Ben-Gurion University of the Negev The Jacob Blaustein Institutes for Desert Research
The Albert Katz International School for Desert Studies
Root water uptake under deficit irrigation: Model calibration
and comparison
Thesis submitted in partial fulfillment of the requirements for the degree of
"Master of Science"
By: Leilah Krounbi
January, 2011
Ben-Gurion University of the Negev The Jacob Blaustein Institutes for Desert Research
The Albert Katz International School for Desert Studies
Root water uptake under deficit irrigation: Model calibration
and comparison
Thesis submitted in partial fulfillment of the requirements for the degree of
"Master of Science"
By: Leilah Krounbi
Under the Supervision of Dr. Naftali Lazarovitch
French Associates Institute for Agriculture and Biotechnology of Drylands
Author's Signature …………….……………………….. Date …………….
Approved by the Supervisor…………….…………….... Date …………….
Approved by the Director of the School ………………. Date ………….…
i
Abstract
Root water uptake research has led to the development of both microscopic and
macroscopic uptake models. The continual refinement of macroscopic models has produced a
range of uptake functions, empirical and mechanistic, compensatory and non-compensatory.
However, little information is available concerning model robustness in varying conditions.
This research evaluates how well two empirical and one semi-mechanistic model predict root
water uptake in optimal and deficit-irrigation conditions, in two soils: loess and fine sand, and
with three crops: wheat, tomato, and sorghum, each characterized by a unique root
distribution.
Models were calibrated by optimizing for unknown soil hydraulic and root water
uptake parameters with a global search algorithm in conjunction with a numerical model.
System variables, including the spatial and temporal soil water potential measured from
cropped, soil columns, served as input data. Model calibration was validated by comparing
the measured transpiration of deficit-irrigated plants, cultivated in a rotating lysimeter system,
to modeled transpiration values. Model performance under varying degrees of compensation
was evaluated, along with the effect of initial soil water potential conditions.
Results from the model calibration for well-watered conditions showed that, while
model robustness was affected by soil and plants, soil texture was the dominant factor. The
soil water dynamics of wheat grown in loess and sand were best predicted when the sink
function of the Richards (1931) equation was solved with the modified Feddes (FD) and van
Genuchten (vG) models, respectively. The same pattern was seen for tomato. The soil water
dynamics of both sand and loess, cropped with sorghum, were best predicted when the Nimah
and Hanks (NH) model was incorporated into the sink function. Water stress was the most
significant in sand columns cropped with sorghum, which may indicate that the accuracy of
NH increases as the soil water potential decreases, possibly due to NH's consideration of the
ii
unsaturated hydraulic conductivity function, which expresses a highly, non-linear dependence
on the soil water potential.
When calibrated models were validated in drought conditions, modeled transpiration
values calculated with vG showed a good fit with measured values for both wheat and tomato.
However, FD overestimated the transpiration of loess-grown wheat regardless of the initial
soil water content, and overestimated the transpiration of loess-grown tomato in an initially
hydrostatic profile. The difference in model performance is attributed to the soil textures for
which they were tested.
The effects of compensation and the root distribution decreased with decreasing water
content, and the initial conditions proved to have the most significant effect on modeled
transpiration results. Transpiration 'overshoot' may be linked to the fact that potential
transpiration values were taken from optimally-irrigated plants, and do not represent the true
potential transpiration of smaller, deficit-irrigated plants.
We conclude that model performance was affected by soil texture, root distribution,
and irrigation regimes. Therefore, a more mechanistic approach, able to account for such
factors, should be incorporated into existing empirical models. In addition, continued testing
should be carried out with the NH model in order to better-understand its limitations and work
toward widening its applicability.
iii
Acknowledgements
I would like to thank my advisor, Dr. Naftali Lazarovitch, for being so generous
with his knowledge and time. This scope of this project was beyond my capabilities, but
Naftali's input throughout the research process, from planning the experiments, to
analyzing the data, to writing the thesis, was enough to get me through. Yuval Shani
was another invaluable person during this project. I could always count on him for
trouble-shooting through technical problems and coming up with ideas to make the
systems more efficient.
The MATLAB modeling in this research was made possible through the help of
Andrew Hinnell, who improved, de-bugged, and whatever else to the DREAM codes in
order to make them operable with our data. The same can be said of Jirka Šimůnek, who
altered the HYDRUS-1D code to include the Nimah and Hanks model along with the
entourage of other models.
I want to thank Liron Summerfield and Eldad Saragusty for the hours spent
sifting roots in the blazing-hot greenhouse during the summer. Nadav Tal, Or Sperling,
Tom Groenveld, and Ram Bhantana also lent helping hands with the physical workload.
My friend and former co-worker Johanna Bachman was a big help in
deciphering the DREAM codes. She was also a good friend during her time here, and it
was a pleasant experience sharing an office with her. I also want to thank my dear
friends Majdalene Azazzma, who took me every Saturday at her desert outpost, Moriah
Groenveld, who was like a sister, and Hannah Bardin, who kept me grounded with her
daily 'checkups'. I also want to thank my father and mother, Mohamad Krounbi and
Nadine Whittaker for their love and support throughout this project.
Lastly, I want to thank God the source of all wisdom and knowledge, the best
advisor and friend, for His grace and strength in every good work.
But you, O LORD, are a shield about me, my glory, and the lifter of my head -Psalms 3:3
This work was partially supported through a scholarship made available through the
ALBERT KATZ INTERNATIONAL SCHOOL FOR DESERT STUDIES FUND, and
through a grant (857-0555-08) from the Chief Scientist of the Ministry of Agriculture,
Israel.
iv
Table of contents
List of figures ...................................................................................................................v List of tables ....................................................................................................................vi Abbreviations and terms.................................................................................................vii 1 Introduction ...................................................................................................................1 2 Scientific background....................................................................................................2
2.1 Water flow through the soil-plant-atmosphere continuum (SPAC)........................ 2 2.1.1 Soil hydraulic properties .................................................................................. 3 2.1.2 The root distribution......................................................................................... 6 2.1.3 Soil-root interface............................................................................................. 8
2.2 Modeling root water uptake .................................................................................. 11 2.2.1 Microscopic root water uptake models .......................................................... 11 2.2.2 Macroscopic root water uptake models.......................................................... 12
2.3 Numerical solutions for macroscopic root water uptake models .......................... 20 2.3.1 Computer software available for modeling water transport and uptake ........ 20 2.3.2 Parameter estimation through inverse modeling............................................ 21 2.3.3 Evaluation of root water uptake model performance through forward and inverse numerical solutions..................................................................................... 26
3 Statement of problem ..................................................................................................28 3.1 Simultaneous model comparison in varying conditions ....................................... 28 3.2 Research hypothesis and objectives ...................................................................... 30
3.2.1 Hypothesis...................................................................................................... 30 3.2.2 Research objectives ........................................................................................ 30
4 Research methods........................................................................................................31 4.1 Column experiments for model calibration........................................................... 31
4.1.1 Experimental system ...................................................................................... 33 4.1.2 Experimental procedure ................................................................................. 36 4.1.3 Model calibration ........................................................................................... 37 4.1.4 Root uptake model comparison...................................................................... 43
4.2 Rotating lysimeter experiments for model verification......................................... 43 4.2.1 Experimental system ...................................................................................... 44 4.2.3 Model verification.......................................................................................... 46
5 Results and discussion.................................................................................................49 5.1 Soil properties ....................................................................................................... 49 5.2 Column experiments ............................................................................................. 51
5.2.1 Wheat ............................................................................................................. 51 5.2.2 Tomato ........................................................................................................... 61 5.2.3 Sorghum ......................................................................................................... 68 5.2.4 Measured vs. modeled soil water potential .................................................... 77 5.2.5 Model comparison.......................................................................................... 82
5.3 Carousel experiments ............................................................................................ 84 5.3.1 Wheat ............................................................................................................. 84 5.3.2 Tomato ........................................................................................................... 91
6 Conclusions .................................................................................................................98 7 References .................................................................................................................101
v
List of figures
Figure 1. The transpiration stream through the soil-plant-atmosphere continuum.. ......... 3
Figure 2. (A) Water content and (B) hydraulic conductivity vs. h of loam and sand . ..... 6
Figure 3. The effect of pz on a sample plant root distribution ........................................... 8
Figure 4. (A) The FD and (B) the vG functions at varying soil water potentials. .......... 15
Figure 5. Ratio of the actual to the potential transpiration as a function of ω ................ 18
Figure 6. Arrangement of growth columns ..................................................................... 32
Figure 7. Illustration of a sample growth column ........................................................... 33
Figure 8. Sample calibration curve of pressure transducers............................................ 34
Figure 9. A sample water balance for wheat, as summed over three days...................... 45
Figure 10. Rotating lysimeter system.............................................................................. 45
Figure 11. Soil water retention curves for (A) fine sand and (B) loess soil.................... 50
Figure 12. The above-ground biomass of wheat ............................................................. 52
Figure 13. The relative root mass with depth of wheat................................................... 52
Figure 14. The measured β(z), and the optimized root distributions, β'(z), of wheat...... 57
Figure 15. The above-ground biomass of tomato plants................................................. 61
Figure 16. The relative root mass with depth of tomato plants....................................... 62
Figure 17. The measured β(z), and the optimized root distributions, β'(z), of tomato .... 66
Figure 18. The dry yield of sorghum............................................................................... 69
Figure 19. The relative root weight with depth of sorghum............................................ 69
Figure 20. The measured β(z), and the optimized root distributions, β'(z), of sorghum . 74
Figure 21. The measured and modeled h over time at varying depths for wheat. .......... 79
Figure 22. The measured and modeled h over time at varying depths for tomato.......... 80
Figure 23. The measured and modeled h over time at varying depths for sorghum....... 81
Figure 24. (A) The transpiration rate and (B) cum. transpiration of wheat with DAS ... 84
Figure 26. The dry wheat yield of lysimeters and the ave. yield and standard dev. ....... 85
Figure 27. The relative wheat root mass with depth ....................................................... 86
Figure 28. Measured cum. Tp and Ta of loess-grown wheat, vs. FD modeled cum. Ta.. . 89
Figure 29. Measured cum. Tp and Ta of sand-grown wheat, vs. vG modeled cum. Ta.. . 90
Figure 30. (A) The transpiration rate and (B) cum. transpiration of tomato with DAS.. 92
Figure 31. The dry tomato yield of lysimeters and the ave. yield and standard dev...... 93
Figure 32. The relative tomato root mass with depth...................................................... 93
Figure 33. Measured cum. Tp and Ta of loess-grown tomatoes, vs. FD modeled Ta....... 95
Figure 34. Measured cum. Tp and Ta of sand-grown tomatoes, vs. vG modeled Ta ....... 96
vi
List of tables Table 1. The soil hydraulic parameters for three soil textures .......................................... 4
Table 2. Experimental timeline and general description................................................. 31
Table 3. Count of optimizations carried out.................................................................... 38
Table 4. (A) Initial parameter ranges for optimizations.................................................. 41
Table 5. Description of treatments in the first rotating lysimeter experiment. ............... 43
Table 6. Description of treatments in the second rotating lysimeter experiment............ 44
Table 7. The hydraulic properties of rock wool [Ben-Gal and Shani (2002)] ................ 46
Table 8. Lysimeters from which HYDRUS-1D input data............................................. 47
Table 9. The particle size distribution measured, soil texture, bulk density (ρb), and
Rosetta Lite SHP predictions for fine sand and loess. .................................................... 49
Table 10. The 95% CI and estimated ML values for uncropped loess and sand SHP.... 50
Table 11. The wheat measured RDP and the total wheat root mass for each treatment . 53
Table 12. The 95% CI, ML values, and an error index for the SHP,⎯Tp, and RDP of the
FD, vG, and NH models with wheat for loess and sand ................................................. 54
Table 13. The 95% CI, ML values, and an error index for the RWUP of (A) the FD, (B)
vG, and (C) NH uptake models with wheat for loess and sand. ..................................... 59
Table 14. The tomato measured RDP and the total root mass for each treatment .......... 63
Table 15. The 95% CI, ML values, and an error indexfor the SHP,⎯Tp, and RDP of the
FD, vG, and NH models with tomato for loess and sand................................................ 63
Table 16. The 95% CI, ML values, and an error index for the RWUP of (A) FD, (B) vG,
and (C) NH uptake models with tomato for loess and sand............................................ 67
Table 17. The measured sorghum RDP and the total root mass for each treatment ....... 70
Table 18. The 95% CI, ML values, and an error index for the SHP,⎯Tp, and RDP of the
FD, vG, and NH models with sorghum for loess and sand............................................. 70
Table 19. The 95% CI and ML values for the RWUP of (A) FD, (B) vG, and (C) NH
uptake models with sorghum, for loess and sand............................................................ 75
Table 20. Model robustness for all six treatments, as evaluated by two error indices.... 82
Table 21. The measured wheat RDP and the total root mass for each treatment............ 86
Table 22. The measured tomato RDP and the total root mass for each treatment .......... 94
vii
Abbreviations and terms
α (L-1) Shape parameter of the van Genuchten (1980) function
β (-) Relative root distribution according to Raats (1974) and Vrugt et al. (2001)
β' (-) Relative root distribution according to Raats (1974) and Vrugt et al. (2001), normalized in HYDRUS-1D
γ (-) Uptake reduction function used in empirical root water uptake functions such as Feddes et al. (1978) and van Genuchten (1987)
θ (L3 L-3) Soil water content θs (L3 L-3) Saturated soil water content θr (L3 L-3) Residual soil water content ρb (M L-3) Bulk density
φ1 (-) Error index evaluating convergence strength based on the ratio of the ML value to the 95% CI range
φ2 (-) Error index representing the sum of squared error of the objective function
ω (-) Weighted stress index for considering compensatory uptake, according to Jarvis (1989)
ωc (-) Critical weighted stress index, above which compensation is partial, according to Jarvis (1989)
B (-) Parameter vector in the objective function H (L) Soil water potential
h0 (L) The soil water potential at which root water uptake commences, according to Feddes et al. (1978)
h1 (L) The soil water potential at which the rate of root water uptake equals the potential transpiration, according to Feddes et al. (1978)
h2 (L) The soil water potential at which the rate of root water uptake decreases from the potential transpiration due to water stress, according to Feddes et al. (1978)
h3 (L) The soil water potential at which uptake ceases, according to Feddes et al. (1978)
h50 (L) The soil water potential at which root water uptake is half of the potential transpiration, according to van Genucthen (1987)
h2H (L) The degree of variation in h2 at high values of the potential transpiration, according to Wesseling and Brandyk (1985) and Šimůnek et al. (1992)
h2L (L) The degree of variation in h2 at low values of the potential transpiration, according to Wesseling and Brandyk (1985) and Šimůnek et al. (1992)
viii
Hroot (L) The root water potential in the Nimah and Hanks (1973) model
Hwilt (L) The soil water potential at which uptake decreases from the potential transpiration, according to Nimah and Hanks (1973)
J (-) Number of optimized parameters in the parameter vector
K (L T-1) Hydraulic conductivity
N (-) Shape parameter of the van Genuchten (1980) soil water retention function
N (-) Number of observations
M (-) Shape parameter of the van Genuchten (1980) function
P (-) Shape parameter of the van Genuchten (1978) function
pz (-) Shape parameter of the Raats (1974) and Vrugt et al. (2001) root distribution function
Q (L T-1) Flux
r2H (L T-1) Value representing the high range of the potential transpiration, according to Wesseling and Brandyk (1985) and Šimůnek et al. (1992)
r2L (L T-1) Value representing the low range of the potential transpiration, according to Wesseling and Brandyk (1985) and Šimůnek et al. (1992)
R (L T-1) Resistance
RRM (M M-1) The gravimetrically-measured relative root mass
∫RM(z) (M) Total root mass over the soil profile
S (L3 L-3 T-1) Sink function of the Richards (1931) soil water flow equation, representing root water uptake
S (L3 L-3 T-1) Average daily uptake of plants as calculated according to a water balance
T (T) Time T (L T-1) Transpiration Ta (L T-1) Actual transpiration Tp (L T-1) Potential Transpiration
pT (L T-1) Potential daily transpiration as calculated with the Fayer (2000) sinusoidal shape function
W (M) Mass of dry roots ΔW (M) Change in lysimeter mass Z (L) Depth
z* (L) Raats (1974) and Vrugt et al. (2001) root distribution parameter representing the depth of maximum rooting density
zm (L) Raats (1974) and Vrugt et al. (2001) root distribution parameter representing the maximum rooting depth
zr (L) Depth of flow domain cumT Cumulative transpiration D Drainage
ix
DAS Days after sowing
DREAM Differential Evolution Adaptive Metropolis (Vrugt et al., 2009)
GA Genetic algorithm I Irrigation
FD The modified, Feddes et al. (1978) root water uptake function with compensation according to Jarvis (1989)
ML Most likely parameter of the last 3,000 simulations of DREAM
NH The Nimah and Hanks (1973) root water uptake function
OF Objective function RDP Root distribution parameters RLS Rotating lysimeter system RMSE Root mean squared error RWU Root water uptake RWUP Root water uptake parameters SHP Soil hydraulic parameters SPAC Soil plant atmosphere conditions SSE Sum of squared errors
vG The modified, van Genuchten (1987) root water uptake function with compensation according to Jarvis (1989)
95% CI 95% confidence interval of the last 3,000 simulations of DREAM
1
1 Introduction
Contemporary agriculture, with its dependence on irrigation, fertilizers, and
pesticides, contributes significantly to water and solute fluxes through the soil,
specifically in arid and semi-arid areas. The quality and quantity of this water as it
passes through the vadose zone is governed primarily by plant water uptake (Vrugt et
al., 2001b; Skaggs and Shouse, 2008; Šimůnek and Hopmans, 2009). Therefore,
accurately predicting the temporal and spatial root water uptake pattern in a drying soil
is important for planning fertigation regimes which both maximize crop production,
while simultaneously preventing soil and groundwater pollution (Segal et al., 2008).
Much of the research concerning root water uptake under water or salt-stressed
conditions relies on empirical models which characterize plant-soil interface fluxes
through a 'black-box' approach (Krounbi and Lazarovitch, 2011). This likely results
from the difficulty in isolating central variables from such a complex system as the soil-
plant-atmosphere-continuum (SPAC) (Hopmans and Bristow, 2002); not only is root
water uptake affected by biological factors such as leaf stomatal aperture and root
membrane signaling, but also by the physical properties of the soil.
But while a new approach to root water uptake modeling is definitely needed in
order to provide the necessary unifying link between plant, root, and soil subsystems, a
thorough, comparative study of the performance of existing models under varying
conditions has yet to be done. This research therefore aims at evaluating how well two
empirical and one semi-mechanistic model predict root water uptake from two
experimental systems consisting of different soil, plant, and irrigation treatments.
2
2 Scientific background
2.1 Water flow through the soil-plant-atmosphere continuum (SPAC)
Water potential gradients serve as the force inducing flow within and between
adjacent compartments in the SPAC (Boyer, 1995; van der Ploeg et al., 2008). This
process was first formulated in the similitude of Ohm's law for steady electric current
flow in a resistance network (Gradman, 1928; van den Honert, 1948; Feddes and Raats,
2004). Accordingly, the steady water flux, q (L T-1), through the roots, stem, and leaves
is a expressed as the product of the difference in the water potential, h (L), between any
two SPAC compartments, and a proportionality factor, C, which defines either the
conductance, K (L T-1), or resistance, R (L-1 T), of the transmitting medium to flow,
1-12
R K, C
)hh(Cq
=
−=
[1]
The definition of the proportionality factor is context dependent, whether soil or
plant, according to the spatial scale of the water-transporting medium (Hopmans and
Bristow, 2002). The overall resistance is defined as the series combination of all
resistances in the SPAC (Campbell, 1985).
Figure 1 highlights the typical potential differences between soil water and
atmospheric water vapor, oftentimes amounting to tens of megapascals (MPa). As the
soil dries, the soil water potential decreases, leading to a subsequent reduction in the
hydraulic conductivity (Segal et al., 2006). To maintain the water potential gradient
powering the transpiration stream, the root water potential must decrease beyond that of
the soil. But while the soil water potential can decrease to very low values, the root
water potential is limited by a critical value, around -1.6 MPa for most agricultural crops
(Koorevaar et al., 1983), below which plants die.
3
Air (-50 MPa)
Leaves (-1.5 MPa)
Roots (-0.3 MPa)
Soil water (-0.03 MPa)
Air (-50 MPa)
Leaves (-1.5 MPa)
Roots (-0.3 MPa)
Soil water (-0.03 MPa)
Figure 1. The transpiration stream through the soil-plant-atmosphere continuum. Representative water potential values are indicated for various compartments. While root water uptake is often equated to transpiration, water flow through the
SPAC may be affected by fluxes between the xylem and adjacent parenchyma cells. The
lag time between the onset of maximum water potential gradients in the leaves and
xylem demonstrates a lack of coordination in water potential changes between SPAC
compartments, pointing to the effect of plant capacitance on the transpiration stream
(Nobel and Jordan, 1983; Williams et al., 1996). In spite of this, a governing assumption
in the uptake models employed for this research is that all the water taken up by roots is
transpired, making variations in the plant water status negligible.
The effectiveness of plants in meeting a continuous evaporative demand through
increased water extraction is determined by three main factors: 1) the water retaining
and transmitting ability of the soil, or the soil hydraulic properties, 2) the root
distribution over the soil profile, and 3) soil to root pathway resistances (Gardner, 1964;
Prasad, 1988; Doussan et al., 1998; Javaux et al., 2008).
2.1.1 Soil hydraulic properties
Water conductivity through the soil is maximal following a saturating rain or
irrigation event. But since plant water uptake occurs at a lower rate than the saturated
4
hydraulic conductivity of most soils (Nobel and Cui, 1992), a plant's ability to utilize
available water over time is dependent upon the water-holding and transporting capacity
of the soil medium, i.e., the soil's hydraulic properties, which are primarily a function of
the soil particle size distribution, or texture (Chen et al., 2003a). This explains why, for
a given amount of precipitation, the plant water status and overall performance vary
between soils (Lamboski et al., 1998; Dexter, 2003; Shani et al., 2009).
The soil hydraulic parameters (SHP) presented in Table 1 parameterize the non-
linear relations between the soil water content, θ(h) (L3 L-3), and the unsaturated soil
hydraulic conductivity, K(h) (L T-1), on the soil water potential, h (L), as presented in
Figure 2A and Figure 2B, respectively.
Soil texture Ks (cm hr-1) α (cm-1) n (-) θr (cm3 cm-3) θs (cm3 cm-3) Sand 29.7 0.145 2.68 0.045 0.43 Loam 1.04 0.036 1.56 0.078 0.43 Clay 0.2 0.008 1.09 0.068 0.38
Table 1. The soil hydraulic parameters (SHP) for three soil textures, as taken from Carsel and Parrish (1988).
van Genuchten (1980) developed a widely-used function expressing this
dependency,
( )[ ]n
1nm,h1)h( mn
rs
r −=α+=
θ−θθ−θ −
[2]
where θs (L3 L-3) is the saturated soil water content, θr (L3 L-3) the residual soil water
content, and α (L-1), n (-), and m (-) represent empirical shape parameters. The Mualem
(1976) model is preeminent in expressing the dependency of K on h,
2mm/1
rs
r
5.0
rs
rs
)h(11)h(K)h(K⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛θ−θθ−θ
−−⎟⎟⎠
⎞⎜⎜⎝
⎛θ−θθ−θ
= [3]
These transformations are necessary for solving the vertical unsaturated soil
water flow problem in a soil matrix, as given by the Richards equation (1931) (Gilding,
1991; Arbogast et al., 1993); this equation is a combination of the conservation of water
mass in one-dimension,
5
with Darcy's law (1865) for vertical water flow, qz (L T-1), in a bare soil matrix,
( ) 1zhq K hz
∂∂
⎛ ⎞= − +⎜ ⎟⎝ ⎠
[5]
The Richards equation (1931) therefore expresses the change in θ over time as equal to
the change in qz over the vertical soil profile,
The uniqueness of each soil texture in terms of its characteristic SHP is
conferred by the soil mineral composition and particle size. Sandy soils are comprised
of primary, inert minerals such as quartz, which cannot adsorb polar water molecules
(Jury and Horton, 2004). In addition, the large 'equivalent radii' of sand particles give
rise to large water-conducting pores, to which a relatively small percentage of water
molecules in the flow stream adsorb. This explains the higher hydraulic conductivity of
sand near the saturation range, Ks (L T-1), when compared to loam, as depicted in Figure
2B, and the drastic drop in the θ of sand when subjected to a slight lowering of h, as
depicted in Figure 2A (Hillel, 1998).
In contrast, loamy soils have a high percentage of negatively-charged, secondary
minerals, readily facilitating direct and indirect bonding with water molecules
(Essington, 2004). These smaller-sized silt and clay particles abounding in loamy soil
bind together to form aggregates which assemble into more heterogeneously sized soil
pores characterized by a high water-retaining capacity. These properties are evident by
the higher θs (Figure 2A), and the much lower saturated hydraulic conductivity of loam,
when compared to sand (Figure 2B).
zqt z
∂∂θ∂ ∂
= − [4]
zzhhK
th z
∂∂∂∂
∂∂θ ⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +
−=1)(
)( [6]
6
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0 0.1 0.2 0.3 0.4 0.5θ (cm3 cm-3)
Soil
wat
er p
oten
tial (
-MPa
)
LoamSandθs Loamθs Sand
1.0E-15
1.0E-11
1.0E-07
1.0E-03
1.0E+01
1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00Soil water potential (-MPa)
Hyd
raul
ic c
ondu
ctiv
ity (c
m s-1
)
LoamSandKs LoamKs Sand
Figure 2. (A) The soil water content and (B) the soil hydraulic conductivity for loam and sand at varying water potentials.
The more gradual change in the soil water content with decreasing soil water
potentials presented in Figure 2A demonstrates that a higher percentage of water
molecules in the flow stream are adsorbed to pore walls in a loamy soil matrix. This
explains why, in a drying soil, the hydraulic conductivity of sand eventually drops far
below that of loam (Figure 2B), as loamy soil may still contain a large amount of small
water-filled pores even when soil pressure values reach the wilting point. Therefore, a
plant growing in loamy soil, as opposed to sand, benefits from a greater amount of
available soil water over a longer period of time due to the enhanced structural and
adsorptive characteristics of silt and clay particles.
2.1.2 The root distribution
In optimal soil moisture conditions, a plant's ability to utilize available water and
nutrients from a soil layer is primarily a factor of the root density in that region
(Marshall et al., 1996). While the bulk of plant roots center around the root crown near
the soil surface, plants allocate carbon stores to increasing the excavating capacity of
roots after untapped soil water and nutrient stores. Thus, by widening and deepening the
root zone, plants are able to meet their ever-increasing 'thirst' throughout the growing
season.
But while the relation between the magnitude of the root zone and root water
uptake is certain, arriving at an accurate representation of root morphology necessary
A B
7
for uptake modeling is not so simple (Raats, 2007), nor is there an established protocol
by which the root distribution should be quantified. Generally speaking, rooting density
decreases with depth, but the manner by which the roots thin out along the soil profile is
a topic open for dispute; not only is root development unique for each plant type, but it
is also influenced by soil texture and the soil water status.
Prasad (1988) assumed a linear root distribution with depth, resulting in linearly
decreasing extraction rates over the soil profile for optimal moisture conditions. Feddes
et al. (1974) and Gerwitz and Page (1974) reported that in most instances, an
exponential function is sufficient for characterizing crop root distribution.
Parameterization of equations for the root density with depth often require geometrical
information on roots which is not easily obtained (Raats, 2007); the root length per unit
root volume, the root surface area per unit root volume, and the root and soil volume
fractions are just some examples of expressions, the variables of which are difficult to
measure.
Vrugt et al. (2001b) proposed a root distribution function based on the
exponential uptake model put forth by Raats (1974),
z*zzp
m
m
z
e]zz1[)z(
−−
−=β [7]
where β(z) denotes the dimensionless spatial root distribution with depth. zm is the
maximum rooting depth (L), z* (L) represents the soil depth with the highest
concentration of roots, and pz (-) is an empirical shape parameter. The advantage of [7]
over the previously mentioned root distribution expressions lies in its ability to
characterize root density profiles with varying degrees of 'linearity' or 'exponentiality', in
accordance with the pz parameter. Figure 3 demonstrates a strong linear correlation of
the β(z) function with depth when 0 < pz < 1, while the curve takes on a distinct
exponential nature when pz > 2. But despite the versatility of [7] conferred by pz, it is the
inability of measuring this parameter which makes arriving at a root distribution profile
8
a statistical matter; parameters z* and zm can be measured in the field, but pz must be
optimized from an initial range of values believed to produce the accurate shape
characterizing a plant's root distribution with depth.
0
25
50
75
100
0 0.2 0.4 0.6 0.8 1
β (-)
Dep
th (c
m) pz = 1
pz = 4pz = 30
Figure 3. The effect of pz on a sample plant root distribution, as calculated with [7]. zm and z* were kept constant at a value of 100 cm and 0 cm, respectively, for all three distributions.
It is important to note that since most uptake functions are one-dimensional, root
density profiles are most commonly presented along the z-axis, necessitating the
assumption of a homogeneous spatial distribution of roots within each two-dimensional
soil layer (Waisel et al., 2002). In reality though, variation is always to be found within
a root system with respect to root diameter, age, and uptake activity (de Jong van Lier et
al., 2008).
2.1.3 Soil-root interface
The transport medium of each sub-compartment in the SPAC is characterized by
differing degrees of conductivity or resistivity. The interface regions between
subsystems also have an effect on transport rates according to their physical and
chemical properties. Numerous studies have been done to isolate the region of greatest
resistance affecting root water uptake in the soil-root system, and are reported in Molz's
(1981) review on soil-plant water transport. He states that while a number of reports
have shown that the soil directly surrounding the roots presents the greatest resistance to
water flow in the soil-root system (Gardner, 1964; Whisler et al., 1968; Nimah and
9
Hanks, 1973a,b; Feddes 1974), others claim the living root tissue as the dominant
resistance region between plant and soil compartments (Taylor and Klepper, 1975;
Nnyamah et al., 1978; Rowse et al., 1978). Much of the uncertainty surrounding this
subject results from the difficulty in measuring soil-root interface potentials and fluxes
(Personne et al., 2003).
Steep water potentials and hydraulic conductivity gradients from bulk soil to
root xylem have been demonstrated through various types of experiments (Duham and
Nye, 1973; Hainsworth and Aylmore, 1989; Schmidhalter, 1997). Under conditions of
moderate to high transpiration rates, dry soil layers characterized by a very low
hydraulic conductivity develop in the rhizosphere, presenting major obstacles to water
uptake. When considering the microscopic, single-root models of water uptake, a drop
in the hydraulic conductivity will result in a steep water potential gradient developing
around an individual root (Philip, 1957; Gardner, 1960), similar to the characteristic
drawdown in the groundwater level in the vicinity of a well during pumping.
Schroeder et al. (2008) recently examined this process for a three-dimensional
plant-scale model, and verified it through numerical means. By using an analytical
solution for the Richards (1931) equation for radial flow, they were able to quantify a
sharp decrease in the hydraulic conductivity near the root surface which greatly affected
water uptake. While slightly evident in relatively wet soils, this effect was especially
pronounced in cases where the radial root hydraulic conductivity was greater than that
of the bulk soil, which occurs at soil water potentials of below -5,000 cm, for clay, sand,
and loam. These results are in agreement with the findings of Newman (1969a,b) who
evaluated experimental evidence showing increased flow resistance in the rhizosphere
near the root surface. He reported that this phenomenon appears most relevant for soils
at soil water potentials lower than -5,000 cm.
10
Experimental data presented by Taylor and Klepper (1975) for water uptake by
cotton roots show that in a wet soil, the hydraulic conductivity of the soil to root
pathway was up to six orders of magnitude smaller than the bulk soil hydraulic
conductivity. They were not able to identify though whether the soil-root interface or the
inner root tissue contained the greatest flow resistance (Rendig and Taylor, 1989).
Herkelrath et al. (1977a,b) postulated that the soil hydraulic conductivity drop at
the soil-root interface resulted from a decrease in the effective area of contact between
root and soil with decreasing water content. During drying periods, soil and root
shrinkage can result in roots diminishing to 60% of their original size, and in much of
the root epidermis detaching from soil particles (Rendig and Taylor, 1989).
The increase in flow resistance independent of the soil water potential, as in the
shrinking root theory, was verified by Zeelim (2009). In his study, Tamarix trees were
grown hydroponically in a manner which enabled him to isolate the effects of the
hydraulic head (equivalent to the soil water potential for a hydroponic system) and the
hydraulic conductivity on root water uptake. Results showed that decreasing the
hydraulic head under conditions of unlimited hydraulic conductivity had no detrimental
effect on root water uptake. He therefore concluded that in an unsaturated soil, water
uptake rates are a function of the hydraulic conductivity and not the soil water potential.
The resistance posed by the root membrane is another factor affecting water
uptake rates. Bio-membranes are essentially permeable to water, allowing passive
diffusion across the lipid bilayer. Nevertheless, they pose a significant barrier to water
flow through the SPAC. For example, when comparing the flow resistance of xylem
stem vessels to that of root membranes, Steudle and Peterson (1998) estimate that a
cylindrical vessel with a diameter of 23 μm would require a length of 24 km to exhibit a
resistance equivalent to that of a single root cell membrane.
11
Aquaporins, or membrane proteins specifying in water transport, have been
shown to greatly increase the water permeability of membranes (Kaldenhoff et al.
2008), facilitating transport of surprisingly large amounts of water along transmembrane
water-potential gradients (Maurel and Chrispeels, 2001). But while facilitated water
uptake by aquaporins can greatly reduce the resistivity of root membranes, their
expression appears tightly coupled to whole plant physiology (Javot and Maurel, 2002).
The literature provides mixed reports on this phenomenon; Lu and Neumann (1999)
report that membrane water channels showed up-regulation under water stress
conditions, while Martre et al. (2001) report the opposite, namely that aquaporin
expression was down-regulated at the onset of water stress. In the latter instance, the
significant reduction in the hydraulic conductivity at the soil-root interface at the onset
of drought stress fits with the results of the previously-mentioned studies (Taylor and
Klepper, 1975; Herkerath et al., 1977a,b; Shroeder et al., 2008).
2.2 Modeling root water uptake
2.2.1 Microscopic root water uptake models
The Ohm's law analogue put forth by Gradman (1928) and van den Honert
(1948) stands as the first attempt to mathematically describe the physical forces
governing water transport through the SPAC (Feddes et al., 1988). It has since been
targeted as a gross simplification of actual plant processes (Moltz, 1981), as SPAC
compartmental resistances, and root resistance in particular, have been reported to vary
with transpiration rates, making the steady state assumption inherent in [1] unlikely,
especially for time scales greater than a day (Hopmans and Bristow, 2002).
The next major breakthrough in modeling root water uptake came with the work
of Philip (1957) and Gardner (1960) on their development of microscopic uptake
models. This approach describes the water uptake process as radial soil water flow
12
across a representative root of uniform geometric and water-absorbing properties. It
approximates the soil-plant flow system as a network of concentric soil cylinders with
cylindrical roots of uniform radius interspersed at regular, definable distances (Hillel,
1998). Microscopic models are parameterized by at least two characteristic lengths
describing the geometry of the soil and root system (Feddes and Raats, 2004).
In his microscopic uptake model, Gardner (1960) calculated the soil water
distribution surrounding a representative root using an analytical solution to the
Richards (1931) equation for radial flow. He approximated the change in water content
of a drying soil over time as a series of steady states (Herkelrath et al., 1977a,b). But
while Gardner's (1960) model has a strong mechanistic basis, its application to field
conditions is limited; the soil water flux and water potential of laboratory column
experiments can be kept constant over time, but field conditions are subject to
environmental elements, and are therefore transient.
A number of other reasons limit the practical application of microscopic models
such as that of Gardner (1960, 1965). First of all, they do not account for the complex
geometries and physical properties of the root system, often varying in time and space
(Doussan et al., 1998). Water permeability, for example, has been shown to vary with
length (Kramer, 1969). In addition, the geometric parameters necessary for character-
izing the root distribution are difficult, if not impossible, to obtain (Molz, 1981).
Consequently, most methods for calculating root water uptake rely on a macroscopic
approach (Hoogland et al., 1981; Vrugt et al., 2001b; Hopmans and Bristow, 2002).
2.2.2 Macroscopic root water uptake models
The macroscopic approach in uptake modeling considers the root system as a
diffuse sink penetrating the soil profile at varying densities between soil layers (Hillel,
1998; Feddes and Raats, 2004). Thus, instead of considering flow to each individual
rootlet, it deals with moisture removal by the root zone as a whole (Nimah and Hanks,
13
1973a,b). Numerous researchers have set to modeling root water uptake from a
macroscopic standpoint (Whistler et al., 1968; Molz and Remson, 1970, 1971; Nimah
and Hanks, 1973a,b; Feddes et al., 1978; van Genuchten, 1987; Clausnitzer et al., 1994),
as it bypasses the geometric complications of analyzing water potential gradients and
fluxes at the microscale.
Yet the macroscopic approach also has its disadvantages, namely the reliance on
spatial averaging of the soil matric and osmotic potentials (Hillel, 1998). In addition,
some researchers seem to be moving back to the microscopic approach, convinced of its
robustness over the macroscopic approach (Personne et al., 2003); the former considers
water flow within the root system (Schroeder et al., 2008) while the latter typically
neglects flow pathways around roots (Šimůnek and Hopmans, 2009).
Quantifying plant water uptake from a macroscopic standpoint is done by adding
a sink term representing root water extraction to the Richards (1931) water mass balance
equation, so that the one-dimensional root water uptake in a given layer of soil, Δz (L),
is represented by,
zz S
zq
t−−=
∂∂
∂∂θ [8]
where Sz is the volume of water taken up by the roots per until volume of bulk soil per
unit time (L3 L-3 T-1). The extraction term depends on numerous factors such as depth in
the soil profile, time, the root and soil water potentials, soil salinity, root characteristics,
and any combination of these variables.
Macroscopic uptake models can be further divided into two types of functions:
1) empirical "black box" functions such as those proposed by Feddes et al. (1978) or van
Genuchten (1987), in combination with Jarvis (1989), and 2) mechanistic functions
derived from the physical properties of the soil-root system, such as that of Nimah and
Hanks (1973a,b). An inherent assumption in all models, as previously mentioned, is that
all water taken up by roots is transpired, so that uptake equals plant transpiration.
14
2.2.2.1 Empirical macroscopic uptake models
According to the empirical approach, the potential root water extraction, Sp, at
soil depth z (L) and time t (T), is expressed as the potential transpiration, Tp (L T-1),
divided over the soil profile depth, zr (L), considering the spatial root distribution, β(z),
which is assumed constant with time,
r
pp z
T)z()t,z(S β=
[9]
Tp is a function of the atmospheric demand, and can be measured through a crop water
balance for optimal conditions, or calculated using either mechanistic or empirical
equations, such as the Penman-Monteith equation (Monteith, 1965; FAO, 1990) or the
Hargreaves formula (Jensen et al., 1997), respectively (Šimůnek and Hopmans, 2009).
Under non-optimal conditions, such as drought or salinity, the actual plant
uptake is less than the potential. The effect of such stress factors can be introduced into
[9] through a dimensionless, uptake reduction function, γ (-), controlled by soil and/or
root water potentials and fluxes (Hoogland et al., 1981; Hopmans and Bristow, 2002;
Feddes et al., 2004). Thus, the relation between the actual uptake, S (L3 L-3 T-1), and Tp
under environmental stress for a given soil depth and time becomes,
p(h(z))Tβ(z)t)S(z, γ= [10]
Feddes et al. (1978) proposed a piecewise linear formula for uptake reduction
due to oxygen and water stress at high and low soil water potentials, respectively. The
equation is parameterized by four critical soil water potentials, h0, h1, h2, and h3 (L),
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
≤≤
<<−−
≤≤
<<−−
=γ
hhorhh,0
hhh,hhhh
hhh,1
hhh,hhhh
)h(
03
0101
0
12
2332
3
[11]
15
According to Feddes et al. (1978), plants are able to transpire at their potential
rate (γ = 1) when the soil water potential is low enough to avoid oxygen stress caused by
water logging h < h1, but high enough to prevent the onset of drought stress, h2 < h. The
drought-stress threshold parameter h2 is a function of the atmospheric demand, such that
a higher atmospheric demand increases the critical pressure head value h2 at which
uptake is reduced from the potential (Figure 4A).
The atmospheric demand in cultivated areas is supplied by stored soil water
primarily through transpiring plants. Therefore, the greater the daily evaporative
demand, the more soil water, characterized as a function of h, plants will need to take up
in order to transpire at their potential rate. Below h2, uptake linearly decreases to zero as
the soil approaches the wilting point h3.
van Genuchten (1987) proposed an alternative, non-linear, S-shaped reduction
function (Figure 4B) requiring only two input parameters,
p
50hh1
1)h(
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=γ [12]
where h50 (L) is the soil pressure head corresponding to a 50% reduction in root water
uptake from the potential, and p (-) represents an empirical shape factor.
h2 high h1h2 low h0h30
0.2
0.4
0.6
0.8
1
Soil water potential (L)
Upt
ake
redu
ctio
n fu
nctio
n γ
(-)
- -
Tp = 1 mm d-1
Tp = 5 mm d-1
h500
0.2
0.4
0.6
0.8
1
Soil water potential (L)
Upt
ake
redu
ctio
n fu
nctio
n γ
(-)
Figure 4. (A) The Feddes et al. (1978) linear stress response function at varying soil water potentials for two potential transpiration values and (B) the van Genuchten (1987) non-linear stress response function at varying soil water potentials.
A B
16
In the Feddes et al. (1978) and van Genuchten (1987) models, each soil layer
contributes to root water uptake independently from neighboring layers in accordance
with the root distribution and γ, which is a function of h [11], [12]. Therefore, as shown
in Figure 4, when the h at a certain soil layer reaches a critical value, h1 < h, h2 > h and
h < 0 cm for the Feddes et al. (1978) and van Genuchten (1987) models, respectively, γ
drops below one, and uptake decreases from the potential.
2.2.2.1.1 Compensatory uptake in empirical macroscopic models
While the negative effects of drought stress on plant transpiration are somewhat
intuitive, research has shown that plants make an effort to maintain potential
transpiration in drying soils (English et al., 1996; Kang et al., 2002; Stikic et al., 2003;
Leib et al., 2006). This is done through compensatory uptake, whereby plants increase
uptake rates from a smaller volume of roots located in wetter soil regions in order to
compensate for the decrease in water uptake from more heavily-rooted, but drier layers
near the soil surface (Arya et al., 1975; Nnyamah et al., 1977; Jarvis, 1989).
Experiments carried out by Molz et al. (1971) and Herkelrath et al. (1977a,b)
with sorghum and wheat, respectively, revealed that the zone of maximum root water
extraction shifts away from the soil surface to the wetter regions of the soil,
characterized by a smaller root density. These results were in contrast with the original
uptake models proposed by both Molz et al. (1971) and Herkelrath et al. (1977a,b)
which underestimated uptake in the sparsely-rooted zones and overestimated uptake in
the root-dense zones closer to the soil surface. Molz et al. (1971) attribute this
discrepancy between field results and model predictions to a root distribution function
which varies with time; they state that the increase in uptake must be attributed to the
addition of roots to wetter regions, deeper in the soil profile. Another more pronounced
example of compensatory uptake is reported by Reicosky et al. (1972), who claim that
17
in the vicinity of a water table, a very small portion of roots, around 20-25%, were able
take up 80-90% of the transpired water.
Both compensatory and non-compensatory models foresee a shift in the zone of
maximum water extraction as the soil dries in accordance with the established
dependence of root water uptake on the soil water status (Herkelrath et al., 1977a,b).
The observed increase in root water uptake in the wetter soil regions under drying
conditions seems to highlight the more dominant role of the soil hydraulic properties
rather than the root distribution, on root water uptake. This explains why the plant water
requirements under drying conditions can be met, as reported by Reicosky et al. (1972),
by a minimal portion of the root zone.
Jarvis (1989) considered the effect of compensation in predicting uptake
patterns by weighting the product of the stress reduction factor, γ, and the root
distribution, β, in [10] by an additional parameter, ω (-) [14]. Figure 5 depicts the range
of the weighted stress index as between zero and one; stress-free conditions in the root
zone are indicated by a ω value of one. For ω values above a critical threshold ωc,
uptake in water-stressed regions of the soil is fully compensated for by increased uptake
throughout the root zone by a factor of 1/ω.
Although ω is applied uniformly over the entire root zone, the contribution to
compensatory uptake is greater from stressed regions, in proportion to the water stress
function and root density. Below ωc, compensation is partial, by a factor of 1/ωc over
the root zone, and the uptake rate, or the actual transpiration, Ta (L T-1), falls below the
potential (Skaggs et al., 2006),
1T))z(h()z(S p ωγβ= [13]
∫ γβ=ωrz
0
dz))z(h()z( [14]
18
Figure 5 highlights the difference in uptake as modeled with a compensatory
(Tac/Tp) and non-compensatory (Ta/Tp) approach. The line representing compensation
illustrates how uptake remains maximal so long as ω values are higher than ωc. With the
non-compensatory approach, ωc = 1, uptake is a monotonically decreasing function with
decreasing ω values, or conversely, increasing water stress.
0
1
0 1
Tac/Tp
(-)
0
1
ω (-)
Ta/Tp
(-)
Non- compensation
Compensation
Figure 5. Ratio of the actual to the potential transpiration as a function of a stress index ω when considering and excluding compensation.
While the Jarvis (1989) model for uptake compensation shows agreement with
experimental evidence (Stikic et al., 2003; Leib et al., 2006), questions have been raised
regarding its conceptual soundness, and further work is needed to develop a model that
quantifies uptake compensation in a more mechanistic fashion. For example, for a ωc
value of zero, the partial compensation index 1/ωc is theoretically undefined.
Furthermore, a conceptual problem arises in instances when the soil water potential and
the resulting water stress are uniform throughout the root zone, for ω > ωc (Skaggs et
al., 2006; Šimůnek and Hopmans, 2009). The increase in uptake in such cases is equal
for each soil layer due the uniformity of water stress with depth, and cannot thus be
described as 'compensating' for the reduced uptake in drier soil regions.
⎪⎩
⎪⎨
⎧
ω<ωωω
≤ω<ω=
cc
c
p
a
,
1,1
TT
[15]
ω (-)
ω 1
19
2.2.2.2 A mechanistic macroscopic uptake model
Nimah and Hanks (1973a,b) developed a macroscopic root water uptake model
which, similar to the Gardner (1960) microscopic model, is based on the physical
properties of the soil-plant system,
According to Nimah and Hanks (1973a,b), root water uptake, S(z,t) (L3 L-3 T-1),
is expressed as a function of the water potential difference between the root, Hroot (L),
and soil, a root distribution function, β(z) (-), and the unsaturated hydraulic conductivity,
K(h) (L T-1), over a soil layer, Δz (L).
The mechanistic nature of the Nimah and Hanks (1973a,b) model considers the
effect of physical processes on root water uptake, such as pressure gradients between the
plant and soil, and variation in the unsaturated hydraulic conductivity. Homaee et al.
(2002b) note the importance of parameterizing root uptake models with consideration to
the soil hydraulic properties. They state, "The water supply by soil to plant roots largely
depends on the soil hydraulic conductivity…Clearly, the hydraulic properties differ
largely from one soil to another and cannot be ignored in root water uptake studies."
In a manner similar to the Feddes et al. (1978) and van Genuchten (1987)
models combined with the Jarvis (1989) model, the Nimah and Hanks (1973a,b) model
inherently accounts for compensatory root water uptake. This is done by altering Hroot to
achieve maximum uptake over the soil profile. A function of plant, climatic, and soil
conditions, Hroot is bounded by the wilting point as the soil dries and by the soil pressure
head at saturation under wet conditions. Only after Hroot decreases below the Hwilt will
uptake decrease from the potential transpiration.
wiltrootp
wiltrootp
root
HH T)t,z(S
HH T)t,z(Sz
))t,z(hH()z()h(K)t,z(S
<<
>=
Δ−⋅β⋅
=
[16]
20
Empirically parameterized macroscopic models have an advantage over the
mechanistic approach in that they do not require complete insight into the physical
processes of root water uptake. This eliminates the need for obtaining parameters
characterizing root processes, such as Hroot, which are difficult to measure in situ
(Homaee et al., 2002b). However, empirical models require calibration for varying
conditions, and the calibration process is often time consuming and error-prone.
New computerized methods are being developed for obtaining model parameters
in a stochastic manner based on measurements of system variables such as the soil water
potential and water content. The implementation of these methods 'normalizes' the
technical difficulty involved in quantifying uptake through either an empirical or a
mechanistic approach, as the manner in which parameters are 'calibrated' is uniform for
both types of models. These new tools therefore shift the focus of modeling root water
uptake away from the need for mathematical simplicity, to better characterizing the
physical processes involved in plant water uptake. Thus, as technology improves, so
should our understanding of the root water uptake process.
2.3 Numerical solutions for macroscopic root water uptake models
2.3.1 Computer software available for modeling water transport and uptake
The Richards (1931) equation upon which macroscopic root uptake models are
based is a highly non-linear, partial differential equation. Due to its complexity, few
analytical solutions have been proposed. Instead, numerical methods have proved the
most expedient for solving the uptake problems. Present-day computing capabilities
have improved the efficiency of numerical modeling (Šimůnek et al., 2008a), putting an
end to difficult and tedious, manual iterative processes, and broadening the scope of the
water flow and uptake problem to one of multi-dimensions (Vrugt et al., 2001b).
Numerous software programs have been developed to numerically solve the
Richards (1931) unsaturated soil water flow equation with a sink term. SWATRE
21
(Belmans et al., 1983), SWIF (Tiktak and Bouten, 1990), HYSWASOR (Dirksen et al.,
1993), WAVE (Vanclooster et al., 1995), and HYDRUS-1D (Šimůnek, 1998) are
among the one-dimensional models. Multi-dimensional soil water flow and root uptake
can be solved with the HYDRUS-2D/3D software packages (Šimůnek et al., 2008b).
All three HYDRUS models use the finite element method in the spatial domain
and the finite difference method in the temporal domain (Šimůnek and Hopmans, 2009),
according to the mass conservative iterative scheme proposed by Celia et al. (1990).
Among the wide range of processes covered by this model is root water uptake as a
function of water stress. Up until recently, only the empirical macroscopic models of
Feddes et al. (1978) and van Genuchten (1987), in combination with Jarvis (1989), were
implemented into HYDRUS-1D. Model parameters for these expressions [Tp, β(z), h]
are relatively easy to obtain (Wu et al., 1999), and experimentally measured, critical soil
water potential values of the Feddes et al. (1978) model, for various crops, have been
amassed into a large data base (Taylor and Ashcroft, 1972; Wesseling et al., 1991),
accessible through the GUI. Recently, the HYDRUS-1D code was updated to include a
solution for the Nimah and Hanks (1973a,b) uptake model.
2.3.2 Parameter estimation through inverse modeling
When solving the root water uptake problem numerically, a reliable method is
needed for not only obtaining initial and boundary conditions, but also for rigorously
identifying model parameters. Parameter uncertainty can lead to errors in model
predictions for water fluxes, water potential values, and uptake distributions. One
example of this was reported by Musters et al. (2000), who showed that uncertainties in
the soil hydraulic and root water uptake parameters in the SWIF model had a much
greater affect on water content values than on root water uptake rates.
Parameter uncertainty intervals are often determined by the quality of data used
for calibration, or in the case of some soil hydraulic parameters, the error margin of
22
specific direct-measurement methods. Musters et al. (2000) calibrated the SWIF model
for a reference, 'true' parameter set using three years of water content data, totaling more
than 125,000 measurements. However, this amount of information is not always
available; and while field measurements can be employed to obtain input parameters,
they are often time-consuming and expensive (Hupet et al., 2005). In addition, many of
the soil hydraulic and root water uptake parameters in macroscopic models must be
inferred inversely, by a trial-and-error process whereby parameter values are adjusted to
produce model results which match the behavior of the real system (Vrugt et al., 2003b).
In inverse modeling, the discrepancy between observations and model-generated
data is expressed by an objective function (OF), oftentimes in the form of the sum of
square errors, such as this example of an OF for the soil water content,
where b is the vector containing j number of parameters, N is the number of
observations, and )(* 1tθ and ),( 1 btθ represent the measured and model-predicted water
content values, respectively, at time t (Vrugt et al., 2001b), at a certain soil layer. The
true parameter set is the b vector corresponding with the OF global minimum across the
j dimensional parameter space.
Inverse modeling was originally done by manually adjusting parameter sets
between successive iterations. The approach is limited in its effectiveness not only
because of its tedious and labor-intensive nature, but also because its success largely
depends on the experience of the modeler (Vrugt et al., 2003b). The 'automatic'
approach to inverse modeling relies on computers, programmed to both numerically
solve, for example, the root uptake problem, and search out the best parameter set or
range for the next iteration.
[ ]∑=
−=N
1i
2jiij )b,θ(t)(t*θ)OF(b [17]
23
A number of both local and global search methods have been developed, two
examples of which are the Levenberg-Marquadt (Marquadt, 1963) non-linear, gradient-
based method employed for the HYDRUS-1D inverse modeling, and the Differential
Evolution Adaptive Metropolis (DREAM) (Vrugt et al., 2009), respectively. When
combined with numerical model solvers, these algorithms assure faster and more
accurate convergence on true parameters sets than manual trial-and-error methods
(Vrugt et al., 2003a). While the latter method has proved successful in optimizing up to
12 model parameters (Vrugt et al., 2001b), the former methods are suitable for the
optimization of only 1 or 2 parameters (Hupet et al., 2005).
Ritter et al. (2003) compared two inverse methods for estimating the hydraulic
properties of a soil profile as an alternative to direct measurements. Soil hydraulic
parameter estimates were directly obtained by fitting a soil water retention curve,
determined using Tempe cells, to the van Genuchten (1987) model. The saturated
hydraulic conductivity was measured using constant head permeameters. The three most
sensitive parameters (θs, θr, n) to the measured soil water content, as determined by a
sensitivity analysis, were inversely optimized for using two methods: a traditional trial-
and-error method, and a global, multilevel coordinate search algorithm combined with
the local Nelder-Mead simplex algorithm (GMCS-NMS). Method robustness was
decided upon based on the magnitude of the root mean squared error (RMSE) between
measured vs. modeled water content values.
Results showed that optimizations performed with the global search method
produced the smallest RMSE values (0.0257) while direct laboratory measurements
produced the largest RMSE values (0.0447). The RMSE value for the trial-and-error
method was in between (0.0319). These results clearly highlight the importance of the
methods by which model parameters are estimated, and demonstrate the robustness of
powerful global search algorithms over both manual inverse and direct methods.
24
The success of an inverse parameter optimization method is determined by how
well problems are posed. Problem posedness can be characterized by three main
aspects: stability, identifiability, and uniqueness (Ritter, 2003). Parameters are termed
unstable when small errors in measured variables or fixed parameters cause large
fluctuations in optimized parameter values. Identifiability and uniqueness are inversely
related, so that if the same output data is produced from two different parameter sets, the
true parameter set is unidentifiable, while when two parameter sets result from a single
set of model response data, the problem is termed non-unique (Russo et al., 1991;
Hopmans and Šimůnek, 1999).
2.3.2.1 Local search algorithms
Ill-posedness can manifest as the convergence of parameter values on local
minima. This is a common problem in local gradient-based search methodologies,
specifically those relying on first-order approximations (Vrugt and Bouten, 2002). Even
with non-linear methods convergence problems arise. In the Levenberg-Marquadt
(Marquadt, 1963) method, for example, the objective function is assumed to behave as a
smooth, convex response surface over the parameter space. The parameter search
therefore progresses in the direction of a decreasing OF until a minimum value is
reached. The problem with this methodology is that the OF response surface is rarely
purely convex, but rather full of small and large local minima as well as continuous first
derivatives (Woehling et al., 2008). This behavior often results from a correlation
between parameters, which is why it is suggested to keep the number of optimized
parameters at a minimum when using local search methods.
Šimůnek et al. (1996) and Hupet et al. (2005) both reported that they
encountered non-uniqueness problems when using local search methods to optimize for
3 soil hydraulic and 2 root water uptake parameters, respectively. Both groups attribute
the type, quantity, and quality of the information in the objective function as the
25
determining factor in the success of the inverse optimization scheme, and not the type of
search algorithm employed, whether local or global.
But while the argument put forth by both Šimůnek et al. (1996) and Hupet et al.
(2005) has been echoed in the literature by other soil hydrologists (Musters et al., 2000;
Zuo et al., 2004), Vrugt et al. (2003a) argue the contrary. Vrugt et al. (2003a) claim that
non-uniqueness problems do not necessarily arise from a lack of information in the
objective function, but rather because the widely-used, local-search optimization
procedures are not powerful enough to converge on a global minimum given the rough,
multi-modal response surfaces characteristic of hydrological models.
2.3.2.2 Global search algorithms
Algorithms which thoroughly exploit the parameter space have an advantage
over local search methods; convergence on a minimum value using the global search
methods results from numerous starting points across the parameter space, while only a
single initial estimate is used to establish convergence with local search methods.
Genetic algorithms (GA) are one example of a global search approach.
The GA search scheme is based on the principle of natural selection and natural
genetics, and the search mechanism is according to defined rules of probability. The
search procedure starts from a randomly selected population of chromosomes, or,
parameter sets. Evolution toward convergence happens stepwise, by generations, in
which chromosomes are chosen to continue on to future generations based on their
fitness. Surviving chromosomes are then broken up and modified to form new
populations used for subsequent iterations. The process continues until either reaching a
maximum number of iterations, or until a sufficient level of precision has been attained.
The global search algorithm DREAM (Vrugt et al., 2009b) carries out
population evolution using GA principles. DREAM avoids converging on local minima
by simultaneously focusing on multiple regions in the parameter space characterized by
26
a high posterior probability. The parameter search evolves from the starting points
through multiple Markov chains which eventually converge upon true values. DREAM
is able to handle non-linear parameter interdependence and multi-modality in the
posterior probability density function, which are both common in hydrological
modeling. It is thus a better alternative to first order or non-linear gradient-based search
methods (Vrugt and Bouten, 2002; Vrugt et al., 2003b; Vrugt et al., 2009b).
Vrugt et al. (2001b) report on the success of genetic algorithms in optimizing for
12 soil hydraulic and root distribution parameters simultaneously for a three-
dimensional waters uptake model. Good agreement was seen between simulated and
measured water content values, with a time-averaged RMSE of 0.018 m3 m-3, which is
comparable to the standard error of the calibration curve for neutron probe readings vs.
gravimetric measurements.
A vast amount of information concerning inverse modeling for soil hydraulic
parameters can be found in the literature (Russo et al., 1991; Ritter et al., 2003;
Lazarovitch et al., 2007; Vrugt et al., 2008). In contrast, few have applied the inverse
method in the context of plant water uptake. Much research is still needed, therefore, to
evaluate the effectiveness of inverse modeling of plant water uptake parameters.
2.3.3 Previous research on the performance of root water uptake models
Numerical models such as the above mentioned HYDRUS -1D, -2D, -3D offer a
number of options for the uptake reduction function under stress conditions. However,
little information is available concerning model preference for various crops and
environmental conditions. One example of a study comparing the effectiveness of
different models in predicting soil water dynamics is that reported by Rasiah et al.
(1992). They assessed the effectiveness of two parameter estimation methods, linear vs.
nonlinear, combined with water stress reduction functions, continuous and
discontinuous, in simulating root water uptake.
27
The continuous reduction function supports the hypothesis that in a profile of
uniform soil water potential, uptake per unit root length decreases with depth (Belmans
et al., 1983; Mahey et al., 1984), according to the root distribution. The second
reduction function was formulated according to a contrary assumption, namely, that root
water uptake does not necessarily decrease with depth, but is discontinuous, and
parameterized at each soil layer by a distinct coefficient.
Rasiah et al. (1992) found a better fit between measured and model-generated
water content values, root water uptake rate distribution, and rooting depth, when using
the discontinuous reduction function with parameters estimated by the non-linear
estimation method. The success of the non-linear method as compared to the first-order
approximation scheme may be due to the non-linearity of the Richards (1931) equation.
The better performance of the discontinuous function over the continuous may indicate
compensatory uptake, as the shape of a compensatory uptake distribution often deviates
from the root distribution curve in a manner which may seem discontinuous.
Homaee et al. (2002a,b) and Skaggs et al. (2006) compared the performance of
different uptake reduction functions for water stress and combined water and salinity
stress, respectively, in reference to experimental data. Homaee et al. (2002a,b) tested
four different water potential-dependent uptake reduction functions for drought stress,
proposed by Feddes et al. (1978), van Genuchten (1987), Dirksen et al. (1993), and
Homaee (1999). The first two functions are described in detail in section 2.2.2.1, while
the latter two are modified versions of the non-linear van Genuchten (1987) expression,
and can be briefly described as single and double-threshold functions, respectively.
Results of the measured Ta/Tp as a function of the soil water potential, as
calculated with all four functions, show that the Feddes et al. (1978) linear expression
had the worst fit with measured data. The van Genuchten (1987) and Dirksen et al.
(1993) functions fit half the data range, while the non-linear two-threshold function
28
showed a reasonable fit to measurements. Nevertheless, a quantitative comparison
between experimental and simulated actual transpiration revealed little difference in
results between the four models.
Skaggs et al. (2006) demonstrated the difficulty of discriminating between the
various uptake reduction functions in terms of their performance against experimental
data. They compared measured and simulated bottom flux values for a threshold-slope
multiplicative water and salt stress reduction function, and an S-shaped reduction
function for salinity. No notable difference was observed in the fit of modeled and
measured data between models. Skaggs et al. (2006) attribute this to the insensitivity of
both models to bottom flux measurements. They conclude that although additional root
zone data might allow for discrimination between functions, different models can
essentially be calibrated to produce the same results. They attribute the inconclusiveness
of their results to the limited physiological basis of the empirical, macroscopic stress
reduction functions.
3 Statement of problem
3.1 Simultaneous model comparison in varying conditions
Research in the field of root water uptake has led to the development of both
microscopic and macroscopic uptake models. The continual refinement of macroscopic
models has produced a range of uptake functions, empirical and mechanistic,
compensatory and non-compensatory, which are dependent on both the soil water
content and root physical properties.
However, little information is available concerning model robustness in varying
conditions. A thorough comparison of the available models is therefore needed in order
to better characterize the role of the soil, root, and soil-root interface parameters in water
29
uptake. Thus, the effects of new water management techniques such as regulated deficit
irrigation and partial root zone drying (Lovey et al., 1997; Stoll et al., 2000), on plant
water uptake can be better predicted using numerical models. In addition, the
effectiveness of inverse search methods for model parameters has not been thoroughly
evaluated for different uptake functions. By assessing the performance of different
uptake models in varying soils and with varying plants, future modelers will be able to
construct a more robust model characterized by a wider performance range.
30
3.2 Research hypothesis and objectives
3.2.1 Hypothesis
The soil hydraulic properties, the root distribution, and the soil-root interface
properties all affect the root water uptake pattern, but to varying degrees. In a vertical
soil profile under drying conditions, a plant transpires at its potential rate by shifting the
zone of maximum uptake to wetter soil regions, further away from the soil surface. And
since the density of plant roots is generally a decreasing function with distance from the
soil surface, fewer roots sustain maximum transpiration as the soil dries
In light of this, we hypothesized that the plant's ability to compensate for the
decreased uptake in one area of the soil by increasing uptake in another, is indicative of
the dominant role that the water retaining and transmitting properties of the soil play in
root water uptake. Among the modified Feddes et al. (1978) [with compensation
according to Jarvis (1989)], modified van Genuchten (1987) [with compensation
according to Jarvis (1989)], and Nimah and Hanks (1973a,b) models, the third model
was predicted as the most robust, due to its consideration of the soil hydraulic
conductivity in the uptake process.
3.2.2 Research objectives
The objectives of the research were:
1. To carry out controlled lab experiments with the purpose of amassing a broad
database of the spatial and temporal soil matric head in the root zone
2. To calculate and calibrate the parameters of the 3 macroscopic root water uptake
models for two soils and three plant types with a global search algorithm
3. To compare the performance of the different models for
root water uptake in well watered and deficit conditions in
two soils and with two plant types
31
4 Research methods
A combination of experimental work and computer modeling was necessary for
the calibration and validation of the root water uptake models mentioned in section
3.2.1. Unknown model parameters were first optimized for using a global inverse search
algorithm; system variables, as measured from cropped soil columns kept in optimal
conditions, were provided as input data. The model calibration was then validated under
drought stress conditions with data from an upscaled experimental system consisting of
24 rotating lysimeters. Both experimental systems were located in a plastic-house at the
Sede Boqer campus of the Ben-Gurion University in Israel.
Date Experiment Soil Plant Irrigation regime
February – May, 2009 Column Fine sand + Loess Wheat (Yuval 1225) Optimal
August, 2009 – May, 2010 Column Fine sand + Loess Tomato (VT-60775) Optimal
May, 2010 – October, 2010 Column Fine sand + Loess
Sorghum (Sorghum bicolor) Optimal
December, 2009 - April, 2010 Rotating system
Fine sand + Loess Wheat (Yuval 1225) Optimal +
deficit
May – September, 2010 Rotating system
Fine sand + Loess Tomato (VT-60775) Optimal +
deficit Table 2. Experimental timeline and general description. 4.1 Column experiments for model calibration
Six soil-filled PVC pipes (Palad, Israel), 15 cm in diameter, and 150 cm in
length, served as growth columns during the first experimental stage. In order to
successfully calibrate root water uptake models for varying soils and root distributions,
a total of three successive column experiments were carried out with two soils, local
loess and artificially manufactured fine sand (Negev Industrial Minerals Ltd, Israel), and
three different plants: wheat (Triticum var. Yuval 1225, Agridera, Israel), tomato
(Solanum Lycopersicum var. VT-60775, Syngenta, Israel), and sorghum (Sorghum
bicolor) (Table 2).
32
A number of preliminary soil checks were performed in order to estimate the soil
hydraulic properties necessary for parameter optimizations. The particle size distribution
of both soils was first measured using the hydrometer method, as outlined by Chen et al.
(2003b). This soil check enabled us to obtain initial SHP estimates from the Rosetta Lite
program (Schapp et al., 2001) accessible through HYDRUS-1D. The bulk density, ρb
(M L-3), shown in Table 9, was controlled by packing a constant mass of soil into a
specified volume, corresponding with 10 cm depth increments. Prior to packing the
columns and lysimeters, soils were sifted with a sieve < 1 mm (Ari J. Levy, Israel).
Soil moisture curves were also measured for both soils with the purpose of
determining θs and θr. The hanging cup method (Vomocil, 1965) was used for sand and
higher matric potentials in loess, while lower matric potentials in loess were measured
with a pressure chamber (5 Bar #1600, Soil moisture, USA). The soil water content in
all instances was determined gravimetrically.
Seeding density varied with each plant, and plants were thinned out as the
experiment progressed. Sorghum was seeded one month later in loess soil than sand due
to technical problems. A total of seven wheat plants, one tomato plant, and one sorghum
stalk were cultivated until maturity in each column, for each experiment. Tomato
flowers were continually removed to allow for continued vegetative growth.
1 W, S
2 3T
4 W, S
5 T
6
SAND
LOESS
Figure 6. Arrangement of growth columns. Wheat and tomato plants were grown in three columns for each soil type, while one column per soil was used with sorghum. Data from one replicate per soil was used for modeling purposes, as indicated by the red letters: W (Wheat), T (Tomato), and S (Sorghum).
33
In the first two experiments, six columns were used for plant propagation,
providing three replicates for each soil type. Nevertheless, data for model calibration
was taken from one column per treatment at the end of each experiment (Figure 6).
4.1.1 Experimental system
The design of the columns, displayed in Figure 7, allowed for the monitoring of
the soil water potential at small time and depth intervals, along the vertical axis. Soil
water potential readings, irrigation, drainage, and the temperature of the greenhouse
were recorded every 15 minutes by a data logger (CR10X, Campbell Scientific Inc.,
USA). The soil water potential along the columns was monitored by 15 high-flow
ceramic cups (Ami, Israel) positioned at 10 cm depth increments along the column.
Figure 7. Illustration of a sample growth column: (A) Column, (B) Tensiometer, (C) Pressure Transducer, (D) Data logger, (E) Multiplexer, (F) Tipping bucket, (G) Irrigation valve (H) Irrigation tube, (I) Water container, (J) Load cell.
We aimed at getting better resolution during the last column experiment with
sorghum by adding 5 more tensiometers toward the root-dense surface layers, where the
greatest information concerning soil water dynamics was believed to be. This gave us a
I
H G
B C
A
D
E
F
J
34
total of 20 tensiometers for both the loess and sand columns for the sorghum
experiment, spaced at 5 cm increments from 5 cm until 55 cm, and 10 cm increments
from 55 cm until 145 cm.
Each porous cup was connected on the outer wall of the column to pressure
transducers (26PCCFA6D, Honeywell, USA) through plastic tubing. A diaphragm
inside the pressure transducers registers the differential pressure (potential) between the
soil water and the atmosphere, as an electrical signal, in units of milivolts. Translating
between milivolts to pressure units of cm H20 required calibration of pressure transducer
at a known water column height. A sample calibration curve is displayed in Figure 8.
y = -22.107x + 0.64R2 = 1
-40
-30
-20
-10
0
10
20
30
40
-1.5 -1 -0.5 0 0.5 1 1.5 2
Volatage (mV)
Wat
er c
olum
n he
ight
(cm
)
Figure 8. Sample calibration curve of pressure transducers.
We discovered during the first two column experiments that tensiometers could
not withstand high suction over time from the root-dense surface layers (0 - 50 cm) and
were continually malfunctioning as a result of air entry around h = -700 cm. We
therefore installed water content sensors for the final experiment with sorghum (5TE,
Decagon Devices, USA) at 4 and 3 locations along the sand-filled and loess-filled
columns, respectively, at 5 cm depth intervals. Water content sensors were installed
directly across from tensiometers in order to assure ongoing measurements at surface
observation nodes under extreme suction. Sensors were calibrated by fitting a van
Genuchten (1980) function to measured θ vs. h data, obtained from water content and
tensiometer sensors, respectively, located at corresponding depths. The Microsoft Excel
35
solver tool was used to isolate the SHP which gave the best fit for each sensor, and
calibration information for both soils is included among the files in Appendix I.
Irrigation was automated according to scheduled times, frequencies, and water
volumes indicated in a computer program (PC208W, Campbell Scientific Inc., USA).
Prior to the irrigation of each column, water was pumped (SF 8290, Flotech, USA) into
a container placed on a load cell (AST 20 kg, Tedea Huntleigh, USA) about 4 m off the
ground, until the mass of the water container reached a pre-determined value. The
amount of irrigation was registered as a reduction in the mass of the container due to
water efflux. Since the load cell registers mass as an electric signal, calibration was
necessary to translate between milivolt units to those of mass. We further assumed a
density of 1 kg L-1 for the irrigation water, thus enabling a direct interchange between
units of mass and volume.
Efflux from the water container was powered by gravity. Six solenoid pilot
control valves (S-Series, Bermad, Israel), positioned along the main irrigation tube,
controlled water flow into columns. Thin plastic tubes branched off from the main
irrigation pipe to individual columns, and each tube was fitted with a single,
uncompensated button dripper (4 L h-1, Netafim, Israel); drippers were positioned at the
soil surface, in the center of each column. A small tube at the base of each column
channeled the drainage water into tipping buckets (Rain-o-matic small, Pronamic,
Denmark). Each time the water collection cup emptied, a pulse was registered by the
data logger. A calibration was therefore performed for each tipping bucket to determine
the volume of water collected before spillage. At the base of each column, a 5 cm layer
of rock wool was inserted prior to soil packing to prevent soil efflux.
The irrigation regime during both column experiments was optimal (I=1.2T),
which is defined as the amount of water needed to assure 100% of the plant
transpiration demand and 20% leaching to prevent salinity stress. For all three column
36
experiments, irrigation was scheduled during the night hours in order to minimize water
fluxes during the daylight hours. We used the following water balance for updating
irrigation according to plant water needs,
N
DIS
N
1iii∑
=
−= [18]
where the average daily uptake,⎯S (L T-1), is equated to the cumulative difference
between the applied irrigation, I (L3), and drainage, D (L3), over N days.⎯S was
calculated for each replicate, for both soil treatments, but changes in irrigation were
made according to the maximum⎯S value of each treatment. Soluble NPK fertilizer
(Deshen-Kol Poly Feed, Haifa, Israel) was applied to the main water tank at a ratio of 1
kg m-3, bringing the electrical conductivity of the irrigation water up to 1 dS m-1.
4.1.2 Experimental procedure
Plants were cultivated until maturity while monitoring irrigation, drainage, and
the change in the soil water potential along the root zone. Once plants were ready for
harvest, the above-ground biomass of each column was removed in order to compare
dry plant yields between soil treatments. The fresh biomass was first weighed (HP-30k,
Prisma, Israel), oven-dried at 64°, and then re-weighed.
The relative root density with depth was also measured at the end of each
experiment. This was done by excavating the soil in 10 cm increments from one column
for each soil treatment. Roots were washed from the soil using a sieve < 1 mm (Ari J.
Levy, Israel), oven-dried at 64°, and weighed (Fx-30001i, A&D company unlimited,
Japan). We considered β(z) (-) to be a function of root mass, expressed as the ratio of the
mass of dry roots extracted from each 10 cm layer mi (M), to the total dry root mass of
the profile MT (M), such that β(z) integrates to unity over the soil profile,
T
i
Mm)z( =β
[19]
37
4.1.3 Model calibration
The modified Feddes et al. (1978) (FD), modified van Genuchten (1987) (vG),
and Nimah and Hanks (1973a,b) (NH) root water uptake models were calibrated for
unknown parameters using the DREAM (Vrugt et al., 2009b) global inverse search
algorithm, run in parallel with HYDRUS-1D (Šimůnek, 1998) (for one-dimensional,
vertical water flow) through a series of MATLAB (The Math Works, 2010) codes. The
program runs HYDRUS-1D forward simulations using parameter values from within
initial ranges provided in the main operating file.
In order to minimize the number of optimized parameters, we attempted to
isolate the SHP by running DREAM with data from uncropped loess and sand, prior to
seeding. In the end, we decided to simultaneously optimize for both the SHP and RWUP
together due to apparent changes in the SHP following plant growth.
4.1.3.1 Inverse parameter search with DREAM
Each model run, or function evaluation, termed ndraw, consists of one
HYDRUS-1D forward simulation followed by an evaluation of the objective function of
model results vs. measured observations, and the progression of parameter values to a
'more likely' set. The DREAM algorithm essentially narrows down initial parameter
ranges to those values confined within the 95% confidence interval, and calculates the
most likely value for each optimized parameter. An explanation of three MATLAB files
which were varied between optimization schemes for different soil and plant treatments
is provided in Appendix III.
A total of 18 optimization schemes were carried out for the calibration of model
parameters for varying soil and plant treatments, as shown in Table 3. Function
evaluations were run step-wise, in a series of 5 x 3,000 simulations due to the
computational limitations of office computers, summing to a total of 15,000 function
evaluations per optimization procedure. The resulting 95% confidence interval (95% CI)
38
parameter ranges generated from the first 3,000 simulations were then entered as the
new initial parameter ranges for the successive set of 3,000 function evaluations.
Count Model Plant Soil 1 Loess 2
Wheat Sand
3 Loess 4
Tomato Sand
5 Loess 6
FD
Sorghum Sand
7 Loess 8
Wheat Sand
9 Loess 10
Tomato Sand
11 Loess 12
vG
Sorghum Sand
13 Loess 14
Wheat Sand
15 Loess 16
Tomato Sand
17 Loess 18
NH
Sorghum Sand
Table 3. Count of optimizations carried out for the calibration of three models, three plants, and two soils.
The effectiveness of DREAM in converging on a global minimum when
optimizing for a large number of parameters lies in its ability to thoroughly search the
parameter space. This is done by carrying out multiple search schemes or 'sequences' in
parallel; the area in which sequences merge together is designated as the global
minimum. Convergence in our optimization runs was through 10 sequences, as shown in
the RunDream sample program file included in Appendix III.
4.1.3.2 The objective function
The objective function used for optimizations was based on the soil water
potential, h, alone. This is in accordance with Hollenbeck and Jensen (1998), who report
that an objective function comprised from multiple system variables does not
necessarily produce a better-posed solution. In a similar manner, Bachmann (2010)
showed the superiority of an objective function based on the water potential alone over a
mixed objective function in parameter optimization schemes.
39
We assured a strong information content of the data included in the objective
function by including tensiometric data from observation nodes at depths characterized
by high soil water dynamics. These high root density regions were within the top 50 cm
from the soil surface. Nevertheless, in order to provide information about the soil water
status over the entire flow domain, data from root-sparse regions was also included.
4.1.3.3 Boundary and initial conditions for HYDRUS-1D
Model parameter calibration and root water uptake calculations were performed
for mature plants with a fully developed root system, and therefore the effect of plant
growth on uptake rates and RWU model parameters was not considered. The soil profile
depth of columns extended to 150 cm, and was discretized at 1 cm intervals. The thin
rock wool layer at the base of columns was not accounted for as a separate material. For
the first two experiments, water dynamics were simulated over a 12.5 hour daylight
period, with print times at each 0.25 hr. interval. The simulation extended to two full
days for the final experiment with sorghum, with print times at each hour. No irrigation
events occurred during the simulation period for all data sets. The input files containing
soil water potential nodal information, initial conditions, and atmospheric boundary
conditions for wheat, tomato, and sorghum experiments are provided in Appendix I.
The soil surface was approximately 2 cm lower than the top of the column and
facilitated water ponding, especially in the loess soil which was characterized by a low
saturated hydraulic conductivity. We therefore chose Atmospheric BC with surface layer
for the upper boundary condition, which allowed for a ponded layer of water. The
bottom boundary condition was characterized as a seepage face. This condition assumes
a zero bottom flux in unsaturated conditions, but imposes a constant pressure head of
zero (h = 0) at the lower boundary when soil is saturated. The lowest permissible matric
potential allowed at the soil surface (Max h at soil surface) was set at -100,000 cm.
40
The hourly Tp rate at each time interval was also required as a boundary
condition. Since plants were irrigated at 120% of their transpiration needs, we assumed
that the actual transpiration of plants equaled the potential. But since columns were not
weighed, we were not able to perform an accurate enough water balance from which Tp
could be extracted. We were also not successful in calculating plant uptake from the
Richards (1931) equation using the tensiometeric data collected at regular time and
space intervals, nor considered it appropriate to use the Penman-Monteith (Monteith,
1965) formula which is meant for canopies stretching over a large fetch and not for
isolated plants. We therefore decided to optimize for the daily plant potential
transpiration⎯Tp, from which the hourly Tp rate could then be calculated according to a
sinusoidal shape function (Fayer, 2000),
18h) (6h, t,
2hr/day 24t2sinT2.75
h18 t6h, t , 0(t)T
pp
⎪⎩
⎪⎨
⎧
∈⎟⎟⎠
⎞⎜⎜⎝
⎛ π−
π
><
= [20]
The root distribution, β(z), was solved with the Raats (1974) and Vrugt (2001b)
function [7], while optimizing for three unknown root distribution parameters (RDF),
z*, zm, and pz. HYDRUS-1D normalizes β(z) according to the following expressing,
∫β
β=β
rz
o
)z(
)z()z(' [21]
to assure that β'(z) integrates to unity over the flow domain (Šimůnek et al., 2008b),
1dz)z('rz
0
=β∫ [22]
The initial pressure head at t = 0 was supplied by tensiometric/water content sensor data.
4.1.3.4 Initial parameter estimates
Parameter ranges for the first set of 3,000 function evaluations, for all models
and treatments, are presented below in Table 4. Altogether, four SHP of the van
Genuchten (1980) [2] and Mualem (1976) [3] equations, three root distribution
41
parameters (RDP) of the Raats (1974) and Vrugt et al. (2001b) model [7],⎯Tp, and the
root water uptake parameters (RWUP) of the FD [11], [13], vG [12], [13], and NH [16]
models were optimized for. Since previous work showed a correlation between the θs
and θr (Bachmann, 2010), we fixed θs at the suggested Rosetta Lite values.
A SHP, RDP, and
Tp for all models/ plants
θs (cm3 cm-3)
θr (cm3 cm-3)
α (cm-1)
n (-)
Ks (cm hr-1)
z* (cm)
zm (cm)
pz (-)
⎯Tp (cm day-
1)
Min - 0.003 0.005 1.06 0.1 0 46 1 0.5 Loess Max - 0.15 0.15 3.00 5.2 45 160 30 3 Min - 0.001 0.01 1.1 15 0 46 1 0.5 Sand Max - 0.1 0.15 6 70 45 160 30 3
B
FD RWUP ωc (-)
h0 (cm)
h1 (cm)
h2H (cm)
h2L (cm)
h2H (cm hr-1)
h2L (cm hr-1)
h3 (cm)
Min 0.001 -5 -30 -750 -12000 0.067 0.021 -20000 Wheat Loess/Sand Max 1 0 -6 -31 -751 0.125 0.063 -12001
Min 0.001 -20 -100 -1000 -3000 0.067 0.021 -10000 Tomato Loess/Sand Max 1 0 -21 -101 -1001 0.125 0.063 -3001
Min 0.001 -20 -100 -10000 -15000 0.063 0.021 -30000 Sorghum Loess/Sand Max 1 0 -21 -101 -10001 0.125 0.063 -15001
C
vG RWUP ωc (-)
p (-)
h50 (cm)
Min 0.001 1 -5000 Loess Max 1 7 -500 Min 0.001 1 -1000
Wheat/ Tomato/ Sorghum Sand Max 1 7 -50
D
NH RWUP Hwilt (cm)
Min -15000 Loess Max -1000 Min -15000
Wheat/ Tomato/ Sorghum Sand Max -500
Table 4. (A) Initial parameter ranges for the SHP, RDP, and⎯Tp, and initial RWUP ranges for the (B) FD, (C) vG, and (D) NH models, for all plants and soils.
The SHP ranges of both soils were chosen around references values predicted by
Rosetta Lite, based on the particle size distribution and bulk density. The lower
boundary of the n range was increased to 1.09 between simulations for the SHP of bare
soils to simulations for both SHP and RWUP in order to avoid convergence problems.
Initial Tp values were set wide enough to account for the effect of column dimensions on
transpiration (explained below in section 5.2.1.3), which increased daily⎯Tp values. The
starting RDP were set wide enough to account for column dimensions (zm) and allow for
either a linear or logarithmic vertical root distribution (pz).
42
Initial FD RWUP parameter ranges were chosen so as to include reference
values from databases for wheat, tomato, and sorghum, as provided by Taylor and
Ashcroft (1972) and Wesseling et al. (1991), respectively. The FD h2 parameter was not
optimized for directly, but was calculated from four optimized parameters, h2H, h2L, r2H,
r2L, according to Wesseling and Brandyk (1985) and Šimůnek et al. (1992).
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
>
<
<<−×−−
+
=
H2pH2
L2pL2
H2p2LPH2L2H2
H2L2H2
2
rT ,h
rT ,h
rTr ),Tr(rrhhh
h [23]
Parameters h2H (H=High) and h2L (L=Low) (L) represent h2 at high and low values for
the potential transpiration, symbolized by r2H and r2L (L T-1), respectively.
Values for p spanned the scale reported on by van Genuchten and Gupta (1993).
The h50 range was chosen so as to include values used by Homaee et al. (2002b) (-2,000
cm) and Segal et al. (2006) (-100 cm) in their research on root water uptake in drought
conditionscarried out with alfalfa grown in potting soil and sunflower grown in a loamy
sand soil, respectively. Similar h50 values were also used by Schoups and Hopmans
(2006) in their evaluation of model complexity and input uncertainty of field-scale water
flow and salt transport for a large variety of crops.
The initial parameter range for Hwilt included the traditional wilting point value
of -15000 cm (Koorevaar et al., 1983) as suggested by Nimah and Hanks (1973a,b).
Hwilt range values were set lower than the Hroot values presented by Shani et al. (2007)
(-6,000 cm) for a variety of field crops, including tomato. The initial h50 and Hwilt ranges
were kept constant for different plants but slightly adjusted between soils to account for
variations in water retention. For example, for similar environmental conditions, uptake
reduction in sand was assumed to preempt that in loess. However, Feddes et al. (1978)
parameters were adjusted only between plant treatments and not between soils.
43
4.1.4 Root uptake model comparison
Two error indices were devised for evaluating model performance under
different treatments. The first, φ1 (-), accounted for the most likely parameter estimates
(ML) in relation to the difference in the minimum and maximum values of the 95% CI
for each parameter,
NML
maxmin
i
N
1i1
∑=
−
=φ [24]
φ1 was averaged over N parameters for each optimization scheme (for one model, plant,
and soil). The second index, φ2 (-), was based on the squared error between measured,
h*, and modeled, h, soil water potential values over time, tj, at observation node, zi.
[ ]1000NN
)t,(zh)t(z*h
21
N
1i
N
1j
2jiji,
2
1 2
××
−=φ
∑∑= = [25]
4.2 Rotating lysimeter experiments for model verification
In the second research stage, root water uptake experiments were upscaled from
six soil columns to a rotating lysimeter system (RLS) (Lazarovitch et al., 2006)
containing 24 square growth containers, 50, 35, 70 cm in height, width, and length,
respectively. Two sets of experiments were carried out with the RLS- one with wheat
(Triticum var. Yuval 1225, Agridera, Israel), and the second with tomato (Solanum
Lycopersicum var. VT-60775, Syngenta, Israel). Seeding density was 40 and 4 plants per
lysimeter, for wheat and tomato, respectively.
Plant Treatment Lysimeter Soil Irrigation regime 1 1 - 3 1.2T 2 4 - 6
Loess 0.5T
4 10 - 12 1.2T Wheat
3 7 - 9 Sand
0.5T Table 5. Description of the four treatments in the first rotating lysimeter experiment.
44
Plant Treatment Lysimeter Soil Irrigation regime 1 1 - 2, 4 - 5 1.2T 2 13 - 16
Loess 0.5T
3 3, 6 - 8 1.2T Tomato
4 9 - 12 Sand
0.5T Table 6. Description of the four treatments in the second rotating lysimeter experiment.
All plants were provided with both optimal (I = 1.2T) and deficit irrigation (I =
0.5T) in order to evaluate the robustness of the calibrated root water uptake models
compared to measured data.
4.2.1 Experimental system
A schematic illustration of the rotating lysimeter is presented in Figure 10 (not to
scale). Irrigation, drainage collection, and rotation speed were controlled through a
series of PC208W computer programs (PC208W, Campbell Scientific Inc., USA),
through a CR10X data logger (CR10X, Campbell Scientific Inc., USA). Each growth
container was positioned on a load cell (Zemic, L6G, Holland) in order to monitor
changes in the mass resulting from root water uptake. Load cells were calibrated prior to
the start of each experiment. Lysimeters were filled with either loess or sand according
to the prescribed treatments listed in Table 5 and Table 6. The particle size distribution,
bulk density, and texture of both soils are listed in Table 9 in section 5.1.
Prior to irrigation, water and fertilizer solution were pumped (SF 8290, Flotech,
USA) up to a weighted bucket (Zemic, L6G, Holland), suspended above lysimeters. The
bucket mass was registered in a data logger before a valve opened which controlled
water flow down to lysimeters. The water flux was controlled by four uncompensated
button drippers (8 L h-1, Netafim, Israel), arranged in two rows, as shown in Figure 10.
1 g L-1 and 0.5 g L-1 of fertilizer were supplied for wheat and tomatoes, respectively.
Irrigation occurred at different times during the morning. The irrigation schedule
was updated multiple times each week according to the maximum⎯S for each treatment,
averaged over N days, and was increased by 20% to prevent salt stress. In addition to
accounting for water inflow (I) and drainage (D), the water balance included changes in
45
load cell readings (ΔW) over the course of each day, making it more accurate than the
water balance used to schedule column irrigation,
N
WDIS
N
1iiii∑
=
Δ−−= [26]
A sample water balance for four wheat treatments is presented in Figure 9.
-3
-1
1
3
5
7
100% 50% 100% 50%
Loess Sand
Treatment
Wat
er (L
)
DrainageIrrigationWeight changeUptake
Figure 9. The components of a sample water balance for wheat, as summed over three days (1/03/10 - 3/03/10). The water values were averaged over replicates for each treatment.
Figure 10. Rotating lysimeter system: (A) main irrigation/fertigation pipe, (B) main irrigation bucket on load cell, (C) irrigation bucket for each lysimeter, (D) lysimeter, (E) four button drippers on the soil surface, (F) drainage pipe filled with rock wool, and (G) drainage water collection bucket.
A B
C
D
E
F
G
46
Excess water seeped out of lysimeters through rock-wool filled drainage pipes
60 cm long, and was collected in small, sealed plastic containers. The effluent was
further channeled, at scheduled times, into a small container on a load cell (Zemic, L6G,
Holland), where the mass (converted to volume) of drainage water was registered.
4.2.2 Experimental procedure
Plants were cultivated until maturity while monitoring the daily and cumulative
transpiration. At the end of experiments, the dry plant yield and the relative root mass
with depth were measured in a manner similar to the column experimental procedure.
4.2.3 Model verification
The performance all uptake models under water stress conditions was evaluated
by comparing the cumulative Ta of the 50% irrigation treatments to the Ta, modeled with
HYDRUS-1D. Model performance with RLS data was assessed graphically, by
comparing the plots of the measured, cumulative Ta (cumTa) of the deficit-irrigation
treatments to the cumTa, as modeled in HYDRUS-1D.
4.2.3.1 Boundary and initial conditions
The lysimeter profile extended 100 cm to the water collection bucket located at
the base of the rock-wool filled pipe, and was therefore divided into two sections, or
materials. The soil-filled lysimeter constituted the upper 40 cm of the profile (material
1), while the rock wool-filled pipe was designated as the bottom 60 cm of the profile
(material 2). The SHP of materials 1 and 2 were taken from the results of DREAM
simulations with column data, and from Ben-Gal and Shani (2002), respectively. The
profile length was discretized at 1 cm intervals.
Medium Ks (cm hr-1) α (cm-1) n (-) θr (cm3 cm-3) θs (cm3 cm-3)
Rock wool 500 0.045 4.167 0.12 0.97 Table 7. The hydraulic properties of rock wool according to Ben-Gal and Shani (2002).
Irrigation and Ta data for HYDRUS-1D simulations were taken from one I=0.5T
treatment replicate, per soil, for both wheat and tomato experiments (Table 8). Tp input
data was taken from the measured transpiration of I=1.2T treatments, averaged over all
47
replicates per soil, per plant type. Ta values were simulated over a period of 20 days for
both wheat and tomato experiments, in time steps varying according to the irrigation and
Tp input data. The HYDRUS-1D time-variable boundary and initial conditions for wheat
and tomato are given in Appendix I.
Unlike the column data used for DREAM simulations, RLS data contained
simultaneous irrigation and transpiration events, as irrigation was often during daylight
hours. The irrigation as measured from water stress treatments was entered as the top
flux input data, while the transpiration of optimal irrigation treatments was used as Tp
input data. The same upper and lower boundary conditions implemented in DREAM for
column data were used for all HYDRUS-1D simulations with RLS data.
Plant Soil Lysimeter for Ta data (I=0.5T)
Lysimeters for Tp data (I=1.2T)
Loess 4 1, 2, 3 Wheat
Sand 7 10, 11, 12 Loess 14 1, 2, 4, 5
Tomato Sand 11 3, 6, 7, 8
Table 8. Lysimeters from which HYDRUS-1D input data was taken. Irrigation amounts were taken from one deficit-irrigation treatment, and the calculated cumTa values of plants in the same lysimeter were compared with modeled results. The calculated transpiration as averaged over all I=1.2T treatment replicates was used as Tp input data.
The RWUP produced from DREAM simulations with wheat and tomato column
data were entered in HYDRUS-1D as uptake model parameters. The effect of
compensation on the modeled cumTa was evaluated by running HYDRUS-1D
simulations with three different ωc values. In addition, the sensitivity of results to the
root distribution function was evaluated by solving for cumTa with two sets of root
distribution data: 1) β'(z) as calculated from optimized RDP (and normalized according
to [21] and 2) β(z) as fitted with solver to the measured RMW of water stress treatments.
We did not measure the soil water potential at the soil surface in lysimeters, and
therefore had to estimate the most probable initial h conditions. We therefore used two
sets of initial conditions for HYDRUS-1D simulations, which served as a type of
'confidence interval' in which we assumed the true initial h values were found. For the
48
first set, the profile was characterized as hydrostatic (z = -h), with -100 cm at the soil
surface and saturation at the base of the profile. For the second set, the soil surface h
was set to -500, which linearly decreased to zero at the base of the soil column. By
using these two sets of initial conditions, we were also able to test the sensitivity of
modeled plant transpiration to stored soil water.
49
5 Results and discussion
5.1 Soil properties
The SHP for fine sand and loess, as predicted with Rosetta Lite, are displayed
below in Table 9 alongside the results of the particle size distribution test and the bulk
density, ρb, (controlled during packing). (Both the particle size distribution and ρb are
required input information for Rosetta Lite neural network predictions.)
Soil % sand
% silt
% clay
Soil texture
ρb (kg m-3)
Ks (cm hr-1)
α (cm-1)
n (-)
θr (cm3 cm-3)
θs (cm3 cm-3)
Loess 74.38 17.33 8.29 Sandy Loam 1320 5.11 0.036 1.56 0.043 0.43
Fine sand 97.50 1.25 1.25 Sand 1500 45.78 0.031 4.06 0.054 0.373
Table 9. The particle size distribution measured with the hydrometer method (Chen et al., 2003), soil texture according to the USDA soil pyramid, bulk density (ρb), and Rosetta Lite (Schapp et al., 2001) SHP predictions for fine sand and loess.
The reference SHP values of Ks and n shown above in Table 9 were realistic for
both soil types. On the other hand, α and θr parameters estimated from Rosetta Lite
seemed uncharacteristic, as values for α are generally much higher for sandy soils than
loess. This is due to the fact that the α parameter alters the vertical 'spreading' of the
water retention curve in such a manner that smaller α values produce a curve
characterized by a wider range of matric potential values at which the soil is saturated,
while larger α values shorten the tension range of water saturation. In addition, θr values
are usually lower in sand compared to loess, as water retention is generally a function of
the clay and silt content of soil.
Nevertheless, the sand used in experiments was a fine-grained medium, and sand
pores are generally more homogeneous than those of loamy soils. Therefore, the sand
water retention may have been increased by the capillary forces of small soil pores
comprising the fine sand matrix. We therefore judged the SHP in Table 9 as reasonable,
and based our initial parameter ranges for optimizations around these values.
The soil moisture curves for sand and loess are presented in Figure 11. We were
not able to reach water content values in the θr range for either soil. Concerning sand,
50
the porous plate at the base of the hanging cup had an air-entry value of -70 cm, and we
did not check sand in a pressure chamber. And although loess was checked in a pressure
chamber, the pressure plate for which we had planned measurements had an air-entry
value of 1 Bar. In addition, the θs values obtained for both loess and sand were higher
than reasonable. We therefore took all SHP values, including θr and θs, from Rosetta
Lite predictions, based on the measured particle size distributions and bulk densities.
BLOESS
ASAND
900
720
540
360
180
0
0
15
30
45
60
75
0 0.1 0.2 0.3 0.4 0.5 0.6
◊ (cm3 cm-3)
-(h) (
cm)
Figure 11. Soil water retention curves for (A) fine sand and (B) loess soil.
A preliminary optimization for the SHP of uncropped loess and sand was carried
out with DREAM prior to solving for the RWUP of the three uptake models. Input data
was based on the soil water dynamics as measured with tensiometers in columns before
plant propagation. The resulting 95% confidence interval (CI) range and most likely
(ML) values for all parameters after 15,000 simulations are presented in Table 10.
Soil 95% CI/ ML Ks (cm hr-1) α (cm-1) n (-) θr (cm3 cm-3)
Min 2.85 0.005 1.06 0.020 Max 4.42 0.007 1.08 0.135 Loess ML 3.35 0.005 1.06 0.048 Min 21.47 0.037 2.80 0.005 Max 39.12 0.044 3.25 0.047 Sand ML 35.82 0.041 3.06 0.004
Table 10. The 95% CI and estimated ML values for the SHP of uncropped loess and sand.
θ (cm3 cm-3)
51
The ML values of Ks, α, and n for a bare loess and sand shown in Table 10 fall
below the reference values for uncropped soils (Table 9). The soil may have been
compacted following irrigation events, thus altering the porosity and the SHP in general.
The loess ML values of α and n converged at the lower boundary of the initial
parameter range (Table 4), possibly indicating that convergence (during the last set of
3,000 simulations) terminated in a local minimum. A correlation between α and n was
ruled out as the reason behind poor convergence, as the response surfaces plotted by
Bachman (2010) for α and n, for loam and loamy sand, showed a clear minimum. Thus,
the initial parameter ranges for α and n were either too wide, or simply bound at the
lower end by too small of values to allow for good convergence.
5.2 Column experiments
The yield, root distribution, and parameter optimization results for three column
experiments carried out with wheat, tomato, and sorghum are presented in the following
sections. All soil water potential readings as well as irrigation and drainage data
collected during experiments, for both soils, are included in Appendix I.
Differences can be seen between SHP values of uncropped soils generated with
data before seeding (Table 10), and SHP values optimized together with RWUP,
generated with data after plant cultivation (Table 12, Table 15, and Table 18). Because
of the apparent effect of root growth on the SHP, we decided not to fix these parameters,
but to optimize for the RWUP and SHP together.
5.2.1 Wheat
5.2.1.1 Plant yield
The dry wheat yield of all soil columns and the average yield of three columns
per soil treatment are presented in Figure 12. The wheat yield of loess column #4, from
which data was taken for simulations, was about 4 times higher than that of sand column
52
#1, from which data was taken. The average yield of loess-grown wheat was also
significantly higher than sand-grown wheat.
0
100
200
300
400
500
600
700
800
1 2 3 4 5 6 AverageColumn
Dry
yie
ld (g
)LoessSand
Figure 12. The above-ground biomass of wheat plants measured from three loess and three sand-filled columns, as well as the average yield for each soil. Data for DREAM simulations was taken from one column for each soil, marked by *. The standard deviation is indicated by error bars.
5.2.1.2 Relative root mass
The relative root mass (RRM) [19] with depth for loess and sand is presented in
Figure 13. The RRM at the soil surface was about 10% higher in loess than sand.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
10
30
50
70
90
110
130
150
Dep
th (c
m)
Relative root mass (g g-1)
Loess- 4Sand- 3
Figure 13. The relative root mass (RRM) with depth of wheat plants extracted from one loess and sand column, the number of which is indicated in the legend.
The Raats (1974) and Vrugt et al. (2001) root distribution function [7] was fitted
to the RRM wheat data with Microscoft Excel solver. The resulting RDP (Table 11),
were used to interpolate the root distribution at each 1 cm increment along the profile.
*
*
53
The 'measured' root distribution, β(z), was then compared to the modeled root
distribution, β'(z), as calculated from the RDP presented in Table 12, in section 5.2.1.3.
Plant Soil z* (cm) zm (cm) pz (-) ∫RM(z) (g) Loess 0.47 150 13.96 5.8
Wheat Sand 0 125 10.9 7.1
Table 11. The wheat RDP as obtained from fitting [7] to the measured RRM. The total wheat root mass ∫RM(z) for each soil treatment is also shown.
The total excavated root mass shown in Table 11 was greater in fine sand than in
loess, presenting an unexpected, inverse correlation with the yield data shown in Figure
12. A number of factors might have contributed to these results. Separating the roots
from the large amount of organic matter in loess soil, such as twigs, leaf litter, and old
roots, proved difficult and time consuming, and the sifting process was often terminated
before the entire fine root mass was collected. A second factor which may have raised
the sand RRM above loess was the failure to completely rinse sand grains from the
dense root mass excavated from the top soil layers. Another possible explanation for the
discrepancy between the root mass and yield data is that the drier conditions of the
sandy soil might have led to a higher root turnover rate. This would mean that much of
the roots sieved from sand were dead, and possibly long inactive in the uptake process.
5.2.1.3 Parameter optimization results
Presented in Table 12 and Table 13 are the results of 6 optimization schemes for
the SHP,⎯Tp, RDP, and RWUP required for calculating the root water uptake of wheat
grown in loess and sand, according to the FD [11], [13], vG [12], [13], and NH [16]
models. An error index, φ1 [24], was used to assess convergence robustness of the last
3,000 simulations (out of 15,000) for each parameter, for all treatments.
The resulting 95% CI and ML values for the SHP were generally representative
of both soil types. Differences can be seen in the SHP of each soil type between the
three models, as well as between values for cropped and uncropped soils (Table 10).
The magnitude of φ1 varied between parameters and treatments (soils and models).
54
Model- Plant
Soil- Column
95% CI/ ML
Ks (cm hr-1)
α (cm-1)
n (-)
θr (cm3 cm-3)
⎯Tp (cm day-1)
z* (cm)
zm (cm)
pz (-)
Min 1.15 0.031 1.09 0.093 2.91 24.6 89.1 6.4 Max 1.82 0.037 1.09 0.1 2.98 25.3 142.8 10.8 ML 1.22 0.036 1.09 0.099 2.98 24.8 97.2 6.7 Loess-4
φ1 0.549 0.167 0 0.071 0.023 0.028 0.552 0.657 Min 41.72 0.075 2.46 0.04 1.2 17.9 89.2 4.3 Max 56.09 0.088 2.59 0.057 1.41 18.8 117 6.1 ML 45.9 0.08 2.55 0.048 1.29 18.7 89.9 4.4
FD- Wheat
Sand-1
φ1 0.313 0.163 0.051 0.354 0.163 0.048 0.309 0.409 Min 0.31 0.027 1.09 0.078 2.86 25 80.9 4.5 Max 0.72 0.029 1.1 0.097 2.98 25.9 125 8.8 ML 0.51 0.029 1.09 0.081 2.97 25 126.7 8.8 Loess-4
φ1 0.804 0.069 0.009 0.235 0.040 0.036 0.348 0.489 Min 52.46 0.074 2.42 0.019 1.36 17 96.3 5.1 Max 59.82 0.084 2.54 0.039 1.55 17.9 123.7 6.8 ML 58.49 0.082 2.5 0.026 1.37 17.8 109.2 5.9
vG- Wheat
Sand-1
φ1 0.126 0.122 0.048 0.769 0.139 0.051 0.251 0.288 Min 0.79 0.038 1.09 0.091 2.93 26.2 86.7 19.2 Max 2.8 0.049 1.09 0.098 2.99 26.7 131.3 28.4 ML 0.99 0.042 1.09 0.098 2.99 26.5 96.3 19.1 Loess-4
φ1 2.030 0.262 0 0.071 0.020 0.019 0.463 0.482 Min 27.37 0.051 2.67 0.041 0.99 12 54.8 12.9 Max 57.51 0.076 3.17 0.071 1.23 12.3 73.8 20.8 ML 30.7 0.055 3.17 0.065 1.12 12.3 55.4 15.5
NH- Wheat
Sand-1
φ1 0.982 0.455 0.158 0.462 0.214 0.024 0.343 0.510 Table 12. The 95% CI, ML values, and an error index, φ1 [24] for the SHP,⎯Tp, and RDP of the FD, vG, and NH models with wheat for loess and sand. Plants were grown in three columns for each soil type, but data for simulations was taken from only one column, the number of which is indicated next to the soil type. Optimizations were carried out with data from full-grown plants.
All ML loess Ks values were lower in a cropped vs. uncropped soil, but higher in
a cropped sandy soil. Values for α and θr of cropped soils were similar between the
three uptake models, but were larger than values of uncropped soils. ML n values for all
models in cropped loess converged at the lower limit of the initial parameter range,
similar to the pattern observed for uncropped loess. We had raised the lower boundary
of the n parameter range to 1.09 between simulations for the SHP of bare soils and
simulations for the SHP and RWUP together, to prevent convergence problems.
However, the new lower boundary was apparently not set high enough, and should have
been increased to at least 1.1.
Of all the SHP values, convergence for Ks was the weakest, as indicated by the
larger φ1 relative to other parameters. In addition, while little difference could be seen in
the ML values for α, n, and θr between models, Ks seemed to be influenced by the
uptake models used for simulations. On the other hand, the n parameter was
55
characterized by the smallest φ1 of all SHP, although this was due to convergence
problems for loess soil. The sand SHP from NH simulations were the closest in
magnitude to the SHP of uncropped sand, while neither of the SHP values for the
cropped loess seemed to agree with those of uncropped loess.
Predicted⎯Tp values were higher for loess-grown plants than sand-grown despite
the fact that all treatments were grown in the campus greenhouse under identical climate
conditions. The variation in⎯Tp between plants grown in varying soils was realistic,
judging by the yield results presented in Figure 12; the dry yield of wheat plants from all
three loess-filled columns was significantly higher than plants grown in sand. In any
case, optimized⎯Tp values for plants grown in both loess and sand, for all models, were
larger than potential transpiration values reported in the literature (Jakobsen and Dexter,
1987; Asseng et al., 1998; Zhang and Oweis, 1999), which centered on 0.5 cm day-1.
One probable explanation for the unusually large transpiration rate for column-
grown plants was the large foliage relative to the small surface area of columns.
Transpiration is calculated in units of length per time under the assumption that the area
of the canopy from which water transpires and the area of soil from which water is
extracted, are equal. This was not necessarily true in our case, as we purposely chose
small-diameter growth columns to better simulate one-dimensional flow. But while
roots were confined to a relatively small growth area, the above-ground biomass
production flourished, as plants experienced no limiting factor to growth such as
shading or resource competition by other plants.
The RDP presented in Table 12 varied slightly between soils and uptake models,
as well as from the measured RDP shown in Table 11. The modeled depth of the
greatest rooting density, z*, was larger for plants grown in loess than sand, although the
measured RRM and RDP presented in Figure 13 and Table 11, respectively, showed
that z* was equal for both soils. In addition, model results predicted z* to fall around the
56
20 cm depth mark, while observations showed that more than 50% of wheat roots lay
within the top 10 cm of the soil profile. The measurements were in agreement with z*
values in the literature (Li et al., 2001; Zhang et al., 2004), which were reported to fall
within the top 10 cm of soil. But while a discrepancy between observed and model-
generated z* values was apparent, DREAM not only converged on similar z* values
between soils for all models, but succeeded in considerably narrowing down the initial
z* parameter range, as indicated by marginal corresponding φ1 values.
The rooting depth, zm, of wheat grown in loess was deeper than in sand, in
agreement with Tennant (1976), who found that the wheat rooting depth in sandy loam
was greater than sand due to the increased water availability in the former. vG
simulations for both soils predicted the deepest zm compared to that predicted with the
other two models. While the modeled zm for sand and loess, for all models except for the
NH simulations with sand, were smaller than the measured zm, values fall around the
lower end of the range reported in the literature 100 < zm < 160 (Li et al., 2001) for a
silty soil located in Saskatchewan, Canada.
The rooting depth of loess vs. sand for all models showed a positive correlation
with both the⎯Tp and yield data (Figure 12). Gulman and Turner (1978) reported on
variations in the root and shoot development among different varieties of tomato grown
in silt and in sandy soil. While the overall dry shoot and root mass were shown to be
higher for silt-grown tomato plants, no significant differences were seen concerning the
effect of soil texture on shoot:root ratios. Brown et al. (1987) reported similar findings
for barley. Thus, we concluded that the shorter rooting depth in sand compared to loess
fits with the smaller dry yield measured from plants cultivated in fine sand.
The higher pz values predicted for loess soil, compared to sand, indicated that
rooting profiles were somewhat more uniform in sand. The relative root distribution was
the least uniform for both media, or most 'logarithmic', with NH simulations. On the
57
contrary, a greater spreading of the total root density along the profile was predicted by
both FD and vG simulations, judging by the smaller pz values (Table 12).
The visual description of 5 relative rooting distributions is displayed below in
Figure 14; the RRM as measured at 10 cm intervals along soil columns is compared
alongside the Raats (1974) and Vrugt (2001) function [7] fitted to observations, β(z),
using parameters in Table 11. Three additional root distributions, calculated using
optimized z*, zm, and pz parameters, and normalized in HYDRUS-1D [21], β'(z), are also
presented for each of the three root water uptake models.
Dep
th (c
m)
ALOESS
125
100
75
50
25
0
125
100
75
50
25
0
0 0.2 0.4 0.6 0.8 1Relative root distribution (-)
RRM
β(z)
FD β'(z)
vG β'(z)
NH β'(z)
BSAND
Figure 14. The measured RRM fitted to [7], β(z), and the optimized, relative root distributions, β'(z), [21] of wheat plants extracted from one (A) loess and (B) sand column, for three models.
The β'(z) of FD and vG simulations are almost identical, but the distribution of
roots between the upper and lower soil layers as predicted by NH simulations is the
closest to β(z).
58
The discrepancy between the modeled and measured RDP for wheat was partly
due to the different scales of resolution achieved through each method; the measured
RRM in Figure 13 was plotted using data points at each 10 cm increment while
DREAM accounts for spatial dynamics at 1 cm intervals. Another factor contributing to
the deviation between modeled and measured RDP was the presence of local minima
over the parameter space into which DREAM converged. But the most likely factor
giving rise to differences between modeled and measured RDP was the fact that the
depth of the greatest rooting mass (our measurements) was not necessarily correlated
with the depth of the greatest root activity (Feddes and Raats, 2004).
In our measurements of the total root mass with depth, no consideration was
given to whether collected roots were active or dead. On the other hand, DREAM
constructs a β'(z) based on spatial water dynamics as affected by root uptake activity. So
while the largest root mass was found in the upper 10 cm of both soils, a good portion
may no longer have been active, and therefore did not contribute to plant water uptake.
Therefore, we judged model predictions as more reliable in supplying information about
the effect of roots on water dynamics than the measured root mass.
Convergence was weaker for the RWUP presented in Table 13 than for the
SHP,⎯Tp, and RDP presented in Table 12, as indicated by the higher φ1 values.
The wide initial parameter ranges for the RWUP may have incorporated various
local minima, making the convergence process difficult. Increasing the number of
simulations might have helped to lower φ1, but the combination of a lack of computer
power and time led us to settle on a total of 15,000 function evaluations, carried out
stepwise in increments of 3,000 runs. The cascade-type manner in which simulations
were carried out may also have hindered potential convergence patterns by decreasing
the parameter space at each step, and possibly shutting out true values. Among the
59
parameters presented in Table 13, stronger convergence was seen for the RWUP of
sand-grown plants over loess.
Table 13. The 95% CI, ML values, and an error index, φ1, for the RWUP of (A) the FD, (B) vG, and (C) NH uptake models with wheat for loess and sand. Optimizations were carried out with data from column # 1 and column # 4 for sand and loess, respectively. The h2 parameter was calculated according to [23]. Parameters h2high, h2low, r2high, and r2low are included in the files in Appendix I.
Uptake compensation, as estimated by both FD and vG simulations for both
soils, was moderately low, judging by the high critical stress parameter ωc ~ 0.6. Since
plants were irrigated nightly at 120% of the estimated daily transpiration according to a
water balance [18], the soil water status in the root-dense upper soil layers during the
morning and early-afternoon hours was often well above h2, and h50. Compensation may
therefore not have been necessary for much of the day for wheat plants.
ML values for h0 showed that plant uptake commenced at higher soil water
potentials in loess, compared to sand, while h1 values indicated that uptake was
A Model- Plant
Soil- Column
95% CI/ ML ωc (-) h0 (cm) h1 (cm) h2 (cm) h3 (cm)
Min 0.16 -4 -26.2 -598.9 -18462.1 Max 0.71 -1.8 -11 -162.4 -13411.5 ML 0.57 -1.8 -21.2 -491.7 -14815 Loess-4
φ1 0.965 1.222 0.717 0.888 0.341 Min 0.4 -3.5 -20.2 -5158.5 -18326.8 Max 0.69 -0.6 -10.4 -3854.9 -14103.8 ML 0.56 -3.3 -18.9 -2348.5 -16109
FD- Wheat
Sand-1
φ1 0.518 0.879 0.519 0.555 0.262
B Model- Plant
Soil- Column
95% CI/ ML ωc (-) p (-) h50 (cm)
Min 0.06 2.5 -4556.6 Max 0.77 6.2 -831.4 ML 0.61 5.8 -832.2 Loess-4
φ1 1.164 0.638 4.476 Min 0.34 2.5 -744.7 Max 0.7 4.9 -274.1 ML 0.68 4.7 -277.1
vG-
Wheat Sand-1
φ1 0.529 0.511 1.698
C Model- Plant
Soil- Column
95% CI/ ML Hwilt (cm)
Min -16681.1 Max -8438.1 ML -15027.8 Loess-4
φ1 0.549 Min -14597.6 Max -9612 ML -12982.7
NH-
Wheat Sand-1
φ1 0.384
60
sustained at less than the potential for a wider matric potential range in loess vs. sand,
due to oxygen stress. We expected the water stress threshold parameter, h2, at which
uptake begins to decrease from the potential, to be lower for loess than sand due to the
increased water-retaining ability of former soil. Results proved otherwise, and DREAM
not only estimated lower h2 values for sand than loess, but also predicted a lower wilting
point h3 for sand than loess, in contradiction to our expectations.
These results are most likely due to the correlation between h2 and⎯Tp; the h2
parameter for loess soil was over 4 times higher than that of sand, and the final⎯Tp value
was twice as high for wheat plants grown in loess than those grown in sand. As can be
seen in Figure 4A and [23], the h2 drought stress threshold parameter was affected by
the atmospheric demand so that the soil matric potential at which plant transpiration
decreased below the potential was higher when the atmospheric demand was higher.
On the other hand, h50 values came out as we expected- lower for loess than
sandy soil. This is most likely due to the higher soil water retention of loess under
drying conditions, in agreement with the ML SHP between loess and sand, for both
uncropped and cropped soils shown in Table 10 and Table 12, respectively.
Convergence for h50 for loess was the weakest of all RWUP, judging by the large φ1
corresponding to the wide 95% confidence interval.
p was much higher than the generally accepted value of p ~ 3 reported in the
literature (van Genuchten, 1987; van Genuchten and Gupta, 1993). The upper boundary
of the initial parameter range was set too high (Table 4C) and optimized values may
have reflected convergence into a local minimum.
Hwilt settled near the traditional wilting point value (Bakker et al., 2007), and was
lower for plants cropped in loess compared to sand (Taylor and Ashcroft, 1972).
61
5.2.2 Tomato
5.2.2.1 Plant yield
The yield from the sand column from which data (#3) was taken was larger than
that of the loess column (#5), as shown below in Figure 15. Nevertheless, average
tomato yields from the two soils were similar, with only a 17% difference between
them, although the average yield and standard deviation of loess-grown plants was
higher than sand. The large standard deviation for tomato plants grown in loess soil was
due to oxygen stress in column 6. The soil packed into this column had been mistakenly
sifted with a <0.5mm sieve which greatly impeded water flow through the soil column
and led to nightly water ponding following irrigation events.
0
50
100
150
200
250
300
350
400
1 2 3 4 5 6 AverageColumn
Dry
yie
ld (g
)
LoessSand
Figure 15. The above-ground biomass of tomato plants measured from three loess and three sand-filled columns, and the average yield for each soil. Data for DREAM simulations was taken from one column for each soil, marked by *. The standard deviation is indicated by error bars.
5.2.2.2 Relative root distribution
The RRM [19] of tomato plants grown in loess and sand is presented in Figure
16. Similar to the wheat RRM, the tomato root density was highest within the top 10 cm
of both soils. Surface root growth was about 30% higher than in sand than loess soil,
and a greater portion of the roots in loess soil were distributed among deeper soil layers.
Both root distribution curves show a sharp decrease between the first and second layers,
after which root mass is fairly equally distributed.
**
62
These results were in agreement with Jackson and Bloom (1990), who reported
that the root biomass of tomatoes grown in a silt loam in Davis, California was
significantly higher near the plant stem than in other soil layers. However, the measured
tomato rooting depth was shallower than values reported by Jackson and Bloom (1990)
and Machado et al. (2003), which were 120 cm and 100 cm, respectively. The tomato zm
was also smaller than the wheat zm in loess and sand by 75%, and 125%, respectively.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
30
50
70
90
110
130
150
Dep
th (c
m)
Relative root mass (g g-1)
Loess- 5
Sand- 3
Figure 16. The relative root mass (RRM) with depth of tomato plants extracted from one loess and sand column, the number of which is indicated in the legend.
The root mass of sand-grown tomatoes was over nine times that of the loess-
grown, as shown below in Table 14. The tomato root mass, specifically in sand, was
also significantly higher than the wheat root mass. Although a constant shoot:root ratio
between treatments may partially explain the higher sand root mass over loess, as the
average yield of sand-grown plants was higher than loess-grown, this does not fully
account for differences in root mass between soils. Problems in separating roots from
the organic matter in loess soil, and a failure to properly wash roots extracted from sand
may have contributed to the disparity in root mass values between soils.
The Raats (1974) and Vrugt et al. (2001) RDP [7] fitted to the measured RRM
are shown in Table 14. Similar to the fitted RDP of wheat roots, the tomato z* was near
the soil surface. However, tomato zm was shallower than the wheat rooting depth, by
over 75% and 100% in loess and sand, respectively.
63
Plant Soil z* (cm)
zm (cm)
pz (-)
∫RM(z) (g)
Loess 0 85 7.0 28.5 Tomato
Sand 5.0 55.0 11.9 264.0 Table 14. The tomato RDP as obtained from fitting [7] to the measured RRM. The total tomato root mass ∫RM(z) for each soil treatment is also shown.
5.2.2.3 Parameter optimization results
The results of DREAM optimizations for the SHP,⎯Tp, RDP, and RWUP of
tomato plants grown in loess and sand are presented below in Table 15 and Table 16.
Model- Plant
Soil- Column
95% CI/ ML
Ks (cm hr-1)
α (cm-1)
n (-)
θr (cm3 cm-3)
Tp (cm day-1)
z* (cm)
zm (cm)
pz (-)
min 0.74 0.008 2.59 0.005 0.51 6.3 59.9 3 max 1.58 0.011 2.97 0.041 0.63 40.2 144.4 23.3 ML 0.75 0.009 2.98 0.016 0.54 33 140.6 4.9 Loess- 5
φ1 1.120 0.333 0.128 2.250 0.222 1.027 0.601 4.143 min 34.4 0.082 2.97 0.033 2.2 32.2 133.8 2.9 max 54.48 0.093 3.2 0.075 2.7 35.5 144.5 3.4 ML 43.8 0.083 3.16 0.039 2.69 34.7 144.5 3.4
FD-Tomato
Sand- 3
φ1 0.458 0.133 0.073 1.077 0.186 0.095 0.074 0.147 min 0.99 0.009 2.57 0.008 0.51 3.2 53.3 4.6 max 3.23 0.013 2.96 0.051 0.63 38.7 141.1 26.1 ML 1.12 0.01 2.96 0.011 0.53 35.9 119.4 6.6 Loess- 5
φ1 2.000 0.400 0.132 3.909 0.226 0.989 0.735 3.258 min 20.09 0.08 2.75 0.048 2.4 31.1 126.2 2.5 max 30.96 0.092 3.01 0.091 2.89 34.3 137.3 3.1 ML 24.21 0.082 2.99 0.067 2.83 33.3 136.5 3.1
vG- Tomato
Sand- 3
φ1 0.449 0.146 0.087 0.642 0.173 0.096 0.081 0.194 min 0.97 0.009 2.68 0.006 0.51 3.4 66.3 4.1 max 1.94 0.012 2.96 0.066 0.66 27.1 150.4 25.5 ML 1.05 0.009 2.94 0.007 0.52 6.9 131.2 23.1 Loess- 5
φ1 0.924 0.333 0.095 8.571 0.288 3.435 0.641 0.926 min 18.47 0.083 2.99 0.015 1.08 1.5 86.1 4.2 max 59.02 0.12 3.82 0.067 1.96 7.6 120.8 6.8 ML 29.35 0.108 3.48 0.062 1.18 6.9 105.2 5.3
NH-Tomato
Sand- 3
φ1 1.382 0.343 0.239 0.839 0.746 0.884 0.330 0.491 Table 15. The 95% CI, ML values, and an error index, φ1, for the SHP,⎯Tp, and RDP of the FD, vG, and NH models with tomato for loess and sand. Plants were grown in three columns for each soil, but data for simulations was taken from only one column, the number of which is indicated next to the soil type. Optimizations were carried out with data from full-grown plants.
The SHP values for all models with both soils shown in Table 15 varied from the
SHP of bare soils (Table 10). Convergence patterns were generally poorer than with
wheat data, as evidenced by the higher φ1. In addition, convergence was weaker with
data of plants grown in loess compared to sand.
Differences in the Ks between loess and sand were representative of the two soil
textures. The effect of the uptake models on the Ks was also evident. ML Ks values for
cropped loess, for all models, were about 3 times lower than the Ks for uncropped loess.
64
ML Ks values for sand from vG and NH simulations were also lower than the Ks of
uncropped sand, but by a smaller margin, while the sand Ks from FD simulations was
larger than that of bare sand. These results, like those of wheat in Table 12, are telltale
of the effect of plant roots on the SHP.
The lower ML α values for loess soil compared to sand verified the greater
ability for water retention at saturation in soils richer in silt and clay particles. Similar to
the results generated with wheat data, ML α values for all three models with both soils
were higher in a cropped vs. uncropped soil, indicating that the soil water retention
ability near saturation decreased following plant growth.
Loess soil is generally characterized by high water retention at large suction
gradients, compared to sand. However, in the case of the tomato experiment, DREAM
predicted higher 95% CI ranges and ML values for θr in sand compared to loess, for all
uptake models, which was actually according to the trend of reference θr values
predicted by Rosetta Lite. In addition to the greater effect of capillary forces on water
retention in fine-grained sand compared to that of coarser-grained mediums, the θr may
be positively correlated with root mass. Therefore, if the water content of roots was
significantly higher than that of the bulk soil at low matric potentials, a greater root
mass, such as in fine sand (Figure 16) would have raised the θr of the medium.
ML n values were similar between loess and sand, and between models, and lay
within the 95% CI range for uncropped sand. The small φ1 for n, relative to the other
SHP, indicated strong convergence for this parameter. Loess ML n values of tomato
simulations were much higher than both ML n values generated from wheat data (Table
12) and ML n values predicted for a bare soil (Table 10), but this was due to
convergence problems encountered with these latter two simulation sets.
65
ML⎯Tp values for sand were more than double those of loess for all uptake
models. Differences in⎯Tp between the two soils were more pronounced in NH
simulations than in FD and vG simulations.
ML⎯Tp values of sand-grown tomatoes, between all models, were much higher
than values reported in the literature, which was also the case with wheat data for loess.
Hanson and May (2006) and Dong et al. (2008) reported that the potential transpiration
of tomatoes grown both in commercial fields and greenhouse conditions, respectively,
averaged between 0.8 and 0.95 cm day-1 at the peak of the season. Interestingly enough,
these values were higher than ML⎯Tp values of loess-grown tomatoes. The disparity
between the modeled⎯Tp of loess- and sand-grown plants, which was over 400% for FD
and vG simulations, was not justified by the average yield data, which differs between
the soils by only 15%; these results may be indicative of parameter correlations, most
likely between⎯Tp, ωc, and h2.
Differences were apparent in the optimized tomato RDP values between models
and soils (Table 15), as well as between the RDP fitted to the measured RRM (Table
14). Good agreement in RDP values for both soil treatments was seen between vG and
FD models, while values produced from NH simulations were significantly different.
Nevertheless, the RDP generated from NH simulations were the closest to those fitted to
the measured RRM.
As was seen with wheat data, ML z* values generated from FD and vG
simulations for both soils showed a deeper location of the greatest rooting density, or
activity, than the measured location of the largest root mass (Table 14). ML z* values
from NH simulations were equal for sand and loess, and were located in the top 10 cm
from the soil surface. Modeled zm values were deeper than the measured, maximum
rooting depth. And while FD and vG simulations estimated zm to be greater in sand than
66
loess, NH simulations predicted a greater rooting depth in loess, in agreement with the
measured tomato zm, as well as the measured and modeled wheat zm.
The relative root distribution curves in Figure 17 present a visual picture of the
effects of differences in the modeled RDP between treatments, and between
observations. ML pz values for all models and soils except NH for loess, predicted a
more uniform distribution of roots with depth than the measured value.
Dep
th (c
m)
ALOESS
125
100
75
50
25
0
125
100
75
50
25
0
0 0.2 0.4 0.6 0.8 1
Relative root distribution (-)
RRMβ(z)FD β'(z)vG β'(z)NH β'(z)
BSAND
Figure 17. The measured RRM, fitted to [7], β(z), and the optimized, relative root distributions, β'(z), [21] of tomato plants extracted from one (A) loess and (B) sand column, for three uptake models
The optimized RWUP for all three uptake models are shown in Table 16. The
estimated degree of compensation in the root water uptake of tomato plants varied
between loess and sand, and between the FD and vG models. The ML ωc values
predicted by FD simulations showed little compensation for loess soil, but showed near
full compensation for sand-grown plants. This trend is reversed in vG simulation
results- compensation was greater in loess than sand. Convergence was stronger at
higher ML ωc values, shown by the much smaller φ1. These results were in contrast to
67
the wheat results presented in Table 13, which estimated equal compensation between
loess and sand for both models.
A Model- Plant
Soil- Column
95% CI / ML
ωc (-)
h0 (cm)
h1 (cm)
h2 (cm)
h3 (cm)
Min 0.21 -17.2 -93.9 -2667.4 -9129.6 Max 0.91 -2.9 -39.1 -1731.2 -4757.4 ML 0.91 -3.9 -92.6 -1319.9 -8317.9 Loess- 5
φ1 0.769 3.667 0.592 0.709 0.526 Min 0.21 -17.5 -28.5 -764.1 -8783.3 Max 0.89 -6.4 -24.2 -414.8 -4998.2 ML 0.28 -11.5 -28.2 -369.6 -6414.2
FD- Tomato
Sand- 3
φ1 2.429 0.965 0.152 0.945 0.590
B Model- Plant
Soil- Column
95% CI / ML
ωc (-)
p (-)
h50 (cm)
Min 0.15 1.4 -4472 Max 0.86 5.6 -786.9 ML 0.29 4.6 -1159.5 Loess- 5
φ1 2.448 0.913 3.178 Min 0.21 3.2 -889.5 Max 0.7 5.4 -433.9 ML 0.55 4.9 -446.1
vG- Tomato
Sand- 3
φ1 0.891 0.449 1.021
C Model- Plant
Soil- Column
95% CI / ML
Hwilt (cm)
Min -15621.1 Max -3816.1 ML -10946.1 Loess- 5
φ1 1.078 Min -14762.3 Max -10853.3 ML -14398.4
NH- Tomato
Sand- 3
φ1 0.271 Table 16. The 95% CI, ML values, and an error index, φ1, for the RWUP of (A) FD, (B) vG, and (C) NH uptake models with tomato for loess and sand. Optimizations were carried out with data from columns #5 and #3, for loess and sand, respectively. h2 was calculated according to [23]. Parameters h2high, h2low, r2high, and r2low are included in the files in Appendix I.
The initial FD RWUP ranges were determined based on plant characteristics
alone, and were not dependent on soil texture. Thus, the magnitude of the ML, FD
RWUP differed between tomato and wheat optimizations. Nevertheless, similar effects
of soil texture on RWUP were seen between results with data from varying crops.
ML values for h0 and h1 were lower for tomato than wheat, in both soils, which
indicated a greater sensitivity of tomato to oxygen stress. The effects of soil texture on
h0 and h1 were similar between tomato and wheat data, but differences in parameter
values between soils were more distinct with tomato data.
68
h2 was about four times lower for loess-grown plants, relative to those grown in
sand, which was in agreement with the lower⎯Tp of loess, according to Figure 5A.
Reverse results were seen with wheat data, namely that⎯Tp and h2 were lower for sand.
ML values for h3 for tomato were about half in magnitude (higher) than those of
wheat, possibly due to the greater resilience of wheat under drying conditions. Tomato
results for h3 show that plant water uptake in loess was sustained under higher tensions
than in sand. This, again, may have resulted from lower⎯Tp rates of loess-grown plants.
ML p values were similar between loess and sand-grown plants. Similar to
wheat results, values were higher than the generally-accepted average value of p ~ 3,
most likely due to an unreasonably high upper boundary for the initial parameter range.
ML values for h50 were around three times lower for loess than sand, which may have
resulted from parameter correlations between h50 and⎯Tp, similar in manner to the
observed correlation between h2 and⎯Tp. All vG RWUP had a higher φ1 for loess than
sand, which was also the case for vG simulations with wheat.
Hwilt was lower for sand than loess, which indicated that there was a larger store
of retained water at low matric potentials in sand. This was contrary to our expectations,
and seemed to indicate a correlation between Hwilt and θr; the ML values of θr were also
higher in sand for all three models.
5.2.3 Sorghum
5.2.3.1 Plant yield
The sorghum yield in fine sand was 40% higher than in loess (Figure 18). These
results were expected since seeding in sand was done one month before loess due to
technical problems with the loess-filled column.
69
0
50
100
150
200
250
300
350
400
Loess SandColumn
Dry
yie
ld (g
)
Figure 18. The dry yield of sorghum plants measured from one loess and one sand-filled column.
5.2.3.2 Relative root distribution
The logarithmic shape of the root distribution with depth [19] seen with wheat
and tomato was also evident with sorghum, as depicted in Figure 19. The RRM of both
soils was maximal within 15 cm from the surface. The sorghum zm in loess soil was
similar to tomato while the zm in sand was similar to wheat.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
5
25
45
65
85
105
125
145
Dep
th (c
m)
Relative root mass (g g-1)
Loess
Sand
Figure 19. The relative root weight (RRM) with depth of sorghum plants extracted from one loess and sand column.
The Raats (1974) and Vrugt et al. (2001) RDP [7] fitted to the measured RRM
are shown in Table 17. Similar to the fitted RDP of wheat and tomato roots, the
sorghum z* was measured in the top 5 cm of the soil surface. The sorghum zm was
shallower in loess than sand, opposite to results for wheat and tomato.
70
Plant Soil z* (cm) zm (cm) pz (-) ∫RM(z) (g) Loess 1.4 75.0 7.0 153.4
Sorghum Sand 4.8 127.0 14.0 205.0
Table 17. The sorghum RDP as obtained from fitting [7] to the measured RRM. The total sorghum root mass ∫RM(z) for each soil treatment is also shown.
The total sand root mass was about 30% greater than loess, which is supported
by the yield data. Assuming a constant root-shoot ratio across soil treatments, plant
yield should be correlated with root growth. Results from wheat, tomato, and sorghum
all showed a greater root mass excavated from sandy soil than loess.
5.2.3.3 Parameter optimization results
Optimizations results for the SHP,⎯Tp, RDP, and RWUP of sorghum plants
grown in loess and sand are presented in Table 18 and Table 19. Significant differences
in the ML parameter values from sorghum optimizations were evident between models
and among soil treatments. In addition, the SHP,⎯Tp, and RDP sorghum data varied
from modeled SHP values of uncropped soils, the measured RDP, and literature⎯Tp
values, respectively, in a similar manner as was seen with wheat and tomato data.
Model- Plant Soil 95% CI/
ML Ks
(cm hr-1) α
(cm-1) n (-)
θr (cm3 cm-3)
Tp (cm day-1)
z* (cm)
zm (cm)
pz (-)
Min 0.41 0.01 1.18 0.034 1.38 20.7 63.3 10.2 Max 1.18 0.014 1.44 0.088 2.6 21.7 126.7 22.9 ML 0.41 0.01 1.41 0.05 2.45 21.5 62.9 11.6 Loess
φ1 1.878 0.400 0.184 1.080 0.498 0.047 1.008 1.095 Min 21.46 0.076 2.85 0.035 1.3 17.6 100.1 2.8 Max 45.82 0.09 2.95 0.062 1.6 21.4 136.2 7 ML 46.78 0.082 2.92 0.044 1.46 20 129.8 5.6
FD- Sorghum
Sand
φ1 0.521 0.171 0.034 0.614 0.205 0.190 0.278 0.750 Min 0.37 0.01 1.26 0.036 1.74 20.6 66.9 11.6 Max 1.54 0.015 1.46 0.081 2.75 21.7 130.6 23.1 ML 0.39 0.011 1.37 0.053 2.39 21.2 131.5 19.9 Loess
φ1 3.000 0.455 0.146 0.849 0.423 0.052 0.484 0.578 Min 38.95 0.037 2.85 0.061 1.56 12.5 113.6 8.5 Max 54.42 0.051 3.29 0.085 2.1 17.7 141.8 17.7 ML 51.08 0.042 2.97 0.084 2.09 17.3 125.9 16.3
vG- Sorghum
Sand
φ1 0.303 0.333 0.148 0.286 0.258 0.301 0.224 0.564 Min 1.78 0.111 1.14 0.025 1.84 18.7 94 11.9 Max 3.32 0.143 1.25 0.082 2.78 19.2 149.1 21.8 ML 2.05 0.143 1.14 0.039 1.94 19.1 122.8 15.7 Loess
φ1 0.751 0.224 0.096 1.462 0.485 0.026 0.449 0.631 Min 35.14 0.089 1.47 0.028 2.56 17.2 93.5 24.3 Max 65.39 0.136 1.76 0.079 2.96 17.7 112.6 29.4 ML 39.35 0.094 1.68 0.05 2.96 17.5 104.7 28.1
NH- Sorghum
Sand
φ1 0.769 0.500 0.173 1.020 0.135 0.029 0.182 0.181 Table 18. The 95% CI, ML values, and an error index, φ1, for the SHP,⎯Tp, and RDP of the FD, vG, and NH models with sorghum for loess and sand. Optimizations were carried out with data from one column per soil, cropped with full-grown plants.
71
Ks values in cropped loess, according to FD and vG simulations, were smaller by
8-fold than the Ks of bare loess (Table 10). ML Ks values of cropped sand were slightly
larger than the Ks for uncropped sand. NH simulations for both soils produced Ks values
closest to those of uncropped soils.
All 9 optimizations for loess, with wheat, tomato, and sorghum data, predicted
lower Ks values following plant cultivation, while 6 out of 9 optimizations for sand
estimated an increase in the Ks after cropping. The latter observation, although less
significant than the former in terms of the change in Ks, has been reported in the
literature. Shirmohammadi and Skaggs (1984) observed a 70% and 40% increase in the
unsaturated hydraulic conductivity in the top 20 cm of fine sand after cropping with
fescue grass and soybeans, respectively. Rasse et al. (2000) saw a 44% increase in the
Ks of a Kalamazoo loam cropped with alfalfa, compared to a bare fallow soil.
But the increase in Ks in cultivated sand vs. bare sand was not attributed to live
roots, but decayed; root decomposition leaves empty macropores which increase the soil
hydraulic conductivity (Mitchell et al., 1995, Green et al., 2003). In fact, live root
growth can lead to a blockage of pores and decrease the soil hydraulic conductivity
(Barley, 1954; Fuentes et al., 2004). Therefore, the observed increase in the sand Ks
after plant cultivation most likely resulted from a higher root turnover and death rate,
possibly due to dry conditions.
A major factor which contributed to the observed decrease in loess Ks following
plant cultivation was soil compressibility (Cockroft et al., 1969; Bruand et al., 1996).
Frictional forces resisting soil compression have been shown to be negatively correlated
with the clay and silt content of soils (Smith et al., 1997), so that loess with its higher
clay and silt content (Table 9) experiences greater compression with root growth than
sand (Dexter, 2004). This means that root growth in loess may have contributed to a
reduction in the soil porosity, subsequently lowering the Ks. If this indeed was the case
72
in cropped loess, θs, which had been fixed for parameter optimizations, would need to
be lowered for simulations with data measured after plant propagation.
Similar to wheat and tomato results, ML α values were higher in both sand and
loess cropped with sorghum, compared to uncropped soils. The increase in α following
plant propagation was higher for loess than sand. ML values for α produced with vG
simulations were closest in magnitude to those of bare sand and loess soils.
Of all the SHP, n values of both soils had the lowest φ1, indicating strong
convergence. Sorghum ML n values were generally lower than values generated from
tomato data, but higher than values estimated with wheat data. The variance in n
between plant treatments might attest to unique rooting patterns and root structures
between plant types which affected the SHP in different ways. Loess ML n values for all
three models were higher than the n of bare loess soil, but this can be linked to
convergence problems with simulations for the SHP of uncropped soils.
Convergence for θr was stronger for sand than loess, as indicated by the smaller
φ1. The θr of both soils, as modeled with sorghum data, were generally higher than
tomato results and lower than θr values generated from wheat simulations. While loess
ML θr values for FD and vG simulations were slightly higher than the θr of uncropped
loess, values for cropped sand were higher by an order of magnitude.
The observed increase in α and θr after plant production in both soils may have
been due to the formation of macropores accompanying root growth (Beven and
Germann, 1982). This might have reduced water retention at saturation leading to an
increase in α, while the water retention capacity of roots under large suction gradients
conversely exceeded that of the soil alone, thus raising θr.
Estimated⎯Tp values were higher for loess-grown plants when optimized with
FD and vG model parameters, but higher in sandy soil when optimized with NH
parameters. NH results were in better agreement with the yield data presented in Figure
73
18. All modeled⎯Tp values deviated from sorghum uptake rates reported in the literature,
which was also the case for wheat and tomato. Blum and Arkin (1984) reported that the
average potential transpiration of sorghum grown in loess in Texas was 0.5 cm day-1,
which was also comparable to the average value for sorghum water requirements
reported by the FAO (Critchley and Siegert, 1991). Accordingly, ML⎯Tp values were up
to 5 times higher in loess, and an average of 4 times higher in sand.
To avoid future discrepancies between the optimized potential transpiration to
values reported in the literature, it is recommended to choose larger-diameter columns.
In addition, when optimizing for⎯Tp, the upper boundary of the initial range should be
set lower, at more realistic values to prevent convergence in a local minimum.
Similar RDP values resulted from all three optimizations, for both soil
treatments. All ML z* values converged at around 20 cm in both soils, with loess values
marginally higher than sand values for all models. Convergence was relatively strong
for z*, with a slightly lower φ1 for FD and vG loess values vs. sand. Sorghum z* values
for both soils were comparable to wheat, but lower than tomato. Overall, a discrepancy
of similar magnitude as that of optimizations with tomato and wheat data could be seen
between measured and modeled z* values.
No apparent pattern was evident concerning the effect of soil texture on either
the rooting depth or the shape of the root distribution function with sorghum data
(Figure 20). ML values for zm were deeper in sandy soil vs. loess when optimized with
FD parameters, but deeper in loess when optimized with both vG and NH uptake
parameters. The sorghum, loess zm as modeled with FD parameters and the sand zm of
vG simulations, were closest to measured values. Nevertheless, zm values for both soils
were significantly smaller than those reported in the literature, 190 cm, and 240 cm for
an oxisol in Australia (Robertson et al., 1993), and a deep loess soil in Kansas, USA
(Stone et al., 2002), respectively.
74
A significant difference in the degree of linearity of the root distribution function
between soils resulted from both the FD and NH simulations; but whereas the pz
predicted with FD parameters was larger in loess than sand, the trend was opposite for
NH simulations (Table 18).
Dep
th (c
m)
ALOESS
125
100
75
50
25
0
125
100
75
50
25
0
0 0.2 0.4 0.6 0.8 1Relative root distribution (-)
RRMβ(z)FD β'(z)vG β'(z)NH β'(z)
BSAND
Figure 20. The measured RRM, fitted to [7], β(z), and the optimized, relative root distributions, β'(z), [21] of sorghum plants extracted from one (A) loess and (B) sand column, for three uptake models.
Differences in parameter values between soils seen in Table 19 highlighted the
effect of soil texture on the RWUP. In addition, the variation in ML values for ωc
between the FD and vG models showed the effect of a linear vs. non-linear uptake
model on results, which was also the case for wheat and tomato optimizations.
ML ωc values were lower in loess than sand, for both FD and vG models,
although a greater extent of compensation was predicted with vG than FD simulations.
Smaller ωc values were accompanied by higher φ1, which may be due to parameter
correlations. Similar results were seen for optimizations with tomato data (Table 16).
75
Table 19. The 95% CI and ML values for the RWUP of (A) FD, (B) vG, and (C) NH uptake models with sorghum, for loess and sand. Optimizations were carried out with data from full-grown plants, grown in one column per soil. The h2 parameter was calculated according to [23]. Parameters h2high, h2low, r2high, r2low, are included in the files in Appendix I.
A correlation seemed apparent between ωc and h3, irrespective of⎯Tp. In
optimizations with wheat, tomato, and sorghum data for both soils, lower ωc values
were accompanied by higher h3 values. One possible explanation for this trend is that
compensation raises the amount of water removed from the soil profile for a given unit
of tension. The extractable soil water is therefore drained at higher matric potentials
when plants employ compensatory mechanisms.
ML h1 and h0 values for both loess and sand were lower than values from wheat
optimizations which indicated a greater sensitivity of sorghum than wheat to oxygen
stress. The sensitivity of sorghum to oxygen stress compared to tomato was not clear
from h0 and h1 results. The sorghum ML h0 value was lower for sand than loess, and
A Model- Plant
Soil 95% CI / ML
ωc (-)
h0 (cm)
h1 (cm)
h2 (cm)
h3 (cm)
Min 0.12 -17.2 -94.4 -10623.2 -28468.2 Max 0.68 -5.3 -36.3 -3720 -17835.4 ML 0.61 -10 -79.1 -7598.5 -23192 Loess
φ1 0.918 1.190 0.735 0.908 0.458 Min 0.21 -16.6 -79.8 -11800.6 -26877.8 Max 0.82 -8 -45 -9240.9 -20924.7 ML 0.81 -12.3 -63.3 -10463 -26827
FD- Sorghum
Sand
φ1 0.753 0.699 0.550 0.245 0.222
B Model- Plant
Soil 95% CI / ML
ωc (-)
p (-)
h50 (cm)
Min 0.187 1.7 -4309.3 Max 0.873 5.3 -699.6 ML 0.28 4.5 -3494.4 Loess
φ1 2.450 0.800 1.033 Min 0.12 4.4 -572.5 Max 0.43 5.5 -208.1 ML 0.33 5.4 -208.5
vG- Sorghum
Sand
φ1 0.939 0.204 1.748
C Model- Plant
Soil 95% CI / ML
Hwilt (cm)
Min -9716.5 Max -4133 ML -9199.8 Loess
φ1 0.607 Min -9790.8 Max -7371.8 ML -9709.8
NH- Sorghum
Sand
φ1 0.249
76
indicated that plant transpiration was hindered over a larger range of matric potentials in
sandy soil. Results for h0 for all plants were unexpected due to the higher θs of loess,
signifying loess' greater water-retaining abilities compared to sand.
Nevertheless, similar to results from both tomato and wheat, h1 was lower for
loess, indicating that oxygen stress was prolonged at lower matric potentials in this
medium. These results seem to indicate a greater heterogeneity in the pore-size
distribution in loess over sand, encompassing both large-diameter pores which are more
readily drained near saturation, as well as small-diameter pores which require lower
matric potentials than sand to release adsorbed water.
Water stress parameters h2 and h3 were lower for sand than loess, most likely
due to their correlation with⎯Tp. Even though the yield in sand was greater (Figure 18),
DREAM estimated that⎯Tp was 75% higher in loess. This would theoretically lead to the
depletion of soil water stores at higher matric potentials in loess soil vs. sand.
p was slightly smaller in loess than sand, but φ1 was larger for loess, as was the
case with tomato data. Overall, p values for all treatments were higher than p values
reported in the literature (van Genuchten, 1987; van Genuchten and Gupta, 1993),
which was the case for both wheat and tomato.
ML values for h50 in loess were about 17 times lower than in sand, which seems
to point to greater water-retaining abilities in loess. The trend for h50 should be similar
to that of FD parameters h2 and h3, but the correlation of these latter two parameters
with⎯Tp led to non-characteristic results.
ML values for Hwilt were similar between sand and loess, and even slightly
higher for loess. This trend was different than what was seen with both tomato and
wheat data, and we had expected a lower Hwilt for loess, similar to results for h50.
77
5.2.4 Measured vs. modeled soil water potential
The robustness of each root water uptake model was evaluated using two error
indices. The first index, φ1 [24], was used earlier in section 5.1 and provides information
about the convergence strength of individual parameters. The second index, φ2 [25],
represents the fitness of the measured soil water potential to modeled-generated results.
Since φ2 was directly related to directly-measured system variables, we considered it
more carefully than φ1 when assessing model performance.
Figure 21, Figure 22, and Figure 23 show the change in measured and modeled h
values over time, at 7 observation nodes. Modeled values are represented by the 95% CI
and ML of the last 3,000 simulations. The φ2, as summed over all 7 nodes, is also
included in each figure.
According to the wheat data presented in Figure 21, root water uptake had no
effect on soil water dynamics below 50 cm in either loess or sand, as the slope of the
graphs beyond this point decreased to zero. Two explanations can be provided for this:
1) either root growth did not extend beyond this point, or 2) downward water fluxes
throughout the day kept the bottom layers at constant h values.
Concerning the rooting depth, the modeled and measured zm for wheat, for both
soils, showed that root growth certainly extended beyond the 50 cm mark (Table 11,
Table 12), leaving us to conclude that the lack of change in h was due to soil water
fluxes following nightly irrigation. Daytime downward water fluxes were also detected
for tomatoes grown in loess, as h values rose after the 10 hr mark (Figure 22A,C,E). In
light of this, we concluded that the soil was not dry enough for either wheat or tomato to
properly calibrate for the FD, vG, or NH models, which are parameterized by variables
estimated as low as -23,000 cm (Table 19A).
Little discrepancy could be seen between the 95% CI and ML curves of the FD
and vG models with wheat and tomato, but differences became clearer with sorghum
78
data for sand (Figure 23B,D,F), and φ2 values were higher for sorghum than for the
other two plants. This might have been due to the extended time of sorghum data (50
hours vs. 12), which enabled us to visualize soil water dynamics for a longer period, as
no irrigation events occurred during the time frame of simulations. h values for sand-
grown sorghum were also the lowest compared to wheat and tomato simulations.
The effect of compensation was somewhat evident in the measured h for
sorghum in sand (Figure 23B, D, F): after h leveled off at the 25 hr. mark at the first
observation node, it continued dropping in the second and third nodes. Eventually, after
50 hours with no irrigation, h at 30, 40, and 60 cm dropped drastically.
79
Figure 21. The measured and modeled h over time at varying depths for wheat plants in two soil media, calculated using three uptake models. (A) FD, loess (B) FD, sand, (C) vG loess, (D) vG, sand, (E) NH loess, (F) NH, sand. The total error, φ2, for all observation nodes is presented for each graph.
FD wheat loess FD wheat sand
0 10 20
-600
-500
-400
-300
-200
-100
Pre
ssu
re h
ead
[cm
]
20 cm
0 10 20
-600
-500
-400
-300
-200
-100
30 cm
0 10 20
-600
-500
-400
-300
-200
-100
40 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
50 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
Pre
ssu
re h
ead
[cm
]
60 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
70 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
80 cm
0 10 20
-200
-150
-100
-50
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20
-200
-150
-100
-5020 cm
0 10 20
-200
-150
-100
-5030 cm
0 10 20
-200
-150
-100
-50
time [hour]
40 cm
0 10 20
-200
-150
-100
-50
time [hour]
pre
ssu
re h
ead
[cm
]
50 cm
0 10 20
-200
-150
-100
-50
time [hour]
60 cm
0 10 20
-200
-150
-100
-50
time [hour]
70 cm
vG wheat loess vG wheat sand
0 10 20
-600
-500
-400
-300
-200
-100
pre
ssu
re h
ead
[cm
]
20 cm
0 10 20
-600
-500
-400
-300
-200
-100
30 cm
0 10 20
-600
-500
-400
-300
-200
-100
40 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
50 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
70 cm
0 10 20
-600
-500
-400
-300
-200
-100
time [hour]
80 cm
0 10 20
-200
-150
-100
-50
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20
-200
-150
-100
-5020 cm
0 10 20
-200
-150
-100
-5030 cm
0 10 20
-200
-150
-100
-50
time [hour]
40 cm
0 10 20
-200
-150
-100
-50
time [hour]
pre
ssu
re h
ead
[cm
]
50 cm
0 10 20
-200
-150
-100
-50
time [hour]
60 cm
0 10 20
-200
-150
-100
-50
time [hour]
70 cm
NH wheat loess NH wheat sand
0 10 20-600
-500
-400
-300
-200
-100
pre
ssu
re h
ead
[cm
]
20 cm
0 10 20-600
-500
-400
-300
-200
-100
30 cm
0 10 20-600
-500
-400
-300
-200
-100
40 cm
0 10 20-600
-500
-400
-300
-200
-100
time [hour]
50 cm
0 10 20-600
-500
-400
-300
-200
-100
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20-600
-500
-400
-300
-200
-100
time [hour]
70 cm
0 10 20-600
-500
-400
-300
-200
-100
time [hour]
80 cm
0 10 20
-200
-150
-100
-50
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20
-200
-150
-100
-5020 cm
0 10 20
-200
-150
-100
-5030 cm
0 10 20
-200
-150
-100
-50
time [hour]
40 cm
0 10 20
-200
-150
-100
-50
time [hour]
pre
ssu
re h
ead
[cm
]
50 cm
0 10 20
-200
-150
-100
-50
time [hour]
60 cm
0 10 20
-200
-150
-100
-50
time [hour]
70 cm
Measured data
95% CI ML
φ2 = 0.545
φ2 = 0.006
φ2 = 0.569
φ2 = 0.006
φ2 = 1.038
φ2 = 0.062
A B
C D
E F
80
Figure 22. The measured and modeled h over time at varying depths for tomato plants in two soil media, calculated using three uptake models. (A) FD, loess (B) FD, sand, (C) vG loess, (D) vG, sand, (E) NH loess, (F) NH, sand. The total error, φ2, for all observation nodes is presented for each graph.
FD tomato loess FD tomato sand
0 10 20-100
-80
-60
-40p
ress
ure
hea
d [
cm]
10 cm
0 10 20-100
-80
-60
-40
20 cm
0 10 20-100
-80
-60
-40
30 cm
0 10 20-100
-80
-60
-40
time [hour]
40 cm
0 10 20-100
-80
-60
-40
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20-100
-80
-60
-40
time [hour]
80 cm
0 10 20-100
-80
-60
-40
time [hour]
100 cm
0 10 20-120
-100
-80
-60
-40
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20-120
-100
-80
-60
-40
20 cm
0 10 20-120
-100
-80
-60
-40
30 cm
0 10 20-120
-100
-80
-60
-40
time [hour]
40 cm
0 10 20-120
-100
-80
-60
-40
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20-120
-100
-80
-60
-40
time [hour]
80 cm
0 10 20-120
-100
-80
-60
-40
time [hour]
100 cm
vG tomato loess vG tomato sand
0 10 20-100
-90
-80
-70
20 cm
0 10 20-100
-90
-80
-70
30 cm
0 10 20-100
-90
-80
-70
40 cm
time [hour]
0 10 20-100
-80
-60
-40
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20-100
-90
-80
-70
time [hour]
80 cm
0 10 20-100
-90
-80
-70
time [hour]
100 cm
0 10 20-100
-90
-80
-70
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20
-100
-80
-60
-40
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20
-100
-80
-60
-40
20 cm
0 10 20
-100
-80
-60
-40
30 cm
0 10 20
-100
-80
-60
-40
time [hour]
40 cm
0 10 20
-100
-80
-60
-40
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20
-100
-80
-60
-40
time [hour]
80 cm
0 10 20
-100
-80
-60
-40
time [hour]
100 cm
NH tomato loess NH tomato sand
0 10 20-100
-80
-60
-40
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20-100
-80
-60
-40
20 cm
0 10 20-100
-80
-60
-40
30 cm
0 10 20-100
-80
-60
-40
time [hour]
40 cm
0 10 20-100
-80
-60
-40
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20-100
-80
-60
-40
time [hour]
80 cm
0 10 20-100
-80
-60
-40
time [hour]
100 cm
0 10 20
-80
-70
-60
-50
-40
-30
pre
ssu
re h
ead
[cm
]
10 cm
0 10 20
-80
-70
-60
-50
-40
-30
20 cm
0 10 20
-80
-70
-60
-50
-40
-30
30 cm
0 10 20
-80
-70
-60
-50
-40
-30
time [hour]
40 cm
0 10 20
-80
-70
-60
-50
-40
-30
time [hour]
pre
ssu
re h
ead
[cm
]
60 cm
0 10 20
-80
-70
-60
-50
-40
-30
time [hour]
80 cm
0 10 20
-80
-70
-60
-50
-40
-30
time [hour]
100 cm
Measured data
95% CI ML
φ2 = 0.096
φ2 = 0.021
φ2 = 0.098
φ2 = 0.021
φ2 = 0.101
φ2 = 0.04
A B
C D
E F
81
Figure 23. The measured and modeled h over time at varying depths for sorghum plants in two soil media, calculated using three uptake models. (A) FD, loess (B) FD, sand, (C) vG loess, (D) vG, sand, (E) NH loess, (F) NH, sand. The total error, φ2, for all observation nodes is presented for each graph.
FD sorghum loess FD sorghum sand
0 25 50
-500
-400
-300
-200
-100
pre
ssu
re h
ead
[cm
]
15 cm
0 25 50
-500
-400
-300
-200
-100
20 cm
0 25 50
-500
-400
-300
-200
-100
25 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
30 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
pre
ssu
re h
ead
[cm
]
35 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
40 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
45 cm
0 25 50-800
-700
-600
-500
pre
ssu
re h
ead
[cm
]
10 cm
0 25 50-800
-700
-600
-500
15 cm
0 25 50-800
-700
-600
-500
20 cm
0 25 50-800
-700
-600
-500
time [hour]
25 cm
0 25 50-800
-600
-400
-200
time [hour]
pre
ssu
re h
ead
[cm
]
35 cm
0 25 50-800
-700
-600
-500
time [hour]
40 cm
0 25 50-800
-700
-600
-500
time [hour]
60 cm
vG sorghum loess vG sorghum sand
0 25 50
-500
-400
-300
-200
-100
pre
ssu
re h
ead
[cm
]
15 cm
0 25 50
-500
-400
-300
-200
-100
20 cm
0 25 50
-500
-400
-300
-200
-100
25 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
30 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
pre
ssu
re h
ead
[cm
]
35 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
40 cm
0 25 50
-500
-400
-300
-200
-100
time [hour]
45 cm
0 25 50
-800
-600
-400
-200
pre
ssu
re h
ead
[cm
]
10 cm
0 25 50
-800
-600
-400
-200
15 cm
0 25 50
-800
-600
-400
-200
20 cm
0 25 50
-800
-600
-400
-200
time [hour]
25 cm
0 25 50
-800
-600
-400
-200
time [hour]
pre
ssu
re h
ead
[cm
]
35 cm
0 25 50
-800
-600
-400
-200
time [hour]
40 cm
0 25 50
-800
-600
-400
-200
time [hour]
60 cm
NH sorghum loess NH sorghum sand
0 25 50
-400
-300
-200
-100
pre
ssu
re h
ead
[cm
]
15 cm
0 25 50
-400
-300
-200
-100
20 cm
0 25 50
-400
-300
-200
-100
25 cm
0 25 50
-400
-300
-200
-100
time [hour]
30 cm
0 25 50
-400
-300
-200
-100
time [hour]
pre
ssu
re h
ead
[cm
]
35 cm
0 25 50
-400
-300
-200
-100
time [hour]
40 cm
0 25 50
-400
-300
-200
-100
time [hour]
45 cm
0 25 50
-800
-600
-400
-200
pre
ssu
re h
ead
[cm
]
10 cm
0 25 50
-800
-600
-400
-200
15 cm
0 25 50
-800
-600
-400
-200
20 cm
0 25 50
-800
-600
-400
-200
25 cm
time [hour]
0 25 50
-800
-600
-400
-200
time [hour]
pre
ssu
re h
ead
[cm
]
35 cm
0 25 50
-800
-600
-400
-200
time [hour]
40 cm
0 25 50
-800
-600
-400
-200
time [hour]
60 cm
Measured data
95% CI ML
φ2 = 1.012
φ2 = 5.721
φ2 = 1.033
φ2 = 5.200
φ2 = 0.808
φ2 = 4.416
A B
C D
E F
82
5.2.5 Model comparisons
A final summary of model performance as evaluated according to convergence
robustness, φ1, and the SSE of modeled vs. measured soil water potential values, φ2, is
presented in Table 20. The two indices were not always 'in agreement'. For example,
while FD appeared the most robust for loess-grown wheat according to φ2, NH had the
lowest φ1 value of the three. In such cases, we chose to rely on φ2 as a more informative
index, as it related directly to system variables affected by root water uptake.
Plant Soil Model φ1 (-)
φ2 (-)
FD 0.496 0.545 vG 0.754 0.569 Loess NH 0.433 1.038 FD 0.458 0.006 vG 0.413 0.006
Wheat
Sand NH 0.393 0.062 FD 1.226 0.096 vG 1.649 0.098 Loess NH 1.763 0.101 FD 0.587 0.021 vG 0.384 0.021
Tomato
Sand NH 0.613 0.040 FD 0.734 1.012 vG 0.940 1.033 Loess NH 0.529 0.808 FD 0.398 5.721 vG 0.481 5.200
Sorghum
Sand NH 0.361 4.416
Table 20. Model robustness for all six treatments as evaluated by two error indices, φ1 and φ2. Highlighted values correspond with uptake models which had the lowest error per treatment.
Differences in φ2 between FD and vG models were negligible for wheat and
tomato. This might have been due to relatively wet conditions at bottom observation
nodes, the error of which was most likely small, and equal for both models. Another
factor which possibly neutralized differences between models was the incorporation of
ω, into both functions, which gave both FD and vG a discontinuous nature.
Rasiah et al. (1992) found that non-linear, discontinuous models were the most
robust in predicting root water uptake, which would point to vG as superior. But the
similarity in the magnitude of φ2 between FD and vG could have arisen from the fact
83
that discontinuity as an uptake model characteristic is preeminent over function type;
uptake can be sustained at the potential rate even past critical water stress thresholds
such as h2 and h50, in accordance with ωc.
Nevertheless, when looking at both indices, it became apparent that each of the
three models outperformed the others in different soil and plant contexts. In fact,
between wheat and tomato data, soil properties seemed more of a factor than the plant
root distribution on RWU predictions; for both wheat and tomato experiments, FD was
the most robust for loess while vG was the strongest in sand. These findings seemed to
support our hypothesis concerning the greater effects of the soil hydraulic properties
than root or plant properties in determining uptake patterns.
According to both φ1 and φ2, NH came out as the best-able to predict uptake
patterns of sorghum grown in both sand and loess, but was not further validated
(sorghum was not grown in the RLS). Differences in φ2 values between uptake models
were the largest in the sorghum experiment with sand. Since the effects of drought stress
were the most significant in this treatment, it is possible that these results contained the
most information regarding model performance. However, this would mean that NH is
the most robust of the three models for predicting uptake patterns, and this is not
consistent with results for wheat or tomato, which show NH as having the highest φ2 for
both soils. It may be that the accuracy of the NH increases as the soil water potential
decreases. The fact that NH considers K when solving for uptake supports this
observation, as K is positively correlated with h.
To summarize, the uptake models in Table 20 characterized by the smallest φ2
(highlighted in bold) were judged as the best able to predict uptake patterns. While all
models were validated with RLS data for wheat and tomato, only the results of the
models characterized by the lowest φ2 in Table 20 are displayed in the following section.
The results for the remaining models are shown in Appendix II.
84
5.3 Carousel experiments
5.3.1 Wheat
5.3.1.1 Plant transpiration over the growth period
The daily water balance from which transpiration was calculated throughout the
duration of both wheat and tomato RLS experiments are presented in Appendix I. The
daily transpiration, T [26], and cumulative transpiration, cumT, of wheat plants with
days after sowing (DAS) are shown in Figure 24A, B, respectively.
A
B
0
25
50
75
100
2
1.5
1
0.5
0
2 28 54 80 106 132
DAS (days)
Tra
nspi
ratio
n (c
m d
ay -1
) h
Treatment 1Treatment 2Treatment 3Treatment 4Start of irr. treatmentsEnd of irrigation
Cum
ulat
ive
tran
spir
atio
n (c
m) h
Figure 24. (A) The transpiration rate and (B) cumulative transpiration of wheat with days after sowing (DAS). Wheat plants were subjected to four treatments: (1) loess I=1.2T, (2) loess I=0.5T, (3) sand I=1.2T, and (4) sand I=0.5T. Differentiation into irrigation treatments and the end of irrigation is marked by the blue and red lines, respectively.
cumT was calculated according to [26] under the assumption that all water taken
up by roots was transpired. Deviations in T between the four treatments [(1) loess
I=1.2T, (2) loess I=0.5T, (3) sand I=1.2T, and (4) sand I=0.5T] became apparent after
the start of irrigation treatments, indicated by the blue line. Treatment 1 had the highest
85
cumT over the experiment, and reached full maturity later than other treatments, at
around 120 DAS. Treatment 3 had the second highest cumT value, followed by
Treatment 2, and lastly Treatment 4. Plants cultivated in the last three treatments
reached maturity between 102-110 DAS.
5.3.1.2 Plant yield
The dry wheat yield of lysimeters and the average yield of each treatment with
the standard deviation are shown below in Figure 26. In agreement with T data shown in
Figure 24, the average yield of plants from Treatment 1 was the highest. Treatments 2
and 3 produced the second-highest yield, as differences between average yield values
were not significant, although the standard deviation of Treatment 3 was high.
Treatment 4 produced the lowest yield, which was expected given the lower amount of
applied water plants received, coupled with the limited water holding capacity of sand.
0
100
200
300
400
500
1 2 3 4 5 6 7 8 9 10 11 12 Ave.
Lysimeter
Dry
yie
ld (g
)
Treatment 1Treatment 2Treatment 3Treatment 4
Figure 26. The dry wheat yield of each lysimeter and the average (ave.) yield and standard deviation for four treatments: (1) loess I=1.2T, (2) loess I=0.5T, (3) sand I=1.2T, and (4) sand I=0.5T. The irrigation and uptake data of Treatments 2 and 4 were taken from lysimeters marked with *, and the average uptake of all replicates of Treatments 1 and 3 were used as Tp input data for model validation in HYDRUS-1D.
5.3.1.3 Relative root distribution
The wheat RRM [19] from one lysimeter for each of the four treatments is
presented below in Figure 27.
*
*
86
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
10
20
30
40
Dep
th (c
m)
Relative Root Mass (g g-1)
Treatment 1- 3Treatment 2- 6Treatment 3- 7Treatment 4- 12
Figure 27. The relative wheat root mass (RRM) with depth. Plants were grown according to four treatments: (1) loess I=1.2T, (2) loess I=0.5T, (3) sand I=1.2T, and (4) sand I=0.5T. The lysimeter from which roots were excavated is indicated in the legend
The wheat roots of Treatment 1 had an uncharacteristic RRM when compared to
the RRM of wheat, tomato, and sorghum from column experiments (Figure 19, Figure
16, Figure 13); the RRM was less than 50% at the surface, sharply decreased at the 20
cm mark, and then instead of gradually tapering off, increased until it was back to 25%
at the bottom of the lysimeter. Old roots and organic matter in loess soil may have
introduced errors into measurements. Nevertheless, the RRM of the remaining three
treatments resembled the logarithmic profile seen from roots of column-grown plants.
The root distribution parameters of [7], when fitted to the measured RRM, are
presented in Table 21, along with the total excavated root mass of per treatment. These
parameters were used to interpolate the root distribution at each 1 cm depth increment.
Model validation was then carried out in HYDRUS-1D with the measured and modeled
wheat RDP (Table 12) to see the effect of the root distribution on plant transpiration.
Plant Soil Irrigation z* (cm) zm (cm) pz (-) ∫RM(z) (g) 100% 0.0 40 2.4 42.8
Loess 50% 4.7 40 5.8 18.9
100% 4.9 40 6.3 38.1 Wheat
Sand 50% 5.1 40 6.7 21
Table 21. The wheat RDP as obtained from fitting [7] to the measured RRM. The total root mass ∫RM(z) from each soil treatment is also included
87
It was important to check both the modeled and measured root distribution on
account of the significant differences between column and lysimeter dimensions; the
soil depth of lysimeters was only 40 cm, while columns extended to 150 cm, which
significantly changed the zm. Only z* values, presented in Table 21, resembled measured
results from the wheat column experiment (Table 11).
The largest root mass was excavated from the lysimeter containing wheat from
Treatment 1. The second-largest root mass were from plants grown under Treatment 3.
5.3.1.4 Model validation
A validation of the uptake models and their respective, optimized parameters
was carried out by comparing the measured and modeled, cumulative transpiration,
cumTa, of deficit-irrigated wheat and tomato over a 20-day simulation period.
Simulations were carried out with data from the last month of the experiment, after
plants had reached maturity and their root systems were considered fully developed.
HYDRUS-1D runs were carried out with two sets of initial soil water potential
conditions: (A) a hydrostatic profile where z = -h, and (B) -500 cm at the soil surface
which increased linearly until zero at the base of the profile, and results are displayed in
two separate graphs for each figure.
Modeled Ta values were calculated using three different ωc values. The first ωc
value was the optimized ML ωc for FD-loess and vG-sand, for wheat and tomato, as
shown in Table 13 and Table 16, respectively. The remaining two ωc values were
chosen so as to allow an evaluation of the full range of compensation, 0.001 < ωc < 1.
The cumulative potential transpiration, cumTp, was also plotted to better assess
compensation effects.
In addition, the optimized ML ωc from FD and vG simulations for loess and
sand, respectively, were tested using two root distribution functions: 1) the measured
root distribution function, β(z), calculated from the parameters fitted to the measured
88
RRM in Table 21 and Table 22, for wheat and tomato grown in the RLS, respectively,
and 2) the modeled root distribution function, β'(z), calculated from the optimized wheat
and tomato RDP of FD-loess and vG-sand simulations, presented in Table 12 and Table
15, respectively. Only β'(z) was used in combination with the other two ωc values tested.
From looking at Figure 28 below, it can be concluded that the cumulative
transpiration of deficit-irrigated plants was overestimated by FD, for all three levels of
compensation, and for both sets of initial conditions. Nevertheless, root water uptake
was less affected by the root distribution and varying levels of compensation in the drier
soil profile (Figure 28B) as indicated by the similarity in the modeled cumTa curves.
For an initially hydrostatic profile (Figure 28A), lowering ωc to 0.001 to allow
for full compensation raised the cumTa to the potential transpiration. Totally removing
the effects of compensation by equating ωc to one, still left a significant gap of 5 cm
between the measured cumTa of deficit-irrigated plants and the modeled cumTa. On the
other hand, when the initial h along the profile was lowered (Figure 28B), full-
compensation still produced the highest cumTa values, but only by a small margin
compared to the curves representing partial (ωc = 0.57) and no (ωc = 1) compensation.
We attributed the high values of modeled transpiration in Figure 28A, compared to
Figure 28B to the excess soil water available to the plant, as 'programmed' into the
system through the initial h conditions.
The difference in the modeled cumTa when ωc = 0.57, generated with both β(z)
and β'(z) was greater when the profile was initially hydrostatic (Figure 28A). The
modeled transpiration calculated with β'(z) closely followed the potential transpiration
curve, while cumTa values dropped within the range of the measured, actual
transpiration when the initial h at the surface was lowered to -500 cm (Figure 28B).
89
B
A
0
7
14
21
28
28
21
14
7
0
0 5 10 15 20
Simulation time (days)
Cum
ulat
ive
tran
spir
atio
n (c
m) j
Cumulative TpCumulative TaFD cumTa, 0.57, β(z) FD cumTa, 0.001, β'(z)FD cumTa, 0.57, β'(z)FD cumTa, 1, β'(z)
Figure 28. The measured, cumulative Tp and Ta of loess-grown wheat, plotted with the cumulative Ta, as modeled with FD over a 20 day period. Simulations were run with two sets of initial conditions (A) a hydrostatic profile (z = -h) and (B) a surface soil water potential of -500 which linearly decreased to zero. Three ωc values were used in simulations, and are specified in the legend. Model evaluations with both the measured, β(z), and modeled, β'(z), relative root distributions were carried out with ωc = 0.57, which is the optimized ML value from FD simulations for loess-grown wheat (Table 13).
The inability of HYDRUS-1D to generate cumTa values with FD which
resembled measurements may be due to the fact that the true initial conditions were
even lower than the second set (-500 cm at soil surface). This reinforces the importance
of installing soil water sensors along the lysimeter profile for future research in order to
assure accurate input data.
But whereas FD overestimated root water uptake for the conditions in which it
was found the most robust during calibration, vG was able to accurately predict uptake
patterns for the conditions in which it was tested, as shown below in Figure 29.
Nevertheless, not even full compensation (ωc = 0.001) was able to bring transpiration
levels up to the potential, most likely due to the meager water-retaining capacity of
90
sand. In initially hydrostatic conditions (Figure 29A), transpiration, as calculated with
β(z), with ωc = 0.68, lined up well with measured cumTa values. When modeled with
β'(z) at the same level of compensation, cumTa increased by about 4 cm at the end the
20 day period. No difference was evident between the effects of partial (ωc = 0.68) vs.
no (ωc = 1) compensation on RWU.
B
A
0
5
10
15
20
0
5
10
15
20
0 5 10 15 20
Simulation time (days)
Cum
ulat
ive
tran
spir
atio
n (c
m)
g
Cumulative TpCumulative TavG cum Ta, 0.68, β(z)vG cum Ta, 0.001, β'(z)vG cum Ta, 0.68, β'(z)vG cum Ta, 1, β'(z)
Figure 29. The measured, cumulative Tp and Ta of sand-grown wheat, plotted with the cumulative Ta, as modeled with vG over a 20 day period. Simulations were run with two sets of initial conditions (A) a hydrostatic profile (z = -h) and (B) a surface soil water potential of -500 which linearly decreased to zero. Three ωc values were used in simulations, and are specified in the legend. Model evaluations with both the measured, β(z), and modeled, β'(z), relative root distributions were carried out with ωc = 0.68, which is the optimized ML value from FD simulations for sand-grown wheat (Table 13).
When the initial pressure head along the soil profile was lowered (Figure 29B),
all effects of compensation and the β(z)/β'(z) on transpiration disappeared, and modeled
cumTa values lined up with the measured cumTa of deficit-irrigated plants.
Both the transpiration 'overshoot' by the FD model for loess soil (Figure 28), as
well as the decrease in the effect of compensation with decreasing h, seen in both Figure
28 and Figure 29 might be linked to incorrect Tp values provided as atmospheric
91
boundary conditions. The transpiration of I=1.2T tomatoes was used to represent the Tp
of deficit-irrigated plants, but since the potential transpiration was essentially a function
of the plant leaf area (Monteith, 1965), a correction factor should have be used to
account for the smaller size of deficit-irrigated plants compared to optimally-irrigated
plants. In addition, if the true Tp values of deficit-irrigated plants was less than the Tp of
optimally irrigated plants, the amount by which full-compensation (ωc = 0.001) should
have increased transpiration would have been less than the margin between the
measured Tp and Ta curves shown in Figure 28, Figure 29, Figure 33, and Figure 34.
5.3.2 Tomato
5.3.2.1 Plant transpiration over the growth period
Figure 30 presents the T [26] and cumT of tomato plants with DAS, respectively.
The cumulative T of tomatoes grown in loess with optimal irrigation at the end of the
experiment was about 20 cm larger than wheat plants grown in the same conditions even
though the length of the experiment was half that of wheat. This indicates that the rate
of water extraction for tomatoes was higher than wheat.
As seen with wheat, deviations in T between the four treatments became
apparent after the start of irrigation treatments, which is indicated by the blue line. Also
in agreement with wheat results, tomatoes grown with Treatment 1 had the highest T
and cumT. Plants grown with Treatment 2 had the second-highest daily T and cumT,
followed by Treatment 3.
92
B
A
105
84
63
42
21
0
5
4
3
2
1
0
6 16 26 36 46 56 66
DAS (days)
Tra
nspi
ratio
n (c
m d
ay -1
) h
Treatment 1Treatment 2Treatment 3Treatment 4Start of irr. treatmentsEnd of irrigation
Cum
ulat
ive
tran
spir
atio
n (c
m)
Figure 30. (A) The transpiration rate and (B) cumulative transpiration of tomato with days after sowing (DAS). Plants were subjected to four treatments: (1) loess I=1.2T, (2) loess I=0.5T, (3) sand I=1.2T, and (4) sand I=0.5T. Differentiation into irrigation treatments and the end of irrigation is marked by the blue and red lines, respectively.
5.3.2.2 Plant yield
The dry tomato yield of 16 lysimeters and the average yield and standard
deviation of the four treatments are shown below in Figure 31. The yield results agree
with the transpiration data, as Treatment 1 produced the largest yield, followed by
Treatment 2. Unlike the wheat results shown in Figure 26, the yield of tomatoes grown
with Treatment 3 was significantly smaller than that of Treatment 2. As expected,
Treatment 4 tomatoes produced the lowest yield, most likely due to the negligible soil
water stores available to plants.
93
0
250
500
750
1000
1250
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Ave.Lysimeter
Dry
yie
ld (g
)
Treatment 1Treatment 2Treatment 3Treatment 4
Figure 31. The dry tomato yield for each lysimeter and the average (ave.) yield and standard deviation for four treatments: (1) loess I=1.2T, (2) loess I=0.5T, (3) sand I=1.2T, and (4) sand I=0.5T. The irrigation and uptake data of Treatments 2 and 4 were taken from lysimeters marked with *, and the average uptake of all replicates of Treatments 1 and 3 were used as Tp input data for model validation in HYDRUS-1D.
5.3.2.3 Relative root distribution
The RRM [19] from 4 tomato treatments is presented below in Figure 32. Each
distribution had a logarithmic shape, similar to the RRM of column-grown plants
(Figure 13, Figure 16, and Figure 19), where the bulk of the roots are located near the
soil surface, and the rest are sparsely distributed among the deeper soil layers. In
Treatments 2 and 4, the roots slightly increased toward the bottom of the lysimeter,
which is apparently a measurement error.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
10
20
30
40
Dep
th (c
m)
Relative root mass (g g-1)
Treatment 1- 4
Treamtent 2- 16
Treatment 3- 6Treatment 4- 11
Figure 32. The relative tomato root mass (RRM) with depth. Plants were grown in four treatments: (1) loess I=1.2T, (2) loess I=0.5T, (3) sand I=1.2T, and (4) sand I=0.5T. The lysimeter from which roots were excavated is indicated in the legend
*
*
94
The measured RRM was used to fit a root distribution function for each
treatment, according to [7]. The resulting RDP are presented in Table 22. Parameter
values were very similar to the wheat RDP shown in Table 21, although the total tomato
root mass was greater than the wheat root mass, for all treatments, which is in
agreement with the differences in yield and T data between the two crops.
Plant Soil Irrigation z* (cm) zm (cm) pz (-) ∫RM(z) (g) 100% 4.6 40.0 5.5 96.1
Loess 50% 1.9 40.0 3.0 73.9
100% 1.6 38.5 2.7 49.9 Tomato
Sand 50% 3.2 40.0 3.8 28.6
Table 22. The tomato RDP as obtained from fitting [7] to the measured RRM. The total root mass ∫RM(z) from each soil treatment is also presented.
Results for the total tomato root mass between all four treatments followed the
transpiration and dry yield results shown in Figure 30 and Figure 31, respectively.
Tomatoes grown in loess soil had a higher root mass than sand-grown plants, while
plants grown with I=1.2T irrigation had more developed root systems than I=0.5T
treatments. The agreement between the tomato root mass and yield data, as well as
between the tomato vs. wheat root and shoot data, may point to a constant shoot:root
ratio between the different treatments, as was seen in the yield and root measurements
of column-grown plants.
5.3.2.4 Model verification
The results of the measured and modeled cumTa of sand- and loess-grown
tomatoes irrigated at 50% of the plant transpiration are shown in Figure 33 and Figure
34. No effect of either compensation levels or β(z)/β'(z) were seen with tomato data.
While no effect of either β(z)/β'(z) or compensation on uptake rates was evident
in Figure 33A,B, the initial water content was seen to significantly affect transpiration
throughout the simulation period. When the profile was initially hydrostatic (Figure
33A), all modeled cumTa series followed the potential transpiration until day 10. But
when h was lowered to -500 cm at the soil surface, modeled cumTa values lined up with
95
measured Ta values. The difference in the location of the modeled cumTa curves
between Figure 33A and Figure 33B is telltale of the effect of initial h conditions, and
the amount of stored soil water enabled by each set of conditions, on simulated results.
B
A
60
48
36
24
12
0
0
12
24
36
48
60
0 5 10 15 20
Simulation time (days)
Cum
ulat
ive
tran
spir
atio
n (c
m)
h
Cumulative TpCumulative TaFD cum Ta, 0.91, β(z)FD cum Ta, 0.001, β'(z)FD cum Ta, 0.5, β'(z)FD cum Ta, 0.91, β'(z)
Figure 33. The measured, cumulative Tp and Ta of loess-grown tomatoes, plotted with the cumulative Ta, as modeled with FD over a 20 day period. Simulations were run with two sets of initial conditions (A) a hydrostatic profile (z = -h) and (B) a surface soil water potential of -500 which linearly decreased to zero. Three ωc values were used in simulations, and are specified in the legend. Model evaluations with both the measured, β(z), and modeled, β'(z), relative root distribution were carried out with ωc = 0.91, which is the optimized ML value from FD simulations for loess-grown tomatoes (Table 16).
The transpiration modeled with vG for sand-grown tomatoes (Figure 34), for
both sets of initial conditions, fit observations well. No effect of either compensation or
β(z)/β'(z) could be seen, which was also the case for loess-grown tomatoes modeled with
FD (Figure 33), and sand-grown wheat modeled with vG (Figure 29).
96
B
A
28
21
14
7
0
28
21
14
7
0
0 5 10 15 20
Simulation time (days)
Cum
ulat
ive
tran
spir
atio
n (c
m) g
Cumulative TpCumulative TavG cum Ta, 0.55, β(z)vG cum Ta, 0.001, β'(z)vG cum Ta, 0.55, β'(z)vG cum Ta,1, β'(z)
Figure 34. The measured, cumulative Tp and Ta of sand-grown tomatoes, plotted with the cumulative Ta, as modeled with vG over a 20 day period. Simulations were run with two sets of initial conditions (A) a hydrostatic profile (z = -h) and (B) a surface soil water potential of -500 which linearly decreased to zero. Three ωc values were used in simulations, and are specified in the legend. Model evaluations with both the measured, β(z), and modeled, β'(z), relative root distributions were carried out with ωc = 0.55, which is the optimized ML value from FD simulations for loess-grown tomatoes (Table 16).
A comparison of Figure 28 vs. Figure 29 and Figure 33 vs. Figure 34 show that
vG was better able to predict uptake over a 20 day period than FD. vG may be a better
uptake model, given its non-linear nature, but a more significant factor affecting model
validation results is the soil texture in which models were tested.
The high Ks and low θr of sand severely limit water-retention. So, even if initial
conditions started off as hydrostatic, plants were not able to utilize water stores due high
downward flux rates, leading to low cumTa values at the end of the simulation period. In
addition, there was little difference in amount of water the plant could take up between
when the initial conditions at the soil surface were -100, or -500, which would explain
why uptake patterns in sand were relative insensitive to initial h conditions.
97
On the other hand, loess soil has a much greater ability for water retention. Thus,
differences in the initial h at the soil surface (-100 or -500 cm) had a significant effect
on the amount of water available to the plant. This not only explains the higher
sensitivity of uptake patterns to initial h conditions of loess-grown plants, but also sheds
light on the reason behind the greater error in the FD-modeled cumTa from measured
data. These results seem to indicate that soil texture played a major role in determining
uptake patterns regardless of the uptake model employed for calculation, and regardless
of plant characteristics.
One last factor which may have affected modeled results for the actual plant
transpiration was inaccurate RWU parameter values generated with DREAM. The soil
water potential data used in the objective function in optimization schemes was
representative of well-watered conditions, and downward fluxes could be detected from
h profiles over time, for wheat and tomato (Figure 21, Figure 22). Model calibrations
therefore should have been performed with data measured in drier conditions, in the
ranges of the critical water stress parameters, h2+h3, h50, and Hwilt, of the FD, vG, and
NH models, respectively.
98
6 Conclusions
Our objectives in this research were to calibrate three root water uptake models
for varying soils and plants using data gathered from cropped soil columns provided
with optimal irrigation, and to validate model performance under drought-stress
conditions with data taken from plants cultivated in a rotating lysimeter system. The
three models evaluated in this research considered compensatory uptake: the modified
Feddes et al. (1978) [with compensation according to Jarvis (1989)] (FD), modified van
Genuchten (1987) [with compensation according to Jarvis (1989)] (vG), and Nimah and
Hanks (1973) (NH). The calibration of models with two soil media, a local loess soil
and fine sand, revealed the effects of the soil hydraulic properties on predicted uptake
patterns. Optimizing for unknown parameters with data from wheat, tomato, and
sorghum experiments, provided insight into how the root distribution affects uptake.
Calibration for unknown model parameters was done with the DREAM
algorithm, in conjunction with HYDRUS-1D, implemented in MATLAB. Model
calibration was then validated by comparing the measured transpiration of deficit-
irrigated plants to model-generated transpiration, solved for in HYDRUS-1D.
Results from the model calibration showed that:
• DREAM algorithm proved robust, but better convergence is possible with less
parameters
o SHP should be measured after plant growth in light of the apparent
changes from before to after cropping
o RDP should be based on the measured root activity instead of root mass
o Growth columns should be weighed to calculate the daily transpiration
and remove⎯Tp from the vector of optimized parameters. This may also
prevent correlations between⎯Tp, h2, and h50.
99
• A large h database was amassed for varying soils and plants. Nevertheless, we
were not able to measure down to values corresponding with critical matric
potentials, h3+h4, h50, and Hwilt due to the high air-entry h of tensiometers. We
therefore suggest employing more robust soil sensors to obtain better quality
data for model calibrations.
• Under well-watered conditions, model robustness appeared affected by soil type
and root distribution, with soil texture as the dominant factor.
o Soil water dynamics of loess- and sand-grown wheat were best predicted
when the sink function of the Richards (1931) equation was solved with
FD and vG, respectively. The same pattern was seen for tomato.
• Soil water dynamics of both sand- and loess- grown sorghum, were best
predicted when NH was incorporated into the sink function of the Richards
(1931) equation. Water stress was the most significant with sorghum.
o The accuracy of NH increased as the soil water potential decreased,
making it the most fitting for use under drought conditions. This may be
due to the ability of NH to account for the soil hydraulic conductivity,
which is strongly dependent on h.
Results from the model validation showed that:
• The parameters of the root distribution function should be calibrated on a system
of the same depth as that with which the model is calibrated, in order to more
accurately model root water uptake.
• The initial soil water potential conditions significantly affected the modeled
transpiration. Therefore, soil water sensors should be installed at varying
positions along the profile in order to provide accurate and unequivocal
information concerning the initial soil water potential with depth.
100
• Under deficit-irrigation conditions, model robustness appeared predominantly
affected by soil texture.
• FD overestimated the transpiration of wheat grown in loess, regardless of the
initial soil water content, and overestimated the transpiration of loess-grown
tomato when the profile was initially hydrostatic.
• The transpiration of wheat and tomato plants in fine sand, as modeled with vG,
demonstrated a good fit with the measured data, although neither compensation
nor stored soil water seemed to have any affect on transpiration.
• The transpiration of optimally-irrigated plants might have been higher than the
true potential transpiration of deficit-irrigated plants. A correction factor should
be introduced to account for the smaller leaf area of water-stressed plants.
Therefore, we conclude:
• In light of the apparent effect of environmental factors such as soil texture and
plant type on root water uptake, observed under both well-watered and deficit-
irrigation conditions, a more mechanistic approach, able to account for such
factors, should be incorporated into existing empirical models. In addition,
continued testing should be carried out with the NH model in order to better-
understand its limitations and work toward widening its applicability.
101
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