## 1. Introduction

[2] Modeling of environmental systems is nowadays used routinely as a tool for research and decision making. With the increasing power of digital computers, more and more scientists use numerical codes to simulate problems in hydrology, soil science, agriculture and in many other fields [*van Dam and Feddes*, 2000; *Simunek*, 2005; *Elmaloglou and Diamantopoulos*, 2009]. In the field of soil science the Richards equation is the standard model to simulate variably saturated water flow in soils [*Vanclooster et al.*, 2004]. The solution of the Richards equation requires knowledge of the soil hydraulic properties, i.e., the soil water retention curve and the unsaturated hydraulic conductivity curve. Since hydrostatic or steady state methods to determine hydraulic properties are time demanding [*Dane and Topp*, 2002], the use of transient methods has become increasingly popular. Typical experimental designs are the one-step outflow (OSO), multistep outflow (MSO), and evaporation methods [*Kool et al.*, 1985; *Hopmans et al.*, 2002; *Schelle et al.*, 2010]. In most cases, such transient experiments are evaluated by inverse modeling.

[3] Inverse modeling involves parameter optimization in all cases [*Durner et al.*, 1999; *Simunek and Hopmans*, 2002; *Vrugt et al.*, 2008]. According to *Durner et al.* [1999], the advantages of parameter identification by inverse simulation are (1) the possibility to estimate simultaneously the parameters defining the retention and conductivity curves, (2) the absence of restrictions with respect to sample size, and (3) the possibility to apply experimental conditions that maximize the information content of the system response. An additional advantage of inverse modeling is the ability to test current process knowledge because limitations in process understanding become obvious, if systematic discrepancies between experimental data and model fits show up, or predictions based on optimal parameters turn out to be wrong under different boundary conditions.

[4] In most applications so far, inverse modeling of unsaturated flow phenomena is based on simulations with the Richards equation. However, comparison of data obtained from transient drainage experiments with data obtained under hydrostatic conditions suggest that more water is stored by the soil matrix at a given pressure head in a dynamic flow experiment compared to a hydrostatic or steady state experiment. *Davidson et al.* [1966] and *Topp et al.* [1967] were the first to report that experimentally determined water retention may differ depending on whether it is determined under transient, static or steady state conditions. During the past decades, numerous researchers have associated such differences with a physical “nonequilibrium” in variably saturated water flow [*Smiles et al.*, 1971; *Vachaud et al.*, 1972; *Plagge et al.*, 1999; *Kneale*, 1985; *Schultze et al.*, 1999; *Wildenschild et al.*, 2001; *Hassanizadeh et al.*, 2002; *O'Carroll et al.*, 2005]. Several causes have been identified which can be responsible for this behavior. In brief, fluid-fluid interface dynamics [*Hassanizadeh et al.*, 2002], entrapment of water [*Topp et al.*, 1967], pore water blockage [*Wildenschild et al.*, 2001], air entrapment [*Schultze et al.*, 1999], dynamic contact angle [*Friedmann*, 1999], air entry value effect [*Wildenschild et al.*, 2001], microheterogeneity [*Mirzaei and Das*, 2007], and large-scale heterogeneity [*Vogel et al.*, 2010] can cause nonequilibrium water flow. Notably, these nonequilibrium phenomena are present even in well sorted, macroscopically homogeneous materials.

[5] Recently, the awareness of the importance to consider dynamic nonequilibrium in water flow has increased and strategies to handle the phenomenon in a quantitative manner have evolved [*Camps-Roach et al.*, 2010; *Sakaki et al.*, 2010; *Vogel et al.*, 2010]. *Schultze et al.* [1999] conducted multistep outflow experiments and noticed that directly after a pressure step, tensiometer readings reached the new equilibrium levels relatively quickly, whereas outflow or inflow of water continued for periods of 24 h or longer. They concluded that the Richards equation could not simulate the observed outflow dynamics, but showed that the simulation of water and air movement by a two-phase flow model with restricted air permeability of the soil at high water saturation could explain the observed phenomena to a considerable extent. On the basis of thermodynamic considerations, *Hassanizadeh and Gray* [1993] proposed that a dynamic capillary pressure can be represented as a linear function of the rate of change in water saturation in the porous medium:

where [*M T*^{–2}*L*^{–1}] is the dynamic capillary pressure, [*M T*^{–2}*L*^{–1}] is the static capillary pressure, [*M T*^{–1}*L*^{–1}] is a material coefficient which defines the time scale necessary to reach equilibrium, and *S _{w}* (dimensionless) is effective saturation.

*O'Carroll et al.*[2005] conducted series of MSO experiments with water and dense nonaqueous phase liquid (DNAPL) and explored the agreement between observed and simulated results using a multiphase flow simulator. They concluded that the inclusion of a dynamic capillary pressure term was necessary to achieve a significant improvement in the agreement between simulated and measured cumulative water outflow data.

*Sander et al.*[2008] presented one- and two-dimensional numerical solutions for a new equation obtained by combining Richards' equation and the nonequilibrium model of

*Hassanizadeh and Gray*[1993]. Moreover, they included a hysteretic capillary saturation function and simulated unstable finger flow through layered soils.

*Berentsen et al.*[2006] developed a numerical two-phase flow model and compared the results with two-phase flow column experiments. They concluded that the experimental data could not be modeled without inclusion of the nonequilibrium effect. Moreover after testing various functions for the dependency of the damping coefficient on water saturation (constant, linear, quadratic, error function and Gaussian) they obtained reasonably good agreement with experimental data when the nonequilibrium capillary pressure term (equation (1)) and a Gaussian function between the coefficient and the saturation was assumed.

[6] *Gerke and van Genuchten* [1993a, 1993b] presented a physically based dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. 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 first-order rate equations. Although such a model is likely to fit outflow and pressure head data obtained from MSO experiments adequately, overparameterization must be expected because different soil hydraulic properties must be assigned and estimated for both the fracture and matrix domain, along with parameters describing the resistance and time scale of the exchange of water between the domains. This is a major disadvantage because the resulting parameter uncertainty limits the applicability of the model for predictive purposes. Furthermore, a conceptual model which distinguishes between a matrix and a fracture domain appears inadequate to describe the nonequilibrium flow phenomena which are observed in MSO experiments conducted on packed sand samples. In a recent study, *Laloy et al.* [2010] studied numerically if one-dimensional models can reproduce the hydrodynamic behavior of structured soils in MSO experiments. The results showed that although the *Gerke and van Genuchten* [1993a] model along with the *Philip* [1968] mobile-immobile model can provide excellent fits to the data, they associate with low predictive capabilities. However, for a macroporous soil, the *Gerke and van Genuchten* [1993a] model provided consistent results for an infiltration experiment.

[7] A much simpler approach to simulate dynamic nonequilibrium in unsaturated flow was proposed by *Ross and Smettem* [2000]. They replaced the local equilibrium relationship between water content and pressure head in the Richards equation with a dynamic water content, which approaches the equilibrium value defined by the soil water retention curve by a first-order kinetics. As a result, the dynamic water content will deviate from the equilibrium value if changes in hydraulic conditions are fast in comparison to the time scale of the equilibration. With their model, they succeeded to describe downward flow through cores of a structured field soil. *Simunek et al.* [2003] used this model to describe upward infiltration data on small undisturbed soil samples that exhibited severe nonequilibrium between measured pressure heads and volumes of infiltrated water. *Vogel et al.* [2010] conducted numerical studies of infiltration into two-dimensional heterogeneous soil profiles with the Richards equation. On the basis of the resulting virtual realities of water content and pressure head dynamics, they analyzed hydraulic nonequilibrium at the infiltration front and tested different effective one-dimensional models. They found that the nonequilibrium concept of *Ross and Smettem* [2000] was able to describe the widening of the infiltration front qualitatively. However, because of remaining inadequacies in the description of their data, they concluded that the linear kinetics of the *Ross and Smettem* [2000] model might not be appropriate for any type of heterogeneity although the general approach to decouple water content from pressure head appears promising.

[8] In this article, observations of nonequilibrium water flow in MSO experiments are investigated and analyzed by numerical inverse simulations. In a first step, the capability of the *Ross and Smettem* [2000] model to match the observations is investigated. In a second step, a new macroscopic dual-fraction model is proposed in which the movement of one fraction of the pore water is described by the Richards equation whereas the motion of the other fraction is described by the *Ross and Smettem* [2000] approach. The new model is compared to the Richards equation, the *Ross and Smettem* [2000] model, and the dual-porosity model of *Philip* [1968]. The feasibility of a simultaneous estimation of equilibrium soil hydraulic functions and physical nonequilibrium parameters by inverse modeling is investigated using synthetic data. Finally, the ability of the new model to describe experimental MSO data is tested against measurements from two real soils.