**1. Introduction**

Marine soft soil is widely distributed throughout the world, and the performance of the soil varies as a result of differences in geological origin, occurrence law, and composition. Therefore, the engineering characteristics that emerged from these factors reflect substantial spatial and temporal variability and regionality. For example, the Canadian coastline exceeds 200,000 km [1], and therefore marine soft soil represented by Leda soft soil is widely distributed in Canada, which has the significant characteristics of marine soft soil [2]. Norway's Drammen marine soft soil also has characteristics of marine soft soil, but Sangrey showed that the engineering properties of Drammen soft soil were different from those of Leda soft soil [2–4]. Other soft soils, such as London soft soil [5], Mexican soft soil [6], Japanese soft soil [7], Busan soft soil [8], Shenzhen soft soil, Shanghai soft soil and so on, all exhibit different marine soft soil characteristics [9–16]. The engineering characteristics of marine soft soil make the problem of consolidation and settlement of marine soft soil foundation complicated, but the one-dimensional consolidation theory by Terzaghi is not suitable for the analysis of complex marine soft soil foundation consolidation problems. The main reason for this is that Terzaghi's

one-dimensional consolidation theory assumes that the drainage boundary is perfectly permeable or perfectly impervious, and the pore water pressure equations were as follows [17]:

$$u(0,t) = 0\tag{1}$$

and:

$$u\_l(h, t) = 0\tag{2}$$

In reality, the sand layer of the top surface of the foundation treatment and the lower layer of the bottom surface are neither perfectly permeable nor perfectly impervious, and they are often somewhere in between. Previous studies showed that permeability of drainage boundary will have a major impact on the final calculation results [18,19]. In view of the unreasonable drainage boundary conditions of consolidation theory, Gray [20] took the lead in conducting research. Based on one-dimensional consolidation theory by Terzaghi, Gray [20] proposed a semi-permeable boundary theory, and the pore water pressure equation was as follows:

$$
\mu\_l(h, t) + h u(h, t) = 0 \tag{3}
$$

Later, Schiffiman and Stein [21] conducted a detailed study on the one-dimensional consolidation problem with a semi-permeable boundary. Huang [22] discussed and promoted the semi-permeable boundary theory. While the one-dimensional consolidation theory of semi-permeable boundary is more practical than one-dimensional consolidation theory by Terzaghi, it is relatively simple, and it assumes that the soil layer is homogeneous elastomer without considering multi-layer soil. According to this, Xie [23] proposed a one-dimensional consolidation theory for a double-layer foundation with a semi-permeable boundary and solved and discussed it. The results showed that the semi-permeable boundary conditions and the stratification of soil can bring the foundation consolidation analysis closer to the actual situation [23]. Based on the research of Xie [23], Hu and Xie [24] studied the one-dimensional consolidation problem of the semi-permeable boundary under the gradual load of a multi-layer elastic foundation and systematically analyzed the layered soil. Hu and Xie [24] studied the effect of semi-permeable boundary conditions, soil properties, and loading rate of external load on pore pressure distribution and total average consolidation. However, the above scholars considered the foundation soil to be a homogeneous elastic soil layer and a static load [23,24]. For this reason, Fang et al. [25], Lin et al. [26], Wang et al. [27], Wang et al. [28], Cai et al. [29], Li et al. [30], Wang and Xia [31], and Zheng et al. [32] carried out consolidation analysis of viscoelastic soil layers under semi-permeable conditions and dynamic load. According to the one-dimensional consolidation equation of the semi-permeable boundary proposed by Gray [20], scholars carried out consolidation analysis under different loading modes and non-Darcy's law conditions [33,34]. However, the one-dimensional consolidation equation based on the semi-permeable boundary proposed by Gray [20] is difficult to solve and not easy to generalize. Combined with the above problems, Mei et al. [35] proposed and solved a one-dimensional consolidation equation for continuous drainage boundary in order to solve the contradiction between the boundary conditions and the initial conditions of the one-dimensional consolidation equation by Terzaghi. The pore water pressure equations by Mei et al. [35] are as follows:

$$u(0,t) = q(t)e^{-bt} \text{ (top drainage)},\tag{4}$$

and

$$u(\mathfrak{D}, t) = q(t)e^{-ct} \text{ (bottom dynamic)}.\tag{5}$$

The continuous drainage boundary is a time-dependent interface boundary that varies between perfectly pervious and impervious conditions. The boundary pore pressure changes exponentially with time. By adjusting the parameters *b* and *c* related to the undisturbed soil, the drainage properties of the top and bottom drainage surfaces of the soil layer can be controlled. For example, *b* can

tend to infinity, and thus the top surface of the soil layer is undrained. Subsequently, scholars carried out research on continuous drainage boundary theory. Zong et al. [36] and Zheng et al. [37] established a generalized Terzaghi's consolidation theory under double-sided asymmetric continuous drainage boundary conditions. Cai et al. [38] developed a subroutine within ABAQUS program to verify the solution of the one-dimensional consolidation equation of the continuous drainage boundary. Wang et al. [39] showed that the continuous boundary has good applicability and extends to the semi-analytical solution of one-dimensional consolidation of unsaturated soil. Feng et al. [40] established the one-dimensional consolidation equation for the continuous drainage boundary and studied the contribution of a soil's self-weight stress. Sun et al. [41] established a general analytical solution for the one-dimensional consolidation of soil for the continuous drainage boundary under a ramp load. Zhang et al. [42] analyzed the excess pore water pressure and the average degree of consolidation under the continuous drainage boundary conditions and discussed the effect of the drainage capacity of the top surface, the smear effect, and the well resistance on consolidation. However, there are few reports on the one-dimensional consolidation theory with the continuous drainage boundary of double-layer soil and its application in marine soft soil engineering.

Domestic and foreign scholar have done a lot of research on the algorithm for the mathematical modeling of consolidated seepage. In 1949, Van Everdingen and Hurst [43] proposed the Laplace transform method. In 1968, Dubner and Abate [44] proposed the Laplace numerical inversion method. Subsequently, Durbin [45] improved the Dubner and Abate algorithm. In 1970, Stehfest [46,47] proposed the Stehfest algorithm because the algorithm is easy to program, has few parameters, has fast calculation, does not involve complex numbers, and has high stability. It has thus been widely used in engineering. It is worth mentioning that the Stehfest algorithm looks like an empirical formula, but it actually has a complicated mathematical background and theoretical derivation. Then, Crump [48] proposed the Crump algorithm, which is based on the Fourier series. Duffy [49] improved the Crump algorithm, avoiding the trigonometric function term. However, because the Crump algorithm does not put forward a method to determine the appropriate attenuation index and truncation term number, it is difficult to apply widely. Therefore, according to the advantages and disadvantages of the above algorithm, an appropriate method should be adopted to carry out the calculation according to the characteristics of the consolidation theoretical model.

Based on an analysis of the above literature, aiming at the characteristics of the marine soft soil double-layer foundation structure and complex drainage conditions, a numerical solution of the improved double-layer foundation consolidation theory considering continuous drainage boundary conditions is presented. In Section 2, the basic equations of the improved consolidation theory is introduced and deduced in detail, and the Laplace transform and Stehfest algorithm are applied in the derivation process, and the improved model is compiled into a program by this paper. In Section 3, the improved model is validated and analyzed by three examples in the literature. This section includes degradation analysis of perfectly permeable boundary conditions and semi-permeable boundary conditions by the improved model. Finally, in Section 4, the improved model is applied to an actual marine soft soil foundation project in Guangxi for application analysis. The settlement and consolidation degree of soft soil foundation are analyzed and compared with the measured data for verification. The conclusions can provide scientific guidance for consolidation analysis of marine soft soil foundation. They also have certain theoretical value and practical significance.
