Fluid flow with fluid pressure linearly changing in time:

p(t, {\bf r}) = \psi({\bf r}) + A \cdot t, \quad A = \rm const

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

The fluid velocity may not be stationary.

In the most general case (both reservoir and pipelines) the fluid motion equation is of fluid pressure and pressure gradient:

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

with right side dependent on time through the pressure variation.

In case of the flow with velocity dependent on pressure gradient only ) the PSS flow velocity will be stationary as the right side of is not dependant on time.

In terms of Well Flow Performance the PSS flow means:

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 also 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:

GLOSSARY > Pseudo Steady State (PSS) fluid flow > PTA Diganostic Plot PSS.png

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