# Asymptotic and numerical analysis of a porous medium model for transpiration-driven sap flow in trees

## TL;DR

This paper presents a comprehensive 3D porous medium model for sap flow in trees, deriving analytical solutions and validating them with numerical simulations and experimental data, to better understand flow regimes and the effects of stem properties.

## Contribution

It introduces an asymptotic analysis framework for a nonlinear sap flow model, highlighting the influence of stem aspect ratio and anisotropy on flow behavior.

## Key findings

- Analytical series solutions for saturation and velocity are derived.
- Flow regimes are characterized by key non-dimensional parameters.
- Model validation shows good agreement with experimental data on Norway spruce.

## Abstract

We develop a 3D porous medium model for sap flow within a tree stem, which consists of a nonlinear parabolic partial differential equation with a suitable transpiration source term. Using an asymptotic analysis, we derive approximate series solutions for the liquid saturation and sap velocity for a general class of coefficient functions. Several important non-dimensional parameters are identified that can be used to characterize various flow regimes. We investigate the relative importance of stem aspect ratio versus anisotropy in the sapwood hydraulic conductivity, and how these two effects impact the radial and vertical components of sap velocity. The analytical results are validated by means of a second-order finite volume discretization of the governing equations, and comparisons are drawn to experimental results on Norway spruce trees.