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:

$$S^e=\{\begin{array}{ll}{\left(\frac{p^b}{s}\right)}^{\lambda }& \text{for:}s\ge p^b\\ 1& \text{for:}s<p^b,\end{array}$$

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:

$$S^w=S^{wr}+(1-S^{wr})\{\begin{array}{ll}{\left(\frac{p^b}{s}\right)}^{\lambda }& \text{for:}s\ge p^b\\ 1& \text{for:}s<p^b,\end{array}$$

The input line takes the form:

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

Theory

The contributions to the Jacobian read:

$$\begin{array}{r}\frac{\partial S^w}{\partial p^c}=-(1-S^{wr})\lambda {\left(\frac{p^b}{s}\right)}^{\lambda }\frac{1}{s}.\end{array}$$

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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. ↩