The transient target current is used to characterize the aluminum laser produced plasma for various fluences and target external polarizations. A double peak structure of the electronic part was experimentally observed, as consequence of the existence of non-Maxwellian electron distribution through the hot and cold electrons. Applying an external voltage on the target, the ionic part becomes oscillatory, with plasma ion frequency. Assuming that particle motions take place on continuous and non-differentiable curves, a fractal hydrodynamic model of plasma expansion is built. In such conjecture, the analytical solutions and numerical simulations allow a reasonable interpretation of our experimental results.