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

Initialization: t = 0

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

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

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

  
	(9)
	s^0_{w\alpha} + s^0_{o\alpha} + s^0}_{g\alpha} = 1

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

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

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

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

  
	(14)
	V^0_{\phi\alpha} = V^0_{w\alpha} + V^0_{o\alpha} + V^0_{g\alpha}

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

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

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

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

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

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

Progression: t → t + 1

Loop Over Step #1 – Step #5

Step #1 – Cross-Cell Face Pressure & Flow

(21)	$\begin{array}{l}\displaystyle s_{w\alpha\beta}=s_{w\alpha} \cdot v_{\alpha\beta} + s_{w\beta}\cdot v_{\beta\alpha}\end{array}$

(22)	$\begin{array}{l}\displaystyle s_{o\alpha\beta}=s_{o\alpha} \cdot v_{\alpha\beta} + s_{o\beta}\cdot v_{\beta\alpha}\end{array}$

(23)	$\begin{array}{l}\displaystyle s_{g\alpha\beta}=s_{g\alpha} \cdot v_{\alpha\beta} + s_{g\beta}\cdot v_{\beta\alpha}\end{array}$

(24)	$\begin{array}{l}\displaystyle k_{rw\alpha\beta} = k_{rw}(s_{\alpha\beta})\end{array}$

(25)	$\begin{array}{l}\displaystyle k_{ro\alpha\beta} = k_{ro}(s_{\alpha\beta})\end{array}$

(26)	$\begin{array}{l}\displaystyle k_{rg\alpha\beta} = k_{rg}(s_{\alpha\beta})\end{array}$

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

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

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

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

or

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

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

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

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

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

(36)	$\begin{array}{l}\displaystyle T_{W\alpha\beta} = [B^t_{w\alpha}]^{-1} \cdot G_{w\alpha\beta}\end{array}$

(37)	$\begin{array}{l}\displaystyle T_{O\alpha\beta} = [B^t_{o\alpha}]^{-1} \cdot G_{o\alpha\beta} + R_{v\alpha} \cdot [B^t_{g\alpha}]^{-1} \cdot G_{g\alpha\beta}\end{array}$

(38)	$\begin{array}{l}\displaystyle T_{G\alpha\beta} = R_{s\alpha} \cdot [B^t_{o\alpha}]^{-1} \cdot G_{o\alpha\beta} + [B^t_{g\alpha}]^{-1} \cdot G_{g\alpha\beta}\end{array}$

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

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

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

(42)	$\begin{array}{l}\displaystyle \rho_{w\alpha} = {\mathring \rho_W} \, \cdot [B^t_{w\alpha}]^{-1}\end{array}$

(43)	$\begin{array}{l}\displaystyle \rho_{o\alpha} = \left( \mathring \rho_O + \mathring \rho_G \cdot R^t_{s\alpha} \right) \cdot [B^t_{o\alpha}]^{-1}\end{array}$

(44)	$\begin{array}{l}\displaystyle \rho_{g\alpha} = \left( \mathring \rho_G + \mathring \rho_O \cdot R^t_{v\alpha} \right) \cdot [B^t_{g\alpha}]^{-1}\end{array}$

(45)	$\begin{array}{l}\displaystyle \delta p_{w\alpha\beta} = p_{cw}(s_{\alpha\beta}) - p_{cw}(s_{\alpha}) - \rho_{w\alpha} \cdot Rg_{\alpha\beta}\end{array}$

(46)	$\begin{array}{l}\displaystyle \delta p_{o\alpha\beta} = p_{co}(s_{\alpha\beta}) - p_{co}(s_{\alpha}) - \rho_{o\alpha} \cdot Rg_{\alpha\beta}\end{array}$

(47)	$\begin{array}{l}\displaystyle \delta p_{g\alpha\beta} = p_{cg}(s_{\alpha\beta}) - p_{cg}(s_{\alpha}) - \rho_{g\alpha} \cdot Rg_{\alpha\beta}\end{array}$

(48)	$\begin{array}{l}\displaystyle L_{W\beta\alpha} = B^{-1}_{w\beta} \cdot G_{w\beta\alpha} \cdot \delta p_{w\beta\alpha}\end{array}$

(49)	$\begin{array}{l}\displaystyle L_{O\beta\alpha} = B^{-1}_{o\beta} \cdot G_{o\beta\alpha} \cdot \delta p_{o\beta\alpha} \; + R_{v\beta} \cdot B^{-1}_{g\beta} \cdot G_{g\beta\alpha} \cdot \delta p_{g\beta\alpha}\end{array}$

(50)	$\begin{array}{l}\displaystyle L_{G\beta\alpha} = R_{s\beta} \cdot B^{-1}_{o\beta} \cdot G_{o\beta\alpha} \cdot \delta p_{o\beta\alpha} \; + B^{-1}_{g\beta} \cdot G_{g\beta\alpha} \cdot \delta p_{g\beta\alpha}\end{array}$

(51)	$\begin{array}{l}\displaystyle p_{\alpha\beta} = \eta_{w\alpha\beta} \cdot \left( T_{w\alpha\beta} \cdot p_\alpha - T_{w\beta\alpha} \cdot p_\beta - L_{w\alpha\beta} + L_{w\beta\alpha} \right)\end{array}$

(52)	$\begin{array}{l}\displaystyle p_{\alpha\beta} = \eta_{o\alpha\beta} \cdot \left( T_{o\alpha\beta} \cdot p_\alpha - T_{o\beta\alpha} \cdot p_\beta - L_{o\alpha\beta} + L_{o\beta\alpha} \right)\end{array}$

(53)	$\begin{array}{l}\displaystyle p_{\alpha\beta} = \eta_{g\alpha\beta} \cdot \left( T_{g\alpha\beta} \cdot p_\alpha - T_{g\beta\alpha} \cdot p_\beta - L_{g\alpha\beta} + L_{g\beta\alpha} \right)\end{array}$

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

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

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

Step #2 – Influx

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

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

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

Step #3 – Balance

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

(61)	$\begin{array}{l}\displaystyle I^{t+1}_{O \alpha} = I^t_{O \alpha} + \delta I^{t+1}_{O \alpha}\; , \; \; \delta I^{t+1}_{O \alpha} >0\end{array}$

(62)	$\begin{array}{l}\displaystyle I^{t+1}_{G \alpha} = I^t_{G \alpha} + \delta I^{t+1}_{G \alpha}\; , \; \; \delta I^{t+1}_{G \alpha} >0\end{array}$

Step #4 – Handle depletion

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

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

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

(66)

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

(67)

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

(68)

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

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

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

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

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

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

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

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

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

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

(78)	$\begin{array}{l}\displaystyle I^{t+1}_{W\alpha} = \frac{V^{t+1}_{W\alpha}}{V^0_{\phi_\alpha}} \; , \quad I^t_{W\alpha} = \frac{V^t_{W\alpha}}{V^0_{\phi_\alpha}}\end{array}$

(79)	$\begin{array}{l}\displaystyle I^{t+1}_{O\alpha} = \frac{V^{t+1}_{O\alpha}}{V^0_{\phi_\alpha}}\; , \quad I^t_{O\alpha} = \frac{V^t_{O\alpha}}{V^0_{\phi_\alpha}}\end{array}$

(80)	$\begin{array}{l}\displaystyle I^{t+1}_{G\alpha} = \frac{V^{t+1}_{G\alpha}}{V^0_{\phi_\alpha}}\; , \quad I^t_{G\alpha} = \frac{V^t_{G\alpha}}{V^0_{\phi_\alpha}}\end{array}$

Step #5 – Reservoir Pressure

(81)	$\begin{array}{l}\displaystyle pp^0_{\alpha} = c_{\phi_\alpha} \cdot p^0_\alpha\end{array}$

(82)	$\begin{array}{l}\displaystyle \tilde B_w(pp) \cdot I^{t+1}_{W\alpha} \, + \, \tilde B_o(pp) \cdot I^{t+1}_{O\alpha} \, + \, \tilde B_g(pp) \cdot I^{t+1}_{G\alpha} - \exp \left( pp-pp^0_{\alpha} \right) = 0\end{array}$

(83)	$\begin{array}{l}\displaystyle p^{t+1}_\alpha = c_{\phi_\alpha}^{-1} \cdot pp\end{array}$

Back to Step #1

Step #6 – PVT properties

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

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

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

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

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

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

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

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

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

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

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

Step #7 – Phase volumes

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

Step #8 – Reservoir saturation

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

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

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

  
	(99)
	s^{t+1}_{w\alpha} + s^{t+1}_{o\alpha} + s^{t+1}_{g\alpha} = 1

Ansatz Flux 2D+

Specify local orthogonal coordinate system:

(100)

$\begin{array}{l}\displaystyle \bold e_{\alpha\beta} = \{ \, \bar e_{1\alpha\beta}, \, \bar e_{2\alpha\beta}, \, \bar e_{3\alpha\beta} \, \}\end{array}$

(101)

$\begin{array}{l}\displaystyle \bar e_{1\alpha\beta} \cdot \bar e_{2\alpha\beta} = 0, \quad \bar e_e_{1\alpha\beta} \cdot \bar e_e_{3\alpha\beta} = 0, \quad \bar e_{2\alpha\beta} \cdot \bar e_{3\alpha\beta} = 0, \quad \bar e_{1\alpha\beta} = \bar e_{3\alpha\beta} \times \bar e_{2\alpha\beta}\end{array}$

(102)

$\begin{array}{l}\displaystyle \bar n_{\alpha \beta} = \bar e_{1\alpha\beta} = \cos \zeta_{\alpha \beta} \, \cdot \, \bar e_x + \sin \zeta_{\alpha \beta} \, \cdot \, \bar e_y\end{array}$

(103)

$\begin{array}{l}\displaystyle \cos \theta_{z\alpha\beta} = \bar n_{\alpha\beta} \cdot \bar e_z = \bar e_{1\alpha\beta} \cdot \bar e_z\end{array}$

(104)

$\begin{array}{l}\displaystyle \bar g = g \cdot \; \bar e_z = g_1 \cdot \; \bar e_{1\alpha\beta} + g_2 \cdot \; \bar e_{2\alpha\beta} + g_3 \; \bar e_{3\alpha\beta} = g \cdot \cos \theta_{z\alpha\beta} \cdot \; \bar e_{1\alpha\beta} + g \cdot \sin \theta_{z\alpha\beta} \cdot \; \bar e_{3\alpha\beta}\end{array}$

(105)

$\begin{array}{l}\displaystyle g_1 = g \cdot \cos \theta_{z\alpha\beta} \; , \; g_2 = 0 \; , \; g_3 = g \cdot \sin \theta_{z\alpha\beta}\end{array}$

(106)

$\begin{array}{l}\hat k = \begin{pmatrix} k_{11} & k_{12} & 0 \\ k_{12} & k_{22} & 0 \\ 0 & 0 & k_v \end{pmatrix}\end{array}$

(107)

$\begin{array}{l}\hat k = \left\{ \begin{split} k_{11} &= k_{\max} \cdot \cos^2 \phi + k_{\min} \cdot \sin^2 \phi \\ k_{12} &= (k_{\max}-k_{\min}) \cdot \sin \phi \cdot \cos \phi \\ k_{22} &= k_{\max} \cdot \sin^2 \phi + k_{\min} \cdot \cos^2 \phi \end{split}\end{array}$

(108)

$\begin{array}{l}\displaystyle \cos \phi = \; \bar e_1 \cdot \; \bar e_{k\max}\end{array}$

(109)

$\begin{array}{l}\bar v = v_1 \cdot \; \bar e_1 + v_2 \cdot \; \bar e_2 + v_3 \cdot \; \bar e_3 = \left( \begin{split} v_1 \\ v_2 \\ v_3\end{split} \right)\end{array}$

(110)

$\begin{array}{l}\displaystyle \hat k * \bar v = (k_{11} \cdot v_1 + k_{12} \cdot v_2) \cdot \; \bar e_1 + (k_{12} \cdot v_1 + k_{22} \cdot v_2) \cdot \; \bar e_2 + k_v \cdot v_3 \cdot \; \bar e_3\end{array}$

(111)

$\begin{array}{l}\displaystyle \hat k * \bar g = k_{11} \cdot g_1 \cdot \; \bar e_1 + k_{12} \cdot g_1 \cdot \; \bar e_2 + k_v \cdot g_3 \cdot \; \bar e_3\end{array}$

(112)

$\begin{array}{l}\displaystyle \hat k * \bar g \cdot \bar e_1 = k_{11} \cdot g_1 = k_{11} \cdot g \cdot \cos \theta_z\end{array}$

(113)

$\begin{array}{l}\displaystyle \hat k * \bar \nabla = k_{11} \cdot \partial_1() \cdot \; \bar e_1 + k_{12} \cdot \partial_1() \cdot \; \bar e_2 + k_v \cdot \partial_3() \cdot \; \bar e_3\end{array}$

(114)

$\begin{array}{l}\displaystyle \hat k * \bar \nabla \cdot \bar e_1 = k_{11} \cdot \partial_1()\end{array}$

(115)

$\begin{array}{l}\displaystyle \big( \hat k * \bar \nabla} p - \rho \cdot \hat k * \bar g \big) \cdot \bar e_1 = k_{11\alpha} \, ( \partial_1 p - \rho \cdot g \cdot \cos \theta_z )\end{array}$

(116)

$\begin{array}{l}\displaystyle \big( \hat k_\alpha * \bar \nabla} p_{f\alpha} - \rho_{f\alpha} \cdot \hat k_\alpha * \bar g \big) \cdot \bar n_{\alpha \beta} = k_{11\alpha\beta} \, ( \partial_1 p_{f\alpha} - \rho_{f\alpha} \cdot g \cdot \cos \theta_{z\alpha_\beta} )\end{array}$

(117)

$\begin{array}{l}\displaystyle \bar u_{f\alpha \beta} \cdot \bar n_{\alpha \beta} = - k_{p\alpha\beta} \cdot \frac{k_{rf\alpha\beta}}{\mu_{f\alpha\beta}} \cdot k_{11\alpha\beta} \, ( \partial_1 p_{f\alpha} - \rho_{f\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} )\end{array}$

(118)

$\begin{array}{l}\displaystyle \bar u_{f\alpha \beta} \cdot \bar n_{\alpha \beta} = - k_{p\alpha\beta} \cdot \frac{k_{rf\alpha\beta}}{\mu_{f\alpha\beta}} \cdot k_{11\alpha\beta} \cdot \left( \partial_1 (p_\alpha + p_{cf\alpha}) - \rho_{f\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} \right)\end{array}$

(119)

$\begin{array}{l}\displaystyle \bar u_{f\alpha\beta} \cdot \; \bar n_{\alpha \beta} = - k_{p\alpha\beta} \cdot \frac{k_{rf\alpha\beta}}{\mu_{f\alpha\beta}} \cdot k_{11\alpha\beta} \cdot \left( \frac{ p_{\alpha\beta} + p_{cf}(s_{\alpha\beta} ) - \left( p_\alpha + p_{cf}(s_{\alpha}) \right) }{R_{\alpha\beta}} - \rho_{f\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} \right)\end{array}$

(120)

$\begin{array}{l}\displaystyle \bar u_{f\alpha\beta} \cdot \; \bar n_{\alpha \beta} = - k_{p\alpha\beta} \cdot \frac{k_{rf\alpha\beta}}{\mu_{f\alpha\beta}} \cdot k_{11\alpha\beta} \cdot R^{-1}_{\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + p_{cf}(s_{\alpha\beta}) - p_{cf}(s_{\alpha}) - R_{\alpha\beta} \cdot \rho_{f\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} \right)\end{array}$

(121)

$\begin{array}{l}\displaystyle \bar u_{f\alpha\beta} \cdot \; \bar n_{\alpha \beta} = - G_{f\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{f\alpha\beta} \right)\end{array}$

(122)

$\begin{array}{l}\displaystyle G_{f\alpha\beta} = k_{p\alpha\beta} \cdot \frac{k_{rf\alpha\beta}}{\mu_{f\alpha\beta}} \cdot k_{11\alpha\beta} \cdot R^{-1}_{\alpha\beta}\end{array}$

(123)

$\begin{array}{l}\displaystyle \delta p_{f\alpha\beta} = p_{cf}(s_{\alpha\beta}) - p_{cf}(s_{\alpha}) - R_{\alpha\beta} \cdot \rho_{f\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta}\end{array}$

(124)

$\begin{array}{l}\displaystyle k_{11\alpha\beta} = k_{\max\alpha\beta} \cdot \cos^2 \phi_{\alpha\beta} + k_{\min\alpha\beta} \cdot \sin^2 \phi_{\alpha\beta}\end{array}$

(125)

$\begin{array}{l}\displaystyle \cos \phi_{\alpha\beta} = \; \bar n_{\alpha\beta} \cdot \; \bar e_{k\max}\end{array}$

(126)

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

(127)

$\begin{array}{l}\displaystyle k_{rf\alpha\beta} = k_{rf}(s_{\alpha\beta})\end{array}$

(128)

$\begin{array}{l}\displaystyle \mu_{f\alpha\beta} = \mu_f(p_{\alpha\beta})\end{array}$

(129)

$\begin{array}{l}\displaystyle s_{f\alpha\beta}=\frac{s_{f\alpha} \cdot V_{\alpha\beta} + s_{f\beta}\cdot V_{\beta\alpha}}{V_{\alpha\beta} + V_{\beta\alpha}}\end{array}$

(130)

$\begin{array}{l}\displaystyle s_{f\alpha\beta}=s_{f\alpha} \cdot v_{\alpha\beta} + s_{f\beta}\cdot v_{\beta\alpha} \; , \quad v_{\alpha\beta} = \frac{V_{\alpha\beta}}{V_{\alpha\beta} + V_{\beta\alpha}} \, , \; v_{\beta\alpha} = 1 - v_{\alpha\beta} = \frac{V_{\beta\alpha}}{V_{\alpha\beta} + V_{\beta\alpha}}\end{array}$

(131)

$\begin{array}{l}\displaystyle \sum_{\beta \in \Gamma_\alpha} V_{\alpha\beta} = V_{\alpha} \; , \; \; \sum_{\alpha \in \Gamma_\beta} V_{\beta\alpha} = V_{\beta}\end{array}$

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

(132)

$\begin{array}{l}\displaystyle \bar u_{w\alpha\beta} \cdot \; \bar n_{\alpha \beta} = - G_{w\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{w\alpha\beta} \right)\end{array}$

(133)

$\begin{array}{l}\displaystyle \bar u_{o\alpha\beta} \cdot \; \bar n_{\alpha \beta} = - G_{o\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{o\alpha\beta} \right)\end{array}$

(134)

$\begin{array}{l}\displaystyle \bar u_{g\alpha\beta} \cdot \; \bar n_{\alpha \beta} = - G_{g\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{g\alpha\beta} \right)\end{array}$

(135)

$\begin{array}{l}\displaystyle U_{W\alpha\beta} = B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{w\alpha\beta} \right)\end{array}$

(136)

$\begin{array}{l}\displaystyle U_{O\alpha\beta} = B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{o\alpha\beta} \right) + R_{v\alpha} \cdot B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{g\alpha\beta} \right)\end{array}$

(137)

$\begin{array}{l}\displaystyle U_{G\alpha\beta} = R_{s\alpha} \cdot B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{o\alpha\beta} \right) + B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha + \delta p_{g\alpha\beta} \right)\end{array}$

(138)

$\begin{array}{l}\displaystyle U_{W\alpha\beta} = B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} \cdot \delta p_{w\alpha\beta}\end{array}$

(139)

$\begin{array}{l}U_{O\alpha\beta} = \begin{split} & B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + \\ & + R_{v\alpha} \cdot B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + R_{v\alpha} \cdot B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta} \end{split}\end{array}$

(140)

$\begin{array}{l}U_{G\alpha\beta} = \begin{split} & R_{s\alpha} \cdot B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + R_{s\alpha} \cdot B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + \\ & + B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta} \end{split}\end{array}$

(141)

$\begin{array}{l}\displaystyle U_{W\alpha\beta} = B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} \cdot \delta p_{w\alpha\beta}\end{array}$

(142)

$\begin{array}{l}U_{O\alpha\beta} = \begin{split} & \left( B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} + R_{v\alpha} \cdot B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \right) \cdot \left( p_{\alpha\beta} - p_\alpha \right) \\ & + B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + R_{v\alpha} \cdot B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta} \end{split}\end{array}$

(143)

$\begin{array}{l}U_{G\alpha\beta} = \begin{split} & \left( R_{s\alpha} \cdot B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} + B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \right) \cdot \left( p_{\alpha\beta} - p_\alpha \right) \\ & + R_{s\alpha} \cdot B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta} \end{split}\end{array}$

(144)

$\begin{array}{l}\displaystyle U_{W\alpha\beta} = T_{W\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + L_{W\alpha\beta}\end{array}$

(145)

$\begin{array}{l}\displaystyle U_{O\alpha\beta} = T_{O\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + L_{O\alpha\beta}\end{array}$

(146)

$\begin{array}{l}\displaystyle U_{G\alpha\beta} = T_{G\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + L_{G\alpha\beta}\end{array}$

(147)

$\begin{array}{l}\displaystyle T_{W\alpha\beta} = B^{-1}_{w\alpha} \cdot G_{w\alpha\beta}\end{array}$

(148)

$\begin{array}{l}\displaystyle L_{W\alpha\beta} = B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} \cdot \delta p_{w\alpha\beta}\end{array}$

(149)

$\begin{array}{l}\displaystyle T_{O\alpha\beta} = B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} + R_{v\alpha} \cdot B^{-1}_{g\alpha} \cdot G_{g\alpha\beta}\end{array}$

(150)

$\begin{array}{l}\displaystyle L_{O\alpha\beta} = B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + R_{v\alpha} \cdot B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta}\end{array}$

(151)

$\begin{array}{l}\displaystyle T_{G\alpha\beta} = R_{s\alpha} \cdot B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} + B^{-1}_{g\alpha} \cdot G_{g\alpha\beta}\end{array}$

(152)

$\begin{array}{l}\displaystyle L_{G\alpha\beta} = R_{s\alpha} \cdot B^{-1}_{o\alpha} \cdot G_{o\alpha\beta} \cdot \delta p_{o\alpha\beta} \; + B^{-1}_{g\alpha} \cdot G_{g\alpha\beta} \cdot \delta p_{g\alpha\beta}\end{array}$

(153)

$\begin{array}{l}\displaystyle U_{W\beta\alpha} = T_{W\beta\alpha} \cdot \left( p_{\beta\alpha} - p_\beta \right) + L_{W\beta\alpha}\end{array}$

(154)

$\begin{array}{l}\displaystyle U_{O\beta\alpha} = T_{O\beta\alpha} \cdot \left( p_{\alpha\beta} - p_\beta \right) + L_{O\beta\alpha}\end{array}$

(155)

$\begin{array}{l}\displaystyle U_{G\beta\alpha} = T_{G\beta\alpha} \cdot \left( p_{\alpha\beta} - p_\beta \right) + L_{G\beta\alpha}\end{array}$

(156)

$\begin{array}{l}\displaystyle T_{W\beta\alpha} = B^{-1}_{w\beta} \cdot G_{w\beta\alpha}\end{array}$

(157)

$\begin{array}{l}\displaystyle L_{W\beta\alpha} = B^{-1}_{w\beta} \cdot G_{w\beta\alpha} \cdot \delta p_{w\beta\alpha}\end{array}$

(158)

$\begin{array}{l}\displaystyle T_{O\beta\alpha} = B^{-1}_{o\beta} \cdot G_{o\beta\alpha} + R_{v\beta} \cdot B^{-1}_{g\beta} \cdot G_{g\beta\alpha}\end{array}$

(159)

$\begin{array}{l}\displaystyle L_{O\beta\alpha} = B^{-1}_{o\beta} \cdot G_{o\beta\alpha} \cdot \delta p_{o\beta\alpha} \; + R_{v\beta} \cdot B^{-1}_{g\beta} \cdot G_{g\beta\alpha} \cdot \delta p_{g\beta\alpha}\end{array}$

(160)

$\begin{array}{l}\displaystyle T_{G\beta\alpha} = R_{s\beta} \cdot B^{-1}_{o\beta} \cdot G_{o\beta\alpha} + B^{-1}_{g\beta} \cdot G_{g\beta\alpha}\end{array}$

(161)

$\begin{array}{l}\displaystyle L_{G\beta\alpha} = R_{s\beta} \cdot B^{-1}_{o\beta} \cdot G_{o\beta\alpha} \cdot \delta p_{o\beta\alpha} \; + B^{-1}_{g\beta} \cdot G_{g\beta\alpha} \cdot \delta p_{g\beta\alpha}\end{array}$

(162)

$\begin{array}{l}\displaystyle p_{\alpha\beta} = \left ( T_{f\alpha\beta} - T_{f\beta\alpha} \right)^{-1} \cdot \left( T_{f\alpha\beta} \cdot p_\alpha - T_{f\beta\alpha} \cdot p_\beta - L_{f\alpha\beta} + L_{f\beta\alpha} \right)\end{array}$

(163)

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

(164)

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

(165)

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

(166)

$\begin{array}{l}\displaystyle \eta_{w\alpha\beta} = \left ( T_{w\alpha\beta} - T_{w\beta\alpha} \right)^{-1} \; \text{if} \; |T_{w\alpha\beta} - T_{w\beta\alpha}| > \epsilon \; \text{else} \; 0, \quad \epsilon = \text{1e-12}\end{array}$

(167)

$\begin{array}{l}\displaystyle \eta_{o\alpha\beta} = \left ( T_{o\alpha\beta} - T_{o\beta\alpha} \right)^{-1} \; \text{if} \; |T_{o\alpha\beta} - T_{o\beta\alpha}| > \epsilon \; \text{else} \; 0, \quad \epsilon = \text{1e-12}\end{array}$

(168)

$\begin{array}{l}\displaystyle \eta_{g\alpha\beta} = \left ( T_{g\alpha\beta} - T_{g\beta\alpha} \right)^{-1} \; \text{if} \; |T_{g\alpha\beta} - T_{g\beta\alpha}| > \epsilon \; \text{else} \; 0, \quad \epsilon = \text{1e-12}\end{array}$

Page tree

Grid-related Computation

Initialization: t = 0

Progression: t → t + 1

Ansatz Flux 2D+

See Also

Page tree

Modified Black Oil Reservoir Flow Grid Computation @ algorithm

Grid-related Computation

Initialization: t = 0

Progression: t → t + 1

Ansatz Flux 2D+

See Also