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Brooks & Corey model

The effective degree saturation \(S^e\) is expressed as a two-parameter power function of the suction \(s\) (originally, the capillary pressure \(p^c\) was used) 1:

\[\begin{equation*} S^e = \begin{cases} \left(\frac{p^b}{s}\right)^\lambda &\mbox{for:}~~~ s \geq p^b \\ 1 &\mbox{for:}~~~ s < p^b, \end{cases} \end{equation*}\]

where the parameter \(p^b\) is the air entry value (bubbling pressure) and is assumed to be related to the maximum size of pores forming a continuous network of flow paths within the soil. The parameter \(\lambda\) is dimensionless and is referred to as the pore size distribution index. One of the shortcomings of the Brooks \& Corey equation is the sharp discontinuity in the derivative at \(s=p^b\).

Note that in numgeo, the version using the degree of saturation \(S^w\) is implemented:

\[\begin{equation*} S^w = S^{wr}+ (1-S^{wr}) \begin{cases} \left(\frac{p^b}{s}\right)^\lambda &\mbox{for:}~~~ s \geq p^b \\ 1 &\mbox{for:}~~~ s < p^b, \end{cases} \end{equation*}\]

The input line takes the form:

*Hydraulic = Brooks_Corey [,Swr]
lambda, p^b

Theory

The contributions to the Jacobian read:

\[\begin{align*} \frac{\partial S^w}{\partial p^c} = - (1-S^{wr}) \lambda\left(\frac{p^b}{s}\right)^\lambda \frac{1}{s}. \end{align*}\]

  1. Royal Harvard Brooks and Arthur Thomas Corey. Hydraulic properties of porous media and their relation to drainage design. Transactions of the ASAE, 7(1):26–0028, 1964. Publisher: American Society of Agricultural and Biological Engineers. ↩