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Earch Sciences Division, Lawrence Berkeley Laboratory, California

JANE C. S. LONG AND PAUL A. WITHERSPOON

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 90. :\0.B4, PAGES 3087-3098, MARCH lO. 1985

meability of the nelwork than would be predicted by Snow's

theory. BOlh lhe degree of interconneclion and lhe heterogen-eilY wil! cause lhe aClual permeabilily to be less than lhatpredicted by Snow's lechnique. This pape r wil! examine the

elfect of degree of interconneclion on the magnitude and

nalure of permeabilily. The elfecl of helerogeneity wil! be dis-cussed in a laler paper.

Several approaches have previously been introduced 10 ac-count for lhe interconnection and helerogeneity elfects which

5110'I"s [1965, 1969J theory ignores. Parsons [1966J, Caldwell

[1971. 1972J, and La Poince and Hudson [1981J used electricalanalog models 10 study flnite fractures. Two-dimensional elec-lrical analog models ali have the drawback that current is

proportional to width af the conduclOr. However, flow in frac-

tures is proportional 10 the aperture cubed. Thus analog

models are restricted to a rather limited range of fractureapenures. but within Iimits lhese models could be used to

e\"aluate lhe permeability tensor for discontinuous fracturenelworks.

In order lO correCl lhe lheoretical lensor predicted using

Snow's method. Rocha and Franciss [1977J have proposed a

fleld method for finding an empirical correction factor. In thismethod a mulliplicalion factor is obtained from the ratio oflhe flow measured in a well lest 10 lhe flow predicted using

Snow's lechnique. This procedure cannot be used to determine

whether or not the rock behaves like a porous medium. nor islhere any evidence lhat such lensors correclly predict flow for

nonradial syslems.Sagar and RlIllchal [1982J extended Snow's [1965. 1969J

lheory for the permeabililY of fractured systems in an attempt

to accounl for finile size of fractures. The aUlhors assumed

thal flow in any fraclUre is independent of flow in the olher

fraclUres and funher lhat flow in each fracture was dependent

~087

The Relationship of the Degree of Interconnectionto Permeability in Fracture Networks

The problem of determining the permeability of a rock mass containing a system of finite fractures ishighly dependent on both the degree of interconnection between fractures and the heterogeneity oftndtvtdual fracture characteristics. This paper examines how degree of interconnection aJfects both mag-nitude and nature of the fracture permeability. The interconnection between given fracture sets is acomplex function of (I) fracture density, i.e., the number of fractures per unit volume, and (2) the fractureextent or size. Unfortunately, neither the density nor the extent of fractures is easily determined fromborehole data. However, fracture frequency can be directly measured in a borehole because it is simplythe number of fractures intersected per unit length of the borehole. The frequency is a measure of the

product of fracture density and size because the probability of a fracture intersecting a borehole isproportional to this product. The effect of the degree of interconnection was investigated by numericallysimulating flow in fracture networks where fracture size and density varied inversely, while the product ofthese two parameters was held fixed. Directional permeabilities of a number of such networks weredetermined. and the hydraulic behavior of each fracture system was compared to that of an ideal porousmedium. The permeability of the rock matrix between the fractures was assumed to be low enouw to benegligible. The results show that as fracture length increases. lhe degree of inlerconnection i;creases.Thus, for a given fracture frequency as measured in a borehole, lhe permeability of the system increasesas fracture length is increased, and density is proportionally decreased. AIso. fracture systems with

shorter. but more dense fractures. behave less like porous media than do syslems with longer but lessdense fractures. Knowledge of fracture frequency and orientation alone is inadequate when determiningpermeability or deciding the important question of whether a given system behaves like a porousmedium. For many cases where these parameters can be measured in the borehole it is criticai to knowfracture length in order to predict the degree of fracture network interconnection. However, for certaincases where fractures are larger than some critical size. it may not be necessary to know fracture lenglhsexactly.

I;-';TRODt.:cTION

Two of the major issues in fracture hydrology are to deter-mine the permeability of fracture networks and to establish

whether or not those networks behave like porous media. In

the pas!. methods developed by 5now [1965. 1969J have beenapplied. In such techniques the orientations and apertures of

lhe fraclures intersected by a borehole are determined in the

field. The fractures are assumed to be infinite in length, andthen an equivalent porous medium permeability tensor can becalculated as an accumulation of the individual fracture per-meabilities.

Because fractures are actual!v flnite. there are three reasonswhy the network permeability \~il! be smal!er than that predic-

ted by Snow's theory. The first is that some of the fractures

may be isolated from the conducting network. Second. a frac-

tUre may be a dead end because it is only connected to thenetwork at one end and therefore does not contribute to the

permeability of the network. Third. if in fact the fracture does

Conducl fluido the contribution of this fracture to the networkpermeability depends on how it is connected to the rest of the

system. For example, flow may be restricted in a situation

where a fracture is connecled to lhe system lhrough a conduitwith a smal!er aperture. In el1ecl. lhe flow is governed by agradient less than the component of lhe fleld gradient in lhe

plane of this particular fraclure. In using Snow's lheory, per-

~eability is calculated by assuming lhal flow in each fractureIS governed by lhe acting componenl of the fleld gradienl in

~he plane of the fracture. Thus a fraclure lhat is partial!ylsolaled by smal!er fractures conlribules less lo the per-

Copyright 1985 by lhe American Geophysical Union.

Paper numba ~B1381.O,- -.)22' 5 004B-1381 S05.00

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MODELING TECHNIQUES

A numerical code has been developed to generate sample

fraClure systems in two dimensions and determine lhe per-meabilily of such systems [Long el aI., 1982: Long, 1983]. Thelwo-dimensional fracture mesh generator (FMG) produces

random realizations of a population of fractures in a square

region called a generation region. Each set of fraclures is gen-

erated independently, and lhen the individual sets are super-imposed (Figure 1). For each set lhe density (number of frac-

tures per unit areal must be supplied to determine the total

number of fraclure centers to be generated (Figure la). Thennormally distributed orientations are randomly assigned lOeach center (Figure lb). The fraclures are randomly truncated

such that the lengths are distributed according to a lognormal

or negative exponential distribution (Figure 1c). Fracturescrossing the boundaries of lhe generation region are truncated

at the boundary. Finally, lognormally distributed apertures

are randomly assigned to each fracture (Figure 1d), and the

seIs are superimposed. This generator produces a fraclure net-work model similar to lhal proposed by Baecher el aI. [1977].

When ali the sets have been generated, a flow region is

selected for tinite element analysis. The flow region is a squarewhich lies enlirely wilhin lhe generation region and is centered

in the generalion region at an arbitrary orientalion. A gradi-ent is applied across the flow region by assigning appropriale

values of head lo lhe fraclures where they inlersect the bound-aries of the flow region (see Long el aI. [1982J for details). Forexample, a head of one is assigned to ali points where frac-

tures intersect the inflow side (side 2). Zero head is assigned to

lhe opposite side which is the outflow side (side 4). The othertwo sides (l and 3) of the flow region are assigned heads which

vary linearly from one to zero (Figure 2). Steady flux into lhe

niques used to study lhe' relalionship between interconnec:. ::n

and permeability are described below.

SUPERIMPOSED

RESU T

SET 2

[JLO~G ASD WlTHERSPOO": FRAcn.;RE hTERCO""ECTIOl' ASD PERMEABIUTI

A

CENTERS

C

LENGTHS

DAPE.RTURES

Fig. 1. Generalion of a fraClure nelwork.

3088

oI I I

---,--- -,----

I I---,---

I

o.25

.50

.75

Boundary conditions applied lO fracture models for per-meability measurement.

o< P 4

O 10 O 10OSIDE 2 " ' : ~ C\ I SIDE 4

" li 11

11 11

-S- - e - . -S- - e - . - e - .'-

Fig. 1.

only on size and orientation in the overall flow field. From

this the authors concluded that "any fracture which does notappear on the boundary of the rock eJement considered is ofno interest in the calculation of the equivalent permeability"

[Sagar and RunchaI, 1982]. While these assumptions are cor-rect for the infinite fracture systems analyzed by Snow, theyare not applicable to discontinuous networks to which theywere applied beca use the effects of interconnection and hetero-geneity are ignored.

In order to study only the effect of interconnection, we haveexamined two-dimensional networks of fractures where ali the

fractures have the same aperture and length. Thus the net-works are homogeneous. and any decrease in permeabilityfrom that which would be predicted by Snow's theory is due

only to the lack of perfect connections between fractures. Since

aperture variation has not been introduced, this study caneasily be compared to a study of percolation effects such as

described by Madden [1983J, EngIman ec aI. [1983J. Robinson[1982]. and others. Percolation theory is primarily used to

determine the probabilily of lhe nelwork being connecled. Thefocus of lhis pape r is the magnitude and nature of per-meabilily in systems wilh various degrees of interconneclion.That is, we determine the magnitude and direclion of the prin-cipal permeabililies (i.e.. lhe permeabilily lensor) which best

explain lhe behavior of lhe network. AI the same time weevaluate how well lhal permeability tensor predicls lhe actual

vagaries in lhe behavior of lhe nelwork. The modeling tech-

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3089

4

3

B

o

2

flow region where flow enters at side 2 and exits at side 4. Fara hamogeneaus, anisotropic medium, II[Kio:)]1/2 versus : x is

an ellipse when plotted in polar coordinates [Marcus and

Evanson. 1961: .'v/arcus. 1962; Bear. 1972]. However, for inho-mogeneous fraetured media, I/[Kg(o:)] 112may not pIot as a

smooth ellipse. In facto the shape of a plot using measuredvalues ofKi:x) for a given test volume af rock may be quiteerratic. This plot can therefore be used as a test of whether or

not the given volume can be approximated as a homogeneousporous medium. If 1 [Kg(a)] 1/2 does not plot at least approxi-mately as an ellipse. then no singIe symmetrie canductivity

tensor can be found to describe the medium. Ir there is no

canduetivity tensor, then flow thraugh the medi um eannot be

analyzed with eantinuum teehniques. Validation of this teeh-nique can be found in the work by Long et ai. [I982J and

Long [1983].

A regression teehnique similar lO that afScheidegger [1954J

is employed to interpret quantitatively the direetional per-meability data by determining a best tit permeability tensor

(Figure 4) that, in general, wiII be anisotropie. In this tech-

(1)

c

LO~G A~D WITHER5POON: FRACTURE INTERCOl'

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(6 )ioL/COS f ) =i.)=LD

N 12 12Number o f in ter sect in(J f r actur es

L 42 42Boreholelenglh

L N/L ,29 .29

t 16 8Mean f r ocr ur e length

I Anum ber cf f roct ures .018 .036

Unlt area

Al co s 8180) .29 .29

A B

This relationship means that for_each set of fractures wem aynot be able to determine i'A and I directly, but we can estimatethe product i.)beca use both i'L and e can be estimated from

direct measurements in the borehole. This product will becalled LD, or the length-density parameter.

Figure 5 illustrates how dilferent values of i oL and igive riseto the same LD. In both Figures 5a and 5b a borehole oflength L penetrates a system of fractures. The fractures in

Figure 5a are twice as long as those in Figure 5b . but there arehalf as many fractures per unit area in Figure 5a as there arein Figure 5b. Both boreholes intersect 12 fractures. and conse-

quently, one would not be able to detect a dilference between

the fracture geometries of the two systems. As will be shownbelow, there is a great deal of dilference in the hydrologic

behavior of these two systems.

ioL = i.) cos e

where i oL is fracture frequency, i.e.. the number of fractures in 'the given set per unit length of sample line: i'A is areal densitywhich becomes the volumetric density in three dimensions; ii~.mean length, which becomes mean area in three dimensions'

and e are the angles between the sample line and the fractur~poles of the given seI. This equation indicates that in twodimensions, the probability of a fracture intersecting a unitlength of borehole is proportional to the areal density and themean fracture length.

Unfortunately, i'A and I are difficult, if not impossible, todetermine from an examination of borehole data. However, ).L

and e can be measured so that if (5) is rearranged with knownson one side and unknowns on the other, we have

- LAL cos

8 o meosuroble borehole poromerer

E

be shown [Robertson, 1970; Baecher er aI.. 1977J that < : prol>.abilistic relationship for fracture frequency i'L can be expressed

b y

Fig. 5. Example of two boreholes of equal length where Figure 50penetrates a system of fractures whose mean fracture length is twicethat of Figure 5b.

(2)

(3)

(4)

-o x

1

(Meosured v~lues ofJ K : l O ),'x /ft

.x

Lo:"lG AND WITHERSPOOS: FRACTURE INTERCO NECTIOJ' Al'o'D PERMEABIUTY

~'

R =I[ K g (: x) - K ijnjnJ2

1 1

JKQ(o )-J(KijOiOj)

X . y

3090

cRjK12 =O

CRlK22 =O

The dilference between the values of directional per-

meability (in the direction nJ calculated using the tensor (Kijnjn) and the actual values calculated from the model (Kg) isconsidered the ' error.'- The mean square erro r can then be

calculated and normalized by dividing by the products of the

principal permeabilities K Iand K2:

nique we solve for the components of Kij by minirnizing the

error R over N dilferently oriented measurements:

where nj is a unit vector in the rxdirection (the direction of the

gradient). We solve for the K lI' K 12' and K22 which minimize

R by simultaneous solution of three equations:

eRiKJl =0

Fig. 4, Pio! of lhe measured \'alues of 1/(Kg)120 and the corre-sponding "bes! fit" ellipse.

This normalized mean square error (NMSE) approaches zero

as the behavior of the fracture system approaches that of ananisotropic, homogeneous porous mediumo The larger NMSE

is, the less the fracture system behaves like a continuum, Theabove techniques have been used here to study the per-

meability of fracture networks with varying degrees of inter-

connection.

BASIS FOR THE LE~GTH-DENSITY STUDY

With the exception of the rather limited number of caseswhere one can map in underground openings, most of what isknown about fracture geometry in the underground is derivedfrom boreholes, It is possible to determine the number of open

fractures that intersect a borehole and estimate their orienta-tion. On the other hand, it is extremely difficult to get a goodestimate of volumetric fracture density, fracture lengths, and

fracture apertures from logging methods or hydraulic tests.The aim of this study was to investigate the elfect on per-meability of some of the fracture geometry parameters that

cannot easily be determined from boreholes. In particular, theelfect of varying fracture length and density is examined. As

will be discussed below, such a study is equi\"alent to an inves-tigation of the degree of interconnectiono

It is not necessary to look at ali possible combinations offracture length and density. For a given set of fractures it can

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3091

Orientation Aperture, em

Standard LD, StandardMean Deviation em-1 Mean Deviation

Set 1 0 20 0.288 0.001 OSet 2 90 20 0.288 0.001 O

TABLE 1. Parameters Used in Ali Length-Density Studies

and therefore permeability continues to increase. An exact

mathematical treatment of the relationship between per-

meability, and ~ for systems with randomly located fractures isapparently not available. However, the length-density studydescribed above wilI provide this relationship beca use if the

product ) =LD is held constant, then C is proportional to 7 .Thus we expect a net increase in permeability when 7 is in-creased and ;'A is decreased such that LD is held constant.Furthermore, since the behavior (as measured by NMSE) is a

function of degree of connection, NMSE should decrease as I

increases. In fact, as will be discussed below, this is exactlywhat we have observed.

A question remains as to the appropriate scale for such

measurements of permeability. TheoreticalIy, in a homoge-

neously fractured rock, permeability is not dependent on scale

as long as the scale af measurement is large enough to include

a statisticalIy representative sample af the fracture network.Pereolation theory can provide a framework for determiningwhether a given sample is large enough to be statisticalIyrepresentative. However, the method of analysis itself doeshave a certain scale dependence. 1f a fracture is truncated by

two boundaries af the flow region, it will canduct fluid wheth-er or not it is connected to the rest of the fracture network (see

Figure 2). The smalIer the sample size and the more poorly

connected the system the more significant this truncation

error is.The scale dependence of percolation frequency has been

studied by Englman et ai. [1983]. These authors defined a

criticaI value of ~, called C * . Infinitely large samples with Cabove ~. have a percolation frequency v equal to one. Thus ali

such samples would be conductive. If C is below C , infinitelylarge samples have v equal to zero. If C > (*, increases in thescale of measurement will increase the percolation frequency.The reason for this is that especialIy at smalI scales of

measurement, it is possible to pick a sample that is between

fracture clusters. Even though the system is welI connected, a

given small sample may not be percolative. This effect declinesassymptopically as the sample size becomes larger than themean space between clusters. T runcation error persists butalso becomes less important as scale of measurement in-creases. We have studied a well-connected sample which con-forms to these observations. In this case, as the scale of

measurement increases, the permeability approaches a con-stant value from above. This case will be presented below.

Englman et ai. [1983J also observed that below the criticai

value, C * , increases in scale of measurement wilI decrease thepercolation frequency. In part this may be due to truncationerro r which is especially significant for small-scale measure-

ments in poorly cannected networks. Such truncation error isan artifact of the method of analysis. A more fundamental

reason for this scale effect is that the measurement happens to

truncate a finite cluster of intersecting fractures. On a larger

scale of measurement one would observe that the cluster ofconnected fractures is not connected to the rest of the system.

(7 )

I <

LONG AND WITHERSPOON: FRACTURE INTERCONNECTION A:-.n PERMEABILlTY

o

A '

c

o

SCQle of meosuremenl

Fig. 6. Eneet of seale of measurement on (a ) pereolation frequeney,(b and ri permeability, and (d) NMSE.

eis essentialIy that fraction of the area of the fracture network

in which there are likely to be intersections between f(actures.In percolation Iiterature, C is calIed the "effective bond oc-cupation probability."

The mean permeability of fracture networksshould alsoincrease as ~ increases. As ~ increases from zero, the fractures

begin to form clusters. When a cluster is large enough totransect the region, then permeability becomes nonzero. As Ccontinues to increase, the size and number of clusters increases

In alI the folIowing examples the value ofLD and the orien-

tatian distribution are the sam~. The values of )'A and 7 arevaried such that the product l)remains a canstant equal to

the chasen value of LD . Intuition tells us that if all else is heldconstant, an increase in fracture length wilI increase the per-meability. Conversely, if alI else is held constant, a decrease infracture density will decrease the permeability. However, it isflOt immediately obvious how permeability will be affected if7is increased by the same factor that lA is decreased. Some

insight can be gained by comparing the parameters used in the. length-density study to those commonly used in percolation

analysis.Englman et ai. [1983J have generated large numbers of frac-

ture networks similar to those studied here. They (and others)have pointed out that the percolation frequency v, which is theratio of percolative networks to the total number, is an in-

ereasing function of a factor C .where

6)

f

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a90

270

75,0175.0

NMSE=.0927

L0121=12 A =,024

27 0

NMSE=.2603

of ;. Figure 6b is a conceptual diagram showing how a series

of individual measurements of permeability might be distrib-uted for a case where ; is above the criticaI value and for acase where ~ is below criticaI. Above criticaI, the expected

value of permeability E(K) is larger than the value measured

on the infinite scale due to the truncation etfect and either

smaller or larger due to the random choice of sample. The

minimum possible value of permeability for any one sample isalways zero, but the theoretical maximum is poorly defined.

The importance of both truncation error and random fluctu-ation declines with increase in scale of measurement. It is not

clear exactly how the expected value of permeability E(K)varies with scale for very smal1 measurements, but with themeasurement technique used here, we expect E(K) to ap-

proach the "infinite" value K", from above because of thepersistent etfects of truncation error. For the case where ( is

below criticaI. measured permeability is always greater than or

equal to the value of permeability for an infinite sample, whichISzero.

Figure 6c is another conceptual diagram showing how wemight expect the plot of E(K) versus scale of measurement to

vary for different values of ;. For values of ~ above criticaI,

E(K) approaches K '" (~) as L increases. However, the particu-

lar value of K", is a function of (: K", (~) increases as (increases. The closer ~ is to ;*,the larger the sample must beto have K ~ K '" (~).Given the fact that a finite number of

90

50,0150.0

270Q

NMSE=.4361

Fig. 7. Permeability results for fraeture lengths 2, 8, 10,and 12.

LONG AND WITHERSPOOI': FRACTURE II'TERCONl'o'ECTIOI' ANO PERMEABIUTI'

TABLE 2. Input Paramelers Csed in the First Series of Length-Density Studies

Fracture Length I Flow Region Number of FracluresISel 1 ando Set 2,. Dimensions L x L, per Set per Unit

Name em em L/I Area i.A, em - 2

L02 2 12.5 x 12.5 6.25 O.J440L08 8 50 x 50 6.25 0.0360LOIO 10 62.5 x 62.5 6.25 0.0288LOl2 12 75 x 75 6.25 0.0240LOl4 14 87.5 x 87.5 6.25 0.0206LOl6 16 100 x 100 6.25 0.0180LD20 20 125 x 125 6.25 0.1440L024 24 150 x 150 6.25 0.0120L064 64 140 x 140 2.34 0.0045

90

12.5112.5

L02l=2 A=.I44

'7 4 < '- 1_{- I \

'\. ' . \ - - -

;,\li{

270

NMSE =3.1410

180

3092

These two etfects are the cause of the scale etfect observed by

Englman el ai. [1983J for systems which were below the criti-

caI value (*. In our study. systems that are clearly below ~*can be identified visual1y. We know that our technique over

estimates the permeability of these systems due to truncationerror alone beca use the true permeability of these systems is

zero.Some difficulties arise for systems where ~ is near (*. As (

approaches ~*.one must examine larger and larger samples in

order to have samples where r is clearly zero or one (see inset,Figure 6a). As ; approaches ;*. the problem becomes that ofdetermining the probability that the system is connected on

the scale of the hydrologic problem itself. Then if the system isconnected. the behavior must be predicted. This is not a trivial

matter because if ; is near ;*. the behavior of the fracturenetwork can depart radical1y from that of an equi\ 'alent

porous medium on scales that are usual1y used to analyze

such problems. Further comments on this topic are given

below, but the analysis of such systems is currently a topic ofresearch.

The above discussion is il1ustrated in Figure 6. Figure 6a isdata replotted from Englman el ai. [1983J. For fracture net-works similar to those examined here, these authors plotted v

as a function of; for ditferent scales of measurement. In orderto examine scale etfect we have replotted the data in terms of vas a function of the scale of measurement L for ditferent values

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3093

90

21 0

NMSEa.OI96

150.0 x 150.0

LD24i=24 .a.012

LD =I.~=0.288

90

27 0

125.0x 125.0

LD20l=20 .=.0144

Except for LD64, the length and width of the flow region L

was kept equal to 6.25 times the fracture length to insure thatan equal proportion of fractures would be truncated in the

flow region of each case. The exception in LD64 is explainedbelow. From previous experience the facto r of 6.25 was ex-pected to yield flow regions where the mean fracture length

was reasonably dose to the vaIue used in generating the net-

works. That is, only a small proportion of the fractures was

truncated by the flow region. Except for LD64 the mean frac-IUre length was always about 85% of the input value. Thus

each of the generated fracture systems was equally repre-sentative.

beca use similar results are expected for all values of LD.Indeed, from percolation theory we expect that with ali elseheld constant the permeability of a fracture system will simply

be proportional to the magnitude ofLD.

Orthogonal fracture sets were used in all studies beca use ifthe sample size is sufficient!y large, the theoretical shape of the

permeability ellipse for orthogonal fractures of constant lengthand aperture is a cirde. Thus the degree of anisotropy would

be known because the systems are isotropic.

In the first series, three parameters were varied in a system-atic manner. As listed in Table 2, these were the fracture

length, the flow region size, and the fracture density A Inputvalues of fracture density and length were calculated using

RESULTSOF THE FIRST SERIES OF

LENGTH-DENSITY STUDlES

Figures 7 and 8 present the results for variations in length

and density from LD2 to LD24. In each figure the top half

shows the networks used to calculate permeability. These flowregions have been reduced by eliminating the dead ends on

the fractures and all isolated fractures. In each figure, only the0 rotation is shown as an example, but for each model a total

of six dilferently oriented flow regions were defined. For eachflow region, each of lhe four sides were sequentially used as

lhe inflow side. Thus permeability was measured every 15

over a range of 360.

900

270

100.0 x 100.0

LD16l=16 .=.018

NMSE' .0305 NMSE a .0177

Fig. 8. PermeabiJity results for fracture lengths 14, 16, 20, and 24.

LONO ANO WlTHERSPOON: FRAC11JRE INTIRCONNECTION ANO PERMEABILITY

270

87.5x87.5

Q

90

NMSE .0665

LD14

1'14 .=.021

180

fractures intersect a line sample, a practical maximum valuefor K x is found for ( equal to infinity by assuming that the

fractures are infinite in length, i.e., using Snow's [1965, 1969Jtechnique. This points out a major dilference in the variationof percolation frequency and permeability with C . Infinite sam-

pIes always have a percolation frequency of unity if ( exceeds

the criticaI value. However, for C > C * , the permeability of

infinite samples remains a function of (, as has also been

observed by S. Wilke et a!. (unpublished manuscript, 1983).Larger values of C yield larger values of permeability. Belowthe criticaI value of (, K xis zero.

Finally, Figure 6d shows how NMSE might vary with scale.An interesting question is whether the NMSE of networkswith ( above C * must always approach zero as the scale ofobservation increases. However, as discussed above, this may

be a moot question because what we need to know is thebeha vior on some specific scale not on the infinite scale. Someexamples will be presented below to support the theoreticaldiscussion given here.

P.-UtA.\{ETERSUSED IN THE LENGTH-DE"SITY STUDY

As mentioned earlier, the length-density study was carried

out using systems where ali fractures had the same aperture

and were uniform in length. The system parameters are givenin Table 1. Table 2 gives the parameters that were varied inorder to study the elfect of increasing length while keeping the

product of length and density constant. The units used in all

these studies are length in centimeters and hydraulic conduc-tivity in centimeters per second.

The length-density parameter LD was arbitrarily held at afixed value of 0.288 cm - 1 for all of the length-density studies

presented. This means that a borehole driven through such a

rock would intersect roughly 288 fractures in 10 m. However.

this approach could easily be scaled to represent a fracturedrock having an LD of say 0.288 m - 1 or about three fractures

for every 10 m of borehole. What is important is relativemagnitude of the results not the absolute magnitude. Fur-

thermore. it can be shown that given the same length distri-

bution. larger values of LD will produce systems of higher

permeability; lower values will produce lower permeability.

Jilferent values of LD were not tried in these simulations

riesibor a

ted

ed

herh e

is

ed .

u-otK)he

p-

eis

h

e0

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LoNG AND WITHERSPOO~: FRACTURE INTERCONNECTJO~ ANO PERMEABILlIT

The bottom halves of Figures 7 and 8 show the per-meability ellipses for each network. Dashed lines connect the

values of I/(K g r 1from values of directional permeability K gcalculated with the model. The area enclosed by the dashedlines is shaded. The smooth ellipse shown by the solid line best

(9 )

0.288=-1-(6.25l)1

=11.25/

fits the data as caJculated by a regression technique. The XXand YY axes are the axes of the first and second principal

permeabilities, respectively. The ellipse provides a simple visu.alization of how well the permeability can be represented by a

symmetric tensor. The closer the shaded area is to the ellipse,the better the representation. However, the ellipse is a pIot of1/(Kg)I/1 and thus is not linear in Kg Care should be used in

interpreting the magnitude of the permeability and the scatter

of the measurements.

Each of the fracture meshes on Figures 7 and 8 are plotted

using a different scale. The scale is varied such that each plot

is the same size on the figure. The fiow region size L is a

constant multi pie of 6.25 times the fracture length I. Thus thefractures in ali plots appear to be the same length. Movingfrom I= 2 t o I= 24 on Figures 7 and 8, the actual fracturelength I and the region size L are increasing, and the areal

density i' A is decreasing. The net result is that the number offractures in each mesh is increasing as I increases. Thus thereis an apparent increase in density which is proportional to I:

Number of fractures in the fiow region =i'A I3

I r ali the meshes in Figures 7 and 8 were drawn to the samescale. it could be seen that there are actually more fractures

per unit area in LD2 than LD8, etc., and that the fractures in

LD2 are shorter than those in LD8, etc.

Drawing ali the meshes to the same size is useful beca usethe '"apparent density" has a strong infiuence on the hydro-

logic behavior of the fracture system. The longer the fracture,the higher the '"apparent density," the higher the permeability,

and the lower the NMSE. In fact, the number of fractures in

the fiow region is proportional to (. This shows graphicallywhy K is a function of ~. Figure 9 is a plot of the values of the

principal permeabilities K l' K1and NMSE versus fracture

length. s I increases from 2 to 24, permeability increases andthe NMSE factor decreases.

It appears that the plot of K as a function of I on Figure 9 is

t= ,288

LI e = 6,25

wr./)

:: Ez

309 4

I

1

- - - - - - - -

1

1I

12 I1I

I

. . . 10 r I

1

" f

I

'" 1

" -1

E I~ I

N

: I I

~ I

II

. , z I

{ I

I

j1III

I 1 1 I, I 1 I 1 , 10 4 6 8 10 12 14 1 6 1 8 2 0 22 24 26

Frocture lenglh (f.) cm

F ig . 9 . P e rm ea bi li ty a nd N MS E v er su s f ra ct ur e l en gt h f or t he f ir st

series of length-density s tudies .

70

20"

60

----- )Peiol esli maledby red uci ng mes h sizeloLli=2.34

504030

Hypolheticalcurve

Tneor,tical assymtope 'Qual tothe permeobil ity predicttd byS n ow ' s t tc h n iQ u t fo r i n f in i te fr o ctu r es

20

Dolo poinls-for Lli'=6,25

lO" 15"

Fraelure lenglh, e (em)Calc ula ted v al ues o f averag e p er meabi li ty f or fr ac tur e l eng th s f raro 2 t o 6 4.

,O

o

Fig. 10.

2.6

2. 4

2.2

2.0u

~ \.8

"E 1.6u~o 1.4

1.2

~ N I I . O+_ NO.8

: . : :-0,6

0.4

0,2

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LONG AND WITHERSPOON: FRACTURE INTERCON~"ECTlON ANO PERMEABILIT'!\

X

Fig. 11 . Different sized flow region sizes at 0 rotation for the REV study for the case ofI =20.

3095

160.140.120.

obtained with a fracture length of 64 and those obtained ear-lier (Figure 9). The permeabilities continue to increase with

fracture length but at a decreasing rate. Using Snow's [1965,1969J theory, one can easily calcula te the permeability of a

parallel system of fractures of inflnite length with the spacingand aperture as used in this work (Table 2). A value of

K =2.35 cm/s was obtained, and this is illustrated on Figure10 by the horizontal solid line. As a further check on this

point, a numerical calculation of permeability was carried outusing our model and assuming a single random case of "infi-nite" fracrures wirh a distributed orientation. The result wasonly slightly higher than that obtained with Snow's approachand is shown on the figure by the horizontal dashed line.

Snow's [1965, 1969J technique assumes that ali fractures are

infinite in length. which means that they are perfectly connec-ted. The permeability predicted using his technique is there-fore a theoretical maximum value to which the numerical re-sults for permeability of discontinuous fracture systems shouldasymptote. It is evident on Figure 10 that the permeabilities

obtained with our model are approaching the maximum value

of Snow as I increases.To show the effect of truncation error. a hypothetical curve

has been drawn through the data on Figure 10. In general, our

numerical approach tends to overestimate permeability be-cause of truncation error. For the case of 1= 2 there is clearly

no connection. so truncation accounts for ali the permeability.As7 increases. tbis error becomes less important.

Figures 7, 8. and 9 could be used to separa te qualitativelyfracture systems which behave like porous media from those

that do noto The closer NMSE is to zero. the more thesefracture systems are like porous media. From a perus aI of

these figures one might conclude that for these fracture sys-

tems a value of NMSE below 0.05 means that the medi um

80. 100.60.40.20.0.

180.

beginning to leveI off at higher values of I. This could beanticipated beca use as I becomes larger, the density becomes

smaller and the effect of increasing length is partially canceledby decreasing density. In this experiment the limit as I goes tozero can be examined, but the limit as I goes to infinity isdifficult to approach. Because the number of fractures per set

in each mesh is equal to 11.251, the number of nodes andelements increases rapidly with fracture length. Therefore sys-

tems with even moderately large values of I could not beprocessed in the computer available to uS. In fact, the case

where I =24 was the largest problem that could be accommo-dated. The apparent leveling off in the curve of K versus Icould not be confirmed under these constraints.

The apparent decrease in the rate of change in permeability

with increase in fracture length is very important. If for certainvalues of I. permeability does not change very much, then it

would appear that permeability can be predicted withoutknowing fracture length exactly. Therefore it was important tofind some way of estimating the trend in permeability for

fracture lengths larger than I =24. Because of computer limi-

tations the permeabiliry of fracture systems with lengrhslonger rhan 24 could nor be calculated the same way as sys-rems wirh lengths less than this value. To circumvent this

problem. ir was decided to reduce the ratio L/I from 6.25 ro2.34. which then made it possible to calculate the permeabiliryof a system with fracture lengths of I=64. This resulted in a

finite element mesh wirh approximately the same size asLD24. The perrneability of this flow region for LlI =2.34 is

expected to be only slightly larger than would be measured fora mesh wirh L/ I =6.25 because the fracrure system is well

connected and the truncarion errors for the two samples arenot significantly differenr.

Figure la shows a comparison of rhe permeability resuIt

)

edlot

a

heng

realof

re

X xpalsu-

yapse,ofi ntter

e

sn

e-

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175

150

F I o . F l eC )i o n125xl25

90

270

H M S E ',0177

-1'1.20 MonttCo,tgRe su l1 s

'-16

/ >f A- - - - - - - - - - - - - - - ~ - - - - -

180

v~ I-IZ

~ :16

IJ:~

I-Z4- - - - - - - - - - - . - - - - - -

[- 20 REV Study

50 75 100 125

Scale 01 measurement (em)

Summaryof aI!permeabilityresults plotted as a functionofscaleofmeasurement.

3,15} .i'Z' 'i0.5

t

el-s

0.4 ''1.20 REVSludJ

90

F i o . F l eC )i o n30x30

w 0,3(/)

: :