3.1. Pore Scale
Pore-scale experiments are rarely performed to study dilution and reactive mixing in solute transport. As indicated by Equation (3), only dynamic diffusion occurring at the pore spaces is included in the transport calculation. The dilution and reactive mixing processes can be explicitly quantified with low uncertainties by applying the detailed velocity data. Nevertheless, a few pore-scale experiments are still conducted to investigate the mixing and reaction kinetics in porous media [40
]. At steady-state, the effects of grain sizes and structures and porous media heterogeneity on dilution and mixing-controlled reactions have been studied [46
]. Here we briefly review the experiments performed at steady-state.
Pore-scale experiments have been commonly performed in two-dimensional flow-through micromodels. The standard lithographic technique is one among the others (e.g., Hele-Shaw, optical lithography, wet, dry and laser or plasma etching, stereo lithography [49
]) frequently used to construct the geometry. The process is briefly described here. Silicon wafers are coated with a photoresist polymer and are selectively exposed to UV light by placing a mask above the wafer. The areas covered by the photoresist polymer are removed utilizing a developer, while the areas not covered by the photoresist polymer are etched using plasma generated from an inductively coupled plasma-deep reactive ion etching system. Flow channels are sealed by anodically bonding a cover sheet of Pyrex glass to etched silicon wafers. Details of the microfabrication process can be found in [47
] and [50
]. One representative example of the pore-scale experimental setup is shown in Figure 1
. The length and width of a typical micromodel can be several centimeters. Solutions are usually injected through different channels with the help of pumps such that the mixing between the reactive solutions can be observed along the flow directions at the steady-state transport. Depending on different solutes used in the experiments, images of fluorescent intensity within the pore space are captured for the analysis of dilution and mixing results. A microscope equipped with an automated stage and a CCD camera (furnished in front of the micromodel) can facilitate the image captions controlled by NIS-Elements imaging software. The images are corrected for nonuniform illumination in post-processing.
Willingham et al. [47
] performed conservative and mixing-controlled reactive pore-scale experiments to investigate the effect of porous media structures on steady-state dilution and reactive mixing. Five different pore structures were considered, including cylinders and horizontal and vertical ellipses with different sizes and arrangements. Fluorophore Oregon Green Bapta 488 (OG5N) in the hexapotassium salt form and Ca2+
were used as solutes due to their fast reaction rate. They found that not only the grain size but also the grain orientation significantly affects mixing and reaction. Furthermore, flow focusing by bringing stream lines closer can increase the transverse concentration gradient thus enhance dilution and reactive mixing. The enhancement is greater in pore structures with longer flow focusing regions and a larger porosity contrast. Similar findings were also reported by Oostrom et al. [46
] in which conservative solute Alexa 488 was used as the tracer in the pore-scale transport experiments.
Pore-scale experiments are essential for the investigation of dilution and reactive mixing of solute transport. However, due to its difficulties of the experimental operation and low reproducibility, very few studies of two-dimensional and three-dimensional pore-scale experiments are reported to our knowledge. Instead, numerical simulations considering pore-scale dilution and reactive mixing have been constantly performed [51
] and some of them are used for the interpretation of the experimental observations in Darcy-scale and field-scale transport problems [28
3.2. Darcy Scale
Darcy-scale experiments have been widely performed to study dilution and reactive mixing of solute transport in the last several decades [3
]. In this section, we will first present the setups and techniques used in the experiments. Then we describe the steady-state dilution and reactive mixing considering conservative and reactive transport in homogeneous and heterogeneous porous media.
The setups performing flow-through experiments consist of one-, two- and three-dimensional systems. Since transverse dispersion is often the focus of the steady-state transport, two- and three-dimensional experiments are of general interest. In two-dimensional experiments only one transverse direction exists, while there are two transverse directions (i.e., horizontal and vertical) in three-dimensional flow-through systems. Three-dimensional experiments can better resemble the natural aquifer. However, due to their operational difficulties, three-dimensional experiments are seldom performed. Figure 2
shows a typical two-dimensional flow-through system. The two-dimensional flow-through chamber normally has a length and a width of around one meter. Note that the length and height of the system should be much larger than the width of the system to better resemble a two-dimensional system.
Steady-state flow can be obtained by the constant head difference between the inlet and the outlet of the flow-through system or constant injection and extraction pumping rates at the inlet and outlet ports [30
]. The former method needs an external injection of a second solution near the inlet and extraction of samples near the outlet. Such a procedure disturbs the flow field inside the porous media and is usually termed as invasive technique. The latter method (i.e., injection and extraction ports facilitate steady-state parallel flow as shown in Figure 2
) is recommended here since the two solutions can be introduced simultaneously through the inlet without a disturbance of flow in the inner porous media. However, a high-resolution and high-precision measurements at the outlet requires many ports (i.e., many pumping channels) and high-precision pumps, thus increasing the labor and budgets of the experiments.
The materials filled in the flow-through systems are commonly glass beads or clean sands in order to exclude the influence of physical or (bio)chemical reactions at the surface of the porous medium. For the purpose of clear observation, glass beads are often used in the two-dimensional experiments. Index-matched solids and liquids phases were also used in the flow-through systems for the purpose of clearer quantification of flow fields and transport [64
]. This technique is often limited to the specific solids and liquids. The liquids are often toxic and hazardous with few exceptions (e.g., hydrogel as solid matrix and water as liquid [67
]). For quantitative analysis of plume distributions, non-invasive imaging methods were widely applied including optical imaging using UV or visible light [68
], dual-energy gamma radiation [70
], X-ray microtomography [72
] and magnetic resonance imaging (MRI) [74
]. Detailed theories and applications of the four imaging methods are referred to [76
]. Optical methods have been most frequently used in the investigation of solute mixing since they are least expensive and easy to handle [77
]. The only limitation is that they cannot facilitate three-dimensional imaging. Rather than the fluorescein intensity captured within the pore space in pore-scale experiments, the intensity captured in Darcy-scale experiments is macroscopic. In two-dimensional experiments, Haberer et al. [79
] proposed a non-invasive optode technique that is based on the dynamic luminescence quenching of a luminophore by molecular oxygen to quantitatively measure the concentration of dissolved oxygen in the flow-through system. Ye et al. [80
] qualitatively analyze solute distributions in three-dimensional inner porous media by freezing and slicing the materials after each experiment. Due to the slicing procedure, quantitative analysis of the plume distribution is not feasible. A direct and non-invasive method is to sample and measure the collected liquids at the outlet ports, however, what happened in the inner domain needs to be interpreted by numerical simulations [17
Mixing-controlled reactive transport was first proposed by numerical modeling assuming instantaneous reactions or double-Monod kinetic reactions between reaction partners [7
]. Later, it was evidenced by laboratory experiments [15
]. Bauer et al. [56
] performed two-dimensional aerobic and anaerobic biodegradation experiments using toluene as the oxidizable compound and found that biodegradation activities were mainly located at the fringe of the toluene plumes.
The influence of the domain dimensionality on solute dilution was studied experimentally by Ye et al. [30
]. Conservative tracers of fluorescein and oxygen were injected at the selected ports at the inlet and measured at the outlet ports. Even though transverse dispersion coefficients estimated from the two-dimensional and three-dimensional experiments are identical, dilution is stronger in three-dimensional systems compared to the two-dimensional systems. This is consistent with intuition since one more transverse direction is provided for dispersion and dilution.
Porous media heterogeneity has been verified experimentally to have a significant effect on dilution and mixing and reaction. In particular, it can facilitate a significant enhancement of dilution and reactive mixing of solute transport [3
]. Groundwater streamlines converge and diverge in heterogeneous porous media, leading to the plume spreading and mixing. The spreading itself cannot enhance dilution or reactive mixing [88
]. However, spreading can facilitate the enlargement of the material surface of the plume and thus increase transverse dispersion, dilution and reactive mixing. The enhancement of dilution and mixing by flow focusing and defocusing in high permeability inclusions was studied in both two-dimensional and three-dimensional systems [3
]. An example of the three-dimensional heterogeneous porous media setup with high-permeability inclusions is shown in Figure 3
]. The distances between the neighboring streamlines are shortened due to the flow-focusing effect, beneficial transverse dispersion of the solutes in the inclusions. A second effect of flow focusing is that it accelerates the flow velocity and reduces the residence time of the solutes in the inclusions. In two-dimensional systems, the former positive effect for dilution and mixing overwhelms the latter one, leading to an enhancement of dilution and reactive mixing. However, the two effects counteract each other in three-dimensional systems. The theoretical analysis shows that there is no enhancement of dilution and reactive mixing by the flow-focusing effect if the transverse dispersion coefficient (i.e., Dt
) is not velocity-dependent. However, Dt
depends on the velocity and grain size (refer to Equation (7)) and the correlation between hydraulic conductivity and grain size prevails over the enhancement of dilution and mixing in three-dimensional systems. Nevertheless, the enhancement of dilution and mixing by flow focusing is stronger in two-dimensional systems compared to three-dimensional systems. The detailed theoretical analysis can be referred to Werth et al. [89
] and Ye et al. [81
]. More importantly, the extent of dilution and mixing enhancement depends on the geometry of the porous medium, particularly the location of the high-permeability inclusions. Only if the plume fringe is focused in the high-permeability inclusions can dilution and mixing be significantly enhanced [3
Macroscopic anisotropy of porous media has also been proved experimentally as an important effect influencing dilution and mixing of solute transport. An example of the experimental setup is shown in Figure 3
]. The streamlines twist creating helical flows in the porous media. Such a complex flow field leads to the deformation of the material surface of the plume and the enhancement of dilution and reactive mixing in porous media. Notice that the helical flow can only happen in three-dimensional flow-through systems. In two-dimensional systems, streamlines can only meander since two streamlines cannot intersect each other. However, due to the difficulties of experimental procedures and measuring techniques, very few experiments considering the effect of macroscopic anisotropic porous media on dilution were performed. Ye et al. [80
] performed steady-state transport experiments using fluorescein as the conservative tracer solute and they measured the concentration at the outlet high-resolution extraction ports. However, plume distributions at the inner porous media were only measured qualitatively by freezing and slicing porous media after steady-state was reached. Nevertheless, sparse experimental data verified the numerical model and the numerical simulation helped interpret concentration distributions, dilution and reactive mixing both at the inner porous media and at the outlet of the flow-through system. More analytical and numerical studies were performed considering the effects of both microscopic and macroscopic anisotropic porous media on the helical flow, solute dilution and reactive mixing, and they showed the same trend as in the experimental outcomes [90
Three-dimensional transverse dispersion and dilution were also studied in natural consolidated rock. Boon et al. [96
] performed steady-state laboratory experiments using cylindrical Berea sandstone core as the material that plume transports. NaI was used as the conservative tracer and the plume distribution was observed by a medical X-ray CT scanner. They found that dilution was enhanced by the flow-focusing effect yet not significantly due to the homogeneity of the Berea sandstone.