Computers and Geotechnics 76 (2016) 83–92

Contents lists available at ScienceDirect

Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Research Paper

Estimation of the REV size for blockiness of fractured rock masses

http://dx.doi.org/10.1016/j.compgeo.2016.02.0160266-352X/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: yuqch@cugb.edu.cn (Q. Yu).

Lu Xia, Yinhe Zheng, Qingchun Yu ⇑School of Water Resources and Environment, China University of Geosciences, Beijing, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 August 2015Received in revised form 19 February 2016Accepted 21 February 2016Available online 5 March 2016

Keywords:Fractured rock massRepresentative elementary volumeRock blockBlockiness

The representative elementary volume (REV) is the premise of the continuous-media method of analysis,and the investigation of the REVs of fractured rock masses is a fundamental area of rock mechanicsresearch. The existence of an REV can be determined based on a variety of physical parameters. Thispaper presents an analysis of the REV from the view of blockiness, which is defined as the percentageof the volume of isolated blocks formed by fractures in the total rock volume. Seventy-seven types of frac-tured rock mass models were developed based on 7 classes of fracture persistence and 11 classes of spac-ing that are suggested by the International Society for Rock Mechanics (ISRM) fracture classification. Rockblocks in each of the 77 types of fractured rock mass models were identified using GeneralBlock to deter-mine the variation in blockiness with model domain sizes, which were changed from 2 to 20 times thefracture spacing. For each model domain size 9 random realizations were carried out to reduce the effectsof randomness. The coefficient of variation (Cv) was then used to quantify the variability of the 9 randomrealizations. The fluctuation in blockiness with the variation in the scale of the model region was alsoinvestigated. In this way, the size of the REV in these models can be calculated using the average andthe variance of the blockiness as indicators of the convergence. The blockiness of these fractured rockmasses can be determined at the REV volume. The results indicate that of the 77 models, 76 REV sizesare between 2 and 20 times the fracture spacing. The fractured rock mass with a wide fracture spacingand very high persistence (WS2–VHP) has a REV size that exceeds 20 times the fracture spacing. Thus,the WS2–VHP model should be investigated further to validate this concept.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of the representative elementary volume (REV) isindispensable to the understanding of fractured rock masses, andthe existence of an REV is the premise of the continuous-mediamethod. For example, the representativeness of the permeabilitytensor and the applicability of the theory of porous media to a rockmass depend on the existence of an REV of the rock mass.

The REV concept is fundamental to the study of fractured rockmasses. The concept has therefore attracted a great deal of atten-tion in the field of geotechnical engineering and has been discussedfrom various points of view in many publications [4,27,32,37,33,42,22,44,51,46]. Bear [4] defined the REV using the concept ofporosity. Fig. 1 presents Bear’s definition of porosity and the repre-sentative elementary volume. The parameters of a rock mass oftenfluctuate dramatically with increases in the volume. The volume atwhich the parameter of interest ceases to vary is defined as theREV.

Long [27] and Wang and Kulatilake [42] discussed the existenceof the REV and its size based on an analysis of the permeability ten-sor. Long [27] indicated that increased fracture densities increasethe possibility that the REV exists. Wang and Kulatilake [42] pre-sented a new method of determining the REV and the three-dimensional hydraulic conductivity tensor of a fractured rockmass. The hydraulic conductivities in different directions do notchange with the block size (in a practical sense) at block sizesgreater than approximately 12.5 m. Wang suggested that the REVof a gneissic rock mass is approximately 12.5 m. Xiang and Zhou[50] adopted a two-dimensional rock model based on the finite-element method to obtain the equivalent elastic modulus of therock mass under uniaxial loading. Based on variations in the equiv-alent elastic modulus with the size of the rock mass, the REV size ofthe rock mass was estimated at 9 m � 9 m. Zhang and Xu [55]defined an index using the generated joint network to determinethe scale of the REV of the rock mass based on the geometric andmechanical parameters of the fractures. Their research indicatesthat the REV scale is approximately three to four times the maxi-mum expected value of the trace lengths of the joint sets in therock mass.

Fig. 1. Variation in porosity as a function of the representative elementary volume [4].

84 L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92

Some models (such as most finite-element models) treat rockmasses as continuums [10,2,3,13,30,15,29], whereas others (suchas those that are based on the discrete element method or discon-tinuous deformation analysis) treat them as isolated blocks[9,39,28,45,25]. For example, Kalenchuk et al. [21] proposed theblock shape characterization method, which is used effectively todescribe and classify the size and shape distributions of any jointedrock mass. Previous studies have encountered two basic problems:how to measure the rock’s level of being an assemblage of isolatedblocks and how to determine whether descriptions and statistics ofrock blocks are representative. Sometimes there are many discon-tinuities in the rock mass and it is treated as an assemblage of iso-lated blocks. In contrast, the total rock mass has very goodintegrity, the number of blocks in the rock mass is quite small,the volume of isolated blocks accounts for only a very small pro-portion of the total rock volume, and the rock mass can be consid-ered a continuum. Treating a rock mass as a continuum or as anassemblage of isolated blocks is not only a problem of model treat-ment, but also a fundamental question in rock mechanics. In fact,most rock masses are neither continuums nor isolated blocks butinstead are somewhere between these extremes. Therefore, furtherstudy is needed to investigate the blockiness level of rock masses.

The existence of an REV and its size can be investigated from theview of blockiness of fractured rock masses. In this study, we usedthe blockiness of a fractured rock mass to measure its level of beingan assemblage of isolated blocks. We adopted the general blockmethod [54] to build models using the GeneralBlock program. Sec-ond, we built 77 fractured rock mass models based on the fractureclassifications of the International Society for Rock Mechanics [19]and calculated the blockiness of each model. Nine random realiza-tions were carried out for each model domain size to reduce theeffects of randomness. The blockiness of each model and the sizeof the REV can be calculated by investigating how the blockinessvaries with the model domain size. Xia et al. [46] had made a com-parison of the theoretical to one actual fractured rock masses of thepowerhouse. Xia et al. [46] found that the blockiness of the sur-rounding rock mass of the underground powerhouse is only 4‰;the total rock mass shows very good integrity, and the number ofblocks in the rock mass is quite small. Therefore, the volume of iso-lated blocks accounts for only a very small proportion of the totalrock volume, and the rock mass can be considered a continuum.The excavation of the underground powerhouse in the ThreeGorges Project verified these conclusions; only a few blocks wereformed by fractures in the rock mass of the undergroundpowerhouse. The calculation results of the theoretical modelsshow that the blockiness of the rock mass with medium persis-tence and very wide spacing is very low. The structure of the rockmass around the underground powerhouse is of this type. Its

blockiness is consistent with the calculation results of the model.Therefore, the results are useful for predicting the stability of rockmasses with different fracture persistence and spacing, and couldserve as a reference for quantifying the quality of rock masses inother projects.

2. Methods

2.1. Concept of blockiness

This study investigated the existence and size of the REV of arock mass based on blockiness. The block percentage (B) servesas an index for the blockiness of the rock mass [46]. It is the per-centage of the volume of isolated blocks formed by fractures inthe total rock volume to measure rock’s level of being an assem-blage of isolated blocks.

If we assume that the model contains n blocks, then the block-iness is calculated using the equation

B ¼Pn

i¼1v i

V� 100% ð1Þ

where V is the total volume of the rock mass and vi is the volume ofblock i. If the blockiness is close to 100%, it indicates well-developeddiscontinuities in the rock and severe rock fragmentation; there-fore, the rock is treated as an assemblage of isolated blocks. Onthe other hand, when the blockiness is close to 0%, it indicates thatthe rock has few discontinuities and therefore is considered to havegood integrity.

2.2. Theory of the general block method

The key block method was proposed by Warburton [43] andGoodman and Shi [16] and was later developed by others [26,18]has been recognized as a powerful tool for addressing block stabil-ity issues. These studies, which involved the identification ofblocks formed by finite fractures, revealed that fractures may ormay not contribute to block formation. This distinguishes themfrom the block generation language model [17,12], in which frac-tures always form a fully connected network, and all fractures con-tribute completely.

This study is based on the theory of the general block method[54,60]. This method is recognized as a generalized procedure foridentifying rock blocks formed by finite fractures around complexexcavations. It was assumed that the study domain could be parti-tioned into a finite number of sub-domains, each of which eitherwas a convex polyhedron or could be approximated as such. Fur-thermore, fractures were finite in size and disc-shaped and were

L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92 85

defined using the location of the disc center, orientation, radius,cohesion coefficient, and friction angle. The fractures may be eitherdeterministic ones obtained from a field survey or random onesgenerated by stochastic modeling. In addition, the rock mass couldbe heterogeneous, i.e., the rock matrix and individual fracturescould have different parameters in different parts. Our procedureinvolved (1) partitioning the model domain into convex sub-domains; (2) removing non-contributive fractures – a fracturewas deemed contributive when it played a part in block formation,i.e., it formed at least one surface of certain blocks; (3) decompos-ing the sub-domains into element blocks that are formed by frac-tures; (4) restoring the infinite fractures into finite disks; and (5)assembling the modeling domain. This procedure facilitates robustcomputational programming and can flexibly address the prob-lems of complex study domains and rock heterogeneity.

2.3. Software and parallel computing

The analysis was conducted using GeneralBlock [47,53], whichis a C++ computer program that is based on the general blockmethod to a fractured rock mass. This program can be used to iden-tify and analyze rock blocks formed by finite fractures and analyzethe blocks of complexly shaped modeling domains, such as slopes,tunnels, underground caverns, or their combinations. GeneralBlockhas been widely used by researchers [57,42,59,46]. Xia [48,49]used the theory of the general block method to develop a 3D blockmodel of the underground powerhouse of the Three Gorges Project.Fig. 2 shows a graphical display of results of the block analysis,specifically, the major blocks in the roof of the undergroundpowerhouse.

Fig. 2. Results of the block analysis using GeneralBlock depicting the major bloc

To characterize the size of the REV, it is important to be able toperform computations with a very large number of blocks. Thetechnology of parallel computing is widely used to speed upnumerical calculation [61,14]. And OpenMP [7,58,38] is the mostcommonly used parallel programming model for shared memory.Using OpenMP directives, it is capable of parallelizing an algorithmof this work. Therefore, we used the general block method to buildmodels of fractured rock masses, and determined their blockinessusing the upgraded GeneralBlock program.

3. Building models of fractured rock masses

The majority of rock masses behave discontinuously, with thediscontinuities largely determining the mechanical behavior. Inrock masses with three groups of disk fractures that intersect eachother, the fractures cut the rock mass into blocks. Individual dis-continuities may further affect the size and shape of the blocks.Therefore, the fracture is the controlling factor for the blockinessof the rock mass, and the geometrical parameters of the fracturehave different influences on the blockiness. If there are many dis-continuities in the rock mass, its blockiness level is very high andit can be treated as an assemblage of isolated blocks. In contrast,the rock mass may exhibits very good integrity when it containsonly a small number of blocks, and its blockiness level is verylow and it can be considered as a continuum.

Blocks are formed by the intersections of discontinuities in arock mass. Therefore, the characterization of discontinuities is animportant part of the engineering characterization of rock masses.When building a three-dimensional model, the fracture parame-ters must be addressed. The discontinuity geometry is primarilycharacterized by the location, orientation, spacing, and persistence

ks in the roof of the underground powerhouse of the Three Gorges Project.

Table1

Spacing,

persistenc

ean

dthree-dimen

sion

alde

nsity(d

3)of

thefracturesin

the77

fracturedrock

mod

els(D

=pe

rsistenc

e;C=sp

acing).

Extrem

elyclos

esp

acing(ECS

)C=0.02

m

Veryclos

esp

acing(V

CS1

)C=0.04

m

Veryclos

esp

acing(V

CS2

)C=0.06

m

Close

spacing

(CS1

)C=0.13

m

Close

spacing

(CS2

)C=0.2m

Mod

erate

spacing(M

S1)

C=0.4m

Mod

erate

spacing(M

S2)

C=0.6m

Widesp

acing

(WS1

)C=1.3m

Widesp

acing

(WS2

)C=2m

Verywide

spacing

(VW

S)C=4m

Extrem

elywide

spacing(EW

S)C=6m

Veryhighpe

rsistence

(VHP)

D=20

.0m

0.15

920.07

960.05

310.02

450.01

590.00

800.00

530.00

240.00

160.00

080.00

05Highpe

rsistence

(HP)

D=15

.0m

0.28

290.14

150.09

430.04

350.02

830.01

410.00

940.00

440.00

280.00

140.00

09Med

ium

persistence

(MP2

)D=10

m0.63

660.31

830.21

220.09

790.06

370.03

180.02

120.00

980.00

640.00

320.00

21Med

ium

persistence

(MP1

)D=6.5m

1.50

680.75

340.50

230.23

180.15

070.07

530.05

020.02

320.01

510.00

750.00

50Lo

wpe

rsistence

(LP2

)D=3.0m

7.07

363.53

682.35

791.08

820.70

740.35

370.23

580.10

880.07

070.03

540.02

36Lo

wpe

rsistence

(LP1

)D=2.0m

15.915

57.95

775.30

522.44

851.59

150.79

580.53

050.24

490.15

920.07

960.05

31Verylow

persistence

(VLP

)D=1.0m

63.662

031

.831

021

.220

79.79

426.36

623.18

312.12

210.97

940.63

660.31

830.21

22

86 L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92

of discontinuities. The block dimensions are typically controlled bythe spacing and persistence of the discontinuities and by the num-ber of sets. However, to obtain workable solutions, only the spacingand persistence of the fractures were considered in building themodels in this study. Knowledge of the fracture density is neces-sary to determine the number of fractures within a specified vol-ume of the rock mass. As the persistence of discontinuitiesincreases for a given fracture density, more blocks are formed inthe rock mass. However, for a constant persistence of discontinu-ities, higher densities result in more blocks of smaller size. In thisstudy, three definitions of fracture density were used: the one-dimensional density (d1), two-dimensional density (d2), andthree-dimensional density (d3). Of these three densities, d1 andd2 can be measured in situ, whereas the three-dimensional densityis more difficult to determine. The fracture shape and size mustalso be addressed. Fractures have been assumed to be disk-shaped, elliptical, or polygonal. Generally, a polygonal model oran elliptical model incorporates more geometric coefficients thanthe more common disk model. Consequently, the disk assumptionwas adopted in this study.

Assuming that a fracture is disk shaped, the relationshipbetween d1 and d3 is

d3 ¼ 4d1

pEðD2Þ ð2Þ

where d1 is the number of fractures encountered by a unit length ofborehole drilled along the mean direction of the fracture unit nor-mal; d3 is the number of fracture centers within a unit volume ofthe rock mass; and E(D2) is the mean value of the squared fracturediameter.

According to this relationship, d3 can be inferred from d1. Underthe assumption that fractures are disc-shaped, the probability dis-tribution of the fracture diameter can be inferred from the distri-bution of the trace lengths [23,56,35], which can be measured onan excavated surface or natural outcrop. However, direct estima-tion of the trace length is often affected by various biases. Thisproblem is sufficiently difficult to require the adoption of a seriesof simplifying assumptions.

In the three-dimensional fracture models in this study, the fol-lowing assumptions were made:

1. The models contain three mutually orthogonal fracture sets.2. All of the fractures of each set are parallel and have the same

persistence in each model.3. The fractures are randomly located in the model domain; i.e.,

the coordinates of the fractures have uniform distributions.

The number of fractures in a rock mass is typically very largeeven at a small scale. Of the large number of fractures, only a verysmall portion can be measured or observed in outcrops, excava-tions or boreholes. Thus, these assumptions were used to increasethe computational efficiency. The geometric parameters of thefractures are usually treated as random variables following astochastic distribution of some form. In this study, the most com-mon and simplest block type was used, which is an approximatelyparallel hexahedron block that is formed by three sets of steeplydipping fractures [52,20,24,1].

ISRM [19] suggested that fractures may be assigned to fiveclasses of persistence and seven classes of spacing. Using theboundaries and median values of each class as the typical values,a variety of fractured rock masses that contain fractures of variouslengths and spacing can be analyzed. For example, low persistencefractures range from 1 to 3 m in length; the boundary values are 1and 3 m, and the median is 2 m. Similarly, wide spacing can rangefrom 600 to 2000 mm, with boundaries of 600 and 2000 mm and a

(a) VCS1 - VLP model dimensions: 0.4 m×0.4 m×0.4 m.

(b) VCS1 - VHP model dimensions: 0.6 m×0.6 m×0.6 m.

(c) VWS - HP model dimensions: 43 m×43 m×43 m.

Fig. 3. Three 3D fracture networks produced by different sets of parameters.

(a) VCS1- VLP model dimensions: 0.4 m×0.4 m×0.4 m.

(b) VCS1- VHP model dimensions: 0.6 m×0.6 m×0.6 m.

(c) VWS - HP model dimensions: 43 m×43 m×43 m.

Fig. 4. Three 3D models of rock masses produced by different sets of parameters.

L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92 87

median of 1300 mm. Table 1 shows the spacing, extent and three-dimensional density of the fractures (d3) in the 77 fractured rockmodels. In this study, the persistence of the fractures was dividedinto 7 classes: very low persistence (VLP), low persistence of twoclasses (LP1 and LP2), medium persistence of two classes (MP1and MP2), high persistence (HP), and very high persistence(VHP). The fracture spacing was divided into 11 classes: extremelyclose spacing (ECS), very close spacing of two classes (VCS1 andVCS2), close spacing of two classes (CS1 and CS2), moderate spac-ing of two classes (MS1 and MS2), wide spacing of two classes(WS1 and WS2), very wide spacing (VWS), and extremely widespacing (EWS). Therefore, based on these parameters in Table 1, 77fractured rock mass models were constructed with fractures of var-ious lengths and spacing. It is likely that these types cover all actualrock masses.

Fig. 3 shows 3 representative 3D fracture networks with differ-ent parameters. Fig. 3(a) shows a 3D fracture network with veryclose spacing (VCS1; C = 0.04 m) and very low persistence (VLP;D = 1.0 m), Fig. 3(b) shows a network with very close spacing(VCS1; C = 0.04 m) and very high persistence (VHP; D = 20 m),and Fig. 3(c) shows a network with very wide spacing (VWS;C = 4 m) and high persistence (HP; D = 15 m).

4. Blockiness analysis and estimation of the REV size

For each of the 77 types of fractured rock mass models, rockblocks were identified using GeneralBlock to determine thevariation in blockiness with the model domain size. Fig. 4 showsthe blocks that are formed by the fracture networks in the modelsfrom Fig. 3. Clearly, the model in Fig. 4(a) contains many rockblocks, with a blockiness of almost 100%. This result indicates

well-developed discontinuities in the rock and severe rock frag-mentation; thus, the rock can be treated as an assemblage of iso-lated blocks. In contrast, Fig. 4(c) shows a rock mass thatcontains few discontinuities and very few blocks and thus has verygood integrity. The blockiness of this rock mass is approximately0%, and the rock displays good integrity.

For each model domain size, the model was realized 9 times toreduce the effects of randomness. Fig. 5 shows the relationshipsbetween the blockiness and domain size, specifically, the relation-ships between B and domain size in twelve typical models. Theblockiness is plotted on the vertical axis, and the domain size isplotted on the horizontal axis. The domain sizes are changed from2 to 20 times the fracture spacing, and the curve represents theaverage of the 9 random realizations. Fig. 5(a) shows the resultsof the model with moderate spacing and very high persistence(MS1–VHP), and Fig. 5(b) shows the results of the model with mod-erate spacing and high persistence (MS1–HP). Fig. 5(c) shows theresults of the model with moderate spacing and medium persis-tence (MS1–MP2), and Fig. 5(d) shows the results of themodel withmoderate spacing and medium persistence (MS1–MP1). Fig. 5(e)shows the results of the model with moderate spacing and low per-sistence (MS1–LP2), and Fig. 5(f) shows the results of the modelwith moderate spacing and low persistence (MS1–LP1). Fig. 5(g)shows the results of the model with moderate spacing and verylow persistence (MS1–VLP), and Fig. 5(h) shows the results of themodel with extremely close spacing and medium persistence(ECS–MP2). Fig. 5(i) shows the results of the model with moderatespacing and medium persistence (MS2–MP2), and Fig. 5(j) showsthe results of themodel with wide spacing andmedium persistence(WS1–MP2). Fig. 5(k) shows the results of the model with widespacing and medium persistence (WS2–MP2), and Fig. 5(l) shows

L/C L/C

(a) MS1 - VHP (b) MS1 - HP

L/C L/C

(c) MS1 - MP2 (d) MS1 - MP1

L/C L/C

(e) MS1 - LP2 (f) MS1 - LP1

L/C L/C

(g) MS1 - VLP (h) ECS - MP2

B B

B B

B B

B B

Fig. 5. Relationships between B and the domain size in 12 typical models (B: blockiness; L: model domain size; C: spacing).

88 L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92

the results of themodel with verywide spacing andmediumpersis-tence (VWS–MP2).

Fig. 5 clearly demonstrates that the law of variation that con-trols the blockiness of the rock mass is the same as that controllingthe porosity that was discussed by Bear [4]. When these rock mass

models are used to model small volumes, the blockiness often fluc-tuates dramatically. The volume at which the blockiness reaches aconstant value is defined as the REV. Further obvious blockinessvariation with the volume exceeding the REV indicates that themedium is heterogeneous. In this study, the fractures in the models

L/CL/C

L/C

(i) MS2 - MP2

L/C

(k) WS2 - MP2

(j) WS2 - MP2

(l) VWS - MP2

B B

B B

Fig. 5 (continued)

B

Stan

dard

dev

iatio

n

L/C

0.1757

Fig. 6. Standard deviation of the blockiness of the 9 realizations of the MS1–LP2model with increasing domain size (B: blockiness; L: model domain size; C:spacing).

L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92 89

are assumed to be uniformly distributed, and the blockiness isassumed to remain constant at volumes that exceed the REV.

Previous studies of the variations of properties with sample sizein porous media have been conducted using both real rocks andnumerical rock models at various scales. These studies focusedon different properties, rock types, measurement techniques, andsample sizes. Several authors [8,34,5,31,6,41] concluded that thecoefficient of variation (Cv) can be used to quantify the variabilityat any particular sample volume for several realizations. Therefore,when Cv remains in the homogeneous range (0 < Cv < 0.5), a repre-sentative effective property has developed, and the correspondingsample volume approximates the REV. Nordahl et al. [34] proposeda method of REV estimation in realistic geological models in whichheterogeneities are explicitly included. They studied the perme-ability as a function of the model volume and used the homogene-ity definition of Corbett and Jensen [8] to define their REV. For eachmodel, the REV was defined as the volume at which the value of Cv

was less than 0.5. The size of the REV was found to depend on boththe property measured (vertical and horizontal permeability) andthe correlation lengths of the lithologic elements.

Therefore, the average and variance of the blockiness valueswere used in this study as indicators of the convergence, and Cv

was used to quantify the variability of the 9 random realizations.The standard deviation of the blockiness values of the 9 realiza-tions must be less than 50% of the average; the domain size usedin the calculations can be defined as above or as the size of theREV. The smallest domain size that complies with these criteriais the minimum REV size.

The moderate spacing and low persistence (MS1–LP2) model isused as an example. Fig. 6 illustrates the standard deviation of theblockiness values of the 9 realizations as the model domain sizeschange from 2 to 20 times the fracture spacing. As shown in thefigure, at 12 times the fracture spacing, the standard deviation ofthe blockiness values of the 9 realizations is approximately 0,

and the variance of the blockiness becomes constant. Therefore,the size of the REV of the fractured rock mass is reached at 12 timesthe fracture spacing, which is 4.8 � 4.8 � 4.8 m3. Fig. 6 shows that

Table 2REVs of the 77 fractured rock mass models (units are multiples of the fracture spacing).

Extremely closespacing (ECS)C = 0.02 m

Very closespacing (VCS1)C = 0.04 m

Very closespacing (VCS2)C = 0.06 m

Close spacing(CS1)C = 0.13 m

Close spacing(CS2)C = 0.2 m

Moderatespacing (MS1)C = 0.4 m

Moderatespacing (MS2)C = 0.6 m

Wide spacing(WS1)C = 1.3 m

Wide spacing(WS2)C = 2 m

Very widespacing (VWS)C = 4 m

Extremelywide spacing(EWS) C = 6 m

Very high persistence (VHP) D = 20.0 m 6 4 4 6 4 10 4 14 – 6 4High persistence (HP) D = 15.0 m 6 4 4 4 8 10 6 14 18 6 2Medium persistence (MP2) D = 10 m 6 4 6 8 8 6 12 18 8 2 2Medium persistence (MP1) D = 6.5 m 6 6 4 8 6 10 18 6 6 2 2Low persistence (LP2) D = 3.0 m 8 6 8 8 12 12 6 2 2 2 4Low persistence (LP1) D = 2.0 m 6 8 6 10 20 6 2 2 2 4 4Very low persistence (VLP) D = 1.0 m 6 4 8 18 6 2 2 2 4 4 4

Table 3Blockiness of the 77 fractured rock mass structural models.

Extremelyclose spacing(ECS) C = 0.02 m

Very closespacing (VCS1)C = 0.04 m

Very closespacing (VCS2)C = 0.06 m

Close spacing(CS1)C = 0.13 m

Close spacing(CS2)C = 0.2 m

Moderatespacing (MS1)C = 0.4 m

Moderatespacing (MS2)C = 0.6 m

Wide spacing(WS1)C = 1.3 m

Wide spacing(WS2)C = 2 m

Very widespacing (VWS)C = 4 m

Extremelywide spacing(EWS) C = 6 m

Very high persistence (VHP) D = 20.0 m 98.72% 98.27% 98.45% 97.94% 97.37% 97.04% 95.23% 90.85% – 5.82% 1.93%High persistence (HP) D = 15.0 m 97.63% 98.29% 98.43% 97.46% 97.53% 96.93% 93.15% 70.20% 15.65% 1.27% 0.31%Medium persistence (MP2) D = 10 m 97.92% 98.26% 97.49% 96.87% 95.23% 93.44% 88.81% 18.19% 5.90% 0.25% 0.07%Medium persistence (MP1) D = 6.5 m 98.24% 97.33% 96.90% 97.45% 94.27% 88.67% 51.83% 6.46% 0.61% 0.11% 0.03%Low persistence (LP2) D = 3.0 m 97.43% 97.12% 96.97% 94.65% 78.78% 17.57% 4.39% 0.26% 0.05% 5.49E�03% 5.22E�07%Low persistence (LP1) D = 2.0 m 97.95% 95.31% 95.25% 88.15% 34.66% 6.84% 0.96% 0.14% 0.02% 1.20E�06% 1.78E�08%Very low persistence (VLP) D = 1.0 m 94.82% 92.19% 91.05% 19.99% 6.23% 0.18% 0.01% 5.13E�03% 1.52E�07% 1.03E�05% 1.74E�07%

90L.X

iaet

al./Computers

andGeotechnics

76(2016)

83–92

L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92 91

the blockiness of the MS1–LP2 model at the size of the REV is17.57%.

For each model, the blockiness varies markedly at volume scalesof less than twice the fracture spacing and tends to plateau at 4times the fracture spacing. The blockiness of most of the modelsbarely varies at volumes exceeding 20 times the fracture spacing.Table 2 presents the REVs of the 77 rock mass models. The sizesof the REVs of 76 of the fractured rock masses are between 2 and20 times the fracture spacing, and no REV volumes are above 20times the spacing. For the one remaining model, the standard devi-ation of the blockiness of the 9 realizations at 20 times the fracturespacing varies between 10% and 20%, although the coefficient ofvariation is less than 0.5. Thus, this model has an REV, but its sizeexceeds 20 times the fracture spacing. The blockiness of this typeof fractured rock mass will be discussed in a future study followingfurther development of the necessary computer hardware andsoftware.

The blockiness of these fractured rock masses can be deter-mined at the volume of the REV, as shown in Table 3. The resultsshow that 36 blockiness values exceed 90%, 3 are between 80%and 90%, 2 are between 60% and 80%, 6 are between 10% and60%, 8 are between 1% and 10%, and the remaining 21 are equalto 0. A blockiness value close to 100% indicates that the rock shouldbe treated as an assemblage of isolated blocks. On the other hand,when the blockiness is close to 0%, the rock is considered to exhibitgood integrity.

5. Conclusions

The presence of a representative elementary volume (REV) isthe premise of the continuous-media method. Therefore, the REVconcept plays a central role in the mechanics and physics of ran-dom heterogeneous materials and in efforts to predict their effec-tive properties. Many researchers have studied the existence andsize of the REV using various methods. We investigated the exis-tence of the REV based on the blockiness, which is defined as thepercentage of the volume of isolated blocks formed by fracturesin the total rock volume.

In this paper, 77 rock mass models were developed based on 7classes of persistence and 11 classes of spacing suggested in theISRM fracture classification. The blocks in each of the 77 types offractured rock mass models were identified using GeneralBlockto determine the variation in blockiness with model domain size,which changes from 2 to 20 times the fracture spacing. For eachmodel, 9 realizations were carried out to reduce the effects of ran-domness. The block calculation is complex and time-consuming.Therefore, a revised GeneralBlock program with parallel comput-ing was developed and used for the block calculations.

The coefficient of variation (Cv) was used to quantify the vari-ability of the 9 random realizations. When the standard deviationof the blockiness values of the 9 realizations is very small andthe variance of the blockiness becomes constant, the size of theREV of the fractured rock masses can be calculated. The blockinessof these fractured rock masses can be determined at the volume ofthe REV.

The results indicate that in 76 of the 77 models, the size of therepresentative elementary volume of the fractured rock mass isbetween 2 and 20 times the fracture spacing. The size of the REVof the fractured rock mass with wide spacing and very high persis-tence (WS2–VHP) exceeds 20 times the fracture spacing. Therefore,the WS2–VHP model should be further investigated to validate thisconcept, and finer classifications of persistence and spacing shouldbe developed to calculate its blockiness in a future study.

This methodology can be used to predict the integrity of rockmasses that have different persistence and spacing and could serve

as a reference for quantifying the quality of rock masses in certainprojects. However, the quality of rock masses is determined byboth the number and size of the blocks. Further, analyzing the rela-tion between the sizes of the blocks and the existence of the REV isvery complicated, and the spacing between adjacent discontinu-ities largely controls the sizes of individual blocks of intact rock[19,36,40,11]. These studies proposed methods to estimate themean block volume by using rock mass discontinuity spacing data.Due to the constant persistence of discontinuities, higher densitiesresult in more numerous, smaller blocks and low integrity,whereas lower densities result in high integrity. In this study, thespacing and persistence of fractures were considered in construct-ing the models to obtain workable solutions. Therefore, only thenumber of blocks was considered when determining the existenceof the REV to increase the efficiency. Moreover, we have alreadybegun handling the size of the blocks when determining the exis-tence of the REV, which will be presented in a follow-up study.

Conflict of interest

The authors declare that there are no conflicts of interest.

Acknowledgements

This study was financially supported by the National NatureScience Foundation of China (No. 41272387), the FundamentalResearch Funds for the Central Universities (No. 2652015001),and the Changjiang Conservancy Commission. The authors aregrateful to the two anonymous reviewers for their reviews, whichhelped to improve the manuscript.

References

[1] Arild P. Measurements of and correlations between block size and rock qualitydesignation (RQD). Tunn Undergr Space Technol 2005;20(4):362–77.

[2] Bao G, Hutchinson JW, Mcmeeking RM. Particle reinforcement of ductilematrices against plastic-flow and creep. Acta Metall Mater 1991;39(8):1871–82.

[3] Biner SB. A finite element method analysis of the role of interface behavior inthe creep rupture characteristics of a discontinuously reinforced compositewith sliding grain boundaries. Mater Sci Eng A Struct Mater Prop MicrostructProcess 1996;208(2):239–48.

[4] Bear J. Dynamics of fluids in porous media. New York: Elsevier; 1972.[5] Blum P, Mackay R, Riley M, Knight J. Hydraulische Modellierung und die

Ermittlung des reprasentativen Elementarvolumens (REV) im Kluftgestein.Grundwasser 2007;12(1):48–65.

[6] Chae BG, Seo YS. Homogenization analysis for estimating the elastic modulusand representative elementary volume of Inada Granite in Japan. Geosci J2011;15(4):387–94.

[7] Chandra R. Parallel programming in OpenMP. San Francisco, CA: MorganKaufmann Publishers; 2001. p. 230.

[8] Corbett PWM, Jensen JL. Estimating the mean permeability: how manymeasurements do we need? First Break 1992;10(3):89–94.

[9] Cundall PA. Formulation of a three-dimensional distinct element model—Part1. A scheme to detect and represent contacts in a system composed of manypolyhedral blocks. Int J Rock Mech Min Sci Geomech Abstr 1988;25(3):107–16.

[10] Dragone TL, Nix WD. Geometric factors affecting the internal-stressdistribution and high-temperature creep rate of discontinuous fiberreinforced metals. Acta Metall Mater 1990;38(10):1941–53.

[11] Elci H, Turk N. Block volume estimation from the discontinuity spacingmeasurements of Mesozoic limestone quarries, Karaburun Peninsula, Turkey.Sci World J 2014:1–10.

[12] Empereur ML, Villemin TO. Database-supported software for 3D modeling ofrock mass fragmentation. Comput Geosci 2003;29:173–81.

[13] Fallah N, Cross M, Taylor G, Bailey C. Comparison of finite element and finitevolume methods application in geometrically nonlinear stress analysis. ApplMath Model 2000;24(7):439–55.

[14] Gao D, Schwartzentruber TE. Optimizations and OpenMP implementation forthe direct simulation Monte Carlo method. Comput Fluids 2011;42:73–81.

[15] Ghavami A, Abedian A, Mondali M. Finite difference solution of steady statecreep deformations in a short fiber composite in presence of fiber/matrixdeboning. Mater Des 2010;31(5):2616–24.

[16] Goodman RE, Shi GH. Block theory and its application to rockengineering. New York: Prentice Hall; 1985.

[17] Heliot D. Generating a blocky rock mass. Int J Rock Mech Min Sci GeomechAbstr 1988;25(3):127–38.

92 L. Xia et al. / Computers and Geotechnics 76 (2016) 83–92

[18] Ikegawa Y, Hudson JA. A novel automatic identification system for threedimensional multi-block systems. Eng Comput 1992;9:169–79.

[19] International Society for Rock Mechanics (ISRM). Suggested methods for thequantitative description of discontinuities in rock masses. Int J Rock Mech MinSci Geomech Abstr 1978;15(6):319–68.

[20] Jason CR, Kelly AR, Engelder Terry. Investigating the effect of mechanicaldiscontinuities on joint spacing. Tectonophysics 1998;295(1–2):245–57.

[21] Kalenchuk Katherine S, Diederichs Mark S, McKinnon Steve. Characterizingblock geometry in jointed rockmasses. Int J Rock Mech Min Sci2006;43:1212–25.

[22] Kibok M, Jing LR. Numerical determination of the equivalent elasticcompliance tensor for fractured rock masses using the distinct elementmethod. Int J Rock Mech Min Sci 2003;40:795–816.

[23] Kulatilake PHSW, Wathugala DNM, Stephansson O. Joint network modelingwith a validation exercise in Strip mine, Sweden. Int J Rock Mech Min SciGeomech Abstr 1993;30(5):503–26.

[24] Kuszmaul JS. Estimating keyblock sizes in underground excavations:accounting for joint set spacing. Int J Rock Mech Min Sci 1999;36(2):217–32.

[25] Li YJ, Zhang D, Fang Q, Yu QC, Xia L. A physical and numerical investigation ofthe failure mechanism of weak rocks surrounding tunnels. Comput Geotech2014;61:292–307.

[26] Lin D, Farihurst C, Starfield AM. Geometrical identification of threedimensional rock block system using topological techniques. Int J Rock MechMin Sci Geomech Abstr 1987;24(6):331–8.

[27] Long JCS. Porous media equivalents for networks of discontinuous fractures.Water Resour Res 1982;18(3):645–58.

[28] Lu J. Systematic identification of polyhedral rock blocks with arbitrary jointsand faults. Comput Geotech 2002;29(1):49–72.

[29] Michael T, Erez G, Eli M. 3D global–local finite element analysis of shallowunderground caverns in soft sedimentary rock. Int J Rock Mech Min Sci2013;57:89–99.

[30] Mondali M, Abedian A, Adibnazari S. FEM study of the second stage creepbehavior of Al6061/SiC metal matrix composite. Comput Mater Sci 2005;34(2):140–50.

[31] Muller M, Siegesmund S, Blum P. Evaluation of the representative elementaryvolume (REV) of a fractured geothermal sandstone reservoir. Environ Earth Sci2010;61(8):1713–24.

[32] Neuman SP. Stochastic continuum representation of fractured rockpermeability as an alternative to the REV and fracture network concepts. In:Proceedings of the 28th US symposium of rock mechanics. Tucson: Belkema,University of Arizona; 1987. p. 533–61.

[33] Noetinger B. The effective permeability of a heterogeneous porous medium.Transp Porous Media 1994;15(2):99–127.

[34] Nordahl K, Ringrose PS, Wen R. Petrophysical characterization of a heterolithictidal reservoir interval using a process-based modeling tool. Petrol Geosci2005;11(1):17–28.

[35] Priest SD. Determination of discontinuity size distribution from scan-line data.Rock Mech Rock Eng 2004;37(5):347–68.

[36] Priest SD, Hudson JA. Discontinuity spacings in rock. Int J Rock Mech Min SciGeomech Abstr 1976;13(5):135–48.

[37] Pinto A, Da CH. Scale effects in rock mechanics. Rotterdam: AA BalkemaPublishers; 1993.

[38] Seo JK, Tak S, Albert SK. OpenMP-accelerated SWAT simulation using Intel Cand FORTRAN compilers: development and benchmark. Comput Geosci2015;75:66–72.

[39] Shi G. Discontinuous deformation analysis, a new numerical model for thestatics and dynamics of block systems. PhD thesis. Berkeley, USA: University ofCalifornia; 1989.

[40] Stavropoulou M. Discontinuity frequency and block volume distribution inrock masses. Int J Rock Mech Min Sci 2014;65:62–74.

[41] Vik B, Bastesen E, Skauge A. Evaluation of representative elementary volumefor a vuggy carbonate rock—Part: porosity, permeability, and dispersivity. JPetrol Sci Eng 2013;112:36–47.

[42] Wang M, Kulatilake PHS. Estimation of REV size and three-dimensionalhydraulic conductivity tensor for a fractured rock mass through a single wellpacker test and discrete fracture fluid flow modeling. Int J Rock Mech Min Sci2002;39(7):887–904.

[43] Warburton PM. Application of a new computer model for reconstructingblocky block geometry analysis single block stability and identifyingkeystones. In: Proceedings of the 5th international congress on rockmechanics, Melbourne; 1983. p. 225–30.

[44] Wu Q, Kulatilake PHSW. REV and its properties on fracture system andmechanical properties, and an orthotropic constitutive model for a jointedrock mass in a dam site in China. Comput Geotech 2012;43:124–42.

[45] Wu JH, Ohnishi Y, Nishiyama S. A development of the discontinuousdeformation analysis for rock fall analysis. Int J Numer Anal Meth Geomech2005;29(10):971–88.

[46] Xia L, Li MH, Chen YH, Zheng YH, Yu QC. Blockiness level of rock mass aroundunderground powerhouse of Three Gorges Project. Tunn Undergr SpaceTechnol 2015;48:67–76.

[47] Xia L, Yu QC, Chen YH, Li MH, Xue GF, Chen D. GeneralBlock: a C++ program foridentifying and analyzing rock blocks formed by finite-sized fractures. IFIP AdvInf Commun Technol ISESS 2015;448:512–9.

[48] Xia L, Li MH, Chen YH, Wang JX, Yu QC. Study of typical blocks in vault of ThreeGorges underground powerhouse. Chin J Rock Mech Eng 2011;30(S1):3089–95.

[49] Xia L. Rock blocks in the rock around the underground powerhouse of Three-Gorges Project. PhDdissertation. Beijing: ChinaUniversity ofGeosciences; 2011.

[50] Xiang WF, Zhou CB. The advances in investigation of representativeelementary volume for fractured rock mass. Chin J Rock Mech Eng 2005;24(2):5686–92.

[51] Yang J, Chen W, Yang D, Yuan J. Numerical determination of strength anddeformability of fractured rock mass by FEM modeling. Comput Geotech2015;64:20–31.

[52] Yang ZY, Chen JM, Huang TH. Effect of joint sets on the strength anddeformation of rock mass models. Int J Rock Mech Min Sci 1998;35(1):75–84.

[53] Yu QC, Xue GF, Chen DJ. General block theory for fractured rockmasses. Beijing: China Water & Power Press; 2007.

[54] Yu QC, Ohnishi Y, Xue GF, Chen DJ. A generalized procedure to identify three-dimensional rock blocks around complex excavations. Int J Numer Anal MethGeomech 2009;33(3):355–75.

[55] Zhang GK, Xu WY. Analysis of joint network simulation method and REV scale.Rock Soil Mech 2008;29(6):1675–80.

[56] Zhang L, Einstein HH. Estimating the intensity of rock discontinuities. Int JRock Mech Min Sci Geomech Abstr 2000;37(5):819–37.

[57] Zhang YT, Xiao M, Ding XL, Wu AQ. Improvement of methodology for blockidentification using mesh gridding technique. Tunn Undergr Space Technol2012;30:217–29.

[58] Zhao GF, Fang JN, Sun L, Zhao J. Parallelization of the distinct lattice springmodel. Int J Numer Anal Meth Geomech 2013;37:51–74.

[59] Zheng YH, Xia L, Yu QC. Unified analysis method of removability and stabilityof rock blocks. In: Frontiers of discontinuous numerical methods and practicalsimulations in engineering and disaster prevention – proceedings of the 11thinternational conference on analysis of discontinuous deformation, Fukuoka,Japan; 2013. p. 383–8.

[60] Zheng YH, Xia L, Yu QC. A method for identifying three-dimensional rockblocks formed by curved fractures. Comput Geotech 2015;65:1–11.

[61] Zsaki AM. Parallel generation of initial element assemblies for two-dimensional discrete element simulations. Int J Numer Anal Meth Geomech2009;33:377–89.