In this paper, the results of permeability and specific surface area
analyses as functions of granulometric composition of various sediments
(from silty clays to very well graded gravels) are presented. The effective
porosity and the referential grain size are presented as fundamental
granulometric parameters expressing an effect of the forces operating on
fluid movement through the saturated porous media. This paper suggests
procedures for calculating referential grain size and determining effective
(flow) porosity, which result in parameters that reliably determine the
specific surface area and permeability. These procedures ensure the
successful application of the Kozeny–Carman model up to the limits of
validity of Darcy's law. The value of effective porosity in the referential
mean grain size function was calibrated within the range of 1.5

The effect of the granulometric composition of granular porous media on its
transmissivity, accumulation and suction parameters is both a permanent
scientific challenge and a practical issue. In hydrogeology, particular
attention is given to hydraulic conductivity. Hazen (1892) and Slichter (1902) have published widely accepted and reputable models for calculating
the hydraulic conductivity of uniform sands using effective grain size. The
term “effective grain”, used for grain diameters in both formulae, could
lead to confusion (Mavis and Wilsey, 1936). However, Hazen's formula uses

The usage of certain forms of mean grain size became inevitable with the development of hydraulic conductivity models that describe relations between the hydraulic conductivity and the specific surface area (Krüger, 1918; Zunker, 1920; Blake, 1922; Kozeny, 1927; Fair and Hatch, 1933). Kozeny (1927) introduced the equation of permeability for the flow model containing a bundle of capillary tubes of even length. Kozeny's permeability formula was later modified by Carman (1937, 1939). Carman redefined specific surface area and presented it as a conversion of mean grain size and the index of porosity and incorporated an effect of tortuosity for the flow around individual grains. The resultant form of the equation is known as the Kozeny–Carman (KC) equation. The verity of the KC formula application results is strongly dependent on the verity of effective porosity and representative grain size. Kozeny (1927) used the harmonic mean grain size of samples. Bear (1972) recommended the same grain size. Koltermann and Gorelick (1995) and Kamann et al. (2007) stated that the harmonic mean performed best in samples with high fine-grain contents. Chapuis and Aubertin (2003) proposed laboratory tests for determining the specific surface area of fine-grained materials for application in the KC formula. Several authors (Al-Tabbaa and Wood, 1987; Dolinar and Otoničar, 2007) have studied applicability of KC formulae for calculation of hydraulic conductivity of fine-grained materials. All of them have concluded that the KC model in its original form does not apply on clays. Dolinar and Otoničar (2007) have also proposed a modified form of the KC equation.

The objective of this article is to research the relationship between average mean grain size and effective porosity in relation to permeability and specific surface area for a wide range of grain sizes and particle uniformities in various soil samples. In the hydraulic conductivity calculations, the Kozeny–Carman equation was used to discover the algorithm for calculating the referential mean grain size. This grain size, along with effective porosity, generates a harmonious parametric concept of the impact of porous media geometrics on its transmission capacity.

For the purpose of this work, data on sandy and gravelly aquifers and clayey-silty deposits were collected. All of the study sites are located in the plains of the Republic of Croatia (CRO) (Fig. 1). The northern parts of the Republic of Croatia are covered by thick quaternary deposits with sandy and gravelly aquifers (Brkić et al., 2010). Covering aquitards are composed of silty-clayey deposits.

Map of northern Croatia with test site locations.

The analyses of non-cohesive deposits were conducted on 36 gravel test samples from 6 investigation boreholes on the Đurđevac well field (marked as GW on Fig. 1); 19 uniform sand test samples from the investigation boreholes on 2 well fields – Beli Manastir (marked as SU1) and Donji Miholjac (marked as SU2); and 28 samples of sand with laminas made of silty material from 2 investigation boreholes on two well fields – Ravnik (marked as FS/SU1) and Osijek (marked as FS/SU2). Appropriate pumping tests were conducted on the test fields to determine the average hydraulic value of aquifers.

Cohesive deposits were investigated at three sites. Soil samples from exploration boreholes (depth 1.0–30.0 m) were laboratory-tested. Analyses on granulometric composition (grain size distribution), hydraulic conductivity and Atterberg limits were conducted. On the first test field (route of the Danube, Sava Canal; marked as CI/MI1), all the aforementioned analyses were conducted for each soil sample. Sixty-five samples of various soil types were analyzed. At the second and third test sites (Ilok, marked as CI/MI2, and Našice, marked as CI/MI3), loess and aquatic loess-like sediments were investigated. Laboratory analyses were conducted on 21 samples from 8 investigation boreholes. Specific analyses at various depths were conducted on the samples from this test site; on account of this, the mean values for the individual boreholes were correlated (Urumović, 2013).

The effects of porosity

Definition sketch of liquid driving and opposed viscous forces for elemental volume.

The forces of pressure and gravity cause the motion of the fluid in the
pores. A pressure force is transferred to

There are four ways to express the specific surface area

In hydrogeology, the specific surface area is often presented with a
conversion of mean grain diameter

Effects of driving (

Evidently, the effective porosity

Many authors present the Kozeny–Carman equation with

The integration of all of the mentioned grain sizes (Eqs. 7–9) in
the sieve residue across the entire sample has a crucial effect on the mean
grain size value. An overview of both the related expert and scientific
literature indicates the use of either the arithmetic mean,

Range and arithmetic mean of specific yield values for 586 analyses in Hydrologic Laboratory of the USGS (from Morris and Johnson, 1967).

Relation between referential mean grain

In a permeability model, the porosity function expressed by porous media
transmissivity factors (Eq.

The value of effective porosity is slightly lower than the value of the
specific yield. This value is related to the referential mean grain size
(

Reliable verification of the analyzed parameter relations for a wide range
of granulometric compositions was conducted using the Kozeny–Carman
equation, and the analyses of the hydraulic conductivity researched deposits in situ
as well as in the laboratory. Hydraulic conductivity

The Kozeny–Carman equation is actually a special form of Darcy's law (in the
case of the unit value of hydraulic gradient). Hence, it should be
applicable across all possible natural samples of porous media. The
hydraulic testing of natural deposits poses a problem in correlation
investigations. Non-cohesive deposits make it almost impossible to ensure
the laboratory testing of the content and distribution of particles or to
consolidate material in its natural and undisturbed state. The average
hydraulic conductivity calculated by analyzing the pumping test data was
used for correlation in the non-cohesive deposits. Test sites were chosen to
fulfill the following criteria: the borehole core must be of a 100 %
natural lithological compound, and the analysis of particle size
distribution must be conducted on the core samples. If the exploration
borehole was located in the vicinity of the tested well, the hydraulic
conductivity of the local scale was used. If there were more boreholes at a
greater distance from the pumped well, the hydraulic conductivity of a
sub-regional scale was determined and used for correlation. Values of the
predicted

The criteria for evaluating the acceptable accuracy of the predicted
hydraulic conductivity, expressed by its correlation with a tested

In the verification process, the results acquired using the KC equation were
matched with the results of the hydraulic tests. The average local

The results of the calculation of hydraulic conductivity using the KC
formula (Eq. 14) for individual samples of sand and gravel were presented
graphically, according to borehole depths. The average values of hydraulic
conductivity for individual pilot fields are presented in the tables. In
this process, the arithmetic

The hydraulic conductivities of samples from various depths are presented for four distinctive aquifers.

First, two aquifers are built of uniform, poorly graded mean- to coarse-grained sand (Fig. 6) lying at different depths. Second, two aquifers are built of well-graded fine- to mean-grained sand (Fig. 7), also lying at different depths.

Average difference (%) between predicted and tested hydraulic conductivity for sandy aquifers.

Predicted hydraulic conductivity calculated using the KC equation for
samples from uniform sandy aquifer (

Predicted hydraulic conductivity calculated using the KC equation for samples from sandy aquifers with thin silty intercalations.

Table 1 gives the average difference between the predicted and tested
(pumping test) hydraulic conductivities. In all cases, the overestimated
value of hydraulic conductivity is a result of using the arithmetic mean
grain size in calculations. The underestimated values of hydraulic
conductivity are a result of using the harmonic mean grain size. The results
are very close to the tested value of hydraulic conductivity because the
geometric mean grain size was used in the KC formula. The applicability of
grain sizes according to the specific sieve size was also analyzed for
median grain size value

The analyses of samples from fine sandy aquifers with silty laminas (Figs. 7–8) resulted in regularly underestimated

Fine sand sample with thin silty intercalations – test field FS/SU1 (Ravnik).

In such specific cases, grain size

The predicted

Gravel core from 23 to 30 m depth from borehole SPB-3 – test field GW (Đurđevac) (see Fig. 10a).

A high-quality drilling core (Fig. 9) from six exploration boreholes and a
particle size distribution data analysis of relevant core samples was used.
All of the boreholes were scattered around the pumped well at test field GW.
Borehole SPB-2 is situated on the border of the well field where a part of
an aquifer of sandy development is located, and hence, the data do not
correspond to a correlated average

Average predicted hydraulic conductivity

Predicted hydraulic conductivity calculated using the KC equation for
samples from the gravelly aquifer (test field GW) –

The highest deviation of the predicted

Numerical results of correlations between tested

Graphical correlation between predicted

The correlation of hydraulic conductivity mean value results for referential
grain sizes

From a practical point of view, an interesting fact is that very good
results are achieved using grain size

The validities of the aquitard's predicted

Graphical correlation between predicted

The graphical correlation (Fig. 12b) illustrates concentrated

The Kozeny–Carman equation was limited to only calculating the hydraulic conductivity of incohesive materials (Kasenow, 1997, 2010). Additionally, the use of the KC equation for calculating the hydraulic conductivities of cohesive materials using particle size has been frequently disputed in numerous papers and reports. The reasons include varied particle size, high proportions of fine fractions in deposits (Young and Mulligan, 2004), electrochemical reaction between the soil particles and water and large content of particles such as mica (Carrier, 2003). All of these factors also affect the effective porosity, and some of them also affect the mean grain size. Is the effect of the aforementioned factors incorporated (and if so, how much) in the size and distribution of effective porosities and referential mean grain sizes?

Relation between of effects of mean grain size

The conducted analyses, as graphically summarized in Fig. 13, confirmed that
the use of (1) geometric mean as a referent mean grain size (Eqs. 12 or 13)
and (2) effective porosity according to Fig. 5 in the Kozeny–Carman
equation forms a model of flow through the porous media. This model is valid
for various soil materials and mixtures with a wide range of hydraulic
conductivity values (from 10

Pearson's correlation analysis was conducted for the numerical and
logarithmic values of predicted hydraulic conductivities

Verification of graphical and numerical correlation between the
tested

A separate sub-group was formed by the non-cohesive material data from all
five CRO test fields by using the referent grain size

The graphical correlation between the tested and the predicted hydraulic
conductivities (Fig. 14) illustrates the universality of the KC model
(when applying referential mean grain size

The following conclusions can be drawn from this study:

The geometric mean size of all particles contained in the sample

The distribution of effective porosities in functions of the referential
grain size

The successful application of the KC flow model confirms its validity in a
range of hydraulic conductivities between 10

The value of the referent mean grain size in cases of analyzed non-cohesive
samples is very close to the value of the grain size

The authors would like to thank Željka Brkić,
Željko Miklin and Ivana Žunić Vrbanek for their perseverance
and help in collecting large amounts of laboratory data used in this study.
This study was supported by the Ministry of Science, Education and Sports of
the Republic of Croatia (Basic Hydrogeological Map of the Republic of
Croatia 1