**1. Introduction**

During the service period of buffer materials for nuclear waste repository [1–3], thermal prefabricated vertical drains [4,5], geothermal energy piles [6], and buried cables and pipelines [7], the temperature of the surrounding soil may change significantly. However, when the soil temperature changes, thermal pore water pressure will be generated in the soil due to the difference of thermal expansion characteristics between the pore fluid and soil particles. In case of geological formations with low hydraulic conductivity, the thermal pore water can cause the loss of effective stress. Furthermore, with the dissipation of thermal pore water pressure, the soil will produce additional thermal volume change. It is found that the magnitude of thermal pore water pressure mainly depends on the stress history. Knowledge of the volume change of saturated soil under different stress states, which is a key factor that needs to be considered in design of thermo-active structures, must accurately estimate the thermal pore water pressure with different degrees of overconsolidation.

A series of studies have investigated the thermal response of saturated clay soils. To explore the evolution law of thermal pore water pressure under undrained conditions, Campanella and Mitchell [8], Ghaaowd et al. [9], Abuel-Naga et al. [10], and Ghabezloo et al. [11] carried out related experimental investigations. These experimental studies found that the magnitude of thermal pore water pressure depends on the compressibility of the soil, the physicochemical coefficient of structural volume change, initial void ratio, initial effective stress, the change in temperature, and thermal expansion coefficients of pore water and soil particles. Furthermore, Abuel-Naga et al. [10], Ghabezloo et al. [11],

**Citation:** Tian, G.; Zhang, Z. A Calculation Method of Thermal Pore Water Pressure Considering Overconsolidation Effect for Saturated Clay. *Appl. Sci.* **2022**, *12*, 6325. https://doi.org/10.3390/ app12136325

Academic Editor: Bing Bai

Received: 20 May 2022 Accepted: 19 June 2022 Published: 21 June 2022

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Burghignoli et al. [12], and McCartney [13] explored the thermally induced volume change of soil by conducting drainage heating tests. Among them, Abuel-Naga et al. [10] used the theoretical microstructure mechanism to explain the thermally induced volume change behavior under different stresses. Based on the principle of particle rearrangement in porous granular materials undergoing thermodynamic process, Bai et al. [14,15] established a thermo-hydro-mechanical constitutive model. This model can accurately describe the irreversible consolidation of normally consolidated saturated soils induced under thermal loading and the aging effect caused by cyclic thermal loading. Subsequently, under the framework of granular thermodynamics, Bai et al. [16] derived a generalized effective stress principle, which can automatically consider the influence of the stress path, temperature path, and soil structure.

It should be noted that both the magnitude of thermal pore water pressure and the sign of the thermally induced volume change are affected by the degree of overconsolidation [10,12,14,17]. With the dissipation of the thermal pore water pressure, normally consolidated clays will exhibit irreversible volume shrinkage, highly overconsolidated clays will undergo elastic thermal expansion, and slightly overconsolidated clays will show a volume change trend that expands first and then shrinks [13,18–20]. Meanwhile, for overconsolidated clays, there is a transition temperature. When the soil temperature exceeds the transition temperature, soil deformation will change from expansion to contraction [18,21–23]. The reason for the different deformation laws of soils with different overconsolidation degrees may be due to the difference in thermal pore water pressure [21]. To explore the effect of overconsolidation ratio on the changes of thermal pore water pressure and thermal volume, constitutive models that can consider the influence of stress history are established based on the Cambridge model or modified Cambridge model [3,10,24,25]. However, because the yield caused by temperature and stress is considered, many parameters are involved in these constitutive models, which makes the solution process extremely cumbersome. Furthermore, these constitutive models cannot obtain an explicit expression of thermal pore water pressure [26], which makes engineering applications very inconvenient. To conveniently and quickly predict the thermal pore water pressure, Campanella and Mitchell [8] proposed a thermo-porous-mechanical model by using concepts of thermoelasticity and linear elasticity. However, for highly overconsolidated soils, the thermo-porous-mechanical model may no longer be applicable. Based on the unified hardening model established by Yao and Zhou [25], Wang et al. [26] established a calculation method of thermal pore water pressure including *OCR*, but when the soil is in highly overconsolidated state, there is a large gap between the prediction results and the experimental results. In addition, the acquisition of thermal parameters in this calculation method requires additional thermal tests.

In summary, various methods for calculating thermal pore water pressure have been developed. However, the overconsolidation effect has not been well explored. To conveniently and quickly predict the thermal pore water pressure in overconsolidated clay and quantify overconsolidation effect on the thermal pore water pressure, by introducing the parameter *Λ* affected by *OCR* and the nonlinear relationship between *OCR* and the thermal pore water pressure, a calculation method of thermal pore water pressure considering overconsolidation effect for saturated clay is proposed. In addition, the calculation method is applied to predict the thermal pore water pressure, and the predicted results are compared with experimental data.
