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Modeling compensated root water and nutrient uptake
Jiří Šimůnek a,∗ , Jan W. Hopmans b
a Department of Environmental Sciences, University of California Riverside, Riverside, CA 92521, USAb Department of Land, Air and Water Resources, 123 Veihmeyer Hall, University of California, Davis, CA 95616, USAa r t i c l e

i n f oa b s t r a c t

Article history:

Plant root water and nutrient uptake is one of the most important processes in subsurface

Received 30 October 2007

unsaturated ﬂow and transport modeling, as root uptake controls actual plant evapotran-

Received in revised form

spiration, water recharge and nutrient leaching to the groundwater, and exerts a major

27 October 2008

inﬂuence on predictions of global climate models. In general, unsaturated models describe

Accepted 13 November 2008

root uptake relatively simple. For example, root water uptake is mostly uncompensated andPublished on line 26 December 2008

nutrient uptake is simulated assuming that all uptake is passive, through the water uptake
pathway only. We present a new compensated root water and nutrient uptake model, imple-

Keywords:

mented in HYDRUS. The so-called root adaptability factor represents a threshold value above

Root water uptake

which reduced root water or nutrient uptake in water- or nutrient-stressed parts of the root

Nutrient uptake

zone is fully compensated for by increased uptake in other soil regions that are less stressed.

Compensated uptake

Using a critical value of the water stress index, water uptake compensation is proportional

Unsaturated water ﬂow

to the water stress response function. Total root nutrient uptake is determined from the total

Numerical modelof active and passive nutrient uptake. The partitioning between passive and active uptake
is controlled by the a priori deﬁned concentration value cmax . Passive nutrient uptake is

HYDRUS

simulated by multiplying root water uptake with the dissolved nutrient concentration, forsoil solution concentration values below cmax . Passive nutrient uptake is thus zero when
cmax is equal to zero. As the active nutrient uptake is obtained from the difference between
plant nutrient demand and passive nutrient uptake (using Michaelis–Menten kinetics), thepresented model thus implies that reduced passive nutrient uptake is compensated for by
active nutrient uptake. In addition, the proposed root uptake model includes compensation
for active nutrient uptake, in a similar way as used for root water uptake. The proposedroot water and nutrient uptake model is demonstrated by several hypothetical examples, for plants supplied by water due to capillary rise from groundwater and surface drip
irrigation.
© 2008 Elsevier B.V. All rights reserved.

1.

Introduction

Root surfaces represent one of the most important phaseboundaries in nature since most mineral nutrients essential for life enter the biosphere and the food chains of theanimal world through the roots of higher plants (Nissen,∗1991). Similarly, root water and nutrient uptake is one of
the most important processes considered in numerical models simulating water content and ﬂuxes in the subsurface,
thus controlling water ﬂow (recharge) and nutrient transport(leaching) to the groundwater, and exerting a major inﬂuence on predictions of climate change impacts (Feddes andCorresponding author. Tel.: +1 951 827 7854; fax: +1 951 827 7854.
E-mail addresses: Jiri.Simunek@ucr.edu (J. Šimůnek), jwhopmans@ucdavis.edu (J.W. Hopmans).
0304-3800/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2008.11.004

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Raats, 2004) on terrestrial ecological systems, driving new
research at understanding roots and their functioning (Skaggs
and Shouse, 2008).
There are two major approaches generally used for thesimulation of root water uptake in vadose zone hydrological models, to be applied at the plot or ﬁeld scale (Hopmans
and Bristow, 2002). Early detailed quantitative studies of water
extraction by plant roots were based on a microscopic or mesoscopic (Feddes and Raats, 2004) approach that considered asingle root to be an inﬁnitely long cylinder of uniform radius
and water-absorbing properties (Gardner, 1960). Water ﬂow
to a root was described using the Richards equation formulated in radial coordinates, with ﬂow into the root driven bywater potential gradients between the root and surrounding soil and proportional to the hydraulic conductivity of thesoil surrounding the root (Mmolawa and Or, 2000) or the root
radial water conductivity parameter (Roose and Fowler, 2004).
Recent numerical modeling studies are increasingly applying the integrated plant root–soil domain approach, whereby
total plant transpiration is computed from solution of water
potential in the combined soil and root domain, solving forboth root and soil water potential (e.g., Doussan et al., 2006;
Javaux et al., 2008). Several models have been suggested thatsimulate individual roots and overall plant root architecture(e.g., Clausnitzer and Hopmans, 1994; Kastner-Maresch andMooney, 1994; Brown et al., 1997; Grant, 1998; Somma et al.,
1998; Biondini, 2001). These models often consider speciﬁc
processes such as biomass allocation to individual roots (e.g.,
Kastner-Maresch and Mooney, 1994; Grant, 1998; Somma etal., 1998; Biondini, 2001), root growth redirection to areas withhigh soil nutrient concentrations (e.g., Somma et al., 1998;
Biondini, 2001), linking of functioning of microbial ecosystems(Brown et al., 1997) and mycorryzal growth (Grant, 1998) to spatial structure of roots, or competition of different plant species
for nutrients (e.g., Biondini, 2001; Raynaud and Leadlley, 2005).
In addition to being more realistic in simulating soil-root interactions at the individual rootlet scale, the main advantage ofthis approach is that it automatically allows for compensation of soil water stress, as root water uptake is controlled bycomputed local water potential gradients and root conductivity for the whole root system. However, because of the lackof relevant soil and root data and the huge computational
requirements for simulation purposes at this microscopic
scale, soil water ﬂow models that consider ﬂow to each individual rootlet or plant root architecture have been limited toapplications at a relatively small scale of a single plant.
Most vadose zone models that are used at the plot or ﬁeld
scale (e.g., Jarvis, 1994; Flerchinger et al., 1996; van Dam et al.,
1997; Fayer, 2000; van den Berg et al., 2002; Šimůnek et al.,2008) utilize the macroscopic approach, whereby the potential
transpiration is distributed over the root zone proportionallyto root density, and is locally reduced depending on soil saturation and salinity status (Molz, 1981). This much more widelyused approach (e.g., Feddes et al., 1974; Bouten, 1995) neglects
effects of the root geometry and ﬂow pathways around roots,
and formulates root water uptake using a macroscopic sinkterm that lumps root water uptake processes into a single termof the governing mass balance equation. A wide variety of root
water uptake reduction functions have been suggested, ranging from a simple two-parameter threshold and slope function(Maas, 1990) or an S-shaped function (van Genuchten, 1987),
to more complex functions that can include up to 5 ﬁtting
parameters such as suggested by Feddes et al. (1978). We referreaders to the review paper by Feddes and Raats (2004) for
more details.
Usually, a compensation mechanism to balance reduced
water uptake from one part of the rhizosphere by increased
uptake in another less-stressed region of the rooting zone,
while simulated in microscopic models, is neglected invadose zone models. There is, however, growing experimental
evidence that plants, especially non-cultural plants, can compensate for water stress in one part of the root zone by taking
up water from parts of the root zone where water is available (e.g., Taylor and Klepper, 1978; Hasegawa and Yoshida,
1982; English and Raja, 1996; Stikic et al., 2003; Leib et al.,2006). The MACRO model (Jarvis, 1994) is an exception among
the more widely used models as it uses a critical value ofthe water stress index, or root adaptability factor, to allow
for compensated root water uptake. This factor represents athreshold value above which root water uptake that is reduced
in stressed parts of the root zone is fully compensated forby increased uptake from other parts. Among the researchmodels, in their ENVIRO-GRO model, Pang and Letey (1998)
used a similar threshold value to compute partial root water
uptake compensation. Similarly, Li et al. (2001) and Bouten(1995) distributed the potential transpiration across the rootzone according to a weighted stress index, which was a function of both root distribution and soil water availability or
water saturation fraction, respectively. A different approach
was used by Adiku et al. (2000), who assumed that plants seek
to minimize the total rate of energy expenditure during root
water uptake. They formulated the root water uptake problem
as a minimization problem and solved it using a dynamic program framework. Their optimized model simulated patternsof water extraction from uniformly wet soil proﬁles, with highest water extraction rates in the section where the root lengthdensity was also highest. For conditions with a dry soil surface, a reduction of root water uptake from the drier near soil
surface zone was compensated for by an increased root activity at greater soil depths, irrespective of root distribution. A
review of compensatory modeling approaches was recently
presented by Skaggs et al. (2006).
Plant nutrient availability and uptake is controlled byboth soil transport and plant uptake mechanisms. A detailed
description of nutrient uptake is often included in agronomic
models that simulate differentiation of plant nutrient demandduring various physiological growth stages (e.g., Parton etal., 1987; Jones and Ritchie, 1990). These models, however,typically greatly simplify soil water ﬂow and nutrient transport towards the root–soil interface. In contrast, vadose zone
models greatly simplify root nutrient uptake, often considering only its passive component and neglecting plant growth
dynamics (e.g., Jarvis, 1994; Flerchinger et al., 1996; van Dam
et al., 1997; Fayer, 2000; van den Berg et al., 2002; Šimůnek etal., 2008).
Solute transport in soils occurs by both mass ﬂow andmolecular diffusion. In the case of non-adsorbing nutrients,
nutrient uptake is controlled mainly by mass ﬂow, as is thecase of nitrate-N (e.g., Barber, 1995). In some cases, mass ﬂow
of speciﬁc nutrients (e.g., Ca2+ and Mg2+ ) may exceed the plant

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requirements, resulting in accumulation of particular ionsor even precipitation of corresponding solids (e.g., CaSO4 ) at
the root surface (Neumann and Römheld, 2002). Alternatively,
nutrients that exhibit low solubility in soil solutions, such
as P, K, NH4 + , and most micronutrients, are rapidly depleted
by root uptake in the soil solution (Neumann and Römheld,
2002), since roots absorb nutrients only in the dissolved state.
A resulting concentration gradient causes nutrient diffusionfrom the bulk soil toward the root surface. A decrease in thesolution concentration also disturbs the equilibrium between
the nutrients in solution and those bound to the solid soil
phase, resulting in their release from the solid phase andreplenishing of the solution concentration (Jungk, 2002). Plants
may utilize various additional strategies to mobilize nutrients, i.e., to release them from their association with thesolid phase, such as modiﬁcation of the chemical composition or association of roots with micro-organisms (Neumann
and Römheld, 2002).
There are many physical, biological, and physiologicalmechanisms that are involved in the nutrient uptake by plantroots (e.g., Jungk, 2002; Hopmans and Bristow, 2002; Darrah
et al., 2006). These can be broadly divided into passive andactive components. While the passive component represents
the mass ﬂow of nutrient into roots with water, the active
component represents a very diverse range of various biological energy-driven processes (e.g., Luxmoore et al., 1978;
Jungk, 2002; Neumann and Römheld, 2002; Silberbush, 2002;
Hopmans and Bristow, 2002) that affect the movement of speciﬁc nutrients from the root’s free space (cell walls) into theplant. The term passive nutrient uptake is thus deﬁned here as
the movement of nutrients into the roots by convective mass
ﬂow of water, directly coupled with root water uptake. The
term active nutrient uptake then represents the movementof nutrients into the roots induced by other mechanisms thanmass ﬂow. These other mechanisms include, for example, speciﬁc ion uptake by electro-chemical gradients, ion pumping
and uptake through ion channels.
Different fractions of different nutrients are supplied by
active and passive mechanisms (Jungk, 2002; Neumann and
Römheld, 2002). Shaner and Boyer (1976) demonstrated thatthe nitrate xylem concentration varied inversely with transpiration rate, and that nitrate uptake is mostly a function of
metabolic rate rather than transpiration rate. Active nitrate
uptake is considered to occur via NO3 − /H+ cotransport, orNO3 − /H+ counter transport via carriers (Haynes, 1986), withthe electrochemical gradient generated by proton pumping.
Although not strictly proven, it is generally proposed that
active uptake dominates in the low supply concentration
range and under stress conditions, whereas passive uptake
becomes more important at higher soil solution concentrations, via mass ﬂow driven by root water uptake andtranspiration (see also Porporato et al. (2003)). Rather thana priori deﬁning the nutrient uptake mechanism, Somma etal. (1998) assumed that passive and active uptake can beconsidered as additive processes, and allowed for a ﬂexible
partitioning between active and passive uptake, with the relative contribution of each to be determined by the model
user, and total nutrient uptake controlled by plant nutrient
demand (Hopmans, 2006). A similar approach was adopted by
Porporato et al. (2003) in their modeling study to evaluate the507

inﬂuence of soil moisture control on soil carbon and nitrogen
cycling.
In addition to soil transport mechanisms, nutrient uptake
is controlled by the spatial distribution of roots, as inﬂuenced
by its architecture, morphology and presence of active sitesof nutrient uptake, including root hairs (e.g., Somma et al.,
1998; Biondini, 2001; Javaux et al., 2008). For nutrients that are
immobile (e.g. P) or slowly mobile (ammonium), a root system must develop so that it has access to the nutrients, byincreasing their exploration volume. Alternatively, the roots
may increase its exploitation power for the speciﬁc nutrient by local adaptation of the rooting system, allowing forincreased uptake efﬁciency of the nutrient. When considering the rhizosphere dynamics of water and nutrient uptake,many more mechanisms may have to be considered, including rhizosphere acidiﬁcation (Pierre and Banwart, 1973) andnitrogen mineralization (Bar-Yosef, 1999).
The root water and nutrient uptake model presented in thismanuscript links soil physical principles with plant physiological concepts, thereby providing for an improved integrationof scientiﬁc principles as needed for an interdisciplinary ecological approach, as compared to most other approaches.
We understand that such integration may polarize different
view points between scientiﬁc ﬁelds, and create misunderstandings of notation and concepts that are unique withineach discipline. However, the mathematical model introduced
here is an honest attempt to cross disciplinary boundaries as
required for advancing the science for the broad and complex study ﬁeld of ecology. The presented subsurface modelingapproach will greatly improve scenario testing for soil–plantsystems, by including plant uptake mechanisms such ascompensated root water and active root nutrient uptake. As
we demonstrate in the example simulations, soil nutrient
concentrations are controlled by the magnitude of partitioning between passive and active plant nutrient uptake. Thisis especially important for natural ecosystems, where soil
nutrient concentrations are generally much lower for mostplant nutrients than in agricultural systems. In addition, natural ecosystems often suffer from environmental stresses(water, nutrient, temperature), and the plant responses to such
limiting factors are highly relevant for understanding theirfunctioning and survival strategies. We note that comparison of model results as shown here at the single-plant scaleis much more difﬁcult for larger ecosystem models, because
of inherent soil heterogeneities, thereby complicating model
input requirements and model calibration (Wegehenkel andMirschel, 2006). Alternatively, less data-intensive and simpler water- and nutrient uptake models may be warranted
for ecosystem-scale analysis of soil environmental constraints(De Barros et al., 2004; van den Berg et al., 2002).
The objective of the presented study is to reformulate themathematical and numerical model of root water and nutrient
uptake, as commonly implemented in vadose zone ﬂow andtransport models (e.g., Jarvis, 1994; Bouten, 1995; Flerchinger
et al., 1996; van Dam et al., 1997; Fayer, 2000; van den Berg et al.,
2002; Šimůnek et al., 2008), by including compensation of local
water and nutrient stresses, and partitioning between passive
and active nutrient uptake. The model should be as general
as possible so that it can be applied to an arbitrary nutrient (either adsorbing or non-adsorbing) without any further

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modiﬁcations by simply selecting the relevant model parameters, while at the same time simple enough so that it can bereadily implemented in vadose zone ﬂow and transport models. Therefore, we do not strive to simulate individual roots
or plant root architecture (e.g., Biondini, 2001 and similar references given above), biomass allocation to roots (e.g., Grant,
1993; Kastner-Maresch and Mooney, 1994), or dynamics of thesoil microbial system and the carbon, nitrogen and phosphorus cycles in the soil (e.g., Parton et al., 1987; Wegehenkel
and Mirschel, 2006). The new root uptake model is implemented into the HYDRUS software packages (Šimůnek et al.,2006, 2008) and several examples are given demonstrating theeffects of the water and nutrient uptake compensation andactive nutrient uptake on root zone soil moisture and nutrient
distribution.

sorption isotherms describing instantaneous sorption of thenutrient, or various kinetic equations describing rate limited
sorption/desorption or precipitation/dissolution. For example,
when the solid phase concentration is expressed by an adsorption isotherm, the left hand side of Eq. (2) can be reformulated
as follows:∂c∂Rc∂b c̄
+
=
,
∂t
∂t
∂t

R=1+

b ∂c̄
 ∂t

(3)

2.

Theorywhere R is the retardation factor [−], deﬁning the partitioningof nutrient between the solid and liquid phases. The retardation factor is closely related to the buffering factor, a term
often used in literature dealing with root uptake. Notice that inboth Eqs. (1) and (2), lower case symbols describing root waterand nutrient uptake represent local values, while upper casesymbols (deﬁned below) represent global values, that is, local
values integrated over the root zone.

2.1.

Water ﬂow and nutrient transport

2.2.

Water ﬂow and nutrient transport in variably saturated porous
media are usually described using the Richards (1931) andconvection–dispersion equations (CDE), respectively:
∂(h)
∂
=
∂t
∂xi





K(h)

∂b c̄
∂
∂c
=
+
∂t
∂xi
∂t



∂h
A
KijA
+ Kiz
∂xj

∂c
Dij
∂xj


−


− s∗ (h)(1)∂(qi c)−  − ra (c, h)
∂xi

(2)

In the Richards equation (1),  is the volumetric water content [L3 L−3 ], h is the soil water potential expressed by pressure
head [L], K is the unsaturated hydraulic conductivity [LT−1 ],KijA are components of a dimensionless anisotropy tensor
KA (which reduces to the unit matrix when the medium isisotropic), t is time [T], xi are the spatial coordinates [L], and s*
is the sink/source term [L3 L−3 T−1 ], accounting for root water
uptake (transpiration). The sink term, s* , represents the volume of water removed per unit time from a unit volume ofsoil due to plant water uptake. In this paper, s* can represent either uncompensated (s) or compensated (sc ) root water
uptake.
In the CDE, b is the bulk density [ML−3 ], c̄ [MM−1 ] andc [ML−3 ] are concentrations in the solid and liquid phases,
respectively; Dij are components of the effective dispersion
tensor [L2 T−1 ], qi is the volumetric ﬂux density [LT−1 ],  is therate of change of mass per unit volume by chemical or biological reactions or other sources (negative) or sinks (positive)[ML−3 T−1 ]. For example, nitriﬁcation and denitriﬁcation are
typical examples of reactions that can be represented by the term. Finally, ra represents the root nutrient uptake [ML−3 T−1 ],which is the sum of the actual (subscript a) passive (pa ) andactive (aa ) root nutrient uptakes. The sink terms ra , pa , andaa represent the mass of nutrient (nutrients) removed perunit time from a unit volume of soil due to the total, passive and active plant nutrient uptake, respectively. The solid
phase concentration, c̄, accounts for nutrient either sorbed tothe solid phase or precipitated in various minerals. This term
is usually mathematically described using linear or nonlinear

Root water and nutrient uptake

In the following we will formulate our root water (s* ) andnutrient uptake (ra ) terms using the macroscopic approach(Fig. 1). We will formulate it for a two-dimensional problem, although both one- or three-dimensional formulations
are trivial simpliﬁcation or expansion of this two dimensional formulation, respectively. Fig. 1 presents a schematic
diagram that illustrates various stressed and unstressed,
compensated and uncompensated, active and passive root
water and nutrient uptakes discussed in the following sections.

2.2.1.

Root water uptake

First, we must deﬁne the magnitude of potential root water
uptake rate, sp , corresponding to the potential transpiration
rate, Tp . Typically, the root water uptake is distributed over
the root zone according to the spatial root distribution (forexample, see Skaggs et al., 2006). The potential transpiration
is determined by the atmospheric demand, as controlled bymeteorological variables, such as net radiation, air temperature, wind speed, and relative humidity, but does not consider
the plant and soil environment. The potential transpiration
rate can be calculated from the meteorological variables using
various process-based or empirical formulas, such as the
FAO-recommended Penman–Montheith combination equation (FAO, 1990) or the Hargreaves formula (e.g., Jensen et al.,1997), respectively.
When distributing the potential water uptake rate equally
over a two-dimensional rectangular root domain, sp becomes:sp (x, z, t) =1
Lt Tp (t)
Lx Lz

(4)

where sp is the potential root water uptake rate [T−1 ] at location (x, z) and time t, Tp is the potential transpiration rate
[LT−1 ], Lz is the depth [L] of the root zone, Lx is the width[L] of the root zone, Lt is the width [L] of the soil surface
associated with the transpiration process, and x and z are spatial coordinates. The distinction between Lt and Lx is needed
for partially plant-covered soil surfaces, and sp reduces to
Tp /Lz when Lt = Lx . Eq. (4) may be generalized by introducing

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509

Fig. 1 – Schematic diagram that illustrates the presented root water (a) and nutrient (b) uptake mechanisms (Tp , Ta , Tac –potential, actual non-compensated, and actual compensated root water uptake rate, respectively; ˛ – water stress response
function; ωc – water stress index; Rp – potential root nutrient uptake rate; Pa – passive root nutrient uptake rate; Ap , Aa , Aac –potential, actual non-compensated, and actual compensated root nutrient uptake rate, respectively; ˛MM – nutrient stressresponse (Michaelis–Menten) function; c – nutrient stress index).

a non–uniform distribution of the potential water uptake rateover a root zone with an arbitrary shape (Vogel, 1987):

where ˝R represents the root zone [L2 ]. From (5) and (6) it
follows that Sp is related to Tp by the expression:sp (x, z, t) = b(x, z, t)Lt Tp (t)1
Tp (t) =Lt(5)

where b(x, z, t) is the normalized water uptake distribution[L−2 ] (Fig. 2). Note that b(x, z, t) is a function of space and time,allowing for plant root growth. The functional description is
ﬂexible, and can accommodate known root distribution functions, such as linear (Feddes et al., 1978), exponential (Raats,1974), or can be more general (Vrugt et al., 2001a,b). The function b(x, z, t) must be normalized to ensure that b(x, z, t)
integrates to unity over the ﬂow domain, i.e.:b(x, z, t) d˝ = 1
˝R

(6)


sp (x, z, t) d˝

(7)

˝R

Notice that the formulation above is given for a twodimensional problem. A similar expression can be derived,with Lt equal to 1 [−] and the soil surface area [L2 ]associated with the transpiration process, for one- andthree-dimensional formulations, respectively. Similarly, thenormalized water uptake distribution is equal either to b(z,t) [L−1 ] or b(x, y, z, t) [L−3 ] for one- and three-dimensional
problems, respectively.

2.2.1.1. Uncompensated root water uptake. For the noncompensated root water uptake model, the actual root water

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pressure head), water uptake is also zero. Water uptake is considered optimal between pressure heads h2 and h3 , whereas
for soil water pressure head values between h3 and h4 (or h1
and h2 ), water uptake decreases (or increases) linearly with h.
For phreatic plants and trees that are able to extract water frombelow the groundwater, values of h1 and h2 can be adjusted,or set to zero in which case the Feddes’ function is similar
to the Maas (1990) model. Parameters of the stress response
function for a majority of agricultural crops can be found invarious databases (e.g., Taylor and Ashcroft, 1972; Wesseling
et al., 1991) and are directly implemented into the GUI ofHYDRUS. As is apparent from inspection of Eq. (7) the actualroot water uptake is equal to potential uptake (and potential
transpiration) during periods of no water stress (˛(h) = 1).
The actual local uncompensated root water uptake, s, isobtained by substituting (5) into (8), or

Fig. 2 – Schematic of the potential water uptake
distribution function, b(x, z, t), in the soil root zone.

uptake, s, is obtained from the potential root water uptake,sp , through multiplication with a stress response function(Feddes et al., 1978), ˛, as follows:s(h, h , x, z, t) = ˛(h, h , x, z, t)b(x, z, t)Lt Tp (t)s(h, h , x, z, t) = ˛(h, h , x, z, t) sp (t)so that the actual transpiration rate, Ta [LT−1 ], is obtained byintegrating (9) over the root domain ˝R , or(8)where the stress response function ˛(h, h ) is a prescribed
dimensionless function of the soil water (h) and osmotic(h ) pressure heads (0 ≤ ˛ ≤ 1). The stress response function
reduces the potential root water uptake due to the moistureand salinity stress. Since extensive literature is devoted to various formulations of the water stress response function (e.g.,
Cardon and Letey, 1992; Feddes and Raats, 2004; Skaggs et al.,2006), we will not discuss these here in detail. For example,
effects of the pressure head and osmotic stresses can be considered to be either additive or multiplicative (van Genuchten,
1987; Cardon and Letey, 1992; Feddes and Raats, 2004) or various functions (e.g., an S-shape function (van Genuchten, 1987)
or a threshold and slope function (Maas, 1990)) representing
these stresses can be used. While some functions require only
two parameters (Maas, 1990; van Genuchten, 1987), the more
complex functions, such as that suggested by Feddes et al.
(1978), require ﬁve parameters.
The presented examples will use the stress response function of Feddes et al. (1978) (Fig. 3). Notice that water uptake isassumed to be zero close to saturation (i.e., wetter than somearbitrary “anaerobiosis point”, h1 ). For h < h4 (the wilting point1
Ta (t) =Lts(h, h , x, z, t) d˝˝R˛(h, h , x, z, t)b(x, z, t) d˝= Tp (t)(10)

˝R

2.2.1.2. Compensated root water uptake. The ratio of actualand potential transpiration of uncompensated root water
uptake is deﬁned as
Ta (t)1=
Tp (t)Tp (t)Lts(h, h , x, z, t) d˝˝R˛(h, h , x, z, t)b(x, z, t) d˝ = ω(t)

=

(11)

˝R

where ω is a dimensionless water stress index (Jarvis, 1989,
1994). Following Jarvis (1989, 1994), we introduce a criticalvalue of the water stress index ωc , or the so-called root adaptability factor. It represents a threshold value, above which theroot water uptake reduced in stressed parts of the root zone
is fully compensated for by uptake from other, less-stressedparts (Fig. 4). Below this critical value, there is a certain reduction of the potential transpiration, although smaller than inuncompensated root water uptake.
Deﬁning compensated actual transpiration rate, Tac , as theratio of Ta /ω, the ratio of the actual to potential transpiration
becomes for the water stress range above the critical water
stress index (i.e., ω > ωc ):Ta (t)Tac (t)==Tp (t)Tp (t)ω(t)Fig. 3 – Schematic of the plant water stress response
function, ˛(h), as used by Feddes et al. (1978).

(9)


˝R

˛(h, h , x, z, t)b(x, z, t) d˝ω(t)sc (h, h , x, z, t) = ˛(h, h , x, z, t)b(x, z, t)Lt

Tp (t)
ω(t)

=

ω(t)
= 1,
ω(t)
(12)

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Fig. 4 – Ratio of the actual to potential transpiration as afunction of the stress index ω (arrows point towards thecorresponding axis; the left axis is for compensateduptake, while the right axis is for uncompensated uptake).

For the water stress range below the critical water stress index(i.e., ω < ωc ) we deﬁne Tac = Ta /ωc , so that
Tac (t)Ta (t)==Tp (t)Tp (t)ωc (t)˝R˛(h, h , x, z, t)b(x, z, t) d˝ωcsc (h, h , x, z, t) = ˛(h, h , x, z, t)b(x, z, t)Lt

=

ω(t)
< 1,
ωc

Tp (t)
ωc(13)

where sc is the compensated root water uptake [T−1 ]. For ωc = 1
and 0, root water uptake is either uncompensated or fully compensated, respectively. We note that division by zero in (13) istheoretically not-deﬁned. Uncompensated root water uptake
is thus a special case of compensated root water uptake whenωc = 1 (Fig. 1a). Combined, both uncompensated and (fully or
partially) compensated models can be deﬁned by
Tac (t)=Tp (t)˛(h, h , x, z, t)b(x, z, t) d˝˝Rmax[ω(t), ωc ]=sc (h, h , x, z, t) = ˛(h, h , x, z, t)b(x, z, t)Lt

ω(t)
≤ 1,
max[ω(t), ωc ]Tp (t)
max[ω(t), ωc ](14)

so that the total actual compensated plant transpiration isequal to1
Tac (t) =Lt=sc (h, h , x, z, t) d˝
˝RTp (t)
max[ω(t), ωc ]˛(h, h , x, z, t)b(x, z, t) d˝

(15)

˝R

Using this approach, water uptake compensation is proportional to the water stress response function. Water uptake
increase (compensation) is maximum in parts of the root zone
where the root water uptake is optimal (i.e., not reduced),equal to zero in parts of the root zone where the pressure headis below the wilting point or above the anaerobiosis point; andproportional to the water stress response for other pressure
head values. In this way, the proposed compensation model isclosely related to models based on minimizing energy of root
water uptake (e.g., Adiku et al., 2000; Van Wijl and Bouten,

5112001). We note that this compensation mechanism spatiallyredistributes root water uptake from stressed to less-stressedregions of the root zone, where water is held with smaller capillary forces and thus more easily available for the plant. Thisis very different than simply modifying the stress response
function by expanding the pressure head intervals for both
unlimited and limited uptake across the whole rooting zone.
However, as pointed out by Skaggs et al. (2006), the presented compensation is conceptually unsound, if the wholeroot domain is equally stressed. For example, the model will
predict a full compensation of root water uptake when theroot zone has the uniform pressure head for which the water
uptake is reduced by less than the critical water stress index.
The predicted root water uptake, which is uniform under suchconditions throughout the root zone, will then deviate, due tocompensation, from the value predicted by the stress response
function. To overcome this conceptual problem one can, forexample, consider compensation of root water uptake to occur(a) only from parts of the root zone which are not stressed at
all or (b) from zones which are stressed less than ω. After suchmodiﬁcation, the updated compensated root water uptake
model will follow the thick line from Fig. 4 only when thepressure head in a part of the root zone is below the wilting point and the rest is either non-stressed or stressed lessthan ω, respectively. For all other conditions, the model will be
between theoretical lines for non-compensated and compensated root water uptake.
The compensated root water uptake model requires as
input the potential evapotranspiration rate Tp , the spatialroot distribution function b(x, z, t), the water stress response
function ˛, and the critical water stress index ωc . Therefore,
the additional new input variable, as compared to traditional
uncompensated root water uptake is the critical water stressindex, ωc . It can be expected that cultural (i.e., agricultural)plants have a relatively high ωc and thus their ability to compensate natural stresses is limited, as compared to natural
plants, especially desert species, that have a low ωc and correspondingly high ability to compensate for natural stresses(i.e., to only take up water and nutrients from those parts of
the root system where they are most available).

2.2.2.

Root nutrient uptake

The presented model allows for both passive and active root
nutrient uptake. Whereas passive uptake describes the mass
ﬂow and root uptake of nutrients dissolved in water taken up
by plant roots, we deﬁne active uptake as all other possible
nutrient uptake mechanisms, including energy-driven processes against electrochemical gradients. The term “passive
uptake” is used here, and throughout the manuscript, to represent ﬂow of nutrients into roots associated with ﬂow of watersupplying the plant transpiration demand. Since transpiration
ﬂow is an active process, the mass nutrient ﬂow is also a process that is actively regulated by plants. The terms “passive”
and “active” are used here mainly to distinguish between these
two mechanisms of root nutrient uptake.
Similarly, as for root water uptake, we deﬁne a potential nutrient demand, Rp [ML−2 T−1 ], that depends on theplant physiological growth stage, supplied by both passive andactive nutrient uptake. Daily nutrient consumption rates are
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stage (after emergence or planting), for example as reported
by Bar-Yosef (1999) for N, P, and K nutrients.
We further assume that passive uptake is the primarymechanism of supplying plants with nutrients, and that active
uptake is initiated only if passive uptake is inadequate. For
their physiological development, plants need to take up waterfrom the root zone, and as nutrients are dissolved in soil water
they can enter the plant by the dissolved water phase pathway. The active uptake will then provide additional nutrientsthat are required beyond what is supplied by passive uptake.
In general, the presented model allows for both passive andactive nutrient uptake mechanisms to occur separately or
simultaneously, as described below. The relative signiﬁcanceof active and passive nutrient uptakes in supplying various
agricultural crops with nutrients can be found in the literature. For example, values for N, P, and K in maize are given by
Jungk (1991).

Passive actual root nutrient uptake for the whole root
domain, Pa [ML−2 T−1 ], is calculated by integrating the local
passive root nutrient uptake rate, pa , over the entire root zone,
or after applying Eq. (14):Pa (t) =

=

=

1
Lt
1
Lt


pa (x, z, t) d˝˝Rs∗ (x, z, t) min[c(x, z, t), cmax ] d˝
˝RTp (t)
max[ω(t), ωc ]˝R

˛(h, h , x, z, t)b(x, z, t) min[c(x, z, t), cmax ] d˝

(19)

Deﬁning Rp as the potential (subscript p) nutrient demand[ML−2 T−1 ], the potential active nutrient uptake rate, Ap[ML−2 T−1 ], is computed from:

2.2.2.1. Uncompensated nutrient uptake model. To clearly differentiate between point and root domain nutrient uptake
rate values, we deﬁne lower case variables to represent pointroot nutrient uptake rates [ML−3 T−1 ], while upper case variables represent nutrient uptake rates [ML−2 T−1 ] over the entire
two-dimensional root zone domain, ˝R . Both point and root
domain nutrient uptakes are assumed to be the sum of their
passive and active components, orra (x, z, t) = pa (x, z, t) + aa (x, z, t)(16)

Ra (t) = Pa (t) + Aa (t)(17)

Ap (t) = max[Rp (t) − Pa (t), 0](20)

Thus, using this formulation, we assume that active nutrient
uptake will be invoked only if the passive root nutrient uptake
term does not fully satisfy the potential nutrient demandof the plant. However, as was discussed earlier, the passive
uptake can be reduced or completely turned off (cmax = 0), thus
allowing the potential active nutrient uptake (Ap ) to be equal
to the potential nutrient demand (Rp ). Once Ap is known,
the point values of potential active nutrient uptake rates, ap[ML−3 T−1 ], are obtained by distributing the potential root zone
active nutrient uptake rate, Ap [ML−2 T−1 ], over the root zone
domain, using a predeﬁned spatial root distribution, b(x, z, t),
as was done for root water uptake in Eq. (5), or

where ra , pa , and aa deﬁne total actual (subscript a) passive
and active root nutrient uptake rates [ML−3 T−1 ], respectively,
at any point, and Ra , Pa , and Aa denote actual total, passive
and active root nutrient uptake rates [ML−2 T−1 ], respectively,
for the root zone domain.
Passive nutrient uptake is simulated by multiplying root
water uptake (compensated or uncompensated) with the dissolved nutrient concentration, for concentration values below
a priori deﬁned maximum concentration (cmax ), or

Using Michaelis–Menten kinetics (e.g., Jungk, 1991) provides
for actual distributed values of active nutrient uptake rates,
aa [ML−3 T−1 ], allowing for nutrient concentration dependency,orpa (x, z, t) = s∗ (x, z, t) min[c(x, z, t), cmax ]aa (x, z, t) =(18)

where c is the dissolved nutrient concentration [ML−3 ] andcmax is the maximum allowed solution concentration [ML−3 ]that can be taken up by plant roots during passive root uptake.
All nutrient dissolved in water is taken up by plant roots when
cmax is large (larger than the dissolved concentration c), while
no nutrient is taken up when cmax is equal to zero, with only
active uptake remaining in that case (Fig. 1b). The maximum
solution concentration for passive root uptake, cmax , thus controls the relative proportion of passive root water uptake tototal uptake. Using this ﬂexible formulation, uptake mechanisms can vary between speciﬁc nutrients. For example, Na
uptake can be excluded by setting cmax equal to zero, passive
Ca uptake can be limited by deﬁning a ﬁnite cmax value, or all
soil solution available P or N is allowed to be taken up passively, by setting cmax to a very large value. Note that the cmax
parameter is introduced as a control model parameter that
does not necessarily have a physiological meaning.

ap (x, z, t) = b(x, z, t)Lt Ap (t)

=

(21)

c(x, z, t) − cminap (x, z, t)
Km + c(x, z, t) − cminc(x, z, t) − cminb(x, z, t)Lt Ap (t)Km + c(x, z, t) − cmin(22)

where Km is the Michaelis–Menten constant [ML−3 ] and cminis the minimum nutrient concentration required for active
uptake to take effect [ML−3 ] (Jungk, 1991), thus assuming that
active nutrient uptake will occur only if the dissolved nutrient
concentration in the soil solution is sufﬁciently high. Other
approaches have been suggested to deal with the concentration threshold (Silberbush, 2002). The Michaelis–Menten
constants for selected nutrients (e.g., N, P, and K) and plant
species (e.g., corn, soybean, wheat, tomato, pepper, lettuce,and barley) can be found in the literature (e.g., Bar-Yosef, 1999).
Though not in agreement with the presented concept of distinguishing between passive and active uptake, the formulation
could be easily modiﬁed so that the Michaelis–Menten kinetics is applied to the sum of the active and passive nutrientAuthor's personal copy513

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from which the total compensated active root nutrient uptake
rate, Aac [ML−2 T−1 ] in the two-dimensional root domain, ˝R ,is calculated, in analogy with the compensated root water
uptake term in Eq. (15), as follows:
Aac (t) ==Fig. 5 – Ratio of actual to potential active nutrient uptake asa function of the stress index  (arrows point towards thecorresponding axis; the left axis is for compensateduptake, while the right axis is for uncompensated uptake).

uptake as it is usually used in the literature (Silberbush,2002).
Finally, total active uncompensated root nutrient uptake
rate, Aa [ML−2 T−1 ], is calculated by integrating the actual
active root nutrient uptake rate, aa , at each point, over theroot domain ˝R , in analogy with the non-compensated root
water uptake term in Eq. (10), or1
Lt

Aa (t) =aa (x, z, t) d˝
˝R= Ap (t)˝Rc(x, z, t) − cminb(x, z, t) d˝Km + c(x, z, t) − cmin(23)

2.2.2.2. Compensated nutrient uptake model. The above nutrient uptake model includes compensation of the passive
nutrient uptake, by way of the root water uptake compensation term, sc , and root adaptability factor, ωc , in Eq. (19). A
similar compensation concept as used for root water uptake in
Eqs. (11) and (12), was implemented for active nutrient uptake
rate, by invoking a so-called nutrient stress index :(t) =Aa (t)Ap (t)(24)

After substitution of the active total root nutrient uptake rate
value from Eq. (23) above, this newly deﬁned nutrient stressindex () is equal to(t) =
˝R

c(x, z, t) − cminb(x, z, t) d˝Km + c(x, z, t) − cmin

(25)

After deﬁning the critical value of the nutrient stress index
c (Fig. 5), above which value active nutrient uptake is fullycompensated for by active uptake in other more-available (lessstressed) soil regions, the local compensated active root nutrient uptake rate, aac [ML−3 T−1 ], is obtained by including thenutrient-stress index function in the denominator of Eq. (22),oraac (x, z, t) =Ap (t)c(x, z, t) − cminb(x, z, t)Lt
Km + c(x, z, t) − cminmax[(t), c ]

(26)

1
Lt


aac (x, z, t) d˝
˝RAp (t)
max[(t), c ]˝R

c(x, z, t) − cminb(x, z, t) d˝Km + c(x, z, t) − cmin(27)

Eq. (20) implies that reduction in root water uptake will
decrease passive nutrient uptake, thereby increasing active
nutrient uptake proportionally. In other words, total nutrient
uptake is not affected by soil water stress, as computed by
the proportion of actual to potential root water uptake. This
is not realistic since one would expect that plant nutrient
requirements will be reduced for water-stressed plants. For
that reason, the uptake model includes additional ﬂexibility,
by reducing the potential nutrient demand Rp [ML−2 T−1 ], in
proportion to the reduction of root water uptake, as deﬁned
by the actual to potential transpiration ratio, or

Tac (t)Ap (t) = max Rp (t)− Pa (t), 0
Tp (t)(28)

In summary, the presented root nutrient uptake model withcompensation requires as input the potential nutrient uptake
rate (demand), Rp , the spatial root distribution function b(x, z,t) as needed for the water uptake term, the Michaelis–Menten
constant Km , the maximum nutrient concentration that can betaken up passively by plant roots cmax , the minimum concentration cmin needed to initiate active nutrient uptake, and thecritical nutrient stress index c . The passive nutrient uptake
term can be turned off by selecting cmax equal to zero. Moreover, active nutrient uptake can be eliminated by specifying azero value for Rp in Eq. (20), or by selecting a very large cminvalue in Eq. (22). It is likely that values of these parameters are
nutrient- and plant-speciﬁc. Similarly as for root water uptake,it can be expected that c for agricultural crops is relatively
high when compared to natural plants that are likely to havemore ability to compensate for soil environmental stresses.
Other parameters, such as cmax will likely need to be calibrated to speciﬁc conditions before the model can be used forpredictive purposes.

3.

Numerical implementation

Both the Richards equation (1) and the convection–dispersion
equation (2) are solved in HYDRUS code (Šimůnek et al., 2006,2008) using the ﬁnite element method in the spatial domain
and the ﬁnite differences method in the temporal domain.
Implementation of the compensated root water and nutrient uptake routines did not require changing the numericalapproach to solve these governing equations. However, a twoor three-step approach was needed to calculate the compensated root water and nutrient uptake with passive and active
components, respectively, for each time step.

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In the root water uptake module, the uncompensated root
water uptake (9) and the dimensionless water stress index ω(11) are evaluated in the ﬁrst iteration step, which is the only
step needed when compensated root water uptake is not considered. The compensated root water uptake (13) is evaluatedin the second step, prior to solving the Richards equation (1).
A multiple-step approach was required to evaluate theroot nutrient uptake term. Only one step is needed, when
only passive or no root nutrient uptake is considered. If noncompensated active root nutrient uptake is considered, atwo-step approach is required. In this case, passive nutrient uptake (18) and potential active nutrient uptake rate Ap(20) are evaluated in the ﬁrst step, whereas uncompensatedactive root nutrient uptake (22) is evaluated during the second step. The additional third step is needed if compensatedactive root water uptake is considered. In that case, the nutrient stress index  (24) is ﬁrst evaluated during the second
step, so that the compensated active root nutrient uptake(26) can be evaluated in the third step, prior to solving theconvection–dispersion equation (2).

4.

ExamplesThe functioning of compensated root water uptake is demonstrated for three examples. While the ﬁrst example considers
a simple one-dimensional soil proﬁle, the second example
applies to an axi-symmetrical three-dimensional water ﬂow
and nutrient transport domain. In the third example wedemonstrate the consequences of the various presented nutrient uptake models, including both passive and active nutrient
uptake, with and without compensation. All examples apply
to the same loamy soil, for which the soil hydraulic properties were taken from Carsel and Parrish (1988). Also, the same
soil water stress response function of Feddes et al. (1978) in
Fig. 3 was used for all examples, with h1 = −10 cm, h2 = −25 cm,
h3,high = −200 cm, h3,low = −800 cm, and h4 = −8000 cm. The true
h3 parameter is interpolated from h3,high and h3,low and Tp (seeFig. 3), as done in the SWATRE code (e.g., Wesseling et al., 1991).

Fig. 6 – Example 1. Root water uptake (top) and soil waterpressure head (bottom) proﬁles for uncompensated (solid
thick lines) and compensated root water uptake with valuesof ωc = 0.75 (solid thin lines) and 0.5 (dashed thick lines).

4.1.
Root water uptake in a one-dimensional soil
proﬁle with groundwater table
The ﬁrst example applies to a one-dimensional 120-cm soil
proﬁle with a 90-cm root depth and the root distribution linearly decreasing with depth. The bottom of the soil is bounded
by a ground water table that is in hydrostatic equilibrium withthe soil proﬁle at the start of the simulation (initial condition). The hypothetical plant is assumed to transpire with apotential transpiration rate, Tp , of 0.4 cm/day, while the water
stress index ωc is varied with values of 1.0 (corresponding touncompensated root water uptake), 0.75 and 0.5, with the latter two values corresponding with increasing compensatedroot water uptake as the index decreases. Since this set ofmodel simulations does not include precipitation or irrigation,water required for transpiration will deplete the soil water proﬁle so that the resulting soil water potential (h) gradient will
partially supply the root zone with water, driven by capillaryrise from the shallow groundwater. Simulations were carriedout for a total period of 50 days.

Fig. 6 presents the depth distribution of root water uptake,s* (day−1 ) and soil water pressure head, h (cm), along thesoil proﬁle for the uncompensated (ωc = 1.0) and compensated(ωc = 0.75 or 0.5) root water uptake scenarios, whereas Fig. 7 isa plot of both actual (cm/d) and cumulative values (cm) of thepotential (Tp ) and actual (Ta ) transpirations, and water ﬂux at
the groundwater table for the three corresponding scenarios
during the 50-day simulation period. As Figs. 6 and 7 show, thenon-compensated root water uptake starts to decrease aroundthe 10th day (Fig. 6) as a result of water stress, with the top
20 cm of the soil proﬁle being fully depleted of water (Fig. 7)
after about 20 days of plant transpiration. For the 30–90 cm
soil depth, capillary rise provides for sufﬁcient water supply
towards the rooting zone, thus allowing the lower root domain
to remain unstressed with root water uptake at the normal
unstressed uptake rate. Root water uptake rates after about
10 days are always either equal or lower than the root water
uptake at the beginning of the simulations.

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515

During this time period, a reduced uptake from the upper partof the soil proﬁle is fully compensated for by an increased
uptake from deeper soil proﬁle. Root water uptake remains
signiﬁcantly higher for the compensated scenarios even when
it is reduced from its potential values at later times. Cumulative actual transpiration during the 50 day period increases
from about 13.7 to 15.0 and 16.7 cm for the uncompensatedand compensated (ωc = 0.75 and 0.5) root water uptake scenarios, respectively. Daily values of Ta at the end of the simulation
period increase from 0.17, to 0.19, and 0.235 cm/day for thesethree scenarios, resulting in an increase in cumulative planttranspiration by almost 40%. Neglecting root water uptake
compensation by plant roots in water-stressed soil conditions
would thus result into signiﬁcant errors in plant transpirationand the soil water balance. The presented interactions of root
water uptake and soil moisture and their spatial variations
conﬁrm experimental studies by Green and Clothier (1995) andAndreu et al. (1997), with the shifting of preferential root water
uptake to the wetter parts of the soil system after irrigation or
the development of water uptake fronts moving downwards as
the soil dries (Doussan et al. (2006). Irrespective of the selectedscenario, near steady-state conditions were attained at theend of the 50-day simulation period, with actual transpiration fully supplied by the capillary rise from the ground watertable. Similar results were obtained when considering more
realistic diurnal plant transpiration variations similar to natural conditions (0.24 Tp during night time (6 pm to 6 am) and2.75Tp sin(2t/tPeriod − /2) during the day).

Fig. 7 – Example 1. The potential transpiration (black), theactual transpiration (red), and the bottom ﬂux (blue) (top)and their cumulative values (bottom) for uncompensated(solid thick lines) and compensated (ωc = 0.75, solid thinlines; ωc = 0.5, dashed thick lines) root water uptake
scenarios. (For interpretation of the references to color inthis ﬁgure legend, the reader is referred to the web versionof the article.)

For the scenarios that considered compensated root water
uptake, a reduction in the uptake from the upper part ofthe proﬁle is initially fully compensated by uptake from thedeeper part of the soil proﬁle. Root water uptake rates gradually increase in deeper parts of the soil proﬁle as the upper
part of the root zone becomes more stressed. The compensation from larger soil depths is larger for lower values of ωc . As aresult, the root water uptake distribution in Fig. 6 moves to theright as time increases; and increasingly so for the more compensated uptake (ωc = 0.5) scenario. Since compensated root
water uptake is greater than the uncompensated root water
uptake at the bottom of the root zone, a larger fraction of thesoil proﬁle is water-depleted, thus capillary rise will need tosupply a smaller fraction of the root zone.
As a result of water stress, Fig. 7 shows that actual transpiration (Ta ) starts to deviate from Tp already after 10 days for theuncompensated scenario, whereas compensated root water
uptake continues to be at a potential rate until days 20 and
27 for scenarios with values of ωc = 0.75 and 0.5, respectively.

4.2.
Root water uptake for an axi-symmetrical
three-dimensional proﬁle with irrigation
In the second example we considered an axi-symmetrical ﬂow
domain for which both depth and radius values are equal to
100 cm. The initial pressure head (t = 0) was −400 cm throughout the soil proﬁle and the potential transpiration rate was
0.04 cm/h. While water needed for transpiration was suppliedby capillary rise from the shallow water table in the ﬁrst example, water was supplied by irrigation for the example two
scenarios. Irrigation water application was conﬁned to a 50cm radius around the center of the simulation domain, at aconstant rate of 0.16 cm/h. Since this irrigated area is 4 times
smaller than the ﬂow domain, the irrigation ﬂux of four times
the value of Tp was exactly equal to the transpiration demand.
Root density was assumed to decrease linearly from its maximum value to zero at the 50 cm soil depth, but to be constantin the radial direction. The water stress index ωc was varied
from 1.0 to 0.4, using increments of 0.2.
Fig. 8 presents both actual and cumulative values of thepotential and actual transpiration rates. While only a limited
reduction in transpiration (0.038 cm/h) was attained for thecompensated root water uptake scenario with ωc = 0.4, root
water uptake for the uncompensated scenario was reduced
by exactly one half (0.020 cm/h). For the other scenarioswith ωc values of 0.6 and 0.8, Ta was reduced to 0.29 and0.24 cm/h, respectively. With the uncompensated root water
uptake scenario being reduced immediately as the initialpressure head was already on the declining part of the water
stress response function (Fig. 3), plant water uptake for thecompensated root water uptake scenarios started to decrease

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Fig. 8 – Example 2. The potential (black line) and actual
transpirations for scenarios with the water stress index ωcequal to 1.0 (pink), 0.8 (blue), 0.6 (green), and 0.4 (red).
Actual values are shown at the top, while cumulativevalues at the bottom. (For interpretation of the references tocolor in this ﬁgure legend, the reader is referred to the web
version of the article.)from potential values after 44, 60, and 120 h for scenarioswith ωc equal to 0.8, 0.6 and 0.4, respectively.
Fig. 9 shows water content proﬁles at the end of thesimulation period for scenarios with uncompensated andcompensated root water uptake (ωc = 0.4). Note that the volumetric soil water content corresponding with an initial h-valueof −400 is equal to 0.156 cm3 cm−3 . The portion of the root zonethat is not irrigated (right side in Fig. 9) is near the wilting point
for both scenarios. Since the water supply rate by irrigation is
identical for both scenarios, whereas the root water uptake issigniﬁcantly larger for the compensated scenario, much morewater is present in the soil proﬁle for the uncompensatedroot water uptake scenario. As a result, the moisture front forthis scenario reached a depth of 75 cm, while it propagatedonly to the 60 cm soil depth for the compensated root water
uptake scenario. Water also moved further laterally in the scenario with uncompensated root water uptake. Although morewater was taken up from the soil proﬁle with compensatedroot water uptake, a larger fraction of the root zone was activein the scenario with uncompensated root water uptake sinceFig. 9 – Example 2. Water content proﬁles for scenarios with
uncompensated (top) and compensated (bottom) (ωc = 0.4)
root water uptake at the end of the simulation period.

more water was available in the larger part of the soil proﬁle.
This example with partial root zone wetting by irrigation
demonstrated that potential plant water demand can be satisﬁed when compensated root water uptake is considered.
Therefore, spatial root distribution is expected to have a signiﬁcant effect on Ta predictions if uncompensated root water
uptake is considered. Insufﬁcient knowledge of the spatial root
distribution may lead to large errors in actual transpiration
calculations. However, spatial root distribution effects may bemuch less important if compensated root water uptake is considered. The compensated root water uptake thus conﬁrms
with recent ﬁndings (e.g., Roose and Fowler, 2004) that root
water uptake is not solely proportional to root density.

4.3.
Coupled root water and nutrient uptake in a
one-dimensional soil proﬁle with groundwater table andno irrigation
The ﬁrst example with compensated root water uptake is further expanded in this third example to evaluate the associatedimpacts on root nutrient uptake, including passive and active

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considerations. To simplify this analysis, nutrient transport
and uptake was coupled with the example 1 scenario that considered uncompensated root water uptake only (thick solidlines in Figs. 6 and 7 with ωc = 1.0). We compared six root
nutrient uptake scenarios: (a) no nutrient uptake, (b) only passive nutrient uptake (i.e., no active uptake), (c) passive anduncompensated active nutrient uptake, (d) passive and compensated active nutrient uptake, (e) uncompensated active
nutrient uptake (i.e., no passive uptake), and (f) compensatedactive nutrient uptake (i.e., no passive uptake).
Calculations were carried out in relative concentrations.
The initial nutrient concentrations in the soil proﬁle and in
the ground water were both assumed to be equal to one. The
potential nutrient demand, Rp , for scenarios that involved
active root nutrient uptake was assumed to be equal to 1(−/cm2 day) as well, whereas its value was set to zero for theexample scenarios without an active root nutrient uptake. For
scenarios that excluded passive root nutrient uptake, the cmax
was set equal to 0, while its value was set to 10 (larger than any
simulated dissolved soil solution concentration) for scenariosthat include passive root nutrient uptake. The critical value ofthe nutrient stress index, c , was assumed equal to 0.5 for scenarios that allowed compensated active root nutrient uptake.
The Michaelis–Menten constant, Km , was set to 0.1, whereas
the minimum concentration, cmin , was assumed zero.
While Figs. 10 and 11 shows concentration proﬁles for thevarious root nutrient uptake scenarios, the actual and cumulative nutrient uptake rates as a function of simulation time
are presented in Fig. 12. It is speciﬁcally noted that scenarios
without any nutrient uptake lead to an increase in nutrient
concentrations in the root zone (Fig. 10, top), and that scenarios that allow active nutrient uptake will lead to a decreasein soil nutrient concentration (Figs. 11) as time progresses.
Dissolved soil solution concentrations gradually increase in
the root zone for the scenario that neglects nutrient uptake(Fig. 10, top), as caused by decreased volumetric water content
by root water uptake. Since water content gradually decreases
while the amount of dissolved ions in the root zone remains
constant, the resulting soil solution concentrations will gradually increase. Concentrations increase initially fastest close
to the soil surface, because the root water uptake is highest
in this region. Concentrations in this zone tend to not further
increase, after the soil water content is depleted, thus limiting
root water uptake by soil water stress. However, as root water
uptake continues from deeper soil regions, concentrations willcontinue to increase there. The increase in nutrient concentrations might eventually affect the soil’s osmotic potential andthus cause salinity stress, in addition, thereby further reducing
root water uptake.
As expected, there will be no change in soil nutrient concentrations when nutrient uptake is only passive (Fig. 10,
bottom), since there are no mechanisms in place that would
cause concentration gradients in the soil proﬁle to develop,
thus eliminating diffusion/dispersion to occur. Since the cmax
value was set to be above soil solution concentrations, water
removed from the root zone by root uptake takes along all
dissolved ions, while water left behind in the root zone willremain at constant concentrations. Yet, the total soil nutrient
mass will decrease, because of the passive nutrient uptake andreduction in water contents.

517

Fig. 10 – Example 3. Depth distribution of nutrient
concentration for nutrient uptake scenarios considering (a)no nutrient uptake (top) and (b) passive nutrient uptake
only (bottom).

Soil solution nutrient concentrations will gradually
decrease throughout the soil proﬁle for those scenarios withactive nutrient uptake (Fig. 11), leading to early nutrient
depletion in the near soil surface regions of the proﬁle. This
depletion is faster for scenarios with compensated root nutrient uptake. We note that there is only a small difference inthe depth distribution of nutrient concentration for scenariosthat consider (both uncompensated and compensated) active
nutrient uptake, irrespective of whether passive nutrient
uptake is allowed.
It is important to realize that the absence of nutrient uptake
will result in an increase in soil solution concentration as aconsequence of root water uptake. No concentration changes
were computed for those scenarios where nutrient uptake issolely by passive nutrient uptake, whereas all scenarios thatinvolved active nutrient uptake led to a signiﬁcant decrease insoil solution nutrient concentration.
Fig. 12 presents the actual and cumulative total plant
nutrient uptakes for scenarios where nutrient uptake wasAuthor's personal copy518

e c o l o g i c a l m o d e l l i n g 2 2 0 ( 2 0 0 9 ) 505–521

Fig. 11 – Example 3. Depth distribution of nutrient
concentration for nutrient uptake scenarios considering
either uncompensated (top) or compensated (bottom) active
nutrient uptake with (solid line) or without (dashed line)
passive nutrient uptake.

considered. For scenarios where active and passive nutrient
uptakes were simultaneously considered, Fig. 12 additionally also shows both active and passive components of thenutrient uptake. When only the passive nutrient uptake isconsidered, total nutrient uptake is directly proportional toroot water uptake as the dissolved soil solution concentrations
remain constant. When both active and passive root nutrient uptakes are considered, the total uptake is signiﬁcantly
larger than when only the passive component is allowed.
The total root nutrient uptake is initially equal to the potential nutrient demand, if nutrient uptake from regions with
sufﬁcient nutrients present can compensate for uptake fromnutrient-depleted soil depths. The passive nutrient uptake issigniﬁcantly smaller when also active uptake is considered,since active uptake leads to lowering of dissolved concentrations that are involved in the passive uptake. The total nutrient
uptake is relatively similar for scenarios that consider the

Fig. 12 – Example 3. The actual (top) and cumulative(bottom) nutrient uptakes for scenarios considering (a) only
passive nutrient uptake (black line), (b) passive anduncompensated active nutrient uptake (red lines), (c)passive and compensated active nutrient uptake (bluelines), (d) uncompensated active nutrient uptake (greenline), and (e) compensated active nutrient uptake (brownline). Thin and dashed lines represent active and passive
nutrient uptake components, respectively. (Forinterpretation of the references to color in this ﬁgurelegend, the reader is referred to the web version of thearticle.)active uptake, irrespectively if the passive uptake is active ornot. When the passive nutrient uptake is neglected, the active
uptake correspondingly increases, resulting to similar results.
As groundwater is the only source of water and nutrients, theroot nutrient uptake eventually decreases with time, similarly
as the root water uptake. Additional irrigation and fertigation
would be needed to sustain water and nutrient uptake at its
potential values.
The presented modeling results conform to many experimental nutrient uptake data that show compensation andenhanced nitrate uptake responses by differential nitrate supply (Robinson, 1994, 1996). Moreover, enhanced local nutrient
uptake to meet total plant nutrient demand supports thenotion that local root uptake rate is not only dependent on thelocal environment, but also on the environmental conditionsof the whole root system (Dunbabin et al., 2002).

Author's personal copye c o l o g i c a l m o d e l l i n g 2 2 0 ( 2 0 0 9 ) 505–521

5.

Summary and conclusions

A compensated root water and nutrient uptake model is presented that is highly ﬂexible. It can accommodate various root
spatial distribution functions, including their temporal variations due to root growth. The water uptake model can be used
for any stress response function accounting for the reductionof the potential root water uptake due to the soil water pressure and osmotic potential stresses.
The root nutrient uptake model considers either activeor passive nutrient uptake or both, is similarly ﬂexible. Few
model parameters will allow for a large number of modeling
options. The model allows for the absence of nutrient uptake,only passive or active nutrient uptake (each uncompensatedor compensated), and any combinations of uncompensated orcompensated passive nutrient uptake with either uncompensated or compensated active nutrient uptake (Fig. 1). Whilethe cmax parameter activates the passive root nutrient uptake,the potential nutrient demand Rp initiates the active nutrient
uptake. Finally, critical values of the water stress index ωc andthe nutrient stress index c activate compensation for root
water uptake and nutrient uptake, respectively.
Presented examples demonstrated several important characteristics of the compensated root water and nutrient uptake
model. Flexibility in modeling compensation of water andnutrient uptake, as presented here, generally reduces plantstress and allows the plant to meet potential transpirationand nutrient demands, irrespective of local soil water andnutrient stresses. Neglecting the possibility of compensation
may cause signiﬁcant errors in the water and nutrient balances, thus affecting biomass production and predicted crop
yields.
Root water uptake is closely related to the spatial root
distribution function when uncompensated water uptake
is considered. However, a much weaker correspondence
between the spatial root distribution and root water uptake
exists for compensated water uptake. Therefore, spatial root
distribution has a signiﬁcant effect on the prediction of actual
plant transpiration when uncompensated root water uptake
is considered. The effect of the spatial root distribution ismuch less signiﬁcant when compensated root water uptake
is allowed. We conclude that a priori knowledge of the spatial root distribution in the soil proﬁle is important when root
water uptake is considered to be uncompensated.
Widely different dynamics of nutrient concentrations inthe soil proﬁle were simulated when a wide range of root nutrient uptake models were applied. While the scenario without
nutrient uptake led to the increase in solution concentrations,the scenario with only passive nutrient uptake maintained
constant soil solution concentrations, whereas all scenariosthat involved active nutrient uptake led to nutrient concentration reductions.
Only small differences in the depth distribution of nutrient
concentrations were determined for scenarios that considered active nutrient uptake, irrespective of whether passive
nutrient uptake was involved. This result indicates that soil
nutrient concentration measurements are unlikely to provide
evidence about the relative importance of active and passive
nutrient uptakes.

519

Acknowledgements
This paper is based on work supported in part by the BARD
(Binational Agricultural Research and Development Fund)
Project IS-3823-06 and by the National Science Foundation
Biocomplexity programs #04-10055 and NSF DEB 04-21530.

references

Adiku, S.G.K., Rose, C.W., Braddock, R.D., Ozier-Lafontaine, H.,
2000. On the simulation of root water extraction: examinationof a minimum energy hypothesis. Soil Sci. 165 (3), 226–236.
Andreu, L., Hopmans, J.W., Schwankl, L.J., 1997. Spatial andtemporal distribution of soil water balance for a drip-irrigatedalmond tree. Agric. Water Manage. 35, 123–146.
Barber, S.A., 1995. Soil Nutrient Bioavailability: A Mechanistic
Approach. John Wiley & Sons, New York.
Bar-Yosef, B., 1999. Advances in fertigation. Adv. Agron. 65, 1–77.
Biondini, M., 2001. A three-dimensional spatial model for plantcompetition in an heterogeneous soil environment. Ecol.
Model. 142, 189–225.
Bouten, W., 1995. Soil water dynamics of the Solling spruce stand,calculated with the FORHYD simulation package. Ecol. Model.
83, 67–75.
Brown, T.N., Kulasiri, D., Gaunt, R.E., 1997. A root-morphology
based simulation for plant/soil microbial ecosystems
modeling. Ecol. Model. 99, 275–287.
Cardon, G.E., Letey, J., 1992. Plant water uptake terms evaluatedfor soil water and solute movement models. Soil Sci. Soc. Am.
J. 32, 1876–1880.
Carsel, R.F., Parrish, R.S., 1988. Developing joint probability
distributions of soil water retention characteristics. Water
Resour. Res. 24, 755–769.
Clausnitzer, V., Hopmans, J.W., 1994. Simultaneous modeling oftransient three-dimensional root growth and soil water ﬂow.
Plant Soil 164, 299–314.
Darrah, P.R., Jones, D.L., Kirk, G.J.D., Roose, T., 2006. Modelling therhizosphere: a review of methods for ‘upscaling’ to thewhole-plant scale. Eur. J. Soil Sci. 57, 13–25, doi:
10.1111/j.1365-2389.2006.00786.x.
De Barros, J., Williams, R., Gaiser, T., 2004. Modeling soil nutrient
limitations to crop production in semiarid NE of Brazil with a
modiﬁed EPIC version. I. Changes in the source code of themodel. Ecol. Model. 178, 441–456.
Doussan, C., Pierret, A., Garrigues, E., Pagès, L., 2006. Water
uptake by plant roots. II. Modelling of water transfer in thesoil root-system with explicit account of ﬂow within the rootsystem—comparison with experiments. Plant Soil 283,
99–117.
Dunbabin, V.M., Diggle, A.J., Rengel, Z., Van Hugten, R., 2002.
Modeling the interactions between water and nutrient uptake
and root growth. Plant Soil 239, 19–38.
English, M., Raja, S.N., 1996. Perspectives on deﬁcit irrigation.
Agric. Water Manage. 32, 1–14.
Fayer, M.J., 2000. UNSAT-H Version 3.0: Unsaturated Soil Waterand Heat Flow Model. Theory, User Manual, and Examples.
Paciﬁc Northwest National Laboratory 13249.
Feddes, R.A., Bresler, E., Neuman, S.P., 1974. Field test of a
modiﬁed numerical model for water uptake by root systems.
Water Resour. Res. 10 (6), 1199–1206.
Feddes, R.A., Kowalik, P.J., Zaradny, H., 1978. Simulation of Field
Water Use and Crop Yield. John Wiley & Sons, New York, NY.
Feddes, R.A., Raats, P.A.C., 2004. Parameterizing thesoil–water–plant root system. In: Feddes, R.A., de Rooij, G.H.,
van Dam, J.C. (Eds.), Proceedings of the Unsaturated Zone
Modelling: Progress, Challenges and Applications.

Author's personal copy520

e c o l o g i c a l m o d e l l i n g 2 2 0 ( 2 0 0 9 ) 505–521

Wageningen UR Frontis Series, vol. 6. Kluwer Academic
Publishers, Dordrecht, The Netherlands, pp. 95–141, x–xi
(Chapter 4).
Flerchinger, G.N., Hanson, C.L., Wight, J.R., 1996. Modelingevapotranspiration and surface energy budgets across awatershed. Water Resour. Res. 32, 2539–2548.
Food And Agriculture Organization of the United Nations, 1990.
Expert Consultation on Revision of FAO Methodologies for
Crop Water Requirements, ANNEX V. FAO Penman–Monteith
Formula, Rome, Italy.
Gardner, W.R., 1960. Dynamic aspects of water availability toplants. Soil Sci. 89 (2), 63–73.
Grant, R.F., 1993. Simulation model of soil compaction and root
growth: I. Model structure. Plant Soil 150, 1–14.
Grant, R.F., 1998. Simulation in ecosys of root growth responses tocontrasting soil water and nitrogen. Ecol. Model. 107, 237–264.
Green, S.R., Clothier, B.E., 1995. Root water uptake by kiwifruitvines following partial wetting of the root zone. Plant Soil 173,
317–328.
Hasegawa, S., Yoshida, S., 1982. Water uptake by dryland rice rootsystem during soil drying cycle. Soil Sci. Plant Nutr. 38,
191–208.
Haynes, R.J., 1986. Uptake and assimilation of mineral nitrogenby plants. In: Haynes, R.J. (Ed.), Mineral Nitrogen in the
Plant–Soil System. Academic Press, Inc., pp. 303–360.
Hopmans, J.W., 2006. Soil properties, processes and associatedroot–soil interactions. In: Porporato, A., D’Odorico, P. (Eds.),
Ecohydrology of Arid and Semiarid Ecosystems: An
Introduction. Springer, pp. 13–29.
Hopmans, J.W., Bristow, K.L., 2002. Current capabilities and futureneeds of root water and nutrient uptake modeling. Adv.
Agron. 77, 104–175.
Kastner-Maresch, A.E., Mooney, H.A., 1994. Modelling optimalplant biomass partitioning. Ecol. Model. 75/76, 309–320.
Jarvis, N.J., 1989. A simple empirical model of root water uptake. J.
Hydrol. 107, 57–72.
Jarvis, N.J., 1994. The MACRO model (Version 3.1), technicaldescription and sample simulations. Reports andDissertations 19. Dept. Soil Sci., Swedish Univ. Agric. Sci.,
Uppsala, Sweden, 51 pp.
Javaux, M., Schröder, T., Vanderborght, J., Vereecken, H., 2008. Useof a three-dimensional detailed modeling approach forpredicting root water uptake. Vadose Zone J. 7, 1079–1088.
Jensen, D.T., Hargreaves, G.H., Temesgen, B., Allen, R.G., 1997.
Computation of Eto under nonideal conditions. J. Irrig. Drain.
123 (5), 394–400.
Jones, J.W., Ritchie, J.T., 1990. Crop growth models. In: Hoffman,
G.J., et al. (Eds.), Management of Farm Irrigation Systems,
ASAE Monograph. AEAE, St. Joseph, MI, pp. 63–85.
Jungk, A.O., 1991. Dynamics of nutrient movement at thesoil–root interface. In: Waisel, Y., Eshel, A., Kafkaﬁ, U. (Eds.),
Plant Roots, The Hidden Half. Marcel Dekker, Inc., New York,
pp. 455–481, Chapter 31.
Jungk, A.O., 2002. Dynamics of nutrient movement at thesoil–root interface. In: Waisel, Y., Eshel, A., Kafkaﬁ, U. (Eds.),
Plant Roots, The Hidden Half. Marcel Dekker, Inc., New York,
pp. 587–616, Chapter 35.
Leib, B.G., Caspari, H.W., Redulla, C.A., Andrews, P.K., Jabro, J.J.,
2006. Partial rootzone drying and deﬁcit irrigation of ‘Fuji’
apples in a semi-arid climate. Irrig. Sci. 24, 85–99.
Li, K.Y., de Jong, R., Boisvert, J.B., 2001. An exponential
root-water-uptake model with water stress compensation. J.
Hydrol. 252, 189–204.
Luxmoore, R.J., Begovich, C.L., Dixon, K.R., 1978. Modelling solute
uptake and incorporation into vegetation and litter. Ecol.
Model. 5, 137–171.
Maas, E.V., 1990. Crop salt tolerance. In: Tanji, K.K. (Ed.),
Agricultural Salinity Assessment and Management. ASCE

Manuals and Reports on Engineering Practice, No. 71, NewYork.
Mmolawa, K., Or, D., 2000. Root zone solute dynamics under drip
irrigation: a review. Plant Soil 222, 163–190.
Molz, F.J., 1981. Models of water transport in the soil–plantsystem: a review. Water Resour. Res. 17 (5), 1245–1260.
Neumann, G., Römheld, V., 2002. Root-induced changes in theavailability of nutrients in the rhizosphere. In: Waisel, Y.,
Eshel, A., Kafkaﬁ, U. (Eds.), Plant Roots, The Hidden Half.
Marcel Dekker, Inc., New York, pp. 617–649 (Chapter 36).
Nissen, P., 1991. Uptake mechanisms. In: Waisel, Y., Eshel, A.,
Kafkaﬁ, U. (Eds.), Plant Roots, The Hidden Half. Marcel Dekker,
Inc., New York, pp. 483–502 (Chapter 31).
Pang, X.P., Letey, J., 1998. Development and evaluation of
ENVIRO-GRO, an integrated water, salinity, and nitrogenmodel. Soil Sci. Soc. Am. J. 62 (5), 1418–1427.
Parton, W.J., Schimel, D.S., Cole, C.V., Ojima, D.S., 1987. Analysis offactors controlling soil organic matter levels in great plans
grasslands. Soil Sci. Soc. Am. J. 51, 1173–1179.
Pierre, W.H., Banwart, W.L., 1973. Excess-base andexcess-base/nitrogen ratio of various crop species and partsof plants. Agron J. 65, 91–96.
Porporato, A., D’Odorico, P., Laio, F., Rodriguez-Iturbe, I., 2003.
Hydrologic controls of soil carbon and nitrogen cycles. I.
Modeling scheme. Adv. Water Resour. 26, 45–58.
Raats, P.A.C., 1974. Steady ﬂows of water and salt in uniform soil
proﬁles with plant roots. Soil Sci. Soc. Am. Proc. 38,
717–722.
Raynaud, X., Leadlley, P.W., 2005. Symmetry of belowgroundcompetition in a spatially explicit model of nutrient
competition. Ecol. Model. 189, 447–453.
Richards, L.A., 1931. Capillary conduction of ﬂuid through porous
mediums. Physics 1, 318–333.
Robinson, D., 1996. Variation, co-ordination and compensation inroot systems in relation to soil variability. Plant Soil 187, 57–66.
Robinson, D., 1994. Tansley Review No. 73. The responses ofplants to non-uniform supplies of nutrients. New Phytol. 127,
635–674.
Roose, T., Fowler, A.C., 2004. A model for water uptake by plantroots. J. Theor. Biol. 228, 155–171.
Shaner, D.L., Boyer, J.S., 1976. Nitrate reductase activity in maize(Zea mays L.) leaves. I. Regulation by nitrate ﬂux. Plant Physiol.
58, 499–504.
Silberbush, M., 2002. Simulation of ion uptake from the soil. In:
Waisel, Y., Eshel, A., Kafkaﬁ, U. (Eds.), Plant Roots, The Hidden
Half. Marcel Dekker, Inc., New York, pp. 651–661 (Chapter 37).
Šimůnek, J., Šejna, M., Saito, H., Sakai, M., van Genuchten, M.Th.,
2008. The HYDRUS-1D Software Package for Simulating theMovement of Water, Heat, and Multiple Solutes in Variably
Saturated Media, Version 4.0, HYDRUS Software Series 3,
Department of Environmental Sciences, University ofCalifornia Riverside, Riverside, California, USA, p. 315.
Šimůnek, J., van Genuchten, M.Th., Šejna, M., 2006. The HYDRUS
Software Package for Simulating Two- and Three-dimensional
Movement of Water, Heat, and Multiple Solutes in
Variably-saturated Media, Technical Manual, Version 1.0, PC
Progress, Prague, Czech Republic, p. 241.
Skaggs, T.H., Van Genuchten, M.Th., Shouse, P.J., Poss, J.A., 2006.
Macroscopic approaches to root water uptake as a function ofwater and salinity stress. Agric. Water Manag. 86, 140–149.
Skaggs, T.H., Shouse, P.J., 2008. Roots and root function:
introduction. Vadose Zone J. 7, 1008–1009.
Somma, F., Clausnitzer, V., Hopmans, J.W., 1998. Modeling oftransient three-dimensional soil water and solute transportwith root growth and water and nutrient uptake. Plant Soil
202, 281–293.
Stikic, R., Popovic, S., Srdic, M., Savic, D., Jovanovic, Z., Prokic, Lj.,
Zdravkovic, J., 2003. Partial root drying (PRD): a new techniqueAuthor's personal copye c o l o g i c a l m o d e l l i n g 2 2 0 ( 2 0 0 9 ) 505–521

for growing plants that saves water and improves the qualityof fruit. Bulg. J. Plant Physiol., 164–171 (Special Issue).
Taylor, S.A., Ashcroft, G.M., 1972. Physical Edaphology. Freeman
and Co., San Francisco, CA, pp. 434–435.
Taylor, H.M., Klepper, B., 1978. The role of rooting characteristics
in the supply of water to plants. Adv. Agron. 30, 99–128.
van den Berg, M., Driessen, P.M., Rabbinge, R., 2002. Water uptake
in crop growth models for land use systems analysis. II.
Comparison of three simple approaches. Ecol. Model. 148,
233–250.
van Dam, J.C., Huygen, J., Wesseling, J.G., Feddes, R.A., Kabat, P.,
van Valsum, P.E.V., Groenendijk, P., van Diepen, C.A., 1997.
Theory of SWAP, version 2.0. Simulation of water ﬂow, solutetransport and plant growth in thesoil–water–atmosphere–plant environment. Dept. Water
Resources, WAU, Report 71, DLO Winand Staring Centre,
Wageningen, Technical Document 45.
van Genuchten, M.Th., 1987. A numerical model for water andsolute movement in and below the root zone. Research Report
No 121, U.S. Salinity Laboratory, USDA, ARS, Riverside,
CA.

521

Van Wijl, M.T., Bouten, W., 2001. Towards understanding threeroot proﬁles simulating hydrologically optimal strategies forroot distribution. Hydrol. Earth Syst. Sci. 5 (4), 629–644.
Vogel, T., 1987. SWMII—Numerical model of two-dimensional
ﬂow in a variably saturated porous medium. Research Rep.
No. 87. Dept. of Hydraulics and Catchment Hydrology,
Agricultural Univ, Wageningen, The Netherlands.
Vrugt, J.A., Hopmans, J.W., Šimůnek, J., 2001a. Calibration of a
two-dimensional root water uptake model. Soil Sci. Soc. Am. J.
65 (4), 1027–1037.
Vrugt, J.A., van Wijk, M.T., Hopmans, J.W., Šimůnek, J., 2001b.
One-, two-, and three-dimensional root water uptake
functions for transient modeling. Water Resour. Res. 37 (10),2457–2470.
Wegehenkel, M., Mirschel, W., 2006. Crop growth, soil water andnitrogen balance simulation on three experimental ﬁeld plots
using the OPUS model—a case study. Ecol. Model. 190,
116–132.
Wesseling, J.G., Elbers, J.A., Kabat, P., van den Broek, B.J., 1991.
SWATRE: Instructions for Input, Internal Note. WinandStaring Centre, Wageningen, The Netherlands.