Dispersion and recovery of solutes and heat under cyclic radial advection

For cyclic injection-extraction wells with various radial flow geometries, we study the transport and recovery of solute and heat. We derive analytical approximations for the recovery efficiency in closed-form elementary functions. The recovery efficiency increases as injection-extraction flow rates increase, dispersion decreases, and spatial dimensionality decreases. In most scenarios, recovery increases as cycle periods increase, but we show numerically and analytically that it varies non-monotonically with cycle period in three-dimensional flow fields, due to competing effects between diffusion and mechanical dispersion. This illustrates essential differences between the spreading mechanisms, and reveals that for a single well it may be impossible to optimize recovery of both solute and heat simultaneously. Whether retardation increases or decreases recovery thus depends on aquifer geometry and the dominant dispersion process. As the dominant dispersion process heavily determines the sensitivity of the recovery efficiency to other parameters, we introduce the dimensionless kinetic dispersion factor S_T, to distinguish whether diffusion or mechanical dispersion dominates. We also introduce the geometric dispersion factor G, which is derived from our full solution for the recovery efficiency and improves upon the concept of the area-to-volume ratio (A/V), often used in analysing well problems. Unlike A/V, G accounts for spatio-temporal interactions between dispersion and flow field geometry, and can be applied to determine recovery efficiencies across a wider range of scenarios. It is found that A/V is a special case of G, describing the recovery efficiency only when mechanical dispersion with linear velocity dependence is the sole mechanism of spreading.