Jan 1, 2002 - Pore Geometry Effect on Gaseous Diffusion and ... know you for your awesome friendship. ...... Illustration of tortuous and straight flow length.
Pore Geometry Effect on Gaseous Diffusion and Convective Fluid Flow in Soils
By ATAC TULI B.S. (Cukurova University, Adana, Turkey) 1987 M.S. (Çukurova University, Adana, Turkey) 1990 M.S. (Wageningen Agricultural University, Wageningen, Netherlands) 1993 DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Soil Science in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved:
Committee in Charge 2001
Abstract In order to better manage and monitor fluid flow and migration of chemicals in gas and liquid phases, a fundamental understanding and knowledge of the saturation dependency of permeability, diffusion and electrical conductivity in a porous medium is required. The initial fundamental formulations and recent research for characterization of transport properties in porous media are discussed. To improve the predictive capability of transport models, physically based formulations that include understanding of the control of pore continuity and tortuosity ( TC ) on transport are proposed. To understand the pore geometry effects of tortuosity and connectivity on the various considered transport processes, the fluid saturation dependency of hydraulic conductivity, air conductivity, gaseous diffusion and bulk soil electrical conductivity was measured for two different soils. While assuming that soils can be characterized by a lognormal pore size distribution function, simultaneous fitting of all 4 transport coefficients to measured data showed that the TC parameter of each transport coefficient is different. Specifically, the value of TC parameters for air conductivity and gaseous diffusion are much higher than for soil hydraulic conductivity. For the bulk soil electrical conductivity, the TC parameters are similar to parameters of gaseous diffusion when they are represented by TC term only. Furthermore, the contribution of both pore size distribution and tortuosity-
connectivity is equally important for characterization of soil hydraulic. The moisture dependency of all 4-transport coefficients can be unique when normalized to their maximum measured values. Since natural soils are especially heterogeneous with soil properties varying in space and time, most uncertainty in the assessment of flow and transport processes in ii
unsaturated soils is at the field scale. Using simultaneous scaling, soil spatial variability of hydraulic functions can be described from a single set of scaling factors, as derived form physical considerations. For soils with a lognormal pore-size distribution physically based scaling leads directly to lognormally distributed scaling factors. Identifying and understanding the effects of key pore characteristic parameters can help the possibility for developing common interrelationships between transport processes and the assessment of spatial variability of these processes in the field scale.
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Acknowledgements The completion of this dissertation is realization of a dream that it started in Netherlands when I was doing my degree of Master of Science in Soil and Water. It would have remained a fantasy, if my advisor and mentor Professor Jan W. Hopmans did not offer a graduate student position in his research group. I really want to thank him for his generous help, advice and support at critical times during my six years presence in the University of California at Davis. I am greatly indebted to Professor Dennis E. Rolston for his strong support and allowing me use his laboratory facilities unlimited times. From Professor Per Møldrup, I am grateful for his constructive comments and criticisms during writing the dissertation. This work could not be improved without his contributions. I wish to express my gratitude to Kearney Foundation of Soil Science and the Agricultural System Program of the NRI-USDA for funding that made my Ph.D. study possible. I am most fortunate to know my wonderful friends Michelle A. Denton, Mutlu Koca, and Yusuf Uludağ. They shared their thoughts and ideas for my concerns during the completion of this work. My friend Çağdaş Arpaç, I enjoyed a lot your company and intellectual conversations during my visits to San Francisco and I felt really lucky to know you for your awesome friendship. I would like to especially thank my friend Alper Yılmaz who helped me to add new perspectives to my life such as Jazz and Classical music and his tremendous support in desperate days of my education and health. To my family for their support and encouragements from so far away, I love you all so dearly! My wonderful daughter Anıl Su Tuli, your birth brought exceptional joy and happiness to my life. To my love, Özlem Tuli, who gave me her endless love and
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support all the time, without you, none of these could happen. Thank you so much and I love you forever!
Atac Tuli January 1, 2002
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Table of contents Title Page................................................................................................................... i Abstract.....................................................................................................................ii Acknowledgements .................................................................................................iv Table of contents .....................................................................................................vi List of symbols ......................................................................................................... x List of figures ........................................................................................................xix List of tables .........................................................................................................xxii 1. Advancements in characterization of transport properties in porous media....... 1 1.1.
Introduction ........................................................................................................ 1
1.2.
Porous media characteristics .............................................................................. 1
1.3.
Water flow in saturated porous media................................................................ 4
1.4.
Water flow in unsaturated soils .......................................................................... 8
1.5.
Soil-water characteristic functions ................................................................... 12
1.6.
Air conductivity................................................................................................ 13
1.7.
Gaseous diffusion in soils................................................................................. 23 1.7.1. Soil-type dependent gaseous diffusivity models for unsaturated, disturbed soils .............................................................................................................. 25 1.7.2. Soil-type dependent gaseous diffusivity models for undisturbed soils..... 31 1.7.3. Gaseous diffusivity models linked to other fluid-phase transport parameters .................................................................................................... 36 1.7.4. Analogy-based gaseous diffusivity model ................................................ 39
1.8.
Electrical conductivity...................................................................................... 42 vi
1.8.1. Electrical resistivity and formation factor................................................. 42 1.8.2. Models for electrical conductivity in unsaturated soil.............................. 46 1.9.
Scaling of soil transport properties................................................................... 49
1.10. Potential links between fluid-phase transport coefficients............................... 53 1.11. Research Objectives ......................................................................................... 55 1.12. References ........................................................................................................ 57 1.13. Appendix 1.1. ................................................................................................... 75 1.14. Appendix 1.2. ................................................................................................... 76
2. Search for an unifying approach to determine transport coefficients of disturbed soils ................................................................................................... 78 2.1.
Abstract ............................................................................................................ 78
2.2.
Introduction ...................................................................................................... 79
2.3.
Theory .............................................................................................................. 83 2.3.1. Effective saturation relations and soil water retention function ............... 83 2.3.2. Relative hydraulic conductivity ................................................................ 86 2.3.3. Relative air conductivity ........................................................................... 87 2.3.4. Gaseous diffusivity ................................................................................... 88 2.3.5. Relative bulk soil electrical conductivity.................................................. 89
2.4.
Materials and Methods ..................................................................................... 90 2.4.1. Soil sample preparation............................................................................. 90 2.4.2. Determination of soil transport coefficients ............................................. 92 2.4.2.1. Soil hydraulic functions………………………………………. 92 2.4.2.2. Air Conductivity……………………………………………… 94
vii
2.4.2.3. Bulk soil electrical conductivity ................................................ 95 2.4.2.4. Gaseous diffusion....................................................................... 97 2.4.3. Analysis of fitting results .......................................................................... 99 2.5.
Results and Discussions ................................................................................. 101
2.5.1. Soil water retention curves...................................................................... 101 2.5.2. Measured transport coefficients.............................................................. 104 2.5.3. Simultaneous optimization of transport coefficients .............................. 108 2.6.
Conclusions .................................................................................................... 119
2.7.
References ...................................................................................................... 121
2.8.
Appendix 2.1. ................................................................................................. 127
3. Simultaneous Scaling of Soil Water Retention and Unsaturated Hydraulic Conductivity Functions Assuming Lognormal Pore-Size Distribution.......... 129 3.1.
Abstract .......................................................................................................... 129
3.2.
Introduction .................................................................................................... 130
3.3.
Theory ............................................................................................................ 133 3.3.1. Scaling of water retention curves............................................................ 134 3.3.2. Scaling of unsaturated hydraulic conductivity function ......................... 136
3.4.
Material and Methods..................................................................................... 138 3.4.1. Experimental ........................................................................................... 138 3.4.2. Scaling method........................................................................................ 140
3.5.
Results and Discussion................................................................................... 142
3.5.1. Optimized individual soil hydraulic functions........................................ 142 3.5.2. Simultaneous scaling of soil hydraulic functions ................................... 146
viii
3.6.
Summary and Conclusions ............................................................................. 153
3.7.
References ...................................................................................................... 155
3.8.
Appendix 3.1. ................................................................................................. 160
3.9.
Appendix 3.2. ................................................................................................. 162
3.10. Appendix 3.3. ................................................................................................. 170
ix
List of symbols a
Air content or air-filled porosity (L3 L-3)
a*
Highest air content value as matching point value (L3 L-3)
ac
Continuous air content (L3 L-3)
amax
Experimentally measured maximum air content (L3 L-3)
amin
Experimentally measured minimum air content (L3 L-3)
af
Fractional air-filled volume (-)
a fc
Air content value at a soil matric head of –100 cm (L3 L-3)
ar
Residual air content (L3 L-3)
av
Volume of larger-pore phase (L3)
a100
Air content at –100 cm soil matric head (L3 L-3)
A
Cross sectional area (L2)
A
Constant value (L2)
b
Slope of the Campbell water retention curve in log-log scale (-)
c1,2
Constants obtained from Campbell water retention curve in log-log scale (-)
c3,4
Constants for empirical transmission coefficient equation (-)
ce
Cementing factor (-)
C
Concentration of gaseous phase (M L-3)
d
Tortuosity parameter of general gas diffusion model (-)
DH
Hydraulic diameter (L)
Dg
Soil-gas diffusion coefficient (L2 T-1)
x
D*
Soil-gas diffusion coefficient at the highest air content (L2 T-1)
Dr*
Matching point diffusivity ( = Dg Do ) (-)
Drg
Gas diffusivity ( = Dg Dsg ) (-)
Dsg
Gas diffusion coefficient value at Sea = 1 (L2 T-1)
Do
Gas diffusion coefficient in free air (L2 T-1)
Dw
Soil water diffusivity (L2 T-1)
Dˆ w
Soil water diffusivity of reference soil (L2 T-1)
EC
Electrical conductivity (Q2 T M-1 L-3)
ECa
Bulk soil electrical conductivity (Q2 T M-1 L-3)
ECb
Electrical conductivity of bulk soil solution (Q2 T M-1 L-3)
ECra
Relative electrical conductivity ( ( ECa ( Sew ) − ECs ) ( ECa ( Sew = 1) − ECs ) ) (-)
EC s
Bulk soil surface electrical conductivity (Q2 T M-1 L-3)
EC w
Electrical conductivity of the saturating solution (Q2 T M-1 L-3)
f
Fluid content (L3 L-3)
f*
The highest fluid content (L3 L-3)
F
Complementary normal distribution function (-)
Fcp
Silt+Sand content ( 0 ≤ Fcp ≤ 1 )
Fg
Geometry factor for gaseous phase (-)
FGE
Geometry factor for electrical conductivity (-)
Fn
Normal distribution function (-)
xi
Fr
Formation resistivity factor (-)
g
Gravitational acceleration (L T-2)
h
Capillary pressure head (L)
hD
Manometer reading (L)
hm
Geometric mean of capillary pressure head in Lognormal retention model (L)
hˆm
Geometric mean of capillary pressure head for reference soil (L)
h
Matric head (L)
he
Air entry pressure head (L)
h init
Initial matric head (L)
hm
Geometric mean of matric head in Lognormal retention model (L)
hˆ
Matric pressure head of reference soil (L)
i
Index for number of soil sample or data points (-)
I
Electrical current (Q T-1)
Ih
Total number of data points of matric head (-)
IK
Total number of data points of hydraulic conductivity (-)
j
Index for number of data points or transport coefficients (-)
k
Permeability (L2)
k fc
Air permeability at a soil matric head of – 100 cm (L T-1)
kK
Permeability for Kozeny (L2)
k KC
Permeability for Kozeny-Carman (L2)
k*
Permeability at highest air content as matching point value (L2)
xii
ko
Dimensionless shape factor (-)
k'
Shape factor (-)
Ka
Air conductivity (L T-1)
K a , cap
Air conductivity of a bundle of the straight capillary tubes (L T-1)
K a , soil
Air conductivity of the soil (L T-1)
K ra
Relative air conductivity ( = K a K sa ) (-)
K sa
Air conductivity value at Sea = 1 (L T-1)
K rw
Relative hydraulic conductivity ( = K w K sw ) (-)
K rw,i
Relative hydraulic conductivity of soil sample i (-)
Kˆ sw
Saturated hydraulic conductivity of reference soil (L T-1)
K sw
Optimized saturated hydraulic conductivity (L T-1)
* K sw
Measured saturated hydraulic conductivity (L T-1)
Kw
Hydraulic conductivity (L T-1)
K w ( S ew )
Unsaturated hydraulic conductivity (L T-1)
Kˆ w (θ or Sˆew ) Hydraulic conductivity of reference soil (L T-1)
K w,cap
Hydraulic conductivity of a bundle of the straight capillary tubes (L T-1)
l
Tortuosity and connectivity parameter (-)
l1
Tortuosity and connectivity parameter for hydraulic conductivity (-)
l2
Tortuosity and connectivity parameter for air conductivity (-)
l3
Tortuosity and connectivity parameter for gaseous diffusion (-)
xiii
l4
Tortuosity and connectivity parameter for bulk soil electrical conductivity (-)
L
Effective length parameter (-)
Le
Tortuous flow length (L)
Lc
Length of soil sample (L)
L st
Straight flow length (L)
m
(= 1-1/n) Parameter of van Genuchten soil-water retention function (-)
n
Fitting parameter of van Genuchten soil-water retention function (-)
na
Gaseous-phase tortuosity parameter (-)
∆P
Pressure difference (F L-2)
Pe
Wetted perimeter (L)
PE
Potential energy (M L2 T-2)
PS
Pore size term (-)
Q
Electrical charge (Q = Coulomb (C))
r
Radius of the largest water-filled capillary (L)
rcrit
Critical pore radius (L)
r fc
Equivalent pore radius at a soil matric head of –100 cm (L)
rm
Geometric mean or median pore radius (L)
rˆm
Geometric mean or median pore radius of reference soil (L)
R
Capillary tube radius (L)
RH
Hydraulic radius (L)
S
Relative water saturation ( = θ θ s ) (-)
Sa
Degree of air saturation (= a φ ) (-) xiv
S a ,ea
Air saturation at emergence or extinction point of the air phase (-)
S a ,max
Absolute maximum air saturation (-)
S a ,r
Residual air saturation (-)
S a,s
Achievable maximum air saturation (-)
S ea
Effective air saturation (-)
S ew
Effective water saturation (-)
Sˆew
Effective water saturation for reference soil (-)
So
Particles surface per unit volume of the particle (L2 L-3)
Sw
Degree of water saturation (= θ φ ) (-)
S w,ea
Water saturation at emergence or extinction point of the air phase (-)
S w,ew
Water saturation at emergence or extinction point of the water phase (-)
S w,max
Absolute maximum water saturation (-)
S w,r
Residual water saturation (-)
S w, s
Achievable maximum water saturation (-)
S'
Particles surface per unit volume of the medium (L2 L-3)
So
Sorptivity (L T-1/2)
Sˆ o
Sorptivity of reference soil (L T-1/2)
t
Time (T)
T
Transmission coefficient (L L-1)
TC
Tortuosity-connectivity term (-)
TrCo
Transport coefficient value (-) xv
TrCo*
Transport coefficient value at the highest fluid content (-)
TrCo j
Jth transport coefficient (-)
u
Soil-specific parameter representing discontinuity of air phase (-)
vc
Average velocity of the capillary tube (L T-1)
v DA
Macroscopic Darcy velocity (L T-1)
v DF
Dupuit-Forchheimer velocity (L T-1)
v gas
Gas flux density (M L-2 T-1)
v HP
Average pore water velocity for Hagen-Poiseuille (L T-1)
vK
Apparent linear velocity for Kozeny (L T-1)
v KC
Apparent linear velocity for Kozeny-Carman (L T-1)
V
Volume of a fluid (L3)
Vp
Electrical potential (L2 M T-2 Q-1)
V p1
Electrical potential at point 1 (L2 M T-2 Q-1)
Vp2
Electrical potential at point 2 (L2 M T-2 Q-1)
Vgas
Amount of diffusing gas (M)
ws
Weighting constant for solid phase (-)
W
Work (M L2 T-2)
x
Distance in soil (L)
z
Soil-specific parameter controlling curve of tortuosity function (-)
α
Fitting parameter of van Genuchten soil-water retention function (L-1)
αi
Scaling factor of soil sample, i (-) xvi
β
Contact angle between water and wall of capillary tube (-)
δ
Constant exponent for pore tortuosity (-)
φ
Total porosity (L3 L-3)
ϕ
Material type constant in Currie (1960b) (-)
γ
Surface tension between the liquid and the air (F L-1)
η
General exponent for pore tortuosity (-)
κ
Empirical constant in Buckingham/soil texture model (-)
λ
Pore size distribution parameter (-)
λi
Characteristic length of soil sample i (-)
λˆ
Characteristic length of reference soil (-)
µ
Dynamic viscosity (M L-1 T-1)
π
Pi = 3.1415
θ
Volumetric water content (L3 L-3)
θ init
Volumetric water content at initial soil matric head (L3 L-3)
θ min
Experimentally measured minimum water content (L3 L-3)
θr
Residual water content (L3 L-3)
θs
Saturated water content (L3 L-3)
ϑ
Constant for particle shape (-)
ρ
Density of fluid (M L-3)
ρw
Density of water (M L-3)
σ
Standard deviation of pore size distribution of Lognormal retention model (-)
σo
Electrical resistivity of the porous media (L3 M T-1 Q-2) xvii
σw
Electrical resistivity of the electrolyte (L3 M T-1 Q-2)
τ
Tortuosity (= ( Le L ) )
ξ
Material type constant in Currie (1960b) (-)
2
xviii
List of figures Figure 1-1. Illustration of tortuous and straight flow length............................................... 6 Figure 1-2. Schematic representation of
(a) soil-water characteristic (b) relative
conductivity of the water (dashed line) and air (solid line) for different scaled saturations (Adapted from Dury et al., 1999). .......................................................... 20 Figure 1-3. Geometrically similar porous medium at microscopic level.......................... 50 Figure 1-4. Possible links between different transport coefficients (Adapted from Moldrup et al., 2001). ............................................................................................... 54 Figure 2-1. Bulk soil electrical conductivity as a function of water content for different solution EC values for the Oso Flaco fine sand. ..................................................... 97 Figure 2-2. Soil water retention data and optimized curves for the Oso Flaco fine sand and Columbia sandy loam soil................................................................................ 101 Figure 2-3. Pore size distribution of Oso Flaco sand and Columbia soil with median radius ( ln rm ) and standard deviation ( σ ) of ln rm ................................................. 103 Figure 2-4. Measured transport coefficients as a function of fluid content for both soils. ................................................................................................................................. 105 Figure 2-5. Measured relative transport coefficients as a function of effective fluid saturation for both soils........................................................................................... 106 Figure 2-6. The measured and simultaneously optimized transport coefficient data for all 6 cases of the Oso Flaco fine sand. ......................................................................... 109 Figure 2-7. The measured and simultaneously optimized transport coefficient data for all 6 cases of the Columbia sandy loam soil. ............................................................... 110
xix
Figure 2-8. Combination of all 4 relative transport coefficients (TrCo j j = 1...4 ) as a function of fluid content and effective saturation. .................................................. 113 Figure 2-9. Sensitivity analysis for relative transport coefficient ( TrCo j ) functions: Saturation dependence of (A) TC term on the tortuosity-connectivity parameter, l , (B) PS term on the effective fluid saturation, (C) relative water phase transport coefficient function on l parameter, and (D) relative air phase transport coefficient function on l parameter. .......................................................................................... 116 Figure 2-10. Sensitivity analysis for relative transport coefficient ( TrCo j ) functions: Saturation dependence of (A) TC term on the tortuosity-connectivity parameter, l , (B) PS term on the effective fluid saturation, (C) relative water phase transport coefficient function on l parameter, and (D) relative air phase transport coefficient function on l parameter. .......................................................................................... 117 Figure 3-1. Value of σ i2 , versus ln hm ,i for both sampling depths. ................................ 144 Figure 3-2. Fractile diagram of the saturated hydraulic conductivity (ln K sw,i ) for both soil depths. ..................................................................................................................... 145 Figure 3-3. Simultaneous PB scaling of subset ln hm,i < 6.0 at 25 cm............................. 147
Figure 3-4. Simultaneous PB scaling of subset ln hm,i ≥ 6.0 at 25 cm............................. 148 Figure 3-5. Soil hydraulic function curves of subsets for reference soils at 25-cm depth. ................................................................................................................................. 150 Figure 3-6. Fractile diagrams of scaling factors distribution obtained by physically-based and conventional method for all subsets of 25- and 50-cm soil depths. ................. 152 Figure 3-7. Simultaneous PB scaling of subset ln hm,i < 6.0 at 50 cm............................. 162 xx
Figure 3-8. Simultaneous PB scaling of subset ln hm,i ≥ 6.0 at 50 cm............................. 163 Figure 3-9. Simultaneous C scaling of subset ln hm,i < 6.0 at 25 cm............................... 164 Figure 3-10. Simultaneous C scaling of subset ln hm,i ≥ 6.0 at 25 cm............................. 165 Figure 3-11. Simultaneous C scaling of subset ln hm,i < 6.0 at 50 cm............................. 166 Figure 3-12. Simultaneous C scaling of subset ln hm,i ≥ 6.0 at 50 cm............................. 167 Figure 3-13. Soil hydraulic function curves of subsets for reference soils of PB scaling method at 50-cm depth............................................................................................ 168 Figure 3-14. Soil hydraulic function curves of subsets for reference soils of C scaling method at 25- and 50-cm depth. ............................................................................. 169
xxi
List of tables Table 2-1. Physical properties of soils used in experiments............................................. 91 Table 2-2. Optimized
( h m , σ , K sw , and l1 ) and measured ( θ r and θ s ) parameters in
multistep outflow. ..................................................................................................... 94 Table 2-3. Case of model parameters used in the simultaneous optimization................ 100 Table 2-4. Optimized σ , h m and tortuosity-connectivity parameters ( TC ), Root Mean quared Error ( RMSE ), and Averaged Residual ( AR j ) for fitted transport coefficient functions.................................................................................................................. 111 Table 2-5. Measured data for the Oso Flaco fine sand. .................................................. 127 Table 2-6. Measured data for the Columbia sandy loam soil. ........................................ 128 Table 3-1. Parameters for reference soil hydraulic function curves (PB-physically based scaling; C-conventional scaling)............................................................................. 143 Table 3-2. WRMSR and total reduction in physically-based (PB) and conventional (C) scaling methods....................................................................................................... 149 Table 3-3. Statistical properties of scaling factors ( Mean(ln α i ) = 0 )............................ 151 Table 3-4. Measured and optimized parameters for soil hydraulic functions used in subset lnhm < 6.0 of 25 cm depth. ...................................................................................... 170 Table 3-5. Parameters for soil hydraulic functions used in subset lnhm ≥ 6.0 of 25 cm depth........................................................................................................................ 173 Table 3-6. Parameters for soil hydraulic functions used in subset lnhm < 6.0 of 50 cm depth........................................................................................................................ 174
xxii
Table 3-7. Parameters for soil hydraulic functions used in subset lnhm ≥ 6.0 of 50 cm depth........................................................................................................................ 176
xxiii
1
1. Advancements
in
characterization
of
transport
properties in porous media
1.1. Introduction Environmental quality is being adversely affected by agricultural, industrial, and municipal activities. Transport models are frequently used by agriculturalists and environmentalists to compare alternative management practices for the purpose of optimizing crop yield and minimizing groundwater pollution by fertilizers and toxic substances. A major difficulty in these modeling efforts has been a substantial lack of information concerning the constitutive relationships governing the multi-fluid flow and transport, in particular, the functional relationships between degree of fluid saturation, permeabilities, diffusion and electrical conductivity (EC) of coexisting phases (Parker et al., 1987; Luckner et al., 1989). In order to better manage and monitor fluid flow and migration of chemicals in gas and liquid phases, a fundamental understanding and knowledge of saturation dependency of permeability, diffusion and electrical conductivity in a porous medium is required.
1.2. Porous media characteristics Transport coefficients depend on the degree of fluid saturation, pore geometry and the combined pore tortuosity-connectivity factor (Bear, 1972). Especially, pore geometry, i.e. the size distribution of pores and the topology of the pore space and its interconnectivities govern flow and transport of water, air, solute, and heat (Vogel and
2 Roth, 1998; Allaire-Leung et al., 2000ab; Vogel, 2000). For example, because soil pores vary in geometry and size, the pore water velocity is highly variable. Larger pores conduct water more rapidly and shear forces at the pore wall cause decreasing water velocity from the center of the pore towards the pore wall. But other factors such as continuity and connectivity of the water phase in the pore space also control water flow. When the soil is saturated, all of the pores are filled with water and the water phase becomes continuous. When the soil desaturates and the air phase replaces the water phase, the cross-sectional area within the water phase decreases, thereby typically increasing the actual path length of the flowing water with comparison to the straight-line path length, owing to the tortuous nature of the pore passages. Although the transport coefficients can usually be measured experimentally in a macroscopic sense, it is desirable to measure the influence of complex geometry of pore space on flow and transport (Carman, 1956; Vogel 2000). To understand and interpret the properties of the complex geometry, Dullien (1992) separated the parameters of the pore structure into two categories: Macroscopic and microscopic pore structure parameters. While macroscopic pore structure parameters represent average behavior of a sample containing many pores, microscopic pore structure parameters are involved with the pore space itself. The more important macroscopic structure parameters are porosity, permeability, specific surface area, formation resistivity factor, and air entry pressure. Characterization of the microscopic parameters is extremely difficult due to the irregular geometry of the pores. Because of the irregular variations of its geometry, the simplified physical quantities such as pore diameter, size, and shape, a model of straight capillaries is generally used to mimic the complex microscopic structure of natural reality. Hence, to
3 adequately describe the macroscopic nature of flow and transport processes in porous media, additional parameters such as pore (or capillary tube) tortuosity and connectivity of the parallel capillaries are introduced.
To emphasize the importance of these
parameters, Childs (1969) stated that the most important single physical attribute of the pore space is pore continuity, defined as a path that can be drawn within the pore space from any point to another point, without leaving the pore space. Unfortunately, the role of pore tortuosity and the connectivity factor is probably the least understood and, therefore, is typically used as a model dependent fitting parameter. The complex geometry of the pore space is extremely difficult to measure, but can be obtained from examining prepared sample thin sections using an optical or electron microscope. Vogel (1997) demonstrated that pore connectivity, as a function of pore size can be determined morphologically using serial sections of impregnated soil samples taken from two soil horizons. From the image analysis of these thin sections, the EulerPoincaré characteristic for pore connectivity index was determined, and used as a measure of connectivity. The pore connectivity was quantified as a function of the minimum pore diameter considered leading to connectivity of the pore space. When morphological pore-size distributions are compared with those obtained from water retention measurements, some discrepancy has been observed due to the differences in pore connectivity. Recent advances in non-invasive and non-destructive measurement techniques such as x-ray computed tomography (CT) and magnetic resonance imaging (MRI) has also provided a means of studying the nature of complex pore geometry and transport processes in soils (Anderson and Hopmans, 1994; Mori, 1999; Clausnitzer and Hopmans, 2000; Zhang et al., 2000).
4
1.3. Water flow in saturated porous media Establishing the exact relationship between transport coefficients and pore geometry appear extremely difficult so that simple models are usually adopted. Bear (1972) stated that microscopic description of the pore space is impossible; therefore, it is commonly assumed that pore space is represented by a bundle of parallel cylindrical capillary tubes, and the size distribution of the equivalent capillary tubes has been used to characterize soil pore size. Among the first fundamental equations to relate steady saturated fluid flow to the size of the pore space is Poiseuille’s law. According to Poiseuille’s law, if a cylindrical capillary tube has a radius, R (L), the volume of water flowing per unit time through that capillary tube under a given water pressure gradient is proportional to the fourth power of the radius, or
π R 4 ∆P Q= 8 µ Lst
[1-1]
where Q is the volumetric flow rate of the water (L3 T-1), µ is the dynamic viscosity (M L-1 T-1), and ∆P is the hydrostatic pressure difference (F L-2) across straight tube length,
L st . Water flow equations are commonly expressed in terms of the volumetric flow rate per unit area or flow velocity. Thus, from Eq. [1-1], the average flow velocity through the capillary tube, vc , can be written as
vc =
Q R 2 ∆P = A 8µ Lst
[1-2]
Any development based on Poiseuille’s law eventually leads to a linear relationship between the velocity and the hydraulic gradient (Bear, 1972). Since the pore
5 space, however, does not consists of a bundle of circular capillaries, but are noncircular and tortuous, the hydraulic radius concept was introduced. It is defined as the ratio of cross-sectional area normal to the flow and wetted perimeter to account for an average pore size (Bear, 1972; Carman, 1956; Dullien 1992). Assuming the pore space in the porous medium is equivalent to a single conduit of which the cross section has a complicated but constant shape, its hydraulic diameter, DH, governing the flow rate is four times the hydraulic radius, R H . For noncircular pores with average pore velocity, Hagen-Poiseuille’s equation can be derived from Eq. [1-2] using a shape factor, hydraulic diameter and average length of flow path
vHP
Q DH2 ∆P = = A 16k o µ Le
[1-3]
where DH is hydraulic diameter (L), Le is the average length of tortuous flow path (L), and ko is a dimensionless shape factor. The basic law governing macroscopic water flow is Darcy’s law, which states that the rate of flow is directly proportional to a pressure gradient or
vDA =
k ∆P µ Lst
[1-4]
where k is defined as the permeability (L2), and is consistent with [1-3]. Introducing the Dupuit-Forchheimer assumption that the water velocity, vDF = v DA φ , and DH =
4φ (for derivation see Appendix 1.1.) into Eq. [1-3], S o (1 − φ )
Kozeny (1927) derived the most widely accepted relationship between water velocity and porous medium properties, assuming that the pore space consists of an equivalent bundle
6 of capillary tubes with a common hydraulic radius, and with an average shape of its cross-sectional area, or
φ 3 ∆P φ 3 ∆P vK = = ko µ S ' 2 Le k ' µ S ' 2 Lst
[1-5]
where, L k'= e Lst
ko and S ' = So (1 − φ )
[1-6]
vK is Kozeny’s apparent linear velocity (L T-1), L e and L st are tortuous and straight flow
length (L), respectively (Figure 1-1), k ' is the dimensionless pore shape factor with values in the range between 2.0-5.0, S ' is the particles surface per unit volume of the medium (L2 L-3), S o is the specific surface of the particle per unit volume of particle (L2 L-3), and φ is total porosity (L3 L-3).
Figure 1-1. Illustration of tortuous and straight flow length.
Comparing Eq. [1-5] with Eq. [1-4], the permeability for Kozeny equation, k K , can be written as
7
kK =
φ3
[1-7]
k ' S o2 (1 − φ ) 2
Later, Carman (1948, 1956) modified the Kozeny’s expression by including the tortuosity effect on the actual pore velocity ( vKC = vK
vKC
φ 3 ∆P = k ' µ S ' 2 Lst
Le ), so that Lst 2
L where now k ' = e ko Lst
[1-8]
yielding the Kozeny-Carman permeability, k KC , k KC =
φ3 k oτ S o2 (1 − φ ) 2
[1-9]
where τ = ( Le Lst ) is the tortuosity. 2
Although later development of Kozeny’s expression accounts for the tortuous flow paths through the pore space, it is valid only for the pores with uniform size, and a certain range of pore shapes with a constant shape factor. With the widely varying radius of capillary bundles and for well-structured bodies, the mean hydraulic radius will not be the correct mean value of permeability calculations. The Kozeny-Carman equation also assumes that tortuosity is not dependent on variations in pore geometry (Carman 1956; Bear, 1972; Hillel 1980). Especially, in unsaturated flow, τ highly depends on the degree of saturation in addition to pore geometry with τ -values increasing as water content decreases. From here on, soil is the porous medium, and water and air flow will be considered only.
8
1.4. Water flow in unsaturated soils Due to the aforementioned limitations of Kozeny-type expressions, an alternative approach for the prediction of permeability in unsaturated soil was derived using poresize distribution. Since no direct methods are available to measure pore size distribution, it is indirectly derived from the soil-water characteristic curve (section 1.5) using the inverse relationship between the largest water-filled pore radius and soil-water matric head, h , (Young-Laplace equation)
r=
2γ cos β hρw g
[1-10]
where, r is the radius of the largest water-filled capillary (L), γ is the interfacial tension between water and the air (F L-1), β is the contact angle, h is the matric head (L), ρ w is the density of water (M L-3), and g is the coefficient of gravitational acceleration (L T-2). Based on the capillary tube analogy, Childs and Collis-George (1948, 1950) related the permeability of a soil to degree of water saturation using pore size distribution as determined from the soil-water characteristic curve. Conceptually, their approach assumes that a soil contains size classes of pores of various radii, which are randomly distributed in the soil, and that the Hagen-Poiseuille equation (Eq. [1-3]) is valid at the level of a single pore. Furthermore, when adjacent planes in the soil domain are brought in contact, the combined permeability across the plane can be computed by statistically means, from the number of pairs of interconnected pores. The permeabilities are controlled by the smaller pore of each pair. Subsequently, Marshall (1958), and Millington and Quirk (1959,1960, 1961) improved the method of the permeability calculation by introducing an arbitrary soil factor at saturation. This prediction was
9 further simplified by Kunze et al. (1968), Green and Corey (1971), and Jackson (1972). Since any of these developments are based on the capillary hypothesis of Eq. [1-10], one would expect that the theory applies more to coarse-textured than fine-textured soils, as the latter may be dominated by film flow, with soil water controlled by adsorptive rather than capillary forces. Moreover, soils generally have very complex pore geometry, and water-filled pores might be naturally discontinuous. Especially, the water trapped inside such discontinuous cavities (dead-end pores) is immobile, not contributing to water flow. The immobile water can also be called residual water, defining the value of saturation at which matric head decreases rapidly with a negligible or zero decrease in saturation (Corey, 1994), and where the unsaturated hydraulic conductivity approaches zero due to inherent discontinuity of the water phase. Pore continuity is the major factor affecting fluid flow in porous media. Vogel and Roth (1998) showed how pore-size distribution and pore connectivity factors are important for governing the shape of the hydraulic functions, i.e. both the soil-water characteristics and unsaturated hydraulic conductivity functions. Also, Wildenschild et al. (2001) and Mortensen et al. (2001) demonstrated the importance of air or water entrapment on soil water retention, using variable-rate outflow experiments. The hydraulic conductivity, K w (L T-1), is not a property of the soil alone but depends on both soil and fluid attributes (Hillel, 1998). The soil characteristics that affect conductivity are the water-filled porosity, distribution of water-filled pores, pore tortuosity and connectivity. The fluid properties that affect the conductivity are viscosity and density. Hence, the relationship between conductivity and permeability is given Kw =
kw ρ g
µ
[1-11]
10 Hence, the permeability is an exclusive property of the porous media and is a function of pore geometry alone. In a stable, rigid porous medium, the permeability is independent of the type of fluids; for example, its value is constant for water, air, or oil. However, this might not be the case for the porous medium with a solid matrix that interacts with the fluid such as water. Using hydraulic conductivity instead of permeability and replacing water pressure by matric head, the Darcy equation [1-4] transforms into the Buckingham-Darcy equation (Buckingham, 1907),
dh vDA = − K (h) + 1 dx
[1-12]
For soils with immobile water, the effective water saturation concept was introduced to study water flow in the mobile water content range only, where the water phase is continuous and hydraulic conductivity is still nonzero: Sew =
θ −θr θs −θr
[1-13]
where θ is the volumetric water content (L3 L-3), θ r is the residual water content (L3 L-3), and θ s is the saturated water content (L3 L-3). Combining the capillary tube assumption with the pore size distribution obtained from the matric head-saturation relationship, Burdine (1953) derived the relative permeability equation that accounted for pore tortuosity using the square of the effective saturation as a correction factor
K w ( S ew ) 2 = S ew K rw ( S ew ) = K sw
dS ew h2 0 1 dS ew 2 0 h Sew
∫ ∫
[1-14]
11 where K w ( S ew ) is the unsaturated hydraulic conductivity as a function of effective water saturation (L T-1), and K sw is the saturated hydraulic conductivity (L T-1), where S ew = 1 . Subsequently, Mualem (1976) introduced a conceptual model similar to that of Childs and Collis-George (1950). In this model, a correction factor was used to account for the eccentricity of the flow path (tortuosity factor) and pore connectivity as a function of water content. Since there is no available method to independently determine this correction factor, he made a similar assumption as Burdine (1953) and Millington and Quirk (1961), representing pore tortuosity and connectivity by a power function of effective saturation, S ew . Applying the Young-Laplace equation [1-10] to the model concept, he obtained
K w ( S ew ) l = S ew K rw ( S ew ) = K sw
dS ew h 0 1 dS ew h 0 Sew
∫ ∫
2
[1-15]
where l accounts for the correlation between pore connectivity and for flow path tortuosity. Due to the accounting for both soil factors, the value of power l depends on specific soil-fluid properties and may vary considerably between soil types (Mualem 1986). Mualem (1976) determined an optimal value of l = 0.5 by minimizing differences between measured and calculated relative conductivity for a wide range of soils. Subsequently, others (Schaap and Leij, 2000; Wösten et al., 1995) have demonstrated that l − values must be allowed variable, and in effect can be smaller than zero, to better fit
measured with predicted K functions, thereby casting doubt that l can be physically interpreted as presented. To simplify the computational efforts for Eqs.[1-14] and [1-15],
12 the soil-water characteristic functions are directly substituted in the integral formulas of K rw ( S ew ).
1.5. Soil-water characteristic functions A commonly used soil-water characteristic function is the Brooks and Corey (1964) S ew = 1.0
for 0 ≤ h < h e λ
h S ew = e for h ≥ h e h
[1-16]
where h e is the air entry pressure head (L), and λ is the pore size distribution parameter. Campbell (1974) proposed a model for soil-water retention function, similar to Brooks and Corey (1964), using water content term instead of effective water saturation,
θ h = he θs
−b
[1-17]
where −b is the slope of the soil-water characteristic curve in a log-log coordinate system. Moreover, Van Genuchten (1980) introduced the soil water retention function S ew =
1 1 + (α h ) n
m
[1-18]
where α is related to the air entry value (L-1) and n represents the width of pore size distribution, and m = (1 − 1 n ) . The α and n both are fitting parameters. The twoparameter lognormal distribution model was derived by Kosugi (1996) assuming that soils can be represented by a lognormal pore size distribution
13
ln h − ln h m Sew = F σ
[1-19]
where h m and σ are geometric mean of matric head and standard deviation of pore-size distribution, respectively. F is the complementary normal distribution function, which is expressed as 1 F ( x) = 2π
∫
∞
x
x2 exp − dx 2
[1-20]
Alternatively, Eq. [1-19] can also be written as 1 ln(h h m ) S ew = erfc 2 σ 2
[1-21]
The closed form analytical expressions for the relative conductivity function are obtained by substituting Eqs. [1-16], [1-18], or [1-19] into Eqs. [1-14] and [1-15]. Chen et al. (1999) presented the closed forms of relative conductivity functions for five different soil-water retention models, including equations [1-16], [1-18], and [1-19], and compared the performance of each model with experimental data.
The similarity
between all K − models is that similar parameters account for tortuosity and connectivity and pore size distribution. As for water flow, air flow is also governed by similar soil physical properties but pore geometry is different for air and water flow.
1.6. Air conductivity Soil air flow as a result of gradients in total air pressure is referred to as advective flow, signifying a transfer of mass from one point to another (Evans, 1965). Air flow is important when differences in pressure develop because of expansion and contraction of the soil gases as a result of changes in barometric pressure and temperature, or by soil air
14 entrapment (Taylor and Ashcroft, 1972). In early studies, air conductivity received a lot of attention in the soil physics literature, as it may be used to characterize soil pore geometry and soil stability as influenced by the geometrical arrangement of solid particles (Ball 1981a). Moreover, it was used to assess the state of aeration, soil structural stability, structural degradation, state of compaction, root penetrability and macropore flow (Reeve, 1953; Ball, 1981b; Groenevelt and Lemonie, 1987; Ball et al., 1988; Blackwell et al., 1990; Lindström et al., 1990; Granovsky and McCoy, 1997; Ball and Smith, 2001). Considerable efforts have been made to measure air permeability in the field and laboratory conditions (Kirkham 1946; Evans and Kirkham, 1949; De Boodt and Kirkham, 1953; Grover, 1955; Maasland and Kirkham, 1955; Janse and Bolt, 1960; Ball et al., 1981; Corey, 1986; Roseberg and McCoy, 1990; Smith et al., 1997, 1998; Ball and Smith, 2001; Iversen et al., 2001; Poulsen et al., 2001). More recently, new interests in the physical understanding of air (gas) permeability of soils has emerged in environmental engineering and vadose zone hydrology because it is intimately related to vapor flow, as relevant in geothermal reservoirs, pneumatic testing, transport of high vapor pressure organic liquids and in vapor extraction for soil remediation (Fisher et al., 1996; Finsterle and Persoff, 1997; Poulsen et al., 1998; Bedient et al., 1999; Poulsen et al., 1999; Massmann, et al., 2000). The air conductivity is estimated based on Darcy’s law (Eq. [1-4]) from measurements of air flow and its measurement is relatively simple when compared to hydraulic conductivity measurements. Since both air and water phase occupy the same pore space, it is intuitively expected that the flow properties of these fluids are interrelated. Therefore, it is conceptually feasible to estimate the conductivity of one fluid
15 from knowledge of the conductivity of the other. Burdine (1953) and Collis-George (1953) derived the relationship between air and hydraulic conductivity, based on the statistical theory of pore size distribution. Throughout their study, the relationship between degree of saturation of air, S a and water, S w is defined by S w + Sa = 1
[1-22]
where S w and S a are defined as S w = θ φ and S a = a φ , respectively, and a denotes the soil-air content (L3 L-3). At any given water content, hydraulic conductivity was calculated by summing contributions from the smallest to the largest water-filled pores. Subsequently, the air conductivity at the identical water content values was calculated from the largest to the smallest air-filled pores. Moreover, in these calculations, all fluid phases are assumed to be continuous within the pore space and their permeability is a function of total phase saturation. Using these assumptions and the relation given in Eq. [1-22], Burdine (1953) derived the relative conductivity for the air phase based on similar principles as used for derivation of Eq. [1-14]: 1
K ra = (1 − S w ) 2
∫ dS ∫h
dS w 2 Sw h
[1-23]
1
w 2
0
Alternatively, Mualem’s (1976) model was expressed for the air phase relative conductivity in the form l K ra = (1 − S w )
dS w Sw h 1 dS w h 0
∫ ∫
1
2
[1-24]
16 where l is a pore tortuosity and connectivity parameter estimated for the hydraulic conductivity by Mualem (1976) to be about 0.5 as an average for many soils. This value was also adopted for the relative conductivity of the air phase. Luckner et al. (1989) used a value of 1 3 for tortuosity and connectivity parameter, l , due to less tortuous flow path of air than flow path of water. Although Eqs. [1-14] and [1-23] were originally derived for relative conductivities of the wetting and non-wetting fluid phase of multi-fluid systems in petroleum engineering applications, they can be directly applied to the unsaturated soils. The closed-form relative conductivity functions as function of non-wetting phase saturation for different soil-water retention functions are combined with Eq. [1-23] or Mualem equation of [1-24] are presented in Chen et al. (1999) and Dury et al. (1999). Simpler predictive models for air (gas) conductivity during variable soil-moisture conditions were developed by Moldrup et al. (1998) for natural heterogeneous soils. First, they tested the simplified Millington and Quirk (1960) model
k a = k * a*
2
[1-25]
where k * is the permeability at the highest air content, a * , as the matching point value, and the modified Brooks and Corey (1966) model 1+ 2 b 2 1+ 2 b − (φ − a ) k a φ = k * a * φ 1+ 2b − (φ − a * )1+ 2b
[1-26]
with a Campbell (1974) water retention model parameter, b, and matching point conductivity measurement to evaluate predictive capabilities of the models in undisturbed
17 soil samples. Moreover, Moldrup et al. (1998) suggested an empirical, soil type dependent model,
k a = k * a*
δ (b )
[1-27]
where δ (b) = c1 + c2b is a function of the Campbell water retention parameter, b, and c1 and c2 are assumed constant between soil types. Actually, values of c1 and c2 are found by optimizing the model against measured diffusivity data. Thus, the assumption of being constant might not be valid for different soil types. Furthermore, they concluded that the Millington and Quirk (1960) model (Eq. [1-25]) requires less soil property information, but gives similar predictions of relative air conductivity as the modified Brooks and Corey model (Eq. [1-26]). In addition, the new soil type dependent empirical model (Eq. [1-27]) predicts air conductivity as a function of air content, while using the measured air conductivity at the highest measured air content as a matching factor and the Campbell retention function parameters, b, as the pore size distribution dependent parameter. The Campbell retention parameters can be obtained from the slope and intercept of log-transformed retention data. This simple model (Eq. [1-27]) for relative conductivity in undisturbed soils as a function of basic soil properties (air content and soil type characteristics) is promising, but does not explicitly relate conductivity and the pore geometry characteristics, which is needed if fluid flow is to be predicted (Vogel, 1997, 2000) or when considering a unifying fluid flow concept. Specifically, prediction of relative conductivities for both wetting and nonwetting phases requires an additional parameter that will account for connectivity and tortuosity of the pore space (Fisher and Celia, 1999; Vogel 2000).
18 Hereafter, water and air designate the wetting and non-wetting fluid, respectively. To evaluate relative air conductivities, Parker et al. (1987) assumed dependence only on total air saturation. In reality, however, considerable amounts of air may be trapped, thereby not contributing to flow, or water blockage can disrupt air connectivity and preventing air flow (Stonestrom and Rubin, 1989ab, Cirpka and Kitanidis, 2001). Since at high water content, the air phase may be discontinuous and includes considerable amounts of trapped air, the relative air conductivity must be function of free air saturation rather than total air saturation (Lenhard and Parker, 1987; Fisher et al., 1997). For example, Luckner et al. (1989) introduced the term “coherency and incoherency” of the fluid phase, to signify the importance of mobility domains in soil systems with two different fluids. While the incoherent fluid phase represents the immobile phase, only the coherent phase allows for nonzero conductivity values as controlled by fluid connectivity. Luckner et al. (1989) included the porosity as the maximum phase saturation in their definition of fluid domain, e.g. for water ( w )
θ −θr φ −θr
[1-28]
φ − a r − θ a − ar = φ − ar φ − ar
[1-29]
S ew =
and for air ( a ) S ea =
In practice, however, the maximum saturation cannot be achieved as a considerable amount of air can be trapped (Stonestrom and Rubin, 1989a). Stonestrom and Rubin (1989b) investigated the air conductivity dependence on water saturation and introduced additional terms to explain their air flow study observations. They defined the water saturation at the emergence point ( S w,ea ) where air flow becomes first detectable while
19 drying. Defining S w =
θ a and S a = , Fisher et al. (1996, 1997, 1998), and Dury et al. φ φ
(1998, 1999) used this new terminology to represent natural end-points for the definition of effective saturation of the water phase, S ew =
S w − S w,ew S w,max − S w,ew
[1-30]
and of the air phase, S ea =
S a − S a ,ea S a ,max − S a ,ea
=
S w,ea − S w S w,ea − S w,r
[1-31]
where S w,ew and S a ,ea are the water and air phase saturations at the emergence point of the water and air phase, respectively, S w,max and S a ,max are maximum water and air saturation, and S w,r is the residual water saturation (Figure 1-2). In general, the respective maximum phase saturations, S w,max and S a ,max have been either accepted as maximum saturation, that is 1, or as achievable saturation, i.e. S w, s . These points represent a range of saturation in which both phases are continuous. Applying the phase approximation given in Eqs. [1-30] or [1-31] and the van Genuchten-Mualem concept (Mualem, 1976; van Genuchten, 1980) led to the emergence point model proposed by Fisher et al. (1996). They used this modeling approach to investigate processes limiting the removal of volatile organic compounds (VOC) during the later stage of soil vapor extraction. Fisher et al. (1997) expanded their phase approximation by applying the Brooks and CoreyBurdine approach (Burdine, 1953; Brooks and Corey, 1964) to compare the efficiency of a parametric model accounting for the discontinuous air phase as compared with the
20
Figure 1-2. Schematic representation of
(a) soil-water characteristic (b) relative
conductivity of the water (dashed line) and air (solid line) for different scaled saturations (Adapted from Dury et al., 1999).
21 classical total air phase model, during wetting and drying processes. Their evaluation of the model showed that the continuous air phase model provided a much better prediction than the model neglecting the discontinuous air phase. For example, shaded areas in Figure 1-2b indicate saturation-ranges, for which one of the phases is discontinuous with corresponding zero relative conductivity values. The emergence point approximation for relative air conductivity provided good agreement with measured relative air conductivity values. Unfortunately, experimental determination of the discontinuous air phase content, and emergence and extinction points are difficult. Even, with the known emergence point, the exponent accounting for tortuosity and connectivity was fixed to 0.5 or used as a fitting parameter as opposed to Eq. [1-24], with l = 1 3 . To relate the emergence point model to pore structure, Fisher et al. (1998) developed procedures to determine soil-water retention, gas conductivity, and pore geometry of undisturbed subsurface fracture-zone samples. Especially, the combination of the thin sectioning technique and image analysis of the undisturbed samples was used to identify the pore structure of the fractured zone samples. Their analysis revealed that air phase continuity was the major governing factor of air phase conductivity. On the other hand, filling the faults with soil particles obstructed the pathways of air phase due to the absorption of water phase on their surfaces, thereby reducing air conductivity. As a consequence, application of the continuous phase model approach as proposed in Fisher et al. (1996, 1997, 1998) and Dury et al. (1998) might be useful for predictive purposes, under limited conditions only. However, in order to improve the predictive capability of these models, physically based methods or new concepts accounting for the tortuosity and connectivity parameter are required.
22 Comparing different air phase relative conductivity models based on Burdine (1953) and Mualem (1976), Dury et al. (1999) showed that an adequate description of the relative air conductivity could only be obtained if the emergence point for air conductivity ( S a ,ea ) is determined. Accordingly, using this emergence point to normalize phase saturation (e.g. Eq. [1-30] and [1-31]), the conductivity models performed better than normalized empirical models such as Corey (1954), Pirson (1958), or Wyllie (1962) models. They found that the retention model chosen for determining the required pore size distribution parameters was not important, whereas the selection of the conductivity model (Burdine and Mualem) was extremely important. In addition to the emergence point, the other most important parameter was the tortuosity and connectivity parameter. The effect of the tortuosity term is determined by the tortuosity and connectivity exponent (l ) , and their magnitude compensates for differences between pore size distribution terms of Burdine and Mualem permeability models. Since the tortuosity exponent is not based on theoretical consideration, it is usually treated as an optimized parameter with a common value (= 0.5 for Mualem and = 2 for Burdine type equations) analogous to water conductivity. However, in theory, the tortuosity and connectivity parameter for air and water conductivity is expected to be different, due to the separate flow paths and connections of the corresponding fluid phases. For example, the l -power of Mualem (1976) was given a value of 1 2 for water conductivity, but a value of 1 3 for air conductivity by Luckner et al. (1989) signifying a large connectivity/tortuosity for the nonwetting fluid in air-water system with the air filling the inside of the pores. Moldrup et al. (1997) also came to a similar conclusion, when comparing solute and gas diffusivity for both disturbed and undisturbed soil
23 samples. Hence, different models accounting for tortuosity and connectivity should be used to predict transport in water and air phases of unsaturated soils for both convective and diffusive transport processes. Especially, the tortuosity involved in diffusive transport process is a purely geometrical porous medium property and represents the average path length of molecules (or particles) that pass through a cross sectional area of porous media at a certain instant.
1.7. Gaseous diffusion in soils Gaseous molecules exhibit random movement as result of their thermal energy. By diffusion, gas molecules move in response to a partial pressure or concentration gradient. Diffusive transport is the major mechanism responsible for the movement of gases in soils, and provides for gaseous interchange between the soil and the atmosphere (Troeh et al., 1982; Rolston, 1986; Arah and Ball, 1994; Kruse et al., 1996). The major soil processes are affected by gaseous diffusion including: (i) soil aeration (Glinski and Stepniewski, 1985); (ii) soil-water vapor movement (Kirkham and Powers, 1972); and (iii) volatilization of organic compounds (Petersen et al., 1994; Amali et al., 1996; Arands et al., 1997). Our modern ideas on diffusion are largely due to two well-known scientists, Thomas Graham and Adolf Eugen Fick (Cussler, 1997). Graham’s research on the diffusion on gases was largely conducted by a straight glass diffusion tube apparatus that was closed by a semipermeable plug at one end. The tube was filled with hydrogen while the other end was immersed in water. His experimental results were simple and definitive, showing that the diffusive flux is proportional to the hydrogen concentration
24 difference between the tube and air. The next major advancement in diffusion theory originated from the work of Adolf Fick. Recognizing that diffusion is a dynamic molecular process, Fick developed the laws of diffusion by means of analogies with Fourier law of heat (Crank, 1975; Jaynes and Rogowski, 1983; Rolston, 1986; Cussler, 1997; Ball and Smith, 2001). Thus, one-dimensional diffusion of gases in soil can be described by
Vgas
∂C = v gas = − Dg At ∂x
[1-32]
where Vgas is the amount of diffusing gas (M), A is the cross sectional area of the soil (L2),
t is time (T), v gas is the gas flux density (M L-2 T-1), C is the bulk soil concentration of gaseous phase (M L-3), x is soil distance (L), and Dg is the so-called soil-gas diffusion coefficient (L2 T-1). However, while useful in soil science, Fick’s law must cautiously be applied to the diffusional processes because it is only appropriate for a couple special cases, such as the diffusion of a trace amount of gas in a multicomponent gas mixture, equimolar, counter-current diffusion on a binary gas mixture, and equimolar, countercurrent diffusion of two gases in a ternary system with the third gas stagnant (Jaynes and Rogowski, 1983; Amali and Rolston, 1993). Extensive efforts have been made to measure the soil gaseous diffusion coefficient in the laboratory (Penman, 1940; Currie, 1960a; Ball et al., 1981; Rolston, 1986) and in situ (McIntyre and Philip, 1964; Lai et al., 1976; Rolston, 1986; Rolston et al., 1991; Ball et al., 1994; van Bochove et al., 1998). However, the measurement of gaseous diffusion coefficient may be time-consuming and difficult. Therefore, researchers introduced mathematical formulations to predict gaseous diffusivity (ratio
25 between gaseous diffusion coefficients in soil, Dg and free air, Do ; i.e. Dg Do ) that include relatively easly measurable soil properties such as total porosity and air-filled porosity or volumetric air content, a = φ − θ .
1.7.1. Soil-type
dependent
gaseous
diffusivity
models
for
unsaturated, disturbed soils Hypothesizing that the cross-sectional area available for diffusion and tortuosity is the controlling soil variable, Buckingham (1904) was among the first to suggest a general relationship between gaseous diffusion coefficient and air-filled porosity, or
Dg Do
= a2
[1-33]
where the exponent 2 accounts for tortuosity. Subsequently, Penman (1940) proposed a universal relationship for variable-saturated porous media,
Dg Do
1 = a
τ
(0 < a < 0.7)
where τ denotes a tortuosity-related constant (approximately equal to
[1-34] 2 ), i.e. τ is
assumed independent of air content, a . Over the range stated, Penman fitted Eq. [1-34] to experimental data, and obtained the well-known linear expression,
Dg Do
= 0.66a
[1-35]
where 0.66 was assumed to represent a universal tortuosity coefficient for all porous materials. Since tortuosity is expected to decrease with an increase in volumetric air content (Hillel, 1980; Xu et al., 1992), the constant value of 0.66 to account for tortuosity
26 is only valid for a limited range of volumetric air content values and causes an overestimation of the diffusion coefficient at low air content values. Marshall (1959) pointed out that the Penman (1940) expression (Eq. [1-35]) did not consider any air phase discontinuity; therefore, he examined the effects of air continuity on the air conductivity. Based on this examination, he concluded that the 1
constant 0.66 in Eq. [1-35] should be replaced by a 2 , assuming that porous media consists of equal-diameter capillary pores, or
Dg Do
=a
3
2
[1-36]
Moreover, to overcome the limitations of Eq. [1-35], Millington (1959) modified the Penman (1940) equation, introducing a tortuosity coefficient as a function of air-filled porosity ( τ = 1 a 3 ), so that 1
Dg Do
=a
4
3
[1-37]
Nevertheless, this expression overestimated the measured diffusion coefficient at high air content values. Similarly, Currie (1960b) linked diffusivity to soil-air content in an empirical way, to fit many porous materials consisting of equal size particles, or Dg Do
= ϕ aξ
[1-38]
where ϕ ( ≤ 1) and ξ ( ≥ 1) are constants for a specific granular material, representing measures of pore shape. In Eq. [1-38], ϕ is a constant, but it is assumed to be function of the total porosity. The equation is not expected to apply for wet materials in which the geometry of the air-filled pores changes with water content.
27 Although Equation [1-38] was recommended for the low porosity range, but for the range over which the larger pores are air-filled, the following empirical equation was used (Currie, 1961) ϑ
a = f Dv av
Dg
[1-39]
where af is the fractional air volume, av is the volume occupied by the larger-pore space, and Dv is the diffusion coefficient when only av is air-filled. Applicability of this equation is difficult because of the estimation of av . By analogy with the unsaturated conductivity, the relationship between gaseous diffusivity and air content was introduced as an exponential function of both air content and porosity (Millington and Quirk, 1960), or Dg D0
=
a
10
3
φ2
[1-40]
The common assumption in this equation is that all air-filled pores contribute equally to diffusion. Recently, Sallam et al. (1984), Xu et al. (1992), Petersen et al. (1994), Washington et al. (1994), Jin and Jury (1996), Moldrup et al. (1996), and Moldrup et al. (2000ab) compared Eq. [1-40] with other diffusivity models. Although Eq. [1-40] was commonly found to significantly underestimate diffusion data, good agreement and predictability was obtained for some soils. Another Millington-Quirk type relationship (Millington and Quirk, 1960), Dg D0
=
a2
φ
2
3
[1-41]
was found to provide significantly better agreement with measured diffusivities in various disturbed soils with varying textures (Jin and Jury, 1996; Washington et al.,
28 1994). On the other hand, Petersen et al. (1994) showed that Eq. [1-41] underestimated the measured diffusion coefficients of several gases, especially at high air content values. However, in the range of low air content values, this model was considered fairly accurate (Sallam et al., 1984). Alternatively, Troeh et al. (1982) proposed a two-parameter model for describing relative diffusivity, Dg
a −u = Do 1 − u
z
[1-42]
where u and z are empirical soil-specific parameters. This model is highly flexible, so that most experimental data can be satisfactorily fitted, when both parameters are optimized simultaneously. Despite its empirical character, the parameter u represents the volumetric air content at which air comprises air pockets or is discontinuous to the air phase. Its value depends on the nature and variability of pore spaces. As air content increases, the volume of trapped air should diminish, and thereby must disappear completely when air content value reaches the total porosity. The parameter z controls the curvature of the Dg (a) -relationship, and represents the tortuosity. Troeh et al. (1982), Petersen et al.
(1994), and Jin and Jury (1996) found that the Troeh model could describe experimental data better than any other model, but also concluded that the parameters u and z do not appear to have an apparent relation to other measurable soil properties such as soil type and bulk density. Hence, the model cannot be used as a general prediction tool, since there was no clear correlation between measurable soil properties and the model parameters.
29 Based on electrical conductivity theory, de Vries (1950) argued that the apparent diffusion coefficient of a gas in soil might be calculated from the weighted mean diffusion coefficient of the constituents that make up the soil, i.e., solids, water, and air. Since the diffusion coefficient of soil solids is zero and that for soil water is often negligible, the diffusion coefficient for soil air can be written based on the proportion of various soil constituents ( a + θ + (1 − φ ) = 1 ), Dg Do
=
a a + ws (1 − a)
[1-43]
where ws and a are the weighting constant for solid and liquid phase, and the volume fraction of air, respectively. The model assumes that ws can be calculated from the known shape of the particles like quartz, feldspars, or clay. de Vries’ (1950) model showed good agreement with Penman’s (1940) measured data for high air content values. Unfortunately, like Penman’s model at low air content values, the model overestimated the measured data due to lack of proper accounting for air phase discontinuity (Xu et al., 1992). Another possible explanation for its overestimation is the assumption that the shape factor of soil solid with surrounding water film is the same as soil solids itself. Especially, at low air content, this is not likely as water fills the whole pore space and not only spreads around the solids. For disturbed (sieved and repacked) soils, Moldrup et al. (2000b) developed a conceptually-based predictive model by introducing a water-induced linear reduction (WLR) term in the models for completely dry sieved repacked soil. This term is equal to the ratio of air content to total porosity. The assumption for introducing this concept was that the diffusivity model modified for wet conditions should equal the diffusivity model
30 for a completely dry soil when a = φ , whereas gaseous diffusion must linearly decrease at the rate equal to a φ . Thus, the WLR concept represents the increased tortuosity in wet soil compared with a dry soil at the same air content value. When comparing with the other models (e.g. Eqs. [1-35] and [1-37]) for dry porous media, Marshall’s (1959) model (Eq. [1-36]), was found to best describe gas diffusion in dry soils. When combined with the WLR term, it yields the following WLR -Marshall model a a 2.5 =a = Do φ φ
Dg
3
2
[1-44]
This model showed overall best performance when using data from the literature. Also, Moldrup et al. (2000b) pointed out that the WLR model is likely not valid for organic soils, as wet organic materials are expected to have a different tortuosity than mineral soils. The aforementioned gaseous diffusivity models, extended from the classical Buckingham (1904) model to the newest WLR model, are generally soil-type independent, except for the inclusion of soil total porosity which is normally a little larger for fine- compared to coarse-textured soils. The models are mostly based on data and can be assumed appropriate for disturbed (sieved, repacked), non-compacted soils only. Recently, a number of studies on intact (undisturbed) soil samples have shown a distinct soil type dependency (Freijer, 1994; Moldrup et al., 1996, 1997, 2000a; Schjønning et al., 1999), indicating the need to include some soil pore characteristic parameters in the predictive gas diffusivity models.
31
1.7.2. Soil-type dependent gaseous diffusivity models for undisturbed soils A distinctive property of the soil is the irregular distribution of its pores (shape and size) due to the irregularity of the soil structure (Dagan 1989). All transport processes including diffusion are governed mainly by the soil’s structure. The characterization of the geometrical properties of soil structure is so crucial that these properties must be taken into account and incorporated into diffusive models for different structural units (Vogel, 1997). Due to the complex geometric reality of the soil structure, it is necessary to make simplifying descriptions of the geometry of an aggregated soil, such as by a soil model consisting of spherical, cylindrical, or flat aggregates (Currie, 1961; Millington and Shearer, 1971; Collin and Rasmuson, 1988; Rappoldt, 1990, Bird and Dexter, 1994). Millington and Quirk (1961) and Currie (1961) developed relative diffusion models based on statistical concepts of cutting and randomly rejoining pores with specific diameters. Further modifications of these concepts have been made to account for material aggregation of a simple geometry (Millington and Shearer, 1971). Collin and Rasmuson (1988) compared the results of these models that account for aggregation with experimental data. They concluded that the model proposed by Millington and Shearer (1971) was superior. In addition, the models were modified to account for the contribution of diffusion in the water-filled pores. In either case, it is important to apply different models for aggregated and non-aggregated soils. Since no experimental data for the inter- and intra-aggregate porosities are available, the models with easily measurable parameters consisting of macroscopic pores with serial jointed tubes were proposed (van Brakel and Heertjes, 1974; Ball, 1981a; Nielson et al., 1984; Freijer, 1994). Analysis of
32 various tube models revealed that a physical relationship existed between relative diffusion coefficient, air content, tortuosity, constrictivity (variation of pore size along its length), and connectivity of the corresponding phase. Recognizing the difficulty of linking soil structural properties to tortuosity, continuity, and connectivity, Arah and Ball (1994) employed a relatively simple functional model dividing the air-filled porosity into three functional categories: (i) porosity contributing directly to diffusion (arterial porosity), (ii) porosity contributing to transverse direction diffusion (perpendicular to concentration gradient) or marginal porosity, and (iii) an entirely disconnected porosity (remote porosity). Using these porosity classes from fitting experimental data, relative diffusivities were calculated in a macroscopic sense (Ball et al., 1981). It was concluded from numerical experiments that the tortuosity and separation of longitudinal and transverse porosity are functionally equivalent and can characterize the diffusion process equally. Due to anisotropy in the nature of tortuosity, which has directional characteristic (tensor), Arah et al. (1994) used porosity information, but contribution of porosities to the diffusion was still questionable. Moreover, like tortuosity, determination of such porosity classes in soil is equally challenging. While these concepts can be defined for seemingly simple structural properties, they are unsatisfactory for complex, three-dimensional pore space (Horgan, 1999). To account and explain the complex geometric influences on diffusion, a network modeling technique based on the capillary tube model in two or three dimensions was adapted using Fick’s law and simple relative diffusivity models (Horgan and Ball, 1994; Steele and Nieber, 1994; Friedman et al., 1995; Horgan, 1999). Unfortunately, these types of models are unclear regarding the physical significance of the fitted parameters.
33 Therefore, they are not immediately applicable to predict relative gaseous diffusion in soils (Ball, 1981a; Freijer, 1994; Moldrup et al., 1996). Moreover, to date, the results obtained from numerical investigations have not been verified systematically from experimental data. To include the effect of soil type, Moldrup et al., (1996) presented and tested simple empirical equations for gaseous diffusivity using tortuosity terms derived from straight capillary tube models for unsaturated hydraulic conductivity based on the soil type dependent parameter of the Campbell soil-water retention function (Eq. [1-17]). Using the measured gaseous diffusion coefficient at the highest air content as a matching point value, Moldrup et al., (1996) proposed the following gaseous diffusivity model, Dg
a = * * D a
δ
[1-45]
where D * is a matching point diffusion coefficient measured at the highest air content, a * , and δ is a soil-dependent exponent that takes into account pore tortuosity,
constrictivity, and connectivity. They adopted three well-known expressions for δ , but they also found that their own expression,
δ = 1.5 +
3 b
[1-46]
fitted experimental gaseous diffusion coefficient data extremely well. The value of 1.5 was considered a mean tortuosity value whereas the 3 b term was defined as a pore water release factor. The latter term was related to connectivity of the air phase, but this can also be linked to the air content value at critical pore size, such as Eq. [1-50], when air phase first starts to connect through the soil pore system (Friedman and Seaton, 1998; Hunt, 2001). Furthermore, Moldrup et al. (1997) combined the Penman (1940) and
34 Millington and Quirk (1960, 1961) relative diffusivity models to develop a general soiltype independent model by introducing a tortuosity parameter, d , to yield: a = 0.66φ Do φ
Dg
12 − d 3
[1-47]
Moldrup et al., (1997) called Eq. [1-47] the Penman-Millington-Quirk (PMQ) model. Fitting this equation to measured data, they showed that a d-value of 3 (high tortuosity) in undisturbed soil and a d-value of 6 (medium tortuosity) for sieved, repacked soils improved predictions, as compared to earlier two-parameter models, such as Eqs. [1-35], [1-40], and [1-41]. Surely, Moldrup et al.’s (1997) conclusion was true that diffusion pathways and tortuosities were very different, which was caused by different diffusion pathways and geometries as well as by soil heterogeneity in both soils, but one might intuitively expect that undisturbed soil samples have lower tortuosity than sieved soils due to their high probability of existence of macropores or shorter pathways for gaseous molecules to travel in distinctive structural pore units. However, macroporosity is generally a small portion the total pore space. Hence, macroporosity effects are likely small for diffusion, but can be large for convective air flow. It is also possible that the parameter, d , does not only represent tortuosity but the combined effect of tortuosity, constrictivity, and connectivity phenomena. Their results implied a significant difference in gaseous diffusivity between undisturbed and disturbed soils and a larger dependency on soil type for the undisturbed soils. Moldrup et al. (1999a) improved the prediction accuracy of the PMQ model as a function of air content including a reference point gaseous diffusion coefficient value at a single soil matric head at –100 cm H2O. Even if the measurement of gaseous diffusion
35 coefficient at single soil matric head greatly reduces time and effort, it may not be the objective for most gaseous transport studies. Therefore, to obtain a better prediction model, Moldrup et al., (1999b) revisited the use of gaseous diffusion in completely dry soil as a reference-point. Thus, Buckingham’s (1904) expression (Eq. [1-33]) at this reference-point was combined with the soil type dependent model. The model was tested with the measured diffusion data with respect to its applicability, and compared with similar models such as Eq. [1-47]. They found that their combined Buckingham-BurdineCampbell (BBC) model a =φ2 Do φ
Dg
2+
3 b
=
a
2+
φ
3 b
3 b
[1-48]
fitted experimental diffusion data well. Alternatively, in case the soil-water characteristic curve or b is not known, the relative diffusivity can be estimated by a purely empirical Buckingham soil texture model, or a =φ Do φ
Dg
κ ( Fcp )
2
[1-49]
where Fcp is the Silt+Sand content, and κ is a empirical parameter that must be determined from existing literature data for different soil textures. Thus, the model requires a thorough calibration before it can be used for predictive purposes. Using a data base of 126 undisturbed soil samples with different soil types, Moldrup et al., (2000a) established a general nonlinear empirical predictive relation between diffusion coefficient and air content value at –100 cm soil matric head ( a100 ) as a matching point (field capacity). Combining this reference point expression with the Burdine-Campbell soil type dependent model proposed by Moldrup et al., (1996, 1997), a
36 simple relative diffusivity model based on the soil-water characteristic curve was developed: a 3 = ( 2a100 + 0.04a100 ) Do a100
Dg
2+
3 b
[1-50]
This new model performed better than other widely used, soil type independent gas diffusivity models, but it requires the soil water data for a minimum of two different soil matric head values, including field capacity.
1.7.3. Gaseous diffusivity models linked to other fluid-phase transport parameters To circumvent laborious and time-consuming measurements, Washington et al. (1994) revisited a simpler empirical model to characterize soil gaseous diffusion and its prediction from air permeability based on common soil properties such as air-filled pore space, continuity and tortuosity. They concluded that the logarithmic-transformed values of both gaseous diffusion and air permeability showed a strong empirical linear relationship. Consequently, a simple measure of air permeability may be used to predict the soil gaseous diffusion coefficient. They showed excellent correlation between diffusion coefficient and permeability, but the logarithmic transformation has to be applied cautiously since regression can easily lead to prediction errors of one order of magnitude or larger. However, it shows that a potential relationship between these two coefficients is possible, albeit not necessarily applicable to all soils. Instead of this empirical relationship, Moldrup et al., (1999a) developed a conceptual model to demonstrate its use for relative diffusivity predictions in undisturbed soils. They
37 suggested including the air permeability at a single soil-water matric head value between –100 and –500 cm as reference value. When combined with the Ball (1981a) tortuous capillary tube model and soil type dependent equivalent tube radius, this yielded the following gas diffusivity model, 8k fc a = 2 Do r fc a fc
Dg
3
[1-51]
where k fc is the air permeability at an air content value, a fc , and rfc is the equivalent airfilled pore radius, respectively, at a soil matric head of –100 cm. To use Equation [1-51], a relationship between rfc and another soil physical property, such as clay content is required. This relationship was purely empirical and applied to only a few soils. The relationship needs further verification, but appears not to be applicable to soils with higher organic matter content. It is clear from the cited literature that a unified relationship between gaseous diffusivity and other soil properties has not received proper attention. Moreover, the dependency of gaseous diffusion on air content needs further investigation, including its dependency on pore size, pore continuity, and tortuosity. Recently, Moldrup et al. (2001) presented three analyses to establish a link between diffusive and convective transport parameters and soil tortuosity, thereby developing a unified and soil-type dependent transport model concept. The analyses were based on the definition of a simplified tortuosity term, assuming that the pores consist of sinusoidal parallel capillary tubes with uniform and similar diameters. They concluded from measurements of solute and gaseous diffusivities that the tortuosity of the water and gaseous phase were different. Tortuosity of the gaseous phase in wet soil was larger than in dry soils, and was larger in
38 undisturbed soils than disturbed soils. Water-phase tortuosity depended on soil-type and related to soil surface area. They showed that relationship between transport coefficients were dependent on the types of transport. Key soil parameters that define these relationships were tortuosity of each fluid phase, soil surface area, soil structure indexes (air content value and equivalent pore diameter at –100 cm matric head), and pores size distribution. While tortuosity governed the relationship between gaseous and solute diffusivity, soil structure indexes were major parameters establishing a relation between gaseous diffusivity and air conductivity. Furthermore, to investigate the behavior of all four transport parameters (solute diffusion, gaseous diffusion, air conductivity, and hydraulic conductivity) with pore size distribution and fluid saturation, the diffusion model (Eq. [1-45]) proposed by Moldrup et al. (1996) was generalized as a constitutive parameter model, η
TrCo f = TrCo* f *
[1-52]
where TrCo is the transport coefficient, f is the fluid content, TrCo* is the measured transport coefficient value at the highest fluid saturation, f * . In this analysis, by relating parameter η (Eq. [1-46]) with the slope of Campbell’s soil-water characteristic curve when plotted on a log-log scale for different transport coefficients, they concluded that both solute diffusivity and hydraulic conductivity showed a faster decline with decreasing water content compared with gaseous diffusivity and air conductivity. This was explained by the higher tortuosity of water-phase. One must keep in mind that Eq. [1-52] is purely empirical. Hence, the parameter η must be determined empirically for different transport coefficients of various soil-types. Although the empirical character of the model gives
39 limited information on the physics of the processes, it is important to seek links between transport processes to develop better predictive models, keeping in mind that careful distinction must be made between disturbed and undisturbed soil samples.
1.7.4. Analogy-based gaseous diffusivity model The analogy-based model concept was first proposed by Mualem and Friedman (1991) to predict electrical conductivity in unsaturated soils. It is based on the idea that a pore geometry factor affects hydraulic conductivity and electrical conductivity similarly. Likewise, Weerts et al. (2000) proposed a conceptual model to predict the gaseous diffusion coefficient assuming pore geometry affects air conductivity and gaseous diffusion equally. Geometry factors considered were the continuous air content, ac , and geometry factor for the gaseous phase, Fg (ac ) . The gaseous diffusivity was defined by Dg Do
= ac Fg (ac )
[1-53]
where the geometry factor of the gaseous phase, Fg (ac ) , was defined as the ratio of the air conductivity of the soil, K a , soil (ac ) and air conductivity of a bundle of straight capillaries, K a , cap (ac ) ( Brooks and Corey, 1966; Parker et al., 1987), or a K a , soil (ac ) Fg (ac ) = = K a , cap (ac ) na c
∫ ∫
1
S
1
S
dS h
dS h2
2
[1-54]
where S is the relative water saturation ( = θ θ s ) and the continuous air content is given by
40 ac = θ s − θ
[1-55]
In Eq. [1-54] na is the tortuosity factor for the gas phase, and it is obtained from fitting of the model with measured gaseous diffusion coefficients as a function of water content. Inserting Eq. [1-54] in Eq. [1-53] yields
Dg Do
= ac1+ na
1
1 dS − 0 h 1 1 dS − 2 0 h
∫ ∫
2
1 dS 0 h S 1 dS 2 0 h
∫ ∫
S
[1-56]
The prediction of relative gas diffusion coefficient for unsaturated soils requires knowledge of the soil-water retention function and tortuosity parameter, na . Thus, since any retention function can be adapted, Weerts et al., (2000) substituted Rossi and Nimmo’s (1994) water retention model into Eq. [1-56] that includes the dry end of the retention function. The resulting model was partly capable of describing measured gaseous diffusivity data for different soil textures. Moreover, they tested the applicability of the gaseous-phase tortuosity parameter for the Mualem (1976) relative hydraulic conductivity model, but concluded that it led to a systematic overestimation of the relative hydraulic conductivity at low water content. Hence, their assumption of equal water-phase and gaseous-phase tortuosity was not correct. However, one must note that it was determined that the value of the denumerator of Eq. [1-54] becomes unrealistically high for matric potential head (h) values near zero (Mualem, 1976; Vanclooster et al., 1994; Weerts et al., 1999). Subsequently in a later study, Weerts et al. (2001) showed that gaseous and waterphase tortuosity parameters were quite similar by comparing gaseous diffusivity and bulk soil electrical conductivity measurements of a sandy soil. They concluded that
41 comparison of all measured data and application of analogy based models make clear that parameters obtained from fittings have a different physical meaning. It was suggested that the tortuosity parameter of water-phase as determined from electrical conductivity measurements is mainly linked with the tortuous pathway, whereas gaseous diffusion processes are more affected by connectivity. Tortuosity is the determining factor for electrical conductivity as the water-phase mostly forms a connected, tortuous path. To support this idea qualitatively, Lattice-Boltzmann simulations were performed to show that predictions with analogy-based models are affected by both tortuosity and connectivity. Connectivity plays major role for diffusion processes for soil in the high water content range. Once pore connectivity is established at the critical water content, tortuosity becomes the major controlling parameter for describing diffusion processes. This double effect of pore geometry on gas diffusion might explain why empirical models generally fail to describe measured gaseous diffusion data in the high water content ranges. However, one should realize that because of insufficient data in both very wet and dry range, further research is needed to investigate the performance of simple empirical models at extreme fluid content ranges. Since the bulk soil electrical conductivity, ECa is also affected by pore space properties, it can be expected that a generalized functional relationship relating pore geometry and tortuosity to ECa might be useful to predict fluid flow in soils. Indeed, it is not at all clear how pore size distribution controls gaseous diffusion and bulk soil electrical conductivity.
42
1.8. Electrical conductivity The soil electrical conductivity is considered an important property that can be used as an indicator of soil salinity in the field. It is influenced by various factors such as water content, tortuosity of pore space and double layer effects (Mualem and Friedman, 1991; Malicki and Walczak, 1999; Weerts, 2000). In experimental studies of solute leaching and transport, changes in solution concentration within a soil profile are usually estimated by interpretation of the measured ECa . The macroscopic form of Ohm’s law governs current flow through saturated porous media and is expressed by ∆V p I = EC A L
[1-57]
where EC is electrical conductivity (Q2T M-1L-3 where Q is the charge), which is the constant of proportionality between the electrical current, I (Amp) = Q T (e.g. coulomb sec ), per unit cross-sectional area, A (L2), and the gradient of the electrical
potential, V p (L-2 M T-2 Q-1) (see Appendix 1.2.). The electric current in Eq. [1-57] is equal to the net amount of charge (Q) that passes through it per unit time (T) at any point. Since the influence of the soil on the transport of the electrical current is contained within the electrical resistivity, which is the reciprocal of electrical conductivity, it may be used as a macroscopic geometric property of the soil (Diedericks and du Plessis, 1996).
1.8.1. Electrical resistivity and formation factor The electrical resistivity of fluid-saturated porous media is often used in studies of flow through porous media, and for interpretation of electrical log and characterization of
43 porous rock. For example, it is used widely as a measure of brine saturation in rocks (Bear, 1972). From measurements of electrical resistivity, Archie (1942) found an empirical relationship between electrical resistivity of saturated porous media, σ o (L3M T-1Q-2) and bulk electrical resistivity of electrolyte, σ w (L3M T-1Q-2),
σ o = Frσ w
[1-58]
where Fr is the formation resistivity factor. Hence, the formation factor is defined as Fr =
σo σw
[1-59]
Since the resistivity is the reciprocal of the electrical conductivity, Friedman and Seaton (1998) suggested that the ratio between electrical conductivity of the saturating solution, EC w , and the bulk electrical conductivity of the porous medium, ECa , can be used for the formation factor, or Fr =
EC w ECa
[1-60]
Archie (1942) determined that the formation factor is a function of pore geometry and varies with porous medium characteristics such as porosity and permeability. The formation factor is description of tortuosity, and is a function of the porosity and air-filled pore space (Wyllie and Spangler, 1952; Hulin, 1993; Suman and Ruth, 1993; Diedericks and du Plessis, 1996). For example, Suman and Ruth (1993) presented an exact relation between the formation factor and tortuosity for a homogeneous porous medium saturated with an electrically conductive fluid. Archie (1942) proposed an expression for the formation factor as a function of porosity, Fr = φ − ce
[1-61]
44 where ce was defined as the cementation factor. According to Archie (1942), ce varies between 1.3 and 2.0 for a variety of sandstone rocks. In most analyses, it is assumed that the formation factor is a parameter reflecting the influence of pore geometry on electrolytic conduction through the pore space (Wyllie and Spangler, 1952). Diedericks and du Plessis (1996) investigated deterministic expressions for the formation factors in terms of porosity for various types of porous structures. They found that measured formation factors varied distinctively between porous materials with different structure. Although water flow and electrical conductivity are quite different processes (i.e. connective versus diffusive), it is assumed that the fluid particle and electrical current follow the same flow paths and thus electrical and hydraulic tortuosities are equal. For example, Avellaneda and Torquato, (1991) introduced the following relationship between hydraulic conductivity and the formation factor, using the effective length parameter, L , K=
µL 2 8 ρ gFr
[1-62]
In contrast, Suman and Ruth (1993) showed that the assumed equivalence of the electrical and hydraulic tortuosities was not valid, although their conclusion was not validated by experiments (Hulin, 1993). In this respect, simplistic approaches with little theoretical justifications may be preferred. For example, Katz and Thompson (1985) related the permeability and formation factor to experimentally determined length scales. Moreover, Friedman and Seaton (1998) determined the relationship between permeability and electrical conductivity of three-dimensional pore networks using the critical path analysis of Ambegaokar et al. (1971). The critical path analysis assumes that the single
45 conductance of a critical pore radius controls the distribution of conductance within the pores. According to critical path analysis, the relationship between transport parameters of the network is determined by this critical pore, so using Eq. [1-60] and the critical pore radius, rcrit (L), as a length scale, the k − EC relationship is similar to Eq. [1-62]: k=
2 rcrit ECa 8 EC w
[1-63]
Using the critical pore radius concept, it would be interesting to investigate, a possible link between ECa and gaseous diffusion, Dg , by combining Eq. [1-51] with Eq. [1-63] for rcrit = rfc . Note, that the electrical conductance of this critical pore is proportional to 4 rcri2 t , while the hydraulic conductance is proportional to the rcrit according to Eq. [1-1]
(Friedman and Seaton, 1998). Wyllie and and Spangler (1952) derived an expression for permeability by combining the fundamental postulates of Kozeny equation (Eq. [1-7]) with the properties of the capillary pressure desaturation curve. Various formulations presented above for the relationship among tortuosity, formation factor and porosity seem sufficient to indicate the difficulties that relate them. Specifically, one must consider difference between geometrical and kinematical tortuosities, e.g. EC vs K (Bear, 1972; Diedericks and du Plessis, 1996). The extension of the concept of tortuosity to the simultaneous flow of immiscible fluids and to the unsaturated flow is required. In this case, tortuosity must be defined and accounted for in each processes. Thus, a proper expression to determine bulk electrical conductivity in unsaturated soils is necessary to provide more insight about this geometric property through the formation factor concept.
46
1.8.2. Models for electrical conductivity in unsaturated soil Considerable effort has been made to determine the relationship between bulk electrical conductivity and conductivity of the solution. Rhoades et al. (1976) proposed a bulk soil electrical conductivity ( ECa ) equation that relates to electrical conductivity of the soil solution ( EC w ) for unsaturated soil ECa = Tθ EC w + EC s
[1-64]
where ECs is the bulk soil surface electrical conductivity, and T is the transmission coefficient. The transmission coefficient accounts for the tortuous nature of the current lines and changes in the mobility of the ions near solid-liquid and liquid-gas interfaces (Rhoades et al., 1976; Bohn et al., 1982). On the basis of experimental results of different soils, they introduced an empirical expression for T as a linear function of volumetric water content, θ , T = c3θ + c4
[1-65]
where c3 and c4 are constants. Rhoades et al. (1989) expanded this model to distinguish between the water and salt present in the “immobile” (fine pores) and “mobile” (large pores) liquid phases. However, in its general form the model has too many parameters that need to be determined. Mualem and Friedman (1991) proposed a conceptual model for prediction of the electrical conductivity of bulk saturated and unsaturated soils. The model was based on the similarity between electrical conductivity and water flow. The assumption is that the geometry factor affecting electrical conductivity is similar to the hydraulic conductivity.
47 Accounting for the complex geometrical arrangement of the pore space, the electrical conductivity of the bulk soil solution, ECb was defined by ECb (θ ) = EC w FGEθ
[1-66]
where FGE is the pore geometry factor and assuming that EC s ≈ 0 . Since the geometry factor cannot be determined directly from experiments, the ratio between the unsaturated hydraulic conductivity, K w (θ ), and the hydraulic conductivity of bundle of straight capillaries, K w,cap (θ ), was used to define the complex pore water geometry
K w (θ ) FGE (θ ) = =θ l K w,cap (θ )
dS 0 h S dS 2 0 h
∫ ∫
S
2
[1-67]
The ratio in Eq. [1-67] was considered as a reduction factor in hydraulic conductivity due to the complex pore water geometry, whereas θ l accounts for the combined effect of the correlation between adjacent pores along the flow direction and pore tortuosity. Substituting Eq. [1-67] into Eq. [1-66] yields
ECb (θ ) = EC w θ l +1
dS 0 h S dS 2 0 h
∫ ∫
S
2
[1-68]
A closed form solution of Eq. [1-68] can be obtained when the soil-water retention function is approximated by a relatively simple mathematical expression. Although Mualem and Friedman (1991) found general agreement with measured data, the model efficiency was only tested for limited experimental data. Weerts et al., (1999) further tested the applicability of the capillary tortuosity model proposed by Mualem and Friedman (1991) to describe experimental data of bulk
48 electrical conductivity, ECa , measurements in unsaturated soils. In their study, the bulk electrical conductivity was defined similar to Eq. [1-64], or ECa = EC w FGEθ + EC s
[1-69]
The closed form of Eq. [1-69] was derived by substituting Eq. [1-67] and the soil-water retention function model proposed by Rossi and Nimmo (1994). The electrical tortuosity parameters estimated from ECa measurements and the closed form of Eq. [1-69] was applied to predict the unsaturated hydraulic conductivity by inverse modeling of soil water flow during the multi-step suction outflow experiment. They concluded that prediction of electrical conductivity is sensitive to the tortuosity parameter, l , and that prediction of hydraulic conductivity was largely dependent on unique estimates of the parameter K s . In contrast, Suman and Ruth (1993) showed that in the case of real soils, the equivalence of tortuosities is more doubtful because the water flow depends on the shape of the pore channels unlike electrical flow. So far we dealt with microscopic soil properties that affect the flow and transport processes at the small soil sample scale, but most uncertainty in the assessment of flow and transport processes in unsaturated soils is at the field scale. Especially, soils are heterogeneous porous media with soil properties varying in space and time. Therefore, we must know the spatial structure of soil physical properties to predict their flow and transport behavior (Kasteel et al., 2000). The scaling approach has been extensively used to evaluate spatial variability of soil properties and to develop a standard methodology to assess the variability of these properties. Using this approach, spatial characteristic of one transport parameter can potentially be used to derive spatial variability of other transport parameters.
49
1.9. Scaling of soil transport properties The concept of scaling approach has been developed from similarity analysis in applied physics by Miller and Miller (1955a,b, 1956). The scaling approach simplifies problems by expressing them in the smallest possible number of reduced variables. With the use of these scaled variables, relationships between physical quantities of different systems can be presented in a compact form. Miller and Miller (1956) developed this approach analyzing the scaling behavior of the Young-Laplace surface tension (Eq. [1-10]) and Navier-Stokes equations in geometrically similar granular system. According to Miller and Miller (1955a,b), geometrically similar porous media have microscopic structures that look identical except for their microscopic characteristic length scale, λi . It is important to note that the characteristic length scale, λi , is applied to entire porous medium, not only to the solid particles (Figure 1-3.) Thus, constant porosity across the geometrically similar soils is the main important assumption in the scaling approach. The key of scaling is the radius of water filled pores, ri , which is also directly related to the radius of curvature of the air-water interface in the capillary pore. So, if two porous medium are geometrically similar, radius of curvature of water film related reciprocally to its characteristic length scale, λi , must be equal to each other. Thus, geometric similarity leads to the relation r1
λ1
=
r2
λ2
=
r3
λ3
=" =
ri
λi
[1-70]
50
Figure 1-3. Geometrically similar porous medium at microscopic level.
Miller and Miller (1955a,b; 1956) showed that for given water content, the matric head for similar soils are related based on by
λ1h 1 = λ2 h 2 = " = λi h i = λˆhˆ
[1-71]
where λˆ and hˆ are the characteristic length (L), and matric head of reference soil (L), respectively. Dividing each length scale by the length scale of reference soil yields
α 1h1 = α 2 h 2 = " = α i h i = hˆ
[1-72]
where α i is the scale factor for soil sample i and αˆ = 1 . From analogy between viscous flow equation (Navier-Stokes) and Darcy equation (Eq. [1-4]), the hydraulic conductivity function K w (θ ) is scaled according to K w,1 (θ )
λ12
=
K w,2 (θ )
λ22
=" =
K w,i (θ )
λi2
=
Kˆ w (θ ) λˆ 2
[1-73]
51 where Kˆ w (θ ) is hydraulic conductivity of reference soil or the mean hydraulic conductivity function. Multiplying both sides with λˆ 2 , Eq. [1-73] becomes K w,1 (θ )
α
2 1
=
K w,2 (θ )
α
2 2
=" =
K w,i (θ )
α
2 i
= Kˆ w (θ ),
where α i =
λi λˆ
[1-74]
Similarly, the soil water diffusivity ( Dw ) and sorptivity ( S o ) can be expressed by scaling factor concept: Dw,i (θ ) = α i Dˆ w (θ )
[1-75]
S io (θ ) = α i1 2 Sˆ o (θ )
[1-76]
Possibly, air conductivity, K a (a) , gaseous diffusion, Dg (a) , and electrical conductivity, ECa (θ ) , can be related to scaling factors.
As it can be seen from Eqs. [1-72] and [1-74], scaling provides a means to relate the hydraulic properties of different soil types or spatial locations using scaling factors. Because of high variation in porosity in field soils, Warrick et al., (1977) introduced the relative water saturation, S (= θ θ s ) , to overcome the restrictive, constant porosity assumption. Applying relative water saturation expression, they provide a more realistic description of spatial variability of the field soils. Values of scaling factors, α i , for both soil-water retention and hydraulic conductivity are generally obtained from best fit by minimizing the sum of squares Ih ( i )
SS =
∑ hˆ j =1
j
− α i h ij
2
[1-77]
where i and Ih(i ) denote the index for soil sample and the total number of h(θ ) data points, respectively, and j is the index for the number of data point for each soil sample,
52 i . Finding scaling factors for the hydraulic conductivity functions, the following sum of
squares is minimized, IK ( i )
SS =
∑ ln K j =1
j w ,i
− ln Kˆ wj − 2 ln α i
2
[1-78]
where K wj ,i refers to the jth hydraulic conductivity value of soil sample, i , and Kˆ wj to the estimated mean hydraulic conductivity or reference soil hydraulic conductivity corresponding to S j or θ j . Warrick et al., (1977) showed that the values of scaling factors for soil-water characteristics function was different than for the hydraulic conductivity function. On the other hand, they emphasized that to predict spatially variable water flow in natural field soils, the ability to characterize the soil-water characteristic and hydraulic conductivity by a single scaling factor set is highly desirable. Therefore, Clausnitzer et al. (1992) provided a method to scale both hydraulic functions simultaneously to yield a single set of scaling factors. Although this is in contrast with the traditional scaling approach, simultaneous scaling yields higher quality data when soil samples within similar soil type are scaled together. This means that variability of hydraulic functions of each soil type can be expressed by a scaling factor distribution. Unfortunately, no physical significance to the set of scaling factors was attributed. Recently, Kosugi and Hopmans (1999) proposed a method of scaling soil-water characteristic functions, using the physically-based water retention model of Kosugi (1996). The pore radius, r , was used as the microscopic characteristic length to scale soil water characteristic curves for soils that are defined by a lognormal pore-size distribution. The physically-based scale factors were directly computed from the parameters describing individual soil water retention functions. This approach provided a theoretical
53 interpretation of scaling factor distributions. Subsequently, using lognormal soil water retention and hydraulic conductivity models (Kosugi, 1996), Hendrayanto et al., (2000) tested different scaling methods to characterize spatial variability of soil hydraulic properties. Among the several tested scaling methods, the conventional simultaneous scaling, which is similar procedure to Clausnitzer et al. (1992) of both soil water retention and hydraulic conductivity curves with a fitted tortuosity parameter was the best method for a forested hill slope soil. Interestingly, they also found that optimizing the tortuosity parameter in the conductivity model improved the scaling results. Although simultaneous scaling with a fitted tortuosity parameter reduced the estimation errors in water retention and conductivity data, the simultaneous scaling procedure in their study applied to conventional fitting approach. It is worthwhile to extend the physically based approach to the simultaneous scaling of soil-water retention and hydraulic conductivity, yielding a physically based, consistent set of scaling factors.
1.10. Potential links between fluid-phase transport coefficients In the previous sections, several flow and transport processes that generally occur in heterogeneous soils, were reviewed separately to understand the effect of macroscopic soil properties on transport parameters involved in these processes. Since estimating effective macroscopic transport parameters is a key component in model development, theoretical considerations and the effect of soil pore characteristics on transport parameters are subject of debate in the literature. In general, the current modeling ability combined with the practical data limitations is not reliable enough to resolve requirements for flow and transport predictions. The core of the problem is establishing
54 the relation between the soil pore characteristics and transport processes. Moreover, the link between each transport process, using key transport parameters, must be considered as a crucial point in the description and understanding of physical processes of interest (Figure 1-4).
Figure 1-4. Possible links between different transport coefficients (Adapted from Moldrup et al., 2001).
The most important soil pore characteristics, which may potentially be used to establish the link are tortuosity and connectivity of the air and water phases. Each transport process can be defined using these characteristics.
For example, tortuosity of air
(gaseous) phase in unsaturated soil is definitely larger than in dry soil and at a certain value of phase saturation (air or water), the water phase tortuosity might be equal or higher than air phase tortuosity because of interaction of water with soil particle surfaces. Thus, by establishing links between the transport parameters in Figure 1-4, one must take
55 into account the differences in pore characteristics between each transport parameter, such as tortuosity parameter of hydraulic conductivity must be different than air conductivity. Although gaseous phase tortuosity might have similar values as water phase tortuosity of hydraulic conductivity or electrical conductivity, this parameter represents more about connectivity of the gaseous phase than its sinuous path length. Furthermore, the differences between transport parameters and pore characteristics can be observed in disturbed compared with the undisturbed soils due to the geometric discrepancies in soil structural units. The observed differences may also depend on the velocity of the transport processes. Solute diffusivity shows similar behavior in disturbed and undisturbed soils but gaseous diffusivity differs considerably. Consequently, air conductivity and hydraulic conductivity in disturbed and undisturbed soils shows different behavior because of different pore size distribution and soil pore geometry. Along the scope of above discussion, one should realize that identifying and understanding the effects of key pore characteristic parameters on transport parameters can help the possibility for developing a common interrelationships between flow and transport processes and better transport modeling.
1.11. Research Objectives The overall objective of this research is to formulate a general framework to characterize transport parameters that play a key role in describing the transport of water, gases and solutes. To accomplish the overall objective, specific objectives were defined to obtain insight into the physical causes governing the dependence of flow and transport
56 coefficients on fluid content and to evaluate potential interrelationships between these coefficients. This research was mainly induced by the lack of available information on the transport characteristics affecting the common flow and transport processes in the soil and from the curiosity of a new approach for predicting one transport coefficient from the other. Consequently, research was initiated and expanded in three parts: 1. To measure transport parameters such as air and water permeability, gaseous diffusion and electrical conductivity, and to evaluate the presence of universal soil transport characteristics 2. To apply a physically based pore size distribution model to predict air and water conductivity, gaseous diffusion, and electrical conductivity as a function of fluid saturation simultaneously. 3. To apply the Miller and Miller (1956) scaling theory to relate soil water retention and unsaturated hydraulic conductivity using the lognormal pore size distribution model.
57
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71 Rossi, C. and J.R. Nimmo. 1994. Modeling of soil water retention from saturation to oven dryness. Water Resour. Res. 30: 701-708. Sallam, A., W.A. Jury, and J. Letey. 1984. Measurement of gas diffusion coefficient under relatively low air-filled porosity. Soil Sci. Soc. Am. J. 48: 3-6. Schaap M.G. and F.J. Leij. 2001. Improved prediction of unsaturated hydraulic conductivity with the Mualem-van Genuchten model. Soil Sci. Soc. Am. J. 64: 843-851. Schjønning, P., I.K. Thomsen, J.P. Møberg, H. de Jonge, K. Kristensen, and B.T. Christensen. 1999. Turnover of organic matter in differently textured soils: I. Physical characteristics of structurally disturbed and intact soils. Geoderma 89: 177-198. Shimamura, K. 1992. Gas diffusion through compacted sands. Soil Sci. 153: 274-279. Smith, E.J., M.J.L. Robin, and D.E. Elrick. 1997. A source of systematic error in transient-flow air permeameters measurements. Soil Sci. Soc. Am. J. 61: 15631568. Smith, E.J., M.J.L. Robin, and D.E. Elrick. 1998. Improved transient-flow air permeameters design: Dampening the temperature effects. Soil Sci. Soc. Am. J. 62: 1220-1227. Steele, D.D. and J.L. Nieber. 1994. Network modeling of diffusion coefficients for porous media: I. Theory and model development. Soil Sci. Soc. Am. J. 58: 13371345. Stonestrom, D.A. and J. Rubin. 1989a. Water content dependence of trapped air in two soils. Water Resour. Res. 25: 1947-1958.
72 Stonestrom, D.A. and J. Rubin. 1989b. Air permeability and trapped-air content in two soils. Water Resour. Res. 25: 1959-1969. Suman, R. and D. Ruth. 1993. Formation factor and tortuosity of homogeneous porous media. Transport in Porous Media 12: 185-206. Taylor, S.A. and G.L. Ashcroft. 1972. Physical Edaphology. The physics of irrigated and nonirrigated soils. W.H. Freeman and Company, San Francisco. Troeh, F.R., J.D. Jabro, and D. Kirkham. 1982. Gaseous diffusion equations for porous materials. Geoderma, 27: 239-253. van Bochove, E., N. Bertrand, and J. Caron. 1998. In situ estimation of the gaseous nitrous oxide diffusion coefficient in a sandy loam soil. Soil Sci. Soc. Am. J. 62: 1178-1184. Vanclooster, M., C. Gonzalez, J. Vanderborght, D. Mallants, and J. Diels. 1994. An direct calibration procedure for using TDR in solute transport studies. In Symposium and Workshop on TDR in Environmental, Infrastructure, and Mining Applications, Spec. Pub., pp. 215-226, U.S. Bur. of Mines, Northwestern Univ., Evanston, IL. van Brakel, J., and P.M. Heertjes. 1974. Analysis of diffusion in macroporous media in terms of a porosity, tortuosity and a constrictivity factor. Int. J. Heat Mass Transfer, 17: 1093-1103. van Genuchten, M Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44: 892-898. Vogel, H.J. 1997. Morphological determination of pore connectivity as a function of pore size using serial sections. European J. Soil Sci. 48: 365-377.
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Comparing
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Lattice-Boltzmann
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Application of electrical resistivity
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75
1.13. Appendix 1.1. Hydraulic radius concept is Hydraulic radius = RH =
A Cross-sectional area normal to flow = Pe Wetted perimeter (surface)
[A1.1-1]
From [A1.1-1], we can write RH =
π R2 1 = R 2π R 2
[A1.1-2]
For circular tube, the hydraulic diameter is assumed to be four times the hydraulic radius, DH = 4 RH = 4
A A. L Void volume =4 = 4 Pe Pe . L Surface area of wetted particles (channels)
[A1.1-3]
Equation [A1.1-3] can be written as Void volumes φ DH = 4 = 4 Total volume Particle surface S' Total volume
[A1.1-4]
Then defining
S ' = So (1 − φ ) =
Particle surface Particle surface Particle volume = Total volume Particle volume Total volume
[A1.1-5]
Substituting Eq. [A1.1-5] into Eq. [A1.1-4] finally gives DH =
4φ S o (1 − φ )
[A1.1-6]
76
1.14. Appendix 1.2. The macroscopic form of Ohm’s law governs current flow through saturated porous media and is expressed by
∆V p I = EC A L
[A1.2-1]
More precisely, the electric current ( I ) is defined as the net amount of charge that passes though per unit time at any point
I=
∆Q ∆t
[A1.2-2]
where Q is the charge (Coulomb). The units for the electrical current is
I=
Q Coulomb (C) = = 1 Amp t sec
[A1.2-3]
In dimensional form, I is defined by I = Q T . In Eq. [A1.2-1], V p is the electrical potential and described by
Vp =
PE Potentail Energy Joules = = = Volt (V) Q Charge Coulomb
[A1.2-4]
The difference in electrical potential ( ∆V p ) by definition is work (W) and can be expressed by
∆V p = V p1 − V p 2 =
W Q
[A1.2-5]
and in dimensional form ∆V p =
F ⋅ L L2 ⋅ M = 2 = Volt(V) Q T ⋅Q
[A1.2-6]
where F is the force (Newton = M L T-2). Substituting all corresponding dimensions into the Eq. [A1.2-1] yields
77 Q L2 ⋅ M 1 = EC T ⋅ L2 T2 ⋅ Q L
[A1.2-7]
Extracting EC from Eq. [A1.2-7] gives its dimensional form
EC =
Q2 ⋅ T M ⋅ L3
[A1.2-8]
78
2. Search for an unifying approach to determine transport coefficients of disturbed soils
2.1. Abstract In order to improve the predictive capability of transport models, physically based methods that include understanding of the control of pore continuity and tortuosity on transport in general is required. Since the models containing the parameters representing tortuosity and connectivity ( TC ) of the fluid-filled pore space phases are generally not derived from theoretical considerations, TC is treated as a single combined fitting parameter for all fluid phases. However, in principle, the tortuosity and connectivity parameters for air and water phase transport processes are expected to be different, due to different flow paths and connections. To understand the effects of TC on various transport processes as affected by pore geometry, the fluid saturation dependency of hydraulic conductivity, air conductivity, gaseous diffusion and bulk soil electrical conductivity was simultaneously measured and modeled for a Oso Flaco fine sand and Columbia sandy loam soil. To compare TC parameters for each of the 4 transport coefficients, a general model expression for prediction of all transport coefficients is presented. This expression consists of two parts: (1) a tortuosity-connectivity ( TC ) term and (2) a pore size ( PS ) distribution term, derived from the physically-based lognormal soil-water retention model. Simultaneous fitting of all 4 transport coefficients to measured data showed that the TC parameter of each transport coefficient was different. Specifically, TC parameters for air conductivity and gaseous diffusion are much higher
79 than for the soil hydraulic conductivity. For the bulk soil electrical conductivity, the TC parameters are similar to parameters of gaseous diffusion when they are represented by TC term only. Furthermore, the contribution of both pore size distribution and tortuosity-
connectivity is equally important for characterization of soil hydraulic conductivity. The TC effect on the air conductivity, gaseous diffusion and bulk soil electrical conductivity
was much higher for the Columbia soil with a wider pore size distribution than the Oso Flaco fine sand. For Oso Flaco sand, the moisture dependency of all 4 transport coefficients is fairly unique when normalized to their maximum measured values.
2.2. Introduction Prediction of flow and transport processes in soil is crucial in many areas of soil and environmental sciences, e.g. water and NAPL’s can move in both the liquid and gas phase, whereas chemical constituents move by either advection or diffusion in both the liquid and gas phase. Although all transport occurs through liquid-filled and/or gas-filled pore space, there is the general lack of information on the constitutive relationships between the various multi-fluid flow and transport processes (Parker et al., 1987; Luckner et al., 1989). While many simultaneous transport models have been proposed, limited experimental studies have been conducted to determine the presence of a general moisture dependency of these relevant transport processes combined. To date, most of the previous experimental and theoretical studies focused on the relationships for two transport properties only. Typical relationships have been presented for the hydraulic conductivity and air conductivity and their mutual dependency on fluid saturation (CollisGeorge, 1953; Parker et al., 1987; Luckner et al., 1989, Dury et al., 1998); air
80 conductivity and gaseous diffusion (Ball, 1981; Washington et al., 1994, Moldrup et al., 1999), and gaseous diffusion and solute diffusion (Friedman et al., 1995; Moldrup et al., 1997) as function of fluid saturation; and the relationship between hydraulic conductivity and bulk electrical conductivity as a function of fluid saturation (Mualem and Friedman, 1991; Vanclooster et al., 1994; Friedman and Seaton, 1998, Weerts et al., 1999). All these published studies investigated the mutual dependency of only two transport coefficients as they vary with fluid content but did not consider other additional soil transport properties. However, since all transport processes occur in the same soil pore space, one would intuitively expect a possible link between these different processes and key soil pore geometry characteristics such as tortuosity and connectivity. Initial efforts were focused on the development of models for describing soil hydraulic functions (Brooks and Corey, 1964; Campbell, 1974; Mualem, 1976; van Genuchten, 1980; Kosugi, 1996). Later, based on capillary tube assumptions, the relationships between hydraulic conductivity and air conductivity were derived using pore size distribution obtained from the soil water retention curve (Burdine, 1953; CollisGeorge, 1953; Mualem, 1976). In all these relations, both the liquid and gaseous phase were assumed to be continuous within the soil pore space, with their relative conductivity solely a function of total phase saturation. However, at high water saturation, the air phase may become discontinuous and include considerable amounts of trapped air (Wildenschild et al., 2001). If the air phase is discontinuous, the relative air conductivity must be considered as a function of free air saturation, rather than total air saturation (Luckner et al., 1989; Stonestrom and Rubin, 1989; Fisher et al., 1997). Using a continuous phase model approach, Fisher et al., (1997) and Dury et al. (1999) expanded
81 their study by applying different conductivity models, and compared the validity of various parametric models while accounting for a discontinuous air phase. Their evaluation showed that the continuous air phase model provided a much better prediction than the model neglecting the discontinuous air phase. Similarly, Wildenschild et al. (2001) suggested taking into account the entrapment of water phase in hydraulic conductivity calculations, which may occur at high water flow rates due to hydraulic isolation of water-filled pores by draining surrounding pores. Meanwhile, in order to improve the predictive capability of flow models, physically based methods that include understanding the control of continuity of the fluid-filled pore system and its tortuosity on transport are required. Since tortuosity and connectivity parameters are not generally derived from theoretical considerations, they are usually treated as a single combined fitting parameter. However, in principle, the tortuosity and connectivity parameters for air and water conductivity are expected to be different, due to their separate flow paths and connections. For example, Luckner et al. (1989) suggested a lower value of the combined tortuosity-connectivity parameter for the relative air conductivity due to the smaller tortuous flow path for the air phase than the water phase. Also, Moldrup et al. (1997) reached a similar conclusion, when comparing solute and gas diffusivity for both disturbed and undisturbed soil samples. Hence, different models accounting for the tortuosity and connectivity should be used for transport characterization of the water and air phases in unsaturated soils, for both convective and diffusive transport. Recently, Moldrup et al. (2001) presented an analysis to link diffusive and convective transport parameters with soil tortuosity, thereby developing a unified and soil-type dependent transport model concept. Their work was based on the definition of a
82 simplified tortuosity term, assuming that soil pores consist of sinusoidal parallel capillary tubes with uniform and similar diameters. Based on analyses of solute and gaseous diffusivity measurements, different tortuosity values for the water and gaseous phase were defined with the water-phase tortuosity related to soil-type and soil surface area. In another study, Weerts et al. (2000) proposed a conceptual analogy model for predicting the gaseous diffusion coefficient, assuming pore geometry affects air conductivity and gaseous diffusion equally. Moreover, they also tested the applicability of the same gaseous-phase tortuosity parameter for the relative hydraulic conductivity model (Mualem, 1976). They concluded that the gaseous tortuosity-connectivity parameter could not be used for predicting relative hydraulic conductivity. In contrast, in later study, Weerts et al. (2001) showed that gaseous and water phase tortuosities for a dense sandy soil were quite similar from comparison of gaseous diffusivity and bulk soil electrical conductivity measurements. However, they determined that this was coincidental, as the physical meaning of the two fitting parameters was different. It was concluded that the tortuosity-connectivity parameters (water-phase) predicted from electrical conductivity measurements were controlled by tortuosity, whereas gaseous diffusion was mostly affected by air phase connectivity (Weerts et al., 2001). Hence, transport properties such as hydraulic conductivity and bulk soil electrical conductivity are mainly affected by the tortuosity-connectivity of the liquid phase. The uniqueness and existence of a single pore geometry parameter is increasingly doubtful for real soils because water flow also depends on the shape and size of the pore channels, unlike electrical current (Suman and Ruth, 1993). Specifically, one may have to consider differences between geometrical and
83 kinematical tortuosity, as these control bulk electrical conductivity, ECa , and hydraulic conductivity, K , respectively (Bear, 1972; Diedericks and du Plessis, 1996). Only few studies have attempted to link the effects of tortuosity and connectivity on transport processes as affected by pore geometry (Suman and Ruth, 1993; Weerts et al., 2001; Moldrup et al., 2001). Thus, an extension of the concept of pore geometry effects on unsaturated flow by mechanisms of convection and diffusion of both fluid phases is required, whereby the tortuosity and connectivity must be accounted for in each transport process separately. The focus of this research was stimulated by the lack of information on the effect of pore geometry on flow and transport in general. Intuition provides the need for a consistent, unifying approach for predicting pore geometry effects on transport by convection and diffusion in both fluid phases. The objectives of the study were (i) to simultaneously measure and model soil transport properties, specifically air and hydraulic conductivity, gaseous diffusion and bulk electrical conductivity, as a function of fluid saturation, and (ii) to evaluate and understand the control of pore space geometry characteristics on each of these transport properties.
2.3. Theory 2.3.1. Effective saturation relations and soil water retention function Since the water and air phase simultaneously occupy the same pore space, both phases are defined in continuous and reduced form. So, the effective saturation, S e , for water phase (w) is defined as
S ew =
θ − θ min θ s − θ min
[2-1]
84 where θ , θ min , and θ s are volumetric water content (L3 L-3), the minimum measured water content (L3 L-3), and saturated water content (L3 L-3), respectively. θ min is defined as the lowest measured water content, instead of the residual water content, θ r . This allows us to make comparisons between different transport coefficients. Similarly, the effective saturation for the air phase (a) can be written as
S ea =
a − amin amax − amin
[2-2]
in which a is the volumetric air content (L3 L-3), amin is the minimum measured air content (L3 L-3), and amax is the maximum measured air content. Defining
amin = φ − θ s amax = φ − θ min
[2-3]
where φ is the total porosity and Equation [2-3] yields the following relationship between the effective saturation of water and air phase
S ea = 1 − S ew
[2-4]
The lognormal retention model proposed by Kosugi (1996) was used for describing soil-water characteristic curves
ln h − ln h m ln h − ln h m 1 Sew = F = erfc σ 2 σ 2
[2-5]
where 1 F ( x) = 2π
∞
∫ x
x2 exp − dx 2
[2-6]
85 The ln h m and σ are the mean and standard deviation of the natural logarithm of matric potential head, ln h , respectively, with h m denoting the median matric potential head. In Eq. [2-6], F is a complementary cumulative normal distribution function, which can be written in terms of the complementary error function (Eq. [2-5]). The advantage of the lognormal distribution model in comparison to many other retention models is that the parameters are physically-based and determined assuming that pore size is lognormally distributed. Linking to the lognormal retention model with its physically-based parameters, the diffusive and convective transport coefficients were defined as a function of corresponding fluid saturation. For example, the Mualem (1976) model was introduced as a convective transport equation for predicting the relative hydraulic conductivity ( K rw ) from knowledge of the soil-water retention curve. Although this model consists of two main parts to characterize the convective transport coefficients, we also adopted this concept to model diffusive transport coefficients. The general expression for all transport coefficients can be written as
TrCo j = TC * PS
[2-7]
where TrCo j is the jth transport coefficient studied in this study, TC is tortuosityconnectivity term ( TC = ( S ew ) or ( S ea ) ) and PS is the pore size term that is defined l
l
by the Mualem (1976) approach in which PS is determined by summing contributions from the smallest to the largest water-filled pores.
86 PS =
dS ew h 0 1 dS ew h 0 Sew
∫ ∫
2
[2-8]
In an analogous way, for air phase, PS can be defined from contributions of the largest to the smallest air-filled pores (Luckner et al., 1989). PS =
dS ew h Sew 1 dS ew h 0 1
∫ ∫
2
[2-9]
Expressing all transport coefficients in the same form gives us an unique opportunity to compare tortuosity-connectivity parameters for each transport coefficient. The transport coefficients examined in the present study are (1) the unsaturated hydraulic conductivity ( K w ), (2) air conductivity ( K a ), (3) the gaseous diffusion coefficient ( Dg ), and (4) the bulk soil electrical conductivity ( ECa ). To evaluate the relationships between these coefficients, each transport coefficient was normalized relative to its highest measured value at the corresponding highest fluid saturation.
2.3.2. Relative hydraulic conductivity Combining the lognormal soil-water retention function (Eq. [2-5]) with Eq. [2-8] and substituting into Eq. [2-7] yields the following functional relationship between relative hydraulic conductivity, K rw , and effective water saturation (Kosugi, 1996) K rw ( Sew ) =
2 K w ( Sew ) l = ( Sew ) 1 F F −1 ( Sew ) + σ K sw ( Sew = 1)
(
)
[2-10]
87 in which K w and K sw are the unsaturated and saturated hydraulic conductivity (L T-1), respectively. The tortuosity-connectivity parameter, l1 , is specific for the water phase of the hydraulic conductivity, and F −1 denotes the inverse function of F defined as in Eq. [2-6] representing a percentage point of the cumulative normal distribution. As pointed out in the foregoing section, the first part of Eq. [2-10] characterizes the combined effects of pore tortuosity and connectivity on the unsaturated hydraulic conductivity, whereas the second part (between brackets) describes the effects of pore size and pore size distribution on K w . Combined, these two terms describe the decrease of the K rw with decreasing water. We hypothesize that all soil transport coefficients are similarly affected by pore geometry, but that the tortuosity-connectivity parameter, l , may have different values and physical meaning between transport coefficients.
2.3.3. Relative air conductivity Assuming the air phase to be continuous within the pore space, its air conductivity is a function of total phase saturation as well. Using these assumptions and Eq.[2-4], Chen et al. (1999) derived a closed-form relative air conductivity function substituting the lognormal soil-water retention function (Eq. [2-5]) and Eq. [2-9] in Eq. [2-7] K ra =
2 K a ( Sea ) l = ( Sea ) 2 1 − F F −1 (1 − Sea ) + σ K sa ( Sea = 1)
(
)
[2-11]
where K a and K sa are the unsaturated and saturated air conductivity (L T-1), respectively, and l2 is the tortuosity-connectivity parameter for the air phase. The presented analysis will use the air conductivity value at the corresponding lowest measured water content,
θ min , as the saturated air conductivity, K sa , where S ea = 1 ( amax = φ − θ min ). Moreover, the
88 possibility of characterizing the relative air conductivity in unsaturated soils using the tortuosity-connectivity part (TC) of Eq. [2-11] only was investigated by using K ra =
K a ( Sea ) l = ( Sea ) 2 K sa ( Sea = 1)
[2-12]
2.3.4. Gaseous diffusivity In the general literature, the gaseous diffusivity term is defined as the ratio between gaseous diffusion coefficients in soil, Dg (L2 T-1), and in free air (Moldrup et al., 1996, Weerts et al., 2001). Because of our description of all measured transport coefficients relative to measured value at the highest fluid saturation, we prefer to treat gaseous diffusion in a similar way. Thus, assuming that gaseous diffusion occurs in the same air-filled pore space as air convection, a similar equation can be adopted for the gaseous diffusivity, but with a different tortuosity and connectivity parameter, l3 , Drg =
Dg ( Sea )
(
)
l = ( Sea ) 3 1 − F F −1 (1 − Sea ) + σ Dsg ( Sea = 1)
2
[2-13]
where Dsg is defined as the soil-gas diffusion coefficient (L2 T-1) where S ea = 1 (relative to θ min being the lowest measured water content). Alternatively, we tested whether the pore size (PS) distribution effect on soil gaseous diffusion is negligible, with gaseous diffusivity defined by the TC-term only, or Drg =
Dg ( Sea ) Dgs ( Sea = 1)
= ( Sea ) 3 l
[2-14]
so that gaseous diffusion is controlled by the air-filled, connected pores, irrespective of their sizes.
89
2.3.5. Relative bulk soil electrical conductivity To describe the relationship between bulk soil electrical conductivity and electrical conductivity of the soil solution ( EC w ) for an unsaturated soil, Rhoades et al. (1976) proposed ECa = T S ew EC w + EC s
[2-15]
in which EC s is the soil surface electrical conductivity (Q2 T M-1 L-3 or dS m-1) and T is the transmission coefficient (L L-1), which accounts for the tortuous nature of the current lines. Mualem and Friedman (1991) proposed a conceptual expression for the transmission coefficient to include its dependence on the geometry of the soil pore system in addition to water content. Assuming that the pore geometry factor affecting bulk soil electrical conductivity is similar to that of soil hydraulic conductivity, Mualem and Friedman (1991) hypothesized that the soil transmission coefficient can be derived from the ratio of soil hydraulic conductivity, K w ( S ew ), to that of a similar soil with pore sizes represented by capillaries, K w,cap ( S ew ), (as defined from Poiseuille’s law), or
T ( S ew ) =
K w ( S ew ) l = ( S ew ) K w,cap ( S ew )
dS ew h 0 S ew dS ew h2 0
∫ ∫
Sew
2
[2-16]
Using Eq. [2-16], it was determined that the value of the denominator becomes unrealistically high for matric potential head (h) values near zero (Mualem 1976; Vanclooster et al., 1994; Weerts et al., 1999). To overcome this anomalous behavior, we suggest using the definition of relative hydraulic conductivity function (Mualem, 1976)
90 instead, with a specific tortuosity-connectivity parameter ( l4 ) for the bulk soil electrical conductivity (Suman and Ruth, 1993) to characterize T . After substitution into Eq. [2-15], this yields l +1 ECa = EC w ( S ew ) 4
2
dS ew h 0 + EC s 1 dS ew h 0 Sew
∫ ∫
[2-17]
The relative electrical conductivity can be expressed as ECra =
2 ECa ( S ew ) − EC s l +1 = ( S ew ) 4 F F −1 ( S ew ) + σ ECa ( S ew = 1) − EC s
(
)
[2-18]
Equation [2-18] has a similar form as the other formulations for relative hydraulic and air conductivity and gaseous diffusivity. In addition to the Mualem and Friedman (1991) type approach to characterize T , we simply formulized T as a power function of effective water saturation, Sew , and examined its capability to estimate the relative bulk soil electrical conductivity function, omitting the contribution of the pore size distribution term in Eq. [2-18]. Substituting this simple form of T into Eq. [2-15] gives ECra =
ECa ( S ew ) − EC s l +1 = ( S ew ) 4 ECa ( S ew = 1) − EC s
[2-19]
2.4. Materials and Methods 2.4.1. Soil sample preparation Two different soils were used for the measurements to test the proposed transport theory. The soil physical properties for both soils are listed in Table 2-1. Prior to packing, the Columbia sandy loam was sieved through a 2 mm-opening sieve and a wet strength
91 fast flow filter paper was glued to one end of 8.25-cm inner diameter and 6-cm long soil cores. A total of 16 Columbia and 20 Oso Flaco soil samples were uniformly packed in these soil cores. The data in Table 2-1 show the range of values of the physical properties for these packed samples. After packing, a two-rod TDR miniprobe was inserted vertically in the center of the each soil sample, and soil samples were subsequently soaked in a 0.01 M CaCl2 solution while maintaining the chloride solution about 1cm below the rim of the soil cores.
Table 2-1. Physical properties of soils used in experiments. Soil Texture Soil
Sand
Silt
Clay
---------- (%) ----------
Bulk Density ( ρb )
Porosity (φ )
θs
* K sw
(g cm−3)
(cm cm−3) (cm cm−3)
(cm h−1)
3
3
Oso Flaco Fine Sand
100
0
0
1.55–1.57
0.41–0.42 0.37-0.42
4.080–5.183
Columbia Sandy Loam
67.6
22.3
10.1
1.37–1.43
0.46–0.48 0.39-0.44
0.012–0.015
K
* sw
: Measured saturated hydraulic conductivity
To construct a two-rod TDR miniprobe, two 0.22-cm diameter stainless steel rods were cut in 5-cm lengths. A groove at one end of each rod served to secure stainless steel clips that connect the rods to a 50-ohm coaxial cable. In addition, the other end of the rods was pointed for easy penetration into the soil samples. After drilling two 1.2-cm spaced guide holes in a 1 by 2-cm, 0.17-cm thick Plexiglas plate, the rods were inserted and glued in place using two-ton epoxy to form the waveguides of a TDR miniprobe. For each TDR measurement, the clips were connected to the TDR miniprobe, but removed in
92 between measurements. The TDR probes, however, were permanently installed throughout all experiments. After saturation, the soil cores were placed on a screen to measure the saturated * ) using the constant head method (Klute and Dirksen, 1986). hydraulic conductivity ( K sw
Upon completion of the saturated hydraulic conductivity measurement, the filter paper was removed and the saturated soil samples were assembled in Tempe pressure cells for estimation of soil-water retention and unsaturated hydraulic conductivity function using the multi-step outflow method (Eching et al., 1994; Tuli et al., 2001b). The samples were resaturated by wetting through the bottom porous membrane assembly, to ensure good contact between the soil sample and the porous membrane with the 0.01 M CaCl2 solution ( ECw = 2.4 dS m -1 ).
2.4.2. Determination of soil transport coefficients 2.4.2.1. Soil hydraulic functions Soil hydraulic functions for each soil type were measured using the multi-step outflow method in combination with parameter optimization (Eching et al., 1994). The magnitudes and number of pressure steps were different for both soils. For the Oso Flaco soil, the external pressure steps were: 0.0035, 0.0039, 0.0041, 0.0043, 0.0045, 0.0049, 0.0052, 0.0055, 0.0058, 0.0065, 0.0080 and 0.02 MPa, whereas for the Columbia soil, pressure steps of 0.0060, 0.01, 0.02, 0.035, and 0.05 MPa were applied. The pressure steps were different because of the lower sensitivity of the retention curve of the sand in the low-pressure range, whereas about equal drainage values were achieved for each of the 4 applied pressure steps of the Columbia soil. The multi-step outflow experiments
93 were started at a specific initial soil matric potential ( h init with corresponding θ init ) to accomplish unsaturation for all soil samples at the beginning of the multi-step outflow experiment (Eching et al., 1994; Hopmans et al., 2002). An initial pneumatic pressure of 0.003 MPa for the Oso Flaco and 0.0045 MPa for the Columbia soil was applied, and hydraulic equilibrium was established first before the first pressure step of the multi-step outflow experiment was applied. Between each successive pressure increment, time was allowed for the soil samples to equilibrate with the applied pressure. After zero drainage rate was achieved, 1-2 selected soil cores were removed from the multi-step outflow setup to measure air conductivity, gaseous diffusion coefficient, and bulk soil electrical conductivity at the assumed equilibrium conditions. After all remaining soil samples were equilibrated with the applied pressure, the next pressure step was applied to the remaining samples in the Tempe cells. The same procedure was followed for subsequent pressure steps until all samples were removed from the multi-step experimental setup. For the Columbia soil, we also included several additional retention points beyond the 0.05MPa pressure step. The additional retention points were collected using a 0.5MPa pressure plate apparatus using external pressures of 0.1, 0.3, and 0.5 MPa (Klute, 1986). Using the additional water content values in combination with the cumulative outflow volumes, water content values with the corresponding soil matric head values of the draining cores were computed. In the parameter optimization procedure, parameters of the soil water retention ( σ , h m ), (Eq. [2-5]) and the unsaturated hydraulic conductivity functions ( K sw ), (Eq. [2-10]) were inversely estimated using SFOPT (Tuli et al., 2001b; Hopmans et al., 2002). The parameter l1 and the measured values of θ s and θ r ( = θ min ) were fixed in the optimization procedure (Table 2-2).
94 Each soil sample removed from the multi-step outflow experimental setup was subsequently used to measure air conductivity, bulk soil electrical conductivity and gaseous diffusion coefficient. Hence, all 4 measurements were conducted for the same soil samples with identical water content values. Soil samples were weighted at the beginning and end of each measurement, to ensure that no significant changes in water content had occurred.
Table 2-2. Optimized ( h m , σ , K sw , and l1 ) and measured ( θ r and θ s ) parameters in multistep outflow.
Soil
hm
σ
θ r = θ min
θs
(θ s - θ min )
(cm3 cm-3)
(cm)
l1
K sw (cm h-1)
Oso Flaco fine sand
55.91
0.285
0.073
0.406
0.333
0.5
0.047
Columbia sandy loam
275.04
1.287
0.098
0.427
0.329
0.5
0.082
2.4.2.2. Air Conductivity The air conductivity of each soil sample was determined using the constant pressure gradient method (Janse and Bolt, 1960). In this method, air conductivity was measured at room temperature (varied between 23–26 oC) in which a constant air pressure (3.5 cm H2O) was applied to the soil sample by a descending float chamber connected to a manometer as air flows through the soil sample. Air conductivity is proportional to the average volume of air (V) passing through the soil sample with cross sectional area (A) in time t under a constant pressure gradient. The calculation of the soil air conductivity is based on the steady Darcy type of equation, or
95 Ka =
V Lc A t hD
[2-20]
where K a is the air conductivity (L T-1), V is the displaced fluid volume (L3), Lc is the length of soil sample (L), hD is the manometer reading (L), and t is time (T). Relative air conductivity values were obtained from the measured air conductivity value at the highest air content of each soil ( amax = φ − θ min where θ min = the water content value at 0.02 and 0.5 MPa pressures for the Oso Flaco and Columbia soil, respectively). Our experiences showed great difficulties to obtain a consistent set of measurement results. Specifically, one must ensure that there is no bypass air flow between the core wall and the soil sample because of soil shrinkage. Air gaps consistently caused misleading results and produced large errors in air conductivity measurements. To avoid these erroneous results, the space between the core wall and the soil sample was sealed with high vacuum grease at both ends. Moreover, the applied air pressure gradient must be small at high water saturation (especially in sandy soils) to prevent redistribution of water during the air conductivity measurements.
2.4.2.3. Bulk soil electrical conductivity Bulk electrical conductivity ( ECa ) and water content of the samples were measured with the TDR method. The TDR method for water content measurement is based on the determination of the velocity of propagation of an electromagnetic (EM) wave along a transmission line (or waveguide) in the soil. Since the propagation velocity is a function of the soil bulk dielectric constant, a relationship between the dielectric constant and water content is established (Topp et al., 1980). Furthermore, the loss of
96 energy of the reflected electromagnetic wave is used as a measure of bulk soil electrical conductivity. As electromagnetic waves propagate along the TDR waveguides in the soil, the signal is attenuated in proportion to the bulk soil electrical conductivity. This reduction in EM wave voltage as a function of electrical conduction serves as a basis for the bulk soil electrical conductivity measurement. Although TDR-measured water content values were available, the gravimetric water content values were used in the present study. All TDR probes were initially calibrated in distilled water to determine their impedance and effective length. For each ECa measurement, the 50-ohm resistant coaxial cable was connected to TDR probes by stainless steel clips. All TDR measurements were conducted with a Tektronix 1502B metallic cable tester. The waveforms were analyzed using the WinTDR99 program which was developed by Or et al. (1998) to compute ECa and θ directly from the analyzed waveforms. The surface conductance of the Oso flaco sand was determined by conducting a separate series of bulk soil electrical conductivity measurements saturated with CaCl2 solutions at 4 concentration levels with EC w values ranging from 0.3 dS m-1 to 2.5 dS m-1 (Figure 2-1). Using these data, the value of surface conductivity, ECs , for the Oso Flaco fine sand was found to be 0.0725 dS m-1, after extrapolation of the data in Figure 2-1 to zero water content. For the Columbia soil, we used the measured bulk soil electrical conductivity value at the lowest water content ( θ min = θ 0.5 MPa ) as the ECs value, yielding a ECs value of 0.186 dS m-1 (using the 0.01 M CaCl2 solution).
97
1
ECw = 0.30 dS m-1 ECw = 0.56 dS m-1 ECw = 1.38 dS m-1 ECw = 2.50 dS m-1
ECa (dS m-1)
0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
θ
Figure 2-1. Bulk soil electrical conductivity as a function of water content for different solution EC values for the Oso Flaco fine sand.
2.4.2.4. Gaseous diffusion The gaseous diffusion coefficient (L2 T-1) of all soil samples was measured with the method described by Rolston (1986). This laboratory method is based on establishing an initial gas concentration within a single diffusion chamber apparatus. Freon-12 (CCl2F2) was chosen as the diffusive gas due to its nonreactivity, low water solubility and its resistance to chemical and biological transformation (Petersen et al., 1994; Jin and Jury, 1996). Before injecting Freon-12 into the gas chamber, the sliding plate of the apparatus was initially positioned to close the chamber. A known amount of initial Freon-12 was injected and was well mixed for 5-10 minutes by a small fan installed inside the chamber. After the Freon-12 was allowed to equilibrate for 20 minutes in the gas chamber, the soil
98 sample was placed inside the sample holder of the sliding plate. High vacuum grease was applied to ensure that no bypass of Freon-12 diffusion occurred between the soil sample and core wall. Specifically, the Columbia soil showed some shrinkage with decreasing water content, causing a small air gap. The sliding plate was subsequently opened with the soil core located directly over the hole of the diffusion chamber. The decrease in concentration of the gas was followed in time by the regular withdrawal of gas samples (75µl) from the chamber with a gas tight syringe. Gas samples were analyzed by gas chromatography. Since the gaseous diffusion measurement took much longer than the air conductivity and electrical conductivity measurement, the diffusion experiments were conducted inside a large Plexiglas container in which water saturated air was circulated to minimize evaporation from the soil samples during the diffusion experiment. At the end of each experiment, samples were placed in an oven to determine the volumetric water content. The diffusion coefficients were calculated by the Taylor’s (1949) approximation. In this approach, the concentration gradient within the soil core is determined from the differences between the gas concentration in the chamber and at the soil surfaceatmosphere interface of the soil core. It is assumed that the Freon-12 concentration is equal to zero at this interface. By plotting these concentration differences as a function of time after initial phase of the experiment, a straight line provides the slope that includes the gaseous diffusion coefficient. With known experimental variables, the gaseous diffusion coefficient, Dg , was calculated, neglecting the change in storage of gas within the soil sample.
99
2.4.3. Analysis of fitting results To
estimate
the
model
specific
tortuosity-connectivity
parameters,
l1 , l2 , l3 , and l4 of all 4 transport coefficients (Eqs. [2-10], [2-11], [2-13], and [2-18] or [2-10], [2-12], [2-14], and [2-19]) were simultaneously optimized by minimization of the objective function (OF) J
OF (b) =
I
∑∑ j =1
y *j ( S e,i ) − y j ( S e,i , b)
2
[2-21]
i =1
where y *j ( S e ,i ) and y j ( S e ,i , b) denote the measured and fitted transport coefficient values at certain fluid content, S e ,i , respectively. The vector b contains the optimized parameter values, the subscripts i and j represents the measurement number and measured transport coefficient, respectively. The model performance for different cases was evaluated by the root mean squared error (RMSE) using J
RMSE =
I
∑∑ j =1
y *j ( S e,i ) − y j ( S e,i , b)
i =1
N −m
2
[2-22]
in which J , I , N and m denote total number of transport coefficients (4 or 5), number of measurements for each transport properties, total number of measurements and optimized parameters, respectively. The individual average fitting error for soil-water retention and each transport coefficient after the optimization was examined by defining the Average Residual ( AR j ) I
AR j =
∑
y * ( S e ,i ) − y ( S e ,i , b )
i =1
I
[2-23]
100 AR j denotes the specific transport coefficient function ( j = 1 for K rw , j = 2 for K ra ,
j = 3 for Drg , j = 4 for ECa , and j = 5 for S ew (h) ). Optimization procedures were carried out using Excel software (Wraith and Or, 1998). We tested 6 different cases for the simultaneous optimizations (Table 2-3). In case 1, we used the σ and h m values equal to those determined from fitting of the soil hydraulic properties using the multi-step outflow data only. In case 2, all parameters used in the transport coefficient functions were optimized (including σ and h m ), allowing the pore size distribution ( PS ) and tortuosity-connectivity ( TC ) term parameters to be equally important for each of the four measured transport coefficients. Cases 3 and 4 were similar with respect to number of optimized parameters to case 1 and 2, but it was assumed that both the gaseous diffusion and bulk soil electrical conductivity formulation
Table 2-3. Case of model parameters used in the simultaneous optimization. Cases
σ
hm
l parameters
K rw (S ew )
K ra (S ea )
D rg (S ea )
EC ra (S ew )
1
Fixed
Fixed
Variable
TC * PS
TC * PS
TC * PS
TC * PS
Variable
TC * PS
TC * PS
TC * PS
TC * PS
Variable
TC * PS
TC * PS
TC
TC
Variable
TC * PS
TC * PS
TC
TC
Variable
TC * PS
TC
TC
TC
Variable
TC * PS
TC
TC
TC
2 3 4 5 6
Variable Variable Fixed
Fixed
Variable Variable Fixed
Fixed
Variable Variable
could be described by the tortuosity-connectivity term ( TC ) only. Cases 5 and 6 were similar to cases 3 and 4, but in addition, the relative air conductivity was described by the tortuosity-connectivity term ( TC ) only.
101
2.5. Results and Discussions 2.5.1. Soil water retention curves The soil hydraulic functions for both soils were determined using inverse modeling according to the parameter estimation technique proposed by Eching et al., (1994). Independently measured and optimized soil-water retention data using the multi-step outflow procedure with equilibrium soil-water retention data points are presented for the Oso Flaco fine sand and Columbia sandy loam soil in Figure 2-2. One must realize that
Soil matric potential head (- cm)
the vertical axis is a log-scale and soil-water retention data were fitted to the retention
10000
Measured Oso Flaco sand Optimized Oso Flaco sand Measured Columbia sandy loam Optimized Columbia sandy loam
1000 100 10 1 0
0.2
0.4
0.6
0.8
1
Sew Figure 2-2. Soil water retention data and optimized curves for the Oso Flaco fine sand and Columbia sandy loam soil.
102 model simultaneously with the unsaturated hydraulic conductivity data. In Figure 2-2, the number of data points corresponds to the number of soil samples. It is also assumed that soil hydraulic properties for all examined samples are identical, despite that bulk density values are slightly varying between samples (Table 2-1). Thus, the measured saturated water content, θ s , value was calculated from the average of the measured saturated water content value of the individual samples. For both soils, there was very good agreement between measured and optimized soil-water retention data. Optimization of the hydraulic parameters was carried out three times, each time with different initial parameter estimates (with arbitrary chosen low, medium, and high values in each parameter domain). The final optimized parameter values in Table 2-2 are presented only if convergence occurred with equal parameter values for each of the three optimizations. Difference in retention parameters and shape and range of the soilmoisture characteristic curve depends on soil texture with optimized parameter values of h m and σ defining pore size distribution (See Eq. [2-5]). Assuming that pore size is lognormally distributed, the median value, h m , corresponds with the soil matric potential head for which Sew = 0.5 , with σ describing the standard deviation of the lognormal pore size distribution (Kosugi, 1996; Tuli et al., 2001a). Since we assume that pore radius ( r ) is associated with soil matric potential head (h) by the capillarity equation ( h = −2γ cos β ρ w gr = −0.149 r ), ln r obeys the normal distribution with N ln rm , σ 2
or N ln hm , σ 2 . Especially, for the Oso Flaco sand, most pores are relatively large (hm = 55.91 cm) and pore size distribution is narrow ( σ = 0.285) (Figure 2-3).
103
1.6 1.4 1.2
f(r)
0.8
Oso Flaco fine sand
1.0 σ = 0.285
0.6 0.4 0.2
b lum o C
0.0 -12
ia
-10
m oa yl d san σ = 1.287
-8
lnrm
-6
-4
-2
lnr
Figure 2-3. Pore size distribution of Oso Flaco sand and Columbia soil with median radius ( ln rm ) and standard deviation ( σ ) of ln rm .
Once the soil matric potential head has exceeded the air entry value of the sand, all large pores are emptied and the pore domain includes a connected air phase. Then, diffusive and advective transports are governed by Fick and Darcy-type laws, respectively. Figure 2-2 shows that the Columbia soil has a similar air entry value as the Oso Flaco, but both the median ( h m = 275.04 cm) and standard deviation ( σ = 1.287) of the Columbia soil are larger, corresponding with smaller pores and a wider pore size distribution (Figure 2-3). Thus, the amount of water held in pores at any given soil matric potential head is higher than for the Oso Flaco, and continued decrease of the soil matric
104 potential head causes a more gradual decrease in water content (Figure 2-2) for the Columbia soil.
2.5.2. Measured transport coefficients All four transport coefficients and relative values as a function of effective fluid saturation are shown in Figures 2-4 and 2-5, respectively (also see Appendix 2.1.). Figure 2-4A presents the optimized unsaturated hydraulic conductivity values at the independently measured water content values using the optimized hydraulic function parameters of Table 2-2. As expected, the Oso Flaco (solid symbols) has a higher unsaturated hydraulic conductivity within the experimental domain than the Columbia soil (open symbols) as controlled by the larger water filled pores at the corresponding water content assuming that the water-filled pores are connected. In contrast to the measured values (Table 2-1), the optimized saturated hydraulic conductivity of the Columbia soil is slightly higher than for the Oso Flaco sand. It should be noted that the Multistep outflow method excludes saturated soil conditions. The optimized saturated value is controlled by the shape of the unsaturated hydraulic conductivity curve and its extrapolation to saturation. Moreover, differences in optimized K sw between soils are determined by the degree of unsaturation achieved ( θ init ) after application of the first pneumatic pressure step (see section 2.4.2.1). The unsaturated hydraulic conductivity, K w ( S ew ) , data are obtained from optimization of the multi-step outflow data, with the selected K w -data originating from the optimized hydraulic conductivity curves at the measured equilibrium water content values. As the water content decreases, the
105 0.10
3.00
(A)
Ka x 103 (cm h-1)
Kw (cm h-1)
0.08 0.06 0.04 0.02 0.00
(B)
2.40 1.80 1.20 0.60 0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
θ 2.00
(C)
0.60
0.3
0.4
0.5
0.3
0.4
0.5
(D)
1.60
ECa (dS m-1)
Dg x 102 (cm2 h-1)
0.75
a
0.45 0.30
1.20 0.80
0.15
0.40
0.00
0.00 0.0
0.1
0.2
a
0.3
0.4
0.5
0.0
0.1
0.2
θ
Oso Flaco fine sand
Columbia sandy loam
Measured Kw
Measured Kw
Measured Ka
Measured Ka
Measured Dg
Measured Dg
Measured ECa
Measured ECa
Figure 2-4. Measured transport coefficients as a function of fluid content for both soils.
106 1.0
(A)
0.8
0.8
0.6
0.6
(B)
Kra
Krw
1.0
0.4
0.4
0.2
0.2
0.0
0.0 0.0
1.0
0.2
0.4
Sew
0.6
0.8
1.0
0.0
1.0
(C)
0.8
0.6
0.6
Drg
ECra
0.8
0.4
0.2
0.0
0.0 0.2
0.4
Sea
0.6
0.8
1.0
Oso Flaco fine sand
0.4
Sea
0.6
0.8
1.0
0.6
0.8
1.0
(D)
0.4
0.2
0.0
0.2
0.0
0.2
0.4
Sew
Columbia sandy loam
Measured Krw
Measured Krw
Measured Kra
Measured Kra
Measured Drg
Measured Drg
Measured ECra
Measured ECra
Figure 2-5. Measured relative transport coefficients as a function of effective fluid saturation for both soils.
107
unsaturated hydraulic conductivity values of the Oso Flaco sand dramatically decreases likely because of the reducing connectivity of the water-filled pores. Although the pores of the Columbia soil are filled with water over a wider pore size range, the smaller pores of the Columbia soil cause lower unsaturated hydraulic conductivity values than for the sandy soil. Similarly, the relative hydraulic conductivity curves show the same trend (Figure 2-5A). However, per definition, both relative hydraulic conductivities at saturation must be equal to 1. Air conductivity ( K a ) and gaseous diffusion coefficient ( Dg ) as a function of air content are presented in Figures 2-4B and 2-4C, respectively. While air conductivity values of both soils are distinctively different, the air dependency of both soils for the gaseous diffusion coefficient is very similar, with Dg values slightly lower for the Columbia soil. It is expected that in addition to the tortuosity-connectivity effect, the effect of pore size distribution is more pronounced for air conductivity (convective process) than gaseous diffusion. Also, Moldrup et al., (2001) showed that pore size can have dramatic effects on air conductivity, but little effect on gaseous diffusion. This is so since convective flow is described by a Poiseuille-type of expression, with flow velocity controlled by pore size. In contrast, diffusion can be considered as being independent of pore size (except at extremely small pore size where Knudsen diffusion occurs). However, diffusion is also controlled by pore tortuosity and connectivity ( TC -term). One must note that the highest measured air conductivity and gaseous diffusion coefficient values were determined at the lowest measured water content values ( amax = φ − θ min ) for all experiments. The relative data of K a and Dg coalescence for both soils when plotted
108 as a function of effective air saturation (Figures 2-5B and 2-5C). On the other hand, at Sea -values < 0.7, the relative values of gaseous diffusion, Drg , and air conductivity, K ra , of the Columbia soil are slightly lower than for the Oso Flaco sand, possibly due to larger TC - values (Figures 2-5B and 2-5C).
Figure 2-4D shows the measured bulk soil electrical conductivity ( ECa ) values of both soils. Throughout the whole water content range, the bulk soil electrical conductivity of the Columbia soil is slightly higher than for the Oso Flaco sand, likely caused by the contribution of the higher soil surface conductance of the soil particles ( EC s are 0.0732 and 0.186 dS m-1for Oso Flaco and Columbia soil, respectively) as caused by the exchangeable ions of the Columbia soil. The EC -values also indicated that while the bulk soil electrical conductivity linearly increases at the low and intermediate water content range, a nonlinear increase is apparent at the near saturation water content range (Figures 2-4D nad 2-5D). Possibly, the larger water-filled soil pores are dominating ECa . When ECa is transformed to relative values as determined by Eq. [2-18], the moisture dependency of ECra for both soils is approximately similar (Figure 2-5D).
2.5.3. Simultaneous optimization of transport coefficients All four transport coefficients were simultaneously optimized for the different optimization scenarios listed as cases in Table 2-3. The optimization results (lines) with the measured data (symbols) are presented as a function of effective air saturation ( Sea ) for each of the 6 cases in Figures 2-6 and 2-7, for the Oso Flaco (solid symbols) sand and Columbia soil (open symbols), respectively. Final optimized parameter values, overall
109
1.0
Case 1
Krw , Kra , Drg , ECra
Krw , Kra , Drg , ECra
Oso Flaco fine sand 0.8 0.6 0.4 0.2 0.0
1.0 0.8 0.6 0.4 0.2 0.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Case 3
Sea
Krw , Kra , Drg , ECra
Krw , Kra , Drg , ECra
Sea
1.0 0.8 0.6 0.4 0.2 0.0
1.0
0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
Sea
Krw , Kra , Drg , ECra
Krw , Kra , Drg , ECra
Sea
Case 5
0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
Sea
Case 4
0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.0
Case 2
1.0
Case 6
0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
Sea
Measured Krw
Optimized Krw
Measured Kra
Optimized Kra
Measured Drg
Optimized Drg
Measured ECra
Optimized ECra
Figure 2-6. The measured and simultaneously optimized transport coefficient data for all 6 cases of the Oso Flaco fine sand.
110
1.0
Case 1
Krw , Kra , Drg , ECra
Krw , Kra , Drg , ECra
Columbia sandy loam
0.8 0.6 0.4 0.2 0.0
1.0 0.8 0.6 0.4 0.2 0.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Case 3
Sea
Krw , Kra , Drg , ECra
Krw , Kra , Drg , ECra
Sea
1.0 0.8 0.6 0.4 0.2 0.0
1.0
0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
Sea
Krw , Kra , Drg , ECra
Krw , Kra , Drg , ECra
Sea
Case 5
0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
Sea
Case 4
0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.0
Case 2
1.0
Case 6
0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0
Sea
Measured Krw
Optimized Krw
Measured Kra
Optimized Kra
Measured Drg
Optimized Drg
Measured ECra
Optimized ECra
Figure 2-7. The measured and simultaneously optimized transport coefficient data for all 6 cases of the Columbia sandy loam soil.
111 fitting results of all transport models as expressed by the Root Mean Squared Error (RMSE) and individual average fitting error (ARj) are presented in Table 2-4. After visual inspection of Fig. 2-6 and from the RMSE values in Table 2-4, it is clear that there are no major discrepancies between different cases for the Oso Flaco sand. With respect to the Columbia soil, there are slight differences in RMSE -values between some cases.
Table 2-4. Optimized σ , h m and tortuosity-connectivity parameters ( TC ), Root Mean quared Error ( RMSE ), and Averaged Residual ( AR j ) for fitted transport coefficient functions. Case
σ
hm
Tortuosity-connectivity parameters*
(cm)
l1
l2
l3
l 4+1
RMSE
AR j K rw (S ew ) K ra (S ea ) D rg (S ea ) EC ra (S ew ) (j= 1)
S ew (h)
(j= 2)
(j= 3)
(j= 4)
(j= 5)
0.0307 0.0308 0.0307 0.0307 0.0278 0.0278
0.0162 0.0163 0.0142 0.0142 0.0142 0.0142
0.0788 0.0784 0.0861 0.0861 0.0861 0.0861
0.0321 0.0279 0.0321 0.0281 0.0321 0.0280
0.0287 0.0289 0.0287 0.0286 0.0290 0.0290
0.0218 0.0220 0.0211 0.0211 0.0211 0.0211
0.0617 0.0362 0.0656 0.0656 0.0656 0.0656
0.0488 0.0488 0.0488 0.0195 0.0488 0.0195
Oso Flaco fine sand 1 2 3 4 5 6
0.29 0.30 0.29 0.29 0.29 0.29
55.91 55.41 55.91 55.36 55.91 55.37
0.500 0.463 0.500 0.495 0.500 0.493
1.008 1.027 1.008 1.011 2.365 2.365
2.180 2.201 3.478 3.478 3.478 3.478
0.264 0.221 3.333 3.333 3.333 3.333
1 2 3 4 5 6
1.29 1.14 1.29 1.56 1.29 1.56
275.04 214.92 275.04 210.08 275.04 210.12
0.484 1.835 0.500 -2.066 0.500 -2.049
2.481 2.407 2.486 2.587 2.713 2.713
3.878 3.810 4.069 4.069 4.069 4.069
-2.844 -2.798 3.949 3.983 3.949 3.983
0.044 0.043 0.047 0.046 0.047 0.046
0.0000 0.0004 0.0000 0.0001 0.0000 0.0001
Columbia sandy loam 0.045 0.039 0.049 0.046 0.049 0.046
0.0001 0.0012 0.0000 0.0042 0.0000 0.0042
* l 1: relative hydraulic conductivity; l 2: relative air conductivity; l 3: gas diffusivity; l 4+1: relative electrical conductivity
Specifically, case 2 (all transport models include the pore size and tortuosityconnectivity terms, (TC+PS)) shows the lowest RMSE -values among all cases (Table 24). With respect to the average residual analysis, the AR j describes average the absolute differences between predicted and measured data. For a j -value of 5, the objective function (Eq. [2-19]) includes the soil-water retention data for cases 2, 4, and 6. Each of
112 the 5 variables is given equal weight, since they all vary between 0 and 1. K rw shows the best agreement among all the 4 transport coefficients for both soils. Furthermore, the largest fitting errors ( AR j ) occurred for the ECra -prediction in all cases for both soils (second to last column in Table 2-4). Since we used the same formulation to characterize the fluid dependency for each transport coefficient, it provides a unique opportunity to compare the tortuosityconnectivity ( TC ) parameters between transport coefficients. Table 2-4 shows that the tortuosity-connectivity parameter values, l , increase from l1 to l3 for all cases. Large l values probably indicate that the relative influence of TC compared to PS is larger for the air than for the hydraulic conductivity, and the same may be valid for air phase diffusivity compared to soil electrical conductivity. The physical meaning of this result is that the tortuosity and connectivity term for the relative air conductivity ( K ra ) and gaseous diffusivity ( Drg ) is more controlling than for the relative hydraulic conductivity ( K rw ) and/or soil bulk electrical conductivity. This was the case for both soils. Especially, gaseous diffusion is dominantly controlled by tortuosity and/or connectivity of the gas-filled pores. The lesser effect of pore size distribution on gaseous diffusion was also demonstrated by the analysis of Moldrup et al. (2001). Moreover, Weerts et al., (2000) showed that the tortuosity-connectivity parameter for gaseous diffusion was significantly different than for K rw , since it could not be used for prediction of K rw . Moreover, the tortuosity and connectivity parameters l2 and l3 of K ra and Drg , respectively, are much larger for the Columbia soil than for the Oso Flaco fine sand due to the longer continuous and tortuous pathways at a certain air content (Table 2-4).
113 Since the fluid-saturation dependency of all four relative transport coefficients is governed by pore geometry of the same pore space, it is worthwhile to investigate whether all transport coefficients follow a common behavior. Figure 2-10 combines all 4 measured relative transport coefficients as a function of the relevant phase saturation. As
Oso Flaco fine sand
Columbia sandy loam 1.0
Krw Kra Drg ECra
0.8 0.6
Krw , Kra , Drg , ECra
Krw , Kra , Drg , ECra
1.0
0.4 0.2 0.0
Krw Kra Drg ECra
0.8 0.6 0.4 0.2 0.0
0.0
0.2
0.4
0.6
Sew , Sea
0.8
1.0
0.0
0.2
0.4
0.6
Sew , Sea
0.8
1.0
Figure 2-8. Combination of all 4 relative transport coefficients (TrCo j j = 1...4 ) as a function of fluid content and effective saturation.
might be expected, all measured data for the Oso Flaco fine sand, when normalized, show a similar behavior providing for a unique characterization of transport, irrespective of whether transport is by diffusion or convection, or whether it occurs in the liquid or air phase. This is likely because Oso Flaco exhibits large changes in water content with a small change in soil matric head values after the air entry value is exceeded (Figure 2-2). This directly shows that the water adsorption and thus, pore size ( PS ) plays a small role for the Oso Flaco. Therefore, it can be concluded that the transport relations mainly are
114 affected by the TC term in the Oso Flaco sand. The PS effect on the hydraulic conductivity is still the highest since relative hydraulic conductivity curve is placed on the lower part of Figure 2-8. Moldrup et al. (2001) also showed that tortuosity and connectivity curves for the air and liquid phase diffusivity for the sieved fine sandy soil were similar. For optimization cases 1 and 2, the smaller TC -parameters ( l1 and l4 + 1 ) of K rw and ECra might imply that the pore size ( PS ) and tortuosity and connectivity ( TC ) parameter of the water phase are equally important for describing these two transport coefficients. Meanwhile, for cases 3, 4, 5, and 6 where ECra is described by the TC term only, the simultaneous optimizations resulted in high values for the tortuosityconnectivity parameter ( l4 ) as compared to corresponding values for cases 1 and 2. The results in these cases agree with the findings of Weerts et al. (2001) where the tortuosityconnectivity parameter value ( l3 ) for the gaseous diffusivity is similar to the relative electrical conductivity ( l4 + 1 ). Alternatively, the high l4 -values for these cases could imply that electrical current follows a highly tortuous pathway with resulting tortuosityconnectivity values equal to that for the gaseous diffusivity (Weerts et al., 2001). On the other hand, these findings totally depend on the how representative ECa is for the liquid diffusivity. A similar unique pattern is not as clear for the Columbia soil, with larger difference between the relative transport coefficients for the liquid and gas phase (Figure 2-8). Likely, this is caused by the dominant effect of pore size distribution on K rw for the Columbia soil that has a much wider pore size distribution. The other three transport coefficients (air conductivity, gas diffusivity, bulk soil electrical conductivity) are, like
115 for the Oso Flaco, almost placed one unique curve. The air conductivity is little higher due to the preferential air flow in connected larger air filled pores. Similar to Oso Flaco, Columbia soil shows high tortuosity-connectivity parameter values ( l2 , l3 and l4 + 1 ) for cases 3 through 6, which represent dominant effect of TC term. In cases 1 and 2, tortuosity-connectivity parameter ( l4 + 1 ) of ECra receives negative values due to compensating the dominance of PS term. Moreover, one should keep in mind that the negative values of tortuosity connectivity parameter have no physical meaning other than the pure optimized value. Therefore, it might be more logical to represent the ECra by TC term only. When considering results of Figure 2-8, the application of a single unique
relation is promising at the transport coefficient range of 0.2-1.0. However, when one is primarily interested in the dry range with TrCo -values smaller than 0.2, prediction errors can be many orders of magnitude. This will become clear when using a log-scale. To illustrate the contributions of the TC and PS terms on the relative transport coefficients (TrCo j j = 1...4 ), sensitivity analyses are presented with different tortuosityconnectivity parameter values in Figure 2-9 and Figure 2-10, for the Oso Flaco sand and Columbia soil, respectively.
While Figure 2-9A and Figure 2-10A illustrate the
dependency of the TC term on the tortuosity-connectivity parameter, l, Figure 2-9B and Figure 2-10B show the contribution of the PS term for both the water and air phase on the relative transport coefficient models, defined by Eq. [2-7] and controlled by pore size distribution parameters σ and h m (Table 2-4). Hence, the optimized parameters σ and h m as obtained from multi-step outflow experiment determine the fluid saturation
116
1.0
2. 0 3.0
l=
4.0
0.4 0.2
l=
0.2
0.6 as e
l=
0.4
0 1.
ate rp ha se
0.5
ph
l=
0.8
W
0.6
Pore Size Term (PS)
0.8
Ai r
l = 0.0
0.0
0.0 0.2
0.4 0.6 Se(w , a)
0.8
1.0
0.0
(C)
1.0
0.8
0.6
0.6
0 1. .0 l = 2.0 0 3. 0
l=
l=
-1
.0
0.4
TrCoj
0.8
0.4
l=
0.0
0.8
1.0
Air phase
0.2
l=
0.2
0.4 0.6 Se(w , a)
(D)
1.0
Water phase
0.2
l= l = 0.5 l 1 l = = 2.0 .0 3 . 0 l= 4.0
0.0
TrCoj
(B)
(A)
1.0
l=
Tortuosity-Connectivity Term (TC)
Oso Flaco fine sand
0.0 0.0
0.2
0.4
Sew
0.6
0.8
1.0
0.0
0.2
0.4
Sea
0.6
0.8
1.0
Figure 2-9. Sensitivity analysis for relative transport coefficient ( TrCo j ) functions: Saturation dependence of (A) TC term on the tortuosity-connectivity parameter, l , (B) PS term on the effective fluid saturation, (C) relative water phase transport coefficient function on l parameter, and (D) relative air phase transport coefficient function on l parameter.
117
Columbia sandy loam (B)
1.4
1.0
l
0.6
.5 =0
l=
0.4
1.0 l=
0.2
2 .0
l=
3.0 4. 0
0.8
0.6 0.4
has e
l = 0.0
0.8
Wa ter p
1.0
Pore Size Term (PS)
l = -0.1
Ai rp ha se
1.2
0.2
l=
Tortuosity-Connectivity Term (TC)
(A)
0.0
0.0 0.0
0.2
0.4 0.6 Se(w , a)
0.8
1.0
0.0
0.2
0.4 0.6 Se(w , a)
0.8
1.0
(D)
(C) 1.0
1.0
Water phase
0.8
0.6
0.6
0.5
l=
1.0
l=
l=
2.0
4.0
3 .0
l=
l=
2.0
l=
0.4 0.2
l=
0.2
l=
TrCoj
0.4 -3 .0
TrCoj
0.0
0.8
Air phase
0.0
0.0 0.0
0.2
0.4
Sew
0.6
0.8
1.0
0.0
0.2
0.4
Sea
0.6
0.8
1.0
Figure 2-10. Sensitivity analysis for relative transport coefficient ( TrCo j ) functions: Saturation dependence of (A) TC term on the tortuosity-connectivity parameter, l , (B) PS term on the effective fluid saturation, (C) relative water phase transport coefficient function on l parameter, and (D) relative air phase transport coefficient function on l parameter.
118 of the PS terms for air and water phase in Figure 2-9B and Figure 2-10B. The sensitivity analysis in Figure 2-10A shows that negative l -values ( l = −0.1 ), which has no physical meaning, can be compensated for by a highly nonlinear PS -function. While comparing Figure 2-9B (Oso Flaco) and Figure 2-10B (Columbia), it is clear that the PS term is highly nonlinear for the Columbia soil, indicating much larger contribution of PS term for the soil with the wider pore size distribution. The dependency of the water-phase relative transport coefficient functions ( K rw and ECra ) on the TC parameter is presented in Figure 2-9C and Figure 2-10C for the Oso Flaco sand and Columbia soil, respectively, whereas the TC -effect on the air-phase relative transport coefficients ( K ra and Drg ) are presented in Figure 2-9D and Figure 2-10D. In either case, these analyses show that the relative transport coefficients decrease faster with decreasing degree of saturation, as the l -parameter values increases. For the Oso Flaco, TrCo j of water and air phases are about
equally sensitive to the tortuosity-connectivity parameters (Figure 2-9C and D). This means that the transport relations are mainly controlled by tortuosity and connectivity of the phases (air conductivity, gas diffusion and bulk soil electrical conductivity) whereas pore size distribution has slight effect only on soil hydraulic conductivity. Meanwhile, for the Columbia soil, the TrCo j of water ( j = 1 ) is not as sensitive to the l parameter as the air phase TrCo j , which is caused by the dominant effect of the PS -term on unsaturated hydraulic conductivity ( j = 1 ) as can be determined from comparison of Figure 2-10B and C. In contrast, for the air phase, the l -parameter is highly sensitive (Figure 2-10D) because of the wide pore size distribution. This also means that air conductivity and gaseous diffusion are dominantly controlled by the tortuosity and connectivity of the air
119 phase, thus contribution of TC term to the transport of air and gas is larger than for hydraulic conductivity. Since the negative l values have no physical meaning, the ECra can be represented by TC term only for both soils.
2.6. Conclusions The four relative transport coefficients ( K rw , K ra , Drg and ECra ) can be simultaneously described with different tortuosity-connectivity ( TC ) parameter values in their formulations. Comparison of optimized with measured data showed that the agreement between models and measured data were good. However, comparison of the optimized air and water phase tortuosity-connectivity parameters indicated that each transport process has different values. Specifically, the TC - parameter of the air phase for relative air conductivity and gaseous diffusivity was much larger than the TC parameter for the water-phase transport coefficients ( K rw and ECra ). These results indicate that tortuosity and connectivity of the air phase is dominantly controlling air conductivity and gaseous diffusion. However, when comparing the two air phase TC parameters ( l2 and l3 ), the tortuosity and connectivity effect of the air phase on air conductivity is less than for gaseous diffusion, likely because of the contribution of pore size distribution on air conductivity. The combined data and analysis also revealed that the contributions of both pore size distribution and tortuosity-connectivity is equally important in characterization of the soil hydraulic and bulk soil electrical conductivity. The dependency of the air conductivity and gaseous diffusion on the tortuosity and connectivity of air phase was much higher for the Columbia sandy loam soil, with a wider pore size distribution than the Oso Flaco fine sand.
120 In summary, in addition to pore size distribution, process specific tortuosity and connectivity parameter values must be used in parametric models predicting transport coefficients for the different fluid phases. Irrespective of the type of transport, the moisture dependency of any transport property in the Oso Flaco sand is fairly unique when normalized using the highest measured value. Although a unified formulation that includes both tortuosity/connectivity and pore size terms was successful in characterizing both diffusive and convective transport in both liquid and gaseous phase, more research is needed to better quantify the magnitude of the TC term a priori. It is emphasized that all measurements and analysis were conducted for disturbed soils only, thereby ignoring soil structure effects on transport processes. Likely, macroporosity may influence the results, magnifying the pore size distribution effects at near saturation for the convective transport properties.
121
2.7. References Ball, B.C. 1981. Modeling of soil pores as tubes using gas permeabilities, gas diffusivities and water release. J. Soil Sci. 32: 465-481. Bear, J. 1972. Dynamics of fluids in porous media. Dover Publications, Inc. New York. Brooks, R.H. and A.T. Corey. 1964. Hydraulic properties of porous media. Colorado Univ. Hydrology paper No.3 Burdine, N.T. 1953. Relative permeability calculations for pore size distribution data. Petroleum Transactions, AIME. 198: 71-78. Campbell, G.S. 1974. A simple method for determining unsaturated conductivity from moisture retention data. Soil Sci. 117: 311-314. Chen, J., J.W. Hopmans, and M.E. Grismer. 1999. Parameter estimation of two-fluid capillary pressure-saturation and permeability functions. Adv. Water Resour. 22: 479-493. Collis-George, N. 1953. Relationship between air and water permeability in porous media. Soil Sci. 76: 239-249. Diedericks, G.P.J. and J.P. du Plessis. 1996. Electrical conduction and formation factor in isotropic porous media. Adv. Water Resour. 19: 225-239. Dury, O., U. Fischer, and R. Schulin. 1998. Dependence of hydraulic and pneumatic characteristics of soils on a dissolved organic compound. J. Contam. Hyd. 33: 3957. Dury, O., U. Fischer, and R. Schulin. 1999. A comparison of relative non-wetting phase permeability models. Water Resour. Res. 35: 1481-1493.
122 Eching, S.O., J.W. Hopmans, and O. Wendroth. 1994. Unsaturated hydraulic conductivity from transient multi-step outflow and soil water pressure data. Soil Sci. Am. J. 58:687-695. Fischer, U., O. Dury, H. Flühler, and M.Th. van Genuchten. 1997. Modeling nonwettingphase relative permeability accounting for a discontinuous nonwetting phase. Soil Sci. Soc. Am. J. 61: 1348-1354. Friedman, S.P., L. Zhang, and N.A. Seaton. 1995. Gas and solute diffusion coefficients in pore networks and its description by a simple capillary model. Trans. Porous Media, 19: 281-301. Friedman, S.P. and N.A. Seaton. 1998. Critical path analysis of the relationship between permeability and electrical conductivity of three-dimensional pore networks. Water Resour. Res. 34: 1703-1710. Hopmans, J.W., J. Simunek, and N. Romano. 2002. Simultaneous determination of water transmission and retention properties - Inverse modeling of transient water flow. In: Topp, G.C. and J.H. Dane, editors. Methods of Soil Analysis. Part I. Third Edition American Society of Agronomy, Monograph No. 9, Madison, WI. Janse, A.R.P. and G.H. Bolt. 1960. The determination of the air-permeability of soils. Neth. J. of Agri. Sci. 8:124-131. Jin, Y. and W.A. Jury. 1996. Characterizing the dependence of gas diffusion coefficient on soil properties. Soil Sci. Soc. Am. J. 60:66-71. Kosugi, K. 1996. Lognormal distribution model for unsaturated soil hydraulic properties. Water Resour. Research 32: 2697-2703.
123 Klute A. 1986. Water retention: Laboratory methods. In A. Klute (ed). Methods of Soil Analysis. Part I. Physical and mineralogical methods. Second edition. American Society of Agronomy, Monograph No. 9, Madison, WI. Klute A. and C. Dirksen. 1986. Hydraulic conductivity and diffusivity: Laboratory methods. In A. Klute (ed). Methods of Soil Analysis. Part I. Physical and mineralogical methods. Second edition. American Society of Agronomy, Monograph No. 9, Madison, WI. Luckner, L., M.Th. van Genuchten, and D.R. Nielsen. 1989. A consistent set of parametric models for the two-phase flow of immiscible fluids in the subsurface. Water Resour. Res. 25: 2187-2193. Moldrup, P., C.W. Kruse, D.E. Rolston, and T. Yamaguchi. 1996. Modeling diffusion and reaction in soils: III. Predicting gas diffusivity from the Campbell soil-water retention model. Soil Sci. 161: 366-375. Moldrup, P., T. Olesen, D.E. Rolston, and T. Yamaguchi. 1997. Modeling diffusion and reaction in soils: VII. Predicting gas and ion diffusivity in undisturbed and sieved soils. Soil Sci. 162: 632-640. Moldrup, P., T. Olesen, T. Yamaguchi, P. Schjønning, and D.E. Rolston. 1999. Modeling diffusion and reaction in soils: VIII. Gas diffusion predicted from single-potential diffusivity of permeability measurements. Soil Sci. 164: 75-81. Moldrup, P., T. Olesen, T. Komatsu, P. Schjønning, and D.E. Rolston. 2001. Tortuosity, diffusivity, and permeability in the soil liquid and gaseous phases. Soil Sci. Soc. Am. J. 65: 613-623.
124 Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12: 513-522. Mualem, M. and S.P. Friedman. 1991. Theoretical prediction of electrical conductivity in saturated and unsaturated soil. Water Resour. Res. 23: 618-624. Or, D., B. Fisher, R.A. Hubscher, and J. Wraith. 1998. WinTDR 98 V4.0 Users Guide. Utah State University, Plants, Soils, and Biometeorology, Logan, Utah. Parker, J.C., R.J. Lenhard, and T. Kuppusamy. 1987. A parametric model for constitutive properties governing multiphase flow in porous media. Water Resour. Res. 23: 618-624. Petersen, L.W., D.E. Rolston, P. Moldrup, and T. Yamaguchi. 1994. Volatile organic vapor diffusion and adsorption in soils. J. Environ. Qual. 23:799-805. Rhoades, J.D., P.A.C. Raats, and R.J. Prather. 1976. Effects of liquid-phase electrical conductivity, water content, and surface conductivity on bulk soil electrical conductivity. Soil Sci. Soc. Am. J. 40: 651-655. Rolston, D.E. 1986. Gas diffusivity. In A. Klute (ed). Methods of Soil Analysis. Part I. Physical and mineralogical methods. Second edition. American Society of Agronomy, Monograph No. 9, Madison, WI. Stonestrom, D.A. and J. Rubin. 1989. Air permeability and trapped-air content in two soils. Water Resour. Res. 25: 1959-1969. Suman, R. and D. Ruth. 1993. Formation factor and tortuosity of homogeneous porous media. Transport in Porous Media 12: 185-206. Taylor, S.A. 1949. Oxygen diffusion in porous media as a measure of soil aeration. Soil Sci. Soc. Am. Proc. 14:55-61.
125 Topp, G.C., J.L. Davis, and A.P. Annan. 1980. Electromagnetic determination of soil water content: Measurements in coaxial transmission lines. Water Resour. Res. 16: 574-582. Tuli, A., K. Kosugi, and J.W. Hopmans. 2001a. Simultaneous scaling soil water retention and unsaturated hydraulic conductivity function assuming lognormal pore-size distribution. Adv. Water Resour. 24:677-688. Tuli, A., M.A. Denton, J.W. Hopmans, T. Harter, and J.L. MacIntyre. 2001b. Multi-step outflow experiment: From soil preparation to parameter estimation. Dept. of Land, Air, and Water Resources Paper No. 100037, University of California, Davis. Vanclooster, M., C. Gonzalez, J. Vanderborght, D. Mallants, and J. Diels. 1994. An direct calibration procedure for using TDR in solute transport studies. In Symposium and Workshop on TDR in Environmental, Infrastructure, and Mining Applications, Spec. Pub., pp. 215-226, U.S. Bur. of Mines, Northwestern Univ., Evanston, IL. van Genuchten, M. Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44: 892-898. Washington, J.W., A.W. Rose, E.J. Ciolkosz, and R.R. Dobos. 1994. Gaseous diffusion and permeability in four soil profiles in central Pennsylvania. Soil Sci. 157: 6576. Weerts, A.H., W. Bouten, and J.M. Verstraten. 1999. Simultaneous measurement of water retention and electrical conductivity in soils: Testing the Mualem-Friedman tortuosity model. Water Resour. Res. 35: 1781-1787.
126 Weerts, A.H., J.I. Freijer, and W. Bouten. 2000. Modeling the gas diffusion coefficient in analogy to electrical conductivity using a capillary model. Soil Sci. Soc. Am. J. 64: 527-532. Weerts, A.H., D. Kandhai, W. Bouten, and P.M. A. Sloot. 2001. Tortuosity of an unsaturated sandy soil estimated using gas diffusion and bulk soil electrical conductivity:
Comparing
analogy-based
models
and
Lattice-Boltzmann
simulations. Soil Sci. Soc. Am. J. 65:1577-1584. Wraith, J.M. and D. Or. Nonlinear parameter estimation using spreadsheet software. J. Nat. Resour. Life Sci. Educ. 27:13-19. Wildenschild, D., J.W. Hopmans, and J. Simunek. 2001. Flow rate dependence of soil hydraulic characteristics. Soil Sci. Am. J. 65:35-48.
1.570* 1.560 1.574 1.568 1.564 1.569 1.567 1.568 1.554 1.565 1.566 1.570 1.571 1.570 1.571 1.574 1.575 1.580 1.576 1.574 1.572
1.0 39.7 43.7 46.8 52.9 52.9 59.0 59.0 66.1 66.1 81.3 203.4 35.6 41.7 45.8 49.8 49.8 55.9 55.9 81.3 203.4
-3
0.418* 0.422 0.417 0.419 0.421 0.419 0.420 0.419 0.424 0.420 0.420 0.419 0.418 0.419 0.418 0.417 0.417 0.415 0.416 0.417 0.418
3
φ
cm cm
* Averaged value of all samples
g cm
cm
-3
ρb
Applied Pressure -3
0.406* 0.406 0.402 0.413 0.417 0.416 0.407 0.415 0.405 0.416 0.406 0.406 0.398 0.409 0.412 0.416 0.416 0.404 0.418 0.401 0.366
3
cm cm
θs
0.406 0.381 0.331 0.318 0.289 0.301 0.211 0.231 0.146 0.156 0.104 0.073 0.363 0.345 0.319 0.282 0.284 0.223 0.231 0.105 0.075
3
-3
cm cm
θ
0.012 0.041 0.086 0.101 0.132 0.118 0.209 0.188 0.278 0.264 0.316 0.345 0.055 0.074 0.099 0.135 0.133 0.192 0.185 0.311 0.343
3
-3
cm cm
a
Table 2-5. Measured data for Oso Flaco fine sand.
1.000 0.925 0.784 0.721 0.628 0.665 0.413 0.462 0.219 0.241 0.092 0.000 0.892 0.809 0.725 0.609 0.615 0.453 0.457 0.102 0.004
S ew 0.000 0.075 0.216 0.279 0.372 0.335 0.587 0.538 0.781 0.759 0.908 1.000 0.108 0.191 0.275 0.391 0.385 0.547 0.543 0.898 0.996
S ea -1
4.700E-02 3.471E-02 1.990E-02 1.526E-02 9.930E-03 1.187E-02 2.840E-03 3.947E-03 4.605E-04 6.019E-04 4.067E-05 2.215E-18 3.056E-02 2.207E-02 1.554E-02 9.044E-03 9.311E-03 3.721E-03 3.839E-03 5.344E-05 8.699E-09
cm h
Kw -1
0.000 0.019 0.170 0.173 0.174 0.145 0.621 0.434 1.482 1.344 2.081 2.687 0.046 0.163 0.182 0.328 0.363 0.756 0.735 2.316 2.605
cm h
K a x 103 -1
0.000 0.002 0.003 0.002 0.010 0.004 0.086 0.056 0.217 0.180 0.345 0.502 0.000 0.007 0.008 0.023 0.025 0.073 0.072 0.340 0.452
2
cm h
D g x 102
0.904 0.600 0.371 0.280 0.307 0.295 0.244 0.267 0.179 0.176 0.139 0.088
-1
dS m
EC b
1.000E+00 7.385E-01 4.235E-01 3.246E-01 2.113E-01 2.525E-01 6.043E-02 8.397E-02 9.798E-03 1.281E-02 8.652E-04 4.713E-17 6.503E-01 4.696E-01 3.307E-01 1.924E-01 1.981E-01 7.917E-02 8.169E-02 1.137E-03 1.851E-07
K rw
0.000 0.007 0.063 0.064 0.065 0.054 0.231 0.161 0.551 0.500 0.774 1.000 0.017 0.061 0.068 0.122 0.135 0.281 0.273 0.862 0.969
K ra
0.000 0.004 0.005 0.005 0.021 0.009 0.171 0.111 0.433 0.360 0.688 1.000 0.000 0.014 0.016 0.046 0.049 0.146 0.144 0.677 0.901
D rg
1.000 0.634 0.359 0.250 0.282 0.267 0.206 0.233 0.128 0.124 0.079 0.018
EC rb
127
2.8. Appendix 2.1.
1.415* 1.421 1.422 1.431 1.434 1.432 1.434 1.433 1.432 1.432 1.432 1.407 1.384 1.389 1.412 1.374 1.368
1.0 61.0 61.0 101.7 101.7 203.4 203.4 355.9 355.9 508.4 508.4 1016.8 1016.8 3050.3 3050.3 5083.9 5083.9
-3
0.466* 0.464 0.463 0.460 0.459 0.460 0.459 0.459 0.460 0.460 0.460 0.469 0.478 0.476 0.467 0.482 0.484
3
φ cm cm
* Averaged value of all samples
g cm
cm
-3
ρb
Applied Pressure -3
0.427* 0.438 0.436 0.444 0.434 0.432 0.429 0.421 0.413 0.393 0.405 0.432 0.432 0.432 0.432 0.432 0.432
3
cm cm
θs
0.427 0.369 0.377 0.323 0.322 0.258 0.258 0.216 0.209 0.198 0.200 0.147 0.147 0.112 0.115 0.099 0.098
3
-3
cm cm
θ
0.039 0.095 0.086 0.137 0.137 0.202 0.201 0.243 0.251 0.262 0.260 0.322 0.331 0.364 0.352 0.383 0.386
3
-3
cm cm
a
1.000 0.798 0.826 0.651 0.667 0.479 0.483 0.365 0.353 0.339 0.332 0.147 0.147 0.042 0.051 0.003 0.000
S ew
Table 2-6. Measured data for Columbia sandy loam soil.
0.000 0.202 0.174 0.349 0.333 0.521 0.517 0.635 0.647 0.661 0.668 0.853 0.853 0.958 0.949 0.997 1.000
S ea -1
7.803E-02 7.379E-03 9.360E-03 2.137E-03 2.454E-03 4.403E-04 4.561E-04 1.246E-04 1.065E-04 8.989E-05 8.129E-05 2.816E-06 2.816E-06 2.626E-08 5.289E-08 3.180E-12 3.867E-23
cm h
Kw
0.000 0.000 0.000 0.011 0.026 0.130 0.123 0.180 0.217 0.309 0.313 0.465 0.490 0.650 0.678 0.709 0.763
-1
cm h
K a x 10
3 -1
0.000 0.005 0.005 0.014 0.019 0.026 0.032 0.083 0.124 0.112 0.118 0.280 0.299 0.509 0.453 0.520 0.575
2
cm h
D g x 10
2
1.773 0.642 0.691 0.592 0.565 0.465 0.459 0.374 0.365 0.370 0.372 0.239 0.256 0.199 0.200 0.189 0.186
-1
dS m
EC b
1.000E+00 9.457E-02 1.200E-01 2.739E-02 3.145E-02 5.643E-03 5.845E-03 1.597E-03 1.365E-03 1.152E-03 1.042E-03 3.609E-05 3.609E-05 3.366E-07 6.779E-07 4.075E-11 4.955E-22
K rw
0.000 0.000 0.000 0.014 0.034 0.170 0.161 0.235 0.284 0.405 0.410 0.610 0.642 0.852 0.888 0.930 1.000
K ra
0.000 0.009 0.009 0.025 0.033 0.046 0.056 0.145 0.216 0.195 0.205 0.487 0.520 0.886 0.788 0.904 1.000
D rg
1.000 0.288 0.318 0.256 0.239 0.176 0.172 0.119 0.112 0.116 0.117 0.033 0.044 0.008 0.009 0.002 0.000
EC rb
128
129
3. Simultaneous Scaling of Soil Water Retention and Unsaturated
Hydraulic
Conductivity
Functions
Assuming Lognormal Pore-Size Distribution∗
3.1. Abstract Using simultaneous scaling, soil spatial variability of hydraulic functions can be described from a single set of scaling factors. The conventional scaling approach is based on empirical curve fitting, without paying much attention to the physical significance of the scaling factors. In this study, the concept of simultaneous scaling of the soil water retention and unsaturated hydraulic conductivity functions is applied to a physically based scaling theory. In this approach, it is assumed that soils are characterized by a lognormal pore-size distribution, which leads directly to lognormally distributed scaling factors. To test this concept, a total of 143 undisturbed soil samples were collected from two soil depths (25 and 50 cm), with each depth divided into two subsets based on the median soil capillary pressure head value, as determined from the lognormal pore-size distribution assumption. Moreover, the theory was compared with the conventional simultaneous scaling method. Both the conventional and physically based simultaneous scaling method performed equally well for all four subsets, as determined from the reduction in weighted root mean squared residual (WRMSR) values after scaling. We
∗
Short version of this chapter has been published in Advances in Water Resources, 24:677-688 (2001).
130 showed that the theoretical interpretation of the lognormal scaling factor distribution was applicable to simultaneous scaling of soil hydraulic functions.
3.2. Introduction Most of the uncertainty in the assessment of water flow in unsaturated soils at the field scale can be attributed to soil spatial variability as caused by soil heterogeneity. The knowledge of the constitutive relationships for the unsaturated hydraulic conductivity, water saturation, and soil water matric potential are essentially required to study water flow as described by the traditional Richards equation. The exact nature of the functional dependence of these flow variables with water content differs among soil types with different particle size compositions and pore size geometry within a heterogeneous field soil. This medium-specific character of soil hydraulic functions can be described by soil hydraulic parameters, so that spatially variable soil hydraulic functions are described by spatially variable soil hydraulic parameters. The scaling approach has been extensively used to characterize soil hydraulic spatial variability and to develop a standard methodology to assess the variability of soil hydraulic functions and their parameters (Warrick, 1990; Nielsen et al., 1998; Sposito, 1998; Warrick, 1998). The single objective of scaling is to coalesce a set of functional relationships into a single curve using scaling factors that describes the set as a whole. The concept of this approach has been developed principally from the theory of microscopic geometric similitude as proposed by Miller and Miller (1956). The procedure consists of using scaling factors to relate the hydraulic properties in a given location to the mean properties at an arbitrary reference point.
131 Various studies (Ahuja et al., 1984; Jury et al., 1987; Snyder, 1996; Granovsky and McCoy, 1997) emphasized that scaling of different soil properties for the same field may result in different statistical properties for each of the computed scaling factor distributions, as for the independent scaling of soil water retention and hydraulic conductivity curves. For example, it has been shown that scale factors for soil water retention and unsaturated hydraulic conductivity functions are not necessarily identical (Warrick et al., 1977; Russo and Bresler, 1980). However, at the same time it was emphasized that the ability to characterize all soil hydraulic functions by a single set of scaling factors is highly desirable when simulating and predicting water flow in spatially variable field soils. Moreover, while adopting the similar media concept, equality of scaling factors of soil water retention and unsaturated hydraulic conductivity functions is expected based on the validity of the capillary and Poiseuille equations. Early scaling studies (Warrick et al., 1977; Simmons et al., 1979; Russo and Bresler, 1980) used an empirical scaling approach, defined as functional normalization, where regression analysis was applied to either soil water retention or unsaturated hydraulic conductivity data (Tillotson and Nielsen, 1984). Alternatively, Clausnitzer et al. (1992) introduced a scaling method whereby both hydraulic functions were scaled simultaneously using the conventional fitting approach, yielding a single set of scale factors without consideration of the physical significance of the resulting set of scaling factors. Meanwhile, it was shown that scaling factors are log normally distributed, as has also been demonstrated by many others (Warrick et al., 1977; Jury et al., 1987; Hopmans et al., 1988; Zavattaro et al., 1999).
132 Difficulties in applying classical Miller similitude scaling to soils with a broad range of particle size distributions and nonuniform porosity has resulted in the development of generalized scaling analysis (Sposito and Jury, 1990; Snyder, 1996), wherein pore size distribution is considered the invariant soil quantity. Hence, using this approach, the characteristic length is associated with pore space (pore size) rather than the geometric arrangement of both pore space and solid matrix, leading to degree of water saturation, S = θ θ s , as an additional scaling factor (Snyder, 1996) for field studies. This approach leads to correlated scaling factor values between those computed from retention and unsaturated hydraulic conductivity functions, which are identical only if soils have identical porosities (geometric similitude scaling). Recently, the pore radius, r , was used as the microscopic characteristic length to scale soil water retention curves for soils that are characterized by a lognormal pore-size distribution (Kosugi and Hopmans, 1998). In their study, the physically based scale factors were computed directly from the physically based parameters describing the individual soil water retention functions (Kosugi, 1996). In a subsequent study (Hendrayanto et al., 2000), the physically based scaling theory was applied to both soil water retention and unsaturated hydraulic conductivity data for a forested hill slope soil. The objective of this study was to extend the physically based scaling approach (Kosugi and Hopmans, 1998) to the simultaneous scaling of soil water retention and unsaturated hydraulic conductivity functions, yielding a single consistent set of scaling factors. In this analysis, the physically based simultaneous scaling factors are compared with those computed by the conventional scaling method.
133
3.3. Theory Assuming a lognormally distributed soil pore radius ( r ) distribution, its probability distribution function (PDF) is defined by
f (ln r ) = dS ew / d ln r , where
S ew = (θ − θ r ) (θ s − θ r ) denotes the effective degree of water saturation, and f (ln r ) is
expressed by the normal distribution N (ln rm , σ 2 ) : 1 f (ln r ) = σ 2π
( ln r − ln rm ) 2 exp − 2σ 2
[3-1]
where ln rm and σ 2 are the mean and variance of log-transformed soil pore radius, ln r , respectively. Hence, rm is the geometric mean or median pore radius for which S e = 0.5 , assuming a lognormal pore size distribution, and σ describes the width of the soil pore radius distribution. The concept of lognormally distributed pore size (Kosugi, 1996) is not new, and was introduced earlier (Gardner, 1956; Hopmans et al., 1988). Pore radius, r, is related to the soil capillary pressure head, h , by the capillary pressure function ( h > 0 for unsaturated soils) h=
2 γ cos β A = ρw g r r
or ln h = ln A − ln r
[3-2]
where γ is the interfacial tension (F L-1) , β is contact angle, ρ w is the density of water (M L-3), g is the acceleration of gravity (L T-2), to yield an approximate value of A = 0.15 cm2. This h notation is used throughout, instead of h denoting the matric
potential head (h < 0 for unsaturated soils). Using Eq. [3-2], the soil water retention function is derived from Eq [3-1], or
134 S ew ( ln h ) =
( ln hm − ln h ) θ −θ r = Fn θ s −θ r σ
[3-3]
where Fn(x) is the normal distribution function defined as 1 Fn ( x) = 2π
∫
x
x2 exp − dx 2 −∞
[3-4]
where ln hm and σ are the mean and standard deviation of ln h , respectively. The capillary pressure head hm is related to the median pore radius rm by Eq. [3-2]. An alternative expression for the retention function given in Eq. [3-3] can be written as S ew (ln h) =
ln h − ln hm 1 erfc 2 σ 2
[3-5]
where erfc denotes the complementary error function. An extensive derivation of the foregoing physically based soil water retention model was presented by Kosugi (1996).
3.3.1. Scaling of water retention curves Based on the similar media concept, the scaling factor, α i , is defined as ratio of a characteristic length, λi , of soil sample i to the characteristic length, λˆ , of a reference soil:
αi ≡
λi λˆ
[3-6]
Assuming that pore-size distribution is an invariant quantity, Kosugi and Hopmans (1998) defined the pore radius r as the microscopic characteristic length, so that Eq. [3-6] becomes
αi =
ri or ln α i = ln ri − ln rˆ rˆ
[3-7]
135 where ri and rˆ are the largest water-filled pore-radii for soil sample i, and the reference soil at equal effective water saturation, respectively. Since rm,i ( S ew = 0.5) is assumed as a suitable representative pore radius to characterize individual soil water retention curves, its value is selected as the macroscopic characteristic length scale (Kosugi and Hopmans 1998), so that Eq. [3-7] becomes rm ,i or ln α i = ln rm ,i − ln rˆm rˆm
αi =
[3-8]
Subject to the constraint that the geometric mean of the set of scaling factors is I
unity (i.e.,
∏α i =1
1/ I i
1 = 1.0 or I
∑ I
ln α i = 0 ), it was shown by Kosugi and Hopmans
i =1
(1998) that ln rˆm is equal to the arithmetic mean of all ln rm ,i -values in the set. Subsequently, using the capillary equation [3-2], ln hˆm and σˆ 2 are computed from 1 ln hˆm = Mean(ln hm,i ) = I
1 σˆ = Mean(σ ) = I 2
2 i
∑ I
ln hm,i
[3-9]
i =1
∑σ I
2 i
[3-10]
i =1
where I denotes the number of soil samples in the set and the individual ln hm ,i and σ i2 values are determined from the fitting of Eq. [3-5] to individual soil water retention data to yield the soil water retention curve for soil sample i, S ew,i , or S ew,i (ln h) =
ln h − ln hm ,i 1 erfc 2 σi 2
Henceforth, the reference soil water retention curve was described by
[3-11]
136
ln hˆm − ln h 1 ln h − ln hˆm = Sˆew (ln h) = Fn erfc σˆ 2 σˆ 2
[3-12]
where ln hˆm ( = ln A − ln rˆm ) represents the mean ln h value for the reference soil. Accordingly, scaling factors for each soil sample, i, can be computed directly from
αi =
hˆm hm ,i
[3-13]
or ln α i = ln hˆm − ln hm ,i . In Kosugi and Hopmans (1998), it was shown that the scaling factors determined from Eq.[3-13], while assuming a lognormal pore size distribution model, are lognormally distributed.
3.3.2. Scaling of unsaturated hydraulic conductivity function Based on the Miller and Miller scaling theory (Miller and Miller, 1956), scaling factors as determined from K w,i ( S ew,i ) = α i2 Kˆ w ( S ew,i ) or ln K w,i = 2 ln α i + ln Kˆ w
[3-14]
are identical as those obtained from soil water retention scaling as defined in Eq. [3-13] using rm ,i as the macroscopic characteristic length scale for both hydraulic functions. Mualem (1976) proposed the following predictive relative hydraulic conductivity model, which can be solved for K rw,i , provided that the soil water retention function, S ew,i (h) , is known
137
K rw,i =
K w,i ( S ew,i ) K sw,i
l = S ew ,i
S ew ,i
∫ ∫ 0
1
0
dS ew,i h dS ew,i h
2
[3-15]
In other words, the relative hydraulic conductivity of soil sample i, K rw,i , is computed from K w,i ( S ew,i ) and the saturated hydraulic conductivity, K sw,i (L T-1). In the predictive unsaturated hydraulic conductivity model, l describes the degree of connectivity between the water-conducting pores. In the subsequent analysis it is assumed that equal to l = 0.5 (Mualem, 1976) although other values can be substituted if so warranted. Combining Eqs. [3-11] and [3-15] yields the functional relationship between K rw,i and S ew,i (See Appendix 3.1.) K rw,i ( S ew,i ) = S
0.5 ew ,i
1 σ i −1 erfc erfc ( 2 S ew,i ) + 2 2
2
[3-16]
where erfc −1 denotes the inverse complementary error function. In the predictive unsaturated hydraulic conductivity model, l describes the degree of connectivity between the water-conducting pores. Similarly, substitution of Eq. [3-12] into the Mualem model yields the hydraulic conductivity function for the reference soil
( )
Kˆ rw Sˆew =
( ) = Sˆ
Kˆ w Sˆew Kˆ sw
0.5 ew
1 σˆ −1 erfc erfc (2 Sˆew ) + 2 2
2
[3-17]
where Kˆ w ( Sˆew ) and Kˆ sw denote unsaturated hydraulic conductivity (L T-1) and saturated hydraulic conductivity (L T-1) of the reference soil, respectively, and Sˆew represents the soil water retention curve of the reference soil (Eq.[3-12]). Writing Eq. [3-14] for the
138 saturated hydraulic conductivity and substitution of the constraint that the geometric 1 mean of the set of scaling factors is equal to unity ( I
∑ I
ln α i = 0 ) leads to the
i =1
convenient result that the saturated hydraulic conductivity of the reference soil (Kˆ sw ) is equal to its geometric mean, or 1 ln Kˆ sw = Mean ( ln K sw,i ) = I
∑ I
ln K sw,i
[3-18]
i =1
Moreover, Eq. [3-14] written for the saturated hydraulic conductivity predicts that K sw is lognormally distributed since the log-transformed scaling factors, ln α i are normally distributed.
3.4. Material and Methods 3.4.1. Experimental To test the proposed simultaneous scaling theory, undisturbed soil samples were collected from seventy two 64 x 64 m plots at two depths (25 and 50 cm) in a 40 ha field. This Long Term Research on Agricultural Systems (LTRAS) project is conducted at the Russell Ranch of the University of California (Chen et al., 1995; Tuli et al., 1999) to study the long-term effects of irrigation and nitrate application to the sustainability of California agriculture. The field includes three different soil series: the Yolo (fine-silty, mixed, non-acid, thermic Typic Xerorothents), the Rincon (fine montmorillonitic, thermic Mollic Haploxeralfs), and the Brentwood (fine, montmorillonitic, thermic Typic Xerocrepts). Within each 64 x 64 m plot, 8.25-cm inner diameter and 6-cm long soil
139 cores were collected using a soil core sampler. The range of values of the main soil physical properties as obtained from these soil cores are (Tuli et al., 1999): bulk density, 1.22-1.66 g cm-3; organic matter, 0.43-1.63 %; saturated hydraulic conductivity, 0.000217.79 cm h-1; saturated water content, 0.32-0.50 cm3 cm-3; sand (63-2000 µm), 11-56 %; silt (2-63 µm), 34-80 %; and clay (< 2 µm), 3-22 %. For each core sample, soil hydraulic functions were measured using the multi-step outflow method in combination with parameter optimization (Eching et al., 1994; Hopmans et al., 2002). The multi-step outflow method and parameter optimization code were slightly modified to improve the overall experimental outcome. Instead of a ceramic porous plate, we used a nylon membrane to support the core sample in the Tempe pressure cell. The low hydraulic resistance of the thin nylon membrane provides high flow rate and bubbling pressure (700 cm), thereby making it more suitable for parameter optimization. Pressure differences across the low resistance membrane are small so that there is no need to specify a hydraulic conductivity of the membrane in the combined flow and optimization code. Consequently, flow is not controlled by the membrane but solely by the soil, thereby improving the parameter optimization procedure. In addition, the optimization code provides for a time-dependent lower head boundary condition, to accommodate for the time-dependent water level in the burette, as measured with pressure transducers (Tuli et al., 2001). We assumed that the soil hydraulic properties are described by the lognormal pore size distribution model, leading to the soil water retention and unsaturated hydraulic conductivity models, described by Eqs. [3-11] and [3-16], respectively. Whereas the saturated water content was fixed to its measured value, the residual water content (θ r ) for
140 each sample i, hm,i , σ i , and K sw,i were considered fitting parameters, to be optimized by the SFOPT software (Tuli et al, 2001). After parameter optimization of the soil hydraulic functions for each soil core, the data was divided into two subsets for each soil depth based on the magnitude of their optimized hm,i-values, thereby grouping the soils into ln hm,i - values smaller or larger than 6.0. This grouping was conducted to satisfy the lognormality assumption of K sw and hm ,i within each subset, as tested using fractile diagram analysis. Rather than the heuristic soil grouping based on the hm -magnitude, we initially attempted to classify soils based on soil texture and related measured soil physical properties (bulk density, measured and optimized saturated hydraulic conductivity), but this analysis failed likely because of other non-measured soil attributes that control flow.
3.4.2. Scaling method In the physically based (PB) scaling method, the functional parameters, ln hˆm ,
σˆ 2 , and Kˆ sw for the reference soil water retention (Eq.[3-12]) and unsaturated hydraulic conductivity (Eq. [3-17]) for each subset (at 25 and 50 cm) were computed directly from Eqs.[3-9], [3-10], and [3-18] using the optimized soil core-specific parameters of the individual soil retention and unsaturated hydraulic conductivity functions. Scaling factors were directly calculated from the retention curve parameter, hm ,i , using Eq.[3-9]. Subsequently, their values were applied to the K w -scaling as well, using Eq. [3-14]. In the conventional scaling approach (C), the parameters, ln hˆm , σˆ 2 , and Kˆ sw for the reference soil water retention (Eq. [3-12]) and unsaturated hydraulic conductivity
141 functions (Eq. [3-17]) and scaling factor values were simultaneously estimated by a least squares fitting procedure provided in the Excel software (Wraith and Or, 1998), minimizing the residual sum of squared differences between the scaled mean curves and the scaled hydraulic data (Hills et al., 1989; Claustnitzer et al., 1992; Kosugi and Hopmans, 1998). Defining appropriate weighting factors, the weighted root mean squared residuals (Weighted RMSR) were minimized from 1 Weigthed RMSR = σ Sew
+
1
σ ln K
w
∑
1
∑ I
i =1
J (i )
I
J (i )
i =1
i =1
i =1
J (i )
j =1
1
2
)
2
J (i )
∑ ∑( I
∑ ∑( I
1
2 S ewj ,i − Sˆew (α i h j )
j =1
ln K wj ,i ( S ewj ,i ) − ln Kˆ w ( S ewj ,i ) − 2 ln α i
)
1
[3-19] 2
where S ewj ,i is the jth effective water saturation of soil sample i with a jth capillary pressure head of h j , Sˆew (α i h j ) is the effective water saturation of reference soil obtained by substituting h = α i h j into Eq. [3-12], ln K wj ,i is the natural logarithm of jth unsaturated hydraulic conductivity of soil sample i, with a jth effective water saturation of S ewj ,i , and ln Kˆ w ( S ewj ,i ) is the natural logarithm of unsaturated hydraulic conductivity of
the
reference soil at S ewj ,i . In this study, we used 11 arbitrarily selected j-classes within the applied external pressure range in the multi-step outflow experiment of 20, 40, 60, 80, 100, 200, 300, 400, 500, 600, and 700 cm. S ew - values selected for the scaling of
142 individual soil water retention and unsaturated hydraulic conductivity data (both PB and C method) correspond with the capillary pressure head values, determined by these applied air pressure increments. To allow for equal weighting between the retention and conductivity data normalizing parameters defined by the inverse of the standard deviations, σ Sew and σ ln K w , as computed from all data within each subset were selected (Clausnitzer and Hopmans, 1995). In the fitting procedure, the geometric mean of scaling factors was constrained to 1 unity, or, Mean(ln α i ) = I
∑ I
ln α i = 0 . This was done to be consistent with the
i =1
physically-based scaling approach.
3.5. Results and Discussion 3.5.1. Optimized individual soil hydraulic functions The average of all fitting parameter values for both subsets and sampling depths are listed in Table 3-1. ( hˆm , σˆ , and Kˆ sw ). These parameters combined define the reference hydraulic functions from which the individual hydraulic functions are scaled by the physically based (PB) scaling method using Eqs. [3-13] and [3-14]. The data clearly show that the two subsets for each soil depth are distinctively different regarding their
σˆ 2 -values, indicating that this grouping resulted in classification of soils with distinct pore size widths. That is, according to the selection of the pore size distribution model for the soil water retention function, σ denotes the standard deviation of the pore size
143 distribution, with larger σ - values characterizing soils with a wider range in waterconducting pore radii (finer-textured soils). An even better illustration of the differences
Table 3-1. Parameters for reference soil hydraulic function curves (PB-physically based scaling; C-conventional scaling). Depth (cm) 25
50
Number of Samples
σˆ 2
ln hˆm
Subset
ln Kˆ s
PB
C
PB
C
PB
C
56
lnh m,i < 6.0
5.02
5.04
2.05
1.51
-2.27
-2.88
16
lnh m,i . 6.0
7.49
7.19
7.00
5.59
-2.54
-3.55
49
lnh m,i < 6.0
5.17
5.19
2.06
1.35
-1.92
-2.71
22
lnh m,i . 6.0
7.51
7.11
7.48
4.68
-2.17
-4.36
between the two subsets is presented in Figure 3-1. Value of, where the apparent coarsertextured soils are identified by ln hm,i - values smaller than 6.0. We were, however, unsuccessful in using soil textural analysis to identify these two soil groupings a priori. Hence, other soil characteristics rather than soil texture alone, such as macroporosity and pore connectivity, may be needed to establish classes of hydraulically-active soils. It may explain why many studies to relate soil water retention and/or unsaturated hydraulic conductivity with simple soil physical properties using pedotransfer function and regression analysis, have been partly unsuccessful (Vereecken, 1989; Williams et al., 1989). In addition to the assumptions proposed by Kosugi and Hopmans (1998) that lead to lognormally distributed scaling factor values, we have shown in Eq. [3-18] that the lognormal pore size distribution model combined with the similar media scaling theory predicts that also the saturated hydraulic conductivity is lognormally distributed.
144 50 cm
18
18
15
15
12
12
σi2
σi2
25 cm
9
9
6
6
3
3
0
0 0
3
6
9
ln hm,i (hm in cm)
12
ln hm,i < 6.0
0
3
6
9
ln hm,i (hm in cm)
12
ln hm,i ≥ 6.0
Figure 3-1. Value of σ i2 , versus ln hm ,i for both sampling depths. Lognormality has been demonstrated by many experimental studies (Nielsen et al., 1973; Kutilek and Nielsen, 1994) and was also suggested by Kosugi and Hopmans (1998) for lognormally distributed scaling factors. Lognormality of the optimized K sw,i was analyzed by (1) construction of fractile diagrams (Figure 3-2.) and (2) applying the Kolmogorov-Smirnov test. This statistical test confirmed that the Ho-hypothesis of ln K sw,i being normally distributed (P ≤ 0.2) could not be rejected for each of the two subsets of each sampling depth. The standard normal deviate presented in the fractile diagrams is a linear transformation of all conductivity data so that they are normalized with a zero mean and unit variance, using normal probability tables. In the case of testing for lognormality, the saturated hydraulic conductivity data must fit along a straight line through the origin. The approximate linear fit to the data indicates that the selected hm - criterion was adequate to yield subsets of lognormally distributed K sw -values.
145
50 cm
4
4
3
3
Standard normal deviate
Standard normal deviate
25 cm
2 1 0 -1 -2 -3 -4
2 1 0 -1 -2 -3 -4
-9
-6
-3
0 3 ln Ksw,i
6
9
ln hm,i < 6.0
-9
-6
-3
0 3 ln Ksw,i
6
9
ln hm,i . 6.0
Figure 3-2. Fractile diagram of the saturated hydraulic conductivity (ln K sw,i ) for both soil depths. Whereas in the physically based (PB) approach, the mean value of ln K sw,i (ln Kˆ sw ) for the reference soil was directly computed from optimized K sw -values of each soil sample i using Eq. [3-18], the ln Kˆ sw in the C-method was optimized by minimization of the residuals in Eq. [3-19]. Since the saturated hydraulic conductivity is among the fundamental soil physical parameters controlling water flow in soil, K sw is expected to be correlated with the soil pore radius distribution parameters hm ,i and σ i2 . For example, coarse textured soils with large pores and narrow pore size distribution will generally have larger conductivity values than the fine textured soils with wider pore size distribution, at the same degree of saturation.
146 Indeed, when comparing ln Kˆ sw - values in Table 3-1, we notice a lower saturated conductivity for the subsets with ln hm ,i ≥ 6.0 (both soil depths), characterizing the finertextured soils with larger values for the median ( hˆm )of the reference soil. However, for the same hm – value, Eq. [A3.1-6] predicts that soils with larger σ values (i.e., a wider pore size distribution) have larger K sw , which contradicts our finding that apparent coarser-textured soils can be identified by smaller σ values. As indicated earlier, the soil saturated hydraulic conductivity depends not only on pore size distribution, but also on the effective porosity and soil pore tortuosity and connectivity (the parameter l in Eq.[3-15]). Moreover, K sw is greatly affected by the presence of macro-pores.
3.5.2. Simultaneous scaling of soil hydraulic functions The scaling results for the two subsets of the 25-cm soil depth based on physically based (PB) scaling are presented in Figure 3-3 and Figure 3-4, respectively. In both figures, both the soil water retention and unsaturated hydraulic conductivity data are scaled simultaneously using scaling factors obtained from the median hm,i - values only (Eq. [3-13]). Solid lines represent the mean hydraulic functions for the reference soil. As would be expected, the scaled soil water retention data for each soil sample coincide with the reference soil water retention curve at S ew = 0.5 , corresponding with the median capillary pressure head values of 151.41 cm ( ln hm ,i < 6.0 ) and 1790.05 cm ( ln hm ,i ≥ 6.0 ), respectively (Table 3-1). Scaling results for the other 3 subsets were similar to those shown in Figure 3-3 and Figure 3-4, and are, therefore, presented in Appendix 3.2.
147
Unscaled
Scaled 10000
WRMSR = 0.437 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
1000
100
10
1
WRMSR = 0.209
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 0.672 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.365 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
Sew
Figure 3-3. Simultaneous PB scaling of subset ln hm,i < 6.0 at 25 cm.
148
Unscaled
Scaled 10000
WRMSR = 0.676 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
1000
100
10
1
WRMSR = 0.306
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 1.464 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew
Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.929 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
Sew
Figure 3-4. Simultaneous PB scaling of subset ln hm,i ≥ 6.0 at 25 cm. The effectiveness of the simultaneous scaling was determined by comparison of the weighted root-mean squared residuals (WRMSR) of the unscaled and scaled soil water retention and unsaturated hydraulic conductivity data separately as well as combined. Values of WRSMR for both scaling methods and all 4 subsets are listed in
149 Table 3-2. When comparing the residual values, it is noted that all subsets showed similar performances with respect to scaling results and total reductions in WRMSR. In
Table 3-2. WRMSR and total reduction in physically-based (PB) and conventional (C) scaling methods. ethods. Soil water retention Physically-Based (PB)
Hydraulic conductivity
Conventional (C)
Physically-Based (PB)
Total Reduction
Conventional (C)
%
Depth (cm)
Subset
Unscaled
Scaled
Unscaled
Scaled
Unscaled
Scaled
Unscaled
Scaled
PB
C
25
lnh m,i < 6.0
0.437
0.209
0.446
0.235
0.672
0.365
0.658
0.265
48.214
54.704
lnh m,i . 6.0
0.676
0.306
0.664
0.301
1.464
0.929
0.826
0.356
42.302
55.880
lnh m,i < 6.0
0.392
0.235
0.396
0.265
0.548
0.474
0.500
0.353
24.569
31.076
lnh m,i . 6.0
0.592
0.414
0.585
0.299
1.089
0.837
0.566
0.395
25.631
39.692
50
general, however, reductions in WRMSR were not so large as expected which suggests that the variances of the individual samples, σ i2 were quite different. As was illustrated in Fig. 2 of Kosugi and Homans (1998), the presented scaling theory is based on the premise that the variance of the pore size distribution of all considered soil samples are identical and equal to that of the scaled mean reference curve. Differences in variances between soil samples is apparent also from inspection of Figure 3-1, demonstrating that the range of variance values is at least as large or larger than the range of hm -values. The smaller reduction in WRMSR for the 50 cm depth is caused by the lower measured variability of the soil hydraulic functions for that depth as compared to the spatial variability of the 25 cm depth. The slightly less effectiveness of the PB-approach is likely the result of scaling factors determined by the scaling of h at a single fixed S ew -value of 0.5 only, whereas the C-scaling method computes scaling factors by selecting the optimum S ew for each sample across the whole S ew -range, thereby being more flexible.
150 The differences in the mean hydraulic functions for the two subsets at the 25 cm soil depth are presented in Figure 3-5. The shapes of the presented reference curves illustrate what was already inferred from comparison of the ln hˆm and σˆ 2 values of the two sets in Table 3-1. The dashed lines represent the hydraulic functions for the sampled soils characterized by ln hm ,i ≥ 6.0 (finer-textured soil samples with large variation in pore sizes), whereas the solid lines describe the references curves for the coarser-textured soil subset with ln hm ,i < 6.0 and smaller pore size variance. The comparison in Figure 3-5 demonstrates that proper identification of different soil groupings is needed to increase the effectiveness of the similar media based scaling approach (See also Appendix 3.2. and Appendix 3.3.). 25 cm Unsaturated hydraulic conductivity (cm h-1)
25 cm
Capillary pressure head (cm)
100000 10000 1000 100 10 1 0
0.2
0.4
0.6
0.8
1
Sew Mean curve for ln hm,i < 6.0
1.0E+002 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
0.8
1
Sew Mean curve for ln hm,i ≥ 6.0
Figure 3-5. Soil hydraulic function curves of subsets for reference soils at 25-cm depth.
151 Finally, the resulting scaling factor data sets were statistically analyzed for their distribution characteristics as well. These results are presented in Table 3-3, whereas the fractile diagrams of scaling factors for all four subsets and both scaling methods are shown in Figure 3-6. Scaling factor distribution has consistently been found to be approximately lognormal (Warrick and Nielsen, 1980; Clausnitzer et al., 1992; Kosugi and Hopmans, 1998). Values for the slope and intercept of the fitted linear relationships in Figure 3-6 characterize the statistical properties (mean and variance) of the lognormal scaling factor distributions. Hence, difference in slope distinguishes between scaling
Table 3-3. Statistical properties of scaling factors ( Mean(ln α i ) = 0 ). Depth
Number of
(cm)
Samples
25
50
Subset
Mean (α i )
Var (α i )
CV, %
Var (ln h m,i )
PB
C
PB
C
PB
C
56
ln h m,i < 6.0
1.16
1.42
0.48
0.60
59.35
64.86
16
ln h m,i . 6.0
3.06
1.37
78.54
1.65
289.42
49
ln h m,i < 6.0
1.12
1.15
0.32
0.44
22
ln h m,i . 6.0
2.01
1.22
12.34
0.74
Var (lnα i ) PB
C
0.30
0.30
0.35
93.81
2.24
2.24
0.63
50.53
57.55
0.23
0.23
0.29
174.57
70.21
1.40
1.40
0.40
factor variability. Since the geometric mean of scaling factors is necessarily zero, linear regression was conducted with the intercept constrained to zero. From comparison of the fitted variance values in Table 3-3, it can be easily determined that the variance of the subset with ln hm ,i values larger than 6.0 (open squares in Figure 3-6) is much greater than for the other subset. This is the case for both soil depths, but less so for the conventional scaling method. Specifically, for the subset ln hm ,i ≥ 6.0 of the PB method, the variance values between samples of the 25 and 50 cm soil depth are 2.24 and 1.40, respectively, and are much larger than any of the other reported values in Table 2 of Hendrayanto et al.
152 Physically based at 50 cm 4
3
3
Standard normal deviate
Standard normal deviate
Physically based at 25 cm 4
2 1 0 -1 -2 -3
2 1 0 -1 -2 -3
-4
-4 -4
-2
0
ln αi
2
4
-4
4
3
3
2 1 0 -1 -2
0 -1 -2
-4
-4 0
ln αi
2
ln hm,i < 6.0
4
1
-3 -2
2
2
-3 -4
0
ln αi
Conventional at 50 cm
4
Standard normal deviate
Standard normal deviate
Conventional at 25 cm
-2
4
-4
-2
0
ln αi
2
4
ln hm,i ≥ 6.0
Figure 3-6. Fractile diagrams of scaling factors distribution obtained by physically-based and conventional method for all subsets of 25- and 50-cm soil depths.
153 (2000). These high variance values, as compared with the dataset of ln hm ,i smaller than 6.0 (solid dots), is caused by the larger variability of the measured soil hydraulic functions, as demonstrated by the larger Var (ln hm ,i ) values in Table 3-3. Since the scaling factors are directly computed from the median capillary pressure head, ln hm ,i , the variance of ln α i obtained by the PB method must be equal to the variance of ln hm ,i (Kosugi and Hopmans, 1998). It is also of interest to note that the large difference between sample variability of the datasets with ln hm ,i ≥ 6.0 corresponds with the largest
σˆ 2 -values (with values of the within sample pore size variability in the range of 7.0-7.5) of the same datasets (Table 3-1). In addition to the results of the fractile diagrams, the lognormal behavior of scaling factors was also confirmed by the Kolmogorov-Smirnov test statistic (P ≤ 0.20). Moreover, the large values for the coefficient of variation (CV) of scale factors in Table 3-3, are another indication that scaling factors are lognormal distributed (Parkin and Robinson, 1992). The CV was calculated for the untransformed scale factors based on 2
lognormal scaling factor distribution, yielding that CV = (eσ − 1) *100% (Warrick et al., 1977; Jury et al., 1987).
3.6. Summary and Conclusions The theory of the physically based scaling method is based on the assumption that pore size distribution as determined from the soil water retention curves is lognormally distributed. If in addition it is assumed that pores are geometrically similar, scaling factors computed from the median pore size or capillary pressure head
154 can be directly applied to express variability of unsaturated hydraulic conductivity functions. Using an experimental data set of soil hydraulic functions, it is demonstrated that scaling factors as determined from individual soil retention curves can be directly used to scale unsaturated hydraulic conductivity data. It is shown that the physically based scaling theory predicts scaling factor and saturated hydraulic conductivity values that are log normally distributed as well. To test the method, a total of 143 undisturbed soil samples at two soil depths (25 and 50 cm) were collected and soil water retention and unsaturated hydraulic conductivity curves were determined from parameter optimization using the multi-step outflow method. Beforehand, the hydraulic data of each soil depth was divided into two subsets to satisfy the lognormality assumptions. The physically based scaling method was compared with a conventional simultaneous scaling method where scaling factors were obtained by minimization of weighted residuals. With respect to the total reduction in WRMSR, both methods performed quite similarly, however, the PB method was slightly less effective, however, much more simple and direct. We conclude that the physically based simultaneous scaling approach can be successfully used to express spatial variability of soil hydraulic functions.
155
3.7. References Ahuja, L.R., J.W. Naney, and D.R. Nielsen. 1984. Scaling soil water properties and infiltration modeling. Soil Sci. Soc. Am. J. 48:970-973. Brutsaert, W. 1966. Probability laws for pore-size distributions. Soil Sci. 101:85-92. Chen, J., J.W. Hopmans, and G.E. Fogg. 1995. Sampling design for soil moisture measurements in small field trials. Soil Sci. 159:155-161. Clausnitzer, V., J.W. Hopmans, and D.R. Nielsen. 1992. Simultaneous scaling of soil water retention and hydraulic conductivity curves. Water Resour. Res. 28:19-31. Clausnitzer, V. and J.W. Hopmans. 1995. Non-linear Parameter Estimation: LM_OPT. General-Purpose optimization code based on the Levenberg-Marquardt algorithm. Land, Air and Water Resources Paper No. 100032, University of California, Davis. Eching, S.O., J.W. Hopmans, and O. Wendroth. 1994. Unsaturated hydraulic conductivity from transient multi-step outflow and soil water pressure data. Soil Sci. Soc. Am. J. 58:687-695. Gardner, W.R. 1956. Representation of soil aggregate-size distribution by a logarithmicnormal distribution. Soil Sci. Soc. Am. Proc. 20:151-153. Granovsky, A.V. and E.L. McCoy. 1997. Airflow measurements to describe field variation in porosity and permeability of soil macrospores. Soil Sci. Soc. Am. J. 61:1569-1576. Hendrayanto, K. Kosugi, and T. Mizuyama. 2000. Scaling hydraulic properties of forest soils. Hydrol. Process. 14:521-538. Hills, R.G., D.B. Hudson, and P.J. Wiring. 1989. Spatial variability at the Las Cruces Trench Site In: van Genuchten, M.Th., F.J., Leij, and L.J. Lund, editors. Proceedings
156 of the international workshop on indirect methods for estimating the hydraulic properties of unsaturated soils. Riverside, California, October 11-13. Hopmans, J.W., J. Simunek, and N. Romano. 2002 Simultaneous determination of water transmission and retention properties - Inverse modeling of transient water flow. In: Topp, G.C. and J.H. Dane, editors. Methods of Soil Analysis. Part I. Third Edition American Society of Agronomy, Monograph No. 9, Madison, WI. Hopmans, J.W., H. Schukking, and P.J.F. Torfs. 1988. Two dimensional steady state unsaturated water flow in heterogeneous soils with autocorrelated soil hydraulic properties. Water Resour. Res. 24:2005-2017. Kosugi, K. 1996. Lognormal distribution model for unsaturated soil hydraulic properties. Water Resour. Res. 32:2697-2703. Kosugi, K. and J.W. Hopmans. 1998. Scaling water retention curves for soils with lognormal pore-size distribution. Soil Sci. Soc. Am. J. 62:1496-1505. Kutilek, M. and D.R. Nielsen. 1994. Soil Hydrology. Cremlingen-Destedt, Germany: Catena Verlag. Jury, W.A., D. Russo, and G. Sposito. 1987. The spatial variability of water and solute properties in unsaturated soil. II. Scaling models of water transport. Hilgardia, 55:33-56. Miller, E.E. and R.D. Miller. 1956. Physical theory of capillary flow phenomena. J. Appl. Phys. 27:324-332. Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513-522.
157 Nielsen, D.R., J.W. Biggar, and K.T. Erh. 1973. Spatial variability of field-measured soilwater properties. Hilgardia, 42: 215-260. Nielsen, D.R., J.W. Hopmans, and K. Reichardt. 1998. An emerging technology for scaling field soil-water behavior. In: Sposito, G., editor. Scale dependence and scale invariance in hydrology. Cambridge, UK: Cambridge University Press, p.136-166. Parkin, T.B. and J.A. Robinson. 1992. Analysis of lognormal data. In: Stewart BA, editor. Advances in soil science, Vol. 20. New York: Springer-Verlag, p. 194-235. Russo, D. and E. Bresler. 1980. Scaling soil hydraulic properties of a heterogeneous field. Soil Sci. Soc. Am. J. 44:681-684. Simmons, C.S., D.R. Nielsen, and J.W. Biggar. 1979. Scaling of field-measured soilwater properties. I. Methodology. Hilgardia, 47:77-102. Snyder, V.A. 1996. Statistical hydraulic conductivity models and scaling of capillary phenomena in porous media. Soil Sci. Soc. Am. J. 60:771-774. Sposito, G. 1998. Scaling invariance and the Richards equation. In: Sposito, G., editor. Scale dependence and scale invariance in hydrology. Cambridge, UK: Cambridge University Press, p.167-189. Sposito, G. and W.A. Jury. 1990. Miller similitude and generalized scaling. In: Hillel D., and D.E. Elrick, editors. Scaling in soil physics: Principles and applications. SSSA special publication number 25, Madison, Wisconsin. Tillotson, P.M. and D.R. Nielsen. 1984. Scale factors in soil science. Soil Sci. Soc. Am. J. 48:953-959. Tuli, A., J.W. Hopmans, D.E. Rolston, and M.J. Singer. 1999. Soil quality assessment in irrigated agriculture: Influence of soil and water management on physical properties.
158 In: Zabel. A. and G. Sposito, editors. Soil quality in the California Environment. Annual Report of Research Projects 1997-1998. Kearney Foundation of Soil Science, Division of Agriculture and Natural Resources, University of California. Tuli, A., M.A. Denton, J.W. Hopmans, T. Harter, and J.L. MacIntyre. 2001. Multistep outflow experiment: From soil preparation to parameter estimation. Department of Land, Air, and Water Resources Paper No. 100037, University of California, Davis. Vereecken, H. 1989. Derivation and validation of pedotransfer functions for soil hydraulic functions. In: van Genuchten, M.Th., F.J., Leij, and L.J. Lund, editors. Proceedings of the international workshop on indirect methods for estimating the hydraulic properties of unsaturated soils. Riverside, California, October 11-13, p. 473-488. Warrick, A.W., G.J. Mullen, D.R. Nielsen. 1977. Scaling field-measured soil hydraulic properties using similar media concept. Water Resour. Res. 13:355-362. Warrick, A.W. 1990. Application of scaling to the characterization of spatial variability in soils. In: Hillel D., and D.E. Elrick, editors. Scaling in soil physics: Principles and applications. SSSA special publication number 25, Madison, Wisconsin. Warrick, A.W. 1998. Spatial variability. In: Hillel, D., editor. Environmental soil physics. New York: Academic Press, p. 655-676. Warrick, A.W. and D.R. Nielsen. 1980. Spatial variability of soil physical properties in the field. In: Hillel, D., editor. Applications of soil physics. New York: Academic Press, p. 319-344.
159 Williams, J., P. Ross, and K. Bristow. 1989. Prediction of the Campbell water retention function from texture, structure, and organic matter. In: van Genuchten, M.Th., F.J., Leij, and L.J. Lund, editors. Proceedings of the international workshop on indirect methods for estimating the hydraulic properties of unsaturated soils. Riverside, California, October 11-13, p. 427-442. Wraith, J.M. and D. Or. Nonlinear parameter estimation using spreadsheet software. J. Nat. Resour. Life Sci. Educ. 27:13-19. Zavattaro, L., N. Jarvis, and L. Persson. 1999. Use of similar media scaling to characterize spatial dependence of near-saturated hydraulic conductivity. Soil Sci. Soc. Am. J. 63:486-492.
160
3.8. Appendix 3.1. After substitution of Eq. [3-11], the numerator of the integral in Eq. [3-15] transforms to
∫
Sew ,i
0
dS ew,i 1 = h σ 2π
∫
( ln h − ln h ) 2 m ,i d ln h exp ( − ln h ) exp − 2 σ i2
∞
ln h
[A3.1-1]
Substituting y for (ln hm,i − ln h) σ i yields
∫
S ew ,i
0
∫
dS ew,i 1 = h hm ,i 2π
ln hm ,i − ln h
σi −∞
which after substitution of z = (σ i − y )
∫
S ew ,i
0
(
2 dS ew,i exp σ i 2 = h hm,i π
)
∫
y2 exp yσ i − dy 2
[A3.1-2]
2 leads to ∞
σi 2
−
ln hm ,i − ln h
exp − z 2 dz
[A3.1-3]
σi 2
Hence, when written in terms of the complementary error function, the numerator of the hydraulic conductivity function becomes
∫
S ew ,i
0
(
)
( ln h − ln hm ,i ) σ exp σ i2 2 1 dS ew,i = + i erfc 2 h hm ,i σi 2 2
[A3.1-4]
Similarly, when applying the substitution rules to the denominator of Eq. [3-15], it transforms to
∫
1
0
dS ew,i 1 = h σ i 2π
∫
∞
−∞
( ln h − ln h ) 2 m ,i d ln h exp ( − ln h ) exp − 2 σ i2
[A3.1-5]
which becomes
∫
1
0
dS ew,i = h
(
exp σ i2 2 hm,i
)
[A3.1-6]
161 From Eq. [3-11], it follows that ln(h hm ,i )
σi 2
= erfc −1 (2S ew,i )
[A3.1-7]
Finally, substitution of Eqs.[A3.1-4], [A3.1-6] and [A3.1-7] in Eq. [3-15] allows direct computation of K rw,i ( S ew,i ) if the retention function, S ew,i , is known or K rw,i ( S ew,i ) = S
0.5 ew ,i
1 σ i −1 erfc erfc ( 2S ew,i ) + 2 2
which is identical to Eq. [3-16].
2
[A3.1-8]
162
3.9. Appendix 3.2.
Unscaled 10000
WRMSR = 0.392 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
Scaled
1000
100
10
1
WRMSR = 0.235
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 0.548 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew
Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.474 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
Figure 3-7. Simultaneous PB scaling of subset ln hm,i < 6.0 at 50 cm.
0.6
163
Unscaled
Scaled 10000
WRMSR = 0.392 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
1000
100
10
1
WRMSR = 0.235
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 0.548 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew
Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.474 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
Sew
Figure 3-8. Simultaneous PB scaling of subset ln hm,i ≥ 6.0 at 50 cm.
164
Unscaled
Scaled 10000
WRMSR = 0.392 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
1000
100
10
1
WRMSR = 0.235
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 0.548 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew
Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.474 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
Figure 3-9. Simultaneous C scaling of subset ln hm,i < 6.0 at 25 cm.
0.6
165
Unscaled
Scaled 10000
WRMSR = 0.664 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
1000
100
10
1
WRMSR = 0.301
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 0.826 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew
Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.356 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
Sew
Figure 3-10. Simultaneous C scaling of subset ln hm,i ≥ 6.0 at 25 cm.
166
Unscaled
Scaled 10000
WRMSR = 0.396 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
1000
100
10
1
WRMSR = 0.265
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 0.500 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew
Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.353 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
Sew
Figure 3-11. Simultaneous C scaling of subset ln hm,i < 6.0 at 50 cm.
167
Unscaled
Scaled 10000
WRMSR = 0.585 Capillary pressure head (cm)
Capillary pressure head (cm)
10000
1000
100
10
1
WRMSR = 0.299
1000
100
10
1 0
0.2
0.4
0.6
0.8
1
0
0.2
Unscaled
1.0E+002
WRMSR = 0.566 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
Sew
0.6
0.8
1
0.8
1
Sew
Unsaturated hydraulic conductivity (cm h-1)
Unsaturated hydraulic conductivity (cm h-1)
Sew
0.4
0.6
0.8
1
Scaled
1.0E+002
WRMSR = 0.395 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
Sew
Figure 3-12. Simultaneous C scaling of subset ln hm,i ≥ 6.0 at 50 cm.
168
50 cm Unsaturated hydraulic conductivity (cm h-1)
50 cm
Capillary pressure head (cm)
100000 10000 1000 100 10 1 0
0.2
0.4
0.6
0.8
1
Sew Mean curve for ln hm,i < 6.0
1.0E+002 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
0.8
1
Sew Mean curve for ln hm,i ≥ 6.0
Figure 3-13. Soil hydraulic function curves of subsets for reference soils of PB scaling method at 50-cm depth.
169
Capillary pressure head (cm)
Unsaturated hydraulic conductivity (cm h-1)
25 cm
100000 10000 1000 100 10 1 0
0.2
0.4
0.6
0.8
25 cm
1.0E+002 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016
1
0
0.2
Sew Unsaturated hydraulic conductivity (cm h-1)
Capillary pressure head (cm)
10000 1000 100 10 1 0
0.2
0.4
0.6
0.6
0.8
1
0.8
1
Sew
50 cm
100000
0.4
0.8
1
Sew Mean curve for ln hm,i < 6.0
50 cm
1.0E+002 1.0E-001 1.0E-004 1.0E-007 1.0E-010 1.0E-013 1.0E-016 0
0.2
0.4
0.6
Sew Mean curve for ln hm,i ≥ 6.0
Figure 3-14. Soil hydraulic function curves of subsets for reference soils of C scaling method at 25- and 50-cm depth.
1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4
1 2 5 6 8 9 1 2 3 4 6 7 9 1 2 4 5 6 7 8 9 1 2 3
1 3 9 11 15 17 19 21 23 25 29 31 35 37 39 43 45 47 49 51 53 55 57 59
Row Column Sample # cm h -1 0.04 16.13 0.07 0.02 1.50 2.05 0.46 0.28 1.40 0.96 0.74 0.52 0.78 0.72 0.84 0.41 2.70 0.96 4.53 4.31 0.12 0.31 2.94 0.02
cm 3 cm -3 0.443 0.470 0.414 0.416 0.434 0.409 0.436 0.383 0.392 0.408 0.413 0.397 0.408 0.361 0.369 0.412 0.388 0.405 0.435 0.414 0.377 0.359 0.373 0.369
Measured K sw Measured θ s cm -1 0.019 0.063 0.011 0.012 0.038 0.014 0.010 0.007 0.009 0.007 0.018 0.018 0.005 0.010 0.010 0.010 0.009 0.011 0.019 0.015 0.006 0.007 0.008 0.005 1.310 2.069 1.472 1.479 1.420 1.453 1.771 2.255 2.252 2.246 1.387 1.605 1.526 1.649 2.060 2.849 2.552 1.722 2.036 1.741 1.908 1.748 1.785 2.077
cm 3 cm -3 0.310 0.296 0.294 0.301 0.300 0.305 0.242 0.277 0.292 0.317 0.231 0.252 0.315 0.166 0.280 0.247 0.281 0.263 0.242 0.271 0.276 0.203 0.274 0.269
cm h -1 0.023 4.580 0.032 0.007 0.283 0.055 0.061 0.168 0.059 0.028 0.160 0.081 0.003 0.143 0.018 0.014 0.057 0.199 0.578 0.048 0.012 0.026 0.027 0.008
Van Genuchten (Optimized) K sw θr n α
Lognormal (Optimized) LOG10h m K sw θr σ 3 -3 h m in cm cm cm cm h -1 2.427 2.144 0.324 0.074 1.407 1.142 0.297 9.081 2.502 1.671 0.299 0.035 2.419 1.777 0.304 0.014 2.060 2.000 0.303 0.700 2.393 1.762 0.311 0.090 2.336 1.365 0.241 0.078 2.330 0.930 0.279 0.182 2.238 0.990 0.291 0.061 2.332 0.957 0.316 0.024 2.344 1.939 0.243 0.414 2.124 1.585 0.250 0.176 2.491 0.945 0.350 0.002 2.417 1.503 0.161 0.165 2.221 1.024 0.282 0.016 2.103 0.681 0.249 0.013 2.168 0.827 0.280 0.057 2.337 1.473 0.258 0.296 1.883 1.290 0.223 1.935 1.929 1.865 0.249 1.595 2.512 1.090 0.280 0.011 2.483 1.293 0.205 0.024 2.412 1.306 0.274 0.028 2.505 1.009 0.268 0.007
Table 3-4. Parameters for soil hydraulic functions used in subset lnhm < 6.0 of 25 cm depth.
170
3.10. Appendix 3.3.
4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 7 7 7 7 7 7 7
4 6 7 8 9 1 2 3 4 6 7 8 9 1 2 4 9 1 2 5 6 7 8 9
61 65 67 69 71 73 75 77 79 83 85 87 89 91 93 97 107 109 111 117 119 121 123 125
Row Column Sample # cm h -1 5.25 0.20 1.32 6.90 15.25 1.78 7.47 10.83 0.18 0.40 1.31 1.84 16.11 0.80 0.12 1.30 0.53 0.09 3.14 8.67 12.01 0.02 0.08 0.10
cm 3 cm -3 0.369 0.383 0.418 0.414 0.442 0.388 0.424 0.431 0.408 0.394 0.386 0.388 0.438 0.418 0.402 0.415 0.393 0.433 0.410 0.423 0.408 0.398 0.415 0.392
Measured K sw Measured θ s cm -1 0.009 0.006 0.014 0.022 0.033 0.015 0.010 0.030 0.022 0.009 0.007 0.010 0.008 0.029 0.032 0.041 0.011 0.008 0.010 0.017 0.012 0.006 0.015 0.011 1.850 2.116 2.005 1.851 1.649 1.922 3.634 1.568 1.562 1.732 1.884 2.009 2.029 1.679 1.289 1.352 1.599 3.233 2.116 2.388 1.807 2.372 1.694 2.253
cm 3 cm -3 0.212 0.298 0.261 0.209 0.234 0.223 0.319 0.253 0.326 0.303 0.308 0.315 0.337 0.269 0.268 0.247 0.304 0.338 0.298 0.322 0.278 0.308 0.326 0.327
cm h -1 0.124 0.008 0.035 0.170 0.402 0.321 0.190 0.428 0.082 0.013 0.007 0.017 0.015 0.162 0.144 0.224 0.007 0.043 0.052 0.031 0.025 0.044 0.014 0.008
Van Genuchten (Optimized) K sw θr n α
Lognormal (Optimized) LOG10h m K sw θr σ 3 -3 h m in cm cm cm cm h -1 2.322 1.228 0.215 0.144 2.414 1.002 0.298 0.007 2.059 1.090 0.266 0.051 1.873 1.301 0.207 0.510 1.800 1.544 0.231 1.536 2.035 1.411 0.209 0.931 2.060 0.611 0.318 0.196 2.024 1.648 0.250 0.579 2.063 1.749 0.322 0.304 2.389 1.332 0.304 0.014 2.431 1.143 0.309 0.006 2.263 1.132 0.315 0.018 2.328 1.098 0.337 0.015 1.950 1.414 0.267 0.154 2.300 2.150 0.275 0.259 2.150 2.117 0.246 0.465 2.261 2.216 0.292 0.305 2.185 0.626 0.339 0.048 2.218 1.139 0.293 0.058 1.683 1.312 0.297 0.434 2.132 1.787 0.255 0.216 2.382 0.883 0.309 0.045 2.117 1.553 0.326 0.046 2.188 1.011 0.326 0.009
Table 3-4. Parameters for soil hydraulic functions used in subset lnhm < 6.0 of 25 cm depth (continued).
171
8 8 8 8 8 8 8 8
1 2 3 4 5 6 7 8
127 129 131 133 135 137 139 141
Row Column Sample # cm h-1 0.56 2.59 3.17 3.23 0.43 4.00 3.04 1.73
cm3 cm-3 0.430 0.387 0.401 0.401 0.389 0.393 0.389 0.411
Measured K sw Measured θ s cm-1 0.008 0.036 0.070 0.026 0.007 0.019 0.046 0.016
α 2.107 1.364 1.310 2.330 2.161 1.997 1.503 2.685
n cm3 cm-3 0.282 0.225 0.202 0.310 0.271 0.299 0.214 0.313
θr cm h-1 0.078 0.357 2.412 0.055 0.057 0.036 0.514 0.019
K sw
Van Genuchten (Optimized)
hm in cm 2.290 2.107 1.968 1.722 2.384 1.941 1.882 1.894
LOG10hm
1.080 2.094 2.306 1.024 0.992 1.213 1.881 0.589
σ
cm3 cm-3 0.279 0.232 0.218 0.302 0.273 0.294 0.210 0.326
θr
K sw cm h-1 0.080 1.384 11.507 0.091 0.058 0.060 1.439 0.023
Lognormal (Optimized)
Table 3-4. Parameters for soil hydraulic functions used in subset lnhm < 6.0 of 25 cm depth (continued).
172
1 1 1 2 2 3 4 5 6 6 6 6 6 7 7 8
3 4 7 5 8 3 5 5 3 5 6 7 8 3 4 9
5 7 13 27 33 41 63 81 95 99 101 103 105 113 115 143
Row Column Sample # cm h-1 9.38 1.59 0.26 0.65 0.28 0.78 0.82 2.67 1.35 0.02 0.02 1.06 0.71 12.28 0.20 2.98
cm3 cm-3 0.424 0.412 0.390 0.414 0.408 0.385 0.376 0.413 0.400 0.430 0.405 0.399 0.391 0.450 0.391 0.426
Measured K sw Measured θ s cm-1 0.006 0.004 0.017 0.036 0.004 0.005 0.016 0.005 0.016 0.023 0.007 0.013 0.011 0.009 0.009 0.018
α 1.111 1.200 1.276 1.101 1.613 2.242 1.159 1.079 1.099 1.073 1.186 1.188 1.332 3.082 1.403 1.115
n cm3 cm-3 0.144 0.041 0.280 0.100 0.0001 0.274 0.080 0.006 0.0001 0.0001 0.066 0.194 0.201 0.368 0.281 0.0003
θr cm h-1 0.116 0.003 0.039 0.674 0.006 0.064 0.098 0.050 0.141 0.124 0.012 0.019 0.004 0.009 0.002 0.169
K sw
Van Genuchten (Optimized) hm in cm 4.129 3.132 2.714 2.703 2.890 2.801 3.062 4.940 3.840 3.839 3.282 2.633 2.649 3.480 2.755 3.208
2.945 1.804 2.819 2.764 1.364 1.655 2.904 3.920 3.380 2.919 2.481 1.762 1.867 2.685 2.433 3.209
σ
cm3 cm-3 0.133 0.179 0.269 0.238 0.002 0.210 0.126 0.001 0.0001 0.0001 0.135 0.278 0.227 0.068 0.271 0.1049
θr
Lognormal (Optimized) LOG10hm
Table 3-5. Parameters for soil hydraulic functions used in subset lnhm . 6.0 of 25 cm depth.
cm h-1 0.097 0.001 1.432 1.486 0.004 0.060 0.791 1.336 0.668 0.011 0.021 0.005 0.004 0.148 0.038 2.817
K sw
173
1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
1 2 3 5 6 9 2 3 8 9 1 3 4 5 6 7 8 1 2 3 4 5 6 7 8
2 4 6 10 12 18 22 24 34 36 38 42 44 46 48 50 52 56 58 60 62 64 66 68 70
Row Column Sample # cm h 0.36 0.13 1.44 0.41 0.10 2.10 1.35 0.17 0.49 0.41 2.95 1.72 0.60 0.40 0.23 0.66 1.66 2.42 0.66 6.92 2.56 0.31 0.09 7.15 1.37
-1
3
cm cm 0.395 0.403 0.384 0.384 0.414 0.374 0.405 0.411 0.432 0.403 0.423 0.411 0.444 0.415 0.410 0.446 0.448 0.490 0.394 0.419 0.393 0.401 0.418 0.457 0.396
-3
Measured K sw Measured θ s -1
cm 0.015 0.037 0.008 0.024 0.011 0.045 0.006 0.009 0.008 0.008 0.009 0.010 0.007 0.007 0.008 0.010 0.034 0.008 0.093 0.041 0.010 0.009 0.017 0.011 0.023 1.300 1.087 2.418 1.289 1.461 1.175 1.979 1.332 1.658 1.674 2.484 2.492 2.511 2.064 1.686 2.109 1.227 1.797 4.446 1.734 2.318 1.806 1.325 2.123 1.754
3
cm cm 0.270 0.001 0.332 0.265 0.292 0.226 0.318 0.323 0.297 0.313 0.189 0.209 0.222 0.283 0.266 0.241 0.069 0.165 0.316 0.194 0.224 0.259 0.198 0.231 0.142
-3
-1
cm h 0.019 0.690 0.013 0.290 0.032 1.097 0.114 0.008 0.015 0.036 0.872 1.025 0.115 0.079 0.146 0.089 1.064 0.096 0.355 2.138 0.418 0.083 0.258 0.156 0.492
Van Genuchten (Optimized) K sw θr n α
Lognormal (Optimized) LOG10hm K sw θr σ 3 -3 -1 hm in cm cm h cm cm 2.530 2.320 0.281 0.135 2.130 0.730 0.330 0.005 2.254 0.841 0.333 0.014 2.369 2.303 0.275 1.678 2.508 1.693 0.297 0.036 2.324 2.582 0.264 6.880 2.407 1.058 0.321 0.118 2.432 1.158 0.360 0.003 2.497 1.481 0.295 0.016 2.467 1.377 0.315 0.037 2.198 0.945 0.184 0.980 2.170 0.967 0.203 1.219 2.277 0.771 0.228 0.109 2.411 1.044 0.281 0.065 2.441 1.272 0.278 0.147 2.239 0.999 0.241 0.079 2.240 2.107 0.175 1.660 2.436 1.363 0.156 0.106 2.128 0.576 0.300 0.073 1.743 1.394 0.199 5.870 2.198 0.992 0.222 0.525 2.354 1.255 0.263 0.096 2.493 2.135 0.213 0.772 1.976 1.454 0.204 1.305 1.947 1.237 0.160 0.964
Table 3-6. Parameters for soil hydraulic functions used in subset lnhm < 6.0 of 50 cm depth.
174
4 5 5 5 5 5 5 5 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8
9 1 2 3 4 6 7 9 5 7 1 2 3 4 5 7 9 1 3 4 5 6 7 8
72 74 76 78 80 84 86 90 100 104 110 112 114 116 118 122 126 128 132 134 136 138 140 142
Row Column Sample # cm h-1 3.47 2.20 4.44 0.35 0.28 0.53 4.78 1.35 0.10 0.21 5.12 0.16 11.03 1.11 17.79 0.08 10.16 2.68 0.60 0.27 0.07 1.32 2.33 9.04
cm3 cm-3 0.403 0.397 0.424 0.427 0.420 0.389 0.406 0.379 0.402 0.415 0.442 0.403 0.420 0.419 0.423 0.417 0.409 0.441 0.399 0.402 0.384 0.323 0.380 0.500
Measured K sw Measured θ s cm-1 0.010 0.012 0.010 0.054 0.006 0.006 0.009 0.006 0.098 0.044 0.009 0.008 0.010 0.010 0.021 0.006 0.124 0.014 0.024 0.011 0.006 0.009 0.027 0.019
α 2.246 2.331 3.816 1.253 2.289 2.020 1.851 1.825 1.887 1.284 3.314 1.532 2.042 2.394 1.748 1.988 1.433 2.060 1.389 1.938 2.153 2.404 1.602 1.765
n cm3 cm-3 0.257 0.219 0.302 0.200 0.307 0.298 0.344 0.325 0.298 0.252 0.360 0.289 0.316 0.359 0.280 0.300 0.215 0.244 0.238 0.319 0.270 0.200 0.181 0.246
θr cm h-1 0.097 0.743 0.211 0.892 0.043 0.006 0.016 0.001 0.006 0.136 0.092 0.007 0.033 0.088 0.117 0.026 8.713 0.598 0.239 0.019 0.050 0.013 0.704 1.602
K sw
Van Genuchten (Optimized) hm in cm 2.162 2.058 2.092 2.249 2.374 2.430 2.356 2.502 2.271 2.136 2.115 2.567 2.236 2.167 1.930 2.476 1.576 2.058 2.233 2.220 2.426 2.217 2.011 2.080
LOG10hm 1.080 1.030 0.637 2.412 0.912 1.024 1.258 1.961 1.049 2.110 0.647 1.490 1.156 1.013 1.597 1.032 2.151 1.189 1.960 1.299 0.968 0.837 1.579 1.320
σ
cm3 cm-3 0.253 0.213 0.290 0.220 0.310 0.299 0.343 0.315 0.302 0.270 0.358 0.296 0.315 0.357 0.264 0.307 0.226 0.235 0.246 0.315 0.273 0.200 0.183 0.240
θr
K sw cm h-1 0.128 1.027 0.091 2.320 0.047 0.005 0.017 0.013 0.004 0.226 0.089 0.006 0.039 0.103 0.576 0.026 47.640 0.926 0.677 0.028 0.049 0.011 1.412 1.860
Lognormal (Optimized)
Table 3-6. Parameters for soil hydraulic functions used in subset lnhm < 6.0 of 50 cm depth (continued).
175
1 1 1 2 2 2 2 2 3 3 5 5 6 6 6 6 6 6 6 7 8 8
4 7 8 1 4 5 6 7 2 9 5 8 1 2 3 4 6 8 9 8 2 9
8 14 16 20 26 28 30 32 40 54 82 88 92 94 96 98 102 106 108 124 130 144
Row Column Sample # cm h 0.79 1.08 0.90 0.21 0.92 5.18 0.55 0.49 0.01 0.07 2.54 0.90 1.15 0.39 0.27 4.50 0.04 0.23 0.04 0.00017 0.60 2.11
-1
cm cm 0.435 0.383 0.436 0.383 0.368 0.420 0.367 0.388 0.379 0.401 0.413 0.392 0.404 0.428 0.399 0.398 0.408 0.412 0.415 0.407 0.407 0.413
3
-3
Measured K sw Measured θ s -1
cm 0.005 0.020 0.046 0.018 0.005 0.018 0.009 0.009 0.007 0.006 0.012 0.006 0.014 0.009 0.030 0.015 0.013 0.008 0.008 0.004 0.004 0.031
α 1.149 1.074 1.110 1.075 1.236 1.188 1.160 1.152 1.164 1.500 1.146 1.819 1.081 1.090 1.057 1.089 1.108 1.250 1.124 1.760 2.100 1.134
n -3
cm cm 0.131 0.109 0.156 0.016 0.240 0.232 0.0001 0.068 0.007 0.251 0.067 0.316 0.031 0.003 0.0001 0.023 0.0001 0.161 0.007 0.314 0.283 0.0001
3
θr cm h 0.007 0.244 0.586 0.215 0.005 0.147 0.127 0.029 0.099 0.023 0.051 0.004 0.095 0.062 0.518 0.060 0.079 0.008 0.025 0.0004 0.004 0.642
-1
K sw
Van Genuchten (Optimized) hm in cm 2.880 3.384 2.802 3.817 2.897 2.828 3.379 2.910 3.593 2.653 3.080 2.623 3.926 4.040 4.270 3.986 3.370 2.827 3.137 2.710 3.730 2.875
LOG10hm 1.480 3.323 3.090 3.825 1.544 2.951 2.555 1.940 2.671 1.455 2.560 1.691 3.697 3.530 3.670 3.174 2.944 1.882 2.272 1.070 3.286 2.829
σ
-3
cm cm 0.310 0.225 0.238 0.110 0.295 0.252 0.042 0.211 0.015 0.268 0.174 0.297 0.116 0.070 0.002 0.0001 0.121 0.224 0.187 0.320 0.025 0.093
3
θr
Lognormal (Optimized)
Table 3-7. Parameters for soil hydraulic functions used in subset lnhm . 6.0 of 50 cm depth.
cm h-1 0.001 1.836 4.248 14.535 0.002 4.164 0.119 0.007 0.112 0.019 0.085 0.007 3.260 0.896 0.589 0.040 0.210 0.005 0.097 0.0003 0.727 1.656
K sw
176
