Simulating the effect of capillary flux on the soil water balance in a stochastic ecohydrological framework

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1 WATER RESOURCES RESEARCH, VOL. 44,, doi: /2008wr006889, 2008 Simulating te effect of capillary flux on te soil water balance in a stocastic ecoydrological framework R. Willem Vervoort 1 and Sjoerd E. A. T. M. van der Zee 2 Received 1 February 2008 revised 25 May 2008 accepted 20 June 2008 publised 19 August [1] Groundwater uptake can play a major role in te survival of vegetation in semiarid areas, but tis as not yet been included in an earlier developed ecoydrological stocastic framework. In tis paper we provide a piecewise linear equation wic includes capillary fluxes from sallow groundwater in te loss function of te ecoydrological stocastic model. Te results indicate tat tis model is able to simulate te capillary fluxes, and te model also reflects te impact of te fluxes on te soil moisture balance. In addition, te results are analytically tractable and allow calculation of te probability density functions of soil water saturation and water stress for different groundwater depts below te root zone. Citation: Vervoort, R. W., and S. E. A. T. M. van der Zee (2008), Simulating te effect of capillary flux on te soil water balance in a stocastic ecoydrological framework, Water Resour. Res., 44,, doi: /2008wr Introduction [2] Recently, Rodriguez-Iturbe and Porporato [2004] developed a framework for te stocastic modeling of te soil water balance and ave coupled tis to a colonization and competition model to describe te dynamics of semiarid (savanna) vegetation systems, i.e., systems wic are primarily controlled by water. Teir stocastic framework currently does not include any groundwater interaction. However, groundwater can be an important driver of vegetation growt and occurrence in many semiarid areas, particularly in relation to groundwater-dependent ecosystems in riparian areas [i.e., Mensfort et al., 1994 Torburn and Walker, 1994 Walker et al., 1993]. Tis raises te question, wic groundwater levels lead to important contributions of groundwater to te soil water balance? [3] Tere are two possible mecanisms for te interaction between groundwater and vegetation: (1) part of te root mass interacts wit te groundwater and water is taken up directly, and (2) capillary fluxes cause water to move into te root zone after wic it is taken up by te vegetation. Wic of te two dominates is not clear from te literature, but in tis study we will concentrate on te capillary fluxes. One way to investigate te importance of te different processes is to describe te process of groundwater interaction using a model, i.e., adapting te stocastic framework to include groundwater uptake. [4] Evaporation and related capillary flow from a relatively sallow groundwater table as been studied for some time [e.g., Gardner, 1958 Pilip, 1957]. Te conceptual model of te system is a omogeneous soil wit a root zone to a dept Z r and a groundwater table at a dept Z below te soil surface (Figure 1). Evaporation and rainfall occur at te soil surface and affect mainly te water storage in te root zone. No ysteresis occurs and te ydraulic relationsips are generally of an exponential or linear form [Salvucci, 1993]. Drainage from te soil store reaces te water table instantaneously, and te soil water profile below te root zone as reaced steady state. Tis means tat te fluctuations in te groundwater table occur at a muc larger time scale tan te fluctuations in te climatic drivers (i.e., years versus days and weeks). Wit some exceptions (suc as transmission losses from a river on a igly permeable bed), tis is reasonable for semiarid systems. Tus, we also assume tat te groundwater table is at a constant level trougout te period of study, and tis will be discussed furter in section 3. [5] Te steady flux of water in te root zone can be described wit te Darcy equation: q ¼ K ð Þ d dz ð1þ were q is te flux (L T 1 ), K() is te ydraulic conductivity function, and d/dz is te potential gradient. For a steady flux, (1) can be integrated to lead to Z ¼ Zð Z¼0Þ 0 d 1 þ q=kðþ : ð2þ 1 Hydrology Researc Laboratory, Faculty of Agriculture, Food and Natural Resources, University of Sydney, Sydney, New Sout Wales, Australia. 2 Soil Pysics, Ecoydrology and Groundwater Management, Environmental Sciences Group, Wageningen University, Wageningen, Neterlands. Copyrigt 2008 by te American Geopysical Union /08/2008WR Equation (2) describes te maximum eigt for wic a designated capillary flux (q) can be supplied for particular soil ydraulic properties and dryness at te soil surface. [6] Full complex analytical solutions for equation (2) can be derived, assuming specific forms of te ydraulic conductivity function [Warrick, 1988]. However, te solutions are not practical for implementation in analytical models 1of15

2 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE since tey cannot be explicitly solved for q, except in te case of q K s (te saturated ydraulic conductivity), and cannot be inverted to provide (q, Z). Approximate models tat do not ave tese disadvantages ave terefore been developed [Eagleson, 1978 Salvucci, 1993]. However, bot are approximate models for equation (2), and wile allowing an analytical description, tey cannot be easily included into te stocastic framework [Rodriguez-Iturbe and Porporato, 2004]. [7] Te main aim of tis paper is, wit an adapted stocastic framework, to study te importance of te capillary fluxes from groundwater to replenis te evapotranspiration demand of te vegetation and to study te resulting canges in te soil water balance in semiarid areas. To acieve tis aim, we develop a new piecewise linear function tat matces te illustration in Figure 1 and q derived from equation (2) for describing groundwater uptake tat fits witin te stocastic framework [Rodriguez- Iturbe and Porporato, 2004] and allows calculation of probability density functions of te annual soil saturation. [8] Similar to a recent paper by Ridolfi et al. [2008], but approacing te same issue from a different direction, tis paper is a first step toward a more complete stocastic model of groundwater surface water interaction. 2. Metods 2.1. Background Teory [9] Te previously derived loss function in te stocastic framework (encefort denoted te RI model ) is defined as [Rodriguez-Iturbe and Porporato, 2004] (Figure 2) rðþ¼ s ¼ E max fz r w ¼ E w fz r 0 s s w s w s w þ ð w Þ þ me bðs sfcþ i K s m ¼ fz r e bð1 sfcþ i: s s w s* s w s < s s w s w < s s* s* < s s fc s fc < s 1 Here s is te soil saturation (0 1), f is te porosity, Z r is te root dept, and b is a parameter to fit te ydraulic conductivity function to te exponential model. Te boundaries are s, te soil type dependent ygroscopic point, and s fc, te field capacity [Rodriguez-Iturbe and Porporato, 2004]. Of te remaining parameters, s* iste soil saturation level at wic te transpiration becomes limited, s w is te wilting point, E max is te maximum evapotranspiration, and E w is te soil evaporation [Rodriguez- Iturbe and Porporato, 2004 parameters and w are te ð3þ Figure 1. Conceptual model for groundwater uptake by vegetation in a semiarid system. root zone dept normalized versions of E max and E w, respectively [Rodriguez-Iturbe and Porporato, 2004]. Te model uses a piecewise linear formulation to enable an analytical solution for te saturation probability density function [Laio et al., 2001 Rodriguez-Iturbe et al., 1999], and te new groundwater uptake function witin tis framework sould terefore be defined along similar lines. [10] Te climate in te RI model is defined by te parameters l 0 and g, wic arise from te Poisson distributed rainfall [Laio et al., 2001 Rodriguez-Iturbe and Porporato, 2004]. Te parameter l 0 is equal to le D/a, were D is te interception dept (cm), a is te mean storm dept, and l is te mean time between rainstorms [Laio et al., 2001 Rodriguez-Iturbe and Porporato, 2004]. Te parameter g is equal to fz r /a or, equivalently, 1/g is te root zone weigted mean storm dept A Simplified Capillary Flux Model Tat Fits Into te Ecoydrological Framework [11] We aim to identify a function for te total losses from te stored soil water (in te root zone) as a function of te soil saturation s. Tis means following equation (3) and wit te total of capillary and drainage fluxes defined as q total, we seek r new =ET+q total as a function of s, were ET is evapotranspiration, wic matces q from equation (2). Rater tan searcing for an analytical solution of equation (2) tat is based on first principles, we approaced te problem from a different end in view of te matematical complexity of te former. [12] We solve equation (2) troug optimization using optim in R [R Development Core Team, 2007]. Tis algoritm uses te Nelder Mead simplex algoritm to find an optimal solution [R Development Core Team, 2007]. Te combinations of q(s,z) must be described wit a piecewise linear function tat fits to tese curves and is appropriate for te stocastic framework. Tis desired 2of15

3 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 2. Grapic representation of te original RI model (equation (3)) and te new (equation (9)) model for a sandy clay loam (Table 1) and E max = 0.55 cm/d and for different groundwater depts (Z) from te soil surface. function replaces equation (3) in te stocastic framework. Te boundaries of suc a function need to be 2þ3=b b q ¼ K s a e s ¼ 0 Z q ¼ 0 q ¼ K s s 2bþ3 s ¼ 1 s ¼ s lim ¼ s d dz ¼ 1 ¼ Z Z 1=b r b were a e and b are parameters in te capillary flux function related to te soil water caracteristic [Eagleson, 1978], b is te bubbling pressure, and K s is te saturated ydraulic conductivity. Te parameter s lim represents te saturation point were te soil sifts from drainage beavior to capillary uptake beavior and is terefore equal to te ydrostatic point. Tis value of s at te ydrostatic equilibrium is a sifting field capacity soil saturation for wic te magnitude depends on te dept of te groundwater and te soil ydraulic parameters. [13] Te suggested form of te function wic predicts te capillary and drainage fluxes (q total ) as a function of te soil saturation s, wic as a similar sape to te q(s, z) curves, is q total ðþ¼ s 8 < m 2 s cr < s s* : m 1 1 e bðs slimþ s* < s s lim m 2 ¼ K sg fz r m 2 m 1 ¼ 1 e bðs* slimþ i: ð4þ ð5þ Here m 2 and m 1 are two constants: m 2 represents te maximum capillary flux for a given groundwater dept and ydraulic properties (encapsulated in G), wile m 1 is equal to m 2 normalized for te reduction in capillary flux wit increased saturation. [14] Te dimensionless parameter G is a function tat describes te relationsip of te capillary flux wit te groundwater dept, te bubbling pressure ( b ), and te ydraulic sape parameters a e and 2 + 3/b [Eagleson, 1978] and is suggested to ave te following functional form [Eagleson, 1978]: G ¼ a e 2þ3=b b : ð6þ Z Z r [15] In equation (5), below s* te actual capillary flux will be driven by te ET demand and can be lower tan te potential capillary flux, wile above s* te capillary fluxes slowly decline wit increased saturation. Basically, te impact of te capillary flux is tat at some value of s te total loss (r new =ET+q total ) actually equals zero (Figure 2). Te soil will never dry out below tis level of soil saturation because at tis point (and below), te potential capillary flux is eiter equal to or greater tan te actual evaporation losses and tus all evaporation demand can be supplied by te capillary flux. We will call tis saturation point s critical (s cr ). In reality, tis means tat below s cr te potential flux will be reduced until te capillary flow matces te actual ET. Tis also implies tat s cr is te minimum soil saturation level te soil will reac for tat particular groundwater level, ET demand curve, and soil type. Tis means s lim and s cr are two important points on te saturation scale, bot of wic are dependent on te 3of15

4 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE groundwater level and te ydraulic properties of te soil. Tey bot represent boundary values at wic point te beavior of te loss function canges. [16] For deep groundwater tables s cr will be equal to s, and for sallow groundwater tables it will be equal to s*. In te function including groundwater uptake, we need to use s lim rater tan s fc as tis point on te saturation curve becomes a variable rater tan a fixed parameter and, as mentioned earlier, is defined as s lim ¼ Z Z 1=b r : ð7þ b In te rest of tis paper we will use s lim in te new piecewise functions. Tis means te new loss function can be defined as 8 s s cr ð m 2 Þ >< s* s cr r new ¼ m 1 1 e bðs slimþ >: þ me bðs slimþ m 2 ¼ K sg fz r m 2 m 1 ¼ 1 e bðs* slimþ i K s m ¼ fz r ½e bð1 slimþ Š : s cr < s s* s* < s s lim s lim < s 1 All oter parameters ave been defined earlier. After integration, te probability density function, p(s), becomes ð9þ 8 l0 ðs* scrþ C s s m 1 cr 2 e gs s cr < s s* ð m 2 Þ s* s cr C m 1 1 e bðs slimþ l0 bð m 1 Þ 1 ½ m 2 Š l 0 bð m 1 Þ p new ðþ¼ s >< e gsþ l0 s s* ð m 1 Þ C 1 þ m l0 s slim eb ð Þ ð ð Þ s* < s s lim b mþ 1 e gsþ l0 ð mþ ðs slimþ eð l0 m 1 Þ s* slim ð Þ ð10þ >: 1 m l 0 2 ð b m 1 Þ s lim < s 1: [17] In te RI model witout capillary flux (equation (3)), te function as two different sections below s*. Tis means tat te function for capillary flux sould also include a section for s w s cr < s* and for s cr < s w, or two sections if m 2 in equation (5) is smaller tan w and only one section if w m 2 <. In te first case te capillary fluxes are very small, unsufficient to even maintain soil moisture above te wilting point s w. We will terefore concentrate first on te situation in wic te capillary fluxes supply sufficient moisture so s cr s w and m 2 > w. We will additionally assume tat m 1 <, wic means s cr < s*, or te capillary fluxes are too small to maintain evapotranspiration at maximum capacity. From te above discussion and te definition of s cr, it also follows tat te total loss of soil moisture below s cr is equal to zero. Tis implies tat s cr can, in tis case, be defined by equating [(s s w )/(s* s w )] wit m 2 : s cr ¼ m 2 ð s* s wþþs w : ð8þ Te parameter C in equation (10) is an integration constant. Tis constant can be derived analytically for te RI model [Laio et al., 2001], but we ave approximated it numerically by using te fact tat te area under a probability density function (pdf) sould equal unity [Laio et al., 2001]. [18] We will now consider te situation m 1 and m 2, wic means te capillary fluxes are so great tat te evapotranspiration is always at its maximum and te soil water saturation never drops below s*. Tis creates a problem in equation (10) as te function can matematically not exist for s cr s < s lim. However, in practice, tis means te term 0 < s s cr would disappear from te pdf (as te soil water storage would never be below s cr and terefore p(s) =0fors s cr ), and te second term of te loss function (s* <s s lim ) simplifies to e b(s s lim ) 8 < e bðs slimþ s* < s s lim r new ¼ : þ me bðs slimþ s lim < s 1 m 2 ¼ K sg fz r K s G m 1 ¼ fz r 1 e bðs* slimþ i ð11þ 4of15 K s m ¼ fz r ½e bð1 slimþ Š :

5 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Te critical groundwater dept (Z cr ) at wic tis will appen can be calculated by setting m 1 equal to and including te definition of G (equation (7)) into te definition of m 1 (equation (9)): Z cr ¼ Z r þ 2 4 ð b E max 1 e b s* s lim K sa e : ð12þ 3 1 Þ 2þ 3 = b 5 As a result, te probability density function (p(s)) for tis situation can be rewritten as 8 C exp ð b ð s s limþ gsþ i >< exp l0 b ð 1 exp ð b ð s s limþþþ s cr < s s lim p new ðþ¼ s C 1 þ m l0 s slim eb ð Þ b m ð Þ 1 >: e gsþ l0 ð mþ ðs slimþ s lim < s < 1: ð13þ [19] Finally, we consider te situation for m 2 < w, and we now ave s < s cr s w. In tese cases K s G is very small as 8 l 0 ðsw scr Þ C s s cr w m 1 2 e gs w m 2 s w s cr Te overall loss function for tis model becomes r new ðþ¼ s 8 m 2 ¼ K sg fz r ð w m 2 Þ s s cr s w s cr >< ð w m 2 Þþð w Þ m 1 e bðs slimþ >: þ me bðs slimþ K s G m 1 ¼ fz r 1 e bðs* slimþ i K s s s w s* s w s cr < s s w s w < s s* s* < s s lim s lim < s 1 m ¼ fz r ½e bð1 slimþ Š : ð15þ After integration, for te resulting p(s) we obtain s cr < s s w C ð w m 2 Þ 1 þ l0 ðs* swþ w s sw 1 w e gs s w < s s* w m 2 s* s w p new ðþ¼ s >< C 1 m l0 1 1 e bðs slimþ bð m 1 Þ 1 1 m l 0 2 ð 1 þ l0 ðs* swþ w w m 2 b m 1 Þ w e gsþ l0 s s* ð m 1 Þ ð Þ s* < s s lim ð16þ C 1 þ m l0 s slim eb ð Þ ð b mþ 1 e gs l0 ð mþ ðs slimþ eð l0 m 1 Þ ðs* slimþ >: 1 m l 0 2 ð b m 1 Þ 1 þ l0 s* sw w w m 2 ð Þ w s lim < s 1: te groundwater tables are deep or ydraulic conductivities are small. Te boundary value s cr is now defined as s cr ¼ m 2 w ðs w s Þþs : ð14þ Te replacement of equation (3) in te stocastic framework now consists of tree cases: equations (9), (11), and (15). Basically, te value of te potential capillary flux (K s G) needs to be compared wit te maximum evapotranspiration rate, after wic te different forms of te probability density function (p(s)) can be calculated using equations (10), (13), or (16) Calculations [20] To compare ow well te suggested new model is able to represent canges in te soil saturation under a varying climatic input and for different soils, water balance 5of15

6 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Table 1. Soil Properties Used in te Simulations a Soil Type Porosity f K s (cm/d) b Y s Y s,s s fc Heavy clay E Medium clay E Ligt medium clay E Sandy clay loam E Loamy sand E a Soil ydraulic data are based on standard Australian soils in Neuroteta [Minasny and McBratney, 2002]. calculations over 10,000 days were performed on te basis of equations (3), (9), and (15). For te water balance calculations, rainfall was generated on te basis of a Poisson distribution of te storm frequency and an exponential distribution of te rainfall amounts [Rodriguez-Iturbe et al., 1984]. Te first 365 days of te water balance calculations were deleted to create a 1 year warm-up period for te derivation of means and variances. [21] Furter, soil and vegetation parameters were derived from te literature [e.g., Porporato et al., 2001] and databases of Australian soils (Tables 1 and 2). Te porosity parameter f was set equal to q s as estimated wit te van Genucten pedotransfer functions in Neuroteta [Minasny and McBratney, 2002], wile for s fc te value at = 100 cm was cosen from te same pedotransfer functions. [22] Some representative climate parameters (Table 3) were calculated from long-term rainfall data for several locations in Australia [Rodriguez-Iturbe et al., 1984], and tis defined te range of possible values for a and l. Maximum evaporation data (E max ) was based on te average data for te listed weater stations in Table 3 (see ttp:// 3. Results and Discussion [23] Te proposed new function (equation (9)) fits te optimized q values quite well (Figure 3). Tis suggests tat equation (9) is similar in beavior to te solution of equation (2) and can be used to simulate te capillary and drainage fluxes for te system in Figure 1 in te stocastic framework. Tere is only a sligt overestimation of te capillary flux close to s lim (Figure 3). Figure 3 clearly indicates ow te point were drainage and capillary fluxes are bot zero (i.e., te ydrostatic point or s lim ) sifts depending on te soil type and te groundwater dept. It also demonstrates tat te capillary fluxes rapidly become very small at deeper groundwater levels (i.e., deeper tan 150 cm below te bottom of te root zone for most soils). [24] A comparison of te water balance results of te two versions of te suggested piecewise linear models (equations (9) Table 2. Vegetation Properties Used in te Simulations Following Porporato et al. [2001] Property Values Z r (cm) 100 D (cm) 0.2 E max (cm/d) 0.5 E w (cm/d) 0.01 y s,s* (MPa) 0.12 y s,sw (MPa) 5 Leaf area index x 2.5 a a From Witeead and Beadle Table 3. Representative Climate Properties for Different Locations in Australia a Location in Australia l (storms/d) a (cm/d) E max (cm/d) Bourke Moree Katerine (NT) b a Rainfall properties were calculated using te metods described by Rodriguez-Iturbe et al. [1984]. b From Hutley et al. [2001]. and (15)) indicates tat tere is only a very small penalty for ignoring te section below s w (Table 4). Te model wit four limits is possibly more accurate for drier climates and deeper groundwater levels, as excursions below s w are more frequent owever, te difference in te variance and te means is small (Table 4). Tis means using equation (9) is appropriate for most situations. [25] Incomparisonwitteoriginalmodel[Rodriguez- Iturbe and Porporato, 2004], te new model predicts lower soil saturations under wetter climates (Figure 4). Tis is because in wetter climates (al =0.7),tedrainageprocess will dominate over te capillary processes, wic means te value of te ydrostatic point s lim becomes important (wic is te point were fluxes are zero). Equations (9) and (15) include a variable s lim wic is a function of te groundwater dept to reflect te ydrostatic point. Tis results in a continuation of te drainage process compared to using a fixed s fc in te original RI model, particularly at deeper groundwater levels. However, tis is a more accurate representation of te real process. A constant s fc value is valid for simulating drainage above deep groundwater tables but will probably overestimate te ydrostatic point for most soils (Table 1 and Figure 3). Tis would result in an underestimation of te actual drainage and tus relatively iger soil saturation values. [26] Overall, te new model clearly demonstrates te effect of groundwater on te soil water balance. Sallower groundwater tables tend to increase te amount of water in te soil, and tis effect is larger for drier climates tan for wetter climates (Figure 4). In addition, te effect dissipates rapidly wit increasing dept of te groundwater table. For te sandy clay loam used in Figure 4, te impact of te capillary fluxes on te soil water balance is minor for a groundwater table at 2 m below te root zone, even under te driest climate (al = 0.18). For soils wit lower K s values, te influence of groundwater will decrease even earlier. [27] Te models presented ere all assume tat te majority of te root water uptake is concentrated in te root zone. Tis is not always te case. Groundwaterdependent vegetation could also ave dimorpic root systems or varying root ydraulics wic means tat te majority of te root water uptake takes place from only a small fraction of te roots close to te groundwater table [Dawson and Pate, 1996]. Tis is not considered in tis study but could be included in extensions on tis work. [28] In tis study, we ave also assumed tat te groundwater level is not directly affected by te daily atmosperic inputs or te evaporation from te vegetation. Tis assumption is only valid if te lateral transmissivity of te aquifer is muc larger tan te vertical transmissivity or if te aquifer system storage is very large compared to te capillary fluxes. Incorporation of te impact of evaporation and 6of15

7 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 3. Comparison of te optimized q capillary values from te integral in equation (2) (solid black lines) wit te new loss function including groundwater interaction ( new model equation (9) dased lines) for different groundwater depts (Z) below te root zone. Data ave been plotted against relative saturation, Se = (s s )/(1 s ), to improve comparison between te soil types. atmosperic inputs is a furter step, and a recent paper as made some progress in tat area [Ridolfi et al., 2008]. In anoter approac te groundwater table could be varied exogeneously (e.g., using a seasonal time series model in relation to te annual rainfall [Salas and Obeysekera, 1992]) were tis variation is ten incorporated into te model in tis paper. In tis case te function G would become related to l and a troug Z, and tis is part of our ongoing researc in tis area Probabilistic Representation of s [29] Te probability density functions (p(s)) also indicate a clear difference between deep to sallow groundwater (Figure 5). Te probability density functions spread out wit sallower water table depts until Z cr is approaced. Te bounding at s cr for te lower end of p(s) is also demonstrated (Figure 5). Te value of s cr will sift toward s lim wit sallower groundwater depts as iger capillary fluxes and tus iger evaporative demand are supported. Heavier soils (suc as te ligt medium clay) will sow a smaller sift as a function of groundwater dept because of lower ydraulic conductivities. In addition, te effect of groundwater on p(s) de more rapidly tan for more igly conductive (large K s ) soils, suc as te sandy clay loam (Figure 6). [30] Wit increasingly sallow groundwater dept, bot s cr and s lim cange. However, initially far from Z cr, because of te nonlinearity of equation (12), s lim will increase muc faster tan s cr, as equation (8) is approximately linear. Tis will cause te probability density function to spread (Figure 6). For groundwater levels closer to Z cr, s lim increases less quickly, causing te probability density function to narrow. Groundwater levels sallower tan Z cr finally generate a steeper and narrower p(s) curve (equation (13)). Tis steepness of p(s) under sallow groundwater tables is partly due to te fact tat p(s) is bounded on te upper end because of te sarp increase in losses from drainage above s lim. Under wetter climate conditions te overall functions broaden because of an increase in excursions above s* for deeper groundwater depts and an increase in te variance relative to te dry climate (see Table 4). Te same narrow probability density functions occur at sallow groundwater levels, as p(s) is pused up against te upper limit. [31] Means and variances of s were calculated numerically by integrating te probability density functions: 7of15

8 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Table 4. Means and Standard Deviations of Soil Saturation s a Z mean ¼ Es ½Š¼ sps ðþds Var ¼ Es 2 Es ½Š 2 ¼ Z s 2 ps ðþds mean 2 : Climate (al) Mean of s New model wit tree limits New model wit four limits Variance of s (10 3 ) New model wit tree limits New model wit four limits Mean of s RI Model New model wit tree limits at 175 cm New model wit tree limits at 300 cm New model wit four limits at 175 cm New model wit four limits at 300 cm Variance of s (10 3 ) RI Model New model wit tree limits at 175 cm New model wit tree limits at 300 cm New model wit four limits at 175 cm New model wit four limits at 300 cm a Te top compares te two new models (wit tree limits, equation (9), and wit four limits, equation (14)) over 10,000 days for different climates (different combinations of al) on a sandy clay loam. Te bottom compares te new piecewise model wit tree limits (equation (9)) and four limits (equation (14)) to te original model witout groundwater (equation (3)) for te same soil at two groundwater depts (175 and 300 cm below te surface) for te same climate combinations and soil. ð17þ Te mean of s increases wit increasingly sallow groundwater levels (Figure 7) and decreases to te mean as predicted witout accounting for groundwater at deeper groundwater depts (RI model). Wile tere is some increase in te mean wit wetter climates, te increase is small relative to te increase in te mean as a result of sallower groundwater tables. Again, te influence of groundwater on te mean soil saturation decreases rapidly as soon as groundwater levels are lower tan 1 2 m below te root zone (depending on te soil type), indicating tat capillary fluxes only affect te soil saturation over only a small range of groundwater depts. [32] In contrast, te variances of s of te new model deviate quite considerably from te variance obtained for te RI model. Overall, te variance increases for wetter climates compared to drier climates, but tis is true for bot models (Table 4 and Figure 8). However, te variances of te new model reveal a maximum wit respect to groundwater dept, wic depends, among oter tings, on climatic forcing (Figure 8). Wit increasingly sallow groundwater depts, te variances of te new model first increase until close to Z cr, after wic te variances steeply decrease to a constant value (i.e., te variance of equation (13)). For deeper groundwater levels te variance of equation (10) also deviates from te original RI model. Tis is due to te canges in te distance between s cr and s lim compared to te RI model wit a fixed s fc. A fixed value of s fc migt be close to te steady state soil saturation value, wic is iger tan te ydrostatic point for deep dwater tables [Salvucci and Entekabi, 1994]. Assuming tat soils in steady state equilibrate at s fc, tis also indicates tat most soils will ave drainage fluxes greater tan zero under steady state conditions above deep water tables Water Stress Calculations [33] Water stress is a useful summary statistic for te growing season, as it incorporates bot te variation and mean of te stress. Basically, two types of stress can be identified: te mean static stress, wic represents te average stress vegetation experiences during a season, and te dynamic water stress, wic takes into account te duration as well as te frequency of te stress [Porporato et al., 2001]. Te water stress calculations and all additional parameters for tose calculations exactly followed Porporato et al. [2001], and te reader is referred to tat paper for furter details. In terms of parameters, following Porporato et al., te sensitivity parameter k was set to 0.5, te degree of nonlinearity (exponent) was set to 3, and te growing season T seas was assumed to be 250 days. [34] Te mean static water stress (Figure 9, top) increases steadily wit increasingly deeper groundwater dept and converges on te no-groundwater case for te same climate. Te difference in te curves between te different soils is small for te mean static water stress. However, for te dynamic water stress (Figure 9, bottom), including te duration of te excursion below s* (denoted by T s* ) and te frequency of te water stress (n s* ) means tat tere are greater differences between te different soils. Tis is because T s* and n s* ave greater values for te sandy clay loam tan for te ligt medium clay for several reasons. Te values of T s* and n s* are strongly driven by te size of soil storage given by f Z r [Porporato et al., 2001] and tus by te porosity differences between te two soils and by te distance between s lim and s*. Te sandy clay loam as a lower porosity and a narrower soil water caracteristic (distance between s lim and s*) wic results in a iger frequency of crossings (n s* ) and a greater duration of te excursions (T s* ). Tis is also reflected in te fact tat te sandy clay loam as a muc larger variance tan te ligt medium clay (Figure 8). Overall, tese results suggest te model migt be used to understand te effect of eiter increasing or lowering groundwater tables on groundwaterdependent ecosystems [Groom et al., 2000]. [35] Te water stress calculations also demonstrate tat groundwater uptake troug capillary fluxes does ave an effect beyond te 1 2 m below te root zone suggested earlier. However, in terms of supplying sufficient water needed for te survival of trees in a semiarid environment, te groundwater levels still need to be relatively close to te root zone. If tis is not te case, ten te process of groundwater uptake is most probably troug single roots accessing te groundwater directly. Te implication is tat te developed model can elp in understanding te mecanism of groundwater uptake by roots and can lead to a better description of groundwater dependency in models. 4. Conclusions [36] Capillary groundwater fluxes influence te soil moisture balance but only in a limited range of groundwater levels. Groundwater levels deeper tan 2 m below te 8of15

9 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 4. Comparison of te water balance for te original model (RI model 0, equation (3), dotted line) witout groundwater uptake and te new model wit tree limits (equation (9)) for two different groundwater depts (Z = 175 cm, solid line, and Z = 300 cm, dased line). Te vegetation is trees (Table 2), and te soil is a sandy clay loam (Table 1). Te different plots sow different combinations of al (climate). 9of15

10 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 5. Probability density functions (p(s)) for two soils (sandy clay loam and ligt medium clay) and different groundwater depts. Te p(s) curves sift to te wet end of te spectrum under te influence of sallower water tables. Tis sift occurs wit sallower water tables for te soil wit te lower K s (ligt medium clay). Curves were calculated for a climate wit l = 0.33 and a = of 15

11 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 6. Probability density functions (p(s)) for four different soils (loamy sand, sandy clay loam, ligt medium clay, and eavy clay), different climates (dry l = 0.21, wet l = 0.45, and a = 1 for bot climates), and different water tables (Z = 150, 175, 250, and 350 cm) for te new model including groundwater interactions (equations (10), (13), and (16)). 11 of 15

12 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 7. Mean of te probability density functions for tree different soils, different climates, and different groundwater levels, indicating an increase in te mean soil saturation at sallower groundwater tables for te new model including groundwater interaction and a sarp decrease in te effect of groundwater on te soil saturation for groundwater tables 2 3 m below te soil surface. 12 of 15

13 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 8. Variance of s for tree different soils (loamy sand, sandy clay loam, and ligt medium clay), different climates, and different groundwater levels, indicating a general decrease in te variance wit groundwater levels from Z cr. Te variances above Z cr, in contrast, are muc lower. Te deviation of te variance for te new model including groundwater interactions compared to te RI model at deep groundwater depts is due to te difference between a constant s fc and a variable s lim. 13 of 15

14 VERVOORT AND VAN DER ZEE: CAPILLARY FLUX AND SOIL WATER BALANCE Figure 9. Mean (top) static and (bottom) dynamic water stress for two different soils (sandy clay loam and ligt medium clay) and different groundwater levels under te Moree climate (Table 3). For te original RI model, te water stress is always a constant independent of Z owever, for te new model including groundwater interactions te water stress decreases wit sallower groundwater tables and stays below te water stress in te original RI model. Oter variables used were T seas = 250 days, te degree of nonlinearity set to 3, and k = 0.5 [Porporato et al., 2001]. bottom of te root zone are unlikely to influence te soil moisture balance even for soils wit ig values of K s. [37] Te new proposed model, wic is based on te RI model but additionally accounts for capillary fluxes from te groundwater, reflects te effect of canging groundwater depts on te soil water storage. Te model consists of tree different cases depending on te balance between te potential capillary flux and te actual evaporative demand. Te probability density functions (p(s)) are bounded on te lower end at s cr, wic represents te point were te evaporative demand equals te capillary flux. Ignoring te s range below te wilting point as little effect on te results, as s cr is generally above s w if tere is any noticeable effect on te soil moisture balance. One key difference between te older model and our model is te inclusion of a variable s lim, wic represents te ydrostatic point, and tis variable replaces te boundary at s fc in te RI model. Tis extension results in drainage continuing at lower values of s in te new model (equations (9), (11), and (15)) compared to te original model (equation (3)), particularly under wetter climates. [38] Te mean of s sifts predictably to a iger value wit sallower groundwater tables owever, te variance sift is less predictable because of te simultaneous canges of s cr and s lim wit canges of groundwater dept. Te new model correctly predicts a decrease in te water stress for vegetation over a sallow dwater table and provides opportunities to better understand groundwater dependency and te mecanisms involved in groundwater uptake by vegetation. [39] Acknowledgments. Te first autor would like to tank A. Porporato and F. Laio for initial assistance and for providing te Matematica code used by Laio et al. [2001] and Porporato et al. [2001]. Tis work was conducted wile te first autor was on sabbatical leave from te University of Sydney and working at te Soil Pysics, Ecoydrology and Groundwater Management Group, funded by te Wageningen Institute for Environment and Climate Researc (WIMEK) at Wageningen University (Neterlands). Te second autor acknowledges support of te International Researc Training Network NUPUS, funded by te German Researc Foundation (DFG, GRK 1398) and Neterlands Organisation for Scientific Researc (NWO, DN ). References Dawson, T. E., and J. S. Pate (1996), Seasonal water uptake and movement in root systems of Australian preatopytic plants of dimorpic root morpology: A stable isotope investigation, Oecologia, 107(1), 13 20, doi: /bf Eagleson, P. S. (1978), Climate, soil, and vegetation: 3. A simplified model of soil moisture in te liquid pase, Water Resour. Res., 14(5), , doi: /wr014i005p Gardner, W. R. (1958), Some steady state solutions of te unsaturated moisture flow equation wit application to evaporation from a water table, Soil Sci., 85(4), , doi: / Groom, B. P. K., R. H. Froend, and E. M. Mattiske (2000), Impact of groundwater abstraction on a Banksia woodland, Swan Coastal Plain, Western Australia, Ecol. Manage. Restor., 1(2), , doi: / j x. 14 of 15

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