Volumetric calculations

(1)	$\begin{array}{l}\displaystyle q_O = q_{Oo} + q_{Og}\end{array}$

(2)	$\begin{array}{l}\displaystyle q_G = q_{Gg} + q_{Go}\end{array}$

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

Following the definition of Solution GOR (Rs) and Vaporized Oil Ratio (Rv) :

(4)	$\begin{array}{l}\displaystyle R_s = q_{Go}/q_{Oo}\end{array}$

(5)	$\begin{array}{l}\displaystyle R_v = q_{Og}/q_{Gg}\end{array}$

so that:

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

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

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

Following the definition of Oil formation volume factor (Bo) , Gas formation volume factor (Bg) and Water formation volume factor (Bw):

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

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

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

so that:

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

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

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

and solving the above system of equations leads to:

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

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

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

Mass calculations

The oil phase $\begin{array}{l}()_o\end{array}$ includes oil component $\begin{array}{l}()_{Oo}\end{array}$ and gas component $\begin{array}{l}()_{Go}\end{array}$ so that the oil phase mass flux is:

(18)	$\begin{array}{l}\displaystyle m_o = m_{Oo} + m_{Go}\end{array}$

The gas phase $\begin{array}{l}()_g\end{array}$ includes gas component $\begin{array}{l}()_{Gg}\end{array}$ and oil component $\begin{array}{l}()_{Og}\end{array}$ so that the gas phase mass flux is:

(19)	$\begin{array}{l}\displaystyle m_g = m_{Gg} + m_{Og}\end{array}$

The water phase $\begin{array}{l}()_w\end{array}$ includes water component $\begin{array}{l}()_{Ww}\end{array}$ only so that the water phase mass flux is:

(20)	$\begin{array}{l}\displaystyle m_w = m_{Ww}\end{array}$

→

(21)	$\begin{array}{l}\displaystyle m_o = \rho_O \cdot q_{Oo} + \rho_G \cdot q_{Go}\end{array}$

(22)	$\begin{array}{l}\displaystyle m_g = \rho_G \cdot q_{Gg} + \rho_O \cdot q_{Og}\end{array}$

(23)	$\begin{array}{l}\displaystyle m_w = \rho_W \cdot q_{Ww}\end{array}$

→

(24)	$\begin{array}{l}\displaystyle m_o = \rho_O \cdot q_{Oo} + \rho_G \cdot R_s \, q_{Oo}\end{array}$

(25)	$\begin{array}{l}\displaystyle m_g = \rho_G \cdot q_{Gg} + \rho_O \cdot R_v \, q_{Gg}\end{array}$

(26)	$\begin{array}{l}\displaystyle m_w = \rho_W \cdot q_{Ww}\end{array}$

→

(27)	$\begin{array}{l}\displaystyle m_o = (\rho_O + \rho_G \cdot R_s) \cdot q_{Oo}\end{array}$

(28)	$\begin{array}{l}\displaystyle m_g = (\rho_G + \rho_O \cdot R_v) \cdot q_{Gg}\end{array}$

(29)	$\begin{array}{l}\displaystyle m_w = \rho_W \cdot q_{Ww}\end{array}$

→

(30)	$\begin{array}{l}\displaystyle m_o = (\rho_O + \rho_G \cdot R_s) \cdot \frac{q_o}{B_o}\end{array}$

(31)	$\begin{array}{l}\displaystyle m_g = (\rho_G + \rho_O \cdot R_v) \cdot \frac{q_g}{B_g}\end{array}$

(32)	$\begin{array}{l}\displaystyle m_w = \rho_W \cdot \frac{q_w}{B_w}\end{array}$

→

(33)	$\begin{array}{l}\displaystyle \rho_o = \frac{\rho_O + \rho_G \cdot R_s}{B_o}\end{array}$

(34)	$\begin{array}{l}\displaystyle \rho_g = \frac{\rho_G + \rho_O \cdot R_v}{B_g}\end{array}$

(35)	$\begin{array}{l}\displaystyle \rho_w = \frac{\rho_W}{B_w}\end{array}$

The total mass flow of all phases:

(36)	$\begin{array}{l}\displaystyle \dot m = \dot m_o + \dot m_g + \dot m_w = (\rho_O + \rho_G \cdot R_s) \cdot \frac{q_o}{B_o} + (\rho_G + \rho_O \cdot R_v) \cdot \frac{q_g}{B_g} + \rho_W \cdot \frac{q_w}{B_w}\end{array}$

→

(37)	$\begin{array}{l}\displaystyle \dot m = (\rho_O + \rho_G \cdot R_s) \cdot \frac{q_O - R_v \, q_G}{1-R_v \, R_s} + (\rho_G + \rho_O \cdot R_v) \cdot \frac{q_G - R_s \, q_O}{1- R_v \, R_s} + \rho_W \cdot q_W\end{array}$

→

(38)	$\begin{array}{l}\displaystyle \dot m = \frac{ (\rho_O + \rho_G \cdot R_s)\cdot (q_O - R_v \, q_G) + (\rho_G + \rho_O \cdot R_v) \cdot (q_G - R_s \, q_O) }{1-R_v \, R_s} + \rho_W \cdot \frac{q_w}{B_w}\end{array}$

→

(39)	$\begin{array}{l}\displaystyle \dot m = \frac{ \rho_O \, q_O \, (1- R_v \, R_s) + \rho_G \, q_G \, (1- R_v \, R_s) }{1-R_v \, R_s} + \rho_W \cdot q_W\end{array}$

→

(40)	$\begin{array}{l}\displaystyle \dot m = \rho_O \cdot q_O + \rho_G \cdot q_G + \rho_W \cdot q_W = \dot m_O + \dot m_G + \dot m_W\end{array}$

→

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

which means that total mass flux of all fluid phases is equal to the total mass flux of all fluid components.

As volatile oil model does not assume water-component exchange between phases the equality (41) can be broken down into two equalities:

(42)	$\begin{array}{l}\displaystyle \dot m_{HC} = \dot m_o + \dot m_g = \dot m_O + \dot m_G\end{array}$

(43)	$\begin{array}{l}\displaystyle \dot m_w = \dot m_W\end{array}$

The total fluid density of Volatile Oil fluid @model is given by following equation (see Multiphase fluid for derivation):

(44)	$\begin{array}{l}\displaystyle \rho = s_o \, \rho_o + s_g \, \rho_g + s_w \, \rho_w\end{array}$

The total fluid compressibility of multiphase fluid is given by following equation (see Multiphase fluid for derivation):

(45)	$\begin{array}{l}\displaystyle c = s_o \, c_o + s_g \, c_g + s_w \, c_w\end{array}$

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Volumetric calculations

Mass calculations

See Also

Page tree

Derivation of the MBO Fluid @model

Volumetric calculations

Mass calculations

See Also