SIMGRO 7.2.25 Theory and model mlementaton P.E.V. van Walsum A.A. Veldhuzen P. Groenendjk Alterra-reort 93. Alterra, Green World Research, Wagenngen
ABSTRACT P.E.V. van Walsum, A.A. Veldhuzen, P. Groenendjk. 204. SIMGRO 7.2., Theory and model mlementaton. Wagenngen, Alterra. Alterra-Reort 93.. 93. SIMGRO s a modellng framework that can facltate the nvestgaton of varous knds of regonal water management roblems. The framework connects to dverse hydrologc model codes lke MODFLOW for groundwater and SOBEK-CF for surface water smulaton. A number of n-house codes make SIMGRO esecally sutable for modellng stuatons wth shallow groundwater levels n relatvely flat areas, lke n delta regons. The metamodel MetaSWAP s based on a quas steady-state soluton of the Rchards equaton. Its mlementaton can nclude cro growth smulaton wth WOFOST, wth feedback to the hydrologc arameters lke root zone deth and leaf area ndex. The SIMGRO ackage also ncludes a smlfed code for the smulaton of surface water rocesses. A secal feature of the framework s the oton for model coulng va a shared state varable that s alternately udated by the connected models. Keywords: ntegrated water management, mechanstc, dstrbuted, dynamc, saturatedunsaturated flow, surface water, dranage; sem-mlct ISSN 566-797 204 Alterra, P.O. Box 47, NL-6700 AA Wagenngen (The Netherlands). Phone: +3 37 474700; fax: +3 37 49000; e-mal: nfo.alterra@wur.nl No art of ths ublcaton may be reroduced or ublshed n any form or by any means, or stored n a data base or retreval system, wthout the wrtten ermsson of Alterra. Alterra assumes no lablty for any losses resultng from the use of ths document.
Contents Introducton 5 2 Plant/sol-atmoshere nteractons 3 2. Prectaton 3 2.2 Evaotransraton 4 2.4. Multle cro coeffcent method 4 2.4.2 Actual transraton 6 2.4.3 Actual ondng and sol evaoraton 7 2.4.4 Actual nterceton evaoraton from canoy 8 2.5 Data summary and coulng to WOFOST 2 3 Sol water 23 3. Introducton 23 3.2 Pondng water and nfltraton 24 3.2. Theory 24 3.2.2 Imlementaton 25 3.3 Unsaturated flow 3 3.3. Theory 3 3.3.2 Imlementaton 38 3.4 Coulng to regonal groundwater flow 43 3.4. Theory 43 3.4.2 Imlementaton 45 3.5 Data summary 5 4 Dranage 52 4. Introducton 52 4.2 Theory 52 4.3 Model mlementaton 58 4.4 Data summary 62 5 Surface water 63 5. Introducton 63 5.2 Theory 63 5.3 Model mlementaton 64 5.3. Schematzaton and hydraulc metafunctons for channel flow 64 5.3.2 Dynamcs of channel flow 67 5.3.3 Water on the sol surface 7 5.4 Data summary 74
6 Water management 75 6. Introducton 75 6.2 Land use 75 6.2. Urban areas 75 6.2.2 Srnkled cros 76 6.3 Surface water 78 6.3. Wers 78 6.3.2 Surface water suly lnks 79 6.3.3 Dscharge ums 80 References 8 Aendx A Steady-state unsaturated flow smulatons 85 Aendx B Cro coeffcents as a functon of LAI 87
Suly caacty Aqutard surface water Surface water system Subsurface rrgaton or dranage Root zone Callary rse or ercolaton Phreat c level Introducton st Aquf er Aqutard nd 2 Aqufer Node ont fnte element grd Hydrologcal base All regonal hydrologc models suffer the defcency that the hydrologc cycle s not fully smulated n all ts rocesses and dmensons. In the frst lace ths concerns the satal lmtaton of a model, whch necesstates the makng of assumtons wth resect to external boundary condtons. Furthermore, most models cover only art of the rocesses wthn a regon. The remedy s to then ntroduce nternal boundary condtons. But these condtons are more often than not a gross schematzaton of realty, lackng essental feedback mechansms. To some degree, that even ales to the commonly used assumton that the rectaton and evaoratve demand of the atmoshere can be seen as external boundary condtons: the larger the basn, the less ths assumton s vald. For comng to grs wth many ssues of ntegrated water management, t s necessary to h a model code that covers the whole (regonal) system, ncludng lant-atmoshere nteractons, sol water, groundwater and surface water. SIMGRO (a dated acronym of SIMulaton of GROundwater, Querner and Van Bakel, 989, Veldhuzen et al., 998) was develoed for that urose. The name SIMGRO was formerly used for referrng to an ntegrated model code, ncludng submodels for the comartments and rocesses as shown n Fgure. Now t s used n the meanng of a modellng framework. Ths framework has been connected to a number of n-house comonents, but also has ossbltes for coulng to other codes. The n-house comonents are ndcated n the form of chater numbers n the overall scheme of regonal water flows gven n Fgure 2. They can be groued as follows: - an SVAT-model that s commonly referred to as MetaSWAP, coverng the lantatmoshere nteractons and sol water; - a smlfed surface water metamodel; - a dranage ackage, for smulatng groundwater dranage wth fast feedback from surface water. Fgure Schematzaton n SIMGRO of the hydrologc system by ntegraton of saturated zone, unsaturated zone and surface water (Querner and van Bakel, 989). Alterra-reort 93._V7.2.25 5 2
6 srnklng 2 rectaton Interceton storage nterceto n 6 srnklng net rectaton/srnklng 3 Deresson storage 3 surface runoff evaotransraton nfltraton Sol mosture storage 3 ercolaton callary rse Groundwater Groundwater storage (mult layer) storage (mult layer) lateral flow subsurface rrgaton dranage 4 Surface water storage 5 surface water flow 5 Groundwater Groundwater storage (mult layer) storage Surface water storage Fgure 2 Schematzaton of regonal water flows by means of transmsson lnks and storage elements. The flows and lnks that can be modelled wth n-house comonents are ndcated wth - references. For groundwater (MODFLOW) and surface water (SOBEK-CF) there are connectors for external models. The current ossbltes for coulng to other codes are: - MODFLOW for groundwater; - SOBEK-CF for surface water. SOBEK-CF can be used n combnaton wth the smlfed surface water metamodel, wth the latter mlemented for the ustream waterways and SOBEK-CF for the larger downstream ones. The current software mlementaton stll nvolves coulngs to secfc external codes. But the way n whch ths has been done makes t a relatvely small ste to arrve at a comonent-based mlementaton that allows connectons to any code that s OenMI-comlant (.e. comlant wth Oen Modellng Interface, HarmonIT, 2005). Ths software mgraton to an OenMI-comlant verson of SIMGRO s lanned for the near future. Alterra-reort 93._V7.2.25 6
Fgure 3 Examle of how the satal schematsatons of an ntegrated model can be constructed. The bottom layer nvolves the unts obtaned from an overlay of the land use and sol mas. The next layer reresents the cells of the groundwater model, followed by the subcatchments of the surface water model n the next layer. The to layer shows how the schematsatons h been combned. If there s just one colour n a groundwater cell, then there s just a sngle SVAT-unt lnked to t. If there are more colours, then two or more SVAT unts are connected. Connectons between models are n SIMGRO defned n terms of mang tables, as s done when usng OenMI technology. Each record of such a table contans a ar of dentfers that defne an d-based lnk between two models. In Fgure 3 an examle s gven of how a model schematsaton can be bult and then coded n the form of these dbased lnks. In the case that a grd cell n the to layer of the fgure contans more than one colour, then two or more SVAT unts are connected to a sngle groundwater cell. Ths s the so-called N: coulng. In the shown schematsaton there s also an N: coulng between the SVAT s and the subcatchments of the surface water model. SIMGRO has two otons for connectons between models: - a flux-based lnk, q-lnk; - a head-based lnk, h-lnk. Flux-based lnks are the tye generally used n combnaton wth OenMI technology. In such a lnk between Model A and Model B, the frst comutes an exchange flux. If ths flux s based on a sngle cycle of model nteracton, then the scheme s called exlct. Wth ths method two tyes of related roblems can occur: - the flux causes Model B to run dry, causng a water balance error f Model A stll assumes that the water s avalable; - the flux causes a large dsturbance of the head smulaton n Model B, whch n the next cycle leads to destablzaton of the model combnaton, nvolvng oscllatory behavour of the flux. Alterra-reort 93._V7.2.25 7
In order to avod water balance errors, a flux-based lnk can be made to nclude a ut and a get oeraton. In the ut-ste the Model A sends a demand to Model B n the form of a flux. That model then checks whether the water s avalable, and determnes the demand realsaton, whch s then returned n a ut by B. The realsaton s then cked u va a get by Model A. The destablzaton roblem can be solved by ncludng an teraton cycle. Such a cycle wll also remedy any water balance errors, f allowed to fully converge. In that case the coulng method s called mlct. In SIMGRO, the ut/get cycle s used for the flux-based coulng between MetaSWAP and a surface water model. In the case of a N: coulng, the demands are frst totalzed. On return of the realsaton, ths s then dstrbuted by SIMGRO over the unts that laced the demand (Fgure 4). The teratve coulng s only used for a flux-based lnk between MetaSWAP and MODFLOW. Sol water MetaSWAP column Tye h-lnk: Tye q-lnk: er teraton er teraton Recharge Bottom flux Srnklng Srnklng demand ut demand ut Storage coeffcent Per t sw Dranage Infltraton demand ut Per t sw Head Demand realzatons Per t sw Runoff/runon Srnklng demand ut Per teraton cycle Head SIMGRO Demand totalzer Realzaton dstrbutor Groundwater MODFLOW MODFLOW: Dranage Infltraton Surface water otons: SurfW SOBEK-CF Fgure 4 Modules wth relatonshs and otons. MetaSWAP s the n-house SVAT (Sol-Vegetaton-Atmoshere Transfer) model of SIMGRO. SurfW s the n-house surface water metamodel. It can be used n combnaton wth the hydraulc model SOBEK-CF. The lnks nvolve the uttng of demands and the rely n the form of a demand realzaton. The left half of the scheme has a tme ste of the groundwater model, t gw, the rght half of the fast rocesses, t sw A so-called h-lnk s a secal feature of the SIMGRO framework, nvolvng the head as a shared state varable of two connected models. The shared varable s alternately udated by Model A and Model B. Ths s done n the followng two substes: - an exlct subste by Model A; - an mlct subste by Model B, usng nformaton obtaned n the frst subste. Alterra-reort 93._V7.2.25 8
In the second subste the assumton s that the fluxes of the frst subste reman unchanged. Under ths assumton, the second subste for Model B only affects Model A n terms of ts water storage. In order to take ths nto account n the second subste, the frst subste also rovdes the relatonsh between the shared state varable and the storage n Model A. The storage relatonsh s ncluded n the set of equatons that s to be solved n the mlct scheme of Model B n the second subste. In an exlct scheme the fluxes between the models are urely determned by the udate of Model A. But n the h-lnk scheme the total fluxes between the connectng models are determned from budget calculatons after Model B has been udated. Ths s what dstngushes the method from a urely exlct one, and justfes callng t semmlct. For the MetaSWAP-MODFLOW coulng the h-lnk s the most used method, because t makes the hreatc level avalable to MODFLOW: the h-lnk can be seen as a q-lnk between the hreatc level and the MODFLOW head, wth a flow resstance that aroaches zero. The MODFLOW head can then be used as the hreatc level n the MODFLOW dranage ackages (RIV, DRN, etc.). For the MetaSWAP-SurfW modellng the h-lnk s used for the runoff/runon smulaton, because the low resstance of ths flow rocess would otherwse requre very small tme stes or a comutatonally demandng teratve mlct scheme to avod destablzaton of the model combnaton. As can be seen from the overvew n Fgure 4, SIMGRO has searate tme stes for the fast rocesses (Δt s ) and for the slow ones (Δt g ). The fast rocesses nclude lant/atmoshere nteractons, flow over the sol surface, dranage wth surface water feedback, and channel flow. Sol water and groundwater flow are modelled as slow rocesses. Tycal tme stes used n the current modellng ractce are Δt s = hour and Δt g = day. In the flow chart wth an overvew of the SIMGRO modellng cycles gven n Fgure 5, the fast tme ste s resent n the to half, and the slow ste n the bottom half. The teraton cycle around the groundwater model s needed because most groundwater models can not handle a nonlnear storage relatonsh. (By contrast, all surface water models h such a faclty.). In the software mlementaton of the coulng scheme, the recharge s assed to the groundwater model along wth the storage coeffcent, for each teraton cycle. But the recharge s n fact only udated once er groundwater tme ste by the MetaSWAP model tself. In the case of a q-lnk, however, the bottom flux s udated for every teraton cycle. Alterra-reort 93._V7.2.25 9
Start Intalsaton of t s = t g t s = t s + t s Demand realsaton To system rocesses er SVAT-unt: - atmosherc nteractons - nterceton, throughfall - cro water utake from root zone - ondng, nfltraton, surface runoff - dranage/nfltraton wth sw feedback Surface water rocesses er trajectory: - levels and flows - sules to to system No Totalzatons over gw tme ste - recharge of sol/groundwater -dranage/nfltraton t s = t g + t g? t g = t g + t g Unsaturated rocesses er SVAT-unt: - ressure heads - ercolaton/callary rse No Formulate groundwater rocesses er cell: - resdual sol/groundwater recharge - tme-varant storage coeffcent - dranage wth gw feedback No Groundwater rocesses er cell: - heads and flows - sules to to system Convergence? Fnalze to system: - water content of sol - storage on sol surface Sto Yes t g = t end? Fgure 5 Overvew of modellng cycles. The tme varable of the fast to-system rocesses t s s synchronzed wth t g of the groundwater rocesses at the begnnng of each groundwater tme ste. The resdual sol/groundwater recharge s the remanng column recharge after dscountng the water needed for udatng the ressure heads. Water management s not shown. Alterra-reort 93._V7.2.25 0
The smulaton of (ossble) changes n water management s the goal of most studes. Gven ts crucal mortance for the ractcal relevance of the model, the management otons are descrbed n a searate secton ( 6). But of course the dstncton between what s consdered water management and what natural functonng deends on the ersectve that one has. Here any tye of land-use nvolvng vegetated sol s consdered to be natural, even though t s some form of agrculture. In the case of land-use, the term water management s reserved for d areas, nvolvng some form of nterference n the cycle, e.g. urban areas. The layout of man-made dtches s descrbed as f t were art of the natural system, whereas t s of course not. In the case of surface water the term management s reserved for the manulaton of the water flows and levels through the use of structures. In the followng sectons a descrton s gven of the used concetualzatons, followed by more detaled descrtons about the way they h been mlemented. To ad the reader n quckly locatng the descrtons, the overvew n Fgure 2 has been furnshed wth hyer-lnks to the subsectons. Alterra-reort 93._V7.2.25
Alterra-reort 93._V7.2.25 2
2 Plant/sol-atmoshere nteractons 2. Prectaton It s well known that the measurement of rectaton usually contans systematc errors due to wnd-effects. The sze of the error s usually n the order of a few % (<5%), but should be accounted for. The natural rectaton can be augmented by srnklng. The total (gross) rectaton s then gven by: P g ( s, n t) P( t) P ( t) () where: P g (t) = gross rectaton rate at tme t (m 3 m -2 d - ) P(t) = natural rectaton rate at tme t (m 3 m -2 d - ) P s,n (t) = net srnklng rate at tme t (Eq. 06) (m 3 m -2 d - ) In the above relatonsh (and n others for the rocesses at or around the sol surface) we use the dmenson [L 3 L -2 ] when the value er-unt-of-area s meant. After multlcaton by the relevant area fracton the satal mean s obtaned n [L]. Not n all cases ths dstncton s relevant; we then use the latter notaton. The tme deendency of the rectaton (and other varables reresentng atmosherc boundary condtons) s raged over the used smulaton nterval. For ths the nterval t s of the 'fast' rocesses s used. The use of tme-raged values s ndcated va the suerscrt ''. So Eq. then becomes: P g P P (2) g s,n where: P g = gross rectaton rate, tme raged for nterval (m 3 m -2 d - ) P = natural rectaton rate, tme raged for the nterval (m 3 m -2 d - ) P = net srnklng rate, tme raged for the nterval (m 3 m -2 d - ) s,n Incomng rectaton (and/or srnklng water) ether falls drectly on the ground surface as free through fall or s nterceted by the vegetaton canoy. Part of that nterceton drectly slashes off at the mact. The art that does not, s subsequently stored on the canoy and evaorate, or else reach the ground surface as stem flow or dros from the les. The nterceton of ranfall s modelled as a dffuse vertcal rocess, wth no clear-cut dvson between vegetaton cover and bare sol. Prectaton that comes nto contact wth the canoy - and that does not drectly slash off - s denoted as the gross nterceted rectaton. It s modelled as a smle fracton of the gross rectaton: P, g ( t) c Pg ( t) (3) Alterra-reort 93._V7.2.25 3
where P,g (t) s the gross nterceted rectaton as a functon of tme (m d - ), P g (t) s the gross rectaton (ranfall lus srnklng rrgaton) (m d - ), and c s the nterceton fracton. Currently t s set equal to the lght fracton that s nterceted. The smulaton of the nterceton evaoraton from the canoy s descrbed below. 2.2 Evaotransraton 2.4. Multle cro coeffcent method An evaotransraton smulaton method should be accomaned by a method that rovdes accetable arameters for the ntended user communty. In our case we mostly use the cro coeffcents gven by Feddes (987) for agrcultural forms of land use. Those coeffcents, however, are ntended for an all n one evaotransraton smulaton method that does not account for varable feedbacks from cro growth. Those feedbacks affect each of the evaotransraton comonents n a secfc manner. We therefore model evaotransraton as four searate and artly nterdeendent terms: cro transraton, canoy nterceton evaoraton, sol evaoraton and ondng evaoraton. The arameters are calbrated usng Feddes (987). As startng ont, we used an adated form of the dual cro coeffcent aroach gven by Wrght (982) and Allen et al. (2005): T E ( K K ) ET (4) s, cb ew 0 where T s the otental cro transraton (m d - ), E s, s the otental sol evaoraton (m d - ), K cb s the basal cro coeffcent (-), K ew s the evaoraton coeffcent of a wet bare sol that s artly shelded by vegetaton (-), and ET 0 s the reference cro evaotransraton (m d - ). The reference cro evaotransraton can be calculated wth the Penman-Monteth method (Wrght, 982; Allen et al., 998) or wth the smlfed Makknk equaton, descrbed by De Brun (98, 987): s K ET0 0. 65 s L v c where s s the sloe of the vaour ressure curve, γ s the sychrometrc constant (kpa K - ), K s the ncomng global radaton (W m -2 ), L v s the latent heat of vaorzaton of water (J kg - ), and c s a converson factor (m d - / kg m -2 s - ). De Brun (987) gves the followng reasons for choosng the Makknk equaton: a. Its behavour s very smlar to that of the Penman formula; b. It s remarkably smle: t requres only ar temerature and global radaton as nut, c. Under dry condtons Makknk s formula erforms better. (5) Alterra-reort 93._V7.2.25 4
Sol evaoraton also ncludes the evaoraton from sol that s artly shelded by a canoy. The bare sol s assumed to be dffuse, meanng that we do not dstngush between sol that s covered by vegetaton and sol that s not, n vew of the varable angle of ncomng radaton. We assume that the net radaton nsde the canoy decreases accordng to an exonental extncton functon of the leaf area ndex (LAI) and that the sol heat flux can be neglected (Goudraan, 977; Belmans, 983): K ew LAI gr e K (6) ew00 where gr s the extncton coeffcent for solar radaton (-), LAI the leaf area ndex, and K ew00 the evaoraton coeffcent of a wet sol recevng full radaton. Rtche (972) and Feddes (978) used κ gr = 0.39 for common cros. More recent aroaches estmate κ gr as the roduct of the dmensonless extncton coeffcent for dffuse vsble lght, κ df, whch vares wth cro tye from 0.4 to., and the extncton coeffcent for drect vsble lght, κ dr : κ κ gr df κ dr (7) If the sol s nundated due to ondng, then the ondng evaoraton s of course the domnant rocess and the bare sol evaoraton s suressed. The otental evaoraton rate of ondng water s calculated wth a cro coeffcent n a fashon that s analogous to that for bare sol. The calculaton of the actual ondng and sol evaoraton s gven n 2.4.3. A wet canoy leads to an ncreased evaotransraton due to nterceton evaoraton, whch lowers resstance for the mosture flow from the canoy to the atmoshere. Wrght (982) accounts for ths lower resstance by adjustng the cro and bare sol coeffcents. We model these effects searately, va the mentoned sol evaoraton method and by exlctly modellng canoy nterceton evaoraton. Interceton evaoraton from a vegetaton canoy s not senstve to sol mosture condtons. Ths makes t relevant to model t searately from the transraton. The smulaton of the actual nterceton evaoraton s descrbed n 2.4.4. Evaoraton from a wet canoy s assumed to be a domnant rocess: as long as t s actve, the transraton s assumed to be zero. Ths assumton s based on the notons that evaoraton has frst choce for usng the energy flux and that due to a hgher ar mosture content the transraton s suressed. To model canoy nterceton evaoraton and ondng evaoraton as domnant rocesses, Eq. 4 s exanded to: T E E E [ K (W ) K K (W ) K ] ET (8), s, ond, cb frac, w ew frac,ond ond 0 where E, s the otental canoy evaoraton rate (m d - ), and K w s the evaoraton coeffcent of a wet canoy (-), wth K w > K cb, E ond, s the otental ondng evaoraton rate (m d - ), K ond s the cro coeffcent of ondng evaoraton (-), W frac, s the tme fracton that nterceton evaoraton s actve (-) and W frac,ond s the tme fracton that ondng evaoraton s actve (-). The value of K ond s reduced for the fracton of lght that reaches the ondng water, n a smlar fashon as s done for a wet sol (converson of K ew00 to K ew ). Alterra-reort 93._V7.2.25 5
The used tme fracton that nterceton evaoraton s actve s comuted as E Wfrac, (9) E,a, where E,a s the actual canoy nterceton evaoraton rate (m d - ), raged over the smulaton tme nterval (m d - ), and E, s the tme-raged otental value (m d - ). The tme fracton that ondng evaoraton s actve s comuted as: W frac, ond E E ond, a ond, (0) where E s the actual ondng evaoraton rate (m d - ), tme raged over the ond, a smulaton tme nterval (m d - ) and E s the tme-raged otental value (m d - ). ond, 2.4.2 Actual transraton In the current model mlementaton the otental canoy evaotransraton s unformly dvded over the root zone deth, as s done n Feddes et al. (978). Prasad (988) would be an alternatve: nstead of a unform dstrbuton the rootng ntensty decreases lnearly wth the deth. For the effect of lmtng sol mosture condtons on the evaotransraton the functon defned by Feddes et al. (978) s used: T α T () a E where: T a = actual evaotransraton rate (m d - ) T = otental canoy transraton rate (m d - ) α E = sol mosture reducton factor (-) T low.0 E low T low α E(-) TE hgh 4 3l 3h 2 (m) Fgure 6 Reducton coeffcent for root water utake, α E, as a functon of sol water ressure head and otental transraton rate T (see T hgh and T low ) (after Feddes et al., 978) 0.0 Alterra-reort 93._V7.2.25 6
Water utake by roots s zero when the sol water ressure head s below 4, whch s assumed to be the wltng ont. Sol water ressure head 3 s called the reducton ont. In between the wltng ont and the reducton ont the evaotransraton rate s lnearly reduced. The reducton ont deends on the otental evaotransraton (see T low and T hgh ). Between ressure heads 2 and 3 the evaotransraton s at ts maxmum ('otental'). Alcaton of the reducton functon for < 2 to the mean root zone ressure head (see 3) would yeld erroneous results due to the non-lnear form of the functon. For ths reason the root zone ressure head s frst downscaled to 0 equal-szed sublayers of the root zone. The functon s then aled to the searate sublayers. As a result of oxygen defcency n the root zone, water utake s hamered for many cros between 2 and. The reducton coeffcent s zero for >. But n we do not aly the reducton for > 2 to the searate sublayers of the root zone lke was roosed by Feddes et al. (978). Alcaton to the searate sublayers s consdered to be nonrealstc because t assumes that the vegetaton has no means to comensate for oxygen defcency n art of the root zone. Esecally under wet condtons wth thck root zones (of trees) the extracton attern wll vary accordng to the season: unform over the whole root zone n stuatons wth a low groundwater level, and only n the uer arts of the root zone n stuatons wth a shallow groundwater level. For we use the ressure head n the to art of the root zone (to 0%) for comutng the reducton coeffcent that s then aled to all of the layers of the root zone n a unform manner. 2.4.3 Actual ondng and sol evaoraton Actual ondng evaoraton s modelled wth a smle bucket model that evaorates at the otental rate (K ond ET 0 ) as long as there s water on the sol surface. The dynamcs of ondng water are descrbed n 3.2. The bare sol evaoraton s a rocess nvolvng the very thn uer sol crust layer. When the layer dres out, a self-mulchng effect occurs, lmtng further loss of mosture. Modellng of the sol water transort at such a detaled scale s beyond the scoe of a metamodel. Instead, we use the method of Boesten and Stroosnjder (986). The bare sol evaoraton for a tme ste s derved from a smle model for the cumulatve values of actual and otental evaoraton snce the start of the dryng erod. The model s centred around the followng assumed relatonsh: ΣE a ΣE a ΣE 2 ΣE for for ΣE ΣE 2 2 2 2 (2) where: β 2 emrcal sol arameter, wth a default value of 0.054 (m ½ ) ΣE a sum of actual evaoraton snce the start of the dryng erod (m) ΣE sum of otental evaoraton snce the start of the dryng erod (m) Alterra-reort 93._V7.2.25 7
The arameter β 2 determnes the length of the otental evaoraton erod, as well as the sloe of the ΣE a versus (ΣE relatonsh n the sol lmtng stage. If there s not an excess n ranfall (P n < E ), the udate of ΣE s done wth: j j ( ΣE ) (ΣE ) ( E Pn ) (3) where P n s the net rectaton that s equal to the drng rate D of the nterceton model, where suerscrt j s for the tme level. Subsequently (ΣE a ) j+ s calculated from (ΣE ) j+ wth Eq. 2 and E a for the tme ste s calculated wth: j j E a Pn ( ΣEa ) (ΣEa ) (4) If there s an excess n ranfall (P n > E ) t can be assumed that the actual evaoraton wll equal the otental value: Ea E (5) The excess ranfall artly (or fully) reverses the dryng out of the sol. To take ths nto account the excess ranfall s subtracted from the cumulatve ΣE a : j j ( ΣEa ) (ΣEa ) ( Pn E ) (6) If ths udate yelds a negatve value for ΣE a t s set to zero, sgnallng a new start for the dryng rocess. In order to mantan consstency n the model the new ΣE s obtaned from the nverse form of Eq. 2, wth theσe a term n the rght-hand sde of the rearranged equaton. 2.4.4 Actual nterceton evaoraton from canoy The nterceton water balance s smulated wth: ds ( t) dt P ( t) E ( t) D ( t) (7),g where S (t) s the water stored n the nterceton reservor (m), E s the canoy evaoraton rate (m d - ), and D (t) s the canoy drng rate (m d - ). It s essental to use an analytcal soluton to the above equaton, to avod numercal errors caused by tme dscretzaton. That s then avoded f the tme ste s not larger than the gaugng nterval of the rectaton and the evaoratve demand. Drng from the canoy starts when the canoy becomes saturated, any excess water drectly drs through. The storage of water n the canoy s assumed to be bounded: S t) S ( t) (8) (, ca Alterra-reort 93._V7.2.25 8
where S (t) s the water stored n the nterceton reservor (m), and S,ca (t) s storage caacty of the nterceton reservor (m). The canoy evaoraton rate s assumed to deend on the canoy saturaton and the otental evaoraton rate as: S ( t) E ( t) [ f,mn ( f,mn ) ] E, ( t), for S ( t) 0 S (9) E ( t) 0, for S ( t) 0,ca where E, (t) s the otental evaoraton rate of a wet canoy (m d - ), and f,mn s the startu fracton of the canoy evaoraton reducton factor (-). If f,mn = 0 the relatonsh s lnear lke n Rutter (97). For a value of.0 the evaoraton rate equals the otental rate as long as there s water n the reservor. The method to obtan the otental rate E, (t) s gven n Eq. (8). In the soluton scheme, the tme varaton of the rectaton rate s handled as a stewse functon. Ths s denoted by the suerscrt. The non-lnear canoy evaoraton rate relatonsh (Eq. 9) requres searate treatment for f,mn equal to unty. Case : Dscrete deendence of evaoraton on canoy saturaton For f,mn equal unty (Eq. 9a), the followng water balance can be made: S S j j S j ( P mn( S,g j, S E j,ca, ) ; ) t S j max( S j,0) (20) j where S (m) s the water stored n the nterceton reservor at tme level j, P,g, (m d - ) s the gross nterceted rectaton rate, tme-raged rate over nterval t, and t (d) s the length of the smulaton tme nterval. In case that the reservor does not become full, the evaoraton follows from the balance: E j j S S )/ ( t P,g (2) where E (m d - ) s the tme raged canoy evaoraton rate. In ths case the drng rate s zero. In case the reservor becomes full, the smulated evaoraton rate equals the otental value durng the whole nterval owng to the dscrete relatonsh of the canoy evaoraton (otental value for a non-zero storage) and the use of tme-raged values: E E (22), The net nterceted rectaton and the dr to the sol surface can now be found from: P,n D ( S P j g S - P j,n )/ t E where P,n (m d - ) s the tme raged net nterceted rectaton and D s tme raged the drng rate. The equatons above can also be used n case the otental canoy evaoraton rate equals zero. (23) Alterra-reort 93._V7.2.25 9
Case 2: Sem-contnuous deendence of evaoraton rate on canoy saturaton If f,mn n Eq. 9a s less than unty, there s a sem-contnuous deendence of the evaoraton rate on the canoy saturaton. After nsertng the exresson for the evaoraton rate as gven n Eq. 9a nto Eq. 7 and rearrangng, we get for an unsaturated canoy (S <S,ca, D =0): ds ( t) E, βs ( t), wth ( f,mn ) and P,g f,mn E, (24) dt S Assumng that β s non-zero (see the above Case 2 for β = 0) and that the nterceton reservor does not become full (S < S,ca, D c =0), the soluton to the dfferental equaton for S (t) s gven by: j j t S ( S )e (25) S j max( S j,0) If the reservor does not become full (S <S,ca ) the evaoraton s now gven by Eq. 2. If S (Eq. 25) yelds a value >S,ca then we frst determne the tme at whch ths haens by solvng for t ca ( t ca t): j t ca S,ca ( S )e (26) j tca ln[( S ) /( S,ca )] where t ca s the tme to fll the reservor durng whch there s no drng from the canoy. The canoy evaoraton can thus be found from a balance lke the one gven n Eq. 2; for the remanng art of the nterval ( t - t ca ) the evaoraton rate s equal to the otental value. So the total canoy evaoraton can be found from:,ca E j j ( ) S tca P,g t tca E, [ S t ] (27) The drng rate then follows from Eq. 23. Alterra-reort 93._V7.2.25 20
2.5 Data summary and coulng to WOFOST For the smulaton erod, the tme seres nformaton of the meteorologcal condtons should be avalable n the form of a ste-functon wth a tme nterval of t s. That concerns both the rectaton and the data needed for smulatng the evaotransraton. For the Makknk method only the values of the reference evaotransraton are needed. If the Penman-Monteth method s used the followng data are needed: - radaton; - temerature; - humdty; - wnd seed. These data can be suled for more than one gaugng staton n the regon. In that case the relevant staton should be secfed for each of the SVAT-unts of the sol water /groundwater model. A method for ntroducng extra satal varaton s to secfy calbraton factors for each of the SVAT s. It s also ossble to secfy all of the tmedeendent data va grds coverng the satal extent of the model area. If the data are only avalable on a daly bass, then some form of downscalng should frst be aled for smulatng the nterceton evaoraton wth some degree of accuracy. Per SVAT-unt the domnant land-use should be secfed. The arameters of the transraton reducton functon are requred for all methods, as are the lght-extncton coeffcents. For modellng the nfluence of the vegetaton on the evaotransraton arameters two methods are avalable: - smle vegetaton model, wth the cro develoment (and deendent arameters) gven er day of a calendar year n a fle; - advanced vegetaton model, wth the dynamc smulaton of the cro develoment wth WOFOST (Van Keulen and Wolf, 986; Van Deen et al.,989; Sut et al. 994). WOFOST and MetaSWAP (Fgure 7) communcate on a daly bass. Cro water stress causes the leaf ores to close artly or comletely, thus mnmzng mosture loss. Ths also reduces the CO 2 assmlaton and cro growth. To model ths reducton, WOFOST requres the relatve transraton as nut,.e. T a /T. Durng erods of nterceton evaoraton the transraton s assumed to be zero; then the relatve transraton s set equal to the α rw -coeffcent for a low value of T. WOFOST returns the root zone deth and LAI; the latter s subsequently used to calculate: - the fracton of radaton reachng the sol surface; - the transraton cro coeffcent; - the ranfall nterceton fracton, the canoy nterceton storage caacty, and the nterceton evaoraton cro coeffcent. The LAI affects the nterceton n varous ways: Alterra-reort 93._V7.2.25 2
- the fracton of ranfall that drectly reaches the sol surface wthout assng through the nterceton reservor s set equal to the fracton of radaton reachng the sol surface; - the nterceton storage caacty of a canoy s assumed to be lnearly deendent on LAI accordng to S,ca = s,ca LAI, where s,ca s the caacty er unt of LAI (mm LAI - ); - the nterceton evaoraton cro coeffcent K w s set to.2 tmes K cb ; ths factor has been set slghtly lower than the rato between the coeffcents for oen water and short grass (.25). The manner n whch WOFOST s used for dervng relatonshs based on LAI s descrbed n Aendx B. Also descrbed s the way n whch the tables for the smle vegetaton model are derved from a 30-year run wth the couled MetaSWAP- WOFOST models. Fgure 7 Coulng MetaSWAP-WOFOST as mlemented n the SIMGRO modellng framework Alterra-reort 93._V7.2.25 22
3 Sol water 3. Introducton As ndcated n the ntroductory chater, water on the sol surface has a multle dentty. Here we descrbe the manner n whch t takes art n the sol column rocesses, ncludng the nfluence of atmosherc nteractons descrbed n the revous secton. For the model formulaton we use a control box 0 that extends uwards from the sol surface. The subsurface sol water dynamcs are descrbed usng two control boxes: for the root zone, the shallow subsol and the dee subsol (Fgure 8). We descrbe the unsaturated zone model n detal n 3.3. The water balances and smulatons are made at the aggregate scale of the control boxes, but the model n fact has contnuous mosture rofles on the background, meanng that at any desred moment the detaled head and mosture content rofles can be made avalable f needs be. Ths dstngushes the aroach from models based on lumng of the vertcal doman. q run P n E T q nf Root zone: control box q,r q c,r Subsol: control box 2 q,2 q c,2 Phreatc level Fgure 8 Flows n the sol water model. Exlanaton of symbols: P n net rectaton; q nf nfltraton; q run - surface runoff; E evaoraton of bare sol; T transraton; q ercolaton; q c callary rse. Alterra-reort 93._V7.2.25 23
3.2 Pondng water and nfltraton 3.2. Theory Water stored on the sol surface has dverse functons and nteractons. Here we confne ourselves to ts role as a reservor and as an ntermedate source for sulyng nfltraton water. The storage on the sol surface s schematzed nto two fractons: - mcro -storage; - macro -storage. The mcrostorage s formed by small deressons n the sol surface at sub-grd scale. When after an nundaton the land falls dry the water n the mcro-storage s retaned. The water n macro-storage can freely move over the sol surface, restrcted only by natural barrers and constructons n the surface water channel network. The schematzaton of the surface water system can be extended to the sol surface, obtaned from an GIS-analyss of the sol surface toograhy. In that case the barrers consst of the saddle onts n the sol surface landscae. Pondng water that s macro -storage on the sol surface has a multle dentty. The stes nvolved n the multle dentty concet are descrbed n 5.3.3 and llustrated n Fgure 29 through Fgure 33. The modellng of nfltraton s notorously dffcult for varous reasons, of whch we menton: - the role of crust-formng at the sol surface; - the comlexty of the rewettng rocess of a dred out sol; - the role of sub-grd scale flow rocesses over the sol surface; water that becomes runoff at one sot can after all nfltrate n a small deresson just a few metres away. We do not attemt to model these asects va a formulaton n terms of a dfferental equaton. Instead we follow a ragmatc aroach nvolvng a mnmum of arameters that stll allow enough freedom for the modeller to control the smulaton along broad lnes. A recent addton (204) s the oton of usng the Curve Number method (USDA SCS, 972). The amount of water n the ondng reservor ( ondng storage ) reacts drectly to ncomng rectaton and exchange wth the surface water model; thus we model the dynamcs of the ondng reservor wth the tme ste of the fast rocesses. The use of a dfferent tme stes for the fast and slow rocesses can lead to numercal artefacts n the model f no recautons are taken. These recautons can take on two dfferent forms: - sreadng of the effect of the revous udate of the groundwater model on the new seres of fast tme stes; - antcaton on the behavour of the model durng the next udate of the groundwater model n reacton to fluxes and state varables of the fast rocesses. Alterra-reort 93._V7.2.25 24
3.2.2 Imlementaton At the begnnng of the frst tme cycle for the fast rocesses wthn a new groundwater tme ste (see Fgure 5) the groundwater above sol surface (f resent) s converted nto ondng water: S j0 ond max{ 0; h j0 h ss } (28) where: j 0 S = ondng storage at the begnnng of the frst fast tme cycle wthn the ond groundwater tme ste (m) h j=0 = groundwater level at tme level j =0 of the fast rocesses,.e. at the end of the revous groundwater tme ste (m) h ss = elevaton of the sol surface (m) The stuaton wth uward seeage to the sol surface s handled n a secal manner. The above equaton s then mlemented n a modfed form, n whch the nfluence of the uward seeage s sread out over all of the fast tme stes wthn the consdered groundwater tme ste. Wthout ths modfcaton the model would roduce artfcal eaks n the surface runoff durng the frst fast tme ste. Infltraton can of course only occur f there s water avalable n the form of net rectaton and/or water stored from the recedng erod. We gve nfltraton of ondng water rorty over evaoraton. So n order to determne the water avalablty for nfltraton the amount of water n the ondng reservor s frst udated wth: S' j+ ond = S j ond + P n Δt s (29) where: j S ond j ond = ondng storage at tme level j of fast rocesses (m) S' = ondng storage after ntermedate udate for net rectaton (m) P n = net rectaton rate, tme raged (m d - ) Δt s = tme ste of fast tme cycle (d) The uer bound for the avalablty of nfltraton water s then gven by: q nf, max,ond S / Δt (30) j ' ond s where q nf,max,ond s maxmum avalable nfltraton rate from ondng water (m d - ) For stuatons wth free ercolaton to the groundwater we use a smle relatonsh for determnng the maxmum nfltraton rate at the sol surface q nf,max,s : q j + nf, max,s qnf,bas + S' ond/ = c to,down (3) where: q = maxmum nfltraton rate at sol surface (m d - ) nf,max,s Alterra-reort 93._V7.2.25 25
q nf,bas = basc nfltraton rate for stuatons wthout ondng (m d - ) c to,down = vertcal flow resstance n to of sol (d) The nfluence of the frst term can be cancelled by secfyng hgh values of q nf,bas n the nut fle. The use of the second term nvolvng c to,down s otonal. In fully saturated condtons and wth a secfed c to,down the nfltraton rate s found from the head dfference between ondng water and groundwater dvded by c to,down. In stuatons wth the head of the groundwater hgher than that of the ondng water the resstance for uward flow s used c to,u. If the resstance s not secfed, then we assume freedom of movement of water through the sol surface n stuatons that are fully saturated. Infltraton can also be hamered by a bottleneck further down n the system,.e. n the sol water and/or groundwater system. Let us frst consder the case wth the groundwater level below the bottom of the root zone. The maxmum amount of storage sace avalable for nfltraton s then estmated wth: S q j =0 nf, max,r = Sr,sat - Sr +( Ea + Ta - nf,max,r = S nf,max,r /Δt s I + q,max )Δt g (32) where: S = avalable storage sace for nfltraton, based on root zone balance (m) nf,max,r S = saturated water content of root zone (m) r,sat j 0 S = water content of root zone at the begnnng of the frst fast tme cycle, r.e. the content at the end of the revous groundwater tme ste (m) E = sol evaoraton of revous fast tme stes,tme-raged over Δt g (m d - ) a T a = transraton of revous fast tme stes, tme-raged over Δt g (m d - ) I = nfltraton durng revous fast tme stes, tme raged (m d - ) q,max = maxmum ossble ercolaton rate (>0) to the groundwater (m d - ) q = maxmum nfltraton rate, based on root zone balance (m) nf,max, r Δt g = tme ste of sol water / groundwater submodels (d) The above exresson takes nto account that evaoraton and/or transraton from the root zone makes storage sace avalable for nfltraton water. The values gven n the above exresson are stll n the rocess of beng udated, durng each fast tme cycle. The tme-raged values are used because t s the volume that determnes the nfltraton sace. The maxmum ossble ercolaton rate to the groundwater s derved from the sol hyscal metafuncton q m gven n 3.3.2.. The ercolaton can be hamered by the stuaton n the subsol. For nvestgatng ths otental bottleneck a balance s made for the whole column: S q nf,max,c nf,max,c S ( E r,sat a S T max{ S j0 r a nf,max,c S I 2,sat / t s S Q,0} j0 2 sgw Q d G ) t g (33) Alterra-reort 93._V7.2.25 26
where: S = maxmum storage sace for nfltraton, based on column balance (m) nf,max,c S 2,sat = saturated water content of shallow subsol (box 2) (m) j 0 S = water content of root zone at the begnnng of frst fast tme cycle (m) 2 sgw Q = extracton for srnklng from groundwater (from frst layer) (m d - ) Q d = dranage / nfltraton (+ = to the column) (m d - ) G = regonal groundwater flow (+=to the column), last known value (m d - ) q = maxmum nfltraton rate, based on column balance (m) nf,max,c The nfltraton s then found from: q mn( q ; q ; q ; q ) (34) nf = nf,max,ond nf,max,s nf,max,r nf,max, c The next udate of the water stored n the ondng reservor s made wth: S" S' q t j ond j ond nf s (35) where: j S" = ondng storage water after udate for nfltraton (m) ond j The value of S" ond s used for determnng the maxmum avalablty of ondng evaoraton water. Subsequently the amount of water n storage s udated wth: S" E t j j ' ond S" ond ond s (36) where: S "' j = ondng storage after udate for evaoraton (m) ond E ond = ondng evaoraton rate (m d - ) The way the runoff s comuted deends on the used combnaton of modellng otons. - the choce whether or not the Curve Number method (USDA SCS, 972) s used; - the choce of method for flow over the sol surface: the resstance method or the ntegrated method nvolvng zero resstance. The Curve Number can only be aled n combnaton wth daly ranfall data. It has been mlemented followng Netsch et al. (20), ncludng the nfluence of the sloe wth the amendment of Wllams (995). In our techncal mlementaton, the runoff that s smulated wth the Curve Number method s converted to extra ondng by reducng the nfltraton. The ondng s then converted to runoff wth the algorthm gven further below. However, deendng on the method used for the flow over the sol surface, t s ossble that not all ondng water becomes runoff durng the consdered tme ste. Ths ondng water wll be a source for nfltraton durng the next tme ste, and thus cometes wth the rectaton for the avalable nfltraton sace. It (after converson to a rate by dvdng through the tme ste of fast rocesses) s therefore added to the gross ranfall for calculatng the Curve Number runoff of the next tme ste. Alterra-reort 93._V7.2.25 27
The Curve Number method s based on daly ranfall, but the runoff tself usually occurs durng a short tme san. So t s not realstc to smulate ondng evaoraton as f the runoff and ondng occur evenly sread over the whole day. Therefore the method s aled after the ondng evaoraton has been smulated. The converson of nfltraton to ondng water takes nto account that the nfltraton can already be lmted by condtons wth near saturaton: S j j "' + + ond= S"' ond +mn( qrun.cn, qnf )Δt s (37) where: q = runoff smulated wth the Curve Number Method (m d - ) run,cn For the flow over the sol surface, the resstance method has the advantage that the tmedelay due to the flow rocess s modelled to some extent. The dsadvantage s that f a low resstance has been secfed the smulaton can become unstable f used n combnaton wth a surface water model, and oscllatons can result. The ntegrated method s uncondtonally stable; the drawback s the mmedacy of transfer of water, whch can be unrealstc. In both methods, runoff can only take lace f the ondng level s above a threshold value. The storage caacty below ths threshold s termed the caacty of mcro ondng storage. If the resstance method s used, then for stuatons where there s no nfluence of the surface water level, the runoff s smly comuted wth: q run =[ S ond, mcro,ca S"' j+ ond ]/ c run (38) where: q run = runoff of the SVAT-unt (+ = to the column) (m d - ) S ond,mcro,ca = caacty of mcro-storage (m) = runoff resstance (d) c run For stuatons wth the surface water level above the runoff threshold, the threshold s relaced by the surface water level (relatve to the sol surface). For stuatons wth runon and the ondng level below the threshold, the ondng level s relaced by the elevaton of the threshold. If the ntegrated method used for the runoff smulaton (see 5.3.3 and Fgure 29 through Fgure 33), the ondng water n the macro-storage s transferred to the surface water model. For ths the storage s frst decomosed nto the two tyes of storage wth: j j S" ' ond, mcro mn{ S"' ond; Sond, mcro, ca} (39) S "' j ond, macro S"' j ond S" ' j ond, mcro where: S " j = ondng mcro-storage (m) ' ond, mcro S = ondng macro-storage (m) "' j ond, macro Alterra-reort 93._V7.2.25 28
The surface water model then comutes hydraulc heads for the channel trajectores and the assocated subcatchments; each subcatchment conssts of one or more SVAT-unts of the to system model. After comletng the smulaton of surface water flow rocesses the newly determned levels are lnked to the resectve SVAT-unts wthn the subcatchments. The modfed amount of water n macro-storage s comuted by: j S" " ond, macro max{ hs ( hss Sond, mcro, ca ); 0} (40) where: j S = ondng water n macro-storage after surface water udate (m) h s h ss "" ond, macro = surface water level (m) = elevaton of the sol surface (m) and the total storage n the ondng reservor s comuted as S "" (4) j j j ond S"' ond, mcro S" " ond, macro where: j S = ondng water after udate for surface water flow (m) "" ond By comarng ths storage to the amount that was transferred to the surface water model, the net runon/runoff of the SVAT-unt can be calculated (>0 for runon): q j j run S"" onds"' ond] / [ t s (42) where: q run = runon/runoff of the SVAT-unt (m d - ) If the consdered fast tme ste nvolves an ntermedate udate wthn the (longer) j groundwater tme ste, the fnal ondng storage S ond s set equal to S "" j ond. The smulaton then contnues wth the next cycle startng from Eq. 29. If the smulaton has arrved at the end of the last fast tme cycle t s followed by a fnalzaton rocedure. In the case that there s hydraulc contact wth the groundwater the ondng storage s comared to that at the begnnng. The dfference s added as recharge to the groundwater model: q jn j0 recha, ond S"" ondsond) / q recha,ond j 0 ond ( t g where: = recharge rate of the vsble groundwater, tme raged over Δt g (m) S = ondng storage at begnnng of groundwater tme ste (m) S" " = ondng storage at end of last fast tme ste n, n = Δt g /Δt s (m) j n ond (43) j n After the groundwater model udate has been erformed, the ondng storage ond s fnalzed by alyng Eq. 28 for the new groundwater level. At ths ont the nfltraton rate can also be fnalzed on the bass of a water balance analyss nvolvng the fnal value of G, the regonal groundwater flow. S Alterra-reort 93._V7.2.25 29
Alterra-reort 93._V7.2.25 30
3.3 Unsaturated flow 3.3. Theory The model schematzaton assumes that unsaturated flow takes lace wthn arallel vertcal columns, wth each column connectng to a smulaton unt of a groundwater model. The hreatc surface acts as a movng boundary between the flow domans of the sol column models and the groundwater model. All lateral exchanges are assumed to take lace n the saturated zone. The man dea of the modelng method for the (unsteady) unsaturated flow s to use steady-state solutons to Rchards equaton as buldng blocks of a dynamc model, a so-called quas steady-state model. The arorate buldng blocks are for each tme level selected on the bass of water balances at the aggregate scale of control volumes for the root zone and the subsol. Put n mathematcal terms, the artal dfferental equaton for the unsteady flow (Rchards equaton) s relaced by two ordnary dfferental equatons: one for the varatons n the vertcal column (usng the steady-state form of the flow equaton) and the other for the varatons n tme (usng a water balance at aggregate scale). The descrbed method s a radcal redesgn of the one resented by de Laat (980) and Wesselng (957). A valdaton of the current method s gven n Van Walsum and Groenendjk (2008). 3.3.. Steady states For one-dmensonal flow n an unsaturated sol wth root water extracton, the steadystate form of the flow equaton can be wrtten as: d dψ K( ψ) τ ψ z z h z z (, ) 0, 0, (44) d d subject to the boundary condtons ψ ( h) 0 (45) dψ K( ψ) dz z 0 q(0) where z = elevaton coordnate, taken ostvely uward (zero at the sol surface) (m) h = groundwater elevaton (m) ψ = ressure head (m) K(ψ) = hydraulc conductvty as a functon of ressure head (m d - ) q(0) = flux densty at the sol surface, taken ostvely uward (m d - ) τ(ψ,z) = deth- and head-deendent extracton term for root water utake (m 3 m -3 d - ) A steady-state rofle s obtaned by secfyng the conductvty arameters of each dstngushed sol layer and by solvng Eq. 44 subject to mosed values for the groundwater elevaton h (Eq. 45), the otental flux densty at the sol surface q ot (0) (Eq. 46), and the otental total root water utake rate T ot of the root zone. A flexble root (46) Alterra-reort 93._V7.2.25 3