# Predicting Swell in Clay-Sand Mixtures Used in Liners
## 1 Introduction  
Studies on the identification and quantification of the swell in expansive clays, which have been carried out by many researchers, are either based on direct measurements of the swell potential and the swelling pressure or estimated from index properties. Most of the formulae established, if not all, are considered of local validity. These variations are mainly due to the different clay origins and mineralogical composition [1; 2; 3]. Field measurements and tests for such soils are very expensive and require long-term monitoring, which can extend to years. There is a lack of a simple field or laboratory test that can give reliable and informative results within a reasonably short period. Geotechnical engineers are concerned about the increase and decrease in the volume of clays following wetting and drying. Bentonite is the name for commercial products that are rich in montmorillonite. Sodium Wyoming bentonite has received the greatest attention due to its ability to absorb large quantities of water and swell more than other types. Modeling studies on the mechanical behavior of swelling clays and/or bentonite have been introduced by many researchers [4; 5; 6; 7; 8; 9; 10]. Understanding the mechanisms of volume changes helps to properly quantify and predict swelling behavior. It is quite common in the literature to come across terms of interlayer swelling, interparticle swelling, and interaggregate swelling. The first refers to the condition in which water is introduced between two layers of montmorillonite, causing the basal spacing to increase. This increase is variable and depends on the state of moisture. For dry commercial bentonite, the basal spacing increase is estimated to be in the order of 12.5 to 20 A [9; 11]. The interlayer swellings can double the volume of dry clay [12]. Interparticle swelling or osmosis is computed based on the double-layer theory, which uses the charge and spaces between particles. Interaggregate swelling occurs when aggregations and clusters are formed within the clay fabric. The model proposed by [PERSON] [11] for the prediction of the swelling behavior of bentonite-sand mixtures is an experimental approach suggesting the swell curve to bepresented in the form of the void ratio against the vertical effective stress. The model is generally based on three sets of data: the relationship between the clay void ratio \((e_{c})\) and the vertical effective stress \((\sigma^{\prime}v)\) for bentonite with an appropriate permeant, the relationship between the sand porosity \(\eta_{s}\) and the vertical effective stress of the sand, and the sand porosity of the mixture in a dry state at an appropriate vertical effective stress. This approach is based on the concept of stress partitioning discussed by [PERSON] [9]. The clay void ratio concept used by [PERSON] [11] and [PERSON] et al. [10] considers a two-phase sand-bentonite gel system (Figure 1). In this case, the clay is described in terms of the clay void ratio, defined as  
\[e_{c}\ =\frac{Vw}{Vc} \tag{1}\]  
The conventional void ratio is shown on the left-hand side of the figure. The sand porosity is the total soil volume occupied by the paste and is given as  
\[\eta s=\frac{(Vw+Vc)}{Vt} \tag{2}\]  
where \(Vt=Vw+Vc+Vs\). [PERSON] [11] assumed that the sand porosity of the dry mixture on a graph of \(\eta_{s}\) versus \(\log\sigma^{\prime}v\) was parallel to the line of the pure sand but displaced from the compression line for the sand only. [PERSON] proposed to obtain the sand porosity for a mixture at a given effective stress on the sand. This value was used to obtain the clay void ratio corresponding to the selected stress level via the following equation:  
\[e_{c}\ =\frac{(\eta s(\text{mixture})-x)}{x\ (1-\eta s(\text{mixture})} \tag{3}\]  
where \(x\) is the volumetric fraction of clay in the mixture.