Precise numerical study on the behaviour of gassy marine soils subjected to thermal and mechanical loadings
Introduction
Gassy soils are widely distributed in river deltas, coastal and marine sediments (Sobkowicz and Morgenstern, 1984, Sills and Gonzalez, 2001), where the gas mainly arises from biogenic and petrogenic process (Thomas, 1987, Grozic et al., 2000, Grozic et al., 2005). The gas phase is discontinuous and exists in discrete occluded bubbles type in gassy marine soils, and the saturation degree (S) is usually higher than 85% (Fredlund, 1976, Nageswaran, 1983, Wang et al., 2018). These gas bubbles show non-negligible influence on the soil properties, e.g. compressibility, permeability and strength (Sills et al., 1991, Wheeler et al., 1991, Rebata-Landa and Santamarina, 2012, Sultan et al., 2012), which affect the bearing capacity of foundations and the instability of structures (Nisbet and Piper, 1998, Haththotuwa and Grozic, 2011).
Several studies have been devoted to studying the characteristics of gassy soils, and reveal that S has remarkable influence on the seepage process (Barden, 1965, Wang et al., 2009). One unique feature of gassy soil is that the presence of gas could either decrease or increase the undrained shear strength of the soil, as compared to its saturated equivalent at a given effective stress, depending on the initial pore water pressure (Thomas, 1987, Wheeler and Gardner, 1989; Hong et al., 2017), i.e., water depth. It is first discovered by Hong et al., 2019, Hong et al., 2020 that the mechanism underlying the unique feature is that the inclusion of gas has modified both dilatancy and yielding of the saturated matrix in the opposite manner under different initial pore water pressures. This has led to formulation of new stress-dilatancy relation and yield function depending on initial pore water pressure and gas contents, and development of constitutive models for gassy clay and sand which predict both damaging and beneficial effect of gas on soil in a unified fashion (Hong et al., 2020, Hong et al., 2021). The above studies are mainly based on the experiments, showing that the gas bubbles migrate, deform, dissolve or precipitate due to the gradient of temperature or external pressure, which change the soil properties. To solve the practical problem theoretically, the governing equations are usually simplified under some assumptions due to the complexity (Barden, 1965, Yin and Ling, 2007). When S is sufficiently high (approximately over 85%), the matric suction can be neglected for the fact that almost the same pressures are captured between the pore water and pore gas (Barden, 1965, Fredlund, 1976). Under this condition, the pore gas and water can be regarded as a compressible mixture (Esrig and Kirby, 1977, Pietruszczak and Pande, 1996), the effective stress equation has the similar form as that of saturated soils and acceptable results have been obtained (Chang and Duncan, 1983, Liu et al., 2016, Finno et al., 2017, Zhang et al., 2018).
Energy facilities, such as heated pipelines, usually cause temperature change during operation and transfer heat to the surrounding soils (Britto et al., 1989, Abuel-Naga et al., 2008, Bourne-Webb et al., 2009, Chen et al., 2011, Ng et al., 2014), which makes the hydro-mechanical coupling processes much more complicated (da Costa et al., 2002, Cui et al., 2018, Shahrokhabadi et al., 2019, Zhu et al., 2020). As shown in Fig. 1, when temperature rises, the bubbles expand, pore water turns liquid into vapor and both the processes of seepage and deformation are influenced accordingly. Considering the complexity of governing equations in thermal-hydro-mechanical (THM) coupling problems of the gassy soil, attempts have been made to study the problem based on the gas–water mixture flow theory (Ghaboussi and Wilson, 1973, Yang and Sato, 2001, Wang and Ai, 2018). The bubble volume is sensitive to temperature change, thus treating pore gas and water as a compressible mixture flow simply may not reflect the thermal effects on excess pore pressure (EPP) and deformation accurately. To date, little study has been reported about the thermal response of gassy soils. Consequently, to describe the behaviour of gassy soils subjected to thermal and mechanical loadings accurately, the expansion and compression characteristics of gas bubbles should be considered in the governing equations.
Due to the complexity of thermal expansion and compressibility of pore gas, water and solid constituents in the governing equations, it is difficult to obtain the explicit analytical solutions by traditional methods. In this paper, a recently developed precise integration method (PIM) is extended to study the aforementioned problems. The term, describing the volume change of bubbles induced by thermal or mechanical load, is firstly introduced into the governing equation. Then, a natural soil layer is divided into 2N layer elements with equal thickness, with N typically chosen as 20. The relational matrices of state space vectors are expressed in term of Taylor series expansion. Accordingly, desired accuracy can be captured, and the solution accuracy is only limited by the accuracy of the computer utilized (Zhong et al., 2004, Wang and Ai, 2015). Based on the recurrence formula for the combination of two adjacent layer elements, each iteration of the combining process reduces the layer elements number by half, that is, a natural layer needs only N iterations of combination. By the aid of integral transform technique, Taylor series expansion and recurrence formula, such problems can be solved by the present extended PIM. The PIM proves to be efficient in solving the quasi-static problems for materials with complex constitutive relations without the necessity to derive the explicit expressions (Zhong et al., 2004, Lin et al., 2014, Wang and Ai, 2015). Verifications against the analytical solutions (Savvidou and Booer, 1989) and results by the finite element method (FEM) (Chaudhry et al., 2019) for THM coupling behaviour of saturated soils were provided. A series of examples were then carried out to discuss the effects of expansion characteristic of gas bubble, saturation degree and water depth on the behaviours of gassy soils.
Section snippets
Governing equations
The THM behaviour of gassy soils consists of heat transfer, seepage of gas–water mixed flow, deformations of soil matrix and gas bubbles. For practical problems, reasonable hypotheses are made in this study, as follows: (1) The skeleton is isotropic and elastic; pore bubbles are micro-bubbles and the size of which is on the same order as that of the soil particles; (2) The gas–water mixed flow obeys Darcy’s law, and heat transfer obeys Fourier’s law; (3) The soil skeleton, gas bubbles and pore
Deduction of ordinary differential matrix equation in the transformed domain
In order to solve the above partial differential equations, the integral transform technique is used to convert them into ordinary differential equations. Laplace transform for time t and Hankel transform for coordinate r are employed (Sneddon, 1972):
Laplace transform and its inverse transform:
Hankel transform and its inverse transform:where is the transform parameter of in Laplace
Numerical results and discussions
This section focuses on investigating the effects of S and H on THM responses. As shown in Fig. 5, A circular heat source Q () with constant power () is buried at the depth of , the center of the heat source and field point are points A and B, respectively. A circular force F () is applied on the soil surface, whose center is the origin of coordinates O. The thickness of the soil is . The parameters used are shown in Table 1 and Table 2. The dimensionless parameters
Conclusion
This paper presents a precise numerical investigation into the THM coupling behaviour of gassy marine soils subjected to thermal and mechanical loadings using the newly developed precise integration method (PIM). Verifications against the analytical solutions and FEM results (by OpenGeoSys) for THM coupling behaviour of saturated soils surrounding a point heat source were provided, followed by a series of examples to discuss the effects of expansion characteristic of gas bubble, saturation
CRediT authorship contribution statement
Bin Zhu: Conceptualization, Writing - original draft, Writing - review & editing, Supervision. Jiasheng Huang: Formal analysis, Methodology, Writing - original draft, Writing - review & editing. Lujun Wang: Conceptualization, Formal analysis, Methodology, Writing - original draft, Writing - review & editing, Supervision. Zhigang Ye: Formal analysis, Writing - original draft.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors gratefully acknowledge the funds from the National Natural Science Foundation of China (Nos: 52078458, 51988101), Zhejiang Provincial Natural Science Foundation of China (Nos: LY21E080026, LCD19E090001) and Fundamental Research Funds for the Central Universities (2020QNA4032).
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