## 1 Introduction  
Soft grounds have presented themselves as a challenge to engineers due to their low bearing capacity, high compressibility, low shear strength, and permeability. The existence of such a challenging soil leads to massive settlements, large differential post-construction settlement, large lateral flow, and slope failure, which subsequently require a tedious evaluation process [1]. The long-established solution for soft ground improvement includes chemical admixture and sand drains. Over the past century, there has been a significant upsurge in soft-ground enhancement methods. Since geosynthetics were introduced, the rapid development of infrastructural projects on soft soils with prefabricated vertical drains (PVDs) has been observed. A substantial number of studies have reported the effects of applied PVDs to mitigate the problem of soft ground. The installation of PVDs is the most economical technique of soft ground improvement since it can accelerate the settlement of the ground foundation and enhance the safety factor of the embankment slope. This technique has recently been considered the most favourable as a result of the mentioned advantages. Several factors affecting the progress of consolidation with vertical drains were examined over the last few decades with the aim of improving the accuracy of the numerical prediction method. The main focus of investigations conducted by several researchers was on the smear effect on consolidation processes (e.g., [2; 3; 4]). If the smear effect was not considered in numerical methods, the time-settlement curve would be far below the measured settlement response [5]. The primary goal of the smear effect is the identification of the parameters employed in order to obtain the equivalent horizontal and vertical permeability. Traditionally, the ratio of horizontal permeability in the undisturbed area to the smeared zone (\(\eta\)) was determined based on the typical literature values. Researchers and designers have been faced with challenges when identifying such parameters, as laboratory processes and experiments prove to be intricate and complex [5; 6]. The proposed \(\eta\) value in the literature is between 1.3 and 10, as reported by a number of researchers [7; 8]. It has been reported that the best way to estimate these values is through back-analysation of the local case histories [6]. Despite this method being widely used, consideration of the solution to equivalent permeability has received little attention [7; 8; 9]. In this regard, it can lead to prediction accuracy problems. In addition, several studies have also used the back analysis method of ground settlement behaviour, but the comparison of the results obtained from the finite element method and measurements in the field is rarely confirmed [10; 11]. Therefore, it is essential to use both methods at once to confirm that the smear effect parameters that were previously studied are adequate. Recently, [PERSON] et al. found that the back-analysis method needs to be adjusted with the analytical approach to determine the permeability ratio in plane strain conditions [12]. [PERSON] et al. used the back-analysis method to predict the settlement without considering the equivalent permeability in developing the finite element model [13]. They also emphasized and suggested that equivalent permeability and smear effect be considered for future studies. On the other hand, [PERSON] and [PERSON] did not succeed in accurately predicting excess settlement using back-analysis and finite element methods because they did not consider several geotechnical parameters, including the appropriate equivalent permeability. In principle, the consolidation of soft soil around the PVD is axisymmetric. In an effort to reduce computational time, plane strain conditions are often used for finite element analysis. For this purpose, several researchers have developed a technique for calculating the equivalence between plane strain and axisymmetric conditions [4; 14].