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Hydrogeological systems are earth systems influenced by water. Their behaviors are governed by interacting processes, including flow of fluids, deformation of porous materials, chemical reactions, and transport of matter and energy. Here, coupling among three of these processes is considered: flow of water, heat, and deformation, each of which is represented by a diffusion-type of partial differential equation. One side of the diffusion-type equation relates to motion of matter or energy, while the other relates to temporal changes of state variables at a given location. The coupling arises from processes that govern motion as well as those that relate to change of state. In this work, attention is devoted to coupling arising from changes in state. Partial derivatives of equations of state constitute the capacitance terms of diffusion-type equations. Of the many partial derivatives that are mathematically possible in physical systems characterized by several variables, only a few are physically significant. Because the state variables are related to each other through an equation of state, the partial derivatives must collectively satisfy a closure criterion. This framework offers a systematic way of developing the coupled set of equations that govern hydrogeological systems involving the flow of water, heat, and deformation. Such systems are described in terms of four variables, and the associated partial derivatives. The physical import of these derivatives are discussed, followed by a description of partial derivatives that are of practical interest. These partial derivatives are then used as the basis to develop a set of coupled equations. A brief discussion is presented on coupled equations from a perspective of energy optimization

Coupled equations for transient water flow, heat flow, and deformation in hydrogeological systems