A global inertial permeability for fluid flow in rock fractures: Criterion and significance

Abstract

Fluid flow in rock fractures is described by Darcy's law and its nonlinear extension, the Forchheimer equation in form of a quadratic polynomial. Since the nonlinearity of the Forchheimer equation, determining its key coefficient of inertial permeability (i.e., the quadratic coefficient representing inertial losses) requires a series of flow tests under different hydraulic gradients. However, how to choose the volumetric flux range for deriving inertial permeability is never scrutinized. Here we report a seemly irrational dependency of inertial permeability on volumetric flux range by massive direct numerical simulations of fluid flow in rock fractures. This variation presents traceability and is closely related to the evolution of microscale eddies and macroscale flow regimes. A concept of global inertial permeability is thus proposed from this traceable variation, which can reproduce the whole flow regime in rock fractures. Two parametrized criterion models are subsequently established for calculating global inertial permeability using the lowest cost and ensuring its accuracy: one in terms of minimum Reynolds number for practical purposes, and the other in terms of minimum eddy volume ratio for mechanistic interpretation. Further discussions indicate that the proposed global inertial permeability is of great importance to non-Darcian flow prediction and flow regime assessment. Moreover, the parameterized criterion models in the form of power-law function for determining global inertial permeability, drawn from 2D fracture simulations, can apply to 3D fractures and large-scale fractured rock aquifers. The findings and results in this study are of great significance for accurately evaluating the hydraulic conductivity of rock fractures, especially at relatively high flow rates, and are useful for many geophysical and industrial applications.