Fracture connectivity from fracture intersections in borehole image logs

$Page 1: Fracture connectivity from fracture intersections in borehole image logs$
Computers & Geosciences 29 (2003) 143–153

Fracture connectivity from fracture intersections inborehole image logs

Sait I. Ozkayaa,*, Joerg Mattnerb

a Baker Atlas Geosciences, P.O. Box 15425, Manama, Bahrainb GeoTechnical Consulting, P.O. Box 20393, Manama, Bahrain

Received 20 September 2001; received in revised form 22 August 2002

Abstract

The connectivity of fractures in subsurface rock formations is a key factor in understanding and predicting fracture

flow in hydrocarbon reservoirs. We present a method to determine average number of fracture intersections per

fracture, l; and fracture length from borehole image logs. Fracture length is estimated from relative frequency of

fractures with partial or complete circumferential traces on borehole image logs.

When all fractures are interconnected fractures are above percolation threshold. Every fracture must intersect at least

two other fractures for percolation, which means that percolation threshold corresponds to l ¼ 2: We introduce the

term sub-percolation threshold at l ¼ 1 to define the transition from isolated fractures to fracture clusters. Fractures

are isolated when lo1 form clusters within sub-percolation range (1olo2) and generate a network above percolation

threshold (l > 2).

Within sub-percolation range, the expected number of fractures, N ; in a cluster is related to l as follows: N ¼2l=ð2 � lÞ: This equation shows that fracture clusters remain small until l reaches the percolation threshold value, at

which point they interconnect in an explosive manner to form a network. Hence, only the areas where l is greater than 2

need to be considered for fracture networks.

Relative frequency of intersecting fractures, which is observed in borehole image logs, depends very much on fracture

size. More than 50% of observed fractures must be intersecting for an average fracture length of 1 m at percolation

threshold. The required frequency for percolation drops to 25% for an average fracture length of 2 m, and to 10% for

10 m average length.

r 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Fracture intersections; Borehole image logs; Connectivity; Percolation and sub-percolation thresholds

1. Introduction

Determining fracture connectivity is a key factor in

predicting and modeling fracture flow in subsurface rock

formations containing economically valuable products

such as water or hydrocarbons. This is true for both

mineralized fractures, which are often considered flow

barriers and non-mineralized fractures, which are

usually considered conduits. Commonly, fracture den-

sity, average length and angular scatter of fractures

determine fracture connectivity. The critical value of

connectivity at which fractures form an infinite network

is called percolation threshold (Berkowitz, 1995). Note

that this ‘‘infinite network’’ is a geometric concept and

exists independently of geomechanical and kinematic

analysis of fracture systems. Hestir and Long (1990)

proposed the following formula as a measure of fracture

connectivity:

z ¼ df LavHðyÞ; ð1Þ

*Corresponding author. Tel.: +973-212-234; fax: +973-212-

345.

E-mail addresses: [email protected]

(S.I. Ozkaya), [email protected] (J. Mattner).

0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 9 8 - 3 0 0 4 ( 0 2 ) 0 0 1 1 3 - 9

where HðyÞ is a measure of angular scatter. df is fracture

density and Lav is the average fracture length. For

uniformly distributed fracture orientation, HðyÞ equals

2=p: The value of z at percolation threshold is 3.6. This

formula, however, applies only to non-natural, con-

ceptual two-dimensional fracture network maps. An

analogous formula does not exist for equivalent three-

dimensional fracture networks. Fracture connectivity

and permeability must be estimated using discrete

fracture networks (DFN) (i.e. Odling, 1992; Odling

et al., 1999; Chen et al., 1999).

In this paper we present a method to deter-

mine fracture connectivity from borehole image logs

and cores using relative abundance of isolated

and connected fractures. We introduce three distinct

cases of fracture connectivity: (i) isolated fractures,

(ii) fracture clusters, (iii) fracture networks. We propose

the term sub-percolation threshold to define the transi-

tion from isolated fractures to fracture clusters. We

show that both the sub-percolation and percolation

threshold values can be obtained from the relative

abundance of isolated to interconnected fractures. The

differentiation of the three cases may have profound

implications on the interpretation of fractured reser-

voirs. The frequency of interconnected fractures also

gives information on the size, volume and surface area

of fracture clusters between sub-percolation and perco-

lation thresholds.

The idea of determining fracture connectivity from

frequency of isolated and connected fractures is

particularly appealing, because isolated fractures can

easily be differentiated from connected/intersecting

fractures on borehole image logs and cores. If we make

the assumption that the observed fracture relationships

are important to the reservoir productivity, some

significant parameters of these fracture systems such as

fracture tributary area, drainage volume, radius of

fracture enhanced zone around a wellbore and perme-

ability enhancement by fractures may be obtained from

borehole image logs.

For the purpose of this paper, only the spatial–

geometric relationships of fractures are examined, with-

out dealing with the geomechanical and kinematic origin

of fractures. The analysis is based on several implicit

assumptions that are listed below.

(1) We assume first that the image analyst has

identified and eliminated all drilling induced

fractures from consideration. It is not al-

ways straightforward to separate induced and

natural fractures (i.e. to differentiate fractures,

which are present in the rock before the well was

drilled from those, which are induced e.g. by the

drilling of the well). The problem becomes com-

pounded when drilling activity enhances natural

fractures.

(2) We also neglect fractures that are smaller than the

borehole diameter, since such small fractures are

usually below the resolution of borehole image logs

and do not contribute significantly to fracture flow.

2. Finding expected number of fracture intersections per

fracture

Let us start with a reference fracture. A borehole may

or may not pass through this reference fracture or may

pass through its edge (Fig. 1). Suppose that the reference

fracture is intersected by another fracture. When a

fracture intersection is encountered by a wellbore

it is usually visible as intersecting or truncating

fracture traces in borehole image logs (Fig. 2A and 3).

However, a borehole may or may not cross through a

fracture intersection (Fig. 2B). Frequency of fracture

intersections in borehole image logs is, therefore, less

than the actual frequency.

It is possible to estimate the actual frequency from

observed frequency of intersecting fractures if additional

Fig. 1. Boreholes (or cores) can have full encounter with

fracture when borehole passes through middle or partial

encounter when borehole passes through edge of fracture.

S.I. Ozkaya, J. Mattner / Computers & Geosciences 29 (2003) 143–153144

information on wellbore diameter, trajectory and

orientation is available. Borehole diameter is always

known. Spatial orientation of fractures and wellbore are

directly measurable.

The probability that a borehole passes through a

fracture intersection given that it has encountered an

intersecting fracture can be estimated from the following

expression (see Appendix A):

p ¼A

B¼

rLð1 þ cos yÞ=8 þ 4r2

ðL=2 þ rÞðL cos y=2 þ rÞðAoBÞ;

p ¼ 1 ðAXBÞ; ð2Þ

where L is fracture length, r is borehole radius and y is

the average angle fractures make with the borehole.

When a fracture is not coin shaped the average of length

and height may be used in place of L: The condition that

p ¼ 1 if AXB has to be placed because the equation is

based on geometric approximations. When the fracture

length is equal or smaller than borehole, the probability

that a borehole ‘‘sees’’ a fracture intersection is regarded

as unity.

Fracture length L is calculated from the relative

frequency of fractures, r; with full and partial circumfer-

ential traces on borehole image logs (see Appendix B). r

is related to length, L; as follows:

r ¼ðL � dÞðL cos y� dÞðL þ dÞðL cos yþ dÞ

; ð3Þ

where d is borehole diameter. L may be regarded as the

average fracture size. If a fracture is rectangular,

fracture width can be calculated from L; if fracture

height is known.

If we assume that fracture intersections follow

Poisson’s distribution we can calculate the expected

number of fracture intersections, l for each fracture

from the following relationship:

Pð0Þ ¼ e�lX

k

Pð0=kÞlk

k!; ð4Þ

where Pð0=kÞ means probability that a wellbore

encounters no fracture intersection given that the

reference fracture is intersected by k other fractures.

Pð0Þ is the probability of having an isolated fracture.

The first few of the conditional probabilities are listed

below.

Pð0=0Þ ¼ 1;

Pð0=1Þ ¼ 1 � p;

Pð0=2Þ ¼ 1 � 2p þ p2: ð5Þ

The conditional probabilities are based on the simplify-

ing assumption that a wellbore encountering one or the

other of the two intersections are independent events, i.e.

Pða,bÞ ¼ PðaÞ þ pðbÞ � Pða-bÞ: ð6Þ

The conditional probability for 3, 4 and higher number

of intersections can be calculated from the general form

of Eq. (5). It is necessary only to find the sum up to 6

intersections. When there are more than 6 intersections

the probabilities are very low and can be neglected. The

probability of having an isolated fracture, Pobsð0Þ; is

estimated by the relative frequency of fractures (12kob)

with no intersections in a wellbore

Pobsð0Þ ¼ 1 � kob: ð7Þ

The relative frequency kob is given by

kob ¼Nc

Nt; ð8Þ

Fig. 2. Fracture intersection on reference fracture may or may

not be encountered by a wellbore depending on location of

wellbore with respect to line of intersection. (A) Borehole

encounters fracture intersection. (B) Fracture intersection is

missed.

S.I. Ozkaya, J. Mattner / Computers & Geosciences 29 (2003) 143–153 145

where Nc is number of fractures with at least one

intersection and Nt is the total number of fractures

observed in a wellbore. Using Pobsð0Þ from Eq. (7) for

Pð0Þ; the expected value l can be calculated from Eq. (4)

by iteration.

3. Percolation and sub-percolation thresholds

Below we introduce three distinct cases of fracture

connectivity:

Case 1: Isolated fracture (Fig. 4).

Case 2: Fracture clusters (Fig. 5).

Case 3: Fracture network (Fig. 6).

We propose the term sub-percolation threshold when

isolated fractures start forming fracture clusters. At

percolation threshold isolated fracture clusters connect

and start forming a continuous three-dimensional

fracture network. The differentiation of the three cases

has profound implications on the interpretation of

fracture flow within a field context and the interpreta-

tion of near wellbore. Both the sub-percolation and

percolation threshold values can be obtained by the

relative abundance of isolated to interconnected frac-

tures.

Fractures reach percolation when the expected

number of intersections per fracture l ¼ 2: Thus, it

becomes possible to determine whether fractures are

above percolation from borehole image logs by finding

the relative frequency of isolated and connected

fractures.

When l is less than 1, fractures are expected to be

isolated. We call this value the sub-percolation thresh-

old. We coin the term sub-percolation range for l values

between these two critical limits (1olo2). Below the

sub-percolation threshold fractures are isolated. Above

the percolation threshold fractures form fully intercon-

nected network.

4. Size of sub-percolation fracture clusters

Fractures occur as clusters within the sub-percolation

range. The frequency of interconnected fractures also

Outcrop of fractured

Mesozoic volcanic rocks

Borehole Image(CBIL)

1m

N S

Fracture intersections

Fra

ctu

res

N

Fig. 3. Outcrop and borehole image log example of intersecting non-mineralized fractures. Intersecting non-mineralized fractures can

form large fluid conductive networks. This type of fractures allows, for example, economical hydrocarbon production from

formations, which would otherwise be considered non-producing. Outcrop shows fractured Mesozoic volcanic rocks in anti-Lebanon

in Syria (see hammer for scale). Image log depicts fractures in same rock formation in Syria at depth of over 2 km. On right, part of

image is interpreted and sketch indicates spatial attitude of intersecting fractures.


gives information on the size, the volume and surface

area of fracture clusters between sub-percolation and

percolation threshold (Fig. 5). The expected number of

fractures in a cluster can be found from the sum of

fractures, N; that are expected to connect to a seed

Isolated fractures (λ<1)

Clusters

(A)

(B)

Fig. 4. Two-dimensional fracture system below sub-percolation

threshold. Fractures are mostly isolated. (A) All fractures, (B)

only intersecting fractures. Only a few small clusters are present

with two to three fractures. Fracture data: average

length=0.291 m, area density=2.32/m2, scan line den-

sity=0.68/m, number of fractures=200, number of intersec-

tions=50, expected number of intersections=0.25.

N=13

N=13

N=8

N=5

N=5

N=51

N=21

N=6

N=17

N=23

Selected clusters (N=number of fractures in cluster)

Fracture clusters (sub-percolation range) (1<λ<2)

(A)

(B)

Fig. 5. Two-dimensional stochastic fracture model with ran-

dom orientation and exponential length distribution. Fractures

are within sub-percolation range and occur mostly in clusters.

(A) All fractures, (B) only some selected fracture clusters.

Fracture cluster size is related to expected number of fracture

intersections. For this example, calculated average number of

fractures per cluster is 2.57. Actual value is 4. Maximum

number of fractures per cluster is 55. Fracture length, density

and expected number of fracture intersections determines this

volume. Fracture data: average length=0.33 m, area den-

sity=6.75/m2, scan line density=2.23/m, number of frac-

tures=503, number of intersections=724, expected number of

intersections=1.44.


fracture, i.e.

N ¼ 2lX

ðl� 1Þn: ð9Þ

When (l� 1) is less than 1 (lo2) this series is

convergent to 1=ð22lÞ: Thus the expected number of

fractures in a cluster is given by

N ¼2l

2 � l: ð10Þ

If fracture length and density are known, this number

can be used to determine the total drainage area and

volume of fracture clusters and the near-wellbore

permeability enhancement by fracture clusters con-

nected to a wellbore.

5. Discussion and conclusion

We presented a method for finding expected value of

fracture intersections per fracture, l; from borehole

image logs and demonstrated how this value indicates

the degree of fracture connectivity. We define two

critical values for l corresponding to fracture percola-

tion (l ¼ 2) and sub-percolation threshold (l ¼ 1).

Fractures are isolated below sub-percolation threshold,

and form interconnected networks above percolation

threshold. Between the two limits, fractures occur as

clusters. We have also provided a simple formula to

determine fracture cluster size at sub-percolation range

in terms of expected number of fracture intersections.

The formula reveals that fracture clusters have little

impact on flow dynamics below percolation threshold

because clusters remain small until percolation is

reached, when they interconnect to form a continuous

network in an exponential manner (Fig. 7). For exam-

ple, the average number of fractures in a cluster does not

exceed a few hundred for l ¼ 1:99: For a fracture

density of 2 fractures per m3 this average cluster volume

is smaller than 15 m3. Fracture clusters remain small

until percolation threshold is reached. Hence, only the

areas where l is greater than 2 will have fracture

networks with fracture clusters elsewhere will be too

small to impact on flow dynamics.

Relative frequency of intersecting fractures, which is

observed in borehole image logs, depends very much on

fracture size (Fig. 8). More than 50% of observed

fractures must be intersecting for an average fracture

length of 1 m at percolation threshold. The required

frequency for percolation drops to 25% for a fracture

length of 2 m, and 10% for 10 m average length. It is

therefore necessary to have an accurate fracture size

estimate in order to determine degree of fracture

connectivity from fracture intersection in borehole

image logs.

It should be noted that the derivations are based

entirely on simplified geometric relationships and do not

take into consideration geomechanical or geological

origin of an observed fracture system. In particular, the

following points must be kept in mind when calculating

fracture connectivity:

Continuous fracture network

Fracture network (λ>2)

(A)

(B)

Fig. 6. Two-dimensional stochastic fracture model with ran-

dom orientation and exponential length distribution. Fractures

are above percolation threshold. (A) Most fractures belong to

continuous network, (B) only fractures that belong to

percolating network. Fracture data: fracture length=0.35 m,

area density=10.9 m2, scan line density=3.83/m, number of

fractures=780, number if intersections=1862, expected num-

ber of intersections=2.38.


(1) If geological fracture connectivity is required, both

cemented and open fractures are to be included in

the analysis. For permeability assessment of

fracture systems, as barriers or conduits, fluid

conductive fractures and non-sealing fractures

should be considered separately.

(2) Partially mineralized fractures create a complex

network, in which connectivity of fluid condu-

ctive segments and sealing segments needs to

be assessed independently. Channeling along

fracture planes creates an additional challenge. In

this case, connectivity of the one-dimensional

tributary channel-network is critical and not the

fracture surface intersection. Two fractures may

intersect, but might fail to have connected flow

pathways.

(3) Fracture characteristics, including length and con-

nectivity are very much dependent on mechanical

layer properties, such as layer thickness, porosity or

dolomite content. The connectivity analysis must

be performed for each mechanical layer.

(4) The average fracture connectivity of an area and

the connectivity of individual fracture clusters are

significantly different. Fracture clusters usually

occur in the vicinity of faults. In order to calculate

the average fracture connectivity of an area for an

100

300

500

0 1 2

λ

N

N= number of fractures in a cluster = average number of fracture intersections per fractureλ

Fig. 7. Fracture cluster size as a function of expected number of fracture intersections. Fracture clusters remain small until percolation

is reached at l ¼ 2; at which point they become interconnected in an exponential manner to form a continuous network.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

Fracture Length (m)

Per

cent

of i

nter

sect

ing

frac

ture

s

Fig. 8. Minimum percent of intersecting fractures that must be observed in borehole image logs or cores for fractures at or above

percolation threshold.


upscale fracture connectivity model, the connectiv-

ity between the clusters must also be calculated.

Appendix A. Fracture intersections from borehole image

logs

Below we derive a formula to remove the sampling

bias and obtain a corrected value for fraction of

intersected fractures from borehole image log/core

observations with the assumption that all fractures are

penny shaped. Consider such a penny shaped reference

fracture of length L; intersected by another fracture

(Fig. 9). Let us extend the line of intersection to the

edges of the reference fracture and call the length of this

extended line s:The edge-to-edge line of intersection may be anywhere

on the reference fracture and will have a length that

varies from zero to L: Since the line of intersection is

randomly located, we can determine the expected value

of s as follows (Fig. 10). The distance of the line of

intersection, a; from the center of the fracture has a

uniform probability distribution between 0 and R ¼ L=2

(Fig. 10A). The half-length of s is a function of a

(Fig. 10B).

S=2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � a2

p: ðA:1Þ

Hence the expected value of s=2 can be calculated from

EðS=2Þ ¼1

R

Z r

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðR2 � a2Þ

pda: ðA:2Þ

This yields

EðS=2Þ ¼ �R

Z p=2

0

sin2 y dy: ðA:3Þ

Evaluation of this integral yields the required result.

EðS=2Þ ¼pR

4¼

pL

8: ðA:4Þ

The reference fractures are interchangeable. The ex-

pected length of s is the same for both fractures, but the

intersection may be located anywhere and value of si

varies from 0 to s (Fig. 11). Since all values have equal

probability, the expected value of si is equal to EðS=2Þwhich is equal to pL=8:

A borehole intersects the reference fracture fully if the

borehole is within the fracture (Figs. 12 and 1, fracture

1). In this case the fracture has a full circumference trace

on image log. If the borehole cuts the reference fracture

at the edge then it is a partial intersection (Figs. 12 and

1, fracture 2). In this case the fracture has an incomplete

circumference trace on image log. If the fracture

diameter is less than the borehole diameter, then we

can only have incomplete circumference trace.

The area within which the borehole intersects the

target fracture is B (Fig. 13). A and B are given,

respectively, by the following:

A ¼ 2rs þ pr2 ðA:5Þ

and

B ¼ pðL=2 þ rÞ2; ðA:6Þ

where r is the borehole radius, s is expected value of

S (See Eq. (A.4)). The probability that a borehole

encounters a fracture intersection given that it has

encountered a reference is given by the ratio of A to B;i.e.

p ¼A

B: ðA:7Þ

So far we have assumed that the borehole is perpendi-

cular to the reference fracture. This condition is rarely

Intersectingfracture

plane

Intersectingfracture

plane

L

Full intersection with respect to referencefracture (si=s)

Partial intersection with respect to referencefracture (si<S)

Fracture boundary

Referencefractureplane

(A)

(B)

s

si

L

Fracture boundary


s

si

Fig. 9. Intersection line between two fractures. For description

purposes one of the fractures is designated as ‘‘reference

fracture’’, the other as ‘‘intersecting fracture’’. Actual intersec-

tion line between two fractures is si: In case of full intersection

of reference fracture, actual intersection line (si) traverses

fracture from boundary to boundary (si ¼ s). Maximum

intersection line equals diameter (L) of reference fracture.


met. More often, the borehole intersects the reference

fracture at an angle y and plan view of the reference

fracture on a plane perpendicular to the borehole is an

ellipse. The length of the two principal axes of this

ellipse, a and b given by (Fig. 14)

a ¼ L cos y; ðA:8Þ

b ¼ L:

The average length of fracture intersection on this plan

view is

sav ¼s þ s cos y

2: ðA:9Þ

The area A and B must be modified, respectively, as

follows:

A ¼ 2rsav þ 4pr2 ðA:10Þ

and

B ¼ pðL=2 þ rÞðL cos y=2 þ rÞ: ðA:11Þ

Accordingly, the probability that a borehole

passes through fracture intersection, given that it has

encountered an intersecting fracture can be estimated

from Eq. (2).

θS/2

a

R=L/2

Referencefracture plane


s1

s2

s3

sn

Planview

Planview

(A)

(B)

Fig. 10. Calculation of expected maximum length of fracture

intersections. (A) Fracture intersection, s; can be located

anywhere on reference fracture. Full intersection length (s)

varies from length zero to diameter of reference fracture (L). (B)

Expected value of s can be calculated as function of distance

from center, assuming this distance has uniform probability

distribution.

S

Ssi

si

Partial intersection case 1 (si<S)

Partial intersection case 2 (si<s)

Intersecti

ng

fractu

re

Intersecti

ng

fractu

re

Referencefracture

Referencefracture

(A)

(B)

Fig. 11. Basis for calculation of expected value for si as

function of s: (A) Partial intersection with respect to reference

fracture, full intersection with respect to intersecting fracture.

(B) Both fractures have partial intersections. Irrespective of

relative fracture sizes, si varies in length from zero to s;randomly. Consequently, it can be shown that expected value of

si equals s=2:


Appendix B. Fracture length from borehole image logs

Consider a coin shaped fracture with diameter L and a

borehole with diameter d : The fracture pole makes an

angle, y with the borehole trajectory. If the center of the

circle with diameter d representing the borehole falls

entirely within the following area, A; where

A ¼pðL � dÞðL cos y� dÞ

4ðB:1Þ

the fracture will have a full circumferential trace (see

Fig. 12), This area is within the fracture projection onto

a plane perpendicular to the borehole (Fig. 14). Now

consider another ellipse with dimension w1; and w2

given, respectively, by

w1 ¼ L þ d ðB:2Þ

and

w2 ¼ L cos yþ d: ðB:3Þ

If the center of the circle that represents the borehole

falls within this ellipse, the borehole intersects the

fracture (Fig. 12). The fracture may have full or partial

circumferential trace on the image log depending on

r

r : borehole radiusL : reference fracture diameter

Full encounterof fracture

Partial encounter of fracture

L/2

Referencefracture

bore-hole

bore-hole

Fig. 12. Borehole has to be located within dotted circle in order

to encounter reference fracture. Partial encounters are located

within outer rim of reference fracture (dotted circle). Borehole

radius (r) plus radius (L=2) of reference fracture define radius of

outer dotted circle. Relative frequency of partial fracture

encounters may be used to estimate fracture length from

borehole image logs.

r : borehole diameters : line of intersection

L : reference fracture diameterr : borehole radius

Area of borehole encounterof fracture intersection (Area A)

Area borehole encountersreference fracture (Area B)

Planview

Plan view

L/2 r

(A)

(B)

Are

a A

Area B

sr

Referencefracture

Fig. 13. Basis for calculation of probability that borehole

encounters fracture intersection on reference fracture given

that there is an intersection. (A) Plan view showing geometric

configuration of area in which borehole encounters fracture

intersection (borehole is perpendicular to reference fracture

plane). (B) Defintion of fracture intersection probability.

Probability that borehole misses fracture intersection, given

that there is an intersection, is 12A=B:


where the borehole encounters the fracture. The area of

this ellipse, A0; is

A0 ¼pðL þ dÞðL cos yþ dÞ

4: ðB:4Þ

The ratio, r of fully intersected fractures to all

intersected fractures is therefore given by

r ¼A

A0: ðB:5Þ

This yields the following expression:

r ¼ðL � dÞðL cos y� dÞðL þ dÞðL cos yþ dÞ

: ðB:6Þ

References

Berkowitz, B., 1995. Analysis of fracture network connectivity

using percolation theory. Mathematical Geology 27,

467–483.

Chen, M., Bai, M., Roegiers, J.C., 1999. Permeability tensors of

anisotropic fracture networks. Mathematical Geology 31,

355–373.

Hestir, K., Long, J.C., 1990. Analytical expressions for the

permeability of random two-dimensional Poisson fracture

networks based on regular lattice percolation and equivalent

media theories. Journal of Geophysical Research 95 (B13),

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Odling, N., 1992. Permeability of natural and simulated

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Fig. 14. Plan view of reference fracture perpendicular to

borehole is elliptical when borehole is not perpendicular to

the reference fracture plane. Area of fracture and fracture

intersection encounter must be modified accordingly.


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Fracture connectivity from fracture intersections in borehole image logs