...
| Well |
|---|
| | total subsurface flowrate of the well |
|---|
| | total well productivity Index |
|---|
| | apparent formation pressure of dual-layer formation | | wellbore pressure loss between the tips of two layers | | true vertical height between the layers tops | | wellbore fuid density | | gravity constant |
|---|
| Layer #1 |
|---|
| | LaTeX Math Inline |
|---|
| body | --uriencoded--p_%7Bwf%7D = p_%7Bwf, 1%7D |
|---|
|
| bottom-hole pr4essure pressure at Layer #1 top |
|---|
| | total subsurface flowrate of the Layer #1 |
|---|
| | formation pressure of the Layer #1 |
|---|
| | productivity Index of the Layer #1 |
|---|
| Layer #2 |
|---|
| | LaTeX Math Inline |
|---|
| body | --uriencoded--p_%7Bwf2%7D = p_%7Bwf%7D + \delta p_2 |
|---|
|
| bottom-hole pr4essure at Layer #2 top |
|---|
| | wellbore pressure loss between the tips of two layers |
|---|
| | total subsurface flowrate of the Layer #2 |
|---|
| | formation pressure of the Layer #2 |
|---|
| | productivity Index of the Layer #2 |
|---|
In many practical cases one can safely assume:
| LaTeX Math Block |
|---|
|
\delta p_2 = \rho \, g \, h |
...
The above equations are valid for both producers producers
and injectors
.
| 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 |
|---|
| p_{wf, 1} = p_{wf} = p_1 - q_1/J_1 |
| LaTeX Math Block |
|---|
| p_{wf,2} = p_{wf} + \delta p_2 = p_2 - q_2/J_2 |
This leads to | LaTeX Math Block |
|---|
| q_1 = J_1 \cdot (p_1 - p_{wf}) |
| LaTeX Math Block |
|---|
| q_2 = J_2 \cdot (p_2 - p_{wf,2}) = J_2 \cdot ((p_2-\delta p_2)- p_{wf}) |
and | LaTeX Math Block |
|---|
| 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 |
|---|
| 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 |
|---|
| q = (J_1 + J_2) \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)) |
|
|
...
