Representation of a new physics-based non-Darcy equation for low-velocity flow in tight reservoirs

There has been a growing interest in the study of low-velocity non-Darcy flow in tight reservoirs. Most existing non-Darcy equations, however, are flawed in model assumption, equation derivation or field practicability. For another, current non-Darcy simulations are based on the traditional methods, which are incapable of describing the fracture-rich feature of tight reservoir geology. In light of this, this paper introduces a new physics-based non-Darcy equation for low-velocity flow from four aspects: equation derivation, experimental validation, flow simulation and flow analysis. The rigorous derivations of the new equation are presented with a modified Buckingham-Reiner equation in which the boundary layer effect and the Bingham fluid behavior are described. The non-Darcy coefficient controlled by the boundary layer parameter and threshold pressure gradient (TPG) is incorporated into Darcy's equation. The equation validations are performed by liquid column experiment and core flow experiment, which yields good matching relationship between modeled values and observed values. The non-Darcy flow simulation is based on the triangle-gridded discrete fracture-matrix model (DFM) which can model arbitrary fractures. The upstream weighting scheme for computing the non-Darcy transmissibility is proposed to improve numerical robustness. Then the scheme is implemented by constructing non-Darcy operators, consisting of half-face upstream weighting operator and gradient operator, and further setting up the Automatic Differentiation (AD)-style discrete equations. The studies on non-Darcy flow in matrix coupled with Darcy flow in fracture are performed by modeling two types of fractured reservoirs. Detailed analyses are illustrated with pressure transient, long-term well bottom pressure and pressure map.