Stochastic model comparison and refinement strategies for gas migration in the subsurface

Abstract

Gas migration in the subsurface, a multiphase flow in a porous-medium system, is a problem of environmental concern and is also relevant for subsurface gas storage in the context of the energy transition. It is essential to know and understand the flow paths of these gases in the subsurface for efficient monitoring, remediation or storage operations. On the one hand, laboratory gas-injection experiments help gain insights into the involved processes of these systems. On the other hand, numerical models help test the mechanisms observed and inferred from the experiments and then make useful predictions for real-world engineering applications. Both continuum and stochastic modelling techniques are used to simulate multiphase flow in porous media. In this thesis, I use a stochastic discrete growth model: the macroscopic Invasion Percolation (IP) model. IP models have the advantages of simplicity and computational inexpensiveness over complex continuum models. Local pore-scale changes dominantly affect the flow processes of gas flow in water-saturated porous media. IP models are especially favourable for these multi-scale systems because using continuum models to simulate them can be extremely computationally difficult. Despite offering a computationally inexpensive way to simulate multiphase flow in porous media, only very few studies have compared their IP model results to actual laboratory experimental image data. One reason might be the fact that IP models lack a notion of experimental time but only have an integer counter for simulation steps that imply a time order. The few existing experiments-to-model comparison studies have used perceptual similarity or spatial moments as comparison measures. On the one hand, perceptual comparison between the model and experimental images is tedious and non-objective. On the other hand, comparing spatial moments of the model and experimental images can lead to misleading results because of the loss of information from the data. In this thesis, an objective and quantitative comparison method is developed and tested that overcomes the limitations of these traditional approaches. The first step involves volume-based time-matching between real-time experimental data and IP-model outputs. This is followed by using the (Diffused) Jaccard coefficient to evaluate the quality of the fit. The fit between the images from the models and experiments can be checked across various scales by varying the extent of blurring in the images. Numerical model predictions for sparsely known systems (like the gas flow systems) suffer from high conceptual uncertainties. In literature, numerous versions of IP models, differing in their underlying hypotheses, have been used for simulating gas flow in porous media. Besides, the gas-injection experiments belong to continuous, transitional, or discontinuous gas flow regimes, depending on the gas flow rate and the porous medium's nature. Literature suggests that IP models are well suited for the discontinuous gas flow regime; other flow regimes have not been explored. Using the abovementioned method, in this thesis, four macroscopic IP model versions are compared against data from nine gas-injection experiments in transitional and continuous gas flow regimes. This model inter-comparison helps assess the potential of these models in these unexplored regimes and identify the sources of model conceptual uncertainties. Alternatively, with a focus on parameter uncertainty, Bayesian Model Selection is a standard statistical procedure for systematically and objectively comparing different model hypotheses by computing the Bayesian Model Evidence (BME) against test data. BME is the likelihood of a model producing the observed data, given the prior distribution of its parameters. Computing BME can be challenging: exact analytical solutions require strong assumptions; mathematical approximations (information criteria) are often strongly biased; assumption-free numerical methods (like Monte Carlo) are computationally impossible for large data sets. In this thesis, a BME-computation method is developed to use BME as a ranking criterion for such infeasible scenarios: The \emph{Method of Forced Probabilities} for extensive data sets and Markov-Chain models. In this method, the direction of evaluation is swapped: instead of comparing thousands of model runs on random model realizations with the observed data, the model is forced to reproduce the data in each time step, and the individual probabilities of the model following these exact transitions are recorded. This is a fast, accurate and exact method for calculating BME for IP models which exhibit the Markov chain property and for complete "atomic" data. The analysis results obtained using the methods and tools developed in this thesis help identify the strengths and weaknesses of the investigated IP model concepts. This further aids model development and refinement efforts for predicting gas migration in the subsurface. Also, the gained insights foster improved experimental methods. These tools and methods are not limited to gas flow systems in porous media but can be extended to any system involving raster outputs.