Search for: All records

Abstract Pollutant transport in discrete fracture networks (DFNs) exhibits complex dynamics that challenge reliable model predictions, even with detailed fracture data. To address this issue, this study derives an upscaled integral‐differential equation to predict transient anomalous diffusion in two‐dimensional (2D) DFNs. The model includes both transmissive and dead‐end fractures (DEFs), where stagnant water zones in DEFs cause non‐uniform flow and transient sub‐diffusive transport, as shown by both literature and DFN flow and transport simulations using COMSOL. The upscaled model's main parameters are quantitatively linked to fracture properties, especially the probability density function of DEF lengths. Numerical experiments show the model's accuracy in predicting the full‐term evolution of conservative tracers in 2D DFNs with power‐law distributed fracture lengths and two orientation sets. Field applications indicate that while model parameters for transient sub‐diffusion can be predicted from observed DFN distributions, predicting parameters controlling solute displacement in transmissive fractures requires additional field work, such as tracer tests. Parameter sensitivity analysis further correlates late‐time solute transport dynamics with fracture properties, such as fracture density and average length. Potential extensions of the upscaled model are also discussed. This study, therefore, proves that transient anomalous transport in 2D DFNs with DEFs can be at least partially predicted, offering an initial step toward improving model predictions for pollutant transport in real‐world fractured aquifer systems. 

Backward models for super-diffusion have been developed to identify pollutant source locations, but they are limited to a single observation and disregard field-measured concentrations. To overcome these limitations, this study derives the adjoint of the space-fractional advection–dispersion equation, incorporating measured concentrations from multiple observation data. Backward probabilities, such as the backward location probability density function (PDF), describe the likely source location(s) at a fixed time before sampling, offering a comprehensive modeling approach for source identification. By applying Bayes’ theorem, the individual PDFs from each observation and its corresponding concentration are combined into a joint PDF, enhancing both the information and reliability compared to the previous single PDF. Field applications show that the improved model enhances accuracy (with PDF peak locations closer to the actual source) and precision (with reduced variance) of backward PDFs for identifying point sources in a natural river and aquifer. The model’s performance is affected by observation count and measurement errors, with double peaks in the backward location PDF possible due to source mass uncertainty. Future refinements, such as incorporating backward travel time analysis and extending applications to reactive pollutants, could further enhance the utility of the conditioned backward fractional-derivative model developed in this study. 

In complex physical systems, conventional differential equations fall short in capturing non-local and memory effects. Fractional differential equations (FDEs) effectively model long-range interactions with fewer parameters. However, deriving FDEs from physical principles remains a significant challenge. This study introduces a stepwise data-driven framework to discover explicit expressions of FDEs directly from data. The proposed framework combines deep neural networks for data reconstruction and automatic differentiation with Gauss-Jacobi quadrature for fractional derivative approximation, effectively handling singularities while achieving fast, high-precision computations across large temporal/spatial scales. To optimize both linear coefficients and the nonlinear fractional orders, we employ an alternating optimization approach that combines sparse regression with global optimization techniques. We validate the framework on various datasets, including synthetic anomalous diffusion data, experimental data on the creep behavior of frozen soils, and single-particle trajectories modeled by Lévy motion. Results demonstrate the framework’s robustness in identifying FDE structures across diverse noise levels and its ability to capture integer order dynamics, offering a flexible approach for modeling memory effects in complex systems.