{"title":"The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow","authors":"Sepehr Arbabi, Muhammad Sahimi","doi":"10.1007/s11242-024-02070-3","DOIUrl":null,"url":null,"abstract":"<div><p>Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient <span>\\({\\varvec{\\nabla }}P\\)</span> and the fluid velocity <b>v</b>. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude <span>\\(\\omega _z\\)</span> of the vorticity is nearly zero. As Re increases, however, so also does <span>\\(\\omega _z\\)</span>, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocity <b>v</b>, given by, <span>\\(-{\\varvec{\\nabla }}P=(\\mu /K_e)\\textbf{v}+\\beta _n\\rho |\\textbf{v}|^2\\textbf{v}\\)</span>, provides accurate representation of the numerical data, where <span>\\(\\mu\\)</span> and <span>\\(\\rho\\)</span> are the fluid’s viscosity and density, <span>\\(K_e\\)</span> is the effective Darcy permeability in the linear regime, and <span>\\(\\beta _n\\)</span> is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.</p></div>","PeriodicalId":804,"journal":{"name":"Transport in Porous Media","volume":"151 4","pages":"795 - 812"},"PeriodicalIF":2.7000,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11242-024-02070-3.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transport in Porous Media","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s11242-024-02070-3","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"ENGINEERING, CHEMICAL","Score":null,"Total":0}
引用次数: 0
Abstract
Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient \({\varvec{\nabla }}P\) and the fluid velocity v. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude \(\omega _z\) of the vorticity is nearly zero. As Re increases, however, so also does \(\omega _z\), and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocity v, given by, \(-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}\), provides accurate representation of the numerical data, where \(\mu\) and \(\rho\) are the fluid’s viscosity and density, \(K_e\) is the effective Darcy permeability in the linear regime, and \(\beta _n\) is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.
期刊介绍:
-Publishes original research on physical, chemical, and biological aspects of transport in porous media-
Papers on porous media research may originate in various areas of physics, chemistry, biology, natural or materials science, and engineering (chemical, civil, agricultural, petroleum, environmental, electrical, and mechanical engineering)-
Emphasizes theory, (numerical) modelling, laboratory work, and non-routine applications-
Publishes work of a fundamental nature, of interest to a wide readership, that provides novel insight into porous media processes-
Expanded in 2007 from 12 to 15 issues per year.
Transport in Porous Media publishes original research on physical and chemical aspects of transport phenomena in rigid and deformable porous media. These phenomena, occurring in single and multiphase flow in porous domains, can be governed by extensive quantities such as mass of a fluid phase, mass of component of a phase, momentum, or energy. Moreover, porous medium deformations can be induced by the transport phenomena, by chemical and electro-chemical activities such as swelling, or by external loading through forces and displacements. These porous media phenomena may be studied by researchers from various areas of physics, chemistry, biology, natural or materials science, and engineering (chemical, civil, agricultural, petroleum, environmental, electrical, and mechanical engineering).