Effective Hydraulic Parameters in Horizontally and...

Effective Hydraulic Parameters in Horizontally and Vertically Heterogeneous Soils forSteady-State Land–Atmosphere Interaction

BINAYAK P. MOHANTY

Department of Biological and Agricultural Engineering, Texas A&M University, College Station, Texas

JIANTING ZHU

Desert Research Institute, Las Vegas, Nevada

(Manuscript received 1 May 2006, in final form 22 November 2006)

ABSTRACT

In this study, the authors investigate effective soil hydraulic parameter averaging schemes for steady-stateflow in heterogeneous shallow subsurfaces useful to land–atmosphere interaction modeling. “Effective” soilhydraulic parameters of the heterogeneous shallow subsurface are obtained by conceptualizing the soil asan equivalent homogeneous medium. It requires that the effective homogeneous soil discharges the samemean surface moisture flux (evaporation or infiltration) as the heterogeneous media. Using the simpleGardner unsaturated hydraulic conductivity function, the authors derive the effective value for the satu-rated hydraulic conductivity Ks or the shape factor � under various hydrologic scenarios and input hydraulicparameter statistics. Assuming one-dimensional vertical moisture movement in the shallow unsaturatedsoils, both scenarios of horizontal (across the surface landscape) and vertical (across the soil profile)heterogeneities are investigated. The effects of hydraulic parameter statistics, surface boundary conditions,domain scales, and fractal dimensions in case of nested soil hydraulic property structure are addressed.Results show that the effective parameters are dictated more by the � heterogeneity for the evaporationscenario and mainly by Ks variability for the infiltration scenario. Also, heterogeneity orientation (hori-zontal or vertical) of soil hydraulic parameters impacts the effective parameters. In general, an increase inboth the fractal dimension and the domain scale enhances the heterogeneous effects of the parameter fieldson the effective parameters. The impact of the domain scale on the effective hydraulic parameters is moresignificant as the fractal dimension increases.

1. Introduction

Moisture flux across the land–atmosphere boundary(through infiltration, evaporation, and plant transpira-tion) is an important component of large-scale hydro-climatic processes. Predicting the mean flux rate for aremote sensing footprint or model grid/pixel is usually aprimary concern in most practical soil–vegetation–atmospheric transfer (SVAT) models. One of the keyland–atmosphere linkages is described by Koster et al.(2004), whose results from a recent model intercom-parison project show that soil moisture anomalies con-sistently produce precipitation anomalies in certain

“hot spot” regions around the globe. The ability to pre-dict soil moisture anomalies and related land surfacefluxes and states requires a comprehensive approachcombining the latest scientific understanding, modelingcapabilities, and available remotely sensed observa-tions.

Land surface models require three types of inputs:initial conditions, atmospheric–soil boundary condi-tions–forcings, and parameters, which are a function ofsoil, vegetation, topography, and other land surfaceproperties. While the quality and availability of re-motely sensed vegetation- and topography-related landsurface parameters have improved significantly overthe last few decades, comparable advances in globalsoil-related parameters at matching scales have not oc-curred. In fact, given that soil moisture is known to bea critical climate variable, it could be argued that ourcurrent approach of using texture-based lookup tables

Corresponding author address: Binayak P. Mohanty, Depart-ment of Biological and Agricultural Engineering, Texas A&MUniversity, 301 Scoates Hall, College Station, TX 77843-2117.E-mail: [email protected]

AUGUST 2007 M O H A N T Y A N D Z H U 715

DOI: 10.1175/JHM606.1

© 2007 American Meteorological Society

JHM606

and/or point-scale pedotransfer functions to estimatesoil hydraulic properties is one of the weakest links incurrent National Aeronautics and Space Administra-tion (NASA) land surface modeling efforts (C. D.Peters-Lidard 2005, personal communication).

From land–atmosphere interaction perspective withmodel grid/pixel scale ranging from hundreds of metersto several kilometers, soil hydraulic properties’ vari-ability may include overlapping small to large (nested)spatial structures due to different intrinsic or extrinsicfactors. For example, microheterogeneity, agriculturaltraffic, and row cropping may induce spatial structuresin soil hydraulic properties at very fine (from centime-ter to meter) scale due to soil structural features andpore-size distribution. On the other extreme, processesfrom variational soil textural deposition patterns acrossa hill slope to land form evolution [e.g., alluvium, gla-cial till; Jenny (1941)] govern the large-scale spatialstructures in soil hydraulic properties at scales fromhundreds of meters to several kilometers. Among oth-ers, Mohanty et al. (1991) and Mohanty and Mousli(2000) measured nested spatial structures for saturatedhydraulic conductivity in a glacial till agricultural fieldin Iowa encompassing random heterogeneity, and twooverlapping structures with distinctly different correla-tion lengths. In addition to horizontal heterogeneity,soil hydraulic properties may vary in the vertical direc-tion in a nested fashion because of surface disturbancesdue to tillage practice, pore-size distribution due tostructural cracks and root development and decay, tex-tural layering, and geology (Mohanty et al. 1994).

Upscaling of soil hydraulic property (Harter andHopmans 2004) is a process that incorporates a mesh ofhydraulic properties defined at the measurement scale(support) into a coarser mesh with “effective/averagehydraulic properties” that can be used in large-scale(e.g., watershed scale, basin scale, regional scale) hy-droclimatic modeling. Soil hydraulic properties havebeen studied extensively at the deep unbounded vadosezone where gravity flow dominates (e.g., Gelhar andAxness 1983; Yeh et al. 1985a,b,c; Montoglou and Gel-har 1987a,b,c; Desbarats 1998; Russo 1992, 1993,1995a,b; Yang et al. 1996; Zhang et al. 1998; Harter andZhang 1999). Common to these analyses is the treat-ment of the flow and transport problem in unboundeddomains assuming a uniform mean head gradient. Thisassumption, coupled with the assumption of small vari-ability of input hydraulic parameters, leads to closed-form analytical solutions for the moments of the flow-dependent variables. The applicability of solutionsbased on the small-perturbation expansion of the flowequation and the assumption of unit head gradient islimited because head variability increases with decreas-

ing saturation, and the mean head gradient may notequal unity over large portions of a bounded flowdomain near the land–atmosphere boundary. In situa-tions of shallow water table, for example, the unit meangradient region may constitute only a small portion ofthe flow domain. The application of hydraulic propertyupscaling schemes to large heterogeneous areas in theshallow subsurface, particularly from a land–atmo-sphere feedback perspective, remains an outstandingissue (Zhu and Mohanty 2002a,b, 2003b).

Kim and Stricker (1996) investigated the indepen-dent and simultaneous effects of heterogeneity in soilhydraulic properties and rainfall intensity on variousstatistical properties of the one-dimensional water bud-get components. Kim et al. (1997) investigated the im-pact of areal heterogeneity of the soil hydraulic prop-erties on the spatially averaged water budget of theunsaturated zone. Zhu and Mohanty (2002a) investi-gated several hydraulic parameter averaging schemesand the mean hydraulic conductivity, in particular theirappropriateness for predicting the mean behavior ofthe pressure head profile and the mean fluxes of (hori-zontally) heterogeneous formations for the steady-stateinfiltration and evaporation. They used two hydraulicproperty models, namely, the Gardner–Russo exponen-tial model (Gardner 1958) and the Brooks–Coreymodel (Brooks and Corey 1964). Zhu and Mohanty(2002b) provided practical guidelines on how the com-monly used averaging schemes (arithmetic, geometric,or harmonic) perform when compared with the effec-tive parameters for steady-state flow in heterogeneoussoils using the widely used van Genuchten (1980) hy-draulic property model. In another study by Zhu andMohanty (2003a), they analytically derived effectivesoil hydraulic parameters that were able to produce themean pressure profile and discharge mean flux for theheterogeneous soils. While the previously mentionedand other recent studies (Zhu and Mohanty 2003b,2004, 2006; Zhu et al. 2004, 2006) addressed the deter-mination of effective hydraulic parameters specificallyrelated to horizontal (areal) heterogeneity of soil andother land surface parameters (e.g., root distribution,surface ponding depth), yet no study has been con-ducted to explore the related issues for a soil with ver-tical heterogeneity (i.e., soil textural layering) and sub-surface heterogeneity because of natural fractal forma-tions in the context of land–atmosphere interaction.The vertical heterogeneity is important because naturalsubsurface formation is typically layered and the under-standing of how the layering might affect vertical mois-ture exchange is a challenging issue. Furthermore, com-parison of effectiveness of the effective soil hydraulicparameters for both vertical and horizontal heteroge-

716 J O U R N A L O F H Y D R O M E T E O R O L O G Y — S P E C I A L S E C T I O N VOLUME 8

neities under different land–atmosphere boundary fluxand water table conditions is warranted for appropriateland surface parameterization in SVAT model grids fordifferent hydroclimatic applications.

In this study we investigate both horizontal and ver-tical soil heterogeneities and address the differences indealing with both heterogeneities for large-scale hydro-climatic processes. One-dimensional models have beenused as approximations of various simplified problemsof steady-state water flow across the land–atmosphereboundary (e.g., equilibrium condition between the pre-cipitation events and ignoring other meteorological fac-tors) under investigation. For one-dimensional analy-ses, two physical scenarios need to be distinguished: 1)vertical layering (heterogeneity), where variations insoil properties are in the vertical directions only (seeFig. 1, bottom) (e.g., Yeh 1989), and 2) vertically ho-mogeneous parallel soil columns with variations of thesoil properties in the horizontal plane only (see Fig. 1,top) (e.g., Dagan and Bresler 1983; Bresler and Dagan1983; Rubin and Or 1993). The objectives of this studywith special emphasis on land–atmosphere interactionare twofold: 1) to investigate the impact of the hetero-geneous hydraulic soil parameters’ statistics on thepixel-scale effective parameters for both horizontal andvertical heterogeneous conditions under verticallyupward/downward steady-state flow scenarios and es-tablish generalized upscaling–aggregation rules basedon the requirement that the equivalent medium dis-charge the mean (evaporation/infiltration) flux of theheterogeneous formation, and 2) to study the signifi-cance of the pixel size (domain scale) and the fractaldimensions of the soil hydraulic parameter fields withnested spatial structure on the effective parameters.

2. Methods

a. Steady-state one-dimensional flow

The unsaturated hydraulic conductivity (K)-capillarypressure head (�) relationship is represented by theGardner model (Gardner 1958)

K � Kse��, �1�

where Ks is the saturated hydraulic conductivity and �is the pore-size distribution parameter inversely relatedto the bubbling pressure. In this study, we define adimensionless form of �, �* � �L where L is the dis-tance to the water table from the soil surface. By using�*, we account for the variations in soil texture (e.g.,sand, silt, clay) and groundwater table depth (e.g., deepwater table in a desert environment, shallow watertable in an agricultural landscape) conditions together

during the effective parameter(s) estimation and scal-ing analyses described later.

The other widely used hydraulic conductivity func-tions can be used as well (e.g., van Genuchten 1980).Our extensive simulations indicated that while the cal-culated effective parameters using the van Genuchtenhydraulic conductivity function differ quantitativelyfrom those using the Gardner function, they follow thesame trend and the conclusion reached for the Gardnerfunction also applies for the van Genuchten model.

FIG. 1. Schematic view of hydraulic parameter heterogeneity:(top) horizontal (areal) heterogeneity; (bottom) vertical hetero-geneity.


Therefore, we only present analysis using Gardnerfunction in this study without loosing the generality ofthe approach for adoption to other soil hydraulic func-tions.

Applying Darcy’s law, general equation relatingpressure head and elevation above the water table forsteady-state vertical flows can be expressed as (e.g.,Zaslavsky 1964; Warrick and Yeh 1990)

zi�1 � zi � ��i

�i�1 Ki�1��

Ki�1�� qd�, �2�

where zi and zi�1 (zi�1 � zi) are the vertical distancesabove the water table, �i and �i�1 are the suction headat zi and zi�1, respectively, Ki�1(�) is the hydraulic con-ductivity function between zi and zi�1, and q is thesteady-state evaporation (positive) or infiltration(negative) rate.

b. Heterogeneous hydraulic parameter fields

Given the values of mean and coefficient of variation(CV) for the soil hydraulic parameters Ks (Ks andCVK respectively), and �* (�* and CVA respec-tively), the cross-correlated random fields of Ks and �*are generated using the spectral method proposed byRobin et al. (1993). Random fields are produced withthe power spectral density function, which was basedon exponentially decaying covariance functions. Asan indicator of correlation between the two randomparameter fields, we used the coherency spectrumgiven by

R�f� ��12�f�

��11�f��22�f��1�2 , �3�

where 11(f), 22(f) are the power spectra of randomfields 1 and 2, respectively, and 12(f) is the cross spec-trum between fields 1 and 2. The norm of the coherencyspectrum, |R|2, may range from 0 to 1, with |R |2 � 1 (i.e.,the correlation coefficient � � 1 in the physical domain)indicating a perfect linear correlation between the tworandom fields. In this study, we used |R |2 � 1 consid-ering the fact that the value of |R |2 did not significantlyalter the results based on our previous/current work.Also, the parameters Ks and �* are assumed to obeythe lognormal distributions as observed by others in thepast (Smith and Diekkruger 1996; Nielsen et al. 1973).Random fields of 10 000 (for a 100 � 100 grid) forhorizontal heterogeneity and 100 (for 100 layers) forvertical heterogeneity are generated. Using this ap-proach, more or less (random) data points do not havesignificant impact on the effective parameters as long asthe mean, variance, and correlation structure remainthe same. In other words, 10 000 (100 � 100) data

points for horizontal heterogeneity scenario and 100points for vertical heterogeneity scenario are proven tobe enough in a statistical sense.

As the mean value of Ks does not affect the effectivehydraulic parameters (demonstrated later), we can useany values for the mean Ks in this study. For the pur-pose of illustrating our more generalized methodologyin terms of a specific example, we use two values of�* � 6 and 1 in generating the random field for �*. Togive a perspective of how these values relate to actualfield soil conditions, we adopt a typical value for siltyclay loam � � 0.01 (1 cm�1) from Carsel and Parrish(1988). Combining �* � 6 or �* � 1 and � � 0.01 (1cm�1) will give a water table depth of about 6 or 1 m,respectively. For describing the spatial variabilities of�* and Ks, we use ranges for their coefficient of varia-tions (CVA and CVK) of 0–0.572 and 0–0.822, respec-tively. The maximum values of CVs (i.e., 0.572 for �*and 0.822 for Ks) used here illustrate that the upscalingmethod developed is appropriate for quite large het-erogeneous field conditions. While we present resultsfor a few selected mean parameter and variability val-ues (i.e., relating to different field conditions), it doesnot mean the procedure developed in the study is lim-ited to these conditions and values. For other field con-ditions, the values of �*, CVA, and CVK need to beadjusted to correspond to the actual field conditions,and the developed procedure will still apply.

c. Description of flux in horizontally heterogeneoushydraulic parameter field

For horizontally heterogeneous soils (Fig. 1a), appli-cation of (2) in combination with the Gardner function(1) in each parallel column (denoted by subscript j)results in the following dimensionless form of the waterflux rate at the ground surface,

qj � KSj

1 � e� *j �1�h�

e� *j � 1, �4�

where �* � �L, h � �L/L are the dimensionless � andsurface pressure, respectively. The L is the depth ofwater table, and �L is the surface pressure head. Notethat in deriving (4) we only consider horizontal hetero-geneity, that is, the hydraulic parameters are constantalong the entire profile from the water table to the soilsurface with values of Ksj and �*j . Using (4) and thegenerated random fields of Ks and �*, we can calculatethe flux for each parallel soil column j, and then themean flux q is calculated by averaging over all the in-dividual column fluxes.


d. Description of flux in vertically heterogeneoushydraulic parameter field

Using Gardner’s model, pressure head distributionacross a vertically heterogeneous (layered) porous me-dium (Fig. 1b) can be obtained from (2):

�*i�1 � �1

�*i�1ln�e��*i�1�z*i �1�z*i��*i � �

q

Ksi�1

� �1 � e��*i�1�z*i�1�z*i ��. �5�

Given the generated random fields of Ks and �* in thevertical direction and the pressure head condition at thesoil surface, h, the flux q can be calculated iterativelyfrom (5) that will satisfy the boundary condition. Set-ting some initial guess for the water flux in the soilprofile q, the iteration procedure starts at the watertable (i.e., z0 � 0 and �*0 � 0) and moves upward untilit calculates the pressure head at the soil surface. Theprocedure is repeated by adjusting the value of q ac-cording to the calculated surface pressure head in theprevious iterations until the calculated surface pressureat the soil surface matches the given boundary condi-tion.

The parallel stream-tube type approach presented inthis study covers an entire range of steady-state verti-cal infiltration and evaporation conditions [i.e., themaximum steady-state dimensionless infiltration (q/Ks)is �1 and the maximum steady-state dimensionlessevaporation rate is over 0.9]. This wide range of param-eter values and flux conditions theoretically correspondto different real-world hydroclimatic scenarios understeady-state conditions typically encountered or ap-proximated between the precipitation events.

e. Effective hydraulic parameter coefficients

Since predicting the mean flux rate (evaporation orinfiltration) is usually a major focus of most practicalsoil–vegetation–atmospheric transfer studies, we use asimple approach to derive “effective” hydraulic param-eters by assuming that the equivalent homogeneousmedium will discharge the same amount of moistureflux across the soil surface as the heterogeneous me-dium. The effective hydraulic parameters are calculatediteratively from the following equation:

Kseff

1 � e�*eff�1�h�

e�*eff � 1� qeff, �6�

where qeff is the effective flux, qeff � q for horizontalheterogeneity case, and qeff � q for vertical heteroge-neity case. Here Kseff and �*eff are the effective param-

eters for Ks and �*, respectively. The left-hand side of(6) is q(Kseff, �*eff). In other words, using the effectivehydraulic parameters will produce the same mean fluxexchange between the subsurface and the atmosphere.For both horizontal and vertical heterogeneous sce-narios, we defined the coefficient of effective param-eters for Ks and �* as a measure of how close the ef-fective parameter is to the arithmetic mean, respec-tively,

ECK �KSeff

Ks, �7�

ECA ��*eff

�*, �8�

where Ks is the arithmetic mean of Ks and �* is thearithmetic mean of �*. Therefore, a coefficient of ef-fective parameter of 1 means an arithmetic mean isappropriate as an effective parameter that will dis-charge the mean flux over the entire heterogeneousregion.

As we mentioned earlier, this study focuses only onestimating effective parameters based on matching themean surface flux (evaporation or infiltration) irrespec-tive of the soil water content or pressure profile distri-bution. Since there are two effective hydraulic param-eters (Kseff and �*eff) in one equation [see (6)], in thefollowing discussion we set one of the effective param-eter as the arithmetic mean and find the other effectiveparameter according to Eq. (6).

3. Heterogeneity effect on effective hydraulicparameters

The mean value of Ks does not alter effective param-eters because of its linear relationship with the verticalflux. On the other hand, for illustrating the impact ofnonlinearity of subsurface flow we used two mean val-ues of the shape parameter �* (6.0 and 1.0, represent-ing a most sensitive �* range) and two values of surfacepressure h (h � 2.0 for evaporation and h � 0.1 forinfiltration). Note, however, while these mean param-eter values are chosen for demonstrating the resultsacross the most sensitive ranges of the selected param-eter(s), our methodology encompasses many possiblesoil hydrologic conditions reflecting real-world sce-narios. To put these two values of �* in a practicalperspective, we use typical values for the hydraulic pa-rameters estimated by Carsel and Parrish (1988) acrossthe soil textural range between sand and clay. The �*values of 1.0 and 6.0 translate into a range between 2.8and 480 cm for the water table depth. Our simulationsindicate that a water table depth below 2.8 cm andabove 480 cm does not significantly affect the effective


parameter values when compared with the results for2.8 cm or for 480 cm. It is noteworthy that as we de-velop a generalized method with dimensionless param-eter �* and use the values from literature (e.g., Carseland Parrish 1988) for demonstration purpose, the ac-tual values of �* should be determined based on themeasured � and groundwater table depth at a particu-lar site of applications.

a. When one parameter is heterogeneous

1) KS HETEROGENEOUS, �* HOMOGENEOUS

For the horizontal heterogeneity case, since the rela-tionship between the surface flux and Ks is linear, theeffective Ks in this case would be simply the arithmeticmean of the Ks field. In other words, the effective co-efficient of Ks (ECK) is 1. For the vertical heterogene-ity case, Fig. 2 shows the ECK versus the coefficient ofvariation of the Ks field (CVK). Results show that ECKdecreases monotonically as the variance of the Ks fieldincreases. It is virtually independent of the surface pres-sure head condition and mean �* values. The ECK isalways smaller than 1, indicating that the vertically het-erogeneous medium is always more limiting to moistureflux than a homogeneous medium with an arithmeticmean Ks value. It is also important to note that direc-tion of flux (infiltration or evaporation) has no influ-ence on the ECK for vertically heterogeneous medium.

2) �* HETEROGENEOUS, KS HOMOGENEOUS

Figure 3 shows the effective coefficient of �* (ECA)versus the coefficient of variation of �* (CVA) whenonly the �* field is heterogeneous (with homogeneous

Ks). Figure 3a is for the case of horizontal heterogene-ity while Fig. 3b is for the case of vertical heterogeneity.For the sake of brevity, we only present the results forthe case of relatively large mean �*, �* � 6.0. Theresults for the case of small mean �* are not shownhere. Generally, results for smaller �* also follow asimilar pattern, but in the case of significant differencesin the patterns for small versus large values of �*, thechanges will be reported in qualitative terms in the dis-cussion that follows. For the horizontal heterogeneitycase, results show that ECA is generally smaller than 1,indicating that the horizontally heterogeneous �* gen-erally favors a larger mean surface flux exchange withthe atmosphere. For the vertical heterogeneity case, re-sults show that ECA is always larger than 1, indicatingthat the vertically heterogeneous �* generally limits thesurface flux exchange with the atmosphere. The vari-ability of the �* field causes ECA to be farther awayfrom 1 for evaporation as compared with infiltration;

FIG. 2. Effective coefficient of Ks (ECK) vs coefficient of varia-tion of Ks (CVK) for vertical heterogeneity when only Ks is het-erogeneous.

FIG. 3. Effective coefficient of �* (ECA) vs coefficient of varia-tion of �* (CVA) when only �* is heterogeneous and �* � 6.0.(a) Horizontal heterogeneity and (b) vertical heterogeneity.


that is, the heterogeneous effect is more significant forthe evaporation case. Our results also indicate that theheterogeneity has more significant impact when themean �* is larger (i.e., relatively deep water table) com-pared to the smaller mean �* (i.e., relatively shallowwater table).

b. When both Ks and �* are heterogeneous

1) FINDING ECK WHILE USING ARITHMETIC

MEAN FOR �*

Figure 4 presents distributions for ECK as functionsof CVK and CVA when both Ks and �* are heteroge-neous under different flow scenarios and variabilityorientations. Figures 4a,b are for the case of horizontalheterogeneity, and Figs. 4c,d are for corresponding ver-tical heterogeneity scenarios with �* � 6.0. Figures4a,c are for the infiltration scenario (h � 0.1), and Figs.4b,d are for the evaporation scenario (h � 2.0). For thecase in Fig. 4a, ECK is close to 1. In other words, hori-zontal variability in both Ks and �* has relatively insig-nificant influence on ECK for infiltration. A homoge-neous medium with arithmetic mean values of Ks and�* would be a good equivalent for the corresponding

horizontal heterogeneous system. But quantitativelyspeaking, both variances (of Ks and �*) make ECKsmaller and usually farther away from the value of 1,indicating that the heterogeneous system deviates morefrom mean behavior and discharges less than theequivalent system with arithmetic mean parameters. Inother words, the effective parameter leads to arithmeticmean (i.e., gets closer to 1) as both CVK and CVAdecrease, reflecting a more homogeneous soil medium.For the evaporation scenario (Fig. 4b), however, ECKbehaves differently and is mainly dominated by the �*variance. Results indicate that the �* variance signifi-cantly increases the surface evaporation as compared tothe equivalent system with arithmetic mean hydraulicparameters. From a physical perspective, this resultsuggests that larger localized evaporative fluxes occurbecause of nonlinear unsaturated hydraulic conductiv-ity distribution across the land–atmosphere interfaceunder relatively deep water table conditions.

Figures 4c,d are for the case of vertical hetero-geneity and matching hydrologic scenarios as for thehorizontal heterogeneity case. Results show that for theinfiltration scenario (Fig. 4c) ECK is dominated by theKs variability. When the flow scenario switches frominfiltration (Fig. 4c) to evaporation (Fig. 4d), ECK is

FIG. 4. Effective coefficient of Ks (ECK) when both Ks and �* are heterogeneous and�* � 6.0. (a) Horizontal heterogeneity, h � 0.1 (i.e., infiltration); (b) horizontal heteroge-neity, h � 2.0 (i.e., evaporation); (c) vertical heterogeneity, h � 0.1 (i.e., infiltration); and (d)vertical heterogeneity, h � 2.0 (i.e., evaporation).


mainly dictated by the �* variability. For the verticalheterogeneity scenario, ECK is always smaller than 1.In other words, the vertical heterogeneity of hydraulicparameters always limits moisture flux both downwardand upward when compared with the equivalent me-dium with the arithmetic mean hydraulic parameters.For the evaporation scenario (Fig. 4d), the evaporativeflux for the vertically heterogeneous formation is se-verely limited where ECK approached zero. Physicallythis may reflect a scenario of a desert environment witha deep groundwater table overlain by layers of low-conductive geological materials. Based on simulationsfor a number of �* values, it has been found that ECKevolved from being entirely CVK dominated for rela-tively shallow water table (small mean �*) and infiltra-tion scenario to being mainly CVA dictated at rela-tively deep water table (large mean �*) and evapora-tion scenario.

2) FINDING ECA WHILE USING ARITHMETIC

MEAN FOR KS

For a particular porous medium, while Ks determinesthe magnitude of (maximum) flux (at saturation), �*governs the flux across the soil water pressure rangeduring wetting or drying events. It can be shown thatthe flux decreases approximately exponentially as thevalue of �* increases, and increases linearly with theincreasing Ks. Figure 5 shows the results for ECA asfunctions of CVK and CVA when both Ks and �* are

considered heterogeneous fields. Figures 5a,b are forthe case of horizontal heterogeneity, while Figs. 5c,dare for the case of vertical heterogeneity. All inputs arethe same between Figs. 4 and 5. Compared with Fig. 4,contour maps in Fig. 5 follow a very similar pattern withan inverse numerical sequence for all the correspondingscenarios. Smaller ECA corresponds to higher ECKand vice versa for the same heterogeneity orientationand hydrologic scenarios.

4. Scale effects on effective hydraulic parameters

Field observations sometimes indicated that soil hy-draulic properties may be correlated over large scalesand their variance increases over domain scales (e.g.,Hewett 1986; Molz and Boman 1993; Wheatcraft et al.1990; Wheatcraft and Tyler 1988). Next, we will exam-ine the effects of these features on the effective hydrau-lic parameters. We first look at the influence of thecorrelation scales of the hydraulic parameters by com-paring the results for the correlation lengths (�) of 5 and100 times of the grid spacing using an exponentiallydecayed covariance function. Figure 6 shows the effec-tive coefficients as functions of the dimensionless sur-face pressure head at the two different correlationlengths of the random hydraulic parameter fields forCVK � 0.8225 and CVA � 0.572. These large CVKand CVA values were selected to investigate the influ-ence of Ks and �* variability. Note, however, that whilethese large values are only chosen to show the signa-

FIG. 5. Effective coefficient of �* (ECA) when both Ks and �* are heterogeneous and�* � 6.0. (a) Horizontal heterogeneity, h � 0.1 (i.e., infiltration); (b) horizontal heteroge-neity, h � 2.0 (i.e., evaporation); (c) vertical heterogeneity, h � 0.1 (i.e., infiltration); and (d)vertical heterogeneity, h � 2.0 (i.e., evaporation).


tures more prominently, result patterns remain un-changed for smaller CVK and CVA. Results demon-strate that the impact of the parameter correlationlengths on the effective coefficients is not significant forall scenarios. Both ECK and ECA are slightly differentin value and follow the same trend at significantly dif-ferent hydraulic parameter correlation lengths (i.e., � �5, 100). Overall, for the case of vertical heterogeneity,the impact of the correlation length is more significantthan for the case of horizontal heterogeneity. Gener-ally, the influence of the correlation length on effectivehydraulic parameters is relatively insignificant com-pared with other inputs, such as the means and vari-ances of the hydraulic parameters.

While we have shown that the correlation length isrelatively insignificant in influencing the effective hy-draulic coefficients, the variances (coefficients of varia-tion) of the hydraulic parameters do influence the ef-fective coefficients significantly. Therefore, the scaleimpact will be mainly reflected in the increase of vari-ance because of an increasing domain scale. For math-ematical simplicity, the soil hydraulic properties in the

horizontal and vertical directions are approximated asfractal phenomena (e.g., Peyton et al. 1994). We as-sume that the hydraulic parameters obey the fractalBrownian motion statistics for which the spectral den-sity has the form of a power law with the power-lawindex related to the fractal dimension (e.g., Hassan etal. 1997). While we use the fractal Brownian motionstatistics as a way to simplify mathematical descriptionand to relate effective hydraulic parameters to the frac-tal dimension, the methodology described in this studycan be extended to any process where increasing hy-draulic parameter variance is related to the domainscale considered. Fractal processes by definition are in-finitely correlated, and their variance increases with thescale of the domain asymptotically to infinity. In realsystems, the domain boundary will determine the limitof heterogeneity. The maximum domain scale in turndetermines the minimum wavenumber (e.g., Zhan andWheatcraft 1996). If we use the minimum wavenumbercutoff, which is related to the maximum scale of thedomain of interest, then the variance (�2) can be re-lated to the scale X as follows:

FIG. 6. Effective coefficients vs dimensionless surface pressure head at different correlationlengths of random hydraulic parameter fields. (a) ECK, horizontal heterogeneity; (b) ECA,horizontal heterogeneity; (c) ECK, vertical heterogeneity; and (d) ECA, vertical heterogene-ity. In the figure legends, � denotes the ratio of the correlation length of the random fields overthe grid length used to generate the random fields.


�2 �S0X2�2�D�

�2 � D��2��2�2�D�, �9�

where S0 is a parameter related to the maximum do-main scale, and D is the fractal dimension. If there is apotential maximum domain scale under consideration,Xmax, we can relate X to Xmax as X � XmaxY, where Yis the dimensionless relative scale. Then the scale-related variance is as follows:

�2 � �max2 Y2�2�D�, �10�

where

�max2 �

S0X max2�2�D�

�2 � D��2��2�2�D�. �11�

Using this variance-scale relationship, we can now re-late the effective coefficients for the hydraulic param-eters to the hierarchical spatial scale structure.

Figure 7 shows the results for ECK versus the rela-tive domain scale (Y) and the fractal dimension of theKs field (Dk) for the case of vertical heterogeneity whenonly Ks is heterogeneous. Results show that ECK de-creases as either the domain scale or the fractal dimen-sion increases. The increase in both the domain scaleand the fractal dimension will shift ECK farther awayfrom 1, indicating an increasing heterogeneous effect.

As discussed earlier, ECK for the case of vertical het-erogeneity is always smaller than 1; in other words, thevertically heterogeneous system limits moisture fluxesin both upward and downward directions.

Figure 8 shows the results for ECA as functions ofthe relative domain scale and the fractal dimension ofthe �* field (Da) when only the parameter �* is het-erogeneous. Figures 8a,b are for the case of horizontalheterogeneity and Figs. 8c,d are for the vertical hetero-geneity. For the infiltration scenario (Figs. 8a,c), ECAchanges slightly within a small range around 1 (i.e.,around arithmetic mean). For the evaporation scenario(Figs. 8b,d), the domain scale and the fractal dimensionhave a more significant influence on ECA. Both thedomain scale and the fractal dimension steer ECAaway from 1. For the horizontal heterogeneity situa-tion, ECA is generally smaller than 1, suggesting thatthe horizontal �* heterogeneity increases the moistureflux where a larger fractal dimension supports largemean moisture. As mentioned earlier, the vertical het-erogeneity constrains/reduces the mean moisture flux(i.e., ECA � 1), especially for the evaporation scenario.The increasing fractal dimension will further reduce themean moisture flux both downward and upward as de-picted by increased ECA values.

Figure 9 illustrates that results for ECK as functionsof the relative domain scale (Y) and the fractal dimen-sion of the Ks field (Dk) when both Ks and �* areheterogeneous and CVA � 0.572. Some of the impor-tant observations include large ECK for the case shownin Fig. 9b indicate that a combination of Ks and �*heterogeneities greatly increases the mean evaporativeflux. For the horizontal heterogeneous situation, a fewsmall �*s dominate the moisture flux and greatly in-crease the mean flux for the heterogeneous formation.Corresponding to Fig. 9b conditions, ECK for the ver-tical heterogeneity case is very small (see Fig. 9d) wherea few large �* will severely limit the mean moistureflux. It can be shown that the flux decreases approxi-mately exponentially with the increasing �*, and in-creases linearly with the increasing Ks. Therefore, it canbe expected that the �* heterogeneity has a more sig-nificant impact on the mean moisture flux in the het-erogeneous soil formations. It can be observed fromour results that the increasing fractal dimension of Ks

(Dk) generally steers ECK farther away from 1 and thusincreases the heterogeneous effects.

Figure 10 displays the results for ECA as functions ofthe relative domain scale (Y) and the fractal dimensionof the �* field (Da) when both Ks and �* are hetero-geneous and CVK � 0.822. Results are presented forsame hydrologic scenarios and heterogeneity orienta-

FIG. 7. Effective coefficient of Ks (ECK) vs relative domainscale (Y ) and fractal dimension of Ks (Dk) for vertical heteroge-neity case when only Ks is heterogeneous.


tions as in the previous cases. Overall, it can be ob-served that ECA is generally larger than 1 with theexception of evaporation for horizontal heterogeneity(Fig. 10b). Since we have used the arithmetic mean forthe Ks field in calculating ECA, the fact that ECA ismostly greater than 1 and ECK is mostly smaller than 1(as seen in Fig. 9) indicates that the heterogeneous for-mation where both Ks and �* are heterogeneous dis-charges generally less than the equivalent homoge-neous formations with the arithmetic mean hydraulicparameters. For the infiltration case, ECA is more pre-

dictable, being consistently larger than 1, and both thedomain scale and the fractal dimension of �* increaseECA. For the evaporation case, our calculations indi-cate that its trend reverses depending on the value ofthe mean �*. For small mean �* (corresponding to arelatively shallow water table), ECA increase as boththe scale and the fractal dimension increase, while forlarge mean �* (corresponding to a relatively deep watertable), it decreases with the scale and the fractal dimen-sion.

Figures 10c,d show the results for the vertical hetero-

FIG. 8. Effective coefficient of �* (ECA) vs relative domain scale (Y ) and fractal dimensionof �* (Da) when only �* is heterogeneous and �* � 6.0. (a) Horizontal heterogeneity, h �0.1 (i.e., infiltration); (b) horizontal heterogeneity, h � 2.0 (i.e., evaporation); (c) verticalheterogeneity, h � 0.1 (i.e., infiltration); and (d) vertical heterogeneity, h � 2.0 (i.e., evapo-ration).


geneity. While the vertical heterogeneity of hydraulicparameters always limits both downward and upwardmoisture fluxes (i.e., ECA is always larger than 1), theinfluence of the domain scale (Y) and the fractal di-mension of �* (Da) is mixed. For the case of infiltration,both the relative domain scale and the fractal dimen-sion of �* slightly decrease ECA (Fig. 10c); that is, theydecrease the heterogeneity effects. For the case ofevaporation, both the relative domain scale and thefractal dimension of �* actually increase ECA (Fig.10d); that is, they increase the heterogeneity effects.

5. Concluding remarks

Based on our comprehensive simulations under hori-zontal and vertical heterogeneity orientation and hy-drologic scenarios from the perspective of land–atmo-sphere interaction modeling, the main conclusionsdrawn from this study are the following:

1) A vertically heterogeneous variably saturated po-rous medium does not discharge as much moistureflux as the equivalent homogeneous medium ofarithmetic mean values for the (Ks and �) hydraulic

FIG. 9. Effective coefficient of Ks (ECK) vs relative domain scale (Y ) and fractal dimensionof Ks (Dk) when both Ks and �* are heterogeneous, �* � 6.0 and CVA � 0.572. (a)Horizontal heterogeneity, h � 0.1 (i.e., infiltration); (b) horizontal heterogeneity, h � 2.0 (i.e.,evaporation); (c) vertical heterogeneity, h � 0.1 (i.e., infiltration); and (d) vertical heteroge-neity, h � 2.0 (i.e., evaporation).


parameters. In other words, a vertically heteroge-neous subsurface soil is more limiting to soil mois-ture flux across the land–atmosphere boundary incomparison to a horizontally heterogeneous soilacross the landscape.

2) When only the Ks field is heterogeneous, the con-cept of effective coefficient of Ks works the best. Forthe case of horizontal heterogeneity, ECK is simply1, since Ks linearly controls the steady-state mois-ture flux. For the case of vertical heterogeneity,ECK decreases only with CVK and is virtually in-

dependent of other hydraulic parameters and hydro-logic conditions.

3) The heterogeneity of the �* field significantly influ-ences ECA and ECK, since it severely limits the fluxboth upward and downward.

4) For the evaporation scenario the effective coeffi-cients are dictated more by the �* heterogeneity,while for the infiltration scenario the effective coef-ficients are mainly controlled by the Ks variability.

5) In general, increase in both the fractal dimensionand the relative domain scale enhances the hetero-

FIG. 10. Effective coefficient of �* (ECA) vs relative domain scale (Y ) and fractal dimen-sion of �* (Da) when both Ks and �* are heterogeneous, �* � 6.0 and CVK � 0.822. (a)Horizontal heterogeneity, h � 0.1 (i.e., infiltration); (b) horizontal heterogeneity, h � 2.0 (i.e.,evaporation); (c) vertical heterogeneity, h � 0.1 (i.e., infiltration); and (d) vertical heteroge-neity, h � 2.0 (i.e., evaporation).


geneity effects of parameter fields on the effectivecoefficients for the hydraulic parameters. The im-pact of the domain scale on the effective coefficientsis more significant as the fractal dimension in-creases.

6) For the case of vertical heterogeneity, �* heteroge-neity dominates the effective coefficients, mainlybecause Ks linearly controls the moisture flux while�* dictates the moisture flux in a highly nonlinearmanner. A few large or small �*s can significantlyreduce or enhance flux, both upward and down-ward.

The above findings, along with our past studies (Zhuand Mohanty 2002a,b, 2003a,b, 2004, 2006; Zhu et al.2004, 2006), will help choose the appropriate equivalentsoil hydraulic parameters for distributed SVAT modelgrids with horizontal and vertical soil heterogeneities.However, further studies are needed to make the pro-posed parallel stream-tube type approach more com-prehensive and realistic in terms of three-dimensionalsubsurface flow and transient atmospheric boundaryconditions.

Acknowledgments. This project is funded by NASA(Grants NAG5-11702 and NNG06GH01G) and NSF(DMS-0621113), and is also supported in part by Sus-tainability of Semi-Arid Hydrology and Riparian Areas(SAHRA) under the STC Program of the National Sci-ence Foundation, Agreement EAR-9876800, Los Ala-mos National Laboratory, the Water Resources Re-search Act, Section 104 research grant program of theU.S. Geological Survey, DRI’s Applied Research Ini-tiative and start-up fund.

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