Ideally balanced water + dead oil 1D waterflood model without gravity and capillary effects.
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where
\displaystyle s= E_D = \frac{s_w - s_{wi}}{1-s_{wi}-s_{or}} | water → oil displacement efficiency |
q | sandface injection rate, assumed equal to sandface liquid production rate |
\phi(x) | reservoir porosity |
\Sigma(x) = h \, D | cross-section area available for flow |
h(x) | reservoir thickness |
D(x) | reservoir width = reservoir length transversal to flow |
\displaystyle f = \frac{1}{1+M_{ro}/M_{rw}} | in-situ fractional flow function |
relative oil mobility | |
M_{wo} = k_{rw}(s_w)/\mu_w | 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:
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where
\displaystyle t_D = \frac{V_\phi \, t}{q} | dimensionless time |
\displaystyle x_D = \frac{x}{L} | dimensionless distance |
L | reservoir length along x-axis |
V_\phi= \phi \cdot h \cdot D \cdot L | reservoir pore volume |
The equation
(4) can be explicitly integrated:
(7) | x_D(s) = \begin{cases}\dot f(s) \cdot t_D, & \mbox{if } s < s^*\\1- \dot f(s) \cdot t_D, & \mbox{if } s \geq s^*\end{cases} |
where
s^* | critical saturation where fractional flow function reaches inflection point: \ddot f = 0 |
\displaystyle \dot f(s) = \frac{d f}{ds} | first derivative of the fractional flow function |
\displaystyle \ddot f(s) = \frac{d^2 f}{ds^2} | second derivative of the fractional flow function |
Algebraic equation (7) can be used to find a solution of (4) in terms of saturation over time and distance: s(t,x).
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Dynamic Flow Model / Reservoir Flow Model (RFM)
[ Production / Subsurface Production / Reserves Depletion / Recovery Methods ]