Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 410390, 9 pages

http://dx.doi.org/10.1155/2014/410390

## Consolidation by Prefabricated Vertical Drains with a Threshold Gradient

^{1}Institute of Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China^{2}Faculty of Architectural Civil Engineering and Environment, Ningbo University, Ningbo 315211, China

Received 6 August 2014; Accepted 1 December 2014; Published 16 December 2014

Academic Editor: Yuming Qin

Copyright © 2014 Xiao Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper shows the development of an approximate analytical solution of radial consolidation by prefabricated vertical drains with a threshold gradient. To understand the effect of the threshold gradient on consolidation, a parametric analysis was performed using the present solution. The applicability of the present solution was demonstrated in two cases, wherein the comparisons with Hansbo’s results and observed data were conducted. It was found that (1) the flow with the threshold gradient would not occur instantaneously throughout the whole unit cell. Rather, it gradually occurs from the vertical drain to the outside; (2) the moving boundary would never reach the outer radius of influence if , whereas it will reach the outer radius of influence at some time; (3) the excess pore pressure will not be dissipated completely, but it will maintain a long-term stable value at the end of consolidation; (4) the larger the threshold gradient is, the greater the long-term excess pore pressure will be; and (5) the present solution could predict the consolidation behavior in soft clay better than previous methods.

#### 1. Introduction

Preloading with sand or prefabricated vertical drains has been successfully applied to accelerate the consolidation of compressible fine-grained soil deposits for decades. Many authors have developed the consolidation theory for the design of a vertical drain [1–10]. Among these theories, the analytic solutions developed by Barron [1] and Hansbo [3] remain popular and widely used because of their simplicity and ease of use.

All of the above-mentioned solutions assume that Darcy’s law is valid. However, several researchers have provided laboratory evidence that Darcy’s law should be corrected for the effect of the threshold gradient in some fine-grained soil. Miller and Low [11] found that the threshold gradient is the hydraulic gradient where no flow occurs. This seems to be supported by experimental observations reported by Byerlee [12]. Hansbo [13, 14] proposed a non-Darcian flow law, which implied that the presence of a threshold gradient value, when the gradient is less than this value, may not cease but may dramatically decrease the flow rate. On the other hand, there are several studies that refute this conclusion [15–18]. For example, having conducted permeability tests on four natural clays, Tavenas et al. [18] concluded that Darcy’s law was valid in soft clays for hydraulic gradients ranging from 0.1 to 50. However, it is nearly impossible to verify the validity of Darcy’s law with very small gradients.

Although there are some inconsistencies among these studies, it is helpful to examine and assess the impact on the consolidation process due to the existence of the threshold gradient, which was demonstrated by previous experiments. In this study, the flow law of the threshold gradient is assumed to be in which is the coefficient of permeability, is the hydraulic gradient, and is the threshold gradient. When , (1) degenerates to Darcy’s law.

Following equation (1), Pascal et al. [19] obtained a numerical solution using the finite difference method to arrive at approximate analytical solutions for one-dimensional consolidation. Xie et al. [20] presented a general approximate analytical solution for one-dimensional consolidation under a time-dependent loading condition and further explored the movement of the seepage boundary in a similar way to (1). Thus, (1) provides an excellent approximation of the effective flow rate. Importantly, it may be simpler to depict the characteristics of the consolidation problem with a non-Darcy flow by using (1).

Various attempts have characterized the influence of the threshold gradient for one-dimension consolidation in clay. However, limited research has focused on the simulation of consolidation behavior using vertical drains that consider the threshold gradient. Hansbo [3] made some approximations and deduced a simple closed-form solution for radial consolidation using vertical drains assuming that Darcy’s law is valid. Following the approach of Hansbo [3], a general analytical solution with a threshold gradient for radial consolidation by vertical drain is developed in this paper. Afterwards, the influence of the threshold gradient on the consolidation process is investigated.

#### 2. Mathematical Model

##### 2.1. Basic Assumptions

Figure 1 presents the schematic representation of an axisymmetric unit cell with the presence of a vertical drain in the center. The vertical drain completely penetrates the soft soil layer, and the top of the layer is freely draining but the bottom is completely impermeable. Thus, the length of the drain well is equal to the thickness of the soft layer . is the equivalent radius of the drainage well; is the radius of the influence zone, which is the radius of the soil cylinder that is dewatered by a drain. is the outer radius of the seepage zone. A widespread surcharge loading of is simulated by the instantaneous application to the upper boundary and this was kept constant during the consolidation process. Additionally, and in Figure 1 indicate the radial coordinates and the depth, respectively.