Microscopic and macroscopic cellular-automaton simulations of fluid flow and wave propagation in rocks

Abstract

Lattice-Gas-Automata (LGA) methods and their extension, Lattice-Boltzmann (LB) methods, have emerged in recent years as alternatives for modeling fluid dynamics and other systems described by partial differential equations. These schemes attempt to create a dynamics of fictitious particles in a lattice, whose macroscopic behavior corresponds to the equations of interest. This dissertation is intended to demonstrate applications of this relatively new approach to simulations of fluid flow and wave propagation in porous media. The basis for several efficient and accurate naturally-parallel computational schemes are introduced for reservoir simulation and seismic modelling. Reservoir simulation algorithms are derived, based on the LB version of the Bhatnagar, Gross, and Krook (BGK) collision model, for single-phase Darcy flow, miscible displacement processes, and immiscible displacement processes, respectively. The simulation results for several, one-dimensional (1D) or two-dimensional (2D) steady or unsteady state flows are shown to be in good agreement with their corresponding analytical solutions. The miscible model is capable of resolving sharp displacement fronts in convection-dominated flows. The miscible model is also shown to be convergent and insensitive to grid orientations and to be capable of handling instabilities due to perturbation in 2-D displacements with unfavorable mobility ratios. Computations of unstable displacements illustrate fingering evolutions very similar to those determined experimentally. The immiscible model is verified with the Buckley-Leverett problem and repeated five-spot pattern by analytical and finite difference methods. Significant reductions in the diffusion of sharp fronts and insensitivity to grid orientation result. The LB approach is developed for simulations of seismic wave propagation in inhomogeneous media. It is demonstrated theoretically and numerically that the macroscopic behavior of the LB acoustic model corresponds to that of a linear acoustic equation. Experiments performed on serial computers indicated that the LB scheme has a better relative performance than the second-order central-differencing scheme. The LB model for elastic waves is motivated by the similarity between the Navier-Stokes equation and the elastic wave equation. The resulting algorithm permits an accurate and robust implementation that involves both free surface and absorbing boundary conditions. Comparisons between numerical results obtained from the LB model and an analytical approach based on the Cagniard technique for Lamb's problem verify the approach. The capabilities of lattice methods for the microscopic simulation and evaluation of various rock parameters, particularly, LGA calculations of permeability and LB determination of seismic wave attenuation, are demonstrated. Models confirm Darcy's law for Poiseuille flow, a given complicated boundary flow and fractal geometry flow. The LGA hydrodynamic model is also applied to the problem of simulating binary fluids and demonstrating the effects of surface tension. Finally, the LB acoustic model is used to investigate seismic wave propagation in media filled with fractal cracks or inclusions. (Abstract shortened by UMI.)