PHYSICAL REVIEW B VOLUME 30, NUMBER 11 1 DECEMBER 1984

Canductivity and permeability af rocks

Po-zen Wong, Joel Koplik, and J. P. Tomanic
Schlumberger Do-ll Research, Old Quarry Road, Ridgefield, Connecticut 06877-4108
(Received 20 July 1984)
The electrical conductivity of salt-water-saturated rocks is modeled by a random resistance net-
work which has a zero percolation threshold. The porosity is varied by a random bond-shrinkage
mechanism. Numerical and analytical calculations of the model in different dimensions show an
Archie's-law behavior: o„= P,
ao where P is the porosity of the rock, and o„and o are the con-
ductivities of the rock and water, respectively. We find that the Archie s exponent m is always
greater than unity and is related to the skewness of the "pore-size distribution" of the rock. Apply-
ing the same model to fluid-flow permeability {k, ) gives k„~P, where m =m {m +1) in one di-
mension, and m =2m in higher dimensions. This power-law form is consistent with the well-
known Kozeny equation and has been frequently suggested by empirical studies. Experimental tests
of the model are performed on artificial rocks, made by fusing small glass beads, as well as real
rocks. From resistivity measurements, we demonstrate that m is larger in samples with a wider
fluctuation of pore sizes, which is qualitatively consistent with the model. From fluid-flow experi-
ments on fused glass beads, we find quantitative support for the m =2m prediction.

I. INTRODUCTION graph of a thin section of a rock would show that the pore
space is multiply connected in a complicated and random
An interesting geometrical feature of rocks is that they way. In Fig. 1 we show micrographs for two sandstones
appear not to have a finite percolation threshold. When and one limestone as illustrations. It should be clear from
their pore space is saturated with salt water, they exhibit these pictures that a more appropriate model of the pore
„)
finite electrical conductivity (o. even when the porosity
(P) is below 1%. An empirical equation that links the
space should involve some kind of random network.
Indeed, such an approach has been widely used in the last
conductivity and the porosity was first proposed by Ar- three decades to simulate the petrophysical properties of
chie and has become known as Archie's law rock formations and to study the behavior of other porous
media. A glance at the literature, however, indicates that
CTr a crw4 a basic understanding has not emerged from these studies.
where 0. is the conductivity of the water, and a and m We shall refer the readers to Refs. 5 and 6 for a survey of
are empirical parameters that vary with the lithology of these studies and not attempt to discuss them here.
the rock formation. Quite often, a is assumed to be unity In more recent years, Sen, Scala, and Cohen have pro-
and m =2. The power-law dependence in this equation posed a self-similar model to explain Archie's law, by con-
resembles the behavior in the usual percolation problem, sidering the geometry of the grain space to be a random
except that it suggests a conduction threshold at / =0. In assemblage of spheres of all radii. In essence, they applied
addition, the exponent m is not entirely universal; dif- Bruggeman's theory which integrated the classical
ferent values have been given by Keller for different kinds Clausius-Mossotti equation for noninteracting dielectric
of formation. spheres embedded in a homogeneous media from the P = 1
For fluid flow through a rock, another empirical law, dilute limit (hence, the model is also known as the "iterat-
known as the Kozeny equation, relates the permeability ed dilute limit" ), and this gives m = —, . This method is
(k, ) to the porosity: attractive in that it intrinsically preserves the pore-space
connectivity for any value of P. Furthermore, other
values of m can be obtained if spheroids with different as-
Sp pect ratios are used. Experimental support for the
iterated dilute result can be found in the work of De La
where So is the specific surface area (i.e., internal surface Rue and Tobias. ' They measured the conductivity of di-
area per unit bulk volume) of the rock and c ( =0.2) is an lute suspensions of glass spheres, polystyrene cylinders,
empirical constant. This equation again has both a
power-law dependence on the porosity and the suggestion m =1.5 in each case.
)
and sand grains in ZnBr2 solutions for P 0. 60 and found

that the pore space is connected at any finite porosity. Using the self-similar model to understand the behavior
Historically, these empirical relationships were justified of rocks presents two difficulties. First, one knows that
by modeling the pore space as a bundle of winding tubes rocks generally have porosities less than 40%, which is far
which do not intersect each other. With that assumption, from the dilute limit in which the assumptions of the
both equations above can be easily derived. Such a model have the most justification. When the porosity is
model is highly unrealistic, however, since any micro- low, the grains are in close contact and the interactions

30 6606 1984 The American Physical Society

6608 PO-ZEN WONG, JOEL KOPLIK, AND J. P. TOMANIC 30
i

with similar grain shapes the exponent m can vary signifi- reduces the local stress, strengthens the wall, and resists
cantly and, conversely, in rocks with very different grain further deformation. Deposition of irregularly shaped
shapes the values of m can be very similar. For example, particles in an irregularly shaped channel can never corn-
resistivity measurements on the three samples shown in pletely block that channel, regardless of how many such
=
Fig. 1 give m 1.94 for the Cotton Valley sandstone and particles are deposited. Furthermore, thin lubricating
m =1. =
65 for the Berea sandstone [assuming a 1 in Eq. films of fluid, if present, will inhibit grain contact. To
(I)], but these two rocks have similar grain shapes. For model such behavior in our network, we randomly choose
the Indiana limestone, which has visibly different grain a tube element and reduce its radius by a fixed factor x,
shapes from the two sandstones, we find m =1.95, essen-
tially the same as that for the Cotton Valley sandstone. r; ~xr;, (3)
These comparisons cannot be explained by a grain-shape where 0 &x & 1 and i is randomly chosen. Since the elec-
effect. trical conductance of a given tube is proportional to its
we focus our attention on the variation of
In this paper, cross-sectional area, it will decrease by a factor x . Simi-
the pore space with porosity and show how the "pore-size larly, the permeability of a cylinder (the ratio of fluid flux
distribution" can influence the conductiuity and permeabil to pressure difference) is proportional to r; and would be
ity In .Sec. II we introduce a tractable random network reduced by a factor x . This shrinking procedure can be
model which exploits the similarity to the bond- repeated indefinitely with the same x to reduce the net-
percolation problem. We consider a lattice of random- work conductance and permeability, and the total volume
sized cylinders, and systematically reduce the volume of of the tubes. The length of the tube is kept unchanged, so
pore space by randomly shrinking their radii. Although that if channel i is chosen n times, its conductance will be
the model is not completely realistic, we will show how it reduced from G; to x "G;, and its permeability from k; to
allows us to qualitatively understand the meaning of x "k;. %'e will neglect the nodes at which the tubes are
Archie's law and the Kozeny equation. Through this im- connected. Although this is unrealistic, we will argue in
proved understanding, we can suggest a modified form for Sec. IV that this and other artificial elements in the model,
the Kozeny equation which relates the permeability to the such as assigning uniform radii to the tubes, shrinking
conductivity. In Sec. III we describe the results of some them by a constant factor, etc , do .not affect the con
simple conductivity and permeability experiments per- clusions that we will deriue from the model It is on. ly im-
formed on artificial rocks (Ridgefield sandstones) made of portant to note at this point that the model has two at-
fused glass spheres, which provide good support for the tractive features: (i) it preserves the network connectivity
theoretical predictions. Some further discussion will be in the P — +0 limit for any x &0 and (ii) the amount of
given in Sec. IV, where we clarify what we mean by change in r; at any shrinking step is dependent on the
"pore-size distribution" and argue that the unrea/istic ele- value of r; at that time. Both of these features are crucial
ments in our model do not affect the main conclusions that in obtaining the behavior of Eqs. (1) and (2). If we con-
we draw from it sider the limiting case x =0, this model coincides with the
usual bond-percolation problem and there will be a finite
percolation threshold.
II; BOND-;SHRINKAGE MODEL
Our model is motivated by the similarity of Eqs. (1)
and (2) to the scaling laws that are characteristic of the A. Solution in one dimension
'

percolation problem, keeping in mind that we want to

make the conduction threshold occur at / =0. We consid-
er a random resistor network on a simple cubic lattice in To see that this model leads to Archie's law and the
d dimensions. Each resistor R; represents a cylindrical Kozeny equation, we first solve it exactly (and trivially) in
fluid-filled tube with radius r; In the us.ual bond- one dimension (1D). We consider X tubes connected in
percolation problem, one chooses a conductance element series and shrink them randomly according to Eq. (3) a to-
at random and sets its radius equal to zero. This pro- tal of M times. Since we keep the tube length I constant,
cedure results in a finite conduction threshold p, (d), for the total volume (or porosity) of this 1D network is pro-
when the concentration of unbroken bonds p is less than portional to the average cross section, and hence the aver-
p„ the network becomes disconnected and ceases to con-
duct. The formation of a sedimentary rock, however, is a
age conductance (G). To calculate P, we note that the
probability for any particular tube to shrink n times is
somewhat different process. There, one imagines, that the simply given by the binomial distribution
rock begins as a packing of unconsolidated grains, which MI 1 X —1
is analogous to a fully connected network, with some ini- — X
(M n)!n! N
tial conductivity and porosity (-40%). In the course of
time, the cross section of any conduction channel can be The average conductance of that tube is
reduced by the pressure on the rock, by further deposition 'M
of smaller particles, or by other mechanisms. The porosi- M —1
G;=G; gx "P(n)=G; 1V+x
ty and the conductivity mill, as result, be reduced simul- n=0
taneously. The probability of the channel becoming com-.
pletely blocked is, however, very small. For example, con- The average conductance of the whole 1D network is
solidation of the wall of the channel under pressure therefore
 i

30 CONDUCTIVITY AND PERMEABILITY OF ROCKS 6609

2— tivity. More importantly, it shows that both quantities

(6) =(6;) =(6;) (6) are governed by-the tube-size distribution and they can be
simply related, at least in one dimension. .
where ( ) denotes the averaging over the network, which B. Numerical results in higher dimensions
is decoupled from the average over the shrinkage proba-
bility P(n) for an individual tube in Eq. (5). The true
If we
consider the same bond-shrinkage model in higher
conductance of the network
since the resistors are in series,
G„„
is not Instead, (6). dimensions (d & 2), we note that while the porosity is still
given by Eq. (6), the network conductance is not given by
Eq. (7). To calculate G„„,
one can apply the Kirchoff's
6„„'
=R„„=N(R;=N(6; ) ) law to finite-size samples and solve the network equations
numerically. We use this approach on simple cubic net-
M
=N(6, ') g x—-'"P(n) works of dimension L '(L+1). The boundary condi-
n=0 tions are (i) constant potential at the two ends of the long
M dimension, and (ii) periodic in the d —1 transverse direc-
—1
=N(G; ') X+x tions. Typically, we assign an initial set of r; s to each
X bond, which has a flat distribution in the range 0& r; ~ 1.
The initial porosity P(:—(r; ) ) is defined to be unity. We
When the number of tubes is large, we have
also let G; =r; and k; =r;, i.e., all the prefactors are de-

= lim In[1+ (x — —1)/N] fined to be unity. The r s are then randomly reduced by
lim the fixed factor x as in Eq. (3). When P is reduced by a
~ ln((G ) /(6 ) ) x in[1+(x 1)/N]
N

x —2 —
m

1
factor of 2, we calculate the network conductance
and permeability k„„.
The process is repeated until P is
G„„
'
x — X2
1.
reduced by about three orders of magnitude. For any set
of values for the sample parameters d, L, and x, we re-
which implies peated the calculation for ten initial sets of r s and aver-
age the results to improve the statistics. Figure 2 shows a
(9)
for an infinite system. Since P ~ (6), we have Archie s 10
law,
—
10 4
G„„ccrc where m = 1
&1 . (10)
X
When x ~0, we have m ~
oo and hence G„~ for any~0 —
10 5
/ &1, which is the correct behavior for 1D percolation. CD

The dependence of m on x indicates that the network con U

C
ductance is sensitiue to the distribution of the tube radii o 10
The reason that m &1 is that the network conductance
and the porosity P depend on the distribution in different
6„
10
ways: G„~is influenced mainly by the narrow tubes and P
is influenced mainly by the wide tubes Our mode. l allows
10-8
us to obtain an exact solution to demonstrate this simple I l I I I

10 10 10 1O'
fact explicitly. Porosity
The above calculation can be easily modified to give the
fluid-flow permeability of the network. Since the permea- 10
bility of a single tube element is k; cc r;, to calculate the 10 '
network permeability in ID, we simply replace x in Eq.
('7) by x . Repeating the steps in Eqs. (8) — (10), we ob- 1O 'y
tain —
10 10 ~

k„„~P where m'=m(m+ I'! - 2m, (11) m

10 11
E
which is analogous to the Kozeny equation [Eq. (2)] ex- 10
cept for the factor 1/So. This factor may be inserted on
10
the basis of dimension analysis because permeability has
the dimension of [length] and I/So=(volume/area) is a 1O
'4
natural unit of length in a porous material. On the other —15-
(b)
10
hand, many studies have suggested the use of other 10 10 10 100
characteristic lengths, such as the average particle size. Porosity
More will be said about this in Sec. IV. The fact that FIG. 2. Numerical results for the 2D 40&41 sample with
m'&m shows that the permeability is more strongly shrinking factor x =0.75: (a) conductivity vs porosity
dependent on the tube-size fluctuation than the conduc- ( m = 1.32), and (b) permeability vs porosity (m'=2. 68).
6610 PO-ZEN WONG, JOEL KOPLIK, AND J. P. TOMANIC 30

set of typical results for a 40X41 sample with x =0.75.

We can see that both G„„andk„have a simple power-
most probable value of a conductance
6mp 2nG
™
6, is

law dependence on P. In this particular case, we find

m =1.32 and m'=2. 68, which differ from the values which is not the same as the auerage value G; in Eq. (5).
predicted by the 1D calculation. The reason is that the amount of change in G; at any par-
To confirm that this finding is not fortuitous, we per- ticular shrinking step is dependent on the value of 6; at
form the same calculation for d =2, 3, 4, 5, using different that time and the central-limit theorem does not apply
sample sizes and x values: L ranges from 5 to 60, under such conditions. (Similar behavior should also
x =0.25, 0.50, and 0.75. Indeed, we find power laws in occur in real rocks, since we expect the smaller channels
each case. The values of m and m' we obtained are tabu- in the rocks to have smaller changes in their cross sections
lated in Table I. Several interesting observations can be regardless of what the reduction process is. ) In the limit
made from these results. that n — +co, fluctuations of order (n)'i are unimportant
(1) m and m' increase with decreasing x, as in the 1D and we can assume G; " appears in Kirchoff's equations.
case, but their values are always smaller than the 1D
values given by Eqs. (10) and (11), which are listed in the
We expect, therefore, the network conductance to be G„„
first row of Table I. 6 I1W G ~P
E
2M/N
(13)
(2) Except in the x =0.25 case where we were unable to Combining this with Eq. (5), we have
obtain reliable values for k„„due to numerical problems,
we find that m'=2m regardless of d and L This . rela-
11Q1
tionship is also different from that in 1D [Eq. (11)]. WWMix In[1+ (x —I)/Xj x —1
(3) Although m and m' seem to decrease with increas-
ing d in Table I, we note that they increase with I. for
(14)
d & 3, which means that our results are affected by the For the permeability exponent m', we note that since
finite-sample size in these higher dimensions. Taking that k, ocr; ~G;, it follows immediately from the above equa-
into consideration, the results suggest that m and m' are tion that
independent of d for d & 2.
(4) In addition to using a flat initial distribution for the k„w~x ~ and hence m'=2m (15)
r s, we have also tried other initial distributions in several The values of m and m' calculated from these equations
cases and found no effect on m and m'. From Eqs. (6)
are given in the last row of Table I. They are in good
and (7), we can see that this is to be expected since the
agreement with the numerical results, especially with
average over P(n) is decoupled from the average over the
those in two (2D), dimensions where we have been able to
initial distribution.
study larger sample dimensions. The agreement worsens
C. Anabjtical cstiHlatcs 1n hlgktcI diIQcnsions with increasing d and decreasing x. The former can be
attributed to the limitation in our sample size as men-
Some insights into the above results can be obtained by tioned above. The latter is probably due to the fact that
analyzing the model in the thermodynamic limit, i.e., when we reduce P by the same amount, M/1V is smaller
when M, X and M/X — + oo. By the central-limit theorem, for a smaller value of x. The crucial step i:n deriving Eqs.
we know that the shrinkage probability P(n) in Eq. (4) be- (14) and (15) is that we assumed G„„cc
G; ~. This approx-
comes a Gaussian distribution centered at n =M/N with imation can have an error of the order of x ", which will
a width 5n = (n ) ' . n is the most probable value of n as lead to a correction in Eq. (14) of the order of
well as its average. By our construction, however, the 5n /n = (M /%) 'i . One expects, therefore, a worse
TABLE I. Numerical values for m and m' obtained from simple cubic networks of different sizes L
in different dimensions d. Three values for the shrinking factor x were tried. The top rom gives the 10
exact values. The bottom row gives the analytical values in higher dimensions.
x =0.25 x =0.5 x =0.75
m'
10 16.00 272.00 4.00 21.00 1.78 4.94
10 2.80 1.8-5 . 3.73 1.32 2.67
20 2.83 5.35 1.85 3.70 1.33 2.67
40 2.91 1.83 3.73 1.32 2.68
60 2. 86 1.86 3.75 1.33 2.70
53 2.31 4.20 1.57 1.23 2.48
10 2.49 4.73 1.68 3.32 1.26 2.48
15 2.55 1.73 3.29 1.25 2.50
54 2.33 4.52 1.55 3.04 1.20 2.39
84 2.37 1.58 3.08 1.23 2.41
55 2.22 4. 17 1.53 2.96 1.19 2.33
ln(x2)
2.96 5.95 1.85 3.70 1.32 2.63
x 2 —1
30 CONDUCTIVITY AND PERMEABILITY OF ROCKS 6611

agreement with the numerical results for a smaller x. In III. EXPERIMENTS

practice, however, we find that the disagreement is much
less than (M/N)
A more rigorous argument for Eq. (13) may be given in To test some of the findings of the model, we per-
terms of upper and lower bounds on the network conduc- formed conductivity and permeability experiments on ar-
tivity. ' ' Suppose for simplicity's sake that we let all tificial rocks made of fused glass beads. The glass beads
the initial bonds have a conductance of unity. In the ther- were obtained from the Ferro Corporation and sifted into
modynamic limit, they will fall in the range three size groups: 44 — 53 pm, 88 —105 pm, and
177 — 210 pm. They were washed first in dilute hydro-
—gn ) chloric acid and then in water. Low-density particles
6 2(n
6 (X 2(n+5n
)
)
GQ were removed by water flow and magnetic particles were
removed by a 6-kOe field. Several melts with different
with an exponentially small number of exceptions. A porosities were made from each of the size groups by fus-
lower bound on the network conductance is obtained by ing them to different degrees. Cylindrical samples 1.5 in.
replacing all 6; ~ GI with 0 and all 6; & GI with GI, and long and 0.75 in. in diameter were cut from each melt. To
thus6„„& bGt. The constant b is a function of the net-
work geometry, and corresponds to the conductance of a
saturate the samples with salt water, they were first placed
in a vacuum to remove the air in the pore space and the
network where most bonds have unity conductance and an water was then let into the evacuated container to fill the
exponentially small fraction have zero conductance, and b pores. The porosity of each sample was determined by
is therefore a finite number. Similarly, an upper bound of measuring its dry weight, wet weight, and buoyancy in
the network conductance follows by replacing all G; &
6„, 6„„&
6„
b'6„.
water. These measurements also give the grain density pg
by infinity and all others by
In the limit n ~
and we have of the sample. For samples with P) 0.03, we find pg in
oo, fiuctuations of order 5n are negligible the range 2. 485+0.010 gm/cm3, in agreement with the
and the two bounds converge to the most probable value, value for bulk glass. From the small spread in the mea-
and we obtain Eq. (13). sured grain density, we can infer that there were essential-
We expect Eqs. (14) and (15) to be correct for an infi- ly no occluded volumes in these samples (less than 1%
nite system in the large-M/X limit, i.e., when P — +0. volume fraction if they exist), i.e., the pore space is com-
They explain all the findings from the numerical calcula- pletely connected and can be fully saturated by the water.
tions. In addition, these analytical results allow us to see In addition, we can also estimate that the accuracy of the
explicitly that m is related to the difference between the porosity measurement is about 0.01. In samples with
most probable conductance 6; " and the average conduc- (0. 03, we find some samples with significantly lower
P
tance 6;. The parameter x is simply a measure of that values of ps, implying the existence of occluded volumes
difference or, in other words, the skewness of the distribu at such low porosities. We did not study those samples
tion. The smaller x is, the more skewed is the distribution, further.
and the larger is m. Applying this knowledge to rocks, we To make conductivity and permeability measurements,
can qualitatively explain the difference of m values for the samples were fitted inside a Hassler collar which is a
the two sandstones in Fig. 1. For the Cotton Valley sand- double-wall cylinder with a length and inner diameter
stone, we see that it has a small number of larger pores matching those of the sample. The inner wall of this de-
which are connected via a large number of much smaller vice is made of neoprene and the outer wall is made of
pores. This implies that the pore-size distribution for this steel. Air can be injected between the two walls to pres-
rock is more skewed than the Berea sandstone and hence surize the neoprene sleeve inward to grasp the sample
its m value should be larger, which is what we observed. tightly. This prevents fluid or current leakage along the
In the next section, we will demonstrate this behavior in wall. Four-terminal ac conductivity measurements were
synthetic rocks in a more systematic way. made by using an impedance meter. The electrodes were
Another interesting prediction of the model is that for made of silver and chlorodized.
)
any d 2, the properties of the network are simply related
to the statistical distribution of the individual elements,
The voltage electrodes
were placed against the ends of the Hassler collar and the
sample, and the current electrodes were far away from
regardless of how they are connected in detail. This leads them. To ensure that the measurements were not affected
to Eq. (15), which relates the conductivity and permeabili- by electrode-polarization effects, we varied the measure-
ty in a very simple way. It is common to use the forma- ment frequency (100 Hz, 1 kHz, and 10 kHz) and voltage
tion factor (F: o~/o„=a/ — ) to' describe the conduc- (50 mV and 1 V), and no significant effects were observed.
tivity of rocks, and Eq. (15) suggests that the permeability In permeability measurements the sample was placed vert-
may follow the law ically with water injected at the bottom and extracted at
the top. %'hen the permeability of the sample is high
[k„) 100 md (millidarcy)], the pressure gradient across
(16) the sample was determined by the difference in water level
between the inlet and outlet reservoirs. When the permea-

[Note that this relationship would be rigorous if a in Eq.

bility is low(k„( 100 md), higher pressures were obtained
by using a piston with a regulator. The flow rate was
(1) were a universal constant, which is not established. ] In determined by measuring the amount of water collected at
the next section, we will show permeability data to sup- the outlet and the time it took to collect it. The former
port this prediction. varied between 0. 1 and 50 cm, the latter varied between 1
 PO-ZEN %ONG, JOEL KOPLIK, AND J. P. TOMANIC 30

and 50 min. The results of these experiments are

described below.

A. Conductivity and formation factor

The main interest in. conductivity measurements is to

determine the formation factov F as a function of porosity,
to test Archie's law. %"e typically measure the sample
conductivity a, for three or four different water conduc-
tivities (o~) which varied between 0.24 and 7.7 'm 0
A 11ncar least-squaI'c fit of 0 ~ vcI sus 0 ~ glvcs a slope
equal to F ' for that sample. A total of 26 samples were
measured and the results are summarized in Fig. 3, where
we plotted F versus P on a log-log scale. The porosity (P)
ranges from 0.023 to 0.399. The dashed line in Fig. 3
represents the theoretical prediction of the spherical grain
self-similar model, which has a =1 and
Ref. 7, data down to /=0. 026 were shown
I
=3/2. ' In
and they
agreed with this prediction. A subsequent study of
Johnson et aI. , ' however, disagreed with those results
with data down to P =0. 10. Our data in Fig. 3 agree with
the latter results, extending the porosity range down to
/=0. 023 and varying the grain size of the sample. We
observe that the data in the porosity range 0.2 &/ &0.4
can be approximated by the self-similar model prediction.
For P &0.2, however, the value of F is always higher than {a)
the prediction, implying a higher value for m.
There are two possible ways to characterize the low-
porosity data. First, it can be approximated by the solid
11nc 1Il F1g. 3, which cofI'csponds to 0 =3.3 and m =2.3,
i.e., both a and m are constants over a large porosity
range. Alternatively, if we assume a =1, we can define
I =d(lno )/d (ln(()) and say that I
increases continuously
with decreasing porosity. The high-porosity data points
correspond to m =1..5 and the low-porosity data points
correspond to m =2. Either way, m is higher for lower
porosity. Such a trend can be qualitatively understood by
studying the microgeometry of the samples. In Fig. 4, we
show the micrographs of two samples with different poro-
sities. We can see that the high-porosity one (/=0. 315)

10 i''4

Q 102
LU
U

E
0 10' =

FIG. 4. Micrographs of two fused-glass-beads samples with

10 different porosities: (a) /=0. 315, and (b) /=0. 061. Thar con-
10 10 10 trasting microgeometry is similar to the two sandstones in Fig.
Porosity 1.
FIG. 3. Formation factor for different fused-glass-beads
samples as obtained by resistivity measurements. The dashed has a more uniform pore-space distribution, much like the
line is the prediction of the self-similar model for spherical Berea sandstone in Fig. 1. In contrast, the low-porosity
=
grains (a 1, m = 2 ). Data below 20% porosity show substan- sample (/=0. 061) shows a larger fluctuation in its pore-
space distribution. Similar to the Cotton Valley sandstone
by the solid line, which corresponds to a =3.3 and
I
=2. 3. I
tial deviation from the prediction. They can be approximated
in Fig. 1, it has a sInall number of large isolated pores
30 CONDUCTIVITY AND PERMEABILITY OF ROCKS 6613

connected via much smaller pores. Following the discus- k„„ccrc ~. If a has different values for high and low
sion in the preceding section, one expects the latter sam- porosities, one would not expect Eq. (16) to hold for all
ples to have higher m values, which is exactly what we porosities. Since the data in Fig. 5 actually cover both
found. Furthermore, since the pore-space distribution P &0.2 and P &0.2, it suggests that one may indeed have
changes continuously with P, it is perhaps more reason- a = 1 at all porosities and consider m as the single param-
able to think of m as also continuously varying with P, eter that characterizes the pore-size distribution, which
rather than being constant over a wide range of P. The varies continuously with P. In a large rock. formation, it
permeability results described below will further support is conceivable that the local porosity varies over a wide
this view. range and the distribution is similar throughout, in which
case Archie's law with a single value of m can apply.
B. Permeability

Since fluid-flow experiments can be performed on the IV. DISCUSSION

same sample that electrical measurements are made, one
can readily test how the permeability is related to the con- The main conclusion that w'e can draw from our results
ductivity or the formation factor, i.e., one can test Eq. is that the scaling behauior of both the conductivity and the
(16). To determine the permeability, we measured the permeability of rocks are determined by the skewness of
flow rate at three or four different pressure gradients their pore-size distribution. The skewness results from the
across the sample. A linear least-square fit of the two rock-formation process which tends to reduce the large
quantities gives a slope equal to k„/p,, where p is the pores by a large amount and the small pores by a small
viscosity of water. Unlike the formation factor which is amount. Such a process will gerierally produce a pore-size
dimensionless and independent of the grain size, k has the distribution which is log-normal, instead of normal.
dimension of (length) and hence must depend on the There is, in fact, ample empirical evidence that both the
grain size. In Fig. S we show a log-log plot of k„versus
F. We can see that the data points fall into three groups described by a log-normal distribution. ' '
pore- and grain-size distributions in rocks are roughly
Our model, in
corresponding to the three different grain sizes we used. essence, shows how this microgeometrical property can be
Within each group, the data can be approximated by a related to the macroscopic properties such as the conduc-
straight line with a slope of — 2, in agreement with Eq. tivity and permeability.
(16). Some data points deviate from the straight lines We use the term pore-space (-size) distribution loosely
when the permeability is below 10 md. They may be due since there is no mathematically precise definition for it.
to the fact that some channels become completely blocked It has a well-defined meaning only in simple models, such
in the low-porosity samples since we know that there are as ours, which approximate the pore space as a network of
occluded volumes when the porosity is below 3%. Anoth- tubes. Such an approximation may be justified when one
er possible explanation will be given in the next section. is dealirig with physicaI properties that are associated with
We should mention, however, that there are difficulties in a length scale that is larger than the characteristic grain
measuring a small permeability. Because the flow rate is size (of order 100 pm), and conductivity and permeability
low, high-pressure gradients across the sample and high are among such properties. On the submicron length
pressure in the Hassler collar must be maintained over a scale, there is some evidence that the pore space of a rock
long period of time. . As a result, there'can be small fluid may have fractal characters, ' in which case it will not be
or air leakages in the measuring system which can lead to appropriate to think in terms of most probable value, aver-
large errors in the results. age Ualue, etc. On the length scale of the grain size, howev
We note that the network model only predicts er, it is reasonable to represent the space between two
grains as a tube with a length comparable to the grain size
104- and a cross-section that varies along the length, like our 1D
53 p. rn Beads
106 p~ Beads network. On a still larger scale, one can assign a uniform
7-210 p, rn Beads
10
effectiue radius to each tube and enuision them being con
I
nected at "nodes" to form a network, like a higher
dimension network. Although we neglected the nodes in
U
10 our model, it is easy to see what they will do. The nodes
are basically pocketlike; they contribute much to the porosi-
JD
ty and little to the conductiuity and permeability In other.
10 words, they add to the skewness of the pore-space distri-
E bution and increase the value of m. Since our model is
CL not intended to predict an absolute value of m, neglecting
10'
the nodes is permissible. Likewise, the artificial process
of shrinking the tubes continuously by a constant factor x
10 is only a way to generate a simple distribution that makes
10' 10 10 10 the network analytically tractable. Had we used a distn-
Formation Factor bution of x values, the conductance distribution will be
FIG. 5. Log-log plot of permeability vs formation factors for more complicated but the scaling-law solution of the
fused-glass-beads samples with different grain sizes. For each model in the $~0 limit should be unchanged. In 1D for
grain size, the relationship k, ~ F is approximately obeyed. example, Eqs. (5) — (10) can be followed through with any
6614 PO-ZEN MONG, JOEL KOPLIK, AND J. P. TOMANIC

skewed distribution which gives different values for 6; pie power laws like Eq. (1 1) with various exponents m'& 3
and 6, . In higher dimensions as long as the conduc- and without the 1/S2o factor have been suggested by dif-
tance distribution is broad (like a log-normal distribution), ferent studies. The variation in m in these studies is con-
the heuristic arguments given in Sec. II should apply re- sistent with the point of view taken here, since we predict
gardless of how the distribution is obtained in detail. the permeability exponent m' to be nonuniversal just like
Conversely, we note that the agreement between the nu- the conductivity exponent m, and the latter is empirically
merical results and Eq. (14) demonstrates the validity of well known to be nonuniversal. The data presented here
those arguments in the $~0
limit. are not sufficient to distinguish whether Eq. (11) or Eq.
The advantage of the analytic arguments is that they al- (17) is more preferable. It will be interesting to perform a
low us to make a simple correlation between the permea- more extensive study that includes measurements of So to
bility and the conductivity exponents: m'=2m, and the test these relationships further.
data on fused glass beads provide good support for this Finally, it is important to make clear that the higher-
prediction. It is interesting to note that the simple dimension result m'=2m in our model is a consequence
power-law expression in Eq. (1 1) for the permeability can of assigning a uniform radii to the tubes. In any real sam-
be rewritten in the form of the Kozeny equation. The ple, the cross-sectional area of a conduction channel varies
surface-to-volume ratio So in our model is proportional to along its length. Provided that this variation is not too
the, average tube radius (r; ).
Following Eqs. (5) and (8), severe (i.e., it does not have a broad log-normal — like dis-
we have tribution), the uniform radii approximation should be
valid. Otherwise, one would expect the one-dimensional
x —1 behavior [Eqs. (10) and (11)] to play an important role. In
S, (x)
the latter case, since the 1D analysis gives m'=m(m
+1) ~ 2m, one expects
'M Zm &m'&m(m+ I) .
We note that in Fig. 5, the deviations of the low-porosity
1
q= lim
1+x
&1. data points, although can be explained otherwise, are con-
ln
1+x —1 sistent with m'~ 2m.

Substituting this result into Eq. (11) gives ACKNOWLEDGMENTS

m"
We are grateful to B. I. Halperin and A. B. Harris for
k, o=
2
where m"=m'+2q, (17) helping us to understand the nomerical results, J. Bana-
So
var, D. L. Johnson, W. Murphy, and D. J. Wilkinson for
Except for the fact that m" is variable, this expression is helpful discussions, and L. P. Kadanoff for encouraging
identical to the Kozeny equation. It should be em- this work. Special thanks are due to our colleagues in the
phasized,however, that there is considerable data on other Rock Laboratory at the Schlumberger-Doll Research
materialswhich do not fit the Kozeny equation and many Center who provided extensive assistance in sample
modified forms have been proposed. In particular, sim- preparation and some of the measurements.

G. E. Archie, AIME Trans. 146, 54 (1942). 827 (1959).

2'. V. Keller, in Handbook of Physical Properties of Rocks, edit- »Q. W, Milton, in Physics and Chemsitry of Porous Media
ed by R. S. Carmichael (CRC, Boca Raton, Florida, 1982). (Schlumberger Doll Research), e-dited by D. L. Johnson and
J. Kozeny, Sitzungsber. Akad. %'iss. Wien 136, 271 (1927). P. N. Sen (AIP, New York, 1984), p. 66.
'P. C. Carman, Flow of Gases Through Povous Media (Academ- V. Ambegaokar, B. I. Halperin, and J. S. Langer, Phys. Rev.
ic, New York, 1956). B 4, 2612 (1971).
5A. E. Scheidegger, The Physics of Flow in Porous Media '3V. K. S. Shante, Phys. Rev. B 16, 2597 (1977).
.

(University of Toronto Press, Toronto, 1974) ~

~4S. Kirkpatrick, in Proceedings of the Seminar on Electrons in
6F. A. L. Dullien, Porous Media, Fluid Transport and Pore Disordered Systems, Kyoto, 1972 {unpublished).
Structure (Academic, New York, 1979). ~5D. L. Johnson, T. J. Plona, C. Scala, F. Pasierb, and H. Koji-
7P. N. Sen, C. Scala, and M. H. Cohen, Geophysics 46, 781 ma, Phys. Rev. Lett. 49, 1840 (1982).
(1981). ' E. Pittman, in Physics and Chemistry of Porous Media
~D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935). (Schlumberger Doll Research), edite-d by D. L. Johnson and
P. N. Sen, Geophysics 46, 1714 (1981); see also K. Mendelson P. N. Sen (AIP, New York, 1984), p. 1.
and M. H. Cohen, ibid. 47, 257 (1982); P. N. Sen, ibid. 49, ~7G. S. Visher, J. Sediment. Petrol. 39, 1074 (1969).
586 (1984). 8D. Avnir, D. Farin, and P. Pfeifer, Nature 308, 261 {1984).
'OR. E. De la Rue and C. W. Tobias, J. Electrochem. Soc. 106,