@wikipedia\backslash frac\{\backslash partial\; s\}\{\backslash partial\; t\}\; +\; q\; \backslash cdot\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; x\}\; \backslash left(\; \backslash frac\{f\}\{\backslash phi\; \backslash ,\; A\; \}\; \backslash right)\; =\; 0s(t=0,x)\; =\; 0s(t,0)\; =\; 1@wikipedia
Ideally balanced water + dead oil 1D waterflood model without gravity and capillary effects.
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| \frac{\partial s}{\partial t} + q \,cdot \frac{\partial }{\partial x} \left( \frac{f}{\phi \, A \Sigma} \right) = 0 |
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| s(t=0,x) = 0 |
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| s(t,0) = 1 |
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...
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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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| anchor | ProxyBL |
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| alignment | left |
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| \frac{\partial s}{\partial t_D} +\frac{\partial f}{\partial x_D} = 0 |
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| anchor | ProxyIC |
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| alignment | left |
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| s(t=0,x) = 0 |
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| anchor | ProxyBC |
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| alignment | left |
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| s(t,0) = 1 |
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where
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| body | --uriencoded--\displaystyle t_D = \frac%7BV_\phi \, t%7D%7Bq%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 |
| reservoir length along -axis |
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| body | V_\phi= \phi \cdot h \cdot D \cdot L |
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| 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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