1. Introduction
In coastal areas, the increase in population and economy induces a surge in land reclamation [
1]. Construction in such low land areas comprised of soft soils is associated with serious post-construction settlement-related problems [
2]. Therefore, settlement control is significant for the soft soil ground, and the ground settlement calculation and forecast play a vital role in all stages of engineering design, construction, and operation [
3].
Terzaghi’s consolidation theory (1923) [
4] states that the consolidation time is proportional to the square of drainage distance. Several techniques were developed to reduce the consolidation time by shortening the drainage distance [
5], such as the vertical drainage wells and utilization of Prefabricated Vertical Drain (PVD). PVD production consumes low energy, leading to fewer emissions and lower costs. Consequently, PVD is considered as an efficient and relatively cheap soft ground improvement technique [
6].
Analytical calculation, numerical simulation, and prediction are commonly adopted in the engineering project design and construction for the settlement evaluation of soft soil ground. However, despite the extensive progress in PVD, several issues regarding the PVD design, installation, and performance are not well understood, including the smear zone dynamics [
7], buckling and clogging of the PVD [
8], and soil inhomogeneity, consequently leading to significant limitations in the existing consolidation and settlement evaluation theories [
9]. For example, the layerwise summation method does not take into account the lateral deformation [
10], and the actual drainage performance of PVDs cannot be accurate when considered with existing analytical calculations during the consolidation and settlement over time.
In addition, numerical simulation has been widely employed as an alternative to avoid the analytical approaches of complex PVD-assisted preloading projects [
11]. However, the prediction precision of numerical approaches is significantly affected by the material constitutive model and parameter selection. Furthermore, these methods are computationally intensive due to complicated boundary and loading conditions [
2]. In particular, proper determination of the smear zone properties of PVDs is challenging [
12]. Moreover, PVD is usually modeled as a solid element in a plane-strain state [
13], resulting in significant computational difficulties and a remarkable difference in the soil-PVD configuration compared to the field [
14]. Therefore, developing a reliable PVD model, considering the drainage consolidation effect without increasing the finite element nodes and excluding the plane-strain equivalent transformation, is greatly needed, but not completely solved yet.
In order to avoid the disadvantages associated with the analytical calculation and numerical simulation, the idea of settlement prediction based on the measured and monitored settlement data has been proposed [
15]. Predictions and forecasts are simple and easy mathematical approaches, and have more accuracy as they fully consider the monitored data, which can reflect the real drainage effect of PVDs and the soil inhomogeneity. Moreover, settlement forecasts can be used for the real-time instruction of construction based on the real-time monitored data [
16].
Most traditional settlement forecasts are based on the Empirical Formula Method (EFM), such as the Three-Point Method (3PM) [
17], Asaoka Method (AM) [
18], Hoshino Nori Method (HNM) [
19], Hyperbolic Method (HM) [
20], and Exponential Curve Method (ECM) [
21], which are widely applied due to their simplicity [
22]. Meanwhile, the Growth Curve Modeling (GCM) is referred to as a logistic curve with an
S-shape having a similar pattern to the ground settlement [
23]. It provides higher accuracy in forecasting settlement [
24], such as Weibull Growth Curve Modeling (WGCM), Pearl Growth Curve Modeling (PGCM), and Gompertz Growth Curve Modeling (GGCM) [
25].
Besides, Grey system theory is commonly used for sequence, topology, and system predictions [
26]. The Grey (1, 1) Model (GM) is a basically and widely used Grey Prediction Model (GPM), especially for sequences with strong exponential regularity [
27]. In addition, the Verhulst Model was aimed to limit the whole development for a real system and effectively describes a specific phenomenon, such as an
S-shaped curve with a saturation region [
28]. The newly combined utilization of Grey and Verhulst models, known as Grey Verhulst Model (GVM), was adopted in the deformation prediction [
29].
Furthermore, Artificial Neural Network (ANN) is a powerful tool for tackling nonlinear problems and has been successfully adopted in optimizations [
30], predictions, and forecasts [
31]. Settlement is a complex nonlinear problem and is often difficult to express using an explicit mathematical expression [
32]. Back Propagation (BP) is the most widely applied model in ANN [
33]. For the BP of ANN (BPANN), in mathematics and computing, the Levenberg–Marquardt (LM) algorithm [
34,
35] is used to solve nonliear least squares problems. Despite the advantages of BPANN, there are still some shortcomings associated with the traditional BPANN with LM algorithm (BPLM) [
33], including: (i) it easily falls into local minimum rather than global optimum; (ii) it has a long training time and slow convergence speed for dealing with a huge amount of data; and (iii) the learning rate must be artificially selected before training. Thus, an improved BPANN with gradient descent of momentum and adaptive learning rate (BPGD) has been proposed [
36].
The above methods are widely and successfully used in settlement predictions and forecasts for various types of soils in different engineering projects, such as in soft soils improved with vacuum-prefabricated vertical drain [
37], foundation pit of artificial fill and silty clay layers [
16], subgrade filled with construction and demolition waste [
10], metro shield tunnel in saturated sand [
38], and the full load-settlement curve of a strip footing [
39]. Although the prediction or forecast accuracy has been significantly improved, there are still errors between the predicted and observed magnitudes of settlements [
40], which may be due to the application limitation and different suitability of each prediction or forecast method for varied situations and time periods. Consequently, it is particularly important to select the appropriate and reliable method according to the different soil properties and project characteristics.
Furthermore, especially, few studies considered settlement forecasting of deep marine soft soil ground improved with Prefabricated Vertical Drain (PVD)-assisted staged riprap filling technique [
41], while studies comparing and optimizing the forecasting methods for such soft ground are lacking. Furthermore, there is no clear mechanism on the settlement of soft soil ground under this technology by reasons of the complexity of influencing factors during construction and limited engineering project data.
Therefore, in this study, by comparing the settlement forecast methods as summarized in
Table 1 [
42,
43,
44,
45], the eight methods from four forecast types were selected considering the creep of soft soils, including HM, ECM, PGCM, GGCM, GM, GVM, BPLM, and BPGD, in which the GVM and BPGD were optimized from GM and BPLM, respectively. Taking Lingni Seawall constructed by PVD-assisted staged riprap filling as an example, forecasts of the short-term, medium-term, long-term, and final settlements at different locations of soft soil ground were performed. Moreover, the forecasting values were compared with each other and with the monitored data. In addition, relative errors and evaluation metrics in regression were analyzed for quantitatively studying the accuracy and reliability of various methods. Finally, the optimal methods of settlement forecasts in different locations and different time periods were studied to provide a reference for the selection of forecast methods for similar engineering projects, and to give a theoretical basis for practical engineering embankment settlement forecasts.
4. Discussion
From the above settlement fitting or training results as shown in
Figure 7,
Figure 8 and
Figure 9, we can find that all the fitting and training results of the eight methods were in good agreement with the monitored data. However, for different time periods in the settlement process and different locations of marine soft soil foundation under embankment construction for this case study, the appropriate and reliable settlement forecast methods were different. The applicability of forecast methods for PVD-assisted staged riprap filling technique can be summarized in
Table 4 for short-term and medium-term settlement forecasting, and for long-term and final settlement forecasting, respectively.
For short-term and medium-term settlement forecasts, at the edge of the berm (Point A), only BPGD was available. This might be due to the fact that the settlement at Point A was not only controlled by the staged loading of the berm, but also obviously affected by the subsequent staged loading of the embankment. Therefore, the forecast methods based on a certain curve regulation, for example, EFMs, GCMs, and GPMs, found it difficult to accurately judge the development trend of settlement at this kind of position. However, the BPGD can feed error rates back through a neural network to make the forecasting more accurate than other methods, and it can solve some shortcomings associated with the BPLM, such as falling into local minimum and the error of artificially choosing the learning rate before training. Moreover, PGCM, GGCM, GM, GVM, and BPGD were available for settlement forecasting at the center of the berm (Point B). In addition, for forecasting at the center of the embankment (Point C), PGCM, GGCM, GVM, and BPGD were available. That is to say, the HM, ECM, and BPLM were not suitable for short-term and medium-term settlement forecasts at all.
For long-term and final settlement forecasts, at the edge of the berm (Point A), ECM, GM, GVM, and BPGD were available. It can be found that the long-term and final settlements at the edge of the berm had a strong exponential regularity. Furthermore, ECM, PGCM, GGCM, GVM, and BPGD were available for Point B at the center of the berm. In addition, at the center of the embankment (Point C), ECM, PGCM, GGCM, GM, GVM, and BPGD were available. It can be observed that, ECM, GVM, and BPGD were appropriate for long-term and final settlement forecasting at all different locations. This is due to the fact that, the ECM and GVM have good exponential characteristics with a saturation region, which can describe the long-term and final settlements of PVD-assisted staged riprap filling foundation in deep marine soft soils well. At the same time, the BPGD was also suitable for long-term and final settlement forecasts at different locations due to its methodological advantages as mentioned above.
By comparing the eight different settlement forecast methods considering the creep of marine soft soils in this case study, the results demonstrated that the applicable conditions of different forecast methods were different, which was mainly due to the different loadings, interaction of foundation soils, and the effect of ground treatment on different foundation positions. Therefore, the settlements at different positions conform to different forms and pattens of variations, and the available forecast methods should be selected according to the specific position and time period. Although these settlement forecast methods have sufficient accuracy, there are still deficiencies in the physical and engineering meanings on forecasting formula and parameters by comparing with analytical calculation and numerical simulation. This will also be a further development direction for the settlement forecast methods in future.
5. Conclusions
From the short-term and medium-term, and the long-term and final settlement results that were forecasted by HM, ECM, PGCM, GGCM, GM, GVM, BPLM, and BPGD methods at different locations (such as at the edge of the berm, and at the center of the berm and embankment) of marine soft soil ground under PVD-assisted staged riprap filling technique, the following conclusions for this case study were obtained.
- (1)
For different time periods during the settlement process and different locations of marine soft soil foundation, the appropriate and reliable forecast methods were different. Only BPGD was appropriate for settlement forecasting at different time periods and at different locations. This may be due to the fact that the BPANN can feed error rates back through a neural network to make the forecasting more accurate than other methods. Furthermore, the BPGD can solve some shortcomings associated with the BPLM, such as falling into local minimum and the error of artificially choosing the learning rate before training.
- (2)
For short-term and medium-term settlement forecasts, the forecast methods can be evaluated by relative errors and evaluation metrics analysis in regression, scilicet when 0 ≥ e > −1%, both the forecasting accuracy and engineering safety are appropriate and reliable. Therefore, for settlement forecasting at the edge of the berm (Point A), only BPGD was available. Moreover, PGCM, GGCM, GM, GVM, and BPGD were available for settlement forecasting at the center of the berm (Point B). In addition, for forecasting at the center of the embankment (Point C), PGCM, GGCM, GVM, and BPGD were available.
- (3)
For long-term and final settlement forecasts, at the edge of the berm (Point A), ECM, GM, GVM, and BPGD were available. ECM, PGCM, GGCM, GVM, and BPGD were available for Point B at the center of the berm. Furthermore, at the center of the embankment (Point C), ECM, PGCM, GGCM, GM, GVM, and BPGD were available. In addition to the BPGD, ECM and GVM were also appropriate for long-term and final settlement forecasts at different locations, because they have an exponential shape with a saturation region, which can well describe the long-term and final settlements.
- (4)
Since this study was a case study of Lingni Seawall, its applicability needs to be further verified based on more engineering cases in order to form a complete methodology and provide a reference for the selection of forecast models for different ground soils under various hydrogeological conditions. For deep soft soil ground with high water content, high compressibility, and high organic content, the forecast method can be selected by referring to the above results, because these soils have creep characteristics and non-negligible secondary consolidation settlement. However, for silty and sandy soil grounds with low water content and low compressibility, the HM may be not suitable, because the predicted values of HM are too large even for soft clay grounds. On the contrary, for silty and sandy soil grounds, BPLM and GVM may be more suitable due to their smaller predicted values for soft clay grounds.
In addition, although BPGD is the most accurate and feasible model for marine soft soil ground, the process of forecast modeling is complex. Therefore, when it is not necessary to establish a unified forecasting model of full-time and full-project monitoring points to describe the settlement process, a simpler forecast method can be selected for each time period and at each monitoring point based on the above results. Furthermore, it is necessary to find a new forecast method or nonlinear logic algorithm with fewer parameters and stronger applicability for this kind of engineering project.
Nevertheless, the goal of case histories is not only to obtain a splendid best fitting, but also to inverse and calibrate the geotechnical parameters, such as compression coefficient and compression index, especially the secondary compressibility coefficient for compressible marine sediments. More accurate forecasted values could provide a new opportunity for more creditable parameter inversion and calibration. Additionally, it is of great significance to study the mathematical meaning of each parameter in each forecasting model, and more importantly, its physical meaning and the analytical relationship between model parameters and soil geotechnical parameters, which will provide an in-depth understanding of the forecasting algorithm and settlement mechanism.