*3.1. Degradation Analysis of Perfectly Permeable Boundary Conditions by the Improved Model*

Xie made remarkable contributions to the theoretical research of the double-layer consolidation model. Therefore, in order to verify the improved model, it is used the case in Xie's paper [51] to conduct the degradation analysis of perfectly permeable boundary conditions by the improved model. When *b* tends to ∞ and *c* tends to 0, the improved equations can be degraded to Xie's double-layer soil consolidation equation (single-sided permeability). In order to verify the conclusion, the corresponding program is compiled by this paper for calculation using the data for the examples of Xie's paper [51]. The parameters of example in paper [51] are shown in Table 2.


**Table 2.** Study data of perfectly permeable boundary conditions.

Question 1: When the load is applied instantaneously, how long will it take for the average consolidation degree of the foundation to reach 60%?

Answer to question 1: According to the data in Table 2, analysis of the double-layer soil consolidation degree is carried out according to the corresponding program compiled by this paper. The results by the proposed model are compared with those of Xie's model. Figure 2 is the solution graph of Question 1. When the load is applied instantaneously, it takes 55 days to reach the average consolidation degree of 60% according to the proposed method, which is basically consistent with the calculation results of Xie's model. Further analysis shows that the relation curves between time and consolidation degree (the *t*–*U* curve) obtained by the proposed method is slightly different from those of Xie's model. This difference shows that the solution by the proposed model is slightly larger in the early stage and is slightly smaller in the later stage. The analysis shows that this reason is based on the Stehfest algorithm. While the algorithm requires fewer parameters and provides higher accuracy, it also leads to some errors in the inversion data. According to Figure 2, the consolidation curve obtained by the proposed method is basically consistent with Xie's method. *J. Mar. Sci. Eng.* **2019**, *7*, x FOR PEER REVIEW 8 of 18 higher accuracy, it also leads to some errors in the inversion data. According to Figure 2, the consolidation curve obtained by the proposed method is basically consistent with Xie's method.

**Figure 2.** Consolidation curve for the instantaneous loading. **Figure 2.** Consolidation curve for the instantaneous loading.

 Xie's method Suggested method

**Figure 3.** Consolidation curve for single-stage constant-speed loading.

*k1*=1.014×10-8 m/s

*h1*=1 m *Ev1*=8 MPa

*b*→∞ *c*→0 (Only the surface is permeable) Single stage constant speed loading for 70 days

0 20 40 60 80 100 120 140 160 180 200

Time (day)

*h2*=9 m *Ev2*=4 MPa

*k2*=2.028×10-8 m/s

Question 2: When the single-stage constant-speed load is 70 days, what is the average

Answer to question 2: Figure 3 is the solution graph of Question 2 and shows that when the single-stage constant-speed load is 70 days, the average consolidation degree of the foundation after 140 days is about 80%, which is consistent with Xie's solution. Similarly, the analysis of the whole curve shows that the solution by the proposed model is slightly larger in the early stage and is slightly smaller in the later stage. The specific reasons for this have already been explained as those above mention. Through the answers to Questions 1 and 2, the double-layer soil continuous drainage boundary consolidation theory based on the Stehfest algorithm is found to have higher

Degree of consolidation (%)

Question 2: When the single-stage constant-speed load is 70 days, what is the average consolidation degree of the foundation at 140 days? Question 2: When the single-stage constant-speed load is 70 days, what is the average consolidation degree of the foundation at 140 days? Answer to question 2: Figure 3 is the solution graph of Question 2 and shows that when the

**Figure 2.** Consolidation curve for the instantaneous loading.

0 10 20 30 40 50 60 70 80 90 100

Time (day)

*h1*=1 m

*h2*=9 m *Ev2*=4 MPa

*Ev1*=8 MPa

*k1*=1.014×10-8 m/s

*k2*=2.028×10-8 m/s

*J. Mar. Sci. Eng.* **2019**, *7*, x FOR PEER REVIEW 8 of 18

higher accuracy, it also leads to some errors in the inversion data. According to Figure 2, the consolidation curve obtained by the proposed method is basically consistent with Xie's method.

*b*→∞ *c*→0 (Only the top surface is permeable)

The instantaneous loading Xie's method Suggested method

Answer to question 2: Figure 3 is the solution graph of Question 2 and shows that when the single-stage constant-speed load is 70 days, the average consolidation degree of the foundation after 140 days is about 80%, which is consistent with Xie's solution. Similarly, the analysis of the whole curve shows that the solution by the proposed model is slightly larger in the early stage and is slightly smaller in the later stage. The specific reasons for this have already been explained as those above mention. Through the answers to Questions 1 and 2, the double-layer soil continuous drainage boundary consolidation theory based on the Stehfest algorithm is found to have higher accuracy. single-stage constant-speed load is 70 days, the average consolidation degree of the foundation after 140 days is about 80%, which is consistent with Xie's solution. Similarly, the analysis of the whole curve shows that the solution by the proposed model is slightly larger in the early stage and is slightly smaller in the later stage. The specific reasons for this have already been explained as those above mention. Through the answers to Questions 1 and 2, the double-layer soil continuous drainage boundary consolidation theory based on the Stehfest algorithm is found to have higher accuracy.
