To investigate the mechanisms by which the structure of different types of porous media affects the development and evolution of eddies, this study designed four different types of porous media, including simple cubic (SC), face-centered cubic (FCC), body-centered cubic (BCC), and randomly packed (RP). The SC is the loosest pore structure, and the arrangement is gradually tight from FCC to BCC, and the corresponding porosity is 0.476, 0.398, and 0.266, respectively. The randomly packed porosity is 0.645. Subsequently, we can determine the coordinates of each sphere once they achieve mechanical equilibrium. The geometric model of the randomly packed porous media is constructed by integrating MATLAB. It is worth mentioning that the particle sizes of pore media in different arrangement modes are consistent, and a total of 5, 8, 10, 15 mm different particle sizes are designed. The different types of porous media models are shown in Fig. 1.

Considering the limitations of traditional physical model experiments, we choose the SC packed model to carry out laboratory seepage and solute transport experiments. According to our previous research (Huang et al., 2013), we found that the wall effect (the flow resistance caused by the tube wall) could be disregarded when the number of spheres in the cross-section of the experimental tubes reached 6 × 6. The schematic diagram and physical diagram of the experimental device are shown in Fig. 2.

The tracer delivery system and imaging system were added on the basis of the seepage experimental device. The experimental tube section is composed of three parts: the inlet and outlet transition section and the porous media section. The total length of the experimental tube section is 150 cm, the pellet filling section is 100 cm, and the length of the inlet and outlet transition section is 25 cm. The transition section is designed to mitigate the effects of inlet and outlet influences. Additionally, a dissipative plate with numerous small holes is installed at the inlet transition section to ensure a more stable flow of water entering the system. The medium section is filled with artificially bonded cubic spheres arranged in a specific pattern, and pressure measurement ports are located 5 cm away from both the inlet and outlet. The measurement system consists of three components: flow, pressure, and temperature measuring devices. The flow rate after stabilization is calculated using the volumetric method. By monitoring the pressure changes between the two pressure measurement ports using a pressure sensor, a quantitative relationship between seepage resistance and flow velocity can be obtained.

To investigate the effects of different influencing factors (such as different flow velocities, particle sizes, and arrangement) on the solute transport process, the selected tracer should not react with the porous media and should have low adsorption properties. Due to its stable chemical properties and good visibility, Brilliant Blue was chosen as the tracer for this solute transport experiment. The photography method involves capturing images or videos of Brilliant Blue concentrations at different time intervals using high-resolution camera (FDR-AX60, SONY). The quantitative relationship between solute concentration and image information can be established by obtaining information from images of a fixed concentration of Brilliant Blue solution (such as grayscale values or RGB values), allowing for the determination of solute concentrations at different times. Compared to the sampling method, the photography method does not disturb the flow field and is easy to operate, which has been widely used. To address the concern regarding light source variation, we implemented a strict protocol: the light source, camera, and column positions were fixed, and each image was corrected using a reference image taken with pure water under identical conditions. Most importantly, the calibration curve was developed under the exact same lighting settings as the experiments, meaning the concentration-grayscale relationship inherently accounts for the specific illumination field, ensuring robust relative measurements. The relationship between the concentration of Brilliant Blue and the grayscale values of the photographs we obtained is shown in Fig. 3, displaying a clear negative power-exponential relationship. Strictly speaking, the observation concerning the increased uncertainty in low-concentration estimation from the calibration curve, which is an inherent limitation of the optical method.

It is very difficult or almost impossible to obtain the 3D seepage field of porous media by traditional physical model experiment. With the rapid development of computational fluid dynamics, numerical simulation has been widely used in the study of groundwater seepage (Banaei et al., 2021; Yang et al., 2019; Yu et al., 2023), which also has a good effect on the simulation of porous media using COMSOL Multiphysics^® (Banaei et al., 2021; Koohbor et al., 2023). The COMSOL Multiphysics^® is a finite element method-based fluid simulation software that can simulate fluid flow in porous media by solving the conservation equations and the Navier-Stokes equations for incompressible fluids. The transport process of solute in porous media was simulated using the advection-diffusion equation coupled with the flow field. And the governing equations are shown as follows: 1ρ∇⋅u=02ρu⋅∇u=μ∇2u-∇p3∂C∂t+∇⋅J+u⋅∇C=04J=-D∇C where ρ is the fluid density, ∇ is the gradient operator, u is the velocity vector in 3D coordinate system, μ is the dynamic viscosity, and p is the total pressure. As for the solute field, C is the injection concentration of the solute, J is the mass flux diffusive flux vector, D is the diffusion coefficient.

Experiments and corresponding numerical simulations were conducted on the SC model under different flow velocities. It is essential to ensure that the dimensions of the numerical model are fully consistent with those of the experimental section. The inlet boundary is set as a velocity boundary, the outlet boundary is set as a pressure boundary, and the model wall is set as a no-slip boundary with zero flow. Besides, the density of water is 998.2 kg m⁻³, and the dynamic viscosity is 1 × 10⁻³ Pa ⋅ s. The concentration at the inlet boundary is set to 0.079 g L⁻¹, the right side of the model is set as the outflow boundary, and the wall is set as a zero-flux boundary. The molecular diffusion coefficient of the solute is set to 2 × 10⁻⁹ m² s⁻¹. Then, the corresponding flow field and solute field are obtained by controlling different inlet velocities. The preprocessing of the model has a significant impact on the results of numerical simulations. The overall accuracy of the model mesh generation is controlled by setting the boundary layer and the number of grid elements. On the one hand, the selected mesh size should not be too large, otherwise it will not be able to reflect the real model structure, on the other hand, the mesh size should not be too small, which will consume much more computer resources, and the calculation results cannot even converge. The COMSOL Multiphysics^® can offer nine levels of grid sizes, and an independent mesh convergence analysis was conducted (as shown in Fig. 4). The results demonstrate that the selected fine-level grid, coupled with boundary layer refinement, ensures grid-independent solutions, accurately resolves near-wall velocity gradients, and optimally balances computational efficiency. In our previous research, The COMSOL Multiphysics^® was also applied to the simulation of the seepage field and solute field in rough conduit media, achieving good results (Li et al., 2024).

To ensure the accuracy of the simulated flow field, we validated the numerical hydraulic gradient against the experimental v–J data detailed in our previous work (Huang et al., 2013). Because the current experimental apparatus is directly built upon this previously validated seepage device and uses identical SC packings, its baseline hydrodynamic behavior is strictly represented by this established dataset, providing a rigorous foundation for the subsequent solute transport modeling. We have compared the numerical simulation results with the experimental results, and the relationship between hydraulic gradient (J) and specific discharge (v) in porous media with SC packing is plotted in Fig. 5a. To avoid confusion, we should point that J indeed represents the mass diffusive flux vector, which is a vector quantity (bold in italics). Conversely, in Fig. 5a, J was used to denote the hydraulic gradient, a scalar quantity.

We can see that the hydraulic gradient in the simulation results is slightly less than that in the experimental results; even so, the simulation results fit well with the experimental results overall. When the particle size of spheres remains the same, the hydraulic gradient increases with the increasing specific discharge. While the specific discharge is somewhere the same, a smaller particle size leads to a greater hydraulic gradient. In other words, the smaller particle size needs to overcome the stronger seepage resistance. Notably, the fitting error between the experimental and simulated hydraulic gradients for the D=5 mm case is slightly higher than that of the larger particle cases (e.g., D=8 mm and D=10 mm). This is likely due to the increased influence of the column wall effect and local structural heterogeneity inherent in smaller particle packings. In these tighter configurations, the ratio of the wall-zone porosity to the bulk porosity creates more significant preferential flow paths, posing a greater challenge for the idealized 3D numerical model to perfectly capture the experimental bulk seepage resistance. In this study, the experimental hydraulic data (v–J curves in Fig. 5a) provided the primary validation for the flow field. Since Brilliant Blue is an inert tracer, its transport is predominantly advection-diffusion under the studied conditions. Therefore, accurately replicating the flow field is the most critical factor for reliable solute transport simulations. The strong agreement between experimental and simulated v–J relationships demonstrates the accuracy of the flow field, thereby validating the foundation for the subsequent solute transport results. To explicitly validate the solute transport model, we compared the simulated breakthrough curves at the column outlet with the experimental BTCs obtained via the photographic tracer method under representative flow conditions, which is plotted in Fig. 5b.

The numerical model successfully captures the macroscopic non-Fickian transport features observed in the physical experiment. Specifically, the simulation accurately reproduces both the early arrival of the solute front and the pronounced late-time tailing. This direct agreement confirms that the coupled advection-diffusion model, built upon the validated flow field, reliably simulates the mass exchange processes between the main flow channels and the eddy zones, thereby providing a robust foundation for the subsequent pore-scale analysis.

Due to the complexity of porous media structure, the streamlines are obstructed during the seepage process, resulting in the formation of numerous eddy zones. However, it is very difficult to directly extract the volume of eddy area in 3D porous media. Different methods for identifying the eddy area in 3D scale was proposed based on the definition of zero flux in eddy zones (Zhou et al., 2019) and the velocity probability density function (PDF), see details in Bijeljic et al. (2013).

Taking the SC model as an example, firstly, we sliced the 3D porous media model using the xz plane (with the x-axis as the flow direction). Multiple cross-sections are set along the y-axis to ensure that the number of slices can effectively cover the entire model, as shown in Fig. 6a, and further export the velocity field data of different 2D sections. We can see that the colors at different positions are different, reflecting different flow velocities through the local magnification of the 2D slice flow field, as shown in Fig. 6b. The flow velocity is lower in the areas where eddies are generated, while the flow velocity at the main flow stream is relatively large. To analyze the flow velocity characteristics of the eddy area and their impact on solute transport, we further obtained the flow velocity and concentration data at different times along the white section line (which includes the main flow stream and the eddy area, see Fig. 6b) according to the numerical simulation results, as shown in Fig. 6c. It can be seen from Fig. 6c that the flow velocity curve exhibits three peak velocity segments and two low velocity segments. Combined with Fig. 6b, we can see that the region with high velocity corresponds to the main flow stream, while the region with low velocity corresponds to the eddy area. The flow velocity of the main flow stream is significantly higher than that of the eddy area, approximately 20 times the velocity in the eddy area. Moreover, we can clearly observe that there is a significant point of mutation in the flow velocity between the main flow stream and the eddy zone (the blue curve in Fig. 6c). In addition, we selected two arbitrary moments (t1, t2) during the solute attenuation process to obtain the solute concentrations in both the main flow stream and the eddy zone. It can be observed that the solute concentrations in the eddy zone are significantly higher than those in the main flow stream. This is due to the relatively low flow velocity in the eddy zone, along with the bending and deflection of streamlines, which allows the solute to be captured by eddies, slowing down the diffusion process into the main flow stream.

To compare the velocity differences at various positions within the flow field, we performed normalization using the average flow velocity (vav), which is actually the inlet flow velocity. We classify the velocities at different positions in 3D space and obtained the velocity probability density function (PDF) and cumulative probability density function (CDF) under different inlet flow velocities, allowing us to compare the changes in the proportion of low-velocity areas under different flow velocity conditions. Based on the velocity difference between the main flow stream and the eddy zone, we can obtain the critical flow velocity (vc), which is the significant point of mutation mentioned above. The process of using the PDF and CDF to quantify the eddy area in a 2D slice, which is then integrated to obtain the 3D proportion, is indeed a key aspect of our analysis, which divided into five steps: Firstly, using the self-developed MATLAB code to process the acquired 2D seepage field data. For a given 2D slice (e.g., an xz plane at a specific y-coordinate, as in Fig. 6a), we export the velocity magnitude at every grid point within the fluid domain. We then calculate the PDF and CDF of these velocity values for that specific slice. The horizontal axis of this plot is the velocity normalized by the average inlet velocity (v/vav). As shown in Fig. 6c, we identify a critical flow velocity (vc) that represents the inflection point separating the low velocities in the eddy zones from the high velocities in the main flow channels. The value of the CDF corresponding to this normalized critical velocity (vc/vav) directly gives the area fraction of that specific 2D slice where the velocity is less than or equal to vc. This area fraction is our quantitative measure of the “eddy area” for that slice. By repeating this PDF/CDF analysis for a dense series of parallel 2D slices spanning the entire model (along the y-axis), we obtain the eddy area fraction for each slice. The eddy area proportion for the entire 3D model is then calculated by integrating (averaging) these 2D area fractions across all slices. This method provides a robust and objective way to quantify the complex 3D eddy volume from 2D slice data. Furthermore, while the vc threshold identification via the PDF or CDF inflection point was established here for relatively uniform packings, its statistical nature suggests potential robustness for heterogeneous natural media. Although a wider particle size distribution would broaden the velocity PDF, the physical transition between advective channels and recirculating zones (eddies) should still manifest as a characteristic inflection in the CDF. This provides a physically-grounded and consistent basis for eddy quantification across diverse and complex porous structures, even where pore-scale geometry is highly irregular.

We selected the SC model with a particle size of 10 mm to conduct numerical simulations at four different flow velocities (including 0.5, 1, 1.5, and 2 cm s⁻¹), and the 2D slice flow fields are shown in Fig. 7. The Reynolds numbers (Re=ρvporedp/μ) range from 113 to 1697, where the average pore velocity (vpore) is used as the characteristic velocity and the particle diameter (dp) as the characteristic length scale. This range encompasses all experimental conditions, providing a standardized metric to characterize the flow. We have chosen to maintain our analysis based on direct flow velocity for the specific objectives of this study. Our focus is on establishing a direct, intuitive link between the hydrodynamic driving force (velocity) and the resulting eddy development and solute transport behavior. The red areas indicate the main flow stream with high velocity, while the blue areas represent the low-velocity regions formed in the porous medium structure.

We can see from Fig. 7 that the flow velocity in the main flow stream is significantly greater than that in the eddy area. When the flow velocity is 0.5 cm s⁻¹, the streamlines are deflected due to obstructions, leading to the formation of eddies; however, the eddy area proportion is quite small. With the continuous increase of flow velocity, we can obviously see that the eddy proportion increases significantly. This aligns with our previous research conclusions that the eddy proportion will ultimately approach a theoretical maximum value (Xiong et al., 2024). When the flow state developed to a fully stable condition, we continuously injected the brilliant blue and conducted solute transport experiments in SC model porous media with particle sizes of 5 and 10 mm, using five flow velocities of 0.2, 0.5, 1, 1.5, and 2 cm s⁻¹. The BTCs under different flow velocities are obtained, as shown in Fig. 8, where the horizontal coordinate was logarithmic and the vertical coordinate normalized the concentration (C/C0). And the corresponding dilution concentration under different inlet flow velocities is defined as the respective initial concentration (C0).

We can see from Fig. 8 that the BTCs for both particle sizes shift to the left as the flow velocity increases, indicating that the penetration time decreases. The residence time of water within the pore space increases at lower flow velocities. This longer contact time allows for a more complete solute equilibration via diffusion between the main flow and the larger, more developed eddy zones present at lower velocities. Consequently, a greater mass of solute is stored in the eddies, which is then released slowly, causing the pronounced tailing. At higher velocities, the reduced contact time and different eddy dynamics limit this equilibration process, leading to less tailing. This correction accurately reflects the interplay between advection timescales and diffusive exchange. Although the tailing of the BTCs is a consistent and robust physical observation, the definitive quantitative analysis of concentrations at very low values (C/C0<0.05) is constrained by the increasing uncertainty of the optical calibration method at these levels. Therefore, the specific values in the late-time tail should be interpreted with appropriate consideration of this methodological limitation. To further quantify the eddy effect at different flow velocities and their impact on the characteristics of the BTCs, we established a quantitative relationship between the times corresponding to the BTC concentrations reaching 5 % and 98 % of the input solute concentration (t5 and t98) (Hou et al., 2018) and the different flow velocities, respectively, as shown in Fig. 9.

It can be seen from Fig. 9 that t5 and t98 show a power-law decreasing relationship with the flow velocity, indicating that eddy effect has a significant influence on the penetration process of solute. With the flow velocity increases, the eddy area proportion continues to increase, while the proportion of the main flow stream decreases. The increase in flow velocity within the main flow stream results in the solute arriving earlier. Additionally, it can be observed that the reduction of t98 is significantly greater than that of t5. Taking the porous medium with a particle size of 5 mm as an example, when the flow velocity increases from 0.2 to 2 cm s⁻¹, the value of t5 decreases by approximately 130 s, while the value oft98 decreases by about 450 s, indicating that the changes of flow velocity have a more pronounced effect on the tailing process of the BTCs.

The SC model porous media consisting of four different particle sizes (including 5, 8, 10, 15 mm) was selected for related numerical simulations. The inlet flow velocity is controlled at a constant 2 cm s⁻¹, and the 2D slice flow field is shown in Fig. 10. In this study, for a given packing arrangement (e.g., Simple Cubic – SC), the porosity remains constant regardless of the particle size. This is a fundamental characteristic of regular, ordered packings; the porosity is determined solely by the geometric arrangement of the spheres. Therefore, when we investigate the effect of particle size (e.g., 5, 8, 10, 15 mm spheres in an SC packing), we are specifically isolating the effect of scaling the pore geometry (i.e., the absolute size of the pores and throats) while the porosity, a measure of the pore volume fraction, is held constant. This allows us to independently analyze the impact of the particle size on eddy development and solute transport.

To facilitate the comparison of flow fields between different particle sizes, we adjusted the scale to ensure that the different particle sizes appear consistent. Then, it is easier to visually compare the velocity differences between the low-speed eddy area and the high-speed main flow stream. In Fig. 10, the blue area represents the low-velocity eddy area, while the red area indicates the high-velocity main flow stream. Besides, we can observe that when the flow velocity remains constant, the larger particle size led to the larger eddy area and larger rotational velocity inside the eddy area. To explore the impact of different eddy developments on solute transport, we obtained the BTCs under various particle size conditions after injecting the solute, as shown in Fig. 11.

We found that the BTCs of different particle sizes show significant differences, exhibiting anomalous early arrival times under the same flow velocity conditions. The BTCs of the porous media with a particle size of 15 mm shows the earliest breakthrough, while the solute penetration in the porous medium with a particle diameter of 5 mm the slowest. Furthermore, as the particle size decreases, there is a diminishing trend in the early breakthrough phenomenon of the solute, which is consistent with the above results of flow fields with different particle sizes (eddy development). In the process of solute transport in porous media, the rapid flow in the main flow stream significantly influences the early arrival of the BTCs. To more intuitively observe the morphology of solute peak, we selected a porous media model with a particle size of 15 mm as an example and obtained the solute distribution at 5 s after solute injection, as shown in Fig. 12a. In addition, we selected two characteristic points (points a and b in Fig. 12a) in the main flow stream and the eddy area, respectively, and obtained their BTCs, as shown in Fig. 12b.

We can see that due to the rapid flow in the main flow channels, multiple distinct solute peak leading edges have been observed. The concentration of solute in the eddy area is significantly lower than that in the main flow stream, and the concentration of solute in the eddy area shows an obvious lag compared with that in the main flow stream. We selected the feature points a and b of the main flow stream and eddy area respectively, and obtained their normalized concentration curves over time, as shown in Fig. 12b. It can be observed that due to the lag in the eddy area, the solute concentration in the main flow stream is always greater than that in the eddy area. The shaded blue area in Fig. 12b represents the process of solute transfer between the main flow stream and the eddy area caused by the concentration difference between the two points. We further magnified this process and plotted the normalized concentration difference between the main flow stream and the eddy area (the feature points a and b). The concentration peaked at 4.1 s, showing a trend of initially increasing and then decreasing, and exhibiting a trailing asymmetry, which indicating that the mass transfer rate between the main flow stream and the eddy area slows down.

To quantify the impact of eddy effect on the degree of heterogeneous distribution of solute under different particle size conditions, the dilution index proposed by Kitanidis (1994) was used to characterize the dilution state of solute plumes during solute transport. The value of the dilution index represents the volume of the solute plume within the pore volume of the porous media. For continuous injection scenarios, the value of the dilution index closer to 1 indicates a higher ratio of the solute plume occupying the entire pore volume of the media. Once the solute is injected into the porous media, due to the inhomogeneity of the medium structure and flow field, the value of the normalized dilution index will gradually increase. Then, the normalized dilution index can reflect the inhomogeneity of the solute distribution in the porous media. Besides, Dou et al. (2018) evaluated the impact of eddy effect on the uneven distribution of solutes in fractured media using the dilution index. The equation for the solute dilution index E(t) in 3D porous media is as follows: 5Et=exp⁡-∫Vpx,y,z,tln⁡px,y,z,tdV6px,y,z,t=cx,y,z,t∫Vcx,y,z,tdV where p (x, y, z, t) is the distribution function of the mass. In this study, the normalized dilution index was used to quantify the influence of eddy effect on the uneven distribution of solute in porous media with different particle sizes. When the flow velocity is 2 cm s⁻¹, the relationship between the normalized dilution index of porous media and the dimensionless time parameter t′ (the pore volume) with different particle sizes was shown in Fig. 13. The t′ represents the time process of solute transport, and the equation is as followed: 7t′=QtAl where Q is the flow flux at the outlet, t is the time of solute transport, and Al is the volume of the pore.

We can see from Fig. 13 that the normalized dilution index varies with the dimensionless time parameter in three stages: Firstly, when t′<0.12, the normalized dilution index increases slowly; when 0.12<t′<1, the normalized dilution index increases sharply; and while t′>1, the normalized dilution index remains almost stable, respectively. In the initial stage of solute entering the porous media, the solute preferentially transfers in the main flow stream, while the transfer to the eddy area is still relatively low. Then, the solute transport between the main channel and the eddy area gradually increases. In the later stage of solute transport, the retention of the solute by the eddy area results in a pronounced tailing effect during the penetration process. In addition, we found that larger particle sizes lead to larger normalized dilution index when t′<1, which is related to the pore structure formed by the SC models with different particle sizes and the development of the eddies. For porous media with larger particle size, the larger volume of a single pore leads to a larger volume of eddy area, resulting in a higher degree of inhomogeneity of the solute distribution. When t′>1, a completely opposite trend is observed.

As we know, larger particle sizes lead to a higher eddy proportion and greater initial flow heterogeneity. However, the larger individual pore bodies in these media also host larger, more coherent eddy structures. While these large eddies initially trap solute and create a highly uneven distribution (low normalized dilution index for t′<1), they also provide a larger volume for diffusive mixing to act upon over time. The mass transfer between the main channel and these sizable eddy zones is governed by diffusion across longer characteristic paths, making it a slower process. Consequently, the system with larger grains requires a longer time to achieve a homogeneous state. After the primary advective pulse has passed (t′>1), this slower, more complete back-diffusion from the large, well-developed eddies eventually leads to a higher degree of homogeneity (a higher normalized dilution index) compared to systems with smaller grains, where the pore structure is more confined and complex, potentially limiting the final extent of mixing. This interpretation, consistent with the mechanisms discussed in studies like Dou et al. (2018), highlights the time-dependent competition between heterogeneity-driven trapping and volume-enhanced diffusive mixing.

In this study, we employed the dimensional dimensionless time parameter t′ (e.g., Fig. 13) strategically. Using actual time as the x-axis in several figures (e.g., Figs. 8 and 11) was intentional, as it allows for a more intuitive understanding of the temporal scale of the observed non-Fickian transport phenomena, such as the absolute time of early arrival and the duration of tailing. This provides a direct, physical sense of the retardation caused by eddies. Meanwhile, the use of dimensionless time in other figures facilitates the comparison of the shape of the BTCs independent of the system's specific volume. We believe this dual approach offers complementary insights.

Different arrangement modes have significant influence on the structure of porous media. This section will describe in detail the eddy evolution of porous media with different arrangement modes (including SC, FCC, BCC and RP), when the inlet flow velocity is 2 cm s⁻¹, the obtained flow field is shown in Fig. 14.

We can see from Fig. 14 that the flow fields of porous media with different arrangement modes are completely different, and different colors respectively represent the main flow stream with high flow velocity (read area) and the eddy area with low flow velocity (blue area). Due to the relatively regular structure of the porous medium in SC model, the pore abdomen can provide a larger development space for eddies, so the volume of the eddy area formed by the SC model is larger than that of the other three arrangements. The spatial positions of eddies in porous media with different arrangement modes are also completely different. As for the RP model, we have observed several very interesting phenomena: The structure is further complicated due to the randomness of the arrangement of the spheres. On the one hand, two distinct preferential flow channels are formed; on the other hand, we observed regions with lower flow velocities, called dead-end-pores (Bordoloi et al., 2022). The identified eddy zones are not stagnant regions but are characterized by active recirculation. As evidenced by the velocity profile in Fig. 6c, the flow velocity within these zones, while significantly lower than in the main flow channels, is distinctly non-zero. This observation is consistent with the findings of Bordoloi et al. (2022), who emphasized that laminar vortices enhance dispersion through their inherent rotational motion. The quantification of these dynamic eddy zones forms the basis for analyzing their role in solute mass transfer and the emergence of non-Fickian transport behavior.

Besides, we also agree that porosity is a fundamental property that profoundly influences flow and transport. We recognize that porosity is not an independent variable here but is a direct consequence of the packing geometry. The substantial difference in porosity (e.g., almost 2 times between SC and BCC) is an inherent and defining characteristic of these distinct arrangements. Therefore, the analysis in Sect. 3.4 inherently addresses the combined effect of the specific pore structure and the resulting porosity. Discussing porosity as a separate, parallel factor to arrangement would create a mismatch in the logical hierarchy of influencing factors, as porosity is an emergent property of the arrangement. Our approach was to treat “arrangement” as a holistic factor that encapsulates both the geometric configuration and the resultant porosity. We believe this provides a more integrated understanding of how systematic changes in the medium's architecture control eddy development and solute transport.

To understand the influence of different arrangement patterns forming eddies on solute transport, we obtained the solute distribution characteristics in various porous media at 3 s after injection, as shown in Fig. 15.

We can see from Fig. 15 that different types of porous media models all present relatively obvious preferential flow channels, although they differ from each other. The Fig. 15a show that the preferential flow channel formed are parallel to each other due to the good connectivity of the pore abdomen in the SC model. The Fig. 15b and c indicate that the difference in solute distribution between the FCC and BCC porous media is primarily manifested in the eddies caused by different pore structures. Compared to the FCC model, an additional sphere present in the pore abdomen of the BCC model occupy more pore space, resulting in a lower eddy proportion, which causes the solute peak in the BCC model to appear more uniform. The Fig. 15d shows that the RP porous media structure is more chaotic, leading to a more uneven solute front with multiple preferential flow paths. Additionally, we further obtained the BTCs of porous media with different arrangements at a flow velocity of 2 cm s⁻¹, especially detailing the solute penetration process at different times in the RP model of porous media, as shown in Fig. 16.

We can see from Fig. 16a that the characteristics of the BTCs of porous media models with different arrangements are completely different when the flow velocity remains consistent. The penetration process of the BCC model is the fastest, followed by the FCC and SC models, while the penetration process of the RP model is the slowest. Among different types of porous media models, the RP model has the highest porosity, followed by the FCC model, while the BCC model has the lowest porosity, which indicate that the structure has a significant impact on the solute penetration process. To compare the effect of eddies on the tail of the BTC, the 1-C/C0 logarithmic coordinate plot is used to process the BTCs (see Fig. 16a). We noticed that the BTCs of the SC model and the RP model are nearly overlapping at the tail, and the higher eddy area proportion lead to the greater impact on the trailing effect. It is worth noting that the numerical results, including the low-concentration tails down to relative concentrations of 0.001, are not significantly affected by numerical dispersion. The observed tailing in the BTCs (e.g., Fig. 16a) is a physical phenomenon resulting from mass exchange between the mobile zone and the eddy (immobile) zones, not a numerical artifact, thereby confirming the reliability of our conclusions regarding tailing behavior.

Specifically, the first arrival time (t5) and the late-stage tailing time (t98) are extracted from the BTCs for each arrangement (SC, FCC, BCC, RP). These metrics provide a direct quantitative measure of the “early arrival” and “tailing” phenomena, respectively. The eddy zone proportion and characteristic breakthrough times for different packing arrangements at an average inlet velocity of 2 cm s⁻¹ are shown in Table 1.

Table 1 has been added summarizing the eddy zone proportion alongside the corresponding t5 and t98 values for each packing structure. This quantitative data reveals a clear and consistent trend. The eddy zone proportion, which is a direct consequence of the pore structure created by the packing arrangement, shows a strong correlation with the characteristic times. The quantitative relationship between packing arrangement, eddy development, and solute transport characteristics is unequivocally demonstrated in Table 1. The results indicate that the pore structure, determined by the specific packing arrangement, exerts a primary control on the proportion of eddy zones. This proportion, in turn, directly governs the key features of non-Fickian transport. A clear positive correlation is observed between the eddy zone proportion and the late-time tailing, quantified by t98. For instance, the SC arrangement, with the highest eddy proportion (0.148), exhibits the most pronounced tailing (t98=8.10 s), whereas the BCC structure, with a minimal eddy proportion (0.008), shows the fastest clearance (t98=3.95 s). While the early arrival time (t5) is generally earlier in structures with fewer flow obstructions (e.g., BCC), the significantly delayed arrival in the RP model underscores the additional influence of flow path tortuosity and connectivity, beyond the mere volume of eddy zones.

This quantitative analysis powerfully substantiates our qualitative arguments. It demonstrates unequivocally that the packing arrangement controls solute transport by determining the development of eddies, which in turn quantifiably governs the degree of both early arrival and tailing observed in the BTCs. This provides a robust, data-driven link between medium structure and non-Fickian transport dynamics.