Constitutive Relations¶
Capillary Pressure - Saturation Functions¶
van Genuchten Saturation Function¶
Capillary pressure is related to saturation by various phenomenological relations, one of which is the van Genuchten (1980) relation
where
where
The quantities
Brooks-Corey Saturation Function¶
The Brooks-Corey saturation function is a limiting form of the van
Genuchten relation for
with
Relative Permeability Functions¶
Two forms of the relative permeability function are implemented based on
the Mualem and Burdine formulations. The quantity
for the Mualem formulation and by
for the Burdine formulation.
Mualem Relative Permeability¶
For the Mualem relative permeability function based on the van Genuchten saturation function is given by the expression
The Mualem relative permeability function based on the Brooks-Corey saturation function is defined by
Burdine Relative Permeability¶
For the Burdine relative permeability function based on the van Genuchten saturation function is given by the expression
The Burdine relative permeability function based on the Brooks-Corey saturation function has the form
Modified Brooks Corey Relative Permeability¶
The modified Brooks Corey relative permeability function can be associated with any saturation function. The liquid relative permeability is defined as
The gas phase relative permeability is defined as
where
Smoothing¶
At the end points of the saturation and relative permeability functions
it is sometimes necessary to smooth the functions in order for the
Newton-Raphson equations to converge. This is accomplished using a third
order polynomial interpolation by matching the values of the function to
be fit (capillary pressure or relative permeability), and imposing zero
slope at the fully saturated end point and matching the derivative at a
chosen variably saturated point that is close to fully saturated. The
resulting equations for coefficients
for chosen points
The conditions imposed on the smoothing equations for capillary pressure