water vapor enhancement factor due to temperature gradient...

明治大学大学院農学研究科

2016 年度

博士学位請求論文

Water vapor enhancement factor due to temperature

gradient in unsaturated soils

不飽和土壌中の温度勾配による水蒸気促進係数

学位請求者農学専攻

サニー・ゴー・エン・ギアップ

Ph.D. Dissertation

March 2017

Water vapor enhancement factor due to temperature

gradient in unsaturated soils

Sunny Goh Eng Giap

Agriculture Program

Graduate School of Agriculture

Meiji University

i

TABLE OF CONTENTS

List of Figures ................................................................................................................................ iv

List of Tables ................................................................................................................................. vii

Chapter 1. General introduction ................................................................................................. 1

Chapter 2. Sensitivity analysis and validation for numerical simulation of water infiltration into unsaturated soil ........................................................................................ 5

2.1 Abstract ............................................................................................................................ 5

2.2 Introduction ...................................................................................................................... 6

2.3 Materials and methods ..................................................................................................... 9

2.3.1 The governing equation of water flow in unsaturated soil,

and its numerical solution ........................................................................................................ 9

2.3.2 Constitutive functions of matric pressure head ( m ) and

hydraulic conductivity ( K ) ................................................................................................... 10

2.3.3 Numerical experiment and the default setting of input

parameters of the flow problem ............................................................................................. 11

2.3.4 Statistical measures ................................................................................................. 11

2.4 Results and discussion .................................................................................................... 11

2.4.1 Simulation results and its accuracy ......................................................................... 11

2.4.2 Sensitivity analysis and simulation model validation ............................................. 12

2.5 Conclusions .................................................................................................................... 15

Chapter 3. Liquid water, water vapor, and heat flow simulation in unsaturated soils ............................................................................................................................ 24

3.1 Abstract .......................................................................................................................... 24

3.2 Introduction .................................................................................................................... 25

3.3 Materials and methods ................................................................................................... 28

3.3.1 The governing equation of water, vapor, and heat transport

in unsaturated soil, and its numerical solution ...................................................................... 28

3.3.2 Input parameters, initial and boundary conditions .................................................. 32

3.3.3 Statistical measure for parameter calibration .......................................................... 33

3.4 Results and discussion .................................................................................................... 34

3.4.1 The comparison of numerical simulation and Heitman data .................................. 34

ii

3.4.2 The uncertainty of hydraulic conductivity and vapor

enhancement factor ................................................................................................................ 36

3.4.3 Limitation and possible improvement .................................................................... 39

3.5 Conclusions .................................................................................................................... 41

Chapter 4. An improved heat flux theory and mathematical equation to estimate water vapor volume expansion-advection as an alternative to mechanistic enhancement factor ................................................................................................... 47

4.1 Abstract .......................................................................................................................... 47

4.2 Introduction .................................................................................................................... 48


4.3.1 Cass experimental setup, parameter estimation, relation to

the present study and limitation ............................................................................................. 50

4.3.2 Theory of heat flux.................................................................................................. 52

4.3.3 The mass balance relation of liquid water and water vapor,

and its effect on water vapor volume expansion-advection .................................................. 55

4.3.4 The presence of volume expansion-advection from the

perspective of ideal gas law ................................................................................................... 56

4.3.5 The presence of air and water vapor volume expansion-

advection................................................................................................................................ 57

4.3.6 Water vapor volume expansion-advection equation ............................................... 58

4.3.7 The relation of mechanistic enhancement factor and water

vapor volume expansion-advection ....................................................................................... 59

4.4 Results and Discussion ................................................................................................... 61

4.4.1 The general trend of thermal conductivities ........................................................... 61

4.4.2 The wet porous media thermal conductivity by heat

conduction, and the sensible heat thermal conductivity by liquid water

movement .............................................................................................................................. 62

4.4.3 The thermal conductivity of water vapor diffusion and

volume expansion-advection ................................................................................................. 63

4.4.4 The water vapor thermal conductivity by latent heat and

sensible heat........................................................................................................................... 64

4.4.5 The water vapor thermal conductivity of the partial

derivative of matric pressure head effects on relative humidity with

respect to temperature ............................................................................................................ 65

iii

4.4.6 Water vapor permeability variables with respect to soil

moisture content and air pressure .......................................................................................... 65

4.4.7 Limitation of previous work, and the possibility of

improvement .......................................................................................................................... 67

4.5 Conclusions .................................................................................................................... 68

Chapter 5. The extended Darcy law and water vapor relative permeability derivation in unsaturated soils ................................................................................. 75

5.1 Abstract .......................................................................................................................... 75

5.2 Introduction .................................................................................................................... 75


5.3.1 The derivation of extended Darcy law, and its relative

permeability and the Klinkenberg effect ............................................................................... 78

5.3.2 The integral form of relative permeability of liquid water

and water vapor ..................................................................................................................... 82

5.3.3 The thermal conductivity governed by Fick’s law and Darcy

law in heat flux equation ....................................................................................................... 87

5.4 Results and Discussion ................................................................................................... 89

5.4.1 Replacing Fick’s law with Darcy law ..................................................................... 89

5.4.2 The water vapor relative permeability, and the Klinkenberg

permeability ........................................................................................................................... 91

5.4.3 Using Darcy law in the thermal water vapor diffusivities ...................................... 92

5.4.4 Limitation, and potential area of improvement....................................................... 93

5.5 Conclusions .................................................................................................................... 94

Chapter 6. Summary and Conclusions ................................................................................... 102

References ................................................................................................................................... 107

iv

List of Figures

Figure 2.1. Comparison of simulated results with Philip's semi-analytical solution. Note: Philip(H) and Philip(K) were from Haverkamp et al. (1977) and Kabala and Milly (1990), respectively. ................................................................. 17

Figure 2.2. The rank of sensitivity coefficient. Note: θ_s and θ_r are the saturated and residual volumetric water content; ∆z, spatial spacing size; ∆t, time-step size; K_s, saturated hydraulic conductivity; θ_L(initial cond.), clay medium initial value of volumetric water content; θ_L(upper bound.), upper boundary of volumetric water content; A, B, β and α are the fitting parameters from Haverkamp, as in Eqs. 2.10 and 2.11. ................................................................................................................ 18

Figure 2.3. The cumulative effect of input parameters, on the absolute residual error at simulation time 105 s. Note: θ_L(initial cond.) (step 2), clay medium initial value of volumetric water content; θ_r (step 3), residual volumetric water content; β (step 4), α (step 5), A (step 7) and B (step 8) are the fitting parameters; K_s (step 6), saturated hydraulic conductivity; ∆t (step 9), time-step size; and ∆z (step 10), spatial spacing size. .......................................................................................................... 19

Figure 2.4. The effect of ∆z (step 10 alone), and cumulative effects of steps 2 to 9 and 2-10 in comparison with Philip(H) and Philip(K).......................................... 20

Figure 3.1. Silt loam (sil-20) simulated: (a) water content and (c) temperature distributions at temperature boundary of 15/-50, 15/-150 and 30/-150, which were 17.5-12.5, 22.5-7.5oC and 37.5-22.5oC, respectively. The (b) water content and (d) temperature distribution at reverse temperature boundary of 22.5/100, i.e. 17.5-27.5oC, after 12, 48 and 96 hours. Note: the initial water content set at 0.20 m3 m-3. The

63.8 10sK m s-1 and zeta � = �� used in the simulation. .............................. 43

Figure 3.2. Sand (s-8) simulated: (a) water content and (c) temperature distributions at temperature boundary of 15/-50, 15/-150 and 30/-150, which were 17.5-12.5, 22.5-7.5oC and 37.5-22.5oC, respectively. The (b) water content and (d) temperature distribution at reverse temperature boundary of 22.5/100, i.e. 17.5-27.5oC, after 12, 48 and 96 hours.

Note: the initial water content set at 0.08 m3 m-3. The 69.8 10sK

m s-1 and zeta � = � used in the simulation.......................................................... 44

Figure 4.1. Water vapor thermal conductivity, and the combined (wet porous media heat conduction and sensible heat by liquid water movement) thermal

conductivity (at 0r oP P P ) versus soil volumetric water content.

Note: the water vapor thermal conductivity was at a pressure ratio of 1.0, i.e. atmospheric pressure, and temperature at 32.5 oC. The water

v

vapor thermal conductivity was corrected for the pressure dependence of the thermal conductivity of air, based on the equation given by Cass. ...................................................................................................................... 71

Figure 4.2. The curve-fitting results of permeability variables as a function of the soil volumetric water content, and the pressure ratio. The permeability variables based on the adapted-improved heat flux theory for (a) the lysimeter sand, and (b) the Portneuf silt loam. Note: the graph (a) and (b) were subjected to Eq. 4.20, before curve-fitting using Eq. 4.22 to determine the permeability variables. Zero permeability variables values were added as supposed zero water vapor volume expansion-advection at fully saturated soil moisture content to improve the curve-fitting results. .............................................................................................. 72

Figure 4.3. The comparison of the experimental and curve-fitted permeability variables based on the adapted-improved heat flux theory for (a) the lysimeter sand, and (b) the Portneuf silt loam. Note: the graph (a) and (b) were subjected to Eq. 4.20, before curve-fitting using Eq. 4.22 to determine the permeability variables. Care should be placed on the fully saturated soil moisture content condition because the permeability variables estimated by Eq. 4.22 could result in negative values. ................................................................................................................... 73

Figure 4.4. (a) The relative permeability ( rk ) of lysimeter sand versus water

saturation estimated using Eq. 4.24, and (b) Enlarged graph on relative permeability value ranged 0-0.3. Note: the other relative permeability models were van Genuchten-Mualem, van Genuchten-Burdine, Corey, and Falta referring to Eqs. 4.25-4.28, respectively. .................... 74

Figure 5.1. Schematic diagram of velocity profile due to fluid flow through a small tube diameter in the porous media. ....................................................................... 96

Figure 5.2. The water vapor thermal conductivity of lysimeter sand versus water saturation at (a) 32.5oC, (b) 22.5oC, (c) Portneuf silt loam versus saturation at 32.5oC, and (d) Kanto loamy soil versus saturation at temperature ranged from 38.5 to 31.5oC. Note: the water vapor thermal conductivity estimated by Darcy law was obtained by subtracting the water vapor thermal conductivity predicted by Fick’s law from the total water vapor thermal conductivity. Equation 5.33 was used in the determination. The total water vapor thermal conductivity dataset derived from Cass et al. (1984) at different pressure ratios, temperatures, and volumetric water contents. ............................. 97

Figure 5.3. The permeability variables ( rnw P Tk k ) versus water saturation at the unity

of pressure ratio and 32.5 degree Celsius. ............................................................ 98

Figure 5.4. The (a) Portneuf silt loam, (b) lysimeter sand, and (c) Kanto loamy soil

permeability variables ( rnw P T ok k k ) data estimated by Eq. 5.34 at

different water saturations, temperatures, and pressure ratios. Note:

vi

the data were curve-fitted by the rnwk and P Tk from Eqs. 5.30 and

5.14, respectively. The equation with and without the temperature as

an independent variable in the expression of P Tk would correspond

to the model e rS T P and e rS P , respectively. From the first to

fourth wave-like curves were respectively the pressure ratios of 1, 0.53, 0.16 and 0.083, as in graph (a), and the each point of the rise and fall of the permeability variables of the wave-like curve, from left to right, correspond to the increasing water saturation. The experimental temperature was 32.5 degree Celsius. In graph (b), the first three wave-like curves correspond to temperatures of 3.5, 22.5 and 32.5 degree Celsius at a pressure ratio of unity. The following three wave-like curves correspond to the similar temperature range, but at the pressure ratio of 0.53, and it continued until the fourth (or last) three wave-like curves at a pressure ratio of 0.083. Graph (c) represented by first to fifth wave-like curves with respective pressure ratios of 1.000, 0.705, 0.495, 0.308 and 0.203. The first wave-like curve permeability variables, from left to right, corresponding to decreasing temperature (42.5 to 31.6 degree Celsius) and increasing water saturation. The rest of the wave-like curves correspond to a similar range of temperature and saturation conditions. The data on graphs (a) and (b) were derived from Cass et al. (1984), while graph (c) was from Tanimoto (2007) derived using DPHP method. .............................................................................. 100

vii

List of Tables

Table 2.1 The coefficients value from Haverkamp et al. (1977) based on the constitutive Eqs. 2.10 and 2.11. These values were used as a base case. ....................................................................................................................... 21

Table 2.2 Statistical measures (μ, μ*, σ) of elementary effect method. They were the mean of elementary effect, the mean of absolute values of the elementary effect, and the standard deviation, respectively. ................................. 22

Table 2.3 Significant digits approximation on input parameter value. ......................................... 23

Table 3.1 Input parameters used for silt loam and sand. ............................................................... 45

Table 3.2 The comparison of water content and temperature root mean square errors (RMSEs) from Heitman et al. (2008) and current simulation, at few temperature boundary conditions. ......................................................................... 46

Table 5.1 The parameter constants ( n and m ) and the Klinkenberg gas slip factor ( b ) of water vapor relative permeability. ........................................................... 101

1

Chapter 1. General introduction

The vapor enhancement factor (VEF) is a well-known but inadequately defined parameter in soil

physics. It was first considered by Philip and de Vries (1957) when they introduced Fick’s law

equation to govern water vapor diffusion in unsaturated soil. But Fick’s law underestimated the

water vapor flux. So, a coefficient, namely, the VEF multiplied the water vapor diffusion

equation to match the discrepancy. Philip and de Vries (1957) hypothesized the VEF as the

combined effects of liquid island and the ratio of the air-filled pores to the overall temperature

gradients. The VEF has been applied in various studies of liquid water, water vapor and heat

flow in unsaturated soil (Banimahd and Zand-Parsa, 2013, Heitman et al., 2008, Milly, 1982,

Milly, 1984, Milly, 1980, Moghadas et al., 2013, Nassar and Horton, 1992, Nassar and Horton,

1997, Noborio et al., 1996, Novak, 2016, Saito et al., 2006, Sakai et al., 2009, Scanlon and Milly,

1994), but its fundamental mechanism is subject to speculation. Shokri et al. (2009) claimed that

the use of capillary flow would negate the need for VEF. Parlange et al. (1998) and Cahill and

Parlange (1998) hypothesized that air volume expansion-contraction convective transport by

diurnal heating and cooling on soil surface is responsible for the VEF. Similarly, Scanlon and

Milly (1994) stated water vapor transport is influenced by thermally driven water vapor

advection, and further added that the great air fluidity in maintaining the relatively constant

atmospheric pressure and in the event of wind gust, it could lead to a net effect of air flow in and

out that transports water vapor. Nevertheless, Ho and Webb (1999) carried out a thorough

investigation on VEF and ultimately concluded its presence. The investigation by Sakaguchi et

al. (2009) and Lu et al. (2011) further confirmed the presence of VEF. However, VEF values

varied from 0.35 in a hydrophobic to 3.5 in a wettable soil, and not even pore-scale models could

2

describe these observations (Horton and Ochsner, 2011). Also, VEF varies in different soil types,

moisture saturation, and ambient temperature (Cass et al., 1984). Clearly, further investigation is

needed to reveal the mechanism controlling VEF.

In this study, the investigation focused on the VEF application and its origin. The VEF

was investigated in two case study situations. The first case study involved the application of

VEF in a simulation of liquid water, water vapor, and heat transport in unsaturated soil. The

second case study estimated to the VEF value indirectly through the use of heat flux theory,

water vapor diffusion based on Fick’s law, and thermal conductivity which was determined by

heat balance equation.

The first case required solving the isothermal liquid water flow equation (Richards, 1931)

and validating the simulated result by Philip’s semi-analytical solution (Chapter 2), before

extending the simulation of liquid water flow to include the effect of water vapor and heat flow

simulation (Chapter 3) in unsaturated soil. A similar validation procedure was employed by

Noborio et al. (1996). The approach set forth necessarily confirmed the mass (water and vapor)

and heat flow simulation developed from a verified solution of liquid water flow simulation. The

Heitman et al. (2008) study has created a controlled experimental temperature boundaries, which

was different from other studies, e.g. Saito et al. (2006) and Sakai et al. (2009). The former study

created an environmental condition successfully isolated from any possible uncertainty encounter

at the boundaries, especially those stated by Scanlon and Milly (1994). Thus, in the current

study, the experimental dataset from Heitman et al. (2008) was re-examined, and the calibration

procedure was improved from using single to using multiple temperature boundaries to

determine whether the VEF, hydraulic conductivity, or their combined effects were responsible

for the discrepancy between the simulation and the physical observation. Furthermore, the

3

outcome of the study would then be able to address the questions of whether or not the

simulation deteriorated over time, whether the VEF could adequately reproduce the water

content distribution after a reverse temperature boundary, whether the calibrated parameters

(VEF and saturated hydraulic conductivity) with the least error in simulation varied when a soil

was simulated at different initial water contents, and whether the search for the least error

simulation generated different calibrated parameters depending on either prioritization on

temperature or water content distributions.

The second case study is based on the work of Cass et al. (1984). The VEF originated as

a correction factor imposed on Fick’s law equation, which itself a constitution of the water vapor

latent heat. Water vapor latent heat combined with wet porous media heat conduction are

assumed as the only two thermal conductivity components used in the simplified heat flux

equation. Cass et al. (1984) published the thermal conductivity values and physical properties of

two soils utilized in the estimation of the VEF. Often, improvement simply identified from the

assumptions used in establishing the theory. Thus, in the current study, similar to the former first

case study, the dataset was re-examined using the heat flux theory from de Vries (1958). The

thermal conductivity contributed by liquid water flow was also considered (Shokri et al., 2009).

Also, the fact that Fick’s law needs the VEF as a correction factor implies an incomplete

phenomenon discovery and so there is an opportunity for improvement. Ho and Webb (1996) and

Ho and Webb (1999) attempted to consider water vapor flux using the Darcy law equation, but

were unsuccessful. In this study, the phase transition volume expansion from liquid water to

water vapor was examined using a mass balance equation. The ideal gas law applied on water

vapor was theoretically investigated under incremental temperature conditions, and the water

vapor advection by air volume expansion concept used in Parlange et al. (1998) was revisited

4

(Chapter 4). The limitation revealed in Chapter 4 was addressed in Chapter 5.

5

Chapter 2. Sensitivity analysis and validation for numerical simulation of

water infiltration into unsaturated soil1

2.1 Abstract

This chapter describes the validation of a FORTRAN code developed for liquid water flow in

unsaturated soil under isothermal conditions to simulate water infiltration into Yolo light clay.

Successful validation of the source code was necessary because the code could be extended to

program the equation governing water vapor and heat flux in unsaturated soil. In this study, the

liquid water flow governing equation, which is Richards’ equation, was approximated by a finite-

difference method. A normalized sensitivity coefficient was used in the sensitivity analysis of

Richards’ equation. The normalized sensitivity coefficient was calculated using one-at-a-time

method (OAT) and elementary effect method based on constitutive equations of matric suction

and hydraulic conductivity. Results from the elementary effect method provide additional insight

into model input parameters, such as input parameter linearity and oscillating sign effect.

Boundary volumetric water content (��(�� . )) and saturated volumetric water content

(��) were consistently found to be the most sensitive parameters corresponding to positive and

negative relations, as given by the hydraulic functions. In addition, although initial volumetric

water content (��(�� . ) ) and time-step size (∆� ), respectively, possessed a great

amount of sensitivity coefficient and uncertainty value, they did not exhibit significant influence

on the model output as demonstrated by spatial spacing size (∆z). The input multiplication of

parameters sensitivity coefficient and uncertainty value was found to affect the outcome of the

model simulation, in which parameter with the highest value was found to be ∆z. This finding

1 This chapter has been published in Goh and Noborio (2015).

6

would imply that the extension of the water flow equation, that is Richards’ equation, to include

water vapor and heat flux, as in Nassar and Horton (1997) and Heitman et al. (2008), would

require a small spatial spacing size to guarantee a good simulation result. Furthermore, validating

the water infiltration source code was used as a necessary step in Noborio et al. (1996) before

modeling two-dimensional model for water, heat, and solute transport in furrow-irrigated soil.

2.2 Introduction

Sensitivity analysis is used for various reasons, such as decision making or development of

recommendations, communication, increasing understanding or quantification of the system, and

model development. In model development, it can be used for model validation or accuracy,

simplification, calibration, coping with poor or missing data, and even to identify important

parameter for further studies (Pannell, 1997).

More than a dozen sensitivity analysis methods are available, ranging from one-at-a-time

(OAT) to variance-based methods (Campolongo et al., 1999, Davis et al., 1997, De Roo and

Offermans, 1995, Fox et al., 2010, Hamby, 1994, Saltelli et al., 2004, Sepúlveda et al., 2014,

Vazquez-Cruz et al., 2014, Vereecken et al., 1990). On a fundamental level, sensitivity analysis is

a tool to assess the effect of changes in input parameter value on the output value of a simulation

model. In this aspect, the sensitivity coefficient, in a normalized form, is given in the following

relationship:

��,� =�� ⁄

�� ⁄ (2.1)

where ��,� is referred to as the normalized sensitivity coefficient for �th input parameter at �th

observation point; �� is the model dependent variable value at �th observation point; and �� is

the �th input parameter value. However, the method does not explore other input space factors

7

in which more than one input parameter is varied. Despite this disadvantage, Saltelli and Annoni

(2010) noted that researchers continual use of the OAT method is due to a few claimed

advantages. Although variance-based method is the best practice, Saltelli and Annoni (2010)

have suggested using the elementary effect method, which is an enhancement of the OAT

method, when computation time is expensive, for instance, in numerical simulation that is

computationally demanding.

The elementary effect method is accomplished through the use of a technical scheme to

generate trajectories. Each trajectory consists of some steps in which each step refers to an

increment or decrement of an input parameter value. The base condition for each trajectory is

different from the others, and it is selected randomly. The random version of trajectory

generation is as following (Morris, 1991, Saltelli et al., 2008, Saltelli et al., 2004):

�∗ = ��,��∗ + (∆ 2⁄ )��2� − ��,��

∗ + ��,��∗ (2.2)

where �∗ is the generated trajectory in the form of a matrix with dimension (� + 1) × �, where

� is the number of independent input parameters; ∆ is the value in [1 (� − 1)⁄ , … ,1 −

1 (� − 1)⁄ ] and � is the number of levels; ��,� is (� + 1) × � matrix of 1's; �∗ is a randomly

chosen base value; � is the low triangular matrix of 1's; � ∗ is the �-dimensional diagonal matrix

in which each element is either +1 or -1, by random generation; and �∗ is the �-by-� random

permutation matrix that each row and column with only one element equal to 1. This method has

been used by some researchers, for instance, Drouet et al. (2011) on nitrous oxide emissions at

farm level, and Chu-Agor et al. (2011) on the vulnerability of coastal habitats to sea level rise.

Campolongo et al. (2011) improved this method into a quantitative approach.

The generated trajectories can be screened to obtain a subset of trajectories with the

greatest geometric distances. The trajectories scanned for maximum geometric distance between

8

all the pairs of points between two trajectories, were determined as follows (Campolongo et al.,

2007):

�� = �∑ ∑ �∑ ��(�) (� ) − ��

(�) (�)��

��

��

�� ≠ �

0

(2.3)

where �� is the distance between a pair of trajectories � and �; ��(�) (�) is the �th coordinate of

the �th point of the �th trajectory; and ��(�) (� ) is the �th coordinate of the �th point of the � th

trajectory.

The sensitivity coefficient of an input parameter in the elementary effect method as ��,

which is the mean of the elementary effect (��). ��

∗ is the mean of absolute values of the

elementary effect, which is to avoid cancellation of difference signs in the mean value. The

sensitivity measures (��, ��∗, �) and ��

� are given by (Saltelli et al., 2008):

��=

��(��∆�)��(��)

∆� (2.4)

�� =�

�∑ ��

�� (2.5)

��∗ =

�

�∑ ��

��

�� (2.6)

�� =

�

��∑ ��

�− ��

�� (2.7)

where ��(��) and ��(�� + ∆�) are the simulation result before and after increment or decrement

of ∆ value, i.e. ∆� which can either positive or negative value; � is referring to the total number of

trajectories; ��

is the elementary effect of � input parameter at � trajectory; and �� is the

standard deviation of � input parameter.

The aim of the current work was to carry out sensitivity analyses on water infiltration into

unsaturated soil as governed by Richards’ equation and to use it as an assessment method to

9

validate the simulation source code with the analytical solution. Thus, the objective of this study

was to use the sensitivity coefficient, under a hypothetical assumption to validate the model

simulation with Philip's semi-analytical solution using literature data. The water infiltration

results from Haverkamp et al. (1977) and Kabala and Milly (1990) were used for the verification.

2.3 Materials and methods

2.3.1 The governing equation of water flow in unsaturated soil, and its numerical solution

The governing equation for transient liquid water flow in soil may be described as (Celia et al.,

1990, Richards, 1931):

��

��=

�

��

��

��

��− ��⃑ � (2.8)

where �� is the volumetric water content (m3 m-3); � is the simulation time (s); � indicates the

vertical distance in the simulation (m); � is the hydraulic conductivity of the medium (m s-1); ��

is the matric pressure head (m); ��⃑ is the vector unit with a value of positive one when it is

vertically downwards.

Equation 2.8 was approximated numerically, and its algebra was implemented in

FORTRAN 2008 using Simply Fortran Integrated Development Environment. The spatial

discretization method used was termed as the cell-centered finite difference (Zheng and Bennett,

2002). To avoid unnecessary redundancy, we only provide the algebra for Eq. 2.8, that was used

for sensitivity analysis in the current study, as follows:

��(�)��(�)

�

∆�=

��

��

��

��

��

∆��(�.�∆��.�∆��)��(��)

�� − ��(�)��

−

��

��

��

��

��

∆��(�.�∆��.�∆��)��(�)

�� − ��(��)�� −

��,�,��

��

��⃑ ��,�,��

��

��⃑

∆�� (2.9)

10

where � indicates the cell-centered number in z-direction in Cartesian coordinate system; ∆� (s)

is the time-step size; ��(�)� (m3 m-3) and ��(�)

�� (m3 m-3) indicate volumetric water contents at

initial time (n) and new time (n+1), respectively; �� ⁄ (m s-1) is the hydraulic conductivity at

the interface between cell � and � + 1 ; �� ⁄ (m s-1) is the hydraulic conductivity at the

interface between cell � − 1 and �; (�� ⁄ )�� ⁄ is the partial derivative of �� with respect

to �� at the interface between the cell � and � + 1; (�� ⁄ )�� ⁄ is the partial derivative of

�� with respect to �� at the interface between the cell � − 1 and �; ∆�� (m), ∆�� (m) and

∆�� (m) are corresponding to the spatial sizes of spacing of cell � + 1 , � and � − 1 .

��(��)�� (m3 m-3), ��(�)

�� (m3 m-3) and ��(��)�� (m3 m-3) are the volumetric water

contents at new time level of cell � + 1, � and � − 1, respectively. An iterative method was used

to solve the mathematical algebra of Eq. 2.9 (Tu et al., 2007). A convergence factor criterion was

used to indicate the condition for the iteration termination process, that is the absolute maximum

difference ��(�)�� − ��(�)

�� for every single cell.

2.3.2 Constitutive functions of matric pressure head ( m ) and hydraulic conductivity ( K )

The constitutive functions implemented were from Haverkamp et al. (1977):

�� = −10�� (��)

��− ��

�

� (2.10)

� = ��

��(��)� (2.11)

where � , � , � and � are the fitted parameters; �� (m3 m-3) is the residual volumetric water

content; �� (m3 m-3) is the saturated volumetric water content; and �� (m s-1) is the saturated

hydraulic conductivity.

11

2.3.3 Numerical experiment and the default setting of input parameters of the flow

problem

Water infiltration into Yolo light clay was used in the numerical experiment. The coefficients

values for the medium (Table 2.1) were based on the constitutive function equations (Eqs. 2.10

and 2.11). The initial condition for the volumetric water content was 0.2376 m3 m-3. Lower

boundary was set to permeable inflow and outflow of water, and the upper boundary was set at

0.495 m3 m-3. After considering the mass balance ratio (Celia et al., 1990) and iteration number,

the time-step size, spatial spacing size, and convergent value were set at corresponding values of

500 s, 1 cm, and 10-12 m3 m-3, respectively.

2.3.4 Statistical measures

A statistical equation was used to determine the goodness of fit between the data and the

simulated results. The equation was known as absolute residual errors (MA) as follows (Zheng

and Bennett, 2002):

�� =�

�∑ |�� − ��|�� (2.12)

where �� is the simulated data at cell �, and �� is the analytical solution at cell �.

2.4 Results and discussion

2.4.1 Simulation results and its accuracy

Based on the conditions as stated in the previous section, water infiltration into Yolo light clay

was simulated for up to 105 s. Philip's semi-analytical solution data were collected from

Haverkamp et al. (1977) and Kabala and Milly (1990), hereafter referred to as Philip(H) and

Philip(K), respectively. Simulation results were compared with the data to verify the simulation

12

(Fig. 2.1). Before referring to any statistical measure, it was evident that the simulation results

slightly under-estimated the water flow infiltration front.

Also, Fig. 2.1 shows that there was a small discrepancy between Philip(K) and Philip(H),

but the former was relatively closer to the simulation results than the latter. Thus, a sensitivity

analysis was carried out to determine the sensitivity coefficient for all input parameters. The

sensitivity analysis results were used to assess the model simulation based on the assumption that

possibly the cumulative effect of input parameters, regarding significant digits approximation,

could be contributing the under-estimation of the volumetric water content of the simulation. The

sensitivity analysis was one of the most important steps in evaluating the effect of input

parameter on simulation results. Moreover, other researchers used sensitivity analysis for model

validation (Gosling and Arnell, 2011, Min et al., 2006, Nathan et al., 2001, Stange et al., 2000).

2.4.2 Sensitivity analysis and simulation model validation

Negligible sensitivity response could be the result of too small a perturbation in size, and

inaccuracy in sensitivity response could be due to too large a perturbation in size (Poeter and

Hill, 1998). The input parameters values were subjected to a perturbation in size of between -5%

and 5%, as suggested by Zheng and Bennett (2002). Seven input parameters, as listed in Table

2.1, and four additional input parameters were tested, i.e. initial volumetric water content

(��(�� . )), boundary volumetric water content (��(�� . )), time-step size

(∆�), and spatial spacing size (∆�), were tested. A depth of 15.5 cm from the ground surface was

used for observation.

The normalized sensitivity coefficients are shown in Fig. 2.2. There were two groups of

sensitivity coefficients, i.e. positive and negative relationships. In the positive relationship group,

13

the boundary volumetric water content had the highest sensitivity coefficient. This was followed

by initial volumetric water content and saturated hydraulic conductivity. The smallest sensitivity

coefficient in the group was the residual volumetric water content. In negative relationship

group, saturated volumetric water content had the highest sensitivity coefficient, and in this

group spatial spacing size and time-step size had the smallest sensitivity coefficients.

For comparison purposes, the elementary effect method was also used to calculate

normalized sensitivity coefficient. Only random generation in �-dimensional diagonal matrix

(� ∗), and Eq. 2.2 was used to generate 50 trajectories. Equation 2.3 was used to screen for four

selected trajectories. The mean of elementary effect was used to calculate the normalized

sensitivity coefficient (Table 2.2). Similar values of � and �∗ indicate linear effects on a few

input parameters in the positive (�, ��, ��(�� . )) and negative (��, �) relationships.

Other input parameters have shown the effect of oscillating sign which result in different values

of � and �∗. Also, both sensitivity analysis methods generated comparable trends of sensitivity

coefficients.

We assumed that a minor deviation in each input parameter, regarding its significant

digits approximation, could contribute some effect to the simulation outcome that could explain

the gap between the simulated results and Philip's semi-analytical solution (Fig. 2.1). In other

words, the parameter values used in the computer simulation by Haverkamp et al. (1977) could

have been different from the exact data that they published. Thus, the sensitivity analysis positive

and negative relationships could be used to determine the hypothetical approximation values in

Table 2.3 for investigation. The cumulative effect begins from step 1 for the base case to step 10

for spatial spacing size. The ��(�� . ) value (0.2376499 m3 m-3) was used as a second

simulation (in step 2) after the base case simulation. A third simulation (in step 3) was applied

14

using �� value as 0.124499 m3 m-3 by keeping the ��(�� . ) value from the second

simulation. In each simulation, the absolute residual error (MA) (Eq. 2.12) was calculated for the

discrepancy between the simulation and Philip's semi-analytical solution. Of the eleven

parameters in Table 2.3, only ∆� and ∆� were parameters without any limit of variation. Thus,

they were reduced by 98% and 90% from 500 s and 1 cm to 10 s and 0.1 cm, respectively. The ��

and ��(�� . ) were not included in the study because decreasing and increasing the

former and latter value would result in simulation failure.

A consistent reduction in MA value from ��(�� . ) to � input parameter was

observed; and a steep slide was observed on the ∆z input parameter simulation (Fig. 2.3). The

reducing ∆� value should lead to a reduction in MA value, but an incremental simulated result

was observed. This may be explained from an elementary effect perspective in that ∆� has

different values of � and �∗, which indicate the capability of sign oscillation (Table 2.3).

Figure 2.4 showed that the simulation of the cumulative effect of steps 2-9, which

combined the effect from ��(�� . ) (step 2) to ∆� (step 9), did not contribute a

significant effect to the advancement of the water infiltration front; it only resulted in a reduction

of 7.8% in MA value, from 0.0254 to 0.02343 m3 m-3. In addition, these eight input parameters

would have to vary in the way as tabulated in Table 2.3 in order to cause the stated percentage

reduction. Therefore, the significant digits approximations might not be the main cause of the

problem in considering the greater effect of ∆� on the advance of the water infiltration front as

shown in Fig. 2.4. A further step to include ∆� in the simulation, that is, the cumulative effect of

steps 2-10, which combined the effect from ��(�� . ) (step 2) to ∆� (step 10), found a

54.7% reduction in MA value of step 9, from 0.02343 to 0.0106 m3 m-3. This indicates that

spatial spacing size was the main cause of the advance of the water infiltration front. Therefore,

15

the simulation was repeated for a last time to assess the effect of spatial spacing size, Step 10

alone, (Fig. 2.4), showing a good agreement between the simulation results and the Philip(K).

This observation could be explained using Eq. 2.1, after rearranging it into the following form,

which we termed it as percentage variation in simulation results:

∆��

��=

∆��

��× ��,� (2.13)

where ∆�� ⁄ is the normalized input parameter value (%); ∆�� ⁄ is the normalized output

parameter value (%); and ��,� is the normalized sensitivity coefficient (%/%). Equation 2.13 was

simply a multiplication of the percentage change in input parameter value from the base case and

the normalized sensitivity coefficient.

From Eq. 2.13, the percentage variation in simulation results from the ∆� and ∆� input

parameters caused an increment of 4.95% and 0.06%, respectively, despite ∆� having the highest

reduction in percentage (-98%) from the base case. This observation can be summarized as

follows: (1) the input parameter with the highest sensitivity coefficient does not imply the

greatest effect on the simulation result, for example, ��(�� . ) ; and (2) the input

parameter with the highest percentage of change also does not imply the greatest effect on the

simulation result, for example, ∆� . Therefore, only the parameter that produces the highest

multiplication of sensitivity coefficient, and greatest percentage change on the input parameter

(or the uncertainty) would cause the most substantial effect on the simulation result.

2.5 Conclusions

The equation governing transient water flow in unsaturated, under isothermal condition, was

approximated numerically by finite-difference solution. It was implemented in FORTRAN

programming language, simulated and verified by Philip's semi-analytical solution on water

16

infiltration into Yolo light clay, using data from the literature.

One-at-a-time (OAT) and elementary effect methods (EE) were used in the sensitivity

analysis. A common trend of sensitivity was observed across the methods in both positive and

negative relationships. The latter method allowed exploration of additional characteristics of

input parameters at different input spaces, such as linearity and sign oscillation effect. The sign

oscillation effect observed on input parameters explained the possibility of its deviation from

those observed in the OAT method at different input spaces, for example, time-step size.

A hypothetical case that was established to study the cumulative effect of input

parameters on the discrepancy between the simulated result and Philip's semi-analytical solution,

regarding significant digits approximation (from base case), was found to be unlikely.

Surprisingly, neither the high normalized sensitivity coefficient of initial volumetric water

content nor the largest percentage changes in time-step size, contributed any substantial impact

on simulation results when compared to spatial spacing size, alone. This observation leads to the

conclusion that the uncertainty of the input parameters and normalized sensitivity coefficients of

the input parameters both control the outcome of the simulation.

Furthermore, the finding of this study indicates the simulation code of Richards’ equation

was indeed working properly, and it could be extended to include water vapor and heat flux

governing equations. The final developed simulation code would have to use a small spatial

spacing size to generate accurate simulation output.

17

Figure 2.1. Comparison of simulated results with Philip's semi-analytical solution. Note:

Philip(H) and Philip(K) were from Haverkamp et al. (1977) and Kabala and Milly (1990),

respectively.

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.20 0.25 0.30 0.35 0.40 0.45 0.50

De

pth

, z (

cm)

Volumetric water content (m3 m-3)

Simulated (10^5 s) Philip(H) @ 10^5s Philip(K) @ 10^5s

18

Figure 2.2. The rank of sensitivity coefficient. Note: θ_s and θ_r are the saturated and residual

volumetric water content; ∆z, spatial spacing size; ∆t, time-step size; K_s, saturated hydraulic

conductivity; θ_L(initial cond.), clay medium initial value of volumetric water content;

θ_L(upper bound.), upper boundary of volumetric water content; A, B, β and α are the fitting

parameters from Haverkamp, as in Eqs. 2.10 and 2.11.

-4.9

7E+

00

-1.0

7E+

00

-5.8

9E-

01

-5.5

0E-

02

-6.1

9E-

04

3.8

5E-

02

8.2

5E-

02

2.0

8E-

01

5.0

8E-

01

6.4

4E-

01

4.7

1E+

00

-11.00

-9.00

-7.00

-5.00

-3.00

-1.00

1.00

3.00

5.00

7.00

9.00

11.00

θ_s B β ∆

z

∆t

θ_r α A

K_s

θ_L

(in

itia

l co

nd

.)

θ_L

(u

pp

er b

ou

nd

.)

No

rmal

ized

sen

siti

vity

co

effc

ien

t (%

/%)

Input parameter

19

Figure 2.3. The cumulative effect of input parameters, on the absolute residual error at

simulation time 105 s. Note: θ_L(initial cond.) (step 2), clay medium initial value of volumetric

water content; θ_r (step 3), residual volumetric water content; β (step 4), α (step 5), A (step 7)

and B (step 8) are the fitting parameters; K_s (step 6), saturated hydraulic conductivity; ∆t (step

9), time-step size; and ∆z (step 10), spatial spacing size.

0.0

25

4

0.0

25

3

0.0

25

3

0.0

24

5

0.0

24

5

0.0

24

5

0.0

24

5

0.0

23

38

0.0

23

43

0.0

10

6

0.010

0.015

0.020

0.025

0.030

0.035

Bas

e C

ase

(Ste

p 1

)

θ_L

(in

itia

l co

nd

.) (

Step

2

) θ_r

(St

ep 3

)

β (

Step

4)

α (

Step

5)

K_s

(St

ep 6

)

A (

Step

7)

B (

Ste

p 8

)

∆t

(Ste

p 9

)

∆z

(Ste

p 1

0)

Ab

solu

te r

esid

ual

err

or

(MA

), (

m3

m-3

)

Input parameter

20

Figure 2.4. The effect of ∆z (step 10 alone), and cumulative effects of steps 2 to 9 and 2-10 in

comparison with Philip(H) and Philip(K).

16

17

18

19

20

21

22

23

0.27 0.29 0.31 0.33 0.35 0.37 0.39

De

pth

, z (

cm)


Philip(H) Base Case Steps 2-9

Steps 2-10 Philip(K) Only Step 10

21

Table 2.1 The coefficient values from Haverkamp et al. (1977) based on the constitutive Eqs.

2.10 and 2.11. These values were used as a base case.

Parameter Value

� 739

�� 0.124 m3 m-3

�� 0.495 m3 m-3

� 4

� 124.6

� 1.77

�� 1.23x10-7 m s-1

Note that �� is the residual volumetric water content, �� is the saturated volumetric water

content, �� is the saturated hydraulic conductivity, and �, �, � and � are the fitting coefficients.

22

Table 2.2 Statistical measures (μ, μ*, σ) of elementary effect method. They were the mean of

elementary effect, the mean of absolute values of the elementary effect, and the standard

deviation, respectively.

μ (%/%) μ* (%/%) σ

�� -6.03E+00 6.03E+00 9.48E-01

� -1.85E+00 1.85E+00 9.38E-01

� -2.07E-01 3.20E-01 3.70E-01

� -4.14E-02 1.25E-01 1.47E-01

∆� -3.35E-02 3.95E-02 4.56E-02

∆� -1.66E-04 5.25E-04 6.21E-04

�� 4.44E-03 3.37E-02 4.02E-02

� 3.13E-01 3.13E-01 7.41E-02

�� 5.24E-01 5.24E-01 6.73E-02

��(�� . ) 8.84E-01 8.84E-01 3.05E-01

Note: �� is the residual volumetric water content, �� is the saturated volumetric water content, ��

is the saturated hydraulic conductivity, ∆� is the spatial spacing size, ∆� is the time-step size,

��(�� ) is the initial value of volumetric water content, and � , � , � and � are the

fitting coefficients.

23

Table 2.3 Significant digits approximation on input parameter value.

Parameter Value

� 739.499 (≈739)

�� 0.124499 (≈0.124) m3 m-3

�� 0.495 m3 m-3

� 3.95 (≈4)

� 124.64 (≈124.6)

� 1.765 (≈1.77)

�� 4.4284×10-2 (≈4.428x10-2) cm hr-1

��(�� . ) 0.2376499 (≈0.2376) m3 m-3

��(�� . ) 0.495 m3 m-3

∆� 10 s, the base case was 500 s

∆� 0.1 cm, the base case was 1 cm

Note: �� is the residual volumetric water content, �� is the saturated volumetric water content, ��

is the saturated hydraulic conductivity, ∆� is the spatial spacing size, ∆� is the time-step size,

��(�� ) is the initial value of volumetric water content, ��(�� ) is the upper

boundary of volumetric water content, and �, �, � and � are the fitting coefficients.

24

Chapter 3. Liquid water, water vapor, and heat flow simulation in

unsaturated soils

3.1 Abstract

Liquid water, water vapor, and heat movement in soil are important for agriculture and

engineering applications. Successfully in simulating the mass and heat transport of soil water and

temperature distributions could be used for land management strategy, and also to reduce the cost

of continuous instrumental measurement. It inspired researcher to program the equation to

simulate the observations. Heitman et al. (2008) attempted to describe the observation after

parameter (��, �) calibration using a single temperature boundary condition out of nine available

boundary conditions, in which each condition has a duration of 96 hrs. Thus, the calibrated

parameters were not optimized. In the current study, the water, vapor, and heat flux equation was

solved by finite difference method and coded to re-examine the experimental data from Heitman

et al. (2008). The calibrated parameters were obtained after taking into account all the imposed

temperature boundary conditions. The results showed that saturated hydraulic conductivity (��)

at its original value produced low error, i.e. root mean square error (RMSE). The zeta (�) value, a

variable that represents vapor enhancement factor, has an uncertain value between 2 and 6, which

corresponds to accuracy prioritization on either temperature or water content, respectively. The

model has potential for further improvement in the inaccuracy noted in the low- and high-

temperature edges of the soil column by implementing the equation from Fayer and Simmons

(1995). The vapor enhancement factor requires further theoretical examination and to

experimentally measure the value, otherwise, it continues as an adjustable parameter without real

physical representation. Furthermore, the reverse temperature boundary simulation caused a

25

significant error revealing the necessity of a new guiding force, for example, vapor buoyancy.

3.2 Introduction

The movement of water, vapor, and heat has important applications in agriculture and

engineering (Saito et al., 2006). In the agricultural sector, the variation of temperature and water

content could influence biological activities, seed germination and plant growth (Banimahd and

Zand-Parsa, 2013). Seed germination can occur with only water supply in the form of water

vapor (Wuest et al., 1999). Also, heat, water, and vapor fluxes have an influence on soil

salinization (Gran et al., 2011). In engineering applications, the coupling of heat, water, and

vapor simulation is necessary for landfill management, nuclear waste repositories, and the

improved detection of buried land mines (Moghadas et al., 2013). Kroener et al. (2014) used the

coupling simulation to study heat dissipation of underground electric power cables, which could

lead to a better design of cable installation. The coupling simulation could also be used for water

balance prediction to estimate potential leachate generation as part of landfill management

(Khire et al., 1997). Furthermore, the coupling simulation has been used to study airflow

pathway development that could be applied to predict vapor intrusion into building basement

(Sakaki et al., 2013).

In soil physics, the governing equation for water, vapor, and heat transport was initially

derived by Philip and de Vries (1957) and de Vries (1958), hereafter known as PdV. The PdV

equation transformed into a matric suction-based equation by Sophocleous (1979). Based on the

work of previous authors and also by Milly (1980) and Milly (1982), the PdV equation was

extended by Nassar and Horton (1997) to include a solute transport mechanism. The extended

PdV form was also used to study furrow-irrigated soil (Noborio et al., 1996). Also, the PdV

26

equation was adapted to simulate coupling of water, vapor, and heat in unsaturated freezing soils

(Hansson et al., 2004, Zhang et al., 2016). Sakai et al. (2009) used the equation to study the

effect of water vapor condensation and evaporation on temperature and water content

distribution in a sandy column, while Kelleners et al. (2016) included a snow water flow

equation in addition to the water, vapor, and heat transport. The PdV equation has also been used

to couple with global climate model prediction to study dry area (Pfletschinger et al., 2014). The

use of PdV equation in diverse applications such as wood drying, concrete drying and fracturing,

sand-water-steam systems, soil during wildfires, and wet porous thermal barriers, was reported in

Massman (2015). Also, Moghadas et al. (2013) integrated the PdV model with ground-

penetrating radar model to improve estimation of soil hydraulic properties. Dong et al. (2016)

attempted to estimate the soil water content, temperature, thermal and hydraulic properties of the

PdV equation by assimilating soil temperatures.

Although the PdV equation has been subjected to continuous improvement over decades,

there is a fundamental problem regarding the model prediction. The problem arises from the

uncertainty of the model parameters. Rose (1968) described the possibility air flux as a possible

phenomenon to explain the vapor enhancement factor due to a temperature gradient, as noted by

Bittelli et al. (2008). The air flux possibility was then investigated by Parlange et al. (1998) and

Cahill and Parlange (1998), and the air volume expansion and contraction by diurnal temperature

variation were claimed to be responsible for additional vapor flux. Or and Wraith (2000)

questioned the findings’ validity because diurnal temperature variation could affect the measured

water content due to time domain reflectometry. Later, Novak (2016) found the PdV equation

without the effect of airflow was sufficient to describe field measurements. Besides, Scanlon and

Milly (1994) noticed other types of uncertainty, such as that the saturated hydraulic conductivity

27

value should be lower in value, air movement resulting from atmospheric pressure fluctuation

due to a wind gust, and the neglected hysteresis in the water retention function.

Apart from all these limitations, Sakai et al. (2009) found a good match between PdV

simulation and experimental observation. The study unveiled an inseparable inverse relationship

between the pore-connectivity coefficient from unsaturated hydraulic conductivity and the zeta

coefficient from vapor enhancement factor. The relationship highlighted the necessity of a low

value of pore-connectivity coefficient at high zeta coefficient, and vice versa. Such calibration

was not an isolated case, as Heitman et al. (2008) found that calibration of the zeta value and the

saturated hydraulic conductivity was required to match the simulation with observation.

The research work by Heitman et al. (2008) originated back in the early work by Nassar

and Horton (1989), when using the destructive (gravimetric) method to determine water content

and the failure of a one-dimensionally designed soil column resulted in two-dimensional

temperature distribution. Subsequently, Prunty and Horton (1994) managed to achieve a one-

dimensional temperature, i.e. free from the ambient temperature interference (ATI), in a 5 cm

column length, but it was too short in length to allow a measuring probe, e.g. TDR, placement in

between. Then, Prunty (2003) designed a soil column length of 10 cm, but was unable to remove

ATI. Later, Zhou et al. (2006) improved the design up to a 10 cm length soil column without

noticeable ATI. Using the improved soil column design, together with the in situ measurement

solution from Ren et al. (2005) for water content and soil thermal property, Heitman et al. (2008)

modeled water, vapor, and heat transport in the closed soil column.

This study re-examines the simulation results from Heitman et al. (2008). The

experimental works by Heitman and his co-workers were relatively thorough, but the model

calibration effort was limited only to a single case of boundary condition, even though multiple

28

sequentially applied boundary conditions, i.e. nine temperature gradients, were available.

Finding optimized parameters would require multiple boundary conditions to be simultaneously

accounted in the model calibration. Thus, the objective of the current study is to calibrate the

model with multiple boundary conditions simultaneously so that either the vapor enhancement

factor, hydraulic conductivity, or the combined effect could be single out as the main culprit for

the uncertainty in simulation output. This work built on the previous chapter with the simulation

code of Richards equation for liquid flow and gravity being extended to include vapor and heat

transport in unsaturated soil. The simulation would implement fine spatial discretization size to

generate reliable simulation results (Goh and Noborio, 2015), as found in the previous chapter.


3.3.1 The governing equation of water, vapor, and heat transport in unsaturated soil, and

its numerical solution

The study used the water content based equation, as in the original PdV equation, to govern

water and vapor fluxes, which can be described as:

�1 +��

��

��

��

��

��−

��

��

��+

��

��

��+

��

��

��

��

��

=�

��(�� + ��)

��

��+ �� + �

��

��

��− ��⃑ � (3.1)

�� = ��

��

��+

��

��

��

�� (3.2)

�� =��

��

��

��

�� (3.3)

�� = ��

�� (3.4)

where Eqs. 3.2, 3.3 and 3.4 are constitutive functions of Eq. 3.1. � is the water vapor

enhancement factor (dimensionless), � is the vapor diffusion coefficient (m2 s-1), Ω is the

29

tortuosity factor (dimensionless), �� is matric suction (m), �� is the volumetric air content (m3

m-3), �� is the liquid water density (kg m-3), �� is the water vapor density (kg m-3), �� is the

volumetric water content (m3 m-3), � is the time (s), � is the temperature (K), � is the vertical

distance (m), �� is the thermal vapor diffusivity (m2 s-1 K-1), �� is the thermal liquid

diffusivity (m2 s-1 K-1), �� is the isothermal vapor diffusivity (m s-1), � is the hydraulic

conductivity (m s-1), and ��⃑ is the vector unit with a value of positive one when it is vertically

downwards. Equation 3.1 was adapted from Heitman et al. (2008), which was originally in

matric suction based form. Although some researchers such as Milly (1982) have expressed the

necessity of using the matric suction based form equation, the water content based equation used

in the current study to examine a soil water content that was far below saturation level, so it

would not be an issue of concern (Heitman et al., 2008).

The heat flux equation was also adapted from Heitman et al. (2008), which originated

from de Vries (1958). The equation can be described as:

{��}��

��+ {��}

��

��=

�

��{�+ ��}

��

��+ {��}

��

��− ��(� − ��)��⃑ � (3.5)

�� = �� + [(� − ��)�� + ��]��

��+

��

��

��

�� (3.6)

�� = ��(�� + ��) + �� + �� (3.7)

�� = (� − ��)(�� − ��) − �� − �� + [(� − ��)�� + ��]��

��

��

�� (3.8)

�� = [�� − (�� − ��)(� − ��)]�� + ��(� − ��)��

+ ��(� − ��)��

�� (3.9)

� = �∗ + �� = �� + �� + ��.� (3.10)

�� = ��(� − ��)�� + ��(� − ��)�� (3.11)

where Eqs. 3.6, 3.8, 3.9, 3.10 and 3.11 are constitutive functions of Eq. 3.5, while Eq. 3.7 is part

30

of Eq. 3.6. �� is the bulk density (kg m-3), �� = 1900 and �� = 730 are the respective organic

and mineral specific heats (J kg-1 K-1), �� and �� are the respective organic and mineral mass

fractions (dimensionless), �� is the reference temperature (273.15 K), �� is the liquid water

specific heat (J kg-1 K-1), �� is the water vapor specific heat at constant pressure (J kg-1 K-1), �� is

the latent heat of vaporization (J kg-1) at the reference temperature, � is the differential heat of

wetting (J kg-1), �∗ is the thermal conductivity of the porous media in the presence of liquid

water (W m-1 K-1), � is the latent heat of vaporization (J kg-1) at the temperature �, and ��, ��

and �� are the fitting parameters.

The water vapor density was derived from the equations of saturated vapor density

(Kimball et al., 1976), matric head effect on relative humidity (Philip and de Vries, 1957), and

water retention curve (Campbell, 1974):

�� = 10�� 19.819−��.�

�+

��

��

��

��[−�(� − ��)]� (3.12)

where � is the gravitational constant (9.81 m s-2), � is the water molecular weight (0.01801528

kg mol-1), � is the ideal gas constant (8.3145 J mol-1 K-1), �� is the saturated water content (m3

m-3), �� and � are the fitting parameters of water retention curve, and � is a constant (2.09 x 10-3

K-1).

Taking the derivative of Eq. 3.12 on matric suction gives:

��

��= 10�� 19.819−

��.�

��

��

��

��

��

��

��[−�(� − ��)]� (3.13)

The water retention curve relation from Campbell (1974) can be taken as the derivative of

the matric suction on water content, together with the matric head effect on relative humidity

from Philip and de Vries (1957):

��

��= −

��

��

��

��[−�(� − ��)] (3.14)

31

Again, Eq. 3.12 can be taken as the derivative of the water vapor density on temperature

to give:

��

��= �

��.�

��−

��

��10��

��

�� 19.819−

��.�

�� (3.15)

Based on the similar derivative procedure as in Eq. 3.14, but this time the derivative of

matric suction with respect to temperature:

��

��= −��

��

��

��[−�(� − ��)] (3.16)

The vapor diffusion coefficient (Kimball et al., 1976), tortuosity factor (Milly, 1984),

unsaturated hydraulic conductivity (Bear et al., 1991, Campbell, 1974), and vapor enhancement

factor (Cass et al., 1984) respectively, are expressed as shown below:

� = 2.29× 10��

��.��.��

(3.17)

Ω = (� − ��)�

� (3.18)

� = ��

�� (��)

�(�) (3.19)

� = � + �� − (� − 1)�� −��1 + � √�⁄ �� (3.20)

where � is the soil porosity (m3 m-3), �� is the saturated hydraulic conductivity (m s-1), �(��) is

the dynamic viscosity at reference temperature (kg m-1 s-1), �(�) is the dynamic viscosity at

temperature �, zeta (�), � and � are respectively 8, 6 and 4, � is the soil particle density (2.6 Mg

m-3), and � is the clay fraction of soil.

Numerical approximation was applied to Eq. 3.1, and the algebra implemented in

FORTRAN 2008. The cell-centered finite difference was used as the spatial discretization

method. The algebra for water and vapor transport (Eq. 3.1) is:

�1 +��(�)

��(�)��

��

��−

��(�)

��(�)��

∆��(�)

�� − ��(�)��

32

+ ��(�)

��(�)��

��

��+ �

��

��

��

��

∆�

=��

��

��

��

��

�.�∆��.�∆��

��

��

��

��

�.�∆��.�∆��

∆��

+

��

��

��

��

��

��

∆��(�.�∆��.�∆��)��(��)

�� − ��(�)��

−

��

��

��

��

��

��

∆��(�.�∆��.�∆��)��(�)

�� − ��(��)�� −

��

��

��⃑ ��

��

��⃑

∆�� (3.21)

where ∆� is the time-step size (s), ∆� is the spatial discretization size (m), � is the middle cell,

� − 1 is the cell located spatially before cell �, � + 1 is the cell after cell �, � −�

� is the interface

between cells � − 1 and �, � +�

� is the interface between cells � and � + 1, � is the initial time

level, and � + 1 is the new time level.

Similar discretization procedure implemented on the heat flux equation of Eq. 3.5:

��(�)

��

�� − �� + ��(�)

��(�)��(�)

�

��

=��

��

��

��

��(�.��.��)��

�� − �� −

��

��

��

��

��(�.��.��)��

�� − ��

+��

�

��

��(��)��(�)

��

��(�.��.��)− �

��

��

��(�)��(��)

��

��(�.��.��)

−��

��

��

��

��

��⃑

��+

��

��

��

��

��

��⃑

�� (3.22)

3.3.2 Input parameters, initial and boundary conditions

All data in this section were retrieved from Heitman et al. (2008). In this study, sand and silt

loam water content (��) and temperature (�) distribution profiles in the vertically closed soil

column were simulated. There were four separate sets of simulation, they were sil-10 (silt loam

at 0.10 m3 m-3), sil-20 (silt loam at 0.20 m3 m-3), s-8 (sand at 0.08 m3 m-3), and s-18 (sand at 0.18

m3 m-3). Ten transient temperature boundary conditions, were imposed on the vertical closed soil

column, they were 17.5-12.5, 20-10, 22.5-7.5, 25-20, 27.5-17.5, 30-15, 32.5-27.5, 35-25, 37.5-

33

22.5, 17.5-27.5oC, in each case corresponding to the top (left) and bottom (right) temperature

boundaries in the 10 cm soil column. The respective boundary conditions in the form of mean

temperature (oC)/temperature gradient (oC m-1) are 15.0/-50, 15.0/-100, 15.0/-150, 22.5/-50,

22.5/-100, 22.5/-150, 30.0/-50, 30.0/-100, 30.0/-150, and 22.5/100. The boundary conditions of

32.5-27.5oC were simulated, but not included in the calibration as in Heitman et al. (2008). Each

transient temperature boundary simulation lasted for 96 hours. The last boundary condition

(17.5-27.5oC) was also held for 96 hours but temperature gradient was reversed, and both �� and

� were recorded at 12, 24, 48 and 96 hours. Hence, there were eight transient boundary

conditions with eight recorded pairs of �� and �, together with another four recorded pairs in the

last boundary condition, to give a total of twelve pairs for each simulation set, e.g. sil-10. The

input parameters are listed in Table 3.1. The study assumed uniform water content and

temperature as in the initial condition. The 10 cm vertical column was spatially discretized into

200 cells, and 100s was used as the time-step size. Nine different temperature boundaries were

equally divided over 864 hours, together with the reverse temperature gradient for 96 hours, to

give a total simulation of the exactly 40 days.

3.3.3 Statistical measure for parameter calibration

Heitman and his co-workers conducted a single boundary condition model calibration for each

soil measured, and then used the single calibrated model to simulate other boundary conditions.

It is noted that the root mean square error (RMSE) used in their study to estimate the error of

simulation from the experiment, was a RMSE at a single boundary condition of the soil, and then

they re-calculated the RMSE again at a few additional boundary conditions. For instance, on silt

loam at 0.20 m3 m-3 water content, the RMSE was used to calibrate parameters at 22.5/-100 and

34

then the calibrated parameters were applied to forecast and re-calculate the RMSE at the

additional boundary conditions of 15/-50 and 22.5/-150. A similar procedure used in calibrating

the silt loam (0.10 m3 m-3) and the sand (0.08 and 0.18 m3 m-3). It should be noted that there were

nine boundary conditions, i.e. twelve pairs of �� and � data, available for each soil type at

specific water content, thus, the calibration approach used by Heitman and his co-workers could

not adequately optimize the parameters. As stated in section 3.2, the simultaneous calibration of

multiple boundary conditions could single out the influence of either the vapor enhancement

factor or hydraulic conductivity on the discrepancy observed between experimental and

simulated results. To allow comparison with the Heitman et al. (2008) simulation, the root mean

square error (RMSE) equation was used as the statistical measure for parameter calibration:

�� = �∑ (��)��

��

� (3.23)

where �� is the simulated data at cell � , �� is the experimental data at cell � , � is the

number of data.

3.4 Results and discussion

3.4.1 Comparison of numerical simulation and Heitman data

At the beginning, the soil water content was homogeneous throughout the vertical soil column.

Then, as soon as the simulation started, the first imposed temperature boundary (17.5-12.5oC)

resulted in the movement of water, vapor, and heat fluxes. The temperature gradient led to mass

and heat fluxes as described by Kroener et al. (2014). The high-temperature region produced

high water vapor concentrations and a flux towards the low-temperature region for water vapor

condensation. The water vapor flux carried a significant amount of heat flux that was necessary

to reduce the temperature in the high-temperature region. As the supply of liquid water from the

35

neighboring low-temperature region declined due to the reduced unsaturated hydraulic

conductivity by declining matric suction (low water content), the soil temperature in the warmer

region started to increase in response to low water content available for vaporization to remove

heat in the form of vapor. The counter balance behavior between temperature gradient and water

content gradient was similarly noted in other studies, for example, Sakai et al. (2009). This

explained the observation of low water content in the high-temperature region and high water

content in the low-temperature region, as in Figs. 3.1 and 3.2.

As noted earlier in section 3.2, the required calibrations were the hydraulic conductivity

and vapor enhancement factor. The saturated hydraulic conductivity (��), Eq. 3.19, and the zeta

(�) coefficient of vapor enhancement factor, Eq. 3.20, were calibrated, as in Heitman et al.

(2008). From now on, it is referred as Heitman. The simulation calibration has to compromise

between the accuracy of water content and temperature, as was also noted in Massman (2015).

The calibrations in Figs. 3.1 and 3.2 were based on the best combination of �� and � to achieve

the lowest root mean square error (RMSE) of water content, at the expense of temperature

accuracy if necessary. For instance, to achieve the lowest temperature RMSE for sil-20, the �

value would have to reduce to 4. A reasonably good match between simulation and Heitman data

for sil-20 was observed for the first eight temperature boundary conditions (Figs. 3.1(a) and

3.1(c) as forward boundary), but a greater discrepancy was found in Figs. 3.1(b) and 3.1(d) after

the reverse boundary condition was initiated. The forward boundary simulation of s-8 water

content (Fig. 3.2(a)) was not as good as for sil-20 (Fig. 3.1(a)), while the reverse boundary

simulation had similar characteristics to that of sil-20. Also, the sil-10 and s-18 could not

adequately reproduce low and high water contents at the close edge regions of high- and low-

temperature, respectively.

36

In the search for the lowest water content and temperature RMSE, a large number of

possible combinations of �� and � were investigated. The outcome of the simulations could be

used to suggest the most probable value for �� and � that would satisfy each soil type at different

water contents was discussed in the following section.

3.4.2 The uncertainty of hydraulic conductivity and vapor enhancement factor

Silt loam at an initial water content of 0.20 m3 m-3 was first simulated for parameter calibration

by Heitman at a single temperature boundary, i.e. 22.5/-100. The calibrated parameters (�� =

0.38 μm s-1, � = 4) were claimed to produce a low RMSE of water content and temperature, and

a reasonably good match with the experimental data. Then, the parameters were used to simulate

at other temperature boundaries, e.g. sil-20 at 15.0/-50 and 22.5/-150, and sil-10 at 22.5/-50 and

22.5/-150. The calculated RMSE for each boundary is shown in Table 3.2. Similarly, parameter

calibration for sand at 0.08 m3 m-3 was performed, initially, by Heitman at the 22.5/-100

temperature boundary. The parameters (�� = 9.8 μm s-1, � = 4) determined from the calibration

were used for simulations at s-8 (15.5/-50, 22.5/-150) and s-18 (22.5/-50, 22.5/-150) boundaries.

In the current study, following the same procedure, simulations for three of out the five

boundaries of silt loam were found to generate lower �� and � RMSE values, i.e. better accuracy,

than those estimated by Heitman (Table 3.2). Similarly, in the current study, simulations for three

out of the five boundary conditions for sand also generated lower RMSE than that found by

Heitman. Only the �� and � RMSE values for the five boundaries were reported by Heitman, but

our findings, at least, indicate the simulation code was performing properly. Note that for each

soil type and water content, e.g. sil-20, there were twelve pairs of �� and � data available for

calibration, as stated in section 3.3.2. The sil-20 calibration was based on three temperature

37

boundaries, while sil-10, s-8 and s-18 were calibrated with two, three and two temperature

boundaries, respectively. Clearly, some other temperature boundaries were not adequately

considered, for instance, the reverse temperature boundary.

After considering all the temperature boundaries, the best fitted �� simulation produced

different calibrated parameters (��, �) for sil-10 (3.8 μm s-1, 16), sil-20 (3.8 μm s-1, 12), s-8 (9.8

μm s-1, 6) and s-18 (5.0 μm s-1, 16). The calibrated parameters could be different when the

priority of best fitting was prioritized towards accurate � fitting. For instance, the sil-20 lowest �

RMSE was found at � = 4 , while maintaining the �� . Moreover, more uncertainty of the

parameters could be found when a selected individual temperature boundary was used as fitting

criterion. For example, sil-20 (15.0/-50) at �� = 3.8 μm s-1 and � = 16 produced 0.013 m3 m-3 of

�� RMSE that was lower than the one shown in Table 3.2. Therefore, it may not be the best

approach to focus on a specific temperature boundary to obtain the best fitting for either �� or �,

and then to further assume it as a good representation at other temperature boundaries.

In this study, different �� and � values were tested for each soil. For sil-10 and sil-20, all

possible combinations of �� values (0.1, 0.38, 3.8 μm s-1) and � values (2, 4, 6, 8, 10, 12, 14, 16),

were simulated. s-8 and s-18 soils were also simulated at all possible combination of �� and �

values, with �� values of 5.0, 6.0, 7.0 and 9.8 μm s-1 and � values that were the same as for the

silt-loam soils. The � values were limited to 2-16 for investigation, after considering the work of

Cass et al. (1984) and Heitman et al. (2008), while the �� values were set to equal to or lower

than the values, from Scanlon and Milly (1994) and Luckner et al. (1989). By taking an average

of the �� RMSE values at different �� for both sil-10 and sil-20, � = 6 was found to give the

most accurate simulation when compared to other � values. This could be written as

�� (sil-10,20) = 6. � = 2 was the best simulation for the average taken over � RMSE

38

values at different ��, i.e. �� (sil-10,20) = 2. �� = 3.8 μm s-1 was the best simulation for

the average of the �� RMSE values at different � in both sil-10 and sil-20, and �� = 3.8 μm s-1

was also valid for the average over the � RMSE values at different �. They can be respectively

simplified as �� (sil-10,20) = ��

(sil-10,20) = 3.8 μm s-1. Performing the

same procedure on s-8 and s-18, it was found �� (s-8,18) = 6 and �� (s-8,18) =

2, while �� (sil-10,20) = ��

(sil-10,20) = 9.8 μm s-1. If the procedure was

repeated to consider both the soils and the water contents, then �� (sil-10,20,s-8,18) = 6

and �� (sil-10,20,s-8,18) = 2. Overall, it could be concluded that � could be varied

between 2 and 6, and the exact �� value from the experiment should be used.

The simulation accuracy deteriorated with time and increasing mean temperature. This

trend was observed for the �� RMSE values from sil-10, sil-20 and s-8, and the � RMSE values

from sil-10 and sil-8. The reverse temperature boundary resulted in a greater deviation from the

exact data (Figs. 3.1(b), 3.1(d) and 3.2(b), 3.2(d)). The average �� RMSE (average of �� RMSE

values at different times) of the reverse boundary simulation was also found greater than the

forward boundaries for sil-10, sil-20 and s-18. When the average � RMSE was taken, a high

RMSE value for reverse simulation was found in sil-20 and s-18. Furthermore, taking the

average of both the soils and the water contents (sil-10, sil-20, s-8, s-18), the average �� RMSE

for forward simulation (0.021 m3 m-3) was less than for reverse simulation (0.0248 m3 m-3).

Similarly, the average � RMSE for forward simulation (0.4800 oC) was less than reverse

simulation (0.5385 oC). If the same procedure is applied to calibrate the forward and reverse

simulation parameters separately, but averaged over soils and water contents, the forward

�� (sil-10,20,s-8,18) = 6 and the reverse �� (sil-10,20,s-8,18) = 16, while the

39

forward �� (sil-10,20,s-8,18) = 4 and the reverse �� (sil-10,20,s-8,18) = 14. The

apparently large error in the reverse boundary simulation that requires a high value of � to

compensate, it suggests that some other water vapor phenomenon may need to be accounted for.

3.4.3 Limitation and possible improvement

The previous section 3.4.2 has shown the difficulty encountered when trying to narrow down the

influence of either hydraulic conductivity or vapor enhancement factors as the cause of the

simulation discrepancy from the real physical data. The examination of the �� value indicates the

exact experimental value should be used rather than reducing its value as conducted in Heitman.

In fact, sil-10 and s-8 showed improvement in the �� RMSE value when the �� values were

increased to 15.8 and 18.8 μm s-1, respectively. These values were excluded from the

investigation because such high values would not be a real physical representation of the

experiment. Another aspect for consideration was the � value used in the vapor enhancement

factor. Irrespective of the soil type over different water contents, the � value changed from 2 to 6

when the priority of accuracy was shifted from � to �� RMSE. When a distinction was made

between forward and reverse boundary, the � value could be varied between 4 or 6 and 14 or 16,

respectively. Such an uncertainty was partly due to the result of the inability of researchers to

properly define the vapor enhancement factor. Hence, it was open to speculation (Cahill and

Parlange, 1998, Parlange et al., 1998, Shokri et al., 2009) and ultimately be one of the parameters

to be calibrated in simulation (Heitman et al., 2008, Sakai et al., 2009). Further effort to better

define the mechanism guiding the vapor enhancement factor so that it was able to be measured

through experimentation would definitely remove the parameter as one of the uncertain

parameters for calibration. The greater � value was required in the reverse than the forward

40

simulation could be partly due to the absence of buoyancy-induced flux. The buoyancy

phenomenon was also neglected in the pore scale mechanisms study (Shahraeeni and Or, 2012).

Moreover, the assumptions establishing the derivations of Eqs. 3.1 and 3.5 could be re-

examined due to the weakness of the mass and heat equations that could possibly be partly

responsible for the discrepancy observed. The equations have similar drawbacks as those stated

in Nassar and Horton (1997) that soil was assumed to be rigid and inert, and they the neglected

hysteresis of the water retention curve. It assumed a local thermodynamic equilibrium that the

water, vapor, solid and air have the same temperature. The equation was limited to one-

dimensional flow in the column test. The force driving mass and heat fluxes is governed by

gravity, water content gradient, temperature gradient, heat conduction, sensible heat flux and the

latent heat flux. Also, the current model assumed soil water vapor in equilibrium with liquid

water so that the phase transition from liquid to vapor was instantaneous (Massman, 2015).

Although not all of those limitations could be significantly affecting the simulation, the

discrepancy observed between the simulation and the physical measure data would remain until

the key factors are further considered.

The obvious key factors that should be addressed first are the unsaturated hydraulic

conductivity and the vapor enhancement factor. The poor fitting of �� at the edges of dry and wet

regions, estimated by the Campbell (1974) equation, could be improved with the Fayer and

Simmons (1995) equation that was modified from van Genuchten (1980) to improve the estimate

at low water content (Tuller and Or, 2001). In order to adequately describe the flow of water in

dry soil, the measurement methods suggested by Sakai et al. (2009) could be applied, such as the

water flow under accelerated gravitational conditions, multistep outflow method, and the

evaporation method. Also, the theoretical re-examination of the vapor enhancement factor could

41

be established from the work of Cass et al. (1984) that built on the theory of heat flux (de Vries,

1958). The Darcy law extended by Klinkenberg (1941) to account for the gas slippage effect as a

possible alternative to Fick’s law should also be examined, as attempted in Ho and Webb (1999).

3.5 Conclusions

The transport equations of water, vapor, and heat were successfully extended from the Richards’

equation. The lesson learned from Chapter 2 was implemented in this study by applying a small

spatial discretization size to improve the model accuracy. The new code could predict

temperature and water content distributions that the overall error, i.e. RMSE, was less than those

reported in Heitman.

The simulation results revealed some previously unknown factors. A single boundary

condition could not provide an adequate calibration when multiple boundaries are available for

the calibration because the multiple boundaries calibration was able to identify the parameters

that provide the best compromise for all the boundaries. The best-calibrated parameters (��, �)

for sil-10, sil-20, s-8 and s-18 were different, and further uncertainty arise when accuracy has to

prioritize either the temperature or the water content distribution. The simulation accuracy

deteriorated over time and with the increasing average temperature, with the exception of the

reverse temperature boundary simulation. The high � value, as part of the vapor enhancement

factor, required for reverse boundary simulation implied that possibly some transport

mechanisms were present such as buoyancy force and water retention hysteresis. Rather than a

single value for the �, as reported in Heitman, the current study revealed that the value was

varied between 2 and 6. The second calibrated parameter, �� was found to be best when

maintained at its original value, whereas Heitman found the value had to be reduced to improve

42

fitting, e.g. 90% reduction of the silt loam �� value. Although the reason for such discrepancy is

not clear, the findings of the current study should prevail because the �� value was justified

based on the lowest error after averaging the twelve soil boundary RMSE values for two

different water contents. Furthermore, the inability of the current model to simulate water content

distribution at dry and wet ends could potentially be improved using the Fayer and Simmons

(1995) equation.

Of all the uncertainties found, the vapor enhancement factor was one of the least known

parameters, primarily because the mechanism involved was unclear and it is continuously open

to speculation (Heitman et al., 2008, Ho and Webb, 1999, Parlange et al., 1998). Also, the

complexity of the theory and experimentation further complicating the investigation. The

published experimental data from Cass et al. (1984) would be an important step to re-examine

the theory, and potentially be used to derive and validate a new mathematical expression to

describe the vapor enhancement factor.

43

(a) (b)

(c) (d)

Figure 3.1. Silt loam (sil-20) simulated: (a) water content and (c) temperature distributions at

temperature boundary of 15/-50, 15/-150 and 30/-150, which were 17.5-12.5, 22.5-7.5oC and

37.5-22.5oC, respectively. The (b) water content and (d) temperature distribution at reverse

temperature boundary of 22.5/100, i.e. 17.5-27.5oC, after 12, 48 and 96 hours. Note: the initial

water content set at 0.20 m3 m-3. The 63.8 10sK m s-1 and zeta (�) = 12 used in the

simulation.

0.05

0.10

0.15

0.20

0.25

0.30

0 0.05 0.1

Wat

er c

on

ten

t (m

3/m

3)

Distance from warm end (m) 15/-50 15/-150

30/-150 Sim (15/-50)

Sim (15/-150) Sim (30/-150)

0.05

0.10

0.15

0.20

0.25

0.30

0 0.05 0.1

Wat

er c

on

ten

t (m

3/m

3)

Distance from warm end (m) R12, 22.5/100 R48, 22.5/100

R96, 22.5/100 Sim (R12, 22.5/100)

Sim (R48, 22.5/100) Sim (R96, 22.5/100)

280

290

300

310

0 0.05 0.1

Tem

per

atu

re (

K)

Distance from warm end (m) 15/-50 15/-150

30/-150 Sim (15/-50)

Sim (15/-150) Sim (30/-150)

290

295

300

305

0 0.05 0.1

Tem

per

atu

re (

K)


R96, 22.5/100 Sim (R12, 22.5/100)

Sim (R48, 22.5/100) Sim (R96, 22.5/100)

44

(a) (b)

(c) (d)

Figure 3.2. Sand (s-8) simulated: (a) water content and (c) temperature distributions at

temperature boundary of 15/-50, 15/-150 and 30/-150, which were 17.5-12.5, 22.5-7.5oC and

37.5-22.5oC, respectively. The (b) water content and (d) temperature distribution at reverse

temperature boundary of 22.5/100, i.e. 17.5-27.5oC, after 12, 48 and 96 hours. Note: the initial

water content set at 0.08 m3 m-3. The 69.8 10sK m s-1 and zeta (�) = 6 used in the

simulation.

0.00

0.05

0.10

0.15

0.00 0.05 0.10Wat

er c

on

ten

t (m

3/m

3)

Distance from warm end (m)

15/-50 15/-150

30/-150 Sim (15/-50)

Sim (15/-150) Sim (30/-150)

0.00

0.10

0.20

0 0.05 0.1Wat

er c

on

ten

t (m

3/m

3)


R96, 22.5/100 Sim (R12, 22.5/100)

Sim (R48, 22.5/100) Sim (R96, 22.5/100)

280

290

300

310

320

0.00 0.05 0.10

Tem

per

atu

re (

K)


15/-50 15/-150

30/-150 Sim (15/-50)

Sim (15/-150) Sim (30/-150)

290

295

300

305

0.00 0.05 0.10

Tem

per

atu

re (

K)


R12, 22.5/100 R48, 22.5/100

R96, 22.5/100 Sim (R12, 22.5/100)

Sim (R48, 22.5/100) Sim (R96, 22.5/100)

45

Table 3.1 Input parameters used for silt loam and sand.

Parameter Silt loam Sand

Wetting differential heat (�, m2 s-2) 4620 200

Bulk density (��, kg m-3) 1200 1600

Organic mass fractions (��) 0.044 0.006

Mineral mass fractions (��) 0.956 0.994

Soil porosity (�, m3 m-3) 0.547 0.396

Soil clay fraction (�) 0.249 0.011

Saturated hydraulic conductivity (��, m s-1) 3.8 x 10-6 9.8 x 10-6

Soil thermal conductivity, Eq. 3.10 fitting parameters

��

��

��

-0.952

-4.31

6.00

-0.394

-5.11

7.23

Water retention curve-fitted parameters

�� (m)

�

-0.13

6.53

-0.03

3.38

Note: parameters compiled from Heitman et al. (2008).

46

Table 3.2 The comparison of water content and temperature root mean square errors (RMSEs)

from Heitman et al. (2008) and current simulation, at few temperature boundary conditions.

��(m�m��), �(K) RMSE

Heitman et al. (2008) Current Simulation

Calibrated at �� = 0.38 μm s-1, � = 4

Silt loam at 0.20 m3 m-3

22.5oC/-100oC m-1 0.013, 0.58 0.010, 0.38

15.0oC/-50oC m-1 0.019, 0.18 0.017, 0.15

22.5oC/-150oC m-1 0.011, 0.91 0.010, 0.45

Silt loam at 0.10 m3 m-3

22.5oC/-50oC m-1 0.014, �� 0.018, 0.10

22.5oC/-150oC m-1 0.017, �� 0.030, 0.78

Calibrated at �� = 9.8 μm s-1, � = 4

Sand at 0.08 m3 m-3

22.5oC/-100oC m-1 0.049, 0.60 0.023, 0.48

15.5oC/-50oC m-1 0.012, 0.14 0.012, 0.23

22.5oC/-150oC m-1 0.023, 1.28 0.021, 0.81

Sand at 0.18 m3 m-3

22.5oC/-50oC m-1 0.009, 0.07 0.010, 0.29

22.5oC/-150oC m-1 0.016, 0.43 0.014, 0.38

Note: �� referring to as not available.

47

Chapter 4. An improved heat flux theory and mathematical equation to

estimate water vapor volume expansion-advection as an alternative to

mechanistic enhancement factor2

4.1 Abstract

The current study extended the theory of heat flux which has been normally used to assess the

mechanistic enhancement factor. The present study improved heat flux theory to include three

new phenomena that were excluded from the simplified heat flux equation. Those three

phenomena were sensible heat by liquid water movement, sensible heat by water vapor

movement, and the effects of partial derivative of matric pressure head on relative humidity with

respect to temperature. The first phenomenon was found to be an important factor in relatively

wet porous media, whereas the third phenomenon was found important in near dry porous media

condition. Moreover, mathematical descriptions were derived to allow direct conversion of the

mechanistic enhancement factor to a water vapor volume expansion-advection term. The study

used the basic mass balance equation, ideal gas law, and water vapor advection by air volume

expansion to describe the water vapor volume expansion-advection mechanism. The water vapor

volume expansion-advection equation was found to be able to describe the influence of soil

moisture content and air pressure. Also, a bell-like curve of water vapor relative permeability

was needed to comply with the Cass et al. (1984) experimental data. A mathematical function

was proposed as a temporary solution. Further investigation of the water vapor relative

permeability mathematical relationship may be necessary to improve the estimation at saturated

soil moisture content, and also, to reveal its theoretical foundation.

2 This chapter has been published in Goh and Noborio (2016).

48

4.2 Introduction

Philip and de Vries (1957) was the first to suggest the presence of a vapor enhancement factor in

soil. A mechanistic enhancement factor ( ) was used to describe the discrepancy between

observed and calculated vapor diffusion by Fick’s law, in the presence of a temperature gradient.

The liquid island effect between solid particles, and the local temperature gradients in the air-

filled pores that were higher than the average temperature gradients of the bulk soil were claimed

to be responsible for the factor. Alternatively, Cary and Taylor (Cary, 1963, Cary, 1964, Cary,

1965, Cary and Taylor, 1962) derived a phenomenological enhancement factor ( ), based on

irreversible thermodynamics, to study similar phenomena. The air porosity and the tortuosity

effect (Penman, 1940) of porous media could be used to relate the mechanistic enhancement

factor to the phenomenological enhancement factor. Therefore, a solution to the mechanistic

enhancement factor alone is sufficient, and it can be translated into the phenomenological

enhancement factor when necessary.

Cass et al. (1984) proposed a simplified theory of the heat flux equation by assuming

only the presence of heat conduction and latent heat transfer, without the presence of liquid water

movement. They applied the assumption to study the effect of the mechanistic enhancement

factor on lysimeter sand and Portneuf silt loam. Their findings demonstrated that their proposed

mathematical function was able to model the variation of the mechanistic enhancement factor

with soil moisture content. Hiraiwa and Kasubuchi (2000) utilized the simplified heat flux theory

to study the effect of temperature on thermal conductivity. Sakaguchi et al. (2009) also used the

theory to estimate vapor flux of the heat pipe phenomenon in Ando soil. Similarly, Lu et al.

(2011) implemented the simplified heat flux theory and the mathematical function to the

estimation of the mechanistic enhancement factor of the silty clay soil and correlated it with the

49

soil moisture content. However, Shokri et al. (2009) found that liquid water movement alone is

sufficient to account for the mechanistic enhancement factor.

Ho and Webb (1996) defended the presence of vapor enhancement diffusion through a

simple mass balance estimation on pore-scale analysis. The analysis estimated that vapor

enhancement diffusion could be ten times the amount of diffusion by Fick’s law. The subsequent

work of Ho and Webb (1999) further established the presence of vapor enhancement diffusion

through experiments with pore scale, fracture aperture, and porous media. The research findings

on pore-scale modeling generated data showing the relationship between the mechanistic

enhancement factor and modeled water saturation. Ho and Webb (1999) proposed that the

mathematical relationship given in McInnes (1981) and Cass et al. (1984) was a preferable

function. The research work of Cass remained valid, and it established a simplified heat flux

theory and a mathematical formulation to relate the mechanistic enhancement factor with soil

moisture content in the study of vapor enhancement diffusion. However, the assumptions used to

establish the simplified heat flux theory are a possible drawback. Clearly, further research is

required to test the validity of the assumptions.

Ho and Webb (1996) have highlighted drawbacks in the vapor diffusion studies. One of

the suggestions for improvement was to use the numerical code TOUGH2 (Pruess, 1991) that

could be applied in geothermal, environmental remediation, and nuclear waste management. The

code consisted of a pressure gradient term used to govern vapor advection that was able to

account for part of the observed vapor enhancement diffusion. The subsequent work of Ho and

Webb (1999) suggested the use of an advection-dispersion model to govern vapor enhancement

phenomenon, but they claimed the dusty-gas model to be more superior in terms of accuracy.

Webb and Pruess (2003) found the diffusive flux of Fick’s law comparable to the dusty-gas

50

model after including the Knudsen effect. A complete description of the dusty-gas model

comprising the Knudsen effect, viscous flow, and Fick’s law is available in Cunningham and

Williams (1980) and Mason and Malinauskas (1983). Furthermore, the research findings of

Cahill and Parlange (1998) and Parlange et al. (1998) found discrepancies between field

observations and the prediction of the Philip and de Vries (1957) model. The discrepancy

resulted from the thermally-driven advection transport mechanism that was governed by

alternating expansion and contraction of the soil air. Clearly, additional investigations of Fick’s

law including the Knudsen effect and advection term are necessary.

In this study, we investigated the assumptions used by Cass et al. (1984) in formulating

the simplified heat flux theory and formulae. From now on it is referred as Cass. The

assumptions have neglected three phenomena. These are sensible heat by liquid water transfer,

sensible heat effect by water vapor transfer, and the partial derivative of matric pressure head

effects on relative humidity with respect to temperature. This study employed a basic mass

balance approach, ideal gas law relation, and air volume expansion-advection mechanism to

justify the necessity of a new mechanism to account for the water vapor enhancement effect. A

mathematical formula was established to relate the mechanistic enhancement factor to the water

vapor volume expansion-advection mechanism.


4.3.1 Cass experimental setup, parameter estimation, relation to the present study and

limitation

Cass set up a research method to allow estimation of thermal conductivities at various soil

pressures (0, 103, 517, and 1034 kPa gauge), moisture contents (0.002, 0.009, 0.021, 0.024,

51

0.054, 0.060, 0.061, 0.108, 0.186, and 0.190 m3 m-3), and temperatures (average temperature of

3.5, 22.5, and 32.5oC). Stainless steel tubes (1.5 mm wall thickness, 47 mm inner diameter, and

230 mm length) were used to contain the sieved and repacked soil samples (lysimeter sand, and

Portneuf silt loam) and they were inflated to the preset gauge pressure. The pressured-steel tube

first was equilibrated in one bath to a preset temperature and then was immersed in another bath.

A temperature difference of 5oC was imposed on each experimental condition. Copper

(Evenohm) constantan thermocouples were used to measure temperatures within and outside the

steel tube. A data logger used to store the data every 12 s until the maximum temperature

difference of 5oC fell to near 0oC. The soil thermal conductivity was estimated by the Riha et al.

(1980) method.

Thermal conductivity values estimated for particular soil moisture content at a few

discreet pressures allowed approximation of the thermal conductivity under different pressure

environments. Extrapolation of a linear line to a vertical intercept on a thermal conductivity

versus pressure ratio ( rP ) graph facilitated the derivation of thermal conductivity at an extreme

pressure condition. The thermal conductivity value estimated at the extreme pressure condition

had significant importance because it represented the heat flux of the media without the presence

of water vapor. Thermal conductivity values at different pressures, except at the extreme

pressure, included the phenomena of three phases. They were the liquid water movement

(sensible heat), wet porous media (liquid water plus dry porous media) heat conduction, and

water vapor heat fluxes (latent heat and sensible heat). The Cass method allowed the isolation of

thermal conductivity of water vapor from the other heat fluxes.

In general, water vapor transfer was assumed to be governed by the diffusion equation.

The assumption required a mechanistic enhancement factor to be applied to the diffusion

52

equation to compensate for the excessively large value contributed by the water vapor flux. In

this study, we re-examined the heat flux theory to reconstruct the temperature gradient heat flux

equation, and also to identify the physical phenomena lacking in the simplified heat flux

provided from Cass. Also, the mathematical equation was derived and validated using the results

of Cass experiments at different pressures and soil volumetric water contents. The present study

used the published data on lysimeter sand and Portneuf silt loam from Cass.

4.3.2 Theory of heat flux

Cass adopted the method of Parikh et al. (1979) to conduct measurements of the thermal

conductivity as a function of temperature, soil moisture content, and pressure. The heat flux

theory used in the calculation of the mechanistic enhancement factor was a simplified version of

the heat flux density flow given in de Vries (1958). From now on it is referred as deVries. The

complete expression of heat flux density given by deVries is as follows, except for the last term

on the right side of the equation:

*h o v v o v L o L L o r

dTq L q c T T q c T T q c T T q

dx (4.1)

where hq is the total heat flux density (J m-2 s-1), * is the thermal conductivity in the porous

media in the presence of liquid water (J m-1 s-1 K-1), oL is the latent heat (J kg-1) of water at

temperature oT (K), vq is the water vapor flux (kg m-2 s-1), vc is the specific heat of water vapor

at constant pressure (J kg-1 K-1), Lc is the specific heat of liquid water (J kg-1 K-1), Lq is the liquid

water flux (kg m-2 s-1), and rq is the liquid water flux by root (kg m-2 s-1).

On the right side of Eq. 4.1, the first term is the heat flux from pure heat conduction, and

the second term is the latent heat flux by water vapor movement. The third term is the sensible

53

heat flux by water vapor movement while the fourth term is the sensible heat flux by liquid water

movement. The fifth term is the sensible heat flux by root water movement. This fifth term was

not important in the vapor enhancement studies because the soils used in Cass experiments were

mainly clay, silt, and sand, excluding the influence of roots, e.g. Caldwell et al. (1998). In Cass’s

research, pure heat conduction and transfer of latent heat by vapor movement were the only flux

terms assumed to govern the heat transfer used in the mechanistic enhancement studies. Cass

excluded the third and fourth terms in his study. In the present study, the first four terms on the

right-hand side of Eq. 4.1 were included in the calculation of the mechanistic enhancement factor.

Water vapor diffusion is governed by Fick’s law (Fick, 1995, Jackson et al., 1963) and it

is an isothermal equation. Under the influence of spatial temperature gradient, Fick’s law can be

rewritten as follows (Cass et al., 1984, Philip and de Vries, 1957):

vv a

d dTq D

dT dx

(4.2)

where is the mechanistic enhancement factor that is used to account for increased vapor

diffusion over the ordinary Fick’s law equation, D is the molecular diffusivity of water vapor in

air (m2 s-1), is the tortuosity factor, dimensionless, a is the volumetric air content of the

media (m3 m-3), and v is the water vapor density (kg m-3).

According to Eq. 4.3, the water vapor density is affected by saturated water vapor density

( s , kg m-3) and matric pressure head effects on relative humidity ( mh , dimensionless). In the

chain rule, the partial derivative of water vapor density with respect to temperature resulted in

Eq. 4.4. Philip and de Vries (1957) introduced the second term on the right side of Eq. 4.4. Later,

Nakano and Miyazaki (1979) proposed the first term of the right side of Eq. 4.4.

v s mh (4.3)

54

v m ss m

hh

T T T

(4.4)

The liquid water flux was governed by the Buckingham-Darcy flux law (Jury and Horton,

2004). Similar to Eq. 4.2, under the influence of temperature gradient the Buckingham-Darcy

flux law became:

mL L

d dTq K

dT dx

(4.5)

where L is the soil liquid water density (kg m-3), K is the hydraulic conductivity (m s-1), and

m is the matric pressure head (m).

By placing Eqs. 4.2-4.5 into Eq. 4.1, and performing a simple mathematical

rearrangement, an improved heat flux equation could be rewritten as follows:

*m m s

h L L a o v s m

d h dTq c T K D L c T h

dT T T dx

(4.6)

Cass measured thermal conductivity at a temperature gradient, and at different soil

moisture contents, and pressures. The method used was based on Riha et al. (1980). At each soil

moisture content and temperature, thermal conductivity values were plotted against the ratio of

the atmospheric pressure to the experimentally imposed pressure values. From the curve-fitted

straight line on the graph, the thermal conductivity at an extreme pressure was determined by

extrapolating to the vertical intercept. The thermal conductivity at extreme pressure should be

given by the first two terms of Eq. 4.6, instead of only the first term as used by Cass. This is

because those two terms are governed by solid and liquid of heat movement.

The simplified heat flux equation proposed by Cass was as follows:

*s

h a o m

dTq D L h

T dx

(4.7)

55

Equation 4.7 excludes the influence of sensible heat in the water vapor movement and

liquid water movement, and the partial derivative of matric pressure head effects on relative

humidity with respect to temperature. In this study, the mechanistic enhancement factor

estimated by Eqs. 4.6 and 4.7 were compared.

4.3.3 The mass balance relationship between liquid water and water vapor, and its effect

on water vapor volume expansion-advection

If a unit mass of liquid water vaporizes fully into the air, the logical assumption is that the

vaporized mass of liquid water is equivalent to the mass of vapor in the air. By knowing that

mass is a product of density and volume, equating both masses of liquid water and water vapor,

the following simple mass balance relation was produced:

vaporwater

vapor water

V

V

(4.8)

It is a known fact that the density of liquid water is greater than the density of water

vapor. For instance, at 20oC, saturated water vapor density is 0.0229 kg m-3, calculated from the

Kimball et al. (1976) equation while the liquid water density is 996.55 kg m-3. The ratio given by

Eq. 4.8, at the temperature, indicates that water vapor volume is 43,578 times greater than the

volume of liquid water. Hence, the volume occupied by water vapor is significantly greater than

the volume of liquid water. The simple mass balance equation indicates that a mass of liquid

water transformed into water vapor would take up a larger volume of pore space than the same

mass in the form of liquid water. Under the constant air pore space in porous media, the

expanding water vapor volume, resulted from vaporizing liquid water, was directly translated to

the volume expansion-advection of water vapor apart from vapor diffusion by Fick’s law.

56

Therefore, the temperature related variation in density and volume, between liquid and vapor

phases, was the reason for water vapor volume expansion-advection. The volume expansion-

advection term could be governed by Darcy law as described in Ho and Webb (1996) and Ho and

Webb (1999).

4.3.4 The presence of volume expansion-advection from the perspective of ideal gas law

The ideal gas law is as follows:

PV nRT (4.9)

where P is the pressure of the gas (Pa), V is the volume of the gas (m3), n is the amount of gas

in moles, and R is the ideal gas constant (J K-1 mol-1).

By taking the partial derivative of Eq. 4.9 with respect to temperature, it resulted in the

following form:

V P nP V nR RT

T T T

(4.10)

where Eq. 4.10 is based on the rationale that the increase in temperature leads to a rise of the

water vapor moles, due to vaporization. The increase in water vapor moles would either be

accompanied by an increase in the volume of gas phase or gas partial pressure or a combination

of both in equilibrium.

Consider a column of soil, under the influence of a temperature gradient, the side of the

column with higher temperature has greater vaporization of liquid water than the other end with

lower temperature. In a soil column under constant pore-space, the evaporation of liquid water to

water vapor would result in high partial pressure of water vapor and water vapor density as

governed by Eq. 4.10. Hence, it implies that partial pressure gradients of water vapor and water

vapor density gradient are present in the soil column. The pressure gradient governing the

57

volume expansion-advection flow would be an adequate approach to represent the vapor

enhancement factor. Clearly, it indicates that the temperature gradient introduced to the system

has resulted in volume expansion-advection of water vapor.

4.3.5 The presence of air and water vapor volume expansion-advection

The mass balance of air equation and the ideal gas equation from the research work of Parlange

et al. (1998) were implemented to support the claim that volume expansion-advection of water

vapor does exist. The use of the mass balance equation of air resulted in the following relation:

air airair air air airu

t t x

(4.11)

where air is the density of air (kg m-3), air is the porosity of air space (m3 m-3), and airu is the

velocity of air (m s-1).

By taking the partial derivative of equation air air airP M RT , converted from Eq. 4.9,

with respect to time, and assuming that pressure does not change with time, it would give the

following relationship:

air air T

t T t

(4.12)

Puting Eq. 4.12 into Eq. 4.11 would generate the following relationship that is the

equation used by Parlange et al. (1998) to estimate air volume expansion:

2

1

1 xair air

air air airxair

Tu x

T t t

(4.13)

They assumed that the air velocity given in Eq. 4.13 governs water vapor flow. Clearly,

air volume expansion caused water vapor advection is present as a result of temperature variation

in the soil.

58

4.3.6 Water vapor volume expansion-advection equation

The equation governing water vapor volume expansion-advection could be described by Darcy

law (Muskat and Wyckoff, 1946):

p vv v

L v v

k gq Pd

dx g

(4.14)

where vq is the water vapor flux (kg m-2 s-1), pk is the gas permeability (m2), g is the gravity

acceleration, 9.81 m s-2, v is the dynamic viscosity of water vapor (kg m-1 s-1), and vP is the

partial pressure of water vapor (kg m-1 s-2).

Equation 4.13 was derived by assuming the pressure of air as a constant value. Hence,

the change of air temperature directly affects the air density without any impact on the air

pressure. The general relationship occurs by allowing the effect of temperature on either water

vapor pressure or water vapor density, or a combination of both. The general form of the ideal

gas law before applying the partial derivative with respect to time is as follows:

v

v v

P RT

M (4.15)

where vM is the molecular weight of water vapor, 0.01801528 kg mol-1.

By applying Eq. 4.15 to Eq. 4.14, it allows any variation of partial pressure and density

of water vapor to have a direct relationship with the temperature of water vapor. Hence, it brings

about the following relationship:

p r L v

v

v v

k k R dTq

M dx

(4.16)

where rk is the relative permeability, dimensionless, and it was applied as in Ho and Webb

(1999).

59

4.3.7 The relation of mechanistic enhancement factor and water vapor volume expansion-

advection

From the simple mass balance equation, the ideal gas law, and air volume expansion, it is evident

that water vapor is not only diffused through porous media, but also that the water vapor volume

expansion-advection mechanism is present. The nature of the experimental setup by Cass

enabled estimation of the mechanistic enhancement factor due to the temperature gradient effect.

Water vapor volume expansion-advection contributed to the enhancement factor. Therefore, the

mechanistic enhancement factor could be detached from the water vapor diffusion term, in Eq.

4.6, to form a new term, i.e. water vapor volume expansion-advection term, as follows:

m sa o v s m

hD L c T h

T T

m so v a o v s m

hf L c T D L c T h

T T

(4.17)

where f is the water vapor volume expansion-advection factor. In simplified form:

1 m sa s m

hf D h

T T

(4.18)

From Eq. 4.7, similarly it is written as:

1 sa mf D h

T

(4.19)

Equation 4.18 is only suitable for newly constructed experimental research while Eq.

4.19 can be used for both new experimental data analysis, and also the existing published data.

To utilize Cass published data, Eq. 4.20 is necessary. The first term in Eq. 4.20 represents the

water vapor diffusion of the simplified heat flux, and the second term represents the water vapor

diffusion of the improved heat flux. The equation used in the current study is as follows:

60

a o m s m s

a s m

o v

D L h hf D h

L c T T T T

(4.20)

In the present study, Eqs. 4.18-4.20 are named as improved, simplified, and adapted-

improved water vapor volume expansion-advection factors, respectively.

The water vapor volume expansion-advection factor ( f ) from the relationship in Eq.

4.16:

p r L v

v v

k k Rf

M

(4.21)

After rearranging Eq. 4.21 it becomes:

p rv v

o L v o

k kf M

k R k

(4.22)

According to Ho and Webb (1999), the permeability is used for the slip or Knudsen

effect, and it has the Klinkenberg factor (b ). The permeability used is as follows:

1 ep o rk k bP (4.23)

where ok is the intrinsic permeability (m2), r oP P P is the pressure ratio where oP is the

atmospheric pressure at 101325 Pa and P is the imposed air pressure when measuring the

thermal conductivity, and e is a fitting parameter.

The relative permeability is a function of soil moisture content. Hence, the following

empirical relationship was proposed to govern the relationship between relative permeability and

soil moisture content:

242

1 2 3

Lr L

L L

k aa a a

(4.24)

The left side of Eq. 4.22 is the dependent variable, while the pressure ratio and soil

61

moisture content are independent variables. The relationship was curve-fitted to determine

variable constants for 1a , 2a , 3a , 4a , b , and e . It is important to note that at zero soil volumetric

water content, the relative permeability, given by Eq. 4.24, would be zero, which would

terminate the water vapor volume expansion-advection. At fully saturated soil moisture content,

the relative permeability would fail to generate a zero value.

van Genuchten-Mualem (Mualem, 1976, van Genuchten, 1980), van Genuchten-Burdine

(Burdine, 1953, van Genuchten, 1980), Corey (Corey, 1954), and Falta (Falta et al., 1989),

advocated other relative permeability equations, shown below, respectively:

20.5 1

1 1 1 1

mm

L Lr

s s

k

(4.25)

2 1

1 1 1 1

mm

L Lr

s s

k

(4.26)

2 2

1 1L Lr

s s

k

(4.27)

3

1 Lr

s

k

(4.28)

4.4 Results and Discussion

4.4.1 The general trend of thermal conductivities

The thermal conductivity of water vapor was lower than the combined thermal conductivities of

wet porous media and liquid water sensible heat. The wet porous media consisted of heat

conduction from liquid water and dry porous media. For instance, in lysimeter sand, the water

62

vapor thermal conductivity had about 0.03-59.6% of the thermal conductivities of wet porous

media and liquid water sensible heat combined, at pressure ratio ( oP P P ) of 1.0. The

variation in water vapor thermal conductivity narrows to 0.03-31.6, 0.03-9.56, 0.03-4.98% at the

respective pressure ratios of 0.53, 0.16, and 0.083. The pressure ratio was given by atmospheric

over air pressure. For example, 1.0 pressure ratio is at atmospheric pressure, whereas the 0.083

pressure ratio is a much higher air pressure than the atmospheric pressure. The high percentage

of water vapor thermal conductivity at low pressure indicated high water vapor transfer. A

similar observation was demonstrated in the Portneuf silt loam. Also, water vapor thermal

conductivity values initially rose at low volumetric water content, before curving downward at

higher soil moisture content observed under various pressure conditions (Fig. 4.1). Campbell

(1985) reported a similar trend for sand, silt loam, and clay soils. However, at extremely high

pressure, both the wet porous media and liquid water sensible heat thermal conductivities

increased with the increasing soil volumetric water content, without falling trend (Fig. 4.1). The

downward curve behavior is consistent with the physical expectation of porous media because as

the soil moisture content increases less air space is available for vapor flux. Hence, at saturated

soil moisture content, the water vapor thermal conductivity should approximate to zero.

4.4.2 The wet porous media thermal conductivity by heat conduction, and the sensible

heat thermal conductivity by liquid water movement

The thermal conductivity at extreme pressure consists of two components, there are thermal

conductivities of wet porous media (liquid water plus dry porous media) and sensible heat by

liquid water movement. They are respectively the first and second terms on the right side of Eq.

4.6. Both thermal conductivities increased with increasing soil moisture content in the porous

63

media. The thermal conductivity by liquid water movement is much less than that of the wet

porous media thermal conductivity, but the increment rate of the former is far greater than the

latter. For instance in a near dry porous region, i.e. 0.014 m3 m-3, the liquid water movement had

125.33 10 % of the wet porous media thermal conductivity value. In the wet region, i.e. 0.295

m3 m-3, it increased to 0.039%. The highest soil moisture content studied was about 65.5% of

fully saturated soil volumetric water content. Hence, at soil moisture content greater than the

studied value, more than 0.04% of liquid water movement over wet porous media thermal

conductivity would be expected. The Portneuf silt loam also demonstrated a similar trend. The

observation indicated that thermal conductivity by liquid water movement is a more important

factor in the wet than in the near dry porous media regions. However, the liquid water movement

thermal conductivity is one of the three factors that were excluded from the simplified heat flux

theory by Cass.

4.4.3 The thermal conductivity of water vapor diffusion and volume expansion-advection

The thermal conductivity of water vapor is composed of two parts - water vapor diffusion, and

water vapor volume expansion-advection. The water vapor diffusion was solved by Fick’s law,

and the water vapor volume expansion-advection by Darcy law was used as the remaining vapor

flux. The observed thermal conductivities of water vapor diffusion and volume expansion-

advection decreased with increasing pressure. The finding is consistent with the earlier

observation that the high pressure condition reduced the availability of water vapor. Therefore,

less water vapor is available for diffusion and volume expansion-advection. In a near dry porous

media region, i.e. 0.014 m3 m-3, the diffused water vapor thermal conductivity from the improved

heat flux equation increased by 7-12% from the values of water vapor diffusion derived from

64

simplified heat flux. At other wet regions, the thermal conductivity of water vapor diffusion was

only increased by 0.3-0.5%. On the other hand, the volume expansion advected water vapor

thermal conductivity at the near dry porous media region was found to decrease by 10-90% from

those conductivities using the simplified heat flux equation. In the wet region, the thermal

conductivity shrank by 0.03-0.3%. When volume expansion-advection and diffusion were

compared, the volume expansion-advection of water vapor thermal conductivity was about 1 to

67% of the values of water vapor diffusion, at the near dry region 0.014 m3 m-3. In the wet

region, it could reach as high as 1600%. In short, the percentage increased with increasing soil

moisture content. Therefore, the diffusion of water vapor thermal conductivity is a major

contributor to water vapor flux only in the near dry porous media region. The importance of

water vapor diffusion was enhanced through the utilization of the improved heat flux theory. The

new terms that contributed to the enhancement of diffusion in the near dry region were excluded

from the simplified heat flux by Cass. The new terms were the sensible heat flux by water vapor

movement, and the partial derivative of matric pressure head effects on relative humidity with

respect to temperature, which were shown in the improved heat flux theory in Eq. 4.6. In the wet

porous regions, the volume expansion-advection of water vapor thermal conductivity was

significantly higher, i.e. it could be up to 16 times higher than those values given by the diffusion

process. Both the lysimeter sand and Portneuf silt loam have manifested similar characteristics as

discussed.

4.4.4 The water vapor thermal conductivity by latent heat and sensible heat

Water vapor movement is influenced by two phenomena - latent heat flux and sensible heat flux.

The water vapor thermal conductivity due to the sensible heat flux has approximately 0.3-0.4%

65

of the latent heat flux of water vapor thermal conductivity. The percentage does not vary with

pressure or soil moisture content, only with temperature because both the latent of vaporization

and the specific heat of water vapor are a function of temperature.

4.4.5 The water vapor thermal conductivity of the partial derivative of matric pressure

head effects on relative humidity, with respect to temperature

Water vapor density is a function of the saturated water vapor density and the matric pressure

head effects on relative humidity (refer to Eq. 4.3). Eq. 4.4 resulted from subjecting Eq. 4.3 to

the partial derivative with respect to temperature. The second term on the right side of Eq. 4.4 is

the original function term used in the simplified heat flux, by excluding the first term of the

equation (refer to Eq. 4.7). Similar to section 4.4.3, in drier porous media regions the first term

has 7-12% of the value of the right side second term of Eq. 4.4, i.e. 0.014 m3 m-3. In wet regions,

it is 65.0 10 - 0.146%. Therefore, the partial derivative of matric pressure head effects on

relative humidity with respect to the temperature term appeared to be more essential in near dry

porous media region than in wet regions. Moreover, in relative terms, the partial derivative term

was the main reason for increased water vapor diffusion in the drier porous media. In other

words, the partial derivative term has a more significant influence than the sensible heat flux

term in the drier region.

4.4.6 Water vapor permeability variables with respect to soil moisture content and air

pressure

In variably saturated porous media, water vapor flux is subjected to the influence of soil moisture

content and air pressure. In the water vapor diffusion process, the variation in soil moisture

66

content directly changes the matric suction head that in turn would affect the relative humidity.

The relative humidity has a direct influence on water vapor density, which is one of the

parameters to affect the water vapor diffusion process. On the other hand, the pressure imposed

on the porous media would have a direct influence on pressure ratio. The pressure ratio would

then change the diffusion of water vapor in the porous media. In terms of water vapor volume

expansion-advection, the soil moisture content would influence the relative permeability (Eq.

4.24) which would then affect the water vapor volume expansion-advection. The pressure would

affect the permeability (Eq. 4.23) that is responsible for the Knudsen effect. In Eq. 4.22 which is

used for curve-fitting the left side of the equation is the dependent variable. The right side of the

equation consists of permeability and relative permeability, that are respectively influenced by

the soil volumetric water content and the pressure ratio as the independent variables. The curve-

fitting results are shown in Fig. 4.2. The permeability variables ( p r ok k k ) predicted from the

simplified and improved heat flux theories possessed concave downward curve characteristics

that were influenced by soil volumetric water content. The characteristic was carried over from

the relationship between water vapor thermal conductivity and soil volumetric water content as

in Fig. 4.1. The concave characteristic was most profound in low pressure regions, while

appeared to be less significant at high pressure. Moreover, the permeability variables decreased

with increasing pressure. The observation is consistent with the fact that the water vapor flux

decreased in high pressure regions because less water can vaporize in a high pressure

environment. The influence of soil volumetric water content and pressure on permeability

variables show a promising relationship. Its trend is similar to the relationship of heat flux, soil

volumetric water content, and pressure as reported by Sakaguchi et al. (2009). Also, the

comparison of experimental and predicted data has demonstrated a reasonably good compliance,

67

with R-squared values of 0.93 as indicated in Figs. 4.3(a)-4.3(b). A consistent relationship was

observed from both lysimeter sand and Portneuf silt loam.

The derived relative permeability values appeared to predominate at lower values when

compared to other models (refer to Fig. 4.4(a)). The other models begin with a unity relative

permeability value at an extremely dry condition. The value reduces with increasing soil

moisture content, and it approximates to zero relative permeability in fully water saturated

porous media. On the other hand, the estimated relative permeability value increases from zero

relative permeability to reach a peak, before it reduces to approximately zero in fully water

saturated porous media (Fig. 4.4(b)). A similar relationship was shown between water vapor

conductivity and matric suction head for sand, silt loam, and clay soils (Campbell, 1985). The

relative permeability given by other models is only applicable to conditions when an external

pressure force is applied to a porous media. In the case of temperature gradient generated relative

permeability, in the absence of external pressure force, the water vapor volume expansion-

advection by temperature-driven vaporization would have to depend on the soil moisture content.

4.4.7 Limitation of previous work, and the possibility of improvement

There were some limitations of the Cass research work that could be considered for further

improvement. For instance, a single measurement of thermal conductivity, at each point of

pressure and water content, and more than two measurements were necessary to establish

standard deviation or error. A temperature gradient of 5oC was imposed on the soil tube, and the

possible effect of different gradients was not investigated. Also, water content at near saturation

was not investigated. The thermal conductivity was measured at only four pressure values to

estimate thermal conductivity at extreme pressure, and so measurement at additional pressure

68

values could further improve the estimation.

Moreover, some factors, related to soil heat capacity and thermal conductivity, are also

worth assessing for future improvement. The findings of the following researchers corroborate

this statement. Bilskie et al. (1998) found variation in α-alumina colloidal suspensions in water

caused changes in soil heat capacity and thermal conductivity measured by DPHP probe. Also,

dissolved compounds like salt decreased thermal conductivity as the concentration increased

(Abu-Hamdeh and Reeder, 2000, Noborio and McInnes, 1993). Apart from the volumetric

proportions of various soil components, both structure, e.g. orientation of soil particles, and

packing, e.g. nature of contacts between soil particles, could affect soil thermal conductivity

(Winterkorn and Fang, 1991). Also, soil porosity, the chemical composition of soil particles and

water structure served as important factors for soil thermal conductivity (Farouki, 1981). The

presence of a vapor enhancement factor was undeniably significant and the proposed governing

equation for water vapor phenomenon, in this study, would be applicable. Since Fick’s law

poorly predicted the vapor flux and required a significant contribution from the water vapor

volume expansion-advection by Darcy law, the possibility of the latter equation to completely

replace the former equation should be examined. The mathematical solution to the water vapor

relative permeability presented in the study was temporary, so a theoretically derived equation

with the ability to handle its value at saturated soil moisture content would require further

examination.

4.5 Conclusions

The resulting thermal conductivity from the liquid water movement sensible heat term possesses

a rapid rate of increment when compared to wet porous media heat conduction that led to thermal

69

conductivity with increasing soil moisture content. This fact suggests that the sensible heat by

liquid water movement is an important factor to consider in highly wet porous media, and in

variably saturation porous media. In relatively dry porous media conditions, the water vapor

diffusion flux was significantly enhanced by the partial derivative of matric pressure head effects

on relative humidity with respect to temperature. Hence, this phenomenon appeared to be

important at the opposite spectrum of the important soil moisture content region of the sensible

heat by liquid water movement. The current proposed improved heat flux theory was established

from the work of de Vries (1958), Cass et al. (1984), and Ho and Webb (1999). Moreover, the

improved heat flux equation would not be limited by the constraint set forth by Cass to only

apply to experimentation work between the relatively dry to moderately wet porous media

conditions. This is because the sensible heat by liquid water movement could account for such

limitation, which is one of three phenomena proposed for the theory and equation improvement.

Also, Shokri et al. (2009) have shown the significance of liquid water movement in evaporation

experiments to account for the excessive vapor flux observed.

Although the vapor enhancement factor had been used by numerous researchers, e.g.

Milly (1982), Shurbaji and Phillips (1995), Noborio et al. (1996), Nassar and Horton (1997),

Saito et al. (2006) and Heitman et al. (2008), until now, an equation to represent its physical

phenomenon has not yet been derived in its fundamental form. The segregation of mechanism is

an important process to allow full understanding of the basic factor governing the phenomena.

As a result, the mathematical description of water vapor volume expansion-advection is

proposed because it represents a more conceivable phenomenon of vapor flux than the

mechanistic enhancement factor imposed-correction on the diffusion equation of water vapor.

The water vapor volume expansion-advection mathematical relation is a function of soil

70

volumetric water content, and air pressure. Further investigation would be needed to derive a

theoretical equation for the water vapor relative permeability that would be able to predict at

saturated soil moisture content. Also, the effect of different average temperatures on water vapor

flux should be included into the equation.

71

Figure 4.1. Water vapor thermal conductivity, and the combined (wet porous media heat

conduction and sensible heat by liquid water movement) thermal conductivity (at 0r oP P P )

versus soil volumetric water content. Note: the water vapor thermal conductivity was at a

pressure ratio of 1.0, i.e. atmospheric pressure, and temperature at 32.5 oC. The water vapor

thermal conductivity was corrected for the pressure dependence of the thermal conductivity of

air, based on the equation given by Cass.

0.2

0.6

1

1.4

0.00

0.10

0.20

0.30

0.40

0.00 0.10 0.20 0.30

Co

mb

ined

th

erm

al c

on

du

ctvi

ty a

t P

r=0

(J

m-1

s-1K

-1)

Wat

er v

apo

r th

erm

al c

on

du

ctiv

ity

(J m

-1s-1

K-1

)


Pr=1.0, T=32.5 Celsius

Without water vapor (Pr=0)

72

(a)

(b)

Figure 4.2. The curve-fitting results of permeability variables as a function of the soil volumetric

water content, and the pressure ratio. The permeability variables based on the adapted-improved

heat flux theory for (a) the lysimeter sand, and (b) the Portneuf silt loam. Note: the graph (a) and

(b) were subjected to Eq. 4.20, before curve-fitting using Eq. 4.22 to determine the permeability

variables. Zero permeability variables values were added as supposed zero water vapor volume

expansion-advection at fully saturated soil moisture content to improve the curve-fitting results.

73

(a)

(b)

Figure 4.3. The comparison of the experimental and curve-fitted permeability variables based on

the adapted-improved heat flux theory for (a) the lysimeter sand, and (b) the Portneuf silt loam.

Note: the graph (a) and (b) were subjected to Eq. 4.20, before curve-fitting using Eq. 4.22 to

determine the permeability variables. Care should be placed on the fully saturated soil moisture

content condition because the permeability variables estimated by Eq. 4.22 could result in

negative values.

0.0E+00

2.5E-06

5.0E-06

7.5E-06

0.0E+00 2.5E-06 5.0E-06 7.5E-06

Pre

dic

ted

kpk r

/ko

Actual data kpkr/ko

R2 = 0.927No. data = 68

0.0E+00

5.0E-05

1.0E-04

1.5E-04

0.0E+00 5.0E-05 1.0E-04 1.5E-04

Pre

dic

ted

kpk r

/ko

Actual data kpkr/ko

R2 = 0.978No. data = 28

74

(a)

(b)

Figure 4.4. (a) The relative permeability ( rk ) of lysimeter sand versus water saturation estimated

using Eq. 4.24, and (b) Enlarged graph on relative permeability value ranged 0-0.3. Note: the

other relative permeability models were van Genuchten-Mualem, van Genuchten-Burdine,

Corey, and Falta referring to Eqs. 4.25-4.28, respectively.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

Rel

ativ

e p

erm

eab

ility

, kr

(dim

en

sio

nle

ss)

Water saturation, θL/θs (m3/m3)

van Genuchten-Mualemvan Genuchten-BurdineCorey

Falta

Estimated relativepermeability

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.2 0.4 0.6 0.8 1.0

Rel

ativ

e p

erm

eab

ility

, kr

(dim

en

sio

nle

ss)

Water saturation, θL/θs (m3/m3)

75

Chapter 5. The extended Darcy law and water vapor relative permeability

derivation in unsaturated soils

5.1 Abstract

This study re-examined the relationship between water vapor relative permeability and soil

moisture saturation. The permeability of water vapor was found to increase with increasing soil

moisture saturation at lower water saturation levels. However, from near mild levels of soil

moisture saturation, the permeability was found to decrease with the increasing soil moisture

saturation. The water vapor relative permeability equation was derived to govern this

observation. Two types of equations used in the derivation were – the integral equations, and the

empirical equation from van Genuchten. The derived equation could adequately describe the

shape of the curvature and its peak location with reasonable accuracy, with an R-squared value

greater than 0.93 for both Portneuf silt loam and lysimeter sand, and greater than 0.8 for Kanto

loamy soil. The vapor enhancement factor imposed on Fick’s law was limited to the effects of

soil moisture saturation, whereas Darcy law could be extended to account for and validate the

ambient pressure and temperature; where very complex environmental conditions could be

considered. In this study, Darcy law was proposed as a replacement for the vapor enhancement

factor imposed-Fick’s law which resulted from the temperature gradient.

5.2 Introduction

Permeability is an important factor for gas, oil and water flow in porous media such as rocks,

industrial products, and soils. Conventionally, gas permeability, under fully saturated gas

concentration, decreases with increasing wetting phase saturation, e.g. liquid water and water

76

vapor respectively as wetting and non-wetting phases. In Chapter 4, the non-wetting phase

relative permeability, e.g. water vapor, appeared to have behaved differently from conventional

gas permeability (Goh and Noborio, 2016). The water vapor relative permeability was derived

from the thermal conductivity data published by Cass et al. (1984). The thermal conductivity

data obtained from the experiment, under which the closed steel tubes subjected to external heat

energy and pressure, was measured at different concentrations of water saturation. The condition

successfully allowed the removal of any external source of the influx of water vapor. Therefore,

the heat flux theory could estimate the thermal conductivity value that can quantify the water

vapor flux. The heat flux theory employed in Cass et al. (1984) was modified from de Vries

(1958). The experimental procedure used was adapted from Parikh et al. (1979), and the

mathematical methods applied to approximate the thermal conductivity data were adopted from

Riha et al. (1980).

In Chapter 4, an improved heat flux theory was proposed. The theory, accompanied by an

empirical relative permeability equation, was put forward to describe the observed relative

permeability data. With regards to the benefits and limitations, discussed in Chapter 4, the

equation successfully captured the curved downward characteristic of relative permeability with

increasing soil moisture saturation under various pressure conditions. However, the equation

failed at fully saturated soil moisture contents. Clearly, more research effort is necessary to

derive a theoretical equation that can fully describe the governing phenomenon. Successful

achievement of this would be a significant step in completing the theory of water vapor flux and

also help to solve the mysteries of the water vapor enhancement factor that have been puzzling

the soil physics community for more than 50 years, since the work of Philip and de Vries (1957).

Also, Horton and Ochsner (2011) acknowledged that the existing theory could not explain the

77

phenomenon that led to large different thermal vapor enhancement factors in soils.

The relative permeability phenomenon, in general, could be governed by capillary,

statistical, empirical and network models (Dullien, 1992, Honarpour et al., 1986). Fatt and

Dykstra (1951) derived an integral equation based on the capillary approach utilizing a bundle of

capillary tubes of various diameters with a fluid path length greater than the sample length. The

equation by Fatt and Dykstra (1951) was similar to Purcell (1949), and was described in Saraf

and McCaffery (1985). Childs and Collis-George (1950) employed a statistical method based on

the probability of two pore sizes encountering each other and subjected the relationship to

integration to generate an effective permeability function in the form of summation or

integration. Marshall (1958), Millington and Quirk (1961) and Mualem (1976) also used a

similar approach. Burdine (1953) derived the integral equation through the hydraulic radius

theory, and Bear (1972) showed an alternative derivation method based on Burdine’s integral

equation. In empirical models, the relative permeability is derived from the combination of either

the capillary or statistical model with an empirical equation to obtain the best approximation of

relative permeability. For instance, van Genuchten (1980) used the statistical model from

Mualem (1976) and proposed a modified equation from Haverkamp et al. (1977) to derive a new

closed-form empirical equation. Corey (1954), Brooks and Corey (1966), Campbell (1974) and

Li and Horne (2006) also utilized a comparable approach. In the present study, we limit the

investigation to the integral equations from the capillary and statistical models, and the empirical

equation used by van Genuchten (1980) which is widely claimed to generate a good compliance

with experimental data. Also, we explored the possibility of using Darcy law to govern water

vapor flux, rather than the Fick’s law alone or the combination of Fick’s law and Darcy law.

78


The study was based on the experimental data from Cass et al. (1984). The experimental setup,

parameter estimation, and limitations were thoroughly discussed in Chapter 4. Some other

details, e.g. thermal conductivity data of lysimeter sand and Portneuf silt loam, and also matric

suction data of lysimeter sand, can be retrieved from Cass et al. (1981) and Cass et al. (1984).

The Portneuf silt loam permeability was derived from Robbins (1977). For comparison, this

study included a Kanto loamy soil dataset from Tanimoto (2007), and the thermal conductivity

was measured with the dual probe heat pulse (DPHP) method. The derivation of the improved

heat flux theory, including those from Cass et al. (1984), and the proposed temporary empirical

equation for water vapor relative permeability was discussed in detail in Chapter 4.

5.3.1 The derivation of extended Darcy law, and its relative permeability and the

Klinkenberg effect

As described by Brooks and Corey (1964), the Navier-Stokes equation could be reduced to a

relationship between pressure force and viscous drag force by assuming that the fluid flow in a

small diameter tube (Fig. 5.1) is incompressible, and has negligible inertia and body force:

' ' '

1 1P u u ur r

x r r r r x x

(5.1)

With further assumptions, such as the symmetrical tube and fluid flow in steady state, i.e.

no variation of flow velocity in 'x -direction (constant velocity profiles 1 and 2), the equation

would be described as:

'

dP d dur

dx r dr dr

(5.2)

where P is the pressure imposed on the fluid system (kg m-1 s-2), is the dynamic viscosity of

79

the fluid (kg m-1 s-1), u is the fluid velocity (m s-1), and r is the radius of the tube where the

fluid flow (m). To solve Eq. 5.2, two times integration was used. The first integration was

derived from the center of the tube where the tube radius was 0r to an unspecified location of

r r . Then Eq. 5.2 would be expressed as:

' 2

dP r du

dx dr

(5.3)

In the second integration, Eq. 5.3 was integrated with respect to the tube radius from an

unspecified location r r to the edge of the tube where or r . Then, Eq. 5.3 is described as:

2 2

' 4o

o

r rdPu u

dx

(5.4)

where ou is the velocity of the fluid at the wall of the tube (m s-1), i.e. slip effect. In the presence

of the fluid slip effect at the edge of the tube, the variable ou became significant. The following

relation was used in Klinkenberg (1941):

o

o

r r

duu c

dr

(5.5)

where c is the proportionality constant, is the mean free path (m), and the derivative is the

velocity gradient (s-1) at the wall of the tube. Putting Eq. 5.5 into Eq. 5.4, and using Eq. 5.3 to

find the velocity gradient at or r , the fluid flow velocity at a location ( r) between 0 and or

could be estimated by:

2 2

'

12

4o o

dPu c r r r

dx

(5.6)

Equation 5.6 is similar to the equation derived by Klinkenberg (1941). The volumetric

flow rate ( q , m3 s-1) of fluid could be estimated by multiplying Eq. 5.6 with 2rdA rdr and

80

then integrating the function with respect to the radius from 0 to or . The volumetric flow rate

would be:

4

'

4 11

8o

o

r c dPq

r dx

(5.7)

Equation 5.7 without the gas slip effect was known as Hagen-Poiseuille’s equation

(Sutera and Skalak, 1993). Moreover, based on the mathematical and theoretical description of

Atkins and De Paula (2006), the mean free path ( ) could be derived through the consideration

of molecules’ mean speed, molecules’ relative mean speed, collision frequency, number of

molecules, volume under consideration, collision cross section ( , m2), and the ideal gas law

equation. The resulting mathematical description is described as:

2

kT

P

(5.8)

where k is the Boltzmann constant (kg m2 s-2 K-1), T and P are the temperature (K) and

pressure (kg m-1 s-2) of the fluid in the tube, respectively. Atkins and De Paula (2006) claimed

that in a sample of constant volume; the pressure was proportional to the temperature, i.e. the

ratio of /T P would remain constant when the temperature increases. Hence, the would be

free from the temperature of a sample of fluid. In the current study, we claimed that the ratio of

temperature to pressure was not entirely constant. The observation from the results of the present

study would be used to support the claim. Hence, a constant b would be used to represent the

variables:

4

2 o o

ckb

r P (5.9)

where b is the Klinkenberg gas slip factor (K-1) that was slightly different from the one

employed in Klinkenberg (1941) and Tanikawa and Shimamoto (2006). Li and Horne (2004)

81

used a different constant derived from Rose (1948). In this study, Eq. 5.9 was sufficient, and

hence Eq. 5.7 could be rewritten as:

4

'

11

8o

r

r dPq bTP

dx

(5.10)

where r oP P P with oP is 101325 kg m-1 s-2 and P is the absolute pressurized air (kg m-1 s-2).

From a single pore space (see the right-hand side of Fig. 5.1), to multiple pores (see the left-hand

side of Fig. 5.1), the total flux (Q , m3 s-1) of fluid could be assumed as the sum of N tubes of

small diameter, hence, Q Nq . Knowing that ' cosx x , and then assuming the tortuosity as

the ratio of the actual travel distance of fluid to the sample path, 'x x , and also taking

porosity as 2 ' 2o oN r x Ax N r A (Dvorkin, 2009), then the total flux of fluid would be

written as:

2

21

8o

r

r A dPQ bTP

dx

(5.11)

Since the mean velocity fluid flowing in the porous media would be given by u Q A ,

and the relative permeability of fluid was provided by the ratio of effective permeability to

saturated permeability as r ok k k , where 2 28ok r , Eq. 5.11 was written as:

1r or

k k dPu bTP

dx (5.12)

Equation 5.12 could be used to govern the water vapor flux:

1rnw o v vv r

v v

k k Pdu bTP

dx

(5.13)

where the subscript v is referring to water vapor. Using the ideal gas law relationship to

governing the water vapor as v v vP RT M , and the improved gas Klinkenberg permeability

82

that includes the effect of temperature 1P T o rk k bTP , Eq. 5.13 was written as:

rnw P T vv

v v

k k R dTu

M dx

(5.14)

where vM is the molecular weight of water vapor (kg mol-1), R is the ideal gas constant (kg m2

s-2 K-1 mol-1), vP is the partial pressure of water vapor (kg m-1 s-2), rnwk is the relative

permeability of water vapor (dimensionless), P Tk is the improved water vapor Klinkenberg

permeability (m2) and v is the dynamic viscosity of water vapor (kg m-1 s-1). Then, Eq. 5.14

extended from the mean velocity of water vapor ( vu , m s-1) to vq (kg m-2 s-1) in porous media,

where v v Lu q , would be given by:

rnw P T v Lv

v v

k k R dTq

M dx

(5.15)

where L is the density of liquid water (kg m-3). The necessity of converting Eq. 5.14 to 5.15

will be explained in a later section 5.3.3.

Equation 5.15 was a derivation of the extended Darcy law, except for the pressure ratio to

the power e , used in Chapter 4 to govern water vapor flux after the Fick’s law. Moreover, the

derivation validated the mathematics employed in Chapter 4. Hereafter, the derived rk variable

from Eqs. 5.11 to 5.12 could be examined further to obtain the integral form for practical

application in estimating the relative permeability value.

5.3.2 The integral form of relative permeability of liquid water and water vapor

In unsaturated soil, the volumetric water content ( L , m3 m-3) was smaller than the soil porosity

given in Eq. 5.11. At the residual water content ( r , m3 m-3), liquid water became immobile. The

83

portion of soil porosity that contributed to liquid water flux would be given by L r . The

equation can be related as ( ) ( )L r s r eS , where s (m3 m-3) is the saturated water

content and eS (dimensionless) is the water saturation. Hence, the effective permeability from

Eq. 5.12 would be given by:

2

2

( )

8s r

o ek r S

(5.16)

If the derivative of the effective permeability ( k , m2) was taken with respect to the water

saturation, and it was followed by integrating differential dk from zero to k , which corresponds

to integrating differential edS from zero to a value of water saturation ( eS ) occupied by the

wetting phase, Eq. 5.16 would be defined as:

2

20

( )

8

eSs r

o ek r dS

(5.17)

Based on the capillary law the matric suction ( m , m) is inversely proportional to pore

radius, i.e. o mr a as given in Mualem (1976), then Eq. 5.17 could be converted into:

2

220

( )

8

eSs r e

m e

a dSk

S

(5.18)

A different form of the capillary equation was provided by Purcell (1949), and ultimately

it only caused variation in the constant a value which would not affect the estimation of the

relative permeability. The relative permeability was given by the ratio of the effective

permeability to the saturated permeability. The saturated permeability could be derived by

following the same steps as those used in deriving Eq. 5.18, except this time the differential dk

would be integrated from zero to sk which corresponds to the differential edS that would be

integrated from zero to unity. Therefore, the relative permeability resulted in the following form:

84

220

1

220

1

1

eSe

m e

rwe

m e

dS

Sk

dS

S

(5.19)

where rwk is known as the wetting phase relative permeability in which the wetting phase refers

to liquid water in porous media. If the variable could be assumed not to vary with water

saturation ( eS ), the following Eq. 5.20 could be derived, which is similar to those from Purcell

(1949) and Gates and Lietz (1950):

20

1

20

eSe

m e

rwe

m e

dS

Sk

dS

S

(5.20)

If the ratio of unsaturated to saturated tortuosity was assumed as a function of water

saturation to the power of two, Eq. 5.19 resulted in the following relationship:

20

2

1

20

eSe

m e

rw ee

m e

dS

Sk S

dS

S

(5.21)

Equation 5.21 is widely known as Burdine’s integral equation (Burdine, 1953). If the

tortuosity squared was assumed as inversely proportional to the pore radius, Eq. 5.19 could be

written as:

30

1

30

eSe

m e

rwe

m e

dS

Sk

dS

S

(5.22)

Equation 5.22 was derived by Fatt and Dykstra (1951), which they found it to have good

85

agreement with experimental data.

Equation 5.16, which was derived from Eq. 5.11, was due to the consideration of a

parameter, i.e. porosity. Another parameter that could be considered in the study is specific

surface area per unit volume ( vS , m-1). The equation could be written as

'(2 ) 2v o oS N r x Ax r . By placing the equation into the effective permeability from Eq.

5.11, and followed by a simple rearrangement, the new relationship could be rewritten as

2 22o vk N r S A . One can use the relation

2 ' 'o o or r r where the square of or is

assumed to be comprised of two different pore spaces. Then, by subjecting the equation to a

similar mathematical operation from Eqs. 5.16 to 5.19, but taking two times differentiation

followed by two times of integration of the effective permeability with respect to water

saturation, the new relative permeability would be:

'

2 '0 0

'1 1

2 '0 0

1

1

e eS Se e

v m e m e

rw

e e

v m e m e

dS dS

S S Sk

dS dS

S S S

(5.23)

If the tortuosity and the specific surface area could be assumed as a function of water

saturation to the power of n , based on Mualem (1976), Eq. 5.23 could be reduced to the

following form:

2

0

1

0

eSe

m enrw e

e

m e

dS

Sk S

dS

S

(5.24)

Hence, it showed that the Mualem (1976) integral equation could be derived by a non-

statistical method. If the power of two was changed to 2.5, and the matric suction power

86

increases to two, and 1n , then the result would be the Timmerman (1982) equation.

In Chapter 4, it was found the water vapor relative permeability curved downward at

higher soil moisture saturation. The empirical equation proposed to govern the relative

permeability provided a good approximation of the experimental data from Cass et al. (1984).

However, the equation failed under fully water saturated conditions, especially when there was

insufficient data at the saturation conditions for estimation. Therefore, it is necessary to find a

better equation. The water vapor relative permeability curvature could be separated into two

parts. The first part would be at lower water saturation where relative permeability would be

limited by the availability of the water vapor itself. The second part would be at higher water

saturation, where the water vapor behaved like a typical non-wetting phase with relative

permeability decreasing as the water saturation increased. To address the curvature described, the

integral equations from Purcell (1949), Burdine (1953), Fatt and Dykstra (1951), Mualem (1976)

and Timmerman (1982) were respectively modified as:

1

2 20

2

1

20

e

e

Se e

Sm e m e

rnw

e

m e

dS dS

S Sk

dS

S

(5.25)

1

2 20

22

2

1

20

1

e

e

Se e

Sm e m e

rnw e e

e

m e

dS dS

S Sk S S

dS

S

(5.26)

1

3 30

2

1

30

e

e

Se e

Sm e m e

rnw

e

m e

dS dS

S Sk

dS

S

(5.27)

87

1

0

21

0

1

e

e

Se e

Sn m e m enrnw e e

e

m e

dS dS

S Sk S S

dS

S

(5.28)

1

2 20

2

1

20

1

e

e

Se e

Sm e m e

rnw e e

e

m e

dS dS

S Sk S S

dS

S

(5.29)

where rnwk is the non-wetting relative permeability which in this study was the water vapor in

porous media. Then, Eqs. 5.25 to 5.29 were solved by the van Genuchten (1980) water

characteristic curve equation, i.e. the relation of e mS , to derive a closed-form equation. The

derived equations from Eqs. 5.25 to 5.29 can be summarized as:

1 1

1 1 1 1

m m

nn m mrnw e e e ek S S S S

(5.30)

where n and m are parameter constants. The power n is the only indicator distinguished

between Eqs. 5.25 to 5.29. The power values as 0, 0.5, 1, 2 or n itself correspond to Purcell

(1949), Mualem (1976), Timmerman (1982), Burdine (1953) and Mualem (1976), respectively.

Equation 5.30 was used in Eqs. 5.32-5.34 to determine the parameter constants of n , m , and the

Klinkenberg slip factor ( b ) by curve-fitting.

5.3.3 Thermal conductivity governed by Fick’s law and Darcy law in heat flux equation

The original Cass heat flux equation was employed with only Fick’s law (Eq. 5.31). The

improved-Cass heat flux equation included both the Fick’s law and Darcy law (Eq. 5.32). In

addition to these two laws, in Chapter 4 a heat flux equation was proposed that included three

88

other phenomena (Eq. 5.33). In Chapter 5, the heat flux equation was examined for improvement

to include only Darcy law (Eq. 5.34). The four equations can be expressed respectively as:

*s

h a o m

d dTq D L h

dT dx

(5.31)

*rnw P T v L s

h o a o m

v v

k k R d dTq L D L h

M dT dx

(5.32)

*m rnw P T v L

L L o v

v v

h

m sa o v s m

d k k Rc T K L c T

dT M dTq

dxhD L c T h

T T

(5.33)

*m rnw P T v L

h L L o v

v v

d k k R dTq c T K L c T

dT M dx

(5.34)

where hq is the total heat flux density (J m-2 s-1), is the vapor enhancement factor, * is the

thermal conductivity of the porous media in the presence of liquid water (J m-1 s-1 K-1), oL is the

latent heat (J kg-1) of water at temperature oT (K), vc is the specific heat of water vapor at

constant pressure (J kg-1 K-1), Lc is the specific heat of liquid water (J kg-1 K-1), D is the

molecular diffusivity of water vapor in air (m2 s-1), is the tortuosity factor (dimensionless), a

is the volumetric air content of the media (m3 m-3), s is the saturated water vapor density (kg m-

3) and mh is the matric pressure head effects on relative humidity (dimensionless), K is the

hydraulic conductivity (m s-1), m is the matric pressure head (m), and T is the temperature

difference between the outside and the middle of the closed steel tube. The first term of Eqs.

5.31-5.34 is the thermal conductivity of the porous media in the presence of liquid water, and the

second term of Eqs. 5.33-5.34 is the thermal conductivity by liquid water movement. The third

89

term of Eqs. 5.33-5.34 and the second term of Eq. 5.32 is the thermal conductivity by water

vapor movement governed by Darcy law. The second term of Eq. 5.31, the third term of Eq. 5.32

and the fourth term of Eq. 5.33 is the thermal conductivity by water vapor movement governed

by Fick’s law.

5.4 Results and Discussion

5.4.1 Replacing Fick’s law with Darcy law

The water vapor thermal conductivity was found to decrease with increasing ambient pressure

( P ), which led to a decreasing pressure ratio as given by r oP P P (Fig. 5.2). The water vapor

thermal conductivity displayed a curved concave downward pattern with increasing water

saturation ( eS ). The water vapor thermal conductivity was also found to increase with rising

temperature (Figs. 5.2(b) to 5.2(a) corresponding to 22.5 and 32.5 oC, respectively). In Chapter 4,

the total water vapor thermal conductivity was separated into two terms – the thermal

conductivities by Fick’s law and Darcy law, as in Eqs. 5.32 and 5.33. In the Chapter 5 study,

Darcy law was introduced to replace the vapor enhancement factor while retaining Fick’s law to

govern water vapor diffusion. Figures 5.2(a) to 5.2(c) clearly show that the water vapor thermal

conductivity estimated by Darcy law equation was in close approximation to the experimentally

derived total water vapor thermal conductivity, while a fairly good approximation was also

demonstrated in Fig 5.2(d). The discrepancy between the experimentally derived total water

vapor thermal conductivity and that estimated by Darcy law reduced as the ambient pressure

increased. The decreased temperature also reduced the discrepancy as shown in the change

between Figs. 5.2(a) and 5.2(b). The relatively small discrepancy value suggests that the total

water vapor thermal conductivity could be described by Darcy law alone, as in Eq. 5.34.

90

Therefore, Darcy law could be used to replace the combined Fick’s law and Darcy law as applied

in Chapter 4 (i.e. Eqs. 5.32 and 5.33) and Fick’s law only used in Cass et al. (1984) (i.e. Eq.

5.31). Also, the vapor enhancement factor was found to be less than unity at volumetric water

content less than 0.059 and 0.15 m3/m3 for the Portneuf silt loam and Kanto loamy soil,

respectively. This below unity value from Eqs. 5.32 and 5.33 would generate inaccurate

estimates which would support the proposal to use Darcy law as a replacement for the heat flux

equation. Furthermore, replacing the Fick’s law temperature gradient term with Darcy law would

not affect the water vapor flux by the matric suction gradient or the moisture content gradient

term of the equation used to govern water and vapor flow in soil. Nassar and Horton (1997) and

Heitman et al. (2008) can be referred to for the partial differential equations of water and vapor

flux that differentiate the terms use for matric suction and temperature gradients. In fact, Fick’s

law was found largely inadequate to describe the temperature gradient resulting in water vapor

flux. Ho and Webb (1996) had estimated that, under a specific environmental condition, a value

of ten times the Fick’s law value was needed to account for the discrepancy from the total water

vapor flux due to the temperature gradient. Also, Cass et al. (1984) had found that the total water

vapor flux in lysimeter sand could be as large as sixteen times the value given by Fick’s law, and

twelve times in Portneuf silt loam. Clearly, the significant discrepancy observed for water vapor

flux due to temperature gradient governed by the Fick’s law equation indicates an insufficiency

and also suggests the need to propose an alternative governing equation. Consequently, the

Darcy law equation was proposed as an alternative replacement to describe the temperature

gradient-resulting water vapor flux. Hence, the heat flux equation would be given as in Eq. 5.34.

Also, the R-squared value obtained from including the water vapor relative permeability (Eq.

5.30) in the heat flux equations (Eqs. 5.32-5.34) was reasonably high. The R-squared value of the

91

proposed heat flux equation (Eq. 5.34, R2=0.933) for lysimeter sand was found to be comparable

to those from the improved-Cass (R2=0.932), and the Chapter 4 heat flux equations (R2=0.933).

The Portneuf silt loam R-squared value was around 0.971 for the three heat flux equations,

whereas the Kanto loamy soil was reasonably good at 0.849 from Eq. 5.34. The R-squared

results imply that Darcy law could practically replace the Fick’s law equation for the

phenomenon caused by the temperature gradient in the soil, without sacrificing much of the

accuracy of estimating the total water vapor flux.

5.4.2 The water vapor relative permeability, and the Klinkenberg permeability

The derived water vapor relative permeability ( rnwk ), i.e. Eq. 5.30, consisted of two parameter

constants – n and m . The constants m and n combined to affect the level of water saturation

that corresponds to the peak in the curve of permeability variables ( rnw P Tk k ). The results indicate

the water saturation peak value for lysimeter sand was less than that for Kanto loamy soil or

Portneuf silt loam, as shown in Fig. 5.3. This observation is consistent with their respective

values of intrinsic permeability of 111.09 10 , 122.91 10 and 134.52 10 m2. Thus, a soil with

high intrinsic permeability possibly indicates an early peak of the permeability variable than for a

soil with lower permeability. Most importantly, Fig. 5.3 illustrates the derived relative

permeability is able to capture zero permeability values at initial and fully saturated soil moisture

conditions. Furthermore, the Klinkenberg gas slip factor ( b ) from the Kanto loamy soil was

found to be greater than that for the Portneuf silt loam and lysimeter sand (Table 5.1). This factor

indicates that increasing the temperature and/or reducing the ambient pressure in the Kanto

loamy soil would have a greater impact on increasing the water vapor slip effect than in the

Portneuf silt loam or lysimeter sand. It was evident from Eq. 5.9 that soil with a greater

92

Klinkenberg gas slip factor ( b ) due to the greater proportional constant ( c ), enhanced the

velocity gradient ( or r

du dr

) of water vapor at the soil particle walls. The phenomena then lead

to increased slip velocity ( ou ), as given by Eq. 5.5. Hence, in theory the Kanto loamy soil could

be more susceptible to the water vapor slip effect as pressure and temperature vary than the

Portneuf silt loam or lysimeter sand.

The effect of including the temperature as an independent variable into the Klinkenberg

permeability ( P Tk ) resulted in an improvement in model prediction. The R-squared value for

lysimeter sand was found to improve from 0.915 to 0.933 (Fig. 5.4(b)), estimated by Eq. 5.34,

and also a similar improvement was observed in the estimations by Eqs. 5.32-5.33. The R-

squared value for the Portneuf silt loam was maintained at 0.971 (Fig. 5.4(a)) because the

thermal conductivity data at 32.5 degree Celsius was the only temperature measured by Cass et

al. (1984). A similar improvement to the lysimeter sand was also noted for the Kanto loamy soil.

Figures 5.4 shows that the proposed heat flux equation of Eq. 5.34, the derived relative

permeability of Eq. 5.30 and the Klinkenberg permeability from Eq. 5.15 were able to describe

the three soils behavior reasonably well at different soil temperatures, pressures, and water

saturations.

5.4.3 Using Darcy law in the thermal water vapor diffusivities

As stated earlier, the Darcy law equation could account for the observed water vapor flux and

could also replace Fick’s law equation. Darcy law could be used to govern the effect of volume

expansion-generating water vapor pressure gradient due to the imposed temperature gradient on

the soil porous media, as described in Chapter 4. Subsequently, it would require modification of

the parameter in one of the terms used in the governing equation of heat and mass balance of

93

liquid water and water vapor in soil media (Heitman et al., 2008, Nassar and Horton, 1997). The

parameter was originally based on Eq. 5.31, as given by:

a m sTv s m

L

D hD h

T T

(5.35)

where TvD is known as the thermal water vapor diffusivities (m2 s-1 K-1). Based on the improved-

Cass (Eq. 5.32) and the Chapter 4 (Eq. 5.33) heat flux equations, the thermal vapor diffusivities

can be summarized as:

rnw P T v a m sTv s m

v v L

k k R D hD h

M T T

(5.36)

In the current study, based on Eq. 5.34, the following expression was proposed:

rnw P T vTv

v v

k k RD

M

(5.37)

Equation 5.36 is an improvement on Eq. 5.35 because it could explain the mystery of the

vapor enhancement factor ( ). The drawback is that it had more mathematical and

computational demand than Eq. 5.35, and also a possible inaccurate estimation of vapor flux at

low water saturation, as pointed out earlier. On the other hand, Eq. 5.37 preserved the quality of

Eq. 5.36 so that it could describe the effects of ambient pressure, temperature, and water

saturation, and maintain its accuracy without the need for the complexity found in Eq. 5.36.

Hence, Eq. 5.37 appeared to be a better solution than Eqs. 5.35 and 5.36. In other words, the first

term of Eq. 5.36 could replace the second term.

5.4.4 Limitation, and potential area of improvement

The limitations of the Cass research work were already discussed in Chapter 4. Moreover, other

factors related to the present study could be considered for future improvement and investigation.

94

The findings of the following researchers corroborate this statement. The network model

considers the interconnectedness of the pore structure and the microscopic flow mechanisms as

described in Dullien (1992). Honarpour et al. (1986) compiled a list of factors that were found to

affect relative permeability – rock properties, wettability, saturation history, overburden pressure,

porosity and permeability, interfacial tension and density, viscosity, and the initial wetting phase

saturation. Also, in addition to that list, Saraf and McCaffery (1985) reported the gravitational

force relating to the Bond Number as a factor that could affect the relative permeability. While

all these factors are relevant, the water vapor flux as a function of water saturation, ambient

pressure and temperature, under the influence of temperature gradient, were successfully

addressed in the present study. Finally, it is important to stress that the mathematical solution

provided in this work was found to model the observed phenomenon well. The curve-fitted

parameters’ ( n , m and b ) exact values and their relative values from one soil to another may

not be their final values; simply because the alternative combination of parameter values, from

curve-fitting solution, could generate similar calculated output. Any experimental effort, in any

possible way, should be directed to estimate the parameter values. Nonetheless, the current

mathematical solution would be a significant step forward for future investigation.

5.5 Conclusions

The water vapor relative permeability relationship showing a concave downward curve pattern

with the increasing soil moisture saturation was successfully derived. The integral equation was

extended from the work of Purcell (1949), Burdine (1953), Fatt and Dykstra (1951), Mualem

(1976) and Timmerman (1982). The empirical equation and derivation method used was based

on the work of van Genuchten (1980). The Klinkenberg (1941) permeability relationship was

95

extended to include the effect of temperature so that both the ambient pressure and temperature

could be incorporated to account for the water vapor slip effect in the soils. Darcy law was found

to be able to replace Fick’s law in governing the water vapor flux in soils, based both on the R-

squared values and graphical examination. One of the advantages is that it could describe the

effect of temperature and pressure on the water vapor slip effect, whereas Fick’s law confined

this to a single unknown vapor enhancement factor which could not account for both the effects.

Also, mathematically and computationally Darcy law did not increase the complexity more than

that from Fick’s law, and it was simpler than the addition of Darcy law to Fick’s law as proposed

in Chapter 4. Moreover, the mathematical relationship derived in the present study should be

able to address challenging ventures such as that stated by Sakaguchi et al. (2009) about the need

to understand ambient pressure variation to reflect the regolith on planets such as Mars.

96

Figure 5.1. Schematic diagram of velocity profile due to fluid flow through a small tube

diameter in the porous media.

97

(a) (b)

(c) (d)

Figure 5.2. The water vapor thermal conductivity of lysimeter sand versus water saturation at (a)

32.5oC, (b) 22.5oC, (c) Portneuf silt loam versus saturation at 32.5oC, and (d) Kanto loamy soil

versus saturation at temperature ranged from 38.5 to 31.5oC. Note: the water vapor thermal

conductivity estimated by Darcy law was obtained by subtracting the water vapor thermal

conductivity predicted by Fick’s law from the total water vapor thermal conductivity. Equation

5.33 was used in the determination. The total water vapor thermal conductivity dataset derived

from Cass et al. (1984) at different pressure ratios, temperatures, and volumetric water contents.

0.00

0.10

0.20

0.30

0.40

0.00 0.20 0.40 0.60 0.80

Wat

er

vap

or

ther

mal

co

nd

uct

ivit

y

(J m

-1s-1

K-1

)

Water Saturation, Se (dimensionless)

Total, Pr=1.000 Darcy, Pr=1.000Total, Pr=0.530 Darcy, Pr=0.530Total, Pr=0.160 Darcy, Pr=0.160Total, Pr=0.083 Darcy, Pr=0.083

0.00

0.10

0.20

0.30

0.40

0.00 0.20 0.40 0.60 0.80

Wat

er v

apo

r th

erm

al c

on

du

ctiv

ity

(J m

-1s-1

K-1

)



0.000

0.100

0.200

0.300

0.400

0.00 0.20 0.40 0.60 0.80 1.00

Wat

er v

apo

r th

erm

al c

on

du

ctiv

ity

(J·m

-1·s

-1·K

-1)



0.000

0.100

0.200

0.300

0.400

0.20 0.40 0.60

Wat

er v

apo

r th

erm

al c

on

du

ctiv

ity

(J·m

-1·s

-1·K

-1)


Total, Pr=1.000 Darcy, Pr=1.000



98

Figure 5.3. The permeability variables ( rnw P Tk k ) versus water saturation at the unity of pressure

ratio and 32.5 degree Celsius.

0.0E+00

1.0E-17

2.0E-17

3.0E-17

4.0E-17

5.0E-17

6.0E-17

7.0E-17

8.0E-17

0 0.2 0.4 0.6 0.8 1

k rn

wk p

-T(m

2)

Water saturation (Se)

Portneuf silt loam Lysimeter sand

Kanto loamy soil

99

(a)

(b)

(c)

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.6E-04k r

nw

k P-T

/ko

Data point

Data

Model Se-Pr-T

Model Se-Pr

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

7.0E-06

8.0E-06

k rn

wk P

-T/k

o

Data point

Data

Model Se-Pr-T

Model Se-Pr

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

k rn

wk P

-T/k

o

Data point

Data

Model Se-Pr-T

Model Se-Pr

100

Figure 5.4. The (a) Portneuf silt loam, (b) lysimeter sand, and (c) Kanto loamy soil permeability

variables ( rnw P T ok k k ) data estimated by Eq. 5.34 at different water saturations, temperatures,

and pressure ratios. Note: the data were curve-fitted by the rnwk and P Tk from Eqs. 5.30 and

5.14, respectively. The equation with and without the temperature as an independent variable in

the expression of P Tk would correspond to the model e rS T P and e rS P , respectively. From

the first to fourth wave-like curves were respectively the pressure ratios of 1, 0.53, 0.16 and

0.083, as in graph (a), and the each point of the rise and fall of the permeability variables of the

wave-like curve, from left to right, correspond to the increasing water saturation. The

experimental temperature was 32.5 degree Celsius. In graph (b), the first three wave-like curves

correspond to temperatures of 3.5, 22.5 and 32.5 degree Celsius at a pressure ratio of unity. The

following three wave-like curves correspond to the similar temperature range, but at the pressure

ratio of 0.53, and it continued until the fourth (or last) three wave-like curves at a pressure ratio

of 0.083. Graph (c) represented by first to fifth wave-like curves with respective pressure ratios

of 1.000, 0.705, 0.495, 0.308 and 0.203. The first wave-like curve permeability variables, from

left to right, corresponding to decreasing temperature (42.5 to 31.6 degree Celsius) and

increasing water saturation. The rest of the wave-like curves correspond to a similar range of

temperature and saturation conditions. The data on graphs (a) and (b) were derived from Cass et

al. (1984), while graph (c) was from Tanimoto (2007) derived using DPHP method.

101

Table 5.1 The parameter constants ( n and m ) and the Klinkenberg gas slip factor ( b ) of water

vapor relative permeability.

Soil type n m b (K-1) R-squared

Lysimeter sand 0.535 1.550 1.883 0.933

Portneuf silt loam 1.280 1.160 116.892 0.971

Kanto loamy soil 3.867 2.542 1310.853 0.849

Note: the permeability variables ( /rnw P T ok k k , dimensionless), calculated from the parameters

n , m and b above, should be multiplied by 10-7 to obtain its original value. Equation 5.34 was

used to determine the parameters.

102

Chapter 6. Summary and Conclusions

The aim of the study was to describe the mechanism governing the vapor enhancement factor, by

investigating in two case studies. The first case study involved the use of the vapor enhancement

factor as a calibration parameter to simulate liquid water, water vapor, and heat flow in

unsaturated soil. The second case study required a re-examination of the theory and equation

used to derive the governing equation of water vapor flux and its constitutive equation.

Therefore, this study was divided into two main stages to address the problem. The first stage of

investigation involved the extension of source code from the simulation of water infiltration in

unsaturated soil (Chapter 2) to simulate water vapor, and heat flow movement, in addition to

liquid water movement (Chapter 3) in unsaturated soil. Key information learned in Chapter 2

was employed in Chapter 3. Also, the outcome of the simulation in Chapter 3 highlighted some

potential areas to improve the uncertainty of the vapor enhancement factor, mainly due to its

undefined mechanism, which lead to the research in the second stage of the overall study. The

second stage proposed an improved heat flux equation, an alternative equation for the vapor

enhancement factor (Chapter 4) and identified a necessary correction for water vapor relative

permeability (Chapter 5). The alternative equation can replace the enhanced-vapor diffusion

equation with a newly derived water vapor relative permeability equation.

In Chapter 2, the source code to simulate water infiltration into an unsaturated soil was

validated using Philip’s semi-analytical dataset. The elementary effect method was used in

addition to the one-at-a-time method in the validation procedure, where the sign oscillation effect

could be identified, for example, the time-step size. A hypothetical case study set up to

investigate the discrepancy observed between the dataset and the simulation revealed that

103

parameters with a high sensitivity coefficient and a high uncertainty range have the greatest

influence on the simulation output. For example, the initial moisture content and time-step size

had the highest sensitivity coefficient and highest uncertainty range, respectively, but neither of

them had the greatest effect on the water infiltration. The factor with the most influence on

model output was spatial spacing size, which implies that the extension of the current source

code to simulate the heat and mass flow of liquid water and water vapor would require a fine

spatial size to generate a reliable simulation result.

The liquid water, water vapor and heat flow source code was developed by extending the

water infiltration source code used in Chapter 2. By using a fine spatial size a reliable simulation

with a low error was obtained, that was able to re-simulate the temperature and water content

distributions based on the experimental conditions used in Heitman et al. (2008). Calibrations

with multiple temperature boundaries was superior to a single temperature boundary because the

optimum determined pair of hydraulic conductivity and vapor enhancement factor values was

more widely applicable and with generally lower error than any other pairs. However, the

observed optimum pair varied with soil type and initial water contents used in the study. For

example, the silt loam at 0.10 and 0.20 m3 m-3 had different optimum pairs, and similarly

different optimum pairs were observed for different soil types at the same water content. The

uncertainty was refined by using different optimum pairs to achieve the required accuracy for

either temperature or water content distributions. The model accuracy deterioration, or increased

error, was enhanced with the simulation time, and with an increasing average temperature

gradient. A higher error in the reverse compared with the forward temperature boundaries

suggested the presence of an unaccounted phenomenon such as buoyancy force and/or water

retention hysteresis. The zeta coefficient used in the vapor enhancement factor ranged between 2

104

for the least error in temperature distribution and 6 for the water content distribution. The values

were obtained irrespective of whether it was based on a single soil type averaged over different

water contents, or averaged over different soils and water contents. The least error value for

saturated hydraulic conductivity was obtained at its original value, without any decrement

necessary. The weak compliance of the simulation at low and high water contents suggested the

need for future improvement with the Fayer and Simmons (1995) equation. Based on a

comparison of both the calibrated parameters, the vapor enhancement factor was the least

understood parameter. This may be due to the theoretical and experimental complexity, which

has deterred adequate investigation by researchers. Hence, the vapor enhancement factor was

investigated in Chapter 4, in particular its theoretical aspects.

In Chapter 4, an alternative equation, based on Darcy law, was proposed to govern the

water vapor enhancement factor and it was validated using published data from Cass et al.

(1984). As part of the solution, an improved heat flux equation was proposed with additional

mechanisms included. A rapid increment rate of thermal conductivity resulting from liquid water

movement sensible heat with increasing water saturation was observed when compared to the

wet porous media heat conduction. The other two added mechanisms were water vapor sensible

heat and the partial derivative of matric pressure head effects on the relative humidity with

respect to temperature. The improved heat flux equation enabled the derivation of an equation for

experimental conditions ranging from relatively dry to saturated moisture conditions, because the

equation accounted for the liquid water flow in wet porous media condition. The Darcy law

equation was proposed to govern the water vapor enhancement factor after considering the

simple mass balance equation, ideal gas law, and the water vapor advection by air volume

expansion. The simple mass balance equation based on the ratio of liquid water to water vapor

105

densities resulted in a clear expansion in estimated water vapor volume. Again, the ideal gas law

described the presence of water vapor volume expansion under the effect of temperature

increment. Hence, not only was water vapor advection caused by air volume expansion, but the

water vapor itself experienced volume expansion as it vaporized. The water vapor density used in

Fick’s law density gradient equation to govern the water vapor flux was right after the water

vapor volume expansion, and thus, an enhanced-Fick’s law equation was required to account for

the under prediction. Water vapor volume expansion-advection, which is regarded as the physical

phenomenon to govern the vapor enhancement factor, would be a more reasonable approach than

the enhanced-water vapor diffusion by Fick’s law. The equation described the effect of water

saturation and air pressure on water vapor flux reasonably well. However, the effect of average

temperature variation on the water vapor flux was not accounted for, and so an equation for

water vapor relative permeability was proposed as a temporary solution. Both the temperature

effect and relative permeability were addressed in Chapter 5.

In Chapter 5, the extended Darcy law equation was derived from Navier-Stokes equation

to obtain the Klinkenberg permeability and relative permeability. The derivation of Klinkenberg

permeability allowed the identification of a theoretical basis for including the average

temperature variable into the equation. Similarly, the derivation of relative permeability was

extended to derive water vapor relative permeability, which was able to describe the concave

downward curve with increasing soil moisture saturation. The integral and empirical equations

were based on the work of previous researchers, and the derivation method was solely based on

the work of van Genuchten (1980). The Fick’s law diffusion equation was found largely

inadequate for describing the observed water vapor flux. Thus, the Darcy law, water vapor

volume expansion-advection equation was proposed to replace the enhanced-vapor diffusion

106

equation. The newly derived Klinkenberg and water vapor relative permeability equations were

able to describe the lysimeter sand and Portneuf silt loam from Cass et al. (1984), and the Kanto

loamy soil from Tanimoto (2007).

107

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