A Nonlinear Version of the Reynolds Equation for Flow in Rock Fractures With Complex Void Geometries. Issue 2 (5th February 2020) - Wang, Zhihe; Xu, Chaoshui; Dowd, Peter; Xiong, Feng; Wang, Hang
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. Plain Language Summary: The flow of fluid through fractured geological media is commonly encountered in many environmental, geotechnical, and resource extraction applications such as underground water systems, petroleum and geothermal energy extraction, and in situ mineral recovery. Most of the existing conceptual models for single‐phase flow in rough rock fractures are restricted to the simple linear flow condition. In this study, a nonlinear flow model is proposed based on a perturbation solution that accounts for flow nonlinearity at pore scale. The overall transmissivity of the fracture is then solved with the obtained pore‐scale transmissivity field. The nonlinearity feature of flow due to surface roughness and inertial effects can be well captured by the proposed flow model. Key Points: A nonlinear version of the Reynolds equation is proposed to account for complex fracture void geometries and flow inertial effects The proposed flow model is based on the perturbation solution for flow in wedge‐shaped cells at pore scale of an entire rock fracture The nonlinear Reynolds equation agrees well with both experiment and simulation results