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One of the Productivity Diagnostics methods based on relation on correlation between pressure drawdown integral  LaTeX Math Inlinebody--uriencoded--\displaystyle G integral:

LaTeX Math Block
anchor1
alignmentleft
G (t) =

...

 \frac{1}{t} \int_

...

0^t \left( p_

...

{wf}(\tau) - p_e(\tau) \right) d\tau

 and and total sandface flowrate cumulatives  LaTeX Math Inlinebody--uriencoded--\displaystyle

LaTeX Math Block
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Q_t(t) =

...

 \frac{1}{t} \int_

...

0^t q_t(\tau) d\tau

where

LaTeX Math Inline
body\tau

production/injection time

LaTeX Math Inline
bodyq_t

total sandface flowrate as function of time 

LaTeX Math Inline
body\tau

LaTeX Math Inline
bodyp_e

drain-area formation pressure as function of time 

LaTeX Math Inline
body\tau
LaTeX Math Inline
body\tau

LaTeX Math Inline
body--uriencoded--p_%7Bwf%7D

bottomhole pressure  as function of time 

LaTeX Math Inline
body\tau

...

LaTeX Math Block
anchorpwf
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G(t) = J^{-1} Q_t(t)

where

LaTeX Math Inline
bodyJ

constant productivity index


In case pressure data is available for a fair interpolation it is recommended to plot sandface cumulatives rather than surface which provides better linearity with pressure integral for Steady-State linearity for stabilized reservoir flow.


Although it is equally applicable to producers and injectors, due to lack of BHP and formation pressure data availability for producers in most practical cases in the past the Hall plot analysis was mostly applied for water injectors.

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