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0038-075C/03/16801-3–14 January 2003Soil Science Vol. 168, No. 1Copyright © 2003 by Lippincott Williams & Wilkins, Inc. Printed in U.S.A.

TECHNICAL ARTICLES

ANALYSIS OF UNSATURATED WATER FLOW IN A LARGE SAND TANK

Britta Schmalz1, Bernd Lennartz2, and Martinus Th. van Genuchten3

1Institute of Water Management and Landscape Ecology, University Kiel, Ol-shausenstr. 40, 24118 Kiel, Germany.2Institute of Soil Science and Plant Nutrition, University Rostock, Justus-von-Liebig-Weg 6, 18051 Rostock, Germany. E-mail: [email protected]

3George E. Brown, Jr. Salinity Laboratory, USDA, ARS, 450 West Big SpringsRd., Riverside, CA 92507-4617,

Received Feb. 27, 2002; accepted Sept. 4, 2002.

DOI: 10.1097/01.ss.0000049727.63732.8a

A realistic, physically based simulation of water and solute movementin the unsaturated soil zone requires reasonable estimates of the water re-tention and unsaturated hydraulic conductivity functions. A variety ofstudies have revealed the importance of how these unsaturated soil para-meters are assessed and subsequently distributed over the numericalmesh on modeling outcome. This study was initiated to acquire experi-mental data about the water flow characteristics of sandy soils to serve asa base for numerical analyses. Specific objectives were to clarify the ef-fects of (i) the invoked procedure for estimating the soil hydraulic para-meters and (ii) using increasingly refined spatial definitions of the hy-draulic properties on simulated two dimensional water content and flowvelocity distributions.

Water flow in and drainage from a large sand tank (approximately 5� 3 m2 at the base, 6 � 5.6 m2 at the top) was investigated using soil hy-drologic and geophysical methods. Numerical analyses of variably satu-rated flow along a two-dimensional cross-section were carried out inattempts to describe the heterogeneous flow fields using the Richardsequation-based HYDRUS-2D code. The unsaturated soil hydraulic prop-erties were described using van Genuchten-Mualem type expressions. In-formation from both in situ and laboratory measurements was employedto obtain parameter estimates.

The observed variability in discharge rate with time was reproducedbest when an average water retention curve was used and the saturatedwater content was set equal to the porosity, whereas cumulative outflowwas predicted best when all van Genuchten hydraulic parameters werefitted to the retention data. Using heterogeneously distributed hydraulicparameters (assuming a layered profile or a random distribution of thesaturated hydraulic conductivity) improve neither predictions of the cu-mulative discharge rate nor the variability in the outflow rate when com-pared with the homogeneous case. Efforts to construct or numericallysimulate heterogeneous flow experiments may, therefore, not always bejustified when water flow in sandy substrates is studied. (Soil Science2003;168:3–14)

Key words: Parameter estimation, modeling, flux field variability, soilwater content.

INFORMATION about soil hydraulic propertiesis needed for predicting and modeling water

and solute movement in the unsaturated zone ofsoils. The use of deterministic models presumeshaving reasonable estimates of water retentionand unsaturated hydraulic conductivity func-tions. Numerous field and laboratory methodshave been developed for estimating unsaturatedsoil hydraulic properties. Kool et al. (1987) andHopmans and Simunek (1999) gave overviews ofparameter estimation techniques, and Feddes etal. (1988) discussed data needs for model inputand validation.

One popular approach has been to use rela-tively simple analytical expressions for the hy-draulic properties, such as the van Genuchten-Mualem equations (van Genuchten 1980).Observed field and/or laboratory soil hydraulicdata are often used to derive parameters in theseexpressions by employing some type of fittingprocedure. Direct, indirect, or inverse methodsmay be used for this purpose. Independently de-termined parameters such as saturated hydraulicconductivity, Ks, can then be used as fixed valuesduring the optimization. Depending on the typeof application, accurate estimates of the saturated,�s, and residual, �r, water contents may also beneeded.

Wessolek et al. (1994) distinguished amongsix optimization procedures for estimating vanGenuchten hydraulic parameters from observedwater retention and hydraulic conductivity data.They selected different values for Ks and also for�s, which was assumed to be equal to porosity.The value of �r was set equal to zero in all runs.A reasonably good description of the hydraulicdata was obtained when setting the residual wa-ter content to zero and the pore connectivity fac-tor, l, to 0.5. A poor fit resulted when the satu-rated water content was equated to the porosityand Ks to its independently measured value.Bohne et al. (1997) used both simultaneous andseparate fitting options to obtain the vanGenuchten parameters. They found that havinggood data in the near-saturation range is a pre-requisite for obtaining reliable parameter sets.Kablan et al. (1989) set �s equal to the largest sin-gle measured water content and found that themaximum water content was only approximately70% of the estimated porosity. Using multistepoutflow data, van Dam et al. (1994) studied theeffect of increasing the number of optimized pa-rameters from three to five in the optimization.Durner (1994) discussed the role of �r and �s, par-ticularly for hydraulic conductivity predictions.

He presented an example where the measuredvalue of � at a pressure head, h, of �1 cm wastaken as �s.Bohne et al. (1993), among many oth-ers, treated �s and �r as unknown parameters. Toavoid unrealistic results for �s, they used the lab-oratory measured ‘saturated’ water content at apressure head of �2 cm as an experimental point.Further discussions of the empirical nature of �sand �r are given by Luckner et al. (1989, 1991)and Nimmo (1991). Hopmans and Overmars(1986) mentioned that, in practice, �r is the watercontent at some large negative value of the soil-water pressure head. Because no data were avail-able in the dry range of the soil water character-istic, they fixed �r to its minimum value of zero.Additional discussions of �r are given by Russo etal. (1991), Simunek and van Genuchten (1996),and Simunek et al. (1998) among others.

In performing parameter optimizations, animportant question is how well defined a fitted pa-rameter set can or must be. Carrera and Neuman(1986) defined the terms uniqueness, identifiabil-ity, and stability in efforts to analyze the extent towhich inverse problems are well posed. Manystudies (e.g., Kool et al., 1987; Abbaspour et al.,1997; Hopmans and Simunek 1999) have raisedconcerns over the difficulties that may be encoun-tered when trying to obtain unique inverse solu-tions for unsaturated flow, with several offeringsuggestions on how to reduce the problem ofuniqueness and the minimum amount of informa-tion needed to guarantee a unique solution.

The uniqueness problem has often been ana-lyzed by studying the behavior of parameter re-sponse surfaces. Bohne et al. (1993, 1997) andRusso et al. (1991) examined infiltration data in this manner. Tension disc infiltrometer datawere similarly investigated by Simunek and vanGenuchten (1996, 1997). In other studies, Toor-man et al. (1992), van Dam et al. (1994), and Ech-ing et al. (1994) analyzed one-step and multi-stepoutflow experiments, whereas Simunek et al.(1998) evaluated evaporation experiments fromthe perspective of parameter uniqueness andminimum data needs.

Field-scale unsaturated flow is further com-plicated by the problem of soil heterogeneity.Soilheterogeneity and the corresponding variabilityin soil hydraulic properties can be considered byaccounting explicitly for the spatial distributionof the parameters or by using effective parame-ters. A variety of statistical, geostatistical, andstochastic approaches have been used for thispurpose. Using two-dimensional numerical sim-ulations, Roth (1995) found that in macroscopi-

4 SCHMALZ, LENNARTZ, AND VAN GENUCHTEN SOIL SCIENCE

cally homogeneous but microscopically hetero-geneous media,water moved primarily through acomplex network of flow channels. The hy-draulic structure of the medium was character-ized by two states, depending on the degree ofsaturation. Results showed that the spatial struc-ture of the scaling factor of the presumed Miller-similar medium (Miller and Miller 1956) was inexcellent agreement with the structure of hy-draulic variables but not with that of the waterflux. Polmann et al. (1991) compared tension andwater content distributions obtained using bothstochastic and deterministic approaches. The sto-chastic theory predicted less vertical flow andsomewhat more horizontal spreading than acomparable deterministic analysis as a conse-quence of the tension-dependent anisotropy ofthe effective conductivity function. The hetero-geneous tension contours from the detailed sim-ulation revealed significant lateral spreading ofthe wetting front in both horizontal and verticaldirections. Differences between the two ap-proaches, however, were too small to indicatewhich one provided a better fit to the data gen-erated in the detailed three-dimensional simula-tion. These differences are expected to increaseover time, particularly in relatively dry situationswhen the effects of tension-dependent aniso-tropy should be larger.

Russo et al. (1994) studied how the degree ofwater saturation may affect solute transport in aheterogeneous soil profile. Simulation resultssuggested that lower saturations enhance solutespreading. The velocity field, and, hence, thespreading of the solute body, were affectedgreatly by the imposed flow boundary conditionsat the soil surface. Quasi steady-state flow pro-duced essentially unidirectional vertical flow,with solute spreading occurring mainly in thelongitudinal direction. During transient flow,however, the flow pattern was much more com-plicated and essentially two-dimensional, therebyenhancing transverse spreading. In a numericalstudy of water flow and solute transport in astrongly heterogeneous medium having differentsaturation scenarios and random fields, Birk-holzer and Tsang (1997) compared two-dimen-sional vertical cross-sections of saturation and ef-fective permeability. They found that the solutetraveled along preferred flow paths or channels.The degree of channeling, the location of chan-nels, and the hydraulic properties along the chan-nels, were found to be a function of the meansaturation of the flow domain. The hydraulicproperties of the channels seemed to be invariant

of the actual location and geometry, thus indicat-ing that they may be an intrinsic characteristic ofsoil heterogeneity and degree of saturation. Kas-teel et al. (2000) predicted effective water flowand solute transport behavior at the column scaleby taking into account the three-dimensionalstructure of the hydraulic properties at the localscale. They determined the local scale hydraulicproperties and the parameter structure indepen-dently and obtained good agreement betweenmeasurements and simulated pressure head andwater content distributions.

The above studies reflect the importance ofthe way in which unsaturated soil parameters areestimated and subsequently distributed over a soilprofile. To gain further insight into this problem,we analyzed, in this study, a large database of in-formation obtained during infiltration in a largesand tank. Specific objectives of the study were:

1. To estimate the soil hydraulic properties fromexperimental data and to test different fittingprocedures, including the effect of using dif-ferent methods for estimating �s.

2. To evaluate the influence of geometry andboundary conditions of the experimental set-up on the unsaturated flow field.

3. To investigate the effect of using different soilhydraulic data sets and different spatial distri-butions of the soil hydraulic properties ontwo-dimensional simulation results. We wereespecially interested in studying the effects ofusing increasingly refined spatial definitions ofthe hydraulic properties on simulated watercontents and flow velocities

MATERIALS AND METHODS

Experimental Set-upInfiltration experiments were carried out us-

ing a large physical sand model having a base of 5m � 3 m and a surface of 6 m � 5.6 m and con-taining three sloped side walls as shown in Fig. 1.The chosen construction with three sloped sidewalls resulted from statistical constraints. All soilhydraulic measuring devices were installed fromthe vertical wall. The tank was filled with a 2-mlayer of homogeneous sand (Hagrey et al. 1999).Measurements of the pressure head (two verticaltensiometer profiles) and the water content (onevertical TDR profile) were conducted at eightdepths. All TDR probes (IMKO systems) werecalibrated before installation using the substrateof the sand tank. Estimated accuracy was 1 Vol%.A three-dimensional view of the sand tank pack-

VOL. 168 ~ NO. 1 WATER FLOW IN A LARGE SAND TANK 5

ing was obtained by ground penetrating radar(GPR) measurements. The lower boundary wasseparated into five compartments to obtain infor-mation about the spatial variability of the dis-charge rate.An irrigation system allowed differentinfiltration intensities to be imposed on the cen-tral part of the surface area (5 m � 3 m). A tentprotected the set-up from natural rainfall. Weperformed 10 infiltration experiments, one ofwhich will be discussed in this study.The selectedexperiment involved an irrigation of 4300 L (287mm) over 14 h, which produced a total dischargeof 3154 L (197 mm) over a 14-day period.

Parameter EstimationThe saturated hydraulic conductivity was cal-

culated from the grain-size distribution of thesand according to the empirical relationship ofHazen (Hölting 1996):

Ks � 41.76 (d10)2 (1)

where Ks is given in mh�1, and d10 is the diame-ter of soil particles (in mm) at 10% of the cumu-lative grain size distribution. The equation is pre-sumably valid for a temperature of 10 �C.

The grain-size distribution was measuredevery 10 cm along three vertical profiles as well asalong two horizontal profiles (25 and 30 samples,respectively). Results revealed that the texture ofthe sand varied slightly with depth. From the topto a depth of 70 cm, the profile was dominated bymedium sand (0.2–0.63 mm), whereas fine sand(0.063–0.2 mm) dominated from 70 cm down tothe bottom of the tank. The saturated hydraulicconductivity estimated using Equation (1) was

found to be 0.47 m h�1 (with a standard devia-tion (SD) of 0.07 m h�1) for the upper part and0.30 m h�1 (SD 0.03 m h�1) for the lower part.The mean hydraulic conductivity of the entireprofile was 0.36 m h�1 (SD � 0.09 m h�1). Themean bulk density measured at different depths(every 20 cm,25 to 30 samples at each depth) was1.48 g cm�3 with a SD of 0.04 g cm�3.

Soil water retention functions were derivedfrom pressure head and water content data mea-sured in the sand tank during the infiltration ex-periments at eight depths (20, 40, 60, 100, 120,140, 160, 180 cm). The retention data were de-scribed with the van Genuchten-Mualem equa-tions (van Genuchten, 1980) as follows:

q(h) � qr � h�0 (2)

q(h) � qs h�0 (3)

K(h) � KsSe1 [1�(1�Se

1/m)m]2 (4)

where � is the volumetric water content, h is thepressure head, , n, m (� 1–1/n), and l (� 0.5) are empirical parameters, Se � (�-�r)/(�s-�r) is the degree of saturation, �r, �s and Ks as definedpreviously.

The drying branches of the in-tank measured�(h) data points were analyzed using two meth-ods. For Method A, all unknown hydraulic para-meters (i.e., �r, �s, , and n) in Eqs. 2 and 3 wereallowed to be adjustable. The parameters werefitted using the RETC parameter estimationcode of van Genuchten et al. (1991). For MethodB, �r and �s were held constant during the fittingprocedure. For this purpose, �r was set equal to 0,

qs � qr[1� �ah�n]m


Fig. 1. Experimental set-up of the sand tank: (1) vertical (access) wall, (2) drainage layer (gravel), (3) discharge reg-istration, (4) tensiometer, TDR-sensors, (5) irrigated area, (6) cross-section.

and �s was set equal to the porosity as calculatedfrom the mean bulk density at each depth and theparticle density (assumed as 2.65 g cm�3) usingthe equation

e � 1 � rb rs�1 (5)

where � is the porosity, �b the bulk density, and �sthe particle density.

Numerical ExperimentsNumerical experiments were subsequently

conducted using the Hydrus-2D software pack-age of Simunek et al. (1996) to obtain two-di-mensional views of the infiltration and redistrib-ution process. The windows-based Hydrus-2Dpackage solves the Richards equation for variablysaturated flow numerically, assuming applicabilityof the van Genuchten-Mualem soil hydraulicfunctions.

Because of the geometric features of the sandtank (Fig. 1), only flow in one particular cross-section was considered. To run Hydrus-2D, weimplemented a relatively fine numerical mesh in-volving 4488 nodes depicting the geometry ofthe sand tank. An atmospheric boundary condi-tion accounting for infiltration and evaporationwas imposed at the soil surface,whereas a seepageface was used at the bottom boundary betweenthe sand and a gravel drainage layer.

We conducted the following simulation runsusing soil hydraulic parameter data sets of in-creasing complexity:

• Homogeneous soil profile: At first we used themeasured and optimized data (Table 1) to sim-ulate water flow in a homogeneous soil. Be-

cause the retention curves varied with depth,three data sets were selected for a more detailedinvestigation: the parameters for depths of 60cm (referred to as A1 and B1), 100 cm (A2 andB2), and 140 cm (A3 and B3). One mean Ks-value (0.36 mh�1) was used for these scenarios.

• Layered soil profile: The GPR measurements re-vealed the presence of a layered system (Fig. 2),apparantly caused by the method in which thetank was packed with sand. This layering wasconsidered in simulation runs (A4; B4) bysetting up a multi-layered distribution of thesoil hydraulic parameters as measured at eightdepths (Table 1), but assuming a two-layereddistribution of Ks as determined from the mea-sured grain-size distributions.

• Random distribution of Ks: Water contents andpressure heads were measured only at selectedlocations (two profiles), which did not allow usto fully characterize their variability in space.In contrast, bulk densities, and grain-size distri-butions were analyzed for each layer and in-cluded 25–30 samples per layer. Standard geo-statistical analyses showed that these propertieswere not spatially autocorrelated. Based on thisresult,we simulated the flow experiment (casesA5 and B5) assuming a random Ks distributionwithout spatial dependency and using vanGenuchten hydraulic parameters as derivedfrom the eight measured �(h)-relationships(Table 1).

• Inverse solution:We also carried out a completeinverse analysis with Hydrus-2D using as ourobjective function the sum of squared differ-ences between measured and simulated averagedischarge rates. For this scenario we again as-sumed a two-layered profile with different Ks-


TABLE 1

Optimized van Genuchten parameters for Method A (�r, �s, and n variable) and Method B (�r�0 and �s�� [m3 m�3]

Depth [cm]Parameters Method

20 40 60 100 120 140 160 180

�r [m3m�3] A 0 0 0 0 0 0 0 0�s [m3m�3] A 0.35 0.27 0.35 0.27 0.27 0.29 0.36 0.32 [m�1] A 2.71 3.51 2.96 2.20 1.65 2.51 2.30 1.97n [�] A 3.60 2.55 2.66 2.96 3.55 3.28 2.63 2.96r2 A 0.99 1.0 0.99 1.0 1.0 0.99 0.99 1.0

�s [m3m�3] B 0.45 0.45 0.43 0.44 0.45 0.46 0.45 0.45 [m�1] B 3.22 8.19 3.88 5.64 4.28 5.35 3.07 5.77n [�] B 3.37 2.02 2.43 2.03 2.03 2.21 2.37 1.63r2 B 0.99 0.97 0.99 0.96 0.95 0.96 0.99 0.98

�r and �s are the residual and saturated water content, and n are the van Genuchten parameters, r2 is the coefficient of de-termination, � is the porosity.

values.Values of �r and Ks were fixed for thesesimulations. We emphasize here that the mainobjective of this study was to present and ana-lyze the independently acquired database; re-sults of the inverse analysis are given here onlyfor comparison purposes.

In the next session we will compare mea-sured and calculated one-dimensional and two-dimensional water content distributions and fluidfluxes densities.

RESULTS AND DISCUSSION

Soil Water Retention FunctionFigure 3 shows measured and fitted water

retention curves at three depths. The measuredwater retention data are typical of the range ofresults we obtained and indicate nonuniform,depth-dependent retention behavior.A compari-son of the fitted van Genuchten parameters(Table 1) showed that the �s-values derived fromporosity (Method B) were much larger than thefitted values obtained using Method A. Using themeasured �s-values also altered the fitted reten-

tion curve near saturation significantly. In con-trast, �r-values for all depths converged to 0 usingboth fitting procedures (A and B). Close exami-nation of the fitted hydraulic parameter and vi-sual inspection of the different retention func-tions did not reveal a clear separation of the soilprofile into two layers as suggested by the ob-served grain-size distributions. As was expected(Bohne et al., 1993), coefficients of determina-tion were slightly better when the number of pa-rameters allowed to float during the fitting pro-cedures was increased (Table 1).

Effective parameters obtained with the in-verse solution using HYDRUS-2D are listed inTable 2. Notice that the estimated �s-values forthe two layers compare closely with those of op-timization procedure A (Table 1). The r2 valuefor regression of observed versus fitted values was 1.0, indicating excellent agreement betweenmeasured and optimized values.


Fig. 3. Water retention curves for three selected depthsand optimized values using Method A (solid lines) andB (dashed lines).

Fig. 2. Nonmigrated ground penetrating radargram: Pre-infiltration (upper arrow: electrode grid in 2-m depthnot reported in this paper; lower arrow: concrete bot-tom of the sand tank; Hagrey et al., 1999). The radar-gram indicates a layered structure of the sand.

Water FluxDarcian Fluxes at the Lower Boundary

The particular design of the sand tank madeit possible for us to analyze the variability of thedischarge rate in space and time. The mean fluxand the corresponding coefficient of variation(CV) were computed from the five outflowcompartments. Values of the CV versus time ofeach simulation run, along with the CVs cal-culated from the measured values, are given inFig. 4. The measured discharge had a large CV(43%) initially, but it then decreased exponen-tially to about 14%. The two modeling ap-proaches (A and B) produced divergent results.Although discharge rates for the A group (moredarkly shaded in Fig. 4) were less variable acrossthe lower boundary at early stages of the experi-ment, and more variable at later times, CVs cal-culated for the B-runs (lightly shaded) were morein line (especially B1) with the measured data.Asexpected, spatial variability in the predicted dis-

charge rate was large when a heterogeneous soilprofile (A5) was considered. Except for A1 afterabout 2 h, distributions versus time of the CVs ofsimulation runs A1-A4 were all comparable. Incontrast to Method A, the CVs for the homoge-neous cases B2 and B3 were larger than those forthe stochastic case B5.

Flow velocity vectors illustrating the two-dimensional flux field within the sand tank andacross its lower boundary are displayed in Fig. 5and are exemplary for case A1 at 2 h after the on-set of discharge. Discharge started at differenttimes, depending on the invoked simulation ap-proach (Table 3).Although discharge had alreadystarted during irrigation when Method A wasused, it occurred after irrigation had ended (at 14h) for the B-scenarios. The velocity vectors inFig. 5 indicate some lateral flow into the nonir-rigated side areas, from which water flowedfunnel-like downward along the sloped sides ofthe tank.

Simulation runs for all Method B scenariosproduced relatively low flow velocities in the en-tire sand body, primarily because irrigation hadalready stopped. The highest velocities occurredat the lower boundary. The velocity vectors wereorientated diagonally and horizontally because ofthe development of a saturated zone at the lowerboundary (characteristic of a seepage face) andconcomitant lateral spreading.

Discharge rates were distributed relativelyuniformly along the lower boundary for all sim-ulation runs, with slightly higher fluxes in themiddle of the tank and along the lower portions


TABLE 2

Parameter sets obtained with the inverse option ofHYDRUS-2D (fixed �r, Ks)

Layer �r [m3 m�3] �s [m3 m�3] [m�1] n [�] Ks [m h�1]

1 0 0.26 2.40 2.51 0.472 0 0.33 4.03 2.30 0.30

�r and �s are the residual and saturated water content, andn are the van Genuchten parameters, Ks is the saturated hy-draulic conductivity.

Fig. 4. Coefficient of variation of Darcian fluxes of the five compartments versus time (the CV is referring to the meanvalue of discharge of every compartment).

of the side walls. The measured fluxes were high-est in the middle compartments, with velocitiesgenerally smaller than those simulated. No in-creases in the measured outflow rates were ob-served near the side walls.

Methods A and B both yielded dischargerates that were not quite identical to the mea-sured fluxes. At the selected time (2 h afterdrainage started) the effect of geometry (slopedside walls) and irrigated area (nonirrigated lateralareas) was apparently higher for the simulationruns than for the experiment. Neither the mag-nitude of the velocity nor its distribution corre-sponded closely with the measured values.Method B, in particular, generated results that de-viated from the observed discharge rates.

Table 4 shows results from a statistical analysisof the mean Darcian fluxes at the lower boundary2 h after the onset of discharge. The mean fluxesof all simulation runs were greater than the mea-

sured values, while the CVs were smaller (exceptB3). Also, the CVs of the B-runs were generallylarger than those for the A group but were fairlysimilar to those of the observed flux.

Cumulative Discharge

The cumulative flux was described best whena homogeneous soil profile (A1) was assumed(Fig.6).When �r and �s were fixed at zero and theporosity, respectively (Method B), the resultingvan Genuchten parameters produced cumulativeoutflow values that differed from the observedcurves.For this case, the onset of outflow was late,and the rate (slope of curve) was always muchsmaller than the measured data (Table 3). This isin contrast to Wessolek et al. (1994), who foundthat using the porosity for �s increased the pre-dicted outflow rates significantly and producedbetter matches with the observed values. Param-


TABLE 3

Onset of discharge and cumulative flux after 336 hours at the lower boundary of selected simulation runs (Methods A and B)

Measured A1 A2 A3 A4 A5Onset of discharge [h] 13.25 14.1 8.1 11.5 11.3 9.9Cum. flux [mm] 197.1 200.3 235.3 204.4 211.6 225.6

Measured B1 B2 B3 B4 B5Onset of discharge [h] 13.25 19.9 23.8 26.4 25.1 18.7Cum. flux [mm] 197.1 165.6 150.0 137.5 145.0 167.2

Fig. 5. Darcy’s velocity vectors 2 hours after the onset of discharge in run A1 (16 h after onset of irrigation).

eter sets derived using Method A yielded flowscenarios that over-predicted the measured dis-charge rate. Our results obtained using bothMethod A and Method B indicate the impor-tance of having good estimates of �s and confirmsimilar findings by Bohne et al. (1993, 1997).

Compared with the three homogeneouscases, the layered soil profile (A4, B4) resulted inslightly lower fluxes.Fluxes for the layered profilewere also lower compared with cases (A5, B5)that used a random distribution of the soil satu-rated hydraulic conductivity. Results obtainedwith the latter two simulations (A5, B5) werewithin the range of cumulative fluxes computedfor the homogeneous soil profiles.

Water Content Distributions

Two-dimensional cross-sections of simulatedwater content distributions 12 h after the onset ofirrigation are depicted in Fig. 7.All 10 simulationruns show similar features: relatively high watercontents in the central part of the tank and at thelower boundary, and lower water contents at the

sides. This was expected since only the centralpart of the physical model was irrigated. Thesloped side walls generated a funnel-like flowregime that resulted in slightly higher water con-tents at and near the lower boundary.

The different scenarios produced an interest-ing range of flow situations.The layered structure(A4, B4) caused discontinuities in the water con-tent across layer boundaries, as well as some lat-eral flow along these boundaries (especially nearthe dry areas). As expected, the simulations witha stochastic distribution of Ks (A5, B5) generateda random structure of the water content (Fig. 5,bottom) as a consequence of local heterogeneity.These results are consistent with those of Roth(1995) and Hammel and Roth (1998) who no-ticed the development of flow channels. In addi-tion, Birkholzer and Tsang (1997) found that theflow patterns, ranging from relatively homoge-neous patterns to strong channeling effects, weresaturation dependent.

For comparison purposes we analyzed statis-tically (descriptive statistics) the water contentcross-sections of the irrigated central part of thesand tank at a depth of 60 cm. Results are shownin Table 5. Compared with the measured watercontent in the center part of the tank (0.25m3m�3), Method A generally gave much lowervalues (means of 0.13 to 0.19 m3m�3), whereasMethod B simulation generated results that weremuch more in line with the observed values(means of 0.23 to 0.28 m3m�3). The calculatedwetting fronts at that time (12 h) had reached dif-ferent depths for the various simulation runs.

The CVs in Table 5 indicate considerablevariability in the water content for the simula-tions, assuming a random Ks distribution (A5,B5)estimated from the measured bulk densities and


TABLE 4

Descriptive statistics of measured and calculated Darcian fluxes at the lower boundary two hours after the start of discharge in each run

Measured A1 A2 A3 A4 A5mean [m h�1] 0.0088 0.0139 0.0163 0.0156 0.0150 0.0163CV [%] 43 28 5 10 11 10min [m h�1] 0.0041 0.0077 0.0151 0.0129 0.0124 0.0141max [m h�1] 0.0149 0.0183 0.0177 0.0178 0.0175 0.0184

Measured B1 B2 B3 B4 B5mean [m h�1] 0.0088 0.0158 0.0185 0.0186 0.0107 0.0239CV [%] 43 39 32 44 18 38min [m h�1] 0.0041 0.0097 0.0088 0.0063 0.0073 0.0132max [m h�1] 0.0149 0.0253 0.0249 0.0278 0.0127 0.0393

CV is the coefficient of variation (n � 5).

Fig. 6. Cumulative boundary flux of selected simulationruns.


Fig. 7. Distribution of water content 12 h after onset of irrigation: Two-dimensional cross-sections and one-dimensional vertical profiles (arrows show position of vertical profile; note: different scaling to show an optimalresolution for each section).

grain-size distributions. Results for these twocases most closely resembled observed values.

CONCLUSIONS

Analysis of the water fluxes and water contentdistributions showed that the quality of the differ-ent simulation runs depended on which parame-ter of the system was used as the criterion (cumu-lative flux, spatial variation in discharge rate,two-dimensional water content distribution, orvariability). None of the simulation approachesstudied reproduced both the measured water bal-ance and the observed flow behavior exactly.From a detailed analysis of the simulation runs, thefollowing conclusions emerged:

1. The selected optimization procedures pro-duced different water retention parametersets. The in-tank measured data covered onlya small range of the entire water retentioncurve. Accordingly, the manner in which thesaturated water content, �s, was estimated wasthe most critical factor affecting all other soilhydraulic parameters and, directly or indi-rectly, the modeled flow regime. Agreementwith observed cumulative water fluxes wasbest when we assumed the presence of a ho-mogeneous soil profile, used a mean water re-tention curve as derived from the sensors at60 cm depth, and used optimized �s and �rvalues. In contrast, observed variabilities in thedischarge rate with time could be reproducedreasonable well with an average water reten-tion curve and using porosity for �s. This sce-nario also produced the most realistic range inmeasured water contents.

2. The geometry of the sand tank and the size ofthe irrigated area affected discharge genera-tion greatly.Lateral flow into the nonirrigatedareas as a result of soil water pressure head gra-dients was facilitated by the layered structureof the sand packing. The sloped side walls in-duced some lateral discharge at the lower

boundary.The lower boundary (seepage face)produced a saturated zone locally, with somelateral flow and nonuniform discharge.

3. Our results demonstrated numerically theoverriding effect of how the soil hydraulicproperties were distributed over the soil pro-file (a homogeneous profile, a layered system,or a profile having a random distribution ofKs). The heterogeneous (layered or random)profiles were implemented on the basis ofGPR information and in-tank measured soilphysical data, assuming such heterogeneousprofiles did not improve our predictions ofwater flow substantially. Thus, our results sug-gests that elaborate efforts to set up heteroge-neous cases, either numerically or physically,may not always be justified when water flowin sandy substrates is to be simulated.

ACKNOWLEDGMENTThis work was supported by the German Re-

search Foundation under grant ME 335/96–2.

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TABLE 5

Descriptive statistics of the observed water contents at 60 cm depth in the central part of the tank 12 hours after onset of irrigation (n � 163)

A1 A2 A3 A4 A5 B1 B2 B3 B4 B5

mean [m3m�3] 0.19 0.13 0.14 0.18 0.14 0.24 0.28 0.27 0.23 0.25CV [%] 5 5 7 5 22 6 6 7 9 15min [m3m�3] 0.15 0.11 0.10 0.14 0.08 0.18 0.19 0.18 0.15 0.16max [m3m�3] 0.19 0.14 0.14 0.19 0.20 0.25 0.29 0.28 0.24 0.32

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