Turbulence in Porous Media: Modeling and Applications

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‘Turbulence in Porous Media’ introduces the reader to the characterisation of turbulent flow, heat and mass transfer in permeable media, including analytical data and a review of available experimental data. Such transport processes occurring a relatively high velocity in permeable media, are present in a number of engineering and natural flows. De Lemos has managed to compile, detail, compare and evaluate available methodologies for modelling simulating purposes, providing an essential tour for engineering students working within the field.
- The hotly debated topic of heterogeneity and flow turbulence has never before been addressed in book format.
- Offers an experimental approach to turbulence in porous media as it discusses disciplines that have been traditionally developed apart from each other.

The hotly debated topic of heterogeneity and flow turbulence has never before been addressed in book format.

Offers an experimental approach to turbulence in porous media as it discusses disciplines that have been traditionally developed apart from each other.

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Product Details

  • ISBN-13: 9780080444918
  • Publisher: Elsevier Science
  • Publication date: 9/7/2006
  • Pages: 368
  • Product dimensions: 0.81 (w) x 6.14 (h) x 9.21 (d)

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Turbulence in Porous Media

Modeling and Applications
By Marcelo J.S. de Lemos


Copyright © 2012 Elsevier Ltd.
All right reserved.

ISBN: 978-0-08-098263-2

Chapter One


Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.

Albert Einstein

1.1 Overview of Porous Media Modeling

Due to its ever-broader range of applications in science and industry, the study of flow through porous media has gained extensive attention lately. Engineering systems based on fluidized bed combustion, enhanced oil reservoir recovery, underground spreading of chemical waste, enhanced natural gas combustion in an inert porous matrix, and chemical catalytic reactors are just a few examples of applications of this interdisciplinary field. In a broader sense, the study of porous media embraces fluid and thermal sciences and materials, and chemical, geothermal, petroleum, and combustion engineering.

Accordingly, applications that are more complex usually require appropriate and, in most cases, more sophisticated mathematical and numerical modeling. Obtaining the final numerical results, however, may require the solution of a set of coupled partial differential equations involving many coupled variables in a complex geometry. This book shall review important aspects of numerical methods, including the treatment of multidimensional flow equations, discretization schemes for accurate solutions, algorithms for pointwise and block-implicit solutions, algorithms for high-performance computing, and turbulence modeling. These subjects shall be grouped into major sections covering numerical formulation and algorithms, geometry, and turbulence.

1.1.1 General Remarks

During the past few decades, a number of textbooks have been written on the subject of porous media. Among them are the works referred to in Muskat (1946), Carman (1956), Houpeurt (1957), Collins (1961), DeWiest (1969), Scheidegger (1974), Dullien (1979), Bear and Bachmat (1990), and Kaviany (1991). Advanced models documented in recent literature try to simulate additional effects such as variable porosity, anisotropy of medium permeability, unconventional boundary conditions, flow dimension, geometry complexity, nonlinear effects, and turbulence. Not all these flow complexities can be analyzed with the early unidimensional Darcy flow model. Recognizing the importance of these applications, the literature has been ingenious in proposing a number of extended theoretical approaches. Below is a short review of basic equations governing fluid flow, followed by a summary of some of the classical models for analyzing transport phenomena in porous media.

1.1.2 Fundamental Conservation Equations

The basic conservation equations describing the flow of a fluid through an infinitesimal volume can be written in a compact form as


where φ is the general variable (not to be confused with the porosity, φ, which will be introduced later), uj is the jth velocity component, ρ is the density, and Γφ and S are the diffusion coefficient and source terms, respectively. The value of φ and its corresponding parameters (Γφ and Sφ) take different forms according to the conserved quantity (mass, momentum, energy, chemical species, turbulent kinetic energy, etc.). The conservation laws recast into the form of Eq. (1.1) appear commonly in many texts devoted to the use of the control-volume approach. It is a convenient way to represent all transport phenomena occurring in a certain flow.

When Eq. (1.1) is written in Cartesian coordinates, all in three dimensions for the case of mass conservation (φ = 1, Γφ = Sφ = 0), one gets

[partial derivative]ρ/[partial derivative]t + [partial derivative](ρu/[partial derivative]x + [partial derivative](ρv/[partial derivative]y + [partial derivative](ρw/[partial derivative]z = 0 (1.2)

For a general variable, Eq. (1.1) can be written in Cartesian coordinates as


where all physical mechanisms and extra terms not included in the total flux components on the right-hand side (rhs) are treated by the source Sφ.

1.1.3 Basic Models for Flow in Porous Media

When the balance equations compacted in Eq. (1.1) are written for the flow through porous media, the medium porosity has to be accounted for. Therefore, the continuity equation (1.2) can be modified and expressed in vector form as

[partial derivative](φρ)/[partial derivative]t = -∇ · ρ[??]D (1.4)

where φ is the medium porosity defined as the ratio of pore volume to total (fluid plus solid) volume, ρ is the density based on total volume, and [??]D is the superficial velocity defined as the volumetric flow rate divided by the unit of total cross-sectional area. The well-known Darcy (1856) law of motion can be given further as

uD = K/µ(∇p - ρ[??]) (1.5)

where p is the pressure based on total area. The quantity K is referred to as the medium permeability, the unit of which is Darcy, defined as the permeability of a porous medium to viscous flow for the flow of 1 ml of a liquid of 1 centipoise viscosity under a pressure gradient of 1 atm/cm across 1 cm2 in 1 s. If, in Eq. (1.4), the porosity φ is assumed to be invariant with time, one gets

φ [partial derivative]ρ/[partial derivative]t = -∇ · ρ[??]D (1.6)

The empirical modification owned to Brinkman (1947) replaces Eq. (1.5) with


where the extra term is intended to account for distortion of the velocity profiles near containing walls.

The flow of fluids through consolidated porous media and through beds of granular solids are similar, both having the general function of pressure drop versus flow rate alike in form; i.e., the transition from laminar flow to turbulent flow is gradual (Perry and Chilton, 1973). For this reason, this function must include a viscous term and an inertia term. Therefore, an extension to Eq. (1.7) to account for the inertia effects was proposed by Forchheimer (1901) in the form


where F is known as the inertia coefficient. For the sake of using a coherent nomenclature throughout this book, another coefficient cF, which is related to F, is the Forchheimer coefficient. The proposition of writing the inertia coefficient as a function of cF is a tentative way to separate the dependence of the medium morphology on experimental values of F for a variety of porous media. The relationship between F and cF will be shown below and, in all simulations to be shown in this book, cF is taken to be a constant with value 0.55. The treatment followed in this book (i.e., the consideration of an additional mechanism for transport in porous media, namely turbulence) is distinct from the approach taken by many authors in the literature, who proposed expressions for cF to fit Eq. (1.8) to experimental data for high values of Reynolds number (Re) (Bhattacharya et al., 2002). Here, on the contrary, cF is assumed to be a constant. The mechanism of macroscopic turbulent transport, explicitly appearing after time and volume simultaneously average the full convective term in Eq. (1.8) (not shown), is modeled separately, giving an alternative way to account for discrepancies when Eq. (1.8) is applied to high Reynolds flow.

In addition, the nonlinear character of Eq. (1.8) has direct implications on its numerical solution. For purely viscous flow, away from any containing walls, the last two terms on the rhs of Eq. (1.8) become negligible, and the resulting equation is again Darcy's equation (1.5). Values of K and F are usually determined experimentally for each type of porous medium. For sphere-pack beds, Ergun (1952) has proposed the following empirical expression:

K = d2φ3/A(1 - φ)2; F = 1:75d/150(1 - φ) (.19)

where d is the diameter of particles or pores in the bed and A is a parameter that depends on the medium morphology. For a medium formed by an array of circular rods displaced in square arrangements, its value is given in Kuwahara and Nakayama (1998) as A = 140. Data on pressure drop as a function of the flow rate for various fluids are generally available from the manufacturer of such porous media.

1.1.4 Extended Models for Flow in Porous Media

Since the 1980s, several studies have been published about the application and extension of the models embodied in Eq. (1.8) for several geometries and different processes. A complete review of all these works would be outside the scope of the present text, so only a few of them will be mentioned.

Boundary and initial effects have been investigated by Vafai and Tien (1981), and various medium porosity studies are presented in Vafai (1984). The flow through packed beds has been accounted for by Hunt and Tien (1988) and by Adnani et al. (1995). The limitations imposed by the use of Eq. (1.8) when applied to several flows of engineering interest have been the subject of the reports by Nield (1991) and Vafai (1995). Knupp and Lage (1995) have extended the early Forchheimer's ideas to the tensor permeability case. Studies of natural convection systems in a porous saturated medium with both vertical and horizontal temperature gradients are reported in Manole and Lage (1995).

The advantages of having a combustion process inside an inert porous matrix are well recognized today. Hsu et al. (1993) points out some of its benefits, including greater burning speed and volumetric energy release rates, increased combustion stability, and the ability to burn gases of a low energy content. Driven by this motivation, the effects on porous ceramic inserts have been investigated by Peard et al. (1993). Turbulence modeling of combustion within inert porous media has been conducted by Lim and Matthews (1993) on the basis of an extension of the standard k-ε model of Jones and Launder (1972). Work on direct simulation of turbulence in premixed flames, for the case when the porous dimension is of the order of the flame thickness, has also been reported in Sahraoui and Kaviany (1995).

Being a multidisciplinary area, studies on flow, heat, and mass transfer in porous media have attracted the attention of many research groups around the world, most of which are concerned with different applications, processes, and system configurations. With the explosion of the World Wide Web on the Internet in the last decade, the reader is encouraged to navigate through interesting sites that summarize the research being conducted at different locations. To mention just a few of them would not do justice to all the important work being carried out presently. The electronic addresses to these sites are readily available through the employment of pertinent key words used in conjunction with the many search engines (database-search facilities) available online.

1.1.5 Models for Petroleum Reservoir Simulation

The recovery and better use of existing oil fields is being considered lately by many countries due to its impact on internal economies. With exploration of new fields becoming prohibitively expensive, and considering further that full experimentation in laboratories is an extremely difficult task, the subject of enhanced oil recovery (EOR) has stimulated many research efforts in the development of mathematical and numerical tools that could analyze existing oil reserves. Simulation of the movement of a different phase (such as water) or of a different component (such as a miscible tracer) introduced through an injection well (Figure 1.1) can provide important technical information to oil companies, aiding their decision-making process to pursue any further extraction from existing wells. Below is a summary of two important mathematical frameworks for analyzing the flow of miscible components and different phases through porous rock embedded in oil.


Excerpted from Turbulence in Porous Media by Marcelo J.S. de Lemos Copyright © 2012 by Elsevier Ltd. . Excerpted by permission of ELSEVIER. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Preface Overview Table of Contents List of Figures List of Tables Nomenclature Part One: Modeling
1. Introduction
2. Governing Equations
3. The Double-Decomposition Concept
4. Turbulent Momentum Transport
5. Turbulent Heat Transport
6. Turbulent Mass Transport
7. Turbulent Double Diffusion Part Two: Applications
8. Numerical Modeling and Algorithms
9. Applications in Hybrid Media References Index
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