Spatial discretization
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| (1) |
\frac{dV_{W \alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{W\alpha\beta}} + q_{W\alpha} |
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| (2) |
\frac{dV_{O\alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{O\alpha\beta} + q_{O\alpha} |
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| (3) |
\frac{dV_{G\alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{G\alpha\beta} + q_{G\alpha} |
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| (4) |
U_{W\alpha\beta} = - \frac{1}{B_w} \cdot \bar u_{w\alpha\beta} \, \bar n_{\alpha\beta} |
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| (5) |
U_{O\alpha\beta} = - \frac{1}{B_o} \cdot \, \bar u_{o\alpha\beta} \cdot \bar n_{\alpha\beta} - \frac{R_v}{B_g} \cdot \, \bar u_{g\alpha\beta} \cdot \bar n_{\alpha\beta} |
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| (6) |
U_{G\alpha\beta} = - \frac{R_s}{B_o} \cdot \, \bar u_{o\alpha\beta} \cdot \bar n_{\alpha\beta} - \frac{1}{B_g} \cdot \, \bar u_{g\alpha\beta} \cdot \bar n_{\alpha\beta} |
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Time Discretization |
Initialization: t = 0 |
| (7) |
s^0_{w\alpha} = s_{wi\alpha} |
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| (8) |
s^0_{o\alpha} = s_{oi\alpha} |
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| (9) |
s^0_{g\alpha} = s_{gi\alpha} |
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| (10) |
s^0_{w\alpha} + s^0_{o\alpha} + s^0}_{g\alpha} = 1 |
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| (11) |
V^0_{\phi\alpha} = V_\alpha \cdot \phi^0_\alpha, \quad \phi^0_\alpha = \phi^0_{i\alpha} |
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| (12) |
V^0_{w\alpha} = s^0_{w\alpha} \cdot V^0_{\phi\alpha} |
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| (13) |
V^0_{o\alpha} = s^0_{o\alpha} \cdot V^0_{\phi\alpha} |
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| (14) |
V^0_{g\alpha} = s^0_{g\alpha} \cdot V^0_{\phi\alpha} |
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| (15) |
V^0_{\phi\alpha} = V^0_{w\alpha} + V^0_{o\alpha} + V^0_{g\alpha} |
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| (16) |
V^0_{W\alpha} = \frac{V^0_{w\alpha}}{B_{wi}} |
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| (17) |
V^0_{O\alpha} =\frac{V^0_o}{B_{oi}} + R_{vi} \cdot \frac{V^0_g}{B_{gi}} |
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| (18) |
V^0_{G\alpha} =\frac{V^0_G}{B_{gi}} + R_{si} \cdot \frac{V^0_o}{B_{oi}} |
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| (19) |
U^0_{W\alpha\beta} = 0 |
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| (20) |
U^0_{O\alpha\beta} = 0 |
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| (21) |
U^0_{G\alpha\beta} = 0 |
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Semi-Implicit Progression: t → t + 1 |
| Loop Over Step #1 – Step #5 |
| Step #1 – Velocity |
| (22) |
U^{t+1}_{W\alpha\beta} = - \frac{1}{B^t_w} \cdot \; \bar u^t_{w\alpha\beta} \cdot \; \bar n_{\alpha\beta} |
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| (23) |
U^{t+1}_{O\alpha\beta} = - \frac{1}{B^t_o} \cdot \; \bar u^t_{o\alpha\beta} \cdot \; \bar n_{\alpha\beta} - \frac{R^t_v}{B^t_g} \cdot \, \bar u^t_{g\alpha\beta} \cdot \; \bar n_{\alpha\beta} |
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| (24) |
U^{t+1}_{G\alpha\beta} = - \frac{R^t_s}{B^t_o} \cdot \; \bar u^t_{o\alpha\beta} \cdot \; \bar n_{\alpha\beta} - \frac{1}{B^t_g} \cdot \, \bar u^t_{g\alpha\beta} \cdot \; \bar n_{\alpha\beta} |
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| Step #2 – Influx |
| (25) |
\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) |
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| (26) |
\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) |
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| (27) |
\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) |
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| (28) |
I^{t+1}_{W \alpha} = I^t_{W \alpha} + \delta I^{t+1}_{W \alpha} |
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| (29) |
I^{t+1}_{O \alpha} = I^t_{O \alpha} + \delta I^{t+1}_{O \alpha}\; , \; \; \delta I^{t+1}_{O \alpha} >0 |
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| (30) |
I^{t+1}_{G \alpha} = I^t_{G \alpha} + \delta I^{t+1}_{G \alpha}\; , \; \; \delta I^{t+1}_{G \alpha} >0 |
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Step #4 – Handle depletion |
| (31) |
\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} |
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| (32) |
\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} |
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| (33) |
\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} |
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| (34) |
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}} |
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| (35) |
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}} |
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| (36) |
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}} |
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| (37) |
- I^t_{W \alpha} = \frac{\delta t}{V^0_{\phi_\alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot \; \tilde U^{t+1}_{W\alpha\beta} \; + \, \tilde q^{t+1}_{W\alpha} \right) |
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| (38) |
- I^t_{O \alpha} = \frac{\delta t}{V^0_{\phi_\alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot \; \tilde U^{t+1}_{O\alpha\beta} \; + \, q^{t+1}_{O\alpha} \right) |
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| (39) |
- I^t_{G \alpha} = \frac{\delta t}{V^0_{\phi_\alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot \; \tilde U^{t+1}_{G\alpha\beta} \; + \, q^{t+1}_{G\alpha} \right) |
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| (40) |
-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^+} + \delta t^*_{W\alpha\beta^-} \cdot\left(\sum_{\beta^-\in\Gamma^-_{\alpha}}A_{\alpha\beta^-}\cdot U^{t+1}_{W\alpha\beta^-} + q^{t+1}_{W\alpha} \right) |
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| (41) |
-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^+} + \delta t^*_{O\alpha\beta^-} \cdot\left(\sum_{\beta^-\in\Gamma^-_{\alpha}}A_{\alpha\beta^-}\cdot U^{t+1}_{O\alpha\beta^-} + q^{t+1}_{O\alpha} \right) |
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| (42) |
-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^+} + \delta t^*_{G\alpha\beta^-} \cdot\left(\sum_{\beta^-\in\Gamma^-_{\alpha}}A_{\alpha\beta^-}\cdot U^{t+1}_{G\alpha\beta^-} + q^{t+1}_{G\alpha} \right) |
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| (43) |
\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}} \; , \quad I^{t+1}_{W \alpha} < 0 \; \Rightarrow \; 0 < \delta t^*_{W\alpha\beta^-} / \delta t < 1 |
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| (44) |
\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}} \; , \quad I^{t+1}_{O \alpha} < 0 \; \Rightarrow \; 0 < \delta t^*_{O\alpha\beta^-} / \delta t < 1 |
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| (45) |
\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}} \; , \quad I^{t+1}_{G \alpha} < 0 \; \Rightarrow \; 0 < \delta t^*_G / \delta t < 1 |
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| (46) |
\tilde q^{t+1}_{W\alpha} = q^{t+1}_{W\alpha} \cdot \delta t^*_{W\alpha\beta^-}/\delta t |
| (47) |
\tilde U^{t+1}_{W\alpha\beta^-} = U^{t+1}_{W\alpha\beta^-} \cdot \delta t^*_{W\alpha\beta^-}/\delta t |
| (48) |
\tilde U^{t+1}_{W\beta^-\alpha} = \tilde U^{t+1}_{W\alpha\beta^-} |
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| (49) |
\tilde q^{t+1}_{O\alpha} = q^{t+1}_{O\alpha} \cdot \delta t^*_{O\alpha\beta^-}/\delta t |
| (50) |
\tilde U^{t+1}_{O\alpha\beta^-} = U^{t+1}_{O\alpha\beta^-} \cdot \delta t^*_{O\alpha\beta^-}/\delta t |
| (51) |
\tilde U^{t+1}_{O\beta^-\alpha} = \tilde U^{t+1}_{O\alpha\beta^-} |
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| (52) |
\tilde q^{t+1}_{G\alpha} = q^{t+1}_{G\alpha} \cdot \delta t^*_{G\alpha\beta^-}/\delta t |
| (53) |
\tilde U^{t+1}_{G\alpha\beta^-} = U^{t+1}_{G\alpha\beta^-} \cdot \delta t^*_{G\alpha\beta^-}/\delta t |
| (54) |
\tilde U^{t+1}_{G\beta^-\alpha} = \tilde U^{t+1}_{G\alpha\beta^-} |
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Step #5 – Reservoir Pressure |
| (55) |
pp^0_{\alpha} = c_{\phi_\alpha} \cdot p^0_\alpha |
| (56) |
\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 |
| (57) |
p^{t+1}_\alpha = c_{\phi_\alpha}^{-1} \cdot pp |
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| Step #6 – PVT properties |
| (58) |
B^{t+1}_{w\alpha} = B_w(p^{t+1}_\alpha) |
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| (59) |
B^{t+1}_{o\alpha} = B_o(p^{t+1}_\alpha) |
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| (60) |
B^{t+1}_{g\alpha} = B_g(p^{t+1}_\alpha) |
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| (61) |
R^{t+1}_{v\alpha} = R_v(p^{t+1}_\alpha) |
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| (62) |
R^{t+1}_{s\alpha} = R_s(p^{t+1}_\alpha) |
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| (63) |
RR^{t+1}_{\alpha} = 1 - R^{t+1}_{v\alpha} \, R^{t+1}_{s\alpha} |
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| (64) |
BB^{t+1}_{o\alpha} = \frac{B^{t+1}_{o\alpha}}{RR^{t+1}_{\alpha}} |
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| (65) |
BB^{t+1}_{g\alpha} = \frac{B^{t+1}_{g\alpha}}{RR^{t+1}_{\alpha}} |
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| (66) |
V^{t+1}_{w\alpha} = B_w(p^{t+1}_\alpha) \cdot V^{t+1}_{W\alpha} |
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| (67) |
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) |
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| (68) |
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) |
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Step #7 – Phase volumes |
| (69) |
V^{t+1}_{\phi\alpha} = V^{t+1}_{w\alpha} + V^{t+1}_{o\alpha} + V^{t+1}_{g\alpha} |
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Step #8 – Reservoir saturation |
| (70) |
s^{t+1}_{w\alpha} = \frac{V^{t+1}_{w\alpha}}{V^{t+1}_{\phi\alpha}} |
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| (71) |
s^{t+1}_{o\alpha} = \frac{V^{t+1}_{o\alpha}}{V^{t+1}_{\phi\alpha}} |
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| (72) |
s^{t+1}_{g\alpha} = \frac{V^{t+1}_{g\alpha}}{V^{t+1}_{\phi\alpha}} |
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| (73) |
s^{t+1}_{w\alpha} + s^{t+1}_{o\alpha} + s^{t+1}_{g\alpha} = 1 |
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Ansatz Flux 2D+
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Specify local orthogonal coordinate system:
| (74) |
\bold e_{\alpha\beta} = \{ \, \bar e_{1\alpha\beta}, \, \bar e_{2\alpha\beta}, \, \bar e_{3\alpha\beta} \, \} |
| (75) |
\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} |
| (76) |
\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 |
| (77) |
\cos \theta_{z\alpha\beta} = \bar n_{\alpha\beta} \cdot \bar e_z = \bar e_{1\alpha\beta} \cdot \bar e_z |
| (78) |
\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} |
| (79) |
g_1 = g \cdot \cos \theta_{z\alpha\beta} \; , \; g_2 = 0 \; , \; g_3 = g \cdot \sin \theta_{z\alpha\beta} |
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| (80) |
\begin{equation}
\hat k =
\begin{pmatrix}
k_{11} & k_{12} & 0 \\
k_{12} & k_{22} & 0 \\
0 & 0 & k_v
\end{pmatrix}\end{equation} |
| (81) |
\begin{equation}
\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{equation} |
| (82) |
\cos \phi = \; \bar e_1 \cdot \; \bar e_{k\max} |
| (83) |
\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) |
| (84) |
\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 |
| (85) |
\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 |
| (86) |
\hat k * \bar g \cdot \bar e_1 = k_{11} \cdot g_1 = k_{11} \cdot g \cdot \cos \theta_z |
| (87) |
\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 |
| (88) |
\hat k * \bar \nabla \cdot \bar e_1 = k_{11} \cdot \partial_1() |
| (89) |
\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 ) |
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| (90) |
\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} ) |
| (91) |
\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} ) |
| (92) |
\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) |
| (93) |
\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) |
| (94) |
\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) |
| (95) |
\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) |
| (96) |
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} |
| (97) |
\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} |
| (98) |
k_{11\alpha\beta} = k_{\max\alpha\beta} \cdot \cos^2 \phi_{\alpha\beta} + k_{\min\alpha\beta} \cdot \sin^2 \phi_{\alpha\beta} |
| (99) |
\cos \phi_{\alpha\beta} = \; \bar n_{\alpha\beta} \cdot \; \bar e_{k\max} |
| (100) |
k_{p\alpha\beta} = \exp \left[ \; c_{k\alpha\beta} \cdot \left( p_{\alpha \beta} - p_{0\alpha \beta} \right) \; \right] |
| (101) |
k_{rf\alpha\beta} = k_{rf}(s_{\alpha\beta}) |
| (102) |
\mu_{f\alpha\beta} = \mu_f(p_{\alpha\beta}) |
| (103) |
s_{f\alpha\beta}=\frac{s_{f\alpha} \cdot V_{\alpha\beta} + s_{\beta}\cdot V_{\beta\alpha}}{V_{\alpha\beta} + V_{\beta\alpha}} |
| (104) |
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}} |
| (105) |
\sum_{\beta \in \Gamma_\alpha} V_{\alpha\beta} = V_{\alpha} \; , \; \; \sum_{\alpha \in \Gamma_\beta} V_{\beta\alpha} = V_{\beta} |
|
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– |
| (106) |
\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) |
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| (107) |
\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) |
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| (108) |
\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) |
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| (109) |
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) |
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| (110) |
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) |
|
| (111) |
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) |
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| (112) |
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} |
|
| (113) |
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} |
|
| (114) |
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} |
|
|
| (115) |
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} |
|
| (116) |
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} |
|
| (117) |
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} |
|
|
|
| (118) |
U_{W\alpha\beta} = T_{W\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + L_{W\alpha\beta} |
|
| (119) |
U_{O\alpha\beta} = T_{O\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + L_{O\alpha\beta} |
|
| (120) |
U_{G\alpha\beta} = T_{G\alpha\beta} \cdot \left( p_{\alpha\beta} - p_\alpha \right) + L_{G\alpha\beta} |
|
|
| (121) |
T_{W\alpha\beta} = B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} |
| (122) |
L_{W\alpha\beta} = B^{-1}_{w\alpha} \cdot G_{w\alpha\beta} \cdot \delta p_{w\alpha\beta} |
|
| (123) |
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} |
| (124) |
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} |
|
| (125) |
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} |
| (126) |
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} |
|
|
| | |
|
| (127) |
U_{W\beta\alpha} = T_{W\beta\alpha} \cdot \left( p_{\beta\alpha} - p_\beta \right) + L_{W\beta\alpha} |
|
| (128) |
U_{O\beta\alpha} = T_{O\beta\alpha} \cdot \left( p_{\alpha\beta} - p_\beta \right) + L_{O\beta\alpha} |
|
| (129) |
U_{G\beta\alpha} = T_{G\beta\alpha} \cdot \left( p_{\alpha\beta} - p_\beta \right) + L_{G\beta\alpha} |
|
|
| (130) |
T_{W\beta\alpha} = B^{-1}_{w\beta} \cdot G_{w\beta\alpha} |
| (131) |
L_{W\beta\alpha} = B^{-1}_{w\beta} \cdot G_{w\beta\alpha} \cdot \delta p_{w\beta\alpha} |
|
| (132) |
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} |
| (133) |
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} |
|
| (134) |
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} |
| (135) |
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} |
|
|
|
| (136) |
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) |
|
|
| (137) |
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) |
|
| (138) |
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) |
|
| (139) |
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) |
|
|
| (140) |
\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} |
|
| (141) |
\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} |
|
| (142) |
\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} |
|
|
