In recent years, ground thermal energy storage has become a topic of interest in the energy community for solar thermal energy storage systems, ground sourced heat pump systems, and data center thermal management systems due to an increase in the energy efficiency of such systems utilizing the ground as a thermal reservoir. The most common method for transferring thermal energy to the ground formation is the geothermal borehole. This dissertation presents the state of the art in geothermal borehole modeling and derives novel analytical functions to model advanced concepts concerning their operation. The novel solutions derived allow a geothermal borehole designer to better understand and design ground energy storage systems. The state of the art in geothermal borehole modeling is the stationary line source solution which is limited to boreholes operating without groundwater flow. Novel solutions for modeling a geothermal borehole with groundwater advection are presented through derivation of a transient moving line source solution as well as a transient moving cylindrical surface source solution. These solutions are applied to model a specific type of open loop geothermal borehole called a standing column well with groundwater advection and are compared to empirical and numerical data for validation. The dissertation then moves into derivation of a property determination method for geothermal boreholes with groundwater advection. The traditional property determination method used to obtain ground formation properties is based on the stationary transient line source method and fails in the presence of groundwater flow. The proposed novel property determination method calculates the thermal conductivity, thermal diffusivity, and superficial flow velocity of groundwater within a ground formation. These methods and solutions are novel tools allowing for geothermal borehole designers to grasp a better understanding of the systems they are designing as well as open other possibilities in design, such as using induced groundwater flow as a means to enhance borehole operation. Preliminary analyses into two advanced concepts for enhancing boreholes using groundwater advection are presented as a precursor to future research. The goal of future research is to design systems that efficiently utilize the ground formation for thermal energy storage while minimizing the capital investment for the system.