## 1 Introduction  
Compaction is considered as a technique for improving the engineering properties of soils, which reduces their permeability and increases their strength [1]. The maximum dry density and optimal moisture content of the soil in both earth filler dams and the foundation backfill for industrial and civil construction projects should be determined. The California Bearing Ratio (CBR) is a critical geotechnical parameter for determining the appropriate thickness of flexible pavements made from subgrade soil. The CBR test is the method most commonly used in developing countries for pavement design [2]. The compaction characteristics and CBR are important due to their implications in transportation construction. Under a given compaction energy, the relationship between the moisture content and dry density (the compaction curve) can be used to represent the compaction characteristics of the soils. The abscissa and ordinate of the peak point of the compaction curve represent the optimal moisture content and maximum dry density, respectively. In early studies, some theories were proposed to explain the compaction characteristics. [PERSON] [3] thought that water has a dual effect of suction and lubrication; as water increases, the suction decreases. Water also lubricates the particle interaction, giving rise to increased dry density up to the maximum. [PERSON] [4] thought that soils undergo hydration, lubrication, swelling and saturation stages of wetting during compaction. Similar to [PERSON], [PERSON] [5] proposed an explanation based on unsaturated soil theory. They thought that soil suction reduces with increasing moisture content. As soil reaches the optimal moisture content, air is trapped and cannot escape, as in the case of the dry density optimum. The compaction energy beyond this moisture content increases air pressure, thereby reducing the effectiveness of compaction and dry density. Many scholars proposed models to quickly determine the compaction curves according to massive data [6; 7; 8]. In the present research, scholars focused on the compaction and CBR characteristics of mixtures of soil and other materials [9; 10; 11]. However, as laboratory CBR tests are laborious and time-consuming, many attempts have been made in the past to predict CBR from simple physical properties such as plasticity index (PI) [12], liquid limit (LL), plastic limit (PL) [13], and strength [12]. The compaction degree K (K is defined as the ratio of dry density to the maximum dry density, as a percentage) of soils to be used as sub-base embankment fillers should exceed 90%, and the CBR value should be at least 3%. Expansive soil is characterized by swelling and shrinkage, over-consolidation and fissures, causing soil particles to be unstable with respect to one another. It is risky to use such material as the filler material when laying the foundations for highways, railways and other infrastructure. Expansive soils, especially strongly and moderately expansive soils, cannot be used directly as embankment fillers [14]. Over the past few years, scholars have tended to use various solid wastes as modifiers for expansive soils, such as lime [15], cement [16; 17], fly ash [18], steel slag [19], and polymers [20]. However, these modifiers cause environmental pollution, are costly, and necessitate complex construction procedures. In certain situations (under hydrological conditions and for applications with lower strength requirements, such as for low-grade roads), weakly expansive soil can satisfy the requirements for use as an embankment filler [21]. In non-cold areas, weakly expansive soil can act as an embankment filler or even roadbed filler in scenarios where it is protected from water (e.g., in a closed-over setup with the expansive soil as a core filler, or with an expansive soil layer wrapped in a geomembrane). However, due to variations in geology and climate, significant differences exist between expansive soils in cold and non-cold areas.