Derivation of the volatile oil sandface/surface flowrates relation

(1)	$\begin{array}{l}\displaystyle \end{array}$

$\begin{array}{l}\displaystyle q_O = q_{Oo} + q_{Og}\end{array}$
$\begin{array}{l}\displaystyle q_G = q_{Gg} + q_{Go}\end{array}$

$\begin{array}{l}\displaystyle q_W = q_{Ww}\end{array}$

(2)	$\begin{array}{l}\displaystyle \end{array}$

$\begin{array}{l}\displaystyle R_s = q_{Go}/q_{Oo}\end{array}$
$\begin{array}{l}\displaystyle R_v = q_{Og}/q_{Gg}\end{array}$

so that:

(3)	$\begin{array}{l}\displaystyle q_O = q_{Oo} + R_v \, q_{Gg}\end{array}$

(4)	$\begin{array}{l}\displaystyle q_G = q_{Gg} + R_s \, q_{Oo}\end{array}$

(5)	$\begin{array}{l}\displaystyle q_W = q_{Ww}\end{array}$

(6)	$\begin{array}{l}\displaystyle q_{Oo}= \frac{q_o}{B_o}\end{array}$

(7)	$\begin{array}{l}\displaystyle q_{Gg} = \frac{q_g}{B_g}\end{array}$

(8)	$\begin{array}{l}\displaystyle q_{Ww} = \frac{q_w}{B_w}\end{array}$

so that:

(9)	$\begin{array}{l}\displaystyle q_O = \frac{q_o}{B_o} + R_v \,\frac{q_g}{B_g}\end{array}$

(10)	$\begin{array}{l}\displaystyle q_G = \frac{q_g}{B_g} + R_s \, \frac{q_o}{B_o}\end{array}$

(11)	$\begin{array}{l}\displaystyle q_W = \frac{q_w}{B_w}\end{array}$

and solving the above system of equations leads to:

(12)	$\begin{array}{l}\displaystyle q_o = \frac{B_o \cdot (q_O - R_v \, q_G)}{1- R_v \, R_s}\end{array}$

(13)	$\begin{array}{l}\displaystyle q_o = \frac{B_o \cdot (q_G - R_s \, q_O)}{1- R_v \, R_s}\end{array}$

(14)	$\begin{array}{l}\displaystyle q_w = B_w \cdot q_w\end{array}$

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