One of the Productivity Diagnostics methods based on relation between pressure integral $\begin{array}{l}G(t) = \int_0^t \left( p_{wf}(\tau) - p_e(\tau) \right) d\tau\end{array}$ and total sandface flowrate cumulatives $\begin{array}{l}Q_t(t) = \int_0^t q_t(\tau) d\tau\end{array}$

where

$\begin{array}{l}\tau\end{array}$	production/injection time
$\begin{array}{l}q_t\end{array}$	total sandface flowrate as function of time $\begin{array}{l}\tau\end{array}$
$\begin{array}{l}p_e\end{array}$	drain-area formation pressure as function of time $\begin{array}{l}\tau\end{array}$ $\begin{array}{l}\tau\end{array}$
$\begin{array}{l}p_{wf}\end{array}$	bottomhole pressure as function of time $\begin{array}{l}\tau\end{array}$

It shows unit slope on log-log plot for stabilized reservoir flow:

(1)	$\begin{array}{l}\displaystyle G(t) = J^{-1} Q_t(t)\end{array}$

where

$\begin{array}{l}J\end{array}$

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 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.

The pressure integral $\begin{array}{l}G(t)\end{array}$ is usually calculated over interpolated values of formation pressure and bottomhole pressure :

$\begin{array}{l}\displaystyle G(t) = \int_0^t \left( p_{wf}(\tau) - p_e(\tau) \right) d\tau = \sum_k \left( p_{wf}(\tau_k) - p_e(\tau_k) \right) \delta \tau_k\end{array}$

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Hall Plot

See Also