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.

The Lattice Boltzmann Method (LBM) is an elegant method to simulate fluid dynamics based on modelling distributions of particles moving and colliding on a lattice. We present examples of two phase flow using the Rothman and Keller (RK) colour gradient Lattice Boltzmann model to study phenomena associated with two phase immiscible fluid flow relevant to water being injected into an oil saturated sandstone. The model involves streaming and colliding two distribution functions (red and blue) representing the number densities of two fluids, where the collision step involves two terms which represents how particle distributions change in each time step due to collision while encouraging colour segregation. We conducted 2D numerical experiments to study the effect of wetting angle on the morphology of flow of a lower viscosity fluid being injected at the left of a simplified model rock matrix that was filled with a higher viscosity fluid. The cases studied involved the injected fluid being non-wetting (wetting angle = 180 degrees), neutral wetting (wetting angle = 90 degrees) and wetting (wetting angle = 0 degrees). These three cases show viscous fingering behaviour with different morphologies for the different wetting angles. For the case of the non-wetting fluid injection, a series of narrow fingers are observed. For the case of neutral wetting, broader and rounded fingers are observed. And for the case of injecting a wetting fluid, a broad but distorted front is observed approaching stable displacement. The results show the importance of the wetting angle on the morphology of viscous fingering. This study demonstrates that the multiphase Lattice Boltzmann Method can simulate phenomenology relevant to studies of enhanced oil recovery such as water injection, and hence, may lead to improved estimates of oil recovery factors.

Recent advances in modeling Rayleigh-Benard convection have demonstrated the existence of turbulent superstructures, whose life and morphology largely varies with Rayleigh (Ra) and Prandtl (Pr) numbers. These structures appear as a two scale phenomena, where small scale rolls organize in larger convection cells, and can be modelled only in 3D on a simulation box characterized by a very large (>10) width/height (W/L) ratio, and sufficiently refined to resolve the boundary layer up to Ra = 108 (>100 divisions in height) and to Ra = 1010 (>200 divisions). To achieve this goal, we use our own 3D Parallel Python implementation of the Lattice Boltzmann Method, tested to run with linear efficiency on thousands of cores. We show the dependency of horizontal fluctuations of RMS of temperature and vertical velocity in the middle of the box and illustrate how the superstructures emerge for W/L ratios of Terrestrial Planets and Super Earths, and quantify the duration of these superstructures and the likely implications for the evolution of their surface features. The effect of the P/T dependent viscosity and thermal conductivity is finally discussed.