Stochastic joint inversion of hydrogeophysical data for salt tracer test monitoring and hydraulic conductivity imaging
Highlights
► Fully coupled inversion of hydrogeochemical and geoelectrical data. ► Hydraulic conductivity tomography from the joint inversion of hydrogeophysical data. ► An electrical field can be passively and remotely measured for salt tracer test. ► Stochastic joint inversion based on the pilot point approach.
Introduction
The steady-state ground water flow and solute transport are mainly controlled by the spatial distribution of the permeability and dispersivity of an aquifer. Permeability can vary over 12 orders of magnitudes and can be very heterogeneous at various scales, which in turn implies a complex pattern for groundwater flow and contaminant transports [1]. At the opposite, porosity and dispersivity does not exhibit such a broad range of variation. The hydraulic conductivity is most commonly estimated from invasive hydrogeological techniques, such as pumping tests. Despite recent advances in hydraulic tomography [2], [3], [4] the resolution of the inverted hydraulic conductivity depends strongly on the density of piezometers [5]. The limited number of available piezometers makes the reconstruction of the hydraulic conductivity field of heterogeneous aquifers a difficult problem in most practical cases [6]. The use of geophysical methods can provide additional complementary information as broadly acknowledged in the last decade [7], [8].
Recently, geophysical tools have benefited from (i) the evolution of the efficiency of numerical methods (for instance the mixed finite element approach) for solving partial differential equations and parallel computing [9], [10], (ii) the development of improved petrophysical models connecting the geophysical signature to the hydraulic properties [11], [12], [13], [14], and (iii) significant improvements in the technology of various sensors and filtering techniques with the possibility to developed multitask sensors. These developments have therefore given birth to a new era of three-dimensional time lapse geophysical imaging for tracking the changes of variables of interest like the moisture content, the salinity, and the pore fluid pressure [7], [9], [10], [11], [12], [15].
Along these lines, the electrical resistivity imaging (ERI) is sensitive to changes in pore water electrical conductivity and temperature and therefore it has been used to track the subsurface migration of conductive tracers (saline or heat tracer tests) with the goal to image the hydraulic conductivity of heterogeneous aquifers [16]. Pollock and Cirpka [17] presented recently a new analysis of the ERI data to image the distribution of the hydraulic conductivity. Their work is based on the analysis of temporal moments of potential electrical disturbances recorded during saline tracer tests. Various geophysical methods with different sensitivity maps can be also used in concert. For instance, Direct Current (DC) resistivity data can be also jointly inverted with Ground Penetrating Radar (GPR) data during salt tracer tests in the vadose zone, to determine the hydraulic conductivity and petrophysical properties such as electrical formation factor, the water content, and the effective grain radius of the sediments [18], [19].
In this paper, we are interested by looking at the value of adding self-potential (SP) measurements in the joint inversion of geophysical data and in situ salt tracer sampling. The goal is still the same: the inversion of the hydraulic conductivity of an heterogeneous aquifer between two wells. The self-potential signals are passively recorded electrical potential signals associated with the occurrence of natural (source) currents in the ground. In the case of a salt tracer test, the source current density is generated by two contributions (i) the gradient in the activity of the salt (diffusion current) and (ii) an electrokinetic coupling (streaming current) directly associated with the flow of the ground water. The occurrence of this second contribution has been recently used to non-intrusively assess ground water flow For instance, Jardani et al. [20] used SP data to reconstruct the hydraulic head variations associated with pumping tests conducted in an alluvial aquifer. Suski et al. [21] showed that SP signals can be used to monitor the variations of the piezometric levels of an unconfined aquifer associated with an infiltration experiment from a ditch. SP data have been combined with the hydraulic head variations recorded during pumping tests to estimate the transmissivity of an heterogeneous aquifer [22], [23] and to image in 3D the hydraulic conductivity [24]. The SP responds associated with the diffusion source current is recognized as an efficient method to delineate contaminated areas [25] and salt tracer tests [26]. We feel that the time-lapse analysis of the self-potential signals associated with a salt tracer test could emerged as a powerful tool in determining quantitatively hydraulic properties.
As a side note, other geophysical methods have been conducted in order to evaluate the hydraulic conductivity of heterogeneous aquifers. For instance, Linde et al. [27] used cross-well ground penetrating radar (GPR) for salt tracer tests. Hyndman et al. [28] presented recently a relationship between the seismic slowness and hydraulic conductivity, which has been successfully used to predict the hydraulic conductivity of an alluvial aquifer from seismic and tracer test data. Hördt et al. [29] proposed the use of spectral induced polarization measurements to image the hydraulic conductivity of a sandy/gravel aquifer.
The big picture is that combining hydrogeophysical and hydrogeological information need to be somehow coupled to reduce the uncertainty associated with the estimation of the hydraulic conductivity. This implies to take into account the uncertainty associated with the in situ measurements as well as with the geophysical data. Two approaches can be used to combine hydrogeological and geophysical data. In the first approach, the hydrological information is used to weakly constrain the inversion of the geophysical measurements [27]. The second approach is to fully couple the inversion of the two types of data [7], [17], [30]. Hinnell et al. [31] devoted an approach using the coupling of hydrological and geophysical data to reconstruct the hydrological parameters using ERI for tracking the infiltration front in vadose zone in a synthetic case study.
In this paper, we are interested by the fully coupled joint inversion of geophysical and hydrogeological data to monitor a salt tracer test and to assess the hydraulic conductivity field. Using salt tracer tests with ERI is a problem that has been received a lot attention recently [32], [33], [34], [35], [36], [37]. The non-uniqueness of the ERI inverse problem (and its sensitivity map) makes this method insufficient by itself. In other words, ERI needs to be combined with other sources of information like in situ measurements of the salt concentration in wells [33], [38], [39]. Irving and Singha [38] introduced a stochastic joint inversion approach of time-lapse cross-well ERI and salt tracer concentration data. Our work is following this idea but adding an additional method, the self-potential method, to the inverse problem. We use the pilot points approach for the joint inversion of ERI, SP, and salt tracer data because this approach reduces the over parametrization of the problem. While previous authors used deterministic methods to choose the values of the pilot points, we place the pilot points on a regular grid and we use a stochastic method based on the Markov chain Monte Carlo, McMC, sampling approach to estimate the values of the model parameters at the pilot points.
Section snippets
Theory
We present in this section the equations governing the physical processes of groundwater flow and saline tracer transport in a heterogeneous unconfined aquifer. We also introduce the semi-coupled equations connecting the salt concentration to the electrical resistivity (to interpret ERI) and to the source current density used to interpret SP data.
Forward modeling in an heterogeneous aquifer
A salt tracer synthetic test involves injecting a known amount of a non-reactive (conservative) salt (e.g., NaCl) into an upgradient well and monitoring the perturbation of the electrical conductivity and self-potential associated with the migration of the salt tracer due to the natural hydraulic gradient plus dispersion and diffusion. As mentioned above, the salt tracer is assumed to be conservative. If this not the case, a sorption isotherm needs to be include in the model and the forward
Description of the algorithm
We choose a stochastic approach to determine the hydraulic conductivity fields m from three data sources (d denotes the data vector) coming from saline concentration, electrical resistivity, and self-potential SP (see Fig. 4). The pilot point method is used as a parameterization technique to reduce the number of the model parameters by identifying a few key-locations (called pilot points) in the aquifer where the values of the hydraulic conductivity can be assessed to reproduce the
Concluding statements
We have presented a synthetic case study investigating the usefulness of adding self potential measurements to electrical resistivity imaging and in situ concentration measurements to reconstruct the hydraulic conductivity field of an heterogeneous aquifer during a salt tracer test. As most of the impedance meters used for ERI are also able to record the self-potential field before the injection of electric current, this approach does not require new equipment with respect to those used
Acknowledgements
We thanks Seine Aval program for funding the project “TidHydrex”. AR thanks the Environment Remediation Science Program (ERSP), US Department of Energy (DOE, award DE-FG02-08ER646559) for funding. We deeply thank Jinsong Chen and three anonymous referees for very constructive comments and the time they have invested in helping us to improve our manuscript.
References (55)
- et al.
Efficient electric resistivity inversion using adjoint state of mixed finite-element method for Poisson’s equation
J Comput Phys
(2006) - et al.
Chemico-electromechanical coupling in microporous media
J Colloid Interface Sci
(2006) - et al.
Geophysical constraints on contaminant transport modeling in a heterogeneous fluvial aquifer
J Contam Hydrol
(2006) - et al.
Seasonal variation of moisture content in unsaturated sandstone inferred from borehole radar and resistivity profiles
J Hydrol
(2002) - et al.
Joint inversion of steady-state hydrologic and self-potential data for 3D hydraulic conductivity distribution at the Boise Hydrogeophysical Research Site
J Hydrol
(2011) - et al.
Vadose zone flow model parameterization using cross-borehole radar and resistivity imaging
J Hydrol
(2002) - et al.
A 3D ERT study of solute transport in a large experimental tank
J Appl Geophys
(2002) - et al.
Imaging and characterization of subsurface solute transport using electrical resistivity tomography (ERT) and equivalent transport models
J Hydrol
(2002) Ionic diffusivity, electrical conductivity, membrane and thermoelectric potentials in colloids and granular porous media: a unified model
J Colloid Interface Sci
(1999)- et al.
FW2_5D: a MATLAB 2.5-D electrical resistivity modeling code
Comput Geosci
(2008)
Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data. 2. Demonstration on a synthetic aquifer
J Hydrol
Estimation of aquifer transport parameters from resistivity monitoring data within a coupled inversion framework
J Hydrol
Stochastic subsurface hydrology
Three-dimensional transient hydraulic tomography in a highly heterogeneous glaciofluvial aquifer–aquitard system
Water Resour Res
Derivation of site-specific relationships between hydraulic parameters and P-wave velocities based on hydraulic and seismic tomography
Water Resour Res
2D characterization of hydraulic heterogeneity by multiple pumping tests
Water Resour Res
Stochastic estimation of aquifer geometry using seismic refraction data with borehole depth constraints
Water Resour Res
Electrical-resistivity tomography of vadose water movement
Water Resour Res
Parallel and explicit finite-element time-domain method for Maxwell’s equations
IEEE Trans Antennas Propag
Geophysics at the interface: response of geophysical properties to solid–fluid, fluid–fluid, and solid–solid interfaces
Rev Geophys
Full waveform inversion of seismic waves reflected in a stratified porous medium
Geophys J Internat
Electrical conductivity in shaly sands with geophysical applications
J Geophys Res
Hydrogeophysics: water and science technology library
Fully coupled hydrogeophysical inversion of synthetic salt tracer experiments
Water Resour Res
Improved hydrogeophysical characterization using joint inversion of cross-hole electrical resistance and ground-penetrating radar traveltime data
Water Resour Res
Cited by (91)
Image-guided structure-constrained inversion of electrical resistivity data for improving contaminations characterization
2024, Computers and GeosciencesContaminant source and aquifer characterization: An application of ES-MDA demonstrating the assimilation of geophysical data
2023, Advances in Water ResourcesCharacterization of DNAPL source zones in clay-sand media via joint inversion of DC resistivity, induced polarization and borehole data
2023, Journal of Contaminant HydrologyHydrogeophysical characterization and determination of petrophysical parameters by integrating geophysical and hydrogeological data at the limestone vadose zone of the Beauce aquifer
2022, Journal of HydrologyCitation Excerpt :Several surface and cross-borehole geophysical methods have been recently used for different hydrological applications. These methods include the ground penetrating radar (GPR) (Dafflon et al., 2011; Lunt et al., 2005; Binley et al., 2001), seismic techniques (Blazevic et al., 2020), electrical resistivity (ER) (Mallet et al., 2021; Johnson et al., 2015; Robinson et al., 2008), self-potential (Abbas et al, 2017; Ahmed et al., 2014; Jardani et al., 2012), induced polarization (Johnson et al., 2010) and nuclear magnetic resonance (Vilhelmsen et al., 2014). Additionally, geophysical methods have been recently used to characterize and quantify processes and interactions in the soil–rhizosphere–atmosphere agricultural ecosystem continuum (Garré et al., 2021).
Monitoring unsaturated water flow using magnetic resonance soundings
2022, Journal of HydrologyCitation Excerpt :Different authors report surface, borehole, and cross-hole geophysical techniques applied to hydrogeological applications. The most frequently used methods are the ground penetrating radar (GPR) (e.g. Dafflon et al., 2011; Lunt et al., 2005), seismic techniques (e.g. Blazevic et al., 2020), electrical resistivity (ER) techniques (e.g. Johnson et al., 2015; Robinson et al., 2008), gravity (e.g. Eppelbaum, 2019), self-potential (e.g. Abbas et al, 2017; Eppelbaum, 2019; Jardani et al., 2013), and induced polarization (e.g. Johnson et al., 2010). Time-lapse geophysical measurements allow monitoring temporal and spatial variations of the water content following natural or artificial infiltration processes (e.g., Uhlemann et al., 2017).