Special Section: MUSIS
Efstathios Diamantopoulos
Wolfgang Durner*
Dynamic Nonequilibrium of Water
Flow in Porous Media: A Review
This review provides an overview on various phenomena, hypothesized causes, and modeling approaches that describe “dynamic nonequilibrium” (DNE) of water flow in soils.
Dynamic nonequilibrium is characterized from observa ons on the macroscale by an
apparent flow-rate dependence of hydraulic proper es or by local nonequilibrium between
water content and pressure head under monotonic imbibi on or drainage histories, i.e.,
not affected by tradi onal hysteresis. The literature indicates that key processes causing
DNE are pore-scale phenomena such as relaxa on of air–water-interface distribu ons,
limited air-phase permeability, dynamic contact angles, and me-dependent we ability
changes. Furthermore, entrapment of water and pore water blockage, air-entry effects,
and temperature effects might be involved. These processes act at different pressure head
regions and on different me scales, which makes effec ve modeling of the combined phenomena challenging. On larger scales, heterogeneity of soil proper es can contribute to
DNE observa ons. We conclude that there is an urgent need for precision measurements
that are designed to quan fy dynamic effects.
Abbrevia ons: DNAPL, dense nonaqueous-phase liquid; DNE, dynamic nonequilibrium; MSO, mul step ou low; REV, representa ve elementary volume; SHP, soil hydraulic proper es; TDR, me domain reflectometry.
Richards’ equation often cannot
describe observa ons of soil water
dynamics, as indicated for example
by an apparent nonuniqueness of
soil hydraulic proper es under transient-flow conditions. We review
observa ons, hypothesized mechanis c causes, and effec ve modeling
approaches for dynamic nonequilibrium of water flow in soils.
E. Diamantopoulos and W. Durner, Ins tut
für Geoökologie der Technischen Universität,
Braunschweig 38106, Germany. *Corresponding
author ([email protected]).
Vadose Zone J.
doi:10.2136/vzj2011.0197
Received 19 Dec. 2011.
© Soil Science Society of America
5585 Guilford Rd., Madison, WI 53711 USA.
All rights reserved. No part of this periodical may
be reproduced or transmi ed in any form or by any
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and retrieval system, without permission in wri ng
from the publisher.
Water flow in the subsurface plays a key role in environmental sciences
such as hydrology, ecology, soil science, or agriculture. Understanding and predicting soil
water dynamics requires proper conceptual modeling of the mechanisms of water retention and flow, and knowledge of the soil hydraulic properties. During the last decades,
tremendous progress has been made in this direction. With numerical models, we are
able to simulate and accurately predict water movement in soils on different scales and for
various boundary conditions (among many others, van Dam and Feddes, 2000; Šimůnek,
2005; Šimůnek et al., 2008). In practical applications, Richards’ equation is currently,
and foreseeably also in the future, the most frequently used conceptual model to simulate
soil water dynamics (Vanclooster et al., 2004). The validity limits of this continuum-scale
description have become evident in various studies. They arise in situations where processes
not considered in the derivation of the Richards equation become relevant. Errors that
result from using this simplifying approach in a given situation are known in a qualitative
sense, but thresholds where it can no longer be used in practical applications remain to be
explored (Ippisch et al., 2006; Narasimhan, 2007).
One of the phenomena that limit the applicability of Richards’ equation is DNE of water
flow in porous media. It is difficult to give a strict definition of the term dynamic nonequilibrium, and previous literature has not always used this term when referring to nonequilibrium phenomena nor is the term used in a unique and coherent manner. In this review, we
define DNE from a phenomenological point of view as the apparent nonuniqueness of the
relationship between measured water content, θ, and pressure head, h, under hydrostatic,
steady-state, or monotonically changing hydraulic conditions. Under these conditions,
the traditional concept of static hysteresis (Funk, 2012) is of no relevance. Our definition
arises from experimental observations in laboratory column studies, where DNE becomes
evident as a flow-rate dependence of soil hydraulic properties under transient-flow conditions or as a drifting θ(h) relationship under no-flux or quasi-static conditions.
The existence of DNE effects has been known since the 1960s, particularly by the work of
Topp et al. (1967). Remarkably, scientific interest in DNE appears to have oscillated since.
We hypothesize that there are multiple reasons for this periodic up and down. First, the
observation of flow-rate dependency of hydraulic properties requires dynamic flow experiments with suitable instrumentation and suitable evaluation techniques because otherwise
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dynamic effects cannot be detected. Second, we believe that the
past inability to treat the mathematical problem on the continuum
scale, which requires the solution of the highly nonlinear partial
differential Richards equation with time-variant or system-statedependent functions for the constitutive relationships, made it
hard to evaluate these experiments quantitatively. Finally, given
the deficits in our current understanding of the phenomenon, it is
impossible to assess the importance of these effects for any practical situation. It is thus currently not clear how important “nonequilibrium” effects are, whether they contribute to the often-observed
discrepancy of hydraulic properties found between laboratory and
field studies, and under which conditions their consideration gives
improved predictions of water flow in porous media. Observations
of DNE are affected by a variety of processes and scale issues, which
we categorize as follows:
1. Pore-scale processes, such as relaxation of water–air
interfaces, dynamic wetting angles, temporal changes
in wettability, dissolution of entrapped air, or slow
redistribution of disconnected water, will lead to DNE
at any macroscopic observation scale. These processes are
reviewed below.
2. Local heterogeneities of porous media on length scales
smaller than the measurement windows of the sensors
will lead to the observation of spatially averaged state
variables. When referring to traditional measurement
instruments in laboratory columns, such as tensiometers
or time domain reflectometry (TDR) probes, these
heterogeneities are of millimeter to centimeter scale, as,
e.g., in aggregated porous media. If measurement windows
become bigger, such as in the emerging geophysical and
remote-sensing-based measurements, local heterogeneities
of this type can be on much larger scales.
3. Finally, heterogeneities can have a larger scale than
the measurement volume of the sensors. In this case,
measurements will be affected by the spatial position
of the sensor installation (Schlüter et al., 2012).
Thus, a direct interpretation of a sensor’s reading
as representative of, e.g., the installation depth is
problematic and any direct relation between measured
state variables obtained from single sensors at different
positions will be subject to the problem of spatial
decoupling. We can expect this type of DNE to be
of relevance for undisturbed samples, lysimeters, or
field measurements, particularly in situations where
preferential flow phenomena are common.
All three categories are superimposed in real soils. Modeling of
DNE effects caused by Type 1 processes in any case needs specific
modeling concepts that go beyond the classical Richards approach.
Flow processes in heterogeneous systems of Type 3 can, in principle,
be described with the Richards equation because they are above the
scale of the representative elementary volume (REV, see below). For
Case 2, the applicability of the Richards equation will depend on
whether the local properties can be resolved by three-dimensional
modeling with a spatial resolution below the measurement scale.
In any case, the use of averaged, measured state variables to define
effective hydraulic properties will not be possible.
Our review aims at giving an overview of observations of DNE,
hypothesized causes, and effective modeling approaches to treat the
phenomena on a macroscopic scale. It is not comprehensive because
it would be impossible to discuss all aspects of nonequilibrium water
flow in one review. Hassanizadeh et al. (2002) reviewed the topic of
DNE water flow in porous media. Since then, advances in sensor
technology, data acquisition systems, automation of experimental
control, and more observations on water flow in unsaturated soils
under transient conditions have given further evidence and insight
to DNE. Perhaps the greatest part of studies on nonequilibrium
water flow in soils has dealt with preferential flow phenomena, referring to macropore flow, finger flow, and heterogeneous flow (van
Dam et al., 1990; Roth, 1995; Jarvis, 2007). A review of nonequilibrium water flow and solute transport in soil macropores was given
by Jarvis (2007). Three more reviews have dealt with model applications in preferential flow studies (Šimůnek et al., 2003; Gerke,
2006; Köhne et al., 2009). Although we discuss heterogeneity as
a reason for nonequilibrium water flow in soils, the focus of this
review is on nonequilibrium phenomena in well-sorted materials or
materials with heterogeneity below the resolution of the measurement instrument (i.e., Types 1 and 2). Similarly, a deeper discussion of the scale dependence of soil hydraulic properties was largely
excluded from this discussion (for more information on this topic,
see Vereecken et al., 2007). Lastly, this review focuses on air–water
systems and we refer only exemplarily to two-phase flow studies
with oil as a nonwetting fluid.
6Theory
Water movement in porous media is described in a “continuum
framework” (Cushman, 1984) by relating the temporal changes
of water content at a point to the spatial gradient of the water flux.
In one-dimensional form for vertical flow, this reads as
∂q
∂θ
=−
∂t
∂z
[1]
where θ is the volumetric water content [L3 L−3], q is volumetric
water flux [L3 L−2 T−1], t is time [T], and z is the spatial coordinate [L] (positive upward); θ and q are defined as averages within
a representative elementary volume (REV) of the porous medium.
The water flux (again for one-dimensional vertical flow) is given by
the Darcy–Buckingham law (Darcy, 1856; Buckingham, 1907):
⎛ ∂h
⎞
q =−K ⎜⎜⎜ +1⎟⎟⎟
⎝ ∂z
⎠
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[2]
where K is the hydraulic conductivity [L T−1], which is described
by a nonlinear relationship with water content θ or pressure head
h [L]. Note that we address the matric potential by the expression pressure head, which is negative for unsaturated conditions
and decreases when a soil becomes drier. Th is is contrary to the
terms capillary pressure, suction, or tension, which define the local
pressure difference between the water and air phases as a positive
quantity and were used in some of the original literature referred
to in this review.
heterogeneities. In a similar way, the unsaturated hydraulic conductivity curve depends on water content, roughness, tortuosity,
and the shape and degree of interconnection of the water-conducting pores in the porous media (Reynolds et al., 2002). By using the
water retention curve estimated under equilibrium conditions, we
automatically assume that these effects have an invariant influence
on the soil water retention curve regardless of whether the water
moves or not. A similar assumption is made for the hydraulic conductivity curve.
Combination of Eq. [1] and [2] leads to the one-dimensional Richards equation (Richards, 1931), which in its h-based form is
Despite the obvious problems that are connected with these conceptual assumptions, the Richards equation has a clear physical
basis. During the past decades, it has been tested against a lot of
experimental data and has proved its applicability for various flow
systems and boundary conditions (among others, Staple, 1969;
Nimmo, 1990; Skaggs et al., 2004). Various observations have been
made, however, that cannot be described by the Richards equation
in the above-mentioned form.
C (h)
⎛ ∂h
⎞⎤
∂h ∂ ⎡
= ⎢ K ( h )⎜⎜⎜ +1⎟⎟⎟⎥
⎢
⎥⎦
⎝
⎠
∂t ∂z ⎣
∂z
[3]
where C(h) = ∂ θ/dh is the specific water capacity [L−1]. Equation [3] is the fundamental model for describing water flow in
the unsaturated zone on the macroscopic scale. Richards’ equation is assumed to be valid if the porous system is rigid, nonswelling, isotropic, and if only isothermal liquid water flow takes place.
Furthermore, a prerequisite is that the air is free to move without
notable pressure gradients in the soil at any system state.
The two constitutive relationships that characterize a porous
medium are the soil water retention curve, θ = f(h), and the unsaturated hydraulic conductivity curve, K = f(h) or K = f(θ). These
functions are commonly referred to as soil hydraulic properties,
SHPs (Durner and Flühler, 2005). The traditional way to estimate
the soil water retention curve is to apply a sequence of equilibrium
states by stepwise draining an initially saturated soil sample to a
sequence of decreasing pressure heads. After hydrostatic equilibrium is attained, the water content is measured. The equilibration
time can differ for every soil, however, from a few minutes to weeks
(Nimmo, 2002). This means that the typical time to obtain a complete water retention curve can be weeks or longer. Similarly, the
traditional way to estimate unsaturated hydraulic conductivity is
based on steady-state flux methods, which can become extremely
time demanding for unsaturated soils. Other methods also exist to
measure the water retention curve and conductivity curve that are
based on transient-flow experiments (Hopmans et al., 2002; Durner
and Lipsius, 2005). To give valid results, these methods require
independence of the hydraulic properties from the flow dynamics.
Typically, the constitutive relationships are parameterized by
simple functions and used with the Richards equation to predict
the water movement in the porous media under various boundary
conditions. According to Hassanizadeh et al. (2002), the water
retention curve is assumed to account for all the effects and processes that influence the equilibrium distribution of fluids, such
as surface tension, the presence of fluid–fluid interfaces, wettability of solid surfaces, grain size distribution, and microscale
6 Observa
ons
Observations of DNE have been published only for laboratory studies (Table 1). The reasons for that are that (i) well-controlled flow
experiments with monotonically changing boundary conditions
can be much better performed in the laboratory. A monotonic
change is a necessary requirement to avoid interference with the
hysteresis problem, which is a major problem under transient conditions in the field; (ii) studies with packed soils in the laboratory can
reduce complications that might arise from soil heterogeneity; and
(iii) noise and bias of water content and matric potential measurements can be minimized in the laboratory, e.g., by keeping temperatures constant, which leads to more accurate and reliable measurement data. Table 1 gives an overview of published experiments
where nonequilibrium has been found. We note that these experiments encompass a rather limited range of length and time scales
and furthermore are in most cases restricted to sandy materials.
Flow-Rate Dependence of Soil
Hydraulic Proper es
We start our review by recalling studies on flow-rate dependences
of hydraulic properties, particularly the water retention curve. Historically, the earliest questions regarding DNE in water flow theory
have been raised in studies testing the diff usion theory (Hassanizadeh et al., 2002). Mokady and Low (1964) were perhaps the first
who wrote that the water retention curve may not be unique. This
hypothesis, however was not supported by their data. Davidson
et al. (1966) conducted imbibition and drainage experiments and
found that more water was removed from soil samples by applying
one single large step of decreasing pressure than a sequence of small
decreases. On the contrary, more water was taken up by the soils
when small pressure steps were applied in the imbibition process.
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Table 1. Experimental studies dealing with dynamic nonequilibrium effects.
Reference
Equilibration time
Length scale
Soil type
Experiment type
Topp et al. (1967)
h
16.6–100
cm
7.6
fi ne sand
drainage for static, steady, and
transient flow + mixed flow
Smiles et al. (1971)
Rogers and Klute (1971)
Vachaud et al. (1972)
Poulovassilis (1974)
Elzeftawy and Mansel (1975)
Stauffer (1977)
Kneale (1985)
Stonestrom and Akstin (1994)
NA†
720
NA
24
NA
NA
NA
2.5–4, 8.3–24
60
100
136
55
5.4–7.6
50–60
15
73
fi ne sand
fi ne sand
fi ne sand
sand
undisturbed fi ne sand
fi ne sand
undisturbed clay loam
sand, sandy loam, silt loam,
and glass beads
imbibition and drainage in horizontal column
drainage
drainage in vertical column
constant flux
drainage
drainage
drainage
constant-flux infi ltration
Plagge et al. (1999)
Schultze et al. (1999)
>6.7
>24
10
15.7
undisturbed silt loam
sand
drainage and imbibition
drainage and imbibition, smooth
boundary conditions
Ross and Smettem (2000)
Wildenschild et al. (2001)
Šimůnek et al. (2001)
O’Carroll et al. (2005)
Bottero et al. (2006)
DiCarlo (2007)
Vogel et al. (2008)
Sakaki et al. (2010)
Weller et al. (2011)
O’Carroll et al. (2010)
Camps-Roach et al. (2010)
Bottero et al. (2011)
Diamantopoulos et al. (2012)
NA
NA
240
>15
12.5
0.05
NA
NA
>50
NA
NA
NA
1 or >4
63–67
3.5
10
9.6
19
40
10
10
10
9.6
20
21
7.2
sandy loam, clay
sand, sandy loam
undisturbed sandy loam
sand
sand
sand
sand
fi ne sand
sand
sand
sand
sand
sand, undisturbed sandy loam
constant-flux infi ltration
multistep outflow
upward infi ltration under tension
multistep outflow (two phase)
two-phase drainage
multistep outflow
multistep and two-step outflow
multistep outflow–inflow
constant-flux infi ltration
multistep outflow (two phase)
multistep outflow–inflow
two-phase drainage
multistep outflow
† NA, not available.
Topp et al. (1967) compared water retention curves (drainage)
of vertical sand columns obtained by static equilibrium, steadystate, and transient conditions. Their experimental observations
are summarized in Fig. 1, which is from their classic study. Water
content measurements in laboratory samples were obtained by the
gamma ray absorption method, and pressure head measurements
were obtained by tensiometry. Water retention data estimated
under hydrostatic equilibrium are shown with solid triangles,
whereas data obtained under steady-state flux conditions are
shown with open triangles. For their unsteady flow experiments,
they used three different experimental runs. In the first run, the
sample was drained from saturation to a pressure head of −56 cm
within 330 min (fi lled squares). In the other two experimental
series, this time span was reduced to 237 (small dots) and 100 min
(large dots), respectively. The key finding was that water contents
observed under static or steady-state conditions were smaller at
a given pressure head than those obtained by dynamic drainage
experiments. They furthermore conducted additional “hybrid”
experiments (“static-unsteady” and “unsteady-static”). They started
with a dynamic drainage experiment. When the pressure head was
equal to −44.5 cm, the pressure change stopped (open squares),
but the water content further decreased to Point B in about 160
h. In the second hybrid experiment, they started with the static
method (open circles) and then they switched to unsteady-state
flow (Point C). Clearly, the slope of the estimated curve changed
at Point C, resulting in a second part of the curve in which water
contents are higher than would have been expected from a continuation of the stepwise static experiment. Their results proved
for the first time that even for a monotonic drainage experiment,
the relationship between water content and pressure head is not
unique but depends on the rate at which the water content changes.
These fi ndings were shortly afterward confi rmed by work done
at Grenoble. Smiles et al. (1971) performed experiments where
a series of imbibition–drainage cycles was applied to a horizontal sand column by imposing different pressure head steps at one
column end. They found the water retention curve not to be a
unique function for drainage but varying with the applied pressure
head and the time taken to achieve equilibrium. They also stated
that the effect seemed not to occur during infi ltration. Vachaud
et al. (1972) confirmed these results for vertical soil columns.
Simultaneously with the study of Smiles et al. (1971), Rogers and
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Fig. 1. Water retention curves estimated by Topp et al. (1967).
Klute (1971) investigated the flow dependence of the hydraulic
conductivity as a function of the water content. They found a ratedependent water content–pressure head relationship but a unique
hydraulic conductivity function, K(θ).
At Cambridge, Poulovassilis (1974) performed experiments to test
the results of Topp et al. (1967). He used a 55-cm-long soil column
filled with sand and percolated water at a constant rate on the top of
the column. When steady-state flow was achieved (constant water
content and pressure head throughout the column), the two ends
of the column were sealed and the column was placed horizontally.
The pressure head was measured for 1 d. He noticed that the pressure head increased appreciably with time while the water content
remained stable. This pressure head increase was more pronounced
for states at medium water contents. In another series of experiments, he left the soil column under constant-flux infiltration for an
additional period of 3 h after steady-state flow was achieved. Then
he sealed both column ends once again and placed the column horizontally. The results showed that the pressure head increase was not
so pronounced as in the first series of experiments.
Elzeft awy and Mansel (1975), in a study similar to Topp et al.
(1967), concluded that the water content at a given pressure head
was higher in the case of unsteady flow than during steady-state
or static equilibrium. At ETH Zürich, Stauffer (1977) performed
drainage experiments in vertical soil columns of quartz sand. He
conducted steady-state and transient experiments and found that
for a certain value of the pressure head, more water was retained
under transient conditions. He also examined the effect of the
estimation method (steady state or transient) in the relative permeability vs. saturation relationship. The results showed a similar
trend as in the case of the water retention curve.
In the following decades, the topic of dynamic effects in soil water
flow found less attention. Some Ph.D. work in Germany that
was directly or indirectly concerned with DNE remained largely
unpublished (Plagge, 1991; Lennartz, 1992; Schultze, 1998).
Plagge et al. (1999) conducted drainage and imbibition experiments by increasing or decreasing the pressure head at the top of
an undisturbed soil column. Their experimental setup allowed the
conduction of evaporation experiments on the same column. They
concluded that the water retention and the saturated hydraulic
conductivity curves were dependent not only on the rate of change
of the water content but also on the type of the applied boundary
condition (variable pressure head or evaporation). Wildenschild
et al. (2001) performed multistep outflow (MSO) and one-step
outflow experiments to investigate the flow-rate dependence of
unsaturated hydraulic properties. The soils used were a sandy and
a silty soil. The results showed that the SHPs of the sandy soil were
flow dependent, whereas the SHPs of the silty soil were not. Sakaki
et al. (2010) measured static and dynamic drainage and imbibition curves in the laboratory. They concluded that, at given water
contents, the pressure heads measured under dynamic drainage
conditions were statistically smaller than expected from the static
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Fig. 2. Continuous outflow–inflow experiment conducted by Schultze et al. (1999): cumulative outflow–inflow data measured at the bottom of the
soil column and measured water content data in two positions inside the soil column (top); and applied boundary condition along with the measured
pressure head in two positions inside the soil column (bottom). The installation depths for both tensiometer and time domain reflectometry (TDR)
sensors were 4.5 and 10.2 cm from the top of the soil column.
capillary curve. On the contrary, for the imbibition curves, the
dynamic pressure head was higher than under static conditions.
In summary, all the studies presented here have clearly shown that
the water retention curve estimated under transient conditions is
different from the water retention curve estimated under steadystate or static conditions. More specifically, for the same pressure
head value, more water is withheld by the soil matrix in the case of
drainage compared with steady-state or static conditions. Similarly,
water content is smaller for the same pressure head under dynamic
conditions than steady-state or static conditions in the case of imbibition. This was proven for different experimental setups and different boundary conditions; however, the materials used in these
studies were mainly limited to sands. Some contradictory results
can be also be highlighted from these studies. Smiles et al. (1971)
and Poulovassilis (1974) concluded that the imbibition curve is
not affected by the DNE. Contrary to that, Sakaki et al. (2010)
found that the dynamic wetting curves also differed statistically
compared with the static curves.
Dynamic Nonequilibrium in Mul step and
Con nuous Ou low–Inflow Experiments
In multistep outflow–multistep inflow (MSO–MSI) experiments,
tensiometer readings have sometimes reached the new equilibrium
levels relatively quickly after a pressure step, whereas outflow or
inflow of water has continued for periods of hours or even days, as
already indicated by the hybrid experiment of Topp et al. (1967).
From the published experimental MSO data, it appears that this is
the rule rather than the exception. Schultze et al. (1999) analyzed
DNE effects occurring in experiments, with a focus on parameter
estimation by inverse modeling. They pointed out that the most
significant deviations between model and observation occurred in
the moisture range near the air-entry point. Moreover, this phenomenon was not limited to the drainage branch but also occurred
during imbibition. They furthermore conducted “continuous” outflow–inflow experiments, where the water pressure at the bottom
of the soil column was changed smoothly from full saturation to an
unsaturated state and back. Figure 2 shows examples of the applied
boundary conditions along with the measured pressure heads and
TDR-measured water contents inside the soil column. Initially the
soil was saturated at its top boundary, corresponding to a pressure
head at the lower boundary being equal to the column length (15.7
cm). During each cycle, the pressure head smoothly changed from
15.7 to −60 cm and remained there for a redistribution period of at
least 24 h. The total time for the first cycle was 432 h. The experiment was repeated three times, increasing the speed of the drainage–imbibition process each time by a factor of four. This resulted
in an accelerated drainage–imbibition process by a factor of 64.
Figure 3 shows the different retention curves obtained by plotting
the TDR data against the pressure head data measured inside the
soil column. There is a small but significant shift toward higher
water contents at a given potential for the fast drainage, which is
in agreement with the results of Topp et al. (1967).
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Fig. 3. Influence of drainage rate on in situ retention curves for the sand
sample of Schultze et al. (1999), obtained by plotting water content
values against tensiometric pressures measured at the same depth (4.5
cm from the top of the soil column) during a continuous outflow experiment (pF is defined as the logarithm of the absolute value of pressure
head in centimeters). The legend shows drainage duration in hours.
Šimůnek et al. (2001) conducted upward infiltration experiments in an undisturbed soil column for a loamy sand soil. The
soil column was equipped with five tensiometers. They noticed
that although the tensiometer readings quickly reached a fairly
constant pressure head, water uptake continued for hours. Th is
is not in accordance with the Richards equation, which predicts
that the inflow would cease when all the tensiometers had reached
a constant pressure head. O’Carroll et al. (2005) explored dynamic
effects in capillary pressure in MSO experiments with water and
also with dense nonaqueous-phase liquid (DNAPL). They tested
whether the traditional multiphase flow simulators could describe
the observed dynamics. Only when they incorporated a dynamic
capillary pressure term was there a significant improvement in the
agreement between simulated and measured cumulative water outflow data. Moreover, the estimated retention curve was in good
agreement with the independently measured static retention curve.
Recently, Diamantopoulos et al. (2012) reported MSO experiments for disturbed (sand) and undisturbed (loamy sand) soil
columns. They found dynamic effects for both soils. Figure 4
shows the experimental results for the undisturbed loamy sand
soil. They recognized two phases in the outflow dynamics. In the
first phase, water drained abruptly from the column directly after
each pressure step, as expected from an equilibrium relationship
with the capillary pressure dynamics, but in a second phase, outflow continued and ceased only slowly. Figure 4 also shows the fitting obtained using the Richards equation with traditional SHPs
that are invariant with respect to the system state. The match to
the pressure head data is good, but the model cannot describe the
outflow data. This reflects the inherent assumption of the Richards
equation that pressure head and water content are tightly coupled
Fig. 4. Observed and simulated cumulative outflow and pressure head
data for a loamy sand soil (Diamantopoulos et al. 2012). From 30 h
on, the equilibration kinetics of pressure head and outflow differed
significantly. The fitted data were calculated using the Richards equation and the Diamantopoulos et al. (2012) dynamic nonequilibrium
(DNE) model. Both models were coupled with the van Genuchten–
Mualem (VGM) model.
through the retention curve. As shown below, this can be dramatically improved by partial decoupling of the water content and pressure head in the modeling. As a side note, it can be further highlighted that ignoring pressure head data from the fitting procedure
creates a danger of getting very different soil hydraulic properties.
The outflow data of Fig. 4 alone could be fitted by the Richards
equation if the hydraulic conductivity function was adjusted to
much lower conductivities.
In summary, the observed DNE in MSO–MSI experiments confirms the results of a higher dynamic water content value in the
case of drainage and a smaller dynamic water content value in the
case of imbibition compared with the water content values for the
same pressure head estimated under static conditions. Moreover,
this kind of experiment contains additional information concerning the equilibration kinetics for water contents. It seems that this
approach is not linear but contains two distinct phases in the case
of outflow, as described by Diamantopoulos et al. (2012). Interestingly, this seems not to be true in the case of pressure head
equilibration, as presented by Poulovassilis (1974) and Weller et
al. (2011). It seems that the pressure head equilibration follows
an exponential function in the case of drainage and in the case
of imbibition.
Dynamic Nonequilibrium in
Evapora on Experiments
Dynamic nonequilibrium in stepwise outflow experiments is
known to occur but is sometimes deemed to be of little relevance
for natural processes because such stepwise changes in boundary conditions do not occur in nature during drainage processes;
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however, DNE is also found under evaporation conditions. We
show some unpublished data regarding this. Evaporation experiments were conducted for soil columns 5 cm in length and 8 cm
in diameter, which were equipped with two tensiometers placed
at 1.25 and 3.75 cm from the bottom to continuously monitor
the pressure head. Th is method is used for estimating the soil
hydraulic properties and further information about the experimental setup can be found in Schindler (1980), Peters and Durner
(2008), and Schindler et al. (2010). The soil used for this study
was a hydrophilic, well-sorted, packed sand. At the beginning of
the evaporation experiment, the pressure head distribution in the
soil was hydrostatic, with zero pressure head at the bottom. In
the first evaporation stage (Shokri et al., 2009), which is of interest here, water evaporated from the soil column to the laboratory
atmosphere at a constant rate because the decrease in unsaturated
hydraulic conductivity due to water loss was fully compensated
by an increase in the hydraulic gradient. Pressure head distributions with depth were approximately linear and close to hydrostatic
because the saturated conductivity was about three orders of magnitude larger than the water flux at the upper boundary, and thus
the hydraulic gradient was almost equal to zero.
We then modified the experiment by introducing an interruption
of the evaporation flux. After allowing the soil column to evaporate for 24 h, we covered it with a cap and set the evaporation flux
to zero for another 24 h. The cap was then removed and evaporation continued. Tensiometer responses for four replicate columns
were almost identical, hence the results for only one soil column
are shown. Figure 5 depicts Stage 1 pressure head evolution at two
depths. We observed the expected decrease in pressure head when
evaporation began. When evaporation was stopped, however, the
tensiometer readings showed a distinct relaxation and the pressure heads increased toward an equilibrium level that was different
from the value under transient conditions. The observed “nonequilibrium” was not a result of vertical water redistribution. This can
quantitatively be proved by inverse modeling (shown in Fig. 5).
The Richards equation predicts that under the given conditions,
the pressure heads will remain constant when evaporation stops.
Bohne and Salzmann (2002) compared SHPs obtained using
equilibrium methods and evaporation experiments. They fitted
the van Genuchten (1980) model to the equilibrium water retention data and then tried to match evaporation experiment data by
fitting only the saturated hydraulic conductivity and the tortuosity
parameter. By doing this, they could not describe the pressure head
evolution in their dynamic experiments. Only when they fitted
both water retention and hydraulic conductivity curves could they
describe the dynamics of the evaporation experiments.
The study of Bohne and Salzmann (2002), along with the new
experimental data presented in this study, shows that DNE effects
occur also in the case of evaporation under laboratory conditions.
We have to note that the flow rate change in these experiments was
Fig. 5. Dynamic nonequilibrium observed in an evaporation experiment
for a well-sorted sand. The soil sample was allowed to evaporate for 24
h and then evaporation flux was stopped for another 24-h period. The
black line shows cumulative evaporation (cm3). A sequence of on–off
cycles was followed until the end of the experiment. The fitted data were
calculated using the Richards equation coupled with the bimodal van
Genuchten (biVGM) approach by Durner (1994).
much slower than those of the previous experiments and that the
equilibration of the pressure head followed an exponential trend,
as was discussed above. Temperature effects on capillary pressure
also need to be accounted for, however, when investigating DNE
in evaporation experiments.
Dynamic Nonequilibrium Effects in Infiltra on
Experiments under Constant-Flux Condi ons
Stonestrom and Akstin (1994) tested the hypothesis that the matric
pressure is a non-decreasing function of time during constant-rate,
non-ponding infi ltration into a homogeneous soil column with
low initial water content. They conducted constant-flux infi ltration experiments in soil columns of 73-cm length with an inside
diameter of 5 cm. The soil columns were equipped with tensiometers at three different depths. Figure 6 shows the evolution of the
pressure head for constant-flux infiltration into initially air-dry soil
columns for the Delphi sand soil and for glass beads with a median
diameter of 80 μm. The pressure head measured by the tensiometers
passed through a maximum value and then decreased steadily as
the wetting front moved farther down the column. This means
that the evolution of pressure head during constant-flux infi ltration was nonmonotonic, which of course cannot be described by
the Richards equation. Since then, various studies have dealt with
the so-called capillary pressure overshoot (DiCarlo, 2005, 2007;
DiCarlo et al., 2010), which has been hypothesized to be responsible (Geiger and Durnford, 2000; Eliassi and Glass, 2001, 2003;
Egorov et al., 2003) for finger flow in homogeneous porous media
and consequently responsible for preferential water flow.
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6 Reasons Proposed for the
Occurrence of Dynamic
Nonequilibrium Effects
Since the study of Topp et al. (1967), almost all the investigators have
proposed physical processes that may be responsible for the observed
DNE in their experiments. Most of the hypothesized causes have
been reviewed by Wildenschild et al. (2001) and Hassanizadeh et
al. (2002). We present the proposed reasons for DNE in soils along
with new insights that have emerged during the last decade.
Air–Water Interface Reconfigura on
When air displaces water (or water displaces air) in a porous
medium, the air–water configuration on the pore scale within the
REV is redistributed, and this redistribution is not instantaneous
but requires a finite time (Barenblatt, 1971; Sakaki et al., 2010). In
this redistribution process, the behavior of interfaces and contact
lines is decisive. When air invades a porous medium, the curvature
of the air–water interface in a pore is unable to smoothly change in
response to changes in capillary pressure. The measured pressure
head will be smaller due to unstable air–water interfaces, and it
will increase as the fluid interfaces reach equilibrium (O’Carroll et
al., 2005). According to O’Carroll et al. (2005), these processes are
not captured when we upscale from the pore to the REV scale and
may contribute on DNE effects in capillary pressure (Kalaydjian,
1992; Hassanizadeh et al., 2002).
Entrapment of Water and Pore
Water Blockage
Fig. 6. Pressure head (ψ) evolution for a constant-flux infiltration
experiment into initially air-dry soil columns as presented by Stonestrom and Akstin (1994). The two materials were Delphi sand and
glass beads with a median diameter of 80 mm. The pressure head was
recorded at three different depths: 2, 5, and 8 cm. Courtesy of AGU.
Recently, Weller et al. (2011) reported the occurrence of DNE
effects in their experiments. They performed constant-flux infi ltration experiments for both drainage and imbibition with a stepwise
change in the water application rate. The aim was to establish a
sequence of zero pressure head gradient conditions inside the soil
column. Th is allows a direct measurement of the soil hydraulic
conductivity, which is equal to the applied flux rate. In the case of
drainage, Weller et al. (2011) noticed that the pressure head measured by the tensiometer dropped immediately after the diminution of the applied flux but then slowly increased again. This second
phase of the tensiometer behavior had a duration of a few days until
equilibrium was reached. Similar behavior was also observed for
increasing flux rates.
Historically, the earliest explanation for the water flow dependence
of the water retention curve was disconnected pendular water rings
(Topp et al., 1967). Harris and Morrow (1964) and Morrow and
Harris (1965) conducted studies in packed large uniform spheres
and found that during drainage some pores remained filled because
they became isolated from the bulk liquid before the local air-entry
pressure was reached. Furthermore, they found that the size of the
rings depended on the rate of drainage. Similarly, Poulovassilis
(1974) explained his experimental results by assuming that during
dynamic drainage some water is left behind in the emptying pores
and some portion of this water is conducted slowly toward the continuous water body by film flow. This would lead to an increase in the
pressure head, recorded by the tensiometers, under stopped flow conditions. The same process explains the two-phase outflow dynamics
often observed in MSO experiments, i.e., the slight ongoing drainage
at constant capillary pressure. Wildenschild et al. (2001) proposed
that in MSO experiments, when a sudden large pressure head is
applied, the soil near the porous plate drains faster than the upper
part of the soil, leading to isolation of conducting paths, which can
retard the whole drainage. Weller et al. (2011) speculated that in
drainage experiments, after a flux reduction, there are pores that
drain more easily followed by a much slower drainage of blocked
pores. This emptying proceeds slowly and caused the slow increase
in water potential observed in their constant-flux experiments.
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Air Entrapment
If the air phase in a porous medium loses its continuity, gas transport takes place only by effective diffusion. Consequently, pressure
changes in the water phase are directly transferred to the entrapped
air phase, and under these conditions the Richards equation, where
capillary pressure is macroscopically related to the ambient atmospheric pressure, is not valid to describe the water dynamics. Smiles
et al. (1971) tested whether the differences between water retention
curves estimated by static or dynamic conditions can be attributed
to restricted air flow. They compared drainage curves obtained by
experiments in a soil column with and without additional lateral
inlets for air. The comparison showed that the air entry near the axis
of the column was adequate to maintain atmospheric pressure. Thus
they concluded that this is not a significant cause for the observed
DNE. Schultze et al. (1999) came to contradicting conclusions.
They conducted multistep and continuous outflow–inflow experiments on soil columns with open and with closed walls and simulated them with one-phase and two-phase flow models. The results
showed that for disturbed and undisturbed soil columns, the air
phase can lose its continuity already at 50 to 70% of water saturation, and dynamic effects consequently occur. Similarly, Wildenschild et al. (2001) considered air entrapment as one reason for flow
dependence of estimated SHP in MSO experiments. Hassanizadeh
et al. (2002) noted that air entrapment could occur in soil column
experiments but was not expected to occur under field conditions.
Nevertheless, field studies have shown that air pressures can rise
above the atmospheric pressure (Linden and Dixon, 1973). In a
recent study, Camps-Roach et al. (2010) conducted drainage experiments and claimed that air entrapment did not result in observed
differences between static and dynamic retention curves.
Air Entry
Wildenschild et al. (2001) proposed another effect that could
contribute to the appearance of DNE in MSO experiments and
that is closely related to air continuity. Based on the experimental
results of Hopmans et al. (1992), who used x-ray tomography, they
stated that for an initially saturated soil sample, a drying front
develops at the top of the sample when drainage initiates. Th is
drying front moves downward until air continuity is established
from the top to the bottom of the sample. Th is behavior cannot
be described by the Richards equation, which predicts that drainage will occur fi rst at the lower end where the pressure drop is
applied. They further assumed that this process contributes to the
rate dependency of estimated soil hydraulic properties for a sandy
soil. This effect, however, can only explain DNE occurring in MSO
drainage experiments.
In evaporation experiments that start from full saturation, airentry effects can be frequently observed in the early state, in particular for unstructured soils. The air that is replacing the evaporating water enters the previously fully saturated soil not in a smooth
and gradual manner, but as small bursts, which leads consequently
to sudden “bounces” of the overall decreasing water pressure (Fig.
7, left). Naturally, this behavior cannot be captured by the Richards equation. If water content is plotted vs. measured tensiometric
water pressures, these nonequilibrium conditions can be visualized
as a shifted water retention curve, most pronounced around the
macroscopic air-entry region of the soils (Fig. 7, right).
Dynamic Contact Angle
It has been known for a long time that the contact angle between
solid–liquid–gas interfaces (advancing and receding) is dependent
on the direction (propagation or withdrawal) and velocity of the
liquid–gas interface (Hoffman, 1975; Friedman, 1999). By using
the dynamic contact angle in the Young–Laplace equation rather
than a static contact angle, Friedman (1999) proposed that the
dynamic contact angle could contribute to the flow dependence
of SHPs measured under transient conditions. Wildenschild et
al. (2001) stated that the dynamic contact angle effect was small
under drainage conditions because when flow velocity increases,
the contact angle should approach zero. In their drainage experiments, Camps-Roach et al. (2010) concluded that the concept of
the dynamic contact angle could be a contributing factor but it is
not the one and only effect that leads to DNE in air–water systems.
O’Carroll et al. (2010) studied theoretically and experimentally
the effect of wettability on dynamic effects in capillary pressure.
They found that the pressure head of the materials with a greater
equilibrium contact angle showed a faster approach to equilibrium
in MSO experiments.
Time-Dependent We ability Changes
The macroscopic wettability of soils is determined by microscale
surface properties of the porous medium. For natural soils, organic
substances play a decisive role for the macroscopic soil wettability. These surface properties can vary with time, dependent on
the degree of water saturation. Th is is in particular pronounced
for imbibition processes, where an initially hydrophobic medium
becomes, on wetting, a more wettable medium (Bachmann et
al., 2011). Th is could be shown with a simple experiment, where
capillary rise increased with an increasing time of contact with
water, whereas the matric potential would be always in hydraulic
equilibrium with the height above the water table. (Bachmann et
al., 2011).
Heterogeneity
Microscale Heterogeneity
Mirzaei and Das (2007) used the term microscale heterogeneities
to refer to microscale lenses of fine sand that have different multiphase flow properties than the surrounding porous medium. These
small-scale heterogeneities occur at a length scale below the REV
scale and have significant effects on the effective SHP. To test the
effect of these heterogeneities in DNE phenomena, Mirzaei and
Das (2007) conducted numerical experiments to investigate how
the microscale heterogeneities affect the dynamics of DNAPL
and water flow in a porous domain. They found that microscale
heterogeneities lead to dynamic effects and that the intensity of
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Fig. 7. Observed tensiometric pressure head data in two depths and cumulative evaporation in the initial phase of an evaporation experiment (left) and
water content vs. measured pressure head data (right) for a sandy soil (top, Durner, unpublished data, 2012) and a silt soil (bottom,(Durner, unpublished data, 2012) (pF is defined as the logarithm of the absolute value of pressure head in centimeters).
heterogeneity increases the emergence of dynamic effects. Experiments dealing explicitly with the effects of microscale heterogeneity have not been yet published, however.
Macroscale Heterogeneity
Heterogeneity on the macroscopic measurement scale is well
known to cause phenomena, including preferential flow, that
cannot be captured with the assumption of quasi-homogeneous
porous medium properties. This problem is less related to the topic
of dynamic effects that emerge from subscale processes, the focus
of this review, but rather to the general question of the existence
of effective hydraulic properties and effective process descriptions
in heterogeneous porous media. It leads, however, to observations
that exhibit dynamic effects if the measurement windows for
pressure heads, water contents, or fluxes are different and do not
perfectly overlap. Th is is the standard case in practical measurements. Despite this general knowledge, it is currently not clear how
strong the heterogeneity effects are and how they depend on various parameters, including the flow regimes.
A special case of macroscale heterogeneity is the distribution of different materials below the measurement scale. This kind of heterogeneity can also provoke DNE in relatively small sample volumes like
undisturbed soil columns. With the assumption of local equilibrium
for pressure head and water content, Manthey et al. (2005) studied
the effect of heterogeneity of local hydraulic properties by using a
two-phase simulator. Based on forward simulations, they examined
the effect of heterogeneity on the occurrence of DNE. The results
showed that the DNE was influenced by a heterogeneous distribution of intrinsic permeability. They also found it to be boundarycondition dependent. Vogel et al. (2008) generated a stochastic, twodimensional, heterogeneous field and simulated a MSO experiment
for this synthetic porous medium. They found a perfect accordance
between the one-dimensional forward simulation using the static
mean hydraulic properties and the two-dimensional simulation.
They concluded from their findings that the MSO approach is not
seriously affected by dynamic effects from this process.
Contrary to that are findings for infi ltration processes. Šimůnek
et al. (2001) suggested that heterogeneity is the reason for the
observed DNE effects in upward infi ltration experiments. They
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postulated that the observed DNE effects are due to the redistribution of water and, more specifically, water being transferred from
larger to smaller pores. Vogel et al. (2010) studied numerically the
effect of large-scale heterogeneity by conducting infi ltration simulations. They found that the larger the correlation length (isotropic
correlation) used for generating the stochastic heterogeneous field,
the larger were the nonequilibrium effects in the infi ltration front.
This is in accordance with the expectation that the smaller the correlation length, the faster is the equilibration of the water potential. Although large-scale heterogeneity is expected to evoke DNE
effects in water flow, it cannot explain the occurrence of these
effects in disturbed and well-sorted materials. Further experiments
or numerical studies are needed for the assessment of large-scale
heterogeneity as a cause of DNE.
Table 2. Modeling approaches for dynamic nonequilibrium effects,
including only models that have been compared with experimental data.
Two-site model with partial Diamantopoulos et al. (2012)
water content equilibration
1
6 Modeling Approaches
Mobile–immobile and
dual-permeability models
Philip (1968)
1
Gerke and van Genuchten
(1993a, 1993b)
1
Modeling of DNE can be done on scales below the traditional
Richards equation, e.g., by pore-scale network models or by
modifications of the Richards equation. The latter approaches are
dominated by formulations that use empirical or physically based
flow-rate-dependent capillary pressures. Alternatively, effective
formulations have been proposed that use a water content that is
kinetically coupled to the pressure head. Dual-continuum models
are a generalization of the latter approaches. Table 2 gives an overview of popular DNE modeling approaches that have been compared with experimental data.
Model
Study
Dimensions
Richards’ equation
coupled with dynamic
capillary pressure
Stauffer (1977)
1
Two-phase flow coupled
with Hassanizadeh and
Gray (1993) model
Richards’ equation coupled
with kinetic water
content equilibration
During the last few decades, pore-scale network modeling has been
established as an alternative approach in modeling two-phase flow
in porous media. It requires the exact microscopic descriptions
of the pore geometry and the physical laws of flow and transport
within the pores (Al-Gharbi, 2004; Al-Gharbi and Blunt, 2005).
Pore-scale network models are very useful in conducting numerical
experiments and analyzing difficult-to-measure quantities such
as interfacial areas or common lines (Held and Celia, 2001). The
analysis of dynamic effects in water flow with the help of pore-scale
network modeling is beyond the scope of this review. For studies
dealing with pore scale network models, see Tsakiroglou and Payatakes (1990), Blunt and King (1991), and Dahle and Celia (1999),
among others. A few studies have also been conducted on DNE
effects with the help of pore-scale network models (Gielen et al.,
2001; Joekar-Niasar et al., 2009).
Con nuum-Scale Models with Dynamic
Capillary Pressure
1
Sander et al. (2008)
1,2
Chapwanya and Stockie (2010)
2
O’Carroll et al. (2005)
1
O’Carroll et al. (2010)
1
Fucik et al. (2010)
1
Ross and Smettem (2000)
1
Šimůnek et al. (2001)
1
and others
and Corey (1964) model with a dynamic capillary pressure. Based
on experimental observations, he proposed that the dynamic capillary pressure, Pcdyn [M T−2 L−1], and the equilibrium (or static)
capillary pressure, Pcstat [M T−2 L−1], be related through (Stauffer,
1978; Manthey et al., 2008)
c
c
Pdyn
− Pstat
=−τ s
Pore-Scale Network Models
Hassanizadeh et al. (2002)
∂S w
∂t
[4]
where τs [M T−1 L−1] is a relaxation parameter, given by
τs =
2
a μ w φ ⎛⎜ Pd ⎞⎟
⎟
⎜
⎟
kλ ⎜⎜⎝ ρ g ⎠⎟
[5]
w
where Sw (dimensionless) is water saturation, a is a dimensionless
scaling parameter, μ w [M T−1 L−1] and ρ w [M L−3] are the viscosity and the density of the wetting phase, φ (dimensionless) is
the porosity, k [L2] is the intrinsic permeability, g [L T−2] is the
gravitational constant, and Pd [M T−2 L−1] and λ (dimensionless)
are the Brooks and Corey (1964) parameters. Based on a much
better matching of experimental data with this dynamic model,
Stauffer (1978) concluded that neglecting the dynamic effects in
the capillary pressure–water saturation relationship can lead to
considerable errors.
Stauffer (1977) Model
Stauffer (1978) simulated drainage experiments by means of a
numerical model using the finite element method. He combined
the Richards equation with a “dynamic” model for the SHPs. To
describe the dynamic capillary pressure–saturation relationship, he
replaced the equilibrium (or static) capillary pressure of the Brooks
Hassanizadeh and Gray (1990) Model
Hassanizadeh and Gray (1990) developed a macroscopic thermodynamic theory to describe two-phase flow in porous media. In
a follow-up study, Hassanizadeh and Gray (1993) stated that the
macroscopic capillary pressure is defined as an intrinsic property
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of the system. Furthermore, a dynamic capillary pressure could
be represented as a linear function of the rate of change in water
saturation in the porous medium:
c
c
Pdyn
− Pstat
=−τ h
∂S w
∂t
[6]
where τh [M T−1 L−1] is a material coefficient that defines the time
scale necessary to reach equilibrium. Equation [6], based on theoretical considerations, is formally equivalent to Eq. [4], which was
based on experimental evidence. It shows that the assumption of a
state-invariant relationship between water pressure, water content,
and hydraulic conductivity is only valid if ∂Sw/∂t equals 0 or simply
if “equilibrium is achieved.” Consequently, the use of hydraulic
functions estimated under static conditions in the Richards equation is questionable if transient water flow takes place. In a similar
approach to Stauffer (1977), Hassanizadeh et al. (2002) proposed
to couple Eq. [6] with the Richards equation. For this case, the onedimensional problem with vertical coordinate z is given by
∂θ ∂ ⎛⎜ ∂ h stat ⎞⎟ ∂ ⎡ ∂ ⎛⎜ ∂θ ⎞⎟⎤ ∂ K
= ⎜K
⎟ + ⎢ K ⎜ τˆ h
⎟⎥ −
∂ t ∂ z ⎝⎜
∂ z ⎠⎟ ∂ z ⎣⎢ ∂ z ⎝⎜ ∂ t ⎠⎟⎦⎥ ∂ z
[7]
where hstat [L] is the static pressure head, given by hstat = hdyn − τˆ h
(∂ θ/∂t) and τˆ h = τh/φ ρ w g.
Numerical solutions of Eq. [7] have been implemented and used
in a variety of applications. Hassanizadeh et al. (2002) simulated
horizontal infi ltration into an initially dry soil. The numerical
results showed that the infi ltration front was retarded for high
values of the material coefficient τh. Sander et al. (2008) used Eq.
[7], and assumed that τh could be given by a simple function of
water content. This was based on the experimental results of Smiles
et al. (1971), which show that τh becomes minimal when the effective saturation approaches zero (Cuesta et al., 2000). It is worth
mentioning that this dependence is in accordance with the theory
of Hassanizadeh and Gray (1990). Sander et al. (2008) also incorporated hysteresis in the capillary pressure–water content relationship in their model. By using numerical models, they were able to
describe the nonmonotonic saturation distribution property of finger-flow evolution (one dimensional) as well as the lateral growth
of fingers observed in experiments (two dimensional). Chapwanya
and Stockie (2010) used the same approach of coupling Richards
equation with Eq. [6] to investigate the dynamics of fi ngered
water flow in initially dry homogeneous soils. They also included
hysteresis in the water retention curve. They concluded that their
model can describe the physics of fingered flow and, moreover, they
showed that for small values of the nonequilibrium parameter τh,
the finger formation was suppressed. In contrast, for relatively large
values of τh, finger flow occurred.
The concept of dynamic capillary pressure was also investigated in
studies related to multiphase flow. O’Carroll et al. (2005), for their
MSO experiments with water and DNAPL, explored the agreement between observed and simulated results using a multiphase
flow simulator. Only if they incorporated Eq. [6] in their model
was there a significant improvement in the agreement between
simulated and measured cumulative water outflow data. Sakaki et
al. (2010) found evidence that the value of τh may be hysteretic,
possibly due to hysteresis in the retention curve. Camps-Roach et
al. (2010) examined the effect of the porous media grain distribution on the dynamic coefficient τh. They found that τh depended
on the grain size. Fucik et al. (2010) developed a one-dimensional
two-phase flow model that can handle flow in two fluids in both
heterogeneous and homogeneous media along with dynamic capillary pressure conditions given by Eq. [6]. The model was validated
against both experimental results and semianalytical solutions.
Based on their simulations, they concluded that the dynamic effect
can be of great importance, especially for heterogeneous media. Bottero et al. (2011) studied DNE effects in two-phase flow by conducting a series of dynamic drainage experiments in a natural sandy soil.
They used tetrachloroethylene as the non-wetting fluid and distilled
water as the wetting fluid. Their results showed once more that the
dynamic water retention curve lies above the static water retention
curve. Moreover, they found no dependence of the dynamic coefficient τh on saturation estimated from the experimental results,
but this can be explained by the fact that they explored a relatively
narrow saturation range (0.50 > Sw > 0.85).
Some recent studies have focused on the effect of fluid properties on
the magnitude of the nonequilibrium parameter τh. Joekar-Niasar
and Hassanizadeh (2011) examined the effect of fluid viscosity on
τh. They found that viscosity strongly affected the variation of τh
with saturation. Goel and O’Carroll (2011) conducted two-phase
flow drainage experiments to examine the effect of fluid viscosity
on the nonequilibrium parameter τh and concluded that the magnitude of τh was strongly dependent on the effective fluid viscosity.
A very interesting topic concerning DNE is the dependency of the
nonequilibrium parameter τh on the scale of observation if local
heterogeneity is the primary cause for DNE. Based on numerical
studies with a two-phase simulator, Manthey et al. (2005) found
τh(Sw) to increase with increasing domain size. A similar result
was obtained also by Bottero et al. (2011) in an experimental study
focusing on DNE at different length scales in two-phase flow. Contrary to that, Camps-Roach et al. (2010), in an experimental study,
found no dependence of τh on the averaged (domain) volume.
Con nuum-Scale Models with Dynamic
Water Content
The Ross and Sme em (2000) Model
To be able to describe local nonequilibrium between water content
and pressure head, Ross and Smettem (2000) proposed a simple
modification of the Richards equation. Basically, they assumed a
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time-dependent evolution of the water content, which approaches
its equilibrium value by a first-order kinetic. They specified the term
∂ θ/∂t of the Richards equation through the differential equation
∂θ
= f ( θ, θ eq )
∂t
[8]
with
f ( θ, θ eq ) =
( θ eq −θ )
[9]
τ
where θ eq [L3 L−3] is the equilibrium water content, θ [L3 L−3]
is the actual water content, and τ [T] is an equilibration time
constant. Note that τ is neither identical to nor the reciprocal of
the relationship parameters τs used by Stauffer (1978), Eq. [4], or
τh used by Hassanizadeh and Gray (1993), Eq. [6]. By using this
simple approach, Ross and Smettem (2000) successfully described
soil water flow during infi ltration experiments in which preferential flow was significant.
Diamantopoulos et al. (2012) Model
Based on the inability to describe experimental observations of
MSO and evaporation experiments with either the Richards equation or the Ross and Smettem (2000) model, Diamantopoulos et
al. (2012) developed an effective one-dimensional nonequilibrium
model that merges the two previous models. Similar to the twosite concept in solute transport (Cameron and Klute, 1977), they
defi ned two fractions of water in the same porous system, one
fraction feq in instantaneous equilibrium with the local pressure
head and another fraction fne for which the equilibration of water
content is time dependent. By assuming that the pressure heads
in the two regions equilibrate quickly relative to the movement
of water in the main flow direction, Diamantopoulos et al. (2012)
arrived at a single equation for the water dynamics:
(1− f ne )
∂θ eq
∂t
+ f ne
⎛ ∂h ⎞ ⎤
∂θ ne
∂ ⎡
= ⎢ K ( h )⎜⎜⎜ ⎟⎟⎟−1⎥
⎢
⎝ ∂ z ⎠ ⎥⎦
∂t
∂z ⎣
[10]
in which
∂θ ne θ eq −θ ne
=
∂t
τ
where θ ne is the water content in the nonequilibrium domain. This
model needs two more parameters than the Richards equation
and one more parameter than the model of Ross and Smettem
(2000). By using this empirical and parsimonious new model, Diamantopoulos et al. (2012) succeeded in describing the observed
DNE effects in MSO experiments and could show that both
nonequilibrium parameters and parameters of the soil hydraulic
functions could be uniquely identified from the experiments by
inverse modeling (Fig. 4).
Dual-Porosity and Dual-Permeability Models
Dual-porosity (domain) models distinguish two different soil
domains, each with its own set of SHPs. If water flow can take
place in both domains, we speak of dual-permeability models.
Dual-porosity and dual-permeability models have been developed
for describing preferential flow in structured soils. They are ideal
for describing DNE in heterogeneous soil materials where a clear
identification of two (or more) different materials can be made
and where the separation in matrix and macropores is the obvious dominant cause for macroscopically observed nonequilibrium
between areal averaged water content and pressure head. An early
example for this class of models is that of Philip (1968), where the
soil consists of two domains: a fracture, macropore, or interaggregate domain and a matrix or intraaggregate domain. Water flow
occurs only in the fracture domain and the matrix domain represents immobile water that is exchanged with the fracture domain
(Šimůnek et al., 2003). A general case of the dual-permeability
models was proposed by Gerke and van Genuchten (1993a, 1993b).
This model involves two coupled continua at the macroscopic level:
a macropore or fractured pore system and a less permeable porous
matrix. In both pore systems, variably saturated water flow is
described by the Richards equation. Transfer of water between
the two domains is described by means of fi rst-order rate equations, being proportional to the difference in effective saturation
of the two regions (Šimůnek et al., 2001) or being proportional
to the pressure head difference between the two domains (Gerke
and van Genuchten, 1993b). By using the Philip (1968) model and
the dual-permeability model of Gerke and van Genuchten (1993a,
1993b), Šimůnek et al. (2001) succeeded in describing nonequilibrium water flow in upward infi ltration experiments. For comprehensive reviews of dual-porosity and dual-permeability models, see
Šimůnek et al. (2003) and Gerke (2006).
6 Discussion
We have defined DNE as an apparent dependence of SHPs on the
flow dynamics. This means that SHPs are different for one material whether water moves or not and whether saturation changes
occur fast or slowly. Furthermore, any process that causes timedependent changes of SHPs leads to dynamic effects. Th is definition includes, for example, phenomena such as the dissolution
of entrapped air or time-dependent wettability as possible causes
of DNE. Finally, numerical studies have shown that heterogeneity can produce DNE. This means that the spatial distribution of
materials with different (time-invariant) SHPs can macroscopically
generate DNE. A generally valid defi nition of DNE thus seems
difficult and we propose to think of DNE as hydraulic state observations that cannot be described by a continuum model (Richards
equation or a two-phase flow model) with a set of unique SHPs.
www.VadoseZoneJournal.org
Our review has shown that there is ample evidence that water flow
phenomena in unsaturated soils occur for which the concept of
unique soil hydraulic properties is not valid. More water is withheld by the soil matrix for a higher flow rate in the case of drainage.
Recent studies have further shown that this difference is statistically
significant (Sakaki et al., 2010; Goel and O’Carroll, 2011). We have
focused in this review on the water retention characteristics, but similar effects have been observed for the hydraulic conductivity curve.
Schultze et al. (1999) found increased hydraulic conductivity values
for faster experiments. Kneale (1985) and recently Weller et al. (2011)
confirmed that DNE effects affect also the hydraulic conductivity
curve. Moreover, DNE effects occur in both drainage and imbibition processes. Although the effect is well recognized, it is not easy to
quantify, and there are still some questions that we try to raise here.
First, it is not known yet what the dominant cause for DNE effects
is in a specific situation. Although various reasons have been proposed to provoke DNE effects, some of them are contradictory
and cannot be considered as the unique reason for nonequilibrium water flow. For example, water entrapment can occur only
in drainage experiments and not in imbibition experiments. As
another example, the effect of a dynamic contact angle is generally very small for drainage experiments. In any case, it is hard to
assess whether a unique physical process or a combination of various processes generates DNE effects in both drainage and imbibition experiments. Furthermore, we do not know in which pressure
head–water content–saturation range each effect acts. Similarly,
we do not know yet if the physical processes that generate DNE
effects are the same among the different experimental setups.
A particular difficulty arises from the interference of dynamic effects
with capillary hysteresis under transient-flow conditions with changing flow directions. Capillary pressure hysteresis is a well-known phenomenon and has been effectively studied during the last decades
(among others, Poulovassilis and Childs, 1971; Mualem and Dagan,
1975; Russo et al., 1989). Both the water retention and hydraulic conductivity curves show hysteresis as a function of the pressure head.
Moreover, hysteresis can significantly influence water flow in variably
saturated porous media (Vachaud and Thony, 1971; Gillham et al.,
1979; Elmaloglou and Diamantopoulos, 2008). The majority of the
studies (and models) concerning hysteresis in the unsaturated zone
deal with static experiments. It has not yet been proven whether we
can model hysteresis under dynamic conditions by coupling existing
hysteresis models and DNE models. A model that accounts for both
processes has been developed by Beliaev and Hassanizadeh (2001);
however, this model has not yet been tested with experimental data.
A closely related matter concerning hysteresis was highlighted by
Hassanizadeh and Gray (1990, 1993). They showed that the inclusion of the specific interfacial area in the capillary pressure–saturation relationship leads to the removal or significant reduction of
hysteresis. There is ample work that supports this statement, including pore-network modeling (Held and Celia, 2001; Joekar-Niasar et
al., 2010) and experimental studies (Cheng et al., 2004; Chen et al.,
2007). On the contrary, Helland and Skjaeveland (2007) showed
that hysteresis does exist also in the capillary pressure–saturation–
interfacial area relationship.
Table 1 summarizes some key characteristics of experimental studies dealing with DNE effects. It shows the equilibration time (if
available) of the water content or the pressure head, the length of
the soil columns, the soil type, and finally the experiment type. It
is obvious from the soil type column of Table 1 that the majority
of the studies (20 out of 24) included a sandy material. Moreover,
17 studies investigated only sandy materials. Are DNE effects more
pronounced in sandy materials? Stauffer (1978) found more pronounced dynamic effects in the case of a fine sand rather than a
coarser sand. Wildenschild et al. (2001) concluded that the rate
dependence of the SHPs has been shown to be of less importance
for a fi ner sandy loam soil. Again to the contrary, other studies
have found nonequilibrium water flow in similar sandy loam materials (Šimůnek et al., 2001). The use of sandy soils may be explained
by the fact that it is very ambitious to study the effect on natural
heterogeneous media if we do not recognize what causes DNE
effects even in well-sorted sandy materials.
During the last few years, a great effort has been made to model
nonequilibrium water flow, as presented above. The majority of
studies have used the model of Hassanizadeh and Gray (1990,
1993) coupled with the Richards equation or with a one-dimensional two-phase flow simulator. For structured soils, many studies
have used mobile–immobile or dual-permeability models for nonequilibrium water flow (Κöhne et al., 2009). As discussed above,
however, the multitude of processes causing DNE for apparently
homogeneous soils is known qualitatively at best. We assume there
is a complex interplay among all the variables that can contribute
to nonequilibrium water flow. In a specific situation, it is hard to
quantify the dominant effects. This has made the development of
physically based models that encompass all causes and can be used
for reliable predictive simulations very difficult if not impossible
until now. For this reason, we believe that effective approaches
treating these phenomena from a macroscopic perspective point
in the right direction for practical problems.
Although modeling has reached a state where we can handle timevariable constitutive relationships and determine the respective
parameters by inverse methods (e.g., Diamantopoulos et al., 2012),
models can help to improve our quantitative understanding of
the relevance of these effects in practical situations only if combined with suitable experiments. These experiments rely on fast
and accurate measurement techniques, and we see a deficiency in
the current standards with respect to this. Obviously, traditional
equilibrium or steady-state measurements will not help to improve
our knowledge of processes that occur under transient conditions.
Due to the nonlinear nature of unsaturated water flow, further
progress in detecting the extent and role of dynamic effects will
depend on the combination of suitable experiments with precise
www.VadoseZoneJournal.org
measurements and their combination with inverse modeling. As
such, systematic experimental studies are needed to quantify the
dependence of DNE effects and the involved parameters on pressure head regions, flow dynamics, and time and length scales.
Another question concerning DNE effects is whether and to what
degree they affect water flow in greater scale (field-scale) problems.
For many of the hypothesized reasons for DNE that have been discussed here, the significance for field-scale flow must be questioned.
For example, air entrapment is not expected to occur under natural
drainage conditions. Also, sudden drops in pressure at the system
boundary, as used in MSO experiments with the aim to maximize
the information content of the measurements, are hardly occurring
under natural conditions. Thus, it has to be clarified whether the
use of SHPs estimated at different flow rates can be used in modeling problems at larger scales.
Dynamic nonequilibrium effects are likely to act in different pressure head regions and on different time scales. The multitude of
processes and the inability to quantify them independently makes
mechanistic modeling of water flow in porous media that includes
all processes currently unfeasible. Effective modeling of the DNE
phenomena is an alternative, but the estimation of effective DNE
parameters requires advanced inverse modeling strategies. We
conclude that further progress will depend on the combination
of suitable experiments and modeling approaches. Additionally,
there is an urgent need for precision measurement techniques that
are designed to quantify dynamic effects in unsaturated water flow.
Acknowledgments
This study was financially supported by the state of Niedersachsen as NTH Project
BU 2.2.7 “Hydraulic Processes and Properties of Partially Hydrophobic Soils.” We
would like to thank S. Majid Hassanizadeh, two anonymous reviewers, and the Associate Editor Jan Vanderborght for their constructive comments on the manuscript.
6 Conclusions
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