There has been some recent interest in exploring applications of fractal calculus in transport models. One of the motivations for this is that such models are able to generate anomalous transport signals. For example, when fractional calculus is employed to define diffusion transport fluxes (heat, mass etc.) the exponent n in the space-time scaling differs from the classical value of n = ½. In this talk we have two objectives. The first objective is to identify physically realizable systems that exhibit anomalous transport behaviors. The second is to arrive at suitable fractional governing equations that can model these systems. To these ends we will build direct simulations of the infiltration of moisture into a porous media containing a distribution of flow obstacles. When the obstacles form a repeating pattern, this problem can be viewed as a limit case of the classical one-phase Stefan melting problem and the measure of the advance of the infiltration length changes with the square root of time. When the obstacles are distributed in a fractal pattern, however, the infiltration shows a sub-diffusive behavior, where the time exponent is less that the square root. Through considering the time scaling of Brownian motion in a fractal obstacle filed we are bale to directly associate this sub-diffusive time exponent to the fractal dimension of the obstacle filed. This in turn, allows us to develop fractional calculus based governing equations, with a closed particular solution, for moisture infiltration into a fractal obstacle field. The talk will close with considerations as to how these findings can be associated with more general Stefan problem that incorporated fractional calculus treatments.