Simulated-annealing-based conditional simulation for the local-scale characterization of heterogeneous aquifers
Introduction
A key control on groundwater flow and contaminant transport in the subsurface is the spatial distribution of hydrological properties. Accurate characterization of these properties is crucial for developing reliable numerical models of flow and transport, which are required to design effective and cost-efficient aquifer remediation and groundwater management strategies. It is well understood that spatial variability needs to be defined at a wide range of scales for effective modeling of hydrological phenomena (e.g., Sudicky and Huyakorn, 1991, Gelhar, 1993, Zheng and Gorelick, 2003, Hubbard and Rubin, 2005). However, conventional hydrological measurement techniques tend to lie at two ends of a spectrum in terms of resolution and sampling volume, leaving a significant gap in a range that is expected to contain particularly critical hydrological information. Whereas pumping and tracer tests tend to yield only gross average properties over a relatively large region, core samples and borehole logs yield high-resolution estimates of aquifer properties, but only along sparse 1-D profiles. Consequently, over the past two decades, much work has been done on the use of geophysical methods for aquifer characterization. Such methods can bridge the gap between the analysis of cores or logs and well tests, and have proven to be extremely useful not only for aquifer zonation but also for estimating the spatial distribution of hydrological parameters (e.g., McKenna and Poeter, 1995, Hyndman et al., 2000, Chen et al., 2001, Tronicke et al., 2002, Hubbard and Rubin, 2005, Kowalsky et al., 2005, Paasche et al., 2006). An important and still largely unresolved issue with the use of geophysical data in hydrological studies, however, is that of data integration. That is, how do we quantitatively integrate geophysical data with an existing database of other measurements to best constrain our knowledge of the spatial distribution of one or several target parameters?
The integration of different types of data for subsurface characterization has been a subject of much investigation in the petroleum industry, and has received increased attention in groundwater studies in recent years. Whereas much data integration and joint inversion work in the past has involved the determination of a single model of the subsurface parameters of interest, a number of recent efforts have focused on the creation of sets of multiple realizations that are consistent with all of the available data, and represent the uncertainty in our knowledge of the spatial distribution of subsurface properties (e.g., McKenna and Poeter, 1995, Bosch, 1999, Avseth et al., 2001, Caers et al., 2001, Mukerji et al., 2001, Ramirez et al., 2005, Hansen et al., 2006). The idea behind such conditional simulation approaches is that they can be used in combination with complex hydrological models to make predictions regarding groundwater flow and contaminant transport within a framework of uncertainty. Work with these methodologies has increased over recent years as a product of continually growing computer capacity, and also the ever increasing realization that “mean models” of subsurface properties do not adequately represent subsurface heterogeneity for reliable flow and transport predictions (e.g., Goovaerts, 1997).
Due to their inherent flexibility with regard to imposing constraints and their conceptual simplicity, simulated-annealing-type conditional stochastic simulations seem to be particularly promising for subsurface data integration (e.g., Deutsch and Wen, 1998, Deutsch and Wen, 2000, Kelkar and Perez, 2002). The simulated annealing (SA) approach is not limited to simple Gaussian statistics, and is able to incorporate any constraint on the output realizations that can be expressed in the form of an objective function, with the caveat that efficiency in terms of computation time and convergence decreases with constraint complexity. With SA, parameter fields satisfying all of the available data are obtained through minimization of a global, generally multi-component objective function, and multiple realizations can be generated by running the algorithm with different initial conditions. It should be emphasized that the variability seen in such multiple realizations depends on the applied constraints and stopping criteria, and as a result the realizations should not be confused with samples drawn from a posterior probability density function. Nevertheless, the SA method still allows evaluation of the variability in flow and transport behavior associated with uncertain data and constraints.
Recently, Tronicke and Holliger (2005) explored the use of SA for hydrogeophysical data integration through a synthetic model study. Starting with a simulated porosity model of a heterogeneous alluvial aquifer, they generated synthetic porosity logs and crosshole georadar traveltime data. These data, along with geostatistical constraints, were then used as conditioning information in a SA-based optimization procedure to generate porosity models that were consistent with all of the available information. In their work, Tronicke and Holliger (2005) pursued the classical SA approach of gradually “organizing” an uncorrelated random initial field through repeated swapping of values in the simulation grid, while adherence to the geophysical and geostatistical data was accomplished through matching the correlation coefficient between the realization and geophysical data to a prescribed value, and matching a prior parametric covariance model, respectively. Although the results obtained using this methodology clearly demonstrated that SA has much potential for hydrogeophysical data integration, we have found that the lateral continuity of the resulting porosity models is in general inadequate, which in turn significantly reduces the predictive value of such models in subsequent flow and transport simulations. Closer inspection indicates that this problem likely arises from the fact that it is inherently difficult with purely stochastic simulations to effectively impose constraints with regard to the underlying deterministic structure of the target parameter, as provided, for example, by high-resolution geophysical data.
In this paper, we present a novel SA-type conditional simulation procedure that aims to address and resolve this issue, as well as to improve the convergence and computational efficiency of the traditional SA method, which are known to be suboptimal as a result of having a relatively complex objective function. To begin, we review the overall methodology and describe an approach to more effectively account for the larger-scale deterministic information contained in geophysical data. Next, we test our conditional stochastic simulation algorithm on a synthetic data set consisting of crosshole georadar data and porosity logs from a highly heterogeneous, realistic aquifer model. Finally, we use our method to integrate field crosshole georadar and neutron porosity log data collected at the Boise Hydrogeophysical Research Site (BHRS) near Boise, Idaho, USA.
Section snippets
Background
Simulated annealing is a directional Monte-Carlo-type optimization procedure, whose central idea is based upon the thermodynamics of a cooling melt. Atoms can move freely throughout a melt at high temperatures, but as the temperature is lowered, their mobility progressively decreases. Eventually, the system reaches its thermodynamic minimum-energy state and the atoms assume fixed positions within a crystal lattice. In SA, there are a large number of possible initial states, but during the
Application to synthetic data
We now apply our SA-type conditional simulation algorithm to a synthetic example. We use the same data that were used in Tronicke and Holliger (2005) such that the potential advantages and limitations of both approaches can be more easily identified. From an initial synthetic subsurface porosity model, crosshole georadar and neutron porosity log data were generated. The goal of the SA procedure was then to integrate these two data types and produce, within the bounds of uncertainty dictated by
Application to field data
We now use our new SA approach to integrate crosshole georadar and porosity log data collected at BHRS near Boise, Idaho, USA, with the objective of simulating the detailed porosity distribution in this heterogeneous alluvial aquifer. The BHRS was established for the purpose of developing methodologies for joint geophysical/hydrological characterization of the 3-D distribution of hydrological parameters in unconsolidated heterogeneous alluvial deposits (Barrash and Knoll, 1998, Clement et al.,
Conclusions
The main objective of this study was to develop a conditional stochastic simulation method that allows for the effective, quantitative integration of high-resolution geophysical data. We have developed a novel SA-based approach that demonstrates strong potential for generating highly detailed and realistic aquifer models honoring different data and prior information. A major advantage of our approach compared with related previous efforts is that deterministic information with regard to the
Acknowledgments
This research was supported by a grant from the Swiss National Fund. We thank CGISS at Boise State University for granting us the permission to work with crosshole georadar and neutron porosity log data collected at the BHRS. We also thank Jens Tronicke at the University of Potsdam for helpful comments and suggestions and access to the results of previous work, as well as the two anonymous reviewers whose suggestions greatly improved this manuscript.
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