Estimating flow parameter distributions using ground-penetrating radar and hydrological measurements during transient flow in the vadose zone
Introduction
Predicting flow phenomena, such as the time a spilled contaminant takes to migrate through the vadose zone and into an aquifer, or predicting soil moisture profiles for agriculture management applications, requires characterization of soil properties such as permeability, porosity, and water retention. Existing techniques allow point values of these parameters to be measured in situ or in the laboratory (using soil cores). However, fluid flow parameters are commonly heterogeneous, and uncertainty in their spatial distributions makes it difficult to model fluid flow and contaminant transport using point measurements alone. Furthermore, point measurements are commonly limited due to their collection being expensive, time consuming and invasive (creating the potential for preferential flow paths). Alternative techniques that allow for the inference of flow parameter distributions are therefore in high demand.
Sequential simulation techniques may be used to generate parameter fields that reflect specified patterns of spatial correlation and preserve point measurements [14], [20], [42], [43]. However, the generation of fields that accurately predict flow given hydrological data and point measurements typically requires the employment of inverse methods. While substantial progress has been made in accounting for multi-dimensional spatial heterogeneity in the saturated zone, methods for the vadose zone are less common and have been mostly limited to one-dimensional cases, usually uniform or layered soil columns [11], [24], [28], [35], [47], [54]. In addition to the problems of ill-posedness and non-uniqueness, endemic to all types of groundwater inverse problems [7], the non-linearity introduced through saturation-dependent flow parameters undoubtedly contributes to the relative shortage of inverse techniques for variably saturated media [42], [45].
As groundwater inverse problems can be made more amenable to solution by incorporating additional types of data [34], integrating geophysical measurements into inverse methods for the vadose zone is a particularly promising area of research, though still in its infancy [4], [25]. Proving increasingly useful for monitoring moisture profiles in the vadose zone are geophysical methods such as ground-penetrating radar (GPR) [1], [5], [15], [21], [22], [26], [51] and electrical resistance tomography (ERT) [53]. Applications are rare in which geophysical methods are used quantitatively in the actual estimation of flow parameters in the vadose zone. Some progress has been made in this direction, such as in the work of Binley et al. [4], who investigated the use of ERT and crosshole GPR, collected during a tracer injection test, to estimate by trial and error the effective value of saturated hydraulic conductivity.
In the present work, we describe a method that allows for estimation of flow parameter distributions in the vadose zone jointly using hydrological and geophysical measurements collected during a transient flow experiment. We consider the special case in which permeability is the only non-uniform flow parameter and its log value may be treated as a space random function (SRF) characterized by a lognormal distribution with known patterns of spatial correlation (i.e., known semivariograms). Through a maximum a posteriori (MAP) inversion framework that employs concepts from the pilot point method, the log permeability distribution and additional flow parameters may be estimated. The employed methodology allows for the generation of multiple parameter distributions that reproduce point measurements, contain the specified patterns of spatial correlation, and are consistent with the hydrological and geophysical measurements. The resulting parameter distributions can be used for hydrological modeling and also to calculate parameter probability density functions (pdfs), which provide a measure of parameter uncertainty.
While additional data types such as capillary pressure and flow rate could easily be included in the method we describe, the measurements considered in this study include only point values of permeability, profiles of water saturation at boreholes (available through methods such as neutron probe logging), and crosshole GPR data (e.g., travel times or GPR-derived estimates of water saturation, as is described below).
The requirements for modeling flow in variably saturated media are described next, as are the GPR measurements used in the current study. Following that is a description of the proposed inversion methodology and its implementation, and then a synthetic example which allows for (1) evaluation of the method's performance under various conditions, (2) consideration of experimental designs, and (3) conclusions to be drawn regarding the joint use of GPR and hydrological measurements for flow inversion.
Section snippets
Modeling flow in the vadose zone
Modeling flow phenomena in the vadose zone requires a forward model that relates fluid flow parameters, such as porosity and permeability, to observational data, such as measurements of water saturation and pressure. For the case of incompressible flow of water in non-deformable porous media, variably saturated flow can be modeled with the Richards' equation, which is given bywhere K and Pc, both functions of water saturation Sw, are the hydraulic conductivity
Ground-penetrating radar measurements
Since GPR methods allow for the collection of non-invasive high resolution data that are sensitive to fluid saturation, they are potentially useful for parameter estimation methods in the vadose zone. However, it should be noted that GPR methods perform best in sites lacking highly electrically conductive materials, such as clay-rich soils [13]. GPR wave attributes such as the electromagnetic (EM) wave velocity and attenuation are governed by electrical parameters including the electrical
Methodology for parameter estimation
In developing an approach for estimating flow parameter distributions using hydrological and geophysical measurements jointly, concepts from the pilot point method [9], [33] are implemented within a Bayesian, maximum a posteriori (MAP) framework. Before describing details of the inverse methodology employed in the present work, some key concepts from the pilot point method are first summarized.
Synthetic example: ponding experiment
A synthetic example involving a ponded infiltration experiment in a two-dimensional vadose zone model (see Fig. 2a) is presented next. The vertical and horizontal dimensions for the modeled domain are 3 and 4 m, respectively, and the nodal spacing is 10 cm. The parameters describing the relative permeability and capillary pressure functions of the soil are spatially uniform, as is the soil porosity. Although effects of hysteresis on relative permeability can substantially impact the
Summary and conclusions
A method that allows transient hydrological and geophysical measurements to be used for estimating permeability distributions and other flow parameters in the vadose zone was described. The MAP method provided a framework within which concepts from the pilot point method could be incorporated. Application of a pilot point procedure allows for the generation of multiple parameter distributions that reproduce point measurements, contain the specified patterns of spatial correlation, and are
Acknowledgements
This work was supported by the US Department of Agriculture NRI grant 2001-35102-09866 Ground-Penetrating Radar: Development of a Vineyard Management Tool, and by US Department of Energy under Contract No. DE-AC03-76SF00098. The authors would like to thank the anonymous reviewers for their constructive feedback and suggestions.
References (54)
- et al.
Vadose zone flow model parameterization using cross-borehole radar and resistivity imaging
J. Hydrol.
(2002) - et al.
Application of the pilot point method to the identification of aquifer transmissivities
Adv. Water Resour.
(1991) - et al.
Impact of measurement errors in stochastic inverse conditional modelling by the self-calibrating approach
Adv. Water Resour.
(2003) - et al.
Maximum-likelihood estimation of unsaturated hydraulic parameters
J. Hydrol.
(1998) - et al.
Ground penetrating radar for determining volumetric soil water content; results of comparative measurements at two test sites
J. Hydrology
(1997) - et al.
Construction of geostatistical aquifer models integrating dynamic flow and tracer data using inverse technique
J. Hydrol.
(2002) - et al.
Estimating moisture contents in the vadose zone using cross-borehole ground penetrating radar: a study of accuracy and repeatability
Water Resour. Res.
(2002) Dynamics of fluids in porous media
(1988)- et al.
Finite-difference modeling of electromagnetic wave propagation in dispersive and attenuating media
Geophysics
(1998) - et al.
High-resolution characterization of vadose zone dynamics using cross-borehole radar
Water Resour. Res.
(2001)
Ray-based synthesis of bistatic ground penetrating radar profiles
Geophysics
Estimation of aquifer parameters under transient and steady state conditions. 2: Uniqueness, stability, and solution algorithms
Water Resour. Res.
Simulation of ground-penetrating radar waves in a 2-D soil model
Geophysics
An analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields
Water Resour. Res.
In situ determination of soil hydraulic properties during drainage
Soil Sci. Soc. Am. J.
Surface penetrating radar
Ground-penetrating radar for high-resolution mapping of soil and rock stratigraphy
Geophys. Prospect.
GSLIB: geostatistical software library and user's guide
Efficient three-dimensional data inversion: soil characterization and moisture monitoring from cross-well ground-penetrating radar at a Vermont test site
Water Resour. Res.
Vadose zone characterization and monitoring: current technologies, applications, and future developments
Stochastic simulation of transmissivity fields conditional to both tranmissivity and piezometric data. 1. Theory
J. Hydrol.
ISIM3D: an ANSI-C three-dimensional multiple indicator conditional simulation program
Comput. Geosci.
Velocity variations and water content estimated from multi-offset, ground-penetrating radar
Geophysics
Field-scale estimation of volumetric water content using ground-penetrating radar ground wave techniques
Water Resour. Res.
Ground-penetrating-radar-assisted saturation and permeability estimation in bimodal systems
Water Resour. Res.
Cited by (127)
Automated calibration methodology to avoid convergence issues during inverse identification of soil hydraulic properties
2022, Advances in Engineering SoftwareCitation Excerpt :Inoue et al. [9] reported a close correspondence between the SHP obtained from the inverse modeling of dynamic transient infiltration experiments with those obtained from steady-state laboratory experiments, where the uniqueness of the inverse model was preserved by considering the dynamically changing pressure head, water content and even tracer concentration. The non-uniqueness of the REVG inverse model is already a very well-known issue, and has been described by a number of publications over the last decades [15–22]. Mous [16] defined criteria for model identifiability based on the sensitivity matrix rank, however numerical computation of the sensitivity matrix, which is defined by the derivatives of the objective function, often involves difficulties in managing truncation and round-off errors.
Coupled full-waveform inversion of horizontal borehole ground penetrating radar data to estimate soil hydraulic parameters: A synthetic study
2022, Journal of HydrologyCitation Excerpt :Although the soil hydraulic parameters cannot be directly measured by GPR, time-lapse GPR measurements can be used to determine soil water content (SWC) dynamics (e.g., Huisman et al., 2003; Klotzsche et al., 2018) that are directly influenced by soil hydraulic properties because of the strong link between SWC and bulk dielectric permittivity (Topp et al., 1980). In the last decades, significant progress has been made with the use of different GPR configurations (surface, off-ground, and borehole GPR) to estimate soil hydraulic parameters (e.g., Busch et al., 2013, Jadoon et al., 2008; Jadoon et al., 2012; Lambot et al., 2006, Kowalsky et al., 2004, Rossi et al., 2015, Rucker and Ferré, 2004, Yu et al., 2021). Borehole GPR has a larger investigation depth and a better control of the vertical resolution than off-ground and surface GPR (Huisman et al., 2003) and thus shows advantages for estimating soil hydraulic parameters, especially at specialized test sites with appropriate boreholes (e.g., Binley et al., 2001; Looms et al., 2008; Kowalsky et al., 2005).
Individual and joint inversion of head and flux data by geostatistical hydraulic tomography
2021, Advances in Water ResourcesTime-lapse magnetic resonance sounding measurements for numerical modeling of water flow in variably saturated media
2020, Journal of Applied GeophysicsCitation Excerpt :Among them, the Ground Penetrating Radar (GPR) and the Electrical Resistivity Tomography (ERT) are the most popular. These methods identify different geological patterns and exploit correlations between the electrical and hydraulic properties of the subsurface (e.g., Kemna et al., 2002; Kowalski et al., 2004; Camporese et al., 2012; Carrière et al., 2013; Høyer et al., 2015; Vereecken et al., 2015; Carrière et al., 2016; Park et al., 2017; Jouen et al., 2018; Power et al., 2018; Saito et al., 2018; Ikard and Pease, 2019). However, other physical properties of soils also affect electrical conductivity.
Streamflow, stomata, and soil pits: Sources of inference for complex models with fast, robust uncertainty quantification
2019, Advances in Water Resources