An Analytical Formulation for Correcting the Relative Permeability of Gas-Water Flow in Propped Fractures Considering the Effect of Brinkman Flow

Gas-water flow in propped fractures can be commonly observed in various practical applications, including hydrocarbon development, geothermal exploitation, contaminant transport, and geological carbon storage. The fluid flow in a propped fracture can be regarded as Darcy type if only the resistance from the propping materials (e.g., cement in natural fractures and proppant-pack in hydraulic fractures) accounts. However, if the fracture width is sufficiently small, the extra resistance from the viscous shear of fracture walls cannot be neglected, resulting in the appearance of Brinkman flow. In practice, during the development of a naturally fractured aquifer or a hydraulicly fractured hydrocarbon reservoir, the fracture width will be significantly reduced as the production proceeds. Therefore, the Brinkman flow can impose a strong effect on the fluid transportation within the fractures. However, the existing study about the two-phase Brinkman flow in propped fractures is still far from adequate. In this work, on the basis of a modified two-phase Brinkman equation, the authors derive a novel analytical formulation to correct the relative permeability of gas-water two-phase flow in propped fractures to account for the Brinkman effect. With the aid of the proposed formulation, the authors carry out a comprehensive investigation of the influence of Brinkman flow on the effective gas-water relative permeability and well performance. The calculated results show that the effect of Brinkman flow on the water phase (or gas phase) transportation is more significant with a larger water (or gas) saturation. A narrower propped fracture is more likely to induce Brinkman flow, thus leading to a lower relative permeability for both water and gas phases. As the propping-material permeability is increased, the fluid transportation bears more severe viscous drag from fracture walls, and the relative permeability will be consequently reduced. Only if the fracture width is significantly reduced during the production, the Brinkman flow demonstrates its influence on the well performance. Otherwise, Darcy's law can provide sufficiently accurate results in characterizing the two-phase flow in propped fractures.