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| \partial_t \bigg [ \phi \ \bigg ( \frac{s_w}{B_w} \bigg ) \bigg ] + \nabla \bigg ( \frac{1}{B_w} \ \mathbf{u}_w \bigg ) =
q_W (\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \bigg ( \frac{s_o}{B_o} + \frac{R_v \ s_g}
{B_g} \bigg ) \bigg ] + \nabla \bigg ( \frac{1}{B_o} \ \mathbf{u}_o
+ \frac{R_v}{B_g} \ \mathbf{u}_g \bigg ) = q_O(\mathbf{r}) |
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| \partial_t \bigg [ \phi \ \bigg ( \frac{s_g}{B_g} + \frac{R_s \ s_o}
{B_o} \bigg ) \bigg ] + \nabla \bigg ( \frac{1}{B_g} \ \mathbf{u}_g
+ \frac{R_s}{B_o} \ \mathbf{u}_o \bigg ) = q_G (\mathbf{r}) |
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| anchor | DarcyW |
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| alignment | left |
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| \mathbf{u}_w = - k_a \ \frac{k_{rw}(s_w, s_g)}{\mu_w} \ ( \nabla P_w - \rho_w \ \mathbf{g} ) |
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| anchor | DarcyO |
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| alignment | left |
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| \mathbf{u}_o = - k_a \ \frac{k_{ro}(s_w, s_g)}{\mu_o} \ ( \nabla P_o - \rho_o \ \mathbf{g} ) |
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| anchor | DarcyG |
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| alignment | left |
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| \mathbf{u}_g = - k_a \ \frac{k_{rg}(s_w, s_g)}{\mu_g} \ ( \nabla P_g - \rho_g \ \mathbf{g} ) |
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| anchor | CapilarOW |
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| alignment | left |
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| P_o - P_w = P_{cow}(s_w) |
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| anchor | CapilarOG |
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| alignment | left |
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| P_o - P_g = P_{cog}(s_g) |
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| anchor | swsosg |
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| alignment | left |
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| s_w + s_o + s_g = 1 |
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(\rho \,c_{pt})_p \frac{\partial T}{\partial t}
- \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \nabla P
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T
- \nabla (\lambda_t \nabla T) = \frac{\delta E_H}{ \delta V \delta t}rho_{\rm inj} \, c_{\rm inj} \, T_{\rm inj} \, q_{\rm inj}({\bf r})\, \delta({\bf r}) |
The disambiguation fo the properties in the above equation is brought in The list of dynamic flow properties and model parameters.
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