### 数学代写|概率论代写Probability theory代考|STAT4028

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|概率论代写Probability theory代考|Influencing Factors on $S W C C$

Several experimental techniques or apparatuses have been developed to measure the SWCC, for example, filter paper method [64], different types of pressure plate apparatuses [54, 65-68] and modified direct shear apparatus [69]. These methods or apparatuses have their own features, and they may be applicable to different cases, for example, measurable range of suction, drying or wetting path.

Based on a large number of experimental results, researchers have found that a number of factors have significant influence on SWCC, such as the type of soil and mineralogy, grain-size distribution, initial void ratio, plastic limit, liquid limit, compaction energy, stress state and stress history, initial water content, clay fraction and temperature. Marinho and Chandler [70] indicated that there was a distinct relationship between the slope of SWCC and the liquid limit. Vanapalli et al. [71] studied the effect of soil structure and stress history on the SWCC based on the tests and found that the beginning of the SWCC was affected greatly. Kawai et al. [72] investigated the dependency of SWCC on initial void ratio. Uchaipichat and Khalili [73] conducted experimental work on the thermo-hydro mechanical behavior of an unsaturated silt. Iyer et al. [74] demonstrated that the initial water content only influenced the initial stage of the $S W C C$, and the specimen thickness had no significant influence on the shape of SWCC. Wang et al. [33] studied the effects of sample dimensions and shapes on the testing duration of SWCC. Elkady et al. [75] found that the SWCC of sand-natural expansive clay mixtures was mainly dependent on the clay content and compaction state.

Among these influencing factors, the effect of void ratio or soil compaction degree is the most significant in geotechnical engineering. Due to the limitations in measuring SWCC, for example, long-term cycle, high cost and discrete data points, the calculation model, which can simulate the effect of void ratio on SWCC, should be established to avoid the tedious experimental work. Considering that the shape and location of SWCC in the space depend on the parameters in SWCC model, some researchers have attempted to construct the relations of model parameters with the void ratio $[76,77]$. There are some researchers trying to introduce the void ratio based on the existing empirical SWCC models [78-80].

## 数学代写|概率论代写Probability theory代考|Empirical Equations

Numerous empirical equations have been proposed to fit the laboratory data, and they contain two or three fitting parameters. Table $1.1$ lists several commonly used equations for $S W C C$. Leong and Rahardjo [81] reviewed and evaluated five popular SWCC equations and found that the Fredlund and Xing [82] equation possesses the best fitting ability. These models can provide the good fitting results with the measured data using the least-squares regression method [82]. They can deduce the whole SWCC using the limited test data, and the estimated results are suitable to be used in the computer model. Zhang and Chen [83] later extended the Fredlund and Xing [63] model and the van Genuchten [84] model to describe bimodal and multimodal SWCCs.

From the view of these empirical equations, some researchers have also investigated the relationships between the associated fitting parameters and the soil basis properties and established the estimation equations. Zapata [86] analyzed the three fitting parameters $(a, k, m)$ in Fredlund and Xing [63] equation using the grain size diameter $D_{60}$ and the product of the percent passing and the plasticity index for different soil types, that is, granular nonplastic materials and plastic materials, respectively. Chin et al. [87] based on regression analysis and one-point SWCC measurement proposed a simplified technique to estimate the $\mathrm{SWCC}$, and the relationships of four fitting parameters $\left(a, k, m, \varphi_{r}\right)$ in Fredlund and Xing equation [63] with basic soil properties were formulated for both coarse- and fine-grained soils, respectively. Torres [88] also obtained the correlation of the fitting parameter $a$ with the grain-size diameter $D_{I O}$ for granular materials, and the other two fitting parameters $k$ and $m$ were found to be related to $a$ and the relevant relation equations were also built.

## 数学代写|概率论代写Probability theory代考|Pedotransfer Functions

The physico-empirical models are built to estimate the water content based on the particle-size distribution and the characteristic of soil-pore structure. Based on the observation of similarity between the shape of water retention curve and particlesize distribution, Arya and Paris [96] first proposed a physico-empirical model to estimate the SWCC using the particle-size distribution and unit weight. The model adopted three assumptions of soil particles and pore structure to deduce the estimated equation of the water content, and an important empirical parameter $\alpha$ was introduced in the model. In order to improve this model, Arya et al. [97] proposed the method of back analysis to determine $\alpha$. Zhuang et al. [98] derived an analytical model using nonsimilar media method to estimate SWCC based on the measured soil physical properties, for example, particle-size distribution, unit weight and bulk density.
This model does not require measuring $S W C C$ in advance; it can make use of most of the available data. However, the physico-empirical models cannot obtain the close predictions for soils where aggregation, cracking and root effects may be pronounced. Otherwise, the model works reasonably well [96].

The fractal theory was first introduced to estimate the SWCC by Tyler and Wheatcraft [99], and then they derived the other fractal model of SWCC based on the Sierpinski carpet [100]. This fractal model considered only the fractal characteristics of the pore space but not the mass. At present, many models have been proposed for the fractal characteristic of SWCC [101-103]. They have been used to describe the water retention behavior and the variation of hydraulic conductivity in soils with different textural structures [104-107].

The disadvantage of the fractal models is that the fractal scaling considers only the effects of tortuosity of pore lengths, but not the effects of other factors on the water retention property, such as organic matter content, packing density, the chemical characteristic of particle surface and fluid property.

## 数学代写|概率论代写Probability theory代考|Pedotransfer Functions

Tyler 和 Wheatcraft [99] 首次引入分形理论来估计 SWCC，然后他们基于 Sierpinski 地毯 [100] 推导出了另一个 SWCC 分形模型。该分形模型只考虑了孔隙空间的分形特征，而没有考虑质量。目前，针对 SWCC 的分形特征提出了多种模型[101-103]。它们已被用于描述具有不同质地结构的土壤中的保水行为和水力传导率的变化 [104-107]。

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## MATLAB代写

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