Space‐fractional advection‐dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data

Abstract

To model the observed local variation of transport speed, an extension of the homogeneous space‐fractional advection‐dispersion equation (fADE) to more general cases with space‐dependent coefficients (drift velocity V and dispersion coefficient D) has been suggested. To provide a rigorous evaluation of this extension, we explore the underlying physical meanings of two proposed, and one other possible form, of the fADE by using the generalized mass balance law proposed by Meerschaert et al. (2006). When the classical Fick's law is replaced by its generalized form in the first‐order mass conversation law, the original fADE with constant parameters extends to the advection‐dispersion equation (ADE) with fractional flux (denoted as FF‐ADE). When the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence, we get the ADE with fractional divergence (FD‐ADE). When the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form, the fADE contains fully fractional divergence (FFD‐ADE). These three fADEs with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by Cushman et al. (1994), while the space‐variable fADEs correspond to the conditional nonlocal theory proposed by Neuman (1993) after specifying the general (Lévy‐type) functional form of the random distribution. The corresponding Langevin Markov models can be found in many cases, where the Lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical, measurable level and resolution. The resulting Lagrangian random‐walk particle tracking methods, along with previous numerical solutions using implicit Euler finite differences, distinguish and elucidate the plume behavior described by these fADEs. The fractional models are applied to fit the tritium plumes measured at the Macrodispersion Experiment test site. When the local parameters gleaned from the hydraulic conductivity (K) distribution are varied even slightly (i.e., a two‐zone model), allowing the use of observed hydraulic conditions at the site, the model fits are well within the variability of the data. The extended fADEs describe the fast and space‐dependent leading edges of measured plumes in the regional‐scale alluvial system, which was underestimated and could not be fully captured by the original fADE with constant parameters. Applications also favor the FD‐ADE model because of the ease of implementation and consistency with previous analysis of the K statistics.