
about 0.7–1 m by 0.3–0.5 m, whereas at Amroth the
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T a b l e
2 . G e o l o g y o f F r a c t u r e T r a c e P o p u l a t i o n s
N a m e ,
L o c a t i o n ,
F i g u r e N o .
R o c
k A g e a n d
F o r m
a t i o n N a m e
S o u r c e ,
T r u n c a t i o n
L i m i t ,
M a p p i n g S c a l e ,
S u r f a c e A r e a
S e t s
P a t t e r n G e o m e t r y
O r i g i n
L l a n t w i t M a j o r , W a l e s ,
U n i t e d K i n g d o m ,
S S 9 5 7 5 6 7 5 4 *
( F i g u r e 6 A )
E a r l y J u r a
s s i c ,
P o r t h k e r r y
F o r m a t i o n ( l i m e s t o n e )
T h i s s t u d y ,
2 0 c m ,
1 : 2 5 ,
4 9 . 5 m
2
J o i n t s c r o s s — 0 7 5
m a s t e r — 1 6 5
L a
d d e r p a t t e r n w i t h a
b i m o d a l t r a c e l e n g t h
d i s t r i b u t i o n
1 6 5 s e t f o r m e
d d u r i n g L a t e
C r e t a c e o u s – E
a r l y M i o c e n e A l p i n e
c o m p r e s s i o n
( N e m c o c k e t a l . ,
1 9 9 5 ) ;
0 7 5 s e t f o r m
e d d u e t o r e l a x a t i o n o r
c o n t r a c t i o n a
t l a t e r s t a g e ( R a w n s l e y
e t a l . ,
1 9 9 8 )
A m r o t h ,
W a l e s , U n i t e d
K i n g d o m ,
S N 1 7 5 6 0 7 2 2 *
( F i g u r e 6 B )
E a r l y W e s
t p h a l i a n
( C a r b o n
i f e r o u s ) ,
E a r l y
W e s t p h a l i a n c o a l
m e a s u r e
s ( s a n d s t o n e )
T h i s s t u d y ,
4 0 c m ,
1 : 2 0 ,
1 6 2 . 0 m
2
( e x p o s e
d
a r e a ) , 1 0 4 . 7 m
2
( s t i p p l e d a r e a )
J o i n t s 2 0 0 , 2 9 0 ,
3 1 6
O r t h o g o n a l p a t t e r n
( 2 0 0 & 2 9 0 ) a n d
o n e y o u n g e r j o i n t
s e t
2 0 0 a n d 2 9 0
s e t f o r m e d d u r i n g
a l t e r n a t i n g r
2
a n d r 3
s t r e s s
d i r e c t i o n s d u
r i n g V a r i s c a n t h r u s t i n g
( D u n n e a n d N o r t h ,
1 9 9 0 )
S l e i p n e r V e s t fi e l d ,
N o r t h S e a ,
( F i g u r e 6 C )
J u r a s s i c , H
u g i n
F o r m a t i o n , r e s e r v o i r
s a n d s t o n e
O t t e s e n E l l e v s e t e t
a l .
( 1 9 9 8 ) , 2 0 0 m ,
u n k n o w n ,
1 1 0 . 7
k m
2
F a u l t s v a r i e t y o f
o r i e n t a t i o n s
S o
m e w h a t p o l y g o n a l
L a t e J u r a s s i c – E
a r l y C r e t a c e o u s
e x t e n s i o n , p o
s s i b l y r e l a t e d t o s a l t
p i l l o w f o r m a t i o n ( O t t e s e n E l l e v s e t e t
a l . ,
1 9 9 8 )
C a r t i e r T r o u g h ,
T i m o r
S e a ( u n s p e c i fi e d )
( F i g u r e 6 D )
U p p e r C r e t a c e o u s t o
H o l o c e n
e s e d i m e n t s
W a l s h e t a l . ( 1 9 9 6 ) ,
4 0 m
d i s p l a c e m
e n t ,
u n k n o w n ,
5 5 5 . 6
k m
2
F a u l t s W S W
S i n g l e s e t i n t e r m s o f
o r i e n t a t i o n
P l i o c e n e – P l e i s t o c e n e n o r m a l f a u l t s
( W a l s h e t a l . ,
1 9 9 6 )
T e l p y n P o i n t , W a l e s ,
U n i t e d K i n g d o m ,
S N 1 8 3 3 0 7 3 1 *
( F i g u r e 6 E )
L a t e N a m
u r i a n
( C a r b o n
i f e r o u s ) ,
U p p e r
S a n d s t o n e G r o u p
T h i s s t u d y ,
4 0 c m ,
1 : 2 5 ,
2 4 7 . 6 m
2
J o i n t s 2 0 0 , 2 6 7 ,
2 9 0 , 3 1 8
O r t h o g o n a l p a t t e r n
( 2 0 0 & 2 9 0 ) ; o t h e r
y o u n g e r s e t s
S a m e a s A m r o
t h
W a r d L a k e ,
C a l i f o r n i a ,
U n i t e d S t a t e s , n o t
s p e c i fi e d ( F i g u r e 6 F )
C r e t a c e o u
s , M t . G i v e n s
G r a n o d i o r i t e
S e g a l l a n d P o l l a r d
( 1 9 8 3 ) , 1 0 0 c m
,
u n k n o w n ,
1 8 3 5 . 6
m 2
( e x p o s e d a r e a ) , 1
1 6 9 . 1
m 2
( s t i p p l e d a r e a
)
J o i n t s 0 1 0 – 0 2 0
S i n g l e s e t
A g e o f j o i n t s h
a s b e e n r e p o r t e d a s
e i t h e r p r e - E o
c e n e o r p o s t - P l i o c e n e
( S e g a l l a n d P
o l l a r d ,
1 9 8 3 ) ; j o i n t s e t s
a r e o f r e g i o n
a l e x t e n t
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Rohrbaugh et al. 2097
P 1 0 0 ,
Y u c c a M t . ,
N e v a d a ,
U n i t e d
S t a t e s , 5 6 2 0 5 1 . 8
f t
E †
7 6 3 4 0 7 . 7
f t N †
( F i g u r e 6 G )
M i o c e n e ,
T i v a C a n y o n
T u f f ( u p p e r l i t h o p h y s a l
z o n e )
B a r t o n e t a l . ( 1 9 9 3 ) , 2 0
c m ,
1 : 5 0 ,
2 3 3 . 2 m
2
( e x p o s e d a r e a ) , 1
2 4 . 1
m 2
( s t i p p l e d a r e a
)
J o i n t s c o o l i n g — 0 5 0 ,
3 2 0 ; t e c t o n i c — 0 1 0 ,
0 4 0 , 3 2 5 , 3 5 9
O r t h o g o n a l s e t o f
c o o l i n g j o i n t s w i t h
t h e 0 5 0 b e i n g
d o m i n a n t ; p o o r l y
d e v e l o p e d t e c t o n i c
j o i n t s
C o o l i n g j o i n t s f o r m e d fi r s t d u r i n g
t h e r m o e l a s t i c
r e l a x a t i o n f r o m
c o o l i n g ; t h e t e c t o n i c j o i n t s p o s t d a t e
c o o l i n g j o i n t s
; b a s e d o n t e r m i n a t i o n
r e l a t i o n s h i p s ,
t h e 3 2 5 t e c t o n i c s e t i s
o l d e s t ,
f o l l o w
e d b y 3 5 9 s e t , w i t h
0 4 0 s e t y o u n g e s t .
( B a r t o n e t a l . ,
1 9 9 3 )
R o c k y C r e e k ,
S o u t h
C a r o l i n a ,
U n i t e d
S t a t e s , 5 1 1 2 1 5 . 3 m
E * * 3 8 2 1 8 4 3 . 5 m
N * * ( F i g u r e 6 H )
C a m b r i a n
, G r e a t F a l l s
M e t a g r a
n i t e
M .
F .
S c h a e f f e r , 1 9 9 8 ,
u n p u b l i s h e d d a t a ,
2 5
c m ,
1 : 1 2 0 ,
1 4 4 6 . 6
m 2
( e x p o s e d a r e a ) , 9
2 2 . 2
m 2
( e x p o s e d , s t i p
p l e d
a r e a )
J o i n t s 0 6 8 , 2 7 4 ,
2 9 2 , 3 5 6
M
u l t i p l e f r a c t u r e s e t s ;
l e n g t h d i s t r i b u t i o n i s
i n d e p e n d e n t o f
o r i e n t a t i o n
J o i n t s a r e a t h i g h a n g l e s a n d o f
t e c t o n i c o r i g i n ; g r e e n s c h i s t f a c i e s
m i n e r a l s a l o n
g t h e j o i n t s 3 0 0 M a ;
t h e l a u m o n i t i t e m i n e r a l i z a t i o n
r e l a t e s t o M e
s o z o i c r i f t i n g 1 5 0 – 2 0 0
M a ( M .
F . S c
h a e f f e r , 1 9 9 9 , p e r s o n a l
c o m m u n i c a t i o n ) .
* B r i t i s h N a t i o n a l G r i d R e f e r e n c e S y s t e m
( U n i v e r s a l T r a n s v e r s e M e r c a t o r s y s t e m ) .
* * U n i v e r s a l T r a n s v e r s e M e r c a t o r s y s t e m
f o r U n i t e d S t a t e s .
† 5 0 0 0 f t g r i d b a s e d o n N e v a d a S t a t e P l a n e c o o r d i n a t e s , N e v a d a S t a t e P l a n e p r o j e c t i o n .
block size for the two dominant sets is about 2.5 m by
1 m. At each pavement, 100 circles with diameters
smaller than the block size and 100 circles with di-
ameters larger than the block size (0.2 and 1.5 m at
Llantwit; 0.5 and 3.0 m at Amroth) were used to es-
timate intensity. The smaller circles yield estimates
with much greater variability than the larger circles
(Figure 8). Therefore, to reduce the variability of es-timates, circles larger than mean block size are rec-
ommended for estimating fracture parameters. Similar
conclusions are applicable to the other estimators and
to fault networks interpreted from seismic cubes. For
example, the greatest estimator error occurs at Sleip-
ner Vest and Cartier Trough for the smallest sampling
circles (Figure 7E, F).
Although these estimators are effective with sam-
ples from two-dimensional surfaces such as rock pave-
ments and maps, they cannot be used effectively with
borehole imagery data because terminations inside theborehole cannot be uniquely constrained and, hence,
m counts are subject to interpretation. Thus, none of
the existing estimation approaches are particularly ef-
fective with borehole data sets at present.
Furthermore, inferring subsurface joint patterns
from boreholes is, in general, problematic, because
boreholes are likely to be smaller than block size in
naturally fractured rocks. Perhaps this issue can be re-
solved by considering microfractures in core specimens
(Laubach, 1997; Ortega and Marrett, 2000), but im-
plementation of the circle-based estimators on micro-fractures inside a solid core is beyond the scope of this
article.
Effects of Pattern Heterogeneity
To consider the effect of pattern heterogeneities and
circle size on estimator variance, intensity estimates for
fractures from the rock pavements were analyzed as a
function of circle size (Tables 2, 3; Figure 6). Fault
trace maps could not be included in the analysis be-
cause their greater size as compared to the joint pat-terns precluded the use of sampling circles with the
same size.
The fracture patterns that were selected for this
analysis encapsulate a wide range of likely fracture ge-
ometries and settings (Table 2; Figure 6). They include
a ladder geometry where cross-joint spacing is con-
trolled by the spacing of master joints (Figure 6A); an
orthogonal pattern of large fractures overprinted by
younger, less persistent joints (Figure 6B); patterns
with clustered fractures due to fault development or
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2098 Fracture Characterization Using Circular Scanlines and Windows
Figure 6. Fracture trace maps. (A) Llantwit Major. (B) Amroth. (C) Sleipner Vest (modified from Ottesen Ellevset et al., 1998). (D)Cartier Trough (modified from Walsh et al., 1996). Continued.
igneous processes (Figure 6E, G); and patterns in crys-
talline rocks that vary from a single fracture set in a
granite (Figure 6F) to a complex array of fractures that
accumulated during a long uplift and erosion history
(Figure 6H). Characteristics of these patterns include
spatial heterogeneity due to different ages of joint sets,
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Figure 6. Continued. (E) Telpyn Point. (F) Ward Lake (modified from Segall and Pollard, 1983). (G) P100 (modified from Barton etal., 1993). (H) Rocky Creek (modified from M. F. Schaeffer, 1998, unpublished data). Light-gray regions in (B), (F), (G), and (H)
represent visually identified homogeneous subdomains, and black regions in (F) and (H) are unexposed. See Table 2 for geologicinformation for trace maps.
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2100 Fracture Characterization Using Circular Scanlines and Windows
Figure 7. Estimates of natural fracture characteristics using 10circular scan lines/windows (diamonds) and areal samples(dashed lines). (A) Density at Llantwit Major. (B) Mean trace
length at Llantwit Major (MJ
master joints; CJ
cross joints).(C) Intensity at Amroth. (D) Intensity at Sleipner Vest. (E) Mean trace length at Sleipner Vest. (F) Density at Cartier Trough. Es- timates are in meters for Llantwit Major and Amroth and inkilometers for Sleipner Vest and Cartier Trough.
varying degrees of persistence, clustering, and the local
heterogeneities arising from the mechanics of joint
origin.
Spatially homogeneous fracture domains were es-
tablished initially by visual inspection. However, afirst-pass visual inspection to eliminate obvious hetero-
geneities was not sufficient to identify homogeneous
domains in all pavements. This insufficiency means
that the circle-based estimators could be tested both
for their performance in homogeneous regions and for
their ability to detect pattern heterogeneity in regions
of proposed homogeneity.
Intensity was sampled for the entire pavements at
Telpyn Point and Llantwit Major and for subdomains
at Amroth, Ward Lake, Yucca Mountain, and Rocky
Creek (Figure 6). To simplify comparison among the
different fracture geometries and sampling schemes,
circle-based intensity estimates within 15% of the areal
value were considered accurate and are marked with Y
(for “yes”) in Table 3. Ranges of low (1 fracture per
m), moderate (1–1.99 per m), and high (2 per m)
were adopted for intensity magnitude to facilitate com-
parison (Table 3). Circle radii of 1–2 m were used toachieve a high likelihood of intersections between frac-
ture traces and circular scan lines as a function of block
size or fracture spacing, while not exceeding pavement
size. Ten circles were used to balance the need for a
large sample with the need to be able to do work in a
timely manner.
The intensity of seven single fracture sets and two
entire fracture patterns was evaluated (Table 3). Four
fracture sets and two fracture patterns yielded accurate
circle-based intensity estimates as compared to the es-
timate for the entire pavement. We interpret theagreement between the two methods as indicating that
these patterns or fracture sets are spatially homoge-
neous. Three fracture sets, however, yielded inaccurate
results. Inspection of these pavements indicated that
the inaccuracy results from spatial heterogeneity. For
example, the 316 set at Amroth is more abundant at
the eastern end of the pavement subdomain (Figure 6),
so, unlike the 200 set, it is heterogeneously distrib-
uted. The cross joints at Llantwit Major are less intense
at the western end of the pavement, which is where
Figure 8. Variability of estimates as a function of circle size with respect to block size (small circles open diamonds; largecircles black diamonds; see text for size details). Variabilityis represented by the coefficient of variation (standard deviationdivided by the mean) for 100 circle counts for each sample.
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Rohrbaugh et al. 2101
Table 3. Accuracy of Intensity Estimates Using Ten Circular Scanlines for a Variety of Natural Fracture Patterns
Estimate Accuracy, Scanline Radius
Natural Fracture Patterns 1.0 m 1.5 m 2.0 m Intensity** (m/m2)
Llantwit Major, cross joint set Y Y N High
14.2 8.8 18.6 2.00
Llantwit Major, master joint set Y
Y
Y
High5.4 12.6 7.7 3.40
Amroth, 316 set N N N Low
49.5 63.1 35.9 0.55
Amroth, 200 set Y Y Y Mod
9.9 10.6 0.1 1.80
Telpyn Point, 290 set Y Y Y Low
15 13.3 2.5 0.50
Telpyn Point, 200 set Y Y Y High
9.0 14.0 3.4 3.60
Ward Lake, 015 set N Y N Mod
50.7 2.9 23.4 1.00
Yucca Mountain, all fracture sets Y Y Y High
7.6 9.7 8.2 2.80
Rocky Creek, all fracture sets Y Y Y High
6.0 4.8 13.7 3.30
*Y(yes)/N(no) indicates whether estimate is/is not within 15% of the areal value; / denotes an over/underestimate.
**Actual percent error for the intensity estimate are shown in the second row for each fracture system. Low/moderate/high intensity indicates values in the ranges
(0:1), (1:2), or (2), respectively.
larger sampling circles are restricted because of pave-
ment width (Figure 6). The subdomain at Ward Lakeeliminates a low intensity region for the fracture set
but contains heterogeneities due to fracture clusters
and covered areas that prevent random circle place-
ment. Thus, the geometry of older joints, joint persis-
tence, and joint clustering, as expected, generate spa-
tial heterogeneity. More important, based on these
results, we believe that a sampling strategy of 10 circles
with a size that exceeds the block size or fracture spac-
ing but is substantially smaller than the minimum di-
mension of a sample area will yield an intensity esti-
mate within 15% of the actual intensity for ahomogeneous fracture pattern.
Minimum Count of m for Reliable Tracelength Estimates
Of the three estimators discussed here, the mean trace-
length estimator exhibits the greatest variability. Low
trace densities, large fracture trace lengths relative to
circle size, or small total sample area can lead to small
counts of fracture endpoints (m). Such small counts
may be an issue for the petroleum industry be-
cause boreholes are commonly small with respect to
fractures. These small counts of m can lead to signifi-cant errors in the mean trace-length estimator (Table
1), because as m tends to zero in the denominator, the
estimate tends to infinity. Thus, it is important to en-
sure that a small m count is an adequate sample of a
fracture pattern with large trace lengths and not a sam-
pling artifact due to insufficient sample size, thereby
producing an overestimate of mean trace length.
To evaluate this issue, five simple synthetic trace
patterns with a single fracture set of intensity 2 m/m2
were examined. The five cases were for trace patterns
with individual traces of length 10, 16, 20, 25, or 50m, respectively. For each case, a suite of 35 sampling
strategies (combinations of circle size and number) was
used, with circle radii ranging from 1 to 20 m and num-
ber of circles ranging from 2 to 10. This combination
of cases and strategies yielded a total of 175 samples in
most of which trace lengths exceeded the circle di-
ameters, consistent with the petroleum industry situ-
ation of fractures larger than borehole diameter.
The accuracy of circle-based estimates of mean
trace length, as determined by comparison with the
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2102 Fracture Characterization Using Circular Scanlines and Windows
Figure 9. Graph illustrating the performance of the mean trace-length estimator as a function of the total number of mcounts in terms of the percentage of accurate estimates for agiven total m count.
input mean trace-length values, is graphed as a func-
tion of m counts in Figure 9. For this graph , an accurate
result was defined as within 15% of the input value.
Using this criterion, virtually all sampling strategies
that yielded m counts of 30 or greater produced ac-
curate estimates. These results (Figure 9) suggest that
for a specified fracture intensity, a threshold value of
m counts exists for which accurate estimates are a nearcertainty.
This analysis could not be extended directly to nat-
ural fracture patterns, because, due to problems of cen-
soring and length bias, no direct technique exists to
accurately estimate mean trace length for comparison
to the results of the circle-based estimator. Although
threshold values of m counts for accurate estimates
could be determined for our synthetic patterns, our
models had Poissonian trace center distributions,
which is unlikely in nature. Still, results from the syn-
thetic patterns should provide guidance, such as in thecase considered here, where a fracture pattern with
large fractures and an intensity of 2 m/m2 that is being
sampled by small circles (or small-diameter boreholes)
needs an m count greater than 30 to yield an accurate
estimate at the 15% level.
U S I N G N E W E S T I M A T O R S W I T HF R A C T U R E G E N E R A T O R S
One way to evaluate fractures as factors in hydrocar-bon transport and storage in reservoirs is to use com-
puter programs to generate representative fracture net-
works in synthetic rock volumes (Dershowitz and
LaPointe, 1994; Swaby and Rawnsley, 1996; Renshaw,
2000). These synthetic networks are then modeled for
fluid flow and storage. The primary contribution that
the circle-based estimators make to such an approach
is providing more accurate estimates for program in-
puts and validating outputs. Intensity estimates can, for
example, substitute for dimensionally equivalent den-
sity or frequency input values in programs. In addition,these estimates of density, size, and intensity provide
tests for determining whether synthetic networks are
truly representative.
C O N C L U S I O N S
1. New circle-based estimators for fracture intensity,
density, and mean trace length virtually eliminate
orientation, censoring, and length biases, which se-
verely limit the effectiveness of the straight scan line
and area methods.
2. Estimator accuracy is improved and variability re-
duced by using circles larger than mean fracture
block size or fracture spacing for a single set.3. A sampling strategy of 10 circles with a diameter
that exceeds block size or fracture spacing but is
significantly less than the minimum dimension of a
sample region yields an intensity estimate within
15% of the actual intensity for a homogeneous frac-
ture pattern.
4. When using small circles to estimate mean trace
length of large fractures, care should be taken to use
sampling strategies that gather a sufficient m count
to eliminate spurious overestimates of fracture size
from small samples. Guidance about the necessarynumber of m counts can be gained from analyzing
synthetic patterns with similar characteristics.
5. The new estimators provide both inputs and inde-
pendent checks of predictions for fracture-genera-
tor programs that model fracture populations in a
rock volume.
6. Fracture characterization protocols that deal with
sampling biases, such as the ones presented in this
article, are likely to yield improved input for explo-
ration and development decisions.
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Rohrbaugh et al. 2103
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