Relationship between percolation–fractal properties and permeability of 2-D fracture networks

$Page 1: Relationship between percolation–fractal properties and permeability of 2-D fracture networks$
International Journal of Rock Mechanics & Mining Sciences 60 (2013) 353–362

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International Journal ofRock Mechanics & Mining Sciences

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Relationship between percolation–fractal properties and permeabilityof 2-D fracture networks

Alireza Jafari 1, Tayfun Babadagli n

Department of Civil and Environmental Engineering, School of Mining and Petroleum, University of Alberta, 3-112 Markin CNRL-NREF, Edmonton, AB, Canada T6G 2W2

a r t i c l e i n f o

Article history:

Received 24 August 2011

Received in revised form

21 December 2012

Accepted 15 January 2013Available online 1 March 2013

Keywords:

Fracture network permeability

Percolation

Fractal properties

Connectivity

09/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.ijrmms.2013.01.007

esponding author.

ail address: [email protected] (T. Babadagli)

ow with Alberta Innovates.

a b s t r a c t

The concepts of percolation theory and fractal geometry are combined to define the connectivity

characteristics of 2-D fracture networks and a new approach to estimate the equivalent fracture

network permeability (EFNP) is introduced. In this exercise, the fractal dimensions of different fracture

network features (intersection points, fracture lines, connectivity index, and also fractal dimensions of

scanning lines in X- and Y-directions), and the dimensionless percolation density of fracture networks

are required. The method is based on the proposed correlations between the EFNP and a percolation

term, (r0–r0c). The first parameter in this term is the dimensionless density, and the second one is the

percolation threshold (a constant number). This term is obtained using the relationships with the

properties of fracture networks mentioned above. It was found that the highest correlation coefficient

between the actual and the predicted EFNP could be obtained using the percolation term, (r0–r0c),calculated from the fractal dimension of fracture lines by the box-counting technique. The method

introduced is validated using different fracture patterns representing a wide range of fracture and

length values. In addition, a correlation between the number of fractures in the domain and the

minimum size of the fracture length is presented to estimate the shortest or minimum fracture length

required for percolation for a given number of fractures in the domain.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Fractures provide high permeability conduits within the hostrock which control the fluid flow through these porous media.Due to the impossibility of locating, measuring and analyzingfractures, predicting fluid flow within the host rocks is stillchallenging despite the extensive work on characterizing frac-tured rocks conducted over the last few decades [1].

Quantification of fracture network properties is a critical task ifone desires relating these characteristics to fluid flow processes. Itis well known that natural fracture patterns exhibit fractalcharacteristics [2–4]. The fractal dimension, which is a non-integer value, shows the tendency of an object to spread or fillin the space where it is located [5]. In a pioneering work, La Pointe[4] reported that implications between the fractal properties of afracture network and its conductivity property exist.

Fracture length, density, aperture, and orientation are theparameters which control the permeability of the fracturenetwork. Rossen et al. [6] indicated that as fracture length anddensity increase, the connectivity of a fracture network will

ll rights reserved.

.

increase. Babadagli [7] speculated that perpendicular fracturesto the direction of the flow probably reduce the permeability.Jafari and Babadagli [8,9] showed that equivalent fracture net-work permeability (EFNP) can be correlated to 2-D fractal proper-ties of the 2-D network and four different fractal and statisticalparameters were observed to show the strongest correlation tothe EFNP [7,8]. Also based on an experimental design analysis,they showed that the single fracture conductivity (or aperture)either does not play a critical role compared to network proper-ties such as fracture density and length (these two parameters arecritical in order to be above the percolation threshold) or has anequivalent importance as these two important fracture networkproperties.

The primary controlling parameter in fracture network perme-ability is connectivity. In fact, fracture connectivity controls manyprinciple properties of fractured reservoirs [1]. For example, Longand Billaux [10] observed that due to low fracture connectivity ata field site in France, roughly 0.1% of fractures contribute to theoverall fluid flow (i.e. permeability). This parameter, however, isdifficult to quantify and include in the fracture network perme-ability correlations. In our earlier studies we included this para-meter defining a connectivity index [7,8]. An alternative is to usethe percolation theory.

The percolation theory is a powerful concept for studyingdifferent phenomena in disordered media, especially for analyzing

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the transport properties of random systems. First attempts onmodeling fracture networks were on lattice networks [11]. However,due to the nature of fracture distribution within host rocks,percolation of fracture networks needed another class of percolationcalled continuum percolation [12]. The necessity of this approachwas justified by Khamforoush et al. [13] as the transport propertiesof fractured networks are affected not only by the individualproperties of fractures but also their connectivity properties. Zim-merman and Bodvarsson [14] proposed a simple procedure toestimate effective grid-block scale conductivity of two-dimensionalfracture networks by applying Kirkpatrick’s effective mediumapproximation [15].

Berkowitz [1] reported that fracture connectivity could beused to study most of the fundamental properties of fracturenetworks [1], which can be mathematically modeled by thepercolation theory [16]. The most important characteristicof the continuum percolation is an appropriate densityparameter equivalent to occupancy probability in the latticepercolation [12].

Conventional methods of fracture network modeling, morespecifically permeability estimation, suggest generating manydifferent–random–realizations of possible fracture network con-figurations and then performing flow simulation for each one.Having a practical technique to quickly estimate the equivalentfracture network permeability using available fracture networkproperties and percolation theory will reduce the computationalwork needed in the conventional characterization and simulationof fractured systems. This has been the motivation for this studyand an approach was introduced to estimate the equivalentfracture network permeability for 2-D systems using the percola-tion and fractal geometry theories.

2. Background on percolation theory and fracture networks

Significant amount of work has been performed to solvenetwork related problems in different areas of science andengineering using the classical theory of percolation since it wasfirst introduced by Broadbent and Hammersley [17]. Scalingequations and universal constants and exponents are well definedfor systematic lattices [18–21]. Fracture networks are randomsystems and the continuum percolation was introduced as analternative to the lattice percolation.

An appropriate density parameter equivalent to occupancyprobability in the lattice percolation was defined in the conti-nuum percolation [11]. Also, Balberg et al. [22] introduced theexcluded volume (in 3D) concept which could play a critical rolein continuum percolation [22]. It is defined as the volume intowhich the center of other similar objects should not be placed inorder not to have any overlap (intersection). However, it shouldbe mentioned that this definition is valid if the objects are placeduniformly in space [23]. Mourzenko et al. [24] combined fracturedensity in a unit cell with excluded volume and ended up with adimensionless density. Since this dimensionless density is relatedto the excluded volume, the macroscopic properties of thefracture networks are independent of fracture shape [24–29].

Mourzenko et al. [12,24] studied a 3-D model containing 2-Dfracture planes (polygons) with power-law size distributionuniformly located in the space. Then, they triangulated each fracturenetwork to solve flow equations and calculate permeability. Theystated that a dimensionless fracture density is an appropriateequivalent parameter in continuum percolation which is indepen-dent of fracture shape. They also proposed a general equation bycombining fracture density weighted by fracture conductivities anda universal function of a dimensionless fracture density which takesinto account fracture shape and size distribution.

King et al. [30] used percolation theory to estimate thedistribution of the shortest path between two wells and also thedistribution of the breakthrough times and associated uncertaintyby simple algebraically calculation rather than the traditionaltime-consuming numerical simulation.

Odagaki et al. [31] investigated the 2-D systems with corre-lated distribution for a model granular system and a lattice gasand found that correlated distribution of percolating objectsaffects the percolation properties in both statistic structure anddynamic processes.

Nolte et al. [32] studied the effect of contact areas of fractureson their flow properties and showed that the effect of tortuosityof the flow path on fluid transport is controlled by the proximityof the distribution of these paths to a percolation threshold. Theyalso presented a model which could be used to determine how farthe systems are located above the percolation threshold.

Very recently, more emphasis was given to the estimation offracture network permeability through the percolation theory.Khamforoush et al. [13] studied the percolation threshold andpermeability of anisotropic 3-D fracture networks. They foundthat the percolation threshold in the X- and Y-directionsdecreases as anisotropy increases but in the Z-direction anopposite trend can be seen. Koudina et al. [29] calculated thepermeability of a 3-D network consisting of polygonal fracturesand studied the data relative to the percolation threshold andfound that permeability exponent, i.e., ‘‘t’’ [where KE(r0 �r0c)t]in 3-D, was in very good agreement with the universal exponentin 3-D reported by Stauffer and Aharony [11,37]; the effect of thefracture shape was also investigated.

Zhang and Sanderson [33] studied the connectivity of 2-Dfractures in terms of fracture density, orientation and length. Theyfound that fracture density is the controlling parameter of thepercolation threshold and fracture connectivity is dependent onlyon this parameter as long as the fractures have much a smaller lengthrelative to the size of the region of interest. Also, at or above thiscritical point, i.e., percolation threshold, the permeability of thesystem increases with an increase in fracture density.

Masihi et al. [34] applied the percolation theory concepts onfractured reservoirs to analyze the reservoir performance andshowed that the performance parameters can be predicted veryquickly by some semi-analytical universal curves. Later, Masihiet al. [35] determined the two scaling curves of connectivity ofisotropic fracture systems which could be used to estimatefracture connectivity and its associated uncertainty very quickly.They also applied the percolation concepts to a field examplefrom the Bristol Channel basin.

The percolation threshold of different simple systems in thepercolation continuum has been determined by differentresearchers; for example, the percolation threshold of randomlyoriented sticks in 2-D in terms of dimensionless density (r0) hasbeen reported to be 3.6 [23]. For analyzing any fracture system,the percolation threshold needs to be estimated theoretically andcompared with the actual density of fractures in the system [36].

On the basis of the review presented above, one may concludethat a definition of a percolation exponent that would eventuallyyield to obtaining the permeability of fracture networks is anecessity. This requires the involvement of different fracturenetwork characteristics in the power law–percolation relation-ship. We use fractal and other statistical characteristics of fracturenetworks for this purpose as described in the following sections.

3. Synthetic fracture patterns

Using a developed code in MATLAB, twenty different fracturepatterns based on predefined ranges for fracture length and

A. Jafari, T. Babadagli / International Journal of Rock Mechanics & Mining Sciences 60 (2013) 353–362 355

density were generated which cover all possible combinations ofvarying these fracture parameters. The randomly oriented frac-tures were distributed within a 100 m�100 m sized domainaccording to a uniform distribution. In these patterns, the numberof fractures per domain (density) and fracture length werevarying parameters. The number of fractures per domain had arange from 50, 100, 150, 200 and 250. Also, fracture length haddiscrete values as 20, 40, 60 and 80 m (Table 1). Properties of eachfracture pattern were defined as average of ten different randomrealizations as given in Table 1.

Then, using a commercial software package (FRACA), theequivalent fracture network permeability (EFNP) of each of thesepatterns was calculated. In this software, fracture conductivity isused instead of aperture for measuring permeability [37], whichis defined as the product of the intrinsic fracture permeability andthe fracture aperture (e) with parallel walls. Also, the intrinsicfracture permeability and conductivity according to Poiseuille’slaw are expressed as e2/12 and e3/12, respectively [38,13]. In all ofthe patterns, constant fracture conductivity, i.e., 1000 mD m, wasassigned to all fractures in the domain. This means that only onefracture, spanning straight across the 100 m�100 m fracturedomain, would give an effective permeability of 10 mD.

Two assumptions were made for the measurement of theEFNP: (1) all fracture reservoirs consisted of only one layer, and(2) all fractures were entirely vertical. These assumptions arebelieved to be necessary to fully focus on the relationshipbetween fractal properties and percolation parameters itself for2-D networks. All the fractures were assigned a constant con-ductivity value of 1000 mD m. Our previous attempt showed thatthe conductivity of a single fracture (aperture) does not play acritical role as compared to the network properties above thepercolation threshold [9]. As our system considered in this studyis above percolation threshold, fracture density and lengthstrongly dominate over the single fracture conductivity, whichjustifies this assumption. A single layer assumption was alsoneeded as the system considered is 2-D.

Each 2-D fracture pattern (in the form of trace map) wasimported and then a model with the above assumptions wasbuilt. The fracture network in the FRACA software was discretizedwith a rectangular grid and then the mass balance equations weresolved by applying a pressure varying linear boundary conditionfor each direction. Finally, the flow rates across the block faceswere computed to calculate the EFNP [37]. The EFNP values for

Table 1Generated synthetic fracture patterns used in this study.

Pattern Fracture length, (m) Number of fractures Permeability (mD)

1 20 50 5.609

2 40 50 77.643

3 60 50 120.310

4 80 50 185.112

5 20 100 27.343

6 40 100 155.048

7 60 100 274.240

8 80 100 384.561

9 20 150 74.672

10 40 150 266.795

11 60 150 424.930

12 80 150 621.084

13 20 200 124.025

14 40 200 395.592

15 60 200 583.337

16 80 200 825.885

17 20 250 161.214

18 40 250 459.049

19 60 250 696.129

20 80 250 1018.037

each pattern are shown in Table 1. Because only very limitednumber of fracture clusters in each pattern at very low fracturelength and density could span the two opposite faces of thedomain, the average value of EFNP over ten different realizationsfor these patterns was fairly low.

4. Fractal properties of the fracture networks

Shortly after Mandelbrot’s well known book on fractals [39] inwhich many different synthetic and natural fractal objects wereintroduced, fractality of natural fracture patterns were tested andreported [2–4].

In this study, we took advantage of the usefulness of fractalgeometry in the quantification of many different properties offracture networks and developed an algorithm to calculate thefractal characteristic of generated fracture patterns (total numberis 200, the different realizations of twenty different configurationsis given in Table 1).

We begin with the classical box counting dimension as firsttested by Barton and Larsen for fracture networks according tothe following relationship [2]:

NðrÞar�D ð1Þ

where NðrÞ is the number of the boxes (grids) containingdifferent fracture features, r is the box sizes and D is the fractaldimension. In this commonly used technique, the fractal dimen-sion is obtained by overlaying a set of different boxes (grids) withdifferent sizes on the fracture network and counting the numberof boxes containing different fracture features (fracture intersec-tion points, fracture lines, etc) for each box size. A plot of thenumber of occupied boxes versus the size of those boxes isdeveloped and then the slope of the straight line fitted to thepoints gives the fractal dimension of that specific feature of thefracture network. The fractal dimension of fracture intersectionpoints and fracture lines was measured using this technique.

Other fracture network characteristics were quantified byapplying different statistical and fractal techniques. To considerthe orientation effect, a square was overlaid on the fracturedomain. Then, a number of imaginary scanning lines in the X-and Y-directions (horizontal and vertical) inside the square weredefined. Next, the number of intersections between these scan-ning lines and fracture lines were counted. This process wasrepeated with different square sizes and at the end, the number ofintersections was plotted against the square size and the slope ofthe best fitted line was calculated [7]. The effects of fractureorientation on the EFNP were taken into account with thismethod. If all fractures are either in the X (horizontal) or Y

(vertical) directions, the fractal dimension using the X- and Y-direction scanning lines would be different from each other; butin randomly uniform distributed fractures, these two fractaldimensions are expected to be close to each other.

A new parameter called the connectivity index was defined asthe total number of intersection point divided by the totalnumber of fracture lines to take into account the connectivity(intersection). The results are presented in Table 2. One maynotice that all the fractal dimensions lie between 1 and 2 whichindicates the fractal nature of a 2-D system and there is mean-ingful relation between the fractal dimensions and fracture lengthas well as number of fractures. Note that twelve different fractaland statistical characteristics of fracture networks were testedagainst permeability and five of them listed in Table 2 wereobserved to show the strongest correlation [8]. For further detailsabout the measurement of these fractal properties, readers arereferred to previous publications [8,9].

Table 2Fractal dimension of different features of the synthetics fracture networks.

Length Number offractures

FD of intersectionpoints

FD of X-scanninglines

FD of Y-scanninglines

ConnectivityIndex

FD of fracturelines

Permeability(mD)

20 50 1.162 1.565 1.592 0.732 1.336 5.609

40 50 1.605 1.180 1.204 3.478 1.501 77.643

60 50 1.747 1.038 1.029 8.096 1.588 120.310

80 50 1.839 0.979 0.969 13.702 1.641 185.112

20 100 1.577 1.424 1.522 1.435 1.504 27.343

40 100 1.843 1.223 1.223 6.530 1.647 155.048

60 100 1.912 1.052 1.059 15.968 1.714 274.240

80 100 1.941 0.993 0.984 27.270 1.761 384.561

20 150 1.778 1.746 1.692 2.225 1.594 74.672

40 150 1.913 1.236 1.237 9.439 1.722 266.795

60 150 1.940 1.039 1.044 24.705 1.774 424.930

80 150 1.967 0.980 0.982 41.930 1.816 621.084

20 200 1.868 1.546 1.530 2.853 1.657 124.025

40 200 1.941 1.220 1.225 13.395 1.769 395.592

60 200 1.963 1.039 1.041 32.229 1.814 583.337

80 200 1.980 0.980 0.980 56.216 1.847 825.885

20 250 1.905 1.444 1.405 3.640 1.701 161.214

40 250 1.957 1.177 1.167 16.858 1.798 459.049

60 250 1.970 1.041 1.029 40.944 1.838 696.129

80 250 1.984 0.986 0.981 70.622 1.866 1018.037


5. Percolation theory and fracture network permeability

Percolation theory is a general mathematical theory which canbe used to describe the connectivity and conductivity (perme-ability) of systems with complex geometry [40]. The advantage ofusing this theory is that many results can be expressed throughsimple algebraic relationships known as scaling law (power laws)[16]. Density of the objects placed randomly in space is directlyrelated to the overall properties of the system of interest [34,35].The easiest model for understanding what the percolation theoryis all about is an infinite lattice consisting of sites and bonds. Now,suppose some of these either sites or bonds are occupied (open toflow) with a probability ‘‘p’’. For small values of this ‘‘p’’, most ofthe occupied sites (or bonds) form isolated clusters (connectedoccupied sites). As this probability increases, number of theseclusters increase and also, some of those grow or merge intoothers and at a particular value known as percolation threshold,one of these clusters spans the entire domain and connects theopposite faces together. This critical value depends on the detailand dimensionality of the lattice or grid. Around this criticalthreshold, the following simple analytical term known as scalinglaw or power law exists [16,34,35]:

PðpÞaðp�pcÞb p4pc ð2Þ

where PðpÞ is the probability that a site or bond connected to thespanning cluster in the domain. The exponent b is known as theuniversal exponent and it is independent of the lattice character-istics, and only depends on the dimensionality of space. Hence,whether the system is lattice or continuum percolation, its valuewould be the same being 0.139 and 0.4 for 2-D and 3-D spaces,respectively.

The above percolation analysis using lattice is called latticepercolation. However, in fracture analyses using this theory,fractures need to be placed randomly and independently withina continuum space. This is called continuum percolation andinterestingly, the same scaling (power) laws with the samecritical exponents as in lattice percolation apply to the continuumpercolation [16].

The major advantage of continuum percolation against thelattice percolation in the analysis of fracture networks is thatfractures can be placed at any location in the domain and are notlimited to any specific points on the lattice. Also, the number of

fracture intersections (connectivity) is not limited to any specific(maximal) number [34]. It has been postulated that there is arelationship between the effective permeability of a system andits percolation properties presented by the following scaling law[40]:

Kef faðp�pcÞm

ð3Þ

where Kef f is the effective permeability, p and pc are occupancyprobability and percolation threshold respectively, and m is auniversal exponent.

The main focus of this study was to improve the relationshipgiven in Eq. (3) to further develop a methodology to obtainequivalent (effective) fracture network permeability. The readilyavailable 2-D fractal and statistical properties of fracturenetworks were used in this exercise.

The most difficult part of using percolation theory in acontinuum system is to define an equivalent term to the occu-pancy probability in lattice percolation. As suggested by [23], thiscould be done by using the excluded volume concept [23]. Thus,using the excluded volume (3-D objects) concept introduced by[22], the excluded area (2-D) of fractures was calculated using thefollowing equation [22]:

Aex ¼2

pl2 ð4Þ

where Aex is the excluded area and l is the length of thefractures in the pattern. This equation is valid for isotropic(randomly and uniformly) oriented fracture patterns [23].

Next, the fracture density of each fracture pattern, which isdefined as the number of fractures per domain area, was calcu-lated using the following relationship suggested by Khamforoushet al. [13]:

r¼Nf r

L2ð5Þ

where r is fracture density, Nf r is the number of fractures inthe domain and L2 is the area of the domain, which was taken as100 m�100 m in this study.

Then, the dimensionless density of each fracture pattern wascalculated as follows [12,22]:

r0 ¼ rAex ð6Þ

Table 3Percolation proeprties of different fracture patterns.

Length Number of Fractures Excluded area (Aex) Fracture density (q) Dimensionless density (q0) (q0 �q0c) Permeability (mD)

20 50 254.777 0.005 1.274 -2.326 5.609

40 50 1019.108 0.005 5.096 1.496 77.643

60 50 2292.994 0.005 11.465 7.865 120.310

80 50 4076.433 0.005 20.382 16.782 185.112

20 100 254.777 0.01 2.548 -1.052 27.343

40 100 1019.108 0.01 10.191 6.591 155.048

60 100 2292.994 0.01 22.930 19.330 274.240

80 100 4076.433 0.01 40.764 37.164 384.561

20 150 254.777 0.015 3.822 0.222 74.672

40 150 1019.108 0.015 15.287 11.687 266.795

60 150 2292.994 0.015 34.395 30.795 424.930

80 150 4076.433 0.015 61.146 57.546 621.084

20 200 254.777 0.02 5.096 1.496 124.025

40 200 1019.108 0.02 20.382 16.782 395.592

60 200 2292.994 0.02 45.860 42.260 583.337

80 200 4076.433 0.02 81.529 77.929 825.885

20 250 254.777 0.025 6.369 2.769 161.214

40 250 1019.108 0.025 25.478 21.878 459.049

60 250 2292.994 0.025 57.325 53.725 696.129

80 250 4076.433 0.025 101.911 98.311 1018.037

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 20 40 60 80 100

FD o

f Int

erse

ctio

n Po

ints

Dimensionless Density (ρ')

Number of Fractures = 50Number of Fractures = 100Number of Fractures = 150Number of Fractures = 200Number of Fractures = 250

Fig. 1. Fractal dimension (FD) of fracture intersection points vs. dimensionless

density for different fracture densities and lengths.

1.21.41.61.82.0

ing

lines



It is necessary to mention that by using an excluded area todimensionless the fracture density, macroscopic properties offracture networks become independent of fracture shape asshown earlier [13,24–29,40–43].

For 2-D randomly oriented fracture sticks, the continuumpercolation threshold in terms of dimensionless density ðr0cÞ hasbeen determined to be 3.6 [23,2]. Since the purpose of this studywas to show any possible correlation between fractal–percolationproperties and equivalent fracture network permeability (EFNP)and not specifically the percolation threshold for these finite-sized fracture patterns, to eventually propose a practical methodto estimate EFNPs, this number was adopted and used throughoutthis study for the evaluation of the percolation properties ofdifferent fracture patterns. The results of the percolation analysisof different patterns are presented in Table 3. In this table, twofracture patterns ended up with a negative (r0 �r0c) value andthese permeability values were ignored. This can be attributed tothe finite size effect of the fracture patterns as explained by Kinget al. [16], i.e., it is possible to have connectivity and hencepermeability, at much lower values than the percolation thresh-old due to the random nature of the process.

0.00.20.40.60.81.0

0 10 20 30 40 50 60 70 80 90 100

FD o

f X-s

cann

(ρ'-ρ'c)

Fig. 2. Fractal dimension of X-scanning lines (in X-direction) vs. the difference

between dimensionless density and dimensionless percolation threshold (r0–r0c)for different fracture densities and lengths.

6. Analysis of the results and discussion

To estimate equivalent fracture network permeability throughEq. (3), we begin with the relationship between fractal andpercolation properties of generated fracture patterns. Fig. 1 showsthat as the fractal dimension of fracture intersection pointsincreases, the dimensionless density (Eq. (6)) increases for allfracture densities. The relationship is not linear and all the curvesconverge as the fractal dimension value approaches a fractaldimension value of two. Note that the dimensionless density isdependent on the number of fractures, fracture length, and alsothe shape and size of the domain. As expected, the fracturepatterns with a higher number of fractures have higher fractaldimensions and after a critical value of the dimensionless density,the fractal dimension of the intersection points do not changeregardless of the fracture density.

In Figs. 2 and 3, the fractal dimension obtained through thescanning lines technique in both the X- and Y-directions is plottedagainst the difference between dimensionless density and thedimensionless percolation threshold (r0 �r0c). The curves in these

plots show the same trend as in Fig. 1 because fractures wereplaced and oriented within the domain randomly and accordingto a uniform distribution and there is not any bias in theirorientation. As can also be inferred from these figures, an inverserelationship exists between the fractal dimension of scanninglines in the X- and Y-directions and (r0 �r0c). If fact, the moreuniformly and randomly oriented the fractures were in thedomain, the more similar the fractal dimensions of scanning linesin the X and Y-directions. In other words, if fractures wereoriented mostly in one direction, the fractal dimension of

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90 100

Con

nect

ivity

Inde

x

(ρ'-ρ'c)


Fig. 4. Connectivity index vs. the difference between dimensionless density and

dimensionless percolation threshold (r0–r0c) for different fracture densities and

lengths.

0.00.20.40.60.81.01.21.41.61.82.0

0 20 40 60 80 100

FD o

f Fra

ctur

e Li

nes

Dimensionless Density (ρ')


Fig. 5. Fractal dimension of fracture lines vs. dimensionless density for different

fracture densities and lengths.

1

10

100

1000

10000

1001010

K, m

D

(ρ'-ρ'c)


Fig. 6. Equivalent fracture network permeability vs. the difference between

dimensionless density and dimensionless percolation threshold (r0–r0c) for

different fracture densities and lengths.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 10 20 30 40 50 60 70 80 90 100

FD o

f Y-s

cann

ing

lines

(ρ'-ρ'c)


Fig. 3. Fractal dimension of Y-scanning lines (in Y-direction) vs. the difference

between dimensionless density and dimensionless percolation threshold (r0–r0c)for different fracture densities and lengths.


scanning lines in the X- and Y-directions would be different andas a result, they show a different trend against (r0 �r0c).

One may note that for a given number of fractures in thedomain, as the fracture length increases, the fractal dimension ofthe scanning lines in the X- and Y-directions decreases, especiallynear the percolation threshold at which (r0 �r0c) will experience acontinuous increase.

The connectivity index was plotted against (r0 �r0c) in Fig. 4.A linear relationship is obvious. Both connectivity index and(r0 �r0c) primarily depend on the number of fractures andfracture length as well as the domain size. For instance, for agiven number of fractures in the domain, connectivity index and(r0 �r0c) both increase by an increase in fracture length. It canalso be noticed that fracture patterns with a higher number offractures have higher (r0 �r0c) values.

Fig. 5 illustrates the change of the fractal dimension of fracturelines with the dimensionless density. The behavior is not linear,especially close to the percolation threshold. One may infer fromthis plot that as the fractal dimension becomes bigger andapproaches the value of two (this means that spatially thefractures are uniformly located within the entire domain), thechances of having a percolating network increases. As a result ofthis, the dimensionless density increases accordingly. Finally, athigher dimensionless densities, the curves converge, representinga case of very well developed fracture networks in the system.

Finally, the equivalent fracture network permeability wasplotted against (r0 �r0c) in Fig. 6. There is a non-linear relation-ship between these two parameters but an obvious trend isobserved. Therefore, the larger the dimensionless fracture densitydifference (due to the higher the number of fractures or the longerfractures in the domain), the higher the equivalent fracturenetwork permeability (EFNP) value, especially around the perco-lation threshold (low values of r0 �r0c). For a given number offractures, as fracture length increases, the (r0 �r0c) and EFNPincrease. At higher values of (r0 �r0c), all curves converge andfollow the same trend. To show this converging trend the pointswere connected, which is represented by solid lines.

The relationship between equivalent fracture network perme-ability vs. the difference between dimensionless density anddimensionless percolation threshold (r0 �r0c) for different frac-ture densities and lengths on these log–og plots could be shownas the following general form:

K ¼ Aðr0�r0cÞm

ð7Þ

where K is the equivalent fracture network permeability, A is theproportionality constant, m is the permeability exponent andfinally r0 and r0c are dimensionless density and percolationthreshold, respectively [36].

One possible application of the joint characterization of 2-Dfractal–percolation properties is that by correlating the fractaldimension of different features to (r0 �r0c) and then (r0 �r0c) tothe EFNP and deriving relationships, it would be possible toestimate the equivalent fracture network permeability of thefracture pattern of interest. In this case, the permeability estima-tion will be conditioned to both fractal and percolation propertiesof the systems which supposedly improve the accuracy of theestimation. Therefore, five parameters including the fractaldimension of intersection points (FDi), the fractal dimension of

Table 7Relationship between the fractal dimension of scanning lines in the X-direction

(FDx) for different numbers of fractures in the domain and (r0 �r0c) along with the

regression coefficient (R2). The first equation is for all fracture densities (number

of fractures).

Equation R2

FDx¼-0.119ln(r0 �r0c)þ1.4671 0.7761

FDx50¼-0.084ln(r0 �r0c)þ1.2126 0.9995

FDx100¼-0.136ln(r0 �r0c)þ1.472 0.9822

FDx150¼-0.139ln(r0 �r0c)þ1.5435 0.9945

FDx200¼-0.146ln(r0 �r0c)þ1.6103 0.9941

FDx250¼-0.131ln(r0 �r0c)þ1.5765 0.9973

Table 8Relationship between the fractal dimension of scanning lines in the Y-direction

(FDx) for different numbers of fractures in the domain and (r0 �r0c) along with the


of fractures).

Equation R2

FDy¼-0.114ln(r0 �r0c)þ1.4468 0.7876

FDy50¼-0.099ln(r0 �r0c)þ1.2411 0.9961

FDy100¼-0.139ln(r0 �r0c)þ1.4817 0.995

FDy150¼-0.128ln(r0 �r0c)þ1.5084 0.9912

FDy200¼-0.142ln(r0 �r0c)þ1.5952 0.9918

FDy250¼-0.122ln(r0 �r0c)þ1.5307 0.9956


fracture lines (FDl), Connectivity index (CI), fractal dimension ofscanning lines in the X-direction (FDx) and the fractal dimensionof scanning lines in the Y-direction (FDy) were plotted against (y).The equations for these correlations were presented in Tables4–8. Note that in each table the first equation is for all fracturecases, i.e., all cases with a different number of fractures were puttogether to derive this equation. The other five equations in thesetables were for different numbers of fractures, as shown byindices. If the number of fractures (in a sense, the fracturedensity) is known, one may select the particular equation corre-sponding to the known value of the number of fractures. Thisobviously increases the accuracy as indicated by a very highregression coefficient (the second column of the Table 4) except inthe case of the connectivity index value, which also yielded a veryhigh value of regression coefficient for the all fracture cases (givenas the first equation in the tables).

Finally, to obtain an equation (as similar to Eq. (7)) forestimating the EFNPs, the equivalent permeability of all fracturedensity cases (Nfr¼50, 100, 150, 200, and 250) was plottedagainst (r0 �r0c) and the result is shown in Fig. 7.

In this plot (fracture patterns with different number offractures), as fracture length increases, the (r0 �r0c) and accord-ingly the equivalent fracture network permeability increases. Twofracture permeability values were obtained as negative due tonegative values of (r0 �r0c), which could be attributed to the finitesize effect on the fracture patterns. Those negative values were

Table 4Relationship between the fractal dimension of intersection points (FDi) for

different numbers of fractures in the domain and (r0 �r0c) along with the


of fractures).

Equation R2

FDi¼0.0452Ln(r0 �r0c)þ1.7758 0.5408

FDi50¼0.0949ln(r0 �r0c)þ1.5632 0.992

FDi100¼0.0575ln(r0 �r0c)þ1.7362 0.9914

FDi150¼0.0336ln(r0 �r0c)þ1.8284 0.9989

FDi200¼0.0284ln(r0 �r0c)þ1.8579 0.9982

FDi250¼0.0221ln(r0 �r0c)þ1.884 0.9903

Table 5Relationship between the fractal dimension of fracture lines (FDl) for different

numbers of fractures in the domain and. (r‘-r‘c) along with the regression

coefficient (R2). The first equation is for all fracture densities (number of fractures).

Equation R2

FDl¼0.0526Ln(r0 �r0c)þ1.5901 0.6842

FDl50¼0.057ln(r0 �r0c)þ1.4764 0.9948

FDl100¼0.0653ln(r0 �r0c)þ1.5232 0.998

FDl150¼0.0381ln(r0 �r0c)þ1.6463 0.978

FDl200¼0.0478ln(r0 �r0c)þ1.6362 0.9993

FDl250¼0.0464ln(r0 �r0c)þ1.6537 0.9999

Table 6Relationship between the connectivity index (CI) for different numbers of

fractures in the domain and. (r0 �r0c) along with the regression coefficient (R2).

The first equation is for all fracture densities (number of fractures).

Equation R2

CI¼0.7003(r0 �r0c)þ2.0397 0.9991

CI50¼0.6662(r0 �r0c)þ2.6196 0.9984

CI100¼0.6755(r0 �r0c)þ2.3845 0.9981

CI150¼0.7018(r0 �r0c)þ1.9868 0.9979

CI200¼0.701(r0 �r0c)þ1.9081 0.9996

CI250¼0.7003(r0 �r0c)þ2.0397 0.9991

K = 88.411(ρ'-ρ'c)0.4602

R2= 0.8494

1

10

100

1000

10000

1001010

K, m

D

(ρ'-ρ'c)

Fig. 7. Equivalent fracture network permeability of all cases vs. the difference

between dimensionless density and dimensionless percolation threshold (r0–r0c).

ignored in this plot. In fact, this could be an indication that somesystems with finite size start percolating with much lowerprobability density (or any other equivalent parameter) than thepercolation threshold determined for infinite systems or even notto percolate at a much higher probability density than thepercolation threshold of infinite systems [16,21], due to therandom nature of fracture networks.

Also, by manipulation of Eq. (3), the following equation can bederived to determine the shortest (or minimum) fracture lengthrequired for a given specific number of fractures and domain sizeto be percolating:

lmin ¼ sqrtðpr0cL2=2Nf rÞ ð8Þ

where lmin is the shortest or minimum fracture length required tobe percolating, L2 is the area of domain and Nfr is the number offractures in the domain. These minimum fracture lengths werecalculated for a different number of fractures but the samedomain size and are given in Fig. 8.

As obvious from Fig. 8, the lower the number of fractures inthe domain, the longer the required fracture length. It is alsoessential to mention that this minimum required fracture lengthdepends on both the number of fractures and also the size of the


domain or system. This observation and Eq. (8) has practicalimportance to determine the shortest length of fracture to have apercolating (or conductive) fracture network system for a givenfracture density (number of fractures).

600

700

800

900

D

Using FD of intersection pointsUsing FD of fracture linesUsing Connectivity indexUsing FD of X-direction scanning linesUsing FD of Y-direction scanning lines

7. Validation of results

To validate the methodology introduced and the equations, aset of new fracture patterns with different random realizationswere considered. Their properties (fractal dimension of intersec-tion points, fractal dimension of fracture line, connectivity index,and also fractal dimension of scanning lines in the X- and Y-directions) were calculated and the equivalent fracture networkpermeability values for each pattern were computed using FRCAsoftware. The results are presented in Table 9.

Then, using the first equation of each table presented in Tables4–8, (r0 �r0c) of each fracture pattern was calculated. Here, weassumed that there is no information available on the number ofthe fractures in each case (pattern) and as a result the firstequation of each table, which represents an average values ofall fracture patterns, were used to estimate (r0 �r0c). Once again,

0

5

10

15

20

25

30

35

40

0 50 100 150 200 250 300Shor

test

Fra

ctur

e Le

ngth

to b

e Pe

rcol

atin

g, m

Number of Fractures

Fig. 8. Minimum required fracture length to be percolating versus number of

fractures.

Table 9Fractal dimension of different features of the fracture networks used for validation.


FD of intersectionpoints

FD of scanning linein X-direction

FDin

20 100 1.573 1.503 1.

20 200 1.875 1.396 1.

40 100 1.835 1.289 1.

40 200 1.950 1.304 1.

60 100 1.917 1.020 1.

60 200 1.969 1.028 1.

80 100 1.953 1.017 1.

80 200 1.972 0.975 0.

Table 10Calculated (r0 �r0c) values using different equations presented in Tables 4 through 8 an

given in Figure 15.


(q0 �q0c)using FDi

K (q0 �q0c)using FDl

K

20 100 0.011 11.239 0.197 41.895

20 200 8.968 242.622 4.074 168.743

40 100 3.686 161.154 3.338 153.959

40 200 47.576 522.925 28.341 412.008

60 100 22.677 371.834 10.760 263.846

60 200 72.350 634.189 71.112 629.174

80 100 50.069 535.360 27.061 403.341

80 200 76.079 649.030 123.890 812.309

if the number of fractures in the domain is known, one can usethe related equation for that specific number of fractures (asshown by indices) instead of using the general equation (the firstequation in each table). In this case, the error of the estimated(r0 �r0c) would be much lower. Next, these values were pluggedinto the equation given in Fig. (7) and the equivalent permeabilityof each pattern was computed. The results are presented inTable 10, for five different fracture network properties. In thistable, the value of (r0 �r0c) obtained using the connectivity indexin the first row was negative; thus, the permeability calculationwas omitted.

A comparison of the predicted and actual (FRACA values)EFNPs is given in Fig. 9. Although the general equation for(r0 �r0c) was used for each case (given as the first equation in

of scanning lineY-direction

Connectivityindex

FD of fracturelines

Permeability(mD)

527 1.730 1.505 27.346

520 2.700 1.664 132.320

275 6.180 1.654 150.122

288 13.530 1.766 413.981

017 16.570 1.715 298.754

020 31.730 1.814 552.036

004 25.550 1.764 367.481

977 57.605 1.844 858.655

d their corresponding equivalent fracture network permeability using the equation

(q0 �q0c)using CI

K (q0 �q0c)using FDx

K (q0 �q0c)using FDy

K

-0.442 - 0.739 76.914 0.497 64.072

0.943 86.050 1.819 116.430 0.525 65.734

5.912 200.292 4.485 176.383 4.499 176.626

16.408 320.384 3.939 166.146 4.034 167.974

20.749 356.932 42.723 497.662 43.400 501.278

42.397 495.911 40.083 483.266 42.329 495.547

33.572 445.410 43.825 503.532 48.834 529.245

79.345 661.706 62.458 592.700 61.598 588.932

0

100

200

300

400

500

0 100 200 300 400 500 600 700 800 900

Pred

icte

d K

, m

Actual K, mD

Fig. 9. Predicted equivalent fracture network permeability vs. the actual perme-

ability obtained using FRACA software.


Tables 4–8), the correlation between the predicted and actualvalues of EFNPs is very strong. It is interesting to note that thebest correlation was obtained when the fractal dimensions offracture lines were used to estimate the (r0 �r0c) values for anyfracture length and density values. Corresponding regressioncoefficients for each case are given in Table 11. To improve theaccuracy of estimation one may choose an average value of EFNPsestimated using these five (or fewer) network properties. If onehas to choose only one property, the fractal dimension of fracturelines obtained through the box counting method would yield themost accurate estimation. The correlation between these twoEFNP values is reasonably good, indicating that the approachpresented can be used for practical purposes.

8. Conclusions and remarks

The relationships between different fracture network proper-ties, i.e., fractal and percolation properties, and their equivalentfracture network permeability were investigated and analyzed. Itwas found that a nonlinear-direct relationship exists between thefractal dimension of fracture intersection points and fracture lines(using the box counting technique) and the dimensionless den-sity. In these cases, at close to percolation threshold, patternswith different number of the fractures have different fractaldimension of intersection points and fracture lines; however, athigher dimensionless density, all curves relating to differentnumbers of fractures diverge into a single value, which is themaximum fractal dimension of two in each case. This is attributedto an increase in the number of the fractures and also the fracturelength in the domain, which eventually causes an increase in bothfractal dimension and dimensionless density.

The connectivity index shows a direct relation against (r0 �r0c)because both are mostly being controlled by the number of thefractures, the fracture length and the domain size. It was alsoshown that there is a non-linear inverse relationship between thefractal dimension of scanning lines in the X- and Y-directions and(r0 �r0c). While (r0 �r0c) continuously increases with an increasein fracture length for a given number of fractures, the fractaldimension of scanning lines in the X- and Y-directions decreasesstarting at the percolation threshold.

Correlations between equivalent fracture network permeabil-ity and (r0 �r0c) for different number of fractures (fracturedensity) were developed and for these finite size fracturepatterns, a permeability exponent (

R) of 0.4602 was found.

The relationship between the number of fractures in thedomain and the minimum size of the fracture length waspresented to have a percolating system for 2-D randomlydistributed fractures. A correlation was introduced, for a givenfracture domain, to quickly estimate the possibility of connectivity(percolation).

A methodology and correlations were presented to estimatethe equivalent fracture network permeabilit‘ty (EFNP) of 2-Dfracture networks using fractal properties of 2-D fracture

Table 11Correlation coefficients for the comparison of

actual and predicted EFNPs (Figure 14) obtained

through five different fractal-statistical properties.

(q0 �q0c) obtained using: R2

FD of intersection points 0.81

FD of fracture lines 0.98

Connectivity index 0.9

FD of X-direction scanning lines 0.61

FD of Y-direction scanning lines 0.59

networks and percolation properties, i.e., percolation densityand threshold. The validation exercise confirmed that the techni-que can be used to estimate the EFNP practically. The fractaldimension of fracture lines (obtained using the box countingtechnique) yielded a more accurate estimation compared to thefractal dimension of intersection points, connectivity index, andfractal dimension of scanning lines in X- and Y-directions used tocalculate the (r0 �r0c) values and the EFNP from it.

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Documents

Relationship between percolation–fractal properties and permeability of 2-D fracture networks