Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

Abstract

Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior with greater efficiency than fully-resolved classical models. In this review article, we first provide a broad overview of fractional-order derivatives with a clear emphasis on the stochastic processes that underlie their use. We then survey three exemplary application areas -subsurface transport, turbulence, and anomalous materials -in which fractional-order differential equations provide accurate and predictive models. For each area, we report on the evidence of anomalous behavior that justifies the use of fractional-order models, and survey both foundational models as well as more expressive state-of-the-art models. We also propose avenues for future research, including more advanced and physically sound models, as well as tools for calibration and discovery of fractional-order models.

Machine-learning of nonlocal kernels for anomalous subsurface transport from breakthrough curves

“…We refer to the latter as nonlocal "dynamic" kernel and provide more details on its choice in Section 3.3. Due to the established relationship, in the case of fractional models, between the nonlocal kernel and the jump rate of the stochastic process associated with the fractional equation [7,26], we assume that the nonlocal kernel is nonnegative. This assumption, enforced as a constraint in the learning procedure described in the following section, also guarantees the well-posedness of problem (5) [7].…”

“…Making accurate large-scale predictions of solute transport in the subsurface is critically important for the efficient management of water resources [25,26] as well as petroleum production, particularly in enhanced oil-recovery (EOR) applications [1,14,24,21]. Subsurface transport is a highly complex phenomenon as it takes place in environments that contain heterogeneities at all scales, requiring the use of fine-scale models at the smallest scales.…”

“…concentration profiles at a specific location as a function of the time. In recent years, substantial effort has been devoted to the development of fractional advection-dispersion equations for describing anomalous transport in heterogeneous environments [3,4,12,19,26], as these models intrinsically yield the desired nonlinear MSD and embed all length scales in their definition. At the same time, fractional models come with several computational challenges, mostly due to their high computational cost and to their nontrivial implementation.…”

An asymptotically compatible coupling formulation for nonlocal interface problems with jumps

“…Nonlocal models have become popular alternatives to model phenomena where classical partial differential equations fail to be descriptive. These phenomena include the presence of discontinuities in the solution (such as fractures in continuum mechanics [36]), the appearance of anomalous diffusion effects (such as super-and sub-diffusion in subsurface transport and turbulence [38]), and the presence of heterogeneities at the small scales that affect the global behavior of a system at the macro scale (such as the effects of micro-scale heterogeneities in wave propagation [43]). As a result, several scientific and engineering fields have benefited from the use of nonlocal equations.…”

Nonlocal Machine Learning of Micro-Structural Defect Evolutions in Crystalline Materials

“…These features are attractive in the solution of problems involving convection-diffusion [16,11], heterogeneous media [17], turbulent flows [2,37,33,3,34], anomalous materials [45], and subsurface dispersion [36,50]. For more applications, please refer to [44] and references therein. The peridynamic theory [38] was proposed as a nonlocal alternative to classical continuum mechanics of solids, with applicability in fracture problems with discontinuities [39,40].…”