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A time-space Hausdorff derivative model for anomalous transport in porous media

  • Yingjie Liang EMAIL logo , Ninghu Su and Wen Chen

Abstract

This paper presents a time-space Hausdorff derivative model for depicting solute transport in aquifers or water flow in heterogeneous porous media. In this model, the time and space Hausdorff derivatives are defined on non-Euclidean fractal metrics with power law scaling transform which, respectively, connect the temporal and spatial complexity during transport. The Hausdorff derivative model can be transformed to an advection-dispersion equation with time- and space-dependent dispersion and convection coefficients. This model is a fractal partial differential equation (PDE) defined on a fractal space and differs from the fractional PDE which is derived for non-local transport of particles on a non-fractal Euclidean space. As an example of applications of this model, an explicit solution with a constant diffusion coefficient and flow velocity subject to an instantaneous source is derived and fitted to the breakthrough curves of tritium as a tracer in porous media. These results are compared with those of a scale-dependent dispersion model and a time-scale dependent dispersion model. Overall, it is found that the fractal PDE based on the Hausdorff derivatives better captures the early arrival and heavy tail in the scaled breakthrough curves for variable transport distances. The estimated parameters in the fractal Hausrdorff model represent clear mechanisms such as linear relationships between the orders of Hausdorff derivatives and the transport distance. The mathematical formulation is applicable to both solute transport and water flow in porous media.

MSC 2010: 26A24; 28A80; 35K57; 76S05

Appendix A. Details of the scale-dependent dispersion model (SDM), and the time-scale dependent dispersion model (TSDM)

The SDM [37] for one dimensional solute transport in porous media is described by

Rc(x,t)t=xD(x)c(x,t)xVc(x,t)x,(8.1)

where c(x, t) is the solute concentration, D = εxV is the diffusion (or dispersion) coefficient, ε the the ratio of dispersivity/distance, x the longitudinal distance, R retardation factor, and V is the flow velocity. When D(x) is constant, the above equation reduces to the traditional constant dispersion equation.

In the TSDM, the governing equation is given by

c(x,t)t=xDc(x,t)xVc(x,t)x,(8.2)

where c(x, t) is the solute concentration, V is the flow velocity, the dispersion coefficient D

D=D0Dx(x)Dt(t)Vn,(8.3)

where D0 is a constant, 1 ≤ n ≤ 2, Dx(x) = xm and Dt(x) = tλ respectively represent the spatial and temporal components of the dispersion coefficient, m and λ are fractal parameters of the media and flow, respectively.

Acknowledgements

The work presented in this paper was supported by the Fundamental Research Funds for the Central Universities (No. 2019B16114), National Natural Science Foundation of China (Nos. 11702085, 11772121), and the China Postdoctoral Science Foundation (No. 2018M630500), and the Opening Fund of MOE Key Laboratory of Groundwater Circulation and Environmental Evolution (No. 20185016112).

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Received: 2019-03-11
Published Online: 2019-12-31
Published in Print: 2019-12-18

© 2019 Diogenes Co., Sofia

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