The Effective Transmissivity of a Plane-Walled Fracture With Circular Cylindrical Obstacles

Stimulated and propped fractures provide the conductive pathways in low matrix permeability rocks. We study the impact of fracture aperture and proppant size and fraction on the effective transmissivity of a stimulated fracture filled with a partial monolayer of proppant grains. The proppant grains are treated as circular cylindrical obstacles, and the fracture walls are planar. The key geometric parameters are the obstacle fraction f and the ratio between the fracture aperture and the obstacle diameter α. We use three‐dimensional Stokes flow numerical simulations to demonstrate that the fracture flow model given by the Reynolds equations may largely overestimate the flow rate. To circumvent the inability of the Reynolds model to fulfill the no‐slip boundary condition at the rims of the obstacles, we use the Brinkman flow model adapted to the case of fracture flow. The relative difference between the effective transmissivities computed with the Stokes and Brinkman models for systems with multiple obstacles is below 10% for fractions up to 0.5. We use the Brinkman model to study the effective transmissivity of a plane‐walled fracture with circular cylindrical obstacles. Systematic numerical simulations show that the normalized effective transmissivity is predominantly dependent on f and α, and the effects of obstacle ordering are minor. The presented numerical results can be used by petroleum engineers for estimating transmissivities of propped fractures under in situ conditions. The model can also be applied to microfluidic systems and for deriving first‐order estimates of the effective transmissivity of rough‐walled, natural fractures with load‐bearing contact areas