The soil parameters that influence water flow and contaminant transport in unsaturated zones are not generally known a priori and have to be estimated by fitting model responses to observed data. The unsaturated soil hydraulic parameters can be (more or less accurately) estimated from dynamic flow experiments (e.g., Hopmans et al., 2002; Vrugt et al., 2003a; Durner and Iden, 2011; Younes et al., 2013). Several authors have investigated different types of transient experiments and boundary conditions suited for a reliable estimation of soil hydraulic properties (e.g., van Dam et al., 1994; Šimůnek and van Genuchten, 1997; Inoue et al., 1998; Durner et al., 1999). Soil hydraulic properties are often estimated using inversion of one-step (Kool et al., 1985; van Dam et al., 1992) or multistep (Eching et al., 1994; van Dam et al., 1994) outflow experiments or controlled infiltration experiments (Hudson et al., 1996).

Kool et al. (1985) and Kool and Parker (1988) suggested that the transient experiments should cover a wide range of water contents to obtain a reliable estimation of the parameters. Van Dam et al. (1994) have shown that more reliable parameter estimates are obtained by increasing the pneumatic pressure in several steps instead of a single step. The multistep outflow experiments are the most popular laboratory methods (e.g., Eching and Hopmans, 1993; Eching et al., 1994; van Dam et al., 1994; Hopmans et al., 2002). However, their application is limited by expensive measurement equipment (Nasta et al., 2011).

Infiltration experiments have been investigated by Mishra and Parker (1989) to study the reliability of hydraulic- and transport-estimated parameters for a soil column of 200 cm using measurements of water content, concentration and water pressure inside the column. They showed that the simultaneous estimation of hydraulic and transport properties yields smaller estimation errors for model parameters than the sequential inversion of hydraulic properties from the water content and/or pressure head followed by the inversion of transport properties from concentration data (Mishra and Parker, 1989).

Inoue et al. (2000) performed infiltration experiments using a soil column of 30 cm. Pressure head and solute concentration were measured at different locations. A constant infiltration rate was applied to the soil surface and a balance was used to measure the cumulative outflow. They showed that both hydraulic and transport parameters can be assessed by the combination of flow and transport experiments.

Furthermore, infiltration experiments were often conducted in lysimeters for pesticide leaching studies. Indeed, lysimeter experiments are generally used to assess the leaching risks of pesticides using soil columns of around 1.2 m depth which is the standard scale for these types of experiments (Mertens et al., 2009; Kahl et al., 2015). Before performing the column leaching experiment, several infiltration–outflow experiments are often realized to estimate the soil hydraulic parameters (Kahl et al., 2015; Dusek et al., 2015).

The key objective of the present study is to evaluate the reliability of different experimental protocols for estimating hydraulic and transport parameters and their associated uncertainties for column experiments. We consider the flow and the transport of an inert solute injected into a hypothetical column filled with a homogeneous sandy clay loam soil. We assume that flow can be modeled by the Richards' equation (RE) and that the solute transport can be simulated by the classical advection–dispersion model. Furthermore, the Mualem and van Genuchten (MvG) models (Mualem, 1976; van Genuchten, 1980) are chosen to describe the retention curve and to relate the hydraulic conductivity of the unsaturated soil to the water content. The estimation of the flow and transport parameters through flow–transport model inversion is investigated for two injection periods of the solute and different data measurement scenarios.

Inverse modeling is often performed using local search algorithms such as the Levenberg–Marquardt algorithm (Marquardt, 1963). The latter is computationally efficient to evaluate the optimal parameter set (Gallagher and Doherty, 2007). Besides, the degree of uncertainty in the estimated parameters, expressed by their confidence intervals, is often calculated using a first-order approximation of the model near its minimum (Carrera and Neuman, 1986; Kool and Parker, 1988). However, as stated by Vrugt and Bouten (2002), parameter interdependence and model nonlinearity occurring in hydrologic models may violate the use of this first approximation to obtain accurate confidence intervals of each parameter. Therefore, in this work, the estimation of hydraulic and transport parameters is performed in a Bayesian framework using the Markov chain Monte Carlo (MCMC) sampler (Vrugt and Bouten, 2002; Vrugt et al., 2008). Unlike classical parameter optimization algorithms, the MCMC approach generates sets of parameter values randomly sampled from the posterior joint probability distributions, which are useful to assess the quality of the estimation. The MCMC samples can be used to summarize parameter uncertainties and to perform predictive uncertainty (Ades and Lu, 2003).

Hypothetical infiltration experiments are considered for a column of 120 cm depth, initially under hydrostatic conditions, free of solute and filled with a homogeneous sandy clay loam soil. Continuous flow and solute injection are performed during a time period Tinj at the top of the column and with a zero-pressure head at the bottom. The unknown parameters for the water flow are the hydraulic parameters: ks (L T-1), the saturated hydraulic conductivity; θs (L3 L-3), the saturated water content; θr (L3 L-3), the residual water content; and α (L-1) and n (-), the MvG shape parameters. The only unknown parameter of the tracer transport is the longitudinal dispersivity, aL(L).

Several scenarios corresponding to different sets of measurements are investigated to address the following questions:

Can we obtain an appropriate estimation of all flow and transport parameters from tracer-infiltration experiments, even though a limited range of water contents is covered (only moderately dry conditions are obtained because of gravity drainage conditions prescribed at the bottom of the soil column)?

For this purpose, synthetic scenarios are considered in the sequel in which data from numerical simulations are used to avoid the uncontrolled noise of experiments that could bias the conclusions.

The paper is organized as follows. The mathematical models describing flow and transport in the unsaturated zone are detailed in Sect. 2. Section 3 describes the MCMC Bayesian parameter estimation procedure used in the DREAM(ZS) sampler. Section 4 presents the different investigated scenarios and discusses the results of the calibration in terms of mean parameter values and uncertainty ranges for each scenario. Conclusions are given in Sect. 5.

We consider a uniform soil profile in the column and an injection of a solute tracer such as bromide, as described in Mertens et al. (2009). The unsaturated water flow in the vertical soil column is modeled with the one-dimensional pressure head form of the RE: ch+SsθθS∂h∂t=∂q∂zq=Kh∂h∂z-1, where h (L) is the pressure head; q (L T-1) is the Darcy velocity; z (L) is the depth, measured as positive in the downward direction; Ss (-) is the specific storage; θ and θs (L3 L-3) are the actual and saturated water contents, respectively; ch (L-1) is the specific moisture capacity; and Kh (L T-1) is the hydraulic conductivity. The latter two parameters are both functions of the pressure head. In this study, the relations between the pressure head, conductivity and water content are described by the following standard models of Mualem (1976) and van Genuchten (1980): Seh=θh-θrθs-θr=11+αhnmh<01h≥0KSe=KsSe1/21-1-Se1/mm2, where Se (-) is the effective saturation, θr (L3 L-3) is the residual water content, Ks (L T-1) is the saturated hydraulic conductivity, and m=1-1/n, α (L-1) and n (-) are the MvG shape parameters.

The tracer transport is governed by the following convection–dispersion equation: ∂θC∂t+∂qC∂z-∂∂zθD∂C∂z=0, where C (M L-3) is the concentration of the tracer, D (L2 T-1) is the dispersion coefficient in which D=alq+dm and al (L) is the dispersivity coefficient of the soil and dm (L2 T-1) is the molecular diffusion coefficient, which is set as 1.04 10-4 cm2 min-1.

The transport Eq. (3) is coupled with the flow Eq. (1) by the water content θ and Darcy's velocity q. The initial conditions are as follows: a hydrostatic pressure distribution with zero-pressure head at the bottom of the column z=L and a solute concentration of zero inside the whole column. An infiltration with a flux qinj of contaminated water with a concentration Cinj is then applied at the upper boundary condition (z= 0) during a period Tinj. Hence, the boundary conditions at the top of the column can be expressed as for0<t≤TinjK∂h∂z-1=qinjθD∂C∂z+qC=qinjCinjfort>TinjK∂h∂z-1=0Cinj=0. A zero-pressure head is maintained at the lower boundary z=L of the column and a zero-concentration gradient is used as the lower boundary condition for the solute transport, namely hz=l=0∂C∂zz=l=0. In the sequel, the infiltration rate and the injected solute concentration are qinj=0.015 cm min-1 and Cinj=1 g cm-3, respectively. The system (Eqs. 1–5) is solved using the standard finite difference method for both flow and transport spatial discretization. A uniform mesh of 600 cells is employed. Temporal discretization is performed with the high-order method of lines (MOL) (e.g., Miller et al., 1998; Tocci et al., 1997; Younes et al., 2009; Fahs et al., 2011). Error checking, robustness, order selection and adaptive time step features, available in sophisticated solvers, are applied to the time integration of partial differential equations (Tocci et al., 1997). The MOL has been successfully used to solve RE in many studies (e.g., Farthing et al., 2003; Miller et al., 2006; Li et al., 2007; Fahs et al., 2009). Details on the use of the MOL for solving RE are described in Fahs et al. (2009).

The vector of unknown parameters that has to be identified by model calibration is ξ=ks,θs,θr,α,n,aL. To analyze the performance of the model calibration procedures, a reference solution is generated by simulating the flow–transport problem (Eqs. 1–5) using the following parameter values (corresponding to a sandy clay loam soil): ks= 50 cm day-1, θs=0.43, θr=0.09, α= 0.04 cm-1, n=1.4 and al=0.2 cm. Four types of variables are extracted from the results of the simulation: the pressure head and water content 5 cm below the top of the column, the cumulative outflow and the solute breakthrough concentration at the outflow of the column. These four data series are modified by adding a normally distributed white noise using the following standard deviations: σh= 1 cm for the pressure head, σθ=0.02 for the water content, σQ=0.1 cm for the cumulative outflow and σC=0.01 g cm-3 for the exit concentration. These perturbations mimic measurement errors and the resulting values of water pressure, water content, cumulative outflow and solute breakthrough concentration are considered as measurements in the following.

The flow–transport model is used to analyze the effects of different measurement sets on parameter identification. For this purpose, we adopt a Bayesian approach that involves the parameter joint posterior distribution (Vrugt et al., 2008). The latter is assessed with the DREAM(ZS) MCMC sampler (Laloy and Vrugt, 2012). This software generates random sequences of parameter sets that asymptotically converge toward the target joint posterior distribution (Gelman et al., 1997). Thus, if the number of runs is sufficiently high, the generated samples can be used to estimate the statistical measures of the posterior distribution, such as the mean and variance, among other measures.

The Bayes theorem states that the probability density function of the model parameters conditioned onto data can be expressed as pξ|ymes∝pymes|ξpξ, where pξ|ymes is the likelihood function measuring how well the model fits the observations ymes, and pξ is the prior information about the parameter before the observations are made. Independent uniform priors within the ranges reported in Table 1 are chosen. In this work, a Gaussian distribution defines the likelihood function because the observations are simulated and corrupted with Gaussian errors. Hence, the parameter posterior distribution is expressed as pξ|ymes∝exp⁡-SShξ2σh2-SSθξ2σθ2-SSQξ2σQ2-SSCξ2σC2, where SShξ, SSθξ, SSQξ and SSCξ are the sums of the squared differences between the observed and modeled data of the pressure head, water content, cumulative outflow and output concentration, respectively. For instance, SShξ=∑k=1Nhhmesk-hmodkξ2, which includes the observed hmesk and predicted hmodk pressure heads at time tk for the number of pressure head observations Nh.

Bayesian parameter estimation is performed hereafter with the DREAM(ZS) software (Laloy and Vrugt, 2012), which is an efficient MCMC sampler. DREAM(ZS) computes multiple sub-chains in parallel to thoroughly explore the parameter space. Archives of the states of the sub-chains are stored and used to allow a strong reduction of the “burn-in” period in which the sampler generates individuals with poor performances. Taking the last 25 % of individuals of the MCMC (when the chains have converged) yields multiple sets of parameters, ξ, that adequately fit the model onto observations. These sets are then used to estimate the updated parameter distributions, the pairwise parameter correlations and the uncertainty of the model predictions. As suggested in Vrugt et al. (2003b), we consider that the posterior distribution is stationary if the Gelman and Ruban (1992) criterion is ≤1.2.

In this section, the identifiability of the parameters is investigated for seven different scenarios of measurement sets (Table 1). In the first scenario, only measured pressure heads and cumulative outflow are used for the calibration. Scenarios 2 to 5 investigate the benefit of adding measured water contents and/or solute outlet concentrations to pressure heads and outflow. The last scenarios (6, 7) investigate the use of measured cumulative outflow and concentration breakthrough at the column outflow because these measurements do not require intrusive techniques. Scenarios 5 to 7 investigate the effects of solute injection duration on the identifiability of the parameters as well.

In all cases, the MCMC sampler was run with three simultaneous chains for a total number of 50 000 runs. Depending on the scenario, the MCMC required between 5000 and 20 000 model runs to reach convergence and was terminated after 30 000 runs. The last 25 % of the runs that adequately fit the model onto observations are used to estimate the updated probability density function (pdf).

In this work, estimation of hydraulic and transport soil parameters have been investigated using synthetic infiltration experiments performed in a column filled with a sandy clay loam soil, which was subjected to continuous flow and solute injection over a period Tinj.

The saturated hydraulic conductivity, the saturated and residual water contents, the Mualem–van Genuchten shape parameters and the longitudinal dispersivity are estimated in a Bayesian framework using the MCMC sampler. Parameter estimation is performed for different scenarios of data measurements.

The results reveal the following conclusions:

All hydraulic and transport parameters can be appropriately estimated from the described infiltration experiment. However, the accuracy differs and depends on the type of measurement and the duration of the injection Tinj, even if the water content remains close to saturated conditions.

This last point has practical applications for designing simple experimental setups dedicated to the estimation of hydrodynamic and transport parameters for unsaturated flow in soils. The setup has to be appropriately equipped to measure the cumulative water outflow (e.g., weighing machine) and the solute breakthrough at the column outflow (e.g., flow through electrical conductivity). The injection should be stopped as soon as the solute concentration reaches the outflow. The accuracy of the estimation of θr, α and n improves by adding pressure measurements inside the column, close to the injection.

These results are of course related to the models and experimental conditions we used. This work can be extended to different types of soils, water retention and/or relative permeability functions to evaluate the interest of coupling flow and transport for parameter identification. This work can also be extended to reactive solutes.