## 2 Case Study
### Modelling an Equivalent Plane Strain  
The methods used for modelling soft grounds treated by PVD in the plane-strain-analysis have been reported in 4 categories. The first category, as [PERSON] et al. and [PERSON] et al. have summarised, is the degree of horizontal consolidation in the plane strain of a unit cell, which has been aligned with the degree of radial consolidation in the axisymmetric model [4,14]. The 1-D drainage element applied in the plane strain analysis may reveal that excessive pore pressure in PVD is equal to zero (u = 0). The second category, which is the effect of the smear zone, is negligible, as the PVD discharge capacity was assumed to be infinite when the quadrilateral macro-element was adopted in numerical methods [15,16]. The third category is an unrealistic method in which the model is developed without improvement through an equivalent vertical permeability [14]. In numerical analyses where PVD was modelled by solid elements of equivalent width and equivalent permeability to soft grounds in the plane strain, a fourth category is frequently applied [11]. In the following study, the fourth category was adopted to develop two numerical models. Model 1 is characterised by an equivalent horizontal permeability, while Model 2 is based on the combination of the equivalent horizontal and vertical permeabilities. Previous studies reveal that the consolidation around PVDs is axisymmetric [17]. However, most numerical analysis methods are based on plane strain. Thus, the equivalent permeability of soft ground needs to be converted from the axisymmetric unit cells to the plane strain in order to allow for numerical analysis [17,18]. [PERSON] et al. proposed the following model, in which the equivalent horizontal permeability's strain is converted from axisymmetric to plane strain before numerical simulation was performed [4]. The model is as follows:  
\[k_{hpl}=\frac{2}{3}\cdot\frac{B^{2}}{D_{e}z}\cdot\frac{1}{\mu}.k_{h}, \tag{1}\]  
In this equation, \(B\) is the width of the plane strain unit cell, \(D_{e}\) is the diameter of the unit cell, \(\mu\) is a factor of the PVD geometry, and \(k_{h}\) is the horizontal permeability of the subsoil. [PERSON] [2] indicated that the value of \(\mu\) can be obtained with the following equation:  
\[\mu=\ln(n/s)+(k_{h}/k_{s})\ln s-0.75, \tag{2}\]  
In this equation, \(n\) is the ratio of the diameter of the unit cell and drainage, \(s\) is the ratio of the diameter of the smear zone and drainage, \(k_{s}\) is the horizontal permeability in the smeared zone, and \(d_{w}\) is the diameter of the smeared zone and the vertical drain, respectively. The diameter unit cell, \(D_{e}\), for PVDs installed with the triangular pattern is calculated via Equation (4). [PERSON] et al. [19] expressed the equivalent drain diameter (\(d_{w}\)) using the following equation:  
\[d_{w}=(w+t)/2, \tag{3}\]\[D_{e}=1.05d, \tag{4}\]  
such that \(w\) and \(t\) represent the width and thickness of the PVD, respectively, and \(d\) is the spacing of the PVDs. The diameter of the smear zone, \(d_{s}\), is calculated by the equations proposed by [PERSON] and [PERSON] [20]:  
\[d_{s}=1.6d_{w}, \tag{5}\]  
Furthermore, [PERSON] et al. [14] expressed the equivalent vertical permeability (\(k_{\text{ve}}\)) equation as follows:  
\[k_{\text{ve}}=\bigg{(}1+\frac{2.5l^{2}}{\mu D_{e}^{2}}\frac{k_{h}}{k_{\text{v}} }\bigg{)}k_{\text{v}}, \tag{6}\]  
where \(l\) and \(k_{\text{v}}\) are the drainage length and the vertical permeability of the subsoil, respectively. The parameters for PVD are listed in Table 2. In order to fit the measured data, the smear effect permeability ratio was varied, and the equivalent horizontal and vertical permeability values were back-calculated and are presented in Tables 3 and 4, respectively. As [PERSON] [21] has suggested, when \(k_{h}\) is equal to \(k_{\text{v}}\), then the \(\eta\) (\(k_{h}\)/\(k_{\text{s}}\)) values are close to unity.