Lattice Boltzmann simulation of convection-diffusion to model viscous fingering with polymer additives

Viscous fingering occurs when a less viscous fluid is injected into a rock matrix saturated with a more viscous fluid. Our past research using the Rothman-Keller (RK) color gradient Lattice Boltzmann Method (LBM) for immiscible two phase flow has allowed us to study viscous fingering morphology and the complex saturation phase space as a function of the fluid’s properties (wettability of the injected fluid and viscosity ratio). In this past work, we found that the primary factor affecting the saturation at breakthrough – when the injected fluid has passed through the entire model – was the viscosity ratio, and the secondary effect was the wettability. Here, we present an extension of our LBM model to enable convection-diffusion to be simulated, thereby allowing us to vary the viscosity of the injected fluid, and mimicking the practice in Enhanced Oil Recovery (EOR) using polymer additives after breakthrough as a means of increasing the viscosity ratio and thus the eventual oil yield. The basic RK multiphase LBM models two fluid number densities moving and colliding on a discrete lattice, where a second collision term is used to model cohesion within each fluid, and contains an extra “recoloring step” to ensure fluid segregation. Here, we model an additional number density representing the concentration of a polymer additive, which affects the viscosity of the injected fluid. The Peclet number – rate of advection to diffusion of the polymer solution – is used to set the diffusion coefficient of the polymer concentration number density and hence, the relaxation time in the LBM for the polymer diffusion process. We present tests to demonstrate the method in which we increase the polymer concentration of the injected fluid after a given time and study the effect on the viscous fingering morphology and saturation evolution. This work demonstrates that the RK color gradient multiphase LBM can be used to study complex viscous fingering behavior associated with injection of water with polymer additives, which can have major scientific and practical significance.