Conventional Advection-dispersion Equation Research Articles

Experimental Research on the Thresholds of Scale-Dependent Dispersivity of Solute Transport

Abstract The conventional advection-dispersion equation (ADE) has been widely used to describe the solute transport in porous media. However, it cannot interpret the phenomena of the early arrival and long tailing in breakthrough curves (BTCs). In this study, we aim to experimentally investigate the behaviors of the solute transport in both homogeneous and heterogeneous porous media. The linear-asymptotic model (LAF solution) with scale-dependent dispersivity was used to fit the BTCs, which was compared with the results of the ADE model and the conventional truncated power-law (TPL) model. Results indicate that (1) the LAF model with linear scale-dependent dispersivity could better capture the evolution of BTCs than the ADE model; (2) dispersivity initially increases linearly with the travel distance and is stable at some limited value over a large distance, and a threshold value of the travel distance is provided to reflect the constant dispersivity; and (3) compared with the TPL model, both the LAF and ADE models can capture the behavior of solute transport as a whole. For fitting the early arrival, the LAF model is less than the TPL; however, the LAF model is more concise in mathematics and its application will be studied in the future.

Influence of Clay on Solute Transport in Saturated Homogeneous Mixed Media

In this study, four homogeneous porous media (HPM1-HPM4), consisting of distinct proportions of sand-sized and clay-sized solid beads, were prepared and used as single fracture infills. Flow and nonreactive solute transport experiments in HPM1-HPM4 under three flow rates were conducted, and the measured breakthrough curves (BTCs) were quantified using conventional advection-dispersion equation (ADE), mobile-immobile model (MIM), and continuous time random walk (CTRW) model with truncated power law transition time distribution. The measured BTCs showed stronger non-Fickian behaviour in HPM2-HPM4 (which had clay) than in HPM1 (which had no clay), implying that clay enhanced the non-Fickian transport. As the fraction of clay increased, the global error of ADE fits also increased, affirming the inefficiency of ADE in capturing the clay-induced non-Fickian behaviour. MIM and CTRW performed better in capturing the non-Fickian behaviour. Nonetheless, CTRW’s performance was robust. 12.5% and 25% of clay in HPM2 and HPM3, respectively, decreased the flowing fluid region and increased the solute exchange rate between the flowing and stagnant fluid regions in MIM. For CTRW, the power law exponent ( β CTRW ) values were 1.96, 1.75, and 1.63 in HPM1-HPM3, respectively, implying enhanced non-Fickian behaviour. However, for HPM4, whose clay fraction was 50%, the β CTRW value was 1.87, implying a deviation in the trend of non-Fickian enhancement with increasing clay fraction. This deviation indicated that non-Fickian behaviour enhancement depended on the fraction of clay present. Moreover, increasing flow rate enhanced the non-Fickian transport based on β CTRW .

Functioning of the multilinear lag-cascade flood routing model as a means of transporting pollutants in the river

AbstractThe purpose of this paper is to use the application of the multilinear lag cascade model as a contaminant transport model through river networks. Monocacy River and Antietam Creek data, which were collected by USGS with different reach lengths and discharge conditions, have been used in the current study. It was found that the multilinear discrete lag-cascade (MDLC) model is capable of reconstructing contaminant breakthrough curves. A complete study was performed to estimate the reach length for use in the accurate simulation, and it was concluded that by assuming a uniform flow through the reach, the length unit should be obtained by applying Pe = 12. Moreover, by using temporal moment matching, explicit relationships for MDLC model parameters (k, n, and τ) and based on conventional advection-dispersion equation (ADE) parameters (D, u, x) were extracted. MDLC parameters of the field breakthrough curves were extracted, and it was found that the increase of Pe number caused an increase in delay time and the number of cascades. However, the residence time was obtained to be fixed. Additionally, by assuming the dispersivity parameter (D/u) is constant, the changes in the MDLC parameters were investigated by velocity variation, and new relationships were proposed to estimate the parameters under different hydraulic conditions. Using presented equations provided in this study for residence time (k), cascade number (n), and delay time (τ), the sensitivity analysis was performed, and it was found that the parameters of velocity (u), dispersion coefficient (D), and velocity (u) have the most important effect in calculation of them, respectively.

Flow transiency on analytical modeling of subsurface solute transport

Groundwater flow velocity and dispersivity might be temporally or spatially variable rather than constant. In this paper, linearly-asymptotically or exponentially distance-dependent dispersivities and temporally exponential flow velocity were coupled to the conventional advection-dispersion equation. The mathematical models were established by considering the case of a coupled time-dependent velocity and scale-dependent dispersivities where one-dimensional (1D) semi-analytical solutions were obtained using the Laplace transform in a finite domain. The solution was verified by comparing it with a numerical solution, based on finite-element COMSOL Multiphysics. The impacts of different parameters of time-dependent flow velocity and scale-dependent dispersivities on breakthrough curves (BTCs) were thoroughly analyzed. The results show that a slight change of time-dependent flow velocity will lead to considerable change of BTCs, meaning that solute transport is sensitive to the temporally variable flow velocity. Secondly, a larger growth rate of the dispersivity in linear-asymptotically distance-dispersivity function can lead to a faster solute transport at early stage, but a lower concentration at late stage; as for the exponentially distance-dependent function, the growth rate of the dispersivity has the same effects on BTCs. Thirdly, it was observed that an increase in final steady velocity (or asymptotic velocity) will amplify the impacts on solute transport due to advection; as for the asymptotic dispersivity, it has similar impacts on the solute transport due to dispersion. Overall, our results show that the effects of time-dependent flow velocity and distance-dependent dispersivities are not negligible when describing solute transport process in subsurface hydrology.

Experimental and modeling evidence of kilometer-scale anomalous tracer transport in an alpine karst aquifer

Karst aquifers are important drinking water resources, but highly vulnerable to contamination. Contaminants can be transported rapidly through a network of fractures and conduits, with only limited sorption or degradation, which usually leads to a fast and strong response at karst springs. During migration, contaminants can also enter less mobile zones, such as pools or water in intra-karstic sediments, or advance from conduits into the adjacent fractured rock matrix. As contaminant concentrations in the main flow path(s) decrease, contaminants may migrate back into the main flow path and reach the karst springs at low (but significant) concentrations over a long time span. This is the conventional interpretation for the oft-observed steep rising limb and the long-tailed falling limb of tracer breakthrough curves in karst systems. Here, field measurements are examined from an alpine karst system in Austria where a series of distinctive, long-tailed breakthrough curves (BTCs) of conservative tracers were observed over distances up to 7400 m. Recognizing that the conventional advection-dispersion equation (ADE) cannot usually quantify such behavior, two other modeling approaches are considered, namely the two-region non-equilibrium (2RNE) model, which explicitly includes mobile and immobile zones, and a continuous time random walk (CTRW) model, which is based on a physically-based, probabilistic approach that describes anomalous (or non-Fickian) transport behavior characteristic of heterogeneous systems such as karst. In most cases, the ADE and 2RNE models do not quantify the low concentrations at longer travel times. The CTRW, in contrast, accounts for the long-tailed breakthrough behavior found in this karst system.

Solute transport in a filled single fracture under non-Darcian flow

Filled single fractures (FSF) are commonly seen in the subsurface but have received little attention in terms of flow and transport. This study uses well-controlled laboratory experiments to investigate flow and transport in a FSF under non-Darcian flow conditions. Flow in the experiments is well described by the Forchheimer or the Izbash equations. Non-Fickian transport is found to dominate with early arrival and long tails. The breakthrough curves (BTCs) of the solute transport are modeled by the conventional advection–dispersion equation (ADE), the ADE with retardation (ADE-R) and the single rate mobile–immobile (MIM) model. The ADE is used as a benchmark for comparison. The choice of ADE-R is to acknowledge the fact that mechanical filtration and intake of contaminant particles in dead or disconnected pore spaces can retard the transport. The choice of MIM is to acknowledge the early arrival and long tails of the BTCs. The ADE model poorly describes the transport; the ADE-R model is better than the ADE in fitting the peak concentration, but it is less satisfactory in fitting the tails of BTCs; the single rate MIM model shows the potential to model the transport behavior over a wide range of discharge. The large dispersivity values found in the experiments are consistent with the prediction of Taylor, who had shown that dispersion under non-Darcian flow condition was much stronger than that under Darcian flow.

Analytical Solutions for Solute Transport in Finite Soil Columns with Arbitrary Initial Distributions

AbstractAnalytical expressions were derived for the solute concentration in soils with a nonuniform initial distribution and a finite outlet condition assuming a zero gradient at the soil outlet. The advection‐dispersion equation (ADE) was solved by successively using a Laplace transform with respect to time, variation of parameters, and two different inversion procedures. Solutions for two resident concentrations and a flux‐averaged concentration were derived for arbitrary and pulse‐type initial distributions. The solutions are applicable for a wide range of conditions. Particular attention was paid to cases involving small Peclet numbers and large dispersive fluxes associated with steep fronts in the initial distribution. Flux‐averaged concentrations may become negative or exceed the initial concentration, while solutions for the resident concentration, derived for first‐ and third‐type inlet conditions, can predict a substantial amount of upstream solute transport due to dispersion. This suggests that the conventional ADE is not always suitable to describe initial value problems where travel times are typically short and large concentration gradients may occur.

On Characterization of Anomalous Dispersion in Porous and Fractured Media

A key characterization of dispersion in aquifers and other porous media has been to map the effects of inhomogeneous velocity fields onto a Fickian dispersion term (D) within the context of the conventional advection‐dispersion equation (ADE). Recent compilations of data have revealed, however, that the effective D coefficient is not constant but varies systematically with the length or timescale over which transport occurs. A natural strategy to encompass this “anomalous” behavior into the context of the conventional ADE is to make D time dependent. This approach, to use D(t) to handle the same anomalous dispersion phenomena, has also been common in the field of electronic transport in disordered materials. In this paper we discuss the intrinsic inadequacy of considering a time‐dependent dispersivity in the conventional ADE context, and show that the D = D(t) generalization leads to quantifiably incorrect solutions. In the course of proving this result we discuss the nature of anomalous dispersion and provide physical insight into this important problem in hydrogeology via analysis of a class of kinetic approaches. Particular emphasis is placed on the effects of a distribution of solute “delay times” with a diverging mean time, which we relate to configurations of preferential pathways in heterogeneous media.

Numerical Studies of Radionuclide Migration in Heterogeneous Porous Media

Burial of hazardous nuclear waste in geological repositories is considered to be in the best environmental interest. However, to minimize potential risk for future generations, an accurate knowledge is required of the time- and space-dependent concentrations of the radionuclides as they migrate from their burial site. Numerical solutions to both the conventional advection-dispersion equation and a new transport equation (containing directional dependence) are utilized to study the migration of radionuclides in heterogeneous media. In particular, layered fractured formations are examined. The new transport formulation, with its directional dependence, yields details in concentration profiles not shown by the advection-dispersion approach.

Observation of Time Dependent Dispersion in Laboratory Scale Experiments with Intact Tuff

The migration of radionuclides through intact tuff was studied using tuff from Yucca Mountain, Nevada. The tuff samples were both highly zeolitized ash-fall tuff from the Calico Hills and densely welded devitrified tuff from the Topopah Springs member of the Paintbrush tuff. Tritiated water and pertechnetate were used as conservative tracers. The sorbing tracers {sup 85}Sr, {sup 137}Cs, and {sup 133}Ba were used with the devitrified tuff only. Greater tailing in the elution curves of the densely welded tuff samples was observed that could be fit by adjusting the dispersion coefficient in the conventional Advection Dispersion Equation, ADE. The curves could be fit using time dependent dispersion as was previously observed for sediments and alluvium by Dieulin, Matheron, and de Marsily. The peak of strontium concentration was expected to arrive after 1.5 years based on the conventional ADE and assuming a linear K{sub d} of 26 ml/g. The observed elution had significant strontium in the first sample taken at 2 weeks after injection. The peak in the strontium elution occurred at 5 weeks. The correct arrival time for the strontium peak was achieved using a one dimensional analytic solution with time dependent dispersion. The dispersion coefficient as a function ofmore » time used to fit the conservative tracers was found to predict the peak arrival of the sorbing tracers. The K{sub d} used was the K{sub d} determined by the batch method on crushed tuff. 23 refs., 9 figs., 2 tabs.« less