An Analytical Method for the Non-Darcy Flow in a Single Fracture

To investigate the seepage behavior in a single fracture, a nonlinear seepage model based on boundary layer theory was developed. An analytical formula for calculating nonlinear flow in a single fracture was derived, providing a new method for assessing non-Darcy flow in single fractures. The flow in this formula consists of two parts: the viscous layer and the inertial layer. When the undetermined coefficient λ is equal to 0.5, the formula is simplified to the traditional cubic law. Under high hydraulic gradient conditions, the formula can be simplified for practical use. COMSOL software was used to explore the relationships between fracture length, fracture width, hydraulic gradient, and flow rate under high hydraulic gradient conditions. Additionally, a semi-empirical formula for calculating non-Darcy flow in single fracture was derived. The flow rates calculated using the proposed method were compared with those obtained from the Forchheimer equation, the Schrauf model and the cubic law. The results confirmed the accuracy and effectiveness of the proposed analytical formula.