### 数学代写|概率论代写Probability theory代考|STAT7614

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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|Estimation of Relative Permeability Function

The soil dry density has significant influence not only on the SWCC but also on other properties of unsaturated soil. In this section, the proposed SWCC model is applied to estimate the relative permeability function of unsaturated soil. The relative permeability function, $k_{r}$, is the ratio of the effective permeability to the saturated permeability for the unsaturated soil $k_{\mathrm{r}}=k_{\mathrm{w}} / k_{\mathrm{s}}$. Several statistical formulae have been proposed to estimate the relative permeability function from $S W C C$ equation by previous researchers. Fredlund et al. $[42,43]$ proposed using the FX equation to predict the coefficient of permeability for unsaturated soil. The formula has the following integration form:
$$k_{\mathrm{r}}(\phi)=\Theta \frac{\int_{\phi}^{\phi_{0}} \frac{\theta(v)-\theta(\phi)}{v^{2}} \theta^{\prime}(v) \mathrm{d} v}{\int_{\phi_{s}}^{\phi_{0}} \frac{\theta(v)-\theta}{v^{2}} \theta^{\prime}(v) \mathrm{d} v}$$
where $v$ is a dummy variable associated with the suction; $\varphi_{0}$ is the suction corresponding to complete dryness, $\varphi_{0}=10^{6} \mathrm{kPa} ; \varphi_{\mathrm{s}}$ is the suction corresponding to the initial saturated water content $\theta_{\mathrm{s}}$ and it has a small finite value, $\varphi_{\mathrm{s}}=0.001 \mathrm{kPa}$,

to avoid singularity in the integration; $\theta^{\prime}$ is the derivative of the volumetric water content function; $\Theta$ is the correction factor used here which can take tortuosity into consideration and increases the flexibility of the permeability estimation equation.
By using Eq. (2.9), both MFX and FX can be used to estimate $k_{\mathrm{r}}$ values. Figure $2.6$ presents the measured and estimated $k_{\mathrm{r}}$ values from MFX and FX for Berlin coarse sand, with the use of the fitting parameters shown in Table 2.4. The SWCC of each soil code was also computed by curve-fitting of Eq. (2.2), and then the SWCC was used in Eq. (2.9) to compute $k_{\mathrm{r}}$, denoted as “fitted curve” in Fig. 2.6. Comparing the fitted curve with the measured $k_{\mathrm{r}}$, it can be seen that the fitted curve does not perfectly agree with the measured data. This disagreement is suspected to be due to the error of measurements or the estimation model, that is, Eq. (2.9) in this case. In order to eliminate the effect of measurement error and model error, the fitted curve is used as a reference to compare with the estimated $k_{\mathrm{r}}$ by MFX and FX. It is found in Fig. $2.6$ that the MFX can well predict the $k_{\mathrm{r}}$ for the soils with different initial porosities, while the estimated $k_{\mathrm{r}}$ by FX deviates substantially from the “fitted curve”.

In order to clearly show the difference between the estimated $k_{\mathrm{r}}$ by MFX and FX, the “fitted curve” is used to compute the relative error of the $k_{\mathrm{r}}$ values. The relative error $(R E)$ is defined as:
$$R E=\frac{k_{\mathrm{r}, \text { estimated }}-k_{\mathrm{r}, \text { fitted curve }}}{k_{\mathrm{r}, \text { fited curve }}} \times 100 \%$$
where $k_{\mathrm{r}}$, estimated is the value estimated by the SWCC of MFX or FX and $k_{\mathrm{r}}$, fited curve is the value in the fitted curve. Figure $2.7$ depicts the relative error of $k_{\mathrm{r}}$ values estimated by the SWCCs of MFX and FX for Berlin coarse sand. It is clearly seen that the $R E$ value of $k_{\mathrm{r}}$ estimated by MFX is much smaller than that by FX for a wide range of suction.

## 数学代写|概率论代写Probability theory代考|Establishment of Relationships

Arya and Paris [14] presented a SWCC model that combined physical hypotheses with an empirical presentation. Based on the shape similarity between the SWCC and the cumulative PSD curve, the Arya and Paris (AP) model was established using the PSD and dry unit weight of a soil. The model considers the water flow paths in the soil as a bundle of capillary tubes and assumes that the size of the soil particles is related to the corresponding pore diameters of the capillary tubes.

Given a set of SWCC data for a soil sample (1) with a known initial void ratio $\left(e^{(1)}\right)$, the fitting SWCC can be obtained using the Fredlund and Xing empirical equation [17]:
$$\theta_{\mathrm{w}}=\frac{\theta_{\mathrm{s}}}{\left{\ln \left[\exp (1)+(\varphi / a)^{n}\right]\right}^{m}}$$
where $\theta_{\mathrm{s}}$ is the saturated volumetric water content; $a, n$ and $m$ are three adjustable parameters; and $\varphi$ is the matric suction of soil. The fitting results for the set of measured SWCC data can be written as $\left(\theta_{w}^{(1)}, \varphi^{(1)}\right)$. Using the dimensionless volumetric water content, $S=\theta_{\mathrm{w}} / \theta_{\mathrm{s}}$, Eq. (3.1) can be re-written as follows:
$$S=\frac{1}{\left{\ln \left[\exp (1)+(\varphi / a)^{n}\right]\right}^{m}}$$
The estimation of the $\operatorname{SWCC}\left(\theta_{\mathrm{w}}^{(2)}, \varphi^{(2)}\right)$ of a soil sample $(2)$ that has the same PSD curve as the sample (1) but a different initial void ratio $\left(e^{(2)}\right)$ is presented as follows.

## 数学代写|概率论代写Probability theory代考|Relationship of Volumetric Water Contents

The particle-size distribution curve is divided into $n$ fractions. The solid particles in each fraction are assumed to be of uniform particle size. Assembling all the fractions, a natural structure sample is formed. The dry density of the sample is assumed equal to that of each assemblage of the $n$ fractions.
The pore volume of each fraction per unit sample mass is given as follows:
$$V_{\mathrm{v} i}=m_{\mathrm{si}} e / \rho_{\mathrm{s}} \quad i=1,2, \ldots, n$$

where $V_{\mathrm{v} i}$ is the soil pore volume of the $i$ th fraction per unit sample mass; $m_{\mathrm{si}}$ is the soil particle mass of the $i$ th fraction per unit sample mass; $\rho_{\mathrm{s}}$ is the soil particle mass density; and $e$ is the initial void ratio. The first fraction of the assemblage has the smallest pore size.

It is assumed that soil water will first fill in the small pores in the soil at certain water contents. Thus, the volumetric water content of the soil sample when the pores of the $i$ th fraction are filled with water can be calculated by the accumulated pore volume:
$$\theta_{\mathrm{w} i}=\sum_{j=1}^{j=i} V_{\mathrm{v} j} / V \quad i=1,2, \ldots, n$$
where $V$ is the sample volume per unit sample mass.
Based on Eqs. (3.3) and (3.4), the volumetric water contents of two soil samples (1) and (2) satisfy the following relationship:
$$\frac{\theta_{\mathrm{w} i}^{(2)}}{\theta_{\mathrm{w} i}^{(1)}}=\frac{e^{(2)} V^{(1)} \sum_{j=1}^{j=i} m_{\mathrm{s} j}^{(2)}}{e^{(1)} V^{(2)} \sum_{j=1}^{j=i} m_{\mathrm{s} j}^{(1)}}$$
where $\theta_{\text {wi }}^{(1)}$ is the volumetric water content when the $i$ th soil pore is filled with water in soil sample (1) with initial void ratio $e^{(1)}$, and $\theta_{w i}^{(2)}$ is that of soil sample (2) with initial void ratio $e^{(2)}$.

Based on the PSD curve, the particle mass in each fraction is the same in both soil samples, that is, $m_{\mathrm{s} j}^{(2)}=m_{\mathrm{s} j}^{(1)}, j=1,2, \ldots, i$. Considering the phase relationships $\rho_{\mathrm{s}} V_{\mathrm{s}}=\rho_{d} V$ and $\rho_{d}=G_{s} \rho_{\mathrm{w}} /(1+e)$, the following equation can be obtained:
$$\frac{V^{(2)}}{V^{(1)}}=\frac{1+e^{(2)}}{1+e^{(1)}}$$
Thus, the ratio of the volumetric water content can be obtained by substituting Eqs. (3.6) into (3.5):
$$\frac{\theta_{w i}^{(2)}}{\theta_{w i}^{(1)}}=\frac{e^{(2)}\left(1+e^{(1)}\right)}{e^{(1)}\left(1+e^{(2)}\right)}$$
The volumetric water content of soil sample (2) can be estimated using the volumetric water content data of soil sample (1) and the initial void ratios.

## 数学代写|概率论代写Probability theory代考|Estimation of Relative Permeability Function

ķr(φ)=θ∫φφ0θ(在)−θ(φ)在2θ′(在)d在∫φsφ0θ(在)−θ在2θ′(在)d在

R和=ķr, 估计的 −ķr, 拟合曲线 ķr, 拟合曲线 ×100%

## 数学代写|概率论代写Probability theory代考|Establishment of Relationships

Arya 和 Paris [14] 提出了一个 SWCC 模型，该模型将物理假设与经验演示相结合。基于 SWCC 和累积 PSD 曲线的形状相似性，利用土壤的 PSD 和干单位重量建立 Arya 和 Paris (AP) 模型。该模型将土壤中的水流路径视为一束毛细管，并假设土壤颗粒的大小与相应的毛细管孔径有关。

\theta_{\mathrm{w}}=\frac{\theta_{\mathrm{s}}}{\left{\ln \left[\exp (1)+(\varphi / a)^{n}\right ]\右}^{m}}\theta_{\mathrm{w}}=\frac{\theta_{\mathrm{s}}}{\left{\ln \left[\exp (1)+(\varphi / a)^{n}\right ]\右}^{m}}

S=\frac{1}{\left{\ln \left[\exp (1)+(\varphi / a)^{n}\right]\right}^{m}}S=\frac{1}{\left{\ln \left[\exp (1)+(\varphi / a)^{n}\right]\right}^{m}}

## 数学代写|概率论代写Probability theory代考|Relationship of Volumetric Water Contents

θ在一世=∑j=1j=一世在在j/在一世=1,2,…,n

θ在一世(2)θ在一世(1)=和(2)在(1)∑j=1j=一世米sj(2)和(1)在(2)∑j=1j=一世米sj(1)

θ在一世(2)θ在一世(1)=和(2)(1+和(1))和(1)(1+和(2))

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