Observations of water flow in unsaturated soils often show “dynamic effects,” indicated by nonequilibrium between water contents and water potential, a phenomenon that cannot be modeled with the Richards equation. The objective of this article is to formulate an effective process description of dynamic nonequilibrium flow in variably saturated soil which is both flexible enough to match experimental observations and as parsimonious as possible to allow unique parameter estimation by inverse modeling. In the conceptual model, water content is partitioned into two fractions. Water in one fraction is in equilibrium with the pressure head, whereas water in the second fraction is in nonequilibrium, described by the kinetic equilibration approach of Ross and Smettem (2000). Between the two fractions an instantaneous equilibration of the pressure head is assumed. The new model, termed the dual-fraction nonequilibrium model, requires only one additional parameter compared to the nonequilibrium approach of Ross and Smettem. We tested the model with experimental data from multistep outflow experiments conducted on two soils and compared it to the Richards equation, the nonequilibrium model of Ross and Smettem, and the dual-porosity model of Philip (1968). The experimental data were evaluated by inverse modeling using a robust Markov chain Monte Carlo sampler. The results show that the proposed model is superior to the Richards equation and the Ross and Smettem model in describing dynamic nonequilibrium effects occurring in multistep outflow experiments. The three popular model selection criteria (Akaike information criterion, Bayesian information criterion, and deviance information criterion) all favored the new model because of its smaller number of parameters.
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 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.
 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. , 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.
 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.  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  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–2L–1] is the dynamic capillary pressure, [M T–2L–1] is the static capillary pressure, [M T–1L–1] is a material coefficient which defines the time scale necessary to reach equilibrium, and Sw (dimensionless) is effective saturation. O'Carroll et al.  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.  presented one- and two-dimensional numerical solutions for a new equation obtained by combining Richards' equation and the nonequilibrium model of Hassanizadeh and Gray . Moreover, they included a hysteretic capillary saturation function and simulated unstable finger flow through layered soils. Berentsen et al.  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.
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.  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  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.
 A much simpler approach to simulate dynamic nonequilibrium in unsaturated flow was proposed by Ross and Smettem . 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.  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.  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  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  model might not be appropriate for any type of heterogeneity although the general approach to decouple water content from pressure head appears promising.
 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  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  approach. The new model is compared to the Richards equation, the Ross and Smettem  model, and the dual-porosity model of Philip . 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.
2. Materials and Methods
2.1. Governing Equations and Numerical Solution
2.1.1. Richards Equation
 Variably saturated water flow in homogeneous, rigid porous media is usually described by the Richards equation. In one spatial dimension it is written as
where [L3L–3] is the water content, t [T] is time, z [L] is the vertical coordinate, positive downward, h [L] is the pressure head and K [L T–1] is the hydraulic conductivity as function of the pressure head. Equation (2) intrinsically assumes an infinite air permeability at all water contents and local equilibrium between the state variables water content, pressure head and hydraulic conductivity. The constitutive relationships are given by characteristic curves, i.e., the retention curve, , and the conductivity curve, K(h). For monotonic drainage processes, as occurring in MSO experiments, the retention curve is nonhysteretic and typically expressed by a simple parametric function. It is immediately clear that nonequilibrium phenomena observed in MSO experiments, i.e., changing water contents at constant pressure heads, cannot be described by this approach irrespective of the selected parameterization of the constitutive relationships. Therefore we will restrict our analysis in this article to the parameterization of van Genuchten  which is described in section 2.2.
2.1.2. Ross and Smettem Nonequilibrium Model
 To model nonequilibrium effects in water flow, Ross and Smettem  proposed to extend the tight coupling of water content and pressure head as expressed by the soil water retention curve by a kinetic expression. They specified the left hand side of equation (2) through the differential equation
where [L3L–3] is the equilibrium water content, [L3L–3] is the actual water content, and [T ] is an equilibration time constant. Note that is neither identical nor the reciprocal of the material coefficient [T–1] in the model of Hassanizadeh and Gray  given by equation (1). Substituting equation (4) into equation (3), Ross and Smettem  proposed to approximate the actual water content at every time step during the numerical solution by
where k + 1 and k represent the new and old time levels in the numerical time discretization and is the length of the time step. The equilibrium water content at the new time level is calculated according to the equilibrium retention curve. As pointed out by Ross and Smettem , this is one of the simplest models for a kinetic approach to equilibrium.
2.1.3. Philip's Dual-Porosity Model
 To describe water movement in aggregated media, Philip  assumed that the soil consists of two domains: a fracture, macropore or interaggregate domain, and a matrix or intra-aggregate domain. Water flow occurs only in the fracture domain and the matrix domain represents immobile water which is exchanged with the fracture domain [Simunek et al., 2003]. The total water content of the soil is equal to the sum of the two water contents in the domains:
where [L3L–3] is the mobile water content (water content in fracture) and [L3L–3] is the immobile water content. The dual-porosity formulation is given by
where [T–1] is the mass transfer rate for water between the interaggregate and the intra-aggregate pores specified as
where is a first-order rate coefficient [T–1] and (dimensionless) and (dimensionless) are the relative saturations of the fracture and matrix domain, respectively. The two relative saturations are given by
where [L3L–3] and [L3L–3] are the saturated water contents in the fracture and in the matrix domain, respectively and their sum is equal to the saturated water content [L3L–3], which is routinely measured in the laboratory. [L3L–3] and [L3L–3] are the residual water content in the fracture and in the matrix, respectively. The dual-porosity model (DPM) model needs three more parameters compared to the Richards equation: , , and . During inverse modeling, the following two linear nonequality constraints have to be accounted for
This requires a nonlinear optimization software which can handle linear nonequality constraints if residual and saturated water contents are estimated simultaneously for one domain. During parameter estimation, the number of degrees of freedom can be reduced by one if one assumes that the residual water content in the fracture domain equals zero [Simunek et al., 2001]. In this case the residual water content of the soil is attributed to the matrix domain only. Note that this reduction in the number of estimated parameters can only be achieved by an additional assumption on the structure of the model which is not necessarily valid in all cases.
2.1.4. New Dual-Fraction Nonequilibrium Model
 For the dual-fraction nonequilibrium (DNE) model proposed in this paper, we follow the usual assumptions for the validity of Richards' equation. The porous medium is homogeneous, its hydraulic properties are described by one equilibrium retention and one conductivity curve, respectively, and water flow is nonhysteretic. However, similar to the two-site concept in solute transport [Cameron and Klute, 1977], there are 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. Following Gerke and van Genuchten [1993a], we formally describe the water flow of the two fractions by two Richards equations:
where the pressure heads and water contents associated with each soil domain are denoted by heq, hne, and , respectively. is the mass transfer term for water [T−1] and fne (dimensionless) is a volumetric weighting factor for the nonequilibrium water fraction given by . In equation (12), and heq are coupled tightly through the equilibrium retention curve, whereas in equation (13), hne and are two independent variables, which are coupled by the kinetic approach of Ross and Smettem . This implies that in the nonequilibrium fraction, the water content will change according to equation (5), independent of the pressure head hne. By assuming that the pressure heads in the two regions equilibrate quickly relative to the movement of water in the main flow direction, equations (12) and (13) can be combined to yield a single flow equation [Dykhuizen, 1987; Peters and Klavetter, 1988; Gerke and van Genuchten, 1993b] reading
2.1.5. Conceptual Differences Between DPM and DNE Models
 Although the DPM and the DNE models have very similar mathematical structure, they differ conceptually. In the DPM model, a part of the total water is regarded as immobile. Water flow is restricted to the fractures (interaggregate or macropores), and water in the matrix (intra-aggregate pores or the rock matrix) does not move at all. Thus, intra-aggregate pores represent immobile pockets that can exchange, retain and store water, but do not permit convective flow [Simunek et al., 2008]. Whereas the concept of mobile-immobile domains has long been applied to solute transport studies [van Genuchten and Wierenga, 1976], it has found limited use in water flow models. The conceptual approach appears particularly suited for transport in media with distinct bimodal porosities, or media that exhibit preferential water flow. However, a discrimination between a matrix and a fracture domain appears questionable for describing nonequilibrium flow in nonaggregated, well-sorted materials which has been reported in the literature and will be treated in section 3. Another conceptual difficulty of the DPM model is that there is no explicit pressure head in the matrix domain. While this poses no conceptual problem for flow modeling, the question arises how to link measured pressure heads to state variables predicted by the model. In principle, the pressure head in the fracture domain could be used but it appears questionable whether a measurement device like a tensiometer will exclusively measure the pressure head in the fracture domain.
 The DNE model is not intended to describe preferential flow phenomena, but focuses on local nonequilibrium between pressure head and total water content that can be caused by the variety of processes discussed in the introduction. Since fne can vary from zero to unity, the model is able to cover the whole range of flow behavior, ranging from equilibrium flow described by the Richards equation (fne = 0) to full nonequilibrium flow given by the Ross and Smettem  model (fne = 1). In case of the DPM, equilibrium flow according to the Richards equation can be simulated if both and are set to zero. However, the case of full nonequilibrium flow where the entire flow domain is assumed to be at nonequilibrium cannot be treated, because water flow is no longer possible if the saturated water content of the matrix approaches the total saturated water content , irrespective of the value of the exchange parameter .
 A further difference of the approaches lies in the number of nonequilibrium model parameters, which is 3 for DPM and 2 for DNE. As discussed earlier, the number of degrees of freedom during parameter estimation using the DPM model can be reduced by setting the residual water content in the fracture domain to zero. However, since this assumption is no a priori justified for all soil types and boundary conditions, all three nonequilibrium parameters of the DPM model should be fitted during parameter estimation in order to explore the full range of nonequilibrium conditions during unsaturated flow.
2.2. Parameterization of Hydraulic Properties
 The unsaturated soil hydraulic properties were parameterized with the van Genuchten–Mualem (VGM) model [van Genuchten, 1980]. The retention function and the hydraulic conductivity function are given by
where , and [L3L–3] are saturated and residual water contents, respectively, [L–1], n (dimensionless), m (dimensionless) and l (dimensionless) are shape parameters, , and .
2.3. Inverse Modeling, Uncertainty Analysis, and Model Selection
 Parameter estimation and uncertainty analysis was carried out in a Bayesian framework using a robust Markov chain Monte Carlo (MCMC) sampler. MCMC methods have been used for inverse modeling of MSO experiments before by Laloy et al. . In Bayesian statistics, the model parameters are treated as random variables and their uncertainty is characterized by means of their joint posterior probability density function (pdf). The posterior density is given by Bayes' theorem and is written as [Gelman et al., 2004]
where is the likelihood function, is the prior distribution of the model parameters, and is a normalizing factor which can be interpreted as the likelihood to observe y but which in equation (17) primarily ensures that the posterior pdf integrates to unity. The likelihood function quantifies the probability of observing the measurement vector y given the vector of model parameters p. In this study, two different data types are used for conditioning the model parameters: cumulative outflow across the lower boundary of the soil column, Q (cm), and pressure head inside the column measured by a tensiometer, h (cm). By assuming independent and normally distributed errors with zero mean, the log likelihood is equal to the one used by Wöhling and Vrugt :
where mQ and mh are the respective number of observations for each data tape, Qi and hi are the observed outflow and pressure head data, and are model-predicted outflow and pressure head data, and are the standard deviations of the error of both data types, and C1 is a constant summarizing all terms which do not depend on p. This constant can be neglected during MCMC sampling because only differences in log likelihoods occur in the Metropolis et al.  algorithm. If one assumes that the two error standard deviations and are known, the log likelihood can be further simplified to
where C2 summarizes all terms which are independent of p and can therefore be ignored.
 The most frequently applied method to characterize the joint posterior density of the model parameters defined by equations (17) and (19) in statistical terms is to sample from it using MCMC techniques [Gelman et al., 2004; Gilks et al., 1996]. The advantage of using sampling techniques to approximate is that the uncertainty of the model parameters and the model predictions can be quantified accurately without the necessity to linearize the model at the mode of the posterior [Vrugt and Bouten, 2002; Laloy et al., 2010].
 We implemented a parallelized version of the DRAM algorithm [Haario et al., 2006] and ran it using four independent chains for all data sets and models tested. Each Markov chain was run on one core of a quadcore PC. A uniform prior distribution was selected for all model parameters because this reflects our lack of knowledge about them prior to carrying out the column experiments. The prior was bounded by predefined minimum and maximum admissible values for each model parameter. For the DPM model, the constraints defined by equations (11a) and (11b) were accounted for by setting the posterior to zero whenever at least one of the constraints was violated. The error standard deviations and were determined by fitting the process models (Richards, Ross and Smettem, DPM and DNE) to the experimental data. This was done by minimizing twice the negative log likelihood given by equation (19)with the shuffled complex evolution (SCE-UA) algorithm of Duan et al.  and computing the maximum likelihood estimates of the error variances of both data types as
where n is the number of estimated parameters. As all process models resulted in different variance estimates, we decided to use the smallest variance estimates (obtained from the DNE model) among all process models for all MCMC simulations.
 The covariance matrix of the jumping distribution in DRAM was scaled by the factor 5.76/n for the stage 1 proposals (adaptive Metropolis, AM) and that of the delayed rejection step was scaled by an additional factor of 0.01 following the suggestion of Haario et al. . Convergence of the four Markov chains was monitored by the scale reduction factor R of Gelman and Rubin  and sampling was stopped after R reached values smaller or equal to 1.1 for all estimated model parameters. We ran the DRAM algorithm long enough to guarantee that a total number of at least 5000 parameter samples was generated. Since we discarded the first half of each chain as burn-in fraction and used only every fourth of the remaining samples for statistical inference, the length of each of the four chain was at least 10,000.
 After sampling with the DRAM algorithm we inspected the posterior pdf of the estimated parameters visually using scatterplots. Quantile-quantile plots were used to check for normality of the marginal densities. The empirical covariance and correlation matrices were computed to analyze the cross correlation of the parameters. The posterior marginal distributions of the parameters were summarized in terms of the standard deviation and the 2.5, 16, 25, 50, 75, 84 and 97.5 percentiles. The 95% uncertainty intervals of the soil hydraulic functions were computed from the parameter samples. This implies that the intervals account for the cross correlation of the parameters defining θ(h) and K(h). The 95% prediction intervals were calculated from the model simulations corresponding to the parameter samples and by drawing samples from the posterior predictive distribution as described by Gelman et al.  and Iden and Durner .
 Quantitative model comparison was based on the root-mean-square error (RMSE) between observations and model predictions and three model selection criteria. The RMSE was calculated separately for cumulative outflow and pressure head data as
 As an increase in the number of estimated parameters leads in most cases to an increased goodness of fit and therefore to smaller values of the RMSE, model selection criteria have been developed which seek to find a balance between model complexity and quality of the fit [Burnham and Anderson, 2002; Ye et al., 2008; Ward, 2008]. In order to additionally account for the number of estimated model parameters, we computed the Akaike information criterion (AIC) for all models as [Ye et al., 2008]
where stands for the mode of the posterior (the maximum likelihood estimate) of the vector of model parameters. The Bayesian information criterion (BIC) was computed as [Ye et al., 2008]
 The most frequently applied Bayesian measure of complexity and fit is the deviance information criterion (DIC) defined as [Spiegelhalter et al., 2002]
where the Bayesian deviance is twice the negative logarithm of the posterior, and pD is the Bayesian complexity or effective number of model parameters defined as
In equations (24) and (25), denotes the mean deviance and is the deviance of the mean. An application of DIC for model selection in environmental modeling is presented by Iden et al. .
 All three information criteria have in common that the best model from a set of competing candidates is indicated by a minimum value of the criteria. Any comparison of models must be based on differences in values between competing models. This is the reason why the constant occurring in equation (19) can be neglected in model selection. Burnham and Anderson  stress that small differences in AIC between different models may be insignificant and as a rule of thumb suggest that models with a ΔAIC (defined in relation to the model with smallest AIC) of 0–2 have substantial support and that those with ΔAIC greater than 10 should be excluded from a further analysis because they fail substantially to describe at least part of the data.
2.4. Inverse Modeling Using a Virtual Experiment and the New Nonequilibrium Model
 In order to assess the principal feasibility of an inverse estimation of hydraulic nonequilibrium parameters for the new DNE model, extended multistep outflow experiments (XMSO) [Durner and Iden, 2011] were simulated to generate synthetic data. The XMSO method combines a saturated percolation and an unsaturated MSO experiment in one single experiment. This is achieved by starting the experiment with a small amount of water ponding on top of the soil column and lowering the pressure head at the bottom to zero to induce a saturated percolation as in a classic falling-head experiment. As shown by Durner and Iden , XMSO increases the estimation accuracy of the saturated hydraulic conductivity of the soil close to water saturation, as compared to MSO experiments, and allows to estimate simultaneously the saturated hydraulic conductivities of soil and porous plate.
 We used a modified code of the Hydrus-1D code [Simunek et al., 2008], and assumed a vertical soil column of 7.2 cm length, which is equal to the column length used for the real experiments (section 2.5). The distance between two nodes of the finite element mesh was 0.05 cm and the resulting mass balance error was smaller than 0.1% for all simulations. The soil hydraulic properties were parameterized with the VGM model given by equations (15) and (16). A loamy sand was selected from the Hydrus-1D soil catalogue to define the model parameters, which are given in Table 1. A hydrostatic pressure head distribution with a pressure head equal to +9.2 cm at the bottom of the soil column was used as initial condition. At the top of the column, an atmospheric boundary condition with surface layer and zero precipitation and evaporation was specified and at the bottom a variable head boundary condition was used, with a pressure head that decreased in six steps from +9.2 to −30 cm within 100 h.
Table 1. True and Estimated (Mode of Posterior) Parameters for the Synthetic Data Sets, Root-Mean-Square Error for Both Data Types in the Likelihood Function, and the Empirical Correlation Coefficient r Between τ and fnea
Values in parentheses represent standard deviations of the parameters computed using the samples from the posterior probability density function. For parameters α, n, and Ks the given standard errors refer to , , and , respectively. The saturated water content was kept constant at 0.41 cm3 cm−3. RMSE, root-mean-square error.
True nonequilibrium parameters used for generating the synthetic data.
True parameters of the VGM model.
 To develop an understanding of the effects of the nonequilibrium parameters, various combinations of parameters fne and were used in the simulation of the virtual experiments (Table 1). We generated cumulative outflow data across the bottom and pressure head data for a depth of 1.8 cm measured from the top of the column. These data were perturbed with normally distributed noise with zero expectation and standard deviations of 0.005 cm for cumulative outflow and 0.25 cm for pressure head, respectively. These values reflect the typical measurement accuracy for the multistep outflow device in our lab. For each data type, 940 synthetic observations were generated. Parameter estimation was carried out using the Bayesian technique outlined in section 2.3. The error standard deviations occurring in equation (19) were set to the values which were used for the perturbation of the synthetic data. The joint posterior pdf of the parameters , α, n, Ks, l, fne, and was approximated by MCMC sampling using DRAM. The saturated water content was kept constant at its known value. This is common practice in multistep outflow experiments because the cumulative outflow provides information on changes in water content only and not on absolute values of [Durner et al., 1999].
2.5. Inverse Modeling of Laboratory XMSO Experiments
 The proposed nonequilibrium model was tested against experimental data from two soil samples. The first sample (S) consisted of a packed sandy soil in a column of 9.4 cm in diameter and 7.2 cm in height, collected from a depth of about 100 cm at a field site near Hannover, Germany. The column was placed on a 0.7 cm thick porous plate with an air entry value of −50 cm. The second sample (L) was an undisturbed loamy sand with the same dimensions as the sand sample, which was collected from the subsoil horizon (35–45 cm) of a Luvisol at a field site close to the city of Braunschweig, Germany [Schelle et al., 2011]. The soil column was placed on a 0.7 cm thick porous plate that was covered by a fine-pored diaphragm. The latter prevented air entry when lowering pressure head during the XMSO experiment below the air entry pressure of the porous plate. Both samples were saturated slowly from the bottom. The initial water content was calculated from the water content at the end of the experiment and the cumulative water loss throughout the experiment. The bulk density, porosity, and initial water content data are given in Table 2.
Table 2. Measured Bulk Densities ρb, Porosities ϕ, Saturated Water Contents θs, and Saturated Hydraulic Conductivities Ks for the Two Soil Samples
Kp denotes the saturated hydraulic conductivity of the porous plate (sample S) and the combination of plate and membrane (sample L).
 As in the case of the virtual experiment, we used the XMSO method as experiment type. For the sand S, the pressure heads applied at the porous plate were 9.2, 0, −5, −10, −15, −20, −25, −30, and −35 cm, whereas for the soil L the pressure heads applied were 10.6, 0, −10, −20, −30, −40, −60, −80, and −100 cm. A tensiometer (T5, UMS Munich) was installed into the soil columns at 1.8 cm depth measured from the top to record the pressure head. As recommended by Durner and Iden , we first separated the data from the saturated percolation part of the XMSO experiments and estimated the saturated conductivities of the soil and the plate/membrane simultaneously by fitting the analytical solution of the saturated flow equation for the composite medium to the measured data by minimizing the nonlinear weighted least squares objective function given by equation (19). In the following inverse simulations, both saturated conductivities were fixed to the resulting values. The saturated hydraulic conductivities of the porous plate (sample S), the combination of plate and membrane (sample L), and the saturated hydraulic conductivity of both soils are given in Table 2.
 Parameter estimation was carried out for both soil samples (S and L) and all models (Richards, Ross and Smettem, DPM, and DNE) using the Bayesian technique outlined in section 2.3. The observed outflow and pressure head data used for the inverse simulations were 294 for each data group, respectively. Table 3 summarizes which parameters were estimated for the different models, which ones kept constant, the total number of model parameters, and the number of estimated parameters (degrees of freedom). The admissible range for all parameters as used for the bounded uniform prior distribution is summarized in Table 4. In case of the DPM model we distinguish two cases. In case 1, referred to as DPM I, the residual water content in the fracture domain is assumed to be zero [Simunek et al., 2001]. As the saturated water content of the soil was determined in the laboratory, the estimated value of the saturated water content in the matrix domain defines the saturated water content in the fracture domain by equation (6). The second case, referred to as DPM II, does not rely on the assumption that equals zero and therefore involves the estimation of one additional parameter.
Table 3. Overview of the Different Process Models Fitted to the Experimental Data, the Estimated Parameters, the Parameters Kept at Constant Values, the Total Number of Parameters, and the Number of Estimated Parameters (Degrees of Freedom)a
Table 4. Minimum and Maximum Parameter Values Defining the Bounded Uniform Prior Distribution Used for Bayesian Inferencea
The parameter transformation used during parameter sampling by MCMC is given in the second column, and 1og10 denotes the decimal logarithm.
θr (cm3 cm−3)
1og10(n − 1)
Ks (cm h−1)
3. Results and Discussion
3.1. Sensitivity Analysis for Nonequilibrium Parameters
Figure 1 illustrates the influence of the nonequilibrium parameter on the outflow patterns and the pressure head dynamics in the MSO experiment (loamy sand), as simulated with the Ross-Smettem model. The first case ( = 0 h) represents equilibrium flow and therefore corresponds to a solution of the Richards equation. The other cases ( = 1, 2, and 3 h) represent nonequilibrium flow. It becomes obvious that the model of Ross and Smettem  is able to reproduce the basic phenomenon that after an applied pressure step, tensiometer readings reach the new equilibrium level relatively quickly, whereas outflow of water continues. Note that the pressure head reaction to changes in the boundary condition is slightly faster for the Ross-Smettem model compared to the equilibrium model. Moreover, the time scale of equilibration which becomes evident in the drainage data is equal for all applied pressure steps. Contrary to our experimental observations (shown later), the water content dynamics during drainage appear to follow an exponential function.
 In MSO experiments with nonequilibrium flow, we find typically a different picture. After a pressure change, a certain amount of water drains quickly and only after this, drainage becomes significantly slower. As illustrated by the dashed lines in Figure 2, this outflow dynamics can be captured by the nonequilibrium model suggested in section 2.1.4 (equation (14)). The two solid bold lines are copied from Figure 1 and represent the limiting cases of the new model, namely full equilibrium flow simulated by the Richards equation (blue) and nonequilibrium flow simulated by the model of Ross and Smettem  with = 3 h (red, entire domain at dynamic nonequilibrium). The first blue dashed line ( = 3 h and fne = 0.25) depicts the drainage of a porous medium in which the domain that is in equilibrium (Richards-type flow) contributes with a factor of 0.75 to the overall flow and the domain that represents the nonequilibrium flow contributes with a factor of 0.25. Two further curves are shown for fne equal 0.5 and 0.75. Evidently, the new model offers greater flexibility and the capability to describe intermediate types of flow, ranging from full equilibrium to complete nonequilibrium flow.
3.2. Inverse Modeling Using Synthetic Data
 The simultaneous estimation of the equilibrium hydraulic functions and the two nonequilibrium parameters by inverse modeling was tested using the synthetic data sets, with the aim to assess the identifiability of both, the hydraulic properties and the nonequilibrium parameters. The inverse modeling leads to a perfect match of the “observed” data for all the combinations between nonequilibrium parameter and parameter fne. This is indicated by the RMSE values of cumulative outflow and pressure head data summarized in Table 1, which are identical to the standard deviations of the normal distribution used for disturbing the data. In addition, the true parameter values are hit in all cases, which indicates an unbiased estimation of soil hydraulic and nonequilibrium flow parameters. Inspection of the scatterplots of the parameter samples and the quantile-quantile plots of the marginals indicated that the joint posterior was well behaved with a single mode and followed a multinormal distribution closely. The standard deviations of the parameters given in Table 1 indicate a very high precision of the estimates. Furthermore, the correlation coefficient between the two nonequilibrium parameters and fne ranged from −0.8 to −0.5. The correlation coefficients between the two nonequilibrium parameters and the parameters of the VGM model which define the equilibrium soil hydraulic functions ranged from −0.51 to 0.50. This indicates a moderate level of parameter interaction for all treated cases. We conclude that the unique inverse identification of all model parameters of the new DNE model from XMSO experiments is feasible.
3.3. Inverse Modeling Using Real Data
3.3.1. Packed Sand
Figure 3 shows the measured and simulated outflow and pressure head data for the XMSO experiment conducted on sample S. The depicted fits are obtained with the Richards equation, the Ross-Smettem model, the DPM with the residual water content in the fracture domain set to zero (DPM I), the DPM model where the residual water content in the fracture domain was fitted (DPM II), and the new DNE model. To emphasize the deviations between fits and measurements, the residuals between observed and calculated values for all models are plotted in Figure 3 (bottom). In Figure 3 (and also in Figure 5), we do not show the outflow data from the initial phase of the experiment where saturated percolation took place, in order to enhance the visibility of the outflow dynamics during the unsaturated MSO part of the experiment. Thus, the corresponding axis begins at a value of 1.0 cm.
 The data in Figure 3 show that after every pressure step, a quick equilibration of the pressure head in the column takes place. A large fraction of water drains quickly from the column directly after each pressure step, and this is followed by a phase with continuing outflow, which ceases only slowly. The comparison of the experimental and fitted outflow curve illustrates that Richards' equation cannot adequately describe these experimental data because it assumes instantaneous local equilibrium between pressure head and water content. The Ross and Smettem nonequilibrium model cannot improve the fit significantly, in particular for the outflow data (Table 5), which appears surprising at first view. The reason for this can be understood by studying Figure 4, which is a magnification of the third pressure step that was selected here because it is the most illustrative part. Figure 4 shows the three fitted curves also shown in Figure 3 and three more forward simulations. For these simulations, all soil hydraulic parameters were kept at the optimized values (obtained with the Ross-Smettem model) and only the equilibration parameter was varied. This blowup shows that the Ross-Smettem model will always approach the equilibrium water content in a smooth manner resembling an exponential function. The outflow dynamics observed in the experiment, however, is structurally different from this. The fit of the Ross-Smettem model leads accordingly to a trade-off between a too slowly simulated outflow in the early phase and an overly high outflow rate in the late phase. The misfit cannot be reduced by altering the parameter determining the equilibration time, as indicated by the dotted lines in Figure 4. Contrary to that, the DNE model and the DPM model match the experimental data very well (Figure 3). It is noteworthy that both versions of the DPM model (7 and 8 degrees of freedom) describe the data equally well.
Table 5. Estimated Parameters (Mode of Posterior) for Both Soils Using the Richards Equation, the Ross and Smettem Model, the DPM, and the New DNE Modela
Here and Ks were kept constant for all simulations at 0.298 cm3 cm–3 and 30.1 cm h−1 for sample S, respectively, and 0.334 cm3 cm−3 and 2.7 cm h−1 for sample L, respectively. For the DPM model, parameter is given instead of . The same holds for the saturated water content in the matrix domain , which replaces parameter fne. The values in parentheses represent the standard deviations of the estimated parameters inferred from the samples of the posterior. For parameters , n, , and the given standard deviations refer to , , , and , respectively.
The values given are either for τ (DNE and Ross and Smettem model) or for ω (DPM models). The (h/h−1) in the column heading indicates the unit for the τ variable is hours (h) (DNE and Ross and Smettem model) and the unit for the ω variable is h−1 (DPM models).
Similarly, the values of fne (DNE model) are dimensionless (−) and those of (DPM models) are in units of cm3 cm−3.
Ross and Smettem
Ross and Smettem
 The upper half of Table 5 shows the estimated parameter values and their standard deviations for all five process models. The nonequilibrium parameters of the new model (DNE) are = 0.71 h and fne = 0.51, with a moderate negative correlation coefficient between and fne of r= −0.56. The correlation coefficients of the two nonequilibrium parameters with the parameters of the VGM model ranged from −0.56 to 0.19 indicating a very moderate cross correlation between the nonequilibrium and equilibrium parameters.
 Interestingly, accounting for nonequilibrium in the inverse identification does not influence the shape of the equilibrium retention function markedly. This is evident from the VGM parameters (Table 5) and is illustrated in Figure 5a, which compares the identified retention curves for the 5 different models (the details of the two DPM curves will be discussed later in this section). The differences in the hydraulic conductivity curve are also small (Figure 5b) indicating that for the sand soil the estimated functions are almost the same irrespective of whether the Richards equation or the DNE model are used. The 95% Bayesian uncertainty intervals depicted in Figures 5a and 5b summarize the joint marginal density of the parameters defining the equilibrium constitutive relationships in an effective way and account for the parameter correlation of the samples from the posterior. It becomes obvious that the soil hydraulic properties can be identified precisely in the range of pressure head covered by the MSO experiment for all three models (Richards, Ross and Smettem, and DNE).
 The two retention curves fitted using the DPM model (7 and 8 degrees of freedom) differ substantially from the other curves and furthermore differ from each other (Figure 5a). The DPM models assume that only the 41% (DPM I) or 50% (DPM II) of the total water content belongs to the mobile (fracture) region. These percentages have been calculated by dividing the mobile saturated water content by the total water content . It appears unlikely that this could be the case for a well sorted material like the packed sand sample. Moreover, for the DPM II model, we find that parameter identification from the MSO outflow problem is ill posed, because there are strong interactions between the parameters , and . This is depicted in Figure 6 which shows scatter plots of the samples drawn from the posterior distribution with the DRAM algorithm for all three possible combinations of the parameters , and . There is a region between 0 and 0.05 cm3 cm–3 for parameters and in which the DPM II model gives almost identical results (Figure 6a). As a result the two parameters are highly correlated (r = −0.93). The same holds for the parameter pairs ( , Figure 6b) and (r = 0.92, Figure 6c). Obviously, there is no unique combination of model parameters which leads to a best fit solution but a large variety of parameter combinations leads to an almost identical good fit of the experimental data. This result is confirmed by the 95% uncertainty interval of the identified soil water retention curves depicted in Figure 5a. Figure 5a also illustrates that the observed uniqueness problem is not restricted to the DPM II model but also occurs for the DPM I model because the identified soil water retention function suffers from considerable uncertainty compared to the ones obtained with the Richards equation, the Ross and Smettem and the DNE model.
 The RMSE values for both cumulative outflow and pressure head data along with the calculated model selection criteria AIC, BIC, and DIC for all models are given in Table 6. According to the RMSE values the DPM I, DPM II, and DNE models describe the experimental data for the sandy soil equally well. As the DNE model yields a minimum value of all three model selection criteria applied (AIC, BIC, and DIC), it is superior to all models including both variants of the DPM model for describing the experimental data.
Table 6. RMSE, AIC, BIC, and DIC Values Calculated for Both Soils Using the Richards Equation, the Ross and Smettem Model, the Philip  Model (DPM), and the New Model (DNE)a
For explanation of the different metrics, see the main text. AIC, Akaike information criterion; BIC, Bayesian information criterion; DIC, deviance information criterion.
Ross and Smettem
Ross and Smettem
3.3.2. Undisturbed Loamy Sand
Figure 7 shows the experimental data and inverse simulation results for sample L, in an analogous manner as Figure 3 for the S sample. In this finer textured soil, the observed equilibration of the cumulative outflow is much slower than that of the pressure head. For example, for the third step (30.8 h < t < 36.8 h) the pressure head equilibrated 0.6 h after the pressure change, at t = 31.4 h, whereas the soil column continued to release water even after 5.4 h, when the next pressure step at the lower boundary was applied. Simulations with the Richards equation and the Ross-Smettem model both yielded a poor description of the outflow data.
 The new model describes both the cumulative outflow and the observed pressure head data very well. It matches the quick equilibration time of the pressure heads, the quick outflow phase occurring directly after every pressure head change, and the following smooth outflow phase. The calculated nonequilibrium parameters were = 3.2 h and fne = 0.78, with a small negative parameter correlation indicated by r = −0.41. The correlation coefficients of the two nonequilibrium parameters with the parameters of the VGM model ranged from −0.65 to 0.64. Although this is slightly higher than for sample S, it still indicates a moderate cross correlation between the nonequilibrium and equilibrium parameters. The RMSE values given in Table 6 confirm a good fit of the DNE model to the observations.
Figures 8a and 8b show a comparison of the hydraulic functions obtained by the inverse simulations, in analogy to Figure 5. The DPM models assume that 17% (DPM I) or 33% (DPM II) of the total water belongs to the mobile region.
Figure 9 shows scatter plots for parameter , , and samples calculated by the DRAM algorithm for the DPM II model. As was the case for sample S, the scatter plots indicate a strong correlation between the parameters which leads to a nonuniqueness of the inverse problem. The correlation coefficient between and was −0.98, that between and was −0.98, and that between and was 0.97. This nonuniqueness resulted in the relatively high standard deviations of the parameters given in Table 5 and the 95% uncertainty intervals in the retention function depicted in Figure 8a. Again, the nonuniqueness of the inverse problem does not only become obvious for the DPM II model, but also for model DPM I. The comparison of the model selection criteria AIC, BIC and DIC given in Table 6 confirms that the DNE model is superior to both variants of the DPM model for the sandy loam soil.
 The retention functions (Figure 8a) for the remaining models indicate that the estimated water content at a given pressure head value is higher if the equilibrium assumption is used. Inverse simulations with free-formed hydraulic functions [Iden and Durner, 2007] (not shown) indicate that this is independent of the underlying model of the hydraulic functions and reflects the fact that the level of the final outflow (true equilibrium, after a long equilibration time) is systematically underestimated by the equilibrium model. However, the magnitude of the difference between the fitted curves is small. Contrary to that, the differences in the identified hydraulic conductivity curves are considerable (Figure 8b). Since the saturated hydraulic conductivity is determined from the percolation phase of the XMSO experiment, the two conductivity curves cannot deviate from each other near saturation, were the VGM model is not flexible enough to allow a difference. However, the two curves start to deviate significantly at lower pressure heads. In summary, the identified unsaturated hydraulic conductivity curve is overestimated, if the Richards equation is used as process model in cases where dynamic nonequilibrium prevails. However, such a far-reaching conclusion must be based on a wider experimental basis and requires future research.
 We have presented an effective model concept to describe nonequilibrium water flow in variably saturated soils. The model partitions the soil water in two fractions. Water content in one fraction of the flow domain is in direct equilibrium with the pressure head, whereas water in the nonequilibrium fraction is approaching its equilibrium value by a first-order kinetic according to the Ross and Smettem  approach. The proposed model needs two additional parameters compared to the Richards equation. One parameter, fne (dimensionless), describes the volume fraction of water where nonequilibrium between water content and pressure head prevails, and the second parameter, [T ], quantifies the equilibration kinetics in that region. The new model includes the Richards and the Ross and Smettem  models as limiting cases.
 The feasibility of identifying the nonequilibrium parameters correctly from multistep outflow experimental data by inverse modeling was investigated with synthetic data perturbed with noise. The results demonstrated that the simultaneous estimation of equilibrium and nonequilibrium parameters from MSO experiments is possible. The obtained parameter estimates were unbiased and the two estimated nonequilibrium parameters showed an acceptably small degree of correlation with each other and with the parameters defining the equilibrium constitutive relationships.
 The new model was tested using MSO data sets from two real soils. Inverse modeling results using Bayesian MCMC sampling lead to an excellent agreement between observations and model predictions. This was not the case for the Richards equation and the nonequilibrium model of Ross and Smettem . The Philip  dual-porosity model described the data equally well as the DNE. However, as this model requires more parameters than the DNE model, model selection based on the three criteria AIC, BIC, and DIC favored the new DNE model for both soil samples. An important disadvantage of the Philip  model was that it suffered from pronounced parameter correlations which lead to considerable uncertainty in the estimated soil water retention curves. Furthermore, our new model circumvents the conceptual problems of the Philip  model given by the difficult interpretation of the soil hydraulic properties and the lack of an explicit treatment of the pressure head in the matrix domain.
 Experimental evidence [Schultze et al., 1999] suggests that the equilibration kinetics of the water content is not the same for the whole range of pressure head, as it was assumed in this article. Furthermore, the kinetics introduced by Ross and Smettem  is a simplified way to describe the equilibration of water content and one cannot expect that this will always hold. Equilibration expressions that consider such dependencies can be easily included in the new conceptual model.
 Future research will be directed toward studies of the nonequilibrium flow in combined MSO-Evaporation experiments [Schelle et al., 2011] and investigations on the possible need to incorporate a water content or pressure head dependence of the nonequilibrium parameters and/or fne. Further work is also needed to assess the influence of nonequilibrium water flow on estimates of equilibrium soil hydraulic properties from dynamic outflow experiments. Moreover, we need to learn under what conditions and for what soils can we assume a Richards type of flow, and when we need to consider hydraulic nonequilibrium in variably saturated water flow studies.
 We thank Birgit Walter for her careful assistance during the experimental work and Dr. Henrike Schelle for providing us with the experimental data of soil L. This study was financially supported by the NTH Project “Hydraulic Processes and Properties of Partially Hydrophobic Soils.”