The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

IF 2.7 3区 工程技术 Q3 ENGINEERING, CHEMICAL Transport in Porous Media Pub Date : 2024-03-14 DOI:10.1007/s11242-024-02070-3
Sepehr Arbabi, Muhammad Sahimi
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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.

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异质多孔介质中从达西流到非线性流的过渡:I - 单相流
通过对纳维-斯托克斯方程进行大量数值模拟,我们研究了流体在无序多孔介质中缓慢流动时从达西定律向非线性流动机制的过渡,在非线性流动机制中,惯性的影响是不可忽略的。多孔介质由砂岩三维图像的二维切片表示。我们研究了孔隙率和雷诺数以及两种边界条件的宽范围问题,并计算了流体流动的基本特征,即涡度强度、孔隙空间的有效渗透率、摩擦阻力以及宏观压力梯度 \({\varvec\{nabla }}P\) 和流体速度 v 之间的关系。结果表明,当雷诺数 Re 低到达西定律成立时,涡度的大小 \(\omega _z\) 几乎为零。然而,随着雷诺数的增大,涡度也会增大,而且涡度从近乎零开始上升的起点与达西定律崩溃的雷诺数相同。我们还证明了宏观压力梯度与流体速度 v 之间的非线性关系,即 \(-{\varvec\nabla }}P=(\mu /K_e)\textbf{v}+\beta _\nrho |\textbf{v}|^2\textbf{v}\)、其中,\(\mu\) 和\(\rho\) 是流体的粘度和密度,\(K_e\) 是线性体系中的有效达西渗透率,\(\beta _n\)是广义非线性阻力。本文提出了这一关系的理论依据,并将其预测结果与福克海默方程的预测结果进行了比较。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Transport in Porous Media
Transport in Porous Media 工程技术-工程:化工
CiteScore
5.30
自引率
7.40%
发文量
155
审稿时长
4.2 months
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).
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