Numerous variations of Invasion-Percolation (IP) models can simulate multiphase flow in porous media across various scales (pore-scale IP to macroscopic IP); here, we are interested in gas flow in water-saturated porous media. This flow occurs either as continuous or discontinuous flow, depending on the 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. Our research compares four existing macroscopic IP models and ranks their performance in these “other” flow regimes. We test the models on a range of gas-injection in water-saturated sand experiments from transitional and continuous gas flow regimes. Using the light transmission technique, the experimental data is obtained as a time series of images in a 2-dimensional setup. To represent pore-scale heterogeneities, we ran each model version on several random realizations of the initial entry pressure field. We use a diffused version of the so-called Jaccard coefficient to rank the models against the experimental data. We average the Jaccard coefficient over all realizations per model version to evaluate each model and calibrate specific model parameters. Depending on the application domain, we observe that some macroscopic IP model versions are suitable in these previously unexplored flow regimes. Also, we identify that the initial entry pressure fields strongly affect the performance of these models. Our comparison method is not limited to gas-water systems in porous media but generalizes to any modelling situation accompanied by spatially and temporally highly resolved data.

Improved understanding of complex hydrosystem processes is key to advance water resources research. Nevertheless, the conventional way of modeling these processes suffers from a high conceptual uncertainty, due to almost ubiquitous simplifying assumptions used in model parameterizations/closures. Machine learning (ML) models are considered as a potential alternative, but their generalization abilities remain limited. For example, they normally fail to predict across different boundary conditions. Moreover, as a black box, they do not add to our process understanding or to discover improved parameterizations/closures. To tackle this issue, we propose the hybrid modeling framework FINN (finite volume neural network). It merges existing numerical methods for partial differential equations (PDEs) with the learning abilities of artificial neural networks (ANNs). FINN is applied on discrete control volumes and learns components of the investigated system equations, such as numerical stencils, model parameters, and arbitrary closure/constitutive relations. Consequently, FINN yields highly interpretable results. To show this, we demonstrate FINN on a diffusion-sorption problem in clay. Results on numerically generated data show that FINN outperforms other ML models when tested under modified boundary conditions, and that it can successfully differentiate between the usual, known sorption isotherms. Moreover, we also equip FINN with uncertainty quantification methods to lay open the total uncertainty of scientific learning, and then apply it to a laboratory experiment. The results show that FINN performs better than calibrated PDE-based models as it is not restricted to choose among a limited set of sorption isotherms.

Bayesian model selection (BMS) and Bayesian model justifiability analysis (BMJ) provide a statistically rigorous framework to compare competing conceptual models through the use of Bayesian model evidence (BME). However, BME-based analysis has two main limitations: (1) it’s powerless when comparing models with different data set sizes and/or types of data and (2) doesn’t allow to judge a model’s performance based on its posterior predictive capabilities. Thus, traditional BME-based approaches ignore useful data or models due to issue (1) or disregards Bayesian updating because of issue (2). To address these limitations, we advocate to include additional information-theoretic scores into BMS and BMJ analysis: expected log-predictive density (ELPD), relative entropy (RE) and information entropy (IE). Exploring the connection between Bayesian inference and information theory, we explicitly link BME and ELPD together with RE and IE to indicate the information flow in BMS and BMJ analysis. We show how to compute and interpret these scores alongside BME, and apply it in a model selection and similarity analysis framework. We test the methodology on a controlled 2D groundwater setup considering five competing conceptual models accompanied with different data sets. The results show how the information-theoretic scores complement BME by providing a more complete picture concerning the Bayesian updating process. Additionally, we present how both RE and IE can be used to objectively compare models that feature different data sets. Overall, the introduced Bayesian information-theoretic framework helps to avoid any potential loss of information and leads to an informed decision for model selection and similarity.