| (1) |
\rho c \cdot \frac{\partial T}{\partial t}
+\sum_{\gamma=w,o,g}\rho_{\gamma}c_{p\gamma} \cdot \bar u_{\gamma}\cdot\nabla T
-\phi\cdot T\cdot\sum_{\gamma=w,o,g}s_{\gamma} \alpha_{T\gamma}\frac{\partial p_\gamma}{\partial t}
-\sum_{\gamma=w,o,g} \left( \rho_{\gamma} \cdot \alpha_{T\gamma} \cdot T - 1 \right) \, \cdot \, \bar u_{\gamma}\cdot\nabla p_\gamma
- \nabla \cdot \left[ \lambda_t \cdot \nabla T \right] = \dot P^{\uparrow} + \dot P^{\downarrow} |
|
| (2) |
\rho c = (1-\phi) \cdot \rho_r \cdot c_r +\phi\,\cdot\!\sum_{\gamma=w,o,g}\rho_{\gamma} \cdot c_{v\gamma} \cdot s_{\gamma} |
|
| (3) |
\lambda_t=(1-\phi) \cdot \lambda_r + \phi \cdot \left ( s_w \lambda_w + s_o \lambda_o + s_g \lambda_g \right) |
|
| (4) |
\mu_{JT} = \frac{1}{c_p} \cdot \left( \alpha_T \cdot T - \frac{1}{\rho} \right) |
|
| (5) |
\alpha_T = - \frac{1}{\rho} \cdot \left( \frac{\partial \rho}{\partial T} \right)_p |
|
| (6) |
\dot P^{\uparrow} =
\sum_{\gamma = w,o,g} \, \left[ \rho_\gamma ({\bf r}) \cdot c_{p\gamma} ({\bf r}) \, T({\bf r}) +p_\gamma({\bf r}) \right] \, q^{\uparrow}_\gamma({\bf r})\, \delta({\bf r})
|
|
| (7) |
\dot P^{\downarrow} =
\sum_{\gamma = w,o,g} \, \left[ \rho^{\downarrow}_\gamma ({\bf r}) \cdot c^{\downarrow}_{p\gamma} ({\bf r}) \, T^{\downarrow} ({\bf r}) +p^{\downarrow}_\gamma({\bf r}) \right] \, q^{\downarrow}_\gamma({\bf r})\, \delta({\bf r}) |
|
|
| (8) |
\rho c \cdot \frac{\partial T}{\partial t} =
-\sum_{\gamma=w,o,g}\rho_{\gamma}c_{p\gamma} \cdot \bar u_{\gamma}\cdot\nabla T
+\phi\cdot T\cdot\sum_{\gamma=w,o,g}s_{\gamma} \alpha_{T\gamma}\frac{\partial p_\gamma}{\partial t}
+\sum_{\gamma=w,o,g} \left( \rho_{\gamma} \cdot \alpha_{T\gamma} \cdot T - 1 \right) \, \cdot \, \bar u_{\gamma}\cdot\nabla p_\gamma
+ \nabla \cdot \left[ \lambda_t \cdot \nabla T \right] + \dot P^{\uparrow} + \dot P^{\downarrow} |
|
| (9) |
\rho c \cdot \frac{\partial T}{\partial t} =
-\sum_{\gamma=w,o,g}\rho_{\gamma}c_{p\gamma} \cdot \nabla( \bar u_{\gamma}\cdot T )
-\sum_{\gamma=w,o,g}\rho_{\gamma}c_{p\gamma} \cdot (\nabla \bar u_{\gamma})\cdot T
+\phi\cdot T\cdot\sum_{\gamma=w,o,g}s_{\gamma} \alpha_{T\gamma}\frac{\partial p_\gamma}{\partial t}
+\sum_{\gamma=w,o,g} \left( \rho_{\gamma} \cdot \alpha_{T\gamma} \cdot T - 1 \right) \, \cdot \, \nabla (\bar u_{\gamma}\cdot p_\gamma)
+\sum_{\gamma=w,o,g} \left( \rho_{\gamma} \cdot \alpha_{T\gamma} \cdot T - 1 \right) \, \cdot \, \nabla \bar u_{\gamma}\cdot p_\gamma
+ \nabla \cdot \left[ \lambda_t \cdot \nabla T \right] + \dot P^{\uparrow} + \dot P^{\downarrow} |
|
| (10) |
\frac{dT}{dt} \cdot \int_V \rho c =
-\sum_{\gamma=w,o,g} \int_A \rho_{\gamma}c_{p\gamma} \cdot ( \bar u_{\gamma}\cdot T ) \cdot d \bar A
-\sum_{\gamma=w,o,g} \int_V \rho_{\gamma}c_{p\gamma} \cdot (\nabla \bar u_{\gamma})\cdot T \cdot dV
+\phi\cdot T\cdot\sum_{\gamma=w,o,g} \int_V s_{\gamma} \alpha_{T\gamma}\frac{\partial p_\gamma}{\partial t} \cdot dV
+\sum_{\gamma=w,o,g} \int_V \left( \rho_{\gamma} \cdot \alpha_{T\gamma} \cdot T - 1 \right) \, \cdot \, \nabla (\bar u_{\gamma}\cdot p_\gamma) \cdot dV
+\sum_{\gamma=w,o,g} \int_V \left( \rho_{\gamma} \cdot \alpha_{T\gamma} \cdot T - 1 \right) \, \cdot \, \nabla \bar u_{\gamma}\cdot p_\gamma \cdot dV
+ \int_A \lambda_t \cdot \nabla T \cdot d \bar A + \int_V \dot P^{\uparrow} \cdot dV + \int_V \dot P^{\downarrow} \cdot dV |
|
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Dynamic Flow Model / Reservoir Flow Model (RFM) / Modified Black Oil Reservoir Flow @model / Subsurface Temperature Dynamic @model
