Plant Soil (2011) 341:233–256DOI 10.1007/s11104-010-0639-0

REGULAR ARTICLE

A one-dimensional model of water flow in soil-plantsystems based on plant architecture

Michael Janott · Sebastian Gayler ·Arthur Gessler · Mathieu Javaux ·Christine Klier · Eckart Priesack

Received: 25 February 2010 / Accepted: 29 October 2010 / Published online: 27 November 2010© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract The estimation of root water uptake andwater flow in plants is crucial to quantify tran-spiration and hence the water exchange betweenland surface and atmosphere. In particular the soilwater extraction by plant roots which provides thewater supply of plants is a highly dynamic andnon-linear process interacting with soil transportprocesses that are mainly determined by the nat-ural soil variability at different scales. To betterconsider this root-soil interaction we extendedand further developed a finite element tree hydro-dynamics model based on the one-dimensional(1D) porous media equation. This is achievedby including in addition to the explicit three-

Responsible Editor: Alexia Stokes.

M. Janott · S. Gayler · C. Klier · E. Priesack (B)Helmholtz Zentrum München, Ingolstädter Landstr. 1,85764, Neuherberg, Germanye-mail: priesack@helmholtz-muenchen.de

M. JavauxUniversité Catholique de Louvain, Croix du Sud,2, bte2, 1348, Louvain-la-Neuve, Belgium

M. JavauxForschungszentrum Jülich, 52425, Jülich, Germany

A. GesslerLeibniz-Zentrum für Agrarlandschafts-Forschung(ZALF), Eberswalderstr. 84, 15374,Müncheberg, Germany

dimensional (3D) architectural representation ofthe tree crown a corresponding 3D characterisa-tion of the root system. This 1D xylem water flowmodel was then coupled to a soil water flow modelderived also from the 1D porous media equation.We apply the new model to conduct sensitivityanalysis of root water uptake and transpirationdynamics and compare the results to simulationresults obtained by using a 3D model of soil wa-ter flow and root water uptake. Based on datafrom lysimeter experiments with young Europeanbeech trees (Fagus silvatica L.) is shown, thatthe model is able to correctly describe transpira-tion and soil water flow. In conclusion, comparedto a fully 3D model the 1D porous media ap-proach provides a computationally efficient alter-native, able to reproduce the main mechanisms ofplant hydro-dynamics including root water uptakefrom soil.

Keywords Transpiration · Plant hydro-dynamicsmodel · Root water uptake · European beech ·Porous media equation

Introduction

An approach to model plant water relations hasdeveloped during the last two decades based ontwo main concepts: the cohesion-tension theory(Tyree and Zimmermann 2002) and the electri-

234 Plant Soil (2011) 341:233–256

cal circuit analogy applied for simulating watertransport in plants using resistances, capacitances,water potentials and flow (Cruiziat et al. 2002).By applying these two concepts, the description oftree hydraulic architecture has made a strong im-provement towards a more realistic and compre-hensive vision of tree water relationships (Cruiziatet al. 2002). Only recently, this approach was fur-ther improved by substituting the electrical circuitanalogy by a porous media description for sapflow in wood that is based on Darcy’s law andincludes a water capacity term to better accountfor the dynamic behaviour of the hydraulic stor-age in trees in a way that mass conservation isconsidered, e.g. by Bohrer et al. (2005), see alsoArbogast et al. (1993), Früh and Kurth (1999),Kumagai (2001) and Aumann and Ford (2002).This avoids calculations of negative water con-tents and unlimited water withdrawal from thetree as occurring with applications of the electricalcircuit model (Chuang et al. 2006). By using athree-dimensional (3D) representation of the treestructure and the physically based representationof tree hydro-dynamics, questions can now beadressed of how the variability in crown architec-ture (i.e. inter-specific, age or ecosystem depen-dent structure) would lead to different transpira-tion responses (Bohrer et al. 2005). In particularby considering the stomatal response to the cor-responding branch water potential, we can thensimulate the stomatal closure, the correspondingchange in leaf water conductance and the resultingactual transpiration rate. Moreover, considering aloss of conductivity with decreasing water poten-tial allows the representation of vulnerability tocavitation at critical highly negative water poten-tial values (Hölttä et al. 2005; Cochard et al. 1999).

Since water transport from the soil throughthe plant into the atmosphere takes place in asoil-plant-air continuum that is interconnected bya continuous film of water, modelling of plantwater transport has to consider both the waterexchange at the leaf-air interface and the waterflux at the soil-root interface. Therefore, also rootwater uptake including root hydraulic architecturehas to be described to integrate the tree hydro-dynamic model into ecosystem models (Doussanet al. 2003), a step still to be taken to replacecurrent representations of plant water trans-

port, e.g. big leaf or resistor-capacitor approaches(Bohrer et al. 2005) that are often coupled toone-dimensional (1D) effective root water uptakemodels (Cowan 1965; Gardner 1960; Nimah andHanks 1973; Feddes et al. 1978; Campbell 1985).New root water uptake models are available thatexplicitly describe root architecture and relatedsoil-plant processes in three dimensions by explic-itly considering the 3D distribution of the uptake(Clausnitzer and Hopmans 1994; Somma et al.1998; Vrugt et al. 2001) and also considering waterflow in the root system (Doussan et al. 2006;Javaux et al. 2008). They can be based on rootgrowth models that allow the integration of a greatdiversity of environmental conditions and theirimpact on root system development (Dunbabinet al. 2002).

Despite the more realistic assumptions for pre-dicting soil-root interactions, a disadvantage ofthe 3D models is their high level of complexity andconsequently their high computational demand.Therefore, to describe and simulate transpirationat the stand level a less complex but still realis-tic approach is needed. By extending the hydro-dynamics model for aboveground parts of treesof Bohrer et al. (2005), we developed a simpleroot water uptake and transpiration model thatcouples a 1D soil water flow model with a 1Dplant xylem water flow model. Notwithstanding its1D character the xylem water flow model takes3D plant architecture into account and can con-sider different root properties for each root nodeand different root types. Moreover, by specifyingdifferent properties of the soil directly surround-ing each root, to a certain extent also horizontalvariability might be represented.

Model development

In our model both the water flow within theplant and the water flow in the soil are describedby applying the 1D porous medium or Richardsequation (Richards 1931). This is based on theassumption that similar to the soil matrix, also theplant xylem can be conceived as a porous medium.The xylem is seen as a bundle of small parallelcapillary pores, which are filled with water andair. A capillary can conduct water only if it is

Plant Soil (2011) 341:233–256 235

completely saturated. With increasing suctionhead more and more pores cavitate and the abilityof the xylem to conduct water decreases. To sim-ulate the xylem water flow and the water uptakefrom the soil by the roots we need a representa-tion of the plant architecture as model input todefine the flow domain and the interfaces betweenthe plant and its environment consisting of the soiland the atmosphere.

Description of plant architecture

The plant architecture is described as a combi-nation of stem, branches, leaves, gross and fineroots including branching roots of several orders

(Fig. 1a). The area of leaves on an outer branchis prescribed by a relative leaf area distribution.Stem, branches and roots are divided into slicesthat are approximated by finite cylinders ofdifferent lengths and diameters (Fig. 1a). Each ofthese cylinders consists of an inner cylinder rep-resenting the heartwood, an outer hollow cylinderfor the sapwood or xylem and further outer hol-low cylinders that include the cambium and thephloem (Fig. 1c). It is assumed that the diametersof these cylinders are maximal diameters in thesense that they do not increase anymore due toan increase in water content of the plant and thata loss of plant water can lead to smaller diameters(Fig. 1d). The flow domain of the xylem is further

Fig. 1 Schematicillustration of the treemodel: a Treearchitecture andconsidered water flows:soil water flow, rootwateruptake, xylem water flowand transpiration. Thevolume Ve of the xylemelement e is given by itslength le and its maximalxylem area sx,max. b Treegraph representing thesolution domain of theRichards equation forxylem water flow.c Descriptions of above-and below-groundcylindrical tree elements.d Water flow through axylem cylinder of lengthle and changing diameterbetween maximaldiameter de,max andminimal diameter de,min

transpiration

xylem water flow

Ve = sx,max le

soil water flow

root water uptake

transpiration

xylem water flow

Ve = sx,max le

soil water flow

root water uptake

a b

root

cambium

cortexendodermis

epidermis

phloem

le

xylem

Ve

xylem

Ve

root

cambium

cortexendodermis

epidermis

phloem

le

xylem

Ve

le

Ve

heartwood

phloemcambium

bark

xylem

above ground

Ve

xylem

le

Ve

heartwood

phloemcambium

bark

xylem

c

above ground

qout

qin

lede,min

de,max

qout

qin

lede,min

de,max

d

236 Plant Soil (2011) 341:233–256

abstracted and represented by a 3D graph consist-ing of 1D elements each standing for an inner orhollow cylinder of xylem tissue (Fig. 1b).

Water flow within the plant

The basic state variable to describe water flowin plants is the hydraulic or water potential ofthe xylem which quantifies the energy status ofthe liquid phase. The hydraulic potential can bedefined as the sum of the xylem potential orxylem hydrostatic potential and the gravitationalpotential assuming that changes of the osmoticpotential are not important for long distance wa-ter flow in the xylem (Früh and Kurth 1999).Hence it is assumed that the only component ofthe xylem potential comes from the energy tomove water isothermally and reversibly into orout of the porous matrix represented by the xylemtissue. Because this potential is determined bythe porous matrix of the xylem in the followingit is called xylem matric potential in analogy tothe soil matric potential (Passioura 1980). If thexylem as the water conducting tissue of the plantis conceived as a porous medium (Siau 1984),water flow can be described by Darcy’s law, therelation between the water potential gradient ofthe xylem and the mass flow of water (Früh andKurth 1999; Chuang et al. 2006). Following furtherFrüh and Kurth (1999) we assume homogeneityof the xylem hydraulic characteristics, the capac-itance or water retention and the axial hydraulicconductivity for each subunit of the plant xylem asrepresented by the xylem cylinders in Fig. 1a. Byfurther assuming that the xylem volumetric watercontents are defined in relation to fixed volumesof the xylem cylinders, we can apply the principleof mass conservation as expressed mathematicallyby the 1D continuity equation (Früh and Kurth1999). For the sake of simplicity, in a first step, weneglect water exchange between xylem, phloem,cambium and heartwood and confine the waterflow model mainly to the longitudinal directionof the stem and branch axes. By adding a sink-source term to represent transpiration and in ourcase also root water uptake, we get the following1D Richards equation as outlined in detail by Frühand Kurth (1999) and Chuang et al. (2006). Thisequation can then be solved on the domain given

by the graph, which represents the plant xylemporous medium by straight lines not only for stemand branches, but also below-ground for roots(Fig. 1b) extending in this way the approach ofBohrer et al. (2005):

∂θx(ψx)

∂t= ∂

∂z

[kx(ψx)

(∂ψx

∂z+ cos αx

)]− Sx (1)

where for all of the tree elements representedby straight lines θx denotes the volumetric watercontent (m3 m−3) of the xylem as function of thexylem matric potential ψx (mm). The xylem matricpotential is given on a weight basis, i.e. it is ex-pressed as hydraulic head (mm). t (s) denotes thetime, z (mm) the axial length of the element andkx(ψx) represents the xylem hydraulic conductiv-ity (mm s−1) defined as function of the xylemmatric potential using the maximal xylem crosssectional area as reference surface of the waterflux. The vertical position is given by the heightabove (positive upward) or the depth below thesoil surface (negative downward). For a branch orroot element αx is the zenith angle (–) and Sx (s−1)the sink or source term, which refers to roots forroot water uptake and outer branches for transpi-ration. In a further model development also theradial exchange of water between the xylem andthe phloem, the cambium and the heartwood ofthe tree might be added to this term.

Hydraulic characteristics of the xylem

Due to the form of the mass conservation princi-ple involved in the formulation of the Richards’equation, xylem water contents are defined inreference to fixed xylem volumes, i.e. in our caseto the maximal xylem volumes. Therefore, if onewants to apply Richards equation and also con-sider xylem desaturation which keeps the xylemvessels still saturated but leads to significantlysmaller xylem volumes, then this desaturation hasyet to be described in relation to the maximalxylem volumes. In this case the usual form of soilwater retention curves, that are nearly constantbetween zero matric potential and air entry value,cannot be applied to describe the xylem water–xylem matric potential relationship on a maxi-mal sapwood volume basis. Hence to account forchanges in xylem volume, in view of measure-

Plant Soil (2011) 341:233–256 237

ments for European beech (Fagus silvatica L.)branches by Oertli (1993), see Fig. 3a, we as-sume a linearly decreasing xylem water retentioncurve between zero matric potential at maximalsaturation and air entry value. Thus we extendthe xylem water retention curve of Chuang et al.(2006) describing for each tree element e of stem,branch or root the dependency of the volumetricxylem water content on the xylem matric potentialat a certain height or depth z (mm) by

θx(ψx) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

θx,a

(ψx

a

)−λ

, if ψx < a

(εx − θx,a)

(a − ψx

a

)+ θx,a, else

(2)

where the maximal volumetric water content ofthe xylem is assumed to be equal to the xylemsapwood porosity εx (m3 m−3) given for each treeelement by the fraction of the maximal volumeof xylem water at saturation to the maximal totalvolume of the xylem sapwood in the tree elementor due to the assumed cylindrical geometry of thexylem elements by the fraction of the maximalwater-filled xylem sapwood area sxw,max,e (m2) andthe maximal xylem sapwood area sx,max,e (m2) ofthe xylem element e. In a similar way we define thexylem water content at air entry θx,a (m3 m−3) byuse of the corresponding water-filled cross sec-tional xylem sapwood area sxw,a (m2) at the xylemmatric potential when cavitation begins to occur:

θx,a(z) = sxw,a(z)

sx,max,e(z)(3)

Assuming that tension changes the volume ofwater conducting sapwood only (Perämäki et al.2001) and neglecting corresponding changes inphloem water content, which are considered tobe comparatively small, then, in combination withthe xylem water retention curve, the xylem watercontent θx,a at air entry can be derived using therelation between xylem matric potential and rela-tive volume change of the xylem element (Steppeand Lemeur 2007) expressed by

1Ve

dVe

dψx= dθx

dψx= 1

E(4)

where E (mm) denotes the elastic modulus of thexylem and Ve the volume of the xylem element. By

evaluating this relation, which is valid in the rangeof xylem water contents between εx and θx,a whenthe vessels are saturated, we get

θx,a = εx + aE

(5)

and hence a value for the xylem water content atair entry can be determined for a given value ofthe elastic modulus. Moreover, inserting Eq. 5 intoEq. 2 we get

θx = εx + ψx

Efor a ≤ ψx ≤ 0 (6)

and from this the actual xylem cross-sectional areasx,e (m2):

sx,e = sxw,e + (sx,max,e − sxw,max,e)

= sx,max,e

(1 + ψx

E

)for a ≤ ψx ≤ 0. (7)

We then can calculate the actual xylem diameterexploiting the assumed cylindrical volume geome-try of the xylem element e (Fig. 1d).

The elastic modulus E (mm), the porosity εx

(m3 m−3), the air entry value a (mm) and theexponent λ (–) that determine the xylem waterretention curve, are model input parameters andassumed to be fixed values for the whole tree.

From the xylem water retention curve we canderive the xylem hydraulic conductivity kx basedon the law of Hagen and Poiseuille for the massflow rate of water in a cylindrical pipe. Accordingto Burdine (1953) by considering capillary bundlesof such pipes, the relative conductivity K(θx) (–)of the volumetric water flux in the capillary bundleas defined by Eq. 11 can be calculated by the fol-lowing integral equation (Hoffmann-Riem et al.1999):

K(θx) = (θx/εx)p

∫ θx/εx

0ψx(θ)−2 dθ

∫ 1

0ψx(θ)−2 dθ

, (8)

Here also other functional forms such asMualem’s equation (Mualem 1976) might be ap-plicable. To account for the tortuosity of soil poresusually p = 2 is assumed, but for the sake of sim-plicity and lack of appropriate data to estimate anadequate value of p we assume p = 0, although

238 Plant Soil (2011) 341:233–256

tortuosity of water films in xylem vessels might oc-cur during embolism. The evaluation of Burdine’sintegral equation for θx ≤ θx,a and the applicationof the Hagen-Poiseuille law to consider the effectof an increase �R of the effective mean hydraulicradius of the capillary bundle for θx > θx,a leads tothe following function of the xylem water contentto represent the xylem hydraulic conductivity:

K(θx) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

(θx

εx

) 2λ+1

, if θx ≤ θx,a

(θx,a

εx

) 2λ+1

+[

1 −(

θx,a

εx

) 2λ+1

]

×(

θx − θx,a

εx − θx,a

)2

, else

(9)

For the saturated case, i.e. for θx > θx,a we con-sider the increase of the mean effective hydraulicradius �R of the capillary bundle with increasingradii of the water-filled tubes resulting in xylemvolume increase, and get for the xylem radii r =r(θx), rmax = r(εx), ra = r(θx,a) (mm) at the respec-

tive xylem water contents θx, εx and θx,a and themean value θm = (εx − θx,a)/2:

�R2 = 1θm

∫ θx

θx,a

r(θ)2 − r2a

r2max − r2

adθ

= 2εx − θx,a

∫ θx

θx,a

θ − θa

εx − θadθ

=(

θx − θx,a

εx − θx,a

)2

(10)

To account for the change of cross-sectionsbetween tree elements, for each tree element atdepth or height z the hydraulic conductivity iscalculated in relation to the water-filled cross sec-tional area of the xylem sapwood (Bohrer et al.2005), normalised by the maximal water-filledcross sectional xylem sapwood area of the stemelement at stem-base, i.e. at height zero sxw,max(0)

(m2):

kx(θx(z)) = sxw,max,e(z)

sxw,max(0)kmax K(θx) , (11)

where kmax (mm s−1) denotes the maximal hy-draulic conductivity of the xylem. kmax is an addi-

Table 1 Inputparameters

θs,res and θs,sat residualand saturated vol. soilwater content, α and nvan Genuchten shapeparameters withm = 1 − 1/n, ksatsaturated soil hydraulicconductivity, kr radialroot conductivity (Steudle2000), kmax maximal axialxylem conductivities ofEuropean beech saplings(Rewald 2008) andstomatal hydraulicconductivity parametersb , c (Lemoine et al. 2002;Bohrer et al. 2005)

Soil or plant Reference parameters Perturbed parameters

Loam θs,res = 0.078 ksat = 2.496 · 103 mm d−1

θs,sat = 0.43α = 0.036 cm−1

n = 1.56ksat = 249.6 mm d−1

Clay loam θs,res = 0.095 ksat = 624.0 mm d−1

θs,sat = 0.41α = 0.019 cm−1

n = 1.31ksat = 62.4 mm d−1

Clay θs,res = 0.068 ksat = 480.0 mm d−1

θs,sat = 0.38α = 0.008 cm−1

n = 1.09ksat = 48.0 mm d−1

Root kmax = 432.0 mm d−1 kmax = 4.32 · 103 mm d−1

kmax = 43.2 mm d−1

kr = 3.0 · 10−6 d−1 kr = 3.0 · 10−5 d−1

Shoot kmax = 1.3 · 103 mm d−1 kmax = 13.0 · 103 mm d−1

kmax = 130.0 mm d−1

Leaf b = 1.0 · 105 mm b = 3.0 · 105 mmc = 3.5

Stem dstem = 10.0 mm

Plant Soil (2011) 341:233–256 239

tional input parameter, which for the sake of sim-plicity is assumed to have fixed values for eithershoot and root. For young European beech treeswe apply in the following different values for kmax

(see Table 1).

Water flow in the soil

To simulate vertical soil water flow we applyagain the 1D Richards equation, an approachfrequently used to model soil-plant-atmospheresystems (Priesack and Gayler 2009; Simunek et al.2008; Kroes and van Dam 2003):

∂θs(ψs)

∂t= ∂

∂z

[ks(ψs)

(∂ψs

∂z− 1

)]− Sw (12)

where θs is the volumetric soil water content(m3 m−3) as a function of the soil matric potentialψs (mm), t (s) denotes time and z (mm) soil depth(here positive downward). ks(ψs) (mm s−1) is thesoil hydraulic conductivity which is given as afunction of the soil matric potential. Both soil hy-draulic property functions θs and ks are describedby parameterisations according to van Genuchten(1980). The last term Sw (s−1) represents the sinkterm of root water uptake (per unit soil depth).Typical boundary conditions of the flow equationare a mixed boundary condition at the soil surfaceto represent atmospheric conditions leading eitherto infiltration, evaporation or water ponding anda free drainage condition at the bottom boundary(Simunek et al. 2008; Kroes and van Dam 2003).

Soil water uptake and transpiration

Water flow between the soil-root interface and theroot xylem is defined by the 1D soil-root volumet-ric flux jr,e (mm3 s−1) which describes the volumeof water exchanged between a root element and asoil layer per unit of time

jr,e = kr sr.e(z) [ψs(z) − ψx(z)] (13)

where kr (s−1) is the radial conductivity of theroot through the root cell structure between rootxylem and soil and sr,e(z) (mm2) the outer surfaceof the root element e within the soil layer atdepth z (mm). ψs(z) and ψx(z) denote the matric

potentials in soil at the root surface and in thexylem (mm) at soil depth z. Instead of applyingonly the radial conductivity kr, we alternativelyconsider the rhizosphere radial conductivity krs

(s−1), defined by

krs(z) =√

kr ks(ψs(z))/ lrs (14)

the geometric mean of the radial conductivitykr of the root and the soil hydraulic conductiv-ity ks at the soil matric potential near the rootsdivided by the radial thickness lrs (mm) of therhizosphere soil around the roots assumed to be2.0 mm (Hinsinger et al. 2005). To account forthe impact of the strong root-soil interaction inthe rhizosphere on the water status of the soildirectly surrounding the roots, we take instead ofthe soil water matric potential ψs a weighted meanψs between xylem and soil matric potentials

ψs(z) = γ1 ψx(z) + γ2 ψs(z)

γ1 + γ2(15)

where γ1 (–) and γ2 (–) are weighting constants.The calculated flux jr,e into or out of the cylin-

drical root element e is then used to calculate thebelow ground water uptake sink or source termsSx and Sw of the Richards equations of xylemand soil water flow. The below ground xylem sinkor source term Sx (mm3 mm−3 s−1) is obtainedby relating the fluxes to the corresponding xylemcylinder volumes Ve = sx,max,e(z) le of the rootelements of maximal xylem sapwood area sx,max,e

(mm2) and element length le (mm):

Sx(z) =∑

e

jr,e(z)

sx,max,e(z)le(16)

The water fluxes between soil and root cylinderslocated in a certain numerical soil layer i are re-lated to the corresponding soil volume �Vi givenby the thickness of the soil layer �zi and the unitarea of a square meter and are summed up togive the sink or source term Sw,i (mm3 mm−3 s−1)of the soil layer i. If the axis of a root cylinderelement e intersects with more than one of thenumerical soil layers, the water flux is divided be-tween these layers proportionally to the fractions

240 Plant Soil (2011) 341:233–256

fe,i (–) of the cylinder axis intersections with theparticular layers:

Sw,i =∑

e

jr,e(z) fe,i

�Vi(17)

Similar to water uptake, also transpiration, theloss of water to the atmosphere, is described bya sink term. At the outer branches, where theleaves are located, the corresponding water fluxis prescribed by the atmospheric evaporative de-mand and by the stomatal hydraulic conductivity,which similar to the xylem hydraulic conductiv-ity is assumed to depend on the xylem matricpotential of the branch element (Bohrer et al.2005). For simplicity the atmospheric water de-mand is prescribed by a constant flux condition orcalculated from the potential evapotranspirationrate (mm s−1) estimated by the FAO Penman-Monteith method (Allen 2000) and is distributedaccording to the leaf area. Therefore, in contrastto the water exchange with the soil, the exchangewith the atmosphere is prescribed and no directfeedback of transpiration to the micro-climate atthe leaf level is considered.

The prescribed flux condition or the potentialtranspiration Tpot (mm s−1) of the total tree is thendistributed to single branch elements accordingto the relative leaf area distribution LADe(z)

(m2 m−2) at each element e, given by the leafarea of all leaves that are connected to the branchelement divided by the total leaf area of the tree.Hence the potential transpiration Tpot,l f,e (mms−1) at the branch element e can be given by:

Tpot,l f,e = Tpot LADe(z) (18)

The stomatal hydraulic conductivity kx,stom

(mm s−1) is modelled in dependence of the xylemmatric potential in the leaf ψx,l f (mm) by

kx,stom(ψx,l f ) = kmax,stom exp(−

[−ψx,l f

b

]c)(19)

following Bohrer et al. (2005) with a stom-atal maximal hydraulic conductivity kmax,stom =1.0 mm s−1 and stomatal hydraulic conductivityparameters b (mm) and c (–) and by additionallyassuming ψx,l f = ψx(zep) at the end point zep ofthe branch element where the leaf is anchored.The actual prescribed transpiration rate Tact,l f,e

(mm s−1) is then calculated from the correspond-ing potential transpiration rate Tpot,l f,e (mm s−1)at the branch element by

Tact,l f,e =min{

Tpot,l f,ekx,stom(ψx,l f )

kmax,stom; qmax,l f

}(20)

for which the maximal evaporation rate from theleaf qmax,l f (mm s−1) at the actual leaf xylem po-tential ψx,l f is given by

qmax,l f = kx(ψx,l f )ψx,l f − ψ0,l f

�zep(21)

where ψ0,l f = −3.0 · 105 mm denotes a potentialthat corresponds to a minimal potential in the leaf,which can be reached under dry air conditions.�zep is the length of the element belonging to theend point where the leaf is anchored.

Finally, the above ground xylem sink term Sx,e

(mm3 mm−3 s−1) which represents transpirationof the outer branch element e is calculated byrelating the volumetric water flux out of the xylemgiven by Tact,l f,e sx,max,e (mm3 s−1) to the xylemvolume Ve (mm3) of the branch element:

Sx,e = Tact,l f,esx,max,e/

Ve = Tact,l f,e/

le (22)

Numerical implementation

The soil and tree water flow equations are cou-pled via the below ground water uptake sinkterms and each solved by iterations using a stan-dard Newton–Picard fixed point iteration scheme(Celia et al. 1990; Priesack 2006) until tree andsoil water contents and potentials converge belowa threshold value (Huang et al. 1996). Althoughboth flow equations may be coupled by a fur-ther fix-point iteration via the water uptake termto account for the non-linearity of the systems,similar to Tseng et al. (1995) we use a single-pass solution scheme without iteration betweenthe two subsystems, since for the considered sim-ulation scenarios the additional iteration does notlead to significantly different results. The solu-tion procedures of both flow equations 1 and 12are based on a finite element discretisation fol-lowing the approach of the HYDRUS-1D model(Priesack 2006; Simunek et al. 2008). Since thesolution method of the Richards equation on a

Plant Soil (2011) 341:233–256 241

graph as represented in Fig. 1b has not been doc-umented in more detail, we describe the appliedfinite element method in Appendix A.

Statistical criteria

Three statistical criteria are used to assess the ad-equateness of simulations. The root mean squareerror RMSE as defined by Mayer and Butler(1993) is used to assess over the whole period ofthe lysimeter experiment the deviation betweenpredicted, Pi, and observed values, Oi, propor-tioned against the mean observed value O:

RMSE = 1

O

√∑ni=1 (Oi − Pi)

2

n(23)

where n is the number of observations duringthe experiment. The lower the value of RMSE,the better is the agreement between simulationand observation. The Modelling efficiency ME

(Willmott 1982) is used to assess the model ade-quacy in describing the observed variability.

ME = 1 −∑n

j=1

(O j − P j

)2

∑nj=1

(O j − O

)2 + ∑nj=1

(P j − P

)2

(24)

where P j and O j are the predicted and observedvalues and O and P are the corresponding meanvalues. A value of ME < 0 indicates that P j =O would be a better estimation for observedvariability than the simulation results. If simu-lated values are completely in accordance withobserved values, ME equals 1.

Finally, a necessary criterion to evaluate theaccuracy of a numerical scheme is given by theglobal mass balance error MB = ∑end

m=1 MB(tm) atthe end of the simulation, where at each time stepm we have for the water balance of the soil-plantsystem:

MB(tm) = 100 ·

∣∣∣∣∣∣∣∣∣∣∣1.0 −

n∑e=1

Ve (θmx,e − θ0

x,e) +k∑

i=1

�Vi (θms,i − θ0

s,i)

m∑j=1

n∑e=1

Ve S jx,e�t j +

m∑j=1

(Q j0 − Q j

k +k∑

i=1

�Vi S jw,i)�t j

∣∣∣∣∣∣∣∣∣∣∣(25)

and where θml denotes the volumetric water con-

tent at time step m of the tree at element l = e orof the soil at node l = i, �Vi is the volume of thesoil layer i and Q j

0 and Q jk denote the volumetric

water fluxes (mm3 s−1) across the boundaries ofthe soil profile.

Simulation scenarios and parameterisation

The new model of water flow in soil-plant systemsis applied to three different scenarios mainly toillustrate that important features of water flowin plants and soils can be simulated. In a firstscenario, the drying soil scenario, we simulate soilwater uptake and transpiration under conditionsof a prescribed constant transpiration rate withoutrefill of soil water. By this scenario the effectsof stomatal conductivity, xylem conductivity, ra-dial root conductivity and soil conductivity on

water uptake and transpiration are considered.In particular the onset of cavitation is examined.In a second scenario, the hydraulic lift scenario,we demonstrate the basic ability of the model tosimulate transport of water through the roots fromwetter to dryer soil regions. In a third scenario,the lysimeter scenario, we test if the model cansimulate daily and seasonal water balances basedon measured input parameters. In this case for thexylem water flow model only parameters of thetree architecture are changed: the stem diameterand the distributions of leaf and root area indexbased on measurements from the lysimeter exper-iment.

In the drying soil scenario, we simulate soil wa-ter uptake and transpiration also for comparisonwith a 3D root water uptake modelling approach.In the scenario, the water is taken up by theroots from a homogeneous soil column of 0.4 m

242 Plant Soil (2011) 341:233–256

Table 2 Soil input parameters of the third scenario

Soil horizon texture Depth (cm) θs,sat (–) α (cm−1) n (–) ksat (mm d−1)

Sandy loam 0–5 0.52 0.11 1.13 1,381Sandy loam 5–30 0.504 0.12 1.14 1,285Silt loam 30–45 0.475 0.1 1.05 1,276Clay loam 45–90 0.395 0.05 1.13 632Sand 90–200 0.417 0.04 1.25 5,000

θs,sat saturated volumetric soil water content, α and n van Genuchten shape parameters with m = 1 − 1/n assuming for theresidual volumetric water content θs,res = 0, ksat saturated soil hydraulic conductivity

depth and of 0.1 m by 0.1 m soil surface, whichis assumed to be initially in hydrostatic equilib-rium with an aquifer located 3.0 m below the soilsurface (Javaux et al. 2008). The soil boundaryconditions are a no flux condition at the bottomof the soil profile and no rainfall or evaporationat the soil surface. For reasons of comparison withthe numerical experiment of Javaux et al. (2008)we use the same parameters to study the effect ofdifferent soil types and to analyse the sensitivityof different soil and xylem hydraulic conductivityvalues (Table 1). For the reference case input pa-rameter values are taken from the second columnof Table 1, and for the sensitivity analysis onlythe conductivity parameters, ksat for the soil, kmax

for the xylem and kr for root water uptake, werechanged as indicated in the third column. Thisperturbation of parameter values corresponds to arealistic variation in observed conductivity valuesfor soil or xylem (Javaux et al. 2008; Doussanet al. 1998). In both cases for the above-groundtree a constant potential transpiration rate of1.0 mm d−1 was assumed and prescribed alongthe tree at the ends of stem or branch elementsaccording to the assumed leaf area distribution.Related to the 0.01 m2 soil column surface thiscorresponds to a volumetric water flux rate of1.0 · 104 mm3 d−1 from the soil to the atmosphere.

In the hydraulic lift scenario, instead of the noflux of the first scenario we consider a Dirich-let boundary condition which prescribes saturatedconditions at the lower boundary of the soil col-umn now assumed to be 1.5 m deep. Furthermore,we now prescribe a daily sinusoidal cycle of thepotential transpiration according to (Childs andHanks 1975; Priesack 2006), but keep the dailyrate at 1.0 mm d−1. All other conditions and para-meters are kept the same as in the loam referencecase of the first scenario.

In the lysimeter scenario, the soil-plant sys-tems are represented by 2–3 years old Europeanbeech trees growing in cylindrical lysimeters of2.0 m depth and 1.0 m2 surface area. Here thesoil boundary conditions are a seepage flow con-dition at the bottom of the soil profile and anatmospheric flow condition at the top soil pre-scribing either infiltration, evaporation or waterponding. The third scenario is based on a lysime-ter study about effects of elevated ozone andbelow-ground pathogen infection on juvenile Eu-ropean beech (Winkler et al. 2009). From thisstudy we take hourly and daily water balance dataof one of the control lysimeters, i.e. lysimeter S3,that has seen neither an ozone nor a pathogentreatment. The input data that characterize thehydraulic properties of the lysimeter soil profileare derived from measurements given in Gayleret al. (2009) and summarized by Table 2. The stemdiameter dstem is 15.8 mm, the xylem conductivityparameter values are the reference values given inTable 1.

In all three scenarios we use a simplified plantarchitecture of a young European beech tree(Fig. 2) obtained using digital photographs andphotogrammetric evaluations. In the second sce-nario we extend a central root to the bottom ofthe soil column to facilitate water uptake fromdeeper soil layers and in the third scenario thestem diameter is increased keeping the diameterfractions between stem and branch elements thesame, whereas the root system is now adapted torepresent the measured root biomass distribution.

We estimate the maximum water-filled crosssectional xylem sapwood area sx,max of each treeelement e from the diameter de (mm) of the treeelement according to Aranda et al. (2005). Foryoung European beech trees we fit the new pa-rameterisation of the xylem water retention curve

Plant Soil (2011) 341:233–256 243

Fig. 2 Simplified plantarchitecture of a youngEuropean beech tree (twodifferent perspectives).Tree xylem hydraulicpotential distribution(between −3,600 mm and−36,000 mm suctionhead) under stillnon-limiting soilconditions at day 30 ofthe reference drying soilscenario simulation

given by Eq. 2 to the measured retention dataof Oertli (1993) for shoots of European beech.This fit results in an elastic modulus value ofE = 1.5 · 106 mm and an air entry value of a =

−3.0 · 105 mm with exponent λ = 0.854 for eachtree element (Fig. 3). Alternatively to the xylemwater retention curve given by Eq. 2 we alsoapply and fit a van Genuchten parameterisation

a b

Fig. 3 a Relative xylem water retention curve: measuredvalues taken from shoots of European beech by Oertli(1993), new parameterisation (solid line) according to Eq. 2and van Genuchten parameterisation (dashed line). b Rel-ative xylem hydraulic conductivity curve: measured valuestaken from shoots of European beech by Magnani andBorghetti (1995): (closed triangles), Cochard et al. (1999):

(open triangles), Lemoine et al. (2002): for a sun branch(empty circles) and a shade branch (f illed circles), con-ductivity curves derived from the new parameterisation(solid line) according to Eq. 9 and derived from the vanGenuchten parameterisation (dashed line) according toMualem (van Genuchten 1980)

244 Plant Soil (2011) 341:233–256

Table 3 Summary of numerical results for the xylem water flow model in case of the soil drying scenario with referenceparameters

δtol (mm) No. elem. (–) CPU (s) MB (%) No. iter. (–) No. steps (–)

10.0 243 3,046 0.05 60,945 21,2191.0 243 3,237 5.0 · 10−3 94,732 22,0040.1 243 3,615 5.0 · 10−4 145,330 22,2220.01 243 3,961 5.0 · 10−5 199,583 22,4140.001 243 4,458 5.0 · 10−6 266,459 22,4430.001 482 13,167 4.3 · 10−6 303,137 24,1320.001 903 26,918 3.6 · 10−6 301,279 23,173

δtol iteration tolerance, i.e. the iteration continues until |ψ j+1,ν+1x − ψ

j,νx | ≤ δtol for the xylem potential ψx at time step j and

iteration number ν at all element nodes, No. elem. is the number of tree elements, MB is the mass balance error of xylemwater flow, No. iter. is the total number of iterations during the simulation, No. steps is the total number of time steps duringthe simulation

(van Genuchten 1980) obtaining the parametersα = 2.8 · 10−6 mm−1, n = 2.77, θsat = εx and θres =0 (Fig. 3).

Results

Numerical accuracy and convergence

All numerical simulations were conducted on anordinary personal computer with a single core 1.5GHz AMD processor. CPU time is given in sec-onds (Table 3). During each simulation the watermass balances are recorded to compare for everyindividual time step the input-output balances of

the whole tree respectively of the soil profilewith the corresponding internal mass change inthe tree or soil. For the simulation of the soildrying scenario using the reference parametersthe global mass balance error of the soil-plantsystem is MB = 1.3 · 10−3%, for the hydraulic liftscenario MB = 3.1 · 10−3% and for the lysimeterscenario MB = 3.2 · 10−4%. Table 3 summarizesresults obtained for the xylem water flow simula-tion in the drying soil scenario using the referenceparameters. By relaxation of the iteration toler-ance δtol it is seen, that already at the criterion ofδtol = 0.01 a rather low global mass balance erroris reached without strongly increasing CPU time,iteration and time step numbers. In contrast, the

a b

Fig. 4 Drying soil scenario (loam): Time course of a tran-spiration rates and b relative xylem water contents θx/εxof the total tree using stomatal conductivity parameterb = 1.0 · 105 mm with (solid line) and without (dotted line

with open circles) reduction to zero conductivity, and b =3.0 · 105 mm with (dashed dotted line) and without (dashedline with f illed circles) reduction to zero conductivity

Plant Soil (2011) 341:233–256 245

CPU time is considerably increased by increasingthe number of tree elements up to 482 or 903by prescribing a maximal axial element length ofonly 20 or 10 mm instead of 100 mm as in thereference case. This increase indicates the firstorder accuracy in space of the applied numericalsolver.

Drying soil scenario

Figures 4 and 5 show the simulated course oftranspiration and root water uptake with related

plant water storage for different parameterisa-tions of the stomatal hydraulic conductivity curve.Because initially, at simulation start, tree watercontent is not in equilibrium with soil water andatmospheric conditions, the tree first looses waterto the atmosphere before the hydraulic gradientsof xylem water potentials get strong enough andforce the roots to take up the available water fromthe soil. Since a constant maximum water fluxis prescribed at the tree leaves and no water isadded to the soil, the water flow in the soil-plantsystem quickly reaches an almost steady state as

a b

c d

Fig. 5 Drying soil scenario: Conductivity sensitivity simulation(loam): Time course of the root water uptake: reference(solid line), 10-fold xylem conductivity kmax (dashed line),0.1-fold xylem conductivity kmax (dotted line), 10-foldradial root conductivity kr (dot-dot-dashed line), 10-foldsaturated soil conductivity ksat (dot-dashed line), a using

the radial root conductivity kr , b using the radial rhizo-sphere conductivity krs with weight coefficients γ1 = 1 =γ2. Soil effect simulation: Loam (solid line) compared toclay loam (dashed line) and clay (dotted line), c using theradial root conductivity kr , d using the radial rhizosphereconductivity krs with weight coefficients γ1 = 1 = γ2

246 Plant Soil (2011) 341:233–256

long as the soil provides enough water to fulfil theprescribed flux. After some time, when the soilgets dry, the root water uptake rate decreases dueto decreased soil hydraulic conductivities and lowsoil water availability. This low water availabilityinduces low xylem water potentials and conse-quently low stomatal hydraulic conductivities suchthat the transpiration rate is reduced. The tran-spiration rate is strongly determined by the as-sumed parameterisation of the stomatal hydraulicconductivity curve which reflects the importanceof stomatal control on xylem water flow. For thelower value of the stomatal hydraulic conductivityparameter b the stomatal hydraulic conductivitydecreases already at higher xylem potentials andhence protects the tree from water losses earlierduring the drying experiment (Fig. 4). If we thenassume that the closure of the stomata would leadto zero stomatal hydraulic conductivity the plantcould almost keep its water content (Fig. 4b, solidline) without reaching the air entry value. But ifwe assume a maximal possible reduction of stom-atal hydraulic conductivity to only 10% of its max-imal value, then the plant internal water is almostcompletely transpired (Fig. 4b, dotted and dashedlines) and cavitation occurs. Generally, a higherb value leads to a higher stomatal hydraulic con-ductivity also at lower xylem potentials. Thereforehigher water losses of tree water storage takeplace, which results in critically low tree watercontents. In our example in three cases xylemwater content falls below the air entry value θx,a =0.8, so that cavitation is induced (Fig. 4b). In thecases where the stomatal hydraulic conductivityremains above a non-zero limit, the transpirationgets finally limited when the plant internal wateris almost exhausted (Fig. 4a).

Due to the evaporative water flux from theleaves to the atmosphere, the simulated xylemmatric potentials show a strong vertical gradientalong the tree height (Fig. 2) having lower po-tentials at the tree crown, where the leaf area ishighest and higher potentials towards the rootsand within the roots which take up the water fromthe soil.

The perturbation of conductivity values givesonly very small differences between the simula-tions of the reference case and the cases withone-tenth- or ten-fold maximal xylem conductiv-

ity (Fig. 5). This small deviation is similar to re-sults obtained by Javaux et al. (2008) when usingthe same reference conductivity values for rootwater uptake simulations (Table 1). Additionally,no difference can be detected if we neglect thesecond term of the relative conductivity curve inthe case θx > θx,a, i.e. if we neglect the impact ofan increased mean radius of the water-filled xylemvessel elements. Moreover, for the small treesthe exact form of the conductivity curve, whichis difficult to evaluate considering the high vari-ability of the measured conductivity data (Fig. 3),may play a less important role, if only the de-crease of the curve with decreasing xylem matricpotential is appropriately described. Because ofthe low sensitivity of maximal xylem conductivityvariation on the simulated actual transpiration incase of the considered small trees, the relativexylem conductivity curve given by Eq. 9 could befurther simplified in the saturated range above theair entry value by omitting the second term ofthe right hand side without significantly changingour simulation results of xylem water flow dy-namics (results not shown). This low sensitivityalso means, that the impact of the elastic changeon the diameter and hence the conductivity ofwater-filled tubes of the capillary bundle repre-senting the xylem can be neglected in our modeldescription of xylem water flow in small Euro-pean beech trees. In this case the low sensitiv-ity of the axial xylem conductivities seems to becaused by their relatively high values comparedto the low stomatal and radial root conductivityvalues.

When compared to the reference case in thecases of increased soil or radial root conductivitythe begin of limited water supply for root wateruptake is 2–5 days delayed (Fig. 5a). Applying theradial rhizosphere conductivity the delay of theonset of limitation of transpiration is 9–10 days(Fig. 5b). In the sequel, the higher conductivitieslead to earlier soil water exhaustion and reducedtranspiration rates below 0.2 mm d−1.

To represent effects of horizontal soil vari-ability we introduce a possibility to account fordifferent soil properties in the direct surroundingsof different roots or root elements. This repre-sentation is achieved by considering the radialrhizosphere conductivity defined by Eq. 14. The

Plant Soil (2011) 341:233–256 247

root water uptake rates for the three soil types ap-plying either the constant radial root conductivitykr or the rhizosphere conductivity krs are shownin Fig. 5c and d. Generally, we observe, that ifwe apply the radial rhizosphere conductivity krs

instead of the radial root conductivity kr in Eq. 13,the simulated root water extraction decreases ear-lier and remains slightly higher when the soil getsdrier (Fig. 5b and d). For kr the root water uptakelimitation appears first for the clay, then for theloam and eventually for the clay loam (Fig. 5c).For krs the picture changes in case of the loam soil,since the time-span until transpiration gets limitedis now shortest for this soil type (Fig. 5d). In bothcases the clay loam soil supplies the transpirationwater demand by far the best, up to 15 d longerthan the clay and loam soils.

The simulated depth distribution of root wateruptake is shown by Fig. 6. Under non limitingsoil water conditions the root demand is met byavailable soil water and the depth distribution ofthe uptake rate almost coincides with the rootarea index distribution. Only in the case of theincreased radial root conductivity available soilwater gets already scarce in the region of highroot area index, i.e. m2 root surface area m−2

soil surface area (Fig. 6a). In this region wateruptake by roots begins to be limited after 15 d,and is already very low after 30 d (Fig. 6b and c).Then water is increasingly taken up from regionsof lower root area index, where soil water is notcompletely exhausted and still can satisfy almosthalf of the prescribed demand (Fig. 6d).

In the cases with increased soil and radial rootconductivity, similar to the 3D root water uptakesimulations of Javaux et al. (2008), the limita-tion of root water uptake is significantly delayedby 2–5 days (Fig. 5a). Furthermore, the findingthat the clay loam best sustains the evaporativedemand of the plant, about 10–15 d longer thanthe other soil types could be confirmed by oursimulations (Fig. 5c and d). But, in contrast tothe 3D water uptake simulations under the sec-ond collar boundary condition (CBC 2) by Javauxet al. (2008), in our 1D simulations water stressoccurred earlier for clay than for loam soil, if theradial root conductivity was used (Fig. 5c). Only ifthe radial rhizosphere conductivity is applied, thesame order of soil types concerning the appear-ance of water stress is found, i.e. transpiration isreduced first for loam, then for clay, and eventu-ally for clay loam soil (Fig. 5d).

a b c d

Fig. 6 Drying soil scenario (loam): Depth distribution ofroot water uptake and soil water content using differentroot or soil parameterizations: reference (solid line), 10-fold xylem conductivity (dotted line), 10-fold radial rootconductivity (dashed line), and 10-fold soil hydraulic con-ductivity (dashed-dotted line). The grey line represents the

depth distribution of the root area index RAI (m2 root sur-face area m−2 soil surface). Figures a–c represent differenttimes t after the start of the drying experiment: a t = 5 d,b t = 15 d, c t = 30 d, and d corresponding volumetric soilwater contents

248 Plant Soil (2011) 341:233–256

Hydraulic lift scenario

Plant roots do not only take up water but can alsorelease water to dry soil (van Bavel and Baker1985; Richards and Caldwell 1987). In this way theplant root system may transport water from moistinto drier, often upper soil horizons and hydrauliclift occurs. Assuming non polar flow across rootmembranes we are interested to see if our pro-posed soil-plant water flow model is in principlecapable to simulate such a hydraulic lift. Underthe conditions of the second scenario with peri-odic daily transpiration cycle the simulated wateruptake rates show a diurnal variation (Fig. 7).During the day when potential transpiration ratesare above zero, water is taken up mainly from theupper soil at 0.0–0.4 m depth (Fig. 7a), where mostof the root biomass is located. During the night,when no transpiration occurs, deep roots take upwater from the bottom of the soil column, wherethe soil is nearly water saturated, replenish xylemwater and additionally release water to the driersoil near the top of the soil column also at 0.0–0.4 m depth (Fig. 7a). Simulations considering atime span of more than 240 days show that waterflow in the soil-plant system gradually reaches aquasi steady state situation, where water is takenup from the lower part of the soil column and

redistributed to the upper soil during the night(Fig. 7b), whereas during the day mainly waterfrom the upper soil layer is taken up by the rootsto accommodate transpiration demand (Fig. 7c).Moreover, compared to a simulation where weassume no water flux from the roots to the soil,the daily transpiration with hydraulic lift is about5% higher during the first 100 days.

Lysimeter scenario

A more realistic scenario is given by the lysimeterscenario. Based on the measured daily precipi-tation, the estimation of daily potential evapo-transpiration by the Penman–Monteith dual cropcoefficient method (Allen 2000; Loos et al. 2007;Gayler et al. 2009) and the partitioning of po-tential evaporation and potential transpirationaccording to Allen (2000), the daily actual evap-otranspiration, the daily changes in water storageand the daily percolation rates were simulated andcompared to the corresponding measurements(Fig. 8). Results show that the model is able toreproduce the actual evapotranspiration and thechanges in water storage quite well, with a modelefficiency ME of 0.901 and a root mean squareerror RMSE of 0.298 for the evapotranspiration,and with the values ME = 0.977 and RMSE =

b c da

Fig. 7 Hydraulic lift scenario: Root water uptake in the soilprofile: a after 120 d within the whole day at night times:0.2 d (solid line), 0.4 d (long dashed line), 1.0 d (dotted line)and at day times: 0.6 d (dashed dotted line) and 0.8 d (shortdashed line); b during night time and c during day time in

each case after 1 d (solid line), 60 d (long dashed line), 120 d(short dashed line), 180 d (dashed dotted line) and 240 d(dotted line). d Depth distribution of the root area indexRAI (m2 root surface area m−2 soil surface)

Plant Soil (2011) 341:233–256 249

a b

c d

Fig. 8 Lysimeter scenario: a Daily precipitation, P(mm d−1, grey bars), leaf area index LAI (m2 leaf sur-face area m−2 soil surface, dotted line); b estimated dailypotential evapotranspiration PET (mm d−1, open circles),together with simulated (solid lines) and measured (grey

bars) daily actual evapotranspiration ET (mm d−1), c dailychanges in water storage �W (mm d−1), and d daily per-colation rates L (mm d−1) of lysimeter S3 (Gayler et al.2009)

Fig. 9 Lysimeterscenario: Simulateddiurnal course oftranspiration (mm d−1,solid line), of root wateruptake (mm d−1, dottedline), of change rate ofxylem water volume perunit soil surface area(mm d−1) of the total tree(dashed line) and ofxylem diameter (mm,grey line) during threedays (24–27 July)

250 Plant Soil (2011) 341:233–256

0.586 for the water storage change. The simulateddates and sizes of the percolation peaks differmore strongly from the measured data giving thevalues ME = 0.845 and RMSE = 1.812, in par-ticular, the simulated percolation peaks seem tooccur always somewhat later than the observedpeaks.

Figure 9 shows the simulated diurnal courseof transpiration, stem diameter, water flux at theroot collar and the rate of change of xylem wa-ter content during three consecutive days of thelysimeter experiment. Transpiration and waterflux at the root collar strictly follow the prescribedsinusoidal potential evapotranspiration. After theonset of transpiration, the rate of change of treewater content decreases due to the onset of tran-spiration but soon increases because root wateruptake begins to refill plant water capacity, beforeit decreases again indicating the end of waterreplenishment of the xylem water by soil watertaken up during the night. Correspondingly, thexylem diameter changes with a maximal diameterdifference of 0.194 mm between day and night.

Discussion

A new model approach is introduced to describewater flow in the soil-plant system. The model ex-tends previous approaches that were developed todescribe hydro-dynamics in above-ground plants(Früh and Kurth 1999; Aumann and Ford 2002;Bohrer et al. 2005) by additionally including thehydraulic root system architecture in combinationwith a soil water flow model. In this way a com-plete water flow model for the whole soil-plantsystem is obtained, which then vice versa extendsexisting root water uptake modelling approaches(Doussan et al. 1998; Javaux et al. 2008) by in-cluding the above-ground water dynamics of theplant. Similar to the soil water flow model, whichis based on the continuum approach as describedby the Richards’ equation, also the plant waterflow is described by this porous media equationfollowing Bohrer et al. (2005). This descriptionis in contrast to more discrete approaches thatconsider the plant as built up of storage com-partments and describe the water flow by waterexchange processes between these compartments

(Perämäki et al. 2001; Steppe and Lemeur 2007).But, besides the advantages of a continuous math-ematical formulation representing mass conserva-tion (Früh and Kurth 1999), due to the similaritybetween the porous media approaches to describewater flow in the soil and the xylem, it seems alsoto be conceptually easier to derive a consistentmodel of hydro-dynamics of soil-plant systemsbased on the continuum approach.

In a first analysis, the new model was appliedto investigate model behaviour and model sen-sitivity by scenarios with young European beechtrees that have already been analysed in two pre-vious studies (Javaux et al. 2008; Gayler et al.2009). Model tests and numerical results show,that the model is successfully implemented andcan efficiently simulate water flow in the soil-plantsystem based on the a simple representation ofplant architecture. Mass balance errors are lowand in an acceptable range. But since the globalmass balance is a necessary but not a sufficientcriterion for having a correct solution (Celia et al.1990), the model needs further analysis, whichmay be based on comparison with exact solutions(Lehmann and Ackerer 1998) or as in our case byevaluating simulations of practical examples usingexperimental data.

In the case of the 3D simulation by Javauxet al. (2008) the effect of the steeper slope of theloam hydraulic properties generates a heteroge-neous distribution of soil water potentials in thehorizontal direction with drier soil near the roots.This distribution leads to a larger conductivitydrop and produces an earlier water stress (Javauxet al. 2008). In the case of our 1D simulationthis effect is mimicked by amplifying the directimpact of the lower xylem potential on the soilmatric potentials directly near the roots resultingin a lower radial hydraulic conductivity betweenroot axis and soil, which is modelled by Eq. 14based on the definition of the radial rhizosphereconductivity krs. If the distribution of radial con-ductivities of the roots is adapted in this way,the main conclusions about effects of soil typeand conductivity values remain, although the rootarchitecture was different in the drying scenariosimulations we compared. But amplitudes or localvalues of considered variables are different due tothe 1D simulation of soil water flow. Therefore, a

Plant Soil (2011) 341:233–256 251

full data comparison cannot be performed, even iflocally the water uptake by the root system maybe of similar magnitude.

Besides the model sensitivity of the radialroot or rhizosphere conductivity, the most sig-nificant model sensitivity on simulated soil wateruptake and transpiration is due to the stomatalhydraulic conductivity. This model behaviour ismainly caused by eventually rather low conduc-tivity values, reflecting the significant role of bothconductivities in controlling tree water flow whichhas been shown in several experimental studies(Steudle 2000; Lemoine et al. 2002; Sperry et al.2003). Moreover, the water has to flow throughnonxylary pathways in the root across the rootcortex and endodermis and in the leaf throughthe mesophyll. These flow paths have low hy-draulic conductivities and can account to a highproportion of the overall root respectively shoothydraulic resistance (Sperry et al. 2003). In partic-ular, the radial root conductivity may be stronglychanged by aquaporin activity (Ehlert et al. 2009),which itself may be controlled by metabolism orbe triggered by environmental factors (Steudle2000). Also changes of leaf hydraulic conductivitymay be mediated by plasma membrane aquapor-ins (Cochard et al. 2007).

Hydraulic lift has been shown to occur in morethan 60 species (Espeleta et al. 2004), “but thereis no fundamental reason why it should not bemore common as long as active root systems arespanning a gradient in soil water potential andthat the resistance to water loss from roots is low”(Caldwell et al. 1998). However, for Europeanbeech trees it seems to be an open questionwhether the radial hydraulic root conductivityfrom inside the roots towards the soil is largeenough to permit considerable water flow to drysoil (Rewald 2008). Therefore our simulations ofhydraulic lift only show that in principle the modelis able to describe this kind of water redistribu-tion from wet to dry soil under almost optimalconditions for hydraulic lift, i.e. assuming watersaturation at the bottom of the soil profile witha resulting simulated leaf xylem matric potentialranging from −40,000 to −70,000 mm. Neverthe-less, the hydraulic lift could be significant in miti-gating plant water stress and need to be accountedfor in models describing root water uptake and

the onset of plant water stress in a water limitedecosystem (Siqueira et al. 2008).

The lysimeter scenario simulations confirm theapplicability of the new model to describe dailywater balance dynamics in soil-plant systems. Alsothe diurnal course of transpiration seems to berealistically described. In particular, the simu-lated diurnal changes of xylem diameter up to0.194 mm (Fig. 9) are within the range of xylemor stem diameter changes that were observedfor young European beech and Norway spruce(Pinus abies L.) trees (Steppe and Lemeur 2007;Perämäki et al. 2001). This agreement results fromthe fact that based on the parameterisation ofthe xylem water retention curve obtained fromthe experimental curve of Oertli (1993), we get avalue for the elastic modulus E which is similarto values found by Steppe and Lemeur (2007) andPerämäki et al. (2001). However, this value mighthave been underestimated, since we assume thatchanges of xylem water potential correspond onlyto changes in the xylem water content neglectingrelated changes in water contents of phloem, cam-bium and heartwood.

Since the new plant water flow model focuseson the water flow in the sapwood tissue in thelongitudinal direction of the stem and branch axes,the model still needs to be extended to describewater flow in the phloem and to account for theradial exchanges of water between xylem, heart-wood, phloem and cambium. In particular theactive uploading and unloading of sugars accord-ing to Münch’s hypothesis (Münch 1928) needsto be considered, because near the sources ofsugars considerable water flow from the xylemto the phloem can occur and vice versa closeto sugar sinks significant amounts of water canflow from the phloem to the xylem. Locally thiswater exchange can strongly influence the xylemwater flow dynamics which also can lead to xylemdiameter changes (Sevanto et al. 2002).

At the present state of development the modelis supposed to be already useful to simulate theimpact of differences in plant architecture andsoil properties on transpiration and water uptakeunder various climatic conditions, in particularconsidering differences of leaf area and root areadistribution or of root and shoot branching. Themodel may be also helpful to describe the impact

252 Plant Soil (2011) 341:233–256

of different distributions of conductivity valuesalong the tree architecture in particular of largertrees to study their role in protecting the xylemagainst cavitation, while maintaining transpira-tional water flow. This description may lead toa better understanding of observed differencesin xylem structures and functions between deepand shallow roots, roots, stem and branches(Nadezhdina 2010; North 2004) and of the rele-vance of nonxylary pathways for water uptake andtranspiration (Sperry et al. 2003). The model mayalso be applied to better describe differences ofstand water budgets between mixed forest standsof different species composition.

Conclusion

The newly proposed model of water flow in soil-plant systems presented in this study combinesapproaches to simulate water flow in the above-ground plant based on the porous media equationwith root water uptake and soil water flow mod-els also based on the porous media equation. Inthis way the model concept exploits the similaritybetween water flow in the soil and the xylemof plants. It could be shown that the model cansimulate water balance dynamics in soil-plant sys-tems with young European beech trees includingsome main known features of water flow in plantsand soil such as the diurnal cycle of plant wa-ter content and xylem diameter change, stomatalcontrol of transpiration, cavitation due to waterstress, sensitivity of root water uptake to radialroot conductivity, hydraulic lift, interaction of hy-draulic soil properties, root distribution and plantwater availability. Due to the simplified repre-sentations of plant architecture and distributionof soil properties that lead to 1D representationsof water flow domains, the model is numericallyvery efficient and may therefore be also applicableto simulate water flow in canopies accounting forspatial and temporal variability of canopy struc-tures based on rather simple representations ofindividual plant architecture.

Acknowledgements We are grateful to the DeutscheForschungsgemeinschaft which funded this study within

the frame of Forschergruppe 788 ‘Competitive mechanismsof water and nitrogen partitioning in beech-dominateddeciduous forests’. We also want to thank an anonymousreviewer whose comments helped to considerably improvethe manuscript and we thank Sebastian Bittner for his helpduring the revision of the manuscript.

Open Access This article is distributed under the termsof the Creative Commons Attribution Noncommercial Li-cense which permits any noncommercial use, distribution,and reproduction in any medium, provided the originalauthor(s) and source are credited.

Appendix A

The finite element method is applied according tothe Galerkin method to discretise the 1D Richardsequation (van Genuchten 1982; Priesack 2006).For this purpose the 3D graph, which representsthe tree is divided into finitely many 1D inter-vals [zi, zi+1] (1 ≤ i ≤ n), the finite elements i (cf.Fig. 1b). For each knot zi, (1 ≤ i ≤ n + 1) thepiecewise linear basis function φi is introduced,which is equal to 1 at knot zi and zero for all otherknots. By this, every continuous, piecewise linearfunction f (z) on the graph can be represented asa superposition of basis functions φi, where thecoefficients of φi are given by the values f (zi) ofthe function f at knot zi. The piecewise linear ap-proximation ψ(t, z) of the unknown ψ = ψ(t, z) ofthe Richards equation therefore can be expressedby the following function:

ψ(t, z) =n+1∑i=1

ψi(t) φi(z) , (26)

where ψi(t) := ψ(t, zi) is defined for all 1 ≤ i ≤n + 1.

The energy method from the calculus of varia-tions leads to the following condition of orthogo-nality for each of the n + 1 basis functions φi, i.e. tothe requirement that the following integrals van-ish on the entire solution domain representedby the graph:

∫

{∂θ

∂t− ∂

∂z

[K

(ψ

) (∂ψ

∂z− cos(α)

)]+ Sx

}φi dz

= 0, for all 1 ≤ i ≤ n + 1 . (27)

Plant Soil (2011) 341:233–256 253

Integration by parts and the insertion of ψ =∑n+1i=1 ψi φi leads to

∫

∂θ

∂tφi dz +

n+1∑j=1

ψ j

∫

Kdφi

dzdφ j

dzdz

= − qb φi

∣∣∣∣∂

+∫

K cos(α)dφi

dzdz

−∫

Sxφi dz (28)

for all 1 ≤ i ≤ n + 1, where ∂ denotes the bound-ary of the domain and qb denotes the waterflux across the boundary. Since we only considerbinary, ternary up to m-ary branching, the basisfunctions φi are in each case at most on m + 1finite elements different from zero and it is ad-vantageous to accomplish the integration elementwise per element e = [ze, ze+1], 1 ≤ e ≤ n:

∑e

∫ e

∂θ

∂tφi dz +

n+1∑j=1

ψ j

∑e

∫ e

Kdφi

dzdφ j

dzdz

= − qbφi

∣∣∣∣∂

+∑

e

∫ e

K cos(α)dφi

dzdz

−∑

e

∫ e

Sxφi dz, (29)

where the summation∑

e needs to be carriedout only for those elements e, that include theknot zi. To further evaluate the integrals (Eq. 29)for the term with time derivative the following

approximation, also denoted as ‘mass lumping’,is taken as definition for all 1 ≤ i ≤ n + 1 (vanGenuchten 1982):

dθi

dt

∫

φi dz ≈∫

∂θ

∂tφi dz (30)

Additionally it is assumed, that the hydraulic con-ductivity K and also the sink term Sx are con-tinuous, piecewise linear functions on , i.e. canbe represented similar to the function ψ by basisfunctions φi (1 ≤ i ≤ n + 1), with Ki(t) = K(t, zi),

and Si(t) = S(t, zi) representing the values at theknot i:

K(t, z) =n+1∑i=1

Ki(t) φi(z) (31)

Sx(t, z) =n+1∑i=1

Si(t) φi(z) (32)

Using these representations the integrals can beexplicitly calculated and, in case of an equidistantdecomposition of into the finite elements e

each of equal length �z, the following generallynon-linear equation system (in matrix notation)results:

Bdϑ

dt+ Aψ = d (33)

for the vectors ϑ , ψ and d and for the matrices Aand B, where we get in case of a mi-ary branching(1 ≤ mi ≤ m) at node zi for the mi + 1 intervals[zpi , zi], [zi, zci,1 ], . . . , [zi, zci,mi

] that contain zi:

Aij =∑

e

∫ e

Kdφi

dzdφ j

dzdz =

∑e

∫ e

∑k

Kkφkdφi

dzdφ j

dzdz

=∫

i

(Kpiφpi

dφi

dzdφ j

dz+ Kiφi

dφi

dzdφ j

dz

)dz +

mi∑k=1

∫ ci,k

(Kiφi

dφi

dzdφ j

dz+ Kci,kφci,k

dφi

dzdφ j

dz

)dz

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−(Kpi + Ki)/(2�z), for j = pi

(Kpi + (mi + 1)Ki + ∑mik=1 Kci,k)/(2�z) for j = i

−(Ki + Kci,1)/(2�z) for j = ci,1...

...

−(Ki + Kci,mi)/(2�z) for j = ci,mi

0 else

(34)

254 Plant Soil (2011) 341:233–256

Bij =∑

e

δij

∫ e

φi dz = 12�z +

mi∑k=1

12�z =

⎧⎨⎩

12(mi + 1)�z for i = j

0 for i �= j(35)

dϑi

dt:= dθi

dt:=

(∫

∂θ

∂tφi dz

)/(∫

φi dz)

‘mass lumping’ (36)

di := −gi − si assuming a zero flux boundary condition qb φi

∣∣∣∣∂

= 0 (37)

gi = −∑

e

∫ e

K cos(αi)dφi

dzdz = −

∑e

∫ e

∑j

K jφ j cos(αi)dφi

dzdz

= − cos(αi)

[∫ i

(Kpiφpi + Kiφi)dφi

dzdz +

mi∑k=1

∫ ci,k

(Kiφi + Kci,kφci,k)dφi

dzdz

]

= −12

cos(αi)

[Kpi − (mi − 1)Ki −

mi∑k=1

Kci,k

](38)

si =∑

e

∫ e

Sxφi dz =∫

e

∑j

S jφ jφidz =∫

i

(Spiφpi + Siφi)φi dz +mi∑

k=1

∫ ci,k

(Siφi + Sci,kφci,k)φidz

= �z6

[Spi + 2(mi + 1)Si +

mi∑k=1

Sci,k

](39)

This nonlinear system of Eq. 33 is then solvedin a standard way using a Newton–Picard itera-tion method (Celia et al. 1990) and elementaryGaussian elimination (Engeln-Müllges and Uhlig1996).

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