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

(1)$$s_e={[1+{\left(\frac{p_c}{p_c^0}\right)}^n]}^{-m},$$

where $p_c$ represents the capillary pressure [Pa], and the effective saturation $s_e$ is defined by

(2)$$s_e=\frac{s-s_r}{s_0-s_r},$$

where $s_r$ denotes the residual saturation, and $s_0$ denotes the maximum saturation. The inverse relation is given by

(3)$$p_c=p_c^0{(s_e^{-1/m}-1)}^{1/n}.$$

The quantities $m$, $n$ and $p_c^0$ are impirical constants determined by fitting to experimental data.

Brooks-Corey Saturation Function

The Brooks-Corey saturation function is a limiting form of the van Genuchten relation for $p_c/p_c^0\gg 1$, with the form

(4)$$s_e={\left(\frac{p_c}{p_c^0}\right)}^{-\lambda },$$

with $\lambda =mn$ and inverse relation

(5)$$p_c=p_c^0{s}_e^{-1/\lambda }.$$

Relative Permeability Functions

Two forms of the relative permeability function are implemented based on the Mualem and Burdine formulations. The quantity $n$ is related to $m$ by the expression

(6)$$m=1-\frac{1}{n},n=\frac{1}{1-m},$$

for the Mualem formulation and by

(7)$$m=1-\frac{2}{n},n=\frac{2}{1-m},$$

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

(8)$$k_r=\sqrt{s_e}{\{1-{[1-{\left(s_e\right)}^{1/m}]}^m\}}^2.$$

The Mualem relative permeability function based on the Brooks-Corey saturation function is defined by

(9)$$\begin{array}{rl}{k}_r& =(s_e{)}^{5/2+2/\lambda }\\ & =(p_c/p_c^0{)}^{-(5\lambda /2+2)}.\end{array}$$

Burdine Relative Permeability

For the Burdine relative permeability function based on the van Genuchten saturation function is given by the expression

(10)$$k_r=s_e^2\{1-{[1-{\left(s_e\right)}^{1/m}]}^m\}.$$

The Burdine relative permeability function based on the Brooks-Corey saturation function has the form

(11)$$\begin{array}{rl}{k}_r& =(s_e{)}^{3+2/\lambda }\\ & ={\left(\frac{p_c}{p_c^0}\right)}^{-(3+2\lambda )}.\end{array}$$

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

(12)$$\begin{array}{rl}{s}_{el}& =\left(\frac{s_l-s_{rl}}{1-s_{rl}-s_{rg}}\right)\\ k_{rl}& =k_{rl,\text{max}}{s}_{el}^{n_l}\end{array}$$

The gas phase relative permeability is defined as

(13)$$\begin{array}{rl}{s}_{eg}& =\left(\frac{1-s_l-s_{rg}}{1-s_{rl}-s_{rg}}\right)\\ k_{rg}& =k_{rg,\text{max}}{s}_{eg}^{n_g}\end{array}$$

where $k_{r\alpha ,\text{max}}$, $n_{\alpha }$ and $s_{e\alpha }$ are the maximum relative permeability, modified Brooks Corey exponent and effective saturation for phase $\alpha$.

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 $a_i$, $i=0-3$, are given by

(14)$$\begin{array}{rl}{a}_0+a_1{x}_1+a_2{x}_1^2+a_3{x}_1^3& =f_1,\\ a_0+a_1{x}_2+a_2{x}_2^2+a_3{x}_2^3& =f_2,\\ a_1{x}_1+2a_2{x}_1+3a_3{x}_1^2& =f_1^{\prime },\\ a_1{x}_2+2a_2{x}_2+3a_3{x}_2^2& =f_2^{\prime },\end{array}$$

for chosen points $x_1$ and $x_2$. In matrix form these equations become

(15)$$\begin{array}{r}\left[\begin{array}{cccc}1& x_1& x_1^2& x_1^3\\ 1& x_2& x_2^2& x_2^3\\ 0& 1& 2x_1& 3x_1^2\\ 0& 1& 2x_2& 3x_2^2\end{array}\right]\left[\begin{array}{c}{a}_0\\ a_1\\ a_2\\ a_3\end{array}\right]=\left[\begin{array}{c}{f}_1\\ f_2\\ f_1^{\prime }\\ f_2^{\prime }\end{array}\right].\end{array}$$

The conditions imposed on the smoothing equations for capillary pressure $f=s_e(p_c)$ are $x_1=2p_c^0$, $x_2=p_c^0/2$, $f_1=(s_e{)}_1$, $f_2=1$, $f_1^{\prime }=(s_e^{\prime }{)}_1$, $f_2^{\prime }=0$. For relative permeability $f=k_r(s_e)$, $x_1=1$, $x_2=0.99$, $f_1=1$, $f_2=(k_r{)}_2$, $f_1^{\prime }=0$, $f_2^{\prime }=(k_r^{\prime }{)}_2$.