Fluid flow with fluid pressure p(t, {\bf r}) is linearly changing in time:
p(t, {\bf r}) = \psi({\bf r}) + A \cdot t, \quad A = \rm const |
The fluid velocity {\bf u}(t, {\bf r}) may not be stationary.
In the most general case (both reservoir and pipelines) the fluid velocity is proportional to pressure gradient and can be written as:
(1) | {\bf u}(t, {\bf r})= - M({\bf r}, p, \nabla p) \nabla p |
with right side dependent on time through the pressure variation.
In case of linear correlation: M({\bf r}, p, \nabla p) = M({\bf r}) the PSS flow velocity will be stationary as the right side of (1) is not dependant on time.
The fluid temperature T(t, {\bf r}) is supposed to vary slowly enough to provide quasistatic equilibrium.
In terms of Well Flow Performance the PSS flow means:
(2) | q_t(t) = \rm const |
(3) | \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: p_e(t) \sim t.
The exact solution of diffusion equation for PSS:
| varying formation pressure at the external reservoir boundary | ||
| varying bottom-hole pressure | ||
| 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