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Article Understanding the Changes in Values of Coarse- and Fine-Grained Porous Media Mixtures

Zuhier Alakayleh 1, T. Prabhakar Clement 2,* and Xing Fang 1,3 ID

1 Department of Civil , Auburn University, Auburn, AL 36849-5337, USA; [email protected] (Z.A.); [email protected] (X.F.) 2 Department of Civil, Construction and Environmental Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA 3 Key Laboratory of Urban Storm Water System and Water Environment, Ministry of Education, Beijing University of Civil Engineering and Architecture, Beijing 100044, China * Correspondence: [email protected]; Tel.: +1-205-348-9033; Fax: +1-205-348-0783

Received: 26 January 2018; Accepted: 7 March 2018; Published: 13 March 2018

Abstract: Low permeability - mixtures are often used to construct hydraulic barriers to prevent contaminated water leaching from landfills and other waste disposal sites from polluting local . In order to engineer effective hydraulic barriers, a proper knowledge of the hydraulic conductivity of clay-sand mixtures is required. While there are several empirical models available in the literature that can be used to predict reductions in hydraulic conductivity values of coarse sand due to the presence of clay and other fine minerals, all these models require measurement of multiple physical properties of the porous media. The resulting empirical expressions have several parameters that need to be individually evaluated using multiple characterization tests. In this study, we propose a single parameter model that can be used to capture the variations in hydraulic conductivity value of different types of porous media mixtures using a scalable modeling framework. Several laboratory tests were conducted to measure the hydraulic conductivity values of a variety of coarse and fine glass bead mixtures. The coarse glass beads were used to simulate sand and small glass beads were used to simulate fine minerals such as and clay. The model results were further validated using the data derived from conducted with natural sand and clay mixtures, and also using multiple literature-derived datasets. Our results show that the proposed model is a useful tool for describing the hydraulic conductivity values of various types of coarse- and fine-grained porous media mixtures.

Keywords: groundwater; hydraulic barrier; hydraulic conductivity; clay-sand mixtures; modeling

1. Introduction Contamination of groundwater systems by landfill leachates is one of the common environmental problems. When an formation is highly permeable with no natural impervious soil layer, engineers use low permeability clay-sand mixture to construct hydraulic barriers that can prevent contaminated wastewater leaching from landfills and other waste disposal areas from polluting local groundwater aquifers [1–5]. The amount of clay required to achieve the desired level of low permeability mixture is evaluated by performing laboratory-scale conductivity tests, and these efforts can be time consuming and cost prohibitive [6–8]. Past studies have shown that the hydraulic conductivity of a clay-sand mixture will decrease with increase in clay percentage [3,8–12]. However, adding excess amount of clay can lead to swelling and shrinkage that can eventually result in cracking and increase the risk of leakage through preferential

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flow paths [2,13]. Also, at high clay levels, the mixture becomes more plastic and extremely difficult to compact [3]. Furthermore, the overall cost of the mixture will increase with increasing in clay content [2,4,8,14]. The most important parameter that controls the groundwater seepage processes through subsurface aquifers is the hydraulic conductivity of the system [9,15]. Without having a proper knowledge of the hydraulic conductivity value, it is impossible to design effective engineering barriers. Therefore, many investigators have conducted studies to develop mechanistic models that can predict the hydraulic conductivity value of clay-sand mixtures. Almost all currently available models are based on empirical formulations that use various physical properties of the materials used to develop the mixture and relate them to the effective hydraulic conductivity value of the mixture. Chapuis [8] introduced an empirical equation to predict the hydraulic conductivity values of soil-bentonite mixtures. Several physical parameters were used to develop the model including , bentonite content, the degree of saturation at the end of the test, grain-size distribution, and the level estimated from the Proctor curve. Permeameter tests were performed to evaluate the conductivity values. Results of the study indicated that there is an inverse relation between hydraulic conductivity and amount of bentonite used in the mixture. No obvious correlation could be observed between the hydraulic conductivity and porosity values of the mixture. However, hydraulic conductivity correlated better with “efficient” porosity, which is related to the pore space available for fast-moving water. Note that the efficient porosity value is different from , since it does not include the portion of immobile water that is retained at the surface of fine particles. A five-variable regression model was developed by Benson et al. [16] to estimate the hydraulic conductivity of compacted soil liners. Results from the regression analysis indicated the five variables that were significantly correlated with the hydraulic conductivity value (analyzed using the natural logarithmic scale); the variables included: compactor weight, plasticity index, percent , initial saturation, and percent clay. The coefficient of determination (R2) of their regression model was 78%. The authors stated that the model can be used to understand the conditions needed to achieve required hydraulic conductivity values, however it should not be used to avoid performing hydraulic conductivity tests in the field or laboratory. Another empirical model for predicting the hydraulic conductivity values was developed by Benson and Trast [17]. This model can determine the hydraulic conductivity of clay-sand mixture based on clay content, plasticity index, initial saturation, and compactive effort. The results from the falling-head hydraulic conductivity test indicated an inverse relationship between hydraulic conductivity and plasticity index, initial saturation, and compactive effort. Mollins et al. [2] used the compaction permeameter falling head permeability test and an indirect test based on consolidation data to measure the hydraulic conductivity values of low and high clay content mixtures. Results showed that hydraulic conductivity values of the clay-sand mixture linearly correlated with the clay when plotted on a logarithmic scale. This study proposed a model that can predict the hydraulic conductivity of the mixture based on the clay content, its properties, sand porosity and tortuosity. Sivapullaiah et al. [9] developed an equation aimed at predicting the hydraulic conductivity of bentonite-sand mixtures based on the void ratio and liquid limit of the mixture. The consolidation cell permeameter test was used to measure the hydraulic conductivity value. A linear relationship was established between the hydraulic conductivity of the mixture (used in the logarithmic scale) and the void ratio. Other studies have also explored the similar type of empirical relationship between hydraulic conductivity to the net void ratio of a mixture [11,18]. In contrast to these studies, Kenney et al. [5] and Castelbaum et al. [19] presented a relation between hydraulic conductivity and void ratio of the bentonite rather than the net void ratio for mixtures present with a sufficient bentonite content to be uniformly distributed to fill all the void spaces between sand particles. Abichou et al. [6] conducted a study aimed at understanding the changes in microstructure and the hydraulic conductivity value of sand-bentonite mixtures at varying bentonite content. Simulated Water 2018, 10, 313 3 of 13 sand-bentonite mixtures were prepared using glass beads, to simulate sand particles, which were then mixed with powdered and granular bentonite. The use of glass beads helped to improve the visual properties of the mixtures. Results showed that pores available for water flow decreased as the bentonite content increased in the mixture, and this resulted in the reduction of hydraulic conductivity. In the case of mixtures prepared with powdered bentonite, the bentonite coated the glass bead particles, swelled, and later filled the pores. Little glass bead particles were coated with bentonite if mixtures were prepared using granular bentonite. In this case, granular bentonite particles occupied the pores between the glass bead particles and then absorbed the introduced water and swelled. In both cases, when sufficient bentonite was available to fill all the pores between glass bead particles, the hydraulic conductivity of the mixture was primarily controlled by the hydraulic conductivity of bentonite. Dias et al. [20] investigated the effect of volume fraction and ratio for binary mixtures of glass beads on the tortuosity coefficient. This was done because of the sensitivity of the tortuosity coefficient in estimating permeability values using the Kozeny–Carman equation that relates permeability with porosity, tortuosity, and . Previous studies correlate tortuosity with porosity using a simplified formula, T = 1/εn, where T is tortuosity, ε is porosity, n is a power factor, which was proposed to be 0.04 by Mota et al. [21]. This simplified formula and the Kozeny–Carman equation were combined together and then a new formula for n factor was developed. Permeability tests were conducted and the data was used to measure the experimental values of n using the new formula. The measured n values were found to range between 0.4 and 0.5 and a model was used to correlate the changes of n with the changes in volume fraction and particle size ratio. The developed model helped to improve the accuracy of permeability values calculated using the Kozeny–Carman equation. Our review indicates that while there are several types of models available for predicting the reduction in the hydraulic conductivity of a due to the presence of clay and other fine minerals, all these models are based on measuring multiple physical properties of the porous media used to develop the mixture. The resulting empirical expressions have several parameters that need to be individually evaluated by multiple soil characterization tests. It will be desirable if one can develop a model based on an effective scaling parameter that can capture the combined effects of multiple soil parameters. Therefore, the objective of this study is to develop an integrated scaling parameter that can be used to fully capture the variations in different types of physical properties into a unified framework. In this study, we have hypothesized that the changes in hydraulic conductivity values of fine and coarse grained mixtures can be corrected to the amount of fine material in the mixture. We collected several sets of laboratory data and also assembled multiple sets of literature-derived data to develop a scalable framework for modeling the changes in hydraulic conductivity values due to the presence of fine material.

2. Materials and Methods

2.1. Synthetic Coarse-Fine and Fine-Coarse Mixtures Three sets of experiments with two using synthetic media and other using natural media were completed in this study. Materials used in the synthetic media experiments were different types of uniform coarse and fine glass beads. The coarse glass beads were used to simulate sand minerals and small glass beads were used to simulate fine minerals such as silt and clay. Our approach is similar to studies by Abichou et al. [6] and Dias et al. [20] that used glass beads to simulate clay-sand mixtures. Figure1 shows the two sets of synthetic media experiments completed in this study. First, a coarse porous medium was mixed with three different sized fine porous media to develop three different dry mixtures. The hydraulic conductivity (Kc) of the coarse porous medium is 920 m/d, and the hydraulic conductivity values (Kf) of the three fine porous media are: 228, 57, and 9 m/d. In the second set of experiments, a fine porous medium was mixed with three different coarse porous media. The hydraulic conductivity of the fine porous medium is 9 m/d, and the hydraulic conductivity values of the three coarse porous media are: 920, 228, and 57 m/d. Water 2018, 10, 313 4 of 13

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Figure 1. Schematic diagram illustrating the two sets of coarse-fine and fine-course porous media Figure 1. Schematic diagram illustrating the two sets of coarse-fine and fine-course porous media mixtures used in our laboratory experiments. K is hydraulic conductivity in m/day, the subscripts c mixtures used in our laboratory experiments. K is hydraulic conductivity in m/day, the subscripts c and f stand for the coarse and fine particles. Each type of mixture was prepared with seven different and f stand for the coarse and fine particles. Each type of mixture was prepared with seven different fine percent levels: 0%, 5%, 10%, 15%, 20%, 25%, and 30%. fine percent levels: 0%, 5%, 10%, 15%, 20%, 25%, and 30%.

Before mixing, the porous media were washed in tap water to remove any dust and then dried in anBefore oven. mixing, Samples the were porous prepared media wereby mixing washed coarse in tap porous water to media remove with any the dust following and then driedamount in an(percent oven. dry Samples weights) were of prepared fine porous by mixing media coarse(0%, 5%, porous 10%, media 15%, 20%, with 25%, the following and 30%). amount An innovative (percent drymixing weights) procedure of fine used porous by Dias media et al. (0%, [22] 5%, was 10%, employed 15%, 20%, to thoroughly 25%, and 30%).mix the An coarse innovative and fine mixing glass procedurebeads. In this used method, by Dias glycerol et al. [ 22is ]used was as employed a binder toto thoroughlyfully mix the mix glass the beads coarse of anddifferent fine glass sizes. beads. After Inpacking this method, the mixture, glycerol the is usedglycerol as a was binder washed to fully ou mixt by the flushing glass beads water of through different the sizes. column. After packing Use of theglycerol mixture, allowed the glycerolus to pack was a uniform washed mixture out by flushingwithin the water column. through the column. Use of glycerol allowedHydraulic us to pack conductivity a uniform tests mixture for withinglass bead the column.mixtures were conducted by closely following the methodHydraulic described conductivity in ASTM-D2423-68 tests for glass[23]. This bead method mixtures uses were the conducted constant head by closely permeameter following test the to methoddetermine described the hydraulic in ASTM-D2423-68 conductivity of [23 ma]. Thisterials method with values uses the greater constant than head 1 × 10 permeameter−5 m/s [24]. Two test × −5 totransparent determine plastic the hydraulic columns conductivity of diameter of1.9 materials cm, but different with values lengths greater were than used 1 to10 constructm/s [24the]. Twopermeameter. transparent The plastic length columns of the short of diameter column 1.9 is cm,25 cm but and different the long lengths column were isused 75 cm. to constructA wire mesh the permeameter.was used at the The bottom length of ofthe the short short column column then is 25the cm soil and sample the long was columnadded over is 75 this cm. mesh A wire and mesh was wascompacted used at gradually. the bottom While of the packing, short column the column then the wa kept sample under was water added to maintain over this fully mesh saturated and was compactedconditions. gradually.To avoid segregation, While packing, the column the column was wasgradually kept under lowered water into to the maintain water bucket fully saturated as it was conditions.packed. After To avoidpacking segregation, the short thecolumn, column a long was graduallytransparent lowered column into was the connected water bucket to the as itshort was packed.column Afterusing packinga rubber the coupling. short column, a long transparent column was connected to the short column usingTap a rubber water coupling.was used as the permeant liquid, and prior to its use, the water was allowed to reach the labTap temperature. water was used This as was the done permeant to avoid liquid, gas andexchanges prior to and its use,air trapping the water while was allowedrunning tothe reach test. theAfter lab the temperature. sample was This compacted, was done a tohigh avoid hydraulic gas exchanges gradient and was air applied trapping to whilewash runningout the glycerol the test. Afterbefore the running sample the was hydraulic compacted, conductivity a high hydraulictests. To ensure gradient consistency, was applied a pump to wash was used out theto introduce glycerol beforethe water running flow theinto hydraulic the sample. conductivity The flow tests.rate, Tosample ensure length, consistency, column a cross-section pump was used area, to introduceand head theloss water were flowmeasured, into the and sample. Darcy’s The law flow was rate, applied sample to length, calculate column the effective cross-section hydraulic area, and conductivity head loss werevalue measured, of the mixture. and Darcy’s law was applied to calculate the effective hydraulic conductivity value of the mixture. 2.2. Natural Clay-Sand Mixtures 2.2. Natural Clay-Sand Mixtures Natural clay was sieved on mesh No. 60 and was used in this study. The hydraulic conductivity Natural clay was sieved on mesh No. 60 and was used in this study. The hydraulic conductivity value of this clay is 1.12 × 10−3 m/d. Fine sand with a hydraulic conductivity value of 46.5 m/d was value of this clay is 1.12 × 10−3 m/d. Fine sand with a hydraulic conductivity value of 46.5 m/d was used. Aged deaired water equilibrated to laboratory conditions was used to avoid gas exchanges used. Aged deaired water equilibrated to laboratory conditions was used to avoid gas exchanges during the test. All the mixtures were prepared under saturated conditions. during the test. All the mixtures were prepared under saturated conditions. Samples were prepared by mixing the sand with varying amount (percent weights) of clay (0%, 5%, 6%, 8%, 10%, 11%, 13%, 15%, 20%, 25%, and 30%). These mixtures were prepared based on the

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Water 2018, 10, x FOR PEER REVIEW 5 of 13 Samples were prepared by mixing the sand with varying amount (percent weights) of clay (0%, 5%, 6%,dry 8%,weight 10%, of 11%,the material. 13%, 15%, To 20%,obtain 25%, complete and 30%). mixing, These deaired mixtures water were was prepared added to based the mixture on the and dry weightit was then of the physically material. stirred To obtain to prepare complete a -mixed mixing, deaired saturated water slurry was (glycerol added towas the not mixture added andin this it wascase). then The physically clay-sand stirredslurry was to prepare left in the a well-mixed mixing pan saturated and covered slurry for (glycerola period of was about not added48 h to inlet thisthe case).clay fully The saturate clay-sand with slurry water. was During left in thethis mixing period, pan water and was covered added for and a period the mixture of about was 48 periodically h to let the claystirred, fully whenever saturate withit is needed, water. During to ensure this complete period, water mixing was and added full saturation. and the mixture was periodically stirred,A falling whenever head it permeameter is needed, to ensurewas used; complete the test mixing procedure and closely full saturation. followed the method described in ASTM-D5084-16aA falling head permeameter [24]. Within the was permeameter, used; the test the procedure clay-sand closely mixture followed was packed the method in between described two insand ASTM-D5084-16a layers. The bottom [24]. Withinsand laye ther permeameter, helped prevent the the clay-sand fine material mixture washing was packed away in betweenthrough twothe sandbottom layers. screen. The The bottom top sandsand layerlayer helped helped prevent us to better the fine compact material the washing mixture. away After through the sample the bottom was screen.packed, Thethe topfalling sand head layer permeameter helped us to test better was compactperformed the to mixture. determine After the the hydraulic sample conductivity was packed, thevalue falling of the head sample. permeameter test was performed to determine the hydraulic conductivity value of the sample. 3. Results 3. Results 3.1. Results of Coarse-Fine Porous Media Mixtures 3.1. Results of Coarse-Fine Porous Media Mixtures Test results for the first set of synthetic coarse-fine porous media mixtures indicated that the Test results for the first set of synthetic coarse-fine porous media mixtures indicated that hydraulic conductivity decreased as the fine percentage increased in the mixture (Figure 2). The the hydraulic conductivity decreased as the fine percentage increased in the mixture (Figure2). reduction in hydraulic conductivity was significant when we started to add fines to the coarse The reduction in hydraulic conductivity was significant when we started to add fines to the coarse material; however, the reductions became less significant when the percentage of the fine was greater material; however, the reductions became less significant when the percentage of the fine was greater than about 15%. Interestingly, the overall hydraulic conductivity of the mixture was almost close to than about 15%. Interestingly, the overall hydraulic conductivity of the mixture was almost close to the hydraulic conductivity of the fine particles when the percent of the fine was above 30%. This trend the hydraulic conductivity of the fine particles when the percent of the fine was above 30%. This trend of decrease in hydraulic conductivity with increasing percent fines is similar to the observations made of decrease in hydraulic conductivity with increasing percent fines is similar to the observations made by others for different types of clay and sand mixtures [5,9,10,12,16]. by others for different types of clay and sand mixtures [5,9,10,12,16].

Figure 2.2. Decrease in the effective hydraulic conductivity values (K values plotted in log scale) of Type-1 (Kc = 920 m/day and Kf = 228 m/day), Type-2 (Kc = 920 m/day and Kf = 57 m/day), and Type-3 Type-1 (Kc = 920 m/day and Kf = 228 m/day), Type-2 (Kc = 920 m/day and Kf = 57 m/day), and (Kc = 920 m/day and Kf = 9 m/day) coarse-fine mixtures. The seven mixtures for each type were Type-3 (Kc = 920 m/day and Kf = 9 m/day) coarse-fine mixtures. The seven mixtures for each type wereprepared prepared by varying by varying the fine the porous fine porous media media conten contentt (percentage (percentage by weight) by weight) as: 0%, as: 5%, 0%, 10%, 5%, 15%, 10%, 20%, 15%, 20%,25%, 25%,and 30%. and 30%.

We usedused aa followingfollowing empiricalempirical equation to describe the reductions in the hydraulic conductivity valuesvalues ofof allall ourour coarse-finecoarse-fine porousporous mediamedia mixtures.mixtures. =− −+ log K(psp ) (logKcf logK ) *exp( * ) logK f (1)

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Water 2018, 10, x FOR PEER REVIEWlog K ( p) = (log Kc − log Kf) ∗ exp(−s ∗ p) + log Kf 6 of(1) 13

wherewhere pp isis the the percent percent of of fine fine (e.g., (e.g., 10 10 for for 10 10 percent percent of offine), fine), K(Kp( )p, ),KKc ,c ,and and KKf f are the hydraulic conductivityconductivity values values of of the the mixtur mixture,e, coarse coarse porous porous media, media, and andfine fineporous porous media, media, respectively. respectively. The modelThe model uses a uses single a single fitting fitting constant constant s, whichs, which is an empirical is an empirical parameter parameter employed employed to scale to the scale results the basedresults on based the percentage on the percentage of fines ofin finesthe mixture. in the mixture. The empirical The empirical model was model fitted was to fittedthe dataset to the datasetshown inshown Figure in 2 Figure and the2 and optimal the optimal values of values the scaling of the scalingparameter parameter for Type for 1, Type2, and 1, 3 2, mixtures and 3 mixtures that best that fit allbest the fit experimental all the experimental data are data 0.04, are 0.05, 0.04, and 0.05, 0.03 and (see 0.03 Figure (see 3), Figure respectively.3), respectively. The nonlinear The nonlinear Solver Add-inSolver Add-inavailable available in Excel in Excelwas used was used to evaluate to evaluate these these fitting fitting parameters. parameters. Hydraulic Hydraulic conductivity conductivity valuesvalues obtained obtained by by Equation Equation (1) (1) in in comparison comparison with with experimental experimental values values for for the the first first set set of of the the three three differentdifferent types types of of mixtures mixtures are are plotted plotted in in Figure Figure 33 asas aa functionfunction ofof percentpercent fine.fine. The root mean square errorerror (RMSE) (RMSE) values values of of log log K K calculated calculated between between the the fitted fitted model model and and experiment are are summarized summarized in in 2 TableTable 11.. TheThe coefficientcoefficient ofof determinationdetermination (R(R2)) values values are are 93.4%, 93.4%, 98.6%, 98.6%, and and 95.4% 95.4% for for Type Type 1, 1, 2, 2, and and 3 3 mixtures,mixtures, respectively. respectively.

FigureFigure 3. 3. ComparisonComparison of of fitted fitted model model results results with with experi experimentalmental data data for for all all three three types types of of coarse-fine coarse-fine

mixtures:mixtures: (a ()a )Type-1 Type-1 (K (Kc =c 920= 920 m/day m/day and and Kf = K 228f = m/day), 228 m/day), (b) Type-2 (b) Type-2 (Kc = 920 (Kc m/day= 920 m/dayand Kf = and 57 m/day),Kf = 57 and m/day), (c) Type-3 and ((Kc)cType-3 = 920 m/day (Kc = and 920 m/dayKf = 9 m/day). and K fThe= 9 effective m/day). hydraulic The effective conductivity hydraulic K valuesconductivity are plotted K values on y axis are plottedin log scale. on y Theaxis cross in log symbol scale. is The used cross to identify symbol the is useddata points to identify used thein thedata three-point points used method in the three-pointdiscussed in method Sectiondiscussed 3.5. in Section 3.5.

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Table 1. Summary of the scaling factor (s) values estimated for all the samples tested in this study and Table 1. Summary of the scaling factor (s) values estimated for all the samples tested in this study and literature-derived experimental datasets. literature-derived experimental datasets. Type of Mixture s Estimated Using the Entire Dataset s Evaluated Using the Three-Point Method 4 Type-1Type coarse-fine of Mixture mixture s 0.037Estimated (RMSE Using 1 = 0.041, the Entire R2 = 93.4%) Dataset s Evaluated 0.041Using (RMSE the = 0.045, Three-Point R2 = 92.3%) Method 4 Type-2 coarse-fine mixture 0.046 (RMSE = 0.041, R2 = 98.6%) 0.050 (RMSE = 0.050, R2 = 97.8%) Type-1 coarse-fine mixture 0.037 (RMSE 1 = 0.041, R2 = 93.4%) 0.041 (RMSE = 0.045, R2 = 92.3%) Type-3 coarse-fine mixture 0.031 (RMSE = 0.138, R2 = 95.4%) 0.026 (RMSE = 0.169, R2 = 93.0%) Type-2 coarse-fine mixture 0.046 (RMSE = 0.041, R2 = 98.6%) 0.050 (RMSE = 0.050, R2 = 97.8%) 2 2 Type-4Type-3 fine-coarse coarse-fine mixture mixture 0.0310.031 (RMSE (RMSE = 0.138, R2 = 95.4%) 95.4%) 0.026 0.026 (RMSE (RMSE = = 0.169, 0.169, R 2R= = 93.0%) 93.0%) Type-5Type-4 fine-coarse fine-coarse mixture mixture 0.0400.031 (RMSE (RMSE = 0.043, 0.138, R2 = 99.9%) 95.4%) 0.026 0.041 (RMSE (RMSE = = 0.169, 0.044, R 2R=2 = 93.0%) 98.7%) Type-6Type-5 fine-coarse fine-coarse mixture mixture 0.0540.040 (RMSE (RMSE = 0.046, 0.043, R2 = 95.6%) 99.9%) 0.041 0.056 (RMSE (RMSE = = 0.044, 0.047, R 2R=2 = 98.7%) 95.4%) Type-6Clay-sand fine-coarse mixture mixture 0.0790.054 (RMSE (RMSE = 0.188, 0.046, R2 = 97.3%) 95.6%) 0.056 0.077 (RMSE (RMSE = = 0.047, 0.191, R 2R=2 = 95.4%) 97.2%) Montmorillonite-sandClay-sand mixture mixture 2 0.0740.079 (RMSE (RMSE = 0.051, 0.188, R2 = 99.9%) 97.3%) 0.077 0.074 (RMSE (RMSE = = 0.191, 0.051, R 2R=2 = 97.2%) 99.9%) 2 2 2 Montmorillonite-sandKaolinite-sand mixture mixture 2 0.0340.074 (RMSE (RMSE = 0.041, 0.051, R2 = 99.7%) 99.9%) 0.074 0.034 (RMSE (RMSE = = 0.051, 0.041, R R=2 = 99.9%) 99.7%) 2 2 2 Bentonite-siltKaolinite-sand mixture mixture 3 0.0300.034 (RMSE (RMSE = 0.134, 0.041, R2 = 99.4%) 99.7%) 0.034 0.032 (RMSE (RMSE = = 0.041, 0.168, R R=2 = 99.7%) 99.1%) Bentonite-silt mixture 3 0.030 (RMSE = 0.134, R2 = 99.4%) 0.032 (RMSE = 0.168, R2 = 99.1%) Note: 1 All root mean square errors (RMSEs) and coefficients of determination R2 of log K were Note: 1 All root mean square errors (RMSEs) and coefficients of determination R2 of log K were calculated using the 2 3 entirecalculated dataset using for each the fitted entire model; dataset2 Data for from each Denson fitted et model; al. [25]; 3 DataData fromfrom Doley Denson et al. et [26 al.]; 4[25];The details Data offrom the three-pointDoley et al. method [26]; 4 are The described details of in Sectionthe three-point 3.5. method are described in Section 3.5.

3.2.3.2. ResultsResults ofof Fine-CoarseFine-Coarse PorousPorous MediaMedia MixturesMixtures TheThe secondsecond set set of experimentsof experiments were were completed completed using using three differentthree different types of types fine-coarse of fine-coarse mixtures, wheremixtures, a uniform where finea uniform porous mediumfine porous was medium mixed with wa threes mixed different with typesthree ofdifferent coarse poroustypes of media coarse at differentporous media dry weight at different percentages dry weight levels. percentages Figure4 shows levels. the increaseFigure 4 in shows hydraulic the increase conductivity in hydraulic values asconductivity we increased values the percentageas we increased of coarse the materialpercentage in theof coarse mixture. material Similar in tothe the mixture. coarse-fine Similar mixtures, to the thecoarse-fine amount mixtures, of fines in the the amount system controlledof fines in the the syst effectiveem controlled hydraulic the conductivity effective hydraulic value of conductivity the mixture. Additionvalue of the of coarsemixture. material Addition into of the coarse fine porousmaterial media into the had fine little porous impact media until had about little 85% impact of coarse until materialabout 85% was of added coarse to material the system, was afteradded which to the the syst systemem, after was which highly the influenced system was by the highly conductivity influenced of theby the coarse conductivity material used of the to coarse prepare material the mixture. used to prepare the mixture.

FigureFigure 4. IncreaseIncrease in in the the effective effective hydraulic hydraulic conductivity conductivity values values (K values (K values plotted plotted in log in scale) log scale) of Type- of 4 (Kc = 920 m/day and Kf = 9 m/day), Type-5 (Kc = 228 m/day and Kf = 9 m/day), and Type-6 (Kc = 57 Type-4 (Kc = 920 m/day and Kf = 9 m/day), Type-5 (Kc = 228 m/day and Kf = 9 m/day), and Type-6 m/day and Kf = 9 m/day) fine-coarse mixtures. The seven mixtures for each type were prepared by (Kc = 57 m/day and Kf = 9 m/day) fine-coarse mixtures. The seven mixtures for each type were preparedvarying the by varyingcoarse porous the coarse media porous content media (percent contentage(percentage by weight) byas: weight)0%, 70%, as: 75%, 0%, 80%, 70%, 75%,85%, 80%,90%, 85%,and 95%. 90%, and 95%.

The empirical model used to describe the coarse-fine Type 1, 2, and 3 mixtures was also used to The empirical model used to describe the coarse-fine Type 1, 2, and 3 mixtures was also used to analyze the fine-coarse Type 4, 5, and 6 mixtures. The fitted values of the scaling parameter for Type analyze the fine-coarse Type 4, 5, and 6 mixtures. The fitted values of the scaling parameter for Type 4, 4, 5, and 6 mixtures are 0.03, 0.04, and 0.05, respectively. Figure 5 shows the experimental and 5, and 6 mixtures are 0.03, 0.04, and 0.05, respectively. Figure5 shows the experimental and modeled modeled hydraulic conductivity values for the three types of fine-coarse mixtures characterized in

Water 2018, 10, 313 8 of 13 Water 2018, 10, x FOR PEER REVIEW 8 of 13 hydraulicthis study. conductivity The coefficient values of determination for the three (R types2) values of fine-coarse are 95.4%, mixtures 99.9%, and characterized 95.6% for the in three this study. types Theof mixtures. coefficient of determination (R2) values are 95.4%, 99.9%, and 95.6% for the three types of mixtures.

FigureFigure 5.5. ComparisonComparison ofof fittedfitted modelmodel resultsresults withwith experimentalexperimental datadata forfor allall threethree typestypes ofof fine-coarsefine-coarse

mixtures:mixtures: ((aa)) Type-4 Type-4 (K (Kc c= = 920 920 m/day m/day and and K Kf f= =9 9 m/day), m/day), ((bb)) Type-5Type-5 (K (Kc c= = 228 228 m/day m/day andand K Kf f= = 99m/day), m/day), andand ((cc)) Type-6Type-6 (K (Kc c= = 5757 m/daym/day and KKff = 9 m/day). The The cross cross symbol symbol is used to identify the datadata pointspoints usedused inin thethe three-pointthree-point methodmethod discusseddiscussed inin SectionSection 3.5 3.5..

3.3.3.3. ResultsResults ofof Clay-SandClay-Sand MixturesMixtures TheThe performanceperformance of of the the model model was was tested tested using using an exploratoryan exploratory dataset dataset collected collected using using natural natural clay andclaysand. and sand. Figure Figure6 shows 6 bothshows experimental both experimental data and data model and results model for results a natural for sand,a natural with sand, saturated with hydraulicsaturated conductivityhydraulic conductivity value of 46.5 value m/day, of 46.5 mixed m/day, with mixed a natural with claya natural with saturatedclay with hydraulicsaturated conductivityhydraulic conductivity value of 1.1 value× 10 of− 31.1m/day. × 10−3 m/day. The behavior The behavior of the of mixture the mixture at varying at varying claycontent clay content was similarwas similar with thewith behavior the behavior of the of different the different types oftype synthetics of synthetic mixtures mixtures discussed discussed in the previous in the previous section. Thesection. optimal The fittedoptimal value fitted of value the scaling of the parameter scaling parameters for the s clay-sand for the clay-sand systemwas system estimated was estimated to be 0.08. to Thisbe 0.08. result This gave result a value gave of a 97.3%value of for 97.3% the coefficient for the coefficient of determination of determination R2. R2.

Water 2018, 10, 313 9 of 13 Water 2018, 10, x FOR PEER REVIEW 9 of 13 Water 2018, 10, x FOR PEER REVIEW 9 of 13

FigureFigure 6. 6.Comparison Comparison of of fitted fitted model model resultsresults withwith experimentalexperimental data data for for the the natural natural clay clay and and sand sand mixture. The cross symbol is used to identify the data points used in the three-point method discussed mixture.Figure The6. Comparison cross symbol of isfitted used model to identify results the with data expe pointsrimental used data in the for three-point the natural method clay and discussed sand in Section 3.5. in Sectionmixture. 3.5 The. cross symbol is used to identify the data points used in the three-point method discussed in Section 3.5. 3.4. Comparison of Model Results with Literature-Derived Experimental Datasets 3.4. Comparison of Model Results with Literature-Derived Experimental Datasets 3.4. ComparisonThe validity of ofModel the model Results in with representing Literature-Derived the changes Experimental in hydraulic Datasets conductivity of the clay-sand The validity of the model in representing the changes in hydraulic conductivity of the clay-sand mixturesThe validityat varying of theclay model content in levelsrepresenting was further the ch testedanges usingin hydraulic data from conductivity previous studies.of the clay-sand Denson mixtures at varying clay content levels was further tested using data from previous studies. etmixtures al. [25] atreported varying the clay hydraulic content conductivitylevels was further of clay-sand tested using mixtures data at from varying previous clay contentstudies. forDenson both Densonmontmorilloniteet al. [25] et al. reported [25] reported and the kaolinite hydraulic the hydraulic clay. conductivity The conductivity purpose of cl ofay-sand their of clay-sand study mixtures was mixtures at to varying investigate at varying clay thecontent clayreduction contentfor both in for bothhydraulicmontmorillonite montmorillonite conductivity and and kaolinite of kaolinite a sand clay. by clay. introducing The The purpose purpose di fferentof oftheir their amount study study was ofwas clay. to toinvestigate Th investigatee hydraulic the the conductivityreduction reduction in in hydraulicofhydraulic the clay-sand conductivity conductivity mixture of of a areduced sand sand by by introducingtointroducing practically differentdi zerofferent when amount amount montmorillonite of of clay. clay. Th Thee hydraulic and hydraulic kaolinite conductivity conductivity content of thewereof the clay-sand about clay-sand 3 and mixture mixture16 percent, reduced reduced respectively. to practicallyto practically Figure zero zero7 whenshows when montmorilloniteboth montmorillonite model results and and and kaolinite experimental kaolinite content content data were aboutfromwere 3 Densonabout and 16 3 percent,andet al. 16 [25] percent, respectively. for montmorillonite respectively. Figure Figure7 and shows kaolinite 7 shows both modelmixedboth model with results sand.results and experimental and These experimental results dataindicate data from Densonthatfrom experimentallyDenson et al. [25 et] al. for [25] montmorillonitemeasured for montmorillonite hydraulic and conductivi kaoliniteand kaolinitety mixed values mixed with at with different sand. sand. These clay These resultscontents results indicate are indicate well that experimentallydescribedthat experimentally by our measured model—Equation measured hydraulic hydraulic conductivity(1). conductivi valuesty values at different at different clay contents clay contents are well are described well bydescribed our model—Equation by our model—Equation (1). (1).

Figure 7. Comparison of our fitted model results with experimental data from Denson et al. [25] for clay-sand mixtures: (a) montmorillonite clay, and (b) kaolinite clay. The cross symbol is used to FigureFigure 7. Comparison7. Comparison of of our our fitted fitted model model resultsresults with ex experimentalperimental data data from from Denson Denson et etal. al. [25] [25 for] for identifyclay-sand the mixtures: data points (a) usmontmorilloniteed in the three-point clay, methodand (b) discussedkaolinite clay.in Section The cross3.5. symbol is used to clay-sand mixtures: (a) montmorillonite clay, and (b) kaolinite clay. The cross symbol is used to identify identify the data points used in the three-point method discussed in Section 3.5. the data points used in the three-point method discussed in Section 3.5.

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Also,Also, thethe datasetdataset fromfrom aa studystudy carriedcarried outout byby DoleyDoley etet al.al. [[26]26] waswas usedused toto testtest thethe validityvalidity ofof thethe proposedproposed model. model. FigureFigure8 8shows shows both both experimental experimental data data and and model model results results for for bentonite bentonite mixed mixed with with locallylocally available available silty silty soil soil in in Brahmaputra,Brahmaputra, India. India. SeveralSeveral hydraulichydraulic conductivityconductivity tests tests werewere performedperformed toto investigate investigate the the effect effect of of bentonite bentonite content content at various at various percentages percentages on the on locally the locally available available soil in soil order in toorder find to the find optimum the optimum percentage percentage that can that be can used be to us geted theto get required the required low hydraulic low hydraulic conductivity conductivity value. Thevalue. study The recommendedstudy recommended mixing mixing the available the available soil with soil 20with to 20 30 to percent 30 percent dry weightdry weight of bentonite of bentonite to modifyto modify and and use use the the available available soil soil aslow as low conductivity conductivity liner liner material. material.

FigureFigure 8.8. ExperimentalExperimental hydraulichydraulic conductivityconductivity valuesvalues forfor bentonite-siltbentonite-silt mixturesmixtures [[26]26] atat differentdifferent bentonitebentonite contentcontent inin comparisoncomparison withwith fittedfitted modelmodel resultsresults usingusing thethe proposedproposed model—Equationmodel—Equation (1).(1). TheThe crosscross symbolsymbol isis usedused toto identifyidentify thethe datadata pointspoints usedused inin thethethree-point three-point methodmethod discusseddiscussed inin SectionSection 3.5 3.5..

3.5.3.5. AA SimpleSimple Three-PointThree-Point MethodMethod toto EstimateEstimate thethe ScalingScaling Parameter Parameter InIn allall ofof thethe aboveabove discussions,discussions, thethe modelmodel fittingfittingexercise exercise was was completed completed using using all all available available data data points.points. BasedBased onon furtherfurther analysisanalysis wewe havehave foundfound thatthat inin orderorder toto approximatelyapproximatelyestimate estimatethe thevalue valueof of thethe model model parameter parameter (the (the scaling scaling factor factors), s one), one would would at leastat least need need three three data data points. points. Fortunately, Fortunately, two oftwo these of these points points are end are members end members (fine and (fine coarse and coarse material) material) and therefore and therefore all we needall we is need the value is the of value one coarse-fineof one coarse-fine mixture. mixture. As an example, As an example, if our model if our ismode usedl is to used describe to describe the Doley the etDoley al. [26 et] al. dataset, [26] dataset, there isthere no need is no to need measure to measure the hydraulic the hydraulic conductivity conductivi valuesty for values many for bentonite-silt many bentonite-silt mixtures mixtures in order toin findorder the to bentonitefind the bentonite percentage percentage that would that yield would the yield required the required low conductivity low conductivity value. Allvalue. one All needs one toneeds measure to measure is the hydraulic is the hydraulic conductivity conductivity value of valu juste one of mixturejust one ofmixture sand and of sand bentonite. and bentonite. In order toIn saveorder the to timesave requiredthe time torequired perform to theperform hydraulic the hy conductivitydraulic conductivity tests, we tests, recommend we recommend using an using optimal an mixtureoptimal withmixture the lowestwith the possible lowest bentonite possible bentonit content requirede content to required achieve to evenly achieve distributed evenly distributed bentonite throughoutbentonite throughout the mixture. the The mixture. hydraulic The conductivityhydraulic cond valueuctivity of this value mixture of this coupled mixture with coupled the hydraulic with the conductivityhydraulic conductivity of the end membersof the end (sand members and bentonite)(sand and arebentonite) sufficient are to sufficient adequately to adequately fit and determine fit and thedetermine scaling factorthe scalings. The fittedfactor model s. The can fitted then model be used can to estimate then be the used hydraulic to estimate conductivity the hydraulic values forconductivity any bentonite values content for any ranging bentonite from 0content to 100 percent.ranging Wefrom applied 0 to 100 this percent. approach We to estimateapplied thethis scalingapproach factor to estimates for all ourthe scaling laboratory factor datasets s for all and our for laboratory all literature-derived datasets and datasets.for all literature-derived The estimated valuesdatasets. of scalingThe estimated parameter values and the of corresponding scaling para RMSEmeter valuesand the are corresponding summarized in RMSE Table1 .values The three are experimentalsummarized datain Table points 1. The used three for thisexperimental approach aredata shown points in used Figures for 3this and approach5–8 for each are dataset;shown in noteFigures these 3 and three 5–8 data for pointseach dataset; are marked note these with athre crosse data symbol. points All are root marked mean with square a cross errors symbol. (RMSEs) All 2 androot coefficients mean square of determinationerrors (RMSEs) R 2anofd log coefficients K in Table of1 weredetermination calculated R using of log the K entire in Table dataset 1 were for fittedcalculated models using developed the entire using dataset the three-point for fitted method; models therefore, developed these using RMSEs the andthree-point R2 values method; can be therefore, these RMSEs and R2 values can be compared with the values of corresponding fitted model

Water 2018, 10, 313 11 of 13 compared with the values of corresponding fitted model developed using the entire dataset. For all different types of mixtures, very little difference was observed between RMSE and R2 values for fitted models developed using the entire dataset and using the three-point method. Therefore, the three-point method is an acceptable approach for estimating the value of the model parameter s.

4. Discussion and Conclusions In typical engineering projects, a soil mixture with a hydraulic conductivity value of at least 1 × 10−7 cm/s is required to construct hydraulic liners [16,27,28]. The percentage of clay and sand to be used vary based on the properties of the materials used to develop the mixture and compaction conditions during the construction of the liner. A laboratory test conducted by Garlanger et al. [12] recommended a minimum bentonite content of 6% to be mixed with locally available material, for a landfill site in central Florida, to achieve a hydraulic conductivity value of 1 × 10−8 cm/s or less. In another study, Gleason et al. [27] stated that a bentonite content of ≤15% is able to achieve a bentonite-sand mixture with a hydraulic conductivity value of less than 1 × 10−7 cm/s. The required bentonite percentage was varied based on the type of bentonite, sodium or calcium bentonite, and sand type. Usually, the bentonite content ranged between 5 and 15 percent [8]. In all these studies, multiple laboratory experiments were conducted to determine the bentonite content required to achieve the target hydraulic conductivity value. The model proposed in this study can estimate the hydraulic conductivity values of coarse-fine mixtures with fine content varying between 0 and 100 percent without using multiple experimental data points. In practical applications, in order to estimate the value of the model parameter (the scaling factor s), one would need at least three data points. In the proposed three-point method these values can include the hydraulic conductivity values of the two end members (pure coarse material and pure fine material), and at least one critical mixture. We recommend using a mixture with the lowest possible fine (or clay material) percentage required to achieve a uniform, well-mixed mixture. A good rule of thumb is to use about 10 to 20 percent of fine material. These three data can be used to fit the model and estimate the value of the scaling parameter s, and then the fitted model can be used to calculate the hydraulic conductivity of mixtures with various amounts of fines. Table1 compares the values of s evaluated using the three-point method against the values estimated using the entire dataset. The refitted s values for all synthetic coarse-fine and fine-coarse mixtures, clay-sand mixtures, and literature-derived experimental datasets using this three-point approach were close to the values estimated using the full set of data. To summarize, in this study, we have investigated the performance of a scalable model that used for predicting the changes in the hydraulic conductivity value of coarse and fine porous media mixtures due to the presence of different amounts of fines. Several laboratory experiments that represented the percentage of fines ranging from 0 to 30 were conducted using simulated coarse-fine and fine-coarse synthetic porous media mixtures. The value of the hydraulic conductivity of the coarse porous media decreased as the percent fine increased in the mixture. The reduction in hydraulic conductivity was significant when we started to add fine material to the coarse material; however, the reductions became less significant when the percentage of fine exceeded about 15%. Typically, the overall hydraulic conductivity value of the mixture was almost close to the hydraulic conductivity of the fine particles when the percent of the fine was above 30%. In the past, others have attempted to predict reductions in the hydraulic conductivity of coarse sand due to the presence of clay material and other fine minerals. However, all available models require measurement of multiple physical properties of the porous media mixture. The resulting empirical expressions have several model parameters that need to be individually calibrated by conducting multiple soil characterization tests. In this study, we have proposed a simpler model that uses a single model parameter to estimate the hydraulic conductivity values of different types of mixtures. The proposed model successfully correlated the changes in hydraulic conductivity values of a variety of coarse-fine mixtures with the amount of fine material in the mixture. We have tested the model performance using several new laboratory datasets, and also using multiple literature-derived datasets. Water 2018, 10, 313 12 of 13

It is important to note that this method is suitable only for modeling artificial mixtures and should not be used to predict the hydraulic conductivity value of undisturbed, heterogeneous natural porous media. The proposed framework is a useful tool for modeling the hydraulic properties of various types of engineered mixtures. The model can help design optimal mixtures without conducting multiple experiments that could be time consuming and cost prohibitive.

Acknowledgments: Zuhier Alakayleh wishes to express his gratitude to Mu’tah University in Jordan for financial support toward his graduate study at Auburn University. Author Contributions: Authors collaborated together for the completion of this work. Zuhier Alakayleh and T. Prabhakar Clement jointly conceived the idea and developed the first draft of this manuscript. Zuhier Alakayleh conducted all the lab work and analyzed the data, and cowrote the manuscript with T. Prabhakar Clement. Xing Fang reviewed the paper and provided useful ideas to improve the work. Conflicts of Interest: The authors declare no conflict of interest.

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