As the energy needs are expected to surpass the energy content found in available fossil-fuel resources in this century, interest in renewable energy sources has increased in the past decade. One area of interest is in geothermal energy harvesting. With these systems, energy is retrieved from the Earth to be used on the surface either directly, such as providing heat to a community, or converted to electrical energy. However, as the fluid moves through the piping from its deepest depth to the surface, energy is transferred from the fluid to the surrounding soil. In conventional deep wells (depths of 4 km or more), this transfer results in a transmission loss of energy, while in more shallow residential geothermal heat-pump systems (depths of 100 m), this transfer is the main energy harnessing mechanism.
We have recently employed some classical mathematical modeling approaches to these systems. For example, with my collaborator T. Baumann at the Technical University-Munich (TUM), we described the temperature attenuation in the fluid from a deep aquifer at a geothermal facility in the Bavarian Molasse Basin . Energy losses depend on the production rate of the facility (potentially up to 30%). Our approach takes advantage of the small aspect ratio between the radius of the well to its length, and that the energy balance is between the axial energy transport in the fluid compared to the radial transport in the soil. We find that the dominant eigenfunction for the radial problem in the fluid captures this balance, and that the corresponding eigenvalue provides the appropriate constant relating the effective axial energy flux with the temperature drop over the length of the well. In the design of these wells, this constant traditionally is prescribed from phenomenological experience.
This approach may be quite useful in the construction of the shallow residential geothermal heat-pump systems. Although operation of these systems is about a third of the cost of conventional heating and cooling systems, they are currently not economically viable, since the installation cost of the wells depends significantly on the well depth required for the power needs of the residence. These systems are used year round, with energy deposited into the soil from the residence in the summer months, and then retrieved in the winter months for heating.
Recently, a group of undergraduate students participated at the NSF-funded Research Experiences for Undergraduates program at Worcester Polytechnic Institute to work on this problem, which was brought to us from the New England Geothermal Professionals Association. With our modeling approach, the eigenvalue and the axial behavior gives a characteristic length for the well, over which an energy attenuation of 1/e is achieved. Hence, three of these characteristic lengths are needed to attain over 90\% of the possible energy available. We are currently extending these approaches to horizontal piping systems.
Department of Mathematical Sciences
Worcester Polytechnic Institute
Worcester, MA 01609
 B.S. Tilley and T. Baumann, “On temperature attenuation in staged open-loop wells”, Renewable Energy, 48 416-423: (2012)