A Percolation-Based Approach to Scaling Infiltration and Evapotranspiration

Optimal flow paths obtained from percolation theory provide a powerful tool that can be used to characterize properties associated with flow such as soil hydraulic conductivity, as well as other properties influenced by flow connectivity and topology. A recently proposed scaling theory for vegetation growth appeals to the tortuosity of optimal paths from percolation theory to define the spatio-temporal scaling of the root radial extent (or, equivalently, plant height). Root radial extent measures the maximum horizontal distance between a plant shoot and the root tips. We apply here the same scaling relationship to unsteady (horizontal) flow associated with plant transpiration. The pore-scale travel time is generated from the maximum flow rate under saturated conditions and a typical pore size. At the field-scale, the characteristic time is interpreted as the growing season duration, and the characteristic length is derived from the measured evapotranspiration in that period. We show that the two scaling results are equivalent, and they are each in accord with observed vegetation growth limits, as well as with actual limiting transpiration values. While the conceptual approach addresses transpiration, most accessed data are for evapotranspiration. The equivalence of the two scaling approaches suggests that, if horizontal flow is the dominant pathway in plant transpiration, horizontal unsteady flow follows the same scaling relationship as root growth. Then, we propose a corresponding scaling relationship to vertical infiltration, a hypothesis which is amenable to testing using infiltration results of Sharma and co-authors. This alternate treatment of unsteady vertical flow may be an effective alternative to the commonly applied method based on the diffusion of water over a continuum as governed by Richards' equation.