The average fluid velocity is deduced from NSE (Navier-Stokes-Equation). Now, an analogy is made with the Darcy's law and eventually, a relation for an intrinsic permeability is deduced as a function of square of fracture aperture size (similar to the concept of square of mean grain size used in a porous medium).
It is known that the fracture transmissivity is extremely sensitive to the fracture aperture thickness as dictated by the cubic law.
Now, it can be noted that -
NSE describes flow through pipes, where the porosity is 100%.
Darcy's approach describes flow through a porous medium - with a porosity much much less than 100% - and that too with the condition that the volumetric flow rate should linearly vary with the pressure gradient.
NSE looks the problem of interest at the microscopic-scale, while Darcy looks the problem of interest at the macroscopic-scale.
Given this background, the analogy between NSE and Darcy stands justified?
Be careful to consider the NSE at a microscopic level. You can approach the continuum limit where the PDE cannot be valid
I consider the best analogy of the Navier-Stokes equation and the Darcy equation is provided by the Brinkman equation, which describes the transition between the two. In particular, here you can (on a formal level) consider limits of the permeability and recover Navier-Stokes and Darcy. But your analogy is also true, that you can recover Darcy (macroscopic) from homogenization of the Navier-Stokes equation (microscopic) as long as we talk about continuum theories.
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