Spatial discretization
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| (1) |
\frac{dV_{W \alpha}}{dt} = - \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot \frac{1}{B_w} \cdot \bar u_w \, \bar n_{\alpha\beta} + q_{W\alpha} |
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| (2) |
\frac{dV_{O\alpha}}{dt} = - \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot \left( \frac{1}{B_o} \cdot \, \bar u_o + \frac{R_v}{B_g} \cdot \, \bar u_g \right) \bar n_{\alpha\beta} + q_{O\alpha} |
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| (3) |
\frac{dV_{G\alpha}}{dt} = - \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot \left( \frac{R_s}{B_o} \cdot \, \bar u_o + \frac{1}{B_g} \cdot \, \bar u_g \right) \bar n_{\alpha\beta} + q_{G\alpha} |
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| (4) |
V = \cup_\alpha \, V_\alpha |
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| (5) |
\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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| (6) |
\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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| (7) |
\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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| (8) |
U_{W\alpha\beta} = - \frac{1}{B_w} \cdot \bar u_w \, \bar n_{\alpha\beta} |
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| (9) |
U_{O\alpha\beta} = - \frac{1}{B_o} \cdot \, \bar u_o \cdot \bar n_{\alpha\beta} - \frac{R_v}{B_g} \cdot \, \bar u_g \cdot \bar n_{\alpha\beta} |
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| (10) |
U_{G\alpha\beta} = - \frac{R_s}{B_o} \cdot \, \bar u_o \cdot \bar n_{\alpha\beta} - \frac{1}{B_g} \cdot \, \bar u_g \cdot \bar n_{\alpha\beta} |
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Time Discretization |
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Initialization: t = 0 |
| (11) |
s^0_{w\alpha} = s_{wi\alpha} |
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| (12) |
s^0_{o\alpha} = s_{oi\alpha} |
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| (13) |
s^0_{g\alpha} = s_{gi\alpha} |
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| (14) |
s^0_{w\alpha} + s^0_{o\alpha} + s^0}_{g\alpha} = 1 |
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| (15) |
V^0_{\phi\alpha} = V_\alpha \cdot \phi^0_\alpha, \quad \phi^0_\alpha = \phi^0_{i\alpha} |
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| (16) |
V^0_{w\alpha} = s^0_{w\alpha} \cdot V^0_{\phi\alpha} |
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| (17) |
V^0_{o\alpha} = s^0_{o\alpha} \cdot V^0_{\phi\alpha} |
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| (18) |
V^0_{g\alpha} = s^0_{g\alpha} \cdot V^0_{\phi\alpha} |
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| (19) |
V^0_{\phi\alpha} = V^0_{w\alpha} + V^0_{o\alpha} + V^0_{g\alpha} |
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| (20) |
V^0_{W\alpha} = \frac{V^0_{w\alpha}}{B_{wi}} |
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| (21) |
V^0_{O\alpha} =\frac{V^0_o}{B_{oi}} + R_{vi} \cdot \frac{V^0_g}{B_{gi}} |
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| (22) |
V^0_{G\alpha} =\frac{V^0_G}{B_{gi}} + R_{si} \cdot \frac{V^0_o}{B_{oi}} |
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Semi-Implicit Progression: t → t + 1 |
| (23) |
V^{t+1}_{W \alpha} = V^t_{W \alpha} \cdot \exp\left[\frac{\delta t}{V^t_{W \alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot U^{t+1}_{W\alpha\beta} \; + \, q^{t+1}_{W\alpha} \right)\right] |
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| (24) |
V^{t+1}_{O \alpha} = V^t_{O \alpha} \cdot \exp\left[\frac{\delta t}{V^t_{O \alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot U^{t+1}_{O\alpha\beta} \; + \, q^{t+1}_{O\alpha} \right)\right] |
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| (25) |
V^{t+1}_{G \alpha} = V^t_{G \alpha} \cdot \exp\left[\frac{\delta t}{V^t_{G \alpha}}\cdot \left(\sum_{\beta\in\Gamma_{\alpha}}A_{\alpha\beta}\cdot U^{t+1}_{G\alpha\beta} \; + \, q^{t+1}_{G\alpha} \right)\right] |
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| (26) |
I^{t+1}_{W\alpha} = \frac{V^{t+1}_{W\alpha}}{V^0_{\phi_\alpha}} |
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| (27) |
I^{t+1}_{O\alpha} = \frac{V^{t+1}_{O\alpha}}{V^0_{\phi_\alpha}} |
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| (28) |
I^{t+1}_{G\alpha} = \frac{V^{t+1}_{G\alpha}}{V^0_{\phi_\alpha}} |
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| (29) |
pp = pp^0_{\alpha} + \ln \Big[ \,
\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} \; \Big] |
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| (30) |
pp^0_{\alpha} = c_\phi \cdot p^0_\alpha \, , \quad pp = c_\phi \cdot p |
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| (31) |
p^{t+1}_\alpha = c_\phi^{-1} \cdot pp |
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| (32) |
B^{t+1}_{w\alpha} = B_w(p^{t+1}_\alpha) |
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| (33) |
B^{t+1}_{o\alpha} = B_o(p^{t+1}_\alpha) |
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| (34) |
B^{t+1}_{g\alpha} = B_g(p^{t+1}_\alpha) |
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| (35) |
R^{t+1}_{v\alpha} = R_v(p^{t+1}_\alpha) |
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| (36) |
R^{t+1}_{s\alpha} = R_s(p^{t+1}_\alpha) |
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| (37) |
RR^{t+1}_{\alpha} = 1 - R^{t+1}_{v\alpha} \, R^{t+1}_{s\alpha} |
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| (38) |
BB^{t+1}_{o\alpha} = \frac{B^{t+1}_{o\alpha}}{RR^{t+1}_{\alpha}} |
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| (39) |
BB^{t+1}_{g\alpha} = \frac{B^{t+1}_{g\alpha}}{RR^{t+1}_{\alpha}} |
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| (40) |
V^{t+1}_{w\alpha} = B_w(p^{t+1}_\alpha) \cdot V^{t+1}_{W\alpha} |
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| (41) |
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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| (42) |
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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| (43) |
V^{t+1}_{\phi\alpha} = V^{t+1}_{w\alpha} + V^{t+1}_{o\alpha} + V^{t+1}_{g\alpha} |
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| (44) |
s^{t+1}_{w\alpha} = \frac{V^{t+1}_{w\alpha}}{V^{t+1}_{\phi\alpha}} |
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| (45) |
s^{t+1}_{o\alpha} = \frac{V^{t+1}_{o\alpha}}{V^{t+1}_{\phi\alpha}} |
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| (46) |
s^{t+1}_{g\alpha} = \frac{V^{t+1}_{g\alpha}}{V^{t+1}_{\phi\alpha}} |
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| (47) |
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:
| (48) |
\bold e_{\alpha\beta} = \{ \, \bar e_{1\alpha\beta}, \, \bar e_{2\alpha\beta}, \, \bar e_{3\alpha\beta} \, \} |
| (49) |
\bar e_{1\alpha\beta} =
\cos \theta_{z\alpha\beta} \cdot \cos \zeta_{x\alpha\beta} \cdot \bar e_x +
\cos \theta_{z\alpha\beta} \cdot \sin \zeta_{x\alpha\beta} \cdot \bar e_y
+ \sin \theta_{z\alpha\beta} \cdot \bar e_z |
| (50) |
\bar e_{2\alpha\beta} =
\cos \theta_{z\alpha\beta} \cdot \sin \zeta_{x\alpha\beta} \cdot \bar e_x +
\cos \theta_{z\alpha\beta} \cdot \cos \zeta_{x\alpha\beta} \cdot \bar e_y
+ \sin \theta_{z\alpha\beta} \cdot \bar e_z |
| (51) |
\bar e_{3\alpha\beta} =
\sin \theta_{z\alpha\beta} \cdot \sin \zeta_{x\alpha\beta} \cdot \bar e_x +
\sin \theta_{z\alpha\beta} \cdot \cos \zeta_{x\alpha\beta} \cdot \bar e_y
+ \cos \theta_{z\alpha\beta} \cdot \bar e_z |
| (52) |
\bar e_{1\alpha\beta} \cdot \bar e_{2\alpha\beta} = 0, \quad \bar e_e_{3\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_{3\alpha\beta} = \bar e_{1\alpha\beta} \times \bar e_{2\alpha\beta} |
| (53) |
\bar n_{\alpha \beta} = \bar e_1 |
| (54) |
\cos \theta_{z\alpha\beta} = \bar e_1 \cdot \bar e_z |
| (55) |
\bar g = g \cdot \bar e_z = \sin \theta_z \, \bar e_1 + \cos \theta_z \, \bar e_3 = ( g_1 = \sin \theta_z, \, g_2 = 0, \, g_3 = \cos \theta_z ) |
| (56) |
A_{\alpha\beta} = D_{\alpha\beta} \cdot h_{\alpha\beta} |
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| (57) |
\hat k * \bar g = \bar e_1 \, (k_{11} \, g_1 + k_{13} \, g_3)
+ \bar e_2 \,(k_{21} \, g_1 + k_{23} \, g_3) + \bar e_3 \, (k_{31} \, g_1 + k_{33} \, g_3) |
| (58) |
\hat k * \bar g \cdot \bar n_{\alpha \beta} = k_{11} \, g_1 + k_{13} \, g_3 = g \cdot ( k_{11} \, \sin \theta_{z\alpha\beta} + k_{13} \, \cos \theta_{z\alpha\beta} ) |
| (59) |
\hat k * \bar \nabla \cdot \bar n_{\alpha \beta} = k_{11} \, \partial_1 + k_{12} \, \partial_2 + k_{13} \, \partial_3 = k_{11} \, \partial_1 \quad \Leftarrow \, \partial_2 \equiv 0, \, \partial_3 \equiv 0 |
| (60) |
\big( \hat k * \bar \nabla} p - \rho \cdot \hat k * \bar g \big) \cdot \bar n_{\alpha \beta} = k_{11\alpha} \, ( \partial_1 p - \rho \cdot g \cdot \sin \theta_{z\alpha\beta} ) |
| (61) |
\big( \hat k * \bar \nabla} p - \rho \cdot \hat k * \bar g \big) = k_{11} \, \partial_1 p - \rho \cdot g \cdot ( k_{11} \, \sin \theta_{z\alpha\beta} + k_{13} \, \cos \theta_{z\alpha\beta} ) |
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| (62) |
\bar u_{\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{r}}{\mu} \cdot k_{11\alpha} \cdot \big( R^{-1}_{\alpha\beta} \cdot (p_{\alpha\beta} - p_\alpha - \dot p_c \cdot (s_{\alpha\beta} - s_{\alpha})) - \rho_\alpha \cdot g \cdot \sin \theta_{z\alpha\beta} \big) |
| (63) |
\bar u_{\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{r}}{\mu} \cdot k_{11\alpha} \cdot \big( R^{-1}_{\alpha\beta} \cdot (p_{\alpha\beta} - p_\alpha - \dot p_c \cdot (s_{\alpha\beta} + \dot p_c \cdot s_{\alpha}) - \rho_\alpha \cdot g \cdot \sin \theta_{z\alpha\beta} \big) |
| (64) |
\bar u_{\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{r}}{\mu} \cdot k_{11\alpha} \cdot \Big( R^{-1}_{\alpha\beta} \cdot \big( \tilde p_{\alpha\beta} - p_\alpha + \dot p_c \cdot s_{\alpha} \big) - \rho_\alpha \cdot g \cdot \sin \theta_{z\alpha\beta} \Big) \, , \quad \tilde p_{\alpha\beta} = p_{\alpha\beta} - \dot p_c \cdot s_{\alpha\beta} |
| (65) |
\bar u_{\alpha \beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{r}}{\mu} \cdot k_{11\alpha} \, ( \partial_1 p - \rho \cdot g \cdot \sin \theta_{z\alpha\beta} ) |
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| (66) |
\bar u_{w\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{rw}}{\mu_w} \cdot k_{11\alpha} \cdot \Big( R^{-1}_{\alpha\beta} \cdot \big( \tilde p_{w\alpha\beta} - p_\alpha + \dot p_{cw} \cdot s_{w\alpha} \big) - \rho_{w\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} \Big) |
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| (67) |
\bar u_{o\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{ro}}{\mu_o} \cdot k_{11\alpha} \cdot \Big( R^{-1}_{\alpha\beta} \cdot \big( \tilde p_{o\alpha\beta} - p_\alpha + \dot p_{co} \cdot s_{o\alpha} \big) - \rho_{o\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} \Big) |
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| (68) |
\bar u_{g\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{rg}}{\mu_g} \cdot k_{11\alpha} \cdot \Big( R^{-1}_{\alpha\beta} \cdot \big( \tilde p_{g\alpha\beta} - p_\alpha + \dot p_{cg} \cdot s_{g\alpha} \big) - \rho_g\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} \Big) |
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| (69) |
U_{W\alpha\beta} =
\frac{k_{p\alpha}}{B_{w\alpha}} \cdot \frac{k_{rw\alpha}}{\mu_{w\alpha}} \cdot k_{11\alpha} \cdot \Big( R^{-1}_{\alpha\beta} \cdot \big( \tilde p_{\alpha\beta} - p_\alpha + \dot p_{cw} \cdot s_{w\alpha} \big) - \rho_{w\alpha} \cdot g \cdot \sin \theta_{z\alpha\beta} \Big) |
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| (70) |
U_{W\beta\alpha} = \frac{k_{p\beta}}{B_{w\beta}} \cdot \frac{k_{rw\beta}}{\mu_{w\beta}} \cdot k_{11\beta} \cdot \Big( R^{-1}_{\beta\alpha} \cdot \big( \tilde p_{\beta\alpha} - p_\beta + \dot p_{cw} \cdot s_{w\beta} \big) - \rho_{w\beta} \cdot g \cdot \sin \theta_{z\beta\alpha} \Big) |
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| (71) |
U_{W\alpha\beta} = U_{W\beta\alpha} |
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