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Dynamic Nonequilibrium of Water Flow in Porous Media: A ReviewThis review provides an overview on various phenomena, hypothesized causes, and mod-eling approaches that describe “dynamic nonequilibrium” (DNE) of water fl ow in soils. Dynamic nonequilibrium is characterized from observa ons on the macroscale by an apparent fl ow-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 aff ected 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 eff ects, and temperature eff ects might be involved. These processes act at diff erent pressure head regions and on diff erent me scales, which makes eff ec ve modeling of the combined phe-nomena 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 eff ects.

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 do-main refl ectometry.

Water fl ow 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 reten-tion and fl ow, 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 diff erent 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). Th e validity limits of this continuum-scale description have become evident in various studies. Th ey 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 fl ow in porous media. It is diffi cult to give a strict defi nition of the term dynamic nonequi-librium, and previous literature has not always used this term when referring to nonequilib-rium phenomena nor is the term used in a unique and coherent manner. In this review, we defi ne 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 defi nition arises from experimental observations in laboratory column studies, where DNE becomes evident as a fl ow-rate dependence of soil hydraulic properties under transient-fl ow condi-tions or as a drift ing θ(h) relationship under no-fl ux or quasi-static conditions.

Th e existence of DNE eff ects has been known since the 1960s, particularly by the work of Topp et al. (1967). Remarkably, scientifi c 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 fl ow-rate dependency of hydraulic properties requires dynamic fl ow experi-ments with suitable instrumentation and suitable evaluation techniques because otherwise

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 tran-sient-flow conditions. We review observa ons, hypothesized mecha-nis c causes, and eff ec ve modeling approaches for dynamic nonequilib-rium of water fl ow 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/vzj2011.0197Received 19 Dec. 2011.

Special Section: MUSIS

Efstathios DiamantopoulosWolfgang Durner*

© Soil Science Society of America5585 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 means, electronic or mechanical, including pho-tocopying, recording, or any informa on storage and retrieval system, without permission in wri ng from the publisher.


dynamic eff ects 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 diff erential Richards equation with time-variant or system-state-dependent functions for the constitutive relationships, made it hard to evaluate these experiments quantitatively. Finally, given the defi cits in our current understanding of the phenomenon, it is impossible to assess the importance of these eff ects for any practi-cal situation. It is thus currently not clear how important “nonequi-librium” eff ects are, whether they contribute to the oft en-observed discrepancy of hydraulic properties found between laboratory and fi eld studies, and under which conditions their consideration gives improved predictions of water fl ow in porous media. Observations of DNE are aff ected 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. Th ese 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 refl ectometry (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 aff ected by the spatial position of the sensor installation (Schlüter et al., 2012). Th us, 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 diff erent 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 fi eld measurements, particularly in situations where preferential fl ow phenomena are common.

All three categories are superimposed in real soils. Modeling of DNE eff ects caused by Type 1 processes in any case needs specifi c 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 defi ne eff ective hydraulic properties will not be possible.

Our review aims at giving an overview of observations of DNE, hypothesized causes, and eff ective 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 fl ow in one review. Hassanizadeh et al. (2002) reviewed the topic of DNE water fl ow in porous media. Since then, advances in sensor technology, data acquisition systems, automation of experimental control, and more observations on water fl ow in unsaturated soils under transient conditions have given further evidence and insight to DNE. Perhaps the greatest part of studies on nonequilibrium water fl ow in soils has dealt with preferential fl ow phenomena, refer-ring to macropore fl ow, fi nger fl ow, and heterogeneous fl ow (van Dam et al., 1990; Roth, 1995; Jarvis, 2007). A review of nonequi-librium water fl ow and solute transport in soil macropores was given by Jarvis (2007). Th ree more reviews have dealt with model appli-cations in preferential fl ow studies (Šimůnek et al., 2003; Gerke, 2006; Köhne et al., 2009). Although we discuss heterogeneity as a reason for nonequilibrium water fl ow in soils, the focus of this review is on nonequilibrium phenomena in well-sorted materials or materials with heterogeneity below the resolution of the measure-ment instrument (i.e., Types 1 and 2). Similarly, a deeper discus-sion 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 fl ow studies with oil as a nonwetting fl uid.

TheoryWater 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 fl ux. In one-dimensional form for vertical fl ow, this reads as

qt z

∂∂θ=−

∂ ∂ [1]

where θ is the volumetric water content [L3 L−3], q is volumetric water fl ux [L3 L−2 T−1], t is time [T], and z is the spatial coordi-nate [L] (positive upward); θ and q are defi ned as averages within a representative elementary volume (REV) of the porous medium. Th e water fl ux (again for one-dimensional vertical fl ow) is given by the Darcy–Buckingham law (Darcy, 1856; Buckingham, 1907):

1hq Kz

⎛ ⎞∂ ⎟⎜=− + ⎟⎜ ⎟⎜⎝ ⎠∂ [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 expres-sion 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 defi ne the local pressure diff erence between the water and air phases as a positive quantity and were used in some of the original literature referred to in this review.

Combination of Eq. [1] and [2] leads to the one-dimensional Rich-ards equation (Richards, 1931), which in its h-based form is

( ) ( ) 1h hC h K ht z z

⎡ ⎤⎛ ⎞∂ ∂ ∂ ⎟⎜⎢ ⎥= + ⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠∂ ∂ ∂⎣ ⎦ [3]

where C(h) = ∂θ/dh is the specifi c water capacity [L−1]. Equa-tion [3] is the fundamental model for describing water fl ow in the unsaturated zone on the macroscopic scale. Richards’ equa-tion is assumed to be valid if the porous system is rigid, nonswell-ing, isotropic, and if only isothermal liquid water fl ow 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 unsat-urated hydraulic conductivity curve, K = f(h) or K = f(θ). Th ese functions are commonly referred to as soil hydraulic properties, SHPs (Durner and Flühler, 2005). Th e 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. Aft er hydrostatic equilib-rium is attained, the water content is measured. Th e equilibration time can diff er for every soil, however, from a few minutes to weeks (Nimmo, 2002). Th is means that the typical time to obtain a com-plete water retention curve can be weeks or longer. Similarly, the traditional way to estimate unsaturated hydraulic conductivity is based on steady-state fl ux 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-fl ow experiments (Hopmans et al., 2002; Durner and Lipsius, 2005). To give valid results, these methods require independence of the hydraulic properties from the fl ow 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 eff ects and pro-cesses that infl uence the equilibrium distribution of fl uids, such as surface tension, the presence of f luid–fluid interfaces, wet-tability of solid surfaces, grain size distribution, and microscale

heterogeneities. In a similar way, the unsaturated hydraulic con-ductivity curve depends on water content, roughness, tortuosity, and the shape and degree of interconnection of the water-conduct-ing pores in the porous media (Reynolds et al., 2002). By using the water retention curve estimated under equilibrium conditions, we automatically assume that these eff ects have an invariant infl uence on the soil water retention curve regardless of whether the water moves or not. A similar assumption is made for the hydraulic con-ductivity curve.

Despite the obvious problems that are connected with these con-ceptual 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 fl ow 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.

Observa onsObservations of DNE have been published only for laboratory stud-ies (Table 1). Th e reasons for that are that (i) well-controlled fl ow 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 condi-tions in the fi eld; (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 measure-ments can be minimized in the laboratory, e.g., by keeping tem-peratures constant, which leads to more accurate and reliable mea-surement data. Table 1 gives an overview of published experiments where nonequilibrium has been found. We note that these experi-ments 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 esWe start our review by recalling studies on fl ow-rate dependences of hydraulic properties, particularly the water retention curve. His-torically, the earliest questions regarding DNE in water fl ow theory have been raised in studies testing the diff usion theory (Hassani-zadeh et al., 2002). Mokady and Low (1964) were perhaps the fi rst who wrote that the water retention curve may not be unique. Th is 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.


Topp et al. (1967) compared water retention curves (drainage) of vertical sand columns obtained by static equilibrium, steady-state, and transient conditions. Th eir 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 f lux conditions are shown with open triangles. For their unsteady fl ow experiments, they used three diff erent experimental runs. In the fi rst 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. Th e key fi nding 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. Th ey furthermore conducted additional “hybrid” experiments (“static-unsteady” and “unsteady-static”). Th ey 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 fl ow (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 con-tinuation of the stepwise static experiment. Th eir results proved for the fi rst 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.

Th ese fi ndings were shortly aft erward confi rmed by work done at Grenoble. Smiles et al. (1971) performed experiments where a series of imbibition–drainage cycles was applied to a horizon-tal sand column by imposing diff erent pressure head steps at one column end. Th ey 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. Th ey also stated that the eff ect 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

Table 1. Experimental studies dealing with dynamic nonequilibrium eff ects.

Reference Equilibration time Length scale Soil type Experiment type

h cmTopp et al. (1967) 16.6–100 7.6 fi ne sand drainage for static, steady, and

transient fl ow + mixed fl ow

Smiles et al. (1971) NA† 60 fi ne sand imbibition and drainage in horizontal columnRogers and Klute (1971) 720 100 fi ne sand drainageVachaud et al. (1972) NA 136 fi ne sand drainage in vertical columnPoulovassilis (1974) 24 55 sand constant fl uxElzeft awy and Mansel (1975) NA 5.4–7.6 undisturbed fi ne sand drainageStauff er (1977) NA 50–60 fi ne sand drainageKneale (1985) NA 15 undisturbed clay loam drainageStonestrom and Akstin (1994) 2.5–4, 8.3–24 73 sand, sandy loam, silt loam,

and glass beadsconstant-fl ux infi ltration

Plagge et al. (1999) >6.7 10 undisturbed silt loam drainage and imbibitionSchultze et al. (1999) >24 15.7 sand drainage and imbibition, smooth

boundary conditions

Ross and Smettem (2000) NA 63–67 sandy loam, clay constant-fl ux infi ltrationWildenschild et al. (2001) NA 3.5 sand, sandy loam multistep outfl owŠimůnek et al. (2001) 240 10 undisturbed sandy loam upward infi ltration under tensionO’Carroll et al. (2005) >15 9.6 sand multistep outfl ow (two phase)Bottero et al. (2006) 12.5 19 sand two-phase drainage DiCarlo (2007) 0.05 40 sand multistep outfl owVogel et al. (2008) NA 10 sand multistep and two-step outfl owSakaki et al. (2010) NA 10 fi ne sand multistep outfl ow–infl owWeller et al. (2011) >50 10 sand constant-fl ux infi ltrationO’Carroll et al. (2010) NA 9.6 sand multistep outfl ow (two phase)Camps-Roach et al. (2010) NA 20 sand multistep outfl ow–infl owBottero et al. (2011) NA 21 sand two-phase drainage Diamantopoulos et al. (2012) 1 or >4 7.2 sand, undisturbed sandy loam multistep outfl ow

† NA, not available.


Klute (1971) investigated the fl ow dependence of the hydraulic conductivity as a function of the water content. Th ey found a rate-dependent 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 fi lled with sand and percolated water at a constant rate on the top of the column. When steady-state fl ow 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. Th e pressure head was measured for 1 d. He noticed that the pres-sure head increased appreciably with time while the water content remained stable. Th is pressure head increase was more pronounced for states at medium water contents. In another series of experi-ments, he left the soil column under constant-fl ux infi ltration for an additional period of 3 h aft er steady-state fl ow was achieved. Th en he sealed both column ends once again and placed the column hori-zontally. Th e results showed that the pressure head increase was not so pronounced as in the fi rst 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 fl ow than during steady-state or static equilibrium. At ETH Zürich, Stauff er (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 eff ect of the estimation method (steady state or transient) in the relative per-meability vs. saturation relationship. Th e results showed a similar trend as in the case of the water retention curve.

In the following decades, the topic of dynamic eff ects in soil water f low 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 experi-ments by increasing or decreasing the pressure head at the top of an undisturbed soil column. Th eir experimental setup allowed the conduction of evaporation experiments on the same column. Th ey 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 outfl ow (MSO) and one-step outfl ow experiments to investigate the fl ow-rate dependence of unsaturated hydraulic properties. Th e soils used were a sandy and a silty soil. Th e results showed that the SHPs of the sandy soil were fl ow dependent, whereas the SHPs of the silty soil were not. Sakaki et al. (2010) measured static and dynamic drainage and imbibi-tion curves in the laboratory. Th ey concluded that, at given water contents, the pressure heads measured under dynamic drainage conditions were statistically smaller than expected from the static

Fig. 1. Water retention curves estimated by Topp et al. (1967).


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 diff erent from the water retention curve estimated under steady-state or static conditions. More specifi cally, 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 imbi-bition. Th is was proven for diff erent experimental setups and dif-ferent 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 aff ected by the DNE. Contrary to that, Sakaki et al. (2010) found that the dynamic wetting curves also diff ered statistically compared with the static curves.

Dynamic Nonequilibrium in Mul step and Con nuous Ou low–Infl ow ExperimentsIn multistep outfl ow–multistep infl ow (MSO–MSI) experiments, tensiometer readings have sometimes reached the new equilibrium levels relatively quickly aft er a pressure step, whereas outfl ow or infl ow 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 eff ects occurring in experiments, with a focus on parameter estimation by inverse modeling. Th ey pointed out that the most signifi cant deviations between model and observation occurred in the moisture range near the air-entry point. Moreover, this phe-nomenon was not limited to the drainage branch but also occurred during imbibition. Th ey furthermore conducted “continuous” out-fl ow–infl ow 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. Th e total time for the fi rst cycle was 432 h. Th e experi-ment was repeated three times, increasing the speed of the drain-age–imbibition process each time by a factor of four. Th is resulted in an accelerated drainage–imbibition process by a factor of 64. Figure 3 shows the diff erent retention curves obtained by plotting the TDR data against the pressure head data measured inside the soil column. Th ere is a small but signifi cant 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).

Fig. 2. Continuous outfl ow–infl ow experiment conducted by Schultze et al. (1999): cumulative outfl ow–infl ow 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). Th e installation depths for both tensiometer and time domain refl ectometry (TDR) sensors were 4.5 and 10.2 cm from the top of the soil column.


Šimůnek et al. (2001) conducted upward infiltration experi-ments in an undisturbed soil column for a loamy sand soil. Th e soil column was equipped with fi ve tensiometers. Th ey 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 infl ow would cease when all the tensiometers had reached a constant pressure head. O’Carroll et al. (2005) explored dynamic eff ects in capillary pressure in MSO experiments with water and also with dense nonaqueous-phase liquid (DNAPL). Th ey tested whether the traditional multiphase fl ow simulators could describe the observed dynamics. Only when they incorporated a dynamic capillary pressure term was there a signifi cant improvement in the agreement between simulated and measured cumulative water out-fl ow data. Moreover, the estimated retention curve was in good agreement with the independently measured static retention curve.

Recently, Diamantopoulos et al. (2012) reported MSO experi-ments 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. Th ey recognized two phases in the outfl ow dynamics. In the fi rst phase, water drained abruptly from the column directly aft er each pressure step, as expected from an equilibrium relationship with the capillary pressure dynamics, but in a second phase, out-fl ow continued and ceased only slowly. Figure 4 also shows the fi t-ting obtained using the Richards equation with traditional SHPs that are invariant with respect to the system state. Th e match to the pressure head data is good, but the model cannot describe the outfl ow data. Th is refl ects the inherent assumption of the Richards equation that pressure head and water content are tightly coupled

through the retention curve. As shown below, this can be dramati-cally improved by partial decoupling of the water content and pres-sure head in the modeling. As a side note, it can be further high-lighted that ignoring pressure head data from the fi tting procedure creates a danger of getting very diff erent soil hydraulic properties. Th e outfl ow data of Fig. 4 alone could be fi tted by the Richards equation if the hydraulic conductivity function was adjusted to much lower conductivities.

In summary, the observed DNE in MSO–MSI experiments con-fi rms 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 concern-ing the equilibration kinetics for water contents. It seems that this approach is not linear but contains two distinct phases in the case of outfl ow, as described by Diamantopoulos et al. (2012). Inter-estingly, 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 ExperimentsDynamic nonequilibrium in stepwise outf low experiments is known to occur but is sometimes deemed to be of little relevance for natural processes because such stepwise changes in bound-ary conditions do not occur in nature during drainage processes;

Fig. 3. Infl uence 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 outfl ow exper-iment (pF is defi ned as the logarithm of the absolute value of pressure head in centimeters). Th e legend shows drainage duration in hours.

Fig. 4. Observed and simulated cumulative outfl ow and pressure head data for a loamy sand soil (Diamantopoulos et al. 2012). From 30 h on, the equilibration kinetics of pressure head and outfl ow diff ered signifi cantly. Th e fi tted data were calculated using the Richards equa-tion and the Diamantopoulos et al. (2012) dynamic nonequilibrium (DNE) model. Both models were coupled with the van Genuchten–Mualem (VGM) model.


however, DNE is also found under evaporation conditions. We show some unpublished data regarding this. Evaporation experi-ments 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 experi-mental setup can be found in Schindler (1980), Peters and Durner (2008), and Schindler et al. (2010). Th e 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 fi rst evaporation stage (Shokri et al., 2009), which is of inter-est 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 distribu-tions with depth were approximately linear and close to hydrostatic because the saturated conductivity was about three orders of mag-nitude larger than the water fl ux at the upper boundary, and thus the hydraulic gradient was almost equal to zero.

We then modifi ed the experiment by introducing an interruption of the evaporation fl ux. Aft er allowing the soil column to evapo-rate for 24 h, we covered it with a cap and set the evaporation fl ux to zero for another 24 h. Th e cap was then removed and evapora-tion 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 pres-sure heads increased toward an equilibrium level that was diff erent from the value under transient conditions. Th e observed “nonequi-librium” was not a result of vertical water redistribution. Th is can quantitatively be proved by inverse modeling (shown in Fig. 5). Th e 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. Th ey fi tted the van Genuchten (1980) model to the equilibrium water reten-tion data and then tried to match evaporation experiment data by fi tting 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 fi tted both water retention and hydraulic conductivity curves could they describe the dynamics of the evaporation experiments.

Th e study of Bohne and Salzmann (2002), along with the new experimental data presented in this study, shows that DNE eff ects occur also in the case of evaporation under laboratory conditions. We have to note that the fl ow rate change in these experiments was

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 eff ects on capillary pressure also need to be accounted for, however, when investigating DNE in evaporation experiments.

Dynamic Nonequilibrium Eff ects in Infi ltra on Experiments under Constant-Flux Condi onsStonestrom 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. Th ey conducted constant-fl ux infi ltra-tion experiments in soil columns of 73-cm length with an inside diameter of 5 cm. Th e soil columns were equipped with tensiom-eters at three diff erent depths. Figure 6 shows the evolution of the pressure head for constant-fl ux infi ltration into initially air-dry soil columns for the Delphi sand soil and for glass beads with a median diameter of 80 μm. Th e pressure head measured by the tensiometers passed through a maximum value and then decreased steadily as the wetting front moved farther down the column. Th is means that the evolution of pressure head during constant-fl ux infi ltra-tion 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 respon-sible (Geiger and Durnford, 2000; Eliassi and Glass, 2001, 2003; Egorov et al., 2003) for fi nger fl ow in homogeneous porous media and consequently responsible for preferential water fl ow.

Fig. 5. Dynamic nonequilibrium observed in an evaporation experiment for a well-sorted sand. Th e soil sample was allowed to evaporate for 24 h and then evaporation fl ux was stopped for another 24-h period. Th e black line shows cumulative evaporation (cm3). A sequence of on–off cycles was followed until the end of the experiment. Th e fi tted data were calculated using the Richards equation coupled with the bimodal van Genuchten (biVGM) approach by Durner (1994).


Recently, Weller et al. (2011) reported the occurrence of DNE eff ects in their experiments. Th ey performed constant-fl ux infi ltra-tion experiments for both drainage and imbibition with a stepwise change in the water application rate. Th e 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 fl ux rate. In the case of drainage, Weller et al. (2011) noticed that the pressure head mea-sured by the tensiometer dropped immediately aft er the diminu-tion of the applied fl ux but then slowly increased again. Th is second phase of the tensiometer behavior had a duration of a few days until equilibrium was reached. Similar behavior was also observed for increasing fl ux rates.

Reasons Proposed for the Occurrence of Dynamic Nonequilibrium Eff ects

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 Reconfi gura onWhen air displaces water (or water displaces air) in a porous medium, the air–water confi guration on the pore scale within the REV is redistributed, and this redistribution is not instantaneous but requires a fi nite 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. Th e measured pressure head will be smaller due to unstable air–water interfaces, and it will increase as the fl uid 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 eff ects in capillary pressure (Kalaydjian, 1992; Hassanizadeh et al., 2002).

Entrapment of Water and Pore Water BlockageHistorically, the earliest explanation for the water fl ow 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 fi lled 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 con-tinuous water body by fi lm fl ow. Th is would lead to an increase in the pressure head, recorded by the tensiometers, under stopped fl ow con-ditions. Th e same process explains the two-phase outfl ow dynamics oft en 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, aft er a fl ux reduction, there are pores that drain more easily followed by a much slower drainage of blocked pores. Th is emptying proceeds slowly and caused the slow increase in water potential observed in their constant-fl ux experiments.

Fig. 6. Pressure head (ψ) evolution for a constant-fl ux infi ltration experiment into initially air-dry soil columns as presented by Ston-estrom and Akstin (1994). Th e two materials were Delphi sand and glass beads with a median diameter of 80 mm. Th e pressure head was recorded at three diff erent depths: 2, 5, and 8 cm. Courtesy of AGU.


Air EntrapmentIf the air phase in a porous medium loses its continuity, gas trans-port takes place only by eff ective diff usion. 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 atmo-spheric pressure, is not valid to describe the water dynamics. Smiles et al. (1971) tested whether the diff erences between water retention curves estimated by static or dynamic conditions can be attributed to restricted air fl ow. Th ey compared drainage curves obtained by experiments in a soil column with and without additional lateral inlets for air. Th e comparison showed that the air entry near the axis of the column was adequate to maintain atmospheric pressure. Th us they concluded that this is not a signifi cant cause for the observed DNE. Schultze et al. (1999) came to contradicting conclusions. Th ey conducted multistep and continuous outfl ow–infl ow experi-ments on soil columns with open and with closed walls and simu-lated them with one-phase and two-phase fl ow models. Th e results showed that for disturbed and undisturbed soil columns, the air phase can lose its continuity already at 50 to 70% of water satura-tion, and dynamic eff ects consequently occur. Similarly, Wilden-schild et al. (2001) considered air entrapment as one reason for fl ow 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 fi eld conditions. Nevertheless, fi eld 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 experi-ments and claimed that air entrapment did not result in observed diff erences between static and dynamic retention curves.

Air EntryWildenschild et al. (2001) proposed another eff ect 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 drain-age will occur fi rst at the lower end where the pressure drop is applied. Th ey further assumed that this process contributes to the rate dependency of estimated soil hydraulic properties for a sandy soil. Th is eff ect, however, can only explain DNE occurring in MSO drainage experiments.

In evaporation experiments that start from full saturation, air-entry eff ects can be frequently observed in the early state, in par-ticular for unstructured soils. Th e air that is replacing the evaporat-ing 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 Rich-ards equation. If water content is plotted vs. measured tensiometric water pressures, these nonequilibrium conditions can be visualized as a shift ed water retention curve, most pronounced around the macroscopic air-entry region of the soils (Fig. 7, right).

Dynamic Contact AngleIt 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 (Hoff man, 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 fl ow dependence of SHPs measured under transient conditions. Wildenschild et al. (2001) stated that the dynamic contact angle eff ect was small under drainage conditions because when fl ow velocity increases, the contact angle should approach zero. In their drainage experi-ments, 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 eff ect that leads to DNE in air–water systems. O’Carroll et al. (2010) studied theoretically and experimentally the eff ect of wettability on dynamic eff ects in capillary pressure. Th ey 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 ChangesTh e 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 wettabil-ity. Th ese 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).

HeterogeneityMicroscale HeterogeneityMirzaei and Das (2007) used the term microscale heterogeneities to refer to microscale lenses of fi ne sand that have diff erent multi-phase fl ow properties than the surrounding porous medium. Th ese small-scale heterogeneities occur at a length scale below the REV scale and have signifi cant eff ects on the eff ective SHP. To test the eff ect of these heterogeneities in DNE phenomena, Mirzaei and Das (2007) conducted numerical experiments to investigate how the microscale heterogeneities aff ect the dynamics of DNAPL and water fl ow in a porous domain. Th ey found that microscale heterogeneities lead to dynamic eff ects and that the intensity of


heterogeneity increases the emergence of dynamic eff ects. Experi-ments dealing explicitly with the eff ects of microscale heterogene-ity have not been yet published, however.

Macroscale HeterogeneityHeterogeneity on the macroscopic measurement scale is well known to cause phenomena, including preferential f low, that cannot be captured with the assumption of quasi-homogeneous porous medium properties. Th is problem is less related to the topic of dynamic eff ects that emerge from subscale processes, the focus of this review, but rather to the general question of the existence of eff ective hydraulic properties and eff ective 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 fl uxes are diff erent and do not perfectly overlap. Th is is the standard case in practical measure-ments. Despite this general knowledge, it is currently not clear how strong the heterogeneity eff ects are and how they depend on vari-ous parameters, including the fl ow regimes.

A special case of macroscale heterogeneity is the distribution of dif-ferent materials below the measurement scale. Th is kind of heteroge-neity 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 eff ect of heterogeneity of local hydraulic properties by using a two-phase simulator. Based on forward simulations, they examined the eff ect of heterogeneity on the occurrence of DNE. Th e results showed that the DNE was infl uenced by a heterogeneous distribu-tion of intrinsic permeability. Th ey also found it to be boundary-condition dependent. Vogel et al. (2008) generated a stochastic, two-dimensional, heterogeneous fi eld and simulated a MSO experiment for this synthetic porous medium. Th ey found a perfect accordance between the one-dimensional forward simulation using the static mean hydraulic properties and the two-dimensional simulation. Th ey concluded from their fi ndings that the MSO approach is not seriously aff ected by dynamic eff ects from this process.

Contrary to that are fi ndings for infi ltration processes. Šimůnek et al. (2001) suggested that heterogeneity is the reason for the observed DNE eff ects in upward infi ltration experiments. Th ey

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, unpub-lished data, 2012) (pF is defi ned as the logarithm of the absolute value of pressure head in centimeters).


postulated that the observed DNE eff ects are due to the redistribu-tion of water and, more specifi cally, water being transferred from larger to smaller pores. Vogel et al. (2010) studied numerically the eff ect of large-scale heterogeneity by conducting infi ltration simu-lations. Th ey found that the larger the correlation length (isotropic correlation) used for generating the stochastic heterogeneous fi eld, the larger were the nonequilibrium eff ects in the infi ltration front. Th is is in accordance with the expectation that the smaller the cor-relation length, the faster is the equilibration of the water poten-tial. Although large-scale heterogeneity is expected to evoke DNE eff ects in water fl ow, it cannot explain the occurrence of these eff ects 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.

Modeling ApproachesModeling of DNE can be done on scales below the traditional Richards equation, e.g., by pore-scale network models or by modifi cations of the Richards equation. Th e latter approaches are dominated by formulations that use empirical or physically based fl ow-rate-dependent capillary pressures. Alternatively, eff ective 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 over-view of popular DNE modeling approaches that have been com-pared with experimental data.

Pore-Scale Network ModelsDuring the last few decades, pore-scale network modeling has been established as an alternative approach in modeling two-phase fl ow in porous media. It requires the exact microscopic descriptions of the pore geometry and the physical laws of fl ow 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 diffi cult-to-measure quantities such as interfacial areas or common lines (Held and Celia, 2001). Th e analysis of dynamic eff ects in water fl ow 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 Pay-atakes (1990), Blunt and King (1991), and Dahle and Celia (1999), among others. A few studies have also been conducted on DNE eff ects with the help of pore-scale network models (Gielen et al., 2001; Joekar-Niasar et al., 2009).

Con nuum-Scale Models with Dynamic Capillary PressureStauff er (1977) ModelStauff er (1978) simulated drainage experiments by means of a numerical model using the fi nite 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

and Corey (1964) model with a dynamic capillary pressure. Based on experimental observations, he proposed that the dynamic cap-illary pressure, Pc

dyn [M T−2 L−1], and the equilibrium (or static) capillary pressure, Pc

stat [M T−2 L−1], be related through (Stauff er, 1978; Manthey et al., 2008)

c c wdyn stat s

SP Pt

∂− =−τ

∂ [4]

where τs [M T−1 L−1] is a relaxation parameter, given by

2w d

sw

a Pk g

⎛ ⎞μ φ ⎟⎜ ⎟τ = ⎜ ⎟⎜ ⎟⎜λ ρ⎝ ⎠ [5]

where Sw (dimensionless) is water saturation, a is a dimensionless scaling parameter, μw [M T−1 L−1] and ρw [M L−3] are the vis-cosity 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, Stauff er (1978) concluded that neglecting the dynamic eff ects in the capillary pressure–water saturation relationship can lead to considerable errors.

Hassanizadeh and Gray (1990) ModelHassanizadeh and Gray (1990) developed a macroscopic thermo-dynamic theory to describe two-phase fl ow in porous media. In a follow-up study, Hassanizadeh and Gray (1993) stated that the macroscopic capillary pressure is defi ned as an intrinsic property

Table 2. Modeling approaches for dynamic nonequilibrium eff ects, including only models that have been compared with experimental data.

Model Study Dimensions

Richards’ equation coupled with dynamic capillary pressure

Stauff er (1977) 1

Hassanizadeh et al. (2002) 1

Sander et al. (2008) 1,2

Chapwanya and Stockie (2010) 2

Two-phase fl ow coupled with Hassanizadeh and Gray (1993) model

O’Carroll et al. (2005) 1

O’Carroll et al. (2010) 1

Fucik et al. (2010) 1

Richards’ equation coupled with kinetic water content equilibration

Ross and Smettem (2000) 1

Šimůnek et al. (2001) 1

Two-site model with partial water content equilibration

Diamantopoulos et al. (2012) 1

Mobile–immobile and dual-permeability models

Philip (1968) 1

Gerke and van Genuchten (1993a, 1993b)

1

and others


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 wdyn stat h

SP Pt

∂− =−τ

∂ [6]

where τh [M T−1 L−1] is a material coeffi cient that defi nes the time scale necessary to reach equilibrium. Equation [6], based on theo-retical 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 equa-tion is questionable if transient water fl ow takes place. In a similar approach to Stauff er (1977), Hassanizadeh et al. (2002) proposed to couple Eq. [6] with the Richards equation. For this case, the one-dimensional problem with vertical coordinate z is given by

stat ˆ hh KK K

t z z z z t z⎡ ⎤⎛ ⎞ ⎛ ⎞∂∂θ ∂ ∂ ∂ ∂θ ∂⎟ ⎟⎜ ⎜⎢ ⎥= + τ −⎟ ⎟⎜ ⎜⎟ ⎟⎜⎜ ⎢ ⎥⎝ ⎠⎝ ⎠∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦

[7]

where hstat [L] is the static pressure head, given by hstat = hdyn − ˆ hτ(∂θ/∂t) and ˆ hτ = τh/φρw 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. Th e numerical results showed that the infi ltration front was retarded for high values of the material coeffi cient τh. Sander et al. (2008) used Eq. [7], and assumed that τh could be given by a simple function of water content. Th is was based on the experimental results of Smiles et al. (1971), which show that τh becomes minimal when the eff ec-tive 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 incor-porated hysteresis in the capillary pressure–water content relation-ship in their model. By using numerical models, they were able to describe the nonmonotonic saturation distribution property of fi n-ger-fl ow evolution (one dimensional) as well as the lateral growth of fi ngers 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 fl ow in initially dry homogeneous soils. Th ey also included hysteresis in the water retention curve. Th ey concluded that their model can describe the physics of fi ngered fl ow and, moreover, they showed that for small values of the nonequilibrium parameter τh, the fi nger formation was suppressed. In contrast, for relatively large values of τh, fi nger fl ow occurred.

Th e concept of dynamic capillary pressure was also investigated in studies related to multiphase fl ow. O’Carroll et al. (2005), for their MSO experiments with water and DNAPL, explored the agree-ment between observed and simulated results using a multiphase fl ow simulator. Only if they incorporated Eq. [6] in their model was there a signifi cant improvement in the agreement between simulated and measured cumulative water outfl ow 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 eff ect of the porous media grain distribu-tion on the dynamic coeffi cient τh. Th ey found that τh depended on the grain size. Fucik et al. (2010) developed a one-dimensional two-phase fl ow model that can handle fl ow in two fl uids in both heterogeneous and homogeneous media along with dynamic capil-lary pressure conditions given by Eq. [6]. Th e model was validated against both experimental results and semianalytical solutions. Based on their simulations, they concluded that the dynamic eff ect can be of great importance, especially for heterogeneous media. Bot-tero et al. (2011) studied DNE eff ects in two-phase fl ow by conduct-ing a series of dynamic drainage experiments in a natural sandy soil. Th ey used tetrachloroethylene as the non-wetting fl uid and distilled water as the wetting fl uid. Th eir 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 coef-fi cient 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 eff ect of fl uid properties on the magnitude of the nonequilibrium parameter τh. Joekar-Niasar and Hassanizadeh (2011) examined the eff ect of fl uid viscosity on τh. Th ey found that viscosity strongly aff ected the variation of τh with saturation. Goel and O’Carroll (2011) conducted two-phase fl ow drainage experiments to examine the eff ect of fl uid viscosity on the nonequilibrium parameter τh and concluded that the mag-nitude of τh was strongly dependent on the eff ective fl uid 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 diff erent length scales in two-phase fl ow. Con-trary 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 ContentThe Ross and Sme em (2000) ModelTo be able to describe local nonequilibrium between water content and pressure head, Ross and Smettem (2000) proposed a simple modifi cation of the Richards equation. Basically, they assumed a


time-dependent evolution of the water content, which approaches its equilibrium value by a fi rst-order kinetic. Th ey specifi ed the term ∂θ/∂t of the Richards equation through the diff erential equation

( )eq,ft

∂θ= θ θ

∂ [8]

with

( ) ( )θ −θθ θ =

τeq

eq,f [9]

where θeq [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 Stauff er (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 fl ow during infi ltration experiments in which preferen-tial fl ow was signifi cant.

Diamantopoulos et al. (2012) ModelBased on the inability to describe experimental observations of MSO and evaporation experiments with either the Richards equa-tion or the Ross and Smettem (2000) model, Diamantopoulos et al. (2012) developed an eff ective one-dimensional nonequilibrium model that merges the two previous models. Similar to the two-site 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 fl ow direction, Diamantopoulos et al. (2012) arrived at a single equation for the water dynamics:

( ) ( )eq nene ne1 1

hf f K ht t z z

∂θ ⎡ ⎤⎛ ⎞∂θ ∂ ∂ ⎟⎜⎢ ⎥− + = −⎟⎜ ⎟⎜⎢ ⎥⎝ ⎠∂ ∂ ∂ ∂⎣ ⎦ [10]

in which

eq nene

tθ −θ∂θ=

∂ τ

where θne is the water content in the nonequilibrium domain. Th is 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, Dia-mantopoulos 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 identifi ed from the experiments by inverse modeling (Fig. 4).

Dual-Porosity and Dual-Permeability ModelsDual-porosity (domain) models distinguish two diff erent soil domains, each with its own set of SHPs. If water fl ow can take place in both domains, we speak of dual-permeability models. Dual-porosity and dual-permeability models have been developed for describing preferential fl ow in structured soils. Th ey are ideal for describing DNE in heterogeneous soil materials where a clear identifi cation of two (or more) diff erent materials can be made and where the separation in matrix and macropores is the obvi-ous 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 interaggre-gate domain and a matrix or intraaggregate domain. Water fl ow occurs only in the fracture domain and the matrix domain repre-sents 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). Th is 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 fl ow is described by the Richards equation. Transfer of water between the two domains is described by means of fi rst-order rate equa-tions, being proportional to the diff erence in eff ective saturation of the two regions (Šimůnek et al., 2001) or being proportional to the pressure head diff erence 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 nonequilib-rium water fl ow in upward infi ltration experiments. For compre-hensive reviews of dual-porosity and dual-permeability models, see Šimůnek et al. (2003) and Gerke (2006).

DiscussionWe have defi ned DNE as an apparent dependence of SHPs on the fl ow dynamics. Th is means that SHPs are diff erent for one mate-rial whether water moves or not and whether saturation changes occur fast or slowly. Furthermore, any process that causes time-dependent changes of SHPs leads to dynamic eff ects. Th is defi -nition 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 heterogene-ity can produce DNE. Th is means that the spatial distribution of materials with diff erent (time-invariant) SHPs can macroscopically generate DNE. A generally valid defi nition of DNE thus seems diffi cult and we propose to think of DNE as hydraulic state obser-vations that cannot be described by a continuum model (Richards equation or a two-phase fl ow model) with a set of unique SHPs.


Our review has shown that there is ample evidence that water fl ow phenomena in unsaturated soils occur for which the concept of unique soil hydraulic properties is not valid. More water is with-held by the soil matrix for a higher fl ow rate in the case of drainage. Recent studies have further shown that this diff erence is statistically signifi cant (Sakaki et al., 2010; Goel and O’Carroll, 2011). We have focused in this review on the water retention characteristics, but simi-lar eff ects 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) confi rmed that DNE eff ects aff ect also the hydraulic conductivity curve. Moreover, DNE eff ects occur in both drainage and imbibi-tion processes. Although the eff ect 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 eff ects is in a specifi c situation. Although various reasons have been pro-posed to provoke DNE eff ects, some of them are contradictory and cannot be considered as the unique reason for nonequilib-rium water fl ow. For example, water entrapment can occur only in drainage experiments and not in imbibition experiments. As another example, the eff ect of a dynamic contact angle is gener-ally very small for drainage experiments. In any case, it is hard to assess whether a unique physical process or a combination of vari-ous processes generates DNE eff ects in both drainage and imbibi-tion experiments. Furthermore, we do not know in which pressure head–water content–saturation range each eff ect acts. Similarly, we do not know yet if the physical processes that generate DNE eff ects are the same among the diff erent experimental setups.

A particular diffi culty arises from the interference of dynamic eff ects with capillary hysteresis under transient-fl ow conditions with chang-ing fl ow directions. Capillary pressure hysteresis is a well-known phe-nomenon and has been eff ectively 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 con-ductivity curves show hysteresis as a function of the pressure head. Moreover, hysteresis can signifi cantly infl uence water fl ow in variably saturated porous media (Vachaud and Th ony, 1971; Gillham et al., 1979; Elmaloglou and Diamantopoulos, 2008). Th e 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). Th ey showed that the inclu-sion of the specifi c interfacial area in the capillary pressure–satu-ration relationship leads to the removal or signifi cant reduction of hysteresis. Th ere is ample work that supports this statement, includ-ing 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 stud-ies dealing with DNE eff ects. 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 fi nally 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 eff ects more pronounced in sandy materials? Stauff er (1978) found more pro-nounced dynamic eff ects in the case of a fi ne 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 fl ow in similar sandy loam mate-rials (Šimůnek et al., 2001). Th e use of sandy soils may be explained by the fact that it is very ambitious to study the eff ect on natural heterogeneous media if we do not recognize what causes DNE eff ects even in well-sorted sandy materials.

During the last few years, a great eff ort has been made to model nonequilibrium water fl ow, as presented above. Th e majority of studies have used the model of Hassanizadeh and Gray (1990, 1993) coupled with the Richards equation or with a one-dimen-sional two-phase fl ow simulator. For structured soils, many studies have used mobile–immobile or dual-permeability models for non-equilibrium water fl ow (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 fl ow. In a specifi c situation, it is hard to quantify the dominant eff ects. Th is has made the development of physically based models that encompass all causes and can be used for reliable predictive simulations very diffi cult if not impossible until now. For this reason, we believe that eff ective 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 time-variable 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 eff ects in practical situations only if com-bined with suitable experiments. Th ese experiments rely on fast and accurate measurement techniques, and we see a defi ciency 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 fl ow, further progress in detecting the extent and role of dynamic eff ects will depend on the combination of suitable experiments with precise


measurements and their combination with inverse modeling. As such, systematic experimental studies are needed to quantify the dependence of DNE eff ects and the involved parameters on pres-sure head regions, fl ow dynamics, and time and length scales.

Another question concerning DNE eff ects is whether and to what degree they aff ect water fl ow in greater scale (fi eld-scale) problems. For many of the hypothesized reasons for DNE that have been dis-cussed here, the signifi cance for fi eld-scale fl ow 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. Th us, it has to be clarifi ed whether the use of SHPs estimated at diff erent fl ow rates can be used in model-ing problems at larger scales.

ConclusionsDuring the last few decades, impressive advances have been made regarding our understanding and our ability to model water fl ow in the vadose zone. Nonequilibrium eff ects in unsaturated water fl ow have received rather little attention in the past, but it seems that with improved measuring and modeling capabilities, it has become increasingly a subject of research for various scientifi c disciplines. Evidence from advanced measurement experiments, which are spe-cifi cally designed to determine hydraulic properties with a high sensitivity, shows that the observed transient water fl ow cannot be described with the standard Richards equation. Dynamic nonequi-librium eff ects are characterized by faster equilibration of the pres-sure head than the outfl ow in MSO experiments and by spurious pressure head drift s in constant-fl ux infi ltration experiments and evaporation experiments with fl ow interruption. More generally, they are expressed by apparent local nonequilibrium between water content and static pressure head at measurement sensor scales.

Various reasons have been hypothesized to cause DNE eff ects. Th ose reviewed here encompass processes below the REV scale, such as relaxation of air–water interfaces, limited air-phase per-meability, dynamic contact angles, heterogeneity of soil proper-ties under the measurement scale, and time-dependent wettability changes. Entrapment of water, pore water blockage, and air entry eff ects were also discussed. Although there is clear evidence that these causes lead to macroscopic DNE observations, it is currently not possible to quantify them in a predictive manner, and a general judgment about the importance of various causes is speculative at the present state of knowledge. Th eir role for fi eld water transport is still unclear. Local heterogeneity of soil hydraulic properties on the macroscopic scale increases complexity because it introduces additionally DNE eff ects for averaged quantities. As a perspective, a better understanding of DNE might help our understanding of scale eff ects and improve the transferability of laboratory-deter-mined hydraulic properties to simulate fl ow processes in the fi eld.

Dynamic nonequilibrium eff ects are likely to act in diff erent pres-sure head regions and on diff erent time scales. Th e multitude of processes and the inability to quantify them independently makes mechanistic modeling of water fl ow in porous media that includes all processes currently unfeasible. Eff ective modeling of the DNE phenomena is an alternative, but the estimation of eff ective 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 eff ects in unsaturated water fl ow.

AcknowledgmentsTh is study was fi nancially 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 As-sociate Editor Jan Vanderborght for their constructive comments on the manuscript.

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