Pseudo Steady State (PSS)

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

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

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

The exact solution of diffusion equation for PSS:

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

where

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

constant productivity index

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

varying formation pressure at the external reservoir boundary

and is shown as unit slope on PTA diagnostic log-log plot

During the PSS regime the formation pressure linearly declines

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