A quantity (usually denoted as G) representing the minimum pressure gradient required to initiate the reservoir flow:
| (1) | \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 {\bf e}_{\nabla p} = \frac{\nabla p}{|\nabla p|} – unit vector along the pressure gradient.
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:
| (2) | {\bf u}= - \frac{k(|\nabla p|)}{\mu} \nabla p |
where k(|\nabla p|) is defined as:
| (3) | \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*} |