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Brooks & Corey model

The Brooks and Corey (1964)1 model is an empirical relationship for the soil-water retention curve. It is particularly known for its simple power-law formulation and its explicit definition of an air-entry pressure. The relationship is defined as a piecewise function, distinguishing between the behavior below and above this pressure threshold.

A key characteristic, and a notable shortcoming of this model, is the sharp discontinuity in the curve's derivative at the air-entry value, which can pose challenges for numerical solvers.

Governing Equation

The model expresses the effective degree of saturation, \(S^e\), as a function of capillary pressure, \(p^c\). The version implemented in numgeo for the total degree of saturation, \(S^w\), is:

\[ S^w(s) = S^{wr} + (1 - S^{wr}) \cdot S^e(p^c) \]

where the effective saturation, \(S^e\), is given by:

\[ S^e = \begin{cases} \left(\frac{p_b}{p^c}\right)^\lambda & \text{for } p^c \ge p_b \\ 1 & \text{for } p^c < p_b \end{cases} \]

Therein,

  • \(p_b\) is the air-entry pressure. This is the minimum capillary pressure required for air to begin entering the largest pores, marking the onset of desaturation. It is also referred to as the bubbling pressure.
  • \(\lambda\) - the pore-size distribution Index. This dimensionless exponent reflects the uniformity of the pore sizes. A smaller \(\lambda\) value corresponds to a wider range of pore sizes, while a larger value indicates a more uniform pore structure.
  • \(S^{wr}\) (optional) is the residual saturation: The degree of saturation at which a further increase in capillary pressure does not produce a significant additional amount of drainage. This parameter is written as Swr in the input.

Jacobian Contribution

The partial derivative of the degree of saturation with respect to suction, \(\frac{\partial S^w}{\partial s}\), is required for the numerical solver. Consistent with the model's piecewise definition, the derivative is:

\[ \frac{\partial S^w}{\partial p^c} = \begin{cases} - (1 - S^{wr}) \lambda \left(\frac{p_b}{p^c}\right)^\lambda \frac{1}{s} & \text{for } p^c > p_b \\ 0 & \text{for } p^c < p_b \end{cases} \]

Note that the derivative is undefined at \(p^c = p_b\), which requires special handling in the numerical implementation.

  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. ↩