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| \frac{\partial s}{\partial t} + q \, \frac{q}{\partialphi }{\partial, x\Sigma} \left(cdot \frac{\partial f}{\phi \, A partial x} \right) = 0 |
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| s(t=0,x) = 0 |
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| s(t,0) = 1 |
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| body | --uriencoded--\displaystyle s= E_D = \frac%7Bs_w - s_%7Bwi%7D%7D%7B1-s_%7Bwi%7D-s_%7Bor%7D%7D |
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| water → oil displacement efficiency |
| sandface injection rate, assumed equal to sandface liquid production rate |
| reservoir porosity |
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| body | A\Sigma(x) = h \, L_ D |
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| cross-section area available for flow |
| reservoir thickness |
| rerservoir reservoir width = reservoir length transversal to flow |
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| body | --uriencoded--\displaystyle f = \frac%7B1%7D%7B1+M_%7Bro%7D/M_%7Brw%7D%7D |
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| in-situ fractional flow function |
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| body | --uriencoded--M_%7Bro%7D= k_%7Bro%7D(s_o)/\mu_o |
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| relative oil mobility |
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| body | --uriencoded--M_%7Bwo%7D = k_%7Brw%7D(s_w)/\mu_w |
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| relative water mobility |
Approximations
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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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| \frac{\partial s}{\partial t_D} +\frac{\partial f}{\partial x_D} = 0 |
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| s(t=0,x) = 0 |
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| anchor | ProxyBC |
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| s(t,0) = 1 |
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where
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| body | --uriencoded--\displaystyle t_D = \frac%7Bq \, t%7D%7BV_\phi %7D |
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| dimensionless time |
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| body | --uriencoded--\displaystyle x_D = \frac%7Bx%7D%7BL%7D |
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| dimensionless distance between injector and producer |
| reservoir length along -axis |
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| body | --uriencoded--V_%7B\phi m%7D= (1-s_%7Bwi%7D-s_%7Borw%7D) \cdot \phi \cdot h \cdot D \cdot L |
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| mobile reservoir pore volume |
See Also
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Petroleum Industry / Upstream / Subsurface E&P Disciplines / Dynamic Flow Model / Reservoir Flow Model (RFM)
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The equation
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can be explicitly integrated:| LaTeX Math Block |
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| 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}
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where | critical saturation where fractional flow function reaches inflection point: | LaTeX Math Inline |
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| body | --uriencoded--\ddot f(s%5e*) = 0 |
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| body | --uriencoded--x%5e*_D= f(s%5e*) \cdot t_D%5e* |
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| "inflection" distance | | LaTeX Math Inline |
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| body | --uriencoded--t_D%5e* |
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| "inflection" time | | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \dot f(s) = \frac%7Bd f%7D%7Bds%7D, \, \, \ddot f(s) = \frac%7Bd%5e2 f%7D%7Bds%5e2%7D |
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| first and second derivatives of the fractional flow function |
Algebraic equation
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can be used to find a solution of | LaTeX Math Block Reference |
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in terms of saturation over time and distance: (see Fig. 1).
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| Fig. 1 – Sample case of Buckley–Leverett reservoir saturation profile capturing the moment,when water front is still on mid-way towards the producing well, sitting at | LaTeX Math Inline |
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| body | x_D = 1 \Leftrightarrow x = L |
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