Determining Soil Hydraulic Properties from One-step Outflow Experiments by Parameter
Estimation: II. Experimental Studies1
J. C. PARKER, J. B. KOOL, AND M. TH. VAN GENUCHTEN2
ABSTRACTUnsaturated hydraulic properties of four soils of varying particle
size distributions were evaluated by determining values of the fiveparameters in van Genuchten's (1980) hydraulic model. Saturated
conductivities (K,) and saturated water contents (9,) were directly
measured and values of residual water content (0,) and the parameters a arid n were evaluated by a nonlinear inversion method tominimize various objective functions. Method I uses an objectivefunction involving sums of squared deviations between measured cumulative outflow with time Q(t) for one-step pressure desorption andnumerically simulated outflow from saturation to a final pressure
head h = —10 m. Method II supplements Qft) data with the measured equilibrium water content 6 at h = — ISO m, while Method
III employs equilibrium 9(h) data only. Method I yields the mostaccurate description of Q(t) and the independently determined hydraulic diffusivity D(6). A fair description of 6(h) is obtained within
the range of the one-step experiment but at lower 6 predictions areless reliable especially for finer-textured soil. Method II extends therange of validity of the predicted 9(h) to lower 9 with generally small
effects on predicted D(9) and Qft). Method III gives the best description of 9(h) but at the expense of accuracy in Q(t) and D(6).
Implications for routine evaluation of soil hydraulic properties are
discussed.
Additional Index Words: unsaturated hydraulic conductivity measurement, water retention measurement, soil water diflusivity, inverse problem, optimization.
Parker, J.C., J.B. Kool, and M.Th. van Genuchten. 1985. Determining soil hydraulic properties from one-step outflow experimentsby parameter estimation: II. Experimental studies. Soil Sci. Soc.
Am. J. 49:1354-1359.

THE PARAMETER estimation approach to determining soil hydraulic properties from transient flow
experiments is based on the premise that assumed
expressions for the soil hydraulic properties accurately
describe the real system and that the inversion prob1
Contribution from the Dep. of Agronomy, Virginia Polytechnic
Institute and State Univ., Blacksburg, VA 24061 and U.S. Salinity
Lab, 4500 Glenwood Drive, Riverside, CA 92501. Received 20 Nov.
1984.
Approved 6 June 1985.
2
Associate Professor and Research Associate, Virginia Polytechnic Institute and State Univ., Blacksburg, VA 24061, and Research
Soil Scientist, U. S. Salinity Lab, respectively.

lem has a unique solution. In Part I of this paper (Koolet al., 1985a), we found that to minimize uniquenessproblems, transient flow data used as input to an inversion procedure should correspond to as broad a
range in soil water contents as possible. For gravity
drainage experiments, such as those considered byZachman et al. (1981, 1982) and Hornung (1983), theminimum pressure head obtained depends directly on
the height of the soil column. For reasonably attainable laboratory column lengths, ranges in water content will accordingly be very narrow for all but thecoarsest textured soils, thus severely limiting the practical applicability of such procedures based on gravity
drainage only.
Dane and Hruska (1983) applied parameter estimation methods to the determination of in situ hydraulic properties using input data consisting of watercontent profiles at different times during gravity drainage of a clay loam soil with a zero flux condition at
the soil surface. Good results were obtained; however,as with the laboratory gravity drainage experiments,
limitations occurred on the attainable range in watercontents and experiment durations had to be rather
long. In their case, observations were extended to 25
d.
Pressure desorption of short soil cores offers an experimentally simple method of obtaining data for parameter estimation suitable to a broad range of soils.
In Part I we investigated numerical constraints on thefeasibility of estimating hydraulic properties by inversion of cumulative pressure outflow data measured
with time from short soil cores. Van Genuchten's(1978, 1980) hydraulic property model was employedwith two of the five model parameters assumed independently measured and three estimated by numerical inversion. This approach was shown to be feasible provided: (i) the pressure increment is large
enough to yield a low final reduced water content, (ii)outflow measurements are extended over an adequate
period of time, (iii) initial parameter estimates are reasonably close to their true values, and (iv) experimental error is relatively small. Additionally, the successof this method will depend on whether the assumed
model accurately describes soil hydraulic properties.

PARKER ET AL.: SOIL HYDRAULIC PROPERTIES FROM OUTFLOW EXPERIMENTS BY PARAMETER ESTIMATION: II. 1355
Table 1. Characterization of soils studied.
Particle size distribution
Textural class

Sand

sat

Clay

Bulk
density

Description

Table 2. Measured values of saturated water contents (0S),saturated conductivities of soil cores (Ks), and of ceramic_____plates (Kp) used in inverse problem solutions.____
Soil

6,

kgm-3
Sandy loam

61

24

15

1550

Silt loam

28

56

15

1570

Sandy clay loam

56

18

26

1530

Clay

21

31

48

1110

K.

Kn

m'm-3

A horizon, granularstructure
A horizon, granularstructure
A horizon, granularstructure
C horizon, massive
weathered shale

Zachman et al. (1982) have shown that when an incorrect model for the soil hydraulic properties is used,
it may still be possible to obtain an apparently acceptable solution to the inversion problem; however,
the hydraulic functions corresponding to the solution
may be in error.
In the present paper, we investigate effects of experimental variables and model accuracy on hydraulicproperty estimation from one-step pressure outflow
experiments on undisturbed soil cores.

7.0 x 10-'

5.5 x 10-'

1.5 x 10-'
1.1 x 10-'2.2 x 10-'8.3 x 10-'
5.5 x 10-'
1.4 x 10-"

Table 3. Parameter values for van Genuchten model determinedfrom transient one-step outflow experiments.
Soil
Sandy loam

Parameter

Final parameter valuest
Method I Method II Method III
1.29
1.511
0.183

1.23
1.301
0.110

0.074

2.50
1.50
0.20

3.35
1.387
0.161

4.71
1.461
0.173

3.46
1.289
0.103

1.00
1.50
0.30

2.15
1.217

2.34
1.225
0.199

0.82
1.275
0.112

0.15
1.485

»r

0.50
1.50
0.40

0.08
1.319
0.127

0.07
1.419
0.109

1

3

a
n
Or

Sandy clay
loam

Initialvaluest
2.50
1.50
0.15

a
n
»r

Silt loam

Clay

METHODS AND MATERIALSSingle undisturbed soil cores of 54-mm diam and 40-mm
length were taken from four locations with a thin-wall hydraulic core sampler for laboratory testing. The soil texture
ranged from sandy loam to clay (Table 1). Cores were assembled in Tempe pressure cells with 5.7-mm thick, —10m air-entry ceramic plates. To ensure reproducibility of themain drying curves, cores were preconditioned by subjecting
them to several wetting and drying cycles between pressure
heads of 0 and —10 m; Final saturation was accomplishedby wetting the cores from the bottom at a pressure head of—0.1 m, followed after equilibration by gradually increasing
the head to h = 0 at the top of the core. A 0.01 M CaCl2
solution was used in all experiments to minimize soil dispersion.
One-step pressure outflow tests were performed for apneumatic head increment h" of 10 m at the top of the cores.
Cumulative outflow volume was measured as a function oftime with a burette arranged to allow adjustment of the outflow free surface to the midheight of the core and to permit
flushing of trapped air from beneath the porous plate. Due
to small but measureable evaporative losses from connecting tubing, "blank" cells were monitored to allow appropriate corrections to be made on outflow measurements.
Further details of the experimental apparatus and techniques are given by Kool et al. (1985b).
After resaturating the cores in the same manner as before,
stepwise desorption experiments were carried out to determine the equilibrium water retention characteristics. Equilibrium outflow volumes were measured as the cores weredesorbed incrementally to pressure heads of —0.1, —0.5,—1.0, —3.0, —6.0, and —10.0 m. After resaturating the soil
and again desorbing to —0.1 m, the cores were removed
from the cells and weighed to permit subsequent gravimetricdetermination of the h = — 0.1-m water content used as areference point for calculation of other equilibrium watercontents from outflow measurements. Oven-dry core masses
were obtained after determining saturated hydraulic conductivities of the cores (Ks) by a falling-head method. The
porous plates were for that purpose replaced by low impedance filters. Hydraulic conductivities of the porous ceramic
plates (Kp) were determined in a similar fashion.
Volumetric water contents 8 at pressure heads of — 30 and— 150 m were determined on air-dried and sieved samples

0.355
0.388
0.402
0.589

Sandy loam
Silt loam
Sandy clay loam
Clay

ill a

a

n
»r
a

n

0.200
0.395

1.53
1.265

t Units of a in m" ,9r in m' m~ , n dimensionless.

of the four soils by pressure desorption using undisturbed
core densities to convert from gravimetric to volumetricwater contents.
RESULTS AND DISCUSSION
We used the cumulative outflow data with time tocalculate the van Genuchten model parameters by numerical inversion of the flow problem as outlined inPart I. Measured values of Ba Ks, and Kp (Table 2)
were input as knowns and unconstrained values of theparameters a, «, and 8r were obtained by inversion
from initial values judged appropriate for the respective sample particle size distributions. Initial parameter values and final fitted values obtained in thismanner, denoted as Method I, are given in Table 3.
All inversion problems were run at least twice withdifferent initial parameter estimates. In cases where
solutions did not converge to the same final estimates,
the estimates that led to the highest r2 values are reported. For the cases shown in the table, convergenceto final values was obtained within 28 to 39 trial solutions.
Comparison of observed and fitted cumulative outflow data (Fig. 1, Method I) indicates that reasonably
accurate flow predictions are obtained. Because theobjective function for inversion utilizes outflow data,
we must turn to independent measurements of hydraulic behavior for corroboration of the model. First,
we compared the transient outflow-derived 8(h)-curvewith conventionally measured static stepwise desorption data. As seen in Fig. 2, reasonable correspondence
is obtained between measured equilibrium 0(h) andthe transient (Method I) derived functions over therange of h from 0 to — 1 0 m obtained during the onestep outflow experiments. For lower h, predicted anddirectly measured 6(h) tend to diverge increasingly,

1356

SOIL SCI. SOC. AM. J., VOL. 49, 1985

ISn

•m

Bi

«n

3
O

O

111

3
u

S I L T LOflM

SRNDY LOflM

-V

0«
0

3

3

TIME V1*)

4 .

TIME

12

9-

«n
•I

Ul

>

6-

z

SflNDY CLflY LORM

u 3-

CLflY

-V
TIME
Fig. 1. Measured cumulative outflow from soil cores ( D datapoints) and predicted outflow using hydraulic properties evaluated by parameter
estimation from outflow data only (————: Method I), from outflow data supplemented by the measured 6 at h = -150 m (—— ——:Method II), and from measured 6 (h) data only (—— - ——: Method III). Data points in parentheses werenot used in inversion procedure.
* indicates Q(t J.
TIME kr"*)

particularly for the finer textured soils. These soils exhibit narrower ranges in water contents over the imposed pressure head range, possibly resulting in nonunique inversion results. Such difficulties could be
reduced by using larger gas pressure increments forfiner textured soils. Conversely, we note that for very
coarse textured soil, a smaller pressure increment may
be necessary to obtain adequate resolution in outflowtime data to permit inversion.
Another approach to reducing prediction errors at
low potentials is to incorporate selected measured 0(h)data into the objective function which is to be minimized as:M

where b is the vector of parameter estimates, Q and

Q are observed and predicted transient outflow volumes, 6 and 9 are observed and predicted water contents at specified h, and w, and v, are weighting factors.
Evaluation of 0 is obtained directly via Eq. [4] of Part
I. In our modified parameter estimation approach, we
employ the same Q(t) data as before with w, = 1 andinclude one (6, h]) pair for h = —150 m (M = 1). To
give 0(/z) values approximately the same weight as Q(t)values we takeParameter values from this modified method, denoted as Method II, are given in Table 3. For the
sandy loam and clay soils, Method II results in better
correspondence between observed and predicted 0(h)for h =^ — 10 m (Fig. 2). Except perhaps for the silt
loam, there is little change in the range —10 </z <0.
Correspondence between observed and calculated out-

PARKER ET AL.: SOIL HYDRAULIC PROPERTIES FROM OUTFLOW EXPERIMENTS BY PARAMETER ESTIMATION: II 1357
10'

10Z

D

5X
u
0D

in

13
10S I L T LOflM

10-'

0.2

0.1

0.3

0.4

10'

102

a 10' i
UJ

I
Id

a

U

cc

0.

ICT

CLflY

10

0.15

0.25

0.35
6 (mV)

0.45

0.55

Fig. 2. Measured 6(h) by static stepwise desorption (datapoints) and predicted 9(h) from parameter values evaluated by Methods I, II, andIII. ————: Method I, —— ——: Method II, —— - ——: Method III.

flow (Fig. 1) is likewise not greatly affected by the additional constraint imposed on the inverse problem.
Besides experimental errors in observed Q(t) and0(/z), discrepancies between observed 0(/z) and thosepredicted by Methods I and II may be due to nonuniqueness of the inverse problems or to deviations ofactual hydraulic properties from the assumed parametric model. To investigate this we carried out theparameter estimation process using only measured
equilibrium 0(/z) for h = 0 to — 150 m in the objectivefunction with no transient flow data (TV = 0). Parameter values from this method, denoted Method III(Table 3), yield 0(/z)-curves which more closely matchdirectly measured 6(h) data than the curves derived
from the transient outflow measurements (Fig. 2).
Comparison of the calculated outflow-time curves inFig. 1 shows that for three of the four soils, Method
III gives predictions of the transient hydraulic behav-

ior that are the same or reasonably similar to the calculations obtained with Methods I and II. However,the predicted outflow curve (Method III) for the sandy
clay loam soil is significantly higher than the observedor directly fitted curves I and II. This suggests that, atleast for this soil, differences in parameter values obtained by Method III vs. those by Methods I or II
should be partially attributed to deviations from theassumed parametric model for the hydraulic properties and not merely to nonuniqueness of the inverse
problem.
The accuracy of the hydraulic model and the appropriateness of the various parameter estimation
schemes may be further evaluated by comparison of
hydraulic diffusivities (D = Kdh/dO) predicted fromparameters estimated by Methods I through III and
those calculated directly from one-step outflow datausing the method of Passioura (1976). In this proce-

1358

SOIL SCI. SOC. AM. J., VOL. 49, 1985
10SILT LOflM
10"

Q ID-* -I
>

510

,-•»

»-l

3
S 10"

o

10"

.O--

0.2

0.3

0.4

10"

CLflY
10"

•3 ID"

10
in

"

10"

0.4

10'

0.1S

0.25

0.35

e tfvi0.45

0.55

Fig. 3. Calculated fl(0) by method of Passioura, 1976 (datapoints) and predicted D(6) from parameter values evaluated by Methods I, II, andIII. ————: Method I, —— ——: Method II, —— - ——: Method III.

dure, the first part of the outflow vs. time curve is not
used. Hence, diffusivities are obtained only for a narrow range of moisture contents. Passioura's method
calculates D(0) from the ratio of second derivative tofirst derivative of the outflow vs. time curve. The procedure is therefore not very accurate when 6 approaches its equilibrium value as both the numerator
and denominator tend to zero. This could explain theaberrant behavior of the calculated D(8) for 0 <0.275
for the sandy loam. Keeping in mind the approximate
nature of the calculated D(6) when comparing thesewith predicted Z)(0)-curves (Fig. 3), correspondence
between the predictions of Methods I and II and Passioura's results is judged to be excellent while for
Method III the results are poorer. Only the sandy clay
loam soil shows an unacceptable prediction of D(0)with Method III. Figure 3 also shows that predictedZ)(0)-curves diverge as 6 decreases, again reflecting thedifferences in fitted 0r values.

SUMMARY AND CONCLUSIONS
Because of experimental difficulties involved in the"direct" measurement of unsaturated hydraulic conductivities (i.e., from data amenable to analytic solution of the inverse problem), the most appealing
method of evaluating unsaturated soil properties is byapplication of pore structure models which yield K(h)from equilibrium 0(/z)-data and the saturated conductivity. The use of parametric formulations for 0(h)which lead to closed-form expressions for K(h) andC(h)= dO/dh, represents a very convenient method ofcharacterizing soil hydraulic behavior. Here we employ van Genuchten's hydraulic model, chosen because it is simple and yet has been found to describewith reasonable accuracy the hydraulic properties ofa variety of soils.
The most common method of estimating parameters in this and similar models is by best-fitting toPARKER ET AL.: SOIL HYDRAULIC PROPERTIES FROM OUTFLOW EXPERIMENTS BY PARAMETER ESTIMATION: II 1359

equilibrium 0(/z)-data only. When deviations between
measured and predicted 0(/z) are used to formulate an
objective function for parameter estimation, any inaccuracies in the hydraulic model will be forced into
the predicted K(h). An alternative approach usingtransient flow data to formulate the objective function
will distribute errors due to model inaccuracies to both0(/z) and K(h) in a manner which optimizes the abilityof the model to describe the entire transient flow process.
We have compared three types of sums-of-squared
deviations objective functions for parameter estimation using observed data consisting of (Method I) cumulative outflow with time from initial saturation following a one-step pneumatic head increment of 10 m,(Method II) cumulative outflow data supplemented by
the measured water content at h = —150 m, and(Method III) equilibrium 6(h) data only without transient flow measurements. Method I yields satisfactory
results within the range of water contents observed in
the transient experiment but cannot be reliably extrapolated to lower 6 especially for fine-textured soils.
Method II extends the range of validity of predicted
properties to low 0 with only minor effects on the predictions at high 6. Method III yields a more accurate
description of 0(/z) than Methods I and II but sometimes at the expense of accuracy in K(h) and in predictions of transient flow, thus reflecting the biases
inherent in the various objective functions employed.
Since the purpose of hydraulic property estimation
is ultimately to enable accurate predictions of flow
behavior, and also considering the tendency toward
widespread use of approximate pore structure models,
employing transient flow data rather than equilibrium
B(h) may be more appropriate for the estimation ofsoil hydraulic properties. Because one-step outflow
tests can be performed in much less time than stepwise equilibrium desorption tests, use of the former
is especially attractive when intensive field-samplingof soil hydraulic properties is needed for validation or
implementation of stochastic flow models.

In this paper we have investigated the numericalinversion of laboratory pressure outflow data to yield
information on hydraulic behavior during monotonic
drying. To more accurately predict transient hydraulicbehavior, it will be necessary to also consider wetting
phenomena owing to hysteresis in soil hydraulic properties. Extension of the parameter estimation approach involving solution of the inverse flow problem
to the analysis of inflow experiments to characterize
hysteretic behavior is currently being investigated. As
with all laboratory analyses, care must be exercised toensure that sample disturbance does not inordinately
effect the results. In subsequent work the extension ofparameter estimation methods to in situ experiments
will be investigated.