PERMEABILITY OF SATURATED SANDS,
SOILS AND CLAYS
B Y P. C. CARMAN
Department of Chemistry, Rondebosch, University of Cape Town
IT has long been known that, in any porous medium, the rate of flow of a
fluid through the mediuna is proportional to the pressure gradient in the
direction offlow(Muskat, 1937). Thus, if Q c.c./sec. are obtained from a
column of length L cm., and cross-sectional area A cm.2, when the pressure difference across the column is AP g./cm.2, then
u = Q = K~
(D'Arcy's law),
(1)
where u is the apparent linear rate offlowin cm./sec, and K is a coefficient. It is also generally recognized (Muskat, 1937) that u is inversely
proportional to the viscosity of the fluid, -q poises, whence
Then Klt which will be called the permeability, depends only on the
nature of the bed, and is independent of the fluid, of the pressure
gradient, and of the dimensions of the bed. In short, in a bed of rigid
particles, the permeability depends only upon the size, shape, and mode
of packing of the particles forming the bed. Many attempts have been
made to calculate Kx theoretically from these variables. The most widely
known and accepted is that of Slichter (1897-8), whose equation for a
bed of uniform spherical particles of diameter d cm. can be put in the
form,
^ = 10-2f,
(3)
where Kz is a function of the porosity e, and varies from 84-3 when
e = 0-26 to 12-8 when e = 046, these being approximately the extreme
limits of e for spheres in regular modes of stacking. The variation in K2
stresses the importance of changes in porosity, and is, indeed, in rough
agreement with experimental values. In recent years, the theoretical
basis of Slichter's equation has been attacked (Graton & Fraser, 1935;
Darapsky, 1912), and, further, it has always suffered from the incon-
P. C. CARMAN
263
venience that no guidance is given for an "effective diameter" to be
substituted for d when particles are neither uniform nor spherical.
In the newer type of treatment (Kozeny, 1927; Carman, 1937, 1938;
Kriiger, 1918), size and shape of particle are included by the specific
surface, /Socm.2/cm.3 For uniform spheres, 80=6/d, and, conversely, for
irregularity of size and shape, one can define, if desired, an effective
diameter dm = 6/S0. In Kozeny's equation (Kozeny, 1927; Carman, 1937),
or, since ^ = 980 cm./sec.2, and k is found experimentally to be 5-0,
1 9 6
w
e3
IA
x
where the effect of porosity is given by the porosity function, e3/(l — e)2.
The derivation of this equation is as follows.
In a circular pipe, Poiseuille's law is given by
ad* AP
/K .
5)
u
<
- s i jz-
Now, if, instead of the diameter de, one uses the hydraulic radius m,
where
_
volume of pipe
~ area of wetted surface'
^'
it is seen that
m = jde2L + TrdeL = -£,
(7)
so that
w
(8)
2 ^ '
In this form the equation becomes roughly applicable to pipes which
are not circular in section. Thus, for rectangular and elliptical pipes, if
one were to replace k0 for the factor 2-0, 1% would vary as follows:
Shape of cross-section
Circle
Ellipse:
(a) major axis = 2 x minor axis
(6) major axis = 10 x minor axis
Rectangles:
(a) width=height, i.e. square
(6) width = 2 x height
(c) width = 10 x height
(d) width is infinite
k0
20
2-13
2-45
1-78
1-94
2-65
3-00
More complex shapes fall within the same limits of k0, the majority
grouping in the region &0=2-0-2-5. Now, the pore space in a granular
2.64 Permeability of Saturated Sands, Soils and Clays
bed may be regarded as a single channel of rather involved shape, but of
constant cross-sectional area, namely, eA, where e is the porosity and A
is the total area of the cross-section. If the cross-sections of the porespace were divided arbitrarily into simpler shapes, there would be a preponderance of elongated shapes, comparable with rectangles and ellipses,
rather than circular, square and triangular shapes. Thus, if equation (8)
were to be applicable, one would expect k^ to be nearer 2-5 than to 2-0.
It is particularly to be noted that the assumption of a constant fractional
free area within a given bed implies a random packing of the particles.
In regular modes of packing, as Graton & Fraser (1935) have remarked,
there is a large and regular variation of fractional free area from crosssection to cross-section; and the average fractional free area in one
direction is different from that in another, so that permeability would
vary with the orientation offlowto the structure of the bed. A completely
random arrangement, from its very nature, must have the same permeability in all directions, and this is also in accord with experimental
observations.
The essential part of the Kozeny theory lies in applying equation (8)
to granular beds. Once one has assumed that this is feasible, further
refinements are readily introduced. First, correction has to be made for
the actual length, Le, of the path taken by the fluid as it passes through
a bed of thickness L. Second, the actual velocity of the fluid in the porespace will not be u, i.e. the apparent velocity over the total area, A,
since the free cross-sectional area is only eA, and, further, it must actually
travel a distance Le when appearing to travel a distance L. Thus, the
actual velocity will be - -y-. Summarizing these points, one obtains
e LJ
e L
2-5 774
K
'
or
and, if one substitutes
AP
u=Kx —=•,
•qL
then
E =€J
where
Jfc = 2-5 &\
i
JT>
.
(12)
Evaluation of m is readily made, since it is merely the ratio of the
P. C. CARMAN
265
pore-space per unit volume of the bed, e, to the surface per unit volume
of the bed, S,
i-e.
m=J
(13)
Since particle volume per unit volume of the bed is (1 — e),
8 = (l-c)S0,
(15)
Experiments by the writer (Carman, 1937) have indicated that, on the
average, a fluid flows at 45° to the axis of the bed, i.e., to the direction in
which flow is measured. This would make LJL — \/2, so that the theoretical value of k is 5-0, in agreement with the value obtained by experiment.
EXPERIMENTAL EVIDENCE FOR KOZENY'S EQUATION
A great deal of evidence for equation (4) has been collected and discussed elsewhere (Carman, 1937), and the writer has also investigated the
equation systematically for variations of viscosity, of particle size, of
particle shape, and for mixtures of particles of various sizes and shapes
(Carman, 1938). These tests cover a range of porosity from 0-26 to 0-9,
which corresponds to a 2000-fold range of the porosity function, e3/( 1 — € ) 2 >
the most characteristic feature of Kozeny's equation. A notable point in
these investigations has been the constancy of k under all conditions.
Theoretically, variations of the order of 20 % might well have occurred,
since both JCQ and LJL might be expected to vary with the shape of the
pore-space. The actual variation in k is within 5 %, so that, for a given
specific surface and porosity, the permeabihty is fixed within this limit of
accuracy. The writer has made use of this fact for the converse prooessof
determining specific surface from permeability measurements.
In the experiments just cited, the least particle size was dm — 89 fi. In
a second part, to be published shortly, verification of Kozeny's equation
has been extended to particles with an effective diameter, dm = 2-7^. It is
sufficient here merely to present a few data to illustrate the method of
approach and the agreement obtained.
In order to obtain an absolute test of Kozeny's equation in the range
dm = 10 fi —100 fi, a powder of spherical glass particles, prepared by the
266 Permeability of Saturated Sands, Soils and Clays
method of Sklarew (1934), was divided into three fractions of three
different sizes, and each examined microscopically. The size analyses,
each based on a count of 1000 particles, are shown in Table I. The total
area of 1000 particles is ir'ENd2, and the total volume, £77 TiNd3, so that
the specific surface, So, is given by So = y^™ , and the average particle
size, dm, by <?m = - V A 7 , . These data were then used to obtain the calcu-
lated values of K1 in Table II, where they are compared with "observed"
permeabilities. It will be noted that agreement was equally good using
both absolute ethyl alcohol and acetone, showing the validity of
equation (4).
Table I. Size analyses of spherical glass particles
Number of particles, N
Size range in /1
Average size ,—
Fraction
in range d
VI
2-5
3-5
7-5
5-10
15
10-20
25
20-30
35
30-40
50
40-60
75
60-90
100
90-110
125
110-140
Totals
Specific surface, <S0 cm. 2 /cm. 3
Average particle size, dm
38
109
331
404
97
19
2
—
1000
2070
29/x
j.
Fraction
II
103
96
125
426
235
15
—
1000
950
63/i
Fraction
in
90
261
307
266
76
1000
653
92 M
Table II. Permeability of spherical glass particles
Material
Fraction I
Fraction II
Fraction III
Liquid
Acetone
Acetone
Alcohol
Alcohol
Acetone
Alcohol
Acetone
Alcohol
Viscosity
17
0-00415
0-00415
0-0154
0-0154
0-00415
0-0154
0-00415
0-0154
Porosity
6
0-338
0-360
0-360
0-370
0-390
0-384
0-375
0-392
* i
(observed)
4-24 x 10-"
4-98 x 10-«
4-95 x 10-"
5-62 x lO- 8
3 1 9 xlO- 6
3 0 9 x 10-*
6-25 x 10- 6
7-41 x lO- 5
(calculated)
4 0 3 x lO-o
5-18 x l O - 6
5-18 x 10~«
5-82 x 10-«
3-46 x 10- 5
3-23 x 10- 5
6-19 x 10- 5
7-5 xlO~ 5
In the case of small, irregular particles, it is not possible to measure
So accurately by examination under the microscope, but, if equation
(4 a) is accurate, it should be possible to calculate So for two powders, by
measuring Kx for each, and then to calculate Kx for any mixture from the
values so obtained.
267
P. C. CARMAN
2
Two quartz powders, tested in this way, gave So=22,500 cm. /cm.3
(or dm = 2-7fj.), and <S0 = 6340 cm.2/cm.3 (or d m =94/i), respectively.
When mixed in equal proportions, the calculated specific surface should
be £ (22,500+ 6340) = 14,400 cm.2/cm.3 Two tests on the permeability of
the mixture gave
# , x 10'
(observed)
2-32
306
t
0-436
0-456
Kt x 10'
(calculated)
2-47
303
In another series of experiments, mixtures of a highly purified grade
of kieselguhr, with a porosity e=0-849, were made with samples of the
second quartz powder, the porosity of which was 0-47. The values of So,
obtained from the permeabilities, were, respectively, 17,200 cm.2/cm.3,
(rfm = 3-5/x), and 6340 cm.2/cm.3, so that a mixture containing a fraction,
x, by volume of the kieselguhr, should have a specific surface given by
So = 6340 x (1 -x) + 17,200z
= (6340 +10,770a;) cm.2/cm.3
Table III. Permeabilities of mixtures of kieselguhr and quartz powder
(volume fraction
of kieselguhr)
000
0-020
0-044
0-123
0-262
0-400
0-495
0-740
0-900
1-00
e
porosity
0-470
0-485
0-530
0-607
0-670
0-721
0-751
0-806
0-831
0-849
e»/(l-e)2
0-37
0-43
0-675
1-45
2-76
4-82
6-74
13-9
201
26-6
Kt x 10*
(observed)
0181
0-203
0-299
0-462
0-631
0-762
0-895
1-31
1-49
1-76
J j X 11
(calculat
0196
0-282
0-484
0-645
0-83
0-97
1-33
1-53
In Table III, there are given observed values of Klt and the values
calculated from these values of So. The two constituents were chosen to
provide a wide range of porosity, and the range of the porosity function,
es/(l — e)2, is shown in Table III. The agreement between the calculated
and the observed values of Kx shows undoubtedly that this function
expresses the true relation between porosity and permeability.
MODIFICATION OP KOZENY'S EQUATION FOR CLAYS
Although, in most cases, experiment is overwhelmingly in favour of
Kozeny's theory, this does not appear to be the case for clays. This is
well shown by an important series of experiments by Zunker (1932), who
investigated the permeabilities of a certain heavy clay soil over a wide
268
Permeability of Saturated Sands, Soils and Clays
range of porosities. The data are given in the first two columns of Table
IV, and, in the second last column, there is the ratio of Kx to the porosity
function, €3/(l —e)2. According to Kozeny's' theory, this should be a
constant, but, instead, it is apparent that Kt decreases very much more
rapidly than the porosity function. Similar observations have been
obtained more recently by Macey (1938), so that the behaviour, in a
greater or less degree, seems to be common to all clays.
Table IV. Zunker's data on permeabilities of a clay soil
, x 10 10
e
9-72
8-94
9-66
7-42
2-89
2-10
1-65
1-23
0-591
0-587
0-582
0-562
0-503
0-479
0-460
0-443
€
= e - [0-262 (1-e)]
0-484
0-479
0-472
0-448
0-373
0-342
0-319
0-298
K1 x 1010
x ( l -c)7e*
7-87
7-47
8-52
7-98
5-62
5-20
4-94
4-38
KT x 1O 1 0
x(l-e)Vei3
14-3
13-9
16-1
15-8
13-8
14-3
14-8
14-4
Owing to its great importance with respect to the permeability of
soils, this behaviour of clays merits a detailed discussion. The evidence
for Kozeny's theory is so strong that it appears reasonable to regard it as
the "normal" law of permeability and therefore to seek some cause for
the "abnormal" behaviour of clays. A first suggestion, which has been
favoured by Terzaghi (1925) and by Macey, is that liquid viscosities in
very narrow, capillary passages are higher than normal, and that viscosity increases as the diameter of the capillary decreases. This would
give a qualitative explanation of the phenomenon, but does not lend
itself to quantitative expression unless the relation between viscosity and
capillary size is known. Such changes of viscosity, however, seem to be
invalidated by the interesting experimental work of Bastow & Bowden
(1935), for these workers have proved that viscosities of simple liquids
show no sign of alteration in rectangular channels down to 1/J, in width.
A more profitable suggestion was made by Zunker himself and later
incorporated by Kozeny (1932) into an extension of his own theory. This
was to assume that a film of water is held stationary at the surface of the
clay particles, with the result that the effective pore space, in which
water is free to flow, is reduced. An indication of the amount of water so
held is given by what Zunker calls the "hygroscopicity " of the clay, i.e.
the weight of water retained by 100 g. of clay (dry weight) when dried in a
desiccator at 20° C. For the clay soil under consideration, the hygroscopicity was 7-53. Since the true density of the fully dried clay was
2'68 g./cm.s, this may be converted more conveniently for our present
P. C. CARMAN
3
269
3
purposes to 0-202 cm. of water per 1 cm. of solid volume. The hygroscopicity represents water bound so firmly by the clay that it has a
negligible vapour pressure at 20° C, and may therefore be regarded as
giving a rough measure, probably on the low side, of the water bound so
firmly that it remains stationary when percolation takes place. If the
two quantities are equated, the free pore space, when the total pore space
is e, must be e1 = [e-0-202 (1—e)], since, in each 1 cm.3 of the bed, there
will be (1— e) cm.3 of solid volume, and hence 0-202 (1—e) cm.3 of
"hygroscopic water".
In modifying his theory, Kozeny (1932) substituted the free pore
space «! for the total pore space e in equation (14), obtaining
Thus, when one substitutes S = (l — e) So, the modified equation becomes
and the ratio of Kz to the new porosity function e13/(l—e)2 should be
constant. Actually, to obtain constancy, as shown in Table IV, it was
necessary to assume that the volume of the stationary film was 1-3 times
that of the hygroscopic water, so that the expression for €x became
[e - 0-262 (1 —e)]. As pointed out above, this is quite reasonable, and
leaves unaffected the main fact that the two quantities are comparable.
The average value of Kx x (1 — e)2/ei3 in the last column of Table IV is
14-7 x 10-10, and, according to equation (17), this is equal to gjkS02 or
196//S02, whence &0 = 3-65 x 105 cm.2/cm.3 and the effective particle size is
dm = 0-164/*. It is thus possible to calculate the thickness of the stationary layer, for every cm.3 of solid volume has a surface, 3-65 x 105 cm.2,
and 0-262 cm.3 of water in the stationary layer. The average thickness of
the layer, 8, is thereby 7-2 x 10-' cm., i.e. 72 A., or about thirty molecular
diameters. At first sight this appears rather excessive, but, in the past,
layers of 1000 and 3000 A. have been postulated to explain the plasticity
of clays (Houwink, 1937). Although such excessive thicknesses are no
longer considered probable, Mattson (1931) has shown that, when water
is added to a dry clay soil, there is a definite heat of adsorption until the
amount of water corresponds to a film of the order of 40 A. in thickness.
Further additions involve no further heat change. This is at least of the
same order as 8.
Exactly in what form the water is retained is still not quite certain,
since it is difficult to conceive of an adsorbed film exceeding 10 A. The
270
Permeability of Saturated Sands, Soils and Clays
more recent advances in the structure of clay minerals may possibly clear
up this matter. Clay minerals are characterized by a layer structure in
which there is a very strong resistance to rupture across the layers, but
very weak forces holding the layers together. It is this which accounts
for the typical shape of clay particles, namely, laminae in which the
length and breadth greatly exceed the thickness. The most typical
mineral in soil clays is beidellite (Marshall, 1936), which, as has been
proved by Hofmann et al. (1934), is capable of a very large and indefinite
degree of hydration between the layers, accompanied by a uni-directional
expansion at right angles to the layers. The following visualization of
Kozeny's modified theory seems therefore a reasonable one. One may
imagine 1 cm.3 of the clay soil completely water free. The porosity is e,
and the particle surface, therefore, S=(l—e)S0. Since the particles are
very thin laminae, this surface will be represented by the two flat sides of
each laminae, for the contribution of the edges will be negligible. Water is
next admitted to fill the pore space. Part of this is adsorbed strongly,
while another part, several times greater, will hydrate the clay and cause
the laminae to increase considerably in thickness. This swelling, together
with the adsorbed layer, corresponds to a reduction in the effective pore
space. Further, since an increase in thickness will not affect the flat
surfaces of the laminae, the value of S will remain practically unchanged,
i.e. it will still be equal to So (1 -e), where So is the specific surface of the
unhydrated clay. In other words, in transforming from equation (14) to
equation (16), Kozeny is justified in assuming that the effective pore
space is reduced from e to elt while the value of S suffers no modification.
This viewpoint cannot be considered unless it can first be shown that
the amount of swelling in each layer falls within reasonable limits. If
the average laminar thickness is t cm., the specific surface So is equal to
2/t, and, conversely, t is equal to 2/S0, i.e. 0-55 x 10"5 cm., or 550 A.
Assuming a spacing of 12 A. between layers for the unhydrated clay, this
gives the average number of layers in a lamina as 45-5. Now suppose the
whole of the water represented by the "stationary film thickness", S, is
used for hydration and swelling. Then, since this film is present on both
sides of each lamina, the amount of swelling in each layer is 28/45-5, i.e.
3-2 A., a value well within the limits suggested by the work of Hofmann
et al. Thus, a sample of montmorillonite, which swelled in water till the
spacing between layers was 19-6 A., gave a spacing of only 9-6 A. on
drying at 350° C.
It is interesting to note that Mattson calculated his value of 40 A. for
the "adsorption film" by assuming /S0 = l-2xl0 6 cm. 2 /cm. 3 . If one
P. C. CARMAN
271
carries out a similar calculation to the above, assuming the water is not
adsorbed but used for hydration, the swelling in each layer amounts to
5-8 A.
DATA OF TERZAGHI ON PERMEABILITY OF CLAYS
Apart from the data of Zunker, the only published data appear to be
some measurements made by Terzaghi (1925), which are printed as a
graph of rather small dimensions. The data given in Table V were read
from an enlargement of this. These figures were read from experimental
points, and not from the curves drawn through them, except in the case
of the highest permeability and porosity for each clay. This last reading
was justified, since, according to the text, the curves were drawn to pass
through experimental points which lay outside the published graph, and
which therefore represented higher porosities and permeabilities than any
reproduced in Table V. The values of K± were calculated from Terzaghi's
permeability coefficients on the assumption that the viscosity of the water
corresponded to a temperature of 10° C. A point worth noting is that
Terzaghi proved the validity of equation (1) (D'Arcy's law) for clays,
since he obtained consistent permeabilities on varying both the thickness
of the clay beds and the pressure gradient across them.
Table V. Terzaghi's data on permeability of clays
Clay
A
B
C
K, x 1010
2-4
101
0-35
0-21
013
005
2-4
0-62
0-35
0-24
2-4
1-35
0-51
0-40
007
Kj x 1010
e2
x (I-*) 2 /* 3
0-56
0-36
2-65
0-287
0-51
1-83
0-214
105
0-46
0-188
0-76
0-442
0-56
0-425
0163
0-27
0-405
0134
1-90
0-369
0-593
0-98
0-262
0-524
0-70
0-225
0-50
0-55
0-203
0-486
0-287
0-42
10-9
0-238
9-4
0-38
5-8
0-186
0-338
5-4
0170
0-325
1-6
0119
0-284
€t for clay A = e-0-455(l-e)
e t for clay B = e-0-55(l-e)
«i for clay C = e-0-23(l-c)
€
AjXll
x (1 -e)
10-0
10-2
10-4
9-8
9-9
7-4
7-8
7-9
7-7
7-7
34
38
35
37
21
As shown in Table V, Terzaghi's data conform very well to Kozeny's
modified theory, except, in the case of clays A and C, for permeabilities
so small that their accuracy was doubtful, both in the original measurement and in the writer's attempt to read the plotted points from Terzaghi's graph.
Journ. Agrio. Sci. xxix
18
272
Permeability of Saturated Sands, Soils and Clays
The values of So for these three clays, calculated from the quantities
in the last column of Table V, are, respectively, 44 x 105 cm.2/cm.s for
clay A, 5-0 x 105 cm.2/cm.3 for clay B, and 2-35 x 105 cm.2/cm.3 for clay C.
The corresponding values of 8 are, therefore, 103, 110 and 99 A. If one
carries out a calculation for the "average amount of swelling per layer",
similar to that given for Zunker's clay soil, the values reached are 5-5 A.
for clay A, 6-6 A. for clay B, and 2-8 A. for clay C. It is to be noted that
clays A and B fall into a different class from clay C, and that the latter
is in the same class as Zunker's clay soil. This is in accord with their
nature, for, whereas clays A and B •were pure plastic clays, clay C was
little more than a "sandy mud", containing 59-1 % of sand, and
Zunker's substance was a soil containing 51-8 % of particles over 2/x
in diameter. The calculation of "average swelling per layer" is based
on the assumption that the material is a pure clay, and it is undoubtedly
true that, were it possible to make allowance for silica particles, and to
calculate on the clay alone, the average swelling per layer would be of the
same order as for the plastic clays.
A distinctive feature of Kozeny's theory is that the permeability
becomes zero while the porosity has still quite a considerable value,
namely, e = 0-207 for Zunker's clay soil, and e = 0-313, e = 0-355 and
e = 0-187, respectively, for clays A, B and C. This is not brought out very
well by Zunker's results, but Terzaghi's data show clearly how rapidly
the permeability falls toward zero as the limiting porosity is approached.
SUMMARY
It is shown that the permeability of a water-saturated sand or fine
powder can be calculated with considerable accuracy, if the porosity and
the specific surface are known. In particular, the Kozeny theory here
discussed leads to a very useful relationship between permeability and
porosity. It is shown that clays do not conform to the theory in its
simple form, but that it may be modified to give a satisfactory representation of the data available. The physical grounds for this modified
theory are discussed in some detail, and it is shown that, while it is open
to criticism, it is at least in harmony with our present knowledge of clays.
An important deduction which follows from the modified theory is
that clays may have zero permeability at quite considerable porosities,
e.g. at e = 0-207 for a clay soil, and e = 0-355 for a plastic clay.
P. C. CARMAN
273
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DARAPSKY, A. (1912). Z. Math. Phys. 60, 170.
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HOFMANN, U., ENDELL, K. & WILM, D. (1934). Z. angew. Chem. 47, 539.
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(Received 8 August 1938)
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