Estimating Fracture Trace Intensity Density and Mean Lenght Using Circular Scan Line and Windows

8/19/2019 Estimating Fracture Trace Intensity Density and Mean Lenght Using Circular Scan Line and Windows

1/16

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; [email protected]

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; [email protected]

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.


2/16

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 fracture 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.


3/16

Rohrbaugh et al. 2091

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 distribution 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).


4/16


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).


5/16


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


6/16


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.


7/16


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


8/16


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


9/16


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


10/16


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,


11/16

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.


12/16


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.


13/16


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


14/16


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.


15/16


R E F E R E N C E S C I T E D

Baecher, G. B., and N. A. Lanney, 1978, Trace length biases in joint surveys: Proceedings of the 19th U.S. Symposium on Rock Me-chanics, v. 1, p. 56–65.

Barton, C. C., E. Larsen, W. R. Page, and T. M. Howard, 1993,Characterizing fractured rock for fluid-flow, geomechanical,and paleostress modeling: methods and preliminary resultsfrom

Yucca Mountain, Nevada: United States Geological SocietyOpen-File Report 93-269, 62 p.Becker, A., and M. R. Gross, 1996, Mechanism for joint saturation

in mechanically layered rocks: an example from southern Israel:Tectonophysics, v. 257, p. 223–237.

Cruden, D. M., 1977, Describing the size of discontinuities: Inter-national Journal of Rock Mechanics and Mining Sciences andGeomechanical Abstracts, v. 14, p. 133–137.

Dershowitz, W. S., and H. H. Herda, 1992, Interpretationof fracturespacing and intensity, in J. R. Tillerson and W. R. Wawersik,eds., Proceedings of the 33rd U.S. Symposium on Rock Me-chanics: Rotterdam, Balkema, p. 757–766.

Dershowitz, W. S., and P. R. LaPointe, 1994, Discrete fracture ap-proaches for oil and gas applications, in R. A. Nelson and S.Laubach, eds., Rock mechanics: Rotterdam, Balkema, p. 19–

30.Dunne, W. M., and C. P. North, 1990, Orthogonal fracture systems

at the limits of thrusting: an examplefrom southwesternWales:Journal of Structural Geology, v. 12, p. 207–215.

Einstein, H. H., and G. B. Baecher, 1983, Probablistic and statisticalmethods in engineering geology, specific methods and exam-ples, part 1: exploration: Rock Mechanics and Rock Engineer-ing, v. 16, p. 39–72.

Ghosh, A., and J. J. K. Daemen, 1993, Fractal characteristics of rock discontinuities: Engineering Geology, v. 34, p. 1–9.

Gilmour, H. M. P., D. Billaux, and J. C. S. Long, 1986, Models forcalculating fluid flow in randomly generated three-dimensionalnetworks of disk-shaped fractures: theory and design of FMG3D, DISCEL and DIMES: Berkeley, California, Lawrence

Berkeley Laboratory, Earth Sciences Division, LBL-19515,144 p.

Jian, X., R. Olea, and Y. Yu, 1996, Semivariogram modeling byweighted least square: Computers and Geosciences, v. 22,p. 387–397.

Kulatilake, P. H. S. W., and T. H. Wu, 1984, The density of discon-tinuity traces in sampling windows (technical note): Interna-tional Journal of Rock Mechanics and Mining Sciences andGeomechanical Abstracts, v. 21, p. 345–347.

Kulatilake, P. H. S. W, D. N. Wathugala, and O. Stephansson, 1993,Joint network modeling with a validation exercise in StripaMine, Sweden: International Journal of Rock Mechanics andMining Sciences and Geomechanical Abstracts, v. 30, p. 503–526.

Kulatilake, P. H. S. W., R. Fiedler, and B. B. Panda, 1997, Box fractaldimension as a measure of statistical homogeneity of jointedrock masses: Engineering Geology, v. 48, p. 217–229.

LaPointe, P. R., 1988, A method to characterize fracture density andconnectivity through fractal geometry: International Journal of Rock Mechanics and Mining Sciences and Geomechanical Ab-stracts, v. 24, p. 421–429.

LaPointe, P. R., 1993, Pattern analysis and simulation of joints forrock engineering, in J. A. Hudson, ed., Comprehensive rock engineering, volume 3—rock testing and site characterization:New York, Pergamon Press, p. 215–239.

LaPointe, P. R., and J. A. Hudson, 1985, Characterization and in-terpretation of rock mass joint patterns: Geological Society of America Special Paper 199, 37 p.

Laubach, S. E., 1997, A method to detect natural fracture strike insandstones: AAPG Bulletin, v. 81, p. 604–623.

Marrett, R., O. J. Ortega, and C. M. Kelsey, 1999, Extent of power-law scaling fornatural fractures in rocks: Geology, v. 27,p. 799–802.

Mauldon, M., 1998, Estimating mean fracture trace length and den-sity from observations in convex windows: Rock Mechanics andRock Engineering, v. 31, p. 201–216.

Mauldon, M., and W. Dershowitz, 2000, A multi-dimensional sys-

tem of fracture abundance measures: Geological Society of America Abstracts with Programs, v. 32, no. 7, p. A474.

Mauldon, M., and J. G. Mauldon, 1997, Fracture sampling on a cyl-inder: from scan lines to boreholes andtunnels: Rock Mechanicsand Rock Engineering, v. 30, p. 129–144.

Mauldon, M., W. M. Dunne, and M. B. Rohrbaugh Jr., 2001, Cir-cular scan lines and circular windows: new tools for character-izing the geometry of fracture traces: Journal of Structural Ge-ology, v. 23, p. 247–258.

Narr, W., 1996, Estimating average fracture spacing in subsurfacerock: AAPG Bulletin, v. 80, p. 1565–1586.

Nelson, R. A., 1985, Geologic analysis of naturally fractured reser-voirs: Houston, Texas, Gulf Publishing, 320 p.

Nemcock, M., R. Bayer, and M. Miliorizos, 1995, Structural analysisof the inverted Bristol Channel Basin: implications for the ge-ometry and timing of fracture porosity, in J. G. Buchanan andP. G. Buchanan, eds., Basin inversion: Geological Society Spe-cial Publication 88, p. 355–392.

Ortega, O. J., and R. Marrett, 2000, Prediction of macrofractureproperties using microfracture information, Mesaverde Groupsandstones, San Juan basin, New Mexico: Journal of StructuralGeology, v. 22, p. 571–587.

Ottesen Ellevset, S., R. Knipe, T. Svava Olsen, Q. J. Fisher, and G.Jones, 1998, Fault controlled communication in the SleipnerVest field, Norwegian continental shelf: detailed, quantitativeinput for reservoir simulation and well planning, in G. Jones,Q. J. Fisher, and R. J. Knipe, eds., Faulting, fault sealing andfluid flow in hydrocarbon reservoirs: Geological Society SpecialPublication 147, p. 283–297.

Peacock, D. C. P., S. J., Harris, and M. Mauldon, in press, Use of curved scan lines and boreholes to predict fracture frequencies:Journal of Structural Geology, v. 25, p. 109–119.

Pickering, G., J. M. Bull, and D. J. Sanderson, 1995, Samplingpower-law distributions: Tectonophysics, v. 248, p. 1–20.

Priest, S. D., 1993, Discontinuity analysis for rock engineering: NewYork, Chapman and Hall, p. 50–54.

Priest, S. D., and J. A. Hudson, 1981, Estimation of discontinuityspacing and trace length using scan line surveys: InternationalJournal of Rock Mechanics and Mining Sciences and Geome-chanics Abstracts, v. 18, p. 183–197.

Rawnsley, K. D., D. C. P. Peacock, T. Rives, and J.-P. Petit, 1998,Joints in the Mesozoic sediments around the Bristol ChannelBasin: Journal of Structural Geology, v. 20, p. 1641–

1661.Reiss, L. H., 1982, The reservoir engineering aspects of fracturedformations: Houston, Texas, Gulf Publishing, 186 p.

Renshaw, C. E., 2000, An example of the use of geological and me-chanical constraints to develop a conceptual model of flow infractured, granitic rock: Geological Society of America Ab-stracts with Program, v. 32, no. 7, p. A64.

Rohrbaugh Jr., M. B., 2000, Estimating joint intensity, density, andmean trace length using circular scan lines and circular win-dows: M.S. thesis, University of Tennessee, Knoxville, 96 p., 1CD-ROM.

Segall, P., and D. D. Pollard, 1983, Joint formation in granitic rock of the Sierra Nevada: Geological Society of America Bulletin,v. 94, p. 563–575.


16/16

Stoyan, D., W. S. Kendall, and J. Mecke, 1995, Stochastic geometryand its applications, 2d ed.: New York, John Wiley and Sons,p. 286–296.

Swaby, P. A., and K. D. Rawnsley, 1996, An interactive 3D fracturemodeling environment: Society of Petroleum Engineers, SPE36004, p. 177–187.

Terzaghi, R. D., 1965, Sources of error in joint surveys: Geotech-nique, v. 15, p. 287–304.

Turner, F. J., and L. E. Weiss, 1963, Structural analysis of meta-

morphic tectonites: New York, McGraw-Hill, 545 p.

Walsh, J. J., J. Watterson, C. Childs, and A. Nicol, 1996, Ductilestrain effects in the analysis of seismic interpretations of normalfault systems, in P. G. Buchanan and D. Nieuwland, eds., Mod-ern developments in structural interpretation, validation, andmodelling: Geological Society Special Publication 99, p. 27–40.

Whitten, E. H. T., 1966, Structural geology of folded rocks: Chicago,Rand McNally, 678 p.

Wu, H., and D. D. Pollard, 1995, An experimental study of therelationship between joint spacing and layer thickness: Journal

of Structural Geology, v. 17, p. 887–905.

Estimating Fracture Trace Intensity Density and Mean Lenght Using Circular Scan Line and Windows

Documents

Transcript of Estimating Fracture Trace Intensity Density and Mean Lenght Using Circular Scan Line and Windows