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Ideally balanced water + dead oil 1D waterflood model without gravity and capillary effects.
\frac{\partial s}{\partial t} + q \cdot \frac{\partial }{\partial x} \left( \frac{f}{\phi \, \Sigma} \right) = 0 |
| | |
where
| water → oil displacement efficiency |
| sandface injection rate, assumed equal to sandface liquid production rate |
| reservoir porosity |
| cross-section area available for flow |
| reservoir thickness |
| reservoir width = reservoir length transversal to flow |
| in-situ fractional flow function |
| relative oil mobility |
| relative water mobility |
Approximations
In many practical applications (for example, laboratory SCAL tests and reservoir proxy-modeling) one can assume constant porosity and reservoir width:
\frac{\partial s}{\partial t_D} +\frac{\partial f}{\partial x_D} = 0 |
| | |
where
| dimensionless time |
| dimensionless distance |
| reservoir length along -axis |
| reservoir pore volume |
The equation can be explicitly integrated:
x_D(s) = \begin{cases}\dot f(s) \cdot t_D, & \mbox{if } s < s^*\\ 2 x^*_D- \dot f(s) \cdot t_D, & \mbox{if } s \geq s^*\end{cases}
|
where
| critical saturation where fractional flow function reaches inflection point: |
| "inflection" distance |
| "inflection" time |
| first and second derivatives of the fractional flow function |
Algebraic equation can be used to find a solution of in terms of saturation over time and distance: (see Fig. 1).
|
Fig. 1 – Sample case of reservoir saturation profile capturing the moment when water front is still on mid-way towards the producing well (sitting at ) |
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Dynamic Flow Model / Reservoir Flow Model (RFM)
[ Production / Subsurface Production / Reserves Depletion / Recovery Methods ]