Saturation Zone Flow

The conservation of mass (water) equation results in the PDE

$a\frac{\partial j}{\partial x} - \frac{\partial D_{sz}}{\partial t} = i - j$     (1)

where

$i$= input from unsaturated zone
$a$=upstream area per unit boundary length draining through the element
$D_{sz}$=saturation zone deficit, and
$j$=$q_{sz}$ (outflow)

The following three assumptions make the solution of Equation (1) simpler.

  1. The flow path discharge $q = aj$ [$\mbox{L}^2$/T] per unit contour width $w$ is proportional to the topographic slope $\sigma$, and
  2. The flow path discharge $q$ is proportional to some function $f$ of soil moisture deficit $D_{sz}$.
  3. $j$ is spatially invariant

Under these assumptions the saturated zone flow $q_{sz}$ can be shown to be:

$q_{sz} = j = \frac{\sigma}{a} T_0 e^{-D_{sz}/m}$     (2)