**2. Calculation Method of Thermal Pore Water Pressure with Different** *OCR*

Campanella and Mitchell [8] developed a calculation model of thermal pore water pressure in saturated clay during an undrained heating test:

$$
\Delta\mu\_T = \frac{n(\kappa\_w - \alpha\_s) + \alpha\_{sT}}{m\_v} \Delta T \tag{1}
$$

in which Δ*uT* (kPa) is the change of pore pressure caused by thermal loading, *n* is the porosity, *mv* (kPa−1) is the compressibility of soil skeleton, Δ*T* ( ◦C) is the change in temperature, *αsT* ( ◦C−1) is the physicochemical coefficient of structural volume change, and *α<sup>w</sup>* ( ◦C−1) and *α<sup>s</sup>* ( ◦C−1) are the volumetric expansion coefficients of pore water and soil particles, respectively.

Under undrained heating conditions, saturated soil will expand along the secondary compression curve, and the compressibility of soil skeleton can be estimated by the isotropic recompression curve and written as

$$m\_{\mathbb{D}} \approx (m\_{\mathbb{D}})\_r = \frac{1}{1 + \varepsilon\_0} \frac{\kappa}{p'} \tag{2}$$

in which *e*<sup>0</sup> is the initial void ratio, *κ* is the slope of the isotropic recompression line, *p* is the mean effective stress, and (*mv*)*<sup>r</sup>* is the compressibility of the soil skeleton determined from the recompression curves.

Substituting Equation (2) into Equation (1) results in

$$
\Delta u\_T = \frac{n(\mathfrak{a}\_w - \mathfrak{a}\_s) + \mathfrak{a}\_{sT}}{\kappa} (1 + \mathfrak{e}\_0) p' \Delta T \tag{3}
$$

For most soils, the values of *<sup>α</sup><sup>w</sup>* and *<sup>α</sup><sup>s</sup>* are usually taken as 1.7 × <sup>10</sup>−<sup>4</sup> ( ◦C−1) and 3.5 × <sup>10</sup>−<sup>5</sup> ( ◦C<sup>−</sup>1), respectively [8,9,12,27].

Since the change of porosity during undrained heating tests is generally negligible and has little effect on the thermal pore water pressure, the porosity (*n*) is approximately equal to the initial porosity (*n*0) [9,28].

The physicochemical coefficient *αsT* is often used to characterize the volume change caused by soil structural rearrangement under unit thermal loading, which is not straightforward to obtain [8,9]. In fact, *αsT* can be estimated from experimental data of Δ*uT* at a given Δ*T*, and the relationship is rearranged as [9]

$$\mathfrak{a}\_{sT} = \left[\frac{\Delta\mu\_T}{p\_0'} \frac{\kappa}{1+e\_0} \frac{1}{\Delta T}\right] - n(\mathfrak{a}\_w - \mathfrak{a}\_s) \tag{4}$$

where *p*<sup>0</sup> (kPa) is the initial mean effective stress.

Because Equation (4) needs to know the experimental data of Δ*uT*, the undrained heating test must be carried out. To avoid complicated heating tests, Ghaaowd et al. [9] proposed an empirical expression for the physicochemical coefficient of normally consolidated clay, which can be written as

$$
\alpha\_{sT} = 1.0 \times 10^{-4} e^{-0.014I\_p} \tag{5}
$$

where *Ip* is soil plasticity index and *e* is the Napierian base.

In addition, through undrained heating tests, Wang et al. [26] found that the thermal pore water pressure of normally consolidated and overconsolidated soil meets

$$
\Delta u\_{Tocr} = \ln(1 + \frac{1.717}{OCR}) \Delta u\_T \tag{6}
$$

where Δ*uTocr* is the thermal pore water pressure of the overconsolidated soil. Substituting Equation (3) into Equation (6) results in

$$
\Delta u\_{Torr} = \ln(1 + \frac{1.717}{OCR}) \frac{n(\alpha\_w - \alpha\_s) + \alpha\_{sT}}{\kappa} (1 + \varepsilon\_0) p' \Delta T \tag{7}
$$

According to Equation (7), it can be seen that *κ* is a key parameter for estimating thermal pore water pressure. However, compared to the slope of the isotropic recompression line, the slope of the isotropic compression line (*Λ*) is more commonly adopted in studies

to calculate thermal pore water pressure. Therefore, the value of *Λ* is used to estimate *κ*. Meanwhile, according to the Cam-Clay model, it can be known that the compression index, *Cc*, and *Λ*, and the rebound index, *C*s, and *κ* satisfy, respectively,

$$
\kappa = 0.434 \text{C}\_{\text{s}} \tag{8}
$$

$$
\Lambda = 0.434 \text{C}\_{\text{c}} \tag{9}
$$

Assuming that the ratio of *Λ* and *κ* is *Λ*, the below equation can be obtained:

$$
\Lambda = \frac{\Lambda}{\kappa} = \frac{\mathbb{C}\_{\mathfrak{c}}}{\mathbb{C}\_{s}} \tag{10}
$$

For normally consolidated clays, *Λ* is generally constant, and the range of *Λ* is about 4.8~10 [9,29–31]. However, the stress history can affect the value of *Λ* [32–34] and then change the magnitude of the thermal pore water pressure. In order to better reflect the influence of stress history on thermal pore water pressure, the relationship between rebound index *Cs* and *OCR* is introduced [32]:

$$\mathcal{L}\_s = 0.0213 \text{ln}(\text{OCR}) + 0.0288 \tag{11}$$

Substituting Equation (10) into Equation (7) results in

$$
\Delta u\_{Tocr} = \ln(1 + \frac{1.717}{OCR}) \frac{n(a\_{\text{i}\text{v}} - a\_{\text{s}}) + a\_{\text{s}T}}{\Lambda} (1 + c\_0) \Lambda p' \Delta T \tag{12}
$$

Equation (12) is the final calculation method of thermal pore water pressure applicable to both normally consolidated and overconsolidated clays. This calculation method can not only consider the direct weakening effect of *OCR*, but also the influence of the variation *Λ* with *OCR* on thermal pore water pressure.

## **3. Validation of the Calculation Method**

The calculation method of thermal pore water pressure, which can consider overconsolidation effect, was applied to predict the variation of thermal pore water pressure, and the predicted results are compared with the experimental data.

## *3.1. Comparison with the Experimental Data in Undrained Heating Test of Wang et al. (2017)*

To investigate the influence of *OCR* on thermal pore water pressure under undrained conditions, Wang et al. [26] used a temperature-controlled GDS triaxial testing apparatus to conduct undrained heating tests on normally consolidated and overconsolidated kaolin clays. During the test, saturated kaolin clay was consolidated at a pressure of 300 kPa. After consolidation was completed, the consolidation pressure was unloaded to 150 kPa, 100 kPa, 75 kPa, 37.5 kPa, 30 kPa, and 10 kPa to obtain soil samples with *OCR* of 1, 2, 3, 4, 8, 10, and 30. Subsequently, the obtained soil samples were used to carry out undrained heating tests. Furthermore, the physical parameters of kaolin clay are shown in Table 1.

**Table 1.** Physical parameters of kaolin clay obtained from the test of Wang et al. (2017).


Through calculation, it has been found that when the clay is in slightly overconsolidated state (1 < *OCR* ≤ 4.3), the *Λ* obtained by Equations (10) and (11) is larger than that of normally consolidated clay. In this case, the value of *Λ* adopts that of normally consolidated clay. However, for highly overconsolidated clays (*OCR* > 4.3), the values of *Λ* are different from those of normally consolidated clays, and when *OCR* of soil is 8, 10, and 30, the calculated values of *Λ* are 5.4, 5.0, and 3.8, respectively. Thus, it can be seen

that for saturated clays, there is a critical threshold of *OCR* that determines whether the stress history will change the calculated value of *Λ*, thereby reducing the thermal pore water pressure. In this paper, the critical threshold of *OCR* is taken as 4.3. Subsequently, *Λ* is brought into Equation (12) to predict the thermal pore water pressure. The predicted results and their comparison with the experimental results of Wang et al. [26] are presented in Figure 1.

**Figure 1.** Comparison of predicted thermal pore water pressure against the results of Wang et al. (2017).

It can be seen from Figure 1 that all experimental data points are basically on the predicting lines. Meanwhile, through calculation, it has been found that except for the predicted results with *OCR* equal to 30, the coefficients of determination (*R*2) of other predicted results are all greater than or equal to 0.89. Moreover, it should be noted that when *OCR* is equal to 30, although the coefficient of determination of the predicted results is only 0.44, the maximum difference between the predicted results and the experimental data is within 1 kPa, which is acceptable in engineering. Therefore, it can be concluded that the calculation method established in this paper can accurately predict the undrained thermal pore water pressure for both normally consolidated and overconsolidated clays.
