1. Introduction
Contamination of groundwater systems by landfill leachates is one of the common environmental problems. When an aquifer 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,
2,
3,
4,
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,
7,
8].
Past studies have shown that the hydraulic conductivity of a clay-sand mixture will decrease with increase in clay percentage [
3,
8,
9,
10,
11,
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 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 porosity, bentonite content, the degree of saturation at the end of the test, grain-size distribution, and the compaction 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 effective porosity, 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 gravel, initial saturation, and percent clay. The coefficient of determination (R
2) 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 void ratio 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 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 particle size 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 grain size. Previous studies correlate tortuosity with porosity using a simplified formula,
, where
T is tortuosity,
ɛ 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 porous medium 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.
Figure 1 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.
Before mixing, the porous media were washed in tap water to remove any dust and then dried in an oven. Samples were prepared by mixing coarse porous media with the following amount (percent dry weights) of fine porous media (0%, 5%, 10%, 15%, 20%, 25%, and 30%). An innovative mixing procedure used by Dias et al. [
22] was employed to thoroughly mix the coarse and fine glass beads. In this method, glycerol is used as a binder to fully mix the glass beads of different sizes. After packing the mixture, the glycerol was washed out by flushing water through the column. Use of glycerol allowed us to pack a uniform mixture within the column.
Hydraulic conductivity tests for glass bead mixtures were conducted by closely following the method described in ASTM-D2423-68 [
23]. This method uses the constant head permeameter test to determine the hydraulic conductivity of materials with values greater than 1 × 10
−5 m/s [
24]. Two transparent plastic columns of diameter 1.9 cm, but different lengths were used to construct the permeameter. The length of the short column is 25 cm and the long column is 75 cm. A wire mesh was used at the bottom of the short column then the soil sample was added over this mesh and was compacted gradually. While packing, the column was kept under water to maintain fully saturated conditions. To avoid segregation, the column was gradually lowered into the water bucket as it was packed. After packing the short column, a long transparent column was connected to the short column using a rubber coupling.
Tap water was used as the permeant liquid, and prior to its use, the water was allowed to reach the lab temperature. This was done to avoid gas exchanges and air trapping while running the test. After the sample was compacted, a high hydraulic gradient was applied to wash out the glycerol before running the hydraulic conductivity tests. To ensure consistency, a pump was used to introduce the water flow into the sample. The flow rate, sample length, column cross-section area, and head loss were measured, and Darcy’s law was applied to calculate the effective hydraulic conductivity value of the mixture.
2.2. Natural Clay-Sand Mixtures
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 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.
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 dry weight of the material. To obtain complete mixing, deaired water was added to the mixture and it was then physically stirred to prepare a well-mixed saturated slurry (glycerol was not added in this case). The clay-sand slurry was left in the mixing pan and covered for a period of about 48 h to let the clay fully saturate with water. During this period, water was added and the mixture was periodically stirred, whenever it is needed, to ensure complete mixing and full saturation.
A falling head permeameter was used; the test procedure closely followed the method described in ASTM-D5084-16a [
24]. Within the permeameter, the clay-sand mixture was packed in between two sand layers. The bottom sand layer helped prevent the fine material washing away through the bottom screen. The top sand layer helped us to better compact the mixture. After the sample was packed, the falling head permeameter test was performed to determine the hydraulic conductivity value of the sample.
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.
Table 1 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. 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.