Estimating flow parameter distributions using ground-penetrating radar and hydrological measurements during transient flow in the vadose zone

https://doi.org/10.1016/j.advwatres.2004.03.003Get rights and content

Abstract

Methods for estimating the parameter distributions necessary for modeling fluid flow and contaminant transport in the shallow subsurface are in great demand. Soil properties such as permeability, porosity, and water retention are typically estimated through the inversion of hydrological data (e.g., measurements of capillary pressure and water saturation). However, ill-posedness and non-uniqueness commonly arise in such non-linear inverse problems making their solutions elusive. Incorporating additional types of data, such as from geophysical methods, may greatly improve the success of inverse modeling. In particular, ground-penetrating radar (GPR) methods have proven sensitive to subsurface fluid flow processes and appear promising for such applications. In the present work, an inverse technique is presented which allows for the estimation of flow parameter distributions and the prediction of flow phenomena using GPR and hydrological measurements collected during a transient flow experiment. Specifically, concepts from the pilot point method were implemented in a maximum a posteriori (MAP) framework to allow for the generation of permeability distributions that are conditional to permeability point measurements, that maintain specified patterns of spatial correlation, and that are consistent with geophysical and hydrological data. The current implementation of the approach allows for additional flow parameters to be estimated concurrently if they are assumed uniform and uncorrelated with the permeability distribution. (The method itself allows for heterogeneity in these parameters to be considered, and it allows for parameters of the petrophysical and semivariogram models to be estimated as well.) Through a synthetic example, performance of the method is evaluated under various conditions, and some conclusions are made regarding the joint use of transient GPR and hydrological measurements in estimating fluid flow parameters 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 byφSwt+K(Sw)ρwgPc(Sw)−K(Sw)ẑ=0,where 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.

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