|
| (1) |
\frac{\partial \, m_W}{dt} + \int\rho_{Ww} \cdot\, \bar u_w \, d \bar \Sigma = \frac{d m^*_W}{dt} |
|
| (2) |
\frac{\partial \, m_O}{dt} + \int \left( \rho_{Oo} \cdot \, \bar u_o + \rho_{Og} \cdot \, \bar u_g \right) d \bar \Sigma = \frac{d m^*_O}{dt} |
|
| (3) |
\frac{\partial \, m_G}{dt} + \int \left( \rho_{Go} \cdot \, \bar u_o + \rho_{Gg} \cdot \, \bar u_g \right) d \bar \Sigma = \frac{d m^*_G}{dt} |
| |
| (4) |
\rho_{Ww} = \frac{m_W}{V_w} = \frac{m_W}{V_W} \cdot \frac{V_W}{V_w} = \frac{\mathring \rho_W}{B_w} |
|
| (5) |
\rho_{Oo} = \frac{m_{Oo}}{V_o}
= \frac{m_{Oo}}{V_O} \cdot \frac{V_O}{V_o} = \frac{\mathring \rho_O}{B_o} |
| (6) |
\rho_{Og} = \frac{m_{Og}}{V_g}
= \frac{m_{Og}}{V_{Og}} \cdot \frac{V_{Og}}{V_G} \cdot \frac{V_G}{V_g} = \frac{\mathring \rho_O \cdot R_v}{B_g} |
|
| (7) |
\rho_{Go} = \frac{m_{Go}}{V_o}
= \frac{m_{Go}}{V_{Go}} \cdot \frac{V_{Go}}{V_O} \cdot \frac{V_O}{V_o} = \frac{\mathring \rho_G \cdot R_s}{B_o} |
| (8) |
\rho_{Gg} = \frac{m_{Gg}}{V_g}
= \frac{m_{Gg}}{V_G} \cdot \frac{V_G}{V_g} = \frac{\mathring \rho_G}{B_g} |
| |
| (9) |
\frac{\partial \, m_W}{dt} + \mathring \rho_W \int \frac{1}{B_w} \, \bar u_w \, d \bar \Sigma = \frac{d m^*_W}{dt} |
|
| (10) |
\frac{\partial \, m_O}{dt} + \mathring \rho_O \int \left( \frac{1}{B_o} \cdot \, \bar u_o + \frac{R_v}{B_g} \cdot \, \bar u_g \right) d \bar \Sigma = \frac{d m^*_O}{dt} |
|
| (11) |
\frac{\partial \, m_G}{dt} + \mathring \rho_G \int \left( \frac{R_s}{B_o} \cdot \, \bar u_o + \frac{1}{B_g} \cdot \, \bar u_g \right) d \bar \Sigma = \frac{d m^*_G}{dt} |
| |
| (12) |
V_W = \mathring \rho_W^{\. -1} \cdot m_W |
|
| (13) |
V_O = \mathring \rho_O^{-1} \cdot m_O |
|
| (14) |
V_G = \mathring \rho_G^{-1} \cdot m_G |
| |
| (15) |
q_W = \mathring \rho_W^{-1} \cdot \frac{d m^*_W}{dt} |
|
| (16) |
q_O = \mathring \rho_O^{-1} \cdot \frac{d m^*_O}{dt} |
|
| (17) |
q_G = \mathring \rho_G^{-1} \cdot \frac{d m^*_G}{dt} |
| |
|
| (18) |
\frac{\partial \, V_W}{dt} + \int \frac{1}{B_w} \, \bar u_w \, d \bar \Sigma = q_W |
|
| (19) |
\frac{\partial \, V_O}{dt} + \int \left( \frac{1}{B_o} \cdot \, \bar u_o + \frac{R_v}{B_g} \cdot \, \bar u_g \right) d \bar \Sigma = q_O |
|
| (20) |
\frac{\partial \, V_G}{dt} + \int \left( \frac{R_s}{B_o} \cdot \, \bar u_o + \frac{1}{B_g} \cdot \, \bar u_g \right) d \bar \Sigma = q_G |
| |
|
| (22) |
V_o =\frac{B_o}{1-R_s \, R_v} \cdot (V_O - R_v \, V_G) |
|
| (23) |
V_g =\frac{B_g}{1-R_s \, R_v} \cdot (V_G - R_s \, V_O) |
|
| (24) |
V_\phi = V_w + V_o + V_g |
|
| (25) |
s_w = \frac{V_w}{V_\phi} |
|
| (26) |
s_o = \frac{V_o}{V_\phi} |
|
| (27) |
s_g = \frac{V_g}{V_\phi} |
| |
| (29) |
V_\phi = V_w + V_o + V_g = B_w \cdot V_W + \frac{B_o - R_s \cdot B_g}{1-R_s \, R_v} \cdot V_O + \frac{B_g - R_v \cdot B_o}{1-R_s \, R_v} \cdot V_G |
|
| (30) |
\phi = V_\phi/V = V_w/V + V_o/V + V_g/V = \phi_0 \cdot \exp(c_\phi \cdot (p-p_0)) |
|
| (31) |
B_w \cdot V_W/V_{\phi0} + \frac{B_o - R_s \cdot B_g}{1-R_s \, R_v} \cdot V_O/V_{\phi0} + \frac{B_g - R_v \cdot B_o}{1-R_s \, R_v} \cdot V_G/V_{\phi0} = \exp(c_\phi \cdot (p-p_0)), \quad V_{\phi0} = V \cdot \phi_0 |
|
|
| (32) |
\bar u_w = - k_p \cdot \frac{k_{rw}}{\mu_w} \cdot \hat k * \big( \bar \nabla} p_w - \rho_w \cdot \bar g \big) |
|
| (33) |
\bar u_o = - k_p \cdot \frac{k_{ro}}{\mu_o} \cdot \hat k * \big( \bar \nabla p_o - \rho_o \cdot \bar g \big) |
|
| (34) |
\bar u_g = - k_p \cdot \frac{k_{rg}}{\mu_g} \cdot \hat k * \big( \bar \nabla p_g - \rho_g \cdot \bar g \big) |
|
| (35) |
k_p = \exp \big[ c_k \, (\,p - p_{\mathrm{ref}}\,) \big] |
| (36) |
\hat k * \bar v =
\bar e_1 \cdot (k_{11} \, v_1 + k_{12} \, v_2 + k_{13} \, v_3)
+ \bar e_2 \cdot (k_{21} \, v_x + k_{22} \, v_y + k_{23} \, v_3) + \bar e_3 \cdot (k_{31} \, v_1 + k_{32} \, v_2 + k_{33} \, v_3) |
in arbitrary coordinate system
\bold e = \{ \, \bar e_1, \, \bar e_2, \, \bar e_3 \, \} |
| (37) |
\rho_w = \frac{ \mathring \rho_W}{B_w} |
|
| (38) |
\rho_o = \frac{ \mathring \rho_O + \mathring \rho_G \cdot R_s}{B_o} |
|
| (39) |
\rho_g = \frac{ \mathring \rho_G + \mathring \rho_O \cdot R_v}{B_g} |
| |
| (40) |
\bar u_w = - k_p \cdot \frac{k_{rw}}{\mu_w} \cdot \big( \hat k * \bar \nabla} p_w - \rho_w \cdot \hat k * \bar g \big) |
|
| (41) |
\bar u_o = - k_p \cdot \frac{k_{ro}}{\mu_o} \cdot \big( \hat k * \bar \nabla p_o - \rho_o \cdot \hat k * \bar g \big) |
|
| (42) |
\bar u_g = - k_p \cdot \frac{k_{rg}}{\mu_g} \cdot \big( \hat k * \bar \nabla p_g - \rho_g \cdot \hat k * \bar g \big) |
| |
| (43) |
\bar u_w = - k_p \cdot \frac{k_{rw}}{\mu_w} \cdot \big( \hat k * \bar \nabla} (p + p_{cw}) - \rho_w \cdot \hat k * \bar g \big) |
|
| (44) |
\bar u_o = - k_p \cdot \frac{k_{ro}}{\mu_o} \cdot \big( \hat k * \bar \nabla (p + p_{co}) - \rho_o \cdot \hat k * \bar g \big) |
|
| (45) |
\bar u_g = - k_p \cdot \frac{k_{rg}}{\mu_g} \cdot \big( \hat k * \bar \nabla (p + p_{cg}) - \rho_g \cdot \hat k * \bar g \big) |
| |
| (46) |
p_w = p + p_{cw}, \quad p_{cw} = \frac{1}{3} \cdot ( -2 \cdot p_{cow} + p_{cog} ) |
|
| (47) |
p_o = p + p_{co}, \quad p_{co} = \frac{1}{3} \cdot (p_{cow} + p_{cog}) |
|
| (48) |
p_g = p + p_{cg}, \quad p_{cg} = \frac{1}{3} \cdot (p_{cow} - 2 \cdot p_{cog} ) |
|
| (49) |
p = \frac{1}{3} \left( p_o + p_g + p_w \right) |
|
| (50) |
\dot p_{cw} = \frac{dp_{cw}}{ds_w} |
|
| (51) |
\dot p_{co} = \frac{dp_{co}}{ds_o} |
|
| (52) |
\dot p_{cg} = \frac{dp_{cg}}{ds_g} |
| |
| (53) |
\hat k * \bar \nabla =
\bar e_1 \, (k_{11} \, \partial_1 + k_{12} \, \partial_2 + k_{13} \, \partial_3)
+ \bar e_2 \,(k_{21} \, \partial_2 + k_{22} \, \partial_2 + k_{23} \, \partial_3) + \bar e_3 \, (k_{31} \, \partial_1 + k_{32} \, \partial_2 + k_{33} \, \partial_3) |
|
| (54) |
\hat k * \bar g =
\bar e_1 \, (k_{11} \, g_1 + k_{12} \, g_2 + k_{13} \, g_3)
+ \bar e_2 \,(k_{21} \, g_1 + k_{22} \, g_2 + k_{23} \, g_3) + \bar e_3 \, (k_{31} \, g_1 + k_{32} \, g_2 + k_{33} \, g_3) |
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Spatial discretization 3D
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| (55) |
\frac{\partial \, V_{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} |
|
| (56) |
\frac{\partial \, V_{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 |
|
| (57) |
\frac{\partial \, V_{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 |
|
| (58) |
V = \cup_\alpha \, V_\alpha |
|
| (59) |
\frac{\partial \, V_{W \alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{W\alpha\beta}} + q_{W\alpha} |
|
| (60) |
\frac{\partial \, V_{O\alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{O\alpha\beta} + q_O |
|
| (61) |
\frac{\partial \, V_{G\alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{G\alpha\beta} + q_G |
|
|
| (62) |
U_{W\alpha\beta} = - \frac{1}{B_w} \cdot \bar u_w \, \bar n_{\alpha\beta} |
|
| (63) |
U_{O\alpha\beta} = - \left( \frac{1}{B_o} \cdot \, \bar u_o + \frac{R_v}{B_g} \cdot \, \bar u_g \right) \, \bar n_{\alpha\beta} |
|
| (64) |
U_{G\alpha\beta} = - \left( \frac{R_s}{B_o} \cdot \, \bar u_o + \frac{1}{B_g} \cdot \, \bar u_g \right) \, \bar n_{\alpha\beta} |
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Spatial discretization 2D
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| (65) |
\frac{\partial \, V_{W \alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{W\alpha\beta}} + q_{W\alpha} |
|
| (66) |
\frac{\partial \, V_{O\alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{O\alpha\beta} + q_O |
|
| (67) |
\frac{\partial \, V_{G\alpha}}{dt} = \sum_{\beta \in \Gamma_\alpha} A_{\alpha\beta} \cdot U_{G\alpha\beta} + q_G |
|
| (68) |
A_{\alpha\beta} = D_{\alpha\beta} \cdot h_{\alpha\beta} |
|
| | | |
Specify local orthogonal coordinate system:
| (69) |
\bold e_{\alpha\beta} = \{ \, \bar e_{1\alpha\beta}, \, \bar e_{2\alpha\beta}, \, \bar e_{3\alpha\beta} \, \} |
| (70) |
\bar e_{1\alpha\beta} =
\cos \theta_{z\alpha\beta} \cdot \cos \zeta_{x\alpha\beta} \cdot \bar e_{x\alpha\beta} +
\cos \theta_{z\alpha\beta} \cdot \sin \zeta_{x\alpha\beta} \cdot \bar e_{y\alpha\beta}
+ \sin \theta_{z\alpha\beta} \cdot \bar e_{z\alpha\beta} |
| (71) |
\bar e_{2\alpha\beta} =
\cos \theta_{z\alpha\beta} \cdot \sin \zeta_{x\alpha\beta} \cdot \bar e_{x\alpha\beta} +
\cos \theta_{z\alpha\beta} \cdot \cos \zeta_{x\alpha\beta} \cdot \bar e_{y\alpha\beta}
+ \sin \theta_{z\alpha\beta} \cdot \bar e_{z\alpha\beta} |
| (72) |
\bar e_{3\alpha\beta} =
\sin \theta_{z\alpha\beta} \cdot \sin \zeta_{x\alpha\beta} \cdot \bar e_{x\alpha\beta} +
\sin \theta_{z\alpha\beta} \cdot \cos \zeta_{x\alpha\beta} \cdot \bar e_{y\alpha\beta}
+ \cos \theta_{z\alpha\beta} \cdot \bar e_{z\alpha\beta} |
| (73) |
\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} |
|
| (74) |
\bar n_{\alpha \beta} = \bar e_1 |
| (75) |
\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 ) |
| (76) |
\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) |
| (77) |
\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} ) |
| (78) |
\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 |
| (79) |
\big( \hat k * \bar \nabla} p - \rho \cdot \hat k * \bar g \big) = k_{11} \, \partial_1 p - \rho \, g \cdot ( k_{11} \, \sin \theta_{z\alpha\beta} + k_{13} \, \cos \theta_{z\alpha\beta} ) |
| (80) |
\big( \hat k * \bar \nabla} p - \rho \cdot \hat k * \bar g \big) \cdot \bar n_{\alpha \beta} = k_{11\alpha \beta} \, ( \partial_1 p - \rho \, g \cdot \sin \theta_{z\alpha\beta} ) |
| (81) |
\bar u_{\alpha \beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{r}}{\mu} \cdot k_{11\alpha \beta} \, ( \partial_1 p - \rho \, g \cdot \sin \theta_{z\alpha\beta} ) |
| (82) |
\bar u_{\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{r}}{\mu} \cdot k_{11\alpha \beta} \cdot \big( R^{-1}_{\alpha\beta} \cdot (p_{\alpha\beta} - p_\alpha) - \rho_\alpha \, g \cdot \sin \theta_{z\alpha\beta} \big) |
|
| (83) |
\bar u_{w\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{rw}}{\mu_w} \cdot k_{11\alpha \beta} \cdot \Big( R^{-1}_{\alpha\beta} \cdot \big( p_{w\alpha\beta} - p_\alpha - \dot p_{cw} \cdot (s_{w\beta} - s_{w\alpha}) \big) - \rho_{w\alpha} \, g \cdot \sin \theta_{z\alpha\beta} \Big) |
|
| (84) |
\bar u_{o\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{ro}}{\mu_o} \cdot k_{11\alpha \beta} \cdot \big( R^{-1}_{\alpha\beta} \cdot (p_{o\alpha\beta} - p_\alpha) - \rho_{o\alpha} \, g \cdot \sin \theta_{z\alpha\beta} \big) |
|
| (85) |
\bar u_{g\alpha\beta} \cdot \bar n_{\alpha \beta} = - k_p \cdot \frac{k_{rg}}{\mu_g} \cdot k_{11\alpha \beta} \cdot \big( R^{-1}_{\alpha\beta} \cdot (p_{g\alpha\beta} - p_\alpha) - \rho_{g\alpha} \, g \cdot \sin \theta_{z\alpha\beta} \big) |
| |
| (86) |
U_{W\alpha\beta} =
\frac{k_{p\alpha}}{B_{w\alpha}} \cdot \frac{k_{rw\alpha}}{\mu_{w\alpha}} \cdot k_{11\alpha \beta} \cdot \big( R^{-1}_{\alpha\beta} \cdot (p_{\alpha\beta} - p_\alpha) - \rho_{w\alpha} \, g \cdot \sin \theta_{z\alpha\beta} \big) |
| | | |
| (87) |
U_{W\beta\alpha} =
\frac{k_{p\beta}}{B_{w\beta}} \cdot \frac{k_{rw\beta}}{\mu_{w\beta}} \cdot k_{11 \beta\alpha} \cdot \big( R^{-1}_{\beta\alpha} \cdot (p_{\beta\alpha} - p_\beta) - \rho_{w\beta} \, g \cdot \sin \theta_{z\beta\alpha} \big) |
| | | |
| (88) |
U_{W\alpha\beta} = U_{W\beta\alpha} |
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