Mobile-immobile Model Research Articles

Hydrologic response of engineered media in living roofs and bioretention to large rainfalls: experiments and modeling

AbstractThe hydrologic response of engineered media plays an important role in determining a stormwater control measure's ability to reduce runoff volume, flow rate, timing, and pollutant loads. Five engineered media, typical of living roof and bioretention stormwater control measures, were investigated in laboratory column experiments for their hydrologic responses to steady, large inflow rates. The inflow, medium water content response, and outflow were all measured. The water flow mechanism (uniform flow vs. preferential flow) was investigated by analyzing medium water content response in terms of timing, magnitude, and sequence with depth. Modeling the hydrologic process was conducted in the HYDRUS‐1D software, applying the Richards equation for uniform flow modeling, and a mobile–immobile model for preferential flow modeling. Uniform flow existed in most cases, including all initially dry living roof media with bimodal pore size distributions and one bioretention medium with unimodal pore size distribution. The Richards equation can predict the outflow hydrograph reasonably well for uniform flow conditions when medium hydraulic properties are adequately represented by appropriate functions. Preferential flow was found in two media with bimodal pore size distributions. The occurrence of preferential flow is more likely due to the interaction between the bimodal pore structure and the initial water content rather than the large inflow rate.

DLVO, hydrophobic, capillary and hydrodynamic forces acting on bacteria at solid-air-water interfaces: Their relative impact on bacteria deposition mechanisms in unsaturated porous media

Experimental and modeling studies were performed to investigate bacteria deposition behavior in unsaturated porous media. The coupled effect of different forces, acting on bacteria at solid-air-water interfaces and their relative importance on bacteria deposition mechanisms was explored by calculating Derjaguin–Landau–Verwey–Overbeek (DLVO) and non-DLVO interactions such as hydrophobic, capillary and hydrodynamic forces. Negatively charged non-motile bacteria and quartz sands were used in packed column experiments. The breakthrough curves and retention profiles of bacteria were simulated using the modified Mobile-IMmobile (MIM) model, to identify physico-chemical attachment or physical straining mechanisms involved in bacteria retention. These results indicated that both mechanisms might occur in both sand. However, the attachment was found to be a reversible process, because attachment coefficients were similar to those of detachment. DLVO calculations supported these results: the primary minimum did not exist, suggesting no permanent retention of bacteria to solid-water and air-water interfaces. Calculated hydrodynamic and resisting torques predicted that bacteria detachment in the secondary minimum might occur. The capillary potential energy was greater than DLVO, hydrophobic and hydrodynamic potential energies, suggesting that film straining by capillary forces might largely govern bacteria deposition under unsaturated conditions.

Critical Role of the Immobile Zone in Non-Fickian Two-Phase Transport: A New Paradigm.

Using a visualization setup, we characterized the solute transport in a micromodel filled with two fluid phases using direct, real-time imaging. By processing the time series of images of solute transport (dispersion) in a two fluid-phase filled micromodel, we directly delineated the change of transport hydrodynamics as a result of fluid-phase occupancy. We found that, in the water saturation range of 0.6-0.8, the macroscopic dispersion coefficient reaches its maximum value and the coefficient was 1 order of magnitude larger than that in single fluid-phase flow in the same micromodel. The experimental results indicate that this non-monotonic, non-Fickian transport is saturation- and flow-rate-dependent. Using real-time visualization of the resident concentration (averaged concentration over a representative elementary volume of the pore network), we directly estimated the hydrodynamically stagnant (immobile) zones and the mass transfer between mobile and immobile zones. We identified (a) the nonlinear contribution of the immobile zones to the non-Fickian transport under transient transport conditions and (b) the non-monotonic fate of immobile zones with respect to saturation under single and two fluid-phase conditions in a micromodel. These two findings highlight the serious flaws in the assumptions of the conventional mobile-immobile model (MIM), which is commonly used to characterize the transport under two fluid-phase conditions.

Comparison of the performance of traditional advection-dispersion equation and mobile-immobile model for simulating solute transport in heterogeneous soils

Comparison of the performance of traditional advection-dispersion equation and mobile-immobile model for simulating solute transport in heterogeneous soils

An upscaled approach for transport in media with extended tailing due to back-diffusion using analytical and numerical solutions of the advection dispersion equation

The mono-continuum advection-dispersion equation (mADE) is commonly regarded as unsuitable for application to media that exhibit rapid breakthrough and extended tailing associated with diffusion between high and low permeability regions. This paper demonstrates that the mADE can be successfully used to model such conditions if certain issues are addressed. First, since hydrodynamic dispersion, unlike molecular diffusion, cannot occur upstream of the contaminant source, models must be formulated to prevent “back-dispersion.” Second, large variations in aquifer permeability will result in differences between volume-weighted average concentration (resident concentration) and flow-weighted average concentration (flux concentration). Water samples taken from wells may be regarded as flux concentrations, while soil samples may be analyzed to determine resident concentrations. While the mADE is usually derived in terms of resident concentration, it is known that a mADE of the same mathematical form may be written in terms of flux concentration. However, when solving the latter, the mathematical transformation of a flux boundary condition applied to the resident mADE becomes a concentration type boundary condition for the flux mADE. Initial conditions must also be consistent with the form of the mADE that is to be solved. Thus, careful attention must be given to the type of concentration data that is available, whether resident or flux concentrations are to be simulated, and to boundary and initial conditions. We present 3-D analytical solutions for resident and flux concentrations, discuss methods of solving numerical models to obtain resident and flux concentrations, and compare results for hypothetical problems. We also present an upscaling method for computing “effective” dispersivities and other mADE model parameters in terms of physically meaningful parameters in a diffusion-limited mobile–immobile model. Application of the latter to previously published studies of systems that exhibit early breakthrough and extended tailing shows that the upscaled mADE model is able to describe the observed behavior with reasonable accuracy given only known physical parameters for the systems without any model calibration.

Enhanced gas recovery with CO2 sequestration: The effect of medium heterogeneity on the dispersion of supercritical CO2–CH4

Reinjection of CO2 into producing natural gas reservoirs is considered as a promising technology to improve gas recovery, mitigate atmospheric emissions and control climate change. However, natural gas and CO2 are miscible at reservoir conditions and could result in CO2 contamination of produced natural gas. This mixing process and consequently the viability of Enhanced Gas Recovery (EGR) projects can be quantitatively determined by reservoir simulations – such simulations require a description of gas dispersion. Here we conduct fluid transport experiments through carbonate and sandstone rock cores at various reservoir conditions to evaluate the effect of medium heterogeneity on the dispersion between supercritical CO2 and CH4, accounting for erroneous contributions from entrance/exit and gravitational effects. Early breakthrough and long-tailed profiles are observed for one of the carbonate cores (Ketton) which is attributed to the existence of intra-grain micro-pores, which results in a persistent pre-asymptotic transport regime. Thus a revised model (Mobile-Immobile Model) was successfully used for this core to obtain dispersion coefficients characteristic of the eventual asymptotic regime. Both heterogeneous carbonate rocks considered exhibit higher dispersion than that observed in previously-measured homogeneous sandstone cores (Honari et al., 2013). The power law describing the dependency of dispersion coefficient on Péclet number at comparatively high interstitial displacement velocities gave an exponent of 1.2 for sandstones and 1.4 for carbonates, consistent with literature predictions (Bijeljic and Blunt, 2006; Bijeljic et al., 2011) based on pore-scale simulations.

Chloride Transport in an Aggregated Clay Soil Column

Soil aggregates as the basic units of soil structure greatly affect soil properties and fertilizer. They have an important theoretical and practical significance in studying the change in solute transport with different aggregate-size soil on the Loess Plateau. Based on a soil column experiment in the laboratory, breakthrough curves (BTCs) of chloride displaced through columns of different aggregate sizes of a clay soil were measured under water-saturated steady-flow conditions. The data were fitted with a convection and dispersion equation (CDE), a mobile-immobile model, and a two-flow-region model. In the CDE, all soil water was assumed to be mobile with a physical equilibrium existing in the system. For the mobile-immobile model, soil water was partitioned into mobile and immobile regions and convective diffusive solute transport was limited to the mobile water region; however, chloride transfer between the two regions was assumed to occur at a rate proportional to the difference in tracer concentration between them. In the two-flow-region model, soil water was divided into two regions based on flow velocities, but neither region had a nonzero flow rate. All three models could fit the solute transport curves in columns packed with all aggregate sizes at a small pore water velocity (0.68 cm/h), but the values of the fitted parameters varied greatly. The Péclet number derived from the mobile-immobile model and the two-flow-region model showed the same changing tendency with aggregate size, increasing as aggregate size increased. The Péclet values derived from the CDE were approximately two orders of magnitude smaller than those obtained by the mobile-immobile model and the two-flow-region model. The mobile water fraction obtained with the two-flow-region model decreased as aggregate sizes increased whereas the mass transfer coefficient ω decreased as pore water velocity increased because of the short resident time of chloride transport through the soil columns.

Preferential percolation quantified by large water content sensors with artifactual macroporous envelopes

For many scientific and practical tasks, it is important to estimate the soil–water percolation fluxes. This paper builds on measurements with large horizontal time-domain reflectometry water content sensors in a loamy Mollisol. The sensors were installed into pre-drilled holes and the gaps between them, and the soil was filled with a slurry of local soil with water. This gave rise to envelopes around them that contained artificial macropores. The sensors reacted to intensive rains by a rapid increase of their readings, often above the native soil's porosity, followed by an almost equally rapid decrease. The paper explores the feasibility of quantifying the rapid percolation, based on these anomalous water content peaks, and demonstrates that this is possible in principle, if the processes are simulated by a suitable model. A two-dimensional dual porosity non-equilibrium (mobile-immobile) model was tried. The envelope around the sensor was modelled as an annulus with higher porosity and hydraulic conductivity, which attracts preferential flow and amplifies the percolation signal. With the model at hand, the flux hydrographs can be derived from model simulations and measured precipitation. For contrast, the Durner equilibrium dual porosity model was tried but was found little suitable. However, even the mobile-immobile model did not perform perfectly. Simulated water contents were similar to the measured ones at some depths but not in the others, and the percolation fluxes were overestimated, compared to cumulative soil–water balance. Efforts to improve model performance were not successful. Hence, the model structure needs to be improved. Copyright © 2015 John Wiley & Sons, Ltd.

Temperature effect on the transport of bromide and E. coli NAR in saturated soils

Summary In this study we investigated the transport of nalidixic acid-resistant Escherichia coli ( E. coli NAR) and bromide (Br − ) through two soils, a sandy loam (SL) and clay loam (CL). Soils were repacked in columns (45 cm length × 22 cm diameter) and subjected to physical (freeze/thaw, and wet/dry cycles) and biological (by earthworms, Eisenia fetida ) weathering for 12 months. Saturated flow conditions were maintained using a tension infiltrometer. Tests were carried out at either 5 or 20 °C. After steady-state flow conditions were established, a suspension containing E. coli NAR and Br − was sprayed onto the surface of soil columns. Leachate was sampled at three depths, 15, 30 and 45 cm. Time to maximum concentration ( C max ) of E. coli NAR was greater for SL at all depths. Both tracers had rapid breakthrough curves (BTCs) shortly after the suspension injection followed by prolonged tailing indicating the presence of preferential pathways and thus soil heterogeneity regenerated after the induced physical and biological weathering. About 40% of the E. coli NAR and 79% of the Br − leached through the entire 45 cm soil columns during the experiments. Leaching with cold water (5 °C) led to lower hydraulic conductivity and flow rate and consequently enhanced bacterial filtration for both soils. Very low values for the detachment coefficient for E. coli NAR at 5 °C suggest an irreversible process of bacterial attachment in heterogeneous soils. BTCs were well described by the mobile–immobile model (MIM) in HYDRUS-1D. Soil texture/structure and temperature had a significant effect on the model’s fitted parameters.

Transport of nitrate and ammonium ions in a sandy loam soil treated with potassium zeolite – Evaluating equilibrium and non-equilibrium equations

The increased use of nitrogen fertilizers in agriculture would cause migration of nitrogen to surface and groundwater; accordingly, would lead to water resources contamination. The objective of this study is to investigate the effect of potassium zeolite on nitrate and ammonium ions sorption and retention in a saturated sandy loam soil in a laboratory condition. The study was conducted as a completely randomized block design with four treatments of 0, 2, 4 and 8 g zeolite per kilogram soil and three replications. Ammonium nitrate fertilizer with concentration of 10 g l−1 was added to soil columns and then leaching was performed. The results show that increasing potassium zeolite to soil causes reduction to the mobility of both nitrate and ammonium and enhancement of the retention of ions in soil. Ions leaching were simulated with convection–dispersion-equation (CDE) and mobile–immobile model (MIM) using HYDRUS-1D code. The results indicate that ammonium ion sorption by soil followed the Freundlich isotherm model. Absorption isotherms and dispersion (De) coefficient were determined through the inverse modeling for both ions. Based on the results, optimized values of Freundlich isotherm were much less than the observed amounts. This shows that the HYDRUS-1D model underestimated the adsorption parameters to predict the ammonium ion mobility in soil macropores. Since soil has been disturbed, the prediction of CDE model was equal to MIM model approximately. Both models showed that as the amount of applied zeolite increases, the dispersion (De) coefficient of nitrate and ammonium ions in the soil increases.

On the mechanism of field‐scale solute transport: Insights from numerical simulations and field observations

AbstractField‐scale transport of conservative and reactive solutes through a deep vadose zone was analyzed by means of two different model processes for the local description of the transport. The first is the advection‐dispersion equation (ADE) model, and the second is the mobile‐immobile (MIM) model. The analyses were performed by means of three‐dimensional (3‐D), numerical simulations of flow and transport considering realistic features of the flow system, pertinent to a turf field irrigated with treated sewage effluents (TSE). Simulated water content and concentration profiles were compared with available measurements of their counterparts. Results of the analyses suggest that the behavior of both solutes in the deep vadose zone of the Glil Yam site is better quantified by the MIM model than by the ADE model. Reconstruction of the shape of the measured solute concentration profiles using the MIM model required relatively small mass transfer coefficient, γ, and relatively large volume fraction of the immobile water θim. This implies that for an initially nonzero solute concentration profile, as compared with the MIM model, the ADE model may significantly overestimate the rate at which solutes are loaded in the groundwater. On the contrary, for an initially zero solute concentration profile, as compared with the MIM model, the ADE model may significantly underestimate solute velocities and early arrival times to the water table. These findings stem from the combination of relatively small γ and relatively large θim taken into account in the MIM model. In the first case, this combination forces a considerable portion of the solute mass to reside in the immobile region of the water‐filled pore space, while the opposite is true in the second case.

Evidence of preferential path formation and path memory effect during successive infiltration and drainage cycles in uniform sand columns

The formation of preferential flow paths in the partially saturated zone, and in naturally structured media, is well known. This study examines non-uniform flow in uniform sand columns under different pressure and infiltration/drainage conditions. Experiments were carried out in a vacuum box, with applied suction set to three different heads, and with infiltration fixed at two different flow rates. Tailing observed in some conservative tracer breakthrough curves suggests the formation of immobile resident water pockets which slowly exchange mass with the flowing water fraction. The applied suction controlled the degree of water immobilization whereas flow rate had minimal effect on the dynamic behavior. Trapping and exchange of water occurred repeatedly during successive infiltration and drainage cycles, implying a (hysteretic) memory effect of the previously formed preferential flow paths. Flow and solute transport modeling suggests that these dynamics can be described by a mobile–immobile model that corroborates measurements suggesting preferential flow path formation. These findings have implications for the natural attenuation of contaminants in the partially saturated zone, but also for the persistence of a contamination source exposed to repeated conditions of infiltration and drainage.

Stochastic modeling of fine particulate organic carbon dynamics in rivers

The majority of particulate organic matter standing stock in streams is < 1 mm in diameter, and the mobile phase is primarily very fine particles. Such fine particles transport downstream in a series of deposition and resuspension events mediated by interactions with coarser bed sediment, yielding fine particle retention over a wide range of time scales. This retention controls the opportunity for biogeochemical processing of particulate organic carbon in streams. We present a conceptual model of particulate organic carbon transport in rivers categorized in three cyclic processes: (i) migration of fine particles from the water column to the underlying and surrounding sediments, (ii) fine particle transport and retention within the bed sediments, and (iii) resuspension of fine particles back to the water column. We developed a stochastic model to describe the transport and retention of fine suspended particles in rivers, including advective delivery of particles to the streambed, transport through pore waters, and reversible filtration within the streambed. We then apply this model to observations of fine particle transport in two small streams, and show that the stochastic mobile-immobile model supports improved interpretation of particulate organic carbon dynamics under base flow conditions. Analysis of in-stream solute and particle data shows that particles engage in multiple deposition and resuspension events during downstream transport, and that long-term retention in the streambed produces extended slow releases to the stream even during base flow conditions. We also show how multiscale stochastic modeling can be used to incorporate local observations of particle retention in predictions of whole-stream particle dynamics.

Evidence of non-Darcy flow and non-Fickian transport in fractured media at laboratory scale

Abstract. During a risk assessment procedure as well as when dealing with cleanup and monitoring strategies, accurate predictions of solute propagation in fractured rocks are of particular importance when assessing exposure pathways through which contaminants reach receptors. Experimental data obtained under controlled conditions such as in a laboratory allow to increase the understanding of the fundamental physics of fluid flow and solute transport in fractures. In this study, laboratory hydraulic and tracer tests have been carried out on an artificially created fractured rock sample. The tests regard the analysis of the hydraulic loss and the measurement of breakthrough curves for saline tracer pulse inside a rock sample of parallelepiped shape (0.60 × 0.40 × 0.08 m). The convolution theory has been applied in order to remove the effect of the acquisition apparatus on tracer experiments. The experimental results have shown evidence of a non-Darcy relationship between flow rate and hydraulic loss that is best described by Forchheimer's law. Furthermore, in the flow experiments both inertial and viscous flow terms are not negligible. The observed experimental breakthrough curves of solute transport have been modeled by the classical one-dimensional analytical solution for the advection–dispersion equation (ADE) and the single rate mobile–immobile model (MIM). The former model does not properly fit the first arrival and the tail while the latter, which recognizes the existence of mobile and immobile domains for transport, provides a very decent fit. The carried out experiments show that there exists a pronounced mobile–immobile zone interaction that cannot be neglected and that leads to a non-equilibrium behavior of solute transport. The existence of a non-Darcian flow regime has showed to influence the velocity field in that it gives rise to a delay in solute migration with respect to the predicted value assuming linear flow. Furthermore, the presence of inertial effects enhance non-equilibrium behavior. Instead, the presence of a transitional flow regime seems not to exert influence on the behavior of dispersion. The linear-type relationship found between velocity and dispersion demonstrates that for the range of imposed flow rates and for the selected path the geometrical dispersion dominates the mixing processes along the fracture network.

Adjoint state method for fractional diffusion: Parameter identification

Fractional partial differential equations provide models for sub-diffusion, among which the fractal Mobile–Immobile Model (fMIM) is often used to represent solute transport in complex media. The fMIM involves four parameters, among which we have the order of an integro-differential operator that accounts for the possibility for solutes to be sequestered during very long times. To guess fMIM parameters from experiments, an accurate method consists in optimizing an objective function that measures how much model solutions deviate from data. We show that solving an adjoint problem helps accurate computing of the gradient of such an objective function, with respect to the parameters. We illustrate the method by applying it on experimental data issued from tracing tests in porous media.

Interplay between resident and infiltrating water: Estimates from transient water flow and solute transport

► Interplay between resident (‘old’) and infiltrating (‘new’) water is investigated. ► Transient unsaturated water flow and solute transport measurements are presented. ► Piston-like displacement of old water is followed by mixing and entrainment. ► The flow regimes are governed by initial water content and soil particle grain size. This study investigates the interplay between resident (‘old’) water and incoming (‘new’) water in homogeneous, partially saturated sand undergoing infiltration, using a series of laboratory column experiments. It was hypothesized that old water pockets, defined both spatially and temporally, may be established during infiltration. This in turn, may affect the flow pattern of the infiltrating new water and/or contravene current geochemical reactions. Sands with three different particle size distributions and three initial water contents were employed. The upper end of each column was irrigated with water containing a conservative tracer, at three different flow rates, while free-drainage conditions were employed at the lower end. Analysis of the resulting 27 infiltration events was based on the Richards equation to describe fluid flow. Subsequently, the mobile-immobile model (MIM) was employed to describe the solute transport; the measurements were reproduced satisfactorily by these models. Results were further analyzed using mass balance considerations. Two regimes were identified: an initial piston-like mechanism that displaces old water, followed by a slow mixing/entrainment of the remaining old water. The relative contributions of these regimes appear to depend on the initial water content and the average grain size. In some cases, up to one-third of the old water fraction remained in the system following five flowthrough pore volumes. Comparison between the measured fractions of old water remaining in the system at the end of each infiltration event (i.e., after five flowthrough pore volumes) and the immobile water content (optimized from the MIM) relative to the initial water content, exhibited a significant linear correlation, with values about threefold higher for the optimized immobile fraction.

Transport by Oscillatory Flow in Soils with Rate‐Limited Mass Transfer: 1. Theory

Oscillatory flow of fluid in a porous medium can generate a one‐way transport of heat or chemicals if there is a gradient of temperature or chemical concentration and a rate‐limited heat or mass transfer between the moving fluid and an immobile phase. For chemical transport in soils, the immobile phase can occur in stagnant porosity, by sorption, or by dissolution of a vapor in the pore water. As a function of oscillation frequency, the transport rate has a broad peak near the value ωτc = 1, where ω is the angular frequency of oscillation and τc is the characteristic equilibration time of the mobile phase. The transport rate is proportional to the gradient and to the square of the amplitude of periodic fluid displacement. A unique diffusivity derived from the analysis enables prediction of the long‐term transport by a diffusion calculation without fluid flow, thereby providing a tool for estimating the removal of contaminant vapors by passive soil vapor extraction (PSVE). We compared predictions of the analytic theory with numerical simulations of PSVE. The mobile–immobile model is also applicable to transport in other cases of oscillatory flow in porous media, including cyclic motion of water or gas due to tidal aquifers or earth tides.

Transport by Oscillatory Flow in Soils with Rate‐Limited Mass Transfer: 2. Field Experiment

This study provides experimental evidence for the analytic model of oscillatory transport in soils, in which a vapor or solute in an oscillating mobile gas or liquid undergoes rate‐limited equilibration with an immobile phase. During a field test of passive soil vapor extraction, the concentrations of volatile organic compounds in the produced gas were monitored at 90‐min intervals. The data were compared with the results of numerical simulations based on the mobile–immobile model of oscillatory transport. A dual‐porosity simulation matched the data when the simulated time for equilibration between the mobile vapor and the immobile phases was one‐third day and the flow was restricted to only 5% of the air‐filled porosity. A purely diffusion calculation without flow, utilizing the exchange diffusivity derived from the mobile–immobile model, predicted total production of extracted vapor during a 1‐yr interval in agreement with the simulation. This illustrates the utility of the exchange diffusivity as an evaluation tool for passive soil vapor extraction.

Solute transport in dual‐permeability porous media

A dual‐advection dispersion equation (DADE) is presented and solved to describe solute transport in structured or layered porous media with different nonzero flow rates in two distinct pore domains with linear solute transfer between them. This dual‐permeability model constitutes a generalized version of the advection‐dispersion equation (ADE) for transport in uniform porous media and the mobile‐immobile model (MIM) for transport in media with a mobile and an immobile pore domain. Analytical tools for the DADE have mostly been lacking. An analytical solution has therefore been derived using Laplace transformation with time and modal decomposition based on matrix diagonalization, assuming the same dispersivity for both domains. Temporal moments are derived for the DADE and contrasted with those for the ADE and the MIM. The effective dispersion coefficient for the DADE approaches that of the ADE for a similar velocity in both pore domains and large values for the first‐order transfer parameter, and approaches that of the MIM for the opposite conditions. The solution of the DADE is used to illustrate how differences in pore water velocity between the domains and low transfer rates will lead to double peaks in the volume‐ or flux‐averaged concentration profiles versus time or position. The DADE is applied to optimize experimental breakthrough curves for an Andisol with a distinct intra‐ and interaggregate porosity. The DADE improved the description of the breakthrough data compared to the ADE and the MIM.

A mobile–immobile model with an asymptotic scale-dependent dispersion function

Summary This study combined the mobile–immobile model (MIM) with an asymptotic dispersivity function of travel distance to embrace the concept of scale-dependent dispersion during solute transport in finite heterogeneous porous media. The proposed MIM with an asymptotic scale-dependent dispersivity (MIMA) was analytically solved by the Laplace transform technique and the extended power series method. The developed semi-analytical solution of MIMA was compared with the Laplace transformed finite-difference numerical solution, and they agreed well with each other. The applicability of MIMA was tested with concentration data from a 1250-cm long and highly heterogeneous soil column. The transport process was also simulated by the MIM with a constant dispersivity (MIMC) for comparison. The simulation results of MIMC were found to depart significantly from the measured breakthrough curves (BTCs) in the column, whereas the fitting of MIMA with measured BTCs improved substantially as it considered that the increase of dispersivity with distance was limited. Our results indicated that MIMA was efficient to capture the evolution of scale-dependent dispersion behavior, and it was more useful for describing anomalous solute transport in heterogeneous porous media. We also generally evaluated the advantages and limitations of MIMA with respect to other up-scaling models for describing solute transport in heterogeneous media, and indicated several issues deserving further investigation for MIMA.