Challenges in the Application of Fractional Derivative Models in Capturing Solute Transport in Porous Media: Darcy-Scale Fractional Dispersion and the Influence of Medium Properties

Table 1

Comparison of the TFDM (2) and the standard CTRW model reviewed by Berkowitz et al. [9]. Note that the standard CTRW defines various empirical memory functions, and the memory function listed below is the most commonly used one (which is a truncated power law similar to the memory function used in the TFDM (2a) and (2b)). In the CTRW model, the parameter 𝜏₂=𝑡₂/𝑡₁ denotes the ratio of the two characteristic times 𝑡₂ and 𝑡₁, and Γ[−𝜉,(𝜏₂)⁻¹] is the incomplete Gamma function.

Comparison	TFDM (2a) and (2b)	The standard CTRW [9]
Physical theory behind the model	Scaling limit of CTRW	The general master equation
Form of the memory function	∫^∞𝑡𝑒^−𝜆𝑟𝛾𝑟^−𝛾−1Γ1−𝛾𝑑𝑟	𝜏2^𝜉exp−𝑡/𝑡21+𝑡/𝑡1^−1−𝜉𝑡1exp[𝜏2^−1]Γ[−𝜉,𝜏2^−1]
Number and value of parameters to capture sub-diffusion	3 (𝛽^∗,𝜆,𝛾), with 0<𝛾<1	4 (𝛽^∗,𝜉,𝑡1,𝑡2), with 0<𝜉<2
Mechanism for sub-diffusion	Diffusion	Slow advection
Modeling super-diffusion	Applicable with 1<𝛼<2	N/A
Modeling the mixed diffusion	Applicable with 0<𝛾<1 and 1<𝛼<2	N/A
Multidimensional extension	Multiscaling index [12]	No multiscaling index
Spatial variability of transport	𝑉 and 𝐷 can vary in space	𝑉 and 𝐷 are spatial averages