LAUSR

Fluid flows through porous media are subject to different regimes, ranging from linear creeping flows to unsteady, chaotic turbulence. These different flow regimes at the pore scale have repercussions at larger scales, with the macroscale drag force experienced by a fluid moving through the medium becoming a nonlinear function of the average velocity beyond the creeping flow regime. Accurate prediction of the transition between different flow regimes is an important challenge with repercussions onto many engineering applications. Here, we are interested in the first deviation from Darcy’s law, when inertia effects become sizeable. Our goal is to define a Reynolds number, $$Re_{\mathrm{C}}$$ReC, so that the inertial deviation occurs when $$Re_{\mathrm{C}}\sim 1$$ReC∼1 for any microstructure. The difficulty in doing so is to reduce the multiple length scales characterizing the geometry of the porous structure to a single length scale, $$\ell $$ℓ. We analyze the problem using the method of volume averaging and identify a length scale in the form $$\ell =C_\lambda \sqrt{\nicefrac {K_\lambda }{\epsilon _\beta }}$$ℓ=CλKλϵβ, with $$C_\lambda $$Cλ a parameter that indicates the sensitivity of the microstructure to inertia. The main advantage of this definition is that an explicit formula for $$C_\lambda $$Cλ is given; $$C_\lambda $$Cλ is computed from a creeping flow simulation in the porous medium; and $$Re_{\mathrm{C}}$$ReC can be used to predict the transition to a non-Darcian regime more accurately than by using Reynolds numbers based on alternative length scales. The theory is validated numerically with data from flow simulations for a variety of microstructures.