Unified Packing Model for Improved Prediction of Porosity and Hydraulic Conductivity of Binary Mixed Soils

Abstract

Binary mixed soils, containing coarse sand particles mixed with variable content of fines (fine sand, silt, or clay) are important for several environmental and engineering applications. The packing state (or porosity) of such sand-fines mixtures controls several important physical properties such as hydraulic conductivity. Therefore, developing an analytical packing model to predict porosity of binary mixed soils, based on properties of pure unmixed sand and fines (endmembers), can contribute to predicting hydraulic conductivity for the mixtures without the need for extensive laboratory measurements. Toward this goal, this study presents a unified packing model for the purpose of predicting the porosity and hydraulic conductivity of binary mixed soils as function of fines fraction. The current model modifies an existing packing model developed for coarse binary mixed soils to achieve three main improvements: (1) being inclusive of wide range of binary mixed soils covering the whole range particle sizes, (2) incorporating the impact of cohesive packing behavior of the fines on binary mixture porosity, and (3) accounting for the impact of clay swelling. The presented model is the first of its kind incorporating the combined impact of all three factors: particle size ratio, fines cohesive packing and swelling, on binary mixtures porosity. The predictions of the modified model are validated using experimental published data for the porosity of sand-fines mixtures from 24 different studies. The model shows significant improvement in predicting porosity compared to existing packing models that frequently underestimate the porosity. By using the predicted porosity as an input in Kozeny–Carman formulation, the absolute mean error in predicting hydraulic conductivity, as function of fines fraction for 16 different binary mixed soils, is reduced by 50% when compared to the use of the previous packing model. The current model provides insights about the endmembers properties (porosity, hydraulic conductivity, and grain size) and fines content required to achieve a certain target desirable porosity and hydraulic conductivity of the mixed soils. This can assist the optimization of soil mixing design for various applications.

1. Introduction

Binary mixed soils are composed of large particles such as coarse sand, mixed with finer particles like fine sand, silt, clay, or bentonite. The packing state (or porosity) of such sand–fines mixtures controls several important physical properties such as hydraulic conductivity and shear strength [1,2,3,4,5,6]. Binary mixed soils have received increasing attention due to their abundance as economically important sediment and for their applications as liners and cutoff walls [7,8,9], in filter design [10,11,12], and in concrete mixing [13,14,15]. Therefore, predicting porosity (or void ratio) and hydraulic conductivity of binary mixed soils can be important for various disciplines including environmental and water resources sciences as well as geotechnical and civil engineering [16,17,18,19,20,21]. In particular, it is very important in practice to determine the fines content that is needed to achieve certain desirable porosity (n) and hydraulic conductivity (K) values in the binary mixture. This can be achieved by preparing binary mixed soils by mixing two endmembers (pure unmixed sand and pure clay for example) at variable fines content and then measuring n and K of the resultant mixtures. Such approach requires extensive laboratory testing and is time consuming. To overcome this issue, one would ideally like to be able to predict n and K of binary mixed soils as function of fines content based on the properties of endmembers and without the need to conduct extensive experimental work. Toward this goal, this study aims to present an approach that can be used to predict n and K of binary mixed soils using input parameters from the pure unmixed endmembers.

Several studies have shown that K of mixed soils can be estimated based Kozeny–Carman formulation which requires n as an input [6,21,22,23,24,25]. Therefore, developing a packing model that can accurately predict n of binary mixed soils can assist the prediction of K for binary mixed soils. Many studies attempted to develop an analytical model for the prediction of binary mixtures porosity based on the properties of unmixed fine and large particles endmembers [7,23,25,26,27,28,29,30,31]. Studies from the field of powder and concrete technology [29,30,31] presented an analytical packing model that is based on ideal packing condition: both fines and large particles maintain their unmixed porosity after mixing. Such ideal packing implies that fine particles effectively fill the pore space between larger particles with no packing disturbance. The ideal packing model is theoretically applicable in the case of significantly low fine particle size ($d_f$) to large particle size ($D_L$) ratio, $d_f$/$D_L$ [29,30,31]. Nevertheless, large body of literature reported that the ideal packing model frequently underestimates the porosity for sand–fines mixtures [2,23,26,27,32,33]. Therefore, several studies (e.g., [23,25,26]) introduced correction parameters into the ideal packing model in an attempt to correct for the deviation from ideal packing case due to increasing $d_f$/$D_L$. The correction parameters are empirically expressed as function of $d_f$/$D_L$ whose increase is a main reason for non-ideal packing as suggested by few studies [23,25,26].

Recent studies however have shown that accounting for particle size ratio ($d_f$/$D_L$) is not enough to explain deviation from ideal packing [7,33]. Two other factors that are relevant here include the cohesive nature of fines [33,34] and the volume expansion of swelling clays [7,35]. Cohesive packing (forming aggregates) of very fine particles is due to inter-particle forces, such as electrostatic forces and van der Waals, which can overcome the impact of gravity force [36,37] resulting in very different packing compared to non-cohesive coarse sands. On the other hand, volume expansion upon saturation of swelling clays such as bentonite can disturb the packing of large particles filling more pore space between them [7,35]. Incorporating such factors into the packing model can be important since fine particles used in binary mixed soils cover wide range of sizes and composition varying from fine sand to silt and clay (including swelling and non-swelling clay).

Existing packing models developed for sand–fines mixtures account for either particle size ratio (e.g., [14,25,26]), fines cohesive packing [33], or swelling impact [7]. The applicability of such models to wide range of binary mixed soils remain unclear leading to uncertain prediction of porosity and consequently errors in hydraulic conductivity estimation. Incorporating all three factors together into a single packing model can be important since fine particles used in binary mixed soils cover wide range of sizes and components varying from fine sand to silt and clay (including swelling and non-swelling clay). This study thus aims to develop a unified packing model that accurately predicts the porosity or void ratio of binary mixed soils as function of fines fraction and using input parameters obtained from the endmembers alone (pure large sand packing and pure fines packing). In doing so, the effects of particle size ratio, fines cohesive packing, and swelling on binary mixtures porosity are all incorporated in one single model. This can ultimately result in improved prediction of hydraulic conductivity for binary mixed soils. Firstly, theoretical background of binary mixture packing is reviewed, highlighting limitations of existing models. Secondly, an existing packing model developed initially for coarse mixed soils is modified to account for all three factors mentioned above: $d_f$/$D_L$, fines cohesive packing and swelling. The modified model is validated using experimental published data for various binary mixed soils characterized by wide range of $d_f$/$D_L$, and fines components including cohesive clays and swelling bentonite. The resultant predicted porosity is then used as an input in Kozeny–Carman formation to highlight improvement in the prediction of hydraulic conductivity over the use of outputs from earlier packing models.

2. Background

2.1. Packing Theory and Model for Binary Mixed Soils

Packing theory for binary mixtures (composed of large particles mixed with finer ones) and porosity prediction for sand–fines mixtures are briefly described here where more details can be found in [25,29,38]. When large particles (such as coarse sands) are mixed with finer particles (such as fine sands or clay), the porosity (or void ratio) of the binary mixture decreases first with fines content before porosity starts to increase with increase of fines fraction (Figure 1). At low fines volume fraction ($V_f$), fine particles fill in the porosity between the larger particles which results in reducing porosity until reaching a minimum porosity point (Figure 1). Such point corresponds to the case where fine particles completely fill the interparticle pores between larger particles [25,38]. The fine fraction required to achieve the minimum porosity (maximum packing density) is frequently referred to as critical fines fraction, $V_f^{*}$. Beyond this point, increasing the fines fraction would require the replacement of solid large particles by the aggregates of micro-porous fines, thus increasing mixture porosity. According to the packing theory of binary mixtures [25,28,29,30,31], an “ideal” packing requires that mixing process does not disturb the original packing and porosity of fine or large particles. That is, the porosity of the pure endmembers does not change after mixing. Such ideal packing is predominantly a theoretical limit that requires significantly low fine-to-large particle size ratio, $d_f$/$D_L$. The porosity of binary mixtures in case of ideal packing has been described using a linear packing model and it represents a lower bound (i.e., most dense packing) as shown in Figure 1. The upper bound represents “no mixing” condition between large and fine particles (Figure 1). In this case, either $d_f$/$D_L$ is one or the two end-members are not mixed together but rather occurs at two separate layers (Figure 1; case C). Nevertheless, the packing in real sand–fines mixtures deviates from both the upper and lower bounds, frequently described as “non-ideal” packing [25,29,33,38]. In this case, the porosity-fines fraction trend frequently plots in between the upper and lower bounds as shown in Figure 1.

The main physical explanation for the deviation of real mixed soils from the ideal packing is the increase of fine-to-large particle, $d_f$/$D_L$ as reported by several studies [6,14,23,25,31,39]. In this case, fines start to disturb the packing of sands and vice versa, creating an extra pore space between the fine particles as well as along the wall of larger sands (Figure 1; schematic B). Several semi-empirical packing models have been developed to correct for the deviation from the ideal packing for binary mixtures of glass beads [6,12,28] and mixed soil [14,23,25,26,27]. This was done by incorporating one or more correction coefficients/functions, expressed as function of $d_f$/$D_L$, into the ideal packing model [14,23,25,26,27]. There are different formulations for non-ideal packing model and correction coefficients but this current work builds upon and modifies the model developed initially by Zhang et al. [25]. This is due to the simplicity of the model in terms of formulation and physical concept. The model uses one simple “mixing-degree” correction coefficient, M, to account for the contribution of increasing $d_f$/$D_L$ to the deviation from ideal packing. Additionally, the authors showed how the model prediction can be used for estimating hydraulic conductivity [25], which is of interest for this study as well.

Zhang et al. [25] showed that the porosity of binary mixture ($n_m$) can be estimated using:

$$n_m=n_L-V_f \left[n_L-n_f-M\left(n_L-1\right)\right],{ \mathrm{when} V}_f<{ V}_f^{*}$$

(1)

$$n_m=V_f n_f+n_L(1-V_f) \left(1-M\right), \mathrm{when} V_f> V_f^{*}$$

(2)

where $n_L$ is the porosity of unmixed large particles (i.e., sands) and $n_f$ is the porosity of unmixed fines. The mixing correction coefficient, M, has been empirically related to the particle size ratio, and can be found using Equation (3) as reported by Zhang et al. [25]:

$$M=0.0363 \left(\frac{D_L}{d_f}\right)+0.2326$$

(3)

In Equation (3), $d_f$ and $D_L$ correspond to the median particle size (d₅₀) of the fines and large particles, respectively. In case d₅₀ is not available, the average between upper and lower sieve values can be used. Since $V_f^{*}$ is not known as a priori, a practical approach to calculate $n_m$ at any $V_f$ is to calculate the porosity using both Equations (1) and (2) and then choose the maximum value at any given $V_f$ [27,28]. It is worth mentioning that when $M$ = 1, Equations (1) and (2) reduce to the ideal packing case while $M$ = 0 represents the no-mixing condition. Since the preparation of mixed soil specimens is frequently done using the mass fraction instead of volume fraction, it is convenient to covert fines volume fraction ($V_f$) to mass fraction ($F_f$) in Equations (1) and (2) which can be achieved through the following equations [7,32]:

$$V_f=\frac{{\rho }_L \left(1-n_L\right) F_f}{{\rho }_f \left(1-n_f\right) \left(1-F_f\right)}, \mathrm{when} V_f< V_f^{*}$$

(4)

$$V_f=\frac{{\rho }_L F_f}{{\rho }_f \left(1-n_f\right)\left(1-F_f\right)+{\rho }_{L }{F}_f}, \mathrm{when} V_f> V_f^{*}$$

(5)

In the equations above, ${\rho }_f$ and ${\rho }_L$ are the solid particle density of fine and large particles, respectively. The packing model described here can also be expressed in terms of void ratio, e, by substituting n with e/(1 + e) in Equations (1) and (2).

2.2. Limitations of Existing Models

Although the above model provided reasonable estimates for the porosity of some binary mixed soils [25], it has some limitations. Firstly, Equation (3) for the determination of M from particle sizes was based on 4 data points only where the values of $d_f/D_L$ did not cover the range between 0.30 and 1. This study however develops an expression for M based on published experimental data on porosity and void ratio of more than 50 sand–fines mixtures from 24 different studies covering the whole range for $d_f/D_L$ up to 1.

Another important limitation of the model above is that it does not apply to mixed soils containing cohesive fines characterized by very small particles (such as clay or fine silt) as highlighted by Zhang et al. [25]. For very small $d_f/D_L$ (<0.05), M is set equal to one rather than using Equation (3) [25]. The model in this case reduces to the ideal packing case which frequently underestimates the porosity of sand–fines mixtures characterized by small $d_f/D_L$ [2,32,33]. Inter-particle forces, such as electrostatic forces and van der Waals, can overcome the impact of gravity force for very fine particles [36,37] resulting in very different packing compared to non-cohesive coarse sands. An earlier study [33] showed that ideal packing model underestimates porosity of sand–clay and sand–silt mixtures due to the cohesive nature of clay/silt which is neither considered in ideal packing model nor in the mixing correction parameter M (Equation (3)). The study showed that a packing model, developed initially for dry binary glass beads powders, can be used to predict porosity in sand–clay mixture [33]. Nevertheless, the model was not tested on saturated mixed soils and it is not applicable to mixtures characterized by $d_f/D_L$ larger than 0.1 [33]. Rather than modifying the glass beads powder packing model, this study incorporates the impact of cohesive packing by modifying Equation (3) in the mixed soils model developed by Zhang et al. [25]. This can help to develop a unified model for binary mixed soils characterized by variable $d_f/D_L$ and cohesive nature.

Another important point considered here when modifying the model above, is to incorporate the impact of swelling and volume expansion upon saturation on packing and porosity. This can be essential for binary mixed soils containing bentonite for example [7,38]. A study done by Xu et al. [7] utilized ideal packing model to model the porosity of sand–bentonite mixtures after incorporating volume expansion through an empirical parameter. However, the ideal packing model developed does not incorporate the impact of particle size ratio ($d_f/D_L$) or fines cohesive packing. Consequently, the use of approach presented by Xu et al. [7] can drastically underestimates the porosity of mixed soils characterized by relatively large $d_f/D_L$ (>0.1) or in the presence of cohesive fines as mentioned earlier. This study thus modifies a non-ideal packing model (Equations (1)–(3)) to incorporate the impact of swelling on packing and porosity/void-ratio.

In summary, this study modifies the model developed by Zhang et al. [25] to achieve the following improvements:

(a)

Developing a correction function that is inclusive of wide range of binary mixed soils from 24 different studies.

(b)

Incorporating the impact of fines cohesive-packing behavior (for clay or fine silt) on the binary mixture porosity

(c)

Modifying the model formulation to account for the impact of swelling (for expansive clays) on mixed soils porosity.

The main aim here is to present a “unified” model that can be used to predict the porosity of sand–fines mixed soils characterized by variable $d_f/D_L$ values while considering the cohesive and swelling nature of fines. The modification and validation of the unified packing model are shown in the next section followed by the use of the resultant porosity for hydraulic conductivity prediction.

3. Packing Model Modification and Validation

3.1. Updated Formulation for Mixing Coefficient M

In the first step for packing model modification, measured porosity (or void ratio) values as function fines fraction of more than 50 binary mixed soils from total of 24 published studies were used to update Equation (3). Out of the total 24 published work, 12 studies [6,17,23,25,30,40,41,42,43,44,45,46] focused on mixed graded sand soils with no cohesive fines (i.e., no clay or fine silt) while the other 12 studies [7,30,32,47,48,49,50,51,52,53,54,55] reported data for mixtures consisting cohesive fines. Equations (1) and (2) were used to find the best fit mixing parameter M that reproduces the porosity-fines fraction experimental trend for each binary mixture. Figure 2 shows the data-derived best fit M versus $d_f/D_L$ where the median particle size (d₅₀) was used in this case to define $d_f$ and $D_L$. The data points are classified based on two groups: one for mixtures with no cohesive fines, and another for mixed soils involving cohesive fines. It is evident from Figure 2 that majority of mixtures containing cohesive particles deviate significantly from the overall trend for non-cohesive mixtures. The results in Figure 2 suggest that the formulation for M in non-cohesive mixtures has to be different from that for mixtures with cohesive components. This section discusses the modified equation for non-cohesive mixtures while the next section addresses the incorporation of cohesive impact. Considering only the data for non-cohesive mixtures (black circles in Figure 2), one may notice that Equation (3) does not match the data well, simply because it was developed based on much smaller number of data points. The updated formulation for M (black curve in Figure 2) is:

$$M={\left(1-\frac{d_f}{D_L}\right)}^{2.7}$$

(6)

Equation (6) does not only fit the data better than Equation (3), but it also honors the two boundary conditions of ideal mixing (M ≈ 1 when $d_f/D_L$ approaches zero) and no-mixing (M = 0 when $d_f/D_L$ = 1). The use of Equation (6) to estimate M resulted in high correlation coefficient (Figure 2; R² = 0.98) and reduced the mean absolute error (MAE) by 77% compared to the use of Equation (3) reported originally by [25]. Figure 3 shows examples for the prediction of porosity or void ratio of binary mixed soils as function of fines fraction. The current modified model (Equations (1), (2) and (6)) provided better prediction of porosity/void ratio compared to the original model (Equations (1)–(3)), especially for mixtures characterized by large $d_f/D_L$ values (Figure 3; blue data points). This is because the original model was developed based on much smaller number of data points that did not cover $d_f/D_L$ larger than 0.3. On the other hand, the current model is developed based on wide range of mixed soils (covering the whole range of $d_f/D_L.$

(Equation (6)) are shown in dashed green and solid black curves, respectively. The correlation coefficient (R²) and mean absolute error (MAE) are shown in black and green colors next to the curves.

3.2. Incorporation of Cohesive Packing Behavior

As highlighted earlier in Figure 2, mixed soils containing cohesive fines such as clay and fine silt require a different treatment compared to completely non-cohesive mixtures. The method adopted here for the incorporation of fines cohesive packing builds upon the approach suggested by an earlier study on glass beads powder from the powder technology and chemical engineering disciplines [34]. The authors introduced the concept of effective particle size ratio, R_e, which is larger than the actual $d_f/D_L$ due to fines aggregation [34]. In the absence of cohesive fines, R_e is equal to $d_f/D_L$ [34]. The authors defined a critical fine particle size value ($d_{cri}$), that separates the cohesive from non-cohesive packing behavior. Below $d_{cri}$, the effect of cohesive inter-particle forces on packing overcome the impact of gravity, thus porosity of fines packing increases drastically with the decrease of particle size. When fines have particle size above $d_{cri}$ (non-cohesive packing), porosity does not change drastically with changes in particle size. In light of the above discussion, Equation (3) is modified to become:

$$M={\left(1-R_e\right)}^{2.7}$$

(7)

where the effective particle size ratio, $R_e$, can be obtained using the following equation [34]:

$$R_e=\{\begin{array}{c}\frac{d_f}{D_L} when d_f> d_{cri}\\ \left(1-x\right){\left(\frac{d_f}{D_L}\right)}^p+x \left(\frac{d_f}{D_L}\right) when d_f< d_{cri}\end{array}$$

(8)

In Equation (8), p is a constant between 0 and 1 that quantifies the degree of cohesive packing for the fines [34]. The decrease in p, means physically an increase in the degree of cohesive packing. This leads to the formation of larger fines cohesive aggregates ($d_f$ increases) and consequently, R_e increases. The parameter x in the equation above describes how much packing disturbance in the binary mixture is caused by the fines cohesive packing and it is given by [34]:

$$x=1-1.543exp\left(-0.697\frac{D_L}{d_{cri}}\right)$$

(9)

From Equation (9), one can infer that the decrease in $D_L$/$d_{cri}$ leads to a decrease in $x$ indicating that cohesive fines significantly disturb the larger particles packing in the binary mixture. As a result, $R_e$ increases for a given p according to Equation (8), resulting in higher M values (Equation (7)) and causing porosity to increase at a given fines fraction. To sum up, the non-ideal packing in the case of mixed soils containing cohesive finds is a result of increasing the effective particle size ratio ($R_e$) which is accounted for in the modified formulation by (a) decrease in p, and/or (b) decrease in x.

The updated formulations above require the determination of $d_{cri}$ and p for binary mixed soils containing sand and cohesive fines. An earlier study [33] showed that $d_{cri}$ = 0.150 mm works well for sandy soils. The only remaining parameter p, can then be estimated using Equations (7)–(9) to find the average value of p that best reproduces the best fit M derived from experimental data in Figure 2 for mixtures containing cohesive particles. All binary mixed soils used here are composed of large particles (i.e., sand) and fine cohesive particles (i.e., clay or silt), characterized by very small $d_f/D_L$ (≤0.1) and $d_f$ (<0.15 mm). The average best fit value for p was found to be 0.25 and the resultant M, estimated from Equations (7)–(9), is compared with the actual best fit values (the ones derived based on experimental data as in Figure 2) as shown in Figure 4. Significant improvement in the prediction of the actual M is observed when incorporating the fines cohesive packing (Equations (7)–(9); R² = 0.98) compared to the formulation for non-cohesive packing (Equation (6); R² = 0.26), as shown in Figure 4. The mean absolute error in predicting M is reduced by more than 90% when incorporating fines cohesive packing (Figure 4). It is worth mentioning that prediction of M using Equations (6) and (7) would match each other when x approaches 1 as $R_e$ would approach $d_f/D_L$according to Equation (8). This is the case for three mixtures in Figure 4 where $D_L$ is very large (>0.750 mm) resulting in very large $D_L$/$d_{cri}$ and consequently x approaches 1 according to Equation (9). Figure 5 shows few examples highlighting how the new formulation incorporating cohesive packing improved the prediction of porosity (or void ratio) for mixed soils. It is evident that the new formulations correct the under-prediction of mixtures porosity encountered when ignoring the impact of fines cohesive packing.

3.3. Incorporation of Swelling

In this section, the impact of swelling (i.e., volume expansion upon saturation) on packing is incorporated in the model. Bentonite is an example of a swelling clay that has much higher swell index compared to non-swelling clay such as kaolinite [7,35]. Xu et al. [7] investigated porosity variations of sand–bentonite mixtures as function of bentonite fraction and compared that with mixtures of sand and non-expansive clay (Figure 6). The results in Figure 6 shows that the presence of expansive fines lowered the critical fines fraction at which minimum porosity is observed, in comparison with the non-expansive clay. The black dashed curve in Figure 6 shows also the prediction of the current model (Equations (1), (2) and (7)–(9)) without incorporating the swelling impact. While the model could accurately predict the porosity and critical fines fraction (0.24) of the mixtures involving non-expansive clay (Figure 6a), the predictions do not match the measurements in the case of sand–bentonite mixtures especially at low bentonite fraction (Figure 6b). Ignoring the impact of swelling results in overestimating both the critical fines fraction (predicted is 0.2 while actual is 0.05) and the minimum porosity value (predicted is 0.20 while measured is 0.14) as shown in Figure 6b. The decrease of minimum porosity and critical fines fraction is due to the expansion of bentonite volume upon saturation, where smaller amount of bentonite can fill more space in between the larger sands compared to the case of non-swelling clay.

In order to account for the impact of swelling, Xu et al. [7] suggested to incorporate a volume expansion parameter, S, in the ideal packing model. The parameter S describes the increase in fines volume due to swelling and it can be estimated based on the swell index of the clays [7]. For the mixtures shown in Figure 6, the non-swelling clay-J has a swell index of 1.6 (mL/2 g) while bentonite has swell index of 17 (mL/2 g) [7]. Taking the non-swelling clay in Figure 6a as a reference (S = 1), the volume expansion parameter for bentonite can be estimated using S = 17/1.6 = 10.6 [7]. The incorporation of S in the ideal packing model improved the prediction of porosity as shown in Figure 7, in comparison with predictions ignoring swelling impact (Figure 6b). The mixtures prepared by Xu et al. [7] are characterized by very small $d_f/D_L$ and very large $D_L$ (0.75 mm), which might explain the reasonable prediction of the ideal packing model although it slightly underestimates porosity at low bentonite fraction (Figure 7). Nevertheless, the model developed by Xu et al. [7] does not take into account particle size ratio (it does not include M) or cohesive packing, and thus its applicability is questioned for mixtures characterized by larger $d_f/D_L$. Therefore, this study alternatively incorporates the parameter S in the present non-ideal packing model as follows:

1

When $V_f< V_f^{*}$, the parameter S multiplies the volume fraction of fines so that the porosity of the mixture is decreased, which can be found using:

$$n_m=n_L-S V_f \left[n_L-n_f-M\left(n_L-1\right)\right]$$

(10)

2

When $V_f> V_f^{*}$, Xu et al. [7] showed that the impact of swelling on the mixture packing can be empirically expressed as an exponential function of S and fines fraction. Based on the experimental data from Xu et al. [7], the best fit swelling correction function was found to be through the following:

$$\alpha =1+\frac{S-1}{S} e^{\left(1-5.5 V_f\right)}$$

(11)

$$n_m=\alpha V_f n_f+n_L(1-\alpha V_f) \left(1-M\right),$$

(12)

The resultant packing model matches the experimental data well as shown in Figure 7. There is a slight improvement over the model developed by Xu et al. [7] primarily due to the different formulation used here for $\alpha$ in addition to accounting for cohesive packing. The main advantage however is that the current unified model can be applied to mixtures with wide range of $d_f/D_L$ containing swelling clays. In contrast, the model presented by Xu et al. [7] does not account for variations in $d_f/D_L$ which is a major source of deviation from ideal packing as mentioned earlier.

4. Discussion

4.1. Unified Packing Model for Practical Applications

Based on the results shown above, a summary of the unified packing model is shown in this section. The model presented here can be used to predict porosity or void ratio of binary mixed soils, as function of fines fraction and based on input parameters obtained from the pure endmembers alone. The model has better performance that the original models because it incorporates the impact of all three factors: particle size ratio ($d_f/D_L$), fines cohesive packing and clay swelling. The required input parameters include $d_f,D_L, n_f,n_L$ and swelling index (for swelling clay only). Other parameters such as p and $d_{cri}$ have been determined earlier to have values of 0.25 and 0.150 mm, respectively, as shown above (Section 3.2) and are substituted in the following equations.

$$n_m=\{\begin{array}{c} n_L-S V_f \left[n_L-n_f-M\left(n_L-1\right)\right] \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} V_f< V_f^{*}\\ \alpha V_f n_f+n_L(1-\alpha V_f) \left(1-M\right), \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} V_f> V_f^{*}\end{array}$$

(13)

$$\alpha =1+\frac{S-1}{S} e^{\left(1-5.5 V_f\right)}$$

(14)

$$S=\{\begin{array}{c} 1 \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{c}\mathrm{l}\mathrm{a}\mathrm{y}\\ \frac{swell index}{1.7} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{s}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{c}\mathrm{l}\mathrm{a}\mathrm{y}\end{array}$$

(15)

$$M={\left(1-R_e\right)}^{2.7},$$

(16)

$$R_e=\{\begin{array}{c}\frac{d_f}{D_L} when d_f> 0.150 mm\\ \left(1-x\right){\left(\frac{d_f}{D_L}\right)}^{0.25}+x \left(\frac{d_f}{D_L}\right) when d_f< 0.150 mm\end{array}$$

(17)

$$x=1-1.543exp\left(-0.697\frac{D_L}{0.150}\right)$$

(18)

For Equation (15), in case swelling index is not available, S = 10 can be used as an approximation for bentonite. The resultant porosity from the model above is expressed as function of volume fraction which can be converted to mass fraction through Equations (4) and (5) as done throughout this study. One application of this model is to predict the value of the minimum possible porosity and the critical fines fraction required to achieve such case. This can save time and effort to optimize mixing and achieve a certain target porosity or void ratio without the need to perform extensive experiments to figure out the right mixing proportions. According to the model presented here, achieving maximum packing density (minimum porosity) for the binary mixture requires (a) very small $d_f/D_L$ where $D_L$ is above 0.750 mm so that $R_e$ is very small and M approaches 1 (Equations (16)–(18)), and (b) small $n_L$ and $n_f$ (Equation (13)). Porosity is also a critical parameter for the determination of hydraulic conductivity, thus the next section discusses how the modeled porosity can be used in the prediction of hydraulic conductivity for binary mixed soils.

4.2. Prediction of Hydraulic Conductivity

The aim here in this discussion is to examine how the use of the above packing model can assist in accurately predicting the hydraulic conductivity, K, for binary mixed soils characterized by variable particle size ratio and fines types (fine sand, silt, or clay). Porosity is a major quantity that controls K where both quantities are linked through the Kozeny–Carman formula (Equation (8)):

$$K=\left(\frac{{\rho }_{fl }g}{\mu }\right)\left(\frac{d_{rep}^2{n}^3}{180{\left(1-n\right)}^2}\right)$$

(19)

In the equation above, $d_{rep}$ is the representative particle size; ${\rho }_{fl }$is fluid density; and $\mu$ is the dynamic viscosity and $g$ is gravitational acceleration. The same equation can be used to estimate intrinsic permeability but without including the terms ${\rho }_{fl }g/\mu$. It is worth mentioning that there are other formulations that relate porosity and hydraulic conductivity but Kozeny–Carman is among the most frequently used with high reliability [6,7,22,23,24,25,40,53,56]. Both porosity and representative particle size need to be determined so that K can be estimated. While porosity of the mixture can be estimated using the packing model presented here (Equations (13)–(18)), $d_{rep}$ of the mixture at any given $V_f$ can be estimated as follows [25]:

$$d_{rep}={\left(V_L D_{repL}^b+V_f d_{repF}^b\right)}^{1/b}$$

(20)

where $D_{repL}$ and $d_{repF}$ are the representative particle size of the sand and fines, respectively. Both $D_{repL}$ and $d_{repF}$ can be calculated through Equation (19) and by using the measured K and n values of the endmembers (i.e., pure unmixed sand and fines). The coefficient b follows a sigmoidal trend from 0 to −1 as $V_f$ increases from 0 to 1 and it can be empirically estimated by the following equation [25]:

$$b=\frac{1}{1+\exp \left[a\left(V_L^{*}-V_f\right)\right]}-1$$

(21)

where $V_L^{*}$ is the critical large particle volume fraction ($V_L^{*}=1-V_f^{*})$and a is an empirical parameter that determines the steepness of the b-$V_f$curve. The following steps were followed to find a: (1) the $d_{rep}$ for binary mixed soils, where K data is available [7,23,25,32,40,48,53,57,58], was calculated using the measured $n_m$ and K, of the binary mixtures and using Equation (19); (2) Equations (20) and (21) were then used to find the best fit parameter a, which reproduces the $d_{rep}$ of the mixtures at variable fines fractions. It was found that a = 10 provides the best estimates for $d_{rep}$ of the different binary mixtures examined in this study.

The practical steps for the calculation of hydraulic conductivity are as follow:

1

Calculating $n_m$ as function of $V_f$ using the packing model presented here (Equations (13)–(18)). The value of $V_f^{*}$ is obtained from the packing model presented above as described earlier.

2

Calculating $D_{repL}$ and $d_{repF}$ through Equation (19) and by using the measured porosity and hydraulic conductivity of the pure endmembers as inputs in the equation.

3

Calculating $d_{rep}$ as function of $V_f$ using Equations (20) and (21) by setting a = 10 and $V_L^{*}=1-V_f^{*}$.

4

Calculating hydraulic conductivity as function of $V_f$ for the binary mixed soil using Equation (19).

The prediction of K using the above approach have been tested using measurements of K for 16 different mixtures from nine different studies [7,23,25,32,40,48,53,57,58]. For mixtures involving clays (very low K), flexible wall permeameter (ASTM D5084) was predominantly used (e.g., [7,57]), while constant head method (ASTM D2434) was used for mixtures involving coarser graded sand (e.g., [23,25]). Figure 8 shows some examples for the prediction of K as function of fines fraction. Figure 8a–c show sand–fines mixtures characterized by small $d_f/D_L$, Figure 8d is for sand–bentonite mixtures, while Figure 8e,f are predominantly for mixtures characterized by larger $d_f/D_L$ (graded sand binary mixtures). The prediction of the current model is shown in bold solid curves while the prediction using the approach presented by Zhang et al. [25] (Equations (1)–(3) and (19)–(21)) are shown in dashed curves. The predictions of the current model outperform that of Zhang et al. [25] especially, for mixtures involving bentonite or large $d_f/D_L$. Figure 8d shows that ignoring the impact of swelling results in overestimation of K by one to two orders of magnitude in the mixtures at low fraction of bentonite. This can be explained by the overestimation of critical fines fraction as well as overestimation of porosity at low bentonite fraction as shown earlier in Figure 6b. Moreover, the packing model presented by Zhang et al. [25] under-predicts porosity of binary mixtures characterized by large $d_f/D_L$ (Figure 3) which consequently results in underestimation of K for such mixtures (Figure 8e,f). Based on the results, K at any given fine fraction of mixed graded sand increases as $d_f/D_L$ increases (Figure 8e,f). This is essentially due to the larger pore sizes present in the case of coarser particles compared to very fine particles.

For engineering purposes, it is important to determine the swelling bentonite content that is required to achieve sufficiently low hydraulic conductivity below the typical regulatory requirement (1 × 10⁻⁷ cm/s) [7,59,60,61]. Such sand–bentonite mixture can be used as barrier/liner material in several engineering applications including landfill, slurry cutoff walls, and radioactive waste disposal [7,8,9,61,62,63,64]. According to the results in Figure 8d, the low K limit is achieved at bentonite percentage of 5%. To investigate this further, Figure 9 shows K as function of bentonite fraction for various mixtures. The curves represent the modeling results incorporating (solid curves) or ignoring (dashed curves) the swelling impact. The curves were constructed based on the endmember data from [59]. The open symbols in Figure 9 correspond to data from different studies where endmembers measurements were not available. All data followed similar behavior where the presented model, incorporating swelling impact, could successfully predict the K of sand-bentonite mixtures. The regulatory limit of K (1 × 10⁻⁷ cm/s) is reached at bentonite percentage that is around 4–5% for medium sand–bentonite mixtures (Figure 9) which is consistent with results in Figure 8d. Such K limit is achieved at slightly lower bentonite fraction for mixtures containing fine sand (Figure 9) which also agrees with the modeling results. On the other hand ignoring the swelling impact in the packing model results in significant overestimation of K and bentonite content required for reaching the regulatory limit.

The mean absolute error in predicting K using the presented modified model is 0.032 cm/s while it is 0.064 cm/s when following the original model developed by Zhang et al. [25]. That is, the use of the current modified model reduces the mean absolute error in predicting K of binary mixed soils by 50%. Moreover, an important advantage of the current model is that it is applicable for wide range of binary mixed soils covering different particle sizes and fines types. It is a “unified” model that takes into account variable factors including: (a) full range of $d_f/D_L$, (b) fines cohesive packing and (c) swelling impact.

Developing an accurate packing model can assist several real life applications such as (a) calculating the primary volume, porosity and permeability of deposited unconsolidated sediments when modeling sedimentary basins [65,66,67], (b) optimizing concrete and aggregates mixing design to achieve a desirable porosity and hydraulic conductivity values [13,14,15,18,19], and (c) designing liners and cut off walls [7,8,9,23,61,62]. For example, the fines fraction as well as the endmembers properties required to achieve a target hydraulic conductivity can be predicted. This can be done without the need for extensive mixing or laboratory measurements. Only properties (n, K, and grain size) of unmixed endmembers need to be measured and used as input parameters in the model through which the porosity and hydraulic conductivity of the binary mixed soil can be predicted at the full range of fines fraction. According to the presented model, minimizing the fines fraction required to reduce K significantly (occurs around $F_f^{*}$) would be ideally achieved by using a swelling clay such as bentonite (Figure 8d). Additionally, the hydraulic conductivity of the mixed soils can be minimized at any given fines fraction by choosing endmembers so that $d_f/D_L$ is small (Figure 8e). Conditions mentioned in Section 4.1 for minimizing porosity can also be considered when targeting a low hydraulic conductivity mixtures.

4.3. Discussion on Input Parameters and Further Potential Development

There are several input parameters that are required by the presented model where their reported values were based on the calibration with various published datasets. Some of the most important parameters include p = 0.25 (Equation (8)), S = 10 (Equations (10) and (11)) and a = 10 (Equation (21)). The use of such values is justified by the good match with experimental data reported for the large number of mixtures used in the calibration and validation process. Details regarding the physical meaning and estimation/calibration of the different parameters can be found in Section 3.2, Section 3.3 and Section 4.2. It is worth mentioning that re-calibration of the parameters could be conducted especially when dealing with mixtures composed of different sediment or particles (like rod-like particles) than the sand-fines mixtures used here. The model also requires measurements of porosity and hydraulic conductivity of the end-members. This section discusses sensitivity of the results to various input parameters values and provide some insights regarding feasible alternatives.

The cohesive packing parameter, p quantifies the degree of cohesive packing of the fines. The values of p that best reproduces the experimental results (Figure 4) ranges from 0.21 to 0.28 with an average of 0.25. This range is considered to be narrow given that p can theoretically vary between 0 and 1. Figure 10 demonstrates the variations of porosity that correspond to such range in p (red curves), where the average p is shown in black solid curve. It is evident from the figure that even the upper and lower limit for p range (0.28 and 0.21) are very close to the overall trend. The maximum difference in the porosity predicted using p = 0.21 and p = 0.28 is 0.02 (2 porosity units) and it can be observed at the critical fines fraction ($F_f^{*}$ = 0.2) as shown in Figure 10. The results in Figure 10 suggest that small perturbation (±0.04) in average reported p dose not impact the porosity prediction significantly (red curves in Figure 10). However, much larger variations (±0.15) would provide unsatisfactory results (green curves in Figure 10) but such values were not observed for any of the mixtures.

Regarding the swelling parameter, S, it can be estimated using the measured swell index of the expansive clay (Equation (15)). In case swell index is not available, S = 10 could be considered as it shows already good match with various experimental results (Figure 8d and Figure 9) where the swell index was not reported. It is worth mentioning that a decrease in S would physically mean less swelling which in turn results in reaching the regulatory K limit (1 × 10⁻⁷ cm/s) and the critical fines fraction at larger fines content.

Regarding the parameter a, it is used to find the representative particle size for the binary mixture (Equations (20) and (21)). The average best fit value was found to be 10 while Figure 11 shows examples for the impact of varying a on the predicted permeability. For mixtures involving very fine particles (e.g., clay), deviation from a = 10 can slightly reduce the accuracy of the predicted permeability (Figure 11a). This is particularly the case for mixtures with fines fraction above 0.2 when using a values that are much lower than 10. In contrast, variations in a do not seem to impact the accuracy of permeability prediction for binary mixtures involving sand graded particles (in the absence of clay) as shown in Figure 11b. It seems that the difference between the endmembers representative particle size (or K) has to be very large (one or two orders of magnitudes), like the case between sand and clay, in order for parameter a to impact the predicted permeability. Otherwise, the predictions are not sensitive to variations in a. The use of a = 10 can ensure reasonable prediction of permeability even in the presence of very fine silt or clay (Figure 8 and Figure 11).

Another important input parameter that the presented model requires is porosity of the pure endmembers ($n_L$ and $n_f$). The advantage of using the measured values for $n_L$ and $n_f$ is that it provides an accurate calibration points for the model which in turns result in accurate prediction. In case that measurements for $n_L$ and $n_f$ are not available, one may use existing empirical relations to predict porosity (or void ratio) of endmembers based on their particle size analysis and morphological characteristics such as roundness [68,69]. Such empirical relations are however developed for uniform sands and cannot be applied to pure fine silt or clay due to the complex packing of the latter. That is, the porosity of endmembers can be predicted using existing empirical relations [68,69] and then used as inputs in the current model when only dealing with binary mixtures composed of graded sands. It is important to mention that such approach can reduce the accuracy of the mixtures prediction due to the error introduced in the predicted $n_L$ and $n_f$ (percentage error varies between 4.9% and 8.5% [68]). Moreover, using empirical relations to find the porosity of endmembers would restricted the model applicability as this is not feasible in the presence of very fine silt or clays. Future work may consider developing relations for predicting the pure clay porosity based on other measurable properties such as plasticity index and particle size and morphology. Such development is outside the focus of this current work which focuses on the porosity of binary mixtures rather than pure uniform packing of endmembers.

The current model also requires permeability or hydraulic conductivity, K, of the pure endmembers as inputs in the model since it is needed for the estimation of representative particle sizes ($D_{repL}$ and $d_{repF}$ in Equations (19) and (20)). Measuring K of the endmembers is probably the most time consuming task in applying the current model but can ensure accurate prediction of representative particle sizes and K for the binary mixtures as function of fines fraction. In case K measurements are not available for endmembers, representative particle size might be estimated based on particle size distribution of the sediment [22,23,70,71]. There are different formulations to predict representative particle size based on particle size distribution but they frequently provide an estimate that is close to the effective grain size D₁₀ [23,71,72,73]. Several studies showed that D₁₀ can be used to obtain representative particle size and thus estimate K for sandy soils [71,72,73]. Following such approach, D₁₀ might be used to find the representative particle size of the endmembers as an alternative to using measured K. In order to evaluate how such an approach impacts the accuracy of K prediction of the binary mixtures, D₁₀ of the pure sand and pure fines (whenever available from the examined published work [7,23,25,32,40,48,53,57,58]) were used to represent $D_{repL}$ and $d_{repF}$, respectively. The prediction of binary mixed soil K in this case does not require K of endmembers as an input. Figure 12 displays a comparison between the accuracy of the model when using K of endmembers as inputs (black circles) and when rather using D₁₀ (red diamonds). Overall, the use of endmembers K to estimate D_rep results in much better prediction of binary mixtures K (R² = 0.88; MAE = 0.032) compared to the use of D₁₀ (R² = 0.18; MAE = 0.11). The results show that using D₁₀ = D_rep works well for mixtures characterized by high K (sand graded mixtures with no clay). Nevertheless, significant underestimation of K is observed for mixtures involving clays (low K) if D₁₀ is used instead of K to define D_rep. This is consistent with the fact the use of D₁₀ and particle size distribution to estimate D_rep and K was developed for sands rather than fines [23,71,72,73]. This results suggests that D₁₀ can be good alternatives to using K of large sand particles to estimate D_rep but such approach does not work for the fines in case of clay or fine silt. Alternative approaches to find representative particle size for clays can be considered in future development.

Porosity (or void ratio) is an important parameter that can impact various properties of binary mixed soils including liquefaction resistance [49], shear strength [4,45], and compressional velocity [32]. Therefore, estimating the void ratios of the binary mixed soils based on the current model can assist and improve the evaluation of such properties. For example, porosity has inverse relationship with compressional velocity, thus correcting the underestimation of porosity for mixtures involving clay (Figure 5) can help avoid the overestimation of compressional velocity as pointed out by [32]. Additionally, the current model can provide information about the critical fines content at which the mixture micro-geometrical arrangement shifts from sand-dominated to fines-dominated regime. Such information can be useful in understanding the liquefaction resistance and shear strength behavior of binary mixed soils as function of fines fraction [45,49]. For instance, liquefaction resistance is controlled by relative density (which is function of void ratios) but the resistance is higher in sand-dominated mixtures compared to fines-dominated mixtures at the same relative density [49]. Nevertheless, this study focuses on the prediction of porosity (or void ratio) and hydraulic conductivity. The use of the developed model and predicted void ratios or hydraulic conductivity in evaluating liquefaction resistance, shear strength or compressional velocity of mixed soils is outside the scope of the current work and can be considered in future development.

It is worth mentioning that the current developed model is applicable for sand-fines mixtures. The formulations and values of input parameter have been calibrated based on mixtures of coarse sands and fines (fine silt, or clay). The model is not intended for application in case of idealized and synthesized particles such as spheres and cylinders that can be of great significance to industrial and chemical engineering. The model formulations (Equations (14)–(18)) and values of input parameters (e.g., M, p, a, and S) would need to be re-calibrated if the current model is to be applied to such particles. There are also other packing models developed specifically for such synthesized particles that can rather be used [10,74,75,76,77,78].

5. Conclusions

This study presents a unified packing model for the purpose of predicting the porosity and hydraulic conductivity of binary mixed soils as function of fines fraction. The model modifies the formulations developed originally by Zhang et al. [25], to achieve three main improvements: (1) developing a correction function that is inclusive of wide range of binary mixed soils covering the whole range of $d_f/D_L$, (2) it takes into account the cohesive packing behavior of the fines (clay or fine silt), and (3) it accounts for the impact of clay swelling. The presented model requires input parameters that can be obtained from the pure unmixed endmember including porosity, hydraulic conductivity and grain size. The model shows significant improvement in predicting porosity compared to existing models that frequently underestimate the porosity especially for mixtures with very small (<0.05) or large (>0.35) $d_f/D_L$. The output of the model can also be used to improve the estimation of hydraulic conductivity of the soil mixtures. It was found that using the current packing model can reduce the mean absolute error in estimating hydraulic conductivity by 50% when compared to the original model developed by Zhang et al. [25]. The current model can be used to optimize concrete and soil mixing design to achieve a desirable porosity and hydraulic conductivity values.

The presented model requires the porosity and hydraulic conductivity of endmembers as input parameters. An alternative could be the use of empirical relations to find porosity and hydraulic conductivity of endmembers as function of particle size analysis and morphological characteristics. Such alternative can provide reasonable results for uniform sand endmembers but further work is needed to develop an approach for the cases involving fine silt or clay. The current study focused on developing an improved packing model for the purpose of predicting porosity and hydraulic conductivity of binary mixed soils. Further work may consider examining the implications of such model for understanding and predicting other important properties including liquefaction resistance and shear strength.