  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). Our
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 of training models (400) sampled from
the prior distribution of K. The associated 400 pairs of simulated predictors
and targets are obtained through forward modelling. Newly collected field data
can then be directly used to predict the approximate posterior distribution of
the corresponding WHPA. The uncertainty range of the posterior WHPA
distribution is affected by the number and position of data sources (injection
wells). Our second objective is to extend BEL to identify the optimal design of
data source locations that minimizes the posterior uncertainty of the WHPA.
This can be done explicitly, without averaging or approximating because once
trained, the BEL model allows the computation of the posterior uncertainty
corresponding to any new input data. We use the Modified Hausdorff Distance and
the Structural Similarity index metrics to estimate the posterior uncertainty
range of the WHPA. Increasing the number of injection wells effectively reduces
the derived posterior WHPA uncertainty. Our approach can also estimate which
injection wells are more informative than others, as validated through a k-fold
cross-validation procedure. Overall, the application of BEL to experimental
design makes it possible to identify the data sources maximizing the
information content of any measurement data.
