Well flow regime with constant rate and constant delta pressure between wellbore and formation does not change in time:

(1)	$\begin{array}{l}\displaystyle q_t(t) = \rm const\end{array}$

(2)	$\begin{array}{l}\displaystyle \Delta p(t) = \| p_e(t) - p_{wf}(t) \| = \Delta p = \rm const\end{array}$

During the PSS regime the formation pressure declines linearly with time: $\begin{array}{l}p_e(t) \sim t\end{array}$ .

The exact solution of diffusion equation for PSS:

(3)	$\begin{array}{l}\displaystyle p_e(t) = p_i + \frac{q_t}{ V_{\phi} \, c_t} \ t\end{array}$

varying formation pressure at the external reservoir boundary

(4)	$\begin{array}{l}\displaystyle p_{wf}(t) = p_e(t) + J^{-1} q_t\end{array}$

varying bottom-hole pressure

(5)	$\begin{array}{l}\displaystyle J = \frac{q_t}{2 \pi \sigma} \left[ \ln \left ( \frac{r_e}{r_w} \right) +S + 0.75 \right]\end{array}$

Fig. 1. PTA Diagnostic Plot for vertical well in single-layer homogeneous reservoir with impermeable circle boundary (PSS).

Pressure is in blue and log-derivative is in red.

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See also