Flow Rate Dependence of Soil Hydraulic Characteristics
D. Wildenschild,* J. W. Hopmans, J. Simunek
ABSTRACT

referred to a study by Harris and Morrow (1964), who
studied pendular rings in packs of relatively large uniform spheres, and found that some of the pores in thedrained sphere pack remained full, as these became
isolated from the bulk liquid before their air-entry pressure was attained. This bypassing of isolated liquid-filled
pores explained the observed higher retained water content. An analogous explanation was suggested by Davidson et al. (1966) who investigated the dependence ofthe retention characteristic on the applied pressure increment during wetting. They reported a noticeable dependence of the equilibrium water content on the sizeof the applied pressure increment such that a higher
water content was measured when small pressure increments were used during water absorption. The increasein water content was attributed to a reduction in the
air volume entrapped during absorption if the soil waswetted at a slower rate. The authors concluded thatthe changes in water content should be treated as an
immiscible displacement process, where the resistanceto movement and spatial configuration of both waterand air need not be single valued with respect to watercontent—a hypothesis stated previously by Nielsen etal. (1962).
Later, Smiles et al. (1971) carried out desorption experiments in a horizontal column of uniform soil andfound that the relationship between the soil water matricpotential and the water content was nonunique throughout the column. Vachaud et al. (1972) continued thework of Smiles et al. (1971) to determine if this same
phenomenon occurred in vertical drainage of a uniform
column of fine sand as well. They compared retention
curves obtained under static and dynamic flow conditions and their results were consistent with those of
Smiles et al. (1971). In particular, Vachaud et al. (1972)
showed that a rate increase of matric potential withtime reduced the volumetric outflow, thereby causing
deviations from the static retention characteristic.
The issues of rate dependence of soil hydraulic properties have since been mostly disregarded. The unsaturated
hydraulic conductivity dependence on flow conditions inparticular has been the focus of few investigations, partly
because of the tedious nature of its measurement; however, recent experiments carried out by Wildenschild etal. (1997), Plagge et al. (1999), Hollenbeck and Jensen(1999), and Schultze et al. (1999) support the notionthat the flow regime may vary notably between different
types of experiments, thereby influencing the estimationof the unsaturated hydraulic properties. In addition to
the water retention characteristic, Plagge et al. (1999)
investigated the influence of both flow rate and boundary condition type on the hydraulic conductivity function and concluded that experiments with larger water
potential gradients tended to increase the unsaturated
hydraulic conductivity. In experiments designed to in-

The rate dependence of unsaturated hydraulic characteristics wasanalyzed using both steady state and transient flow analysis. One-stepand multistep outflow experiments, as well as quasi-static experiments
were performed on identical, disturbed samples of a sandy and aloamy soil to evaluate the influence of flow rate on the calculated
retention and unsaturated hydraulic conductivity curves. For the sandy
soil, a significant influence of the flow rate on both the retention andunsaturated hydraulic conductivity characteristic was observed. At agiven matric potential, more water was retained with greater applied
pneumatic pressures. Matric potential differences of 10 to 15 cm (forgiven saturation) and water content differences of up to 7% (for given
potential) could be observed between the slowest and the fastest
outflow experiments, predominantly at the beginning of drainage.
The hydraulic conductivity also increased with increasing flow rate
for higher saturations, while a lower hydraulic conductivity was observed near residual saturation for the higher flow rates. We observed
a continuously increasing total water potential gradient in the sandy
soil as it drained, especially for high-pressure transient one-step experiments. This indicates a significant deviation from static equilibrium,
as obtained under static or steady-state conditions. For the finer textured soil, these flow-rate dependent regimes were not apparent. A
number of physical processes can explain the observed phenomena.
Water entrapment and pore blockage play a significant role for thehigh flow rates, as well as lack of air continuity in the sample during
the wettest stages of the experiment.

The two basic soil hydraulic characteristics controlling flow in unsaturated porous media are the retention characteristic, ␪(h), and the unsaturated hydraulicconductivity characteristic, K(h). Commonly, thesecharacteristics are measured under static equilibrium or
steady-state conditions and are subsequently applied toboth steady-state and transient flow analyses, thereby
assuming that the retention characteristic is not affectedby nonequilibrium conditions. Thus, both equilibrium
and steady-state measurements are routinely used toanalyze transient flow phenomena and vice versa.
However, a number of experiments presented in thesixties and early seventies suggested that these assumptions might not be entirely justifiable. When comparing
drying water retention data obtained by equilibrium,
steady state, and transient methods, Topp et al. (1967)
found that more water was retained in a sand at a given
matric potential for the transient flow case than for thestatic equilibrium and steady-state cases. The authors
D. Wildenschild, Earth and Environmental Sciences, Lawrence Livermore National Laboratory, P.O. Box 808, L-202, Livermore, CA
94550; J.W. Hopmans, Hydrology, Dep. of Land, Air and Water Resources, 123 Veihmeyer Hall, Univ. of California, Davis, CA 95616.
J. Simunek, U.S. Salinity Laboratory, USDA-ARS, 450 Big Springs
Rd., Riverside, CA 92507. D. Wildenschild, presently at Dep. of Hydrodynamics and Water Resources, Technical University of Denmark.
Received 25 Jan. 2000. *Corresponding author (wildenschild1@llnl.gov).
Published in Soil Sci. Soc. Am. J. 65:35–48 (2001).

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SOIL SCI. SOC. AM. J., VOL. 65, JANUARY–FEBRUARY 2001

vestigate the influence of pore scale dead-end air fingers
on relative permeabilities for air sparging in soils, Clayton (1999) found that the measured air permeabilities
decreased with increasing displacement rate. Clayton attributed the displacement-rate dependent behavior to thedevelopment and subsequent breakthrough of dead-end
air fingers. Most recently, Friedman (1999) concluded
that the influence of flow velocity on the solid–liquid–gas
contact angle could also explain the phenomena observedby Topp et al. (1967).
With the introduction of new and faster techniques fortransient measurement of the hydraulic characteristics
such as the one-step (Kool et al., 1985) and multi-step(van Dam et al., 1994; Eching et al., 1994) outflow methods, the question of the validity of these measurements
has since become increasingly important. As many researchers now apply these faster techniques to determine the hydraulic characteristics of soils, it is important
to examine the influence of the boundary conditions on
the measurement results for these experiments. In manycases, recorded data of cumulative outflow as a functionof time is combined with soil water matric potential
head measured with a tensiometer at a point inside
the sample (Eching et al., 1994) to facilitate inverse
estimation of the hydraulic parameters. If the watercontent is dependent not only on the matric potential,
but is also influenced by outflow rate, it needs to betaken into consideration when estimating the soil hydraulic properties. Outflow procedures are, however,not the only methods to be affected by this phenomenon. For the traditional static methods such as the pressure plate extraction method (Klute, 1986), soil water
retention may be a function of the rate of flow between
equilibrium points as well.
As improvements in measurement techniques, as wellas the implementation of dynamic measurement methods, have become available, the aim of the present study
was to continue the above referenced work. Using thecurrently available improved measurement techniques,
we investigated the rate dependence of unsaturated hydraulic characteristics for two soils in short laboratory
columns. Thus, the objective of this study was to investigate the influence of flow rate on soil hydraulic characteristics using both a direct and an inverse estimation
method for two soils with different pore size distributions. The direct estimation is based on Darcy’s Law(steady state), while the inverse estimation relies on
numerical solution of Richards’ equation and as such isa transient approach.
MATERIALS AND METHODS
Experimental Setup
A diagram of the flow cell and associated gas and waterflow controls is shown in Fig. 1. The samples were packed ina pressure cell, 3.5-cm high and 7.62-cm diam. In addition to
the top air inlet and the bottom water outlet, an extra outlet
at the bottom was used to flush air bubbles from underneaththe porous membrane. All connections consisted of quick disconnect fittings (Cole-Parmer, Delrin, 1/4-inch NPT, 0635972) so that the cell could be detached for weighing. Weighing
allowed for determination of the sample water content during

Fig. 1. Laboratory setup for outflow experiments.

the drainage experiment. Two tensiometers were inserted 1.1
and 2.4 cm from the bottom. The ports were offset laterally
to minimize disturbance of flow between the tensiometers andto the overall flow in the sample. The tensiometers were madefrom 0.72-cm diam., 1-bar, high-flow tensiometer cups (Soil
Moisture Corp. 652X03-BIM3), epoxyed to 1/4-inch diam.
acrylic tubing. A short piece of smaller diameter brass tubing
was used to support the connection of the tensiometer cup
with the acrylic tubing. The tensiometers extend ≈2 cm into
the sample. Two 15-psi transducers (136PC15G2, Honeywell,
Minneapolis, MN) were used to monitor the matric potential
head at the two locations. In addition, a 1-psi differential
transducer (26PCAFA1D, Honeywell, Minneapolis, MN) was
mounted to monitor the difference in matric potential between
the two tensiometers, thus allowing computation of the hydraulic gradient. Two additional ports were added on opposite
sides of the cell to vent the sample with CO2 and allow fullwater saturation at the start of the outflow experiment.
The bottom outlet was connected to a burette for measuring
the outflow as a function of time. The burette was mounted
such that outflow water drained at atmospheric pressure. A
1-psi transducer (136PC01G2, Honeywell, Minneapolis, MN)
was attached at the bottom of the burette to measure drained
cumulative water volume. The upper boundary condition was
controlled using regulated pressurized N. The N was bubbled
through a distilled water reservoir before entering the pressure
cell to minimize evaporation loss in the cell. Two layers of 1.2micron, 0.1-mm thick nylon filters (Magna Nylon Membrane
Filters, Micron Separations Inc., Westborough, MA) wereused as a porous membrane at the bottom of the sample. We
combined two nylon filters for a single porous membrane to
reduce possible leaks thereby maintaining a bubble pressureof at least 700 cm during the outflow experiments. With asaturated hydraulic conductivity of ≈7.0 ⫻ 10⫺6 cm/s, the hydraulic resistance of the thin nylon membrane was low compared with other commonly used porous membranes, thereby
minimizing water pressure differences across the porous membrane during drainage of the soil core. The experiments were

WILDENSCHILD ET AL.: FLOW RATE DEPENDENCE OF SOIL HYDRAULIC CHARACTERISTICS

Table 1. Bulk physical properties of the investigated soils.
Soil type

Sand

Silt

Clay

Bulk density

Columbia
Lincoln

63.2
88.6

%
27.5
9.4

9.3
2.0

g/cm3
1.45
1.69

conducted for two soils of varying textural composition, aLincoln sand obtained from the EPAs RS Kerr Environmental
Research Laboratory in Ada, OK, and a Columbia fine sandy
loam collected along the Sacramento River near West Sacramento, CA. Soil properties for both soils are listed in Table
1 (Liu et al., 1998). The Columbia and Lincoln soil were sieved
through 0.5- and 0.6-mm sieves, respectively, prior to packing.
Each sample was packed only once for each series of experiments to minimize packing effects on the results. The soil was
packed in the pressure cell in small increments and the cell
tapped between each successive addition. Initial experimentswith the Columbia soil showed some settling after a few wetting and drying cycles; therefore, the Columbia soil sample
was vibrated for ≈1/2 hour after packing to obtain a wellsettled sample.
Three different types of outflow experiments were performed. First, for the one-step experiments a single high pneumatic pressure was imposed on the soil sample to induce
outflow. Second, multistep outflow experiments were carriedout using identical procedures as for the one-step experiments
except that a varying number of smaller pressure increments
were applied instead of one large pressure step. Between eachsuccessively increasing pressure increment, time was allowed
for the sample to equilibrate or for outflow to cease. In addition to the gas pressure induced outflow scenarios, a syringe
pump procedure (Wildenschild et al., 1997) was used in a third
series of experiments in which the soil sample was drained at
a constant low flow rate, simulating quasi-static conditions at
any time during the drainage experiment. A drainage rate of
0.5 ml/hr was used for the syringe pump measurements inboth soils.
All experiments were started at an initial condition of
hbottom ≈ ⫺2 cm. Prior to each drainage experiment, the soil
samples were resaturated using the same procedure every time(including the use of CO2 to dissolve trapped air) to maintainidentical initial saturation values between the different drainage experiments. Otherwise, observed differences could be
attributed to varying initial conditions. The standard deviations of the initial sample weights representing variations ofinitial saturation were 0.962 g for six replicates of the Columbia
soil and 0.732 g for 10 replicates of the Lincoln soil, representing water content variations of 0.0075 and 0.0050 cm3 cm⫺3for the Lincoln soil and the Columbia soil, respectively.

37

head gradients are most likely to occur. Also, the presence ofthe tensiometers in the sample could cause some disturbanceof the flow field. As mentioned earlier, we believe that thesmall size and the offsetting of the tensiometers justify theassumption of one dimensionality.
The unsaturated hydraulic conductivity was estimated directly from Darcy’s Law using various computational procedures. The tensiometer pair provided the hydraulic gradientin the center of the soil core as a function of drainage time,
and when combined with the outflow rate provided the unsaturated hydraulic conductivity as a function of matric potential
head or volumetric water content. To account for water flux
density differences between the upper and lower parts of thesoil sample during outflow, we used the method of Wendrothet al. (1993) to estimate the unsaturated hydraulic conductivity.
The soil sample was divided into three compartments asoutlined in Fig. 2. The first (top) compartment (l1) extended
between the surface of the soil sample and the center between
the two tensiometers, and was represented by the matric potential head measured in the first (top) tensiometer, h1. The
second compartment (l2) extended from the center between
the two tensiometers to the center between the lower tensiometer and the bottom of the soil sample. This second compartment was represented by the matric potential head measured
with the second (lower) tensiometer, h2. The third compartment (l3) was defined by the bottom of the soil sample andthe center between the lower tensiometer and the bottom of
the sample. The matric potential head between the bottom of
the sample and the lower tensiometer was assumed to belinearly distributed. Thus, this compartment was represented
by the weighted mean of the matric potential head within thecompartment h3 ⫽ hbottom ⫺ (hbottom ⫺ h2)/4. For each compartment, the water content at each time step was calculated fromthe representative matric potential head values (i.e., ␪1, ␪2,and ␪3).
For direct estimation of K(␪) or K(h ), the fluxes between
the compartments were computed in two ways: either (1) fromthe top to the bottom, or (2) from the bottom to the top.
1. During the drainage experiments, the flux between compartments 1 and 2 (q12) was computed from water storagechanges between measurement times using ␪1, while thefluxes between compartments 2 and 3 (q23) were computed from time rate of water storage changes using ␪1and ␪2. Subsequently, K12(h ) and K23(h ) or K12(␪) andK23(␪) were estimated, assuming the Darcy equation to
be valid, while substituting the representative matric po-

Direct Estimation Using Darcy’s Law
The retention characteristics for the soils were establishedfrom the average of the two tensiometer readings and themeasured cumulative outflow volumes. The cumulative outflow was converted to sample average water content using
soil porosity and assuming initial fully saturated conditions.
At the conclusion of each experiment, the sample was weighed
to determine final water content values, from which initialwater content values were verified. This approach may introduce potential errors, since the measurements were carriedout during transient conditions, thereby leading to possible
depth variations in matric potential head, while a single core
averaged water content was estimated from the cumulative
outflow data. This could be of particular concern for the fast,
one-step experiments where depth dependent matric potential

Fig. 2. Compartments and flux boundaries for the calculated hydraulic conductivities.

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SOIL SCI. SOC. AM. J., VOL. 65, JANUARY–FEBRUARY 2001

Table 2. Inversely estimated parameters for the Lincoln soil. (1.) estimated parameters: (␣, n, Ks, ␪r), (2.) estimated parameters: (␣, n,
Ks, ␪r, l).
Date

SSQ

␪r

␣

0611a
0611b

0.049
0.041

0.097
0.110

0.019
0.019

0531a
0531b

0.020
0.013

0520a
0520b

Ks (cm/h)

l

R2

4.803
7.240
Onestep 0–125 mbar

2.883
1.934

⫺0.282

0.993
0.996

0.077
0.075

0.020
4.730
0.020
4.476
Multistep 0–50–100 mbar

2.019
2.324

0.637

0.999
0.999

0.051
0.029

0.041
0.095

0.023
3.104
0.022
5.882
Multistep 0–25–35–62–80–100 mbar

0.911
0.363

⫺0.788

0.998
0.997

0524a
0524b

0.014
0.010

0.050
0.072

0.021
3.749
0.021
4.420
Syringe pump (0.5 ml/h)

0.588
0.413

⫺0.104

0.999
0.999

0607b

0.008

0.043

0.022

1.349

⫺0.0870.998

nOnestep 0–250 mbar

tential head values (h1 and h2 for q12, and h2 and h3 forq23). Finally, K3 values were calculated directly from thebottom compartment drainage rate and matric potential gradients.
2. Alternatively, using the drainage rate from the bottom
compartment, q3, K3(h ) was estimated from the Darcy
equation, using h2 and hbottom to estimate the representative matric potential head gradient. Subsequently, K32(h )values and K21(h ) values were calculated from fluxes
between compartments 2 and 3 (q32) and compartments
1 and 2 (q21). These fluxes were estimated from q3, after
subtraction of the time rate of soil water storage changesof the representative compartments (obtained from timechanges of ␪1, ␪2, and ␪3), and using the appropriate matricpotential gradients.
To simplify the direct estimation results, we will only present data using a selection of experiments and conductivity
estimations for each soil. For the Lincoln soil these experiments are one-step (0–250 mbar and 0–125 mbar), multistep(0–50–100 mbar and 0–25–35–62–80–100 mbar) and the quasistatic experiment. For the Columbia soil, we are presenting
estimated data for one-step (0–500 mbar), multistep (0–250–
500 mbar and 0–125–250–375–500 mbar), and the quasi-static
experiment. In addition, we have chosen only to present data
for the middle (K23 and K32) and lower part (K3) of the sample.
This was done partly for simplicity, but also because the estimates from the upper part (K12 and K21) were affected by thelarge gradients present towards the end of the experiments.
For the multistep experiments in particular, this meant thatthe curves were discontinuous between individual pressure
steps. Also, the estimates from the middle part of the sample(K23 and K32) provide a more correct basis for comparison
with the estimates based on the drainage rate (K3), as opposedto the estimates from the upper part of the sample (K12 andK21).
Unfortunately, it was not possible to estimate the hydraulicconductivity from the syringe pump experiments because thevery small hydraulic gradients prevented accurate K estimation.

Inverse Estimation Based on Richards’ Equation
The unsaturated hydraulic properties were inversely estimated using HYDRUS-1D (Simunek et al., 1998), which numerically solves Richards’ equation in one dimension using aGalerkin-type linear finite element scheme. Minimization isaccomplished using the Levenberg-Marquardt nonlinear

3.402

weighted least squares approach. The objective function was
defined as the average weighted squared deviation normalized
by the measurement variances of particular measurement sets.
For additional details about inverse modeling, its applicationfor estimation of soil hydraulic properties and the definitionof the objective function, refer to Simunek et al. (1998), andHopmans and Simunek (1999). For these experiments, theobjective function contained matric potential head readings
for the two tensiometers and cumulative outflow data measured as a function of time. The expressions of van Genuchten(1980) were used to parameterize the hydraulic functionsSe ⫽(␪ ⫺ ␪r)⫽ [1 ⫹ (␣|␺|)n]⫺m␪s ⫺ ␪r
K ⫽ KsSle[1 ⫺ (1 ⫺ Se1/m)m]2[1]

where Se is effective saturation, ␪s, ␪r and ␪ are full, residual
and actual water contents, ␣, n and l are constants, m ⫽ 1 ⫺
1/n, ␺ is matric potential, and K and Ks are unsaturated andsaturated hydraulic conductivities. The parameters optimized
were either (␣, n, Ks, ␪r) or (␣, n, Ks, ␪r, l ) where l is the
exponent in Mualem’s (1976) equation. The exponent is commonly fixed at a value of 0.5, however, the fit to the data
was improved if the l parameter was optimized as well. The
saturated water contents (␪s) were fixed at 0.37 and 0.45 cm3
cm⫺3 for the Lincoln and Columbia soils, respectively. The
unsaturated hydraulic conductivity was not fixed at its measured value in the optimizations because the inverse solutions
converged readily without Ks data. The variation in optimized
Ks values (Table 2 and 3) is less than an order of magnitude,
and as such could not be improved with a laboratory Ks measurement, which usually has an estimation error of similar
magnitude. Also, we were interested in investigating dynamic
phenomena, so including Ks data measured at steady-state
conditions might confuse the issue.
Optimized parameters as well as squared residual (R2) andsum of squared residual (SSQ) values for the different optimizations are listed in Table 2 and Table 3 for the Lincoln andColumbia soil, respectively. As seen in Table 2, the sum ofsquared residuals generally decreases with decreasing flow
rate. For the Columbia soil in particular (Table 3), the SSQ
values are significantly reduced when the exponent is allowed
to vary (Case 2). For the Lincoln soil (Table 2) we observe
an increase in the ␣ values with decreasing flow rate, while
the optimized n values are less sensitive to the flow rate. The
increase in n values is not seen for the Columbia soil. These
inversely obtained results reflect the trends observed for the39

WILDENSCHILD ET AL.: FLOW RATE DEPENDENCE OF SOIL HYDRAULIC CHARACTERISTICS

Table 3. Inversely estimated parameters for the Columbia soil. (1.) estimated parameters: (␣, n, Ks, ␪r), (2.) estimated parameters: (␣,n, Ks, ␪r, l).
Date

SSQ

␪r

␣

0528a
0528b

0.045
0.001

0.040
0.186

0.010
0.007

0531a
0531b

0.039
0.012

0.036
0.187

0604a
0604b

0.029
0.008

0.041
0.186

Ks (cm/h)

l

R2

1.634
3.655
Onestep 0–500 mbar

1.050
0.140

⫺1.118

0.995
0.999

0.010
1.604
0.007
3.582
Multistep 0–125–250–375–500 mbar

1.005
0.143

⫺1.147

0.993
0.998

1.128
0.187

⫺1.095

0.999
0.999

n

Multistep 0–250–500 mbar

0.011
0.010

other estimation methods, which are discussed in the following.

1.561
2.830

Fig. 3 shows the estimated average retention dataobtained using both the direct (symbols) and inverse
estimation (lines) approaches. We only present optimization data for the cases where l was optimized. Fig. 3a–d
present the estimated retention data for each individual
experiment, while Fig. 3e and 3f compare all directly
estimated data (Fig. 3e) and optimized curves (Fig. 3f),
respectively. It is evident from both Fig. 3e and 3f thatsoil water retention for the Lincoln soil is influenced by
the drainage rate, regardless of the estimation method.
In general, soil water retention increases as the numberof pressure steps decreases, with the largest retention
and residual water content for the single step experiment (0–250 mbar), and the lowest retention and residual water content for the quasi-static syringe pump andlow-pressure multi-step outflow experiments. The differences were ≈7% by volume for a given matric potential in the early stages of the experiment. As we progressfrom the highest (0–250 mbar) to the lowest (quasistatic) flow rate, we also note a decrease in the measured
air-entry value of ≈15 cm. Similar trends can be observedin the inversely estimated curves, suggesting that thedirectly estimated (measured) curves are representativeof the average behavior of the sample. In contrast, noapparent rate dependence was observed for the finetextured Columbia soil (Fig. 4a–e). Generally, our measured curves for both the Lincoln and Columbia soil
compare favorably with the curves measured by Chenet al. (1999).

the inversely estimated curves for the two soils. Generally, the hydraulic conductivities computed using thetwo different estimation approaches (top down, K23 orbottom up, K32) are very similar, thereby corroborating
the K estimation methods. As mentioned previously,
only data for the bottom (K3) and middle (K23 and K32)
compartments are presented. In Figures 5a and 5b, wenote a slight underestimation of the hydraulic conductivity, relative to the optimized conductivity curves, forthe fast experiments, however, the data for the slower
experiments (Fig. 5c and 5d) matched the optimized
curves. A similar underestimation was determined for
the Columbia soil (Fig. 6a–6c).
The estimated unsaturated hydraulic conductivity
data for the different outflow experiments for the Lincoln soil are compared in Fig. 5e (direct, K3) and 5f(inverse). The inversely estimated curves show an increase in hydraulic conductivity with increasing flowrate at high saturations. At high saturations at the startof the outflow experiments, differences in optimized
hydraulic conductivity curves between the slow and fast
outflow experiments are approximately one order ofmagnitude, with the highest conductivity values for the
faster experiments (0–250 and 0–125, Fig. 5f). This trend
is not so clearly apparent from comparison of the direct
estimates (Figure 5e), because of lack of data points
at high saturations for the fast outflow experiments;
however, we would expect a similar behavior. As the
sample water content decreases, this trend reverses withthe highest unsaturated conductivity values (both optimized and directly computed) for the low flow-rate experiments. Similar trends were not observed for the
Columbia soil (Fig. 6d and 6e) for which almost identicalcurves were obtained for the different drainage rates
for both the inversely and directly computed estimates.

Unsaturated Hydraulic Conductivity

Physical Processes Controlling Outflow

The directly and inversely estimated unsaturated hydraulic conductivities are shown in Fig. 5 and 6 for the
Lincoln and Columbia soil, respectively. Fig. 5a–d andFig. 6a–c show a selection of the estimated unsaturated
conductivity data for each individual experiment for the
Lincoln and Columbia soils, respectively. Fig. 5e and6d compare the directly estimated conductivity data as
estimated using the bottom compartment drainage rates
and matric potential gradients (K3) for the Lincoln andColumbia soil, respectively. Fig. 5f and 6e present all

A number of different physical processes might contribute to the rate-dependent results observed in our
experiments. We suggest the following processes or a
combination thereof, which systematically can explain
the observed phenomena.

RESULTS AND DISCUSSION
Soil Water Retention Characteristics

1. Entrapment of water. This is a plausible mechanism at high flow rates. We hypothesize that water
entrapment occurs through hydraulic isolation ofwater-filled pores by draining surrounding pores.
The larger the drainage rate, the less opportunity

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SOIL SCI. SOC. AM. J., VOL. 65, JANUARY–FEBRUARY 2001

Fig. 3. Directly and inversely estimated retention curves for the Lincoln soil as a function of applied pressure (a) 0–250 mbar, (b) 0–125 mbar,(c) 0–50–100 mbar, (d) 0–25–35–62–80–100 mbar, (e) directly estimated curves for all experiments, (f) inversely estimated curves for all
the experiments.

exists for all pores to drain concurrently. It may
occur throughout the soil sample and will increasewater retention, but will decrease unsaturated hydraulic conductivity, as the mobile portion of thesoil water is decreased. The highest flow rates occur for one-step experiments and for coarse-textured soils, and hence water entrapment is likely

to be more prevalent in coarse soils with large
head gradients.
2. Pore water blockage. When applying a sudden
large pressure step to a near-saturated or saturated
soil sample, the large matric potential head gradients near the porous membrane result in fasterdrainage of the pores at the bottom of the sample

WILDENSCHILD ET AL.: FLOW RATE DEPENDENCE OF SOIL HYDRAULIC CHARACTERISTICS

41

Fig. 4. Directly and inversely estimated retention curves for the Columbia soil as a function of applied pressure (a) 0–500 mbar, (b) 0–250–500
mbar, (c) 0–125–250–375–500 mbar, (d) directly estimated curves for all experiments, (e) inversely estimated curves for all the experiments.

than in the overlying soil, thereby isolating the
conductive flow paths and impeding further drainage and equilibration of the soil. Consequently, thesample’s unsaturated hydraulic conductivity will
be reduced. Pore blockage is likely to occur formaterials with a uniform pore size distribution (likethe Lincoln sand). It is assumed that air is available

to replace the draining water at the bottom of thesample, hence it can only occur if there is macroscopic air continuity across the whole sample (airentry value of soil must be exceeded).
3. Air entrapment. This mechanism was discussed by
Schultze et al. (1999) who used a two-fluid numerical code to model their results, thereby accounting

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SOIL SCI. SOC. AM. J., VOL. 65, JANUARY–FEBRUARY 2001

Fig. 5. Directly and inversely estimated unsaturated hydraulic conductivity curves for the Lincoln soil as a function of applied pressure (a) 0–250
mbar, (b) 0–125 mbar, (c) 0–50–100 mbar, (d) 0–25–35–62–80–100 mbar (only the curves for the middle and bottom compartments are shown),(e) directly estimated curves for all experiments for the bottom part of the sample (K3), (f) inversely estimated curves for all experiments.

for spatial and temporal changes in air pressure in
the sample. When a soil core is drained either byincreasing the gas phase pressure or by decreasing
the water phase pressure, it is generally assumed
that air is available to replace the draining water;
however, for a sample holder that is open to air
only at the top of the soil sample, this assumptionof air phase continuity throughout the sample is
not necessarily valid. According to Corey and
Brooks (1999), initial drainage for such or similar
conditions will occur as a result of depression ofinterfaces at the sample holder boundaries or bya slight expansion of entrapped gas, instead of air
replacing draining water. As pointed out by Schultze
et al. (1999), air entrapment can increase water
retention. Air entrapment is more likely if air permeability is low at high water saturation. As aresult, soil water retention will be nonunique, andis determined by the rate of pressure step changes.
Also, as was demonstrated by Schultze et al.
(1999), the trapped air will effectively decrease
cumulative drainage and drainage rate, and result
in reduced unsaturated hydraulic conductivity values at near saturation. Regarding the use of either

WILDENSCHILD ET AL.: FLOW RATE DEPENDENCE OF SOIL HYDRAULIC CHARACTERISTICS

43

Fig. 6. Directly and inversely estimated unsaturated hydraulic conductivity curves for the Columbia soil as a function of applied pressure (a)0–500 mbar, (b) 0–250–500 mbar, (c) 0–125–250–375–500 mbar, (d) directly estimated curves for all experiments for the bottom part of thesample (K3), (e) inversely estimated curves for all experiments. (Only the curves for the middle and bottom compartments are shown).

air phase pressure or water phase suction to induce
outflow, Eching and Hopmans (1993) measured
retention curves using both suction and air pressure experiments and found relatively small differences between the two approaches; however, we
acknowledge that there are additional complexities
regarding the suction vs. pressure issue for instanceas discussed by Chahal and Young (1965) andPeck (1960).
4. Air-entry value effect. This phenomenon is present
when dynamic flow experiments are carried out

while the soil in the pressure cell (with porous
membrane) is saturated as occurs when the air
entry value of the soil is not exceeded prior toapplying the first pressure step. The lack of continuity in the gas phase will cause piston-type flow,
with drainage occurring from the top, rather thanthe bottom (Hopmans et al., 1992). This process
is mostly effective if the soil has a distinct air entry
value. It will increase unsaturated K near saturation, however, it is not clear how it will affect water
retention. Also, according to Corey and Brooks

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SOIL SCI. SOC. AM. J., VOL. 65, JANUARY–FEBRUARY 2001

(1999), matric potentials cannot be measured forsaturations above ≈85%, because of the general
disconnection of the gas phase at these high saturations. Consequently, one would expect nonuniquesoil water retention curves, controlled by imposedboundary conditions and pore connectivity characteristics.
5. Dynamic contact angle effect. Friedman (1999) hypothesized that the advancing or receding solid–
liquid–gas contact angle in a capillary tube is dependent on the velocity of the propagating orwithdrawing liquid–gas interface. Hence, in dynamic experiments, the static contact angle has tobe replaced with a dynamic contact angle. Although in concept possible, Friedman agrees thatsuch explanation is probably for fluids with low
interfacial tensions such as NAPL water. For draining soil samples, a reduction of the contact angle
in the range between 0 and 30⬚ has a limited influence on the matric potential, as increasing flow
velocities would decrease that contact angle to aminimum value of zero. Hence, the contact angle
effect will be small in any case for draining soils.
Moreover, although the contact angle effect can
explain the increasing water retention with increasing flow velocities, it can not explain the increasing
unsaturated hydraulic conductivity with increasingwater velocities.

Observed Phenomena in the Context
of the Suggested Processes
We suggest that the marked effect on the curves for
the Lincoln soil can be attributed to three or four of
the above stated mechanisms. The first mechanism iscaused by disconnection of flow paths at higher flow
rates, so that water is being trapped in dead-end porespace, Process 1. For the low-pressure outflow experiments, water in individual pores remains connected tothe bulk water, thereby allowing water to drain as thewater potential decreases during multi-step outflow experiments. A similar explanation has been suggested
earlier by Smiles et al. (1971) and Topp et al. (1967).
In addition to water entrapment, we believe that Processes 2, 3 and 4 contribute to the measured rate-dependent soil hydraulic characteristics. Initial drainage of thelower part of the sample (Process 2) was observed in
one-step outflow experiments using x-ray computer assisted tomography (CT) by Hopmans et al. (1992) and
we are currently observing similar patterns in preliminary x-ray CT experiments of the Lincoln soil. Thismechanism is enhanced for soils with a narrow poresize distribution, and hence is expected to be more pronounced for the sandy Lincoln soil. Similarly, the airentry value effect (Process 4) is likely to influence ourresults, since the air entry value of the Lincoln soil is≈⫺20 cm; that is, drainage occurs only if matric potential
head values are smaller than ⫺20 cm; however, theinitial condition of the soil was only ≈⫺2 cm, so thatthe first pressure increment was applied when the Lincoln soil was still saturated. This is particularly critical

for the Lincoln soil experiments with large pressure
increments (one or two steps) where early drainage most
likely did not conform to Richards type of flow, whichassumes that the nonwetting or air phase is continuous
throughout the soil sample (Hopmans et al., 1992). Also,
Schultze et al. (1999) stated that the air phase is discontinuous during drainage until a significant amount ofwater has left the pore system and an emergence point
saturation is reached, at which point the gas permeability jumps to a finite value.
The influence of flow rate on the retention characteristic was not apparent for the finer textured Columbia
soil (Fig. 4a–e) using different applied pressures; however, we hypothesize that a similar behavior might haveoccurred if much higher gas pressures had been applied
to drain the Columbia soil than used in the reported
experiments. Chen et al. (1999) showed that the airpermeability is higher for the Columbia soil than for
the Lincoln soil, at the same degree of saturation. Therefore, some of the above processes, which are controlled
by the low air permeability and the lack of air phase
continuity of the Lincoln soil, might not be of importance for the Columbia soil. Another reason for the
relatively small deviations among the retention curves
for the Columbia soil, could be the small matric potential
gradients present later in the experiment (i.e., at lower
saturations). Fig. 7 and 8 illustrate matric potential gradients as estimated directly from the two tensiometers
for the Lincoln and Columbia soils, respectively. As
seen from Fig. 8, the directly estimated matric potential
gradients in the Columbia soil (solid lines) are large atfirst following an applied pressure increment, butquickly drop to ≈⫺1.0, as is expected if hydraulic equilibrium is attained. The periodic return of the matric potential gradients to near zero for the Columbia soil is theresult of the total soil water potential approaching static
conditions after each of the applied pressure increments;
however, the return to static equilibrium after each applied pressure increment does not always occur for the
sandy soil, especially not for the high-pressure single
step experiments as illustrated in Fig. 7. The continuously increasing matric potential gradient (solid lines)
is indicative of a soil water regime not tending towards
static equilibrium. Our proposed mechanism, Process 2,
can explain this phenomenon. After a large pressure
increment is applied, the bottom section of the sandy
soil is drained, thereby emptying the largest, highly conductive pores. For a soil with a narrow pore size distribution (Lincoln soil), drainage of the largest pores prevents
continued drainage and equilibration of the soil above
the initially drained portion, and effectively results in
the increasing total water potential gradients with progressing drainage as observed in Fig. 7. For the two slower
outflow experiments (0–50–100 mbar and 0–25–35–62–
80–100 mbar), a similar behavior as for the finer textured soil is observed, and the gradients return to unity,
shortly after the pressure step is applied.
In general, the numerical model simulated total matric potential heads similar to what was observed, evenfor the fast outflow experiments (compare solid withdashed lines in Fig. 7 and 8). Extended forward model-

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45

Fig. 7. Matric potential gradients measured and optimized between the two tensiometers in the Lincoln soil.

ing of the 0–250 mbar experiment for the Lincoln soil
showed that static equilibrium is not approached until
2000 hours or ≈83 days have passed (see inset of Fig.
7). Generally, when performing retention curve measurements on sandy materials it is assumed that thesecan be done relatively fast, whereas the more clayeymaterials require time-consuming experiments; however, our results indicated otherwise.
At low saturations, we observed low unsaturated hydraulic conductivities for the fast outflow experiments
for the Lincoln soil, Fig. 5e and Fig. 5f. We suggest that
this is due to water being either trapped (Process 1) orblocked from the main water body (Process 2), thereby
becoming immobile and effectively not contributing toflow. We hypothesize that this water becomes entrappedand immobile in the early stages of the experiment whenrelatively high flow rates prevail, particularly in thoseexperiments where large one-step pressure steps wereused to induce outflow. In Fig. 9, the cumulative outflow
volumes are plotted as a function of time for the Lincoln
soil, showing that the maximum flow rates occur initiallywhen the first pressure increment is applied. We assume
that these maximum flow rates occur near the nylon
membrane at the bottom of the sample. Specifically, the0–250 and 0–125 drainage experiments induce these highflow rates in the first few minutes of the experiment(see inset of Fig. 9). The initially trapped or blockedwater will remain trapped as the soil continues to drainbecause it is disconnected from the flowing water phase.
Another possible reason for the negligible effect of theoutflow rate on the hydraulic characteristics of the Columbia soil is its dominance of smaller pores following
our proposed Process 1. Single water-filled pores areless likely to be isolated from the main flow path in
the Columbia soil, compared with the coarser textured
Lincoln soil.
At high saturations, we observed higher conductivi-

ties for the faster experiments (Fig. 5e and 5f). Similar
results were found by Plagge et al. (1999) and Schultze
et al. (1999), and we believe that the main cause of theseincreased conductivities derives from our proposed
Process 4. If the air pressure is suddenly increased (orthe water pressure decreased as in the case of the experiments of Plagge et al. (1999) and Schultze et al. (1999)
without first exceeding the air entry value of the soil,
thereby providing air passage into the soil, water willdrain from the sample relatively fast and the unsaturated
hydraulic conductivity will be overestimated. However,at the later stages of drainage, the water entrapment
effect and pore blockage becomes the controlling factor
in the hydraulic conductivity estimation, and a crossoverof the curves in Fig. 5e and 5f is observed. Apparently,
this crossover occurs at a lower saturation in the in-Fig. 8. Matric potential gradients measured and optimized between
the two tensiometers in the Columbia soil.

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SOIL SCI. SOC. AM. J., VOL. 65, JANUARY–FEBRUARY 2001

Fig. 9. Cumulative outflow rates as a function of time for the Lincoln soil. Early time data is shown in inset.

versely estimated curves (between water contents of
0.10–0.15 cm3 cm⫺3), while the directly estimated curves
show a crossover point at a water content of ≈0.20 cm3cm⫺3. The inversely estimated data is based on the assumption of Richards flow and air phase continuity,which could be the cause of this difference between the
direct and indirect unsaturated hydraulic conductivity
calculations.
The crossover of the hydraulic conductivity curves is
not observed for the finer textured Columbia soil in
Figure 6e, likely due to its more poorly defined air-entry
value (Process 4), wider pore-size distribution (Process 1
and 2), and higher air permeability (Process 3). Also,
the measurements of the finer-textured Columbia soil
do not show the continuously increasing nonequilibrium
gradients as measured for the Lincoln soil (compare
Fig. 7 and 8), and therefore the hydraulic characteristicsof this soil are less likely to be dependent on flow rate.
The lower initial flow rates and their smaller variation
among the Columbia experiments are illustrated in Fig.
10 (note that the inset is plotted on the same scale forthe two soils).

CONCLUSIONSIn conclusion, we believe that the rate-dependent
phenomena observed for the coarse Lincoln soil in theseexperiments are a consequence of a combination of flow
processes, and are a result of differences in pore size
distribution and pore connectivity between the two soils.
Process 4 mainly affects the unsaturated hydraulic conductivity curve at high water contents, while the accompanying high flow rates trap (Process 1) and block (Proc-

ess 2) water, thereby affecting soil water retention. Allof these phenomena are mostly prevalent in the coarse
textured soil and for the high flow rates. As the soil
desaturates, Process 4 is no longer a factor, however, itis the entrapped and blocked water that controls thehydraulic properties, thereby reducing the unsaturated
hydraulic conductivity in the lower water content range.
Our explanations can be applied to results presented
by others. For example, both Plagge et al. (1999) and
Schultze et al. (1999) found increased hydraulic conductivities with increasing flow rates that can be explained
from Process 4. Also, the results of Topp et al. (1967)
can be explained using the air entrapment argument(Process 4), as their drainage experiments were initiatedat full saturation while the soil’s air entry value was≈⫺35 cm. In the experiments of Smiles et al. (1971), airaccess was secured at any time because the sample
holder included ventilation holes. However, their experiments confirm Process 1 to be effective. Water retention is highest closest to the water inlet (largest fluxes)
and for experiments with the largest head gradients(Fig. 4 and 6). The results of Smiles et al. (1971) wereconfirmed by Vachaud et al. (1972), who showed thatthe largest deviations from the static water retention
curve occur when large head gradients are applied(Process 1 and 2).
Based on our investigation it is critical to consider
the method by which the hydraulic properties for unsaturated soils are determined, thus keeping in mind the
purpose of the characterization. For the coarse textured
Lincoln soil we have shown that the use of hydraulicparameters obtained under relatively high outflow conditions may not accurately represent slow flow phenom-

WILDENSCHILD ET AL.: FLOW RATE DEPENDENCE OF SOIL HYDRAULIC CHARACTERISTICS

47

Fig. 10. Cumulative outflow rates as a function of time for the Columbia soil. Early time data is shown in insert.

ena, as mostly experienced in the field. The lack of static
equilibrium in the coarser soil material may affect bothinverse and direct estimations. Therefore, the choice ofboundary conditions and estimation method must be
carefully evaluated.
The rate dependence of the hydraulic characteristics
has been shown to be of less importance for the finer
textured Columbia soil, at least for the experimental
conditions investigated in this study. The higher air permeability for the Columbia soil at a given saturation,
along with its wider pore-size distribution and a morepoorly defined air entry value are likely reasons for thelack of rate dependence observed for the Columbia
soil. Also, the induced flow rates in the Columbia soil
experiments were not as high as those measured for the
Lincoln soil. Therefore, it is possible that some of theflow rate effects on the soil hydraulic characteristics are
effective for the Columbia soil as well, if higher flow
rates were used by applying much larger water pressure
gradients than used in this study.
We realize that other processes such as those statedfor instance by Demond and Roberts (1991), Hassanizadeh and Gray (1993), Clayton (1999), and Friedman(1999) may also be contributing to the observed flowrate dependent soil hydraulic properties. Regardless,
ignoring these rate-dependent phenomena might leadto significant errors, especially in situations where hydrological models are used with hydraulic parametersthat are obtained at considerably different flow rates.
ACKNOWLEDGMENTS
Thanks to J.L. McIntyre for invaluable help in the laboratory. The following foundations are gratefully acknowledged

for financial support: COWI Foundation, Kaj og Hermilla
Ostenfeld’s Foundation, A/S Fisker & Nielsen Foundation,
Otto Mønsted’s Foundation, G.A. Hagemann’s Foundation,
and Civil Engineer Åge Corrits Grant. This work was performed under the auspices of the U.S. Department of Energy
by Univ. of California Lawrence Livermore National Laboratory under contract no. W-7405-Eng-48.

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