Abstract

A lattice-Bollzmann automaton (LBA) is an efficient and accurate alternative to traditional methods in computational fluid dynamics such as finite differences or finite elements. In the lattice---Boltzmann approach, the non-equilibrium Bollzmann- transport equation is solved in terms of the particle distribution al each element of a lallice, and all other parameters (density, velocity, energy, etc) are derived from this spatial distribution as it evolves in time.

A square LEA has been defined for the simulation of two-dimensional waterflooding involving two immiscible, incompressible fluids in an isothermal environment which incorporates pre-defined boundary conditions. combinations of porosity and permeability, and source-sink geometries. The numerical performance of the LBA has been thoroughly tested for several simple waterflooding examples which have analytical solutions. The advantages of the lattice-Boltzmann simulator include its computing efficiency, minimal numerical dispersion, insensitivity to grid orientations and easy implementation of complex reservoir geometries.

Introduction

During the past decade there has been a veritable revolution in our perception of the physical world. Fractals, chaos, cellular automata and neural networks, in association with fast computer graphics, have been the instigators of this revolution. Armed with today's computing power, it is possible to lest hypotheses, perform trial-and-error experiments and display complicated patterns which a decade age; were simply not practical. Some of these new concepts and techniques havebeen applied to solutions of important engineering problems and, as well, have provided new insights into our understanding of the physical processes governing these problems. One example related to petroleum reservoir engineering is multiphasetransport in porous media which is governed by physical processes on different length scales. When applied to reservoir simulations, it is important that the numerical solutions representative of the physical processes being modelled are independent of the underlying grid. With the introduction of computer tomography, one can gain insight into geological structure in the form of porosity and permeability on the laboratory scale. With present computers, it is possible to incorporate such information in numerical computation. In this paper, we develop and implement such a numerical procedure based on the lattice-Boltzmann method.

The LBA is a numerical method for simulating a number of fluid and fluid-like systems that is motiva1ed by kinetic theory1,2. It may be viewed either as a discretization of a simplified Boltzmann equation using a symmetric lattice or as a Lagrangian finite-difference scheme for the Navier-Stokes equation. Is is the kinetic theory point of view that is emphasized here. The lattice--Boltzmann method represents the state of the fluid at a computational node using a set of real numbers which are called velocity populations and are analogous to the microscopic density function of the Boltzmann transport equation, The populations are streamed from one lattice site to another in discrete lime steps and are relaxed towards local equilibrium between every Streaming. The relaxation step or collision operation conserves mass and momentum (and energy for thermal models) just like a particle collision in kinetic theory. As a derivative of lattice-gas cellular automata3 (which are natural parallel processors), lattice-Boltzmann methods are well-suited for simulating flow problems with complex geometries in a parallel computing environment

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