The fracture flow equation and its perturbation solution

Abstract

[1] 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 zeroorder 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.

Effects of the geometry of two‐dimensional fractures on their hydraulic aperture and on the validity of the local cubic law

“…As the condition R e > 15 was found in all the geometries that we considered (Figures , and ), One can generalize this result for 2D fractures where the aperture varies along the flow direction.Furthermore, our numerical analyses show that the relative error does not change as a function of R e in the case of parallel walls. This is in agreement with the analytical results of Basha and El‐Asmar () and Nizkaya () who found that inertial terms are null in the first and second order terms of the N S expansion solution. On the other hand, this error is maximum when the phase shift between the two walls Δ x is maximum.In all cases, the relative error between the L C L and N S solutions for higher R e increases with the increase of ε .…”

“…H h and H 0 were compared to estimate the conditions under which the L C L , developed with respect to the hydraulic aperture H h , can be replaced by the C L , developed with respect to the mean aperture H 0 . Analytical expressions of H h based on Reynolds equation (or the zeroth order solution of the N S equations) exist for specific geometries, such as fractures having mirror‐symmetric walls (Zimmerman et al ), parallel and shifted walls (Basha & El‐Asmar, ), and non‐identical shifted walls (Nizkaya, ). These expressions show that H h deviates from H 0 and depends on the fracture geometry.…”

Roughness decomposition and nonlinear fluid flow in a single rock fracture

“…However, the assumption of the Cubic Law and its local validity may be acceptable only when the flow is largely laminar, and might not be able to provide adequate support for investigating the impacts of surface roughness on energy and mass transport processes in natural fractures, as reported in literature with both laboratory experiments and numerical simulations (e.g. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). …”

“…The issue of inertial term was investigated by Basha and ElAsmar [11] in which a steady state NSE was solved analytically, using perturbation solutions, for idealized rock fracture geometries formed by regularly shaped sinusoidal wavy 2D smooth profiles with different relative positions or two smooth and nonparallel planar or slightly curved profiles. The derived perturbation solutions consist of a leading-order approximation of Reynolds lubrication approximation and higher-order terms representing effects of amplitude and frequency of the asperities and the inertial term.…”

Experimental study of the effect of roughness and Reynolds number on fluid flow in rough‐walled single fractures: a check of local cubic law

“…Unfortunately, the full NSE is often too difficult to solve, either analytically or numerically (Zimmerman and Yeo, 2000). As a result, only a handful of studies have been reported on the numerical simulation of the NSE in an SF (Brush and Thomson, 2003; Al‐Yaarubi et al , 2005; Cardenas et al , 2007) and even fewer studies on the analytical solutions of the NSE in an SF (Hasegawa and Izuchi, 1983; Basha and El‐Asmar, 2003). Therefore, various approximations are usually made, which reduce the NSE to a simplified equation.…”