Introduction

Hillel (2003) made an effective simile to illustrate the cohesion-tension theory of water flow through plants: “plants can be compared to a wick in an old fashioned Kerosene lamp” moving water from the soil to the atmosphere. The same author also recognized the crudeness of this simplification, since plants can actively regulate transpiration (Ep) through the stomata (Hillel 2003). Globally, the magnitude of water flow through plants is paramount: more than half of the global precipitation falling onto the earth is returned to the atmosphere through plant transpiration (Jackson et al. 2000). As critical actors in the water cycle, a better understanding of the processes involved in root water uptake and Ep are crucial to efficiently understanding and managing global water resources (Green et al. 2006). This challenge can be achieved with the aid of models which serve to study not only the complicated interactions between plants and their environment, but to highlight the knowledge gaps that need to be filled.

When modelling Ep, one should be aware of the scales at which the different elements composing the plant system work. At the leaf level, variations in Ep through changes in the stomatal conductance (g s ) are mediated by shifts in the microclimate conditions like temperature, radiation or vapour pressure deficit (VPD); but the loss of water from the leaves must be matched by the uptake from the roots, which in turn will depend on their spatial distribution and on the soil water content (Clothier and Green 1997; Williams et al. 2001). Hence, to correctly simulate Ep it is important to integrate the process that modulate g s at the leaf level with those related to the fluxes of water through the SPAC (Tuzet et al. 2003).

SPAC models in the literature which have set up equations describing the demand of water from the canopy and its supply provided by the roots have demonstrated that all the elements contributes, in some degree, to the regulation of Ep; particularly when some level of water stress is present (Williams et al. 1996; Sperry et al. 1998; Tuzet et al. 2003; Deckmyn et al. 2008). Nevertheless, while in those models the canopy is generally treated with a high level of detail, the soil-root system is simulated with a single 1 dimensional compartment divided into several layers and applying a constant value for the root specific radial resistance.

A simple one-dimensional approach may be suitable when the soil is uniformly wetted and the roots are evenly distributed. However, in many field conditions and particularly for crops watered with localised irrigation systems, very large variations in soil water content (θ soil ) throughout the root zone are present, generating a significant variability in root density as roots will tend to grow faster in portions of the soil whose conditions are favourable to their function (Klepper 1991). In this case, the use of one-dimensional models to compute root water extraction would fail to account for preferential water uptake from the wetted areas or the differences in root activity (Clothier and Green 1997). To illustrate this, one can think in a tree crop, drip irrigated, in a Mediterranean climate characterised by hot dry summers. Generally, at the end of the summer the area of soil influenced by the wet bulb will be near field capacity while the rest of the soil will be close to permanent wilting point, meaning that the same root system will be facing two soil contrasting situations. One in which water is easily available and root density is high and another in which the soil water potential is low.

To overcome this problem one could use models that work in more than one dimension. Doussan et al. (1998) for example, developed a detailed three dimensional maize root water uptake model combining an explicit representation of the root system with the catenary hypothesis, allowing them to study the so called “hydraulic architecture of the root system”. However, the computational requirements and its complex parametrization reduce its applicability (Couvreur et al. 2012).

A solution between the simple one dimensional approach to compute water extraction and the more complicated explicit models is proposed. The soil is not only divided into different layers, but also into compartments to capture local variations in θ soil and root density, using an approach which is simple enough to be easily implemented with measurable parameters, but with a sufficient degree of complexity to correctly compute the different water uptake from each single compartment.

The present paper describes a SPAC model for trees which couples a sun/shade approach for the canopy and applies a multi-compartment solution to the soil-root system. Plant transpiration, water potential at leaf and root collar, photosynthesis and stomatal conductance are calculated when heterogeneous conditions of soil water content or root density are present. Outputs of the model have been compared with the results of a split-root experiment using olive ‘Picual’ (Olea europea L.) and almond ‘Marinada’ (Prunus dulcis L.) grafted onto GF677 rootstock (Prunus dulcis x Prunus persica), where the fluxes of water extracted from each split part of the root system were continuously recorded using sap flow and time domain reflectometry (TDR) probes. The model performance and the main conclusion from the experiments are discussed.

Model description

Since van den Honert (1948), transport of water through the plant has been depicted as an electric circuit analogy composed of a set of resistances in parallel or in series and the differences in water potential as the driving force. For steady-state conditions, Ep, can be calculated as the difference in soil water potential (Ψ s ) minus water potential at the xylem collar (Ψ c ) and divided by the resistance of the soil (R s ) plus the resistance of the root (R r ); or Ψ c minus leaf water potential (Ψ l ) divided by the resistance of the xylem vessels (R x ), if no capacitance is considered.

$$ Ep=\frac{\Psi_c-{\Psi}_l}{R_x}=\frac{\Psi_s-{\Psi}_c}{R_s+{R}_r} $$
(1)

This simple equation represents the basis on which most of the SPAC models are developed (Williams et al. 1996; Sperry et al. 1998). On the right hand of Eq. 1 a supply function of the system describing the water withdrawn by the roots is given, while the left hand side describes the demand function that accounts for the water being transported through the trunk and branches. The following paragraphs of this section describes how the sun/shade and the soil multi-compartment approach have been integrated into the supply and demand parts of Eq. 1, and how both parts have been linked to computing total tree Ep. Figure 1 provides a visual scheme of the model. A detailed description of the mathematical derivations and the structure of the model are presented in supplementary material 1.

Fig. 1
figure 1

Schematic representation of the model. The figure represents the sun/shade approach for a tree with two soil layers and two soil compartments with different Lv and θ soil . The dotted line placed at the tree collar represents the virtual separation between soil compartments. Ψ s , R s and R r represents soil water potential, soil resistance and root resistance, numbers corresponds to the layer and compartment number. Ψ c is the water potential at the collar. R x corresponds to the resistance of the stem xylem. Finally, Ψ l , g co2 and Ep represents leaf water potential, stomatal conductance for CO2 and transpiration. The suffix sun and shade indicates if the symbols correspond to the sunlit or the shaded area

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The supply function

To derive the uptake capacity of a group of roots, Gardner (1960) defined them as being infinite long cylinders, uniformly distributed and withdrawing water in the radial direction from a volume of soil surrounding them. The volume of soil could be inferred from root length density (Lv) (Gardner 1960; Cowan 1965; Newman 1969). Gardner’s equation was derived for the path from the mid space between two consecutive roots towards the root surface:

$$ {\Psi}_s-{\Psi}_r=\frac{q}{4\pi k} \ln\;\left(\frac{{b^{\prime}}^2}{a_{root}^2}\right) $$
(2)

Where Ψ r is water potential at root surface (kPa), k is the unsaturated soil water conductivity (kg s−1 m kPa−1), q is the water flux from the soil to a single root ( kg m−1 s−1 ),b’ is the half distance between roots (m) and a root is the root radius (m). If roots are assumed to be evenly distributed in the soil, b’ can be derived from the root length density Lv (m m−3) like (Newman 1969):

$$ b^{\prime }=\sqrt{\frac{1}{\pi Lv}} $$
(3)

The combination of Eqs. 2 and 3 with the expressions for the flux of water coming to a group of roots based on the root length density proposed by Cowan (1965) and the differences in soil and root surface water potential developed by Campbell (1985b) led to the following expression for the resistance of the soil surrounding a root:

$$ {R}_s=\frac{ \ln\;\left(\frac{1}{\pi Lv\;{a}_{root}^2}\right)}{Lv\kern0.3em d\;4\pi k} $$
(4)

In Eq. 4, Lv refers to the root length density of the absorbing roots. However, it is often difficult to describe an absorbing root because, although the highest rates of water uptake are found in the thin, unsuberized portions, the contributions of the woody suberized root parts can be significant in trees with large root systems (Kramer 1969). In the present model, only roots with a diameter less than 1.4 mm are considered to be active in water uptake. A detailed description of the derivation of Eq. (4) is provided in the supplementary material 1.

The other resistance present in the supply function can be derived by simply upscaling the specific radial hydraulic resistance of a root (r r ) to the entire root system using Lv and d:

$$ {R}_r=\frac{r_r\left(\theta, T\right)}{Lv\ d} $$
(5)

The value of r r integrates, for a root section in the radial direction, the degree of permeability of the different tissues arranged in series that conform the root cylinder (Steudle and Peterson 1998). This, however, is usually a fixed value in most of the models. Nevertheless, permeability might well vary due to changes in the root environment like temperature or soil water content by: modifications in the degree of suberification of the exo and endodermis, aquaporine activity, or a loose contact between the root surface and the soil, among others (Herkelrath et al. 1977; North and Nobel 1992; 1997; Steudle and Peterson 1998; Steudle 2000). In the present model, r r is obtained as a function of the temperature using the approach developed by García-Tejera et al. (2016), and then modified according to θ soil using Bristow’s model (Bristow et al. 1984).

Tree Ep must equal the sum of all the fluxes coming from each soil layer (i) of each soil compartment (j) if tree capacitance is not taken into account. Once the total resistance from root and soil are derived using Eqs. 2 and 3, the water withdrawn by the roots from each soil layer (i) at each soil compartment (j) is obtained by applying to the supply function the water potential gradient between the soil (Ψ si,j ) and the root xylem (Ψ rxi,j ). The integration of all those fluxes to obtain tree Ep would read:

$$ Ep={\displaystyle \sum}\frac{\Psi_{si,j}-{\Psi}_{rxi,j}}{R_{si,j}+{R}_{ri,j}} $$
(6)

Equation 4 can be further simplified. Unless cavitation is present, root xylem resistance is negligible compared to R s and R r (Sperry et al. 1998; Tyree and Zimmermann 2002). Considering a negligible root xylem resistance necessarily implies a common xylem water potential throughout the root xylem network; in other words, Ψ c is assumed to be the same throughout the whole conductive root system. With this simplification in mind, Eq. 4 can be rearranged as a function of Ψ c as:

$$ {\Psi}_c=\frac{{\displaystyle \sum}\frac{\Psi_{si,j}}{R_{si,j}+{R}_{ri,j}}-Ep}{{\displaystyle \sum}\frac{1}{R_{si,j}+{R}_{ri,j}}} $$
(7)

The demand function

The specific hydraulic conductance of the xylem has been estimated by adding up the conductivities of the conduits found in a cross-section of wood using a modified version of the Hagen–Poiseuille equation, in which the quarter-power diameters are substituted by mean vessel diameter and vessel density (Tyree and Ewers 1991). The specific hydraulic resistance of the xylem is then calculated as the inverse of the specific hydraulic conductance. However, the use of the Hagen-Poiseuille equation to model water transport through the stems consistently underestimates the specific resistance measured experimentally on wood segments. This deviation has been attributed to the resistance of inter-conduit pit pores, as sap has to cross a porous membrane to flow from one conduit to the next. Some studies including several angiosperm species have estimated that, on average, those pits contribute 56 % to total xylem specific resistance regardless of wood porosity (ring or diffuse) (Wheeler et al. 2005; Hacke et al. 2006). The present model uses the Hagen–Poiseuille equation to compute specific conduit resistance. Pit specific resistance is then obtained by multiplying 1.27 (56 % of the total xylem specific resistance) to the specific conduit resistance. Finally, the total specific hydraulic resistance (r t ) is calculated by adding up conduit and pit specific resistances. Upscaling to total xylem hydraulic resistance (R x ) can be done by considering the path from the mean root depth (Z root ) to shoot height (Z shoot ) and the cross-sectional area of sapwood per square meter of soil (SWA) as:

$$ {R}_x=\frac{r_t\left({Z}_{root}+{Z}_{shoot}\right)}{\ SWA} $$
(8)

In computing the demand function, tree canopy is discretized into sunlit and shaded leaves to upscale from leaf to tree photosynthesis (dePury and Farquhar 1997). If the canopy is divided into sunlit and shaded leaves, total transpiration must be computed as the sum of Ep from each leaf class

$$ Ep={E}_{psun}+{E}_{pshade} $$
(9)

For well coupled canopies, like forests or tree crops, Ep can be directly related to the stomatal conductance using a simplified version of the Penman-Monteith equation, assuming that, the aerodynamic conductance is much higher than the stomatal conductance due to the “roughness” of the canopy surface (Villalobos et al. 2000; Orgaz et al. 2007). In that case, Ep is called “imposed” and is estimated as a function of the vapour pressure deficit (VPD) and the stomatal conductance for CO2 (g co2 ) (Jarvis and McNaughton 1986):

$$ {E}_{psun}={g}_{co2 sun}1.6\frac{VPD}{P}LA{I}_{sun} $$
(10)

Where P is atmospheric pressure and LAI sun and g co2sun are leaf area index and stomatal conductance for CO2 of the sunlit leaves. LAI sun is computed hourly as a function of the zenith angle, the G projection function and the solar radiation reaching the canopy, assuming an spheroidal canopy shape. Equation 8 (and the following Eqs. 9 and 10) refer only to sunlit leaves for the sake of concision; those for shaded leaves are analogous.

Equation 8 can be further developed using the adaptation of Leuning’s equation proposed by Tuzet et al. (2003), in which, g co2 is related to leaf assimilation, the concentration of CO2 at the substomatal cavities (C i ), the light compensation point (Γ), and reduced by leaf water potential through an empirical function (see Tuzet et al. (2003) for a detailed description). For a sunlit leaf, Tuzet’s equation reads:

$$ {g}_{co2 sun}={g}_0+\frac{m{A}_{sun}^{\prime }}{C_{isun}-\Gamma}f\left({\Psi}_{lsun}\right) $$
(11)
$$ f\left({\Psi}_{lsun}\right)=\frac{1 + \exp \left[{s}_f{\Psi}_f\right]}{1+ \exp \left[{s}_f\left({\Psi}_f-{\Psi}_{lsun}\right)\right]} $$
(12)

In Eq. 9, the symbol g 0 is the night time conductance (i.e. for zero gross assimilation) and m is a proportionality factor between photosynthesis and stomatal conductance, while in Eq. 10 Ψ f is a reference water potential which marks the initial value of Ψ lsun at which g co2sun is affected and s f modulates the rate of the reduction. The present model uses gross assimilation (A’) computed using the Farquhar et al. (1980) approach instead of net assimilation as was originally proposed by Tuzet (2003). Parameters for Farquhar’s model are assumed to change with temperature following the exponential function developed by Bernacchi et al. (2001).

The modification of the Leuning’s equation proposed by Tuzet et al. (2003), recognize the effect over the stomatal conductance of the link between the demand imposed by the atmosphere and the supply of water provided by the roots through the xylem. In the expression described in Eq. 12, the empirical reduction function describes the sensitivity of the stomata to leaf water potential, including implicitly, in the parameters Ψ f and s f , the leaf morphological adaptations of the different species and the environmental conditions experienced during growth (Tuzet et al. 2003).

Coupling supply and demand

In Eq. 1 Ψ c appears in the demand and the supply functions. On the other hand, in Eq. 5, an expression to obtain Ψ c was developed for the supply part. Now substituting Eq. 5 in the demand function of Eq. 1 and rearranging for Ψ l yields:

$$ {\Psi}_l=\frac{{\displaystyle \sum}\frac{\Psi_{si,j}}{R_{si,j}+{R}_{ri,j}}-Ep}{{\displaystyle \sum}\frac{1}{R_{si,j}+{R}_{ri,j}}}-Ep{R}_x $$
(13)

For a canopy discretized into sunlit and shade leaves, Eq. 11 turns into:

$$ {\Psi}_{lsun}=\frac{{\displaystyle \sum}\frac{\Psi_{si,j}}{R_{si,j}+{R}_{ri,j}}}{{\displaystyle \sum}\frac{1}{R_{si,j}+{R}_{ri,j}}}-{g}_{co2 sun}1.6\frac{VPD}{P}0.018\left(\frac{{\displaystyle \sum}\frac{1}{R_{si,j}+{R}_{ri,j}}+{R}_x}{f_{sun}}\right)LA{I}_{sun} $$
(14)

Where f sun is the ratio LAI sun /LAI and f shade would be 1- f sun . The coefficients 0.16 and 0.018 transform the conductance for CO2 in mmol m−2 s−1 to conductance for H2O in kg m−2 s−1As in Eqs. 8, 9 and 10, Eq. 11 is formally the same for Ψ lshade , simply changing the values of the corresponding parameters to those of the shaded leaves.

Equation 12 assumes that the tree is split into two parts according to the fractional area of sun and shade leaves. In other words, the model assumes that each leaf class will be sustained by a proportion of roots and trunk equal to f sun and f shade , respectively.

To find the values for Ψ l and g co2 for each leaf class an iterative procedure is followed using C i as a convergence criterion. Once an equilibrium point for C i is attained, values for Ep and Ψ c are then obtained from Eqs. 8, 7 and 5. The appendix section at the end of the paper provides a detailed description of the iterative procedure to match Ψ l and g co2 . A symbol list, with the units, a brief description and their acquisition source is also supplied.

Split root experiment

Model results were compared to the data obtained from an experiment with two 1-year old almond ‘Marinada’ grafted onto GF677 rootstock and two 2-year old olive ‘Picual’. The four trees were received in the spring of 2012. The root system of the cuttings was carefully divided into two equal halves and then placed in individual 25-l pots. As a result of the dividing process, each cutting occupied two pots with one half of the root system in each.

The cuttings stayed in the pots until the winter of 2013. At the end of that season, plants were transferred to specially designed lysimeters with transparent walls (Fig. 2). The lysimeters consisted of a metal frame of 0.2 × 0.8 m and with a depth of 0.8 m, covered with 6 transparent polymethyl methacrylate sheets (4-mm thick) on all the vertical walls. The volume of each lysimeter was divided by another plastic wall into two isolated parts of 0.2 × 0.4 × 0.8 m. At the bottom, each part had its own drainage collector system, while at the top a plastic cover was fitted to prevent evaporation from the soil substrate. The cuttings were placed on top of the division sheet with the split roots on the two sides of the lysimeter. The substrate used was a sterilised mix of 50:50 % sand:vermiculite.

Fig. 2
figure 2

Lysimeter design

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Measurements

The experiment was conducted in a shade house covered with an anti-trip mesh at the facilities of the Institute of Sustainable Agriculture (IAS) in Cordoba, Spain (37°52′N, 4°49′W) from May to September, 2014. In one tree at a time, an irrigation sequence was imposed consisting of two phases, one with full irrigation (F, irrigation given each morning to both sides of the root systems until the occurrence of steady drainage), followed by dry one half (D, one side of the root system receiving water like F and the other one receiving none). Throughout the irrigation cycles, the flux of water coming from each half and the soil moisture variations in the lysimeter were recorded every 10 min. Whereas for almond trees only one irrigation sequence was imposed, olive trees received two consecutive sequences, inverting the irrigated and dried sides during phase D namely D1 and D2. Figure 3 presents a diagram of the different irrigation phases.

Fig. 3
figure 3

Diagram showing the different irrigation phases for the experiment with olive trees. F represents phases where both sides were watered while D1 and D2 are the phases in which one side was dried out

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To impose different levels of stress, olive tree no. 1 was submitted to a period without irrigation (i.e. neither side receiving water during 1 day) at the end of D1 and at the beginning of D2; also, phase F between D1 and D2 was shorter than for olive tree no. 2. Almond trees were subjected to the same water stress level as olive tree no. 2, with at least one side of the root system always receiving water.

Irrigation in each half was applied using 3 emitters with a discharge rate of 4 l h−1. Variations in soil water content were monitored with 8 TDR probes per half (model CS645, Campbell Scientific Inc. Logan UH, USA) distributed in two columns of four sensors at 0.10, 0.30, 0.50, 0.70 m depth from the soil surface and spaced 0.2 m between them. TDR probes were 75 mm long and were previously calibrated using the method suggested in Regalado et al. (2001). All the 16 TDR probes were monitored at a 10-min frequency with a multiplexed TDR (model TDR100 + two SDMX50, Campbell Scientific Inc. Logan UH, USA). Total soil moisture variation in the lysimeter was also monitored continuously at the same frequency of the TDR using a +/− 1 g precision scale (model Kern DSK150, Kern & Sohn GmbH, Balingen, Germany).

The water flux through each half of the root system was recorded using two sap flow sensors per plant, installed in each of the two branches of the split collar at the beginning of the experiment. The sap flow measurement system was based on the Compensated Heat Pulse method (Swanson and Whitfield 1981) and the Calibrated Average Gradient method at low convective velocities (Testi and Villalobos 2009). The probes consisted of three stainless steel rods 2 mm in diameter, one being a linear heater and the other two temperature sensors, installed 10 and 5 mm down- and upstream of the heater, respectively. Each temperature sensor contained two thermocouple junctions that were sampled separately to obtain heat-pulse velocities at 5 and 15 mm depths below the cambium. The system was controlled by a datalogger (CR1000, Campbell Scientific Inc., Logan, UT, USA) and sampled at 10-min intervals. Heat-pulse velocities were corrected for wounding effects (Green et al. 2003) and then converted into sap flux densities assuming reported values of solid and liquid fractions in the sapwood of olive and almond (López-Bernal et al. 2014). Finally, sap flow rates were obtained by integrating sap flux densities across the radius of the collar branches (Green et al. 2003). The two collar branches of each plant originated from splitting the root system, so that the exact anatomy of their conductive system could differ substantially from the assumptions normally made in the flux density integration of normal trunks, due to their expected cicatrisation tissue. In order to cope with the associated methodological artefacts, the final flux in each branch was obtained as the apparent flux from sap flow measurements times a calibration coefficient (specific for each probe) derived from mass balance calculations using total mass from the scale, θ soil variations on each side from TDR differential values and the drainage collected. On the one hand, the scale records allowed the derivation of the total amount of water extracted by the tree; while on the other hand, TDR probes and the collected drainage from each half permitted the calculation of the fraction of water extracted from each half of the root system. By multiplying the total flux extracted from the roots by the fractions derived from the TDR probes and the drainage, the real flux for each half was obtained. The calibration coefficient for each sap flow probe was computed like the ratio of the apparent and the real flux. The calibration procedure was performed at the beginning of the experiment (phase F) and the calibration coefficients remained stable during the whole experimental period. The lowest calibration coefficient was 0.3 and the highest was 0.8. This large variability among probes could be probably related to different cicatrisations responses after splitting the collar.

Temperature and relative humidity inside the shade house were monitored every 10 min using a sheltered temperature and humidity probe (HMP45AC, Vaisala, Wantaa, Finland). Global radiation data were collected from a silicon cell pyranometer (model SKS 1110, Skye Instruments Ltd, Llandrindod Wells, Powys, UK) at 1.7 m height from an automatic weather station at the IAS facilities. Radiation data from the weather station were corrected for the shade house according to mesh transmissivity values provided by the manufacturer.

Root dynamics during the irrigation cycles were monitored using the root intersection method in the lysimeters with olive trees. The two frontal transparent sheets were divided into squares of 5 × 5 cm and roots in contact with the transparent sheet of each square were counted at the beginning of the irrigation cycle (phase F), at maximum stress (end of phase D1) and several days after phase D1 (second phase F).

In olive tree no.2, measurements of leaf water potential at the end of phase D2 (maximum stress) were made at 7:00, 9:00 and 11:00 GMT using a pressure probe chamber (model 3000, Soil Moisture Equipment, California, USA).

At the end of the experiment, the trees were extracted from the lysimeters and processed as follows. All the leaves were removed to measure leaf area using a leaf area meter (LI-3100 Area Meter, Licor inc, Nebraska USA). Roots were carefully extracted and then washed on a 1 mm mesh sieve to remove the substrate and then dried in an oven at 60 °C for 3 days, after taking a subsample of 10 % in mass from each side of the root system. The subsamples were digitally scanned using a commercial scanner (HP Scanjet G3110) and the resultant image files were then analysed with root image analysis software (WinRhizo, Regent Instruments Inc, Quebec City, Canada) to obtain the total root length of the subsample and average root radius (a root ). Once analysed, the subsamples were dried in the oven at the same temperature and for the same time as the rest of the root systems. The dry weight of the subsamples was obtained using a four decimal precision balance (model AV104, Mettler Toledo, Greifensee, Switzerland) while root dry mass of the total root systems was measured using another balance (Kern FCB 8 K0.1, KERN & SOHN GmbH, Balingen Germany). Finally, the average specific root length (SRL) for each tree, used to convert from root biomass to Lv, was calculated as the ratio of mean root length to mean root dry mass of the corresponding subsamples.

Substrate hydraulic properties

The soil water retention curve was derived from six samples of soil substrate with different water contents, using a Dewpoint Potentiometer (WP4-T, Decagon Devices, Washington, USA); a more detailed description of the procedure is available at http://www.decagon.com/education/ . For the saturated water conductivity (k s ), the Philip-Dunne permeameter method was used following the protocol developed by Muñoz-Carpena et al. (2002). The data collected were then fitted using a curve fitting software package (Curve Fitting toolbox Matlab R2010b, MathWorks Inc, Massachusetts, USA) applying the empirical relationships proposed by Campbell (1985a), which represent the water release characteristic curve and the unsaturated soil water conductivity.

Model parameters and calibration

Because the model was tested using small trees (maximum measured leaf area of 4.97 m2); it was assumed that the fraction of shaded leaves was negligible compared to that of sunlit ones. Therefore the simulations were made considering that all the leaves were illuminated. The parameters for the biochemical photosynthesis model of Farquhar et al. (1980) for olive trees were obtained from Diaz-Espejo et al. (2006) while for almond trees, reported values from Egea et al. (2011) for the summer period were used.

All the parameters of the Tuzet model, except m and Γ, that were obtained from Moriana et al. (2002), were derived directly from this experiment. Tuzet’s empirical relation (Eq. 10) was calibrated in one tree of each species using the recorded sap flow values from the first day of the experiment (when the two parts of the lysimeters were well watered). The calibrated parameters were used on the remaining days and for the experiment with the other tree of the same species. For the same tree and the same period as for the calibration of the Tuzet’s function, the value for g 0 was obtained from the conductance derived by inverting Eq. 9, and using the calibrated sap flow records at 2 h before dawn. Soil water content was daily initialized using recorded TDR values.

Statistical analysis

Regression analyses were performed comparing: measured Ep from each tree, calculated as the sum of the calibrated fluxes recorded from the sap flow probes inserted on each side of the split collar, against the Ep derived from the model. The efficiency factor of the model was computed as:

$$ EF=1-\frac{{\displaystyle \sum }{\left( Xmeas- Xmod\right)}^2}{{\displaystyle \sum }{\left( Xmeas-\overline{X} meas\right)}^2} $$
(15)

Where Xmeas and Xmod correspond to measured and modelled values. EF was calculated using the same data set as for the regression analyses.

Results

Model parameters

Table 1 presents a summary of: Lv, leaf area (LA), a root , VD, Φ v and specific root length (SRL) for the four trees used in the experiment. In relation to olive trees, the almond plants presented a lower LA and a similar or higher Lv. The mismatch in LA:Lv relation between the two species is related to the differences in SRL, which were almost double for almond.

Table 1 Root and shoot characteristics, Lv 1 and Lv 2 : root length densities for each lysimeter compartment. SRL, LA and a root are: specific root length, leaf area and average root radius. VD, Φ v are the vessel density and vessel diameter used during the simulations. Finally r r 20 is the value for the root specific hydraulic resistance at 20 °C
Full size table

Calibration of Tuzet’s empirical parameters (Table 2), gave different values for almond and for olive trees. The sensitivity factor (s f ) was smaller for almond than for olive, meaning a lower susceptibility of g co2 to Ψ l . The reference water potential (Ψ f ), however, was very similar for both species with a reduction threshold for g co2 close to zero. Unlike Tuzet’s function parameters, olive g 0 was one order of magnitude lower than that for almond.

Table 2 Values for the parameters used in the photosynthesis and stomatal conductance models Parameters for photosynthesis obtained from (Diaz-Espejo et al. 2006) for olive trees and from (Egea et al. 2011) for almond trees are: the degree of curvature of the response to PAR of the electron transport rate (θ), the quantum efficiency (α) and values of activation energy (ΔH a ) and the scaling constants (c) for: Michaelis constant for CO2 (K c ) and O2 (K o ), CO2 compensation point (Γ), maximum catalytic activity of Rubisco in the presence of saturating amounts of ribulose biphosphate and CO2 (V cmax ), maximum ratio of electron transport (J max ), and the rate of CO2 evolution in the light resulting from process other than photorespiration (R d ). Parameters for stomatal conductance have been obtained from the recorded sap flow values through a calibration procedure, those are: a proportionality factor between photosynthesis and stomatal conductance (m), the stomatal conductance for null net photosynthesis (g 0 ) and the parameters for the empirical model of Tuzet, namely, reference water potential (Ψ f ) and a sensitivity factor (s f )
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Soil substrate parameters for Campbell’s function are presented in Table 3. Two things characterized the substrate used in the experiment. One was the high k s , and the other the low value of air entry potential (Ψe), when compared to data for different soil types from the literature (Campbell 1985a; Campbell and Norman 1998)

Table 3 Parameters to describe soil substrate hydraulic properties measured during the experiment. Symbols represent: depth layer (d), soil water content at permanent wilting point (θ ll ), field capacity (θ ul ) and saturation (θ sat ), air entry soil water potential (Ψ e ), saturated hydraulic conductivity (k s ) and a shape factor (b)
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Model test

The model was tested against total tree Ep computed as the sum of the calibrated fluxes recorded from the two sap flow sensors installed in each tree (Fig. 4). To facilitate the visibility of the data, only hourly values of modelled versus observed Ep are presented. Regression analysis produced R2 values of around 0.9 for all trees except for almond no. 2 (R2 = 0.82), and EF close to 1 (Table 4). According to Fig. 4, the model tends to under-predict almond Ep in the mid-range, while maximum Ep is slightly over-predicted; this is more evident for almond no. 2. For olive trees, instead, an over prediction of the mid-range values of Ep was observed.

Fig. 4
figure 4

Measured transpiration (Ep meas) computed as the sum of the calibrated fluxes from each side of the lysimeter against the transpiration obtained with the model (Ep mod) on an hourly basis for olives and almonds. Dots represent tree number one and crosses refers to tree number two in both species

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Table 4 Coefficients for linear regressions, root mean square error (RMSE), model efficiency factor (EF) and regression coefficient (R2)
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In Fig. 5 the time series of modelled and recorded root water extraction fluxes from each split root branch of the two olive trees are presented. The model was able to capture the tendencies in both parts of the split root system not only during the periods in which both sides were irrigated (phases F), but also when one side of the root system was dried out (phases D). In general, for the four trees during phase D, the model tended to under-predict the flux of the dried side while the flux of the wetted one was overestimated (see Fig. 5 for olive trees and supplementary material 2 for almond trees). We also found a lag in the recovery of the flux after phase D of the side that had been previously dried (see doy 181 and 182 in olive tree no. 2 as an example). Also noticeable was the deviation in olive tree no. 1, in the modelled water extraction flux from one of the two root branches on the last 4 days of the experiment, with an over prediction as high as 44 % with respect to the measured flux.

Fig. 5
figure 5

Time span of measured and modelled fluxes for olive trees no. 1 and no. 2 at different irrigation phases. Open triangles and vertical lines are the fluxes measured from sides one and two, while continuous black and grey lines correspond to modelled fluxes from sides one and two

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Root dynamics

Root counts on both sides of the two olives trees highlighted root dynamics when a reduction in soil water content occurred in one part of the root system (Fig. 6). During the period of maximum stress, at the end of phase D1, a decrease in the root counts on the water-deprived side was observed for both olives trees, with a reduction of 80 and 40 % with respect to the value at the beginning of the drying cycle for olive trees no. 1 and no. 2. Surprisingly, for the same period, the irrigated side also underwent a 20 % reduction in both olive trees. Counts after phase D1, gave different results: while there was an increase in root appearance in olive tree no. 2; olive tree no. 1, which experienced a higher reduction in root counts on the dry side at the time of maximum stress (Dry (D)), was not able to reach the values at the beginning of the counting period (Irri (F2)).

Fig. 6
figure 6

Root counts on both sides of the lysimeter for olive no. 1 (black) and no. 2 (grey) during their first drying phases, expressed as a percentage (%) of the initial (F1). Counts were made after the beginning of phase D1 (F1), at the end of phase D1 (D) and several days before the end of phase D1 (F2). The dried and the irrigated side are indicated as “Dry” and “Irri”

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Leaf water potential estimations

Under the assumption that root clumping could be responsible of the differences between measured and modelled Ψ l , the measured Lv was substituted by an effective Lv (Lv’) to account for the clumping effect. To do so, the measured Lv was reduced until modelled Ep and Ψ l matched the measured values. Figure 7 presents measured and modelled Ψ l at the end of phase D2 in olive tree no. 2, using the two Lv values, the measured and the effective. When measured root biomass was used, a 12 bar difference between measured and modelled Ψ l was observed. However, when root biomass was reduced by 40 % and Tuzet’s empirical function was recalibrated, the model was able to match measured Ψ l and Ep.

Fig. 7
figure 7

Measured (points) and modelled (lines) leaf water potential (Ψ l ). The solid line corresponds to modelled leaf water potential for measured Lv while the dotted line is the leaf water potential when effective Lv is assumed to be 60 % of the measured value (Lv’)

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Discussion

Model performance

The fluxes during the different irrigation phases were successfully simulated (Fig. 5). Nevertheless, some deviations were observed, particularly after the end of the drying phases in both olive trees and also at the end of phase D2 in olive tree no. 1. Those biases are probably associated with different phenomena. The deviation observed in olive tree no. 1 at the end of the experiment, might well be related to the root dynamics observed during the different phases of the experiment (Fig. 6). Unlike olive tree no. 2, olive tree no. 1 showed a reduction in root counts on both sides after phase D1 (Fig. 6, column Dry F2 and Irri F2). This meant a lower Lv value than that used in the model at the start of the simulation period. As a result, Lv in the model was higher during the recovery period after phase D2, leading to an overestimation of the modelled fluxes. On the other hand, the divergence observed immediately after the end of D phases, particularly in olive tree no. 2 (doy 181 and 182, Fig. 5) was related to a faster redistribution of water between layers than that predicted by the model (see supplementary material 2). If the water applied to a previously dried side is less uniformly distributed by the model, increases in θ soil associated with irrigation would be restricted to the top layers thus reducing the amount of roots with access to water and, as a result the predicted fluxes would be lower. However, those deviations have had a minor effect on the simulation of tree Ep, as can be deduced from the high R2 (around 0.9) and the high efficiency factor (close to 1) for the four trees presented in Table 4.

For field conditions one would expect those errors to be smaller. Lifespan of roots growing in the field is prolonged during periods of water stress thanks to a redistribution of the water throughout the root system from the wetted zones to the driest areas, allowing the maintenance of the membrane integrity and root turgor (Bauerle et al. 2008). On the contrary, in the set up used in the present experiments, the root system was completely divided into two parts preventing the redistribution of water from the wet to the dry compartments, leading to a faster root mortality (Bauerle et al. 2008).

A mismatch between measured and modelled leaf water potential for olive tree no. 2 at the end of phase D2 was observed (Fig. 7). One might argue that the absence of capacitance in the model was responsible for the differences observed. However under the climate conditions experienced during the experiment, with a high evaporative demand and clear days, the inclusion of the capacitance would only have served to improve the simulation of Ep at the beginning and end of the day, but it would not have had any implications in the estimation of daytime Ψ l especially at noon. The mismatch observed between measured and modelled Ψ l is more related to the underestimation of tree resistance. Xylem failure could be a possible cause of the mentioned underestimation since the model does not include cavitation, but Hacke et al. (2000) demonstrated that the Ep of a tree with a root area/leaf area ratio (A r :A l ) below 10 and growing in a sandy soil would be limited by rhizosphere resistance, meaning that Rs + Rr would be the main resistances controlling the tree water relations. The A r :A l ratios computed from Table 1 for olive and almond trees were 7.64 and 4.52. Then again, Torres-Ruiz et al. (2013) working on olive trees observed that a xylem potential of −2 MPa would induce less than a 20 % loss of conductivity. The measured Ψ l were around −2.1 MPa meaning that the xylem potential during the experiment should be higher to that value. Those evidences led us to think that cavitation, if it occurred, played a minor role.

Another possibility could be related to the clumping of roots. Roots growing in a confined media tend to overlap producing clumping (see supplementary material 2). Root clumping has important effects on the determination of rhizosphere resistance since for its computation, it is assumed that roots are evenly distributed occupying soil cylinders of the same volume (Passioura 1988; Tardieu et al. 1992). To cope with this problem, Passioura (1988) proposed the use of an effective Lv (Lv’) as a substitute for the observed Lv. This author assumed that clumped roots would behave as a single one, with access to a given volume of soil; so that measured Lv should be reduced to Lv’ to account for clumping when computing Rs and Rr. In our experiment, the use of an Lv’ obtained from a 40 % reduction of the Lv measured led us to match modelled Ψ l and Ep with the data observed for olive tree no. 2 after the recalibration of Tuzet’s parameters (Fig. 7). Unfortunately, Ψ l was only measured for olive no tree. 2, so it was impossible to determine the Lv’ for the other trees studied. However, that fact only had implications in the determination of Ψ l and not in the simulation of the water extracted from each side of the split root system and tree transpiration (Figs. 4 and 5).

It is important to remember that the experiment was performed on a lysimeter with a relatively low soil volume compared to a plant in the field. Hence, one would expect that some of the observed bias in the model should be reduced under field conditions. For instance, plants growing in the field will have a lower Lv than the values observed in the present experiment, meaning that the effect of root clumping should be smaller, and the possible errors in the determination of Ψ l might be reduced. A more detailed description of the artifacts related to pot experiments is given by Passioura (2006).

The importance of a variable radial resistance

During the drying phases, the model was able to reproduce the reduction in flux of the dried side as water was withdrawn, and the maintenance or the increase on the wetted side. The same extraction patterns have been reported in other experiments with different species, in which only a fraction of the root system was watered (Green and Clothier 1995; Ameglio et al. 1999). Ameglio et al. (1999), studying partially irrigated walnut trees, observed a reduction in the flux and an increase in the total hydraulic resistance of the dried compartment. The author proposed two possible explanations for the observed increase: one related to a rise in Rs and another related to an increment in Rr. When the soil dries out, there is a sharp decrease in k, so Rs increases (Eq. 2). In addition, changes in root morphology or the presence of air gaps between the root surface and the soil have been reported as the soil dries, with Rr increasing as a consequence (Herkelrath et al. 1977; Bristow et al. 1984; North and Nobel 1992; Stirzaker and Passioura 1996; North and Nobel 1997). It is generally accepted that Rr is the major resistance of the plant if no cavitation occurs (Tyree and Zimmermann 2002). When θ soil is reduced below a certain value, Rs is greater than Rr, and controls root water uptake (Gardner 1960). However, as Newman (1969) demonstrated, this is only true when Lv is low (less than 5000 m m−3); at higher values (more than 10,000 m m−3), Rs would hardly be of any significant importance, even when the soil is completely dry. To illustrate this idea, one can use the Lv data obtained from the experiment with olive tree no. 1 to compute Rs and Rr (Eqs. 2 and 3). While Rr at θ ul is 3.3 · 106 s m2 kPa kg−1 the value for Rs at θ ll is 9.7 · 104 s m2 kPa kg−1, two orders of magnitude lower. The observed reduction in Lv at the end of phase D1 (Fig. 6, Dry (D) column) does not change the aforementioned relation between Rs and Rr when the soil is dried out. Although a decrease in Lv would not affect both resistances to the same extent, the two orders difference is still maintained between them even with the reduced Lv observed (data not shown). Looking at the numbers in the example, it would seem that, given the high Lv values measured (Table 1), Rr is the main resistance in the soil part on both sides, even under dry conditions.

The Bristow equation (supplementary material 1) includes the effect of a drying soil on r r and its incorporation into the model significantly improved the simulations during the drying phases. Values of Lv above the threshold of 10,000 m m−3are commonly observed in the field (Jackson et al. 1996), which means that situations in which Rr controls root water uptake are not restricted to potted plants. Yet, the Bristow equation is an empirical function, so the development of a more mechanistic model should be a priority in future studies.

The model presented here does not include a chemical signaling like ABA in the control of stomatal conductance. In theory, the ABA generated in the roots as a response to a drying soil, would amplify the feedback of water potential on stomatal conductance, thereby reducing transpiration (Tardieu and Davies 1992). However, that reduction was not observed when one side was dried out (Fig. 8). The daily relative flux (the ratio of measured flux and reference evapotranspiration) of the wetted side increased during drying phase D2 meaning that g s was not affected. The decay observed at the beginning of phase D2 is related to a 1 day interruption of the irrigation on both sides whereupon the wetted side recovered almost instantly after that day without irrigation. Same results were observed by Holbrook et al. (2002) on split-root tomato plants. The author used a wild tomato cultivar and an ABA deficient mutant. After imposing a sequence of drying cycles like the ones performed in this experiment for olive and almond trees, the authors did not observed any difference in the g s response to a drying soil between the wild cultivar and the mutant. Besides, Torres-Ruiz et al. (2015) working on olive trees observed that ABA variations after imposing dry periods could not explain the observed changes in g s . The evidences reported here do not imply the absence of ABA during phase D2, but suggest a possible lack of control over the stomata under the conditions of the present experiment.

Fig. 8
figure 8

Relative measured fluxes from olive tree no. 1 from 3 days after the beginning of phase D2 until the end of the experiment. The relative values correspond to the fluxes measured with sap flow probes divided by the reference evapotranspiration (ET0). The circle indicates the beginning of phase D2 in which the irrigation was cut off on both sides of the lysimeter. Filled dots relates to the watered side whilst the empty dots represent the side that has been dried out

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Species-related parameters and their implications

Table 1 presents a summary of the parameters measured for the two species studied that showed a large difference in specific root length while the root radius was almost the same. The root bulk density, i.e. the ratio of dry mass and root volume (ρroot , g cm−3 ) can be calculated as:

$$ {\rho}_{root} = \frac{1}{\pi\ {a_{root}}^2SRL} $$
(16)

The value for olive (2.1 103 kg m−3) doubled the one for almond (1.1 103 kg m−3) in parallel with differences in root radial resistance. García-Tejera et al. (2016) found the same difference in rr for the two species used here at 20 and 25 °C. The relation between root density and r r might be related to the deposition of compounds like lignin in the exo- and endodermis of the root, thus reducing its permeability (North and Nobel 1992). Future work should test the feasibility of using root density to predict root radial resistance for different species or genotypes, which could be used as a selection index in breeding programs. The model presented here couples the water supply and demand of the tree so it could be used to study the implications of the variability in the parameters between and within species in water uptake under limited water supply for different current and future environments.

Conclusions

The SPAC model presented here can simulate tree transpiration when a contrasting water content situation exists in the soil by using a soil multi-compartment approach. The model was able to simulate the tendencies in the fluxes of both sides when different levels of stress were applied to olive and almond trees. The inclusion of a variable Rr was essential for simulating the water extracted from each soil compartment, and it was necessary to account for root clumping (through an effective root length density) to match the measured Ψ l and Ep. The model integrates the whole plant response to water availability using a mechanistic approach to describe the supply and demand functions using measurable parameters. This provides a useful tool for exploring the interaction between the plant and the environment in the cases in which a non-uniform water content is present in the root zone. The inclusion of physiological and anatomical parameters of the roots may be useful as a help in breeding programmes for tree crops.