Fluid flow with fluid pressure
is linearly changing in time:
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p(t, {\bf r}) = \psi({\bf r}) + A \cdot t, \quad A = \rm const |
The fluid velocity
may not be stationary.
In the most general case (both reservoir and pipelines) the fluid motion equation is given by fluid velocity proportional to pressure gradient:
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{\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 motion equation (
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| body | M({\bf r}, p, \nabla p) = M({\bf r}) |
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) the
PSS flow velocity will be stationary as the right side of
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is not dependant on time.
The fluid temperature
is supposed to vary slowly enough to provide
quasistatic equilibrium.
In terms of Well Flow Performance the PSS flow means:
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q_t(t) = \rm const |
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\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:
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| p_e(t) = p_i - \frac{q_t}{ V_{\phi} \, c_t} \ t |
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varying formation pressure at the external reservoir boundary
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| p_{wf}(t) = p_e(t) - J^{-1} q_t |
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varying bottom-hole pressure
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| J = \frac{q_t}{2 \pi \sigma} \left[ \ln \left ( \frac{r_e}{r_w} \right) +S + 0.75 \right] |
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constant productivity index |
and develops a unit slope on PTA diagnostic plot and Material Balance diagnostic plot:
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / PSS Diagnostics
Steady State (SS) well flow regime
