The effect of non-Fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer

Abstract

Solute transport in a fractured porous confined aquifer is modelled by using an equation with a fractional-in-time derivative of order γ , which may vary from 0 to 1. Accounting for non-Fickian diffusion into the surrounding rock mass, which is modelled by a fractional spatial derivative of order α , leads to the introduction of an additional fractional-in-time derivative of order α /(1+ α ) in the equation for solute transport. Closed-form solutions for solute concentrations in the aquifer and surrounding rocks are obtained for an arbitrary time-dependent source of contamination located at the inlet of the aquifer. Based on these solutions, different regimes of contaminant transport in aquifers with various physical properties are modelled and analysed.

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

“…Our computations show that the difference between the values of Q 1 computed by Equations (32) and (33) is negligibly small within the relatively long time interval from 0 to 1/Λ and β ≥ 0.1 [23]. Thus, Equation (33) can be used as a good approximation for Q 1 , the exact value of which is given by Equation (32).…”

“…The fracture does not contain any porous inclusions, which is different from the concept in the previous study [23]. A schematic sketch of a fracture and rock matrix is presented in Figure 1.…”

“…This paper provides the analytic derivation of the function Q 1 (t) in its most general form, so that the majority of the well documented models can be obtained as particular cases of the presented model. For example, in the absence of radioactive decay, the solution presented in this paper reduces to the solution obtained by Fomin et al [23] for Λ = 0 and Q 1 = Q 0 . When mass flux is Fickian with Λ = 0 and α = 1, the solution was obtained by Tang et al [12], which also follows as a particular case from the solution obtained in this paper.…”

“…Applying the technique of the group analysis of differential equations [31], the solution of the formulated boundary-value problem (18)- (21) can be obtained in the following form [23]:…”

“…Liu et al [22] considered the time fractional advection-dispersion equation, and the solution was obtained by using variable transformation. Fomin et al [23] studied mass transport in a fractured-porous aquifer (i.e., aquifer filled with porous blocks) and modeled the effects of interaction with porous blocks in the aquifer by temporal fractional derivatives. Numerical study shows that varying the variations of order of fractional derivatives enables the description of different power law decays obtained from a homogeneous porous medium to a fractured medium [24].…”