Heat extraction by circulating cold water through a geothermal reservoir could potentially induce earthquakes of large magnitudes. In this work, we explore the role of water injection inside a normal fault in triggering seismic slip, under different temperature and rate scenarios of fluid injection into fault damage zones. The resulted non-uniformity of thermal drawdown along the fault damage zones significantly affects the magnitude and timing of induced earthquakes. A non-dimensional expression integrating fault configuration (e.g. length and thickness) and injection condition (e.g. rate and temperature) is used to describe the relationship between the injection schedule and the resulting fault seismic slip event. As the dimensionless parameter increases, suggesting a transition from heat conduction to convection, the dimensionless event timing also grows nonlinearly. The perturbation of fault stress field induced from localized thermal cooling process is pronounced, compared to decoupled hydro-mechanical scenarios. The stress field perturbation in the system due to thermal cooling is characterized through a Coulomb friction ratio analysis for evaluating the stress changes along the fault plane and a tensor-based stress perturbation analysis for quantifying the stress changes in the damage zones and host rocks. The thermal influence acting on local patches along the fault strike not only advances the timing of seismic slip, but also increases the magnitude of induced seismic events, by unloading the fault to prompt seismic rupture. The injection temperature has a significant impact on facilitating the onset of seismic slip, i.e. attempting to accelerate the timing and increase the magnitude of fault reactivation. The injection rate variation will affect the timing by changing the pore pressure field and heat transfer manner. Prior to the onset of fault reactivation, the thermal unloading response increases fault permeability by decreasing normal stress, such that more permeable channels in the fault allow fluid to diffuse. Produced plastic shear strain due to fault slip provide extra positive contributions to increased normal aperture through shear dilation, thereby the fault permeability increases significantly by around two orders of magnitude.