Determination of hydraulic conductivity applying empirical formulae. and physical modeling

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1 Determination of hydraulic conductivity applying empirical formulae and physical modeling Seyed Habib Musavi-Jahromi Associate Prof., School of Water Sciences Engineering, Shahid Chamran University, Iran Phon: Reza Shiravand MSc candidate in hudraulic structures, School of Water Sciences Engineering, Shahid Chamran University, Iran Corresponding author: Abstract To have a realistic approach to hydraulic conductivity is a very important matter in different fields of sciences related to water and soil. In this study, several empirical formulae were applied to determine the hydraulic conductivity of 4 uniform sandy samples in order to what extend they are applicable they are. The results obtained from applying of empirical measurements were compared with the results obtained from laboratory observations conducted by the constant head experiment in the laboratory. Results show that USBR and Slitcher are more accurate than other formulae. In addition, applying a right value for sorting coefficient in Terzaghi equation can increase the accuracy of this equation. Keywords: Hydraulic conductivity, Empirical formulae, Constant head experiment 1. Introduction The amount of hydraulic conductivity usually depends on the soil particles. That means, with increase in particle size, hydraulic conductivity also increases. Although this may not be accurate for non-uniform soils. To create a relationship between hydraulic conductivity and soil particle distribution, can be a useful method to predict the hydraulic conductivity of soils which any experiments have not been conducted to determine this coefficient. This fact is absolutely important to the matters related to water sciences- for example drainage design, construction of dams and coastal dikes and so on. There are different ways to calculate the amount of hydraulic conductivity which can be categorized including; "Field methods such as pumping from wells", "Laboratory methods such as constant head and falling head experiments" and "Application of empirical formulae". Taking lots of money and time are two worst things about field measurements. Accurate estimation of hydraulic conductivity in the field is limited by the lack of precise knowledge of aquifer geometry and hydraulic boundaries (Uma et al. 1989) which are other limitative restrictions. On the other hand, laboratory measurements, presents formidable problems in the sense of obtaining representative samples and, very often, long testing times (Odong 2007). In the third method, researches try to create a relationship between hydraulic conductivity and grain size distribution. This method does not have the two others method s limitative restrictions as well. That means, the expenditure of conducting this method is much less than two other s method. Most importantly, since information about the textural properties of soils or rock is more easily obtained, a potential alternative for estimating hydraulic conductivity of soils is from grain-size distribution. 15 ISSN X

2 2. Material and method Numerical studies which have been done are on the basis of experimental works which have been made. There are several empirical equations suggested by researchers. Most widely accepted ones are Kozeney-carman equation, the equations suggested by Terzaghi(1964) and also hazen (1982). Any of thes e equations, has some limitative criteria which differ from other s equations criteria. General form of all these equations, can be ment ioned as bellow (Vukovic and Soro 1992) K = g ν. C. f n. d e (1) Where K is the coefficient of hydraulic conductivity, g is acceleration due to gravity; ν is kinematic viscosity, C is sorting coefficient, f n is a porosity function and d e is effective grain diameter of the soil particles. The values of C and f(n) depend on the equation wished to use.. Some of the most widely accepted empirical formulae have been presented bellow. Different equations with particular limitative criteria will be explained Hazen: K = g 6 ν [ n 0.26 ] (2) Hazen formula was originally developed for determination of hydraulic conductivity of uniformly graded sand but is also useful for fine sand to gravel range, provided the sediment has a uniformity coefficient less than 5 and effective grain size between 0.1 and 3mm. uniformity coefficient can be expressed as bellow U = d 60 (3) Where d 60 and d 610 presents the grain diameter in (mm) which for 60% an% of the sample respectively, are finer than. Kozeney-carman: K = g ν [ n 3 (1 n) 2] 2 (4) The equation is one of the most widely accepted and used derivations of permeability as a function of the characteristics of the soil medium (Odong, 2007). The equation however is not appropriate for soil with effective size above 3 mm or clayey soils (Carrier, 2003). Brayer: K = g 6 ν 10 4 log (5) U This formula is often considered most useful for materials with heterogeneous distributions and poorly sorted grains with uniformity coefficient between 1 and 20, and effective grain size between 0.06mm and 0.6mm. Slitcher: K = g 1 ν 10 2 n (6) This formula is most applicable for grain-size between 0.01mm and 5mm. Terzaghi: K = g C ν t ( n ) 2 2 (7) 1 n Where the C t = sorting coefficient and > C t > In this study, an average value of C t is used. Terzaghi s formula is most applicable for large-grain sand (Cheng and Chen 2007). USBR: K = g 4.8 ν 10 4 d d 20 (8) USBR formula calculates hydraulic conductivity from the effective grain size ( d 20 ), and does not depend on porosity. The formula is most suitable for medium-grain sand with uniformity coefficient less than 5 (Cheng and Chen 2007). g The value of from the above equations is taken as for water at 20 C in this study (Schwartz and Zhang, 2003). υ 2.1 conducting the experiments In order to examine the validity of equations mentioned in above with the experimental observations, 4 different sands with various grain size diameters were considered. To calculate the amount of hydraulic conductivity base on constant head experiment, a box was used which had 33 cm width and 35 cm height. The length of the box- and actually the seepage length- was 16 cm. This box was located in the sub-branch of an experimental flume. To get assure that the particles of a particular sample does not displace anymore, soil was compacted in submerge condition and then left for about 48 hours under a constant head. Constant head was created by a header in the main branch of the experimental flume. A flow, which was in balance with the seepage, entered into the flume intended to keep the head constant by a half-inch hose. Figure 1 shows a typical flume in which the experiments were done. The water seeped through the box, was gathered by a measurement cylinder. Simultaneously, time of the gathering was recorded to calculate the discharge. To calculate the values of K, Dupiut assumptions were applied. According to his equation, K can be calculated as bellow 16 ISSN X

3 K = 2QL w. (9) Where K is hydraulic conductivity, Q is the discharge of seepage, L is the length of the seepage(which equals to 16cm), 1 and 2 are the hydraulic potential in front of and behind the sample respectively, w is the width of the box (which is 33 cm) and h is the elevation of water- or the constant head which equals to 1. In order not to collapse the sample in the box, a kind of lacing with large breaches was used. At the same time, a sort of cotton was used. These were applied due to keep the seepage length constant and avoid the sample from sliding. It is necessary to mention that applying of lacing and cotton did not affect the hydraulic conductivity of samples. During the experiments, h 1 was equal to the head performed by the header which its height was 21 cm. h 2 was zero, because of the quick drainage behind the sample. Except Brayer equation and USBR equation, other empirical equations, need a coefficient such asf(n), which is an index of porosity. To find this value, ring experiment was conducted for all the samples and G s was calculated in the laboratory. Figure 3 displays the grain size distribution curves of 4 studied samples. As it can be seen, these samples are uniform. Information about grain size distribution and the characteristics of the sample have been presented in table soil classification From the grain-size distribution curves, soil samples were classified according to particle size us ing a standard British Soil Classification System, detailed in BS 5930: Site Investigation. In this system, soils are classified into named basic soil-type groups according to size, and the groups further divided into coarse, medium and fine sub - groups. The classifications based on the grain-size distribution curves were as follows Sample 1 comprised 63% fine gravel and 37% coarse sand and therefore classified as sandy gravel. Sample 2 comprised 17% fine gravel and 83% coarse sand and therefore classified as gravely sand. Sample 3 comprised 4% coarse sand, 86% medium sand an% fine sand and therefore classified as medium sand. Sample 4 comprised 66% medium sand and 34% fine sand and therefore classified as medium sand. 3. Results and discussion Table 2 shows the values of equations mentioned before in comparison with the experimental values. According to the data from table 2, for sample 1 all the empirical formulae have overestimated hydraulic conductivity except USBR. Among these equations, USBR has the most accuracy and then Slitcher and Terzaghi respectively. Kozeney-Carman and Hazen have the least accuracy. The same trend can be expressed for sample 2 for the most accurate ones. But it can be seen that Kozeney-Carman has the least accuracy for this sample. For samples 1 and 2, Brayer formula is not applicable. For sample 3, in contrast with other equations, Slitcher and USBR have underestimated. Slitcher has approached closely to the experimental value. Terzaghi, Brayer, Hazen and Ko zeney- Carman are after it. USBR has the least precision for samples 3. Except USBR, which has underestimated for sample 4, other equations have overestimated. Slitcher is the most precision and after it, USBR, Terzaghi, Brayer, Hazen and Kozeney-Carman give accurate answers respectively. Differences are given in percentage in table 3. In contrast with USBR for samples 1 and 2, and Slitcher for sample 3, which underestimate, other equations overestimate the value of hydraulic conductivity. As it can be seen, the accuracy of empirical equations for higher diameters is lower than the accuracy for smaller diameters. By selecting a correct amount for C t in Terzaghi s equation, the estimation becomes more accurate. 4. Conclusion According to what aforementioned, following conclusions can be drawn: If correct method does not be applied, then the hydraulic conductivity may be calculated underestimated or overestimated. The accuracy of empirical equations for large diameters is lower than small diameters. All of the equations overestimate for samples 1 and 2 except USBR equation which is much closer approach in comparison with other equations. Selecting a right value for C t in Terzaghi equation, helps to estimate closely to the real value of hydraulic conductivity. 17 ISSN X

4 Therefore, the most suitable formulae for the estimation of hydraulic conductivities in this study were as follows Sample 1: USBR and Slitcher Sample 2: USBR Sample 3: Terzaghi Sample 4: Slitcher Acknowledgement The authors would like to acknowledge Shahid Chamran University of Ahwaz and the Centre of Excellence on Operation Management of Irrigation and Drainage Networks for their support. Also this acknowledgement is extended to Mis M.Farahani for her coopration in this research program. References Boadu, F. K Hydraulic Conductivity of Soils from Grain-Size Distribution: New Models. Journal of Geotechnical and Geoenvironmental Engineering, DOI: /j tb00587.x. Carman, P. C Fluid Flow through Granular Beds. Trans.Inst.Chem. Eng., 15,150. Carman, P.C Flow of Gases through Porous Media. Butterworths Scientific Publications, London. Carrier, W.D Goodbye, Hazen; Hello, Kozeny-Carman. Journal of Geotechnical and Geoenvironmental Engineering DOI: /~ASCE! ~2003!129:11~1054! Cheng, C., and Chen, X Evaluation of Methods for Determination of Hydraulic Properties in an Aquifer- Aquitard System Hydrologically Connected to River. Hydrogeology Journal. 15: DOI: /s z Cirpka, O. A Environmental Fluid Mechanics I: Flow in Natural Hydrosystems. Freeze, R. A., and Cherry, J. A Groundwater. Prentice Hall Inc., Englewood Cliffs, New Jersey. Odong J (2007). Evaluation of Emp irical Formulae fo r Determination of Hydraulic Conductivity based on Grain- Size Analysis. J. Am. Sci., 3(3): Popoff M, W iedmann J, De Klasz I (1986). The Upper Cretaceous and Gongila and Pindiga Formations, northern Nigeria: Subdivisions, age, stratigraphic correlations and paleogeographic Implications. Eclogae. Geol. Helv., 79: Popoff M (1988). Gondwana to the South Atlantic: The connections of the Benue Trough basins with the North East brësilion ĺ opening up the Gulf of Guinea Cretaceous interior. J. Afr. Earth Sci., 7: Todd, D. K., and Mays, L.W Groundwater Hydrology. John Wiley & Sons, New York. Vukovic, M., and Soro, A Determination of Hydraulic Conductivity of Porous Media from Grain -Size Composition. Water Resources Publications, Littleton, Colorado Mansur, C. I., and Kaufman, R. I. ~1962!. Dewatering. Foundationengineering, G. A. Leonards, ed., McGraw- Hill, New York, Uma KO, Egboka BCE, Onuoha KM (1989). New statistical grain-size method for evaluating the hydraulic conductivity of sandy aquifers. J. Hydrol., 108: Doi: /j Yenika ME, Uma KO, Obiefuna GI (2003). A case study of shallow aquifer in Jimeta-Yola Metropolis, Northeastern Nigeria. Water Resources: J. Nig. Assoc. Hydrogeol. (NAH), 14: Terzaghi, K., and Peck, R. B. ~1964!. Soil mechanics in engineering practice, Wiley, New York. Terzaghi, K., Peck, R. B., and Mesri, G. ~1996!. Soil mechanics in engineering practice, Wiley, New York. Hillel, D. ~1980. Fundamentals of soil physics, Academic Press, New York. 18 ISSN X

5 Table 1. Characteristics of samples characteristic d 20 d 30 d 50 d 60 2 d 30 Sample Sample Sample Sample C c = d U = d n Table 1 illustrates the characteristics of considered samples in the present article. The samples in the table are in descending order corresponding to diameter. Table 1. values of hydraulic conductivity applying empirical formulae Sample d10 d20 n Hazen Kozeny-Carman Brayer Slitcher Terzaghi USBR m/day Experimental values N.A N.A Table 2 displays the values of hydraulic conductivity applying empirical equations in comparison to experimental ones. Table 2. difference of empirical with experimental values of K in percentage Sample Experimental values Hazen Difference in percentage Kazeney-Carman Difference in percentage Brayer N.A N.A Difference in percent N.A N.A Slitcher Difference in percentage Terzaghi Difference in percentage USBR Difference in percentage N.A= Not Applicable, Minus (-) = underestimate 19 ISSN X

6 Table 3 presents the difference of applying empirical formulae with experimental observations in percentage; that means difference in percentage has been calculated as bellow Difference in Percentage = EM.Value EX.Value EX.Value 100 Where, EM is a particular empirical value and EX is experimental observation. Where water enters A Main branch of flume Wing walls A Figure 1. Schematic plan of the experimental flume Figure 1 shows a typical flume in which the experiments were conducted. When the water enters into the flume by a half inch hose, the header does not allow water to pass it. So the elevation of water in the main branch increases up to 21 cm what the height of the header is. Two stuff next to the drainer were used as wing walls in order to gather all the seeped water through the sample. Figure 2. Longitudinal profile of sub branch along A-A in the Fiigure 1 Figure 2 displays a schematic longitudinal profile along A-A in the Figure 1. As it can be seen, 2 behind the sample can not be formed due to quick drainage. 20 ISSN X

7 WD% Archives Des Sciences Vol 65, No. 5;May sample 1 sample 2 sample3 sample Diameter(mm) Figure 3. Grain size distribution curves of samples Figure 2 displays the grain size distribution curves of the 4 studied samples. It is obvious that all the samples were uniform. Further informat ion about these samples has been presented in table ISSN X

Study on Estimation of Hydraulic Conductivity of Porous Media Using Drag Force Model Jashandeep Kaur, M. A. Alam

26 IJSRSET Volume 2 Issue 3 Print ISSN : 2395-99 Online ISSN : 2394-499 Themed Section: Engineering and Technology Study on Estimation of Hydraulic Conductivity of Porous Media Using Drag Force Model Jashandeep