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Quantitative predictions of natural and induced phenomena in fractured rock is one of the great challenges in the Earth and Energy Sciences with far-reaching economic and environmental impacts. Fractures occupy a very small volume of a subsurface formation but often dominate fluid flow, solute transport and mechanical deformation behavior. They play a central role in CO2 sequestration, nuclear waste disposal, hydrogen storage, geothermal energy production, nuclear nonproliferation, and hydrocarbon extraction. These applications require predictions of fracture-dependent quantities of interest such as CO2 leakage rate, hydrocarbon production, radionuclide plume migration, and seismicity; to be useful, these predictions must account for uncertainty inherent in subsurface systems. Here, we review recent advances in fractured rock research covering field- and laboratory-scale experimentation, numerical simulations, and uncertainty quantification. We discuss how these have greatly improved the fundamental understanding of fractures and one's ability to predict flow and transport in fractured systems. Dedicated field sites provide quantitative measurements of fracture flow that can be used to identify dominant coupled processes and to validate models. Laboratory-scale experiments fill critical knowledge gaps by providing direct observations and measurements of fracture geometry and flow under controlled conditions that cannot be obtained in the field. Physics-based simulation of flow and transport provide a bridge in understanding between controlled simple laboratory experiments and the massively complex field-scale fracture systems. Finally, we review the use of machine learning-based emulators to rapidly investigate different fracture property scenarios and accelerate physics-based models by orders of magnitude to enable uncertainty quantification and near real-time analysis.

This paper presents a physics-informed machine learning (ML) framework to construct reduced-order models (ROMs) for reactive-transport quantities of interest (QoIs) based on high-fidelity numerical simulations. QoIs include species decay, product yield, and degree of mixing. The ROMs for QoIs are applied to quantify and understand how the chemical species evolve over time. First, high-resolution datasets for constructing ROMs are generated by solving anisotropic reaction–diffusion equations using a non-negative finite element formulation for different input parameters. The reactive-mixing model input parameters are: time-scale associated with flipping of velocity, spatial-scale controlling small/large vortex structures of velocity, perturbation parameter of the vortex-based velocity, anisotropic dispersion contrast, and molecular diffusion. Second, random forests, F-test, and mutual information criterion are used to evaluate the importance of model inputs/features with respect to QoIs. We observed that anisotropic dispersion contrast is the most important feature and time-scale associated with flipping of velocity is the least important feature. Third, Support Vector Machines (SVM) and Support Vector Regression (SVR) are used to construct ROMs based on the model inputs. The constructed SVR-ROMs are then used to predict scaling of QoIs. We also present estimates and inequalities on the QoIs, which inform that the species decay, mix, and produce in an exponential fashion. These inequalities also inform that a radial basis function is the most suitable kernel for the SVM/SVR models for QoIs. It is observed that R 2 -score for SVR-ROMs on unseen data is greater than 0.9, implying that the SVR-ROMs are able to predict the reaction–diffusion system state reasonably well. Finally, in terms of the computational cost, the proposed SVM-ROMs are O (1 0 7) times faster than running a high-fidelity finite element simulation for evaluating QoIs. This makes the proposed ML-based ROMs attractive for reactive-transport sensing and real-time monitoring applications as they are significantly faster yet reasonably accurate. • ML-ROMs are built to predict reactive-mixing under heterogeneity and anisotropy. • Accuracy of our ROMs on unseen data is greater than 90%. • Proposed ROMs are ≈ 1 0 7 times faster than high-fidelity numerical simulations. [ABSTRACT FROM AUTHOR]

Analysis of reactive-diffusion simulations representing complex mixing processes requires a large number of independent model runs. For each high-fidelity model simulation, the model inputs are varied and the predicted mixing behavior is represented by temporal and spatial changes in species concentration. It is then required to discern how the model inputs (such as diffusivity, dispersion, anisotropy, and velocity field properties) impact the mixing process. This task is challenging and typically involves interpretation of large model outputs representing temporal and spatial changes of species concentration within the model domain. However, the task can be automated and substantially simplified by applying Machine Learning (ML) methods. In this paper, we present an application of an unsupervised ML method (called NTF k) using Non-negative Tensor Factorization (NTF) coupled with a custom clustering procedure based on k -means to reveal the temporal and spatial features in product concentrations. An attractive and unique aspect of the proposed ML method is that it ensures the extracted features are non-negative, which is important to obtain a meaningful deconstruction of the mixing processes. The ML methodology is applied to a large set of high-resolution finite-element model simulations representing anisotropic reaction-diffusion processes in perturbed vortex-based velocity fields. The applied finite-element method ensures that spatial and temporal species concentration are always non-negative, even in the case of high anisotropic contrasts. The simulated reaction is a fast irreversible bimolecular reaction A + B → C , where species A and B react to form species C. The reactive-diffusion model input parameters that control mixing include properties of the velocity field (such as vortex structures), anisotropic dispersion, and molecular diffusion. We demonstrate the applicability of the ML feature extraction method to produce a meaningful deconstruction of model outputs to discriminate between different physical processes impacting the reactants, their mixing, and the spatial distribution of the product C. The presented ML analysis allowed us to identify additive temporal and spatial features that characterize mixing behavior. The application of the proposed NTF k approach is not limited to reactive-mixing. NTF k can be readily applied to any observed or simulated datasets that can be represented as tensors (multi-dimensional arrays) and have separable latent signatures or features. • NTF k provides an elegant way to extract hidden features in irreversible fast bimolecular reaction-diffusion systems. • Features extracted using NTF k are non-negative which is an important constraint to ensure physically meaningful decomposition of the reactive-mixing process. • Added benefit of our method is data compression along with minimal loss of information. [ABSTRACT FROM AUTHOR]

• Machine learning (ML) can understand reactive-mixing phenomena. • ML predicts quantities of interest (QoIs) of reactive-mixing phenomena. • 20 Machine learning emulators are used to predict reactive-mixing phenomena. • Linear and Bayesian emulators predict QoIs with accuracy <80%. • Ensemble and MLP ML emulators predict QoIs with accuracy >99%. Mixing phenomena are important mechanisms controlling flow, species transport, and reaction processes in fluids and porous media. Accurate predictions of reactive mixing are critical for many Earth and environmental science problems such as contaminant fate and remediation, macroalgae growth, and plankton biomass growth. To investigate the evolution of mixing dynamics under different scenarios (e.g., anisotropy, fluctuating velocity fields), a finite-element-based numerical model was built to solve the fast, irreversible bimolecular reaction-diffusion equations to simulate a range of reactive-mixing scenarios. A total of 2,315 simulations were performed using different sets of model input parameters comprising various spatial scales of vortex structures in the velocity field, time-scales associated with velocity oscillations, the perturbation parameter for the vortex-based velocity, anisotropic dispersion contrast (i.e., ratio of longitudinal-to-transverse dispersion), and molecular diffusion. The outputs comprised concentration profiles of reactants and products. The inputs and outputs from these simulations were concatenated into feature and label matrices, respectively, to train 20 different machine learning (ML) models intended to emulate system behavior. These 20 ML emulators, based on linear methods, Bayesian methods, ensemble learning methods, and multilayer perceptrons (MLPs), were trained to classify the state of mixing and predict three quantities of interest (QoIs) characterizing species production, decay (i.e., average concentration, square of average concentration), and degree of mixing (i.e., variances of species concentration). Unsurprisingly, linear classifiers and regressors failed to reproduce the QoIs; however, ensemble methods (classifiers and regressors) and the MLP model accurately classified the state of reactive mixing and the QoIs. Among ensemble methods, random forest and decision-tree-based AdaBoost faithfully predicted the QoIs. At run time, trained ML emulators produced results ≈ 10 5 times faster than the finite-element simulations. Due to their low computational expense and high accuracy, ensemble and MLP models are excellent emulators for these numerical simulations and great utilities in uncertainty quantification exercises, which can require 1,000s of forward model runs. [ABSTRACT FROM AUTHOR]