One-dimensional Incompressible Two-phase Flow Problem For A Class Of Numerical Methods

In this article, two numerical methods are given for one dimensional two-phaseincompressible flow in porous media: finite volume element method and characteristicfinite volume element method. On the original grid, piecewise linear polynomial func-tion space is used as the trial function space in the two methods. And on the dualgrid, piecewise constant function space is used as the test function space. Finally errorestimates in the L~2-norm and H~1-norm are proved.The problem of the multi-dimensional incompressible two-phase flow displacementin porous media is governed by a system of two equations. One of elliptic form for thepressure and the other of parabolic form for the concentration of one of the fluids. Theproblem is one of the most important basic fields of oil reservoir numerical simulation.J.Douglas. et al. developed this research field firstly. They have presentedmany famous numerical methods, such as the characteristic finite difference method, thecharacteristic finite element and the characteristic mixed finite element method etc., andhave completed theoretical analysis and numerical experiments which have become thebasic theories and experiments on oil reservoir numerical simulation. In thisarticle we are interested in the study of a cutting-off domainΩin multi-dimensional space.Choose cutting-off planeI = [a,b], then we will consider the problem of one dimensionaltwo-phase incompressible flow in porous media in I, the numerical simulations will bemore accurate.The work consists of three chapters.Chapter One is the introduction, which mainly introduces the mathematic modelof the discussed problem and the trial function space and the test function space.Chapter Two mainly introduces finite volume element method and the analysis oferror estimate.Chapter Three mainly introduces characteristic finite volume element method andthe analysis of error estimate.