A quantity (usually denoted as
) representing the minimum pressure gradient required to initiate the reservoir flow:| LaTeX Math Block |
|---|
|
\begin{equation*}
\begin{cases}
{\bf u}= - \frac{k}{\mu} ( \nabla p - G \, {\bf e}_{\nabla p} ), & |\nabla p| > G,
\\
{\bf u}= 0, & |\nabla p| \leq G .
\end{cases}
\end{equation*} |
where where
| LaTeX Math Inline |
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| body | {\bf e}_{\nabla p} = \frac{\nabla p}{|\nabla p|} |
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|
– unit vector along the pressure gradient
.and
is threshold pressure gradient (TPG) and represents the minimum pressure gradient required to initiate the reservoir flow At high flow velocities and pressure gradients the model is reducing to Darcy equation.
This model can be reformulated in terms of non-linear permeability model:
...
where
os is defined as:
| LaTeX Math Block |
|---|
|
\begin{equation*}
\begin{cases}
k(|\nabla p|) = k_0 \, ( 1 - \frac{G}{|\nabla p|} ), & |\nabla p| > G,
\\
k(|\nabla p|) = 0, & |\nabla p| \leq G .
\end{cases}
\end{equation*} |
