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Comparison of the Instantaneous Profile Method and inverse modelling for the prediction of effective soil hydraulic properties.


Investigations into water movement in soils require knowledge of the hydraulic conductivity function K([theta]) or K(h) and the soil water retention [theta](h) function, where [theta] is the volumetric soil water content and h is the soil matric pressure head. Owing to their relative importance in many disciplines, including environmental engineering, soil physics (Hopmans et al. 2002), and agricultural and environmental issues (Vachaud and Dane 2002), numerous methods are being developed and improved to effectively determine soil hydraulic properties. These properties are difficult to measure and therefore require the use of both direct and indirect methods to adequately describe the flow and transport processes. Several field and laboratory methods for such determinations exist, each having their own limitations. In-situ determinations of K([theta]) are generally preferred owing to the potential problem of relating K([theta]), determined on undisturbed soil cores in the laboratory, to actual K([theta]) data in the field. Despite the application of various methods, erroneous results are frequently observed (Klute and Dirksen 1986; Dirksen 1991) since there is no single approach that can be generalised for all applications. Problems arise due to the heterogeneity of porous media, the inherent spatial variability of soil (Tseng and Jury 1993), and lack of validity in the assumptions used in the various methods. The dynamic nature of the soil physical processes and the strong non-linearity of soil hydraulic functions also pose serious problems. As a result of these limitations, measured and predicted values often disagree strongly. In particular, an understanding of the spatial variability of soil hydraulic properties has become an essential factor for predicting transient water flow in the field. Therefore, comprehensive evaluation of different approaches is necessary before they can be accepted as tools for predicting hydraulic properties in heterogeneous media.

The 1-dimensional flow of water in a porous medium can be represented by the following general flow equation (Klute 1973):

(1) C(h) [partial derivative]h / [partial derivative]t = [partial derivative][[KAPPA](h) * ([partial derivative]h / [partial derivative]z + 1)] / [partial derivative]z

where C(h)= [partial derivative][theta]/[partial derivative]h is the water capacity function, t is the time, and z is distance positive above soil surface. The main problem associated with the solution of Eqn 1 is the determination of the soil hydraulic functions for a particular soil. Traditionally, these properties are measured directly in the laboratory or in the field (e.g. instantaneous profile method). Laboratory measurements often lead to hydraulic properties that are not representative of the field (Hopmans et al. 2002). The increasing importance of having accurate and reliable data has also accentuated the need for the development of accurate methods for describing soil hydraulic characteristics (Hopmans et al. 2002). Direct field methods have been extensively used to measure these properties, e.g. plane of zero flux (Arya 2002), constant flux vertical time domain reflectrometry (Parkin et al. 1995), and the instantaneous profile method (IPM) (Vachaud and Dane 2002). Although still most reliable, these methods have proven to be expensive and laborious and also require the imposition of restrictive initial and boundary conditions for arriving at analytical or semi-analytical solutions. This has prompted attempts to find a more efficient way of determining these properties. An alternative approach is the application of what is known as the inverse procedure, whereby the hydraulic functions K([theta]) and/or [theta](h) are indirectly predicted from measurable easily determined properties such as water content or pressure heads by optimising and estimating the model parameters.

Kool and Parker (1988) in their analysis of inverse modelling, listed several advantages connected with this approach: (i) it allows for some flexibility in initial and boundary conditions, (ii) parameters determined in this way give the optimal reproduction of the transient flow event by a numerical model, and (iii) the availability and use of computers makes it even more convenient to apply. The inverse procedures are equally applicable to field experiments even under non-trivial boundary conditions (Kool and Parker 1988; Hopmans et al. 2002) and large-scale spatially distributed properties (Vrugt et al. 2004).

There has been an increasing interest in several models which describe the non-linear soil hydraulic functions K([partial derivative]) and/or [partial derivative](h) analytically and not all are generally applicable to all conditions, van Dam et al. (1994) tested the applicability of inverse modelling for outflow experiments and found that multi-step outflow data yield unique estimates of K([partial derivative]) and [partial derivative](h), while l-step outflow data yield non-unique solutions. Chen et al. (1999) also tested 7 different soil hydraulic models in outflow experiments and concluded that only 4 of these models were able to describe the outflow data successfully. Several investigators have proposed theoretical pore-size distribution models that predict the hydraulic conductivity from more easily measured soil water retention data (Millington and Quirk 1961; Mualem 1976, 1992; van Genuchten 1980; Arya et al. 1999b; Chaudhari and Batta 2003). These models have assumed mathematical expressions for hydraulic conductivity and water retention characteristics, thus enabling easy computation of soil hydraulic properties. A commonly used model is that proposed by van Genuchten (1980):

(2) [S.sub.e] = [theta] - [[theta].sub.r] / [[theta].sub.s] - [[theta].sub.r] = [[1 + [([alpha]|h|).sup.n]].sup.-m]

(3) [KAPPA] = [[KAPPA].sub.s][S.sup.[tau].sub.e][[1 - [(1 - [S.sup.1/m.sub.e]).sup.m]].sup.2]

(4) m = 1 - 1/n

where [S.sub.e] is the effective saturation (-); [[theta].sub.r] and [[theta].sub.s], are residual and saturated soil water content ([cm.sup.3]/[cm.sup.3]), respectively; [tau] is the tortuosity (-); [K.sub.s] is the saturated hydraulic conductivity (cm/h); K is the unsaturated hydraulic conductivity (cm/h); [alpha] is the inverse of the air entry value (1/cm); and n is a parameter which depends on the width of the pore size distribution (-). These empirical relations are used to predict soil hydraulic properties during a transient flow process. An improved prediction will be realised if the model parameters ([tau], [alpha], n, [[theta].sub.r], [[theta].sub.s], [[KAPPA].sub.s]) are well estimated using a nonlinear least square parametric estimation technique. Typically the number of parameters needed to describe the soil hydraulic functions varies between 4 and 7 (Durner et al. 1997) and may reach 8 for bimodal hydraulic models (Zurmuhl and Durner 1998).

Dane and Hruska (1983) found that this model gave a good description of water retention and/or conductivity data for a large number of soils. While the water retention data showed good agreement between predicted and measured values, the comparison for the conductivity data was less satisfactory. It was argued that this was because of the overestimation of the saturated hydraulic conductivity attributed to soil structure or macropores because of their influence to water flow regimes near saturation (van Genuchten et al. 1999; Chaudhari and Batta 2003). It therefore becomes essential to comprehensively evaluate different approaches before they can be accepted as suitable for predicting soil hydraulic properties (Tseng and Jury 1993).

The objective of this study was to compare the instantaneous profile method and the inverse modelling in effectively predicting soil hydraulic properties during lysimeter drainage experiments for sandy and loamy soils. The inverse modelling was also evaluated for its effectiveness following the optimisation of parameters in various datasets including water content, pressure head, and/or a combination of both.

Materials and methods

Materials and experimental set up

In this study, determinations of soil hydraulic properties were carried out using a lysimeter drainage experiment for 1-dimensional transient flow for 2 distinct soils with contrasting textural properties (Table 1). Collected samples of sand and loam soils were filled into 2 lysimeters, each with a cross-sectional area of 1 [m.sup.2] and depth of 1.20 m, and were set up for the drainage experiment. Both lysimeters had holes drilled in the sides, for positioning of tensiometer cups attached to mercury manometers, and an outlet or holes at the bottom through which water could drain.

Soil water content and pressure heads were measured during gravity drainage at selected depths following flooding of the lysimeters with tap water. Soil water contents were measured using time domain reflectrometry (TDR), while soil water potentials were measured using the tensiometers. Prior to the taking of measurements, the lysimeters were flooded until water ran out through the outlet to ensure uniform distribution of water. Before flooding, measurements of [theta] and h were made in order to estimate residual water content Or from measured low water content under dry conditions for the soil, which is also one of the model parameters. The experiment took place in a period (i.e. winter time) when there was small amount of evaporation. The lysimeters were also shielded against wind in order to prevent evaporation, while the soil surface was covered by plastic sheets to maintain a zero flux as top boundary condition and maintain a constant temperature during the experiment. The bottom boundary condition was achieved with a lysimeter with free drainage. Lysimeters had holes or outlets at the bottom to allow water to drain freely under gravity. Measurements and monitoring of soil water content and pressure heads continued over a period of approximately 2 weeks, from which reliable data with higher resolution were obtained in the first 10 days.

The unknown parameters are estimated by minimising the difference between observed and fitted values as illustrated below:

(5) [Min.sub.b]O(b) = [k summation over [i = 1] [[w.sub.i]([Q.sub.1]-[Q.sub.2]*).sup.2]

where, [Min.sub.b]O(b) is the objective function; [Q.sub.i] is observed attribute like soil water content, pressure heads or cumulative infiltration; [Q.sub.i]* is the model predicted values of attributes corresponding to observed values and for a particular set of estimated parameters; k is the number of observed values; and [w.sub.i] is the weighting factor for ith observation. The weighting factor weighs the observed values depending on their accuracy and correlation with other values. Additional weighting can be assigned to individual data (Hollenbeck and Jensen 1998a). The best optimised parameter set could be judged by the lowest sum of squares SSQ such that the difference between the measured and the predicted data is minimal, when there is no longer a change in the sum of squares.

Tensiometry or water potential measurements

Tensiometer ceramic cups with mercury manometers were installed horizontally and sealed into the holes drilled in the sides at the depths 0.15, 0.25, 0.35, 0.45, 0.55, and 0.75m. These were tilted slightly upwards to enable air to escape from the system. The cups were connected to a mercury reservoir and the system was flushed with deaerated water to avoid air entrapment. In all cases, holes of comparable sizes to the tensiometers were made to ensure good contact between tensiometer cups and soil to provide rapid adjustment to changes in soil water status. The first reading was taken as soon as water disappeared from the surface of the soil. Initial observations were made at small time intervals, since the changes in drainage were relatively faster at initially high water contents. Subsequently the time intervals were increased to about 3 days. The water potential or pressure heads at selected depths were computed according to h = -12.61 + [z.sub.1] + [z.sub.2], H = h + z, where h is pressure head in the ceramic cup (cm), l is mercury column length (cm), [z.sub.1] is the depth of tensiometer cup below soil surface (cm), [z.sub.2] is the height of mercury level above soil surface (cm), and H is the hydraulic head (cm).

Water content measurements

The TDR method was used to determine water contents simultaneously with water potential at selected depths. The Tektronix TDR cable tester (model 1502B) system consisted of a pulse generator, voltmeter, 4-5-m-long coaxial cable feeder, TDR probes, and personal computer for data logging. TDR indirectly measures soil water content over a quasi-elliptical area of approximately 10[cm.sup.2], and is therefore suitable for high resolution soil water regime measurements. The determination of water content by this method follows from the notion that there exists a unique relationship between volumetric water content [theta] ([cm.sup.3]/[cm.sup.3]) and soil permittivity or water dielectric constant [epsilon](-). Several studies have indicated the value of TDR as a non-destructive method of soil water measurements (Topp et al. 1980, 1994) and do not require frequent calibrations in most field measurements (Lane and Mckenzie 2001).

The following empirical relationship for most mineral soils was used to estimate water content (Topp et al. 1980): [theta] = -0.053 + 0.029[epsilon] -5.5 * [l0.sup.-4][[epsilon].sup.2] + 4.3 * [10.sup.- 6][[epsilon].sup.3]. The field exercise revealed the Topp equation to be superior to the laboratory derived equations and other published empirical equations (Lane and Mckenzie 2001). The TDR probes were installed horizontally at 0.10-m increments for the following selected locations or depths: 0.15, 0.25, 0.35, 0.45, 0.55, and 0.75 m. A good contact was ensured between cable probes and soil by pushing the probes until the base touched the soil.

Data analysis

Instantaneous profile method

The more commonly used field method to measure K([theta]) and [theta](h) is the IPM (Rose et al. 1965; Van Bavel et al. 1968) and has undergone several modifications (Libardi et al. 1980). The IPM, although laborious, has proven to be reliable and is still considered a standard method in terms of precision of estimation when applied correctly (Vachaud and Dane 2002). The technique has been applied by Normand et al. (1997) in the leaching of nitrate below the root-zone and by Hutchinson and Bond (2001) in routine measurement of soil water potential gradient near saturation. The IPM method analyses replicated soil water content and tensiometry measurements of soil profiles during transient flow events according to the following continuity or mass balance equation for soil water flux q(z,t) ([cm.sup.3]/h) at a certain depth z (cm) and time t (h) such that:

(6) [partial derivative][theta]/[partial derivative]t = [partial derivative]q/[partial derivative]z


(7) q(z,t) = -[SIGMA] [[partial derivative][theta]/[partial derivative]t] [partial derivative]z and L([theta]) = q(z,t)/ [partial derivative]H/[partial derivative]z

Soil water flux and hydraulic conductivity were calculated for the layers of soil water contents, and pressure heads were measured. Soil water contents for each layer were plotted against time. Soil water gradients ([partial derivative]q/[partial derivative]z) were determined for selected depths by interpolation from graphs of [theta] v. time t. Particular attention was paid to the first few days of the experiment when measurements were quite sensitive to water changes. Hydraulic heads H were also plotted against time t, and hydraulic gradients ([partial derivative]H/[partial derivative]z) were determined at the same selected times for soil water gradient. Finally, hydraulic conductivity K([theta]) was calculated for each depth and together with water retention [theta](h) presented graphically.

Inverse modelling

The second part of the data analysis involved optimisation of the soil hydraulic functions and was carried out by applying van Genuchten's (1980) parametric model as modified by Kool and Parker (1987) to transient flow experiments. Any initial and boundary conditions may be employed, thus enabling greater flexibility in the experimental setup than by traditional methods. Saturated hydraulic conductivity ([K.sub.s]) and saturated soil water content ([[theta].sub.s] were estimated from values measured just after flooding. Other initial parameters (n, [alpha], [tau]) were estimated from the van Genuchten-Mualem model spreadsheet (van Genuchten 1980) such that there was a best correspondence between the experimental or observed and predicted [theta](h) functions. The set of parameters for the best fit was taken as the initial values. These parameters were used as initial parameters in optimising soil hydraulic functions in the inverse model. The parameter [alpha] is sensitive to pore size distribution, which is estimated from the water retention using [partial derivative]P = (2[sigma]/[r.sub.p]), where [partial derivative]P is the pressure difference (Pa) across an air-water interface, [sigma] is surface tension of water (J/[m.sup.2]), and rp is the radius of a circular capillary tube (m).

Results and discussion

Instantaneous profile method

In all cases, both water content and tensiometry data showed good responses indicating that the methods were valid for the lysimeter drainage experimental set up. Volumetric water content as a function of time and depth [theta](z,t) was collected from the water content measurements, while hydraulic head as a function of time and depth H(z,t) was obtained from the tensiometry measurements at different times and depths in a draining profile. The H(z) profile, especially for the wetter part and initial times, showed some similarity with those generated in other studies (deBoer and Rice 1968; Ahuja et al. 1980), which reported a cubic spline curve flexible fitting for the H v. z curve. For computational analysis, [theta](z,t) and H(z,t) curves were plotted as shown in Figs 1 and 2, respectively, for sandy and loamy soils. Figure 1 shows a monotonic decrease of water content with time reflecting a draining profile, while Fig. 2 shows a more or less linear (though with cubic spline-like shape) relationship between hydraulic head and depth. The faster drainage in the sandy soil is reflected by the sharp drop in water content with time, while in the slower draining loamy soil there was a slower response in the changes. Vachaud and Dane (2002) commented that that the IPM is not applicable to heavy (or non-drainable) soils.


Flux (q) was calculated by applying the continuity principle, Eqns 6 and 7. Hydraulic conductivity (K) was computed from ([partial derivative]H/[partial derivative]z) at selected times and by application of Darcy's equation: q = K[partial derivative]H/[partial derivative]z. The conductivity data together with water retention were computed to describe the hydraulic functions K([theta]) and [theta](h) for both soils and are shown in Figs 3 and 4. respectively. The data processing was quite sensitive, more especially for estimating slopes [partial derivative][theta]/[partial derivative]t for the driest part of the profile (where slopes were almost flat for a loam soil), and this might account for the more variable hydraulic conductivity data in a loam soil. The IPM data also showed highly scattered K([theta]) data (Fig. 4) for both soils with regression coefficient ([R.sup.2]) of 0.57 and 0.29, respectively, for sandy and loamy soils for a semi-logarithm plot of K([theta]) data. High scatter data accounted for errors associated with qualitative estimations of [partial derivative][theta]/[partial derivative]t and [partial derivative]H/[partial derivative]z in predicting hydraulic properties, while the relatively higher scatter data in loamy soil is attributed to its poor drainage (Vachaud and Dane 2002). Yseng and Jury (1993) also reported that accuracy of K([theta]) data may be affected by errors such as measurement of [theta] and h values at various locations and also due to limited number of observations in space and time, producing poor estimates of hydraulic gradients and time derivatives of integrated water content at specified times and positions.


Inverse modelling

In arriving at theoretical predictions of soil properties, 3 datasets, water content (WC) only, pressure heads (P-h) only, and a combination of water content and pressure heads (WC & P-h), were used. The objective was to see which of the data [theta](z,t) or H(z,t) and/or a combination resulted in the most accurate estimates of the soil hydraulic functions when inverse modelling was applied. The optimised parameters as verified for their statistical accuracy for each dataset are tabulated in Table 2 for the sandy and loamy soils for [r.sup.2], the correlation coefficient, and SSQ, the lowest sum of squares for optimised parameters in the various datasets for measured attributes including water content and pressure heads.

Comparative analysis: IPM and inverse modelling

The results generally showed a better curve fitting for a sandy soil than for the loamy soil for both water retention and hydraulic conductivity functions. It also showed better agreement of experimental data with data from parameters estimated using both soil water data and pressure head data. The fast soil water drainage in the sandy soil seems to provide better data resolution compared with the slow-draining loamy soil. Similar results were obtained by Knopman and Voss (1987) when they concluded that more accurate model parameters are obtained from data with rapid change in soil water contents.

Sandy soil

The experimental (IPM) water retention curve (Fig. 3) fitted well with predicted (inverse modelling) data from optimisation of both water content and pressure heads. The optimisation of pressure heads did not produce much difference, while the water content showed a large difference (more significantly for the wetter profile) between predicted and experimental data. Comparison of predicted and measured data was possible in the common wetness range only (Chaudhari and Batta 2003). Similarly Vachaud and Dane (2002) commented that the IPM yields a direct field estimation of K([theta]) data in wet range. This suggests that though optimisation of water content was well accomplished, it does not necessarily mean that the predicted soil hydraulic functions will fit well with the functions obtained from the IPM method. The hydraulic conductivity function was treated in the same way as water retention. As presented in Fig. 4, predictions from all treatments fitted well with experimental data, except for the difference from exclusive optimisation of pressure heads.


Loamy soil

Figures 3 and 4 indicate that predicted and experimental hydraulic properties agree better when both pressure heads and soil water contents are used to estimate the hydraulic parameters, especially at near saturation. There was some deviation after the soil water content drained to 24% by volume. The predicted curves from either soil water contents or pressure heads data do not agree. Curves from pressure heads show some similarity with the curve from the measured data during the initial stages but later diverge as the soil dries. The predicted curve using soil water content data does not show any agreement with experimental data. Using data from a tension infiltration experiment, Simunek and van Genuchten (1997) found that optimisation with water content was the best practical set up for predicting soil hydraulic parameters.

The hydraulic conductivity curve (Fig. 4) calculated from the instantaneous profile method does not agree well with other curves. It only shows some similarities with the curve predicted using soil water content and pressure head data during the initial stages of drainage. The hydraulic conductivities calculated using IPM were always higher than the 3 sets of predicted hydraulic conductivities. Dane and Hruska (1983) showed the same trend in their drainage experiment, that is, less comparable results for the hydraulic conductivity curve than for the water retention curve. This difference could be explained by the relative insensitivity of unsaturated flow to saturated hydraulic conductivity [K.sub.s], especially near saturation (Kool and Parker 1987; Durner 1994). Furthermore, Kool and Parker (1988) indicated less sensitivity of hydraulic conductivity (K) and parameter n to both drainage and infiltration processes.


Summary and conclusions

The study has demonstrated that the IPM method, though laborious and time-consuming, is still applicable as there was a good fit for the hydraulic conductivity and water retention data with inverse modelling. The results have also indicated that good parameter estimation can be obtained from the combination of soil water content and pressure head data. Soils with relatively high hydraulic conductivity influence the data resolution positively, because of the rapid change in soil water content. Predicted hydraulic conductivity and measured hydraulic conductivity in most cases do not show good comparison because of the low sensitivity of the unsaturated transient flow to the saturated hydraulic conductivity. Parameter estimation of sandy soil with relatively high hydraulic conductivity shows rather good results; hence, the inverse modelling looks quite attractive compared with traditional methods (IPM). From this investigation it was shown that the predicted and experimental data fitted well for both the hydraulic conductivity and water retention data. Though there was a discrepancy in water retention curve for exclusive optimisation of water content, some theoretical considerations were still held. The results showed that the sandy soil gave a much better resolution and close agreement for fitted and measured properties than the loamy soil, and hence the applicability of the inverse modelling approach for the sandy soil is appropriate.


This work was carried out during my MSc studies at the University of Wageningen, The Netherlands. I would like to thank my MSc project supervisors; Drs C. Dirksen and J. C. van Dam for their guidance and supervision, l am also thankful to the anonymous reviewers for their constructive comments on the manuscript.


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Oagile Dikinya Department of Environmental Science, The University of Botswana, Private Bag 0022, Gaborone, Botswana. Current address: School of Earth and Geographical Sciences, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Email:
Table 1. Particle size distribution (by mass) for
sandy and loamy soils

Soil % Sand % Silt % Clay

Sandy soil 86 6 8
Loamy soil 42 37 21

Table 2. Optimised parameters with statistical indicators for sandy
soil and loamy soil

Soil data Model parameters

 [alpha] n [tau] [[theta] [[theta] [[KAPPA]
 (1/cm) (-) (-) .sub.s] .sub.r] .sub.s]

 ([cm.sup.3]/ (cm/h)

Sandy soil

WC only 0.2500 2.9552 0.0001 0.346 0.035 1.453
WC & P-h 0.0429 3.0525 0.0001 0.346 0.035 1.653
P-h only 0.0287 2.7052 0.0040 0.346 0.035 0.500

Loamy soil

WC only 0.0080 2.394 0.5000 0.375 0.050 0.050
WC & P-h 0.2397 2.1089 0.2463 0.375 0.050 0.311
P-h only 0.0171 2.2228 0.0001 0.375 0.050 0.348

Soil data Statistical criterion

 [r.sup.2] SSQ

 Sandy soil

WC only 0.9340 0.013
WC & P-h 0.9853 0.025
P-h only 0.9305 303.5

 Loamy soil

WC only 0.821 0.012
WC & P-h 0.979 0.015
P-h only 0.850 1475
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Author:Dikinya, Oagile
Publication:Australian Journal of Soil Research
Date:Sep 1, 2005
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