## 2 Materials and Methods
### The Models of Elementary Volume of Cohesive/Non-Cohesive Soils  
According to their structure, composition and physical properties, soils are divided into two general groups: dense and loose. Loose soils are of particular interest from a geotechnical point of view, including sandy and clayey soils. In this study, the sandy soils were considered non-cohesive and clayey soils--as cohesive.  
Non-cohesive sandy soils with a solid matrix are characterized by a negligible change in porosity, while fluid saturation of the porous space can vary significantly. The fluid saturation coefficient, to a certain extent, affects the properties of the sandy soils as the basement for structures. Non-cohesive onshore soils in situ usually have a notable fraction of pore fluid even when located above the water table, whereas the amount of pore space without fluid (with air) can be presumed as negligible at offshore sites or below the water table at onshore sites.  
In favor of the above statements is the fact that in single-grained soils like sand, water percolates quickly through the soil, while in massive soils, such as clays, water movement is relatively slow. This is also due to the pore size, as capillary suction increases as pore size decreases, which is why water is held more tightly in micro-process present in clay-dominated soils [35]. This leads to the fact that the clayey soils are often expansive, i.e., natural clay deposits exhibit significant volume changes due to water content variation, while remaining perpetually saturated (i.e., degree of saturation is one). This process is governed by the attraction of bipolar water molecules to the negatively charged clay particles [36; 37]. In turn, the combination of adsorptive and capillary forces in both cohesive and non-cohesive soil makes it hard to gain zero moisture in natural conditions [35].  
Therefore, it is supposed that elementary volume of non-cohesive soil: first, has constant porosity; second, is a three-phase (air-fluid-solid) medium at onshore sites above the water table and two-phase (fluid-solid) medium at either offshore sites or below the water table at onshore sites (Figure 2).  
For cohesive clayey soils, as their fluid saturation coefficient is close to unity, the total water content and the degree of cohesion caused by it are more significant. The amount  
Figure 2: Schematic representation of the elementary volume of cohesive and non-cohesive soil with different moisture values.  
of water contained in cohesive soils depends primarily on the volume of the pore space. Thus, it is supposed that elementary volume of cohesive soils: first, has variable porosity; second, is a two-phase (fluid-solid) medium both at onshore and offshore sites (Figure 2).  
The Equation (6) for averaging the density of porous saturated soils is suitable for cohesive soils when the whole pore space is always filled by the fluid. For non-cohesive soils, it is more suitable to use the following equation for averaging the density:  
\[\rho=\rho_{m}(1+W) \tag{10}\]  
where \(\rho_{m}\) is a density of the solid matrix and \(W\) is a moisture coefficient defined as follows:  
\[W=\frac{S_{r}e\rho_{f}}{\rho_{S}}\left\{\begin{array}{cc}e\
eq const,&S_{r}= 1=const,&for\ cohesive\ soils\\ e=const,&S_{r}\
eq const,&for\ non-\ cohesive\ soils\end{array}\right. \tag{11}\]  
where \(e\) is a void ratio, \(S_{r}\) is a saturation factor (saturated pore volume fraction).  
The governing equation of motion (9) can be rewritten as follows:  
\[\left\{\begin{array}{cc}\left(\phi\rho_{f}+(1-\phi)\rho_{s}\right)\frac{ \partial^{2}d}{\partial t^{2}}+\mu\frac{\partial d}{\partial t}=\frac{\partial \tau}{\partial z}&for\ cohesive\ soils\\ \rho_{m}(1+W)\frac{\partial^{2}d}{\partial t^{2}}+\mu\frac{\partial d}{\partial t }=\frac{\partial\tau}{\partial z}&for\ non-\ cohesive\ soils\end{array}\right. \tag{12}\]