Kriging groundwater solute concentrations using flow coordinates and nonstationary covariance functions

Interpolation of solute concentration measurements often yields disappointing results, especially when it fails to incorporate some knowledge relative to the underlying physics of groundwater flow and solute transport. Concentration maps, however, are required in several applications such as plume monitoring, evaluation of total dissolved mass, design of hydraulic containment and pump-and-treat systems, source identification and natural attenuation assessment. Kriged concentration maps can be improved by a coordinate transformation based on natural flow coordinates, as this allows to consider variations in the local anisotropy direction produced by heterogeneous groundwater flow. On the other hand, this does not consider nonstationarity induced by local dispersion. This paper presents a flexible kriging approach that combines a coordinate transformation based on groundwater flow and a class of nonstationary (NS) covariance functions that can be parameterized to account for the evolution of concentration correlation and variance with travel time/distance. We also propose an alternative flow coordinate transformation that does not rely on the stream function, enabling the computation of approximate 3-D flow coordinates. Nonstationary covariance parameters are estimated using maximum likelihood, and candidate parametrizations are compared by the likelihood ratio test and Akaike Information Criterion. The approach is tested on simple 2-D and 3-D synthetic plumes shaped by heterogeneous flow and local dispersion. Results show that: (1) NS covariance parameterizations improve significantly the likelihood compared to the stationary models, (2) the combined coordinate transformation and NS covariance approach enhances kriging of solute concentrations. Although the flow fields are considered known in the case studies, the approach can be modified to incorporate flow uncertainty and lends itself to various useful extensions, such as mass flux estimation.

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