Abstract
Fractional calculus provides efficient physical models to quantify non-Fickian dynamics broadly observed within the Earth system. The potential advantages of using fractional partial differential equations (fPDEs) for real-world problems are often limited by the current lack of understanding of how earth system properties influence observed non-Fickian dynamics. This study explores non-Fickian dynamics for pollutant transport in field-scale discrete fracture networks (DFNs), by investigating how fracture and rock matrix properties influence the leading and tailing edges of pollutant breakthrough curves (BTCs). Fractured reservoirs exhibit erratic internal structures and multi-scale heterogeneity, resulting in complex non-Fickian dynamics. A Monte Carlo approach is used to simulate pollutant transport through DFNs with a systematic variation of system properties, and the resultant non-Fickian transport is upscaled using a tempered-stable fractional in time advection–dispersion equation. Numerical results serve as a basis for determining both qualitative and quantitative relationships between BTC characteristics and model parameters, in addition to the impacts of fracture density, orientation, and rock matrix permeability on non-Fickian dynamics. The observed impacts of medium heterogeneity on tracer transport at late times tend to enhance the applicability of fPDEs that may be parameterized using measurable fracture–matrix characteristics.
Highlights
Fractional calculus, defined by non-integer order derivatives and integrals, has been applied to problems involving non-Fickian or anomalous dynamics for almost three decades [1,2]
The ensemble average of breakthrough curves (BTCs) for all 100 realizations for each discrete fracture networks (DFNs) scenario are shown in Figure 2, along with the best-fit solutions using the tt-fractional advection–dispersion equation (fADE) model (10) and (11)
An additional fourth scenario explored the impact of fracture orientation on transport dynamics, by changing mean fracture set orientation to −45◦ and 45◦ for a 60 fracture DFN
Summary
Fractional calculus, defined by non-integer order derivatives and integrals, has been applied to problems involving non-Fickian or anomalous dynamics for almost three decades [1,2] Despite their vast potential, both theoretical development and real-world applications of fractional partial differential equations (fPDEs) have been commonly constrained by the lack of understanding of how earth system properties influence non-Fickian transport dynamics, especially for the hydrologic sciences [3]. Both theoretical development and real-world applications of fractional partial differential equations (fPDEs) have been commonly constrained by the lack of understanding of how earth system properties influence non-Fickian transport dynamics, especially for the hydrologic sciences [3] This major challenge has historically reduced fPDEs to curve-fitting mathematical exercises, instead of routine hydrological modeling tools [4]. After Benson et al [11] first introduced the spatial fractional advection–dispersion equation (fADE) to capture super-diffusive transport in sand tanks and a relatively homogenous aquifer, the fADEs had been applied to model anomalous transport in saturated, heterogeneous porous media [12,13,14,15] and Earth surfaces, such as natural rivers [16,17,18]
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