A wellbore or borehole is a hole drilled in the ground to extract or explore the earth's natural resources. For extraction of geothermal heat, most geothermal reservoirs use an injection well and a production well where water is fed in and received respectively. Though effort is made to ensure efficient production, the production well is usually associated with some heat loss to the surrounding rock. An investigation of this heat loss is important for optimizing the efficiency of a geothermal reservoir. In this work, the heat equation for a wellbore surrounded by rock is formulated and solved to estimate the fluid temperature. The model takes conduction and convection into account as mechanisms for heat exchange between wellbore and surrounding rock formation. With the model, various investigations are made possible - the effect of the borehole diameter could be studied with insights into the new proposed earth energy extraction system (Triple E System). The Triple E system is a concept that uses preheating of injection fluid in a wellbore with ultra-slim diameter to overcome the limitations of conventional geothermal systems. The model could also be coupled with a geothermal reservoir model and further extended to oil reservoir wellbores especially in permafrost regions where geothermal gradient is significant.