AUTHORS

Orlando J. Ortega � Shell InternationalExploration and Production, 200 North DairyAshford, Houston, Texas 77079;orlando.ortega@shell.com

Orlando J. Ortega received his M.S. degree andhis Ph.D. from the University of Texas at Austin.He is a structural geologist specializing in fracturesfor Shell Exploration and Production in Hous-ton, Texas.

Randall A. Marrett � Department of Geo-logical Sciences, John A. and Katherine G.Jackson School of Geosciences, University ofTexas at Austin, Austin, Texas 78712-1101

Randall A. Marrett is an associate professor inthe Department of Geological Sciences, JacksonSchool of Geosciences, University of Texas atAustin. Quantitative analysis of fracture systemsis a major area of his research. He receivedhis Ph.D. from Cornell University.

Stephen E. Laubach � Bureau of EconomicGeology, John A. and Katherine G. JacksonSchool of Geosciences, University of Texas atAustin, Austin, Texas 78713-8924;steve.laubach@beg.utexas.edu

Stephen E. Laubach is a senior research scientistat the Bureau of Economic Geology, where heconducts research in structural diagenesis. HisPh.D. is from the University of Illinois.

ACKNOWLEDGEMENTS

This study was partially supported by the Na-tional Science Foundation Grant EAR-9614582,the Texas Advanced Research Program Grant003658-011, and the industrial associates of theFracture Research and Application Consortium:Anadarko, Bill Barrett Corp., BG Group, ChevronTexaco, Conoco Inc., Devon Energy Corpora-tion, Ecopetrol, EnCana, EOG Resources, Huber,Instituto Mexicano del Petroleo, Japan NationalOil Corp., Lariat Petroleum Inc., Marathon Oil,Petroleos Mexicanos Exploracion y Produccion,Petroleos de Venezuela, Petrobras, Repsol-YPF-Maxus, Saudi Aramco, Shell, Schlumberger,Tom Brown, TotalFinaElf, Williams Exploration& Production. We thank Faustino Monroy San-tiago, Julia Gale, Jon Olson, and Bob Gold-hammer for valuable discussion and Alvar Bra-athen, Ross Clark, Wayne Narr, and AmgadYounis for thoughtful reviews.

A scale-independent approachto fracture intensity and averagespacing measurementOrlando J. Ortega, Randall A. Marrett, andStephen E. Laubach

ABSTRACT

Fracture intensity, the number of fractures per unit length along a

sample line, is an important attribute of fracture systems that can

be problematic to establish in the subsurface. Lack of adequate con-

straints on fracture intensity may limit the economic exploitation

of fractured reservoirs because intensity describes the abundance of

fractures potentially available for fluid flow and the probability

of encountering fractures in a borehole. Traditional methods of

fracture-intensity measurement are inadequate because they ignore

the wide spectrum of fracture sizes found in many fracture systems

and the consequent scale dependence of fracture intensity. An al-

ternative approach makes use of fracture-size distributions, which

allow more meaningful comparisons between different locations

and allow microfractures in subsurface samples to be used for

fracture-intensity measurement. Comparisons are more meaning-

ful because sampling artifacts can be recognized and avoided, and

because common thresholds of fracture size can be enforced for

counting in different locations. Additionally, quantification of the

fracture-size distribution provides a mechanism for evaluation of

uncertainties. Estimates of fracture intensity using this approach for

two carbonate beds in the Sierra Madre Oriental, Mexico, illustrate

how size-cognizant measurements cast new light on widely ac-

cepted interpretation of geologic controls of fracture intensity.

INTRODUCTION

Sampling problems represent a fundamental challenge to subsur-

face fracture characterization because complete sampling of large

conductive fractures that dominate fluid flow in the subsurface is

GEOHORIZONS

AAPG Bulletin, v. 90, no. 2 (February 2006), pp. 193–208 193

Copyright #2006. The American Association of Petroleum Geologists. All rights reserved.

Manuscript received April 1, 2005; provisional acceptance August 5, 2005; revised manuscript receivedAugust 16, 2005; final acceptance August 25, 2005.

DOI:10.1306/08250505059

unfeasible. Large fractures in the subsurface typically

are widely spaced and have much larger dimensions

than the diameter of a borehole. Consequently, the

probability of encountering such fractures in a given

layer are small, and even where such fractures are in-

tersected, only fragmentary data can be collected (Narr,

1991). Large fractures commonly are more open than

smaller equivalents, so core recovery of these fractures

is typically incomplete (Laubach, 2003; Laubach et al.,

2004). Horizontal boreholes enhance the likelihood

of encountering large fractures but at most intersect

only a fraction of the population because of the finite

strike lengths of fractures.

One approach for overcoming limitations to sam-

pling fractures in the subsurface is to treat abundant

microfractures as proxies for related macrofractures in

the same rock volume. For example, orientations of

microfractures may provide estimates of macrofrac-

ture orientations (Laubach, 1997; Ortega and Marrett,

2000). Likewise, microfracture abundance is directly

related to macrofracture abundance in many cases

(Marrett et al., 1999; Ortega and Marrett, 2000). This

approach relies on systematic measurement of micro-

fracture attributes in samples retrieved from the sub-

surface. A more traditional approach for predicting

the abundance of subsurface fractures is to understand

how geologic parameters like mechanical-layer thick-

ness control variations in fracture intensity. For exam-

ple, measurable characteristics of sedimentary layers

might be used to predict the intensity of natural frac-

tures in the subsurface for identifying and ranking ex-

ploration and exploitation targets.

For the purposes of this article, fracture intensity

is the number of fractures encountered per unit length

along a sample line oriented perpendicular to the frac-

tures in a set. Nelson (1985) summarized some widely

accepted ideas of geologic controls on fracture inten-

sity: (1) composition, (2) texture (including grain size

and porosity), (3) structural position, and (4) stratig-

raphy (bed thickness). Compositional controls on frac-

ture intensity have been studied in laboratory experi-

ments (e.g., Handin et al., 1963) and with systematic

measurements of fracture spacings in outcrop (Das

Gupta, 1978; Sinclair, 1980). Core studies show that

progressive diagenesis causes mechanical properties to

evolve with marked effects on fracture-intensity pat-

terns (Marin et al., 1993). Price (1966) found an inverse

relation between rock strength and porosity, and

Nelson (1985) proposed that lower porosity rocks of

similar composition should have more abundant frac-

tures than higher porosity rocks. Perhaps the most

widespread paradigm of geologic controls on fracture

intensity is that of bed thickness (or mechanical-layer

thickness). Bogdonov (1947)was apparently the first to

document the intuitive result that fracture intensity

increases (i.e., fracture spacing decreases) with decreas-

ing bed thickness, but numerous other authors have

subsequently arrived at similar conclusions (Price, 1966;

Ladeira and Price, 1981; Huang and Angelier, 1989;

Narr and Suppe, 1991; Gross, 1993; Mandal et al.,

1994; Gross and Engelder, 1995;Wu and Pollard, 1995;

Becker and Gross, 1996; Narr, 1996; Pascal et al., 1997;

Ji and Saruwatari, 1998; Bai and Pollard, 2000).

However, a common flaw shared by all previous

empirical investigations quantifying controls on fracture

intensity is a lack of an explicit accounting for fracture

size. Althoughmany joints in outcrop apparently have a

narrow size range, opening-mode fractures formed in

the deep subsurface vary significantly in size, in some

cases ranging across at least five orders of magnitude,

with larger fractures typically being progressively less

abundant (Gillespie et al., 1993; Marrett et al., 1999).

Cumulative frequency is commonly related to fracture

size in the formof a power law (e.g.,Marrett et al., 1999).

Moreover, geomechanical models of fracture growth

indicate that many combinations of rock properties and

loading paths should result in fracture populations

having a wide spectrum of sizes (Olson, 1997, 2003).

As we demonstrate below, cumulative frequency

is a direct measure of fracture intensity, so there is an

implicit covariance of fracture intensity with fracture

size. Consequently, it is meaningless to quantify frac-

ture intensity without specifying what range of frac-

ture sizes are represented, and it is risky to compare

fracture-intensity measurements from different loca-

tions if the results have not been normalized to a com-

mon range of fracture sizes.

In this contribution, we present a new method

that explicitly accounts for variation of fracture inten-

sity according to fracture size. Applying this method

to several different locations permits meaningful com-

parison of resulting fracture-intensity measurements

and quantification of variation with independent geo-

logic parameters. Moreover, the approach is amenable

to systematic observations at any scale, so it can be

used for microfracture-based subsurface studies, thus

circumventing fracture-sampling challenges.

We use examples from the Sierra Madre Oriental,

northeastern Mexico, to illustrate the new approach.

The SierraMadreOriental is a northeast-directed, thin-

skinned fold-thrust belt of early Tertiary (Laramide)

age (Figure 1), which exposes a thick sequence of

194 Geohorizons

Mesozoic carbonate strata that, near Monterrey, have

been deformed into isoclinal anticlines (e.g., Marrett

and Aranda-Garcıa, 1999). Excellent outcrops in this

area provide an ideal opportunity to study sedimen-

tary and stratigraphic controls on fracture intensity.

This platform-to-basin carbonate system is the subject

of numerous sedimentologic and stratigraphic studies

(e.g., Goldhammer et al., 1991, and references there-

in). The fractures studied in the Lower Cretaceous

carbonate strata have high aperture-to-length ratios

and are mostly calcite and/or dolomite filled, so they

represent veins. The mineral-fill patterns and shapes of

these fractures closely match those found in core, and

except for late-stage calcite fill, these outcrop fractures

are good analogs to fractures in producing fields (Lau-

bach, 2003). Relative timing constraints suggest the

veins predominantly formed early in the burial history

of the rocks (Marrett and Laubach, 2001) and signifi-

cantly predate local Laramide-age folding.

FRACTURE DATA COLLECTION

The essential fracture documentation technique of this

study was to measure fracture attributes along linear

traverses (scan lines). Fracture orientation, morphol-

ogy, crosscutting relationships, composition and tex-

ture of fracture fill, and mechanical-layer thickness

were recorded for each fracture set of the beds studied.

Opening displacement (or kinematic aperture; here-

after, aperture) of each fracture, intercepted by a frac-

ture set–perpendicular scan line, was also recorded.

Beds were chosen so that they would represent con-

trasting sedimentary facies, various bed thicknesses,

and different degrees of dolomitization (Figure 2). Me-

chanical bed thickness ranges from less than 10 cm

(4 in.) to nearly 2 m (6.6 ft). Dolomite content ranges

from 0 to 100%, but most beds are either highly do-

lomitized or only weakly dolomitized. Dolomite con-

tent was measured in digital photomicrographs of

Figure 1. Location of the study area on a map of the geologic provinces of Mexico. Basins adjacent to the study area producehydrocarbons from units that are equivalent to those studied. Tectonic elements are conceptual. Modified from Benavides (1956).

Ortega et al. 195

thin sections stained with alizarine red, but we did not

distinguish dolomite resulting from different diagenet-

ic events. The beds studied were classified into six

different sedimentary facies on the basis of dominant

texture and allochem content, along with the inter-

pretation of the depositional environment. However,

fracture-intensity analyses were done independently

for two broadly defined sedimentary textures: mud-

supported and grain-supported facies. Here, we pres-

ent and illustrate the methodology; the complete

fracture analysis of the Sierra Madre Oriental carbon-

ate rocks will be presented elsewhere.

Fracture Set Determination

Patterns of fracture attributes, like fracture morphol-

ogy, timing, and orientation, can be used to establish

fracture sets. Although fracture orientation is com-

monly used alone for distinguishing fracture sets, in

many cases, fracture timing is mostly diagnostic. In-

terpretations of fracture timing are most based on

crosscutting relationships. Crosscutting principles sug-

gest that a fracture set that systematically crosscuts

another fracture set must postdate the other set. Al-

though crosscutting relationships do not constrain the

absolute time of fracture formation, they can be used to

separate groups of fractures that formed contempora-

neously and probably under similar remote stress con-

ditions. Fracture timing can also be constrained in rela-

tion to products of other geologic processes, such as

diagenesis and tectonism. Fractures that systematically

abut against other fractures can be interpreted as youn-

ger (e.g., Gross, 1993). Statistically based abutting re-

lationships between fracture sets have been used to

support fracture timing relationships based on orien-

tation and crosscutting relations.

Fracture Size Data

Fracture size was quantified by measuring apertures

along scan lines drawn perpendicular to each fracture

set in bed-perpendicular exposures (Figure 3). More

than 14,000 fracture apertures from 42 beds compose

the fracture-size database for the Sierra Madre Ori-

ental. Outcrop limitations generally prevented the col-

lection of sufficient fracture-aperture measurements

for a reliable determination of fracture intensity of

weakly developed fracture sets (Bonnet et al., 2001).

However, an effort was made to obtain fracture-

intensity data from low-fracture-intensity beds to doc-

ument approximate fracture intensity and to facilitate

comparison with other beds.

Accurate aperturemeasurement of these fractures is

possible because they are filled by secondary mineraliza-

tion. To collect fracture-aperture measurements, we

used a logarithmically graduated comparator (Figure 4).

Using this toolwith a hand lens allows documentation of

fracture apertures as small as about 0.05mm (0.002 in.)

on sufficiently exposed outcrops in the field. The com-

parator contains lines with increasing width starting

at 0.05 mm (0.002 in.) and ending at 5 mm (0.2 in.).

Increments in line width represent approximately uni-

formmultiples of each other and are thus evenly spaced

Figure 2. Variation of facies, bed thickness, and dolomitecontent among layers studied. The facies definition includestextural groups and depositional environment interpretations.Some histograms exclude multifacies beds and/or chert beds.Total dolomite content is based on thin-section point counting.

196 Geohorizons

whenplottedon a logarithmic axis. In thisway, apertures

were measured with consistent accuracy as viewed in

a log-log graph.

NORMALIZED FRACTURE INTENSITY

A complete inventory of apertures along a scan line

provides a more complete estimate of fracture inten-

sity than methods that do not account for size (e.g.,

Bogdonov, 1947; Ladeira and Price, 1981; Narr and

Suppe, 1991; Wu and Pollard, 1995; Narr, 1996). In

our data set, we find that fracture-size distributions

follow power-law scaling. However, the new method

is equally applicable to any type of cumulative size

distribution.

Fracture intensity, fracture density, average frac-

ture spacing, and fracture frequency all describe spatial

abundance of fractures. Fracture frequency is a general

term that subsumes fracture intensity and fracture den-

sity. Fracture intensity has units of inverse length and

is a ratio of the number of fractures, sum of fracture

lengths, or sum of fracture surface areas to the length,

area, or volume of observation, respectively (Mauldon

et al., 2001). The literature contains numerous exam-

ples of fracture-intensity determinations, mostly from

outcrops (e.g., Corbett et al., 1987; Huang and An-

gelier, 1989; Narr and Suppe, 1991; Gross, 1993) and

also from the subsurface (Corbett et al., 1987; Lorenz

and Hill, 1991; Narr, 1996) and laboratory tests (Rives

et al., 1992, 1994;Wu and Pollard, 1992, 1995;Mandal

Figure 4. Fracture-aperture comparator. Versions used in thefield have been calibrated by microscope. The version reproducedin this article is affected by printer limitations and paper qualityand should not be used without recalibration. Reproductionusing photocopying methods will further reduce the accuracyof the line widths.

Figure 3. Segment of the scan line foraperture-data collection in bed 3, La Es-calera Canyon. The mechanical bound-aries are indicated by dashed lines andcorrespond to the systematic terminationsof fractures. Two scan lines were con-structed for this bed, one for each ofthe two fracture sets present. The scanline shown (white traces) is approximatelyperpendicular to fracture set B (inclinedwith respect to bed boundaries).

Ortega et al. 197

et al., 1994). Fracture intensity (F) is most commonly

determined fromone-dimensional observation domains

(i.e., scan lines) by dividing the number of fractures (N)

by the total length (L) of the scan line:

F ¼ N

Lð1Þ

However, the most common indication of fracture

abundance reported in the literature is the average

spacing (S ) between fractures along a scan line (Huang

and Angelier, 1989; Narr, 1991; Gross, 1993; Ji and

Saruwatari, 1998). Average spacing is the inverse of the

fracture intensity:

S ¼ 1

N

XNi¼1

Si ¼L

N¼ 1

Fð2Þ

where Si is the spacing between the nearest neighbor

fractures along the scan line.

Some fracture spacing data sets follow a normal

distribution (Ji and Saruwatari, 1998), but most are

better modeled by negative exponential (Priest and

Hudson, 1976; Pineau, 1985), log-normal (Sen and

Kazi, 1984; Narr and Suppe, 1991; Rives et al., 1992;

Becker and Gross, 1996; Pascal et al., 1997), or gamma

distributions (e.g., Huang and Angelier, 1989; Gross,

1993; Castaing et al., 1996). Sampling bias might ex-

plain variations in the best mathematical model for

fracture spacing distributions, but Rives et al. (1992)

and Ji and Saruwatari (1998) have proposed alterna-

tive explanations.

Although numerous measurements of fracture in-

tensity or average spacing have been made previously,

these estimates typically do not explicitly account for

fracture size. Thus, data used to infer relations be-

tween bed thickness and fracture intensity or spacing

are commonly unclear about the size of fractures used

for spacing determinations. In some cases where nu-

merous smaller fractures are obvious, a rule is speci-

fied about the fracture size (bed-normal dimension) to

be measured relative to bed thickness; that is, only

fractures that span a given bed thickness are to be

measured (for example, Laubach and Tremain, 1991;

Laubach et al., 1998 for fractures in coal). However,

in most instances, fractures counted could be all that

were visible to the naked eye, perhaps only those that

reached the boundaries of the mechanical layer, or

merely those that are prominent (Huang and Angelier,

1989; Lorenz and Hill, 1991; Castaing et al., 1996).

Moreover, the size of fractures visible on an outcrop

varies according to exposure quality and fracture-fill

characteristics, which typically have not been accounted

for. Furthermore, if we changed the scale of observa-

tion, for example, from outcrop scale to microscopic

scale, estimates of fracture intensity or spacing would

typically be different. This lack of rigor potentially

introduces bias in the calculation of fracture intensity

or spacing and, consequently, may undercut conclu-

sions regarding relationships between fracture intensity

and geologic parameters.

In the subsurface, the bed thickness cannot be

generally measured, except under rare circumstances

(e.g., Marin et al., 1993). However, in most subsurface

cases, there will be insufficient fractures having the

size of interest (i.e., large conductive fractures) to de-

rive a statistically significant estimate of spacing.

Calculation of Normalized Fracture Intensity

Use of cumulative-frequency fracture-size distributions

addresses the potential scale bias inherent to average

fracture intensity or spacing determinations because

they yield a measure of fracture abundance that ex-

plicitly varies according to their size. This method

effectively quantifies fracture intensity for detection

thresholds corresponding to all fracture sizes considered.

It has the advantage of facilitating a comparison between

data sets collected at different scales of observation or

levels of resolution. The following example illustrates

the advantages of using cumulative-frequency fracture-

size distributions over the traditional method to mea-

sure and compare fracture abundance.

Figure 5 shows a schematic cross section of a frac-

tured layer in which fracture abundance is to be de-

termined along a 1-m (3.3-ft)-long scan line at three

different scales of observation. This example represents

real aperture data collected along a scan line in a do-

lomitized mudstone bed (bed 3) that is approximately

8 cm (3.1 in.) thick, studied in La Escalera Canyon,

northeasternMexico. The fractures have been random-

ly located in the cross section because, for the following

analysis, the actual location of the fractures is not im-

portant. First, only fractures with apertures greater

than 0.5 mm (0.02 in.) are counted (Figure 5A). Frac-

ture size spans an order of magnitude from 0.5 to 5mm

(0.02 to 0.2 in.) in aperture, measurable with a 0.5-mm

(0.02-in.) graduated ruler. At this scale of observation

(24 fractures in 1000 mm [39 in.]), we find that the

fracture intensity is approximately 24 fractures/m

198 Geohorizons

(7.3 fractures/ft). However, if we included all fractures

with apertures down to 0.05 mm (0.002 in.) along one

part of the scan line (28 fractures in 200 mm [8 in.])

measurable with a hand lens and the aperture compar-

ator (Figure 4), then the fracture intensity is 140 frac-

tures/m (42.7 fractures/ft) (Figure 5B) and nearly an

order of magnitude higher than the first estimate. Fur-

thermore, microfracture-aperture data from a thin sec-

tion of this bed (Figure 5C) indicate that the intensity

of fractures wider than 0.005 mm (0.0002 in.) (10 frac-

tures in 12 mm [0.47 in.]) is nearly another order of

magnitude higher, about 830 fractures/m (253 frac-

tures/ft). This example illustrates that fracture inten-

sity varies with changes in resolution (which typically

accompany changes in observation scale) because the

number of fractures observed in a domain will increase

as the threshold size for fracture detection decreases.

Consequently, fracture intensity (and average spacing)

is onlymeaningful if the fracture-detection threshold is

quantified.

Cumulative-frequency fracture-size distributions

provide a measure of fracture intensity (or average

spacing) that explicitly accounts for fracture size and

permits a comparison of data collected at different lo-

cations and/or observation scales. For purposes of com-

paring any two data sets, normalization of fracture

intensities is effectively achieved by choosing a com-

mon fracture-detection threshold. To estimate the

Figure 5. Schematic cross section of bed 3, La Escalera Canyon. Fractures of set A are solid white lines, and fracture aperture is thenumber next to each fracture. The scan line is represented by a dashed line that runs perpendicular to the fractures. (A) Fracturesshown all have apertures greater than 0.5 mm (0.02 in.), the minimum size of fractures that span the mechanical layer. The averagespacing between fractures of this size is 40 mm (1.57 in.). (B) The first 20% of the scan line (shaded area in A) showing the fractureswith apertures of 0.05 mm (0.002 in.) or greater. The average fracture spacing at this scale of observation is 7 mm (0.27 in.). (C) Scanline at the microscopic scale (black area in B) showing fractures having apertures of 0.005 mm (0.0002 in.) or larger in a thin section.The average fracture spacing at the thin-section scale is about 1.2 mm (0.04 in.). All fractures in the thin section are invisible atoutcrop scale, with the exception of one fracture 0.115 mm (0.0045 in.) wide.

Ortega et al. 199

cumulative-frequency fracture-size distribution for a

scan-line data set, we followed these steps:

1. List all measurements of fracture size (aperture is

used to quantify fracture size in the examples pre-

sented here), and sort them from the largest to the

smallest.

2. Number the sorted fracture sizes, such that the

largest is number 1 and successively smaller frac-

tures receive higher numbers as fracture size de-

creases (i.e., generate a list of cumulative numbers).

3. Simplify the list by eliminating all fractures having

the same size (given the resolution of measure-

ments), except for the one with the largest cumu-

lative number.

4. Normalize the cumulative numbers by the length of

the scan line. This will generate estimates of cumu-

lative fracture intensity (i.e., number of fractures of

a certain size or larger per unit length of scan line).

This parameter is a measure of the fracture inten-

sity as a function of detection threshold and can be

used to compare and contrast fracture intensities

from different beds and/or observation scales.

5. Plot cumulative fracture intensity versus fracture

size to provide a graphical display of the distribu-

tion. Several fracture-size distributions can be ade-

quately modeled using simple mathematical laws.

Power-lawdistributions plot as lines in a log-log plot.

Exponential fracture-size distributions plot as a line

in a log-linear graph.

6. Obtain the parameters of the bestmodel for the distri-

bution observed (e.g., coefficient and exponent of the

power law, as determined by least-squares regression).

The size distribution for the example discussed

above (Figure 5) is shown in Figure 6. A power law

adequately models these data across four orders of

magnitude variation in aperture. The power law relates

fracture size to the intensity of fractures having a certain

size or larger or, alternatively, to the average spacing of

fractures of a certain size or larger. The coefficient rep-

resents the value of the power law for a fracture size of

one (and, therefore, depends on the units chosen to

quantify fracture size). The exponent of the power law

represents the slope of the power-law line in a log-log

plot. This exponent is always negative, because by defi-

nition, the cumulative number of fractures must in-

crease or remain constant as fracture size decreases.

Obtaining the normalized fracture intensity or aver-

age spacing for a given fracture-detection threshold is

straightforward. From the cumulative-frequency frac-

ture-size distribution plot (Figure 6), we read the cor-

responding frequency (number of fractures per unit

length of scan line) for a given fracture size using the

power-law distribution model as a guide. We can also

obtain an average spacing estimate using the power law,

referring to an ordinate axis graduated with inverse fre-

quency values (i.e., average spacing). For the exercise

discussed previously, we can now obtain the fracture in-

tensity and average spacing for any scale of observation.

The arrows in Figure 6 indicate the scale depen-

dence of fracture abundance. Figure 7 illustrates the

range of possible fracture-intensity results based on

counting fractures without accounting for their sizes.

This approach generates a single estimate of average

spacing and intensity that is only valid for some specific

threshold of fracture aperture that remains unknown.

Figure 6. Normalized fracture intensityfrom cumulative fracture-size distributionfor fracture set A in bed 3, La EscaleraCanyon. Fracture abundance can be char-acterized as intensity (cumulative fre-quency) or average spacing (inverse ofcumulative frequency). Open and filledsquares are field data. The open squaresare data subject to truncation bias notused in regression. The dashed line isa power-law regression. In the formula,m�1 and mm�1 refer to the left-handand right-hand scales.

200 Geohorizons

Consequently, average spacing or intensity estimates

cannot be compared confidently with measurements

fromother scan lines because fracture-detection thresh-

olds might differ because of variable outcrop quality,

different fracture morphologies, or different sizes of

fractures that span a layer. Additionally, there is no

information about sampling bias possibly affecting the

data. If identical fracture sizes are chosen for fracture

spacing measurement at different locations, then con-

ventional approaches and this population-based ap-

proach will give the same results. However, this con-

dition is unlikely to occur without a conscious effort to

maintain a consistent fracture size threshold from loca-

tion to location. Figure 7 indicates ranges of fracture

intensity that might be inferred for different observa-

tion scales or resolution levels.

Sampling and Topologic Artifacts

Fault-scaling studies dominate the literature on frac-

ture scaling (Ackermann and Schlische, 1997; Marrett

et al., 1999; Bonnet et al., 2001, and references therein).

Much less abundant is the literature on opening-mode

fracture scaling. However, there is now agreement

about the power-law nature of many opening-mode

fracture-size distributions (Gudmundsson, 1987;Wong

et al., 1989; Heffer and Bevan, 1990; Barton and Zo-

back, 1992; Gillespie et al., 1993; Hatton et al., 1994;

Sanderson et al., 1994; Belfield and Sovich, 1995; Clark

et al., 1995; Gross and Engelder, 1995; Johnston and

McCaffrey, 1996; Marrett, 1997; Marrett et al., 1999;

Ortega and Marrett, 2000; Bonnet et al., 2001). Most

work on this aspect of fracture population statistics has

been done for fracture length. However, fracture-length

determinations are fraught with uncertainty, most of

which arise from the difficulties of clearly defining what

is a throughgoing fracture where fractures branch or

intersect (Ortega and Marrett, 2000). Studies of

fracture-aperture distributions are less common in the

literature, perhaps because of the lack of appropriate

tools for effective fracture-aperturemeasurement in the

field. However, considerable progress has been made in

recent years using new methods for accurate fracture-

aperture measurements, which have generated some

Figure 7. Representation of the fracture-counting method for fracture-intensity calculation in a log-log graph of fracture-sizedistribution. Thick lines represent the example explained in the text, and lower detection ranges are approximate for the CupidoFormation. The counting method is fraught with errors introduced by inconsistent limits of minimum fracture size counted for thecalculation. Patterned areas are determined by the approximate minimum size detectable at various observation scales. Note therange of possible fracture intensities and overlapping areas.

Ortega et al. 201

well-constrained fracture aperture-size distributions

(Ortega et al., 1998; Marrett et al., 1999).

Marrett et al. (1999) showed examples of opening-

mode fracture and fault-size distributions that follow

consistent power laws across three to five orders of

magnitude in size. Some of these distributions included

microscopic-scale aperture data from sandstones and

carbonate rocks. However, fracture-size distributions

commonly show deviations from an ideal power law in

the small and large fracture-size parts of the distribu-

tion. Deviations from power-law behavior can be ex-

plained by consideration of sampling and topologic and

statistical artifacts, which can be assessed from patterns

of fracture-size distributions. To use scaling methods,

these artifacts need to be understood and minimized.

Truncation Artifacts

Deviations of small fractures from the scaling shown

by larger fractures of a population have been long

recognized as truncation artifacts (Baecher and Lan-

ney, 1978; Barton and Zoback, 1992; Pickering et al.,

1995). Truncation artifacts arise from variable sam-

pling completeness for fracture sizes near the limits of

resolution of the observation tool (e.g., smallest frac-

ture size accurately measurable with the naked eye

varies from place to place in outcrop exposures). Al-

though a mathematical description of truncation arti-

facts has not yet been proposed, the part of a size dis-

tribution affected by truncation artifacts (commonly the

smallest fractures) typically shows a convex-upward

curve that decreases in slope with decreasing fracture

size. Truncation artifacts can be minimized by artifi-

cially imposing a threshold size for fracture measure-

ment that can be readily observed at all locations in a

sampling domain. Although smaller fractures might be

visible (at least locally), fractures of such size might not

be reliably measured everywhere in the sampling

domain and lead to an incomplete sample.

Censoring Artifacts

Censoring artifacts (Baecher and Lanney, 1978; Laslett,

1982; Barton and Zoback, 1992; Pickering et al., 1995)

have been best described for fracture-length distribu-

tions obtained from two-dimensional domains (i.e.,

maps). Censoring occurs where some or all of the

largest fractures in a population are incompletely sam-

pled, or only minimum estimates of their sizes can be

made. Censoring of fracture-length measurements can

result from fracture traces that continue beyond the

limits of an observation domain, so their true sizes

cannot be determined. Censoring of fracture-aperture

measurements can happen in core-based studies be-

cause the largest aperture fractures commonly result

in incomplete core recovery, so some of the largest

fracture apertures are preferentially missed. Enhanced

erosion at large fractures in outcrops can produce a

similar effect. Inclusion of minimum estimates of frac-

ture length in a size distribution acknowledges the

presence of such fractures but degrades accuracy of the

length distribution. Omission of large fractures in a

size distribution produces a more pronounced effect.

Censoring results in an apparent homogeneity of

large fracture sizes compared with the wider range of

sizes for uncensored fractures. Censoring dispropor-

tionately affects data from the largest fractures be-

cause the probability of censoring is proportional to

fracture size. Censoring effects on a fracture-size distri-

bution typically show a convex-upward curve that in-

creases in slope with increasing fracture size.

Topologic Artifacts

Topologic artifacts are best illustrated by referring to a

specific example. Assume that several layer-perpen-

dicular fractures are randomly distributed in a layer of

rock, and that these fractures follow a power-law dis-

tribution of sizes. The volume of rock will contain

some fractures that span the thickness of the layer and

other fractures whose tip lines are partly or totally

embedded within the layer. In a layer-parallel two-

dimensional slice of this volume, the probability of

sampling a fracture that does not reach both layer

boundaries depends on the fracture height, but a com-

plete sampling of fractures that span the layer thickness

will be obtained because they will intersect any layer-

parallel two-dimensional slice. Thismeans that a three-

dimensional sampling of spanning fractures is effectively

obtained from a two-dimensional observation domain.

However, fractures that do not span the layer thickness

will show lower apparent abundance than their true

abundance in three dimensions. The cumulative size

distribution will reflect this change in apparent abun-

dance as a discrete decrease in the slope of the power-

law distribution for nonspanning fractures, which typi-

cally are most abundant (Marrett, 1996).

A similar reasoning can be extended to one-

dimensional sampling domains. A scan line perpen-

dicular to the fractures in the layer will sample a sub-

set of all fractures in the volume. The probability that

a scan line will sample a fracture is proportional to the

fracture surface area. In this case, not only fracture

height but also fracture length and the geometry of the

fracture surfacewill determine the sampling probability.

202 Geohorizons

Fractures that span the layer and the limits of the study

volumewill reflect three-dimensional sampling. Fractures

that do not reach the bed boundaries or the lateral bound-

aries of the study region will reflect one-dimensional

sampling.

Mechanical Boundary Effect

The effect of mechanical-layer thickness on fracture-

height distribution is trivial and can be explained using

topologic arguments. However, the effects of mechan-

ical-layer boundaries on fracture-length and aperture

distributions are poorly known. Ortega and Marrett

(2000) showed an example in which uncensored

fractures in fracture-length distributions deviate from

ideal power-law distributions with exponents that can-

not be explained by topologic artifacts. It is possible

that limits to fracture propagation affect the relation-

ship between aperture and length; however, no system-

atic study of fracture-aperture– and fracture-length–

scaling variations below and above the length scale of

the mechanical layer has been done. If the length and/

or aperture growth rates are perturbed when fractures

reach the bed boundaries, then nontopologic effects

might result, and these might be difficult to distinguish

from topologic effects.

Undersampling of Large Fractures

Power-law scaling predicts that the largest fractures

are least common in a fractured volume of rock. Con-

sequently,most size distributions are poorly constrained

at the large-scale end of the distribution. In addition,

the spatial distribution of fractures will affect sam-

pling probabilities. For example, if fractures are highly

clustered and a short scan line intersects a fracture

swarm, then the size distribution may show an anom-

alously high number of fractures for the size of the

observation domain. Fracture clusters commonly in-

clude the largest fractures of a population (Gomez,

2004). Mechanical modeling of en echelon fracture

arrangements suggests that the enhanced growth of

clustered large fractures is favored in this type of spatial

distribution (Olson and Pollard, 1991; Olson, 2004).

More typically, scan lines shorter than the half-spacing

between clusters will undersample large fractures. Con-

sequently, spatial distribution of fractures affects esti-

mates of a large-fracture intensity. The calculation of

intensity or average spacing of small fractures is more

reliable, particularly if long scan lines and large amounts

of data are collected.

Quantification of Uncertainties

Measuring fracture intensity for a volume of rock is

challenging, but estimating uncertainties is even more

difficult. In the previous sections, we discussed how

the scales of observation and fracture size affect esti-

mates of fracture intensity. The use of fracture-size

distributions promises a solution to the influence of

fracture size on fracture intensity by choosing a com-

mon fracture size as the basis of fracture-intensity

comparisons. However, sampling, topology, number

of observations, and spatial architecture of fracture

arrays can limit the accurate attainment of the under-

lying size distribution for a given fracture system. For

example, we expect a progressively more reliable ap-

proximation of the fracture intensity to be obtained as

we collect progressively more fracture data by ex-

tending an observation scan line. Extending the scan

line reduces possible spatial distribution effects on

fracture-size distributions and improves the likelihood

of adequately sampling the largest fractures.

One estimate of the uncertainty in fracture-intensity

determinations is the variance of consecutive fracture-

intensity estimates as data collection progresses along

a scan line. To evaluate this uncertainty, the normal-

ized fracture intensity can be calculated for progres-

sively larger parts of a complete scan line, choosing a

minimum fracture size to be considered for fracture

intensity (this does not have to be the threshold size

for fracture measurement). The standard deviation of

the normalized fracture-intensity estimates can then

be calculated for every new fracture encountered along

the scan line, using all of the estimates obtained up to

that point along the scan line. The normalized fracture-

intensity estimate and its uncertainty can be plotted

in a graph of the normalized fracture frequency ver-

sus the fracture number along the scan line (Figure 8).

Such graphs show that when a fracture of the mini-

mum size or larger is found along an observation scan

line, normalized abundance increases if the last spac-

ing is smaller than the average spacing up to that point.

However, if the last spacing is greater than the average

spacing up to that point, the normalized abundance de-

creases.Where only fractures smaller than theminimum

size are found, fracture-intensity estimates decrease be-

cause of the increase in scan-line length. The standard

deviation of successive normalized fracture intensities

provides a measure of the variability of fracture in-

tensity. One standard deviation around the expected

fracture-intensity estimate encompasses approximately

68% of the normalized intensity estimates determined

Ortega et al. 203

up to a given spot along the scan line (Koch and Link,

1971).

For the estimates of intensity uncertainties in

Figure 8, the first 20 fracture-intensity estimates were

ignored because the normalized fracture intensity is

unstable when a small number of fractures are used.

We assumed that the normalized fracture intensity up

to a given point in the scan line is independent of pre-

vious determinations, and thus, the expected value is the

last normalized intensity obtained. We justify the use

of this method because we expect a progressively more

representative fracture-intensity determination as the

scan line progresses, whereas a mean fracture-intensity

determination is a worse approximation. Estimation of

uncertainty (i.e., dispersion) using one standard devi-

ation around the expected value is conservative.

The determination of total fracture intensity in a

bed requires accounting for all fracture sets present in

the bed. We do this by simply adding the scan-line–

based estimates of fracture intensity for each set. The

propagation of errors for the addition of normalized

fracture intensities must be estimated. If Q is the sum

of the fracture intensities for all sets (e.g., sum of in-

tensities for sets A, B, C, etc.), we can obtain the ap-

proximate uncertainty sQ as

sQ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiXni¼1

s2i

sð3Þ

where s is the standard deviation, and i represents eachof the n fracture sets present in the bed (Young, 1962).

Anotherway of estimating the combined uncertainty of

the sum of fracture intensities (sall) is by weighting the

intensity of each fracture set (Fi) by the inverse of the

uncertainty associated with each of the estimate (1/si).This method of weighting aims to decrease the impact

of the least confident estimates commonly associated

with problematic size distributions or limited data and

provides a better approximation to the uncertainty of

fracture intensity in the bed. The formula used to cal-

culate this dispersion is

sall ¼

Pni¼1

Fi

Pni¼1

Fisi

ð4Þ

Uncertainty estimates using equations 3 and 4 prove

to be similar in most cases, but consistently higher es-

timates result from equation 3. The uncertainties re-

ported below and in Figure 8 were obtained using

equation 4.

COMPARISON OF FRACTURE INTENSITY INTWO LAYERS

For a comparison of the results shown in Figures 6

and 8, we provide analogous results from another

layer studied in the Sierra Madre Oriental. Bed 7 is a

carbonate mudstone approximately 60 cm (24 in.)

thick and only slightly dolomitized (10% of the total

volume of rock) studied near Iturbide, northeastern

Mexico. This bed is thicker than bed 3. Thicker beds

Figure 8. Uncertainty (68% confidenceinterval) for fracture-intensity estimateof set A in bed 3, La Escalera Canyon. Onlyfractures having an aperture of 0.2 mm(0.008 in.) or larger were included in thefracture-intensity estimate. The graphsuggests that the fracture-intensity aver-age does not change much after collectingthe first 100 fractures. The uncertaintyin the intensity determination diminishesprogressively toward the end of the scanline, suggesting that there are no majorchanges in the spatial distribution of frac-tures in the scan line measured. Largechanges in fracture intensity at the begin-ning of the scan line are expected forsparse data.

204 Geohorizons

are expected to have a wider average spacing than

thinner beds (Bogdonov, 1947; Price, 1966; Ladeira

and Price, 1981; Huang and Angelier, 1989; Narr

and Suppe, 1991; Gross and Engelder, 1995; Wu and

Pollard, 1995; Becker and Gross, 1996; Pascal et al.,

1997; Ji and Saruwatari, 1998; Bai and Pollard, 2000),

and consequently, thicker beds have lower fracture

intensity than thinner beds. Bed 7 is also less dolomi-

tized than bed 3. Sinclair (1980) found that dolostones

have higher fracture intensity than limestones. Both of

these considerations indicate that bed 7 is expected to

have lower fracture intensity than bed 3. The fracture-

aperture distribution obtained frommeasurements along

a scan line perpendicular to set C in bed 7 produces a

fracture-size distribution that can be adequately mod-

eled by a power law across more than one order of

magnitude variation in aperture (Figure 9). Set C is

the most abundant fracture set in bed 7, and the

fracture set studied in bed 3 is also the most abundant

in bed 7.

For comparing fracture intensity in the two layers,

we use 0.2 mm (0.008 in.) as the minimum aperture

considered for fracture-intensity estimation. Aper-

tures as small as 0.05 mm (0.002 in.) were measured

in both layers, but apertures of 0.2 mm (0.008 in.)

were well constrained for all layers in our Sierra Madre

Oriental database. Fracture intensity in bed 3 is nearly

five times higher than the fracture intensity in bed 7

(Figure 10). Confidence intervals for fracture intensi-

ties in the two beds do not overlap after 50 or more

fractures have been observed, suggesting that fracture

intensity in bed 7 is significantly lower than in bed 3.

The uncertainty of fracture intensity is slightly higher

for bed 3 than for bed 7, suggesting that fractures

having apertures of 0.2 mm (0.008 in.) or larger are

slightly more clustered in bed 3 than in bed 7. The

variation of fracture intensity and uncertainty in bed 7

suggest that several apertures 0.2 mm (0.008 in.) or

larger were encountered near the beginning of the scan

line. After encountering 50 fractures, the fracture in-

tensity progressively decreases, and the fracture inten-

sity changes little after sampling 100 fractures, al-

though it continues decreasing very slightly until the

end of the scan line is reached.

The difference in the fracture intensity obtained

from beds 3 and 7, where 229 and 197 fractures were

measured, respectively, implies considerably different

scan-line lengths. The scan line for bed 3 was 1.7 m

(5.5 ft) long, but a scan line 6.2 m (20.3 ft) long was

needed to encounter a comparable number of frac-

tures in bed 7.

CONCLUSIONS

Fracture-intensity estimates for a single rock volume

vary according to the observation scale because of varia-

tion of the smallest fracture size included. A compari-

son of fracture intensities among different volumes of

rock requires fracture-intensity estimates that share a

common minimum size of fracture. The method de-

scribed here provides a scale-independent approach

to fracture intensity that uses fracture-size distributions

Figure 9. Aperture distribution offracture set C in bed 7, Iturbide. Theaperture distribution can be ade-quately modeled by a power law formore than an order of magnitudevariation in aperture. The coefficientof this power law is approximatelya factor of 8 smaller than the coef-ficient in the power-law regressionfor bed 3, suggesting that this bed haslower fracture intensity than thefirst bed. Filled squares are fielddata used for regression. In the for-mula, m� 1 and mm�1 refer to theleft-hand and right-hand scales.

Ortega et al. 205

and permits choosing the minimum fracture size such

that sampling artifacts are minimized. The technique

can be used to estimate the fracture intensity and po-

rosity in the subsurface.

Uncertainty of fracture-intensity estimates is ob-

tained by calculating the standard deviation of fracture

abundance as collection of data progresses. A graph of

the variation of fracture abundance and uncertainty as

data collection progresses provides information on the

clustering of fractures larger than the normalization

size. This graph also illustrates whether the number of

data collected are sufficient for a stable estimate of

fracture intensity.

The use of scale-independent fracture-intensity es-

timates allows a reassessment of the geologic controls

on fracture intensity that have hitherto been broadly

accepted. Because previous methods do not account

for fracture size, it is likely that many fracture-intensity

estimates reflect arbitrary (or at least unexamined)

choices or assumptions about which fractures to mea-

sure. Moreover, this variation might have been sys-

tematically related to the very geologic controls under

examination. For example, if smaller fractures aremore

readily visible in one lithology than another, then biased

fracture-intensity estimates can falsely indicate a pat-

tern of geologic control. Lithologic controls on fracture

intensity need to be reexaminedwith appropriate scale-

independent estimators.

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