Bayesian Evidential Learning combined with experimental design : the case of wellhead protection area prediction

Decisions related to groundwater management such as sustainable extraction of drinking water or protection against contamination can have great socio-economic impacts. Ideally, a complete uncertainty analysis should be performed to foresee all possible outcomes and assess any risk. Uncertainties arise from our limited understanding of the physical processes involved and the scarcity of measurement data, whether directly or indirectly related to the physical parameters of interest. In this contribution, we predict the wellhead protection area (WHPA, target), the shape and extent of which is influenced by the distribution of hydraulic conductivity (K), from a small number of tracing experiments (predictor). The first objective is to make stochastic predictions of the WHPA within the Bayesian Evidential Learning (BEL) framework, which aims to find a direct relationship between predictor and target using machine learning. This relationship is learned from a small set (200 samples) of training models sampled from the prior distribution of K, for which forward modeling is run to obtain the 200 pairs of simulated predictor and target. Newly collected field data can be directly used to predict the posterior distribution of an unknown WHPA, avoiding the classical step of data inversion. The uncertainty range of any prediction is affected by the number and position of data sources (injection wells). The second objective is then to extend BEL to identify the optimal design of data sources in order to minimize the prediction uncertainty of the WHPA. This can be done explicitly, without averaging or approximation because the BEL model, once trained, allows any new input data to readily compute the uncertainty, estimated with the Modified Hausdorff Distance (MHD) and the Structural Similarity (SSIM) index metrics. We demonstrate that increasing the number of injection wells effectively reduces the uncertainty on the WHPA prediction, because the breakthrough curves store information on a larger area of the K field surrounding the pumping well, and that some injection wells are more informative than others, as validated through the k-fold cross-validation procedure. Overall, the application of experimental design combined with BEL makes it possible to identify the data sources maximizing the information content of any measurement data within limited budget constraints, and at low computational costs.