Fluid flow with fluid pressure gradient  and velocity  are not changing in time:

\nabla p(t, {\bf r}) = \nabla p({\bf r})


{\bf u}(t, {\bf r}) = {\bf u}({\bf r})


The fluid temperature   is supposed to vary slowly enough to provide quasistatic equilibrium.



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

Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / PSS Diagnostics

Steady State (SS) well flow regime