Spreading of gravity currents in porous media has traditionally been investigated analytically by means of similarity solutions under the Dupuit-Forchheimer approach. We present a novel formulation to analyze the axisymmetric propagation of single-phase gravity currents induced by the release of a time-variable volume of fluid in a porous domain. Our approach is based on a first order expansion of the velocity potential that allows for the presence of vertical Darcy velocities. Coupling the flow law with mass balance equations leads to a PDE which admits a self-similar solution for the special case in which the volume of the fluid fed to the current increases at a rate proportional to t^3. A numerical solution is developed for rate proportional to t^α with α \neq 3. Current profiles obtained with the first order solution have a finite height at the origin. Theoretical results are compared with two experimental datasets, one having freshwater and the other air as an ambient fluid. In general, experimental current profiles collapse well onto the numerical results; the first order solution shows a marked improvement over the zeroth order solution in interpreting the current behaviour near the injection point. A sensitivity and uncertainty analysis is conducted on both the first order and zeroth order theoretical model. The sensitivity analysis indicates that the flow process is more sensitive to porosity variations than to other parameters. The uncertainty analysis of the present experimental data indicates that the diameter of glass beads in an artificial porous medium is the source of most of the overall uncertainty in the current profile.