A theory of dipole flow is developed to model flow induced by a vertical circulation well consisting of injection and extraction chambers in a single borehole. Included in the theory are an analytical description of the kinematic flow structure around a vertical circulation well and the drawdown in the well chambers. Using Stokes' stream function, simple criteria are derived to determine the region of intensive recirculation. This region extends (from the dipole center) approximately five distances between chamber centers in the radial direction and two distances between chamber centers in both vertical directions. The vertical scale does not depend on anisotropy of hydraulic conductivity, and the radial scale is corrected for anisotropic aquifers. Based on these estimates, criteria are given for the selection of the appropriate aquifer model to employ in five settings, including infinite, semi-infinite confined, semi-infinite unconfined, finite confined, and finite unconfined aquifers. Applications of dipole flow theory are given for analytical estimation of the capture zone for recirculation wells and for simultaneous measurement of horizontal and vertical hydraulic conductivity in uniform anisotropic aquifers using steady state measurements of drawdown in the well chambers.