Estimation of two-phase relative permeabilities based on treelike fractal model and pressure drop around gas bubbles in porous media

In order to estimate relative permeabilities of brine-gas, a semi-analytical model based on fractal theory is proposed. The model simplifies porous media as treelike nanotubes, the single nanotube's flow has been branched out into whole pore network of core sample through treelike fractal theory. Two-phase flow description in sole nanotube, based on a setting of gas bubbles pushing continuous water phase ahead, solves pressure drop around brine and gas bubbles by Stokes' viscous force, Laplace equation and Fairbrother-Stubbs bubble film thickness formula, Bretherton bubble pressure drop. By substituting the other three classic formulas into Bretherton expression, the iteration equation of pressure drop around a single gas bubble is acquired, the number of gas bubbles is multiplied and the gas phase pressure drop is calculated. Through merging small bubbles into a large bubble, the relative permeabilities of two phases can be solved by introducing Poiseuille's law and Darcy's law. After getting the fundamental data, pore size distribution data and viscosity of two phases, relative permeability curves can be drawn, revealing comprehensive elements including wettability, fluid viscosity, saturation, morphological characteristics. The calculated curves cross each other on the right half of the x-axis, meet the features of real production, flowback difficulties of hydraulic liquid and well production dropping quickly, also show identical trend with experimental data. Through a series of sensitivity analysis, to some extent, the influence on relative permeabilities is compared of different fractal models, different film thickness, bubble length, water sorption layer thickness, elastic deformation of pores.