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| LaTeX Math Block |
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| q = q_1 + q_2 |
| | LaTeX Math Block |
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| p_{wf} = p_e - q/J |
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| LaTeX Math Block |
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| J = J_1 + J_2 |
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| LaTeX Math Block |
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| p_e = \frac{J_1 \cdot p_1 + J_2 \cdot (p_2- \rho \, g \, hdelta p_2)}{J_1 + J_2} |
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where
| Well |
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| | total subsurface flowrate of the well |
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| | total well productivity Index |
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| | apparent formation pressure of dual-layer formation | | true vertical height between the layers tops | | wellbore fuid density | | gravity constant |
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| Layer #1 |
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| | LaTeX Math Inline |
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| body | --uriencoded--p_%7Bwf%7D = p_%7Bwf, 1%7D |
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| bottom-hole pr4essure pressure at Layer #1 top |
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| | total subsurface flowrate of the Layer #1 |
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| | formation pressure of the Layer #1 |
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| | productivity Index of the Layer #1 |
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| Layer #2 |
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| | LaTeX Math Inline |
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| body | --uriencoded--p_%7Bwf2%7D = p_%7Bwf%7D + \rho \, g\, hdelta p_2 |
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| bottom-hole pr4essure at Layer #2 top |
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| | wellbore pressure loss between the tips of two layers |
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| | total subsurface flowrate of the Layer #2 |
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| | formation pressure of the Layer #2 |
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| | productivity Index of the Layer #2 |
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In many practical cases one can safely assume:
| LaTeX Math Block |
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\delta p_2 = \rho \, g \, h |
where
| wellbore fuid density |
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| gravity constant |
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| LaTeX Math Inline |
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| body | h = TVDSS_1 - TVDSS_2 |
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| true vertical height between -th layer and reference layer | LaTeX Math Inline |
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| body | --uriencoded--k_%7Breff%7D |
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|
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The above equations are valid for both producers producers
and injectors
.
| Expand |
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| borderColor | wheat |
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| bgColor | mintcream |
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| borderWidth | 7 |
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| LaTeX Math Block |
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| p_{wf, 1} = p_{wf} = p_1 - q_1/J_1 |
| LaTeX Math Block |
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| p_{wf,2} = p_{wf} + \delta p_2 = p_2 - q_2/J_2 |
This leads to | LaTeX Math Block |
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| q_1 = J_1 \cdot (p_1 - p_{wf}) |
| LaTeX Math Block |
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| q_2 = J_2 \cdot (p_2 - p_{wf,2}) = J_2 \cdot ((p_2-\delta p_2)- p_{wf}) |
and | LaTeX Math Block |
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| q = q_1 + q_2 = q_1 = J_1 \cdot (p_1 - p_{wf})+ J_2 \cdot ((p_2-\delta p_2)- p_{wf}) |
| LaTeX Math Block |
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| q = - (J_1+J_2)\cdot p_{wf} + J_1 \cdot p_1 + J_2 \cdot (p_2-\delta p_2) |
or | LaTeX Math Block |
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| q = J \cdot (p_e - p_{wf}), \ {\rm where} \ J = J_1 + J_2 \ {\rm and} \ p_e = J^{-1} \cdot (J_1 \cdot p_1 + J_2 \cdot (p_2-\delta p_2)) |
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See Also
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Petroleum Industry / Upstream / Production / Subsurface Production / Subsurface E&P Disciplines / Field Study & Modelling / Production Analysis / Productivity Diagnostics
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