Solute Transport In Porous Media Research Articles

Relation between critical exponent of the conductivity and the morphological exponents of percolation theory

A central unsolved problem in percolation theory over the past five decades has been whether there is a direct relationship between the critical exponents that characterize the power-law behavior of the transport properties near the percolation threshold, particularly the effective electrical conductivity σe, and the exponents that describe the morphology of percolation clusters. The problem is also relevant to the relation between the static exponents of percolation clusters and the critical dynamics of spin waves in dilute ferromagnets, the elasticity of gels and composite solids, hopping conductivity in semiconductors, solute transport in porous media, and many others. We propose an approach to address the problem by showing that the contributions to σe can be decomposed into several groups representing the structure of percolation networks, including their mass and tortuosity, as well as constrictivity that describes the fluctuations in the driving potential gradient along the transport paths. The decomposition leads to a relationship between the critical exponent t of σe and other percolation exponents in d dimensions, t/ν=(d−Dbb)+2(Dop−1)+dC, where ν, Dbb, Dop, and dC are, respectively, the correlation length exponent, the fractal dimensions of the backbones and the optimal paths, and the exponent that characterizes the constrictivity. Numerical simulations in two and three dimensions, as well as analytical results in d=1 and d=6, the upper critical dimension of percolation, validate the relationship. We, therefore, believe that the solution to the 50-year-old problem has been derived. Published by the American Physical Society 2024

Three-dimensional solute transport in finite and curved porous media with surface input sources

In this paper, an analytical solution for three-dimensional solute transport in porous media between two curved surfaces is investigated. It is assumed that the groundwater velocity and dispersion coefficient vary with time and position. Groundwater velocity is not considered to be horizontal. The components of dispersion coefficient along the axes are considered to be proportional to the square of corresponding the position variable. The dispersion coefficient components along axes are proportional to the corresponding component of groundwater velocity in temporal aspects while former is squarely proportional to letter one in position components. It is assumed that the sources originate from two curved surfaces. The nature of the source on the two surfaces is the same, but there may be a variation in potential. Initially, the aquifer's domain is supposed to be uniformly polluted. The Laplace Integral Transformation Technique (LITT) is used to obtain analytical solutions. Numerical examples are given to demonstrate the effects of various factors on the solute concentration profile in a system where advection and dispersion play important roles.In addition, the sub-case of horizontal flow is also discussed. The model is extremely useful in analyzing and dealing with widespread surface sources of groundwater pollution in simulated agricultural fields or urban dumping areas.

Anomalies of solute transport in flow of shear-thinning fluids in heterogeneous porous media

Solute transport and mixing in heterogeneous porous media are important to many processes of practical applications. Most of the previous studies focused on solute transport in flow of Newtonian fluids, whereas there are many processes in which the phenomenon takes place in flow of a non-Newtonian fluid. In this paper, we develop a computational approach to evaluate and upscale dispersion of a solute in flow of a shear-thinning (ST) fluid in a heterogeneous porous medium. Our results indicate that the dispersivity is a non-monotonic function of the Péclet number and the shear rate, and this behavior is accentuated by the heterogeneity of the pore space and spatial correlations between the local permeabilities. As a result, solute transport in ST fluids deviates significantly from the same phenomenon in Newtonian fluids. Moreover, the shear-dependence of the dispersivity strongly influences the fate of solute transport in porous media at large length scales, including larger effluent concentration at the breakthrough point, which also occurs much faster than Newtonian fluids. To provide further evidence for the numerical findings, we compare dispersion in flow of a power-law fluid in a single tube with the same in a bundle of such tubes. Our results emphasize the shortcomings of the current theories of dispersion to account for the role of fluid rheology in solute mixing and spreading.

Evaluating nickel removal efficacy of Filtralite under laboratory conditions: Implications for sustainable urban drainage systems

Our study deals with the critical issue of heavy metal pollution in urban runoff; we focused particularly on the threat posed by heavy metals such as nickel to living organisms and water resources owing to their toxicity at low concentrations and non-biodegradability. In this study, we evaluated the efficacy of Filtralite, a filtration medium with excellent hydraulic properties, in removing nickel from water through laboratory experiments and advanced simulation tools. We analysed the geochemical processes involved in nickel removal, including precipitation and adsorption. Importantly, we implemented an innovative approach for numerical modelling using HP1, by combining HYDRUS-1D and PHREEQC for modelling solute transport in porous media and geochemical reactions, respectively. The experimental results revealed that the Filtralite-laden column achieved a removal efficiency of 93.5 % for injected nickel. We observed a significant increase in the effluent pH from 7.0 to 11.5 during the interaction with Filtralite, affecting the solubility of the nickel phases. The model showed the highest nickel concentration in the initial column layers because of rapid chemical precipitation and adsorption upon contact with Filtralite. High-resolution scanning electron microscopy images confirmed the presence of amorphous precipitates around the soil particles, aligning with the changes in pH. This study contributes significantly to environmental engineering and urban water management by introducing an innovative numerical model in HP1 to accurately simulate the removal of nickel using Filtralite. These findings validate the efficiency of Filtralite in extracting nickel from aqueous solutions. Thus, Filtralite is a strong candidate for inclusion in sustainable urban drainage systems.

Mechanisms of Non-Fresh Groundwater Presence at Water Tables in Highly Permeable Coastal Aquifers.

Coastal aquifers with high hydraulic conductivities on the order of 10-2 m s-1 have unconventional salinity distributions with the presence of non-fresh groundwater at the water table over a wide swath near the coast. This study aims to unravel the mechanisms underlying the phenomenon via numerical simulations for variably saturated, density-driven flow and solute transport in porous media. The simulation results indicate that the existence of non-fresh groundwater at the water table is attributed to the upward mass flux in the saturated zone near the coast, which transports solute from deeper groundwater toward the water table. With high hydraulic conductivity, the upward mass flux becomes prominent at shallower elevations because of the high Darcy flux and the shallow saline groundwater. The upward mass flux has two main drivers, upward advection by the upward flow component and transverse dispersion by the seaward flow component. The advective mass flux dominates over the transverse dispersion in the deep part of the saturated zone where only groundwater with sea water salinity exists. In contrast, the transverse dispersion becomes more pronounced than the upward advection in the shallow saturated zone just beneath the water table and in the unsaturated zone immediately above the water table. Our findings help interpret the unconventional salinity distributions observed and elucidate the unique dynamics of groundwater flow and solute transport in highly permeable coastal aquifers.

Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media

Simulating solute transport in heterogeneous porous media poses computational challenges due to the high-resolution meshing required for traditional solvers. To overcome these challenges, this study explores a mesh-free method based on deep learning to accelerate solute transport simulation. We employ Physics-informed Neural Networks (PiNN) with a periodic activation function to solve solute transport problems in both homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that rely on large training datasets, PiNNs use strong-form mathematical models to constrain the network in the training phase and simultaneously solve for multiple dependent or independent field variables, such as pressure and solute concentration fields. To demonstrate the effectiveness of using PiNNs with a periodic activation function to resolve solute transport in porous media, we construct PiNNs using two activation functions, sin and tanh, for seven case studies, including 1D and 2D scenarios. The accuracy of the PiNNs’ predictions is then evaluated using absolute point error and mean square error metrics and compared to the ground truth solutions obtained analytically or numerically. Our results demonstrate that the PiNN with sin activation function, compared to tanh activation function, is up to two orders of magnitude more accurate and up to two times faster to train, especially in heterogeneous porous media. Moreover, PiNN’s simultaneous predictions of pressure and concentration fields can reduce computational expenses in terms of inference time by three orders of magnitude compared to FEM simulations for two-dimensional cases.

The effects of heterogeneity on solute transport in porous media: anomalous dispersion

Mixing of solute in sub-surface flows, for example in the leakage of contaminants into groundwater, is quantified by a dispersion coefficient that depends on the dispersivity of the medium and the transport velocity. Many previous models of solute transport data have assumed Fickian behaviour with constant dispersivity, but found that the inferred dispersivity increases with the distance from the point of injection. This approach assumes that the dispersion on either side of the advective front is symmetric and that the concentration there is close to half of the injected value. However, field data show consistent asymmetry about the advective front. Here, motivated by experimental data and a fractal interpretation of porous media, we consider a simplified heterogeneous medium described by a dispersivity with a power-law dependence on the downstream distance from the source and explore the nature of the asymmetry obtained in the solute transport. In a heterogeneous medium of this type, we show that asymmetry in solute transport gradually increases with the increase in heterogeneity or fractal dimension, and the concentration at the advective front becomes increasingly different from 50% of that at the inlet. In particular, for a sufficiently heterogeneous medium, the concentration profiles at late-time approach a non-trivial steady solution and, as a result, the concentration at a downstream location will never reach that at the inlet. By fitting the Fickian solution to our results, we are able to connect the parameters from our model to those found from experimental data, providing a more physically grounded approach for interpreting them.

Numerical solution of non-linear advection-dispersion equation in a finite porous domain

This study presents a numerical simulation of the one-dimensional concentration-dependent convection-dispersion equation in a finite heterogeneous porous formation. The solution of the convection-diffusion equation with variable coefficients is obtained with the help of MATLAB pdepe solver. The groundwater flow velocity depends on the pollutant concentration, and the dispersion coefficient is proportional to the groundwater flow velocity. The effects of the zero-order production term and the first-order decay are also considered. The aquifer is assumed to be heterogeneous and finite, with sources concentrated in the flow direction. It is assumed that the porous media and the pollutant are chemically non-reactive. Initially porous domain is considered not solute free. The model assumes a uniform continuous input point source and a variable input point source released from the left end of the aquifer domain. The obtained results graphically describe the importance of the dispersion coefficient and other relevant parameters for solute transport in porous media. The developed numerical solution is verified with an analytical solution, and it is found that they are in good agreement.

A benchmark for variably saturated variable-density variable-viscosity flow and solute transport in porous media

In natural environments, fluid density and viscosity can be affected by spatial and temporal variations of solute concentration, for example, due to saltwater intrusion in coastal aquifers, leachate infiltration from waste disposal sites, and upconing of saline water from deep aquifers. Potentially unstable situations may arise in which a dense fluid overlies a less dense fluid. This situation can produce instabilities manifested by dense plume fingers moving vertically downwards counterbalanced by vertical upward flow of the less dense fluid. The resulting free convection increases solute transport rates over large distances and times relative to constant-density flow. Unstable brine flow is further complicated if the porous medium is variably saturated. The results from a laboratory experiment of variably saturated variable-density flow and solute transport from Simmons et al. (2002) are used as the physical basis to define a new mathematical benchmark. This benchmark aims at realistically reproducing the experimental fingering patterns. Random hydraulic conductivity fields were used in the simulations as a numerical perturbation method to realistically mimic the observed dense plume fingering. The HydroGeoSphere code coupled with PEST are used to calibrate the parameter set that defines the benchmark. A grid convergence analysis is performed to obtain the adequate spatial and temporal discretizations. The new mathematical benchmark is useful for model comparison and testing of variably saturated variable-density flow in porous media. Simmons CT, Pierini ML, Hutson JL (2002) Laboratory investigation of variable-density flow and solute transport in unsaturated–saturated porous media. Transp Porous Media. 47(2): 215–244, 10.1023/A:1015568724369.

Scale Dependence of Dispersion Coefficient for Solute Transport in Porous Media Using Image Analysis

Scale dependence of dispersion coefficient (D) in the advection-dispersion equation (ADE) for solute transport in porous media was investigated by a series of experiments using image analysis. A hexahedral plexiglass box sized 200×8×1.5 cm (L×W×H) was set and packed with glass beads as porous media. The solute transport under different conditions was simulated by changing the particle size of glass beads, flow rate, and detection scale using Bright Blue as tracer. The image analysis method was used to dynamically monitor and identify the spatiotemporal variation of solute concentration distribution. The results showed that image analysis can effectively monitor and identify the solute concentration in porous media, as indicated by an R2 value of 0.9890. There is an obvious linear relationship between hydraulic gradient (J) and velocity (v) in porous media under different experimental conditions. The ADE model is suitable for solute breakthrough curve (BTC) with good fitting accuracy, and can effectively reflect the concentration variation during solute transport. The key parameters controlling the solute transport were analyzed. D has abnormal diffusion (i.e., non-Fickian phenomenon) and scale dependence, and BTCs had a long tail, which becomes more obvious with the increase of flow rate, medium particle size, and transport scale.

Lattice Boltzmann method for solute transport in dual-permeability media

Dual-permeability models are currently under active investigation for predicting the flow and transport of pollutant concentration in porous media. The basic idea of these models is usually to separate the medium into two distinct flow regions with separate hydraulic and solute movement properties. Most of the dual-permeability approaches presented to date assume the reaction terms to be independent of space. For local reaction processes in soils, however, the depth effects are important and should be incorporated into the dual-permeability models. To reveal the transport phenomenon in such case, the best way is to obtain an exact solution with a suitable analytical approach, but it may be hard or even impossible for this general scenario. As such, this paper seeks an alternative method, and propose a lattice Boltzmann method for solute transport in dual-permeability media. The Chapman–Enskog analysis shows that the present model can recover the governing equations correctly. The capability and accuracy of this model is first verified by two problems: the non-reactive mass transfer in dual-permeability porous media, and the solute transport in a depth-dependent soil. With this model, we further studied the contaminant transport in dual-permeability soil with depth-dependent reaction rates. The numerical results show that when the exchange coefficient is relatively smaller, there are usually two peaks for the breakthrough curves (BTCs), whereas it degenerates to a single peak as the exchange coefficient increases. In addition, we note that the distributions of the concentration in different domains are nearly the same for a larger exchange coefficient. As for the pore water velocity, it is found that when the difference in the pore water velocity in the two domains is significant, the BTC exhibits a bimodal distribution. However, there is only one peak for the BTCs when the above difference is insignificant. Further, as compared with the constant-reaction rate cases, we observe that the BTCs obtained for the media with depth-dependent rates lie in the intermediate zone defined by the above two extremes. Finally, we adopt the current approach to simulate the solute transport in an Andisol and observe that the difference between the numerical and experimental results is insignificant, indicating our approach is capable in modeling solute transport in porous media.

Prediction of local concentration fields in porous media with chemical reaction using a multi scale convolutional neural network

The study of solute transport in porous media is of interest in many chemical engineering systems. Some example applications include packed bed catalytic reactors, filtration devices, and batteries. The pore scale modeling of these systems is time consuming and may require large computing resources, for this reason computational fluid dynamics (CFD) simulations are not practical if a large number of simulations is required, like in multiscale modeling, where a model at a large scale calls for pore scale simulations. It has been shown that neural networks can be trained with a dataset of flow simulations and then predict fields orders of magnitude faster, and with less computational resources, in new domains. However, it is crucial to provide the neural network with an effective description of the domain and the undergoing operating conditions to be able to train models that generalize accurately in unseen samples. Therefore, research is needed to employ neural networks in new complex systems. The appropriate training of a network for predicting coupled flow and solute transport processes is an outstanding problem due to the complex interplay between geometry and operating conditions. In this work, we train a multi scale convolutional neural network (MSNet) with a diverse dataset of simulations of transport and chemical reaction in porous media to predict the local concentration fields in images of porous media. Our dataset contains a wide diversity of sphere pack arrangements under different operating conditions (Péclet and Reynolds numbers). We train a robust model by employing different input descriptors that represent the medium and the different operating conditions of each system. Our trained model is able to provide nearly instantaneous predictions, compared to around twenty hours of the CFD workflow, with less than 3.5% error on new geometries and transport conditions. Thus the model could be easily integrated in a multiscale workflow where fast response is needed.

Verification of TRANSPORT Simulation Environment coupling with PHREEQC for reactive transport modelling

Abstract. Many types of geologic subsurface utilisation are associated with fluid and heat flow as well as simultaneously occurring chemical reactions. For that reason, reactive transport models are required to understand and reproduce the governing processes. In this regard, reactive transport codes must be highly flexible to cover a wide range of applications, while being applicable by users without extensive programming skills at the same time. In this context, we present an extension of the Open Source and Open Access TRANSPORT Simulation Environment, which has been coupled with the geochemical reaction module PHREEQC, and thus provides multiple new features that make it applicable to complex reactive transport problems in various geoscientific fields. Code readability is ensured by the applied high-level programming language Python which is relatively easy to learn compared to low-level programming languages such as C, C++ and FORTRAN. Thus, also users with limited software development knowledge can benefit from the presented simulation environment due to the low entry-level programming skill requirements. In the present study, common geochemical benchmarks are used to verify the numerical code implementation. Currently, the coupled simulator can be used to investigate 3D single-phase fluid and heat flow as well as multicomponent solute transport in porous media. In addition to that, a wide range of equilibrium and nonequilibrium reactions can be considered. Chemical feedback on fluid flow is provided by adapting porosity and permeability of the porous media as well as fluid properties. Thereby, users are in full control of the underlying functions in terms of fluid and rock equations of state, coupled geochemical modules used for reactive transport, dynamic boundary conditions and mass balance calculations. Both, the solution of the system of partial differential equations and the PHREEQC module, can be easily parallelised to increase computational efficiency. The benchmarks used in the present study include density-driven flow as well as advective, diffusive and dispersive reactive transport of solutes. Furthermore, porosity and permeability changes caused by kinetically controlled dissolution-precipitation reactions are considered to verify the main features of our reactive transport code. In future, the code implementation can be used to quantify processes encountered in different types of subsurface utilisation, such as water resource management as well as geothermal energy production, as well as geological energy, CO2 and nuclear waste storage.

On the Numerical Approximation of Mobile-Immobile Advection-Dispersion Model of Fractional Order Arising from Solute Transport in Porous Media

The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.

Fully Implicit Form of Differential Quadrature Method for Multi-Species Solute Transport in Porous Media

Solute transport problems, including sequential multi-species transport phenomena, frequently occur in soil systems. The goal of this paper is to present a novel one-dimensional numerical model with a fully implicit form of differential quadrature method for solving multi-species solute transport equations. The analytical results of three multi-species solute dispersion problems with three- and four-chain members are used to analyse the developed model. Simultaneously, the outcomes of the developed model are compared with the performance of the fully implicit fourth-order finite difference method. Finally, the accuracy of the established model is discussed and evaluated. According to the numerical experiments, the derived model is very useful and widely applicable.

Enhanced solute transport in porous media due to pH-dependent adsorption and transverse dispersion

The process of pH-dependent adsorption/desorption controls the transport of solutes in reactive porous media, the greater the adsorption the slower the solute migration. Earlier works show that in the presence of longitudinal hydrodynamic dispersion and pH-dependent adsorption, a fast solute transport phenomenon may arise. However, the effect of transverse dispersion on this fast transport phenomenon has not been investigated, yet.In this paper, we report an experimental and modeling work on the fast transport of strontium (Sr2+) through a reactive porous medium made of sand and hydrous ferric oxide (HFO) coated sand, where both longitudinal and transverse dispersion were important. A reactive transport model for an incompressible fluid was developed combining surface complexation with mass conservation equations for a solute and the acidity (difference between the concentration of total proton and hydroxide ions). The model was used to design and describe experiments run in a column-flood system and in a two-dimensional (2D) bead-pack. Results show that when an alkaline solution containing Sr2+ is injected into a 2D bead-pack containing HFO-coated sand and equilibrated with an acidic solution, a Sr2+ plume forms which comprises: (i) a retarded front due to the adsorption of Sr2+ onto the porous medium; (ii) a fast wave (or pulse) ahead of the retarded front due to longitudinal dispersion and traveling at the average fluid velocity; and (iii) a continuous leakage from the retarded front due to transverse dispersion.The results of this work confirm that pH-dependent adsorption/desorption process in conjunction with longitudinal dispersion causes fast wave, i.e., a solute migration which is faster than expected if the interplay between pH and dispersion is neglected. Furthermore, transverse dispersion in the presence of a pH gradient gives rise to a plume composed of a leakage from the retarded front connected to the fast wave.

Surface complexation reactions in sandy porous media: Effects of incomplete mixing and mass-transfer limitations in flow-through systems

Although mixing and surface complexation reactions are key processes for solute transport in porous media, their coupling has not been extensively investigated. In this work, we study the impact of mass-transfer limitations on heterogeneous reactions taking place at the solid-solution interface of a natural sandy porous medium under advection-dominated flow-through conditions. A comprehensive set of 36 column experiments with different grain sizes (0.64, 1.3 and 2.3 mm), seepage velocities (1, 30 and 90 m/day), and hydrochemical conditions were performed. The injection of NaBr solutions of different concentrations (1–100 mM) led to the release of protons via deprotonation reactions of the quartz surface. pH and solute concentration breakthrough curves were measured at the outlet of the columns and the propagation of pH fronts in the column setups was tracked inside the porous medium with non-invasive optode sensors. The experimental results show that the deprotonation of the reactive surfaces, resulting from their interactions with the injected ionic species, strongly depends on the hydrodynamic conditions and differs among the tested porous media despite their apparent similar surface properties. Reactive transport modeling was used to quantitatively interpret the experimental results and to analyze the effects of mass-transfer limited physical processes on surface complexation reactions, propagation of pH fronts, transport of major ions and spatio-temporal evolution of surface composition. A dual domain mass transfer formulation (DDMT) combined with a surface complexation model (SCM) allowed capturing the effects of incomplete mixing on the surface reactions and to reproduce the experimental observations collected in the experiments with high flow velocities. The SCM was parameterized with a single set of surface complexation parameters, accounting for the similar surface properties of the porous media, and was capable of describing the surface complexation mechanisms and their impact on the hydrochemistry over the large range of tested ionic strengths.

A Numerical Model with Finite Difference Schemes for Multi-Species Solute Transport in Porous Media

A precise forecast of contaminant and solute transport has an inevitable role in the management of water resources. In accordance with this purpose, in this paper, a novel one-dimensional numerical model for the transport of a decay chain through homogeneous porous media is proposed. To develop the suggested model, two different schemes of the finite difference method, namely the Lax-Wendroff scheme and Fourth-Order scheme, are used. The verification and validation of the established model are examined by the analytical results of three multi-species solute dispersion problems with three- and four-chain members. The total mean square error, L2- and L∞-norms are applied to assess the results. Although analyses show that both schemes provided reliable results, the numerical results of the Lax-Wendroff scheme are more accurate.

Experiment and Simulation of Non-Reactive Solute Transport in Porous Media.

To more accurately predict the migration behavior of pollutants in porous media, we conduct laboratory scale experiments and model simulation. Aniline (AN) is used in one-dimensional soil column experiments designed under various media and hydrodynamic conditions. The advection-dispersion equation (ADE) and the continuous-time random walk (CTRW) were used to simulate the breakthrough curves (BTCs) of the solute transport. The results show that the media and hydrodynamic conditions are two important factors affecting solute transport and are related to the degree of non-Fickian transport. The simulation results show that CTRW can more effectively describe the non-Fickian phenomenon in the solute transport process than ADE. The sensitive parameter in the CTRW simulation process is , which can reflect the degree of non-Fickian diffusion in the solute transport. Understanding the relationship of with velocity and media particle size is conducive to improving the reactive solute transport model. The results of this study provide a theoretical basis for better prediction of pollutant transport in groundwater.

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.