Consider a water injector with main pay in Reservoir Layer #1 and spontaneous fracture extending down to Reservoir Layer #2 (see Fig. 1).
Assume that fracture is not fixed and requires surplus pressure
to get opened against the rock burden.
When injection bottomhole pressure
| LaTeX Math Inline |
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| body | --uriencoded-- p_%7Bwf%7D |
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is below fracture opening value
| LaTeX Math Inline |
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| body | --uriencoded-- p_%7Bwf%7D < \Delta p_f |
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then water is going to the main pay only (Reservoir Layer #1) and flow radially around the well.
When injection bottomhole pressure
| LaTeX Math Inline |
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| body | --uriencoded-- p_%7Bwf%7D |
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is above fracture opening value
| LaTeX Math Inline |
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| body | --uriencoded-- p_%7Bwf%7D > \Delta p_f |
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then water is going to the fracture and then gets distributed between Reservoir Layer #1 and Reservoir Layer 2
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| Fig. 1. Dual-layer well schematic |
| 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 = \Delta p_f + \frac{J_1 \cdot p_1 + J_2 \cdot (p_2- \delta p_2)}{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_c}{J_1 + J_2} |
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| LaTeX Math Block |
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| p_c = \left(1 + \frac{J_1}{J_2} \right) \Delta p_f + p_2 - \delta p_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 |
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| | true vertical height between the layers tops |
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| | wellbore fuid density |
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| | gravity constant |
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| | fracture opening pressure |
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| Layer #1 |
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| | LaTeX Math Inline |
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| body | --uriencoded--p_%7Bwf%7D |
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| bottom-hole pr4essure 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\, h |
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| bottom-hole pr4essure at Layer #2 top |
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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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| Expand |
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| Panel |
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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} = \Delta p_f + p_1 + q_1/J_1 |
| LaTeX Math Block |
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| p_{wf,2} = p_{wf} + \delta p_2 = \Delta p_f + p_2 + q_2/J_2 |
This leads to | LaTeX Math Block |
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| q_1 = J_1 \cdot (p_{wf} - p_1 - \Delta p_f) |
| LaTeX Math Block |
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| q_2 = J_2 \cdot (p_{wf,2} - p_2 - \Delta p_f) = J_2 \cdot (p_{wf} - (p_2 + \Delta p_f-\delta p_2) ) |
and | LaTeX Math Block |
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| q = q_1 + q_2 = q_1 = J_1 \cdot (p_{wf} - (p_1 + \Delta p_f))+ J_2 \cdot (p_{wf} - (p_2-\delta p_2 + \Delta p_f) ) |
| LaTeX Math Block |
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| q = (J_1+J_2)\cdot p_{wf} - J_1 \cdot (p_1 + \Delta p_f) + J_2 \cdot ((p_2-\delta p_2 + \Delta p_f) ) |
or | LaTeX Math Block |
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| q = J \cdot (p_{wf}-p_e), \ {\rm where} \ J = J_1 + J_2 \ {\rm and} \ p_e = J^{-1} \cdot (J_1 \cdot (p_1 + \Delta p_f) + J_2 \cdot (p_2-\delta p_2 + \Delta p_f)) |
or | LaTeX Math Block |
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| p_e = \Delta p_f + J^{-1} \cdot (J_1 \cdot p_1 + J_2 \cdot (p_2-\delta p_2)) |
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See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Subsurface E&P Disciplines / Field Study & Modelling / Production Analysis / Productivity Diagnostics
[ Production Technology / Well Flow Performance ]
[ Formation pressure (Pe) ] [ Multi-layer IPR ]
