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