[Paper Review] A new framework for experimental design using Bayesian Evidential Learning: the case of wellhead protection area

Robin Thibaut, Eric Laloy+1 more|arXiv (Cornell University)|May 12, 2021

Groundwater flow and contamination studies67 references33 citations

TL;DR

This paper proposes a Bayesian Evidential Learning (BEL) framework that directly links tracer breakthrough curves to wellhead protection area (WHPA) predictions, enabling fast, calibration-free stochastic uncertainty quantification and optimal experimental design. By training on 400 forward-modeled realizations, BEL predicts full posterior WHPA distributions and identifies the most informative injection well locations—validated via k-fold cross-validation with a 250-sample test set—reducing computational cost while maintaining accuracy.

ABSTRACT

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.

Motivation & Objective

• To develop a computationally efficient method for stochastic prediction of wellhead protection areas (WHPA) under subsurface uncertainty.
• To apply Bayesian Evidential Learning (BEL) to bypass traditional model calibration and directly link tracer data (predictors) to WHPA (target) predictions.
• To identify optimal injection well locations for tracer experiments that minimize posterior WHPA uncertainty.
• To validate the informativeness of specific data sources using k-fold cross-validation and uncertainty metrics.
• To demonstrate that a small training set (400 models) is sufficient for reliable WHPA prediction and experimental design.

Proposed method

• Train a BEL model on 400 forward-simulated realizations of hydraulic conductivity fields and corresponding tracer breakthrough curves and WHPA shapes.
• Use Canonical Correlation Analysis (CCA) in reduced-dimensional space to learn a direct, non-linear mapping between breakthrough curves (predictors) and WHPA (target).
• Apply the trained BEL model to new field data to directly compute the full posterior distribution of the WHPA without iterative inversion.
• Quantify posterior uncertainty using Modified Hausdorff Distance (MHD) and Structural Similarity (SSIM) indices as data-utility functions.
• Perform k-fold cross-validation with varying test set sizes (100 and 250 samples) to validate robustness and identify optimal data set size.
• Evaluate the information content of individual injection wells by comparing MHD and SSIM metrics across folds and data configurations.

Experimental results

Research questions

• RQ1 Can Bayesian Evidential Learning (BEL) be used to predict the full posterior distribution of a wellhead protection area (WHPA) from tracer breakthrough curves without model calibration or inversion?
• RQ2 Which injection well locations provide the highest information content for reducing WHPA prediction uncertainty?
• RQ3 What is the minimum number of test samples required to reliably rank the informativeness of data sources in WHPA experimental design?
• RQ4 How does the size of the training dataset (e.g., 400 vs. 1000 models) affect the robustness of WHPA prediction and experimental design outcomes?
• RQ5 Can BEL-based experimental design outperform traditional methods like Bayesian Model Averaging or surrogate modeling in terms of computational efficiency and accuracy?

Key findings

• A training set of 400 models is sufficient for both accurate WHPA prediction and robust experimental design using BEL.
• The most informative injection wells are consistently ranked as numbers 4, 5, and 6 (downstream wells), with well 6 showing the narrowest uncertainty bounds across all k-fold splits.
• Well 1 (upstream) is consistently the least informative, with the broadest uncertainty intervals, indicating low information content.
• A test set of at least 250 samples is required to achieve consistent and reliable ranking of data source informativeness across k-fold cross-validation.
• The use of MHD and SSIM as data-utility functions enables direct, computationally efficient uncertainty quantification without full posterior sampling.
• The BEL framework avoids the high computational cost of Markov Chain Monte Carlo or surrogate modeling while maintaining predictive accuracy and enabling explicit experimental design.