Numerical simulations of the thermal response of porous rock surrounding a well extracting geothermal energy were performed. The rock surrounding the well was chosen as Limestone (Salem) and was homogenous throughout the surrounding control volume. The porosity of the rock ranged from 0.05 to 0.30, and the pores were assumed to be spherical in shape, evenly distributed throughout the control volume, and to be closed, i.e. fluid transport effects were negligible. The depth of the water table in the control volume ranged from 100 m to 4000 m below the surface, and pores above the water table were saturated only with air while pores at or below the depth of the water table were saturated with liquid water. The temperature profile of the well was held constant throughout the simulation, with a temperature difference from the initial rock temperature profile ranging from 5 K to 10 K. In the simulations the well had a diameter of 0.3 m, and extended 3000 m below the surface, and the control volume extended from the surface to 3750 m below the surface and from the well wall outward 25 m radially, and the simulations ran for a time span of 10 years. The simulation model was developed from the radial coordinate form of the heat equation and accounted for heat transfer radially and vertically, and the change in thermo-physical properties caused by rock porosity and the fluid contained within the pores.
The simulations showed two distinct regions of change in the temperature profile of the rock, the region at the depth of the well and the region below the depth of the well. In the region below the well, there was no change in the temperature profile over time. These results indicate that heat transfer in the control volume took place almost exclusively in the radial direction, with notable temperature change extending radially from the well a relatively small distance. In the well region, two distinct sub-regions were found to exist when the depth of the water table was located in the well region, one region above the water table where the pores were saturated with air, and one below where the pores were saturated with water. Both regions showed a similar response, a transient temperature change starting at the well wall and travelling radially outward through the control volume over time, with the amplitude of the transient decreasing over time. The temperature change in the air filled region was found to be higher than the change in the water filled region before the transient, while temperature change in the water filled region was higher after the transient. Changes in the water table depth were found to only affect the location of the air filled and water filled regions, with the temperature response of these regions the same regardless of the water table depth. Increased porosity was found to cause decreased change in the temperature profile, with the effects more noticeable in the water filled region. The well temperature difference was found to cause proportional changes in the temperature profile change with increased well temperature difference resulting in increased temperature profile change.
The heat transfer rate from the surrounding rock to the well was studied over time with initial heat transfer rates for a well temperature difference of 10 K of 120 to 190 kW decaying exponentially over time to steady-state heat transfer rates of 70 – 40 kW. The heat transfer rate was found to be proportional to the well temperature difference and had no other affect on the heat transfer rate over time. Increased porosity was determined to decrease the heat transfer rate while increasing the time to reach steady-state, while increasing water table depth was found to decrease both the heat transfer rate and time to reach steady-state. The time to reach steady-state ranged from 5.7 years to 7.3 years, and the effect of porosity on time to steady-state was found to be greater at decreased water table depths, with water table depth having a larger affect than porosity.
From the thermal response of the system, it was determined that the temperature change in the rock extended a relatively small distance from the well, allowing the wells to be located relatively close together. By operating these wells simultaneously, the geothermal energy extraction would be additive allowing for greater overall rates of energy extraction. Conversely, if the wells were operated sequentially, geothermal energy could be extracted for a greater period of time.