If a heat source in a wellbore is kept at a constant 200 C how far will a 10 C temp increase be conducted?

A 6.25" horizontal wellbore which is 100 meters long.


The wellbore is in a sandstone formation with a 30% porosity. The porosity is filled with 12 API gravity oil which has a 25% water saturation. The normal temperature of the formation surrounding the wellbore is about 30 Celsius. The heat source is along the entire length of the 100 meter wellbore.

If a heat source in a wellbore is kept at a constant 200 C how far will a 10 C temp increase be conducted?
This is a poorly posed question.





You don't say how long in duration the wellbore is kept at 200C. If you keep it at that temperature indefinitely, there is no limit on the distance to which the +10C isotherm will propagate (at least in the lateral and downward directions -- presumably the Earth's surface should be modeled as being a constant-temperature boundary. In this case, the steady-state vertical temperature distribution directly above the centerline of the borehole will be a linear temperature gradient between 200C and the surface temperature.)





Furthermore, it is probably inappropriate to consider just conductive heat transfer in this problem. I have implicit;y assumed that the heated pipe is buried at a depth at which the boiling points of the liquid phases are above 200C. If not, then the fluid will boil, and much of the heat transfer will then be occur by advection and recondensation of the vapor phase.





Even in the absence of boiling, I would guess that with 30% porosity, the connected porosity would be pretty high, and therefore permeability of the formation would be high enough to support large-scale convection. In that case, convection of the porosity-occupying fluid will be a significant mode of heat transfer. A realistic simulation goes far beyond what can be addressed here, and requires a fairly sophisticated thermohydrologic code capable of handling multiple fluid phases.