In this case, the position the temperatura and the input flow are important.
Basically you need to define the input of water mass flow (m3/s) for the hydrodynamic part and could define a flux of heat (k dT/dx diferent of zero) to describe the injection of cold water in your equation of heat conservation (transport of advective difussive heat).
Imagine that the geo-thermal well is circular and contains stagnant liquid of depth H from the base. Take the temperature of the liquid to be UNIFORM. You can position the cold water injector anywhere along the circular surface that shears the top layer of the liquid and give FLOW RATE and TEMPERATURE of the cold water. The injector acts as a SOURCE. Hence this can be modelled using Hydrodynamic governing equations coupled with temperature.
I suppose, the main idea of this simulation could deal with HEAT ENERGY PREDICTION and THERMAL MODELING in the GEOTHERMAL WELL