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

q_t(t) = \rm const


\Delta p(t) = | p_e(t) - p_{wf}(t) | = \Delta p = \rm const


During the PSS regime the formation pressure declines linearly with time: .


The exact solution of diffusion equation for PSS:


p_e(t) = p_i + \frac{q_t}{ V_{\phi} \, c_t} \ t



varying formation pressure at the external reservoir boundary



p_{wf}(t) = p_e(t) + J^{-1} q_t



varying bottom-hole pressure



J = \frac{q_t}{2 \pi \sigma} \left[ \ln \left ( \frac{r_e}{r_w} \right)  +S + 0.75 \right]



constant productivity index


and develops a unit slope on PTA diagnostic plot  and Material Balance diagnostic plot:

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.



See also


Steady State (SS) well flow regime