Investigation of cross-interactions of coupled thermal-hydraulic-mechanical model using stochastic simulations

Abstract

The fundamental study of coupled thermal–hydraulic-mechanical (THM) systems is an important issue in multi-physical fields. This study developed a THM model in which the hydraulic and mechanical effects are fully coupled and the temperature change serves as the perturbation source of the system. A porothermoelasticity model was applied to investigate the cross-interactions among the temperature, change in pore water pressure, and displacement of a buffer material in the near-field and a host rock in the far-field with the conceptual model in a nuclear waste repository. The results show that the size scale of the domain of interest and the THM properties play important roles in the coupled THM system. The changes in pore water pressure depended on the combined effect of the deformation of the porous space and pore water controlled by the thermal expansion coefficients of the solids and fluids. Stochastic analyses show that the uncertainties of variables varied spatio-temporally due to the thermal influences and approached zero due to the presence of a stable condition. The cross-interactions between displacement and change in pore water pressure induced by the thermal effect were complex and can be determined from statistical moment analyses.

Introduction

The cross-interactions among thermal, hydraulic, mechanical, and chemical (THMC) processes are an important issue in the safety assessment of the final disposal site for spent nuclear fuel (Lennart and Jan, 1999, Liu et al., 2011, Rutqvist et al., 2011, Steefel et al., 2010, Stephanson, 2004, Zheng et al., 2011); geothermal energy resource production (Jacquey et al., 2016, Vallier et al., 2018, Zhao et al., 2015), and carbon dioxide sequestration (Fan et al., 2019, Fan et al., 2019, Li et al., 2006). For the final disposal site for spent nuclear fuel, the heat generated from the nuclear canister acts as a driving force that disturbs the engineered barrier system in the near-field. Solid and fluid cross-interactions in the buffer material affect the transport behavior of nuclides, which is the main safety assessment target for disposal sites. Heat is continuously transmitted to the far-field, changing the equilibrium state of the host rock, and the transportation characteristics change through a hydraulic-mechanical (HM) coupled process. In geothermal energy resource production, a high-temperature discharge and a low-temperature recharge of the fluid cycle, which are accompanied by a pore fluid pressure disturbance, are commonly used. Chemical precipitation due to a change in fluid pressure may decrease production efficiency, which is an important issue in this field (Borgia et al., 2012, Cui et al., 2016, Lerm et al., 2013). The main driving force is the pore fluid pressure variation, which may induce rock fracture that triggers seismicity (Kümpel, 2012, Talwani and Acree, 1985). Pore fluid includes water and vapor and makes the coupled THMC system very complex. In carbon dioxide sequestration, fluidized carbon dioxide is recharged into a reservoir beneath the ground surface. Through chemical reactions, fluidized carbon dioxide can transform into a rock type, stabilizing the carbon dioxide (Gislason et al., 2010, Matter et al., 2011, Matter et al., 2009). At a sequestration site, carbon dioxide can exchange between gas and liquid phases with a water component, making the fluid component complex. The main driving force in the THMC system is pore fluid pressure. The thermal conditions are relatively simple and mainly affected by the geothermal gradient and chemical reactions. Considering that cross-interactions are very complex in a multi-phase fluid, a conceptual model of the final disposal site for spent nuclear fuel is here chosen as the case study under the assumption of full saturation.

The DECOVALEX (DEvelopment of COupled models and their VALidation against EXperiments in nuclear waste isolation) project has investigated coupled thermal–hydraulic-mechanical (THM) and THMC processes using numerical modeling and compared the results from different models (Garitte et al., 2017, Garitte et al., 2017, Nguyen et al., 2005, Pan et al., 2016, Tsang, 2009, Tsang et al., 2009, Birkholzer et al., 2005a, Birkholzer et al., 2005b, Birkholzer et al., 2005c, Birkholzer et al., 2019d). A THM model was initially used and a THMC model is being developed. Because the coupled behavior is complicated and varies for site-specific materials, understanding the cross-interaction between THM variables, for example using covariance analysis, can provide abundant information for risk quantification in safety assessment.

THM modeling provides the state changes of pore water pressure, stress (displacement), and temperature (Booker and Savvidou, 1984, Booker and Savvidou, 1985, Palciauskas and Domenico, 1982, Smith and Booker, 1996). According to Cheng (Cheng, 2016), there are three types of THM model, namely the thermally uncoupled, fully coupled, and complete THM models. The complete model uses the coupled constitutive equations of the THM system to develop the governing equations. A fully coupled model modifies the complete model by simplifying the interactions; nevertheless, the governing equations with THM variables are still solved simultaneously. A thermally uncoupled model solves the THM system by neglecting the heat variation due to the HM effects, and thus the thermal variable can be uncoupled. The diffusion equation of temperature was solved independently in the thermally uncoupled model and then the temperature solution was substituted into the THM equations to obtain the solutions of displacement and change in pore water pressure. Each model has its own advantages and limitations. Details are provided in the study of Cheng (Cheng, 2016).

Poroelasticity theory, proposed by Biot (Biot, 1941), can be used to analyze the deformation and pore water pressure changes in a coupled HM system. It is a rigorous theory for describing the cross-interactions between solid and fluid phases (Cryer, 1963, Schiffman et al., 1969, Wang and Hsu, 2009, Wang and Hsu, 2013) and serves as the basis for analyzing a THM system. An extension of poroelasticity theory that considers the thermal effect, called porothermoelasticity theory, was proposed by Biot (Biot, 1977). It considers thermal, hydraulic, and mechanical factors as variables. Various porothermoelasticity analysis models have been proposed, such as the thermally uncoupled model (Booker and Savvidou, 1984, McTigue, 1986, Palciauskas and Domenico, 1982, Schiffman, 1971) and the fully coupled model (Bear and Corapcioglu, 1981, Derski and Kowalski, 1979, Kurashige, 1989, Smith and Booker, 1993). For example, McTigue (McTigue, 1986) developed a thermally uncoupled model and provided a fundamental study of analytical solutions in the porothermoelasticity system using a deterministic method. Because of its simplicity and capability to model complex THM problems, the thermally uncoupled THM model was adopted in this study. A conceptual model of the final disposal site for spent nuclear fuel proposed in the H12 report by the Japan Nuclear Cycle Development Institute (JNC) (Japan Nuclear Cycle Development Institute, 2000) and the SNFD2009 report by Taiwan Power Company (Taiwan PowerCompany, 2010) was adopted. Models of the buffer material in the near-field and the host rock in the far-field were investigated in this study to explore the cross-interactions in different materials and on different scales. The fractures and matrix of the host rock are conceptually treated as an equivalent porous medium for the far-field case.

Due to multiscale heterogeneity and insufficient site characterization, the stochastic approach is widely used to account for the uncertainty in groundwater problems (Dagan, 1989, Dagan and Neuman, 1997, Gelhar, 1993, Neuman et al., 1987, Rubin, 2003, Zhang, 2002) and the inversion of the hydraulic parameters, e.g., (Tsai et al., 2017, Yeh et al., 1996). Sensitivity analyses of important parameters in the THM coupling process are required prior to the uncertainty assessment. “Global sensitivity analysis and uncertainty quantification can help not only in obtaining a better physical understanding but also in the identification of meaningful target validation corridors for modelers to aim at” (Buchwald et al., 2020). Several approaches for the stochastic analysis of an HM system have been recently proposed, including Monte Carlo simulations (Ferronato et al., 2006, Frias et al., 2004, Wang and Hsu, 2009, Wang and Hsu, 2017), moment differential equations (Wang and Hsu, 2009, Wang and Hsu, 2013, Wang et al., 2015), and the spectral method (Chang and Yeh, 2015, Chang and Yeh, 2016). Stochastic studies have been applied to analyze THM systems (Buchwald et al., 2020, Nguyen-Tuan et al., 2016, Wang and Hsu, 2017, Watanabe, 2011, Watanabe et al., 2010, Watanabe et al., 2009). Parameter identification and uncertainty analysis have been the main targets of previous studies. A lack of understanding concerning the cross-interactions of state variables still hinders the evaluation of safety assessments. An investigation of interactions has been attempted using a stochastic concept (Wang and Hsu, 2017). However, Wang and Hsu (Wang and Hsu, 2017) examined the mean and variance behaviors in a buffer material model without considering the parameters under full saturation and identical temperature conditions. The present study rigorously selects suitable parameters, adds a host rock model, and conducts parameter sensitivity analyses and covariance assessments to investigate the cross-interactions in a THM system. We more thoroughly investigate the cross-interactions in the coupled THM system to obtain more accurate results.

The remainder of this paper is organized as follows. Porothermoelasticity theory is introduced in Section 2. Details of the model settings and validation are provided in Section 3. The results and a discussion of the sensitivity and covariance analyses are presented in Section 4. Finally, conclusions are given in Section 5.