The quality of statistical calibration of hydraulic and transport soil properties is studied for infiltration experiments in which, over a given period, tracer-contaminated water is injected into an hypothetical column filled with a homogeneous soil. 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 Markov chain Monte Carlo (MCMC) sampler. The impact of the kind of measurement sets (water content, pressure inside the column, cumulative outflow and outlet solute concentration) and that of the solute injection duration is investigated by analyzing the calibrated model parameters and their confidence intervals for different scenarios. The results show that the injection period has a significant effect on the quality of the estimation, in particular, on the posterior uncertainty range of the parameters. All hydraulic and transport parameters of the investigated soil can be well estimated from the experiment using only the outlet concentration and cumulative outflow, which are measured non-intrusively. An improvement of the identifiability of the hydraulic parameters is observed when the pressure data from measurements taken inside the column are also considered in the inversion.

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

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)?

What is the optimal set of measurements for the estimation of all the parameters? Can we use only non-intrusive measurements (cumulative outflow and concentration breakthrough curve) or are intrusive measurements of pressure heads and/or water contents inside the column unavoidable?

Is there an optimal design for the tracer injection?

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

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:

The tracer transport is governed by the following convection–dispersion
equation:

The transport Eq. (3) is coupled with the flow Eq. (1) by the
water content

The vector of unknown parameters that has to be identified by model
calibration is

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

The Bayes theorem states that the probability density function of the model
parameters conditioned onto data can be expressed as

Prior lower and upper bounds of the uncertainty parameters and reference values.

Bayesian parameter estimation is performed hereafter with the DREAM

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).

The data sets obtained from solving the flow–transport problems (Eqs. 1–5)
using the parameters given in Sect. 2 are shown in Fig. 1. The pressure
head at 5 cm from the top of the column (Fig. 1a) increases from its initial
hydrostatic negative value (

The breakthrough concentration curve (Fig. 1e) shows a sharp front, which starts shortly after 3000 min. Note that if the injection of both water and contaminant are stopped once the solute reaches the output. For an injection period of 3000 min, the breakthrough curve exhibits a smoother progression (Fig. 1f).

The data considered as measurements, which are used as conditioning information for model calibration, are also shown in Fig. 1. In Fig. 1b, the water content seems to be more affected by the perturbation of data than the pressure head and cumulative outflow. This phenomenon is due to the relative importance of the measurement errors of the water content often observed with time-domain reflectometry probes and to the weak variations of the water content during the infiltration experiment. The perturbation of the breakthrough curve is relatively small because of the low added noise since output concentrations can be accurately measured. The perturbations of the pressure head and cumulative outflow seem weak because of the large variation of these variables during the experiment.

MCMC solutions for the transport scenario 1. The diagonal plots represent the inferred posterior probability distribution of the model parameters. The off-diagonal scatterplots represent the pairwise correlations in the MCMC drawing.

MCMC solutions for transport scenario 2 (see Fig. 2 caption).

The uncertainty model parameters are assumed to be distributed uniformly over the ranges reported in Table 1. This table also lists the reference values used to generate data observations before perturbation. Seven scenarios are considered, corresponding to different sets of measurements for the estimation of the hydraulic and transport soil parameters (Table 2).

Measurement sets and injection periods for the different scenarios.
The pressure head

The MCMC results of the seven studied scenarios are given in Figs. 2–8. The “on-diagonal” plots in these figures display the inferred parameter distributions, whereas the “off-diagonal” plots represent the pairwise correlations in the MCMC sample. If the draws are independent, non-sloping scatterplots should be observed. However, if a good value of a given parameter is conditioned by the value of another parameter, then their pairwise scatterplot should show a narrow sloping stripe. The sensitivity of parameters is obtained by comparing prior to posterior parameter distribution. A significant difference between the two distributions for a parameter indicates high model sensitivity to that parameter (Dusek et al., 2015).

MCMC solutions for transport scenario 3 (see Fig. 2 caption).

MCMC solutions for transport scenario 4 (see Fig. 2 caption).

MCMC solutions for transport scenario 5 (see Fig. 2 caption).

MCMC solutions for transport scenario 6 (see Fig. 2 caption).

MCMC solutions for transport scenario 7 (see Fig. 2 caption).

To facilitate the comparison between the different scenarios, Figs. 9–14 show the mean and the 95 % confidence intervals of the final MCMC sample that adequately fit the model onto observations for each scenario, and Table 3 summarizes the pairwise parameter correlations.

Summary of the pairwise parameter correlations.

Figure 2 shows the inferred distributions of the parameters identified with
the MCMC sampler using only the pressure and cumulative outflow measurements
(scenario 1). The parameters

The parameter

The dispersivity coefficient

The MCMC results in Fig. 3 show that water content measurements throughout
the experiment (scenario 2) allow the estimation of both the residual and
saturated water contents. The parameter

When the concentration measurements are also considered in the inversion
(scenario 3), the results depicted in Fig. 4 show very significant
correlations between

Posterior mean values and 95 % confidence intervals of the saturated hydraulic conductivity for the different scenarios.

Posterior mean values and 95 % confidence intervals of the saturated water content for the different scenarios.

Posterior mean values and 95 % confidence intervals of the residual water content for the different scenarios.

In the inversion procedure of scenario 4, the measurements of the water
content are not considered. This scenario leads to the same quality of the
estimation for the parameters

The pressure head, cumulative outflow and concentration measurements are
used in the estimation procedure of scenario 5, but the injection period is
now reduced to

In scenario 6, the pressure head measurements are removed and only
non-intrusive measurements (

The last scenario (scenario 7) is similar to the previous one, but the
injection period is reduced to

Posterior mean values and 95 % confidence intervals of the shape
parameter

Posterior mean values and 95 % confidence intervals of the shape
parameter

Posterior mean values and 95 % confidence intervals of dispersivity for the different scenarios.

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

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

The use of concentration measurements at the column outflow, in addition to traditional measured variables (water content, pressure head and cumulative outflow), reduces the hydraulic parameter uncertainties, especially those of the saturated water content (comparison between scenarios 2 and 3).

The saturated hydraulic conductivity is estimated with the same order of accuracy, independent of the observed variables.

The estimation of the dispersivity is sensitive to the injection duration.
Scenarios 5 and 7 with

A better identifiability of the soil parameters is obtained using

Using only non-intrusive measurements (cumulative outflow and output concentration) yields satisfactory estimation of all parameters (scenario 7). The uncertainty of the parameters significantly decreases when the injection of water and solute is maintained for a limited period (comparison between scenarios 6 and 7).

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

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.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

The authors are grateful to the French National Research Agency, which funded this work through the program AAP Blanc – SIMI 6 project RESAIN (no. ANR-12-BS06-0010-02). Edited by: H. Cloke Reviewed by: four anonymous referees