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Fluid flow with fluid pressure gradient
is linearly changing in time:| LaTeX Math Block |
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p(t, {\bf r}) |
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= \psi({\bf r}) + A \cdot t |
The fluid velocity
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 are not changing in time:
| LaTeX Math Block |
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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 correlation:
| LaTeX Math Inline |
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| body | M({\bf r}, p, \nabla p) = M({\bf r}) |
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the
| LaTeX Math Block |
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{\bf u}(t, {\bf r}) = {\bf u}({\bf r}) |
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| borderColor | wheat |
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| borderWidth | 10 |
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| In the most general case (both reservoir and pipelines) the fluid velocity is proportional to pressure gradient and can be written as: | LaTeX Math Block |
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| {\bf u}(t, {\bf r})= - M({\bf r}, p, \nabla p) \nabla p |
with right side not dependent on time in stationary flow: | LaTeX Math Block |
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| \frac{\partial {\bf u}(t, {\bf r})}{\partial t}= 0 |
which leads to | LaTeX Math Block Reference |
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. |
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and velocity
are not changing in time:| LaTeX Math Block |
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{\bf u}(t, {\bf r}) = {\bf u}({\bf r}) |
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