- Email: [email protected]
Numerical study of critical behaviour of deformation and permeability of fractured rock masses
Marine and Petroleum Geology 04 "0887# 424Ð437
Numerical study of critical behaviour of deformation and permeability of fractured rock masses Xing Zhang \ D[ J[ Sanderson Department of Geology\ Southampton Oceanography Centre\ University of Southampton\ Empress Dock\ Southampton\ SO03 2ZH\ U[K[ Received 16 October 0886^ revised 7 May 0887^ accepted 16 May 0887
Abstract The connectivity of fractures in rock masses is determined using a numerical simulation method[ There is a continuous fracture cluster throughout a fractured rock mass if fracture density "d# is at or above a threshold fracture density dc\ Fractal dimension "Df# is used to describe the connectivity and compactness of the largest fracture clusters[ Df increases with increasing fracture density[ Percolation theory is used to determine the universal law\ Df Af"d−dc#f\ which describes the critical behaviour of connectivity of fractures in rock masses[ The results from numerical modeling show that the deformability of fractured rock masses increases greatly with increasing fracture density "i[e[\ fractal dimension#\ and the critical behaviour of deformability can be described by Bs A"d−df#s[ Also\ the overall permeability of a fractured rock mass occurs at or above a critical fracture density "dc# and increases with increasing fracture density[ The critical behaviour of permeability can be described by q Ap"d−dc#p[ The critical behaviour of connectivity and permeability of naturally fractured rock masses is examined using the universal forms[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords] Fractured rock ^ Critical behaviour ^ Connectivity ^ Deformability ^ Permeability
0[ Introduction Fractures of all sizes have a dominant e}ect on many physical properties of the upper crust\ such as the mech! anical\ hydraulic\ thermal and seismic properties[ It is desirable for many projects\ including hazardous waste disposal\ e.cient oil recovery from fractured reservoirs\ thermal energy extraction and ore deposition study\ to have a thorough understanding of the properties of frac! tured rock masses[ The deformability and permeability of fractured rock masses may increase super!linearly with the increase in fracture density\ being greatly dependent on the degree of fracture connectivity and related charac! teristics\ including aperture\ density\ spacing\ length\ orientation and wall mechanical properties "e[g[\ Engl! man et al[\ 0872 ^ Shimo and Long\ 0876 ^ Rasmussen\ 0876 ^ Zhang et al[\ 0881 ^ Zhang\ 0882#[ More and more studies show that the connectivity of fractures plays a key role in the deformability and permeability of frac! tured rock masses "e[g[\ Balberg\ 0875 ^ Gueguen and Dienes\ 0878 ^ Balberg et al[\ 0880 ^ Zhang et al[\ 0882 ^ Zhang and Sanderson\ 0883 ^ 0884#[ In this study\ a 1!dimensional numerical model is used Corresponding author[
to investigate the connectivity of fractures in a rock mass[ Based on 04 natural fracture patterns and 3 groups of simulated fracture patterns\ the critical stage or per! colation threshold at which connectivity shows a sudden phase change\ is determined[ The relations between frac! ture connectivity and other parameters\ such as fracture density\ length\ orientation and con_guration are inves! tigated\ and fracture density shows the most important role in controlling fracture connectivity where fracture length is relatively small in relation to a region of interest[ The relationship between fracture connectivity and frac! ture density may be described in a form of power law\ in which the fracture connectivity is indicated with a fractal dimension and the power law is characterized by a critical fracture density and an exponent[ Distinct element methods are used to model the defor! mation and permeability of those fracture networks at di}erent phase changes[ It is demonstrated that fracture connectivity has a very important e}ect on deformation and permeability\ and deformation and permeability pre! sent a sudden increase if a connected fracture network exists[ The deformation and permeability of a fractured rock mass can be predicted to some degree based on fracture density which dominantly controls fracture con! nectivity[ The results suggest that the relation between fracture density and deformation or permeability may be
S9153Ð7061:87:,*see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved PII ] S 9 1 5 3 Ð 7 0 6 1 " 8 7 # 9 9 9 2 9 Ð 9
425
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
described with a power law where fracture density is above a critical value[ 1[ Modeling of 1!dimensional fracture networks Fractures are commonly observed as traces intersecting an outcrop\ and the size of a fracture can be described by its trace length in 1!dimensional problems[ In this study\ a numerical model of fracture systems\ based on self! avoiding random generations\ has been developed[ The aim of this work is to stimulate fracture maps under conditions where parameters such as the length\ density and pattern of fractures can be controlled[ The model involves the following procedures ] "a# The simulated region is a square of unit size[ The fracture trace!lengths vary from a lower limit of 9[4Ð 3) of the side of the square\ up to 14)[ The fractures are small in relation to the size of the simulated region and the resulting pattern is scale invariant[ "b# Trace!lengths are sampled from a power!law dis! tribution where the number of fractures "N# of length "L# has a form NaLE "Segall and Pollar\ 0872 ^ Barton and Hsieh\ 0878 ^ He}er and Bevan\ 0889 ^ Jackson and Sanderson\ 0881#[ The exponent can be varied and\ together with limits of fracture sizes\ controls the length distribution[ "c# Fracture orientation is de_ned as the angle that the fracture!trace makes to a reference axis[ Angles may be selected randomly in the range of 9Ð079>\ or with a normal distribution\ with sets of fractures being de_ned by their mean orientation and standard devi! ation "here termed dispersion angle#[ "d# The coordinates of the centre of a fracture are ran! domly selected from a uniform distribution within the square[ A procedure of self!avoiding generation is used\ such that new fractures are selected only if they are located at a minimum distance to previously generated fractures[ The minimum distance may be varied\ but is usually set at 9[4) of the size of the square[ "e# Fractures are generated sequentially\ according to the above rules[ As more fractures are added the density of the fractures increases[ Fracture density is de_ned as the total length of the fracture!traces per unit area[ Four types of fracture patterns have been simulated within a square of 09×09 m ] "i# Group A consists of two sets of parallel\ orthogonal fractures with the same lower limit of 9[1 m and exponent of 0[1 "Fig[ 0#[ "ii# Group B consists of two sets of fractures "as Group A#\ but the dispersion angle of each set is selected as 04>[ The mean orientation of the _rst set is parallel to the x!axis\ and the second set is parallel to the y! axis "Fig[ 1#[
"iii# Group C is similar to group B\ but with a dispersion angle of zero and with di}erent lower limits\ 9[93 m and 9[05 m\ for the fracture trace lengths in the x! and y!directions\ respectively "Fig[ 1#[ "iv# Group D is similar to group A\ but with sets of angles of ¦:− 59> to the x!axis "Fig[ 1#[ These fracture patterns\ together with some natural examples\ were then used as input to distinct element models in order to study deformation and ~uid ~ow[ 2[ Density\ connectivity and fractal dimension Within a fractured rock mass\ some fractures are iso! lated "unconnected#\ whilst others intersect[ At low frac! ture densities\ most are isolated\ although some may be connected locally[ As the fracture density increases\ more fractures become connected and relatively large clusters form "Fig[ 0a#[ At this stage\ no continuous cluster develops throughout the fractured rock mass\ since the largest cluster within the simulated square area does not intersect all the boundaries[ With a further increase in fracture density\ more fractures and clusters become con! nected\ until the largest cluster intersects all the bound! aries of the simulated square area "Fig[ 0b#[ At this critical point\ it is possible to de_ne a critical fracture density[ If more fractures are added to the fractured rock mass\ the largest cluster will grow to cover the whole area\ and the rock mass becomes highly fragmented "Fig[ 0c#[ The connectivity of the largest fracture cluster can be described with its fractal dimension "Df# by box!counting "Zhang and Sanderson\ 0883#[ Df is the slope of a best _t straight line on a plot of a log ðN"s#Ł against Log ð"0:s#Ł\ where N"s# is the number of boxes of size "s# which con! tain at least one fracture belonging to the largest cluster[ The relation between N"s# and "s# for the two fracture patterns in Figs 0b and c is shown in Fig[ 2[ The critical fracture cluster "Fig[ 0b# has a critical fractal dimension\ Dfc 0[161\ and the cluster with a higher density "Fig[ 0c# has Df 0[314[ The fractal dimension of the largest cluster must be between 0 "the dimension of a single fracture crossing the region# and 1 "the dimension of the square area itself#[ The four groups of simulated fracture patterns have been used to investigate the connectivity in relation to fracture density[ Figure 3 shows the relations between fracture density and the fractal dimensions of the largest fracture clusters "connectivity of fractures#[ Fractal dimensions "Df# increase with fracture density "d# where fracture density is above a critical fracture density "dc#[ The critical fractal dimension "Dfc# varies little with change in the model parameters\ such as the lower limit of fracture lengths "Ll#\ power!law exponent "E# and dispersion angle "Ad#[ The critical fractal dimensions lie in a narrow range from 0[11Ð0[27 "average 0[29# for those critical clusters with variations in the lower limit of length
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
426
Fig[ 0[ Simulated fracture patterns and the corresponding largest clusters of group A\ at di}erent fracture densities[ "a# For a low fracture density\ the largest fracture cluster does not intersect all the boundaries^ "b# At the critical density "percolation threshold\ dc\ the largest cluster intersects all the boundaries ^ "c# Above the critical density\ more fractures connect to form the largest cluster and the rock is fragmented[
Fig[ 1[ Simulated fracture patterns with di}erent con_guration[ Group B consists of two sets of fractures with dispersion angle of 04> for each set[ Group C consists of two sets of fractures with di}erent fracture lengths and the di}erence in trace lengths results in an anisotropic factor of 0[707 "Zhang and Sanderson\ 0884#[ Group D consists of two sets of fractures with an angle of 59> between them and this results in an anisotropic factor of 2[33[ Anisotropic factor indicates the anisotropy in geometry of a fractured rock[
from 9[994Ð0[4 m\ in dispersion angle of fracture direc! tion from 9Ð49> and in exponents from 0[1 and 0[7[ This indicates that the fractal dimension of a critical fracture
cluster is dominated by fracture density and is largely independent of other geometrical parameters\ although there is some random variation[ A number of _eld studies
427
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
and orientation\ but can be controlled by appropriate choice of the relative fracture density[ 3[ Critical behaviour of fractured rock masses
Fig[ 2[ Fitting fractal dimension of largest clusters for two fractured rocks using box!counting method[ The largest cluster having the critical density "Fig[ 0b# has a fractal dimension "Dfc of 0[161 and the largest cluster having a higher density "Fig[ 0c# has a fractal dimension "Df# of 0[341[
Percolation theory was _rst used to investigate poly! merization "e[g[\ Flory\ 0830 ^ Stockmayer\ 0832#[ Recently\ percolation models have been intensively applied to examine geometric properties and transport phenomena observed in porous rocks "e[g[\ Englman et al[\ 0872 ^ Balberg\ 0875 ^ Charlaix et al[\ 0876 ^ Gueguen and Dienes\ 0878 ^ Balberg et al[\ 0880 ^ Berkowitz and Balberg\ 0881#[ However\ little direct use of percolation theory has been made in the study of the geometry and physical properties of fractured rock masses[ Percolation theory can be used to investigate many macroscopic properties of a system that are determined by the statistic properties of the system elements[ Special properties of a system\ which emerge at a critical value of the system elements\ are known as percolation phenomena[ An advantage of percolation theory is that it provides universal forms which describe the overall properties of a system\ which are independent of its local geometry[ Generally\ the overall properties "Gr# of a sys! tem can be determined by a value "V# describing some statistic property of the elements with a power law of the form Gr"V−Vc#r
"0#
where Vc is the threshold of V and r is some exponent which can be found from theory and:or computer simu! lation and:or experiment[ Such a form also characterises the critical behaviour of some special properties of a system\ which emerge where the value of V is at or above the critical value of Vc[ In this study\ the overall properties "connectivity\ deformability and permeability# of frac! tured rock masses have been investigated in relation to the fracture density "d# of the fractured rock masses[ 3[0[ Universal law of connectivity With a form similar to equation "0#\ the universal law of connectivity of fractures in rock masses can be described with the following equation ] Fig[ 3[ Relationship between fracture density "d# and the fractal dimen! sion "Df# for four groups of simulated fracture patterns with di}erent con_gurations[
have examined the fractal nature of many natural fracture networks over a wide range of scales\ lithologies and geological settings\ as reviewed in Berkowitz and Hadad|s paper "0886#[ In their study\ they also found that the fractal dimension of a synthetic fracture network is rela! tively insensitive to parameters such as fracture length
Df af"d−dc#f
"1#
where Df is the fractal dimension of the largest cluster of a fractured rock mass\ which describes the connectivity and compactness of the fractures ^ d is fracture density\ with a value "dc# at the percolation threshold ^ f is an exponent determining the universal law\ and af is a constant[ Figure 4 shows the relations between fracture density and fractal dimension "connectivity#\ based on the results of the four groups of fracture patterns in Figs 0 and 1[
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
428
Fig[ 4[ Universal laws of connectivity for four groups of fracture patterns with di}erent con_gurations[
The power law exponent " f # has a nearly constant value "9[001Ð9[016# for di}erent geometries\ but the critical density of fractures varies from 3Ð4[4 m:m1[ Hence\ the percolation thresholds "critical density of fractures\ dc# of di}erent con_gurations must be determined separately[ The interest of percolation theory arises from the fact that a given fracture pattern has a certain exponent of f ^ thus\ one can predict the connectivity of a fractured rock mass by _nding f from some similar small fracture net! works in local regions[ In other words\ it is possible to predict the connectivity if the fracture density and the con_guration of the fracture system of the rock mass is known[ Note that the _tted lines through the data points in Fig[ 4 represent average slopes[ Actually\ the data show a slightly concave!upwards curvature\ so di}erent parts of the data may _t di}erent slopes[ Obviously\ the data
at small values of "d−dc# describe the nature of critical behaviour well\ in the neighbourhood of dc[ However\ an average slope can be used to describe the relation between fracture connectivity and fracture density over a rela! tively wide range of fracture density[ Fifteen natural fracture patterns "Fig[ 5#\ which form continuous fracture clusters\ were sampled in the English Lake District\ and tested to _nd the universal law for connectivity[ Each fracture pattern was sampled within a square of 1×1 m\ with all fractures greater than 9[0 m in trace length[ In each of the samples\ the fracture pattern consists of 2Ð3 very steeply inclined fracture sets[ Hence\ the maps of sub!horizontal exposures provide a reason! able 1!dimensional characterization of the fracture network[ The same procedures have been used to calculate the fractal dimensions of the largest cluster for each network[
439
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 5[ Fifteen fracture patterns sampled from exposures in the English Lake District[ Each pattern has an area of 1×1 m and only those trace lengths greater than 9[0 m are shown in diagrams[
The universal exponent " f # for connectivity has a rather low value of 9[095 "Fig[ 6# ^ the points in Fig[ 6 are scattered due to local variation in fracture con_guration and density[ 3[1[ Critical behaviour and deformability Numerical experiments\ using the distinct element method "UDEC i[e[\ Universal Distinct Element Code\ Version 1[9\ 0882#\ have been carried out to study the deformability of fractured rock masses[ The fractured rock mass is treated as a series of elastic blocks with displacement possible along and across fractures when loaded "Cundall et al[\ 0867 ^ Last and Harper\ 0889#[ Table 0 lists the material properties used[ The normal and tangential displacements between two adjacent blocks are determined directly from block geometry and block centroid translation and rotation[ The force!dis! placement law relates incremental normal and shear for! ces "DFn\ DFs# to the amount of incremental relative displacement "Dun\ Dus# ]
DFn KnDun
"2#
DFs KsDus
"3#
Where Kn and Ks are the contact normal and shear sti}! ness respectively[ Such force!displacement relationships allow the evaluation of shear and normal forces between the intact blocks in a deformed region of fractured rock[ Simulations of biaxial compressive tests have been run\ with _xed con_ning stress "s1# of 9 MPa or 9[2 MPa and di}erent deviatoric stresses "s0Ðs1#[ This enables the study of the deformability of fractured rocks with di}erent geometric and mechanical parameters[ The rock and frac! ture parameters are selected to represent a tight sandstone "e[g[\ Barton et al[\ 0874 ^ Yoshinaka and Yamabe\ 0875 ^ Last and Harper\ 0889#[ Figure 7 shows the schematic diagram of the loading and monitoring system of the numerical experiments[ The displacements in two direc! tions are measured at eight points so that average defor! mation in axial and lateral directions could be obtained[ Rocks with a fracture density below the percolation threshold "dc# show only a small amount of deformation\
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
430
Fig[ 6[ The universal law of the natural fracture patterns for connect! ivity[ Fig[ 7[ Schematic illustration for loading and monitoring system of numerical experiments[ Two deformation components in the x! and y! directions at eight points were measured[
Table 0 Material parameters used in this modeling Value
Units
Block property Density Shear modulus Bulk modulus
1499 04[3 22[2
kg:m2 GPa Gpa
Fracture property Tensile strength Cohesion Friction angle Dilation angle Joint normal sti}ness Joint shear sti}ness
9 9 14 9 099 59
MPa MPa Degree Degree GPa:m GPa:m
Young|s modulus 39 GPa^ Poisson|s ratio 9[2
mainly contributed by the elastic deformation of intact blocks[ Where fractures barely form a completely con! nected cluster "Fig[ 8a# i[e[\ at the percolation threshold\ the critical fractal dimension is 0[29\ the larger defor! mation occurs due to shear displacements and openings along a few fractures[ At higher fracture density\ the connected fracture cluster\ with a fractal dimension of 0[28\ covers more of the simulated area\ and the resulting rock mass is more fragmented "Fig[ 8b#[ Deformation of the fractured rock increases abruptly and is pre! dominantly contributed by shear displacements and openings rather than elastic deformation of the blocks[ The stressÐstrain behaviour of three models with
di}erent fracture densities is shown in Fig[ 09[ At the critical fracture density "dc 4[44 m:m1\ Fig[ 8a#\ the stress!strain curve still shows an approximately elastic behaviour\ since the deformation is mainly contributed by the elastic deformation of the intact blocks[ Above the critical fracture density "Df 0[23\ d 5[4 m:m1#\ the stressÐstrain behaviour presents softening\ i[e[\ the ratio of stress to strain drops at a deviatoric stress of 1[6 MPa\ and ~ow begins at a deviatoric stress of 2[5 MPa\ with the development of the shear and normal displacements along fractures[ At still higher fracture density "Df 0[28\ d 6[44 m:m1\ Fig[ 8b#\ yield occurs at a lower deviatoric stress level "0[7 MPa#\ and ~ow occurs at 1[3 MPa\ since more fractures contribute displacement to the defor! mation of the fractured rock[ At di}erent levels of con_ning stress "9 and 9[2 MPa#\ the relationship between deformation and fracture den! sity is summarized in Fig[ 00[ Although large deformation occurs above the percolation threshold "dc#\ a sudden increase occurs when the fracture density exceeds the second threshold of fragmentation "df#\ independent of con_ning stress level[ The second threshold is at a value of 5[4 m:m1 "an associated fractal dimension of 0[24# and corresponds to the fragmentation of a rock mass[ The following form ] Bs as"d−df#s
"4#
is used to describe the critical behaviour of deformability
431
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 8[ Geometry before "left# and after loading "right# of fractured rock masses "a# at a critical fractal dimension of 0[29 and critical fracture density of 4[14 m:m1 ^ after loading\ some displacements and openings can occur along a few fractures ^ "b# at a higher fractal dimension of 0[28 and fracture density of 6[66 m:m1[ After loading\ more displacements and openings occur along many fractures[
for the fractured rock masses "Fig[ 01#[ Here Bs is the deformation ")# of the fractured rock masses\ df is the second threshold of fragmentation\ s is an exponent determining the universal law\ and as is a constant[ Although the value of con_ning stress has little e}ect on the second threshold\ the universal exponent "s# is signi_cantly di}erent\ about 9[53 for a zero con_ning stress and 9[80 for a con_ning stress of 9[2 MPa[ This means that without a con_ning stress\ deformation may develop early[ 3[2[ Critical behaviour of permeability Numerical modeling of ~uid ~ow through fractured rock masses has been carried out using UDEC\ based on
the geometric models of groups AÐD[ In the ~ow model\ it has been assumed that ~ow is through fractures and the matrix rock is impermeable[ The ~ow!rate through a fracture "qj# is calculated by the cubic law "e[g[\ Snow\ 0857 ^ Witherspoon et al[\ 0879 ^ Zhang and Sanderson\ 0885a# ] qj "0:01m#a2"DH:DL#
"5#
where m is dynamic viscosity ^ and a is fracture aperture[ For each fracture pattern\ ~ow!rates under a hydraulic pressure di}erence of 09 kPa "or a hydraulic gradient of 0 kPa:m# are calculated in both the x or y!direction\ as shown in Fig[ 02a[ It can be shown\ both theoretically and in the labora!
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
432
Fig[ 09[ StressÐstrain behaviours of 2 fractured rock masses with di}erent fractal dimensions "i[e[\ di}erent fracture densities#[ For the fractured rock mass with a fracture density at percolation threshold of 4[14 m:m1 and Dfc of 0[29 "Fig[ 8a#\ the stress!strain presents an approximate elastic behaviour[ With fracture density at the second threshold "corresponding to fragmentation# of 5[4 m:m1 and Df of 0[23\ the stress!strain behaviour presents softening and ~ow at relatively low deviatoric stresses "1[6 MPa and 2[5 MPa#[ At a higher fracture density of 6[66 m:m1 and Df of 0[28 "Fig[ 8b#\ the strain!softening and ~ow occur at lower deviatoric stresses "0[7 and 1[3 MPa#[
Fig[ 00[ Axial and lateral deformation of fractured rock masses undergoing di}erent con_ning stresses against fracture density[
tory\ that the stress applied to a fractured rock can have a signi_cant in~uence on its e}ective permeability due to the closure of aperture\ and a varying aperture is likely to exist in a fracture network "Snow\ 0857 ^ Charlaix\ 0876 ^ Zimmerman and Bodvarsson\ 0885 ^ Zhang and
Sanderson\ 0885a ^ 0885b#[ The sti}ness of fractures "nor! mal sti}ness and shear sti}ness# are two important par! ameters which control the permeability of fractured rock masses[ However\ description of fracture properties still appears to be a problem\ and widely accepted quan!
433
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 01[ Universal laws of axial and lateral deformability of fractured rock masses with di}erent con_ning stresses[ The value of con_ning stress has little e}ect on the second threshold\ but the universal exponent " f # is signi_cantly di}erent\ about 9[53 for a zero con_ning stress and 9[80 for a con_ning stress of 9[2 MPa[
titative procedures are limited "Bandis\ 0889#[ Exper! imental measurements show that fracture sti}ness is related to fracture topography\ the properties of intact rock\ and is a highly non!linear function of the e}ective stress normal to a fracture "Cook\ 0881#[ Clearly the properties of fractures and the stress state surrounding them should have signi_cant e}ects on their permeability[ Investigating the e}ects of fracture property and stress state is out of the scope of the paper\ and here all fractures used for ~uid ~ow modeling are assigned a constant hydraulic aperture of 9[4 mm[ Hence\ the ~ow!rates of
fractured rock masses are controlled by the fracture con! nectivity only[ Figures 02b and c show the ~ow!rates and ~ow direc! tions\ under a head gradient in the y!direction\ through the fracture patterns in Figs 0b and c ^ a thicker line indicates a higher ~ow!rate[ The ~ow!rate through the fracture pattern in Fig[ 0a is zero because there is no continuous network connecting all the boundaries[ Figure 03 shows that relations between fracture density "d# and the average ~ow!rates for the four groups of fracture patterns[ Clearly\ ~ow!rates increase with
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
434
Fig[ 02[ Illustrations for ~uid ~ow modeling[ "a# Hydraulic boundary conditions used in the modeling[ Two simulations have been carried out for each fracture patterns in two orthogonal directions with a constant hydraulic gradient of 0 kPa:m ^ "b# Flow!rates in the y!direction through the fracture patterns in Figs 0b and c ^ "c# Directions and paths of ~uid ~ow in the y!direction through the fracture patterns in Figs 0b and c[ Only those ~ow!rates which exceeded 4×09−5 m2:s are shown in the diagrams of Figs 02b and c[
increasing fracture density above the percolation thr! eshold "dc#[ From Equation "0# again\ the critical behav! iour of permeability for fractured rock masses can be described with q ap"d−dc#p
"6#
where q is the average ~ow!rate\ p is an exponent deter! mining the universal law\ and ap is a constant[ Figure 04 shows the results for four groups of fracture patterns[ The exponent "p# varies between 0[936 and 0[257 for di}erent fracture patterns\ with a critical density of 3Ð4[4 m:m1[ Di}erent models "con_gurations# of frac! ture patterns have a signi_cant e}ect on p[ It is important to note that the value of p is around 0[2 for isotropic patterns of fractures "groups A and B#[ For a two dimen! sional model of lattice patterns\ the universal exponent of conductivity of a network is 0[2 "e[g[\ Stau}er\ 0874 ^ Sahimi\ 0876#[ Also\ Berkowitz and Balberg "0881# simu! lated a 1!dimensional porous medium and obtained a universal exponent of the overall conductivity of 0[2[ It seems that for randomly oriented\ isotropic fracture
patterns with fracture length being much less than the region of interest\ the universal exponent "p# is about 0[2[ However\ anisotropy "Groups C and D# of fracture patterns has a major e}ect on both the universal exponent and percolation threshold for permeability "also see Zhang and Sanderson\ 0884#[ The natural fracture patterns are also used to test the critical behaviour of permeability "Fig[ 05#[ As expected\ the universal exponent "p# for permeability has a rela! tively low value of 0[045 because of the relatively large length of fractures in the natural fracture patterns[
4[ Conclusions Numerical modeling methods have been used to inves! tigate connectivity of 1!dimensional fractured rock masses against various fracture parameters such as frac! ture density\ length\ orientation and con_guration[ Frac! ture density plays the most important role in forming the critical fracture cluster at which a completely connected
435
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 03[ Relationship between fracture density "d# and the average ~ow!rates for four groups of simulated fracture patterns with di}erent con_gurations[
Fig[ 04[ Universal laws of permeability for four groups of fracture patterns with di}erent con_gurations[ The universal exponent "p# varies from 0[936Ð0[257 with a percolation threshold between 3Ð4[4 m:m1[ Di}erent models "con_gurations# of fracture patterns have a signi_cant e}ect on the value of exponent "p#\ particularly the anisotropy of a fractured rock mass "Zhang and Sanderson\ 0884#[
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
436
characteristics of local samples which are relatively easy to obtain[ The quantitative conclusions of this study are ]
Fig[ 05[ Universal laws of the natural fracture patterns for permeability[ The universal exponent "p# for permeability has a relatively low value of 0[045 because of the relatively large length of fractures in the natural fracture patterns[
network barely forms[ Connectivity is relatively insen! sitive to the other fracture parameters\ if fracture length is relatively small in relation to the region of interest[ There is a percolation threshold at a fracture density below which no connected network exists[ A fractal dimension is used to measure the connectivity and com! pactness of the largest fracture cluster where fracture density is above the critical one\ and it increases with the increase of fracture density[ Distinct element methods have been applied to simu! late deformation and ~uid ~ow of fractured rock masses[ Based on percolation theory\ the critical behaviour "con! nectivity\ deformability and permeability# of fractured rock masses can be described using universal laws where the critical fracture density can be determined[ Per! colation theory is also applied to predict the connectivity and permeability of natural fracture patterns[ Comparing with those simulated fracture patterns\ the natural sam! ples are more scatter of the points\ particularly in the ~ow tests[ However\ their connectivity and permeability can still be described with a universal law form to some degree[ Based on the four groups of simulated fracture patterns and _fteen natural fracture patterns\ it seems that per! colation theory can be used to investigate the connect! ivity\ deformation and permeability\ and universal laws may provide a way to understand the overall properties of the fractured rock masses by considering the statistical
"0# The critical fractal dimension "Dfc# of the critical frac! ture clusters has a nearly constant value\ and is a key parameter in the determination of the critical behaviour of connectivity[ Dfc lies between 0[11 and 0[27 for the variation in the size distribution "E 0[1 to 0[7# ^ the dispersion angle of fracture orientation "Ad 9Ð49># ^ and the lower limit of trace length "Ll 9[994Ð0[4 m for a sample size of 09×09 m#[ "1# The critical behaviour of connectivity can be described with a universal form Df af"d−dc#f\ on the basis of percolation theory[ For simulated frac! ture patterns with di}erent con_gurations\ the uni! versal exponent " f # varies between 9[001Ð9[016[ Anisotropy of fracture patterns has a signi_cant e}ect on f and the critical fracture density must be deter! mined separately[ "2# Distinct Element modeling of simulated fracture pat! terns shows that there are sudden changes in the deformability of the rock mass at the percolation threshold "dc# and the second threshold of frag! mentation "df#[ The deformability of rock mass increases above the percolation threshold "dc#\ but large deformation occurs at or above the second thr! eshold of fragmentation "df# of 5[4 m:m1[ The critical behaviour of deformability can be determined with Bs as"d−df#s\ and the universal exponent "s# is sig! ni_cantly di}erent\ about 9[53 for a zero con_ning stress and 9[80 for a con_ning stress of 9[2 MPa[ "3# At or above the percolation threshold\ the per! meability increases with fracture density[ The critical behaviour can be determined with q ap"d−dc#p\ and the universal exponent "p# varies between 0[936Ð 0[257[ Anisotropy of fracture patterns has a major e}ect on the universal exponent "p#[ Acknowledgements Funding for part of this study was provided by the Natural Environment Research Council "grant GR8:0132#[ Natural fracture networks were sampled in conjunction with British Geological Survey and Nirex Ltd[ "U[K[#[ The authors would like to thank Professor D[ G[ Roberts and an anonymous reviewer for detailed comments and constructive suggestions which improved the manuscript[
References Balberg\ I[ "0875#[ Connectivity and conductivity in 1!D and 2!D frac! ture systems[ In R[ Englman and Z[ Jaeger "Eds[# ^ Proceedin`s of International Conference on Fra`mentation\ Form and Flow in Fractured Media\ 7 "pp[ 78Ð090#[ Annals of Israel Physical Society\ Bristol\ U[K[ ] Adam Hilger[
437
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Balberg\ I[\ Berkowitz\ B[\ + Drachsler\ G[ E[ "0880#[ Application of a percolation model to ~ow in fractured hard rocks[ Journal of Geophysics Research\ 85"B5#\ 09904Ð09910[ Bandis\ S[ C[ "0889#[ Mechanical properties of rock joints[ In N[ Barton and O[ Stephansson\ "Eds[#\ Rock Joints "pp[ 014Ð039#[ Barton\ C[ C[\ + Hsieh\ P[ A[ "0878#[ Physical and hydraulic!~ow proper! ties of fractures[ 17th International Geological Congress\ T274[ Barton\ N[ R[\ Bandis\ S[\ + Bakhtar\ K[ "0874#[ Strength\ deformation and conductivity coupling of rock joints[ International Journal of Rock Mechanics\ Minin` Sciences + Geomechanics Abstracts\ 11\ 010Ð039[ Berkowitz\ B[\ + Balberg\ I[ "0881#[ Percolation approach to the prob! lem of hydraulic conductivity in porous media[ Transportation of Porous Media\ 8"2#\ 164Ð175[ Berkowitz\ B[\ + Hadad\ A[ "0886#[ Fractal and multifractals of natural and synthetic fracture networks[ Journal of Geophysics Research\ 091"B5#\ 01194Ð01107[ Charlaix\ E[\ Guyon\ E[\ + Roux\ S[ "0876#[ Permeability of a random array of fractures of widely varying apertures[ Transport in Porous Media\ 1\ 20Ð32[ Cook\ N[ G[ W[ "0881#[ Jaeger Memorial Dedication Lecture!Natural joints in rock ] Mechanical\ hydraulic and seismic behaviour and properties under normal stress[ International Journal of Rock Mech! anics\ Minin` Sciences + Geomechanics Abstracts\ 18\ 087Ð112[ Cundall\ P[ A[\ Martin\ J[\ Beresford\ P[\ Last\ N[ C[\ + Asgian\ M[ "0867#[ Computer modeling of jointed rock masses[ US Army En`in! eers\ WEST Technical Report[ N!63!7[ Englman\ R[\ Gur\ Y[\ + Jaeger\ Z[ "0872#[ Fluid ~ow through a crack network in rocks[ Journal of Applied Mechanics\ 49\ 696Ð00[ Flory\ P[ J[ "0830#[ Molecular size distribution in three dimensional polymers[ Journal of American Chemical Society\ 52\ 2972Ð2099[ Gueguen\ Y[\ + Dienes\ I[ "0878#[ Transport properties of rocks from statistics and percolation[ Mathematical Geolo`y\ 10"0#\ 0Ð02[ He}er\ K[ J[\ + Bevan\ T[ G[ "0889#[ Scaling relationship in natural fractures ] Data\ theory and application[ Society of Petroleum Engineers[ SPE\ 19870\ 256Ð265[ Jackson\ P[ + Sanderson\ D[J[ "0881#[ Scaling of fault displacements from the Badajoz!Cordoba shear zone\ SW Spain[ Tectonophysics\ 109\ 068Ð089[ Last\ N[ C[\ + Harper\ T[ R[ "0889#[ Response of fractured rock subject to ~uid injection[ Part I ] Development of a numerical model[ Tec! tonophysics\ 061\ 0Ð20[ Rasmussen\ T[ C[ "0876#[ Computer simulation model of steady ~uid ~ow and solute transport through three!dimensional networks of variably saturated\ discrete fractures[ In D[ D[ Evans and T[ J[ Nicholson "Eds[#[ Flow and transport throu`h unsaturated rock "pp[ 096Ð003#[ Geophysical Monograph 31\ American Geophysical Union[ Sahimi\ M[ "0876#[ Hydrodynamic dispersion near the percolation thr! eshold ] Scaling and probability densities[ J[ Phys[ A[ Math[ Gen[\ 19"07#\ L0182ÐL0187[
Segall\ S[ W[\ + Pollard\ D[ D[ "0872#[ Joint formation in granite rock of the Sierra Nevada[ Geolo`ical Society of America Bulletin\ 83\ 452Ð464[ Shimo\ M[\ + Long\ J[ "0876#[ A numerical study of transport par! ameters in fracture networks[ In D[ D[ Evans and T[ J[ Nicholson "Eds[#[ Flow and Transport throu`h unsaturated rock "pp[ 010Ð021#[ Geophysical Monograph 31\ American Geophysical Union[ Snow\ D[ T[ "0857#[ Rock fracture spacing\ openings and porosities[ J[ Soil Mech[ Found[ Div[\ ASCE\ 83"SM0#\ 62Ð80[ Stau}er\ D[ "0874#[ Introduction to Percolation Theory[ London ] Taylor and Francis[ Stockmayer\ W[ H[ "0832#[ Theory of molecular size distribution and gel formation in branched!chain polymers[ Journal of Chemical Physics\ 00\ 34Ð44[ Universal Distinct Element Code\ Version 1[9 "0882#[ ITASCA CON! SULTING GROUP\ INC[ Minneapolis\ Minnesota\ U[S[A[ Witherspoon\ P[ A[\ Wang\ J[ S[ Y[\ Iwai\ K[ + Gale\ J[ E[ "0879#[ Validity of cubic law for ~uid ~ow in a deformable rock fracture[ Water Resour[ Res[\ 05\ 0905Ð0913[ Yoshinaka\ R[\ + Yamabe\ T[ "0875#[ Joint sti}ness and the defor! mation behaviour of discontinuous rock[ International Journal of Rock Mechanics\ Minin` Sciences + Geomechanics Abstracts[\ 12\ 08Ð17[ Zhang\ X[\ Harkness\ R[ M[\ + Last\ N[ C[ "0881#[ Evaluation of connectivity characteristics of naturally jointed rock masses[ En`in! eerin` Geolo`y\ 22\ 00Ð29[ Zhang\ X[ "0882#[ Shear resistance of jointed rock masses and stability calculation of rock slopes[ Geotechnical and Geolo`ical En`ineerin`\ 00\ 096Ð013[ Zhang\ X[\ Sanderson\ D[ J[\ Harkness\ R[ M[ + Last\ N[ C[ "0882#[ Connectivity and instability of fractured rock masses[ Proceedin`s of 1nd International Conference on Non!Linear Mechanics "pp[ 818Ð 821#[ Beijing[ Zhang\ X[\ + Sanderson\ D[ J[ "0883#[ Fractal structure and defor! mation of fractured rock masses[ In Kruhl\ J[ H[ "Eds[#\ Fractal and Dynamic Systems in Geoscience "pp[ 26Ð41#[ Springer!Verlag[ Zhang\ X[\ + Sanderson\ D[ J[ "0884#[ Anisotropic features of geometry and permeability in fractured rock masses[ En`ineerin` Geolo`y\ 39\ 54Ð64[ Zhang\ X[\ + Sanderson\ D[ J[ "0885a#[ E}ects of stress on the two! dimensional permeability tensor of natural fracture networks[ Geo! phys[ J[ Int[\ 014\ 801Ð813[ Zhang\ X[\ + Sanderson\ D[ J[ "0885b#[ Numerical modeling of the e}ects of fault slip on ~uid ~ow around extensional faults[ J[ Struc[ Geol[\ 07\ 098Ð008[ Zimmerman\ R[ W[\ + Bodvarsson\ G[ S[ "0885#[ E}ective trans! missivity of two!dimensional fracture networks[ International Jour! nal of Rock Mechanics\ Minin` Science + Geomechanics Abstracts\ 22\ 322Ð327[
Numerical study of critical behaviour of deformation and permeability of fractured rock masses Xing Zhang \ D[ J[ Sanderson Department of Geology\ Southampton Oceanography Centre\ University of Southampton\ Empress Dock\ Southampton\ SO03 2ZH\ U[K[ Received 16 October 0886^ revised 7 May 0887^ accepted 16 May 0887
Abstract The connectivity of fractures in rock masses is determined using a numerical simulation method[ There is a continuous fracture cluster throughout a fractured rock mass if fracture density "d# is at or above a threshold fracture density dc\ Fractal dimension "Df# is used to describe the connectivity and compactness of the largest fracture clusters[ Df increases with increasing fracture density[ Percolation theory is used to determine the universal law\ Df Af"d−dc#f\ which describes the critical behaviour of connectivity of fractures in rock masses[ The results from numerical modeling show that the deformability of fractured rock masses increases greatly with increasing fracture density "i[e[\ fractal dimension#\ and the critical behaviour of deformability can be described by Bs A"d−df#s[ Also\ the overall permeability of a fractured rock mass occurs at or above a critical fracture density "dc# and increases with increasing fracture density[ The critical behaviour of permeability can be described by q Ap"d−dc#p[ The critical behaviour of connectivity and permeability of naturally fractured rock masses is examined using the universal forms[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Keywords] Fractured rock ^ Critical behaviour ^ Connectivity ^ Deformability ^ Permeability
0[ Introduction Fractures of all sizes have a dominant e}ect on many physical properties of the upper crust\ such as the mech! anical\ hydraulic\ thermal and seismic properties[ It is desirable for many projects\ including hazardous waste disposal\ e.cient oil recovery from fractured reservoirs\ thermal energy extraction and ore deposition study\ to have a thorough understanding of the properties of frac! tured rock masses[ The deformability and permeability of fractured rock masses may increase super!linearly with the increase in fracture density\ being greatly dependent on the degree of fracture connectivity and related charac! teristics\ including aperture\ density\ spacing\ length\ orientation and wall mechanical properties "e[g[\ Engl! man et al[\ 0872 ^ Shimo and Long\ 0876 ^ Rasmussen\ 0876 ^ Zhang et al[\ 0881 ^ Zhang\ 0882#[ More and more studies show that the connectivity of fractures plays a key role in the deformability and permeability of frac! tured rock masses "e[g[\ Balberg\ 0875 ^ Gueguen and Dienes\ 0878 ^ Balberg et al[\ 0880 ^ Zhang et al[\ 0882 ^ Zhang and Sanderson\ 0883 ^ 0884#[ In this study\ a 1!dimensional numerical model is used Corresponding author[
to investigate the connectivity of fractures in a rock mass[ Based on 04 natural fracture patterns and 3 groups of simulated fracture patterns\ the critical stage or per! colation threshold at which connectivity shows a sudden phase change\ is determined[ The relations between frac! ture connectivity and other parameters\ such as fracture density\ length\ orientation and con_guration are inves! tigated\ and fracture density shows the most important role in controlling fracture connectivity where fracture length is relatively small in relation to a region of interest[ The relationship between fracture connectivity and frac! ture density may be described in a form of power law\ in which the fracture connectivity is indicated with a fractal dimension and the power law is characterized by a critical fracture density and an exponent[ Distinct element methods are used to model the defor! mation and permeability of those fracture networks at di}erent phase changes[ It is demonstrated that fracture connectivity has a very important e}ect on deformation and permeability\ and deformation and permeability pre! sent a sudden increase if a connected fracture network exists[ The deformation and permeability of a fractured rock mass can be predicted to some degree based on fracture density which dominantly controls fracture con! nectivity[ The results suggest that the relation between fracture density and deformation or permeability may be
S9153Ð7061:87:,*see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved PII ] S 9 1 5 3 Ð 7 0 6 1 " 8 7 # 9 9 9 2 9 Ð 9
425
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
described with a power law where fracture density is above a critical value[ 1[ Modeling of 1!dimensional fracture networks Fractures are commonly observed as traces intersecting an outcrop\ and the size of a fracture can be described by its trace length in 1!dimensional problems[ In this study\ a numerical model of fracture systems\ based on self! avoiding random generations\ has been developed[ The aim of this work is to stimulate fracture maps under conditions where parameters such as the length\ density and pattern of fractures can be controlled[ The model involves the following procedures ] "a# The simulated region is a square of unit size[ The fracture trace!lengths vary from a lower limit of 9[4Ð 3) of the side of the square\ up to 14)[ The fractures are small in relation to the size of the simulated region and the resulting pattern is scale invariant[ "b# Trace!lengths are sampled from a power!law dis! tribution where the number of fractures "N# of length "L# has a form NaLE "Segall and Pollar\ 0872 ^ Barton and Hsieh\ 0878 ^ He}er and Bevan\ 0889 ^ Jackson and Sanderson\ 0881#[ The exponent can be varied and\ together with limits of fracture sizes\ controls the length distribution[ "c# Fracture orientation is de_ned as the angle that the fracture!trace makes to a reference axis[ Angles may be selected randomly in the range of 9Ð079>\ or with a normal distribution\ with sets of fractures being de_ned by their mean orientation and standard devi! ation "here termed dispersion angle#[ "d# The coordinates of the centre of a fracture are ran! domly selected from a uniform distribution within the square[ A procedure of self!avoiding generation is used\ such that new fractures are selected only if they are located at a minimum distance to previously generated fractures[ The minimum distance may be varied\ but is usually set at 9[4) of the size of the square[ "e# Fractures are generated sequentially\ according to the above rules[ As more fractures are added the density of the fractures increases[ Fracture density is de_ned as the total length of the fracture!traces per unit area[ Four types of fracture patterns have been simulated within a square of 09×09 m ] "i# Group A consists of two sets of parallel\ orthogonal fractures with the same lower limit of 9[1 m and exponent of 0[1 "Fig[ 0#[ "ii# Group B consists of two sets of fractures "as Group A#\ but the dispersion angle of each set is selected as 04>[ The mean orientation of the _rst set is parallel to the x!axis\ and the second set is parallel to the y! axis "Fig[ 1#[
"iii# Group C is similar to group B\ but with a dispersion angle of zero and with di}erent lower limits\ 9[93 m and 9[05 m\ for the fracture trace lengths in the x! and y!directions\ respectively "Fig[ 1#[ "iv# Group D is similar to group A\ but with sets of angles of ¦:− 59> to the x!axis "Fig[ 1#[ These fracture patterns\ together with some natural examples\ were then used as input to distinct element models in order to study deformation and ~uid ~ow[ 2[ Density\ connectivity and fractal dimension Within a fractured rock mass\ some fractures are iso! lated "unconnected#\ whilst others intersect[ At low frac! ture densities\ most are isolated\ although some may be connected locally[ As the fracture density increases\ more fractures become connected and relatively large clusters form "Fig[ 0a#[ At this stage\ no continuous cluster develops throughout the fractured rock mass\ since the largest cluster within the simulated square area does not intersect all the boundaries[ With a further increase in fracture density\ more fractures and clusters become con! nected\ until the largest cluster intersects all the bound! aries of the simulated square area "Fig[ 0b#[ At this critical point\ it is possible to de_ne a critical fracture density[ If more fractures are added to the fractured rock mass\ the largest cluster will grow to cover the whole area\ and the rock mass becomes highly fragmented "Fig[ 0c#[ The connectivity of the largest fracture cluster can be described with its fractal dimension "Df# by box!counting "Zhang and Sanderson\ 0883#[ Df is the slope of a best _t straight line on a plot of a log ðN"s#Ł against Log ð"0:s#Ł\ where N"s# is the number of boxes of size "s# which con! tain at least one fracture belonging to the largest cluster[ The relation between N"s# and "s# for the two fracture patterns in Figs 0b and c is shown in Fig[ 2[ The critical fracture cluster "Fig[ 0b# has a critical fractal dimension\ Dfc 0[161\ and the cluster with a higher density "Fig[ 0c# has Df 0[314[ The fractal dimension of the largest cluster must be between 0 "the dimension of a single fracture crossing the region# and 1 "the dimension of the square area itself#[ The four groups of simulated fracture patterns have been used to investigate the connectivity in relation to fracture density[ Figure 3 shows the relations between fracture density and the fractal dimensions of the largest fracture clusters "connectivity of fractures#[ Fractal dimensions "Df# increase with fracture density "d# where fracture density is above a critical fracture density "dc#[ The critical fractal dimension "Dfc# varies little with change in the model parameters\ such as the lower limit of fracture lengths "Ll#\ power!law exponent "E# and dispersion angle "Ad#[ The critical fractal dimensions lie in a narrow range from 0[11Ð0[27 "average 0[29# for those critical clusters with variations in the lower limit of length
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
426
Fig[ 0[ Simulated fracture patterns and the corresponding largest clusters of group A\ at di}erent fracture densities[ "a# For a low fracture density\ the largest fracture cluster does not intersect all the boundaries^ "b# At the critical density "percolation threshold\ dc\ the largest cluster intersects all the boundaries ^ "c# Above the critical density\ more fractures connect to form the largest cluster and the rock is fragmented[
Fig[ 1[ Simulated fracture patterns with di}erent con_guration[ Group B consists of two sets of fractures with dispersion angle of 04> for each set[ Group C consists of two sets of fractures with di}erent fracture lengths and the di}erence in trace lengths results in an anisotropic factor of 0[707 "Zhang and Sanderson\ 0884#[ Group D consists of two sets of fractures with an angle of 59> between them and this results in an anisotropic factor of 2[33[ Anisotropic factor indicates the anisotropy in geometry of a fractured rock[
from 9[994Ð0[4 m\ in dispersion angle of fracture direc! tion from 9Ð49> and in exponents from 0[1 and 0[7[ This indicates that the fractal dimension of a critical fracture
cluster is dominated by fracture density and is largely independent of other geometrical parameters\ although there is some random variation[ A number of _eld studies
427
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
and orientation\ but can be controlled by appropriate choice of the relative fracture density[ 3[ Critical behaviour of fractured rock masses
Fig[ 2[ Fitting fractal dimension of largest clusters for two fractured rocks using box!counting method[ The largest cluster having the critical density "Fig[ 0b# has a fractal dimension "Dfc of 0[161 and the largest cluster having a higher density "Fig[ 0c# has a fractal dimension "Df# of 0[341[
Percolation theory was _rst used to investigate poly! merization "e[g[\ Flory\ 0830 ^ Stockmayer\ 0832#[ Recently\ percolation models have been intensively applied to examine geometric properties and transport phenomena observed in porous rocks "e[g[\ Englman et al[\ 0872 ^ Balberg\ 0875 ^ Charlaix et al[\ 0876 ^ Gueguen and Dienes\ 0878 ^ Balberg et al[\ 0880 ^ Berkowitz and Balberg\ 0881#[ However\ little direct use of percolation theory has been made in the study of the geometry and physical properties of fractured rock masses[ Percolation theory can be used to investigate many macroscopic properties of a system that are determined by the statistic properties of the system elements[ Special properties of a system\ which emerge at a critical value of the system elements\ are known as percolation phenomena[ An advantage of percolation theory is that it provides universal forms which describe the overall properties of a system\ which are independent of its local geometry[ Generally\ the overall properties "Gr# of a sys! tem can be determined by a value "V# describing some statistic property of the elements with a power law of the form Gr"V−Vc#r
"0#
where Vc is the threshold of V and r is some exponent which can be found from theory and:or computer simu! lation and:or experiment[ Such a form also characterises the critical behaviour of some special properties of a system\ which emerge where the value of V is at or above the critical value of Vc[ In this study\ the overall properties "connectivity\ deformability and permeability# of frac! tured rock masses have been investigated in relation to the fracture density "d# of the fractured rock masses[ 3[0[ Universal law of connectivity With a form similar to equation "0#\ the universal law of connectivity of fractures in rock masses can be described with the following equation ] Fig[ 3[ Relationship between fracture density "d# and the fractal dimen! sion "Df# for four groups of simulated fracture patterns with di}erent con_gurations[
have examined the fractal nature of many natural fracture networks over a wide range of scales\ lithologies and geological settings\ as reviewed in Berkowitz and Hadad|s paper "0886#[ In their study\ they also found that the fractal dimension of a synthetic fracture network is rela! tively insensitive to parameters such as fracture length
Df af"d−dc#f
"1#
where Df is the fractal dimension of the largest cluster of a fractured rock mass\ which describes the connectivity and compactness of the fractures ^ d is fracture density\ with a value "dc# at the percolation threshold ^ f is an exponent determining the universal law\ and af is a constant[ Figure 4 shows the relations between fracture density and fractal dimension "connectivity#\ based on the results of the four groups of fracture patterns in Figs 0 and 1[
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
428
Fig[ 4[ Universal laws of connectivity for four groups of fracture patterns with di}erent con_gurations[
The power law exponent " f # has a nearly constant value "9[001Ð9[016# for di}erent geometries\ but the critical density of fractures varies from 3Ð4[4 m:m1[ Hence\ the percolation thresholds "critical density of fractures\ dc# of di}erent con_gurations must be determined separately[ The interest of percolation theory arises from the fact that a given fracture pattern has a certain exponent of f ^ thus\ one can predict the connectivity of a fractured rock mass by _nding f from some similar small fracture net! works in local regions[ In other words\ it is possible to predict the connectivity if the fracture density and the con_guration of the fracture system of the rock mass is known[ Note that the _tted lines through the data points in Fig[ 4 represent average slopes[ Actually\ the data show a slightly concave!upwards curvature\ so di}erent parts of the data may _t di}erent slopes[ Obviously\ the data
at small values of "d−dc# describe the nature of critical behaviour well\ in the neighbourhood of dc[ However\ an average slope can be used to describe the relation between fracture connectivity and fracture density over a rela! tively wide range of fracture density[ Fifteen natural fracture patterns "Fig[ 5#\ which form continuous fracture clusters\ were sampled in the English Lake District\ and tested to _nd the universal law for connectivity[ Each fracture pattern was sampled within a square of 1×1 m\ with all fractures greater than 9[0 m in trace length[ In each of the samples\ the fracture pattern consists of 2Ð3 very steeply inclined fracture sets[ Hence\ the maps of sub!horizontal exposures provide a reason! able 1!dimensional characterization of the fracture network[ The same procedures have been used to calculate the fractal dimensions of the largest cluster for each network[
439
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 5[ Fifteen fracture patterns sampled from exposures in the English Lake District[ Each pattern has an area of 1×1 m and only those trace lengths greater than 9[0 m are shown in diagrams[
The universal exponent " f # for connectivity has a rather low value of 9[095 "Fig[ 6# ^ the points in Fig[ 6 are scattered due to local variation in fracture con_guration and density[ 3[1[ Critical behaviour and deformability Numerical experiments\ using the distinct element method "UDEC i[e[\ Universal Distinct Element Code\ Version 1[9\ 0882#\ have been carried out to study the deformability of fractured rock masses[ The fractured rock mass is treated as a series of elastic blocks with displacement possible along and across fractures when loaded "Cundall et al[\ 0867 ^ Last and Harper\ 0889#[ Table 0 lists the material properties used[ The normal and tangential displacements between two adjacent blocks are determined directly from block geometry and block centroid translation and rotation[ The force!dis! placement law relates incremental normal and shear for! ces "DFn\ DFs# to the amount of incremental relative displacement "Dun\ Dus# ]
DFn KnDun
"2#
DFs KsDus
"3#
Where Kn and Ks are the contact normal and shear sti}! ness respectively[ Such force!displacement relationships allow the evaluation of shear and normal forces between the intact blocks in a deformed region of fractured rock[ Simulations of biaxial compressive tests have been run\ with _xed con_ning stress "s1# of 9 MPa or 9[2 MPa and di}erent deviatoric stresses "s0Ðs1#[ This enables the study of the deformability of fractured rocks with di}erent geometric and mechanical parameters[ The rock and frac! ture parameters are selected to represent a tight sandstone "e[g[\ Barton et al[\ 0874 ^ Yoshinaka and Yamabe\ 0875 ^ Last and Harper\ 0889#[ Figure 7 shows the schematic diagram of the loading and monitoring system of the numerical experiments[ The displacements in two direc! tions are measured at eight points so that average defor! mation in axial and lateral directions could be obtained[ Rocks with a fracture density below the percolation threshold "dc# show only a small amount of deformation\
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
430
Fig[ 6[ The universal law of the natural fracture patterns for connect! ivity[ Fig[ 7[ Schematic illustration for loading and monitoring system of numerical experiments[ Two deformation components in the x! and y! directions at eight points were measured[
Table 0 Material parameters used in this modeling Value
Units
Block property Density Shear modulus Bulk modulus
1499 04[3 22[2
kg:m2 GPa Gpa
Fracture property Tensile strength Cohesion Friction angle Dilation angle Joint normal sti}ness Joint shear sti}ness
9 9 14 9 099 59
MPa MPa Degree Degree GPa:m GPa:m
Young|s modulus 39 GPa^ Poisson|s ratio 9[2
mainly contributed by the elastic deformation of intact blocks[ Where fractures barely form a completely con! nected cluster "Fig[ 8a# i[e[\ at the percolation threshold\ the critical fractal dimension is 0[29\ the larger defor! mation occurs due to shear displacements and openings along a few fractures[ At higher fracture density\ the connected fracture cluster\ with a fractal dimension of 0[28\ covers more of the simulated area\ and the resulting rock mass is more fragmented "Fig[ 8b#[ Deformation of the fractured rock increases abruptly and is pre! dominantly contributed by shear displacements and openings rather than elastic deformation of the blocks[ The stressÐstrain behaviour of three models with
di}erent fracture densities is shown in Fig[ 09[ At the critical fracture density "dc 4[44 m:m1\ Fig[ 8a#\ the stress!strain curve still shows an approximately elastic behaviour\ since the deformation is mainly contributed by the elastic deformation of the intact blocks[ Above the critical fracture density "Df 0[23\ d 5[4 m:m1#\ the stressÐstrain behaviour presents softening\ i[e[\ the ratio of stress to strain drops at a deviatoric stress of 1[6 MPa\ and ~ow begins at a deviatoric stress of 2[5 MPa\ with the development of the shear and normal displacements along fractures[ At still higher fracture density "Df 0[28\ d 6[44 m:m1\ Fig[ 8b#\ yield occurs at a lower deviatoric stress level "0[7 MPa#\ and ~ow occurs at 1[3 MPa\ since more fractures contribute displacement to the defor! mation of the fractured rock[ At di}erent levels of con_ning stress "9 and 9[2 MPa#\ the relationship between deformation and fracture den! sity is summarized in Fig[ 00[ Although large deformation occurs above the percolation threshold "dc#\ a sudden increase occurs when the fracture density exceeds the second threshold of fragmentation "df#\ independent of con_ning stress level[ The second threshold is at a value of 5[4 m:m1 "an associated fractal dimension of 0[24# and corresponds to the fragmentation of a rock mass[ The following form ] Bs as"d−df#s
"4#
is used to describe the critical behaviour of deformability
431
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 8[ Geometry before "left# and after loading "right# of fractured rock masses "a# at a critical fractal dimension of 0[29 and critical fracture density of 4[14 m:m1 ^ after loading\ some displacements and openings can occur along a few fractures ^ "b# at a higher fractal dimension of 0[28 and fracture density of 6[66 m:m1[ After loading\ more displacements and openings occur along many fractures[
for the fractured rock masses "Fig[ 01#[ Here Bs is the deformation ")# of the fractured rock masses\ df is the second threshold of fragmentation\ s is an exponent determining the universal law\ and as is a constant[ Although the value of con_ning stress has little e}ect on the second threshold\ the universal exponent "s# is signi_cantly di}erent\ about 9[53 for a zero con_ning stress and 9[80 for a con_ning stress of 9[2 MPa[ This means that without a con_ning stress\ deformation may develop early[ 3[2[ Critical behaviour of permeability Numerical modeling of ~uid ~ow through fractured rock masses has been carried out using UDEC\ based on
the geometric models of groups AÐD[ In the ~ow model\ it has been assumed that ~ow is through fractures and the matrix rock is impermeable[ The ~ow!rate through a fracture "qj# is calculated by the cubic law "e[g[\ Snow\ 0857 ^ Witherspoon et al[\ 0879 ^ Zhang and Sanderson\ 0885a# ] qj "0:01m#a2"DH:DL#
"5#
where m is dynamic viscosity ^ and a is fracture aperture[ For each fracture pattern\ ~ow!rates under a hydraulic pressure di}erence of 09 kPa "or a hydraulic gradient of 0 kPa:m# are calculated in both the x or y!direction\ as shown in Fig[ 02a[ It can be shown\ both theoretically and in the labora!
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
432
Fig[ 09[ StressÐstrain behaviours of 2 fractured rock masses with di}erent fractal dimensions "i[e[\ di}erent fracture densities#[ For the fractured rock mass with a fracture density at percolation threshold of 4[14 m:m1 and Dfc of 0[29 "Fig[ 8a#\ the stress!strain presents an approximate elastic behaviour[ With fracture density at the second threshold "corresponding to fragmentation# of 5[4 m:m1 and Df of 0[23\ the stress!strain behaviour presents softening and ~ow at relatively low deviatoric stresses "1[6 MPa and 2[5 MPa#[ At a higher fracture density of 6[66 m:m1 and Df of 0[28 "Fig[ 8b#\ the strain!softening and ~ow occur at lower deviatoric stresses "0[7 and 1[3 MPa#[
Fig[ 00[ Axial and lateral deformation of fractured rock masses undergoing di}erent con_ning stresses against fracture density[
tory\ that the stress applied to a fractured rock can have a signi_cant in~uence on its e}ective permeability due to the closure of aperture\ and a varying aperture is likely to exist in a fracture network "Snow\ 0857 ^ Charlaix\ 0876 ^ Zimmerman and Bodvarsson\ 0885 ^ Zhang and
Sanderson\ 0885a ^ 0885b#[ The sti}ness of fractures "nor! mal sti}ness and shear sti}ness# are two important par! ameters which control the permeability of fractured rock masses[ However\ description of fracture properties still appears to be a problem\ and widely accepted quan!
433
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 01[ Universal laws of axial and lateral deformability of fractured rock masses with di}erent con_ning stresses[ The value of con_ning stress has little e}ect on the second threshold\ but the universal exponent " f # is signi_cantly di}erent\ about 9[53 for a zero con_ning stress and 9[80 for a con_ning stress of 9[2 MPa[
titative procedures are limited "Bandis\ 0889#[ Exper! imental measurements show that fracture sti}ness is related to fracture topography\ the properties of intact rock\ and is a highly non!linear function of the e}ective stress normal to a fracture "Cook\ 0881#[ Clearly the properties of fractures and the stress state surrounding them should have signi_cant e}ects on their permeability[ Investigating the e}ects of fracture property and stress state is out of the scope of the paper\ and here all fractures used for ~uid ~ow modeling are assigned a constant hydraulic aperture of 9[4 mm[ Hence\ the ~ow!rates of
fractured rock masses are controlled by the fracture con! nectivity only[ Figures 02b and c show the ~ow!rates and ~ow direc! tions\ under a head gradient in the y!direction\ through the fracture patterns in Figs 0b and c ^ a thicker line indicates a higher ~ow!rate[ The ~ow!rate through the fracture pattern in Fig[ 0a is zero because there is no continuous network connecting all the boundaries[ Figure 03 shows that relations between fracture density "d# and the average ~ow!rates for the four groups of fracture patterns[ Clearly\ ~ow!rates increase with
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
434
Fig[ 02[ Illustrations for ~uid ~ow modeling[ "a# Hydraulic boundary conditions used in the modeling[ Two simulations have been carried out for each fracture patterns in two orthogonal directions with a constant hydraulic gradient of 0 kPa:m ^ "b# Flow!rates in the y!direction through the fracture patterns in Figs 0b and c ^ "c# Directions and paths of ~uid ~ow in the y!direction through the fracture patterns in Figs 0b and c[ Only those ~ow!rates which exceeded 4×09−5 m2:s are shown in the diagrams of Figs 02b and c[
increasing fracture density above the percolation thr! eshold "dc#[ From Equation "0# again\ the critical behav! iour of permeability for fractured rock masses can be described with q ap"d−dc#p
"6#
where q is the average ~ow!rate\ p is an exponent deter! mining the universal law\ and ap is a constant[ Figure 04 shows the results for four groups of fracture patterns[ The exponent "p# varies between 0[936 and 0[257 for di}erent fracture patterns\ with a critical density of 3Ð4[4 m:m1[ Di}erent models "con_gurations# of frac! ture patterns have a signi_cant e}ect on p[ It is important to note that the value of p is around 0[2 for isotropic patterns of fractures "groups A and B#[ For a two dimen! sional model of lattice patterns\ the universal exponent of conductivity of a network is 0[2 "e[g[\ Stau}er\ 0874 ^ Sahimi\ 0876#[ Also\ Berkowitz and Balberg "0881# simu! lated a 1!dimensional porous medium and obtained a universal exponent of the overall conductivity of 0[2[ It seems that for randomly oriented\ isotropic fracture
patterns with fracture length being much less than the region of interest\ the universal exponent "p# is about 0[2[ However\ anisotropy "Groups C and D# of fracture patterns has a major e}ect on both the universal exponent and percolation threshold for permeability "also see Zhang and Sanderson\ 0884#[ The natural fracture patterns are also used to test the critical behaviour of permeability "Fig[ 05#[ As expected\ the universal exponent "p# for permeability has a rela! tively low value of 0[045 because of the relatively large length of fractures in the natural fracture patterns[
4[ Conclusions Numerical modeling methods have been used to inves! tigate connectivity of 1!dimensional fractured rock masses against various fracture parameters such as frac! ture density\ length\ orientation and con_guration[ Frac! ture density plays the most important role in forming the critical fracture cluster at which a completely connected
435
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Fig[ 03[ Relationship between fracture density "d# and the average ~ow!rates for four groups of simulated fracture patterns with di}erent con_gurations[
Fig[ 04[ Universal laws of permeability for four groups of fracture patterns with di}erent con_gurations[ The universal exponent "p# varies from 0[936Ð0[257 with a percolation threshold between 3Ð4[4 m:m1[ Di}erent models "con_gurations# of fracture patterns have a signi_cant e}ect on the value of exponent "p#\ particularly the anisotropy of a fractured rock mass "Zhang and Sanderson\ 0884#[
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
436
characteristics of local samples which are relatively easy to obtain[ The quantitative conclusions of this study are ]
Fig[ 05[ Universal laws of the natural fracture patterns for permeability[ The universal exponent "p# for permeability has a relatively low value of 0[045 because of the relatively large length of fractures in the natural fracture patterns[
network barely forms[ Connectivity is relatively insen! sitive to the other fracture parameters\ if fracture length is relatively small in relation to the region of interest[ There is a percolation threshold at a fracture density below which no connected network exists[ A fractal dimension is used to measure the connectivity and com! pactness of the largest fracture cluster where fracture density is above the critical one\ and it increases with the increase of fracture density[ Distinct element methods have been applied to simu! late deformation and ~uid ~ow of fractured rock masses[ Based on percolation theory\ the critical behaviour "con! nectivity\ deformability and permeability# of fractured rock masses can be described using universal laws where the critical fracture density can be determined[ Per! colation theory is also applied to predict the connectivity and permeability of natural fracture patterns[ Comparing with those simulated fracture patterns\ the natural sam! ples are more scatter of the points\ particularly in the ~ow tests[ However\ their connectivity and permeability can still be described with a universal law form to some degree[ Based on the four groups of simulated fracture patterns and _fteen natural fracture patterns\ it seems that per! colation theory can be used to investigate the connect! ivity\ deformation and permeability\ and universal laws may provide a way to understand the overall properties of the fractured rock masses by considering the statistical
"0# The critical fractal dimension "Dfc# of the critical frac! ture clusters has a nearly constant value\ and is a key parameter in the determination of the critical behaviour of connectivity[ Dfc lies between 0[11 and 0[27 for the variation in the size distribution "E 0[1 to 0[7# ^ the dispersion angle of fracture orientation "Ad 9Ð49># ^ and the lower limit of trace length "Ll 9[994Ð0[4 m for a sample size of 09×09 m#[ "1# The critical behaviour of connectivity can be described with a universal form Df af"d−dc#f\ on the basis of percolation theory[ For simulated frac! ture patterns with di}erent con_gurations\ the uni! versal exponent " f # varies between 9[001Ð9[016[ Anisotropy of fracture patterns has a signi_cant e}ect on f and the critical fracture density must be deter! mined separately[ "2# Distinct Element modeling of simulated fracture pat! terns shows that there are sudden changes in the deformability of the rock mass at the percolation threshold "dc# and the second threshold of frag! mentation "df#[ The deformability of rock mass increases above the percolation threshold "dc#\ but large deformation occurs at or above the second thr! eshold of fragmentation "df# of 5[4 m:m1[ The critical behaviour of deformability can be determined with Bs as"d−df#s\ and the universal exponent "s# is sig! ni_cantly di}erent\ about 9[53 for a zero con_ning stress and 9[80 for a con_ning stress of 9[2 MPa[ "3# At or above the percolation threshold\ the per! meability increases with fracture density[ The critical behaviour can be determined with q ap"d−dc#p\ and the universal exponent "p# varies between 0[936Ð 0[257[ Anisotropy of fracture patterns has a major e}ect on the universal exponent "p#[ Acknowledgements Funding for part of this study was provided by the Natural Environment Research Council "grant GR8:0132#[ Natural fracture networks were sampled in conjunction with British Geological Survey and Nirex Ltd[ "U[K[#[ The authors would like to thank Professor D[ G[ Roberts and an anonymous reviewer for detailed comments and constructive suggestions which improved the manuscript[
References Balberg\ I[ "0875#[ Connectivity and conductivity in 1!D and 2!D frac! ture systems[ In R[ Englman and Z[ Jaeger "Eds[# ^ Proceedin`s of International Conference on Fra`mentation\ Form and Flow in Fractured Media\ 7 "pp[ 78Ð090#[ Annals of Israel Physical Society\ Bristol\ U[K[ ] Adam Hilger[
437
X[ Zhan`\ D[J[ Sanderson:Marine and Petroleum Geolo`y 04 "0887# 424Ð437
Balberg\ I[\ Berkowitz\ B[\ + Drachsler\ G[ E[ "0880#[ Application of a percolation model to ~ow in fractured hard rocks[ Journal of Geophysics Research\ 85"B5#\ 09904Ð09910[ Bandis\ S[ C[ "0889#[ Mechanical properties of rock joints[ In N[ Barton and O[ Stephansson\ "Eds[#\ Rock Joints "pp[ 014Ð039#[ Barton\ C[ C[\ + Hsieh\ P[ A[ "0878#[ Physical and hydraulic!~ow proper! ties of fractures[ 17th International Geological Congress\ T274[ Barton\ N[ R[\ Bandis\ S[\ + Bakhtar\ K[ "0874#[ Strength\ deformation and conductivity coupling of rock joints[ International Journal of Rock Mechanics\ Minin` Sciences + Geomechanics Abstracts\ 11\ 010Ð039[ Berkowitz\ B[\ + Balberg\ I[ "0881#[ Percolation approach to the prob! lem of hydraulic conductivity in porous media[ Transportation of Porous Media\ 8"2#\ 164Ð175[ Berkowitz\ B[\ + Hadad\ A[ "0886#[ Fractal and multifractals of natural and synthetic fracture networks[ Journal of Geophysics Research\ 091"B5#\ 01194Ð01107[ Charlaix\ E[\ Guyon\ E[\ + Roux\ S[ "0876#[ Permeability of a random array of fractures of widely varying apertures[ Transport in Porous Media\ 1\ 20Ð32[ Cook\ N[ G[ W[ "0881#[ Jaeger Memorial Dedication Lecture!Natural joints in rock ] Mechanical\ hydraulic and seismic behaviour and properties under normal stress[ International Journal of Rock Mech! anics\ Minin` Sciences + Geomechanics Abstracts\ 18\ 087Ð112[ Cundall\ P[ A[\ Martin\ J[\ Beresford\ P[\ Last\ N[ C[\ + Asgian\ M[ "0867#[ Computer modeling of jointed rock masses[ US Army En`in! eers\ WEST Technical Report[ N!63!7[ Englman\ R[\ Gur\ Y[\ + Jaeger\ Z[ "0872#[ Fluid ~ow through a crack network in rocks[ Journal of Applied Mechanics\ 49\ 696Ð00[ Flory\ P[ J[ "0830#[ Molecular size distribution in three dimensional polymers[ Journal of American Chemical Society\ 52\ 2972Ð2099[ Gueguen\ Y[\ + Dienes\ I[ "0878#[ Transport properties of rocks from statistics and percolation[ Mathematical Geolo`y\ 10"0#\ 0Ð02[ He}er\ K[ J[\ + Bevan\ T[ G[ "0889#[ Scaling relationship in natural fractures ] Data\ theory and application[ Society of Petroleum Engineers[ SPE\ 19870\ 256Ð265[ Jackson\ P[ + Sanderson\ D[J[ "0881#[ Scaling of fault displacements from the Badajoz!Cordoba shear zone\ SW Spain[ Tectonophysics\ 109\ 068Ð089[ Last\ N[ C[\ + Harper\ T[ R[ "0889#[ Response of fractured rock subject to ~uid injection[ Part I ] Development of a numerical model[ Tec! tonophysics\ 061\ 0Ð20[ Rasmussen\ T[ C[ "0876#[ Computer simulation model of steady ~uid ~ow and solute transport through three!dimensional networks of variably saturated\ discrete fractures[ In D[ D[ Evans and T[ J[ Nicholson "Eds[#[ Flow and transport throu`h unsaturated rock "pp[ 096Ð003#[ Geophysical Monograph 31\ American Geophysical Union[ Sahimi\ M[ "0876#[ Hydrodynamic dispersion near the percolation thr! eshold ] Scaling and probability densities[ J[ Phys[ A[ Math[ Gen[\ 19"07#\ L0182ÐL0187[
Segall\ S[ W[\ + Pollard\ D[ D[ "0872#[ Joint formation in granite rock of the Sierra Nevada[ Geolo`ical Society of America Bulletin\ 83\ 452Ð464[ Shimo\ M[\ + Long\ J[ "0876#[ A numerical study of transport par! ameters in fracture networks[ In D[ D[ Evans and T[ J[ Nicholson "Eds[#[ Flow and Transport throu`h unsaturated rock "pp[ 010Ð021#[ Geophysical Monograph 31\ American Geophysical Union[ Snow\ D[ T[ "0857#[ Rock fracture spacing\ openings and porosities[ J[ Soil Mech[ Found[ Div[\ ASCE\ 83"SM0#\ 62Ð80[ Stau}er\ D[ "0874#[ Introduction to Percolation Theory[ London ] Taylor and Francis[ Stockmayer\ W[ H[ "0832#[ Theory of molecular size distribution and gel formation in branched!chain polymers[ Journal of Chemical Physics\ 00\ 34Ð44[ Universal Distinct Element Code\ Version 1[9 "0882#[ ITASCA CON! SULTING GROUP\ INC[ Minneapolis\ Minnesota\ U[S[A[ Witherspoon\ P[ A[\ Wang\ J[ S[ Y[\ Iwai\ K[ + Gale\ J[ E[ "0879#[ Validity of cubic law for ~uid ~ow in a deformable rock fracture[ Water Resour[ Res[\ 05\ 0905Ð0913[ Yoshinaka\ R[\ + Yamabe\ T[ "0875#[ Joint sti}ness and the defor! mation behaviour of discontinuous rock[ International Journal of Rock Mechanics\ Minin` Sciences + Geomechanics Abstracts[\ 12\ 08Ð17[ Zhang\ X[\ Harkness\ R[ M[\ + Last\ N[ C[ "0881#[ Evaluation of connectivity characteristics of naturally jointed rock masses[ En`in! eerin` Geolo`y\ 22\ 00Ð29[ Zhang\ X[ "0882#[ Shear resistance of jointed rock masses and stability calculation of rock slopes[ Geotechnical and Geolo`ical En`ineerin`\ 00\ 096Ð013[ Zhang\ X[\ Sanderson\ D[ J[\ Harkness\ R[ M[ + Last\ N[ C[ "0882#[ Connectivity and instability of fractured rock masses[ Proceedin`s of 1nd International Conference on Non!Linear Mechanics "pp[ 818Ð 821#[ Beijing[ Zhang\ X[\ + Sanderson\ D[ J[ "0883#[ Fractal structure and defor! mation of fractured rock masses[ In Kruhl\ J[ H[ "Eds[#\ Fractal and Dynamic Systems in Geoscience "pp[ 26Ð41#[ Springer!Verlag[ Zhang\ X[\ + Sanderson\ D[ J[ "0884#[ Anisotropic features of geometry and permeability in fractured rock masses[ En`ineerin` Geolo`y\ 39\ 54Ð64[ Zhang\ X[\ + Sanderson\ D[ J[ "0885a#[ E}ects of stress on the two! dimensional permeability tensor of natural fracture networks[ Geo! phys[ J[ Int[\ 014\ 801Ð813[ Zhang\ X[\ + Sanderson\ D[ J[ "0885b#[ Numerical modeling of the e}ects of fault slip on ~uid ~ow around extensional faults[ J[ Struc[ Geol[\ 07\ 098Ð008[ Zimmerman\ R[ W[\ + Bodvarsson\ G[ S[ "0885#[ E}ective trans! missivity of two!dimensional fracture networks[ International Jour! nal of Rock Mechanics\ Minin` Science + Geomechanics Abstracts\ 22\ 322Ð327[