This work derives the fracture flow equation from the two-dimensional steady form of the Navier-Stokes equation. Asymptotic solutions are obtained whereby the perturbation parameter is the ratio of the mean width over the length of the fracture segment. The perturbation expansion can handle arbitrary variation of the fracture walls as long as the dominant velocity is in the longitudinal direction. The effect of the matrix-fracture interaction is also taken into account by allowing leakage through the fracture walls. The perturbation solution is used to obtain an estimate of the flow rate and the fracture transmissivity as well as the velocity and the pressure distribution in fractures of various geometries. The analysis covers eight different configurations of fracture geometry including linear and curvilinear variation as well as sinusoidal variation in the top and bottom walls with varying horizontal alignment and roughness wavelengths. The zero-order solution yields the Reynolds lubrication approximation, and the higher-order equations provide a correction term to the flow rate in terms of the roughness frequency and the Reynolds number. For sinusoidal and linear walls, the mathematical analysis shows that the zero-order flow rate could be expressed in terms of the maximum to minimum width ratio. For equal widths at both ends of the fracture, the first-order correction is zero. For sinusoidal fractures, the flow rate decreases with increasing Reynolds number and with increasing roughness amplitude and frequency. The effect of leakage is to create a nonuniform flow distribution in the fracture that deviates significantly from the flow rate estimate for impermeable walls. The derived flow expressions can provide a more reliable tool for flow and transport predictions in fractured domain.