improved calibration of time domain reflectometry soil water content measurements

Improved Calibration of Time Domain ReflectometrySoil Water Content Measurements

C. Dirksen* and S. Dasberg

ABSTRACTTime domain reflectometry (TDR) is becoming a widely used method

to determine volumetric soil water content, O, from measured effectiverelative dielectric constant (permittivity), e, using the empirical 0(c)Topp-Davis-Annan calibration equation. This equation is not ade-quate for all soils. The purpose of this study was to compare the Toppcalibration equation with a theoretical (Maxwell-De Loor) and anempiricial (fitting exponent a) mixing model for the fourcomponents: solid phase (s), tightly bound water (bw), free water, andair. Water content permittivity were measured, gravimetrically andby TDR, on packed columns of 11 soils ranging from loess to purebentonite. Measured specific surfaces were S = 25 to 665 m2 g-1 andbulk densities p,, = 0.55 to 1.65 g cm'3. Topp yielded accurate c(0)values only for the four soils with pb>1.30 g cm-3, including illite (S= 147 m2 g~')- Maxwell-De Loor gave similar accuracy for sevensoils, including attapulgite (S = 270 m2 g"1, Pb = 0.55 g cm-3),assuming a monomolecular tightly bound water layer (thickness 5 =3 x 10-'° m; fl̂ . = 8 p,,S), e^, = 3.2, and e, = 5.0. The «(fl) curveof these soils had the same shape as Topp. Two gibbsite soils withdissimilar curves required ,̂w = 3.2 and e, = 16 to 18, and twosmectite soil materials required e^ = 30 to 50 and e, = 5.0, to obtaingood fits. Deviations from Topp appear generally due more to thelower Pt and thus higher air volume fraction at the same O associatedwith fme-textured soils than to tightly bound water with low E. Botheffects, as well as apparent anomalous behavior such as decreasingeffective c with increasing e,, can be accomodated by the Maxwell-De Loor equation. This makes it a better calibration equation thanTopp. The empirical a model is sensitive to the unpredictable valueof a and cannot accomodate anomalous behavior.

TME DOMAIN REFLECTOMETRY is rapidly developinginto a major method to measure soil water con-

tent. With TDR, the soil relative dielectric constant(relative permittivity), e, is measured in the frequencyrange of about 10 MHz to 1 GHz. This parameter ismainly dependent on the water content, owing to thevery high value of water's relative dielectric constant.An empirical relation between the soil e and volu-

C. Dirksen, Dep. of Water Resources,Wageningen AgriculturalUniv., Wageningen, the Netherlands; S. Dasberg, Institute of Soilsand Water, AKO, Volcani Center, Bet Dagan, Israel. This studywas carried out at Wageningen Agricultural University. Received28 Apr. 1992. * Corresponding author.

Published in Soil Sci. Soc. Am. J. 57:660-667 (1993).

metric water content, 6, was, at an early stage ofdevelopment, claimed to be independent of soil type,soil density, soil temperature., and salinity (Topp etal., 1980):

e = 3.03 + 9.30 + 14602 - 76.70s [1]

Since the required probes are also relatively inexpen-sive and easy to make, TDR is indeed an attractivemethod for measuring 6. Recently, the Topp relation-ship (Topp et al., 1980) was built into software ofcommercial systems to translate pulse travel time di-rectly into 6, while fast automatic monitoring of mea-suring networks left unattended in remote sites is nowalso feasible (Baker and Allmaras; 1990; Heimovaaraand Houten, 1990; Herkelrath et al., 1991). The sameis being developed for simultaneous measurements ofsoil salinity (Dalton, 1992; Dalton et al., 1984; Das-berg and Dalton, 1985; Dalton and van Genuchten,1986).

If Topp were indeed universal, it would make TDRessentially a direct method for measuring absolute soilwater contents without need for calibration for differ-ent soils. However, even in their original paper, Toppet al. (1980) stated that for vermiculite and for organicsoils the e (6) relationship was different, showing littleresponse in the O < 6 < 0.1 region. Such discrepantbehavior has also been documented, for organic soil,by Herkelrath et al. (1991) and, for fine-textured soils,by Smith and Tice (1988) and Dasberg and Hopmans(1992).

The effect of soil texture on the e (ff) relation hasbeen known for some time in the remote sensing lit-erature, especially in the microwave frequency range,1 to 5 GHz (Dobson et al., 1985; Wang and Schmugge,1980; De Loor, 1990). Recent data on the dielectricresponse of soils in the frequency range 1 to 50 MHzshowed the marked soil dependence of this response(Campbell, 1990).

In view of this situation, we investigated the e (ff)relationship for TDR measurements on soils with dif-ferent clay contents and clay mineralogies. Our aimwas to obtain a single relationship that, in order to besuitable for calibration purposes, should involve onlysoil properties or parameters that are generally known

DIRKSEN & DASBERG: IMPROVED CALIBRATION OF TIME DOMAIN REFLECTOMETRY MEASUREMENTS 661

or can be estimated easily with sufficient accuracy.Earlier attempts to model the e (6) relation assumedfour component systems of solid phase, air, free water,and bound water, in which the bound water has alower e than the free water due to its proximity toactive colloidal surfaces. Two main approaches forthese mixing models are:

1. A theoretical model based on the Maxwell equa-tion, describing a homogeneous mixture of one or moresubstances randomly distributed in a medium with dif-ferent e (De Loor, 1964, 1990). The basic equationof this model was rewritten for a four-component sys-tem with plate-shaped soil particles as the host me-dium by Dobson et al. (1985):

3es + 2(6 -20bw(ebw - es)

Table 1. Composition of soils used in this study.

€ —

w - es)2(<t> - 0)(£a - £s)

3 + e ~ - 1)

[2]

where <£ = porosity, and subscripts bw, fw, a, and srefer to bound water, free water, air, and solid phase,respectively. This theoretical model contains onlyphysical parameters without additional fitting param-eters.

2. An empirical a model, proposed by Birchak etal (1974):

e- = 1 -

+ ^bw^bw [3]

The curve-fitting parameter a can be interpreted as ameasure of the geometry of the medium with relationto the applied electric field. For two-phase systems,it can be shown that - 1 < a < 1 (Roth et al., 1990).These authors found an average value a = 0.50 fora variety of soils, assuming three phases (no boundwater). Dobson et al. (1985) found an average valuea = 0.65 for four-phase systems.

We compared relative dielectric constants measuredwith TDR on a variety of soil materials at a range ofwater contents with calculated e values according tothe theoretical and empirical mixing models (Eq. [2]and Eq. [3], respectively), as well as Topp (Eq. [1]).

MATERIALS AND METHODSEleven soils were used in this study. Four Dutch soils were

available at our laboratory: Groesbeek loess (Typic Hapludalf),Wichmond valley bottom sandy loam (Typic Haplaquent),Munnikenland fluvial silty clay loam (Typic Haplaquept), andY-Polder marine silty clay (Typic Haplaquept). Four clay soilswere obtained from the International Soil Reference and In-formation Centre (ISRIC) at Wageningen: the A and B hori-zons of a Brazilian humic Ferralsol (Typic Acrortox), a FrenchMediterranean red soil (Typic Rhodoxeralf), and a Kenyanpellic Vertisol (Typic Pellustert). Three pure clay minerals wereobtained from the Department of Soil Science and Plant Nu-trition of Wageningen Agricultural University: bentonite(smectite, from Osage, WY), illite (Grundite Co.), and atta-pulgite (source unknown). Attapulgite, also called palygor-skite, is a clay mineral with high water-absorbing capacityrarely occurring in soils with a fibrous morphology. The clay,

SoilOrganic

Clay Silt Sand matter Clay minerals

Groesbeek 10 70 20 0.95WichmondFerralsol-AFerralsol-BMunnikenlandMediterraneanY-PolderVertisol

14635740404586

31263356344210

5511103

27134

4.3005.00.44.61.4

smectite + vermiculitegibbsite + kaolinitegibbsite + kaoliniteillite + kaoliniteillite + kaoliniteillite + kaolinitesmectite

silt, sand, and organic matter contents and the major types ofclay minerals determined by X-ray diffraction are given inTable 1.

All determinations were made in duplicate. Air-dry soil ma-terial (where appropriate, first sieved to <2.0 mm) was step-wise brought to the desired water contents using an atomizedwater spray assembly (Dirksen and Matula, 1992). After soiland water were thoroughly mixed at each water content, twoacrylic cylinders (5-cm diam., 12.5 cm long) were packed toa bulk density as uniformly as possible, and their weight andvolume were determined. The TDR measurements were car-ried out with a TDR cable tester (Model 1502B, TektronixHolland N.V., Hoofddorp, the Netherlands) with a primaryfrequency range of 10 MHz to 1 GHz. For all measurementswe used the same three-rod wave guide (10 cm long, 1.0-cmdistance between inner and outer rods, 0.2-cm rod diameter),directly connected to the cable tester by a 3.5-m length of 50fl coaxial cable. The waveform obtained on the cable testerwas transferred and stored on an IBM-compatible PC, accord-ing to the program developed by Heimovaara and Bouten (1990),for later retrieval and analysis. After the TDR measurements,the soil was removed from the cylinders and samples weretaken for gravimetric water content determinations, from whichvolumetric water contents were calculated based on the averagebulk density of the packed cylinders, p,,. These steps wererepeated 8 to 12 times for each soil, resulting in 0 incrementsof 0.02 to 0.03 cm cm-3. The TDR waveforms were analyzedwith the program of Heimovaara and Bouten (1990). The exactpoint in the TDR waveforms corresponding with the entranceof the sensor into the soil—the apparent length of the (one)waveguide used for all TDR determinations—was calculatedfrom TDR measurements in air and in water (T.J. Heiovaara,1991, personal communication).

The following determinations were also made on all soils:hygroscopic water content (air dryness), water content at wilt-ing point (1.5 MPa, pressure plate), and specific surface withethylene glycol monoethyl ether, which is adsorbed on claymineral surfaces in a similar way as water (Carter et al., 1986).

RESULTS AND DISCUSSIONExperimental Data

The basic experimental data are the values of relativedielectric constant and bulk density measured in the packedsoil columns as a function of their volumetric water con-tents. These values are plotted in Fig. 1 against the leftand right y axes, respectively, for all 11 soils in orderof increasing measured specific surface, 5, listed in Ta-ble 2. The total number of data sets (packed columns persoil), n, and maximum water content, 0max, are given inTable 3.

A major problem in applying the four-component mix-ing models, Eq. [2] and [3], is assigning values to thevolume fraction of tightly bound water, 6,,w, and its av-erage relative dielectric constant, ebw. The models can

662 SOIL SCI. SOC. AM. J., VOL. 57, MAY-JUNE 1993

32

c 24

coo

16

a. Qroesbeek b. Wichmond c. Ferralsol-A

2.0

1.5 E^J^>

cCD

•

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.50.0

32

24

<ncOo

d. Ferralsol-B e. Munnikenland f. Mediterranean

2.0

1.5 Eo"B

1-° %c<D

T3

0.5 =

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

32

c 24rawcoo

2.0

16

g. Y-Polder h. Illite

0.0 0.1 0.2 0.3 0.4 0.5

32

24

coo

16

J»(D

. Vertisol

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6

""•""I 2 ° water content - cm3/cm3

t;

k. Bentonite

1.5 E

>.c

0.5

0.00.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

water content - cm3/cm3 water content - cm3/cm3

Fig. 1. Dielectric constants vs. soil water content. Comparison between time-domain reflectometry data, Topp, and Maxwell-DeLoor model. For Soils a, b, e, f, g, h, and i: initial parameter values are (number of molecular water layers / = 1, bound waterpermittivity e^ = 3.2, soil phase permittivity e, = 5); for Soil c: e, = 18; for Soil d: e, = 16; (For Soil j: %,w = 30; and forSoil k: «,,„ = 50. Measured bulk densities of packed soil columns are also given.


Table 2. Measured soil parameters and volume fraction ofmonomolecular layer of tightly bound water.

soilor puremineral

GroesbeekWichmondFerralsol-AFerralsol-BMunnikenlandMediterraneanY-PoIderIlliteAttapulgiteVertisolBentonite

bulkdensityt

(ft,)g cm-3

1.491.361.141.131.131.381.081.300.550.920.94

specificsurface

(S)m'g-'

254161617993

107147270428665

boundwaterf(<O

0.0110.0170.0210.0210.0270.0390.0350.0570.0450.1180.187

hygro-scopicwater

cm3 cm"3

0.0170.0220.0250.0280.0350.0430.0400.0500.0390.1180.114

wiltingpoint§

0.0770.1200.2600.2450.1770.1850.2000.2900.3200.375

—

t Average of packed columns (for variation, see Fig. 1).t Monomolecular layer, 0bw = 8 p,, 5.§ 1.5 MPa pressure plate determinations.

only accommodate one average value for each parame-ter. The volume fraction is determined by the magnitudeand properties of the soil specific surface (colloidal ac-tive surface area per unit mass), which depend on claycontent and clay mineral type. Related theoretical (Guret al., 1978) and experimental (Israelachvili and Pashley,1984) evidence point to a gradual (exponential) increasein e with distance from the mineral surfaces. In general,the volume fraction of tightly bound water covering themineral surfaces can be approximated by

[4]

where / is the number of molecular water layers of tightlybound water, and 8 = 3 x 10~10 m is the thickness ofone molecular water layer. The lower limit of tightlybound water would appear to be a monomolecular layer(/ = 1). This lower limit, estimated according to Eq.[4], as well as other related measured soil parameters,are listed in Table 2. Except for bentonite, Table 2 showsa close correspondence between the estimated bound waterand the hygroscopic water content. Banin and Amiel(1970) also found a high correlation between S and thehygroscopic water content, and suggested use of the lat-ter as a predictor for S by assuming a monomolecularwater layer covering all mineral surfaces. The water con-tent at wilting point, which could be another measure ofbound water, has much higher values.

The relative dielectric constant of free water is e,̂ =81 at 18 °C, and it decreases by 0.37 per 1 °C (Wob-schall, 1977). The lower limit of the dielectric constantof the tightly bound water is likely to be similar to thatof ice, or ebw = 3.2. Dobson et al. (1985) found goodcorrespondence between theory and experiment for e^values between 20 and 40, but this was for more than amonomolecular layer.

ToppThe Topp empirical equation is plotted in each seg-

ment of Fig. 1. For the majority of soil materials, the fitis unsatisfactory. These fits can be quantified by com-paring each pair of calculated (Topp) and measured evalues (TDK) individually by means of a linear regres-sion (forced through the origin) according to:

^calculated = Ae,'measured [5]

The resulting coefficients of linear regression, A, andconstants of determination, R2, according to Eq. [5] arepresented in Table 3. These results show that Topp pre-dicts e within the (arbitrary) conditions 0.91 < A < 1.09and R2 > 0.98 only for Groesbeek, Wichmond, Medi-terranean, and illite, while Y-Polder falls just outsidethese limits. Topp values are, on the average, too large(A > 1.00), except for Groesbeek. This proves the needfor other (additional) calibration functions.

Initial Set of Parameter ValuesFaced with the fact that the mixing models can ac-

comodate only one average value for the volume fractionand dielectric constant of the bound water, we startedour analysis, for lack of additional information, by as-suming a monomolecular water layer (/ = 1) with thedielectric behavior similar to ice (ebw = 3.2) coveringS. The remainder of the water, if any, was assumed freewith e ,̂ = 81. For the relative dielectric constant of thesoil materials we assumed es = 5, which is an averageof published values in the TDR frequency range (Alhartiand Lange, 1987; Roth et al., 1990; Topp et al., 1980).

Maxwell-De Loor ModelThe results of the linear regressions according to Eq.

[5] for the relative dielectric constants measured withTDR and calculated with the theoretical Maxwell-De

Table 3. Linear regression results according to Eq. [5] for full number («) of data sets, with 0max the highest occurring watercontent. Initial parameter values / = 1, ,̂w = 3.2, and e, = 5; a was selected such that 0.99 < A < 1.01.

Topp Maxwell-De LoorSoil

GroesbeekWichmondFerralsol-AFerralsol-BMunnikenlandMediterraneanY-PolderIlliteAttapulgiteVertisolBentonite

n

no.2115222012221221262034

e™,cm3 cm~3

0.3450.3320.4370.4240.4030.4330.4410.4730.5330.3400.371

A

0.9951.0671.1181.0331.1941.0671.1041.0871.7201.2871.067

IP A

0.9900.9900.7260.617 (0.9860.982

.087

.111

.007).930.032.040

0.967 0.9280.993 0.9320.978 1.0680.757 0.6700.693 0.412

B?

0.9900.9960.8110.7180.9920.9830.9870.9820.9780.9170.963

A

1.0011.0041.0011.0000.9940.9960.9930.9970.9940.9950.730

a modelR2

0.9840.9850.6980.5240.9850.9820.9570.9890.9840.9160.956

a

0.520.490.540.610.500.600.600.670.390.811.00


Loor model (Eq. [2]) for the initial set of parameter,values 1=1, ebw = 3.2, and es = 5, are given in Table3. Using the same criteria used above for Topp, Max-well-De Loor gave a satisfactory fit for seven soils. Be-sides the four soils for which Topp gave good predictions,Y-Polder also fell well within the selected criteria, whilethe improvement for Munnikenland and attapulgite wasquite significant. The results for Groesbeek and Wich-mond were slightly worse than with Topp. The predictedMaxwell-De Loor values for these seven soils are con-nected by solid line segments in Fig. 1. These seven"regular" soils show a rather smooth increase of e withwater content, similar to Topp. The TDR values of theFerralsols, Vertisol, and bentonite exhibit a slower in-crease with water content than Topp to about 6 = 0.25cm3 cm~3; beyond that approximate water content, theyincrease much more rapidly. As a result, neither of thetwo equations predict the experimental e values for these"irregular" soils correctly.

Bulk densities differed not only between soils (Table2 gives average values), but also between packed col-umns of the same soil. Generally, bulk densities firstdecreased and then increased with water content (Fig. 1,secondy axis). Since bulk density determines the volumefractions of solid phase, air, and water, its variationsaffected both the measured (TDR) and calculated Max-well-De Loor values. This explains the irregularities inthe Maxwell-De Loor plots in Fig. 1, which are partic-ularly prominent for Mediterranean and Ferralsol-A. Forthe same reason, the goodness of fit of the mixing modelswith the TDR values cannot be judged by the degree ofdeviation from a smooth curve such as Topp. Instead, itcan be judged properly by comparing each pair of TDRmeasured and calculated e values individually. This oc-curs by the linear regressions according to Eq. [5].

Influence of Specific Surface and Bulk DensityThe differences between TDR values and Topp (Fig.

1) do not increase monotonically with the magnitude ofthe specific surface, and thus with the volume fractionof bound water, as we initially expected. Most pro-nounced in this respect is attapulgite, which exhibits thelargest difference with Topp, though its specific surfaceis only 40% of that of bentonite. The attapulgite columnscould be packed only to very low bulk densities (averagePb = 0.55 g cm~3). At the highest water content, stillalmost one-half the porosity was filled with air, with itsvery low dielectric constant ea = 1. This suggests thatthe maximal deviations of the TDR results from Toppobserved for attapulgite are not so much due to its largespecific surface, as to its very low bulk densities. Thiswas confirmed by comparing hypothetical Maxwell-DeLoor curves. For instance, the calculated curve (not shown)for a soil with the specific surface of illite (S = 147 m2

g-1) and the bulk density of attapulgite deviated almostas much from Topp as attapulgite (5 = 270 m2 g"1).

A general impression of the sensitivity of the dielectricconstant for specific surface (i.e., bound water fraction)can be obtained from calculated Maxwell-De Loor curvesfor hypothetical soils with p^, = 1.0 g cm~3 and 5 = 0,100, 200, 400, and 600 m2 g-1 (Fig. 2). The differencebetween coarse soils (5 = 0) and swelling clay soils (5= 600 m2 g~') increases with increasing water content.For the chosen bulk density, all curves are located below

0.0 0.1 0.2 0.3 0.4 0.5

water content-cm3/cm3

Fig. 2. Influence of specific surface on dielectric constant vs.soil water content relationship according to the Maxwell-De Loor model for hypothetical soils with bulk density pt,= 1.0 g cm-3, and specific suiface 5 = 0, 100, 200, 400,and 600 m2 g"1, respectively. The Topp equation is alsoshown.

Topp. For coarse soils (5 = 0)., the calculated curve (notshown) is close to Topp when p,, = 1.35 g cm~3. Thesensitivity for bulk density can be evaluated on the basisof calculated curves for hypothetical soils such as in Fig.3 for 5 = 150 m2 g-1 and ̂ == 0.6, 0.9, 1.2, 1.5, and1.8 g cm~3. Similar plots show that Topp agrees closelywith combinations in the range S = 0 to 100 m2 g ~!

and pb = 1.35 to 1.5 g cmr3. These values representthe typical range of bulk densities and specific surfacesand support the generally found validity of Topp for soilswith low clay contents. As the texture of natural soilsbecomes finer, this not only increases the specific sur-face, but also generally results in lower bulk density.Both factors have a lowering effect on the overall di-electric constant and thus increase the deviation fromTopp.

Curtailed Data SetsThe theoretical Maxwell-De Loor model assumes that

the water consists of (disk-shaped) foreign inclusionsembedded in a homogenous isotropic dielectric medium(mineral phase). As the water content increases beyondabout one-third of the total volume, this assumption be-comes less valid and the accuracy of the model predic-tions may decrease (De Loor, 1964, 1990). Figure 1shows that the Maxwell-De Loor values tend to deviatefrom the TDR data at medium to high water contents.Therefore, we performed linear regressions omitting thehighest remaining water content one by one until thevalue A = 1.00 was approached most closely or thehighest remaining water content was less than about 0.3.Curtailed data sets yielded near-perfect results for Med-


32

c(0tocooO

oJ2?0)=5

24

16

S = 150 m2/g

0.0 0.1 0.2 0.3 0.4 0.5

water content-cm3/cm3

Fig. 3. Influence of bulk density oh dielectric constant vs. soilwater content relationship according to the Maxwell-De Loormodel for hypothetical soils with specific surface S = 150m2 g-', and bulk density pt, = 0.6, 0.9, 1.2, 1.5, and 1.8 gcm~3, respectively. The Topp equation is also shown.

iterranean and Y-Polder; the remaining number of datapoints, remaining maximum water content, and new val-ues of A and R2 are given in Table 4. For the other five"regular" soils, curtailing the data set did not yield im-proved fits. The excellent A values for the Maxwell-DeLoor model with the full data sets of the.Ferralsols provedto be fortuitous; they changed drastically when only watercontents up to 6 = 0.326 cm3 cm~3 were considered.Of course, this was reflected in the low R2 values for thefull data sets. For the "irregular" soils, the regressionsfor the curtailed data sets are still not satisfactory.Empirical a Model

The analysis with the theoretical Maxwell-De Loormodel and the initial set of parameters is straightforward.For the empirical a model, however, the value of a muststill be selected. We adjusted a until A differed < 0.01from unity (near A = 1.00, a and A vary by roughlythe same amount, while R2 remains the same). Thesebest-fit values of A, R2, and a are given in Table 3. Forbentonite, the theoretical maximum value a = 1.00 stilldid not yield A = 1.00. The R2 values of the Ferralsolsare too low to consider the fit by the a model acceptable.For the seven "regular" soils, a varied between 0.39and 0.81. Even with the extra fitting parameter, we foundthat generally the a model gave a slightly smaller R2

value than the Maxwell-De Loor model for the same Avalue. Since the value of a is not known, the a modelpresents no advantage over the Maxwell-De Loor modeland will not be considered further.

Adjusted Parameter ValuesThus far, the TDR data have been analyzed by using

the initial set of parameter values. With it, the theoretical

Table 4. Linear regression results according to Eq. [5] forcurtailed data sets and adjusted parameter values. 0ma> isthe remaining highest water content for the curtailed numberof data sets n; e^ and e, are dielectric constants of boundwater and solid phase, respectively; / is number of molecularlayers of bound water.

curtailment Maxwell-De LoorSoUMediterraneanY-PolderFerralsol-AFerralsol-BVertisolBentoniteIlliteY-PolderAttapulgiteWichmondGroesbeek

adjustment—— .

.̂ = 18^ = 16

«w = 30«k- = 50«w = 20«w = 40; = 1.51 = 2.5/ = 3.0

n181016161419—....——

"max

0.3990.2730.3260.3200.2450.249————.. —.._.——

A1.0101.0051.0070.9930.9950.9990.9900.9921.0010.9981.001

fl2

0.9860.9860.9160.8080.9170.9110.9820.9840.9780.9940.990

Maxwell-De Loor model yielded good to excellent re-sults for seven "regular" soils, but it could not followthe abrupt changes for the four "irregular" soils. Forthe Ferralsols, the Maxwell-De Loor model with initialvalues matched the TDR results only at the low and highwater contents; for the Vertisol it came closely only atintermediate water contents, and for the bentonite therewas no match at all. The TDR data are consistent andthere is no reason to doubt their accuracy. Therefore, inview of the good results of the Maxwell-De Loor modelwith the "regular" soils, the initial parameter valueswere adjusted to look for better fits. Since the model hasa theoretical basis, this could also lead to possible ex-planations for the different behavior of the four soils.The results discussed below are compiled in Table 4.

For the Ferralsols (Fig. Ic and Id), the different be-havior is most likely due to the main constituent of theclay fractions, gibbsite. This Al hydroxide can be ex-pected to have dielectric behavior similar to metal ox-ides, which have dielectric constants of about es = 15(Weast, 1979, p. E-58). Keeping the thickness and di-electric constant of the bound water layer the same, butvarying the third of the initial values, es, and using onlywater contents up to 6 — 0.326, the Maxwell-De Loormodel yields A = 1.00 within 0.01 for Ferralsol-A andFerralsol-B at es = 18 and es = 16, respectively. Thedata for 6 > 0.326 cm3 cm~3 were deleted from thelinear regressions, but Fig. Ic shows that Maxwell-DeLoor with es = 18 predicts the very steep increase of ewith 0 for Ferralsol-A quite well (last two data points).This is also shown in the 1:1 plot in Fig. 4, which mayserve as a general illustration for the linear regressionsaccording to Eq. [5]. The significance of the value R2

= 0.916 for this soil can be judged with this figure,keeping in mind that the six highest water contents wereexcluded from the linear regression. The value R2 =0.808 for Ferralsol-B is not satisfactory. Variations ofall other combinations of model parameter values leadto worse fits for both soils.

It should be emphasized that the overall e of the mix-ture was decreased by increasing es. This anomalous be-havior is inherent in the Maxwell-De Loor model withits complex influence of es in both numerator and de-nominator. The a model yields an increase in overall ewith an increase in e, for the same a. It can accomodate


OO

35

28

9 21

"53X 14

co 7

+ +

14 21 28 35

e - T D RFig. 4. Comparison of time-domain reflectometry data of

Ferralsol-A soil and Maxwell-De Loor values for numberof molecular water layers 1 = 1, bound water permittivity

,̂w = 3.2, and solid phase permittivity e, = 18 (same asFig. Ic). Goodness of fit obtained by linear regressionaccording to Eq. [5] for n — 16, represented by A and R2

values in Table 4.

the anomalous behavior only by slightly negative a val-ues (a = —0.10 and —0.03, for practically the sameA and R2 values), without any apparent physical reason.

For the Vertisol and the bentonite, the Maxwell-DeLoor fits to the TDK data could be improved only byincreasing values of ebw; changing more parameter val-ues did not improve the predictions. Dobson et al. (1985)reported that a comparison between model predictionsand measured data indicated dielectric constant of boundwater of the order of 20 to 40. Omitting again the higherwater contents (6 > 0.25 m3 m~3), values^ = 1.00 areobtained for ebw = 30 and 50, respectively, with R2 >0.91. The Maxwell-De LOOT results for these parametervalues are plotted in Fig. Ij and Ik. The value ebw =50 for bentonite falls outside the range found by Dobsonet al. (1985); however, ebw = 40 yielded only.4 = 0.88.The fit of illite also improved by changing to e^ = 20.The Y-Polder fit became nearly perfect for ebw = 40,but also by just eliminating the two highest water con-tents (Table 4). Y-Polder stands out for its low bulkdensity. With the same use of a as a fitting parameteras before, the a model yielded comparable fits for theVertisol and bentonite with the regular value of a =0.44

The Maxwell-De Loor fit for attapulgite did not im-prove by curtailing the data set, even though the maxi-mum water content was 0.533. The fit becomes nearlyperfect across the entire water content range with / =1.5. To show this, the water content scale for attapulgitein Fig. 1 is different from all others. The fits for Wich-mond and Groesbeek also became nearly perfect bychanging / to 2.5 and 3.0, respectively. These soils yieldedconsistently too-high values for the initial parameters (Fig.la and lb).

Thus far, we changed only one parameter value with

respect to the initial values. Changing more than onevalue, such as increasing / and e,,w, did not improve fits.We tried, in particular, to obtain better fits for the abruptchanges in slope for the "irregular" sols, which seemsto be associated with a change from bound water to freewater. Each of the "legs" could be matched quite well,but not both of them. In view of the limitation for theMaxwell-De Loor model, we chose in the first place tomatch at the smaller water contents.

SUMMARY AND CONCLUSIONSFigure 1 presents the basic experimental data obtained

on the packed soil columns: gravimetrically determinedwater contents and bulk densities, and relative dielectricconstants (permittivities) measured by TDR. All otherpresented results can be derived from these basic data,the specific surface areas listed in Table 2, and the equa-tions. Because a was used as an additional fitting pa-rameter, the a model generally yielded values close tothe Maxwell-De Loor model. To keep Fig. 1 clear, noneof the a model results are plotted. The merits of theMaxwell-De Loor model and a model can be evaluatedon the basis of Tables 3 and 4. Although measurementsand analyses on a larger assortment of soils would berequired to establish these more firmly we feel that ourresults make it possible to draw the following tentativeconclusions.

Topp (Eq. [1]) was found to be valid for the soils withlow clay contents (specific surface) and typical bulk den-sities (pb = 1.35-1.50 g cm~3). Deviations from Toppappear to be more due to lower bulk densities associatedwith fine-textured soils than to bound water with a lowrelative dielectric constant. The Maxwell-De Loor model(Eq. [2]) can account for both factors with average val-ues of the volume fraction and dielectric constant of thetightly bound water. This model has no fitting parame-ters and a TDR calibration curve can be derived fromthe easily measured soil properties, bulk density, andspecific surface. We tested the fit to TDR data by linearregression per data pair according to Eq. [5]), thus takingbulk density variations within each soil into account. Theresults in Table 3 suggest that, for most naturally oc-curring soils, the apparent dielectric behavior of tightlybound water can be calculated from the specific surface,assuming a monomolecular water layer (thickness 3 x10~10 m) with a relative dielectric constant ebw = 3.2,and taking es = 5 for the soil material. If the specificsurface is not known, the easily measured hygroscopicwater content can be used as an estimate of the boundwater fraction (Table 2). If the soil contains high amountsof smectitic clays, or unusual components such as gibb-site, the model parameters need to be adjusted. Whetherthis can be done on the basis of the specific knowledgeto which the curve-fitting results point remains to beseen. We have generally not been able to match theMaxwell-De Loor model with the TDR data at higherwater contents, particularly for the "irregular" soils. Thismay be due to the fact that the underlying assumptionsof this model are no longer strictly valid if water occu-pies more than about one-third of the total volume.

Where the A values were about the same for the Max-well-De Loor model and the a model, the R2 values forthe latter were generally slightly lower. Furthermore, the


fits with the a model are very sensitive to the value ofa. In this study, a could be fitted to the TDK data; itsvalues varied from -0.10 to 0.81. Normally, a is notknown for any given soil, nor can it be measured directlyor inferred from other soil properties. Finally, the a modelcannot inherently accomodate apparent anomalous di-electric behavior as found for the Ferralsols. These ar-guments lead us to the conclusion that the a model isinferior to the Maxwell-De Loor model, especially foruncommon soils.

A similar study has recently been published by Rothet al. (1992). The experimental results for their mineralsoils are generally in agreement with our data, but theircurve for "other capillary-porous media" was locatedabove the Topp curve (see their Fig. 5), while our sim-ilarly shaped curves for the "irregular" soils are locatedbelow that curve. A more significant difference is that,in the data analysis they restricted themselves to poly-nominal regressions similar to the Topp equation.

ACKNOWLEDGMENTSWe gratefully acknowledge the International Soil Reference

and Information Centre (ISRIC) at Wageningen and the De-partment of Soil Science and Plant Nutrition of WageningenAgricultural University for providing soil materials. We alsothank ISRIC for the assistance with the soil specific surfacemeasurements, and Dr. J.D.J. van Doesburg of the Departmentof Soil Science and Geology of Wageningen Agricultural Uni-versity for performing the x-ray diffraction determinations.

improved calibration of time domain reflectometry soil water content measurements

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