Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2768952, 7 pages

https://doi.org/10.1155/2017/2768952

## Segmental Modification of the Mualem Model by Remolded Loess

^{1}Department of Airport Engineering and Architecture, Air Force Engineering University, Xi’an 710038, China^{2}School of Economics and Management, Chang’an University, Xi’an, Shaanxi 710064, China

Correspondence should be addressed to Le-fan Wang; moc.qq@770378923

Received 1 November 2016; Accepted 26 December 2016; Published 15 January 2017

Academic Editor: Konstantinos Karamanos

Copyright © 2017 Le-fan Wang 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

The measured diffusion coefficient and soil-water characteristic curve (SWCC) of remolded loess were used to modify the Mualem model for increasing its accuracy. The obtained results show that the goodness of fit between the Mualem model and the variable parameter-modified Mualem method comparing with the test results was not high. The saturation of 0.65 was introduced as the boundary to divide the curve of the measured diffusion coefficient into two segments. When the segmentation method combined with the variable parameter method was used to modify the Mualem model, the fitting correlation coefficient was increased to 0.921–0.998. The modified parameters and corresponding to remolded loess were calculated for different dry densities. Based on the exponential function between and dry density and the linear relation between and dry density, the segmentally modified Mualem model was established for remolded loess by considering variation in dry density. The results of the study can be used for directly determining the unsaturated infiltration coefficient and for indirectly determining the SWCC through diffusion coefficient.

#### 1. Introduction

The unsaturated infiltration coefficient is an important content in the study of unsaturated soil and is commonly predicted by using the soil-water characteristic curve (SWCC). Childs and Collis-George [1], Marshall [2], Xu [3], and Mei et al. [4] have achieved lots of beneficial results in the field of unsaturated infiltration coefficient prediction. Mualem [5] proposed a model, widely known as the Mualem model, for unsaturated infiltration coefficient prediction by studying 45 groups of undisturbed soil and remolded soil in 1976, which has been widely applied.

The saturated infiltration coefficient is an important parameter of the Mualem model. Van Genuchten and Nielsen [6] believed that mainly reflects the characteristic of moisture flow in soil pores incurred by external force, while unsaturated seepage is mainly caused by the matrix suction inside the soil mass. Schaap and Leij [7], by analyzing 235 groups of soil sample, stated that the saturated infiltration coefficient in the Mualem model should be replaced by the modified infiltration coefficient and that is one order of magnitude smaller than .

Another important parameter of the Mualem model was originally fixed at 0.5 [5]. However, Yates et al. [8] found the value can be a variable greater than −3.31 by reanalyzing the data of Mualem in 1976. Likewise, Schuh and Cline [9] obtained the value between −8.73 and 14.80 through analyzing 75 groups of data. It is found in these papers [6–9] that setting all the parameters of the Mualem model as fixed values reduces the model’s accuracy in predicting the unsaturated infiltration coefficient. Valiantzas [10] combined Brooks-Corey and Van Genuchten closed-form models improving prediction of conductivity values near the residual water content and saturation. Hu et al. [11] developed a new soil-water characteristic curve model for deformable soils. Zhang et al. [12] improved the accuracy of saturated hydraulic conductivity in the Mualem model.

The unsaturated infiltration coefficient can also be of either the SWCC method or the diffusion coefficient method [13]. Wang et al. [14] and Yao et al. [15] used this method to analyze the infiltration characteristic of loess. Guided by the idea of integrating the above two methods of determining the unsaturated infiltration coefficient, this study conducted a horizontal one-dimensional (1D) seepage test to measure the diffusion coefficient and used SWCCs of remolded loess of different dry densities to modify the Mualem model. Subsequently, this paper proposes a dry density-based, remolded loess-modified Mualem model.

#### 2. Theoretical Basis

##### 2.1. Mualem Model and Proposed Modification of Its Variable Parameters

The Mualem model for predicting the unsaturated infiltration coefficient is presented as follows:where is the saturated infiltration coefficient and is the matrix suction corresponding to the effective saturation .

On the other hand, Hoffmann-Riem proposed a universal model for predicting the unsaturated infiltration coefficient in 1999 as follows:

The present paper set and in (2) in correspondence with the Mualem model. This is equivalent to introducing two variable parameters, and , in the original Mualem model to obtain a diffusion coefficient-modified Mualem model.

##### 2.2. Determination of the Diffusion Coefficient by Horizontal 1D Seepage Test [13]

Without considering the action of gravity or the deformation of soil mass, the differential equation and boundary conditions for moisture movement in the horizontal 1D seepage test without free head are shown as follows:where is the volumetric moisture content (cm^{3}·cm^{−3}), is the diffusion coefficient (cm^{2}·min^{−1}), is the distance from the water inlet (cm), and are, respectively, the initial volumetric moisture content of soil sample and the volumetric moisture content at water inlet (cm^{3}·cm^{−3}), and is the infiltration time (min).

After measuring the distribution of the volumetric moisture content of soil columns at time , can be calculated according to the Boltzmann parameter as . The diffusion coefficient can be calculated as follows:

The relationship among the diffusion coefficient , the SWCC , and the infiltration coefficient is given as

After measuring the diffusion coefficient and the SWCC, the unsaturated infiltration coefficient can be calculated from (7).

##### 2.3. SWCC Model

In order to modify the Mualem model, the first step is to determine the SWCC model. For this purpose, the present paper adopted the Brook and Corey model, as shown below:where is the shape parameter of the SWCC, is the saturated volumetric moisture content, is the residual volumetric moisture content, and is the air entry value.

#### 3. Issues Related to the Modification of Mualem Model

##### 3.1. Calculation Method of Modified Parameters

The unsaturated infiltration coefficient can be calculated after measuring the SWCC and the diffusion coefficient in the test. The Mualem model was modified directly based on the measured diffusion coefficient in this paper to reduce the steps of calculating the unsaturated infiltration coefficient and to avoid the accumulation of errors in the calculation process.

A relational expression for the diffusion coefficient and the volumetric moisture content can be obtained, through transformation, by substituting (7) and (9) into (5):

After obtaining the parameters of (10) through curve fitting the diffusion coefficient and volumetric moisture content measured in the horizontal 1D seepage test, we can substitute the SWCC parameters to calculate the modified parameters and of the diffusion coefficient-modified Mualem model.

##### 3.2. Requirements on Horizontal 1D Seepage Test

The diffusion coefficient-modified Mualem model, as discussed in Section 3.1, actually combines the two methods of indirectly determining the unsaturated infiltration coefficient. As indicated by (10), the diffusion coefficient of soil sample was zero when the volumetric moisture content was equal to or lower than the residual moisture content. In the horizontal 1D seepage test, the diffusion coefficient corresponding to the initial moisture content of soil sample was zero, so the combination of these two methods must satisfy

The residual moisture content is an important parameter of the SWCC, so, in the horizontal 1D seepage test, the initial moisture content of soil sample should be precontrolled to be approximately equal to the residual moisture content.

#### 4. Modification of the Mualem Model

##### 4.1. Test Introduction

The loess used in the test was collected from Heping Town in Ningxia Province at a depth of 2-3 m. The horizontal 1D seepage test was carried out in an organic glass tube 186 mm in inner diameter and 1,000 mm in length, and the soil sample was placed in the organic glass tube in layers of 5 cm thickness and was compacted until the set dry density was achieved. A Mariotte bottle was used to supply water for the seepage test, and the free head was controlled at 10 cm. According to the literature [16], the SWCC was determined in the horizontal 1D seepage process with the preembedded TDR moisture meter and Fredlund thermal conduction suction probe jointly. After 10–12 hours of water supply and seepage, the soil sample was sliced and dried to calculate the diffusion coefficient.

The loess used in the test was naturally air-dried loess, and its physical parameters are shown in Table 1. According to the residual moisture content of remolded loess in [9] and the method of measuring the residual moisture content in [17], it was confirmed that the initial moisture content of soil sample had satisfied relevant requirements. Figure 1 shows the relationship between the diffusion coefficient and saturation and Figure 2 shows the measured SWCC.