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**An integro-differential model for non-Fickian tracer transport in porous media: validation and numerical simulation.**
*(English)*
Zbl 1398.76216

Summary: Diffusion processes have traditionally been modeled using the classical parabolic advection-diffusion equation. However, as in the case of tracer transport in porous media, significant discrepancies between experimental results and numerical simulations have been reported in the literature. Therefore, in order to describe such anomalous behavior, known as non-Fickian diffusion, some authors have replaced the parabolic model with the continuous time random walk model, which has been very effective. Integro-differential models (IDMs) have been also proposed to describe non-Fickian diffusion in porous media. In this paper, we introduce and test a particular type of IDM by fitting breakthrough curves resulting from laboratory tracer transport. Comparisons with the traditional advection-diffusion equation and the continuous time random walk are also presented. Moreover, we propose and numerically analyze a stable and accurate numerical procedure for the two-dimensional IDM composed by a integro-differential equation for the concentration and Darcy’s law for flow. In space, it is based on the combination of mixed finite element and finite volume methods over an unstructured triangular mesh.

### MSC:

76S05 | Flows in porous media; filtration; seepage |

76M10 | Finite element methods applied to problems in fluid mechanics |

65R20 | Numerical methods for integral equations |

45K05 | Integro-partial differential equations |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

### Keywords:

tracer transport; porous media; non-Fickian; integro-differential; CTRW; validation; numerical simulation### Software:

CTRW
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\textit{J. A. Ferreira} and \textit{L. Pinto}, Math. Methods Appl. Sci. 39, No. 16, 4736--4749 (2016; Zbl 1398.76216)

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