Grid-related Computation

(1)	$\begin{array}{l}\displaystyle \epsilon = \text{1e-12}\end{array}$

(2)	$\begin{array}{l}\displaystyle v_{\alpha\beta} = \frac{V_{\alpha\beta}}{V_{\alpha\beta} + V_{\beta\alpha}}\end{array}$

(3)	$\begin{array}{l}\displaystyle COS2_{\alpha\beta} = \left( \bar n_{\alpha\beta} \cdot \; \bar e_{k\max} \right)^2\end{array}$

(4)	$\begin{array}{l}\displaystyle Rg_{\alpha\beta} = R_{\alpha\beta} \cdot g \cdot \sqrt{1-COS2_{\alpha\beta}}\end{array}$

(5)	$\begin{array}{l}\displaystyle Rk_{11\alpha\beta} = R^{-1}_{\alpha\beta} \cdot \left( k_{\max\alpha\beta} \cdot COS2_{\alpha\beta} + k_{\min\alpha\beta} \cdot (1 - COS2_{\alpha\beta}) \right)\end{array}$

(6)	$\begin{array}{l}\displaystyle \tilde B_w = \tilde B_w(T,p) = B_w\end{array}$

(7)	$\begin{array}{l}\displaystyle \tilde B_o = \tilde B_o(T,p) = \frac{B_o - R_s \cdot B_g}{1-R_s \, R_v}\end{array}$

(8)	$\begin{array}{l}\displaystyle \tilde B_g = \tilde B_g(T,p) = \frac{B_g - R_v \cdot B_o}{1-R_s \, R_v}\end{array}$

(9)	$\begin{array}{l}\displaystyle c_\phi(p) = \left[ \, \text{table interpolation} \, \right ]\end{array}$

or

Dobrynin Pore compressibility-pressure @model :

(10)	$\begin{array}{l}\displaystyle \displaystyle c_\phi(p) = c_{\phi i} \cdot \frac{ \ln \left( \frac{ p_n }{p_{\rm max}} \right) }{ \ln \left( \frac{p_{ni}}{p_{\rm max}} \right) }\end{array}$

(11)	$\begin{array}{l}\displaystyle p_n = p_{\rm min} + 1.75 \cdot \phi^{0.51} \cdot (p_{\rm max} - p)\end{array}$

(12)	$\begin{array}{l}\displaystyle p_{ni} = p_{\rm min} + 1.75 \cdot \phi^{0.51} \cdot (p_{\rm max} - p_i)\end{array}$

Wide pressure range: p_min = 1 MPa < p < p_max = 200 MPa

Initialization: t = 0

(13)	$\begin{array}{l}\displaystyle s^0_{w\alpha} = s_{wi\alpha}\end{array}$

(14)	$\begin{array}{l}\displaystyle s^0_{o\alpha} = s_{oi\alpha}\end{array}$

(15)	$\begin{array}{l}\displaystyle s^0_{g\alpha} = s_{gi\alpha}\end{array}$

(16)	$\begin{array}{l}\displaystyle V^0_{\phi\alpha} = V_\alpha \cdot \phi^0_\alpha, \quad \phi^0_\alpha = \phi^0_{i\alpha}\end{array}$

(17)	$\begin{array}{l}\displaystyle V^0_{w\alpha} = s^0_{w\alpha} \cdot V^0_{\phi\alpha}\end{array}$

(18)	$\begin{array}{l}\displaystyle V^0_{o\alpha} = s^0_{o\alpha} \cdot V^0_{\phi\alpha}\end{array}$

(19)	$\begin{array}{l}\displaystyle V^0_{g\alpha} = s^0_{g\alpha} \cdot V^0_{\phi\alpha}\end{array}$

(20)	$\begin{array}{l}\displaystyle V^0_{W\alpha} = \frac{V^0_{w\alpha}}{B_{wi}}\end{array}$

(21)	$\begin{array}{l}\displaystyle V^0_{O\alpha} =\frac{V^0_o}{B_{oi}} + R_{vi} \cdot \frac{V^0_g}{B_{gi}}\end{array}$

(22)	$\begin{array}{l}\displaystyle V^0_{G\alpha} =\frac{V^0_G}{B_{gi}} + R_{si} \cdot \frac{V^0_o}{B_{oi}}\end{array}$

(23)	$\begin{array}{l}\displaystyle I^0_{W\alpha} = \frac{V^0_{W\alpha}}{V^0_{\phi\alpha}}\end{array}$

(24)	$\begin{array}{l}\displaystyle I^0_{O\alpha} = \frac{V^0_{O\alpha}}{V^0_{\phi\alpha}}\end{array}$

(25)	$\begin{array}{l}\displaystyle I^0_{G\alpha} = \frac{V^0_{G\alpha}}{V^0_{\phi\alpha}}\end{array}$

(26)	$\begin{array}{l}\displaystyle p^0_{w\alpha} = p^0_\alpha + p_{cw}(s^0_\alpha)\end{array}$

(27)	$\begin{array}{l}\displaystyle p^0_{o\alpha} = p^0_\alpha + p_{co}(s^0_\alpha)\end{array}$

(28)	$\begin{array}{l}\displaystyle p^0_{g\alpha} = p^0_\alpha + p_{cg}(s^0_\alpha)\end{array}$

(29)	$\begin{array}{l}\displaystyle U^0_{W\alpha\beta} = 0\end{array}$

(30)	$\begin{array}{l}\displaystyle U^0_{O\alpha\beta} = 0\end{array}$

(31)	$\begin{array}{l}\displaystyle U^0_{G\alpha\beta} = 0\end{array}$

Progression: t → t + 1

Loop Over ▶ – ◀

▶

Cross-Cell Face Saturation

(32)	$\begin{array}{l}\displaystyle s^t_{w\alpha\beta}=s^t_{w\alpha} \cdot v_{\alpha\beta} + s^t_{w\beta}\cdot v_{\beta\alpha}\end{array}$

(33)	$\begin{array}{l}\displaystyle s^t_{o\alpha\beta}=s^t_{o\alpha} \cdot v_{\alpha\beta} + s^t_{o\beta}\cdot v_{\beta\alpha}\end{array}$

(34)	$\begin{array}{l}\displaystyle s^t_{g\alpha\beta}=s^t_{g\alpha} \cdot v_{\alpha\beta} + s^t_{g\beta}\cdot v_{\beta\alpha}\end{array}$

Cell Relative Permeability

(35)	$\begin{array}{l}\displaystyle k_{rw\alpha\beta} = k_{rw}(s^t_{\alpha\beta})\end{array}$

(36)	$\begin{array}{l}\displaystyle k_{ro\alpha\beta} = k_{ro}(s^t_{\alpha\beta})\end{array}$

(37)	$\begin{array}{l}\displaystyle k_{rg\alpha\beta} = k_{rg}(s^t_{\alpha\beta})\end{array}$

Fluid Phase Viscosity

(38)	$\begin{array}{l}\displaystyle \mu_{w\alpha\beta} = \mu_w(p^t_{w\alpha\beta})\end{array}$

(39)	$\begin{array}{l}\displaystyle \mu_{o\alpha\beta} = \mu_o(p^t_{o\alpha\beta})\end{array}$

(40)	$\begin{array}{l}\displaystyle \mu_{g\alpha\beta} = \mu_g(p^t_{g\alpha\beta})\end{array}$

Phase Densities

(41)	$\begin{array}{l}\displaystyle \rho_{w\alpha} = {\mathring \rho_W} \, \cdot I\!B^t_{w\alpha}\end{array}$

(42)	$\begin{array}{l}\displaystyle \rho_{o\alpha} = \left( \mathring \rho_O + \mathring \rho_G \cdot R^t_{s\alpha} \right) \cdot I\!B^t_{o\alpha}\end{array}$

(43)	$\begin{array}{l}\displaystyle \rho_{g\alpha} = \left( \mathring \rho_G + \mathring \rho_O \cdot R^t_{v\alpha} \right) \cdot I\!B^t_{g\alpha}\end{array}$

Capillary and Gravity Cell Centre-to-Face Pressure Drop

(44)	$\begin{array}{l}\displaystyle \delta p_{w\alpha\beta} = p_{cw}(s^t_{\alpha\beta}) - p_{cw}(s^t_{\alpha}) - \rho_{w\alpha} \cdot Rg_{\alpha\beta}\end{array}$

(45)	$\begin{array}{l}\displaystyle \delta p_{o\alpha\beta} = p_{co}(s^t_{\alpha\beta}) - p_{co}(s^t_{\alpha}) - \rho_{o\alpha} \cdot Rg_{\alpha\beta}\end{array}$

(46)	$\begin{array}{l}\displaystyle \delta p_{g\alpha\beta} = p_{cg}(s^t_{\alpha\beta}) - p_{cg}(s^t_{\alpha}) - \rho_{g\alpha} \cdot Rg_{\alpha\beta}\end{array}$

Multiphase Cross-Cell Face Pressure

(47)	$\begin{array}{l}\displaystyle p^t_{\alpha\beta} = s^t_{w\alpha\beta} \cdot p^t_{w\alpha \beta} + s^t_{o\alpha\beta} \cdot p^t_{o\alpha \beta} + s^t_{g\alpha\beta} \cdot p^t_{g\alpha \beta}\end{array}$

(48)	$\begin{array}{l}\displaystyle k_{p\alpha\beta} = k_{\alpha\beta}(\phi^t_{\alpha \beta})\end{array}$

or

(49)	$\begin{array}{l}\displaystyle k_{p\alpha\beta} = \exp \left[ \; c_{k\alpha\beta} \cdot \left( p^t_{\alpha \beta} - p_{0\alpha \beta} \right) \; \right]\end{array}$

(50)	$\begin{array}{l}\displaystyle kRk_{11\alpha\beta} = k_{p\alpha\beta} \cdot Rk_{11\alpha\beta}\end{array}$

(51)	$\begin{array}{l}\displaystyle G_{W\alpha\beta} = \frac{k_{rw\alpha\beta}}{\mu_{w\alpha\beta}} \cdot kRk_{11\alpha\beta}\end{array}$

(52)	$\begin{array}{l}\displaystyle G_{o\alpha\beta} = \frac{k_{ro\alpha\beta}}{\mu_{o\alpha\beta}} \cdot kRk_{11\alpha\beta}\end{array}$

(53)	$\begin{array}{l}\displaystyle G_{g\alpha\beta} = \frac{k_{rg\alpha\beta}}{\mu_{g\alpha\beta}} \cdot kRk_{11\alpha\beta}\end{array}$

(54)	$\begin{array}{l}\displaystyle T_{W\alpha\beta} = I\!B^t_{w\alpha} \cdot G_{w\alpha\beta}\end{array}$

(55)	$\begin{array}{l}\displaystyle T_{O\alpha\beta} = I\!B^t_{o\alpha} \cdot G_{o\alpha\beta} + R_{v\alpha} \cdot I\!B^t_{g\alpha} \cdot G_{g\alpha\beta}\end{array}$

(56)	$\begin{array}{l}\displaystyle T_{G\alpha\beta} = R_{s\alpha} \cdot I\!B^t_{o\alpha} \cdot G_{o\alpha\beta} + I\!B^t_{g\alpha} \cdot G_{g\alpha\beta}\end{array}$

(57)	$\begin{array}{l}\displaystyle L_{W\alpha\beta} = I\!B^t_{w\alpha} \cdot G_{w\alpha\beta} \cdot \delta p_{w\alpha\beta}\end{array}$

(58)	$\begin{array}{l}\displaystyle L_{O\alpha\beta} = I\!B^t_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + R_{v\alpha} \cdot I\!B^t_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta}\end{array}$

(59)	$\begin{array}{l}\displaystyle L_{G\alpha\beta} = R_{s\alpha} \cdot I\!B^t_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + I\!B^t_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta}\end{array}$

(60)	$\begin{array}{l}\displaystyle \eta_{w\alpha\beta} = \left ( T_{w\alpha\beta} - T_{w\beta\alpha} \right)^{-1} \quad \text{if} \; \|T_{w\alpha\beta} - T_{w\beta\alpha}\| > \epsilon \quad \text{else} \; 0\end{array}$

(61)	$\begin{array}{l}\displaystyle \eta_{o\alpha\beta} = \left ( T_{o\alpha\beta} - T_{o\beta\alpha} \right)^{-1} \quad \text{if} \; \|T_{o\alpha\beta} - T_{o\beta\alpha}\| > \epsilon \quad \text{else} \; 0\end{array}$

(62)	$\begin{array}{l}\displaystyle \eta_{g\alpha\beta} = \left ( T_{g\alpha\beta} - T_{g\beta\alpha} \right)^{-1} \quad \text{if} \; \|T_{g\alpha\beta} - T_{g\beta\alpha}\| > \epsilon \quad \text{else} \; 0\end{array}$

Cross-Cell Face Pressure

(63)	$\begin{array}{l}\displaystyle p^{t+1}_{w\alpha\beta} = \eta_{w\alpha\beta} \cdot \left( T_{w\alpha\beta} \cdot p_\alpha - T_{w\beta\alpha} \cdot p_\beta - L_{w\alpha\beta} + L_{w\beta\alpha} \right)\end{array}$

(64)	$\begin{array}{l}\displaystyle p^{t+1}_{o\alpha\beta} = \eta_{o\alpha\beta} \cdot \left( T_{o\alpha\beta} \cdot p_\alpha - T_{o\beta\alpha} \cdot p_\beta - L_{o\alpha\beta} + L_{o\beta\alpha} \right)\end{array}$

(65)	$\begin{array}{l}\displaystyle p^{t+1}_{g\alpha\beta} = \eta_{g\alpha\beta} \cdot \left( T_{g\alpha\beta} \cdot p_\alpha - T_{g\beta\alpha} \cdot p_\beta - L_{g\alpha\beta} + L_{g\beta\alpha} \right)\end{array}$

Cross-Cell Face Flux

(66)	$\begin{array}{l}\displaystyle U^{t+1}_{W\alpha\beta} = T_{W\alpha\beta} \cdot \left( p^{t+1}_{w\alpha\beta} - p_\alpha \right) + L_{W\alpha\beta}\end{array}$

(67)	$\begin{array}{l}\displaystyle U^{t+1}_{O\alpha\beta} = T_{O\alpha\beta} \cdot \left( p^{t+1}_{o\alpha\beta} - p_\alpha \right) + L_{O\alpha\beta}\end{array}$

(68)	$\begin{array}{l}\displaystyle U^{t+1}_{G\alpha\beta} = T_{G\alpha\beta} \cdot \left( p^{t+1}_{g\alpha\beta} - p_\alpha \right) + L_{G\alpha\beta}\end{array}$

Cell Influx

(69)	$\begin{array}{l}\displaystyle \delta I^{t+1}_{W \alpha} = \frac{\delta t}{V^0_{\phi_\alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot U^{t+1}_{W\alpha\beta} \; + \, q^{t+1}_{W\alpha} \right)\end{array}$

(70)	$\begin{array}{l}\displaystyle \delta I^{t+1}_{O \alpha} = \frac{\delta t}{V^0_{\phi_\alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot U^{t+1}_{O\alpha\beta} \; + \, q^{t+1}_{O\alpha} \right)\end{array}$

(71)	$\begin{array}{l}\displaystyle \delta I^{t+1}_{G \alpha} = \frac{\delta t}{V^0_{\phi_\alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot U^{t+1}_{G\alpha\beta} \; + \, q^{t+1}_{G\alpha} \right)\end{array}$

Cell Fluid Balance

(72)	$\begin{array}{l}\displaystyle I^{t+1}_{W \alpha} = I^t_{W \alpha} + \delta I^{t+1}_{W \alpha}\end{array}$

(73)	$\begin{array}{l}\displaystyle I^{t+1}_{O \alpha} = I^t_{O \alpha} + \delta I^{t+1}_{O \alpha}\end{array}$

(74)	$\begin{array}{l}\displaystyle I^{t+1}_{G \alpha} = I^t_{G \alpha} + \delta I^{t+1}_{G \alpha}\end{array}$

Handle Fluid Component Depletion

(75)	$\begin{array}{l}\displaystyle \text{if} \; I^{t+1}_{W \alpha} < 0 \; \Rightarrow \; I^{t+1}_{W \alpha} = 0 \; , \; \; \delta I^{t+1}_{W \alpha} = - I^t_{W \alpha}\end{array}$

(76)	$\begin{array}{l}\displaystyle \text{if} \; I^{t+1}_{O \alpha} < 0 \; \Rightarrow \; I^{t+1}_{O \alpha} = 0 \; , \; \; \delta I^{t+1}_{O \alpha} = - I^t_{O \alpha}\end{array}$

(77)	$\begin{array}{l}\displaystyle \text{if} \; I^{t+1}_{G \alpha} < 0 \; \Rightarrow \; I^{t+1}_{O \alpha} = 0 \; , \; \; \delta I^{t+1}_{G \alpha} = - I^t_{G \alpha}\end{array}$

(78)

$\begin{array}{l}\displaystyle \delta t^*_{W\alpha\beta^-} =\frac{-I_{W\alpha}^t\cdot V_{\phi_{\alpha}}^0-\delta t \cdot\sum_{\beta^+\in\Gamma^+_{\alpha}}A_{\alpha\beta^+} \cdot U^{t+1}_{W\alpha\beta^+}}{\sum_{\beta^-\in\Gamma^-_{\alpha}}A_{\alpha\beta^-}\cdot U^{t+1}_{W\alpha\beta^-} + q^{t+1}_{W\alpha}}\end{array}$

(79)

$\begin{array}{l}\displaystyle \delta t^*_{O\alpha\beta^-} =\frac{-I_{O\alpha}^t\cdot V_{\phi_{\alpha}}^0-\delta t \cdot\sum_{\beta^+\in\Gamma^+_{\alpha}}A_{\alpha\beta^+} \cdot U^{t+1}_{O\alpha\beta^+}}{\sum_{\beta^-\in\Gamma^-_{\alpha}}A_{\alpha\beta^-}\cdot U^{t+1}_{O\alpha\beta^-} + q^{t+1}_{O\alpha}}\end{array}$

(80)

$\begin{array}{l}\displaystyle \delta t^*_{G\alpha\beta^-} =\frac{-I_{G\alpha}^t\cdot V_{\phi_{\alpha}}^0-\delta t \cdot\sum_{\beta^+\in\Gamma^+_{\alpha}}A_{\alpha\beta^+} \cdot U^{t+1}_{G\alpha\beta^+}}{\sum_{\beta^-\in\Gamma^-_{\alpha}}A_{\alpha\beta^-}\cdot U^{t+1}_{G\alpha\beta^-} + q^{t+1}_{G\alpha}}\end{array}$

(81)	$\begin{array}{l}\displaystyle \tilde q^{t+1}_{W\alpha} = q^{t+1}_{W\alpha} \cdot \delta t^*_{W\alpha\beta^-}/\delta t\end{array}$

(82)	$\begin{array}{l}\displaystyle \tilde U^{t+1}_{W\alpha\beta^-} = U^{t+1}_{W\alpha\beta^-} \cdot \delta t^*_{W\alpha\beta^-}/\delta t\end{array}$

(83)	$\begin{array}{l}\displaystyle \tilde U^{t+1}_{W\beta^-\alpha} = \tilde U^{t+1}_{W\alpha\beta^-}\end{array}$

(84)	$\begin{array}{l}\displaystyle \tilde q^{t+1}_{O\alpha} = q^{t+1}_{O\alpha} \cdot \delta t^*_{O\alpha\beta^-}/\delta t\end{array}$

(85)	$\begin{array}{l}\displaystyle \tilde U^{t+1}_{O\alpha\beta^-} = U^{t+1}_{O\alpha\beta^-} \cdot \delta t^*_{O\alpha\beta^-}/\delta t\end{array}$

(86)	$\begin{array}{l}\displaystyle \tilde U^{t+1}_{O\beta^-\alpha} = \tilde U^{t+1}_{O\alpha\beta^-}\end{array}$

(87)	$\begin{array}{l}\displaystyle \tilde q^{t+1}_{G\alpha} = q^{t+1}_{G\alpha} \cdot \delta t^*_{G\alpha\beta^-}/\delta t\end{array}$

(88)	$\begin{array}{l}\displaystyle \tilde U^{t+1}_{G\alpha\beta^-} = U^{t+1}_{G\alpha\beta^-} \cdot \delta t^*_{G\alpha\beta^-}/\delta t\end{array}$

(89)	$\begin{array}{l}\displaystyle \tilde U^{t+1}_{G\beta^-\alpha} = \tilde U^{t+1}_{G\alpha\beta^-}\end{array}$

Reservoir Pressure

(90)

$\begin{array}{l}\displaystyle \tilde B_w \left( p^{t+1}_\alpha + p_{cw}(s^t_\alpha) \right) \cdot I^{t+1}_{W\alpha} \, + \, \tilde B_o \left( p^{t+1}_\alpha + p_{co}(s^t_\alpha) \right) \cdot I^{t+1}_{O\alpha} \, + \, \tilde B_g \left( p^{t+1}_\alpha + p_{cg}(s^t_\alpha) \right) \cdot I^{t+1}_{G\alpha} - \exp \left( c_{\phi_\alpha}(p^{t+1}_\alpha) \cdot (p^{t+1}_\alpha - p^0_\alpha) \right) = 0\end{array}$

(91)	$\begin{array}{l}\displaystyle p^{t+1}_{w\alpha} = p^{t+1}_\alpha + p_{cw}(s^t_\alpha)\end{array}$

(92)	$\begin{array}{l}\displaystyle p^{t+1}_{o\alpha} = p^{t+1}_\alpha + p_{co}(s^t_\alpha)\end{array}$

(93)	$\begin{array}{l}\displaystyle p^{t+1}_{g\alpha} = p^{t+1}_\alpha + p_{cg}(s^t_\alpha)\end{array}$

PVT properties

(94)	$\begin{array}{l}\displaystyle B^{t+1}_{w\alpha} = B_w ( T^{t+1}_\alpha, \, p^{t+1}_{w\alpha})\end{array}$

(95)	$\begin{array}{l}\displaystyle B^{t+1}_{o\alpha} = B_o ( T^{t+1}_\alpha, \, p^{t+1}_{o\alpha})\end{array}$

(96)	$\begin{array}{l}\displaystyle B^{t+1}_{g\alpha} = B_g ( T^{t+1}_\alpha, \, p^{t+1}_{g\alpha})\end{array}$

(97)	$\begin{array}{l}\displaystyle I\!B^{t+1}_{w\alpha} = \left[ B^{t+1}_{w\alpha} \right]^{-1}\end{array}$

(98)	$\begin{array}{l}\displaystyle I\!B^{t+1}_{o\alpha} = \left[ B^{t+1}_{o\alpha} \right]^{-1}\end{array}$

(99)	$\begin{array}{l}\displaystyle I\!B^{t+1}_{g\alpha} = \left[ B^{t+1}_{g\alpha} \right]^{-1}\end{array}$

(100)

$\begin{array}{l}\displaystyle R^{t+1}_{v\alpha} = R_v(p^{t+1}_\alpha)\end{array}$

(101)

$\begin{array}{l}\displaystyle R^{t+1}_{s\alpha} = R_s(p^{t+1}_\alpha)\end{array}$

(102)

$\begin{array}{l}\displaystyle RR^{t+1}_{\alpha} = 1 - R^{t+1}_{v\alpha} \, R^{t+1}_{s\alpha}\end{array}$

(103)

$\begin{array}{l}\displaystyle BB^{t+1}_{o\alpha} = \frac{B^{t+1}_{o\alpha}}{RR^{t+1}_{\alpha}}\end{array}$

(104)

$\begin{array}{l}\displaystyle BB^{t+1}_{g\alpha} = \frac{B^{t+1}_{g\alpha}}{RR^{t+1}_{\alpha}}\end{array}$

(105)

$\begin{array}{l}\displaystyle \tilde B^{t+1}_{w\alpha} = \tilde B_w (T^{t+1}_\alpha, p^{t+1}_{w\alpha})\end{array}$

(106)

$\begin{array}{l}\displaystyle \tilde B^{t+1}_{o\alpha} = \tilde B_o (T^{t+1}_\alpha, p^{t+1}_{o\alpha})\end{array}$

(107)

$\begin{array}{l}\displaystyle \tilde B^{t+1}_{g\alpha} = \tilde B_{g} (T^{t+1}_\alpha, p^{t+1}_{g\alpha})\end{array}$

Fluid Component Volumes

(108)

$\begin{array}{l}\displaystyle V^{t+1}_{W\alpha} = V^0_{\phi\alpha} \cdot I^{t+1}_{W\alpha}\end{array}$

(109)

$\begin{array}{l}\displaystyle V^{t+1}_{O\alpha} = V^0_{\phi\alpha} \cdot I^{t+1}_{O\alpha}\end{array}$

(110)

$\begin{array}{l}\displaystyle V^{t+1}_{G\alpha} = V^0_{\phi\alpha} \cdot I^{t+1}_{G\alpha}\end{array}$

Reservoir Phase Volumes

(111)

$\begin{array}{l}\displaystyle V^{t+1}_{w\alpha} = B_w(p^{t+1}_\alpha) \cdot V^{t+1}_{W\alpha}\end{array}$

(112)

$\begin{array}{l}\displaystyle V^{t+1}_{o\alpha} = BB^{t+1}_{o\alpha} \cdot \left( \, V^{t+1}_{O\alpha} - R^{t+1}_{v\alpha} \cdot V^{t+1}_{G\alpha} \, \right)\end{array}$

(113)

$\begin{array}{l}\displaystyle V^{t+1}_{g\alpha} = BB^{t+1}_{g\alpha} \cdot \left(\, V^{t+1}_{G\alpha} - R^{t+1}_{s\alpha} \cdot V^{t+1}_{O\alpha} \, \right)\end{array}$

(114)

$\begin{array}{l}\displaystyle V^{t+1}_{\phi\alpha} = V^{t+1}_{w\alpha} + V^{t+1}_{o\alpha} + V^{t+1}_{g\alpha}\end{array}$

Reservoir Saturation

(115)

$\begin{array}{l}\displaystyle s^{t+1}_{w\alpha} = \frac{V^{t+1}_{w\alpha}}{V^{t+1}_{\phi\alpha}}\end{array}$

(116)

$\begin{array}{l}\displaystyle s^{t+1}_{o\alpha} = \frac{V^{t+1}_{o\alpha}}{V^{t+1}_{\phi\alpha}}\end{array}$

(117)

$\begin{array}{l}\displaystyle s^{t+1}_{g\alpha} = \frac{V^{t+1}_{g\alpha}}{V^{t+1}_{\phi\alpha}}\end{array}$

◀

Page tree

Grid-related Computation

Initialization: t = 0

Progression: t → t + 1

See Also

Page tree

Modified Black Oil Reservoir Flow Grid Computation @ algorithm

Grid-related Computation

Initialization: t = 0

Progression: t → t + 1

See Also