Gravity currents are predominantly horizontal flows driven by density differences. In the context of porous media, they occur in environmental applications such as carbon sequestration and geothermal energy recovery, where flow arises from the density contrast between two fluids, such as liquefied CO2 and denser brine. While there has been substantial research on gravity currents in porous media, most of the emphasis has been on studying two-dimensional rectilinear flows due to their ease of visualization and analytical modeling. Consequently, only a limited number of studies have investigated axisymmetric gravity currents. Moreover, earlier research on gravity currents has often assumed the presence of a sharp interface between the gravity current and the ambient fluid, which is not a realistic assumption. In this study, we developed a depth-averaged model of mass and concentration to account for the dispersive entrainment of ambient fluid by the axisymmetric gravity current. Our findings demonstrate that, under the condition of a constant and continuous flux release from a point source, the buoyancy of the current exhibits self-similar behavior, while the height and spread of the gravity current increase in a non-self-similar manner, proportional to t1⁄2 (where t represents time). Similarly, the depth-averaged concentration of the gravity current also displays non-self-similar behavior. Through this study, we have eliminated the assumption of a sharp-surface interface and established that an axisymmetrically flowing gravity current can be effectively modeled using a dispersive entrainment approach.