A Nonlinear Version of the Reynolds Equation for Flow in Rock Fractures With Complex Void Geometries

Abstract

This study presents a nonlinear Reynolds equation (NRE) for single‐phase flow through rock fractures. The fracture void geometry is formed by connected wedge‐shaped cells at pore scale, based on the measured aperture field. An approximate analytical solution to two‐dimensional Navier‐Stokes equations is derived using the perturbation method to account for flow nonlinearity for wedge‐shaped geometries. The derived perturbation solution shows that the main contributors to the determination of general flow behaviors in local wedges are the degree of aperture variation relative to mean aperture, the ratio of aperture variation to wedge length, the Reynolds number, and the degree of wedge asymmetry. The transmissivity of the entire fracture is then solved with a field of local cell transmissivity that varies along both longitude and latitude directions on the fracture plane. The performance of the proposed NRE is tested against flow experiments and flow simulations by solving numerically the three‐dimensional Navier‐Stokes equations for three cases of rock fractures with different void geometries. Results from the proposed model are in close agreement with those obtained from simulations and experiments. As the Reynolds number increases, the pressure difference obtained from the NRE demonstrates the same nonlinear behavior as that obtained from the simulations. Overall, the mean discrepancy between the proposed model and flow simulations is 5.7% for Reynolds number ranging from 0.1 to 20, indicating that the proposed NRE can capture the flow nonlinearity in rock fractures.

A Novel Procedure for Coupled Simulation of Thermal and Fluid Flow Models for Rough-Walled Rock Fractures

“…Among these approximates, fractures are found to be well described by a series of connected wedges. Wang et al [37,38] used a wedge-shaped fracture to obtain the perturbation solution of pressure ∆p and flowrate Q under the pressure boundary condition. The perturbation parameter ε is selected as the relative aperture variation along the wedge length l in his study.…”

“…This section provides a brief description of the approach rather than a detailed derivation. A more detailed perturbation derivation can be viewed in Wang et al [37,38]. By perturbation analysis, the pressure difference can be made dimensionless and expressed as expanded series with a small parameter ε:…”

Modelling of Coupled Hydro-Thermo-Chemical Fluid Flow through Rock Fracture Networks and Its Applications

“…Many studies have attempted to include pore-scale modifications to improve the standard RE by considering mid-surface variation and applying various averaging schemes for local transmissivity. In a departure from previous approaches, we proposed a non-linear version of the Reynolds equation (NRE) based on a two-dimensional (2-D) perturbation solution (PS) that can accurately estimate the transmissivity at local wedge-shaped cells [7]. The derived PS is an approximate analytical solution to the 2-D NSE, and can be seen as a further extension to the widely adopted CL assumption used locally in the RE [2].…”

“…The final PS can be found by inserting the derived stream function into the auxiliary condition. More details can be found in our previous work [7]. The equivalent hydraulic aperture b T of the wedge can be defined by multiplying the mean aperture b m by an extra modification factor:…”

“…Different fracture intersections and their influences on fluid flow further complicate the issues [5]. There are numerous published works in which different correction factors were used for the roughness of fracture walls and for non-linear flow behaviours at high Reynolds numbers [1,[6][7][8]. Even with this two-dimensional simplification, the solution can still become intractable if there is a significant number of fractures in the system, which is often the case in large-scale engineering applications such as groundwater flow modelling, enhanced geothermal systems, or in-situ recovery of minerals.…”