Detailed physical models describing root water uptake (RWU) are an important tool for the
prediction of RWU and crop transpiration, but the hydraulic parameters
involved are hardly ever available,
making them less attractive for many studies. Empirical models are more readily used because of their
simplicity and the associated lower data requirements. The purpose of this study is to evaluate the
capability of some empirical models to mimic the RWU distribution under varying environmental conditions
predicted from numerical simulations with a detailed physical model. A review of some empirical models
used as sub-models in ecohydrological models is presented, and alternative empirical RWU models are proposed.
All these empirical models are analogous to the standard Feddes model, but differ in how RWU is partitioned
over depth or how the transpiration reduction function is defined. The parameters of the empirical models are
determined by inverse modelling of simulated depth-dependent RWU. The performance of the empirical models and
their optimized empirical parameters depends on the scenario. The standard
empirical Feddes model only performs
well in scenarios with low root length density

The rate at which a crop transpires depends on atmospheric conditions, the
shape and properties of the boundary between crop and atmosphere, root system
geometry, and crop and soil hydraulic properties. The study and modelling of
the involved interactions is motivated by the importance of transpiration for
global climate and crop growth

In many one- and two-dimensional problems, macroscopic RWU is modelled as a
sink term in the Richards equation, whose dependency on water content or
pressure head is usually represented by simple empirical functions (e.g.

Recently,

Other physical RWU models include

Empirical RWU models are more readily used because of their relative
simplicity and lower data requirements. On the other hand, their empirical
parameters do not have a clear physical meaning and cannot be independently
measured. Their limitations under varying environmental conditions are not
well established. For the case of the

The general purpose of this study was to evaluate the ability of some
empirical models to mimic the dynamics of RWU distribution under varying
environmental conditions performed in numerical experiments with a detailed
physical model proposed by

RWU and crop transpiration are linked through the principle of mass
conservation for water flow in the soil–plant–atmosphere pathway:

In a macroscopic modelling approach, RWU is calculated as a sink term

Different expressions for

By upscaling earlier findings about water flow towards a single root in the
microscopic scale, disregarding plant hydraulic resistance

In the

Root water uptake (RWU) as a function of soil pressure head

Figure

Figure

In order to account for RWU pattern changes due to heterogeneous soil water
distribution (the so-called “compensation”), several empirical models have
been developed over the years. These models follow the general framework of
the

In principle, any definition of

The physical basis of

Substitution of Eqs. (

Equation (

As an alternative to the

Decades before

These models may better represent RWU and compensation than the

In describing soil water availability, the matric flux potential

The RWU can then be obtained by inserting Eq. (

Table

All these models were embedded as sub-models in the SWAP ecohydrological
model

The values of the

Summary of empirical models used in this study.

Values of the parameters of the

Mualem–van Genuchten parameters for three soils of the Dutch
Staring series

Boundary conditions for drying-out simulations were no rain/irrigation and a
constant atmospheric demand (potential transpiration) over time. The
simulation continued until simulated crop transpiration by the physical RWU
model approached zero. Soil evaporation was set to zero, making soil water
depleted only due to RWU or bottom drainage. The free drainage (unit
hydraulic gradient) at the maximum rooting depth was the bottom boundary
condition. The soil was initially at hydrostatic equilibrium with a water
table located at 1 m depth. We performed simulations for two levels of
atmospheric demand given by potential transpiration (

Soil data for three top soils from the Dutch Staring series

Three levels of root length density were used, according to the range of
values normally found in the literature. We considered low, medium and high
root length density for average crop values equal to 0.01, 0.1 and 1.0
cm cm

Root length density distribution over depth calculated by
Eq. (

Figure

Parameters of the root water uptake models estimated by
optimization and their respective constraints

The parameters of the empirical RWU models were estimated by solving the
following constrained optimization problem:

Equation (

The optimizations were performed for the drying-out simulation only. This
guaranteed that RWU predictions from SWAP corresponded to the best fit of
each empirical model to the

In the growing season simulation, all models were evaluated by simulating the
transpiration of grass with weather data from the De Bilt weather station,
the Netherlands (52

The values of the empirical parameters of each RWU model corresponding to the
type of soil and root length density were taken from the optimizations
performed in the drying-out experiment. Each parameter was estimated for two
levels of

As in the drying-out simulations, the bottom boundary condition was free drainage. Initial pressure heads were obtained by iteratively running SWAP starting with the final pressure heads of the previous simulation until convergence.

Time–depth root water uptake (RWU, d

In this section we first focus on the behaviour of the

The leaf pressure head

Optimal parameters of each empirical model for all scenarios in the drying-out experiment.

Maximum possible transpiration

For the high

In the field, transpiration rate and root length density are related to each
other: a high transpiration rate only occurs in a high leaf area, and a high
leaf area implies a high root length density. Thus, even under very dry and
hot weather conditions, a crop with a low

In this section, we evaluate the empirical RWU models (models and their
abbreviations are listed in Table

Time–depth root water uptake (RWU) pattern and relative
transpiration (

The RWU patterns simulated by VLM and the empirical models for the sandy soil
and high

Time–depth root water uptake (RWU) pattern and relative
transpiration (

When reducing RWU for a period depending on

The RWU patterns predicted by the JMf and JMm models can be very different,
as shown by Fig.

For high

The proposed models (PM and PMm) are capable of predicting RWU patterns
similar to VLM. For the low

JMII does not mimic well the RWU pattern predicted by VLM for the high

Comparing RWU predictions from JMf and JMII, the Jarvis-type
models are affected by the definition of

Box plot of the coefficient of determination

The fact that JMII is more sensitive to both

The performance of the empirical models was analysed by the coefficient of
determination

Statistical indices for the evaluated scenarios of each model are concisely
shown by the boxplots in Fig.

Among the models that account for RWU compensation, JMf and JMII performed
worst, especially in the high

Best models for the evaluated scenarios (root length density

In predicting transpiration, all models accounting for compensation performed well, except JMf. It can be noticed that JMII performed much better in predicting transpiration than RWU. As for the RWU, all models performed worse in high R scenarios than in low R scenarios.

As the evaluated models differ regarding the number of empirical parameters
(from 0 to 2), it is important to use a statistical measure that accounts for
this and penalizes the models with more parameters. The Akaike's information
criteria (AIC) is a suitable measure for such a model comparison. The
selection of the “best” model is determined by an AIC score, defined as

Time–depth root water uptake (RWU) pattern and relative
transpiration (

The optimal values of the empirical parameters of all models (except JMII
that has no empirical parameters) for all scenarios but the high

The optimal

In order to interpret the parameters in Table

Values for

The optimal parameters of the proposed models follow their logical relation
to

High correlation parameters might result in uncertainties and a non-unique
solution of the optimization problem. In general, the correlation parameter
coefficients were low, except for some scenarios in which high correlation
coefficients between

The empirical models fitted only to RWU, since the primary interest is to evaluate the model's capability to predict the RWU patterns under different scenarios. RWU is not easily obtained in real conditions, making the use of physical RWU models a great advantage. On the other hand, plant transpiration, one of the main outputs in RWU models, is more easily measured. Thus, one might consider to fit the models to the temporal course of (relative) plant transpiration or to fit the models simultaneously to both plant transpiration and RWU, for which a rather complicated optimization scheme would be required.

We addressed this issue by fitting the models to the course of relative
transpiration for some scenarios. The procedure was the same as explained in
Sect.

By evaluating the RWU models under real weather conditions during a
relatively dry year and considering the same soil types and crop
characteristics as for the drying-out experiment, it was possible to use the
calibrated parameters for specific soil type and root length density. This
evaluation is important to analyse whether our calibration of the empirical
models with a single drying-out experiment results in consistent predictions
for other circumstances. Models were not evaluated for the low

Time course of actual cumulative plant transpiration

Figure

Comparing cumulative

An overall analysis of model performance is shown in Fig.

According to the AIC, PM, PMm and JMm are best in predicting RWU. Regarding

In general, the proposed models as well as JMm showed better performance than
the other empirical models. It should be noted, however, that these models
are based on

Box plot of the coefficient of determination

Several simple RWU models have been developed over the years, and we outlined
some of these models and also proposed alternatives. Some of these models
were embedded as sub-models in the SWAP eco-hydrological model

Best models for the evaluated scenarios (root length density

The widely used

The JMf model provides good predictions only for low and medium

The proposed models can predict RWU patterns similar to those
obtained by the

Regarding the ability of the models in predicting plant transpiration, all models accounting for compensation have good performance. The AIC indicates that JMII is the “best model”. This model is also more suitable for blind predictions, as no empirical parameters need to be estimated.

The simulations of a growing season with grass confirmed these findings, suggesting that an experiment of soil drying-out for two levels of potential transpiration, as performed, is adequate for analysing the performance of RWU models and retrieving their empirical parameters by defining the objective function in terms of RWU.

It should be noticed that the predictions from the

Model, input data and optionally modeling results are available from the corresponding author upon request.

The first author thanks CAPES (the CAPES Foundation, Ministry of Education of Brazil) and CNPq (National Council of Technological and Scientific Development, Brazil) for the PhD scholarship. The authors are grateful to N. Jarvis, who acted as a reviewer and who, together with two anonymous reviewers, gave valuable comments and suggestions during the review process. Edited by: N. Romano Reviewed by: three anonymous referees