Modeling Tree Water Flow as an Unsaturated Flow through a Porous Medium

Abstract

The electric circuit analogy has had a profound influence on how tree physiologists measure, model and think about tree water flow. For example, previous models that attempt to account for changes in saturation use the electric circuit analogy to define capacitance as the change in saturation per change in pressure. Given that capacitance is constant, this relationship implies that subjecting a block of wood to a pressure of −2.5 MPa for 2 min results in the same change in saturation as subjecting the same block to the same pressure for 2 days. Given the definition of capacitance, it is unclear how the electric circuit analogy could be used to predict changes in saturation separately from changes in pressure. The inadequacies in the electric circuit analogy discussed in this paper necessitate a new theory of tree water flow that recognizes the sapwood as being a porous medium and explicitly deals with the full implications of the unsaturated flow occurring in the sapwood. The theory proposed in this paper combines the Cohesion theory with a mathematical theory of multiphase flow through porous media. Based on this theory, both saturated and unsaturated tree water flow models are presented. Previous partial differential equation models of tree water flow based on the electric circuit analogy are shown to be mathematically equivalent to the model of saturated porous flow. The unsaturated model of tree water flow explicitly models the pressure profile and the rates of change in saturation and specific interfacial area (a measure of how the water in the unsaturated sapwood is partitioned between mobile and immobile components). The unsaturated model highlights the differences between saturated and unsaturated flow and the need to measure the variables governing tree water flow at higher spatial and temporal resolutions.