A mechanical picture of fractional-order Darcy equation

•Fractional-order Darcy equation is discussed in the paper.•A physical model providing anomalous relation among pressure and flux has been reported.•Appropriate bounds of the fractional operators involved have been provided.•Different kind of diffusion have been introduced.

In this paper the authors show that fractional-order force-flux relations are obtained considering the flux of a viscous fluid across an elastic porous media. Indeed the one-dimensional fluid mass transport in an unbounded porous media with power-law variation of geometrical and physical properties yields a fractional-order relation among the ingoing flux and the applied pressure to the control section. As a power-law decay of the physical properties from the control section is considered, then the flux is related to a Caputo fractional derivative of the pressure of order 0⩽β≤10⩽β≤1. If, instead, the physical properties of the media show a power-law increase from the control section, then flux is related to a fractional-order integral of order 0⩽β≤10⩽β≤1. These two different behaviors may be related to different states of the mass flow across the porous media.