A governing equation for fluid flow in rough fractures

Fluid flow in a rock fracture bounded by two irregular surfaces is complex even under a laminar how regime. The major factor causing deviation of predicted fracture flow behavior from the ideal parallel plate theory is the nature of nonparallel and nonsmooth geometry of fracture surfaces. Important questions on the validity of the cubic law and the Reynolds equation for complicated fracture geometries have been studied by many researchers. The general conclusion from these efforts is that the cubic law is valid provided that an appropriate average aperture can be defined. Many average apertures have been proposed, and for some cases, some work better than others. Nonetheless, to date, these efforts have not converged to form a unified definition on the fracture aperture needed in the cubic law, which stimulates the current effort to develop a general governing equation for fracture flow from a fundamental consideration. In this study, a governing equation stemming from the principle of mass conservation and the assumption that the cubic law holds locally is derived for incompressible laminar fluid flow in irregular fractures under steady state conditions. The equation is formulated in both local and global coordinates and explicitly incorporates two vectorial variables of fracture geometry: true aperture and tortuosity. Under the assumption of small variations in both tortuosity and aperture, the governing equation can be reduced to the Reynolds equation. Two examples are provided to show the importance and generality of the new governing equation in both local and global coordinate systems. In a simple fracture with two nonsmooth and nonparallel surfaces, the error in permeability estimation can be induced using the Reynolds equation with the apparent aperture and can reach 10% for a 25 degrees inclination between the fracture surfaces. In a fracture with sinusoidal surfaces, the traditional method can cause significant errors in both permeability and pressure calculation.