Mathematical Problems in Engineering

Volume 2019, Article ID 5359076, 12 pages

https://doi.org/10.1155/2019/5359076

## Consolidation Analysis of Ideal Sand-Drained Ground with Fractional-Derivative Merchant Model and Non-Darcian Flow Described by Non-Newtonian Index

School of Civil Engineering, Zhengzhou University, Zhengzhou, Henan 450001, China

Correspondence should be addressed to Zhongyu Liu; nc.ude.uzz@uilyhz

Received 1 June 2019; Accepted 19 July 2019; Published 7 October 2019

Academic Editor: Fazal M. Mahomed

Copyright © 2019 Zhongyu Liu 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

To further investigate the rheological consolidation mechanism of soft soil ground with vertical drains, the fractional-derivative Merchant model (FDMM) is introduced to describe the viscoelastic behavior of saturated clay around the vertical drains, and the flow model with the non-Newtonian index is employed to describe the non-Darcian flow in the process of rheological consolidation. Accordingly, the governing partial differential equation of the ideal sand-drained ground with coupled radial-vertical flow is obtained under the assumption that the vertical strains develop freely. Then, the numerical solution to the consolidation system is conducted using the implicit finite difference method. The validity of this method is verified by comparing the results of Barron’s consolidation theory. Furthermore, the effects of the parameters of non-Darcian flow and FDMM on the rheological consolidation of ground with vertical drains are illustrated and discussed.

#### 1. Introduction

In the southeastern coastal and inland areas of China, soft clay foundations exist widely. Consolidation with vertical drains has been proved to be an effective method for the reinforcement of such soft soil foundations and has always been an important research topic in the field of soil mechanics. In 1948, Barron [1] established the basic differential equation for axisymmetric consolidation and derived the analytical solution under the extreme assumptions of equal strain and free strain, which has been widely applied in practice to predict the consolidation behavior. Yoshikuni and Nakanodo [2] perfected the definition and assumption of free strain, introduced the flow continuity condition of well column, and deduced the theoretical solution of sand-drained ground considering the well resistance under the condition of free strain. Hansbo [3] and Xie and Zeng [4] developed Barron’s theory, respectively, deduced the analytical solutions that satisfy the basic equation of radial consolidation and definite conditions, which further perfected the analytical theory of consolidation with sand drains under the condition of equal strain. Since then, a group of researchers [5–12] have conducted a series of studies on the consolidation problems of sand-drained ground and, respectively, provided the consolidation solutions of sand-drained ground under different conditions, which took into account the influence of variable load, partially penetrated vertical drains, stratification of the ground, well resistance, smearing effect, and other factors. However, in those above studies, the deformation of the soil skeleton was modeled as a linear elastic material, and the rheological properties of soft clay were neglected. Therefore, there sometimes exists notable difference between the theoretical predictions of consolidation and the measured values.

In order to consider the rheological properties of the soil layer, many researchers use various component rheological models, which are the combinations of Hooke’s elastic element (spring) and Newton’s viscous element (dashpot), to simulate the deformation of soil skeleton structure and establish the corresponding rheological consolidation theory considering the viscoelasticity of soft soil, but most of them have been focused on the one-dimensional (1D) consolidation [13–17]. For the consolidation with vertical drains, it can be traced back to the 1960s in China, Qian et al. [18], extended the Barron’s theory to the viscoelastic soil based on the standard Merchant model (SMM). Subsequently, Zhao [19] used the generalized Voigt model to derive the widely applicable analytical solution of sand-drained ground which considered the viscoelasticity of the soil. Liu et al. [20] gave the analytical solutions of viscoelastic theory under the condition of free strain and equal strain based on the SMM and established a relatively perfect viscoelastic consolidation theory considering the influence of well resistance and smearing. Wang and Xie [21], Liu et al. [22], and other researchers also considered the influences of different loading forms, unpenetrated soil layers, and semipermeable boundaries based on the SMM, and further revealed the influence of viscoelastic parameters on the consolidation mechanism. Nevertheless, in the past few decades, many authors [23, 24] pointed out that the fractional-derivative model (FDM) is more suitable for describing the viscoelastic behavior of various real materials. Gemant [25] verified the applicability of the fractional-derivative constitutive model to viscoelastic soil through experiments, which provided the basis for the development of fractional order in geotechnical engineering. Subsequently, some researchers [26–31] began to use the spring-pot element (or Abel’s dashpot) in which the first derivative of strain with respect to time is replaced by the fractional-derivative, the fractional-derivative kelvin model or Merchant model, in which the dashpot element is replaced by the spring-pot element, to fit the rheological experimental data of different soft soils. Liu and Yang [32], Liu et al. [33], and Wang et al. [34, 35] also introduced the fractional-derivative kelvin model or Merchant model to study the 1D consolidation of viscoelastic soil. Zhang et al. [36] put forward the fractional-derivative merchant model (FDMM) to study the interaction between rectangular plates and viscoelastic soil layers. However, to the author’s knowledge, the consolidation of viscoelastic soil with vertical drains has been seldom simulated using a fractional-derivative model.

In addition, many permeability tests [37, 38] indicate that water flow in soft soil may deviate from Darcy’s law under low hydraulic gradients. Hansbo [37] established a segmented function model to describe non-Darcian flow. Liu and Jiao [39] also introduced this flow model to modify the traditional consolidation theory with sand drains. However, Hansbo’s model is represented by a piecewise function whose cut-off point is not well determined, and its parameters are relatively more. It is believed that the behavior of non-Newtonian fluid of pore water in soft clay is the main factor for the occurrence of non-Darcian phenomenon. Therefore, Swartzendruber [40] proposed a continuous exponential flow equation describing non-Darcy law, and the expression iswhere is the flow velocity; is the hydraulic gradient; and are the slope and intercept of the asymptote of the -*i* curve, respectively, and is similar to the permeability coefficient of Darcy’s law; and is the non-Newtonian index. If becomes zero, non-Darcian flow equation described by equation (1) will degenerate into Darcy’s law.

The mathematical description of this flow model with non-Newtonian index is relatively simple, and it has fewer parameters and easy to determined. Moreover, it has a good fitting with the experimental data of Hansbo [37]. Based on equation (1), Li et al. [41] preliminarily analyzed the influence of its parameters on the 1D consolidation behavior of soft soil layer.

In order to investigate the applicability of the fractional viscoelastic model in the consolidation theory of sand-drained ground, the FDMM based on the fractional derivative defined by Caputo [42] is introduced to describe the rheological properties of saturated clay. And the flow model with the non-Newtonian index, i.e., equation (1), is considered to describe the relationship between flow velocity and hydraulic gradient. Then, under the assumption of free vertical strain, the partial differential governing equation of the ideal sand-drained ground is derived, and its numerical solution is obtained by the implicit finite difference method. At last, the influences of the parameters of non-Darcian flow and FDMM on the rheological consolidation process are investigated.

#### 2. Fractional-Derivative Merchant Model

A spring-pot element (see Figure 1) is defined by Koeller [43], and it describes the stress-strain relationship as follows:where *E* is the elastic modulus; is the viscous time of the viscoelastic body; *F* is the viscosity coefficient; and represents the fractional differentiation of Caputo’s definition, which is defined aswhere is the fractional order, *t* is the elapsed time after loading, and is the Gamma function. If or , the spring-pot element is, respectively, degenerated into a spring element with elastic modulus *E* or a dashpot element with viscosity coefficient *F*. Therefore, if takes a value between 0 and 1, the spring-pot element will represent the mechanical behavior of the viscoelastic materials.