【作者】 柴振华；

【导师】 施保昌； 郭照立；

【作者基本信息】 华中科技大学 ， 热能工程， 2009， 博士

【摘要】 非线性渗流或非达西渗流作为渗流力学中的一个重要分支，由于其在农业、能源、冶金、化工、材料、环境、生命科学以及医学等众多领域的广泛应用而引起国内外学者的普遍关注，并发展成为国际上研究的一个热点问题。通过对大量相关文献的分析可以看出，非线性渗流问题自身的复杂性大大限制了人们对其背后物理机理的认识，以至于对于某些问题至今也未能给出较为合理准确的数学模型或物理公式。目前对非线性渗流的研究主要基于三类方法：一是实验方法，作为一种传统的研究方法，其在渗流力学中仍然发挥着重要的作用，但是，在实际中该方法又受到很多条件的限制，比如实验周期长、花费大、受实验环境的影响等。二是解析或近似方法，这类方法通常会对问题先进行一定的简化或给出一些合理的假设，在此基础上，进一步通过数学的手段来求得非线性渗流的解析解或近似解。目前，这类方法中最具代表的就是近年来发展起来的基于分形理论的解析方法。另外一类就是数值方法，随着计算机技术的飞速发展，该方法越来越受到人们的广泛关注，并已成为解决实际的问题的一种重要手段。与实验方法相比，数值方法依赖于前期建立的数学模型，但它具有周期短、耗费小、不易受环境影响等特点；同时，该方法也可以弥补解析或近似方法所存在的不足。目前对非线性渗流的数值研究主要是借助于传统的数值方法(如有限差分法，有限体积法，有限元方法等)来数值求解孔隙尺度上的Navier-Stokes方程或表征体元(REV)尺度体积平均的Navier-Stokes方程。作为一种研究非线性渗流问题的有效手段，数值研究近年来备受关注，其应用范围也越来越广泛，但多孔介质的复杂结构使得这些方法在研究渗流问题时会遇到如下困难：(1)边界处理复杂，破坏数值方法自身的稳定性；(2)多孔介质中的微尺度效应难以体现；(3)计算量较大，并行效率低。鉴于此，发展高效的数值方法，并利用其研究非线性渗流规律，建立完善的数学描述是揭示非线性物理机理的理论基础。格子Boltzmann方法(Lattice Boltzmann Method，LBM)作为一种新兴的微观方法，它最初起源于格子气自动机(Lattice Gas Automata，LGA)，但后来也被证实可以从连续的Boltzmann方程经过差分离散得到。格子Boltzmann方法与以连续微分方程为基础的宏观计算流体力学方法有着本质的不同，它是基于流体微观模型和细观动理论方程的方法，因此不受连续假设的限制，这为其能够模拟多孔介质中微细管道中的流动、刻画滑脱效应(Klinkenberg效应)提供了坚实的理论基础。此外，与传统的计算流体力学方法(基于Navier-Stokes方程的数值方法，比如，有限差分法、有限体积法、有限元方法等)相比，LBM还具有许多独特的优势，如计算效率高、边界条件容易实现、具有天然并行性等，更为重要的是其自身的微观特性使得它成为研究多孔介质中非连续流动提供了一种新的有效手段。本文首先发展了LBM的相关理论，在此基础上，利用LBM对单相非线性渗流力学中的一些前沿问题进行了探讨。论文的主要工作包括以下两个方面：(1)在格子Boltzmann方法的相关理论方面，论文首先提出了一种处理微尺度流动的反弹与Maxwell漫反射组合边界条件，并详细分析了该边界条件在单松弛模型与多松弛模型中的离散效应；其次，论文还给出了一种求解Poisson方程的新的格子Boltzmann模型，该模型可以有效克服已有模型的不足。这些理论的发展将为推动LBM在渗流领域的应用奠定必要的基础。(2)在非线性渗流的数值研究方面，论文首先系统研究了高速非线性渗流，详细分析了低雷诺数下产生非线性现象的物理机理，并进一步并给出了改进的数学模型；其次，基于(1)中的理论结果，论文详细研究了气体通过多孔介质的滑脱效应，利用文中发展的一些理论结果揭示了滑脱效应产生的物理根源；最后，论文还进一步考察了微尺度多孔材料中的电渗流，系统分析了各种因素对电渗流速度分布的影响，这些结果对实际中的工程问题具有重要的指导意义。总之，本文利用格子Boltzmann方法研究了孔隙尺度和REV尺度的非线性渗流，为推动该方法在渗流领域中的应用作出了许多有益的尝试。同时，针对针对高度非线性的Poisson-Boltzmann方程和复杂的多孔介质，我们分别构造了新的模型和边界条件；此外，我们还对格子Boltzmann方法进行了大量的数值实验，验证了该方法用于模拟渗流问题的可靠性。这些工作为后续研究的开展奠定了必要的基础。更多还原

【Abstract】 Nonlinear or non-Darcy flow is an important part of flow in porous media,it attainsan increasing attention and becomes a hot topic due to its wide applications in agriculture,energy,metallurgy,chemical engineering,material science,environment engineering,lifesciences,medicine etc.According to a large number literature published previously,it isfound that,owing to the complexity inherent in nonlinear flow in porous media,the physicsmechanics behind the nonlinear phenomena is not understood clearly,and there is no rea-sonable or accurate formula that can be used to describe this nonlinear effect.Up to now,there may be three methods that can be used to study nonlinear flow in porous media,one isexperimental method,which,as a traditional method,still plays an important role in inves-tigating flow in porous media,but there are also some constrains that limit its applicationsin practice,for example,the time and money spent on experiments are usually very large,and what is more,the practical experiments are usually affected by surrounding conditions.The second is analytical or approximate method,coupling with some assumptions or sim-plicities,it can be used to derive some analytical or approximate solution of nonlinear flowin porous media.The last one is numerical method,which becomes an important approachand attains increasing attention in solving many practical problems with the developmentof computer,compared with experimental method,it depends on mathematical model ofproblems,but it also has many merits,for instance,the expense of numerical method is less,and the surrounding effects can be reduced significantly,meanwhile,it also can be used tocircumvent some shortcomings inherent in analytical or approximate methods.Now,the numerical researches on nonlinear flow in porous media are to solve theNavier-Stokes equations at pore scale or REV scale with the traditional numerical methods(for example,finite difference method,finite volume method,finite element method andso on.).In the past several years,the numerical research,as an effective method to studynonlinear flow,attained great attention,and was applied in more and more fields.However,these numerical methods will face some problems when they are used to simulate flow inporous media:(1) The treatment on boundary condition is very complex,which leads to the fact thatthe stability of numerical method becomes much worse;(2) It is a hard work to discover the micro-scale effect for gas flow in porous media;(3) The computational expense is very large and the parallel efficiency is very low.Therefore,it is desirable to develop some advanced numerical methods,which can be used to investigate nonlinear flow in porous media,and further to built mathematic formula,this is also the theoretical premise to explain nonlinear physical mechanics.As a new mesoscopic method,Lattice Boltzmann method (LBM) originates from Lat-tice Gas Automata (LGA),but later it is shown that this method also can be derived fromcontinuum Boltzmann equation with proper discrete scheme.Unlike the traditional compu-tational fluid dynamics methods,LBM is derived from kinetic theory,which leads to LBMis not constrained by continuum hypothesis,it is just this reason that LBM can be used tosimulate pore scale flow with slippage effect in porous media,and successfully display slip-page effect.In addition,compared to traditional numerical methods (the methods to solveNavier-Stokes equations,for example,finite difference method,finite volume method,finiteelement method etc.),LBM has some distinguished characteristics,such as high efficiencyof computation,easy implementation for complex boundary conditions,fully parallelismand so on,what is more,the microscopic characteristic makes it be an effective approach tostudy flow with slippage effect in porous media.In this thesis,the theory of LBM is firstdeveloped,and then,some new and open problems of single phase nonlinear flow in porousmedia are studied with LBM.This thesis is composed of two parts:(1) In term of the theory related to LBM,a new boundary condition,i.e.,combinedboundary condition of bounce back and Maxwell diffusion,is first proposed,simultaneously,a detailed analysis on discrete effects of this boundary condition in single relaxation modeland multi-relaxation model is further presented,then,a novel model,which can be used tocircumvent shortcomings in previous models,is provided to solve the Poisson equation.Wewould like to point out that the development to LBM will provide a necessary premise forfollowing research on flow in porous media.(2) In term of the numerical research on nonlinear flow in porous media,the high-velocity nonlinear flow in porous media is first studied,in this subsection,we not onlypresent a more precise formula which can be used to describe high-velocity nonlinear flowin porous media,but also give a detailed analysis on physical mechanism for nonlinear phe-nomena which are usually observed at low Reynolds number,then,the slippage effect ofgas through porous media is also investigated,the physical mechanism of slippage effect isrevealed with the aid of the new theory developed in this thesis,finally,the electro-osmoticflow in porous media is also studied,a detailed analysis on effects of many factors on distri-bution of velocity is presented in this part,furthermore,the results derived in this part playan important role in solving problems in practice.In conclusion,we used LBM to investigate nonlinear flow in porous media at REVand pore scale,and made many valuable efforts to accelerate the applications of LBM inflow through porous media.Simultaneously,we propose a new lattice Boltzmann model to solve nonlinear Poisson-Boltzmann and a new combination boundary condition to complexporous media.In addition,a large number of numerical tests are conducted to verify thismethod in studying flow in porous media.This work can be viewed as a necessary basis forfuture studies.更多还原