Fluid flow in a charged porous medium generates electric potentials called streaming potential (SP). The SP signal is related to both hydraulic and electrical properties of the soil. In this work, global sensitivity analysis (GSA) and parameter estimation procedures are performed to assess the influence of hydraulic and geophysical parameters on the SP signals and to investigate the identifiability of these parameters from SP measurements. Both procedures are applied to a synthetic column experiment involving a falling head infiltration phase followed by a drainage phase.

GSA is used through variance-based sensitivity indices, calculated using sparse polynomial chaos expansion (PCE). To allow high PCE orders, we use an efficient sparse PCE algorithm which selects the best sparse PCE from a given data set using the Kashyap information criterion (KIC). Parameter identifiability is performed using two approaches: the Bayesian approach based on the Markov chain Monte Carlo (MCMC) method and the first-order approximation (FOA) approach based on the Levenberg–Marquardt algorithm. The comparison between both approaches allows us to check whether FOA can provide a reliable estimation of parameters and associated uncertainties for the highly nonlinear hydrogeophysical problem investigated.

GSA results show that in short time periods, the saturated hydraulic conductivity

Results of parameter estimation show that although the studied problem is
highly nonlinear, when several SP data collected at different altitudes
inside the column are used to calibrate the model, all hydraulic

Flow through a charged porous medium can generate an electric potential (Zablocki, 1978; Ishido and Mizutani, 1981; Allègre et al., 2010; Jougnot and Linde, 2013), called streaming potential (SP). SP signals play an important role in several applications related to hydrogeology and geothermal reservoir engineering as they are useful for examining subsurface flow dynamics. During the last decade, surface SP anomalies have been widely used to estimate aquifers' hydraulic properties (Darnet et al., 2003). Interest in SP is motivated by its low cost and its high sensitivity to water flow. Either coupled or uncoupled approaches can be used for hydraulic parameter estimation from SP signals (Mboh et al., 2012). In the uncoupled approach, Darcy velocities (e.g., Jardani et al., 2007; Bolève et al., 2009) are obtained from the tomographic inversion of SP signals and are then used for the calibration of the hydrologic model. In the coupled approach, anomalies related to the tomographic inversion are avoided by inverting the full coupled hydrogeophysical model (Hinnell et al., 2010).

The SP signals have been widely studied in saturated porous media (Bogoslovsky and Ogilvy, 1973; Patella, 1997; Sailhac and Marquis, 2001; Richards et al., 2010; Bolève et al., 2009, among others). Fewer studies focused on the application of the SP signal in unsaturated flow despite the large interest in such nonlinear problems (Linde et al., 2007; Allègre et al., 2010; Mboh et al., 2012; Jougnot and Linde, 2013). Hence, in this work we are interested in the SP signals in unsaturated porous media. Our main objective is to investigate the usefulness of the SP signals for the characterization of soil parameters. For this purpose, we evaluate the impact of uncertain hydraulic and geophysical parameters on the SP signals and assess the identifiability of these parameters from the SP measurements.

The impact of soil parameters on SP signals is investigated using global
sensitivity analysis (GSA). This is a useful tool for characterizing the
influential parameters that contribute the most to the variability of model
outputs (Saltelli et al.,1999; Sudret, 2008) and for understanding the
behavior of the modeled system. GSA has been applied in several areas, for risk assessment for groundwater pollution (e.g., Volkova et al., 2008),
nonreactive (Fajraoui et al., 2011) and reactive transport experiments
(Fajraoui et al., 2012; Younes et al., 2016), for unsaturated flow
experiments (Younes et al., 2013), natural convection in porous media
(Fajraoui et al., 2017) and seawater intrusion (Rajabi et al., 2015; Riva et
al., 2015). To the best of our knowledge, GSA has never been used for SP
signals in unsaturated porous media. Hence, in the first part of this study,
GSA is performed on a conceptual model inspired from the laboratory
experiment of Mboh et al. (2012) in which SP signals are measured at different
altitudes in a sandy soil column during a falling-head infiltration phase
followed by a drainage phase. Four uncertain hydraulic parameters, saturated
hydraulic conductivity

In the second part of this study, we investigate the identifiability of
hydrogeophysical parameters from SP measurements. For this purpose, parameter
estimation is performed using two different approaches. The first is a
Bayesian approach in which model parameters are treated as random variables
and characterized by their probability density functions. With this
approach, the prior knowledge about the model and the observed data is
merged to define the joint posterior probability distribution function of
the parameters. In the sequel, Bayesian analysis is conducted using the
DREAM

The present study is set out as follows. Section 2 presents the hydrogeophysical model and the reference solution. Section 3 reports on the GSA results of SP signals. Then, Sect. 4 discusses results of parameter estimation with both MCMC and FOA approaches for the two investigated scenarios.

Illustration of the experimental device.

The test case considered in this work is similar to the laboratory
experiment developed in Mboh et al. (2012), involving a falling-head
infiltration phase followed by a drainage phase (Fig. 1). This experiment is
representative of several laboratory SP experiments (Linde et al., 2007;
Allègre et al., 2010; Jougnot and Linde, 2013, among others). Quartz sand is
evenly packed in a plastic tube with an internal diameter of 5 cm to a
height of

The total electrical current density

Hence, the conservation equation

The streaming current density

Hence, the combination of the previous Eqs. (1)–(4) leads to the following
partial differential equation governing the SP signals:

The water velocity

Although several numerical techniques have been developed for the
solution of the multidimensional Richards equation (e.g., Fahs et al., 2009;
Belfort et al., 2009; Younes et al., 2013; Deng and Wang, 2017), the
standard finite volume method is used here for the spatial discretization of
the one-dimensional Richard's equation (Eq. 6). The integration of this
equation over the finite volume

Then, the temporal discretization of the obtained nonlinear ODE/DAE system
(9–10) is performed with the method of lines (MOL) using the DASPK (Brown et
al., 1994) time solver. The MOL is suitable for strongly nonlinear systems
since it allows high-order temporal integration methods with formal error
estimation and control (Miller et al., 1998; Younes et al., 2009; Fahs et
al., 2009, 2011). In the current study, the relative and absolute local
error tolerances are fixed to 10

Numerical simulations are performed assuming typical MVG hydraulic
parameters for the sandy soil with (according to Carsel and Parrish, 1988)

Reference SP signals. Solid lines represent the reference SP solution and dots represent the sets of perturbed data serving as conditioning information for model calibration.

Based on these hydraulic and geophysical parameters, a reference (mesh-independent) solution is obtained using a uniform mesh of 235 cells of 0.5 cm
length. Data are generated by sampling the output SP signals every 10 min
during 1800 min. Figure 2 shows that the SP signals have an almost linear
behavior in the saturated falling-head phase. During the drainage phase,
they have a nonlinear behavior and approach zero voltage for the dry
conditions occurring toward the end of the experiment. The SP signals are independent Gaussian random noises with a standard deviation of
2.73 10

The aim of GSA is to assess the effect of the variation of parameters on the model output (Mara and Tarantola, 2008). Such knowledge is important for determining the most influential parameters as well as their regions and periods of influence (Fajraoui et al., 2011). The sensitivity of a model to its parameters can be assessed using variance-based sensitivity indices. These indices evaluate the contribution of each parameter to the variance of the model (Sobol', 2001). Polynomial chaos theory (Wiener, 1938) has been largely used to perform variance-based sensitivity analysis of computer models (see for instance, Sudret, 2008; Blatman and Sudret, 2010; Fajraoui et al., 2012; Younes et al., 2016; Shao et al., 2017; Mara et al., 2017). It can be stated that the PCE method is a surrogate-based approach. However, we argue that this method employs ANOVA (analysis Of variance) decomposition and hence can be considered as a spectral method (such as the Fourier amplitude sensitivity test; Cukier et al., 1973; Saltelli et al., 1999). Indeed, with this method, the sensitivity indices are directly obtained from the PCE coefficients without needing to run the surrogate model.

Let us consider a mathematical model with a random response

Equation (12) is similar to an ANOVA representation of the original model (Sobol'
1993), from which it is straightforward to express

In the sequel, a PCE is constructed for each SP signal at each observable
time. The number of coefficients for a full PCE representation is

Reference values, lower and upper bounds for hydraulic and geophysical parameters.

The SP responses are considered for uniformly distributed parameters over
the large intervals shown in Table 1. These intervals include the reference
values reported in Mboh et al. (2012). The sensitivity indices of the six
input parameters

Time distribution of the SP variance at 5 cm

Figure 3 depicts the temporal distribution of the streaming potential variance, represented by the blue curve, and the relative contribution of the parameters, represented by the shaded area. This figure corresponds to the temporal ANOVA decomposition for sensor 1 (5 cm from the soil surface) and for sensor 4 (77 cm from the soil surface). Interactions between parameters are represented by the blank region between the variance curves and the shaded area. Note that because a Dirichlet boundary condition with zero SP is maintained at the outlet boundary, the variance of the SP signal is zero at the bottom and reaches its maximum value near the soil surface. Hence, the variance is higher for the first sensor, located 5 cm from the soil surface (Fig. 3a) than for sensor 4, located 77 cm from the soil surface (Fig. 3b).

The first-order sensitivity index

The SP signals at different altitudes exhibit similar behavior (Fig. 3). In
the following, we comment on the results of sensor 1 (Fig. 3a). Because

During the second period

During the third period

Figure 3b shows similar behavior for sensor 4 located 77 cm from the
soil surface. The results in Table 2b indicate that the total variance
observed at

In summary, the GSA applied to SP signals identifies the influential
parameters and their periods of influence and shows that

the parameter

the parameter

the parameters

the parameter

the relevance of the parameter

Calibration of computer models is an essential task since some parameters
(like the Mualem–van Genuchten-shaped parameters

As important as the determination of the optimal parameter sets are the
associated 95 % confidence intervals (CIs) to quantify the uncertainty of the
estimated values. The determination of CIs is not straightforward if the
observed model responses are highly nonlinear functions of model parameters
(Christensen and Cooley, 1999). In the sequel, the parameter estimation is
performed using two approaches: the popular FOA approach and the Bayesian
approach based on the MCMC sampler. Contrarily to
FOA, the MCMC method is robust since no assumptions of model linearity or
differentiability are required. Furthermore, prior information available for
the parameters can be included. MCMC provides not only an optimal point
estimate of the parameters but also a quantification of the entire parameter
space. Several MCMC strategies have been developed for Bayesian sampling of
the parameter space (Gallagher and Doherty, 2007; Vrugt, 2016). In
a groundwater and vadose zone modeling context, the most widely used of these
strategies is the Metropolis–Hastings algorithm (Metropolis et al., 1953;
Hastings, 1970). It proceeds as follows (Gelman et al., 1996).

Choose an initial candidate

A new candidate

Calculate

If

Resume from (ii) until the chain

Many improvements have been proposed in the literature to accelerate the
MCMC convergence rate (e.g., Haario et al., 2006; ter Braak and Vrugt, 2008;
Dostert et al., 2009, among others). Vrugt et al. (2009a, b) developed
the DREAM MCMC sampler based on the differential evolution–Markov chain
method of ter Braak (2006) to improve sampling efficiency. DREAM runs
multiple Markov chains in parallel and uses subspace sampling and outlier
chain correction to speed up MCMC convergence (Vrugt, 2016). Laloy and Vrugt
(2012) developed the DREAM

In the following, both MCMC and FOA approaches are employed for the inversion of the highly nonlinear hydrogeophysical problem using SP measurements.

Hydrogeophysical parameters are estimated using the DREAM

The DREAM

We assume that the saturated water content has been initially measured with
a fair degree of accuracy. However, instead of fixing its value (as in Kool
et al. , 1987, van Dam et al., 1994, and Nützmann et al., 1998, among
others), we assign a Gaussian distribution to

MCMC solutions in which all SP data are considered for the calibration. 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 in which calibration is performed using only SP data located 5 cm from the surface. The diagonal plots represent the posterior probability distribution of the parameters. The off-diagonal scatterplots represent the pairwise correlations in the MCMC drawing.

Figure 4 shows the results obtained with MCMC when the SP data of the five
sensors are used for the calibration. The “on-diagonal” plots in this figure
display the posterior parameter distributions, whereas the “off-diagonal”
plots represent the correlations between parameters in the MCMC sample.
Figure 4 shows nearly bell-shaped posterior distributions for all
parameters. A strong correlation is observed between

From the obtained MCMC sample, it is straightforward to estimate the posterior 95 % confidence interval of each parameter. This as well as the mean estimate value of each parameter obtained with both MCMC and FOA approaches are reported in Table 3.

Estimated mean values (bold), confidence intervals (CIs) and size of the posterior CIs (italic) with MCMC and FOA approaches for scenario 1.

Estimated mean values (bold), confidence intervals (CIs) and size of the posterior CIs (italic) with MCMC and FOA approaches for scenario 2.

The results of this table show that the parameters are well estimated from
the SP measurements since (

The posterior CI of the parameter

Further, the results of Table 3 show that FOA and MCMC approaches yield similar mean estimated values. Moreover, very good agreement is observed between FOA and MCMC uncertainty bounds. Concerning the efficiency of the two calibration methods for this scenario, the FOA approach is by far the most efficient method since it requires only 95 s of CPU time. The MCMC method was terminated after 16 000 model runs, which required 14 116 s. The convergence was reached at around 12 000 model runs. The last 4000 runs were used to estimate the statistical measures of the posterior distribution. Recall that contrarily to FOA, MCMC can include prior information available for the parameters and allows a quantification of the entire parameter space.

In this scenario, the number of measurements used for the calibration is strongly reduced. Only SP measurements from sensor 1 (located 5 cm below the soil surface) are considered.

The results of MCMC are plotted in Fig. 5. The correlation observed
between

The results obtained with MCMC and FOA approaches depicted in Table 4 show
the following.

The FOA approach yields accurate mean estimated values similar to MCMC results for all parameters.

The MCMC and FOA mean estimated values are close to the reference solution
and to the previous scenario. The maximum difference is observed for

The MCMC CIs for the parameters

Due to the reduction of the number of data used for model calibration in scenario 2, the MCMC CIs for the parameters

The FOA method yields accurate CIs for the parameters

Concerning the efficiency of the calibration methods, the FOA required approximately 174 s of CPU time, and the MCMC required many more runs to reach the convergence than in the previous scenario. Indeed, the sampler was used with 50 000 runs (35 000 runs were necessary to reach the convergence).

In this work, a synthetic test case dealing with SP signals during a drainage experiment has been studied. The test case is similar to the laboratory experiment developed in Mboh et al. (2012), involving a falling-head infiltration phase followed by a drainage phase. GSA and Bayesian parameter inference have been applied to investigate (i) the influence of hydraulic and geophysical parameters on the SP signals and (ii) the identifiability of hydrogeophysical parameters using only SP measurements. The GSA was performed using variance-based sensitivity indices which allow the contribution of each parameter (alone or by interaction with other parameters) to the output variance to be measured. The sensitivity indices have been calculated using a PCE representation of the SP signals. To reduce the number of coefficients and explore PCE with high orders, we used the efficient sparse PCE algorithm developed by Shao et al. (2017), which selects the best sparse PCE from a given data set using the Kashyap information criterion (KIC).

The GSA applied to SP signals showed that the parameters

Parameter estimation has been performed using MCMC and FOA approaches to
check whether FOA can provide a reliable estimation of parameters and
associated uncertainties for the highly nonlinear
hydrogeophysical problem investigated. All hydraulic (

When the number of SP measurements used for the calibration is considerably reduced (i.e., data are scarce), the MCMC inversion provides larger uncertainty regions of the parameters. The FOA approach yields accurate mean parameter values (in agreement with MCMC results) but inaccurate and even unphysical CIs for some parameters with large uncertainty regions.

No data sets were used in this article.

AY framed the research question, worked on sensitivity analysis and finalized the manuscript. JZ and FL worked on parameter estimation and numerical model development. MF performed simulations, analyzed the results and reviewed the manuscript.

The authors declare that they have no conflict of interest.

The authors acknowledge the financial support from the Tunisian–French joint
international laboratory NAILA (