ASSESMENT OF SELECTED EMPIRICAL FORMULAS FOR COMPUTATION OF SATURATED HYDRAULIC CONDUCTIVITY

This paper deals with the comparison of empirical formulas used for computation of saturated hydraulic conductivity values Kd. The disturbed samples of bed silts were obtained from the Komárňanský channel at the Žitný ostrov (ŽO), Slovakia. The bed silts were extracted from three different vertical parts of silt top, middle and bottom part of silt layer in each selected cross-section profile of the Komárňanský channel. Because the samples are disturbed only the empirical formulas based on the grain size analyses were used. The measurements of silting and the extraction of bed silt samples were carried out in 2019. These measurements were used for calculation of saturated hydraulic conductivity values Kd. In the previous study we calculated the values of saturated hydraulic conductivity for disturbed samples Kd according to Bayer – Schweiger; Špaček I and Špaček II empirical formulas. In this current paper we used other empirical formulas based on the grain size analyses. We selected Hazen I.; Bayer; USBR and Orechova formulas which were in the past used in the software Geofil. These valid values Kd reached from 2.00 x10 to 9.07 x 10 m s. We used the number of valid computed results (count) of Kd to determine the formula ́s ability to give results meeting the validity requirements. The recommended formula for calculation of Kd of bed silts in Komárňanský channel based on this criterium is Hazen I., which range is 1.16 x 10 to 7.25 x 10 m s.


Introduction
The bed silt permeability impacts water flow between surface water in the channel and surrounding groundwater in the scope of their interaction at this area. The permeability of bed silts is expressed by value of their saturated hydraulic conductivity. Therefore, it is important to obtain the values of saturated hydraulic conductivity. Engineering practice often requires the investigation of ground water movement, volumes in storage and computation of the amount of infiltrated water into or from the aquifer. Hydraulic engineers, hydrologists and hydrogeologists have been studying this topic for decades with a variety of conclusions. Thus, we focus on using a simple and useful method for quantifying hydraulic conductivity. Our approach includes the assessment of results obtained from selected empirical formulas. A number of empirical formulas for saturated hydraulic conductivity determination are being used in engineering practice. Most of the ground water textbooks reference formulas of institutions and scholars such as Hazen, Beyer, Sauerbrei, Kozeny, USBR, Pavchich, Schlichter, Terzaghi, Kruger, Zunker, Zamarin, Boonstra and de Ridder, Špaček, Palagin, Schweiger, Carman-Kozeny, Seelheim, Orechová, Zieschang and others (Dulovičová and Velisková, 2005, user manual of commercial software GeoFil, Říha et al., 2018). Most of the empirical formulas are based on laboratory or field experiments. The structure of these formulas ranges from a simple function of grain size d10, d15, d17, d20, d50 or d60 to the most complex exponential equations with a number of other input data and parameters, which need to be computed through additional equations. However, many textbooks do not describe the exact conditions under which a formula was derived, nor the range of its application. Unfortunately, the values of saturated hydraulic conductivity documented in the literature do not always include the sizes of databases. Computed values exhibit a wide range of results which differ by factors of ten, hundred, thousands or more. Decisions about which formula can justify a result are often subjective. Thus, results might not always be in agreement with the values computed from formulas or values reported in the literature. In the current literature research papers usually focus on a wide variety of saturated hydraulic conductivity related topics. Habtamu et al. (2019) evaluate saturated hydraulic conductivity with different land uses of disturbed and undisturbed soil, they developed an equation which replaces the time taking in-situ saturated hydraulic conductivity measurement. Duong et al. (2019) clarify the effects of soil hydraulic conductivity and rainfall intensity on riverbank stability using a GeoSlope analysis. Říha et al. (2018) present the verification of validity of various published porosity functions and empirical formulae with the use of the experimental data obtained from the glass beads. Wang et al. (2018) present an alternative model to predict soil hydraulic conductivities, in their study model testing with 24 soil data sets was successful in predicting conductivities over arrange of moistures. Ren and Santamarina, (2018) present an analysis of hydraulic conductivity of sediments as a function of void ratio. Hwang et al. (2017) compare saturated hydraulic conductivities of sandy soils to characterize properties of water retention. Ghanbarian et al. (2017) propose scale dependent pedotransfer functions to estimate saturated hydraulic conductivity more accurately than seven other frequently used models. (Gadi et al., 2017) studied spatial and temporal variation of hydraulic conductivity and vegetation growth in green infrastructures, using infiltrometer and a visual technique. Yusuf et al. (2016) studied hydraulic conductivity of compacted laterite, treated with iron ore tailings. Hussain and Nabi (2016) used seven empirical formulas to calculate hydraulic conductivity, based on grain size distribution of unconsolidated aquifer materials. Kutílek (1978) calculated the value of saturated hydraulic conductivity by empirical formulas coming out from grain size analysis. In this current paper we also used a way of several empirical formulas for saturated hydraulic conductivity determination for bed silts on Komárňanský channel.

Location and site description
Žitný ostrov (ŽO) is the area between two branches of the Danube River -Small Danube and Danube - Fig. 1 and it is a component of the Danube Lowland. This part was created by sediments transport from upper part of the Danube River (Čelková, 2014). This area formed as a flat plain with only small differences in altitude. Its average slope (about 0.25‰) was one of the reasons for building channel network here (Kováčová, 2017). The longitudinal slopes of single channels of channel network are also very low. This fact had impact to production of bed silts on the channel bottom. The thickness and structure of bed silts influence mutual interaction between groundwater and water level in channel network (Baroková and Šoltész, 2014). As important characteristics influencing this interaction was determined the permeability of silts, expressed by saturated hydraulic conductivity value of silts. Komárňanský channel which is the subject of this study, is shown in Fig. 2. Komárňanský channel is a drainage subarea of Váh River drainage area. At Fig. 3 is shown the view at Komárňanský channel.

Material and methods
Komárňanský channel is the largest one of three main channels of ŽO channel network. This channel was built in the late 19th century for drainage primary, now is used also for irrigation function. Komárňanský channel is supplemented from the Váh river over pumping station Komárno -Nová Osada and it connects with Chotárny channel through a manipulating objects northwest of the Okoč village. The last measured length of Komárňanský channel was about 28 km. The channel width was in range 10-29 m, the measurements of channel depth registered maximal values up to 2.7 m (according to located cross-section profiles). The values of saturated hydraulic conductivity in aquifers nearby this channel Kfs were 0.40-3.4 x 10 -3 m s -1 (Mišigová, 1988). The last measurements of silting of whole Komárňanský  Sediment sampling was conducted using the 04.23 Sediment Core Sampler, a rod operated type Beeker. This instrument collects the samples of sediments in 1 m long acrylic tube as shown in Fig. 6a, b. The silt sample was taken from each selected cross-section profile, then from each whole sample a part from top, middle and bottom layer was extracted and thus 24 samples of silts were obtained. Next, the granularity analyses for each disturbed sample were performed, which was a base for saturated hydraulic conductivity computation.

Determination of saturated hydraulic conductivity from granularity analysis
As it was mentioned above several empirical formulas for determination of hydraulic conductivity from granularity exist, but it is possible to apply only a few of them because their limited validity, which will be discussed below.
Therebefore we used for calculation of saturated hydraulic conductivity of bed silts Kd at Komárňanský channel the relationships by Beyer-Schweiger and Špaček (Špaček, 1987). In these relationships the value of saturated hydraulic conductivity Kd is function of d10 particle diameter in 10% of soil mass and d60particle diameter in 60% of soil mass. Both these parameters were determined from granularity curves of all extracted    (Dulovičová, et al., 2020). We were interested to try using also other empirical formulas based on the grain size analyses. Our selection was following: 1. -the formula according to Hazen I.; 2.
formula according to Orechova (all these formulas were in the past used in commercial software GeoFil (User's manual of software set GeoFil) and also were published in the past (e.g. Dulovičová and Velísková, 2005 The valid values of saturated hydraulic conductivity from disturbed samples of silts along the Komárňanský channel Kd according to these 4 formulas were calculated and summarized in Table 2.

Results and discussion
Sometime in the field it is not possible to extract undisturbed samples of bed silts or sediments. However, it is necessary to find out the rate of their permeability. One way is the determination of bed silt permeability using granularity analyses. As mentioned above, according to publication (Dulovičová et al., 2020), the computation of saturated hydraulic conductivity uses the empirical formulas by Beyer-Schweiger, Špaček I. and Špaček II. As an input data were used the measurements from year 2019. These formulas are based on particle diameter d10 and d60. These two variables control the validity of application (Šurda et al., 2013 (Dulovičová et al., 2020).
In this study we decided to use other formulas for calculation of saturated hydraulic conductivity Kd which are according to Hazen I., Bayer, USBR and Orechova. These formulas contain the variables d10, d17, d20 and d60.
Conditions of validity of these formulas are also function of d10, d15, d17, d20 and d60. The calculated values of saturated hydraulic conductivity Kd according to Hazen I., Bayer, USBR and Orechova formulas are summarised in Table 2. This computation produced 96 calculated values Kd, but only 57 of them were the valid values due to the condition of the validity. These values Kd reached from 2.00 x 10 -10 to 9.07 x 10 -06 m s -1 .
In the case of application Bayer-Schweiger formula we obtained only 4 valid Kd values: in rkm 9.0 and 25.0 were these values from top layer, in rkm 28.0 from middle layer and in rkm 20.0 from bottom layer of silt, they changed from 10 -5 to 10 -8 . Using by Špaček I. formula we obtained 20 valid Kd values: in rkm 2.0, 9.0, 12.0, 23.0, 25.0 and 28.0 from all three layers (top, middle and bottom), they varied from 10 -6 to 10 -7 m s -1 at which 10 -7 predominated. In rkm 7.0 and 20.0 the valid values were obtained only from top layer of silt and their range was also from 10 -6 to 10 -7 m s -1 . Using by Špaček II. formula we obtained only 5 valid Kd values: in rkm 7.0 from all three layers, in rkm 20.0 from middle and bottom layer, these values varied from 10 -4 to 10 -5 m s -1 , with dominance 10 -5 . We used the count of valid results computed from each individual formula as a criterion for recommending the formula.  For illustration the Fig. 7a   The illustration of statistical evaluation of data range.

Conclusion
In this paper 7 empirical formulas were used for calculation of saturated hydraulic conductivity of bed silts along Komárňanský channel from 24 extracted silt samples. In the previous study (from year 2020) we used 3 formulasaccording to Bayer -Schweiger, Špaček I. and Špaček II. The current study introduces 4 formulasaccording to Hazen I., Bayer, USBR and Orechova. All 7 formulas are based on the granularity analyses, with the inputs d10, d15, d17, d20 and d60.
The resultant values are presented in Table 1 the values calculated by Beyer-Schweiger and Špaček I., II. formulas and in Table 2 the values calculated by Hazen I., Bayer, USBR and Orechova formula. The valid values of saturated hydraulic conductivity of bed silts according to Hazen I., Bayer, USBR and Orechova reached from 2.00 x 10 -10 to 9.07 x 10 -06 m s -1 . We used the number of valid computed results (count) of Kd to determine the formula´s ability to give results meeting the validity requirements. The recommended formula for calculation of saturated hydraulic conductivity of bed silts in Komárňanský channel for current study is Hazen I., due to the 23 computed valid results out of total number of results 24. The range of valid values is 1.16 x 10 -8 to 7.25 x 10 -06 m s -1 .
In the next level of our research is needful to compare these results with the values obtained from undisturbed samples of bed silts determined by the laboratory falling head method. All obtained information about values of bed silts saturated hydraulic conductivity can be used for numerical simulation models and simultaneously they supplement insufficient information for future groundwater level regulation in surroundings of the Komárňanský channel or other channels at the ŽO area.