A proxy model of watercut in producing well with reservoir saturation s=\{ s_w, \, s_o, \, s_g \} and reservoir pressure p_e:
| (1) | {\rm Y_{wm}} = \frac{1 - \epsilon_g}{1 - \frac{M_{ro}}{M_{rw}} \cdot \frac{B_w}{B_o} }, \quad \epsilon_g = \frac{A}{q_t} \cdot M_{ro} \cdot \left[ \frac{\partial P_c}{\partial r} + (\rho_w-\rho_o) \cdot g \cdot \sin \alpha \right] |
where
B_w(p_e) | Water formation volume factor | B_o(p_e) | Oil formation volume factor | s | Reservoir saturation \{ s_w, \, s_o, \, s_g \} |
|---|---|---|---|---|---|
M_{rw}(s) | Relative water mobility | M_{ro}(s) | Relative oil mobility | p_e | Current formation pressure |
\rho_w | Water density | \rho_o | Oil density | g | Standard gravity constant |
q_t | Total sandface flowrate | A | Cross-sectional flow area | \alpha | Deviation of flow from horizontal plane |
P_c(s) | capillary pressure |
If saturation does not vary along the flow substantially then capillary effects are vanishing so that \displaystyle \frac{\partial P_c}{\partial r} = \dot P_c \cdot \frac{\partial s_w}{\partial r} \approx 0.
If flow is close to horizontal then gravity effects are vanishing too and \sin \alpha \approx 0.
In these cases (1) simplifies to:
| (2) | {\rm Y_{wm}} = \frac{1}{1 - \frac{M_{ro}}{M_{rw}} \cdot \frac{B_w}{B_o} } |
The models (1) and (2) can also be used in gross field production analysis assuming homogeneous reservoir saturation:
| (3) | s_w(t) = s_{wi} + (1-s_{wi}-s_{or}) \cdot \rm RF(t)/E_S |