Modelling solute transport in homogeneous and heterogeneous porous media using spatial fractional advection-dispersion equation

Abstract

This paper compared the abilities of advection-dispersion equation (ADE) and spatial fractional advection-dispersion equation (sFADE) to describe the migration of a non-reactive contaminant in homogeneous and heterogeneous soils. To this end, laboratory tests were conducted in a sandbox sizing 2.5 × 0.1 × 0.6 m (length × width × height). After performing a parametric sensitivity analysis, parameters of sFADE and ADE were individually estimated using the inverse problem method at each distance. The dependency of estimated parameters on distance was examined. The estimated parameters at 30 cm were used to predict breakthrough curves (BTCs) at subsequent distances. The results of sensitivity analysis indicated that average pore-water velocity and dispersion coefficient were, respectively, the most and least sensitive parameters in both mathematical models. The values of fractional differentiation orders (α) for sFADE were smaller than 2 in both soils. The scale-dependency of the dispersion coefficients of ADE and sFADE was observed in both soils. However, the application of sFADE to describe solute transport reduced the scale effect on the dispersion coefficient, especially in the heterogeneous soil. For the homogeneous soil, the predicting results of ADE and sFADE were nearly similar, while for the heterogeneous soil, the predicting results of sFADE were more satisfactory in comparison with those of ADE, especially when the transport distance increased. Compared to ADE, the sFADE simulated somewhat better the tailing parts of BTCs and showed the earlier arrival of tracer. Overall, the solute transport, especially in the heterogeneous soil, was non-Fickian and the sFADE somewhat better described non-Fickian transport.

The boundary element method applied to the solution of the anomalous diffusion problem

“…Recalling that 5) and discretising time t into intervals of duration ∆t, one can consider the integral in equation (3.5) time-step by time-step, over each of which we assume u to be constant, i.e.…”

“…Many applications of fractional calculus may be found in the literature. Examples include modelling the transmission and spread of the Zika virus in Brazil and internationally to 32 countries [3], the vibration analysis of a beam or plate resting on a viscoelastic soil foundation [4], and the simulation of solute transport through porous media [5]. A recent survey of these, and many more, applications can be found in Sun et al [6].…”

Numerical Simulation of Solute Transport in Saturated Porous Media with Bounded Domains

“…There is a large body of literature demonstrating that solute transport in porous media follows anomalous (non‐Fickian) transport type. The anomalous transport, resulted from heterogeneity of a porous medium, has been widely observed at laboratory and field scales (e.g., Benson 1998; Benson et al 2001; Huang et al 2006; Zhang et al 2009; Martinez et al 2010; Garrard et al 2017; Moradi and Mehdinejadiani 2018, 2020). Increasing evidence over the past two decades or more has revealed that advection‐dispersion equation (ADE), a Fickian‐based solute transport model, cannot explain well the solute transport process in porous media (e.g., Benson et al 2001; Huang et al 2006; Gao et al 2009; Garrard et al 2017; Lu et al 2018; Moradi and Mehdinejadiani 2018).…”

“…The anomalous transport, resulted from heterogeneity of a porous medium, has been widely observed at laboratory and field scales (e.g., Benson 1998; Benson et al 2001; Huang et al 2006; Zhang et al 2009; Martinez et al 2010; Garrard et al 2017; Moradi and Mehdinejadiani 2018, 2020). Increasing evidence over the past two decades or more has revealed that advection‐dispersion equation (ADE), a Fickian‐based solute transport model, cannot explain well the solute transport process in porous media (e.g., Benson et al 2001; Huang et al 2006; Gao et al 2009; Garrard et al 2017; Lu et al 2018; Moradi and Mehdinejadiani 2018). To capture the anomalous transport type, nonlocal solute transport models, including stochastic averaging of the classical advection–dispersion equation (SA‐ADE) method (Cushman 1987), multi‐rate mass transfer (MRMT, Haggerty and Gorelick 1995), continuous time random walk (CTRW, Berkowitz et al 2006), and fractional advection‐dispersion equations (FADEs, Zhang et al 2009) were proposed.…”

Numerical simulation of solute transport and structural analysis for groundwater connection medium based on the tracer test

“…Bianchi and Pedretti () proposed a novel approach to link solute transport behaviour to the physical heterogeneity of the aquifer, which they fully characterized with two measurable parameters. Moradi and Mehdinejadiani () established solute transport models in homogeneous and heterogeneous porous media using spatial fractional advection‐dispersion equation. Besides, the publications (Feehley et al , ; Dentz et al , ; Larsbo et al , ; Bijeljic et al , ; ; ; Anwar, ; Zhang et al , ; Gharehbaghi, ) also consider the solute transport.…”