JambhekarMaster(1)

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Universität Stuttgart - Institut für Wasser- und

Umweltsystemmodellierung

Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

Prof. Dr.-Ing. Rainer Helmig

Master’s Thesis

Forchheimer Porous-media Flow Models -

Numerical Investigation and Comparison with

Experimental Data

Submitted by

Vishal A. Jambhekar

Matrikelnummer 2550192

Stuttgart, 26. November 2011

Examiner: Prof. Dr.-Ing Rainer Helmig

Supervisors: Dipl.-Ing. Philipp Nuske and Dipl.-Ing. Katherina Baber



To my parents

To my parents



To my parents

I hereby acknowledge that I have prepared this master’s thesis independently, and that only

those sources, aids and advisors that are duly noted herein have been used and / orconsulted.

Signature:

Date:



Contents

1 Introduction 1

1.1 Applications and motivation . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Fundamentals of Porous Media Flow 4

2.1 Scales - the continuum approach . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Local equilibrium in porous media . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Effective parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Porosity (φ): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Saturation (S ): . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.3 Intrinsic permeability(K ): . . . . . . . . . . . . . . . . . . . . . . 7

2.3.4 Capillary pressure (P c): . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.4.1 Micro-scale capillarity ( pc) . . . . . . . . . . . . . . . . . 82.3.4.2 Macro-scale capillarity ( pc) . . . . . . . . . . . . . . . . 9

2.3.5 Relative permeability (kr,α): . . . . . . . . . . . . . . . . . . . . . 11

2.4 Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.2 Momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2.1 Darcy law . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2.2 Forchheimer law . . . . . . . . . . . . . . . . . . . . . . 15

2.4.3 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.3.1 Local thermal equilibrium . . . . . . . . . . . . . . . . . 17

2.4.3.2 Local thermal non-equilibrium . . . . . . . . . . . . . . 17

2.5 Multiphase non-Darcy flow . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Modified Ergun equation . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 Barree-Conway equation . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2.1 Barree-Conway model for single and multiphase flows . . 24

i



2.5.2.2 Barree-Conway approach for relative permeability-saturation

relationship . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 DuMuX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 ITLR Experiment 293.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Isothermal experiment . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2 Non-isothermal experiment . . . . . . . . . . . . . . . . . . . . . . 32

3.1.3 Motivation for the current work . . . . . . . . . . . . . . . . . . . 32

4 Results and Discussion 34

4.1 Intrinsic permeability and Forchheimer coefficient . . . . . . . . . . . . . 34

4.1.1 Linear regression analysis for intrinsic permeability K and nonlin-

ear regression analysis for Forchheimer coefficient β . . . . . . . . 354.1.1.1 Linear regression analysis for intrinsic permeability K . 35

4.1.1.2 Nonlinear regression for Forchheimer coefficient β . . . . 37

4.1.2 Nonlinear regression analysis for both intrinsic permeability K and

Forchheimer coefficient β . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.3 Apparent permeability . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.3.1 Constant Forchheimer coefficient β for complete range of

experimental data . . . . . . . . . . . . . . . . . . . . . 44

4.1.3.2 Forchheimer coefficient β for limited Re ranges . . . . . 444.1.3.3 Linear Forchheimer coefficient β (Re) . . . . . . . . . . . 47

4.2 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Incompressible isothermal case . . . . . . . . . . . . . . . . . . . . 50

4.2.2 Non-isothermal case . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2.1 With local thermal equilibrium . . . . . . . . . . . . . . 56

4.2.2.2 With local thermal non-equilibrium . . . . . . . . . . . . 65

5 Conclusion 68

ii



List of Figures

1.1 Applications of porous media : (a) Catalytic converter for automobile

exhaust system. (b) Porous heat exchanger for air cooled condensers. (c)

Cooling pores in a gas turbine blade. . . . . . . . . . . . . . . . . . . . . 1

2.1 Micro-scale to macro-scale transition [30] . . . . . . . . . . . . . . . . . . 5

2.2 Definition of the REV [21] . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Capillary forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Capillary pressure-saturation relationship [21] . . . . . . . . . . . . . . . 11

2.5 Deviation of experimental data from Forchheimer linear equation [23] . . 23

2.6 Relative permeability-saturation relationship[3] . . . . . . . . . . . . . . 26

2.7 Corrected relative permeability-saturation relationship [3] . . . . . . . . . 26

3.1 Photograph of ITLR experimental setup [30] . . . . . . . . . . . . . . . . 30

3.2 Porous structure used in ITLR experiments [29] . . . . . . . . . . . . . . 30

3.3 Schematic representation of ITLR experimental setup [30] . . . . . . . . 31

4.1 Linear regression with Darcy law . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Linear Darcy regression for intrinsic permeability K and nonlinear Forch-

heimer regression for Forchheimer coefficient β . . . . . . . . . . . . . . . 38

4.3 Nonlinear regression with Forchheimer law . . . . . . . . . . . . . . . . . 41

4.4 Apparent permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 Constant Forchheimer coefficient β for the complete range of experimental

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6 Forchheimer coefficient β for limited Re ranges . . . . . . . . . . . . . . 45

4.7 Apparent permeability K app for limited Re ranges . . . . . . . . . . . . . 45

4.8 Forchheimer coefficient β as a function of Reynolds number (Re) . . . . 47

4.9 Apparent permeability β as a function of velocity . . . . . . . . . . . . . 48

4.10 Incompressible isothermal model domain . . . . . . . . . . . . . . . . . . 51

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4.11 Pressure distribution across porous domain for an incompressible isother-

mal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.12 Velocity distribution across porous domain for an incompressible isother-

mal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.13 Friction coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.14 Model domain non-isothermal case . . . . . . . . . . . . . . . . . . . . . 56

4.15 Unphisical heating along edges . . . . . . . . . . . . . . . . . . . . . . . . 58

4.16 Pressure distribution across porous domain for a compressible non-isothermal

flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.17 Temperature distribution across porous domain for a compressible non-

isothermal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.18 Velocity and density distribution across porous domain for a compressible

non-isothermal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.19 Evolution of pressure (non-isothermal model with local thermal equilibrium) 61

4.20 Evolution of temperature (non-isothermal model with local thermal equi-

librium) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.21 Evolution of velocity (non-isothermal model with local thermal equilibrium) 63

4.22 Evolution of temperature (non-isothermal model with local thermal non-

equilibrium) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

iv



List of Tables

2.1 Exponents for relative permeability kr and relative passability ηr for the

liquid phase (l) and the gas phase (g) [36, 37] . . . . . . . . . . . . . . . 22

4.1 Experimental data for isothermal case (ITLR) . . . . . . . . . . . . . . . 36

4.2 Forchheimer coefficient β for different subsets of the experimental data . 38

4.3 Percentage error for the calculated pressure gradients using different Forch-heimer coefficients given in Table 4.2 . . . . . . . . . . . . . . . . . . . . 39

4.4 Forchheimer coefficient β for different subsets of the experimental data . 40

4.5 Percentage error of the calculated pressure gradients using different in-

trinsic permeabilities and Forchheimer coefficients given in Table 4.4 . . . 42

4.6 Forchheimer coefficient β for limited Re ranges . . . . . . . . . . . . . . 46

4.7 Comparison of percentage errors of calculated pressure gradients using

different approaches for Forchheimer coefficient β . . . . . . . . . . . . . 49

4.8 Ergun coefficient C E for different Forchheimer coefficient β approaches . 504.9 Experimental and numerical velocity data . . . . . . . . . . . . . . . . . 53

4.10 Experimental and numerical friction coefficients . . . . . . . . . . . . . . 54

4.11 Boundary conditions non-isothermal case . . . . . . . . . . . . . . . . . 57

4.12 Material parameters and input data . . . . . . . . . . . . . . . . . . . . . 57

4.13 Experimental and numerical wall temperatures (non-isothermal model

with local thermal equilibrium) . . . . . . . . . . . . . . . . . . . . . . . 64

4.14 Experimental and numerical wall temperatures (non-isothermal model

with local thermal non-equilibrium) . . . . . . . . . . . . . . . . . . . . . 67

v



Nomenclature

λ lumped thermal condictivity [W/mK ]

T transition constant [1/m]

β Forchheimer coefficient [1/m]

ηr,α relative passability of phase α [-]

η intrinsic passability [m]

α thermal diffusivity of the fluid phase [m2/s]

h convective heat transfer coefficient [W/m2K ]

λf thermal condictivity of the fluid phase [W/mK ]

λs thermal condictivity of the porous matrix [W/mK ]

g gravity vector [m/s2]

K intrinsic permeability tensor [m2]

Kf hydraulic conductivity tensor [m/s]

v seepage velocity vector [m/s]

vf Darcy or Forchheimer velocity vector [m/s]

µ dynamic viscosity [kg/ms]

ν kinematic viscosity [m2/s]

φ solid matrix porosity [-]

σ surface tension [N/m]

density [kg/m3]

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C E Ergun coefficient [-]

c pf specific heat capacity of the fluid phase [J/kgK ]

d capillary diameter [m]

d p averaged particle diameter [m]

g gravity [m/s2]

h piezometric head [m]

hf specific enthalpy of fluid [J/kg]

K intrinsic permeability [m2]

K f hydraulic conductivity [m/s]

K app apparent passability [m2]

kr,α relative permeability for phase α [-]

L characteristic length [m]

pc capillary pressure [P a]

pn pressure of the non-wetting phase [P a]

pw pressure of the wetting phase [P a]

q fs exchange energy between the solid matrix and the fluid phase [W/m3]

sv specific interfacial area [1/m]

S α saturation of the fluid phase α [-]

T temperature [K ]

uf specific internal energy of fluid [J/kg]

vf Darcy or Forchheimer velocity [m/s]

z elevation head [m]

Nu Nusselt number [-]

vii



Pr Prandtl number [-]

Re Reynolds number [-]

viii





1 Introduction

1.1 Applications and motivation

Fluid flow and transport processes through porous structures is a topic of great interest

in various scientific and technical fields. In particular, in engineering applications such as

catalytic converters (used for reducing toxicity of exhaust emissions from automobiles engines,

see Figure 1.1a), condensers (used as heat exchanger for cooling condensers, see Figure1.1b) and gas turbines (used for cooling gas turbine blades, see Figure 1.1c), fluid flow

through porous media in the high-velocity regime becomes relevant. In heating and cooling

applications, in addition to high-velocity flow, developing a deep understanding about non-

isothermal flow and related heat-transfer processes becomes crucial.

Motivated by real world engineering applications and scientific interest, many scientists have

channeled research efforts towards developing a detailed understanding about these flow

and transport processes by means of experimentation and numerical analysis. Authors like[22, 36, 4, 32, 37] used the Forchheimer law (see Section 2.4.2.2) [19] to describe high

velocity flow. They supported their choice by stating that the Forchheimer law accounts for

high velocity inertial effects.

(a) (b) (c)

Figure 1.1: Applications of porous media : (a) Catalytic converter for automobile ex-haust system. (b) Porous heat exchanger for air cooled condensers. (c)Cooling pores in a gas turbine blade.

1



Mayer et al. [30] observed that metallic porous media offer an effective solution in many

heating and cooling applications by the virtue of their inherent properties such as large

surface area-to-volume ratio, high permeability and thermal dispersion of the fluid due to

the complex flow within pore channels. Pressure loss and convective heat transfer in these

metallic structures have been investigated for the above mentioned engineering applications.

At the Institute of Aerospace Thermodynamics / Institut für Thermodynamik der Luft- und

Raumfahrt (ITLR), University of Stuttgart, experiments are carried out to analyze and opti-

mize a uniform metallic porous structure for cooling applications. The pressure drop across

the structure and temperatures at different locations are measured to validate the numerical

models for both isothermal and non-isothermal flow systems.

In the scope of this master’s thesis, the existing numerical model for a single-phase isothermal

Darcy (creeping) flow is modified for high velocity non-Darcy flow by using the Forchheimer

equation. The Forchheimer flow model is also extended to allow the description of non-

isothermal flow with local thermal equilibrium and with local thermal non-equilibrium.

The ultimate goal of the current work is to develop satisfactory isothermal and non-isothermal

Forchheimer flow models. We achieve this by means of linking experimental data with

numerical studies. Determination of accurate effective parameters from the experimental data

is an important component of the current work. Thus, we review the Darcy and Forchheimer

laws in the context of determination of the intrinsic permeability K and the Forchheimer

coefficient β and validate our numerical models against experimental measurements.

1.2 Objectives

Given below are the objectives of the current work

• Detailed analysis of the experimental data and determination of the intrinsic perme-

ability K and the Forchheimer coefficient β .

• Implementation of the Forchheimer model for a single-phase isothermal flow.

• Implementation of the Forchheimer model for a single-phase non-isothermal flow as-

suming local thermal equilibrium.

• Extension of the single-phase non-isothermal Forchheimer model for local thermal non-

equilibrium.

2



• Setup of relevant numerical examples with appropriate boundary conditions.

• Attempt to validate above mentioned isothermal and non-isothermal models by com-

paring numerical simulations with experimental results.

• Detailed literature review for multiphase Forchheimer flow.

1.3 Structure of the thesis

• Chapter 2: In this chapter, the theoretical background about the different scales, effec-

tive parameters, equilibrium criteria and balance equations necessary for the description

of a complex porous media flow system is explained. In addition, this chapter also pro-

vides a brief introduction to the multiphase Forchheimer flow (see Section 2.5) and

DuMuX, an open source simulation software for flow and transport processes in porousmedia (see Section 2.6).

• Chapter 3: The experimental setup and procedure are discussed in detail in this chapter.

• Chapter 4: Analytical determination of the intrinsic permeability K and the Forch-

heimer coefficient β are discussed in detail in this chapter. The determined intrinsic

permeability K and the Forchheimer coefficient β are used for numerical simulations

and the results are compared with corresponding experimental data.

3



2 Fundamentals of Porous Media

Flow

2.1 Scales - the continuum approach

According to [32], the treatment of the problem of flow through a porous structure is largely

dependent on the scale considered. At a small scale, one looks at just a few small pores

(micro-scale or pore-scale). It is therefore convenient to use conventional fluid mechanics

approach to describe the flow phenomenon in the fluid-filled spaces. However, when the scale

is large (macro-scale), the field of vision includes a large number of pore spaces. In such a

case, the complicated flow paths and the need to describe complex spatial resolution of the

porous structure rule out the possibility of using the conventional fluid mechanics approach.

Hence, a volume averaging (continuum) approach is used [32].

The finite scale an engineer would look at is the molecular scale. The continuum mechanicsbased approach is used for transition from the molecular scale to the micro-scale [1]. The

consideration of a continuum corresponds to replacing the molecular properties by averaged

properties over a large number of molecules. According to [21, 13], the consideration of a

continuum at the macro-scale is a fundamental concept of fluid mechanics. For example, air

is used as the fluid phase at ITLR for experiments and numerical simulations. Here, rather

than looking at the movement of every single molecule, the overall air-flow is observed using

averaged fluid properties such as density and viscosity µ. In the context of the current

work, the term “phase” is used to differentiate between physical continua separated by a

sharp interface (e.g., solid and fluid phase).

Initially, the continuum approach is used to transfer the molecular properties to the micro-

scale or pore-scale in order to resolve the flow in pore spaces. In the context of the current

work, micro-scale can be represented by the dimension of a unit cell forming the uniform

porous structure (see Figure 2.1). At ITLR, numerical simulations are performed at the pore-

4



scale using conventional fluid mechanics approach. However, for reasons discussed previously,

for a large scale problem, the possibility of using the conventional approach is ruled out. Thus,

further volume averaging is needed to describe the flow properties on the macro-scale.

Figure 2.1: Micro-scale to macro-scale transition [30]

Figure 2.2: Definition of the REV [21]

The Figure 2.1 shows the different scales involved in the averaging process. The micro-scale

properties are averaged over a representative elementary volume (REV) in order to obtain a

macro-scale description of the system with effective parameters such as porosity φ, saturation

S and intrinsic permeability K (see Section 2.3). The macro-scale is also referred to as the

5



REV-scale. It can be seen from Figure 2.1 that at the REV-scale, detailed spatial resolution

of solid matrix and fluid phase is lost, and effective volume averaged parameters (effective

parameters) are available.

For the process of volume averaging, proper selection of REV size is very crucial, as thevolume-averaged quantities need to be independent of the REV-size. Figure 2.2 shows that

the selected REV should not only be smaller than the flow domain, but it also should be

larger than a single pore in the porous medium. A very small REV leads to oscillations due

to existence of inhomogeneities at the micro-scale. On the other hand, a very large REV

leads to fluctuations caused by macroscopic heterogeneities of the medium [21].

2.2 Local equilibrium in porous media

The local thermodynamic equilibrium in porous media mainly consists of thermal, chemical

and mechanical equilibria as follows:

Thermal equilibrium:

A system is said to be in local thermal equilibrium, if at any given point of the system all the

phases exist at the same temperature

T = T s = T f [K ].

Here, T s and T f are the temperatures of the solid matrix and fluid phase respectively.

Chemical equilibrium:

A system is said to be in chemical equilibrium, if the potential for exchange of chemical

components across different phases or within a phase is zero. In other words, there is no

exchange of components within a phase or between different phases.

Mechanical equilibrium:

When multiple fluid phases are present in the system, mechanical equilibrium refers to the

existence of equal pressure on either side of the phase boundary (e.g., a lake surface).

However, in the context of the porous media flow, one must account for the pressure jump

6



at the fluid phase boundaries due to capillarity i.e., the capillary pressure (see Section 2.3.4)

[21].

2.3 Effective parameters

2.3.1 Porosity (φ):

As discussed earlier, porous media consists of interconnected voids and a solid matrix. Poros-

ity is defined as the ratio of the volume of pores in REV to the volume of REV:

φ = Volume of pores in REV

Volume of REV [−]. (2.1)

Here, 1 − φ is the volume fraction of the solid matrix. For the current study porosity is

assumed to be constant i.e. the solid matrix is assumed to be a rigid structure. Porosity is

a dimensionless parameter.

2.3.2 Saturation (S ):

With the macro-scale approach and adaptation of REV for multiphase flow problems, a

new effective parameter called saturation is introduced at the macro-scale. This parameter

accounts for the existence of different fluid phases in a given REV. In other words, it accounts

for the fractions of the pore space occupied by different fluid phases. The saturation of a

fluid phase α is the ratio of the volume of fluid phase α in REV to the volume of pores in

REV and is given as follows:

S α = Volume of fluid phase α in REV

Volume of pores in REV [−]. (2.2)

Saturation, like porosity φ, is also a dimensionless parameter.

2.3.3 Intrinsic permeability(K ):

The intrinsic permeability K of a porous medium represents its ability to allow the fluid to

flow through. It is a macro-scale property. The intrinsic permeability tensor K is a part of

the definition of the hydraulic conductivity tensor Kf . “Hydraulic conductivity is a macro-

scale parameter which accounts for the influence of viscosity and adhesion at the soil grain

7



surfaces” [21]. The hydraulic conductivity tensor Kf for a single-phase flow is given as:

Kf = Kg

µ

m

s

, (2.3)

where K is the intrinsic permeability tensor, and µ are the density and viscosity of the fluidrespectively and g is the gravitational acceleration. The intrinsic permeability tensor is a

second order tensor with nine components indicating permeabilities in different directions:

K =

K xx K xy K xz

K yx K yy K yz

K zx K zy K zz

[m2]. (2.4)

Intrinsic permeability K is only dependent on the porous structure and can be same or differ-

ent in different directions, depending on whether the porous matrix is isotropic or anisotropic.

2.3.4 Capillary pressure (P c):

2.3.4.1 Micro-scale capillarity ( pc)

For a multiphase flow through porous media, a certain force acts at the interfacial area

between different fluid phases. This force is called surface tension and is strongly influenced

by the solid and fluid properties. The surface tension caused by the interaction of the solid

and the different fluid phases leads to capillary pressure ( pc) [21]. Capillary pressure is defined

as difference between the pressure of the non-wetting ( pn) and wetting ( pw) phases at the

interface as shown in Figure 2.3:

pc = pn − pw [P a]. (2.5)

Here, wetting and non-wetting phases are relative terms. For a two-phase flow system, the

wetting phase is the phase which has higher affinity with the solid phase. The Laplace

equation for the capillary pressure is as follows:

pc = 4σ cos θ

d = pn − pw [P a], (2.6)

8



Figure 2.3: Capillary forces

where σ is the surface tension for the fluid phase, θ is the angle of contact between the fluid

phase and the solid phase and d is the pore diameter.

2.3.4.2 Macro-scale capillarity ( pc)

From the detailed study of capillary effects, it is observed that the fluid saturations have a

strong influence on the capillary pressure [21]. The macro-scale capillary effects are taken into

account based on the fundamental correlation between the capillary pressure and saturation

of the wetting phase and the non-wetting phase.

Typical examples of multiphase flow system are: Imbibition - injection of a wetting phase into

a porous medium to displace a non-wetting phase and draining - injection of a non-wetting

phase into a porous medium to displace a wetting phase.

For example, during a draining process, as the saturation of the wetting phase decreases,

the wetting phase retreats into the smaller pores in the porous medium. It is noticed that a

higher capillary pressure is needed for further displacement of the wetting phase. In this way,

the required capillary pressure keeps increasing as the wetting phase moves into finer pores of the porous matrix. The capillary pressure required for the displacement of the wetting phase

can be expressed as a function of its saturation (see Equation 2.7). Detailed discussion on

the macro-scale capillary pressure-saturation relationship can be studied in [21].

pc = pc(S w). (2.7)

9



Many scientists came up with ideas to describe capillary pressure-saturation relationship.

According to [21], the best known capillary pressure-saturation relationships for air-water

systems could be found in the literature from Leverett (1941) [25], Brooks and Corey (1964)

[10] and Van Genuchten (1980) [20].

• Brooks and Corey:

The capillary pressure-saturation relation proposed by Brooks and Corey is discussed below:

S e( pc) = S w − S wr

1 − S wr

=

pd

pc

λ

for pc ≥ pd, (2.8)

where λ is the Brooks Corey parameter, pc is the capillary pressure, pd is the entry pressure,

S e is the effective saturation, S w is the wetting phase saturation and S wr is the residual

saturation of the wetting phase (w). Here, S wr refers to the saturation of the detached part

of a phase which is held back within the porous medium. Detailed description of residual

saturation can be found in [21]. λ indicates the material grain size distribution and it ranges

between 0.2 and 0.3. The Brooks-Corey relation is valid only for capillary pressure greater

than entry pressure pd (see Figure 2.4). Here, entry pressure pd is the minimum pressure

needed for the non-wetting phase to enter the porous medium.

According to Brooks and Corey [10], the capillary pressure as a function of wetting phase

saturation S w is given as:

pc(S w) = pdS − 1

λe ; for pc ≥ pd. (2.9)

• Van Genuchten:

The capillary pressure-saturation relation proposed by Van Genuchten is as follows:

S e( pc) = S w − S wr

1 − S wr

= [1 + (α

· pc)n]m for pc > 0, (2.10)

where m, n and α are Van Genuchten parameters, such that m = 1 − (1/n). m and n and

are dimensionless parameters whereas, α has a dimension of [1/P a]. The Van Genuchten

relation is valid for all capillary pressures greater than zero.

10



According to Van Genuchten [20], the capillary pressure as a function of wetting phase

saturation S w is given as:

pc(S w) = 1

α(S

− 1

me − 1)

1

n ; for pc > 0. (2.11)

Figure 2.4: Capillary pressure-saturation relationship [21]

Figure 2.4 shows the capillary pressure-saturation relationship for an air-water flow system

by Van Genuchten [20] and Brooks-Corey [10].

2.3.5 Relative permeability (kr,α):

Hydraulic conductivity of a porous medium is already discussed in Section 2.3.3. However,

for a multiphase system, the hydraulic conductivity is defined as follows:

Kf = Kkr,ααg

µα

m

s

, (2.12)

where K is the intrinsic permeability tensor, α and µα are the density and viscosity of the

fluid phase α respectively and g is the gravitational acceleration. The relative permeability

kr,α is a dimensionless parameter and it depends on the fluid phase saturation S α [21].

The relative permeability kr,α accounts for the dependence of effective permeability K kr,α

on saturation S α through the relative permeability-saturation relationship. The best known

11



relative permeability-saturation relationships are proposed by Brooks and Corey (1964) [10]

and Van Genuchten (1980) [20].

• Brooks and Corey:

The relative permeability-saturation relationship proposed by Brooks and Corey [21] for a

two phase air-water system is as follows:

kr,w = S 2+3λλ

e and (2.13)

kr,n = (1 − S e)2(1 − S 2+λλ

e ), (2.14)

where, as discussed in Section 2.3.4.2, λ and S e are the Brooks-Corey parameter and effectivesaturation respectively (see Equation 2.8). Here, kr,w and kr,n are the relative permeabilities

of the wetting phase (water) and the non-wetting phase (air) respectively.

• Van Genuchten:

The relative permeability-saturation relationship proposed by Van Genuchten [21] for a two

phase air-water system is as follows:

kr,w = S e[1 − (1 − S

1

m

e )m

]2

and (2.15)

kr,n = (1 − S e)γ [1 − S 1

me ]2m, (2.16)

where, as discussed in Section 2.3.4.2, m and S e are the Van Genuchten parameter and

effective saturation respectively (see Equation 2.10). The parameters and γ describe the

connectivity of pores. Generally, = 12

and γ = 13

[21].

2.4 Balance equations

The choice of the continuum approach for the current work necessitates the definition of the

laws for conservation of mass, momentum and energy.

12



2.4.1 Mass balance

The continuity equation or the mass balance equation ensures that the overall change of

mass within a continuum is zero. The equation for conservation of mass for a free flow

system is as follows:∂f

∂t + ∇ · (f v) − q = 0, (2.17)

where f is the density of the fluid, t is time, v is the velocity vector of the fluid and q is

the external source or sink.

Equation 2.17 indicates that “the rate of change of mass per unit control volume, fluxes

across the faces of the control volume and the potential sources and sinks must balance” [1].

In the context of the current work, porosity is included in Equation 2.17, as the continuity is

only considered for the fluid flow through porous matrix:

φ∂f

∂t + ∇ · (f φv) − q = 0. (2.18)

Neglecting the source or sink term q and using the relation between the Darcy / Forchheimer

velocity vector vf and seepage velocity vector v given by:

v = vf

φ , (2.19)

equation 2.18 can be rewritten as follows:

φ∂f

∂t + ∇ · (f vf ) = 0. (2.20)

The velocity vector vf is referred to as Darcy velocity vector in Section 2.4.2.1 and Forch-

heimer velocity vector in Section 2.4.2.2.

Assuming an incompressibile fluid, a rigid porous medium and no source or sink (q = 0),

Equation 2.20 is reduced to the following mass balance equation:

∇ · (vf ) = 0. (2.21)

13



On the other hand, an assumption of a compressible fluid, a rigid porous medium and no

source or sink (q = 0) leads to the following mass balance equation:

φ∂f

∂t + ∇ · (f vf ) = 0. (2.22)

2.4.2 Momentum balance

This section discusses the macro-scale momentum balance equations for the fluid flow

through porous media. According to [32], the Darcy law (see Section 2.4.2.1) is used to

describe slow or creeping flows and the Forchheimer law (see Section 2.4.2.2) is used for the

description of high velocity flows. Both the Darcy law and the Forchheimer law are obtained

experimentally in order to describe the flow through a porous medium and have become the

macro-scale momentum equation of choice in literature [21, 32]. According to [13], these

equations for momentum description allow the decoupling of the continuity and momentum

balance.

2.4.2.1 Darcy law

A French scientist, Henry Darcy in his work on the investigation of hydrological systems for

water supply in the city of Dijon performed steady-state unidirectional flow experiments for

a uniform sand column [14]. From his experimental observations, he proposed the Darcy law

as follows:

vf = −Kf · ∇h, (2.23)

where vf is the Darcy velocity vector, Kf is the hydraulic conductivity tensor as explained

in Section 2.3.3 and ∇h is the gradient of the piezometric head h given by:

h = p

f g + z [m], (2.24)

where pf g

is the pressure head and z is the elevation head. From Equation 2.24 and Equation

2.3, the Darcy law can be rewritten as follows:

vf = −K

µ · (∇ p + f g∇z ), (2.25)

where µ is the dynamic viscosity of the fluid , ∇ p is the applied pressure gradient and K is

the intrinsic permeability tensor of the porous matrix. For the current study, gravitational

14



effects are neglected and as the porous matrix is isotropic, the intrinsic permeability is treated

as a scalar. Thus, the Darcy law boils down to:

vf = −K

µ∇ p. (2.26)

According to [21], Bear and Bachmat (1986) [6] derived the Darcy law. In the derivation,

they have neglected inertial or time dependent effects. Thus, the Darcy law is only valid for

slow (creeping) flows (Re << 1). Here, the Reynolds number (Re) is defined as the ratio of

the inertia force to the viscous force:

Re = Inertia Force

Viscous Force =

f vf L

µ [−], (2.27)

where f , vf and µ are the density, velocity, and dynamic viscosity of the fluid respectively.For the current work, the characteristic length (L= 1/901 [m]) is obtained from the specific

interfacial area (sv= 901 1m

) [30]. The specific interfacial area sv is defined as the area of

contact between the solid and fluid phase per unit volume.

2.4.2.2 Forchheimer law

An Austrian scientist Phillip Forchheimer (1901) [19] in his work “Wasserbewegung durch

Boden”, investigated fluid flow through porous media in the high velocity regime. During this

study, he observed that as the flow velocity increases, the inertial effects start dominating theflow. In order to account for these high velocity inertial effects, he suggested the inclusion of

an inertial term representing the kinetic energy of the fluid to the Darcy equation [38]. The

Forchheimer equation is given as follows:

∇ p = − µ

K vf − βf v

2f . (2.28)

Here, the parameter β is called the Forchheimer coefficient and vf stands for the Forchheimer

velocity. The Forchheimer equation in the vector form is given below [22, 38, 32]:

∇ p = − µ

K vf − βf |vf |vf , (2.29)

where vf is the Forchheimer velocity vector.

Theoretical evaluation of the Forchheimer coefficient β is cumbersome. Thus, for most

practical applications, this parameter is obtained from the best fit to the experimental data.

15



Ergun and Orning (1949) [16] worked for the investigation of fluid flow through packed

columns and fluidized beds. Based on this work, Ergun (1952) [15] proposed an expression

for the Forchheimer coefficient β :

β = C E

√ K 1

m

, (2.30)

where C E is called Ergun constant and it accounts for inertial (kinetic) effects. K is the

intrinsic permeability (see Section 2.3.3). From Equation 2.30 and Equation 2.29, we get

the Ergun equation. This equation is also referred to as Forchheimer equation with Ergun

expression for Forchheimer coefficient [22, 32]:

∇ p = − µ

K vf − C E √

K f |vf |vf . (2.31)

The Ergun coefficient C E is strongly dependent on the flow regime. For slow flows, C E is

very small. Thus, the second term on the right hand side of Equation 2.31 is very small and

can be neglected. This reduces the Forchheimer equation to the Darcy equation.

As the flow velocity increases, inertial effects also increase and the flow adapts to the Forch-

heimer flow regime [34]. These inertial effects are accounted for by the Ergun coefficient C E

and the kinetic energy of the fluid f |vf |vf [38]. However, according to [31, 4, 2], a constant

Ergun coefficient C E is valid as long as the fluid flow is laminar. Thus, in the high velocity

flow regime, the Ergun coefficient C E needs to be adapted to reflect the experimental inertial

effects.

2.4.3 Energy balance

As discussed in Section 1.1, in the scope of the current work, numerical models are imple-

mented for both isothermal and non-isothermal flows. The non-isothermal model is imple-

mented for two possible thermodynamic scenarios - namely, with local thermal equilibrium

and with local thermal non-equilibrium. For the first scenario, the system is not necessarilyin thermal equilibrium globally. However, thermal equilibrium exists locally. This means that

at any given point in the system, different phases exists at same temperature. In such a case,

only one energy equation is required for the description of the temperature of each phase at

any given point in the system. For the other scenario, there exists no thermal equilibrium.

That is, at any given point in the system, different phases exist at different temperatures.

16



Thus, different energy equations are required for the description of the temperature of each

phase. In what follows, these two scenarios are discussed in detail.

2.4.3.1 Local thermal equilibrium

The behavior of non-isothermal flow system is strongly dependent on the flow velocity. For

a slow flow system, as different phases (e.g., solid phase and fluid phase) are in contact for

a sufficient period of time, the possibility for the system to exchange energy locally and to

establish local thermal equilibrium always exists. In such a case, only one energy equation

is sufficient for the description of temperature of all phases at any given location within the

system. For a single-phase flow through porous matrix, the energy balance equation is given

as follows:

∂

∂t (φf uf ) + ∂

∂t ((1 − φ)sc psT ) + ∇ · (f hf vf ) −∇ · (λ∇T ) − q = 0. (2.32)

Here, uf , hf , c ps and T stands for the internal energy of the fluid, the enthalpy of the fluid,

specific heat capacity of the solid matrix and temperature respectively. q is the external

source or sink. λ is averaged thermal conductivity of the solid matrix and the fluid phase

and is given by Equation 2.33 [11, 9, 8] :

λ = (1 − φ)λs + φλf

W

m K

, (2.33)

where λs and λf are the thermal conductivities of the solid matrix and the fluid phase

respectively.

2.4.3.2 Local thermal non-equilibrium

For high velocity flow, the interaction between different phases is rapid. Thus, different

phases cannot exchange sufficient amount of energy to establish local thermal equilibrium.

Therefore, at any given location in the system, different phases exist at different temperatures.

In such a case, one needs different energy equations for the description of the temperature

of each phase. In the context of the current work, the energy equations for the solid matrix

and the fluid phase are given by Equation 2.34 and Equation 2.35 respectively:

∂

∂t((1 − φ)sc psT s) −∇ · ((1 − φ)λs∇T s) − q s = 0 and (2.34)

17



∂

∂t(φf uf ) + ∇ · (f hf vf ) −∇ · (φλf ∇T f ) − q f = 0, (2.35)

where q s and q f represent the exchange energy with other phases or external source/sink. In

the context of the current work, q s = −q f = q fs is the exchange energy between the solidmatrix and the fluid phase.

The idea now, is to obtain an expression for exchange of energy between different phases.

For this purpose, we present some fundamentals of heat transfer.

In the context of the current work, transfer of thermal energy in the solid matrix and the

fluid phase takes place by means of two different processes: Firstly, by means of conduction

- involving the transfer of thermal energy between different parts of a system due to temper-

ature difference. Secondly, by means of convection - involving the transfer of thermal energy

due to the motion of molecules. Convection occurs only in fluids, while conduction takes

place in solids as well as fluids.

In order to determine the dominance of conductive or convective heat transfer, a new dimen-

sionless number called Nusselt number is used. Nusselt number is defined as follows:

Nu = Convective Heat Transfer Coefficient

Conductive Heat Transfer Coefficient

=hL

λf

[

−]. (2.36)

Here, h is the convective heat transfer coefficient and λf L

is the conductive heat transfer

coefficient.

The Nusselt number can also be described in terms of other dimensionless numbers, namely

- Reynolds number and Prandtl number:

Nu = Nu(Re, Pr). (2.37)

For the definition of Reynolds number see Equation 2.27. Prandtl number is defined as the

ratio of viscous diffusion rate to the thermal diffusivity:

Pr = Viscous Diffusion Rate

Thermal Diffusivity =

ν

α [−]. (2.38)

18



Here, ν stands for the kinematic viscosity and the thermal diffusivity α is the ratio of thermal

conductivity λf to the volumetric heat capacity f c pf :

α = λf

f

c pf

m2

s , (2.39)

where c pf is the specific heat capacity of fluid. For a single-phase flow system, the exchange

of energy between the solid matrix and the fluid phase is given by following equation:

q fs = hsv(T s − T f ).

W

m3

, (2.40)

where q fs is the heat exchange between the solid and the fluid phase, T s is temperature

of the solid, T f is temperature of the fluid, h is convective heat transfer coefficient and sv

is specific interfacial area. From the definition of Nusselt number (see Equation 2.36), theconvective heat transfer coefficient h can be expressed as:

h = λf

L Nu(Re, Pr)

W

m2K

. (2.41)

Various problem-specific Nusselt correlations are available in literature. The following Nusselt

number correlation used in the current work is taken from [27]:

Nu (Re, Pr)=0.023 Re0.8Pr0.33. (2.42)

2.5 Multiphase non-Darcy flow

This section provides a detailed overview of the different approaches available for modeling

multiphase non-Darcy flow. These approaches would be useful for extension of the current

work to high velocity multiphase flow in the future. The Forchheimer flow (see Section

2.4.2.2) is also referred to as non-Darcy flow in literature [12, 17, 4, 5, 3]. Realizing the

importance of the non-Darcy multiphase flow in industrial applications, many authors [7, 17,

36, 3] have come up with different modeling approaches.

Schäfer and Lohnert (2006) [35] mentioned that most of the multiphase dryout models for

nuclear research are based on the Ergun equation (see Section 2.5.1). Bennethum and Giorgi

(1997) [7] derived the generalized Forchheimer equation for the description of single phase and

multiphase flows. Ewing et al. (1999) [17] developed a numerical model for the description

19



of the non-Darcy multiphase flow through isotropic porous media. Barree and Conway (2007)

[3] proposed a new Forchheimer type equation for the description of a multiphase non-Darcy

flow (see Section 2.5.2). Wu et al. (2011) [39] supported and discussed the Barree-Conway

approach for modeling multiphase non-Darcy flow through porous media.

The classical approaches for capillary pressure-saturation relationship discussed in Section

2.3.4 and the relative permeability-saturation relationship discussed in Section 2.3.5 are valid

for the Darcy flow regime (Re << 1). For a multiphase non-Darcy flow, the capillary forces

are neglected for the sake of simplicity and only relative permeability kr,α is taken into account

[35, 37]. The relative permeability-saturation relationships for a multiphase non-Darcy flow

are discussed in Section 2.5.1 and Section 2.5.2.2.

2.5.1 Modified Ergun equationAccording to [36, 35], the modified Ergun equation is based on the Ergun equation (see

Equation 2.31) and is used to describe multiphase non-Darcy flows. The modified Ergun

equation for each phase α in a multiphase non-Darcy flow is given by:

∇ pα = −αg− µα

kr,αK vfα − α

ηr,αη|vfα|vfα, (2.43)

where ∇ pα is the pressure gradient, α is the density, µα is the dynamic viscosity, K is

the intrinsic permeability, kr,α is the relative permeability, vfα is the Forchheimer velocityvector, η is the intrinsic passability and ηr,α is the relative passability for phase α. Comparing

Equation 2.43 with Equation 2.31 one can define intrinsic passability η as the ratio of the

square root of intrinsic permeability to the Ergun coefficient C E :

η =

√ K

C E =

1

β [m]. (2.44)

Modified Ergun equation for the representation of the multiphase non-Darcy flow differs

from the Ergun equation (see Equation 2.31) in terms of interpretation of the intrinsicpermeability K and intrinsic passability η. In the case of the modified Ergun equation, the

intrinsic permeability and passability are interpreted in terms of spatial parameters as follows

[28, 36, 35, 37, 26]:

K = d2

pφ3

A(1 − φ)2 and (2.45)

20



η = d pφ3

B(1 − φ), (2.46)

where φ stands for porosity and d p is the averaged diameter of particles in the porous medium

[35]. According to the definition of the modified Ergun equation, A = 150 and B = 1.75

[36, 37].

During multiphase flow through a porous medium, the flow area for one phase is reduced

due to the existence of other phases. “Relative permeability kr,α and relative passability ηr,α

are the parameters which quantify the effect of the reduced flow area for each phase” [37].

We have already discussed relative permeability kr,α in Section 2.3.5. Relative passability

ηr,α for a certain phase α is a part of the definition of effective passability ηα = η ηr,α. Fora gas-liquid non-Darcy flow system, relations between effective and intrinsic permeabilities

and effective and intrinsic passabilities are given as follows:

For the gas phase,

K g = K kr,g(S g) and (2.47)

ηg = ηηr,g(S g). (2.48)

For the liquid phase,

K l = K kr,l(1 − S g) and (2.49)

ηl = ηηr,l(1 − S g). (2.50)

Here, S g is saturation of the gas phase. The relative permeability and passability for a gas-

liquid flow system are given as kr,g = S ng , kr,l = (1 − S g)n,ηr,g = S mg , ηr,l = (1 − S g)m

[36, 37]. Inserting the expression for the relative permeability and passability in Equation

2.43, we get the modified Ergun equation to estimate the pressure drop caused by each phase

21



in a non-Darcy gas-liquid flow system as follows:

∇ pg = −gg − µg

S ng K vfg − g

S mg η|vfg|vfg and (2.51)

∇ pl = −lg− µl(1 − S g)nK

vfl − l(1 − S g)mη

|vfl|vfl, (2.52)

where ∇ pg is the pressure gradient for the gas phase and ∇ pl is the pressure gradient for the

liquid phase. Here, vfg and vfl are the Forchheimer velocity vectors for the gas and liquid

phases respectively.

Table 2.1: Exponents for relative permeability kr and relative passability ηr for the liquid

phase (l) and the gas phase (g) [36, 37]kr,l kr,g ηr,l ηr,g

Exponent n n m m

Lipinski 3 3 3 3Reed 3 3 5 5Theofanous 3 3 6 6

According to [36], commonly used values of the exponent n and m in Equation 2.51 and

Equation 2.52 for relative permeability kr,α and relative passability ηr,α are proposed by

Lipinski, Reed and Theofanous and are given in Table 2.1. Detailed discussion on the selectionof exponents for relative permeability kr,α and relative passability ηr,α functions can be found

in [36, 37] .

According to [35], Tung and Dhir (1988) accounted the interfacial drag between different

flow phases and added a new term to the Ergun equation as given below:

∇ pg = −gg − µg

S ng K vfg − g

S mg η|vfg|vfg +

F iS g

and (2.53)

∇ pl = −lg− µl

(1 − S g)nK vfl − l

(1 − S g)mη|vfl|vfl − F i

(1 − S g), (2.54)

where F i N m3

is the volumetric interfacial drag force [35].

22



Alternative to the modified Ergun equation, Barree and Conway proposed a new Forchheimer

type formulation for the description of Darcy and Forchheimer flow regime. Their approach

is discussed below.

2.5.2 Barree-Conway equation

Barree and Conway (2004) performed an experimental analysis of the non-Darcy flow through

porous media [4]. They represented the Forchheimer equation in a form similar to the Darcy

equation as given below:

∇ p = − µvf

K app, (2.55)

where vf is the Forchheimer velocity vector and K app is called apparent permeability and is

defined as below:1

K app= 1

K

1 + β Kf |vf |

µ

. (2.56)

Thus, the Forchheimer equation can be rewritten as follows:

∇ p = −µvf

K

1 + β

Kf |vf |µ

. (2.57)

Figure 2.5: Deviation of experimental data from Forchheimer linear equation [23]

As shown in Figure 2.5, the experimental data of Barree and Conway (thick blue line) did not

follow the linear apparent permeability K app (thin blue and red lines) given by Equation 2.56

for a constant Forchheimer coefficient β (slope). Thus, Barree and Conway (2004) argued

23



that the Forchheimer coefficient β and thus, the apparent permeability K app must vary with

the flow rate [4]. Barree and Conway also stated that a general model for a non-Darcy flow

can be obtained by giving up on the expectation for a constant Forchheimer coefficient β [4].

The literature from Kaviany (1991) [22] and Nield & Bejan (2006) [32] discussed in Section

4.1.3 also supports this argument.

From Equation 2.56, Barree and Conway suggested that the apparent permeability K app can

also be given as follows:

kapp = K

1 + Re, (2.58)

where the Reynolds number Re (see Equation 2.27) is evaluated based on the characteristic

length β K [m]. There exists no direct relationship for the interpretation of the Forchheimer

coefficient β and it has to be determined from the experimental data.

2.5.2.1 Barree-Conway model for single and multiphase flows

Barree and Conway developed a new Forchheimer type equation for the representation of

single and multiphase non-Darcy flows. They proposed an alternative relationship for the

apparent permeability based on the Log-Dose equation [4] as follows:

K app,α = kr,α + K − kr,α

(1 + ReF α )E , (2.59)

where, K app,α is the apparent permeability, kr,α is the relative permeability and Reα is the

Reynolds number for phase α. The exponents F and E are selected such that Equation

2.59 follows the experimental data. Most of the literature based on the Barree-Conway

approach uses F = 1 [4, 3, 39]. According to [3], the Barree-Conway equation for apparent

permeability is given as follows:

K app,α = K

kmr,α +

(1 − kmr,α)

(1 + α

|vfα

|/µα

T )E

, (2.60)

where T 1m

is called the transition constant and the minimum permeability ratio kmr,α is

the ratio of the relative permeability kr,α to the intrinsic permeability K [39]. Substituting

Equation 2.60 in Equation 2.55, we get the general Barree-Conway relationship for single

24



and multiphase non-Darcy flows:

∇ pα = − µvfα

K

kmr,α + (1−kmr,α)

(1+α|vfα |/µα T )E

, (2.61)

where ∇ pα is the pressure gradient for phase α. Substituting E = 1 and the minimum

permeability ratio kmr,α = 0 in Equation 2.61, we get the Forchheimer equation for a single

phase flow. Here, the transition constant T is related to the Forchheimer coefficient β as

T = 1βK

1m

.

Barree and Conway (2007) [3] stated that their model takes into account the Reynolds

number for each phase dependent on its intrinsic velocity (vfα) and the possibility of defining

phase Reynolds number distinguishes their work from the previous ones. Wu et al. (2011)

[39] presented a mathematical and numerical model to implement the Barree-Conway model

for multiphase non-Darcy flows and also compared the numerical results with experiments.

2.5.2.2 Barree-Conway approach for relative permeability-saturation

relationship

The relative permeability-saturation relationship for a nitrogen-water non-Darcy porous media

flow is predicted by many authors [33, 12, 3]. Barree and Conway (2007) tested nitrogen-

water non-Darcy flow through various poppants (porous matrices) at different pressure gra-dients [3]. Given below is the brief summary of their experimental work.

Barree and Conway experimentally evaluated the relative permeability-saturation relationship

for a multiphase non-Darcy flow system. The long solid matrix, initially fully saturated with

water was drained using nitrogen gas at different inlet pressures. Every single time prior to

gas injection, it was carefully ensured that the porous matrix is completely saturated with

water.

A typical relative permeability-saturation relationship for a non-Darcy gas-water flow system

is shown in Figure 2.6. Barree and Conway stated that the measured data cannot be used to

calculate the relative permeability as long as the injected gas breaks through the other end

of the porous medium. The break through point was observed to be at around 38% of the

gas saturation. Figure 2.6 shows the calculated gas and water relative permeability only after

the injected gas breaks through the other end. “Directly measured data for both gas and

25



Figure 2.6: Relative permeability-saturation relationship[3]

Figure 2.7: Corrected relative permeability-saturation relationship [3]

26



water curves is affected by the non-Darcy flow, proportional to the Reynolds number (Re).

Using individual phase saturation and velocity, the relative permeability curve of each phase

is corrected for non-Darcy effects” [3]. Figure 2.6 also shows corrected curves for non-Darcy

relative permeability-saturation relationship of gas (magenta) and water (green) phases [3].

For details on non-Darcy corrections, please refer to [3, 24].

Barree and Conway performed five tests at different injection pressures and observed that after

application of non-Darcy corrections, the relative permeability data for a phase collapses to a

consistent data set (see Figure 2.7). This data set is extrapolated beyond the measured values

in order to predict relative permeabilities for both lower and higher saturations. For additional

data and further details regarding the experiments performed by Barree and Conway, please

refer to [3].

The modified Ergun equation discussed in Section 2.5.1 is a commonly used model in nuclear

research for multiphase dryouts [36, 35, 37]. The acceptance of modified Ergun equation for

the description of a multiphase flow through a well structured porous matrix is convenient

as the intrinsic permeability (see Equation 2.45) and passability (see Equation 2.46) can be

calculated in terms of spatial parameters. However, for a complex porous matrix, the intrinsic

permeability K and passability η need to be determined using the experimental data.

On the other hand, according to [3, 39], the Barree-Conway model is valid for both Darcy

and Forchheimer flow regimes. The Barree-Conway equation is a new approach mainly used

in the petroleum industry. For this approach, even though there is no need to calculate

the Forchheimer coefficient β , mathematical regression of the experimental data is anyway

required to determine the transition constant T and the exponent E [4, 3].

The current work is restricted to single phase flow through porous medium and uses the

Forchheimer equation with Ergun interpretation of the Forchheimer coefficient (see Equation

2.31).

2.6 DuMuX

DuMuX (DUNE for multi-{phase, component, scale, physics, . . . } flow and transport in

porous media) is an open-source software for simulating the flow and transport processes

in porous media [18]. DuMuX is built on top of DUNE (Distributed and Unified Numerics

27



Environment). The main purpose of the DuMuX software is to provide a sustainable and

consistent framework for the implementation and application of model concepts, constitutive

relations, discretizations, and solvers for porous media applications. DuMuX can also be

referred to as an additional DUNE module as it inherits its functionalities from the DUNE

core modules.

DuMuX mainly consists of two different sorts of module implementations, fully-coupled

modules and decoupled modules. Fully-coupled modules describe the flow system using

a strongly coupled system of equations, which can be mass balance equation, energy balance

equation and balance equations for different phases. However, decoupled modules consist

of a pressure equation which is iteratively coupled to a saturation equation, energy balance

equations etc.

As discussed in Chapter 1, Forchheimer models for single-phase isothermal and non-isothermal

flow through porous media are implemented for numerical simulations in DuMuX for each

of the following thermodynamic assumptions:

• Isothermal single-phase Forchheimer flow.

• Non-isothermal single-phase Forchheimer flow with local thermal equilibrium.

• Non-isothermal single-phase Forchheimer flow with local thermal non-equilibrium.

Balance equations for the conservation of mass, momentum and energy for the above men-

tioned models are given in Section 2.4. Relevant numerical examples with appropriate bound-

ary conditions are set up for each model and numerical simulations are performed for the

same. Detailed description of numerical examples and comparison of numerical simulations

with the experimental results are given in Chapter 4.

28



3 ITLR Experiment

As discussed in Section 1.1, a metallic porous medium offers an effective solution in many

heating and cooling engineering applications. Motivated by this, an experimental analysis

was performed in order to understand the convective cooling behavior of a well-structured

homogeneous porous material at ITLR. The objective of this analysis is to develop a good

understanding about the convective heat transfer process and to use this knowledge to

enhance the efficiency in cooling applications.

3.1 Experimental setup

Mayer et al. [30] conducted the experimental analysis for the current work at ITLR. In order

to determine the pressure loss and the overall heat transfer, an experiment was set up as

shown in Figure 3.1. The porous matrix for this work is a uniform honeycomb like cylindrical

structure as shown in Figure 3.2. This cylindrical porous medium is held horizontally with

the help of a metallic structure. On its surface, the porous cylinder is wound with a heatingcoil along its complete length and the heating coil in turn is covered with a thick layer

of insulating material in order to minimize the heat loss. During experiments, the porous

structure is exposed to a forced convective flow.

The schematic representation of the experimental setup is shown in Figure 3.3. The experi-

mental setup consists of three major parts as follows.

Air Supply: Air at high pressure is supplied with the help of a compressor. A valve is used

to regulate the air pressure and flow rate. The mass flow is measured using a Venturi nozzle.

Test Section: The test section is a circular pipe filled with the porous medium. The porous

cylinder has a diameter, D = 30 mm and a total length, L = 295 mm. The porous cylinder

is made up of Ni-alloyed steel. Thermocouples with a diameter of 0.25 mm are embedded

29



Figure 3.1: Photograph of ITLR experimental setup [30]

Figure 3.2: Porous structure used in ITLR experiments [29]

30



Figure 3.3: Schematic representation of ITLR experimental setup [30]

into the tube wall at the inlet and the outlet of the porous cylinder. The thermocouples arealso distributed at regular intervals along the tube length in order to measure the surface

temperature of the porous structure.

Data Acquisition System: The energy flow into the specimen is monitored and recorded

by a single-phase energy meter. Pressure measurement modules with differential pressure

transducers are used to determine the pressure differences between the inlet and the outlet

of the specimen. Measurements of fluid and wall temperatures were taken using a data

acquisition unit. In the measurement procedure, the wall heat flux and the mass flow rate

are held constant.

The inlet of the horizontally placed cylindrical assembly is connected to the air supply unit

which continuously supplies air at the desired inlet pressure. The outlet is positioned such

that the air can escape directly into the atmosphere after traveling through the porous

matrix. The pressure and the temperature are monitored by the data acquisition system.

Two different experiments were conducted at ITLR for the current work and are discussed

below.

3.1.1 Isothermal experiment

In this experiment, the complete system is assumed to be isothermal. That is, the tempera-

ture at different locations in the porous domain is assumed to be the same and equal to the

atmospheric temperature. During the experiment, compressed air at high pressure is injected

through the inlet and is released into the atmosphere at the outlet. As the experiment is

31



isothermal, no heat flux is supplied to the porous structure. The velocity and pressure loss

across the cylindrical porous medium are measured.

Mayer et al. performed pore-scale computational fluid dynamics (CFD) simulations to eval-

uate a CFD model against the experimental data [30]. In the current work, the isothermalexperimental data is used for the determination of intrinsic permeability K and Forchheimer

coefficient β and to validate the REV-scale isothermal Forchheimer numerical model (see

Section 4.2.1).

3.1.2 Non-isothermal experiment

During this experiment, compressed air at high pressure is injected through the inlet and is

released into the atmosphere at the outlet. Once the flow is established through the porous

medium, a constant heat flux is applied at the surface of the cylindrical porous structure.

Upon reaching the steady-state, velocity and pressure are measured across the cylindrical

porous matrix. Using thermocouples, surface temperature is measured at different locations

along the length of the porous cylinder.

Similar to the isothermal case, Mayer et al. also performed pore-scale simulations to evaluate

a CFD model against the non-isothermal experimental data [30]. In the current work, the

experimental data is used to validate the REV-scale non-isothermal Forchheimer numerical

model (see Section 4.2.2).

3.1.3 Motivation for the current work

At ITLR, experiments are performed to analyze the heat-transfer properties of a uniform

porous medium. In addition to the experimental analysis, pore-scale CFD simulations are

also performed for the reduced domain for both isothermal and non-isothermal cases [30].

Even with the reduced domain, the complexity of flow paths and the porous structure make

it very difficult to perform a detailed pore-scale numerical investigation [30]. Moreover,

computational cost and time for the pore-scale CFD simulation has always been an issue.

Thus, for a large scale application, with the need to describe a complex porous structure and

limited computational resources, it would be practically impossible to perform a pore-scale

CFD simulation. This motivates the use of the volume averaged (REV-scale) approach for

the current work.

32



In the scope of the current work, as discussed in Section 2.6, the isothermal and non-

isothermal models are implemented as a part of DuMuX. Numerical simulations are per-

formed and an attempt is made to validate the implemented DuMuX models against the

experimental data from ITLR (see Section 4.2).

33



4 Results and Discussion

As mentioned in Chapter 1, the ultimate goal of the current work is to develop thermody-

namic models as discussed in Section 2.6 and validate them against experimental data. For

this purpose, several numerical simulations have been carried out and are discussed in this

chapter. Firstly, we look at and compare various approaches for the determination of intrinsic

permeability K and Forchheimer coefficient β in Section 4.1. Secondly, in Section 4.2, the

numerical results for an isothermal case are compared with the isothermal experimental data(see Section 4.2.1) and the numerical results for a non-isothermal case are compared with

the non-isothermal experimental data (see Section 4.2.2).

4.1 Intrinsic permeability and Forchheimer coefficient

As discussed in Section 2.4.2.2, for high velocity flow through porous media, the momentum

is described by the Forchheimer equation. In order to perform numerical simulations with the

Forchheimer model determining accurate values of intrinsic permeability K and the Forch-heimer coefficient β for the flow system becomes very crucial. The intrinsic permeability

K and the Forchheimer coefficient β for a flow system are either determined analytically

by fitting the experimental data with the Forchheimer equation or by using some standard

relationships in terms of spatial parameters (see Section 2.5.1). For the current work, both

intrinsic permeability K and Forchheimer coefficient β are determined by fitting the experi-

mental data for the isothermal case (see Table 4.1) with the Forchheimer equation given by

Equation 2.28.

Initial attempts to fit both intrinsic permeability K and Forchheimer coefficient β over the

complete range of the isothermal experimental data (see Table 4.1) using nonlinear regression

analysis with the Forchheimer equation did not lead to a physically meaningful intrinsic

permeability K . In order to overcome this issue, it was decided to use following three

approaches.

1. Determine the intrinsic permeability K by performing linear regression analysis of the

34



experimental data with the Darcy equation and use this intrinsic permeability K for

the nonlinear regression analysis with the Forchheimer equation to calculate the Forch-

heimer coefficient β (see Section 4.1.1).

2. Determine both intrinsic permeability K and Forchheimer coefficient β by performingnonlinear regression analysis for subsets of the experimental data (Re < 180) with the

Forchheimer equation (see Section 4.1.2).

3. Use intrinsic permeability K from the above approach and adapt the Forchheimer

coefficient β in order to account for high velocity (Re > 180) inertial effects [22, 32]

(see Section 4.1.3).

4.1.1 Linear regression analysis for intrinsic permeability K andnonlinear regression analysis for Forchheimer coefficient β

For this approach, the idea is to first calculate the intrinsic permeability K by performing

linear regression analysis for different subsets of the experimental data given in Table 4.1

with the Darcy equation (see Equation 2.26). The calculated intrinsic permeability K is used

for the nonlinear regression analysis of different subsets of the experimental data with the

Forchheimer equation (see Equation 2.28) in order to determine the Forchheimer coefficient

β .

4.1.1.1 Linear regression analysis for intrinsic permeability K

For the current work, the available experimental data is well beyond the Darcy range, i.e,

(Re >> 1). Thus, in order to determine the intrinsic permeability K , linear regression

analysis is performed only for the set of first three experimental data values in Table 4.1.

The Darcy system of equations for the linear regression is given below:

[−∇ pi] = 1

K [µivfi] for i = {1, 2, 3}, (4.1)

where the subscript i indicates the experiment number given in Table 4.1. The intrinsic

permeability K calculated by linear regression is given below:

K = 2.8 × 10−8 [m2]. (4.2)

35





100 200 300 400 500 600 700 800 900 1000 −12

−10

−8

−6

−4

−2

0 x 10

4

Re [−]

∇ p

[ P a / m ]

ITLR Exp

Linear Regression

Figure 4.1: Linear regression with Darcy law

The intrinsic permeability given by Equation 4.2 and the experimental data for velocity and

viscosity (see Table 4.1) are used with Equation 4.1 for the back calculation of the corre-

sponding pressure gradients. The calculated pressure gradients ∇ pCalc (black plus marks)

and the experimental pressure gradients (blue stars) are plotted against the flow Reynolds

number (Re) as shown in Figure 4.1. The Reynolds number for each experimental data is

determined using Equation 2.27 and is given in Table 4.1.

From Figure 4.1, one can clearly observe that the calculated Darcy pressure gradients (blackplus marks) immediately start diverging from the experimental pressure gradients (blue stars)

and follow an inclined line. Moreover, the experimental pressure gradient drops nonlinearly

with increasing Reynolds number (Re). From [19, 15, 22, 5, 32], one can say that, this

nonlinear drop in the pressure gradient is caused by the high velocity inertial effects. As

discussed in Section 2.4.2.2, these inertial effects are accounted for by the kinetic energy

term in the Forchheimer equation.

4.1.1.2 Nonlinear regression for Forchheimer coefficient β

In this section, the Forchheimer coefficient β is determined by performing nonlinear regression

analysis of the experimental data with the Forchheimer equation (see Equation 2.28). Here,

the intrinsic permeability K is given by Equation 4.2. The system of equations for nonlinear

Forchheimer regression is as follows:

[−∇ pi] =

µivfi iv2fi

1/K β

T for i = {1, 2, 3, . . , 17}, (4.3)

37



Table 4.2: Forchheimer coefficient β for different subsets of the experimental data

Expt. dataset i Forch. coeff. β [1/m] Ergun coeff.C E

{1, 2, 3} β 1= 1608.66 C E 1= 0.2499

{1, 2, . . , 6

} β 2= 1573.28 C E 2= 0.2444

{1, 2, . . , 9} β 3= 1757.76 C E 3= 0.2731{1, 2, . . , 17} β 4= 1902.16 C E 4= 0.2955

100 200 300 400 500 600 700 800 900 1000 −14

−12

−10

−8

−6

−4

−2

0 x 10

4

Re [−]

∇ p

[ P a / m ]

ITLR Exp Forch coeff β

1

Forch coeff β2

Forch coeff β3

Forch coeff β4

Figure 4.2: Linear Darcy regression for intrinsic permeability K and nonlinear Forch-heimer regression for Forchheimer coefficient β

where the subscript i indicates the experiment number given in Table 4.1. The nonlinear

regression analysis is performed for different subsets of the experimental data (i.e., i =

{1, 2, 3}, i = {1, 2, 3, . . , 6}, i = {1, 2, 3, . . , 9} and i = {1, 2, 3, . . , 17}).

The Forchheimer coefficients obtained for different subsets of the experimental data are given

in Table 4.2. These Forchheimer coefficients β are used along with intrinsic permeability

K given by Equation 4.2, the experimental data for velocity and viscosity in Table 4.1 and

Equation 4.3 to back calculate the corresponding pressure gradients. The calculated pressuregradients ∇ pCalc are plotted against the Reynolds number as shown in Figure 4.2. It appears

from Figure 4.2, that the calculated pressure gradients match well with experimental ones,

especially for lower Reynolds numbers (Re). However, it is clear from Table 4.3 that with

the current approach, the pressure gradients at lower Reynolds numbers are predicted poorly.

38



Table 4.3: Percentage error for the calculated pressure gradients using different Forch-heimer coefficients given in Table 4.2

∇ pExpt[Pam

] % Error∇ p(β 1) % Error∇ p(β 2) % Error∇ p(β 3) % Error∇ p(β 4)

181.660 46.27 45.00 51.64 56.84

651.760 34.27 32.52 41.66 48.812120.25 18.87 16.95 26.97 34.814993.44 13.00 10.97 21.54 29.828004.28 13.23 11.11 22.19 30.86

12263.25 9.260 7.151 18.17 26.8019448.21 4.580 2.500 13.35 21.8424294.88 2.060 3.010 10.71 19.0930284.81 3.080 1.940 8.660 16.9640987.76 7.390 2.450 8.250 16.6349150.42 8.140 5.150 5.320 13.52

58746.00 5.060 7.040 3.290 11.3868519.69 3.350 5.380 5.200 13.4878375.66 7.910 9.860 0.260 8.19088688.64 6.600 8.580 1.730 9.800

100100.93 9.300 11.23 1.180 6.670116934.34 11.08 12.98 3.090 4.640

Here, the percentage error for calculated pressure gradient ∇ pCalc is defined as follows:

%Error(∇ p): = 100 · ∇ pExpt

−∇ pCalc

∇ pExpt

, (4.4)

where ∇ pCalc and ∇ pExpt stand for the calculated and experimental pressure gradients re-

spectively.

In Section 4.1.3, the apparent permeability K app for the current approach is determined using

the intrinsic permeability K given by Equation 4.2 and the Forchheimer coefficient β 4 and

compared with the apparent permeability K app of the approach discussed in Section 4.1.2.

4.1.2 Nonlinear regression analysis for both intrinsic permeability

K and Forchheimer coefficient β

As discussed earlier in this chapter, determination of both intrinsic permeability K and Forch-

heimer coefficient β using the quadratic Forchheimer regression analysis for the complete set

of the experimental data given in Table 4.1 did not lead to a physically meaningful intrinsic

39



permeability K . Thus, nonlinear regression analysis is repeated for different subsets of the

experimental data. It is observed that the quadratic regression analysis of the experimental

data with Reynolds number Re < 180 leads to a reasonable and physically meaningful value

of the intrinsic permeability K and the Forchheimer coefficient β . The system of equations

for the nonlinear Forchheimer regression is as follows:

[−∇ pi] =

µivi iv2i

1/K β

T for i = {1, 2, 3, 4, 5, 6}, (4.5)

where the subscript i indicates the experiment number for Re < 180 given in Table 4.1.

Here, the nonlinear regression analysis is performed for different subsets of the experimental

data (i.e., i = {1, 2, 3}, i = {1, 2, 3, 4}, i = {1, 2, 3, 4, 5} and i = {1, 2, 3, 4, 5, 6}).

Table 4.4: Forchheimer coefficient β for different subsets of the experimental data

Expt. dataset i Intr. perm. K [m2] Forch. coeff. β [1/m] Ergun coeff.C E

{1, 2, 3} K 5 = 6.0687 × 10−8 β 5= 1608.66 C E 5= 0.3962{1, 2, 3, 4} K 6 = 5.7589 × 10−8 β 6= 1593.02 C E 6= 0.3822{1, 2, 3, 4, 5} K 7 = 4.1427 × 10−8 β 7= 1510.57 C E 7= 0.3074

{1, 2, 3, 4, 5, 6} K 8 = 5.7299 × 10−8 β 8= 1573.28 C E 8= 0.3766

The Intrinsic permeability K and the Forchheimer coefficient β calculated from the nonlinear

Forchheimer regression of the above mentioned experimental datasets are given in Table 4.4.The Ergun coefficient C E is calculated for each Forchheimer coefficient β using Equation

2.30. It is observed that the intrinsic permeability K for all the datasets mentioned in Table

4.4 has the same order of magnitude as obtained by Mayer et al. [30].

The intrinsic permeability K and the Forchheimer coefficient β for each dataset given in

Table 4.4 are used along with Equation 4.5 and the experimental data for density, velocity

and viscosity given in Table 4.1 for the back calculation of pressure gradients ∇ pCalc. The

calculated pressure gradients are plotted against Reynolds number (Re) as shown in Figure

4.3. The dotted line connecting the calculated pressure gradients indicate the quadratic

behavior of the Forchheimer equation for the corresponding intrinsic permeability K and

Forchheimer coefficient β pair in Table 4.4.

Table 4.5 shows the percentage error for calculated pressure gradients ∇ pCalc. Here, the

percentage error is determined using Equation 4.4. Comparing percentage errors in Table

40



100 200 300 400 500 600 700 800 900 1000 −12

−10

−8

−6

−4

−2

0 x 10

4

Re [−]

∇ p

[ P a / m ]

ITLR Exp

Forch coeff β5

Forch coeff β6

Forch coeff β7

Forch coeff β8

Figure 4.3: Nonlinear regression with Forchheimer law

4.3 and Table 4.5, it is clear that the current approach fits better over the complete range of

the experimental data, especially for low Reynolds numbers (Re < 180). Thus, from Table

4.3 and Table 4.5, the best fitted intrinsic permeability K and the Ergun coefficient C E are

chosen for numerical simulations as follows:

K = K 8 = 5.73 × 10−8m2 and (4.6)

C E = C E 8 = 0.3766. (4.7)

The numerical results are compared with the experimental data in Section 4.2.

From Figure 4.3 and Table 4.5 it is also clear that at higher Reynolds numbers (Re > 180)

the experimental data starts diverging from the current approach. Similar behavior is also

observed by [22, 5, 32] and is discussed in Section 4.1.3.

4.1.3 Apparent permeability

As discussed in Section 2.5.2, Barree and Conway [4] in their work stated that if one gives up

on the expectation of a constant Forchheimer coefficient β , any porous media flow regime can

be described using the Forchheimer equation. They represented the Forchheimer equation

by a Darcy-like equation (see Equation 2.55) , where the apparent permeability given by

41



Table 4.5: Percentage error of the calculated pressure gradients using different intrinsicpermeabilities and Forchheimer coefficients given in Table 4.4

∇ pExpt[Pam

] % Error∇ p(β 5) % Error∇ p(β 6) % Error∇ p(β 7) % Error∇ p(β 8)

181.660 6.9189 5.5958 5.8862 5.1192

651.760 1.4013 1.7924 6.6393 0.93052120.25 0.0870 0.2641 0.4089 1.26954993.44 0.4754 0.0233 1.2985 1.06388004.28 3.2408 2.6558 0.4181 1.5052

12263.25 1.3640 0.7096 2.0713 0.442219448.21 1.4409 2.1465 5.3562 3.285824294.88 3.1797 3.9012 7.2606 5.028230284.81 4.4719 5.2096 8.7130 6.328840987.76 4.1592 4.9333 8.6960 6.065649150.42 6.4367 7.2086 11.000 8.3184

58746.00 7.9362 8.7108 12.551 9.807068519.69 5.9489 6.7547 10.783 7.878678375.66 10.200 10.977 14.883 12.05288688.64 8.6869 9.4873 13.531 10.584

100100.93 11.163 11.949 15.935 11.017116934.34 12.692 13.473 17.458 12.526

Equation 2.56 can be rewritten as follows:

1

K app =

1

K + β

|vf

|µ . (4.8)

The apparent permeabilities K app are determined using the intrinsic permeability K and the

Forchheimer coefficient β for approaches discussed in Section 4.1.1 and Section 4.1.2. The

apparent permeabilities K app for experiments are calculated using Equation 2.55 and the

experimental data given in Table 4.1. Figure 4.4 shows a plot of the apparent permeability

for the approaches discussed in Section 4.1.1 and Section 4.1.2.

In Figure 4.4, slope of each data-line represents the Forchheimer coefficient β for the corre-

sponding approach. It is very clear from Figure 4.4, that the approach discussed in Section

4.1.1 under-predicts the intrinsic permeability K and thus the apparent permeability K app

over the complete range of the experimental data.

It can also be seen from Figure 4.4, that for higher flow velocities, the apparent permeability

K app for experiments (blue stars) starts behaving nonlinearly and diverges from the approach

42



0 1 2 3 4 5 6

x 105

0

2

4

6

8

10

12x 10

8

ρ.v/µ [m−1

]

1 / K

a p p

[ m − 2 ]

ITLR Exp Data

Forch Regression (K & β) (1−6)

Darcy Regession (K) & Forch Regression (β)

Figure 4.4: Apparent permeability

discussed in Section 4.1.2. Barree and Conway (2005) [5] explained such behavior by stating

that, the Forchheimer coefficient β should vary with the flow rate in order to account for

inertial nonlinearities. Kaviany (1991) and Nield & Bejan (2006) [22, 32] stated that, the

quadratic Forchheimer equation with a constant Forchheimer coefficient β is only valid up

to a specific Reynolds number (for the current work Re ≈ 180), and for higher Reynolds

numbers (Re > 180), the Forchheimer coefficient β should be adjusted in order to account

for high velocity inertial effects. In the scope of the current work, an attempt is made to

account for the inertial effects using one of the following approaches:

1. A constant Forchheimer coefficient β is defined for the complete range of the experi-

mental data (see Section 4.1.3.1).

2. The complete set of the experimental data in Table 4.1 is divided into different ranges

on the basis of the Reynolds number (Re) and a constant Forchheimer coefficient β is

determined for each range (see Section 4.1.3.2).

3. For the high velocity flow regime (Re > 180), based on the experimental data, theForchheimer coefficient β is described as a linear function of the Reynolds number (see

Section 4.1.3.3).

43



4.1.3.1 Constant Forchheimer coefficient β for complete range of experimental

data

For this approach, the intrinsic permeability K is set to a constant value given by Equation

4.6. Using calculated apparent permeabilities K app for experiments, intrinsic permeability K

and the experimental data along with Equation 4.8, a constant Forchheimer coefficient β

(slope) is determined for the complete range of the experimental data as shown (green line) in

Figure 4.5. The red line and the blue stars in Figure 4.5 describe the apparent permeabilities

K app for the nonlinear regression approach (see Section 4.1.2) and experiments respectively.

0 1 2 3 4 5 6

x 10

5

0

2

4

6

8

10

12x 10

8

ρ.v/µ [m−1

]

1 / K

a p p

[ m − 2 ]

ITLR Exp Data

Forch Regression (1−6)

Forch Coeff β Comp Range

Figure 4.5: Constant Forchheimer coefficient β for the complete range of experimentaldata

Intrinsic permeability K given by Equation 4.6, Ergun coefficient C E = 0.4904 determined

using the Forchheimer coefficient β for the current approach and Equation 2.30 are used for

numerical simulations. Pressure gradients ∇ pCalc are back calculated using Equation 2.29

and the experimental data from Table 4.1. The percentage errors for the calculated pressure

gradients are determined using Equation 4.4 and compared with other approaches in Table4.7.

4.1.3.2 Forchheimer coefficient β for limited Re ranges

For this approach, the intrinsic permeability K is set to a constant value given by Equation

4.6. Using calculated apparent permeabilities K app for the experiments, intrinsic permeability

44



0 100 200 300 400 500 600 700 1000

1500

2000

2500

Re [−]

β [ 1

/ m ]

ITLR Exp

Forch Coeff Nonlinear Regression

Forch Coeff (Re<180)

Forch Coeff (180<Re<340)


Forch Coeff (Re>475)

Figure 4.6: Forchheimer coefficient β for limited Re ranges

0 1 2 3 4 5 6

x 10 5

0

2

4

6

8

10

12 x 10 8

ρ.v/µ [m−1]

1 / K a p p [ m − 2 ]

ITLR Exp

Forch Regression

App Perm (Re<180)

App Perm (180<Re<340)

App Perm (340<Re<475)

App Perm (Re>475)

Figure 4.7: Apparent permeability K app for limited Re ranges

45



K and the experimental data along with Equation 4.8, corresponding Forchheimer coefficients

β are determined and are shown (blue stars) in Figure 4.6.

The calculated Forchheimer coefficients β are divided into different ranges on the basis of

Reynolds number (Re) and an average Forchheimer coefficient β is determined for eachReynolds number range and is shown (horizontal magenta lines) in Figure 4.6. Using the

average Forchheimer coefficients β , intrinsic permeability K and the experimental data along

with Equation 4.8, apparent permeabilities K app are back calculated for each Reynolds num-

ber range and are shown (inclined magenta lines) in Figure 4.7.

Figure 4.6 and Figure 4.7 show a plot of the Forchheimer coefficient β against the Reynolds

number (Re) and a plot of the corresponding apparent permeability K app equation (see

Equation 4.8) respectively. It can be seen from Figure 4.6 and Figure 4.7 that the Forchheimercoefficient β for nonlinear regression approach (red line) discussed in Section 4.1.2 is constant

for all flow velocities. Whereas, for the current approach, the Forchheimer coefficient β and

therefore the apparent permeability K app are adjusted in order to account for nonlinear inertial

effects at higher Reynolds numbers (Re > 180).

Table 4.6: Forchheimer coefficient β for limited Re ranges

Reynolds number (Re) Forch. coeff. β Ergun coeff. C E

Re < 180 β 9= 1573.28 C E 9= 0.3766180 < Re < 340 β 10= 1615.25 C E 10= 0.3866340 < Re < 475 β 11= 1709.87 C E 11= 0.4093Re > 475 β 12= 1772.92 C E 12= 0.4244

The Forchheimer coefficient β obtained for the different ranges of Reynolds numbers are given

in Table 4.6. For each Reynolds number range, the Ergun coefficient C E is calculated using

Equation 2.30 and the corresponding Forchheimer coefficient β . The intrinsic permeability

K given by Equation 4.6 and the Ergun coefficients C E for different Reynolds number ranges

from Table 4.6 are used for numerical simulations. Pressure gradients ∇ pCalc for the current

approach are back calculated using Equation 2.29 and the experimental data from Table 4.1.

The percentage errors for the calculated pressure gradients are determined using Equation

4.4 and compared with other approaches in Table 4.7.

46



0 100 200 300 400 500 600 700 1000

1500

2000

2500

Re [−]

β [ 1

/ m ]

ITLR Exp Data



Forch Coeff Fit (180<Re<340)

Forch Coeff Fit (340<Re<475)

Forch Coeff Fit (Re>475)

Forch Coeff f(Re)

Figure 4.8: Forchheimer coefficient β as a function of Reynolds number (Re)

4.1.3.3 Linear Forchheimer coefficient β (Re)

Similar to Section 4.1.3.1 and Section 4.1.3.2, for this approach, the intrinsic permeability K

is set to a constant value given by Equation 4.6. As discussed in Section 4.1.3.2, Forchheimer

coefficients β for experiments (blue stars), nonlinear Forchheimer regression approach (red

line) and Forchheimer coefficient β for limited Re ranges approach (magenta line) are shown

in Figure 4.6. It is clear from Figure 4.6, that for Re > 180, the Forchheimer coefficient β

for experiments varies linearly with the flow rate. Thus, as suggested by [22, 32], for the high

velocity flow regime (Re > 180), a linear relationship between the Forchheimer coefficient β and the flow Reynolds number (Re) is determined (see Figure 4.8) as follows:

β (Re) = 0.6364 Re + 1560.3. (4.9)

It can be seen from Figure 4.8, that the linear description of the Forchheimer coefficient

β (cyan line) accounts for nonlinear inertial effects at higher flow velocities ( Re > 180).

Equation 4.9 for the Forchheimer coefficient β allows the cubic behavior of the Forchheimer

equation and follows the experimental data for Re > 180. A cubic form of the Forchheimer

equation is given below:

∇ p = µvf

K

1 + β (Re(|vf |))

K|vf |µ

. (4.10)

47



0 1 2 3 4 5 6

x 10 5

0

2

4

6

8

10

12 x 10

8

ρ.v/µ [m−1]

1 / K

a p p

[ m − 2 ]

ITLR Exp Data





Forch Coeff (Re>475)

Forch Coeff f(Re)

Figure 4.9: Apparent permeability β as a function of velocity

Using Equation 4.8, the intrinsic permeability K given by Equation 4.6, the Forchheimer

coefficient β given by Equation 4.9 and the experimental data (see Table 4.1), the appar-

ent permeabilities K app are calculated for the high velocity flow regime (Re > 180). The

calculated apparent permeabilities (cyan line) are plotted as shown in Figure 4.9.

Pressure gradients ∇ pCalc for the current approach are back calculated using Equation 2.29

and the experimental data from Table 4.1. The percentage errors for the calculated pressure

gradients are determined using Equation 4.4 and compared with other approaches in Table

4.7. From Figure 4.9 and Table 4.7, it is very clear that the current approach describes the

experimental pressure gradients better than any other approach discussed in Section 4.1.1,

Section 4.1.2 and Section 4.1.3.

The Ergun coefficient C E is determined from Equation 2.30 and Equation 4.9 as follows:

C E =√

K (0.6364 Re + 1560.3). (4.11)

For the current approach, a constant Ergun coefficient C E = 0.3766 (see Equation 4.7) for

Re < 180 and a linear Ergun coefficient C E (see Equation 4.11) for Re > 180 are used along

with intrinsic permeability K given by Equation 4.6 for numerical simulations.

48



T a b l e 4 . 7 : C o m p a r i s o n o f p e r c e n t a g e e r r o r s o f c a l c u

l a t e d p r e s s u r e g r a d i e n t s u s i n

g d i ff e r e n t a p p r o a c h e s f o r F o r c h h e i m e r c o e f -

fi c i e n t β

∇ p E x p t

[ P a

m

]

N o n l i n e a r r e g r e s s i o n

∇ p

( β c o m p

)

∇ p

( β r a n g e s

)

∇ p

( β ( R e

) )

∇ p

( β 5

)

∇ p

( β 6

)

∇ p

( β 7

)

∇ p

( β 8

)

1 8 1 . 6

6 0

6 . 9

1 8 9

5 . 5

9 5 8

5 . 8

8 6 2

5 . 1

1 9 2

1 0 . 8

3 0

5 . 1

1 9 2

5 . 1

1 9 2

6 5 1 . 7

6 0

1 . 4

0 1 3

1 . 7

9 2 4

6 . 6

3 9 3

0 . 9

3 0 5

2 . 4

2 8 6

0 . 9

3 0 5

0 . 9

3 0 5

2 1 2 0 . 2

5

0 . 0

8 7 0

0 . 2

6 4 1

0 . 4

0 8 9

1 . 2

6 9 5

3 . 7

5 6 9

1 . 2

6 9 5

1 . 2

6 9 5

4 9 9 3 . 4

4

0 . 4

7 5 4

0 . 0

2 3 3

1 . 2

9 8 5

1 . 0

6 3 8

5 . 7

2 0 5

1 . 0

6 3 8

1 . 0

6 3 8

8 0 0 4 . 2

8

3 . 2

4 0 8

2 . 6

5 5 8

0 . 4

1 8 1

1 . 5

0 5 2

9 . 2

3 7 2

1 . 5

0 5 2

1 . 5

0 5 2

1 2 2 6 3

. 2 5

1 . 3

6 4 0

0 . 7

0 9 6

2 . 0

7 1 3

0 . 4

4 2 2

7 . 6

5 4 5

0 . 4

4 2 2

0 . 2

1 0 5

1 9 4 4 8

. 2 1

1 . 4

4 0 9

2 . 1

4 6 5

5 . 3

5 6 2

3 . 2

8 5 8

5 . 0

2 4 6

0 . 8

2 0 2

0 . 8

8 6 7

2 4 2 9 4

. 8 8

3 . 1

7 9 7

3 . 9

0 1 2

7 . 2

6 0 6

5 . 0

2 8 2

3 . 3

1 4 1

2 . 5

9 6 3

1 . 6

7 3 2

3 0 2 8 4

. 8 1

4 . 4

7 1 9

5 . 2

0 9 6

8 . 7

1 3 0

6 . 3

2 8 8

2 . 0

6 5 5

3 . 9

1 8 7

1 . 8

8 4 3

4 0 9 8 7

. 7 6

4 . 1

5 9 2

4 . 9

3 3 3

8 . 6

9 6 0

6 . 0

6 5 6

2 . 5

7 0 6

3 . 6

3 3 7

0 . 3

6 8 6

4 9 1 5 0

. 4 2

6 . 4

3 6 7

7 . 2

0 8 6

1 1 . 0

0 0

8 . 3

1 8 4

0 . 2

1 4 6

0 . 5

6 1 5

0 . 8

5 7 2

5 8 7 4 6

. 0 0

7 . 9

3 6 2

8 . 7

1 0 8

1 2 . 5

5 1

9 . 8

0 7 0

1 . 3

1 5 4

2 . 1

5 4 2

1 . 1

3 3 1

6 8 5 1 9

. 6 9

5 . 9

4 8 9

6 . 7

5 4 7

1 0 . 7

8 3

7 . 8

7 8 6

0 . 8

8 7 4

0 . 0

4 1 8

2 . 5

8 7 3

7 8 3 7 5

. 6 6

1 0 . 2

0 0

1 0 . 9

7 7

1 4 . 8

8 3

1 2 . 0

5 2

3 . 6

3 1 7

3 . 5

5 8 5

1 . 1

2 0 5

8 8 6 8 8

. 6 4

8 . 6

8 6 9

9 . 4

8 7 3

1 3 . 5

3 1

1 0 . 5

8 4

1 . 9

5 8 3

0 . 5

7 5 6

2 . 0

2 9 7

1 0 0 1 0 0 . 9

3

1 1 . 1

6 3

1 1 . 9

4 9

1 5 . 9

3 5

1 1 . 0

1 7

4 . 5

8 1 1

2 . 1

4 6 9

0 . 4

0 0 2

1 1 6 9 3 4 . 3

4

1 2 . 6

9 2

1 3 . 4

7 3

1 7 . 4

5 8

1 2 . 5

2 6

6 . 1

7 7 3

3 . 8

2 4 8

0 . 4

3 7 9

49



4.2 Test cases

After analyzing the experimental data for the determination of the intrinsic permeability K

and the Forchheimer coefficient β , the immediate objective is the validation of implemented

DuMuX

models by setting up a relevant numerical example for each model. Thus, a test casewith appropriate boundary conditions is setup for an isothermal model (see Section 4.2.1)

and a non-isothermal model (see Section 4.2.2).

4.2.1 Incompressible isothermal case

During incompressible isothermal experiments at ITLR, different pressure gradients are ap-

plied across the porous cylinder and the corresponding flow velocities are recorded (see Table

4.1). These pressure gradients are used as boundary conditions for numerical simulations

with the corresponding DuMuX

model and numerical results are compared with the experi-mental data. Nitrogen is used as a working fluid for the numerical simulations in the current

work. The incompressible behavior of the fluid is ensured by fixing the density and viscosity

of the fluid phase as given in Table 4.1.

A set of simulations is performed using the intrinsic permeability K given by Equation 4.6

and the Ergun coefficients C E for different Forchheimer coefficient β approaches discussed

in Section 4.1.2 and Section 4.1.3 (see Table 4.8).

Table 4.8: Ergun coefficient C E for different Forchheimer coefficient β approaches

Approach for Forchheimer coefficient β Ergun Coefficient C E

Nonlinear Forchheimer regression 0.3766Forchheimer coefficient β for complete data range 0.4904

Forchheimer coefficient β for limited Re ranges

0.3766 (Re < 180)0.3866 (180 < Re < 340)0.4093 (340 < Re < 475)0.4244(Re > 475)

Forchheimer coefficient β = β (Re) 0.3766 (Re < 180)√ K (0.6364 Re + 1560.3) (Re > 180)

50



Model domain:

Figure 4.10: Incompressible isothermal model domain

For simulating incompressible isothermal flow, due to the symmetric shape of the porous

domain and the unidirectional flow velocity, only one quarter of the cylinder is modeled. For

this domain, a uniform mesh with hexahedral elements is created using IcemCFD 12.1 as the

meshing tool. The dimensions of the mesh are the same as that of the porous domain, i.e.,

length (L) = 295 mm and diameter (d) = 30 mm. The model domain is shown in Figure

4.10. Boundary conditions, input data and results for the incompressible isothermal case are

discussed below.

Boundary conditions:

For the incompressible isothermal case, the flow is only governed by the pressure difference

applied across the cylinder. Thus, a Dirichlet boundary condition for pressure is applied at the

inlet and the outlet of the domain. The Dirichlet boundary condition at the inlet is calculated

from the pressure gradient data provided by ITLR. Whereas, the Dirichlet boundary condition

at the outlet is set to the atmospheric pressure (100000 P a). The other boundaries are set

to no flow (Neumann) boundary condition. As the problem is isothermal, the temperature

throughout the domain is set to a constant atmospheric temperature, i.e., 300 K .

Input data:

• The intrinsic permeability K for the solid matrix is obtained earlier in this chapter as

K = 5.73 × 10−8 m2 (see Equation 4.6).

51





Table 4.9: Experimental and numerical velocity data

∇ pExpt [Pam

] vf Expt.

ms

vf

ms

(β 8) vf

ms

(β comp) vf

ms

(β ranges) vf

ms

(β (Re))

181.660 0.2404 0.2423 0.2571 0.2423 0.2571651.760 0.5337 0.5283 0.5269 0.5283 0.5432

2120.25 1.0059 1.0073 0.9869 1.0072 1.03144993.44 1.5788 1.5785 1.5694 1.5785 1.60538004.28 2.0372 2.0096 1.9477 2.0096 2.0315

12263.25 2.4992 2.4899 2.4687 2.4899 2.500919448.21 3.0885 3.1228 3.0135 3.1016 3.109524294.88 3.4055 3.4751 3.3507 3.4515 3.443430284.81 3.7527 3.8559 3.7152 3.7239 3.799640987.76 4.3199 4.4318 4.2662 4.2798 4.325649150.42 4.6284 4.8068 4.6248 4.6416 4.664758746.00 4.9618 5.1954 4.9965 5.0167 5.0094

68519.69 5.3539 5.5486 5.3319 5.3551 5.307378375.66 5.5293 5.8629 5.6348 5.6009 5.584988688.64 5.8606 6.1622 5.9210 5.8437 5.8287

100100.93 6.0607 6.4615 6.2071 6.1272 6.0783116934.34 6.3725 6.8534 6.5819 6.4987 6.3922

Figure 4.12b, that for an incompressible isothermal flow, the pressure is linearly distributed

and the flow velocity is constant along the axis of the porous cylinder.

The numerical data for velocity is collected for the different sets of simulations performed us-

ing Ergun coefficients C E for different approaches mentioned in Table 4.8. The experimental

data for the pressure gradient and the corresponding experimental and numerical data for

velocity are given in Table 4.9.

The experimental and the numerical velocity data presented in Table 4.9 is used for the

evaluation of the friction coefficient f . The friction coefficient of a porous medium is defined

in terms of porosity φ and specific interfacial area sv as follows [30]:

f = ∇ pφ3

f |vf |2sv. (4.12)

Table 4.10 shows calculated friction coefficients and corresponding percentage errors belong-

ing to the experimental and numerical velocities given in Table 4.9. Here, the percentage

53





10 100 1000 100000.1

1

Re [−]

f [ − ]

ITLR Exp


Forch Coeff Comp Expt Data

Forch Coeff Re Ranges

Forch Coeff Funct of Re

ITLR Num

Figure 4.13: Friction coefficient

error of the friction coefficient f is defined as follows:

% Error (f ) := 100 ·

f Expt − f Num

f Expt

, (4.13)

where f Expt and f Num are the experimental and numerical friction coefficients respectively.

Figure 4.13 shows a plot of the friction coefficient f against the Reynolds number (Re).From Figure 4.13, Table 4.9 and Table 4.10, it is clear that, all of the approaches, viz.,

Quadratic Forchheimer regression (see Section 4.1.2), Forchheimer coefficient β for the

complete range of experimental data (see Section 4.1.3.1), Forchheimer coefficient β for

limited Re ranges (see Section 4.1.3.2) and Forchheimer coefficient as a linear function of

Reynolds number (β = β (Re)) (see Section 4.1.3.3) fit with the experimental data within

an acceptable tolerance. However, the approach discussed in Section 4.1.3.3 gives the best

fit with a maximum deviation of 2 − 3% from the experimental data.

It is natural to assume that after using properly fitted intrinsic permeability K and Forch-

heimer parameter β , the numerical results should go well with experiments. However, one

needs to take into account that the determination of a proper intrinsic permeability K and

Forchheimer parameter β was one of the most challenging tasks in the current work. The

Forchheimer coefficient β is a function of both the porous media structure and the flow

regime, which makes it problem-specific [32]. Thus, it was very important to cross check the

55



selected intrinsic permeability K and Forchheimer coefficient β by validating the numerical

results against the experimental data.

It was observed that the DuMuX simulations for an incompressible isothermal flow through

a porous medium takes considerably less time compared to the pore-scale CFD simulationsperformed at ITLR. Moreover, the pore-scale CFD simulations are performed for a small

section of the experimental porous domain [30]. DuMuX takes approximately 150 sec of

CPU time for a simulation of an isothermal case using the macro-scale approach. Whereas,

the pore-scale CFD simulation performed by Mayer et al. (2010) [30] requires a comparatively

larger computation time. Thus, one can say that, in addition to better numerical results (see

Figure 4.13), the macro-scale approach for an isothermal flow system is very advantageous

in terms of computational cost and time, especially for large scale problems.

4.2.2 Non-isothermal case

4.2.2.1 With local thermal equilibrium

For heating and/or cooling applications it is very important to analyze the porous structure for

its heat-transfer properties. Experimental analysis of heat transfer properties of a cylindrical

porous structure is performed at ITLR. The experimental data from ITLR is used as the

base, and an attempt is made to validate a non-isothermal DuMuX model with local thermal

equilibrium against this experimental data. Nitrogen is used as a working fluid for simulations

in the current work. The model domain, boundary conditions and results for the non-

isothermal case are discussed below.

Model domain:

Figure 4.14: Model domain non-isothermal case

56



Unlike in the isothermal case, the complete cylindrical domain is modeled for the non-

isothermal case. The model domain was extended to a length L = 400 mm whereas, the

diameter is maintained the same as in the isothermal case i.e d = 30 mm. The extended

domain is used to simulate the atmosphere, i.e., the model domain consists of 295 mm of

porous matrix and 105 mm of atmosphere (atmosphere domain) as shown in Figure 4.14 bywhite and red colors respectively. The reasons for modeling the complete cylinder and using

an extended domain are explained below in “challenges”. A uniform mesh with hexahedral

elements is created using IcemCFD 12.1 as a meshing tool for this cylindrical model domain.

Boundary conditions:

Table 4.11: Boundary conditions non-isothermal caseBoundary condition Value

Inlet pressure 101131.443 P aOutlet pressure 100000.000 P aSurface heat flux 22059.088 W/m2

Inlet temperature 299.749 K Outlet temperature 299.749 K

For the non-isothermal case, a Dirichlet boundary condition is applied for pressure and tem-

perature at the inlet and the outlet of the domain (see Table 4.11). A constant heat flux(Neumann) boundary condition is applied only at the surface of the porous matrix. Here,

the outlet boundary condition refers to the outlet of the model domain and not the porous

domain. Only one set of the experimental data as shown in Table 4.11 is available for the

non-isothermal model validation.

Table 4.12: Material parameters and input data

Parameters Value Source

λa 0.028 W/m K Expert’s opinion

λs 15 W/m K ITLRc ps 510 J/kg K Expert’s opinion 7900 kg/m3 ITLRφ 0.558 ITLRK 5.73 × 10−8m2 Current workβ 0.3866 Current work

57



Input data:

Apart from boundary conditions, other additional input data and material parameters are

needed for the simulations as shown in Table 4.12. This data was either provided by or

discussed with an expert from ITLR.

Some challenges:

Extension of the model domain and addition of the atmospheric part to the model domain

is a modeling trick. This trick is used in order to be able to simulate the flow at the outlet

of the porous cylinder. Before using the extended domain, an attempt was also made to use

the existing outflow boundary condition available in DuMuX. However, it was found that

this boundary condition fails for energy closure with the Forchheimer approach.

Use of the extended domain for the atmospheric part also produced a need to describe

parameters such as intrinsic permeability K and porosity φ for this domain. The intrinsic

permeability and the porosity values of the extended domain are K = 1 × 10−5m2 and

φ = 1 respectively. With the usage of permeabilities orders of magnitude different for the

porous medium and the atmospheric part, an unphysical jump in the flow properties viz.,

velocity, pressure and temperature was observed at the permeability junction during DuMuX

simulations. This jump existed due to an averaging problem while writing the output at the

permeability junction. This problem was countered by using the Piola transformation for

evaluation of velocity fluxes while writing the output.

Figure 4.15: Unphisical heating along edges

Initially, only one quarter of the cylinder was modeled for the non-isothermal case (see Figure

4.15). However it was observed that, when the heat flux is applied at the surface of the

58



porous matrix, the long surface edges of the domain heat up more than the rest of the

porous medium, which is unphysical. Hence, it was decided to model the entire cylindrical

domain.

One of the major problems with the non-isothermal model was its slow convergence. It wasobserved that the DuMuX model converges very slowly with the given boundary conditions.

Thus, once a linear pressure field is developed across the porous medium, the heat flux

boundary condition is switched on, i.e., the heat flux is switched on at time t = 100 s. This

helped to achieve faster convergence.

Results and discussion:

Typical distributions of the pressure and the temperature across the porous cylinder for a non-

isothermal case is as shown in Figure 4.16 and Figure 4.17 respectively. It can be observed

from Figure 4.16b, that unlike in the incompressible isothermal case, in the non-isothermal

case, the pressure distribution is nonlinear over the solid matrix. This nonlinearity in the

pressure distribution is due to the compressible behavior of the fluid (i.e Nitrogen) and the

temperature-dependency of the fluid properties.

It can also be seen from Figure 4.16b, that the pressure-drop across the atmospheric part of

the model domain is very small. This low value of the pressure drop across the atmospheric

part of the model domain is due to very high permeability and porosity in this part. In order

to compensate for this pressure drop across the atmospheric part, the Dirichlet boundary

condition for the inlet pressure is adjusted in such a way that the pressure drop across the

porous cylinder matches with that of the experiment.

Figure 4.17b shows a plot of the distribution of temperature along the length of the model

domain. From Figure 4.17b, it is observed that the slope of the bulk temperature (brown

line) and the slope of the surface temperature (black line) are equal. It is very important

to know that this behavior of the slopes of the bulk and the surface temperatures is alsoobserved for the experimental data at ITLR.

Figure 4.18 shows the distribution of the velocity and density over the length of the porous

medium. From Figure 4.18 and Figure 4.17b, it is observed that the velocity of the fluid

within the porous matrix increases with increase in temperature and the resulting decrease

in fluid density.

59



(a) Qualitative pressure distribution (b) Quantitative pressure distribution

Figure 4.16: Pressure distribution across porous domain for a compressible non-isothermal flow

(a) Qualitative temperature distribution (b) Quantitative temperature distribution

Figure 4.17: Temperature distribution across porous domain for a compressible non-isothermal flow

(a) Velocity distribution (b) Density distribution

Figure 4.18: Velocity and density distribution across porous domain for a compressiblenon-isothermal flow

60



(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.19: Evolution of pressure (non-isothermal model with local thermal equilib-rium)

61



(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.20: Evolution of temperature (non-isothermal model with local thermal equi-librium)

62



(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.21: Evolution of velocity (non-isothermal model with local thermal equilibrium)

63



Table 4.13: Experimental and numerical wall temperatures (non-isothermal model withlocal thermal equilibrium)

Dist. from inlet [mm] Expt. temp T Exp [K ] Num. temp T Num [K ]

0 299.75 299.75

43 439.24 452.0568 454.72 526.9684 464.63 573.83

105 477.63 634.01203 538.32 892.92235 558.15 968.39295 595.29 1061.49

We have already discussed that for a non-isothermal model, the heat flux boundary condition

for the surface of the cylinder is switched on at time t = 100 s, i.e., once a linear pressurefield is developed across the porous medium. In Figure 4.19, Figure 4.20 and Figure 4.21,

this can be observed. It can also be inferred, that the fluid behaves incompressibly as long

as the heat flux boundary condition is switched off. This incompressible behavior might also

exist due to the low pressure difference applied across the porous cylinder which might not

be the case for higher pressure differences. However, in the current work, no conclusion can

be made in this regard due to limited experimental data.

For the non-isothermal model with local thermal equilibrium, comparison of numerical and

experimental wall temperatures along the length of the porous cylinder is given in Table

4.13. We have already discussed from Figure 4.17b that for the current problem, the slope

of the bulk temperature is equal to the slope of the surface temperature, as expected from

the experimental observations at ITLR. However, from Table 4.13, it is very clear that the

temperatures at the surface of the porous cylinder are considerably over-predicted by the

current model.

There could be various reasons for this over-prediction of the surface temperature. Some

of the possible reasons are: unavailability of exact thermodynamic properties of the porousmaterial and the fluid (see Table 4.12) and selection of nitrogen as fluid phase for numerical

simulations instead of air (used for experiments at ITLR). However, the most probable reason

is the selection of an improper thermodynamic model. As discussed in Section 2.4.3.1, the

assumption of local thermal equilibrium is very well suited for the Darcy flow regime. As the

flow velocity for the ITLR experiments is beyond the Darcy flow regime (see Table 4.1), it

64



is most likely that the phases are not in contact for a sufficient period of time and no local

thermal equilibrium exists within the system (see Section 2.4.3.2).

Thus, it is concluded that the assumption of local thermal equilibrium is not appropriate to

numerically simulate the non-isothermal experiments performed at ITLR. In order to simulatethese experiments, local thermal non-equilibrium is assumed (see Section 4.2.2.2).

4.2.2.2 With local thermal non-equilibrium

As discussed in Section 4.2.2.1, the possibility of existence of local thermal non-equilibrium

motivates the use of a non-isothermal model with local thermal non-equilibrium for the cur-

rent work. The implementation of non-isothermal model with local thermal non-equilibrium

in DuMuX is discussed in Section 2.6. Similar to the model discussed in Section 4.2.2.1, an

attempt is made to validate the current model against the experimental data. The modeldomain, boundary conditions and input data for the current numerical problem are same as

that of the problem for non-isothermal model with local thermal equilibrium. The numerical

results for this model are discussed below.

Results and discussion:

Figure 4.22 shows the evolution of the temperature field for the fluid (black) and the solid

(magenta) phases. The plots for the evolution of the pressure and velocity fields for thiscase are not presented here as they are similar to that of the non-isothermal case with local

thermal equilibrium (see Figure 4.19 and Figure 4.21). It can be clearly observed from Figure

4.22, that the wall heat flux boundary condition for the porous domain is switched on after

time t = 100 s, (i.e., once a linear pressure field is developed across the porous medium).

For the non-isothermal model with local thermal non-equilibrium, comparison of experimental

and numerical temperatures of the fluid phase, along the length of the model domain surface

is given in Table 4.14. Comparing Table 4.13 and Table 4.14, it is very clear that the

assumption of local thermal non-equilibrium has definitely produced better numerical results.

However, it can also be observed from Table 4.14, that even with the assumption of local

thermal non-equilibrium, the numerical fluid temperatures deviate from the experiments in

the range of 8 − 20%.

65



(a) 0.0 s (b) 100.0 s

(c) 200.0 s (d) 300.0 s

(e) 400.0 s (f) 500.0 s

(g) 600.0 s (h) 700.0 s

Figure 4.22: Evolution of temperature (non-isothermal model with local thermal non-equilibrium)

66





5 Conclusion

In this work, the intrinsic permeability K and the Forchheimer coefficients β are determined

by performing nonlinear regression analysis of the experimental data from ITLR in Section

4.1.1, Section 4.1.2 and Section 4.1.3. From the numerical results discussed in Section 4.2.1,

it is concluded that an appropriate intrinsic permeability K (see Equation 4.6) is selected and

depending on the desired precision, different approaches can be used for the determination

of the Forchheimer coefficient β .

From the Forchheimer equation (see Equation 2.28), we know that the inertial effects are

directly proportional to the flow kinetic energy. From the detailed analysis of the experimental

data, it is observed that, in order to imitate the inertial effects, one has to given up on the

expectation of a constant Forchheimer coefficient β . Various approaches to determine the

Forchheimer coefficient β for different flow regimes are explained in Section 4.1.3.

As discussed in Section 4.2.1, fitting the Forchheimer coefficient β to the experimental data

produced better numerical results. However, before proceeding to the non-isothermal models,

it was extremely important to get the momentum balance right. From Section 4.2.1, it is

very clear that the Forchheimer coefficient β approach discussed in Section 4.1.3.3 precisely

accounts for nonlinear inertial effects, especially in the high velocity flow regime (Re > 180)

and provides the best fit to the experimental data with a maximum deviation of 2 − 3%.

Thus, it is concluded that the REV-scale approach provides an effective, efficient and eco-

nomical solution for the numerical modeling of fast isothermal single-phase flow through

porous media, especially for large scale applications.

From the comparison of non-isothermal numerical results with the corresponding experimental

data, it is concluded that the assumption of local thermal equilibrium is not appropriate as it

fails to accurately describe the non-isothermal experimental data (see Section 4.2.2.1). The

assumption of local thermal non-equilibrium definitely describes the experimental results more

accurately (see Section 4.2.2.2). Also, with this assumption, the numerical results deviate

68



from the experiments in a reasonable range of 8 − 20%. However, for further improvement

of results, radial mesh grading (with finer mesh at the surface of the cylinder), modification

of the outflow boundary condition in DuMuX, use of air as the fluid phase and a precise

Nusselt number (Nu) correlation for the exchange of energy between different phases are

recommended.

69



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