TOUGH+Millstone has been developed for the analysis of coupled flow, thermal and geomechanical processes associated with the formation and/or dissociation of CH4-hydrates in geological media. It is composed of two constituent codes: (a) a significantly enhanced version of the TOUGH+HYDRATE simulator, V2.0, that accounts for all known flow, physical, thermodynamic and chemical processes associated with the behavior of hydrate-bearing systems undergoing changes and includes the most recent advances in the description of the system properties, coupled seamlessly with (b) Millstone V1.0, a new code that addresses the conceptual, computational and mathematical shortcomings of earlier codes used to describe the geomechanical response of these systems. The capabilities of TOUGH+Millstone are demonstrated in the simulation and analysis of the system flow, thermal and geomechanical behavior during gas production from a realistic complex offshore hydrate deposit. In the first paper of this series, we discuss the physics underlying the T+H hydrate simulator, the constitutive relationships describing the physical, chemical (equilibrium and kinetic) and thermal processes, the states of the CH4+H2O system and the sources of critically important data, as well as the mathematical approaches used for the development of the of mass and energy balance equations and their solution. Additionally, we provide verification examples of the hydrate code against numerical results from the simulation of laboratory and field experiments.