Fluid flow with fluid pressure
linearly changing in time:| LaTeX Math Block |
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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:
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{\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
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| body | {\bf u} = {\bf u}({\bf r}, \nabla p) |
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) the PSS flow velocity will be stationary as the right side of | LaTeX Math Block Reference |
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is not dependant on time.
In terms of Well Flow Performance the PSS flow meansWell flow regime with constant rate and constant delta pressure between wellbore and formation does not change in time:
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q_t(t) = \rm const |
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During the PSS regime the formation pressure also declines linearly with time:
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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:
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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
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Also
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Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / PSS Diagnostics
[ Steady State (SS) well fluid flow regime ]
