Solving Coupled Partial Differential Equations in Porous/Fractured Geomaterials

Exploring efficient and robust algorithms to solve the simultaneous Partial Differential Equations (PDEs) is essential to model the prevalent multiphysics processes in deep rock engineering activities, such as the thermo-hydro-mechanical coupling in nuclear waste disposal and geothermal exploitation. In this study, the staggered and monolithic solution schemes are developed in the context of poroelastic and fractured geomaterials, and the applicability of the two solution schemes is analyzed in detail. It is found that the degree of coupling between primary variables plays a pivotal role in determining the performance of the staggered and monolithic solution schemes. Two-way coupling can only be tackled by the monolithic solution scheme, while the staggered solution scheme is very robust and efficient to deal with one-way coupling. This study provides an overarching principle to solve simultaneous PDEs: Understand the degree of coupling between primary variables, and use the staggered and monolithic solution schemes to address one-way and two-way couplings, respectively. 1. INTRODUCTION Rock mechanics, as an applied geoscience, has its roots deeply entangled with various rock engineering activities. The pioneers and early generations of rock engineering practitioners established the framework and theory of rock mechanics mainly from on/near ground surface projects, such as rock foundations, rock slopes, and shallowly buried tunnels. In the last two decades, rock engineering activities started to go deeper and deeper to serve increasingly complex engineering purposes. Examples include nuclear waste disposal, hydraulic fracturing, geothermal exploitation, CO2 sequestration, and injection-induced earthquakes. The above new rock engineering activities pose great challenges to the existing rock mechanics theory, but at the same time, they provide a fertile ground for rock mechanics to extend and prosper. Future rock mechanics will interact more closely with computer science, seismology, petroleum engineering and many other disciplines to further highlight its interdisciplinary attributes. A common feature of rock engineering at a great depth is the potential coexistence of in-situ stress, high temperature, fluid pressure in pores/fractures, and chemical reactions. These factors are coupled together, and each of them plays a considerable role in the entire system. Therefore, deep rock engineering is a typical multiphysics system. It is not trivial to replicate such a complex working environment in the laboratory, neither can the field tests be readily conducted at such a depth. Numerical simulation, however, provides an insightful, economical and effective approach to understand the multiphysics system.