Porosity and Permeability in Sediment Mixtures
by Patrick J. Kamann1, Robert W. Ritzi2, David F. Dominic1, and Caleb M. Conrad1
Abstract
Porosity in sediments that contain a mix of coarser- and finer-grained components varies as a function of the
porosity and volume fraction of each component. We considered sediment mixtures representing poorly sorted
sands and gravely sands. We expanded an existing fractional-packing model for porosity to represent mixtures in
which finer grains approach the size of the pores that would exist among the coarser grains alone. The model well
represents the porosity measured in laboratory experiments in which grain sizes and volume fractions were systematically changed within sediment mixtures. Permeability values were determined for these sediment mixtures
using a model based on grain-size statistics and the expanded fractional-packing porosity model. The permeability
model well represents permeability measured in laboratory experiments using air- and water-based permeametry
on the model sediment mixtures.
Introduction
Grain size, shape, and packing are key parameters
affecting porosity and permeability in unconsolidated
clastic sediment. Our interest is in poorly sorted sands
and gravely sands. In idealized sediment comprising uniform spheres with uniform packing, porosity is not a function of grain diameter, but permeability is a function of
the square of grain diameter (Hubbert 1940). However, in
mixtures of finer and coarser grains, both porosity and
permeability are related to grain diameters, volume fractions, and packing arrangements of the components.
We begin with a focus on porosity. In the following
section, we review prior work, including the fractionalpacking model for porosity in binary sediment mixtures
of Koltermann and Gorelick (1995). This empirical model
represents mixtures in which finer grains are much smaller than the pore spaces between coarser grains, as might
be true in sandy gravels. We derive the model from first
principles, and thereby make clear the physical meaning
of certain parameters and the underlying assumptions.
1Department of Earth and Environmental Sciences, Wright
State University, Dayton, OH 45435
2Corresponding author: Department of Earth and Environmental Sciences, Wright State University, Dayton, OH 45435;
(937) 775-3455; robert.ritzi@wright.edu
Received October 2006, accepted January 2007.
Copyright ª 2007 The Author(s)
Journal compilation ª 2007 National Ground Water Association.
doi: 10.1111/j.1745-6584.2007.00313.x
With that understanding, we extend the Koltermann and
Gorelick model to represent mixtures in which the finer
grains are similar in size to the spaces between coarser
grains, as in poorly sorted sands. We then present the results of laboratory experiments in which porosity was
determined in model sediments representing poorly sorted
sands and gravely sands, and compare porosity computed
with the modified fractional-packing model to the experimental results.
Shifting the focus to permeability, we follow the
approach of Koltermann and Gorelick (1995) of determining permeability for sediment mixtures by using grainsize statistics, our fractional-packing model for porosity,
and a modification of the Kozeny-Carmen equation. We
compare permeability computed using this approach to
measurements we made with water-based and air-based
permeameters. The results collectively illustrate relationships among porosity, permeability, and fractional-packing parameters. Furthermore, the positive comparisions
between the models and the experimental data help build
further confidence in these approaches for modeling
porosity and permeability in sediment mixtures.
Sediment Mixtures and Fractional-Packing
Models for Porosity
Particle-packing studies, as summarized by Koltermann and Gorelick (1995), have led to the following
Vol. 45, No. 4—GROUND WATER—July–August 2007 (pages 429–438)
429
knowledge about porosity in sediment mixtures: (1) the
porosity of a sediment mixture of two components is
determined by the size ratio and the percentage of each
component; (2) the porosity of a sediment mixture is less
than a linear combination of the porosity of the mixture
components; (3) the average grain diameter does not
determine porosity; (4) a porosity minimum occurs when
the volume of the finer-grained component equals the
pore volume of the coarser-grained component and the
finer grains completely fill the pore space between the
coarser, supporting grains; and (5) porosity values
decrease when mixtures of sediments are compacted (see
also Paxton et al. 2002).
Koltermann and Gorelick (1995) considered the case
in which the finer-grained component has a diameter
much smaller than the spaces among the coarser-grained
component, as illustrated in Figure 1. In this case, Clarke
(1979) identified two types of ideal packing. In ideal
coarse packing, the percentage of finer grains is less than
the porosity of the coarser grains taken alone and groups
of finer grains reside within the pore spaces among the
self-supported, coarser grains. Thus, the volume of the
mixture is equal to the volume of the coarser-grained
component. Each group of finer grains retains the porosity of the premixed, finer-grained sediment. In the second
type, ideal fine packing, finer grains are self-supported
and coarser grains are suspended throughout the finergrained matrix. Here, the volume of pore space that
would exist in the coarser-grained component alone is
lost. Note that in either type of packing, the volume of the
mixture is less than the sum of the volumes of the two
components before mixing them.
Here, we define parameters that are used to derive
the ideal packing models from first principles, and that
are further used to explore and expand the Koltermann
and Gorelick (1995) model. The total volume of the
mixture is VM and the total volume of voids within the
mixture is VVM. The volume of the premixed coarser component is VC, and VVC and VSC are the volumes of the
voids and solids, respectively, within it. The volume of
the premixed finer component is VF, and VVF and VSF are
the volumes of voids and solids, respectively, within it.
In ideal coarse packing, VM is assumed to equal VC,
corresponding to the ideal that the mixture is supported
by contacts among the coarser grains and the finergrained component occurs entirely within the pore spaces
created by the coarser grains. Accordingly, the composite
porosity, /, is determined by how much VVC is reduced
by VSF. Thus,
/¼
VVM
ðVVC 2 VSFÞ
¼
; VF , VVC; VM ¼ VC
VM
VM
ð1Þ
Note that:
ðVVC 2 VSFÞ
VVC
VF VSF
¼
2
;
VM
VC
VM VF
VF , VVC; VM ¼ VC
ð2Þ
where VVC/VC is the porosity of the premixed, coarsergrained component, /c. The coarse packing model also
assumes that each group of finer grains residing within
coarse pores retains the original packing arrangement and
has the same internal porosity that existed in the finergrained component before mixing. Therefore, VSF/VF =
1 2 /f, where /f is the porosity of the premixed finergrained component. We define rf as VF/VM, the ratio of
the premixed volume of the finer-grained component to
the postmixed volume of the sediment. (We use this symbol instead of Koltermann and Gorelick’s symbol c to
avoid confusion with references to the coarser component.) Using these symbols, Equation 2 can be written as:
/ ¼ /c 2 rf ð1 2 /f Þ; rf , /c ; VM ¼ VC
Figure 1. (A) Conceptual illustration of ideal packing. In
ideal coarse packing, groups of finer grains fit within pore
space between coarser grains. In ideal fine packing, individual coarser grains are supported within a matrix of finer
grains. (B) Variation in porosity as a function of the finergrained content. (C) Variation in permeability as a function
of the finer-grained content (modified from Koltermann and
Gorelick 1995).
430
P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
ð3Þ
Figure 1B shows that as rf increases, / reduces toward a minimum, /min, which occurs when VF = VVC so
that the finer-grained component exactly fills all pores
within the coarser-grained component. Substituting VF for
VVC and VC for VM in Equation 2 and rearranging gives:
VF
VSF
/¼
12
ð4Þ
VC
VF
and back substitution of VVC for VF gives:
/¼
VVC VVF
;
VC VF
VF ¼ VVC;
VM ¼ VC:
ð5Þ
Thus, at its minimum, the composite porosity is
equal to the product of the porosities of each component,
/min ¼ /c /f ;
rf ¼ /c ;
VM ¼ VC
ð6Þ
In ideal fine packing, the coarser grains are dispersed
and each is supported by a matrix of finer grains. Accordingly, the void spaces of the premixed coarser component
are not preserved in the composite mixture and the mixed
total volume is given by:
VM ¼ VF 1 VSC;
ð7Þ
VF > VVC
Because the pore spaces of the composite mixture
exist only within the finer-grained component, the porosity is as follows:
/¼
VVF
;
ðVF 1 VSCÞ
ð8Þ
VF > VVC
and because 1/(VF 1 VSC) = 1/VM = rf/VF:
/ ¼ rf / f ;
rf > /c ;
VM ¼ VF 1 VSC
both packing types may be present, especially for rf in the
middle of its range (Clarke 1979).
Koltermann and Gorelick (1995) presented a fractional-packing model (Koltermann and Gorelick model
hereinafter), which fits experimental data much better
than the ideal model because it represents the existence of
both. The model is defined piecewise over rf:
/ ¼ /c 2 rf yð1 2 /f Þ 1 ð1 2 yÞrf /f ;
/ ¼ /c ð1 2 yÞ 1 rf /f ;
0.7
rf /c
ð11Þ
The equations are constructed with a weighting parameter, y, such that the value of porosity for ideal coarse and
fine packing are returned when y = 1. The y is empirically
defined as a piecewise-linear triangle function:
ymin 2 1
y ¼ rf
ð12Þ
1 1; rf , /c
/c
ð9Þ
Equations 3, 6, and 9 show how porosity varies with
rf under ideal packing. Together, they can be considered
the ideal packing model.
The ideal packing model underpredicts porosity of
sediment mixtures (Koltermann and Gorelick 1995). The
greatest underprediction occurs at the porosity minimum.
This is shown in results presented in Figure 2 and also
in results from other experiments on sediments made of
uniform spheres and of angular grains (Furnas 1929;
McGeary 1961; Shakoor and Cook 1990). This underprediction results from the assumption that only one type
of packing occurs at each value of rf, whereas in reality,
rf , /c ð10Þ
y ¼ ðrf 2 1Þ
1 2 ymin
1 1;
1 2 /c
rf > /c
ð13Þ
which varies with rf, from 1 at each endmember (rf = 0,1)
to ymin at rf = /c (corresponding to /min). The physical
meaning of y will become clearer subsequently. In practice, the model is defined experimentally by determining
the parameters /c, /f, /min, and then computing ymin as
follows:
ymin ¼ 1 1 /f 2
/min
/c
ð14Þ
The Koltermann and Gorelick model is plotted in
Figure 2. It matches the porosity data for sediment mixtures better than the ideal model.
0.6
Fractional
Packing
Model
Porosity
0.5
Deriving the Koltermann and Gorelick Model
from First Principles and Extending It
0.4
0.3
Ideal Packing
Model
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1.0
Clay Content by Weight
Figure 2. Comparison of porosities computed with the Koltermann and Gorelick and the ideal packing models to
experimental data. The fraction of the finer-grained component, in this case clay, was quantified by weight instead of by
volume. Clay content by weight can be related to rf via
Equations 2a and 2b in Koltermann and Gorelick (1995).
The dashed curves represent porosity computed with the
ideal packing model. The ideal model captures some of the
features of the data, such as the porosity minimum. However, it systematically underpredicts porosities. The solid
curves represent porosity computed with the Koltermann
and Gorelick model. Source: Koltermann and Gorelick
(1995).
The empirical Koltermann and Gorelick model is
based on the premise that a mixture contains both regions
with coarse packing (type A regions) and regions with
fine packing (type B regions). Equations 10 and 11 of the
model can be derived by carefully defining all the components of volume within these regions of such a mixture,
writing the exact equation for the mixture, and then
examining what approximations and corresponding assumptions are needed to derive Equations 10 and 11 from
the exact equation. Doing so allows us to clarify the physical meaning of certain parameters in the Koltermann and
Gorelick model, to identify underlying assumptions, and,
importantly, to extend the model.
The total volume of type A regions is VA. Within
type A regions, the total volume of the coarser-grained
component is VCA, and this component has a volume of
voids (before consideration of finer grains occupying
them) of VVCA. Furthermore, within type A regions, the
total volume of the finer-grained component is VFA, and
this component has a volume of voids of VVFA and
P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
431
a volume of solids of VSFA. The total volume of type B
regions is VB. Within type B regions, the total volume
of the finer-grained component is VFB, the total volume
of voids of the finer-grained component is VVFB, and the
volume of solid coarser grains is VSCB. Because type B
regions have fine packing with each coarser grain individually supported in a finer-grained matrix, VB = VFB 1
VSCB. For mixtures with both types of regions, the porosity is exactly given by:
/¼
VVCA 2 VSFA VA
VVFB
VB
1
ð15Þ
VA
VM
VFB 1 VSCB VM
Equation 10, applicable when rf < /c, can be derived
from Equation 15. This is done by expanding Equation 15
as three terms:
VVCA VA
VSFA VA
/¼
2
VA VM
VA VM
VVFB
VFB 1 VSCB
1
ð16Þ
VFB 1 VSCB
VM
and then making assumptions to achieve equivalency
between each of these terms and each of the three respective terms of Equation 10. The first term of Equation 10
is derived by assuming that though region B exists, its
volume is small enough so that VA/VM ’ 1. The first
term in Equation 16 then becomes VVCA/VA, which
equals VVCA/VCA because all fines are within pores of
the coarser component, which, in turn, is /c. The second
term of Equation 10 is derived by rewriting the second
term of Equation 16 in the following steps:
VSFA VA
VFA
VVFA VFA VA
¼
2
VA VM
VFA
VFA VA VM
¼ ð1 2 /f Þ
VF VFA
¼ ð1 2 /f Þrf y
VM VF
ð17Þ
It is clear that y represents VFA/VF, the volume fraction of the finer-grained component that exists within
region A, when rf < /c. The third term in Equation 10 is
derived by assuming that VSCB is very small and therefore can be neglected. In that case, the third term of Equation 16 can be rewritten as follows:
VVFB VFB
VFB VF
¼ /f
¼ /f ð1 2 yÞrf
VFB VM
VF VM
ð18Þ
Thus, to derive Equation 10 from exact considerations of volume, it must be assumed that VA/VM ’ 1 and
VSCB ’ 0.
When rf > /c, the Koltermann and Gorelick model is
given by Equation 11, which has two terms. Each of these
terms is an approximation of each of the respective two
terms of Equation 15. The first term of Equation 15 is
approximated as follows:
VVCA 2 VSFA VA VVC
’
ð1 2 yÞ
VA
VM
VC
ð19Þ
and the second term is approximated as follows:
432
P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
VVFB
VB
VF VVF
’
¼ rf / f
VFB 1 VSCB VM VM VF
ð20Þ
Making these approximations requires assuming that
VSFA and VSCB are both negligible, that VA = VC, that
VB = VF and accordingly VFB = VF, and that VM = VF
1 VC, as is true when the components are unmixed. It
also requires that y represent a different physical attribute
over rf > /c in Equation 11 than it represents over rf < /c
in Equation 10. It is apparent through these comparisons
that in Equation 11, y is equal to 1 2 VA/VM, which is
equal to VB/VM, which, under these approximations, is
equal to VF/VM, which is, in turn, equal to rf. Thus, in
Equation 11, rf and y are the same, and the rf and (1 2 y)
terms are weights that add to unity (which are not true for
Equation 10).
The piecewise-linear function Koltermann and Gorelick used for y over rf < /c in Equation 12, representing
VFA/VF, cannot be defined from first principles following the aforementioned methods because there is no basis
for defining what volume fraction of the finer-grained
component should reside in type A or type B regions.
However, the function explains the results in Figure 2
fairly well and thus serves as a good empirical model.
To summarize, the Koltermann and Gorelick model
is conceptually based on the idea that different types of
packing (type A and type B regions) can occur within
a sediment mixture. It contains empirically structured
equations. To define the model for a specific sediment
mixture, one must know the porosity of each of the premixed components and the minimum porosity that would
occur among possible rf.
Considering Poorly Sorted Sands
The ideal packing model and the Koltermann and
Gorelick model both assume that the diameter of the finer
grains, df, is much smaller than the size of pore spaces
between the coarser grains, dcp, and thus that in type A regions, many finer grains are able to fit within each pore
space between coarser grains. This assumption cannot be
made for many natural sediments. For example, spheres
with a diameter, d, of 0.385 mm (equivalent to medium
sand used below) in perfect cubic packing can contain
a spherical grain as large as 0.414d, or 0.159 mm in diameter (Kamann 2004). Let dcp represent the size of the
largest such sphere to fit within the pore space between
coarser grains. If fine sand spheres with a diameter
greater than 0.159 mm are mixed with this medium sand,
a single fine sand grain could not fit within an undisturbed pore between medium sand grains. Clearly an aggregate of fine sand grains, with their original pore space
intact, could not fit within such a pore space either.
We noticed in some early work that the Koltermann
and Gorelick model did not represent mixtures very well
if the diameter of the finer grains, df, was as large as or
larger than dcp. Furthermore, the minimum porosity in
such mixtures did not occur where rf = /c (as shown in
Figure 3). Equation 10 is a function of rf and y, and Equation 12 shows rf is also a function of y. When Equation 12
is substituted in Equation 10, it is clear that porosity is a
parabolic function of rf. The first and second derivatives
of the parabolic function show it will have a minimum at
21
rf ¼ 0:5/c ð/f 2 1Þð/f 2/min /21
c Þ . As shown in Figure 3, if this minimum occurs at rf < /c, the polynomial
can swing down to a spurious minimum, below what is
specified for /min in Equation 14. Thus, we sought a fractional-packing model that better represented these mixtures and that did not have a parabolic formulation.
Our approach to developing an alternate fractionalpacking model is still conceptually based on the idea that
multiple types of packing can occur within a sediment
mixture. When rf < /c and df > dcp, then ideal coarse
packing cannot occur because each finer grain mixed
into the coarser component causes separation of the surrounding coarser grains. The packing of coarser grains
will be disturbed. In this case, and also when df ’ dcp, the
void space of the premixed finer-grained component is
largely lost. We will refer to such cases as disturbed
coarse packing.
We allow that there may be a number of region types
possible in a mixture: regions with (1) ideal coarse packing or (2) disturbed coarse packing, and regions with (3)
ideal fine packing, (4) coarser grains only, and (5) finer
grains only. There is no way to define the relative volumes of each region type a priori from physical first principles. However, it again seems reasonable to assume, as
Koltermann and Gorelick did, that the most significant
departure from ideal packing porosity will occur when rf
has intermediate values, and the least departure will be at
extreme values of rf, where the sediment is virtually of
one component.
A triangle function is used as in the Koltermann
and Gorelick model, as follows. First, n is defined as the
premixed volume of finer grains per premixed volume
of sediment, VF/(VC 1 VF). If the components were
combined but not mixed (so that regions of finer and
coarser grains were entirely segregated), then rf = n. If
the sediment is stirred, then regions of types (1) to (3)
can occur and, consequently, the volume of the mixture
reduces and rf becomes greater than n. We define rv as
a ratio
rv ¼
rvmax 2 1
rv ¼
n 1 1;
n/min
rv ¼
Modified
Model
/c 2 /min rf
/ ¼ /c 2
;
n/min
rv
Porosity
n¼
0.4
ð23Þ
where n/min is the premixed volume fraction of fines at
which /min occurs. A piecewise-linear fractional-packing
model for porosity is then posed as follows:
0.3
0.2
n n/min
ð22Þ
rf
21 ;
rv
n n/min
n n/min
ð24Þ
ð25Þ
To define the model for a specific sediment mixture,
one must experimentally know /c, /f, and /min, just as in
the Koltermann and Gorelick model. The rvmax is computed from the VC, VF, and VM at which /min occurs.
When df ’ dcp and /c ’ /f, then /min occurs where
n ’ /c, rather than at rf ’ /c as in the Koltermann and
Gorelick model. We can understand this from physical
first principles starting with:
Mean
Porosity
0
n n/min
rvmax 2 1
ð1 2 nÞ 1 1;
1 2 n/min
/ 2 /min
/ ¼ /f 1 f
1 2 n/min
K&G
Model
ð21Þ
To relate this expression to the triangle weighting
function, y, in the Koltermann and Gorelick model, recall
that y equals unity under ideal coarse packing. The
amount that y departs below unity represents the degree
of nonideal packing. The greatest departure below unity is
at ymin. In a similar vein, under ideal coarse packing, rv
will equal (VF 1 VC)/VC, which is at a maximum when
VF = VVC (Equation 5). The amount that rv actually departs below this value represents the degree of nonideal
packing. The greatest departure below this value is at
rvmax. Accordingly, we define the following triangle function for rv:
0.4
rf
VC 1 VF
¼
VM
n
0.6
0.8
1
rf
Figure 3. Comparison of fractional-packing models to
porosities of mixtures of fine and medium sand. The mean
porosity determined in the laboratory among replicate
mixtures is plotted with bars representing 62 standard
deviations. The parameters used in the models are the
mean values from the experiments, with /c = 0.407, /f =
0.411, /min = 0.328 at rf = 0.407 in the Koltermann and Gorelick (K&G) model, and /min = 0.328 at n = 0.410 in the
modified model.
VF
VVF 1 VSF
¼
VF 1 VC
VF 1 VC
ð26Þ
Consider the case where each pore among coarser
grains is large enough to accommodate approximately one
finer grain. Therefore, /min should occur when a finer
grain obstructs every pore among coarser grains (VSF ’
VVC). Thus:
n¼
VVF 1 VVC
VVF
VF
VVC
VC
¼
1
VF 1 VC
VF VF 1 VC
VC VF 1 VC
ð27Þ
P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
433
Furthermore, if /c ’ /f, Equation 27 can be rewritten as follows:
n¼
VVC
VF
VVC
VC
1
VC VF 1 VC
VC VF 1 VC
ð28Þ
Thus,
VVC
VF
VC
1
n¼
¼ /c
VC VF 1 VC
VF 1 VC
ð29Þ
Experiments Measuring Porosity in
Sediment Mixtures
We combined grains from two different size categories (pairing from among four categories: fine sand,
medium sand, coarse sand, and pebbles) to create binary
sediment mixtures as simple models for poorly sorted
sands and sandy gravels. So that grains were fairly
uniform and spherical in each size category, glass sandblasting beads were used for the sand-sized components
and marbles were used for the pebble-sized component.
The sand blasting beads were obtained as presorted fine,
medium, and coarse sand. These beads were further
sorted in the laboratory with sieves. Our model fine sand
was restricted to those grains retained between sieves with
0.148 and 0.177 mm openings, medium sand between
sieves with 0.350 and 0.420 mm openings, and coarse
sand between sieves with 0.590 and 0.710 mm openings.
The marbles representing the pebble size category were
already uniform in size with a diameter of 10 mm.
Table 1a compares the grain diameters to the approximate ‘‘spherical’’ pore sizes under cubic packing (largest
pore sizes) for the sediments used. In the mixtures with
only sands, df is approximately equal to dcp. Thus, disturbed coarse packing will occur.
Samples were created for each endmember category
and for mixtures of two categories. The categories can
be paired in six combinations (Table 1b). Samples were
created for each mixture combination with n differing by
10% increments, spanning the range between 0% and
100%. Additional mixtures were created with n at 25%
and 75%, and with n = /c. Each sample was created with
100 mL of premixed sediment (VF 1 VC). To create
a sample, the appropriate volume of each component
was measured in graduated cylinders. These components
were added together in a larger container and stirred
with a rod 15 cm long and 0.05 cm in diameter until
they were thoroughly mixed (see Conrad [2006] for experiments on mixing time required to achieve a homogenous mixure). After the sediments were mixed, they
were then poured into a graduated cylinder and VM was
recorded.
To measure /, water was added to the mixture within
a graduated cylinder. To avoid trapped air bubbles, the
graduated cylinders were tilted as water was slowly poured in. This allowed the water to flow down one side of the
glass to the bottom of the graduated cylinder and force
out air in pore space above it. The volume required to saturate the sediment mixture was recorded as the volume
of void space within the sediment, and / was computed
as VV/VM. This procedure for creating a sample and
measuring / was repeated three times for each n. Additional detail on the procedure and results of these experiments is given in Kamann (2004).
The average / was 41.1% for fine sand, 40.7% for
medium sand, 39.0% for coarse sand, and 40.7% for pebbles. Each value is plotted in Figure 4 as endmembers,
with bars indicating two standard deviations. Note that all
four grain categories had / that were close among
repeated measures, close among categories, and closer to
the theoretical / for spherical grains in cubic packing
(47.65%), than in rhombohedral packing (25.95%). The
mean of the percent volume reductions from VC 1 VF to
VM in each sediment mixture is given in Kamann (2004).
The pebble mixtures had the greatest reductions, typically
10% to 30%. The greatest volume reduction generally
occurred at the /min.
In Figure 4, the / measurements are plotted with the
modified fractional-packing porosity model given in
Equations 24 through 25. The average measured /min for
mixtures containing pebbles is close to the ideal packing
model. The greatest departure from the ideal model
occurred in the medium and coarse sand mixture where
average measured /min is 36.2% and the ideal /min is
15.9%.
The model compares well to laboratory porosity
measurements. However, the model deviates from measurements in mixtures of fine sand and pebbles and
medium sand and pebbles. Here, the model tends to overpredict measured porosity when rf > /c just as
the Koltermann and Gorelick model does in Figure 2.
The linear interpolations in Equations 12 through 13
(Koltermann and Gorelick model) and Equations 22
Table 1a
Table 1b
Sizes of Grains and ‘‘Spherical’’ Pore Spaces
Under Cubic Packing
Median Grain
Diameter (mm)
Pebbles
Coarse sand
Medium sand
Fine sand
434
10
0.650
0.385
0.163
df/dcp for Various Mixtures
Coarser Component
dcp
(mm)
4.142
0.269
0.159
0.068
P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
Finer Component
Fine sand
Medium sand
Coarse sand
Medium
Sand
Coarse
Sand
1.025
—
—
0.606
1.431
—
Pebbles
0.039
0.093
0.157
Figure 4. Comparison of porosities computed with the expanded fractional-packing model to experimentally determined
porosities. For the experimental data, the mean is plotted with bars representing two standard deviations.
through 23 (new model) do not capture all aspects of
fractional packing when df dcp .
Permeability in Sediment Mixtures
k¼
Method Based on Grain-Size Statistics
Hubbert (1940) showed that for spherical grains of
uniform diameter, permeability is proportional to the
square of grain diameter:
k ¼ vd 2
ð30Þ
where v is a proportionality coefficient (dimensionless).
Kozeny (1927) and Carman (1937) related v to porosity
and took median grain size, dm, as representative in sediments with nonuniform grains giving:
k¼
2
dm
/3
180 ð12/Þ2
and Gorelick (1995) modified the Kozeny-Carman equation to consider the fractional packing of grains in sediment mixtures:
ð31Þ
The use of dm works best in sediment that is well sorted. To better represent sediment mixtures, Koltermann
2 3
dfp
/fp
180ð12/fp Þ2
ð32Þ
where dfp is the representative grain diameter, dependent
on fractional packing, and /fp is the porosity of the sediment mixture calculated with the fractional-packing model.
Koltermann and Gorelick (1995) found that in sediment mixtures with rf < /c, the geometric mean of grain
diameter works well for dfp; with rf > /c, the geometric
mean causes overprediction of k, but the harmonic mean
of grain diameter works well for dfp.
Figure 1C illustrates how k predicted by the Koltermann and Gorelick model changes with change in the
percentage of finer grains. As rf increases from 0 toward
that at which a porosity minimum occurs, k decreases
slightly from that of the coarser-grained component, kc.
As rf nears that of the porosity minimum, k sharply
decreases toward kmin. As rf increases past that of the
P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
435
porosity minimum, k gradually increases to that of finer
grains, kf, at rf = 1.
Using Equation 32, k was computed for each of the
six sediment mixtures with rf ranging from 0 to 1. For
these computations, /fp was computed with Equations
24 and 25. Based on the findings of Koltermann and
Gorelick (1995), for pebble mixtures, the geometric mean
of median grain diameters for each component (Table 1a)
was used to calculate dfp when rf < /c, and the harmonic
mean of grain diameters was used when rf /c. For mixtures without pebbles, the geometric mean of the median
grain diameters was used to calculate dfp when n < /c,
and the harmonic mean of grain diameters was used when
n /c. The computed k for each grain mixture is plotted
against rf in Figure 5.
Experimental Determination of Permeability
Permeability was measured using both air- and waterbased methods. Samples were loaded into a constant-head
permeameter tube 31.3 cm long and 8.3 cm in diameter
with two manometer tubes spaced 15.3 cm apart along its
axis. Air permeameter measurements were collected as the
constant-head permeameter tube was filled. This was done
by closing one end with a cap and glass-wool filter packing, securing the tube with the long axis vertical and capped end on bottom, and filling it with premixed sediment
at 5 cm increments for a total of six increments. At each
increment, an air permeameter measurement, a, was made
at three locations on the sediment surface, and at each
location, three measurements were taken. The air permeameter tip seal had an inside diameter of 4 mm and an
outside diameter of 26 mm. More details on the compressed-air permeameter instrument and a model used to
correct for non-Darcian flow and other effects are given in
Conrad (2006). After the measurements were completed
on an increment, the next increment of sediment was added and the process was repeated until the tube was filled.
This resulted in a total of 54 measurements with the air
permeameter on the entire column of sediment. The geometric mean of the nine measurements for each increment
was computed. To determine the average k of the entire
sediment column, the harmonic mean of the geometric
means from each increment was calculated giving Æaæ, the
air-based measure for a column of sediment.
Figure 5. Comparison of permeability computed with the expanded fractional-packing model (Equations 24, 25, and 32) to
air- and water-based experimental measurements. Note that the scales of the ordinate axes for (A) to (C) differ from those for
(D) to (F). There are three x and Æaæ values plotted at each rf. This cannot be discerned in all the plots because, in most cases,
the three values varied only slightly.
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P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
When the permeameter tube was filled, the top was
packed with glass-wool and sealed with a cap. It was
then clamped in place and connected to a system of hoses
that provided for water flow into one end, discharge from
the other end, and head measurements from the manometers. Reynold’s numbers during water flow were well
within the Darcian regime (Conrad 2006). We measured
head in each manometer of the permeameter to within
0.001 m, measured flow rate to within 0.001 m3/s, and
measured water temperature to within 0.1C. These measurements and tables giving q and l as a function of temperature were then used to compute k with Darcy’s law,
giving a water-based measurement, x. Each time the permeameter was filled, these water-based measurements
were repeated three times under different gradients,
giving three x values. The mean of these values was
computed, giving x, the water-based measure for a column of sediment.
After completing the air- and water-based measurements on a sediment column, the sediment was emptied
from the tube and dried. The procedures for filling the
permeameter tube and taking air- and water-based measurements were then followed again using the dried sediment, giving a total of three air-based and three waterbased determinations of k for each sediment mixture.
Using these procedures, permeability was determined
using sediment samples representing each of the four
grain-size categories, and samples representing the six
combinations of any two categories. Generally, sediment
samples were created representing each combination with
n varied to represent a percentage of finer grains that was
below, around, and above /c (e.g., 0.25, 0.35, 0.40, 0.75).
However, in combinations that include pebbles, when n <
/c, the finer grains do not remain evenly dispersed in the
mixture but settle to create a layer where they completely
fill the pores among the coarser grains, overlain by a layer
where they are mostly absent. Because it was not possible
to maintain uniformity in these combinations when n <
/c, they were not tested. Furthermore, the 100% pebble
sediment has permeability above the measurement range
of the air-based permeameter (the compressed air backpressure is not measurably different from ambient air
pressure) and the water-based permeameter (a difference
in head is not measurable across the manometers). Thus,
the experiments were limited to gravely sands and poorly
sorted sands. Different instruments would be required for
the determination of the permeability of the pebbles.
permeability values for the calculation. The lines in Figure 5 representing the permeability model are computed
using only the median grain size of the endmembers and
the expanded fractional-packing model for porosity. Thus,
the model lines and the experimental permeability data
presented in Figure 5 are completely independent of each
other over the full range of rf. The differences between
the model and the experimental results are viewed as
generally small, and the model is judged to represent the
permeability of the sediment well.
Discussion
The expanded fractional-packing model for / generally compared well to the / measurements among all sediment mixtures. The results represented cases in which
df/dcp is approximately 0.04, 0.10, 0.15, 0.6, 1.0, and 1.4.
The original Koltermann and Gorelick model represents
only the first three of these. Use of the expanded model
does not require knowledge of df/dcp and can be used in
all cases.
When k values are determined from the modified
Kozeny-Carmen model, which uses the expanded fractional-packing model for /, they correspond closely to airand water-based k measurements among the sediment
mixtures we examined. The results increase confidence
in determining k using a method based on grain-size
statistics.
All sediments used in these experiments consisted of
highly spherical glass beads, and sediment mixtures were
of bimodal, nearly binary, grain sizes. These sediments
provide an approximation of natural sands and gravely
sands. However, grains in natural deposits have more
angularity and a more complex distribution of grain sizes.
Nevertheless, the sediment mixtures used in these experiments did contain complex packing arrangements,
which likely included finer grains filling pores of coarser
grains (ideal coarse packing), single finer grains occupying expanded space between coarser grains (disturbed
coarse packing), coarser grains occupying expanded
space between finer grains, and regions of only coarser
grains and of only finer grains. This conceptual framework for packing arrangements seems generally applicable to angular grains that are either much smaller than
or close to the size of pores within the coarser-grained
component.
Conclusions
Results
The experimentally determined permeabilities are
plotted in Figure 5 together with permeability computed
with Equation 32 and the expanded fractional-packing
model. Note that there is an important difference between
this figure, which compares measured and modeled
permeability, and Figure 4, which compares measured
and modeled porosities. The fractional-packing model for
porosity required using the endmember and minimum
porosities to compute porosity. However, the model for
permeability (Equation 32) does not require knowing any
In this research, we expanded on work by Koltermann and Gorelick (1995) and examined how porosity
and permeability are related to attributes characterizing
sediment mixtures. We measured porosity and permeability
in model sediment mixtures, while systematically varying grain sizes and proportions of size fractions. While
Koltermann and Gorelick (1995) considered mixtures in
which the finer grains are much smaller than the pore
spaces between coarser grains, we also considered mixtures in which the finer grains are not. Furthermore, we
experimentally measured permeability using air- and
P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
437
water-based methods as a basis for evaluating permeability computed using the Koltermann and Gorelick version
of the Kozeny-Carmen equation (based on grain-size
statistics) with our expanded fractional-packing model
for porosity. The analysis of these results has led to the
following significant conclusions:
1. The expanded fractional-packing model performed well
in representing porosity of the sediment combinations
used in these experiments. This model represents porosity
in sediments where the diameter of finer grains is much
smaller than the diameter of pores among coarser grains,
as does the Koltermann and Gorelick (1995) fractionalpacking model. The model also represents porosity in
sediments where the diameter of finer grains is approximately equal to the diameter of pores among coarser
grains.
2. In sediment mixtures where the diameter of finer grains is
approximately equal to the diameter of pores between
coarser grains and the porosities of the two components
are similar, a porosity minimum occurs when the volume
fraction of finer-grained component in the premixed
sediment equals the porosity of the coarser-grained
component.
3. When permeability values calculated from grain-size statistics account for fractional packing, they correspond
closely to experimentally determined permeability measurements among the sediment mixtures we examined. This
correspondence indicates that methods based on grainsize statistics give representative results for the permeability of poorly sorted sands and gravely sands.
Acknowledgments
This research was partly supported by the National
Science Foundation under grant NSF-EAR 00-01125.
Any opinions, findings and conclusions, or recommendations expressed in this article are those of the authors
and do not necessarily reflect those of the National Science Foundation. This research was also partly supported by a Wright State University Research Incentive
Grant. This support is gratefully acknowledged. We
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P.J. Kamann et al. GROUND WATER 45, no. 4: 429–438
appreciate the constructive review comments of Christine
Koltermann and two anonymous reviewers. Finally, we
thank David Moulton and Daniel Tartakovsky for providing the geometric factor value (via the method of Tartakovsky et al. 2000) for the air permeameter used in these
experiments.
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