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k = 1014.24 \cdot {\rm FZI}^2 \cdot \frac{(\phi_f -\phi_{f0})^3}{( 1 - \phi_f+\phi_{f0})^2}

{\rm FZI} = \frac{1}{\sqrt{F_S} \, S_{gV} \, \tau }

where

In case of proppant-filled fracture the Flow Zone Indicator can be approximated as:

...

For the fluid-filled fracture (

k_f =  \frac{w_f^2}{12}

and in pressure transient analysis behaves as the infinite-value permeability due to a high contrast with reservoir permeability. 

It can be formally interpreted as the extreme case of finite-conductivity fracture with the following trends the Flow Zone Indicator can be approximated as:

{\rm FZI} =\rightarrow \frac{w_f}{2 \, \sqrt{F_S}}

F_S \approxrightarrow 253.56 \, \frac{\phi_{f0}^2}{(1-\phi_{f0})^3}

which leads to simple correlation for fracture permeability:

...

...

}

See Also

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Petroleum Industry / Upstream / Well / Well-Reservoir Contact (WRC)  / Hydraulic Fracture