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Lattice-Boltzmann Fluid Dynamics a versatile tool for multiphase and other complicated flows Shiyi Chen, Gary D. Doolen, and Kenneth G. Eggert 98 Los Alamos Science Number 22 1994 Number 22 1994 Los Alamos Science 99 Lattice-Boltzmann Fluid Dynamics ong-term research efforts at the method yields a good approximation to projects of this type helped motivate Laboratory often produce results the standard equations of fluid flow, the the oil companies to ask that Congress Lthat prove highly valuable to Navier-Stokes equations, in the limit of substantially increase DOE support for U.S. industry. The example described long wavelengths and low frequencies. oil and gas research and make those here is a new approach to modeling the Also, recent generalizations of the funds available for significant new col- flow of multiphase fluid mixtures, such method have extended its applicability laborations between the oil and gas in- as oil and water, through very compli- to multiphase flows, chemically react- dustry and the National Laboratories. cated geometries. This new modeling ing flows, diffusion and thermohydro- The result is ACTI—the Advanced technique, called the lattice-Boltzmann dynamics. Computational Technology Initiative— method, evolved out of ideas that have United States oil companies have an initiative that will receive $30–50 been intensely investigated since 1985, expressed considerable interest in the million in funding in 1995. when Laboratory scientists and others lattice-Boltzmann method for address- In this article we will first sketch the discovered that very simple models of ing problems in oil recovery. For ex- basic ideas of the lattice-Boltzmann discrete particles confined to a lattice ample, we are collaborating with the method and its general applicability. can be used to solve very complicated Mobil Exploration and Producing Tech- We will then explain how the lattice- flow problems. nical Center on using the lattice-Boltz- gas and lattice-Boltzmann methods are This lattice method can be regarded mann method to simulate the flow of adapted to describe the dynamics at the as one of the simplest microscopic, or oil and water through oil-bearing sand- interfaces between two immiscible flu- particle, approaches to modeling macro- stone at a scale and an accuracy never ids such as oil and water and present scopic dynamics. It is based on the before possible. Mobil provided Los some simulations of phase separation. Boltzmann transport equation for the Alamos with 5-micron-resolution sand- Finally, we will mention a few of the time rate of change of the particle dis- stone geometries for use in the simula- new directions into which lattice-Boltz- tribution function in a particular state. tions. (Typical pore diameters are tens mann research is moving. Specific ap- The Boltzmann equation simply says of microns.) The goal of the work is to plications to the flows in oil reservoirs that the rate of change is the number of simulate, at the scale of individual are discussed in the companion article. particles scattered into that state minus pores in the rock, what happens when the number scattered out of that state. water is pumped though the rock to The method is fully parallel (the force out oil. Viscosities, surface ten- Lattice Methods for Modeling same calculations are performed at sions, contact angles, and surfactant ef- Continuum Dynamics every lattice site) and local (only near- fects can be varied to determine how by particles interact with each other). well this method of oil recovery can be Lattice methods, including the lat- It is therefore easily programmed and made to work in specific circumstances. tice-gas method and its derivative, the runs efficiently on parallel machines. The collaboration is described by Mobil lattice-Boltzmann method, present pow- Complex boundary conditions are in- and Los Alamos scientists in the com- erful alternatives to the standard “top- corporated in a straightforward way and panion article, “Toward Improved Pre- down” and “bottom-up” approaches to cause the calculational speed to de- diction of Reservoir Flow Perfor- modeling the behavior of physical sys- crease by only a few percent. The mance—Simulating Oil and Water tems. The “top-down” approach begins Flows at the Pore Scale.” The work with a continuum description of macro- Illustration on previous spread: A lattice-Boltz- has included the development of soft- scopic phenomena provided by partial mann simulation of flow past a slab shows the ware to calculate relative permeabilities differential equations. The Navier- complicated flow patterns that can be modeled of oil and water in complex geometries. Stokes equations for incompressible with this technique. The lattice is 1024 3 256 The software received an R&D-100 fluid flows are an example. Numerical sites; the thickness of the slab is 32 sites, which gives the flow a Reynolds number of 960. Wind- award for 1993. techniques, such as finite-difference and tunnel boundary conditions are used—the flow The lattice-Boltzmann work is a su- finite-element methods, are then used to velocity at the entrance (left), the top, and the perb example of applying the extraordi- transform the continuum description bottom boundaries is fixed. The top image shows nary computer power and expertise in into a discrete one in order to solve the contours of equal vorticity; the bottom image shows the pressure distribution of the flow. The computational physics available at Los equations numerically on a computer. simulation was done using the Connection Ma- Alamos to problems in petroleum pro- The “bottom-up” approach is based chine 2 at the Advanced Computing Laboratory. duction. The Laboratory’s success at on the microscopic, particle description 100 Los Alamos Science Number 22 1994 Lattice-Boltzmann Fluid Dynamics lattice1.adb 7/26/94 provided by the equations of molecular dynamics; here the position and veloci- ty of each atom or molecule in the sys- tem are closely followed by solving Newton’s equations of motion. This microscopic description is straightfor- ward to program on a computer but simulations using the largest computers presently available are limited to very small systems (ten million particles) and very short times (a few picosec- Figure 1. Allowed Velocities at a Lattice Site onds). Therefore molecular dynamics The arrows indicate the magnitudes and directions of the allowed velocities ei at a lat- simulations are more suitable for under- tice site in a three-dimensional lattice-Boltzmann simulation. The lattice has a cubic standing the fundamental interactions structure. Six arrows point to nearest-neighbor sites. Eight arrows point along body that underlie macroscopic material diagonals. The sphere at the lattice site represents zero velocity, e0 5 0. Lattice-Boltz- properties than for modeling macro- mann simulations based on a proper equilibrium particle distribution and this minimum scopic dynamics. set of velocities preserve the desired isotropy of fluid properties. Intermediate between the two schemes are the lattice-Boltzmann and attain a speed of 20 gigaflops on a 512- mohydrodynamics, magnetohydrody- lattice-gas methods, which might be processor CM-5 Connection Machine. namics, the dynamics of liquid crystals considered mesoscopic approaches. Typical simulations of flow through and the design of semiconductors. The lattice methods begin from a parti- porous media include 100 million lat- These models are validated by precise cle description of matter: A gas of par- tice sites and run for 5000 timesteps comparisons with other numerical algo- ticles exists on a set of discrete points and therefore require only a few hours rithms, analytic results, and experiments. that are spaced at regular intervals to on that machine. When applied to peri- form a lattice. Time is also divided odic geometries, the three-dimensional into discrete timesteps, and during each lattice-Boltzmann model is able to The Lattice-Boltzmann Method timestep particles jump to the next lat- achieve the same spatial resolution as tice site and then scatter according to conventional methods in half the time. In the lattice-Boltzmann method, simple kinetic rules that conserve mass, Also, because boundary conditions— space is divided into a regular lattice momentum, and energy. This simpli- even complex ones—are imposed local- (for example, a simple cubic lattice) fied molecular dynamics is very care- ly, lattice methods simulate flows in and real numbers at each lattice site fully crafted to include the essentials of both simple and complex geometries represent the single-particle distribution the real microscopic processes. Conse- with almost the same speed and effi- function at that site, which is equal to quently the macroscopic, or averaged, ciency. They are therefore suitable for the expected number of identical parti- properties of lattice simulations obey, simulating flows in the extremely com- cles in each of the available particle to a good approximation, the desired plex geometries of porous media states i. In the simplest model, each continuum equations. Despite their ori- whereas conventional methods are not. particle state i is defined by a particle gin in a particle description, lattice Finally, developing code for lattice velocity, which is limited to a discrete methods are essentially numerical methods is considerably faster and easi- set of allowed velocities. During each schemes for studying averaged macro- er than for traditional schemes. discrete timestep of the simulation, par- scopic behavior. They nevertheless re- General lattice models already exist ticles move, or hop, to the nearest lat- tain the advantages of a particle descrip- for solving the equations of fluid flow, tice site along their direction of motion, tion, including clear physical insight, the wave equation, and the diffusion where they “collide” with other parti- easy implementation of boundary condi- equation. Special versions of these cles that arrive at the same site. The tions, and fully parallel algorithms. models have been developed for simulat- outcome of the collision is determined Because lattice methods are entirely ing flow through porous media, turbulent by solving the kinetic (Boltzmann) local, they are extremely fast.

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