## 1 Introduction  
The simple classification of soil into coarse- and fine-grained is a conventional approach in soil mechanics. However, most natural soils and manmade fills are mixtures of coarse- and fine-grained soils [1, 2]. Recent studies on sand-fine mixtures have shown that the void ratio may not be an appropriate state variable to describe the mechanical behaviour of these soil mixtures [3]. For sand-fine mixtures where sand is the major component and plays a dominant role in the soil skeleton, experimental studies have shown that an addition of a small amount of fines in sand decreases its void ratio, but the stress-strain behaviour essentially remains the same, implying that the void ratio is no longer a suitable state variable to represent the stress-strain response of sand-fine mixtures [4]. The main reason for such behaviour is the inactive contribution of fines to the force chains of the sand matrix. To properly describe the mechanical behaviour of sand-fine mixtures, a new density parameter, the intergranular void ratio \(e_{g}\)[5, 6], has been proposed by considering fines as voids:  
\[e_{g}=\frac{e+f_{c}}{1-f_{c}} \tag{1}\]where \(e\) is the void ratio, and \(f_{\rm c}\) is the fines content as a percentage, presenting the volume of fines when the volume of soil mixture is equal to 1. Similarly, for sand-fine mixtures where fines are the major components and play a dominant role in the soil skeleton, sand grains may not take part in the force chains of the fine matrix. In this case, the volume of sand can be ignored. Based on this assumption, [PERSON] et al. [7; 8] further proposed an interfering void ratio \(e_{f}\) defined by neglecting the volume of sand grains to describe the mechanical behaviour of sand-fine mixtures:  
\[e_{f}=\frac{e}{f_{\rm c}} \tag{2}\]  
where the fines content \(f_{\rm c}\) (the volume of fines) is equal to the volume of soil mixture when the volume of sand grains is ignored. Past studies on sand-fine mixtures mostly focused on quartz sands [9; 10; 11; 12; 13; 14; 15]. However, in the submarine sediment, carbonate sand (CS) is widely distributed along the continental shelves and the coastlines. The engineering properties of carbonate sand have received widespread attention and have been intensively studied [16; 17; 18; 19; 20; 21; 22]. Recent explorations have reported that a large amount of natural methane hydrate was found in the submarine fine-grained sediments containing carbonate materials in the northern South China Sea [23; 24; 25; 26; 27]. Methane hydrate is formed and preserved in locations where there is an ample supply of methane, as well as where the temperature and pressure requirements have met the stability conditions [28; 29]. The occurrence of natural methane hydrate has been verified through core drilling at locations all over the world, and the large number of methane hydrate reserves highlights it as a potential future energy source [30; 31; 32; 33; 34; 35; 36]. As an unconventional energy source, the exploitation of methane hydrate has attracted attention worldwide [37; 38]. By increasing temperature and reducing the pressure, methane hydrate can decompose into methane gas and water. Multiple approaches have been proposed and employed to extract methane from hydrate-bearing sediments, such as depressurisation, thermal stimulation, gas exchange, and other innovative methods [39; 40; 41; 42]. However, geological disasters and engineering accidents could occur from the production of methane hydrate, such as subsea landslides, seabed settlements, mining holes, and production platform collapse [43; 44; 45]. Thus, understanding the mechanical behaviour of methane hydrate-bearing sediments is essential for their safe extraction. Further investigation into the sediment drilled from the northern South China Sea showed that the methane hydrate content was significantly influenced by the amount of carbonate material found in the fine-grained sediment [46; 47; 48; 49].