Volumetric flowrate of the fluid phase across the well-reservoir contact

In most popular practical case of a 3-phase fluid model this will be:

Oil sandface flowrate	$\begin{array}{l}q_o\end{array}$
Gas sandface flowrate	$\begin{array}{l}q_g\end{array}$
Water sandface flowrate	$\begin{array}{l}q_w\end{array}$
Total sandface flowrate	$\begin{array}{l}q_t\end{array}$

In case of Volatile Oil Reservoir the relation to surface flowrates $\begin{array}{l}\{ q_O, \, q_G, \, q_W \}\end{array}$ and mass flowrates $\begin{array}{l}\{ \dot m_O, \, \dot m_G, \, \dot m_W \}\end{array}$ is given by following equations:

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

(2)	$\begin{array}{l}\displaystyle \rho_o = \frac{ \dot m_o}{q_o}= \frac{\rho_O + \rho_G \, R_s}{B_o}\end{array}$

(3)	$\begin{array}{l}\displaystyle \dot m_o = \rho_o \cdot q_o = (\rho_O + \rho_G \, R_s) \cdot \frac{ q_o}{B_o} = \cdot \frac{(\rho_O + \rho_G \, R_s) \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}\end{array}$

(4)	$\begin{array}{l}\displaystyle q_g = \frac{ B_g \cdot ( q_G - R_s \, q_O)}{1- R_v \, R_s}\end{array}$

(5)	$\begin{array}{l}\displaystyle \rho_g = \frac{\dot m_g}{q_g}= \frac{\rho_G + \rho_O \, R_v}{B_g}\end{array}$

(6)	$\begin{array}{l}\displaystyle \dot m_g = \rho_g \cdot q_g = (\rho_G + \rho_O \, R_v) \cdot \frac{q_g }{B_g} = \frac{ (\rho_G + \rho_O \, R_v) \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}\end{array}$

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

(8)	$\begin{array}{l}\displaystyle \rho_w =\frac{\dot m_w}{q_w}= \frac{\rho_W}{B_w}\end{array}$

(9)	$\begin{array}{l}\displaystyle \dot m_w = \rho_w \cdot q_w = \rho_W \cdot \frac{q_w}{B_w} = B_w \cdot q_W\end{array}$

(10)	$\begin{array}{l}\displaystyle q_t = q_o + q_g + q_w\end{array}$

(11)	$\begin{array}{l}\displaystyle q_t = \frac{B_o - B_g \, R_s}{1-R_v \, R_s} \cdot q_O +\frac{B_g - B_o \, R_v}{1-R_v \, R_s} \cdot q_G + B_w \cdot q_W\end{array}$

(12)	$\begin{array}{l}\displaystyle q_t = \frac{B_o - B_g \, R_s}{1-R_v \, R_s } \cdot \frac{\dot m_O }{\rho_O} +\frac{B_g - B_o \, R_v}{1-R_v \, R_s } \cdot \frac{\dot m_G }{\rho_G} + B_w\cdot \frac{\dot m_W}{\rho_W}\end{array}$

(13)	$\begin{array}{l}\displaystyle \dot m = \dot m_o + \dot m_g + \dot m_w = \dot m_O + \dot m_G + \dot m_G\end{array}$

(14)	$\begin{array}{l}\displaystyle \rho_t = \frac{\dot m}{q_t} = \frac{\dot m_O + \dot m_G + \dot m_G}{ \frac{B_o - B_g \, R_s}{1-R_v \, R_s } \cdot \frac{\dot m_O }{\rho_O} +\frac{B_g - B_o \, R_v}{1-R_v \, R_s } \cdot \frac{\dot m_G }{\rho_G} + B_w\cdot \frac{\dot m_W}{\rho_W} }\end{array}$

Derivation

Derivation of the volatile oil sandface/surface flowrates relation

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See Also

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Sandface flowrates

See Also