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FUSSIM2: brief description of the simulation model andapplication to fertigation scenarios

Marius Heinen

To cite this version:Marius Heinen. FUSSIM2: brief description of the simulation model and application to fertigationscenarios. Agronomie, EDP Sciences, 2001, 21 (4), pp.285-296. �10.1051/agro:2001124�. �hal-00886124�

Review article

FUSSIM2: brief description of the simulation modeland application to fertigation scenarios

Marius HEINEN*

Alterra, Green World Research, Department of Water and the Environment, PO Box 47, 6700 AA Wageningen, The Netherlands

(Received 11 July 2000; revised 29 January 2001; accepted 13 February 2001)

Abstract – Standard theories for water movement, solute transport and uptake by roots can be applied to closed, recirculating crop-ping systems, even when coarse substrates are used. Some of these systems, such as bedding systems or potting plants, are vulnerableto accumulation of solutes in the root zone, which may be harmful to crop development. A two-dimensional simulation model for theabove-mentioned processes is briefly described. This model can be used, amongst other possibilities, to search for good fertigationstrategies in order to minimise solute accumulation in the root zone. This modelling approach can largely reduce the number ofexperiments. Some simulated scenarios showing different distributions of the electrical conductivity in a layer of 15-cm coarse sandare presented. This study is not meant to be an extensive search for the best fertigation strategy but to demonstrate how the simula-tion model can help in finding alternative strategies.

electrical conductivity / fertigation / root uptake / sand bed / two-dimensional simulation model

Résumé – FUSSIM2 : brève description du modèle de simulation et application aux scenari de fertirrigation. Les théories cou-rantes pour le mouvement de l’eau, le transport de solutés et le prélèvement par les racines peuvent être appliquées étroitement à unsystème de culture avec recirculation, même lorsqu’un substrat grossier est utilisé. Un certain nombre de ces systèmes tels quelescultures en planches ou en pots sont vulnérables à l’accumulation de solutés dans la zone racinaire, ce qui peut être dommageablepour le développement de la culture. Un modèle bidimensionnel de simulation pour les processus ci-dessus mentionnés est décritbrièvement. Ce modèle peut être utilisé, parmi d’autres possibilités, pour rechercher de bonnes stratégies de fertirrigation de manièreà minimiser l’accumulation de solutés dans la zone racinaire. Cette approche basée sur une modélisation peut permettre de réduire lenombre d’expériences. Un certain nombre de scenari simulés montrant différentes distributions de la conductivité électrique dans unecouche de 15 cm de sable grossier sont présentés. Cette étude n’a pas l’intention d’être une recherche exhaustive de la meilleure stra-tégie de fertirrigation mais de démontrer comment le modèle de simulation peut aider à trouver les stratégies alternatives.

conductivité électrique / fertirrigation / prélèvement racinaire / lit de sable / modèle 2D

1. INTRODUCTION

Around 1990 the Dutch glasshouse horticultural sec-tor, although small in area, was regarded as a majorsource of environmental pollution. Large amounts ofwater with dissolved nutrients were applied to the crop,

more than required for optimal uptake. In the case ofnon-protected cultivation, the excess amount of waterand nutrients leached to the groundwater or open watersystems. De Willigen and van Noordwijk [14] and vanNoordwijk [37] stated that 40–80% of all nutrientsapplied to tomatoes and cucumber grown on rockwool

Agronomie 21 (2001) 285–296 285© INRA, EDP Sciences, 2001

Communicated by Christian Gary (Avignon, France)

* Correspondence and reprintsm.heinen@alterra.wag-ur.nl

M. Heinen286

without recirculation leached from the root zone.Sonneveld [31] estimated that the total leaching from theDutch glasshouse horticultural sector amounted to 6×106 kg⋅y–1 for N and K each. Protected cultivation, orrecirculation cropping systems offer possibilities toreduce the leaching to the environment. The design andmanagement of such systems requires further researchfocusing upon the interactions between substrate proper-ties, crop requirement, and fertigation. This can be donevia experimental (trial-and-error) research, or via a com-bination of modelling plus experimental research. Such amodel should incorporate existing knowledge of process-es occurring in the rooting medium, and helps to under-stand what is happening inside the root zone. Examplesof such combined modelling plus experimental researchcan be found in [17, 18, 26].

For some cropping systems, e.g. potting plants [26] orbedding systems [18], accumulation of solutes near thesurface will occur. Accumulation is due to two process-es: (1) water evaporates at the surface leaving the solutesbehind, and (2) the standard concentration of the nutrientsolution is larger than the uptake concentration – definedas the ratio of amount of solute taken up to the amount ofwater taken up – resulting in accumulation of solutes inthe zones of high root densities, i.e. in the top layer ofthe root zone. For example, the concentration of nitrogenin the nutrient solution for growing tomato in rockwoolwith recirculation is 12 mmol⋅L–1, while a typical uptakeconcentration for this situation is 9.6 mmol⋅L–1 [32]. Toohigh salt concentrations, reflected by electrical conduc-tivities EC, negatively influence growth [21, 22].Schwarz et al. [29] showed for a sand bed cropping sys-tem (see Sect. 2.1) that less roots were present in zoneswith high EC. These authors experienced EC values inthe range 2 dS⋅m–1 to 7 dS⋅m–1 in the root zone. Basedon a limited set of data, these authors concluded that foroptimal crop growth in their sand bed cropping system,drippers should be present next to each plant and that ahigh leaching fraction should be applied. The questionremains, however, if other and better methods exist toobtain more uniform solute distributions in the rootingzone. In other words, are alternative fertigation strategiespossible?

The aim of this paper is to present briefly a simulationmodel for describing water movement, solute transportand uptake of water and nutrients by roots in a root zone,and to show, using this simulation model, that differentfertigation strategies lead to different EC distributions. Itis not an in-depth study to obtain the best strategy, but itdemonstrates how a simulation model can help in defin-ing fertigation strategies. No experimental tests havebeen performed to validate the fertigation scenarios.Only one cropping system, i.e. the sand bed cropping

system of Heinen [18], is considered. However, the sameapproach was used by Otten [26] for potted plants usinga peat-perlite substrate, and Heinen [18] used the samemodel as presented below for a rockwool system.

2. MATERIALS AND METHODS

2.1. Sand bed cropping system

For crops with a high planting density small volumeroot zones (such as slabs) are not suitable. Therefore,proper closed recirculation cultivation systems requirelarge scale root zones. Heinen [18] made use of a sandbed system in which several growth conditions could beapplied and tested. Heinen [18] gathered experimentaldata from this system to validate the simulation modeldescribed below (example given at the end of Sect. 2.2.1). Figure 1a shows a schematic cross-sectionalview of the sand bed cropping system. This cross-section

Figure 1. Schematic representation of the sand bed croppingsystem (a) and an enlargement of half the root zone betweentwo drains (after [18]).

FUSSIM2 and fertigation scenarios 287

is perpendicular to the plant rows, drip lines and drains.By representing the true three-dimensional system in thistwo-dimensional (x, z) way, it is assumed that there areno gradients of processes and quantities along the plantrows, drip lines and drains. In what follows a unit lengthin the third direction, i.e. ∆y = 1, is assumed. In this waythe soil compartment has a volume, so that volumetricquantities have units of cm3 in stead of cm2. Due to sym-metry, only half the distance between two drains (Fig. 1b) needs to be considered for the use of the simu-lation model. Coarse sand – median particle size 0.6 mm– in a layer of 15 cm was used as the rooting medium.The drain distance was 80 cm. In Figure 1b the drain,with a radius of 2 cm, is located in the lower left corner.Three crop rows can be distinguished (Fig. 1b): row 1directly above the drain, row 3 midway between twodrains, and row 2 between rows 1 and 3. The volume ofthe rooting medium considered is 600 cm3 (40 × 15 × 1).The drip irrigation tubes run parallel to the plant rows,with a dripper (0.075 cm⋅min–1) next to the plants. Theirrigation tubes could be opened or closed with valves.The drip line between crop rows 2 and 3 was alwaysopen, while the drip line between rows 1 and 2 waseither open or closed. The nutrient solution was appliedwith the drip irrigation system (fertigation).

In this study lettuce was used as the crop. Heinen [18]showed that lettuce growth was well described by alogistic growth function. In this paper simulations weredone for the more or less linear growth phase of twelvedays in the middle of the six weeks growth period. Alsonutrient uptake during this phase was constant. Schwarzet al. [29] determined the final, two-dimensional rootdistribution in the sand bed system. Highest root lengthdensities were found near the planting position, and rootlength density decreased with increasing distance fromthe plant. No accumulation of roots at the bottom wasobserved. Assuming that root growth can also bedescribed by a logistic growth function (related to theabove-ground growth), two-dimensional root develop-ment was modelled in this study.

The composition of the nutrient solution was as fol-lows: NO3 21 mmol⋅L–1 (no NH4), K 11.2 mmol⋅L–1, P 4 mmol⋅L–1, Ca 4.6 mmol⋅L–1, Mg 1 mmol⋅L–1, S1.4 mmol⋅L–1, Cl 0.5 mmol⋅L–1, and Na 0.03 mmol⋅L–1.These concentrations were also used as the initial con-centrations in the root zone.

2.2. The FUSSIM2 simulation model

2.2.1. Water movement

Richards [28] combined the continuity equation andDarcy’s law [13] to obtain the general, governing flow

equation for incompressible water movement in non-deformable porous media. In two dimensions theRichards equation is given as

(1)

where θ is the volumetric water content (mL⋅cm–3), t isthe time (d), x is the horizontal co-ordinate (cm), z is thevertical co-ordinate oriented positive downwards (cm), Kis the hydraulic conductivity (cm⋅d–1), h is the pressurehead (cm), and Sw is a sink strength for water here con-sidered as root uptake of water (mL⋅cm–3⋅d–1). For con-venience, in this paper volume of water is designatedwith mL while volume of substrate is designated as cm3.Equation (1) is non-linear due to the non-linear constitu-tive relationships between h, θ and K. Here the waterretention characteristic θ(h) is given by the vanGenuchten function [36]

(2)

where S is the effective saturation (dimensionless), θr isthe residual θ, θs is θ at saturation, and α (cm–1), n(dimensionless) and m (dimensionless) are curve shapeparameters. The hydraulic conductivity characteristic isgiven by Mualem [23] (with m = 1 – 1/n)

, (3)

where Kr is the relative hydraulic conductivity (dimen-sionless), Ks is K at saturation (cm⋅d–1), and λ is a curveshape parameter (dimensionless). Topp [34] showedexperimentally that the θ(h) relationship is not unique,i.e. it is hysteretic, while the K(θ) has negligible hystere-sis. Hysteresis is described by supplying both the maindrying and the main wetting curves, and intermediatescanning curves are modelled with the modified, depen-dent domain model of Mualem [24]. The main dryingand main wetting curves are both described by equation(2) with, except for the α parameter, the same parame-ters.

Equation (1) with equations (2) and (3) is solvednumerically using the “control volume” finite elementmethod [19, 27], in which the mass-conservative conceptof Celia et al. [11] is used. The numerical solution givesfor any t the θ(x,z) and h(x,z) distributions. From thesethe water flux densities q between any locations can becomputed from Darcy’s law. The water results are thenbeing used to solve the solute transport equation.

( ) ( ) ( ) 2/111[ −−==

mm

sr SS

K

SKSK λ ]

,( ) ( ) ( )>

≤+=

−−=

01

01

1

h

hh

hhS mn

rs

rαθθ

θθ {

( ) ( ) ( ) ( ) ( )wS

z

K

z

hK

zx

hK

xt− ,

∂∂−

∂∂

∂∂+

∂

∂∂∂=

∂∂ θθθθθθ

M. Heinen288

The numerical solution can only be obtained whenproper initial and boundary conditions are supplied. Forthe sand bed system (Fig. 1b) the boundary conditionsare as follows:

• left and right: because of symmetry there is no flowacross these boundaries (qx = 0);

• bottom: impermeable (qz = 0), except at the drainlocation where free outflow occurs when at that loca-tion the substrate is saturated (h remains fixed then ath = 0); otherwise, no flow occurs across this part ofthe bottom (qz = 0);

• top: known water flux densities, either evaporation (qz < 0) of water and/or infiltration (qz > 0) of water atthe drip locations.

Heinen [18] parameterised and validated the modelfor water flow. The hydraulic parameters were deter-mined as: top layer, 0–5 cm: θr = 0.01573, θs = 0.326, α (drying) = 0.06069 cm–1, α (wetting) = 0.11745 cm–1,n = 4.98171, Ks = 1256 cm⋅d–1, and λ = 0.52581; bottomlayer, 5–15 cm: θr = 0.02311, θs = 0.311, α (drying) =0.05312 cm–1, α (wetting) = 0.09466 cm–1, n = 4.90919,Ks = 1256 cm⋅d–1, and λ = 0.52581. Figure 2 presentsexperimental and simulated time courses of θ at threedepths in the sand bed system, and of cumulativedrainage. The comparison between experimental andsimulated data is good.

2.2.2. Solute transport

As the sand is considered as inert, all solutes are dis-solved in the liquid phase and no adsorption is consid-ered (see [20] for a simple example of adsorption). Intwo dimensions the governing convection-dispersionequation for solute transport is then given by

, (4)

where c is the concentration (mol⋅L–1), q is the waterflux density (mL⋅cm–2⋅d–1), Ss is the sink strength of thesolute (mol⋅cm–3⋅d–1), and θD is the dispersion-diffusiontensor (cm2⋅d–1) given by [5, 30]

, (5)

where aL is the longitudinal dispersivity (cm), aT is thetransversal dispersivity (cm), |q| is the absolute value of

q (mL⋅cm–2⋅d–1), δij is the Kronecker delta (δij = 1 if i = j,δij = 0 if i ≠ j), τ(θ) is a tortuosity factor (dimensionless;see below), and D0 is the diffusion coefficient in freewater (cm2⋅d–1) and is available in the literature or can becomputed from [1]

(6)

where µ is the ionic mobility of the ion (cm2⋅s–1⋅V–1; see Tab. I), kB is the constant of Boltzmann (1.381×10–23 J⋅K–1), T is the absolute temperature (K), nis the valence of the ion (dimensionless), e is the electroncharge (1.6022×10–19 C), and 86400 is a time units con-version (s⋅d–1). The tortuosity factor τ(θ) is given by abroken-line relationship [3]

(7)( ) ( )<

+

≥+

=l

l

l

l

ff

ff

θθθ

θθ

θθθ

θτ21

21{

864000 neTk

D Bµ= ,

( ) ( ) zxjiDqq

aaaD ijji

TLijTij ,,,0 =+−+= δθτθδθq

q

szxzzxzxx

zx

Sx

cD

zz

cD

zz

cD

xx

cD

x

z

cq

x

cq

t

c

−∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

−∂

∂−=

∂∂

θθθθ

θ

Figure 2. Measured (symbols) and simulated (lines) timecourses of volumetric water content at three depths in the sandbed system (a) and of cumulative drainage from the sand bed (b).

FUSSIM2 and fertigation scenarios 289

where θl is the water content where the two linear linesintersect (mL⋅cm–3), f1 is the slope of the second linearline (dimensionless), and f2 is the intercept of the secondlinear line (dimensionless). For sand [3] we obtained θl =0.12, f1 = 1.58 and f2 = –0.17. Heinen [18] calibrated aLand aT using a measured breakthrough curve. A goodagreement was obtained for aL = 2.0 cm and aT = 0.2 cm(Fig. 3). The simulated results were hardly influenced bythe value of aT [18]. Therefore, the commonly used ratioaL/aT = 10 [5] was adopted.

The above equations can be applied to all individualions; in this study NO3, K, P, Ca, Mg, S, Cl and Na wereconsidered. However, in that case one cannot use a sepa-rate D0 (Tab. I) as this would lead to separation ofcharge. Therefore, a geometric averaged D0 was used.

The continuity equation (4) with equations (5)–(7) isexplicitly, numerically solved using the most recentknown output of the water parameters (θ and q). As forwater, proper initial and boundary conditions arerequired. For the sand bed system (Fig. 1b) the boundaryconditions are as follows:

• left and right: because of symmetry there is no flowacross these boundaries (qsx = 0);

• bottom: impermeable (qsz = 0), except at the drainlocation where solute drains with the water (qsz= qzc),with c the (simulated) solute concentration at the drainlocation;

• top: no flow of solutes across the top boundary,except known inflow during fertigation at the driplocations (qsz = qzc), with c being the concentration ofthe nutrient solution (as given at the end of Sect. 2.1).

2.2.3. Electrical conductivity

When multiple ions are present in the solution, theelectrical conductivity EC can be computed. Based onthe theory of ionic mobility and conductivity of the indi-vidual ions, the EC (dS⋅m–1) follows from [1, 2, 10]

(8)

where F is the Faraday constant (96485 C⋅mol–1), ms isthe number of solutes, ρs is the density of water(kg⋅dm–3), ci

* is the concentration in mol⋅kg–1, ni is thevalence of ion i , µ i is the ionic mobility of ion i(cm2⋅s–1⋅V–1; see Tab. I), and fi is the activity coefficientof ion i. The computation of f is given in Appendix 1.

2.2.4. Root uptake of water

De Willigen and van Noordwijk [14] considereduptake of water by a single root. Each root is surroundedby a cylinder of substrate with a radius R1 = (πLrv)

–0.5

(cm), where Lrv is the root length density (cm⋅cm–3).Flow of water from the bulk substrate towards the root surface q2 (mL⋅cm–2⋅d–1) must equal flow across the root surface q1 (mL⋅cm–2⋅d–1). Integrated over the

Σ=

=sm

iiiiis fncFEC

1

* µρ

Table I. Values for the diffusion coefficient in free water D0 (cm2⋅d–1), the ionic mobility µ (10–4 cm2⋅s–1⋅V–1) [2, 10] and the iondiameter d (10–10 m) [25] for ions present in nutrient solutions.

NO3– H2PO4

– SO42– Cl– NH4

+ K+ Ca2+ Mg2+ Na+

D0 1.64 0.759 0.92 1.76 1.64 1.69 0.685 0.610 1.15 µ 7.40 3.42 8.27 7.91 7.61 7.62 6.16 5.50 5.19 d 3.0 4.0 4.0 3.0 3.0 3.0 6.0 8.0 4.0

Figure 3. Measured (symbols) and simulated (lines) break-through curves for the sand bed system for several values ofthe longitudinal (aL) and transversal (aT) dispersivities. On thex-axis is the volume applied relative to the volume present inthe sand bed, and on the y-axis is the concentration of the draineffluent relative to the concentration of the irrigation water.

M. Heinen290

whole root system this uptake must equal the actual tran-spiration Ta (mL⋅cm–2⋅d–1). q1 is given by [12]

(9)

where ∆z is the thickness of the layer considered (cm),K1 is the hydraulic conductance of the root (cm⋅d–1), hrsis h at the root-substrate interface (cm), hr is the pressurehead inside the root (cm), σ is the solute reflection coef-ficient (dimensionless), ho,rs is the osmotic pressure headoutside the root (cm), and ho,r is the osmotic pressurehead inside the root (cm). q2 is based on the steady-ratedistribution of the matric flux potential around the rootaccording to

(10)

where ρ is the normalised radius defined as R1/R0 with R0 being the root radius (cm), φ– is the matric flux poten-tial in the bulk substrate surrounding the root (cm2⋅d–1)and is defined as the integral of the K(h) relationship(which can be obtained by substituting Eq. (2) into Eq. (3); it is computed for h of the bulk substrate sur-rounding the root as obtained from Eq. (1)), φrs is φ atthe root-substrate interface, and G0(ρ) is a geometryfunction defined as

(11)

The actual transpiration rate Ta is assumed to be a func-tion of the potential transpiration rate Tp and hr. Forexample, Campbell [8, 9] gave the following reductionfunction

(12)

The root water uptake model is solved by finding hr, hrsand Ta for given h so that ∫q1 = ∫q2 = Ta. It is assumedthat σ, ho,rs and hor are explicitly known; ho,rs is estimat-ed from the EC as ho,rs = –400EC [35], with EC beingthe EC of the bulk substrate surrounding the root. Onlyfor situations of limited uptake, the present nutrientuptake model (as described in next Sect. 2.2.5) yields theconcentration profile around the root. For non-limitingsituations (as occurred in the scenario calculations laterin this paper) the present model does not yield concen-tration profiles around the root. The concentration ofmajor ions around the root will be less than in the bulksubstrate, while for other salts (e.g. Cl and Na) it may be

higher. A priori, the concentrations and thus EC at theroot-substrate interface is unknown. Therefore, it wasestimated to be equal to that of the bulk substrate.

Due to the non-linear relationship between h and φand between Ta and hr the solution has to be found itera-tively. This model is solved for each location in the rootzone yielding the local value of Sw as required in equa-tion (1). Sw is given by the local q1 (=q2) times the sur-face area representative for this position. Sw is explicitlycomputed from the water status obtained at the previoustime step.

In this study data of Heinen [18] were used: R0 =0.017 cm, K1 = 3.6×10–6 cm⋅d–1, a = 10, and hr,1/2 =10000 cm, and σ = 0.9 and ho,r = 0 (worst case situation).Heinen [18] also measured water use and estimated howthis can be partitioned to Tp and Ep (see Sect. 2.3). Thetwo-dimensional Lrv distribution was obtained asdescribed at the end of Section 2.1.

2.2.5. Root uptake of nutrient

Nutrient uptake is assumed to be primarily dictated bythe demand of the crop. The actual uptake rate will beequal to the demand unless the substrate cannot re-sup-ply enough nutrient to the root surface by means of massflow and dispersion-diffusion. Based on a steady-rateapproximation of the concentration profile around a root,de Willigen and van Noordwijk [15, 16] gave an expres-sion for the maximum possible uptake rate Ssm (hereextended for non-zero-sink conditions)

(13)

where Ssm is the maximum possible uptake rate(mol⋅cm–2⋅d–1), D is the diffusion coefficient (cm2⋅d–1), c–

is the mean concentration in the bulk substrate surround-ing the root (mol⋅L–1) being c as used in equation (4),cmin is the minimum concentration at the root surface atwhich uptake can take place (mol⋅L–1), n is the dimen-sionless uptake of water given by

(14)

and G(ρ,n) is the geometry function given as

(15)( )( ) ( )( )

( )( )( )

( )( )−++−+

−+−+−+−

+= +

+

+ 142

11

12

112

12

112

1),(

22

42

22

22222

ν

ν

ν

νν

ρννρ

ρννρρ

νρρρ

ννρG

.

DLz

q

rvπν

∆−=

41 ,

( )( ) ( )( )−+

−−−∆=+

minrvsm ccG

DzLS11

1,

12

222

ρνρ

νρρπ

ν

,

.

1

2/1,

1

−

+=a

r

rpa h

hTT

][

.( )−

+−=1

ln431

21

)(2

42

0 ρρρρρG

,( )( ) ( )rsrv G

Lzq φφρ

ρπ −−∆=0

2

21

( )( ),rorsorrsrv hhhhKzLq ,,11 −+−∆= σ

FUSSIM2 and fertigation scenarios 291

For n = 0 (transport by diffusion only), G(ρ,n) reduces toG0(ρ) as given by equation (11).

As long as required uptake is less than Ssm the uptakeSs (in Eq. (4)) equals the demand, otherwise Ss =Ssm/∆z⋅Ssm is explicitly computed from the solute andwater status obtained at the previous time step. In thisstudy the zero-sink situation, i.e. cmin = 0, is used. As thedemand of the crop the experimentally obtained uptakefrom Heinen [18] were used (see also end of Sect. 2.1). The constant required uptake rates were: N 2.808 mmol⋅plant–1⋅d–1, K 1.656 mmol⋅plant–1⋅d–1, P 0.1728 mmol⋅plant–1⋅d–1, Ca 0.18 mmol⋅plant–1⋅d–1, Mg 0.1008 mmol⋅plant–1⋅d–1, S 0.05976mmol⋅plant–1⋅d–1, Cl 0.05976 mmol⋅plant–1⋅d–1, and Na 0.04968 mmol⋅plant–1⋅d–1.

2.3. Scenarios

Some scenario simulations of examples of possiblefertigation strategies are presented. It is not meant to bean extensive search for the best fertigation strategy but todemonstrate how the simulation model can help in find-ing alternative strategies. The following aspects are usedin this study.

• Number of drip points: 1 or 2; 1: at (x, z) = (0, 30 cm);2: at (x, z) = (0, 10 cm) and (x, z) = (0, 30 cm). A sin-gle drip line could mean savings in the purchase ofdrip lines, but the question is if this would yield unac-ceptable ECdistribution in the root zone;

• Irrigation (fertigation) when a certain threshold cumu-lated potential evapotranspiration (ΣETp) has beenreached: ΣETp = 0.2 cm or 1.6 cm. In the experimentsof Heinen [18] and Schwarz et al. [29] a thresholdvalue of about ΣETp = 0.2 cm was applied. However,Heinen [18] argued that more than 2.0 cm of water iseasily available for uptake. Therefore, a second largerthreshold value was chosen to see how this will affectthe ECdistribution;

• Two different “leaching fractions” (LF) are used (i.e.the ratio of amount of desired drainage and the totalamount of irrigation water): LF = 0.25 or 0.5. Thismeans that 1.33 or 2.0 times ΣETp is to be supplied,respectively. Higher LF is advised in order to removeaccumulated salts, while a lower LF is desired forpractical reasons (volume of drainage vessel, volumeof drainage water that needs to be disinfected);

• The concentration of the irrigation water is either thestandard concentration (C, as given at the end ofSect.2.1), or one-third of this standard concentration(C/3). It is common practice to have C at a high levelso that nutrients never become depleted. As explainedin the introduction this results in accumulation of

nutrients in the root zone. Therefore, it was decided todetermine what happens if the concentration of theirrigation water is less than C;

• The number of times a nutrient solution is supplied asirrigation water: (a) each irrigation nutrient solutionwith concentration C is applied, or (b) each third irri-gation nutrient solution with concentration C isapplied while during the other two irrigations purewater (C = 0) is supplied.

In Table II the scenarios are listed that are considered.Scenario 1 is comparable to the experimental conditionsof Heinen [18]. Simulations were carried out for a periodof twelve days, during the more or less linear growthphase of the crop (see Sect. 2.1). For completeness, thepotential transpiration and potential evaporation for thesetwelve days were as follows: potential transpiration:0.211, 0.198, 0.253, 0.195, 0.120, 0.120, 0.082, 0.082,0.075, 0.075, 0.087 and 0.087 cm⋅d–1; potential evapora-tion: 0.023, 0.026, 0.041, 0.038, 0.035, 0.035, 0.035,0.035, 0.047, 0.047, 0.079 and 0.079 cm⋅d–1.Evapotranspiration was assumed to occur during 14 hours of the day.

The main focus is on the EC. Bernstein [6] gave yieldreductions for lettuce as a function of the EC in the satu-ration extract (ECe): 10%, 25% and 50% yield reductionat ECe = 2, 3, and 5 dS⋅m–1, respectively. ECe can beestimated from EC as ECe = EC/1.6 [33]. Presumably,yield reduction will occur at uniform ECe. The initialconcentration (see Sect. 2.1) results in EC = 2.76 dS⋅m–1

Table II. Summary of scenarios considered: number of drip-pers, threshold evapotranspiration value ΣETp (cm), leachingfraction LF, frequency of use of nutrient solution as irrigationwater (1/1: each irrigation a nutrient solution is used; 1/3: eachthird irrigation a nutrient solution is used while pure water isused the other two times), and the concentration C of the nutri-ent solution (C: standard nutrient solution; C/3: one-third of theconcentration of the standard nutrient solution).

Scenario Number of ΣETp LF Frequency Concentrationnumber drippers

1 1 0.2 0.25 1/1 C 2 1 0.2 0.5 1/1 C 3 1 1.6 0.25 1/1 C 4 1 1.6 0.5 1/1 C 5 2 0.2 0.25 1/1 C 6 2 0.2 0.25 1/3 C 7 1 0.2 0.25 1/3 C 8 2 0.2 0.5 1/3 C 9 1 0.2 0.25 1/1 C/3

10 1 0.2 0.5 1/1 C/3

M. Heinen292

and ECe = 1.725 dS⋅m–1. This is presumably the optimalEC for lettuce. When the average EC in the root zoneincreases yield reduction is likely to occur, e.g. due toosmotic hindering of water uptake. When the averageEC in the root zone decreases, nutrient uptake maybecome limited.

3. RESULTS AND DISCUSSION

The results that will be shown are: visualisation offlow during an irrigation event (Sect. 3.1; scenario 1),time course of concentration at selected locations duringan irrigation event (Sect.3.2; scenario 1), and for fertiga-tion strategies 1, 2, 3, 5, 6, and 7 spatial EC distributionsat one moment – i.e. just before the start of a fertigationevent – are shown (Sect.3.3). The results for scenarios 4,8, 9 and 10 are very much similar to those of scenarios 3,6, 7 and 7, respectively, and, therefore, are not shown.Because the actual spatial EC distributions change intime, a snap-shot of the distribution itself does not saytoo much. Paper, however, does not allow one to showpatterns of change in time. Therefore, an animation pro-gram is available that shows how the spatial EC distribu-tion changes in time (Sect.3.4).

3.1. Visualisation of flow during a fertigation event

Figure 4 shows for four typical moments during anirrigation event how water flows through the root zone.The arrows represent the vectors constructed from thelocal average qx and qz data. The patterns for a solute arealmost identical, indicating that mass transport is domi-nating.

Prior to the start of irrigation (Fig. 4a) water movespredominantly upwards. The small horizontal compo-nents are due to movement towards regions with thehighest Lrv. Most roots are in the upper layers and belowthe plants, and at the surface water evaporates. Thus,upward movement of water results.

When irrigation starts water moves vertically down-wards below the drip location. Figure 4b shows the pat-tern just before drainage starts. Drainage starts onlysome time after the start of irrigation because first re-supply of water must occur before the bottom becomessaturated.

When at the drain location the substrate becomes satu-rated, drainage occurs. A saturated zone develops at thebottom (shaded area in Fig. 4c). Figure 4c pertains to thetime just before irrigation halts. The main flow path isfrom the dripper location towards the drain. Note thatthere is almost no flow occurring in the upper left corner

(“dead” corner), so that there only minor refreshment ofnutrient solution will occur.

After irrigation stops, drainage will continue for awhile until the substrate at the drain location becomesunsaturated. In Figure 4d the situation is shown justbefore drainage will stop; there is still a small saturatedzone present (shaded area in Fig. 4d). After drainagestops, a similar situation to that in Figure 4a will develop.

3.2. Time course of the concentration of a solute during a fertigation event

In Figure 5 the simulated c distribution of a solute,e.g. N, at times t = 0 min and t = 160 min (comparable to

Figure 4. Simulated net water flux density distribution at fourtimes during an irrigation event: (a) just before start of irriga-tion I with evapotranspiration ET (t = 0 min), (b) during I andET and just before start of drainage D (t = 142 min), (c) justbefore I halts (with I, ET, D) (t = 160 min), and (d) just beforeD stops (with ET, D) (t = 170 min). The sizes of arrows repre-sent relative sizes, and parts of the arrows outside the root zoneare not shown for clarity reasons. In (b) and (c) an arrow withsize of 10 x-axis units represents q = 0.2 mL⋅cm–2⋅min–1; thearrows in (a) and (d) are enlarged 500 and 5 times, respective-ly. The shaded areas in (c) and (d) represent the extent of thesaturated zone.

FUSSIM2 and fertigation scenarios 293

the times in Figs. 4a and 4c) are shown. As discussed inthe previous section, refreshment of the upper left corneris not complete. In Figure 6 time courses of c at four dif-ferent (x, z) locations inside the root zone are shown.Large fluctuations of c occur near the surface of the rootzone, while these are small in the middle and near bot-

tom of the root zone. In Figure 6 the horizontal line at c= 15 mmol⋅L–1 represents the concentration of the irriga-tion water.

Accumulation of solutes near the surface are due totwo processes: (1) water evaporates at the surface leav-ing the solutes behind, and (2) the standard concentrationof the nutrient solution is larger than the uptake concen-tration – defined as the ratio of amount of solute takenup and amount of water taken up – resulting in accumu-lation of solutes in the zones of high Lrv, i.e. in the toplayer of the root zone.

3.3. Snap-shots of EC distributions at different fertigation strategies

In Figure 7 snap-shot distributions of simulated ECare shown for scenarios 1, 2, 3, 5, 6, and 7 (counter-clockwise starting from the upper left corner). Scenario 1is similar to the experimental conditions used by Heinen[18] and will serve here as the reference situation.Applying a twice as large LF (scenario 2) does not resultin a clear improvement with respect to scenario 1.Increasing LF results in an extension of the period ofboth irrigation and drainage, and from Figure 4c andFigure 4d we know that water predominantly movesfrom the dripper towards the drain, so that the upper leftcorner is hardly refreshed.

Because ample water is available in the root zone,there is no need to irrigate frequently [18]. When wewait eight times longer between irrigation events, the ECat the surface increases drastically (scenario 3). This

Figure 5. Simulated concentration distribution of a solute(mmol⋅L–1) in the root zone: (a) just before start of irrigation (t = 0 min), and (b) just before irrigation stops (t = 160 min).

Figure 6. Simulated time courses of the concen-tration of a solute (mmol⋅L–1) at four locations(x, z) in the root zone during a period of tendays. The in-set shows the position of the fourlocations in the rectangular sand bed system.

M. Heinen294

implies that in many cases for the sand bed system thestart of irrigation is determined by EC control rather thanby a shortage of water.

The presence of a second dripper results in a some-what different EC distribution (scenario 5) than for sce-nario 1. But this is not yet a big improvement.

When only every third irrigation is carried out with anutrient solution (scenarios 6 and 7) then the EC at thesurface does not increase too much. The average EC inthe root zone (in Fig. 7) is close to the EC of the nutrientsolution. From Figure 7 it could be concluded that sce-nario 6 is the best scenario. However, Figure 7 is just asnap-shot. If a series of snap-shots (using the animationprogram of Sect. 3.4) is considered, then it follows thatscenario 6 yields sometimes too low EC values in theroot zone. Probably a better strategy would be to use anutrient solution as irrigation water every other irrigationevent instead of each third irrigation event.

Scenarios 1 and 5 are comparable to the experimentalconditions used by Schwarz et al. [29], except thatSchwarz et al. [29] used LF = 0.5 for the case with twodrippers. These authors concluded that for optimal cropgrowth drippers should be present next to each plant andthat a high LF should be applied. This first conclusion isin agreement with the findings in this paper, while theeffect of LF in this paper (comparing scenarios 1 and 2)is only minor.

3.4. Animation software

For a proper interpretation of the effect of a scenarioon the EC distribution in the root zone, the spatial ECdistribution as a function of time must be considered.This is hard to show on paper. Therefore, an animationsoftware program is available to view the developmentof the spatial EC distributions. A demonstration versionof this program is available from the author1 whichincludes the data for all scenarios considered in thispaper.

4. SUMMARY AND CONCLUSIONS

Standard theories for water movement, solute trans-port and uptake by roots of water and nutrients can alsobe used for coarse porous media, used as root zone sub-strates in glasshouse horticulture. In this paper a simula-tion model (FUSSIM2) was briefly described. Watermovement is modelled by the Richards equation with theconstitutive relationships between h, θ and K given bythe van Genuchten and Mualem functions. The Richardsequation is implicitly solved using the control volumefinite element method. The convective-dispersive solutetransport equation is explicitly solved. No adsorption isconsidered as the substrate used in this study is assumedto be inert. Other chemical (equilibrium) processes are

Figure 7. Simulated EC distribu-tions for scenarios 1, 2, 3, 5, 6,and 7 (see Tab. II). The patternsrepresent a snap-shot just beforethe start of an irrigation. The rec-tangular shape represents the rootzone of the sand bed with thedrain located in the lower left cor-ner, and the drippers are indicatedwith dots at the substrate surface.

1. download via http://www.alterra.wageningen-ur.nl/animat/animat.zip;via E-mail: m.heinen@alterra.wag-ur.nl

FUSSIM2 and fertigation scenarios 295

disregarded, such as precipitation, changes in water den-sity due to increased concentrations, and no micro-scalesolute distributions around the root are simulated. Micro-scale distributions are only temporarily used for thedetermination of the maximum possible uptake of nutri-ents by a model presented by de Willigen and vanNoordwijk. Nutrient uptake will result in a change of thebulk concentration in the substrate surrounding the root.Water uptake is more or less treated analogously tonutrient uptake.

With the help of such a simulation model the effectsof different management strategies can be simulated, sothat in a later stage promising strategies can be tested inpractice. Here, the study is restricted to presenting someillustrative examples of alternative fertigation strategiesfor a system used by Heinen [18]. Although in the rootzone of this system there is ample water available, fre-quent irrigations are required to control the EC distribu-tion in the root zone. For this system and the scenariosstudied it is concluded that a dense grid of drippers, e.g.one dripper per plant, is required. Refreshment of thesolution in so-called dead corners in the root goes slow-ly, so that irrigating with high leaching fractions did notresult in a more homogeneous ECdistribution.

Although this simulation has some (chemical) short-comings, it can serve well to get ideas of the dynamics ofwater and nutrients in the root zone.

Acknowledgements: This research was carried out withinthe DLO research program 256 “Nutrient and WaterManagement in Protected Glasshouse Cultivation” financed bythe Direction of Science and Knowledge Transfer of the DutchMinistry of Agriculture, Nature Management and Fisheries.

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APPENDIX 1. ON THE ACTIVITY COEFFICIENT

The activity coefficient f depends on the total soluteconcentration or ionic strength I (mol⋅kg–1) of the solu-tion according to

(A-1)

The relation f(I) is given by the Debye-Hückel theory

(A-21)

with

(A-3a,b)

where di is the effective diameter of ion i (m; see Tab. I),Na is the number of Avogadro (6.022 × 1023 mol–1), e isthe electron charge (1.6022 × 10–19 C), ρs is the densityof water (kg⋅m–3), ε0 is dielectric permittivity of vacuum(8.8542 × 10–12 C⋅V–1⋅m–1), εr is the relative dielectricpermittivity of water (dimensionless), Rg is the universalgas constant (8.3144 J⋅mol–1⋅K–1), and T is the absolutetemperature (K). For water ε = 78.54 (–) and ρs = 1000 kg⋅m–3, so that at T = 298 K, A = 0.51(mol⋅kg–1)0.5 and B = 3.3 × 109 (mol⋅kg–1)0.5⋅m–1. Forvery dilute solutions (I < 0.005 mol⋅kg–1 according to[4], or I < 0.01 mol⋅kg–1 according to [10]) or in case di = 0, the reduced Debye-Hückel formula is

(A-4)

When for all ions the same ion diameter of di = 3 × 10–10 mis used, the Günterberg approximation of the Debye-Hückel formula is

(A-5)

For high values of I (I > 0.1 mol⋅kg–1), the Davies exten-sion of the Debye-Hückel theory is [4]

(A-6)( ) InI

IAnf i

ii

22

1.01

log ++

−= .

( )I

IAnf ii +

−=1

log2

.

( ) IAnf ii2log −= .

( ) ( )10ln8

25.1

0

32

TR

eNA

gr

sa

εεπ

ρ=

TR

eNB

gr

sa

εε

ρ

0

2=, and ,

( )IBd

IAnf

i

ii

+−=

1log

2

Σ=

=sm

iii ncI

1

2*5.0 .