# Recent Advances in Experimental Studies of Steady-State Dilution and Reactive Mixing in Saturated Porous Media

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## Abstract

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## 1. Introduction

## 2. Theory

**u**is the velocity vector. The first term at the left part of Equation (1) turns out to be the pressure gradient. At the Darcy scale, the combination of the continuity equation and the Darcy’s law leads to the governing steady-state flow equation in saturated porous media:

**q**is the specific discharge vector,

**x**is the vector of spatial coordinates, and

**K**is the hydraulic conductivity tensor with the expression depending on the porous media (i.e., isotropic or anisotropic) and the orientation of the Cartesian coordinate system.

_{aq}is the aqueous diffusion coefficient, and r is the reaction rate which equals zero for conservative solute transport. At the Darcy scale, advection-dispersion-reaction equation describes solute transport. For steady-state reactive transport, such an equation reads as:

**v**is the seepage velocity vector (i.e.,

**v**=

**q**/θ), θ is the porosity, and $D=\frac{v\otimes v}{v\xb7v}({D}_{l}-{D}_{t})+I{D}_{t}$ is the dispersion tensor, $v\otimes v$ and $v\xb7v$ are the tensor and scalar product of

**v**itself, D

_{l}and D

_{t}are the longitudinal and transverse dispersion coefficient, and

**I**is the identity matrix. For steady-state transport of continuously emitted plumes, transverse dispersion is of key importance [18]. Therefore, the transverse dispersion coefficient (i.e., D

_{t}) is highly relevant to the resulted concentration distributions in the experiments [9].

_{p}) and a mechanical dispersion coefficient (i.e., D

_{mech}). The pore diffusion coefficient is often derived from empirical correlations [20,21] since molecular diffusion can only occur in pore spaces and the size of the pores as well as the tortuous nature of the pores are unknown in most practical cases. Under saturated conditions, D

_{p}can be approximated as:

_{t}is the transverse dispersivity, and v is the absolute value of the seepage-velocity vector. α

_{t}is a property of porous media, which in principle, can be measured [25]. A more accurate empirical parameterization of the mechanical dispersion based on an earlier statistical model [26] has been proposed and validated by two and three-dimensional well-controlled laboratory experiments [17,27,28,29,30], reads as:

_{aq}is the grain Péclet number, d is the grain size, δ is the ratio between the length of a pore channel and its hydraulic radius, and β is an empirical exponent that accounts for the degree of incomplete mixing within the pore channels [31,32]. A less-than-one β value was found in the previous experiments [17,27,29,30] and it is suggested to keep β as a constant value [17,27]. The introduction of β ≠ 1 is based on the hypothesis that the physical law for the growth rate of the volume occupies by a moving elementary mass of solute in a porous medium might be non-linear. Equation (7) implies a non-linear relationship between the mechanical dispersion coefficient and the flow velocity and a compound-dependency of the mechanical dispersion term [17,27]. Such a phenomenon has been observed in earlier experimental studies [33,34,35,36,37,38,39].

## 3. Laboratory Experiments

#### 3.1. Pore Scale

^{2+}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.

#### 3.2. Darcy Scale

_{t}) is not velocity-dependent. However, D

_{t}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 [3,81].

## 4. Quantification of Dilution and Reactive Mixing

#### 4.1. Conservative Transport

#### 4.1.1. Moment Analysis

_{y}) of the system and it varies along the height (H

_{z}). In this context, the normalized first transverse spatial moment (i.e., m

_{1}) quantifies the location of the gravity center as a function of the longitudinal distance and can be calculated as:

_{2C}) represents the spreading of the concentration distribution about its gravity center and its value can be calculated as:

_{1,y}and m

_{1,z}are the normalized first transverse spatial moments at transverse longitudinal and vertical directions, m

_{2C,y}and m

_{2C,z}are the normalized second central transverse spatial moments at transverse longitudinal and vertical directions, and c

_{y}and c

_{z}represent the average concentrations along the transverse horizontal and vertical directions, respectively:

_{2C}in two-dimensional steady-state transport experiments. The value of m

_{2C}decreases once the plume enters into high-permeability inclusions. This phenomenon is not consistent with the natural rule that dilution decreases the peak concentration and increases the entropy of a conservative plume. A few other experiments also verify that the moment analysis can bias our understanding of dilution and mixing in heterogeneous porous media [104].

#### 4.1.2. Flux-Related Dilution Index

_{Q}) can be written as:

_{x}is the specific discharge in the longitudinal direction, and p

_{Q}is the flux-weighted probability density function of the solute mass defined as:

_{Q,inlet}) and outlet (i.e., E

_{Q,outlet}) of the flow-through system. In both three-dimensional and two-dimensional systems, all the symbols are above the black line in Figure 4, indicating a larger dilution at the outlet compared to the inlet (i.e., E

_{Q,outlet}> E

_{Q,inlet}). Furthermore, dilution is enhanced by the heterogeneity and anisotropy of the porous media, which is reflected by the flux-related dilution index between homogeneous and heterogeneous anisotropic setups in Figure 4.

#### 4.2. Reactive Transport

#### 4.2.1. Plume Length

_{t}. In cases that one reactive solution (A) is injected as a line source or square source and the other reactive compound (B) was injected in the ambient solution, the effective transverse dispersion coefficient (i.e., D

_{t,eff}) can be estimated from the observed plume length of A in two- and three-dimensional systems respectively, defined as [80,105]:

_{crit}is the critical mixing ratio according to [105]. Two such equations were derived from the analytical solutions of the advection-dispersion equation in homogeneous setups. Therefore, D

_{t,eff}might be considerably larger than the local transverse dispersion coefficient when the plume length is measured in a heterogeneous porous media setup.

#### 4.2.2. Mass Flux

## 5. Summary and Outlook

- In most performed experiments, glass beads and clean sands were used as the porous medium to exclude the external effects such as the irregular shapes of the grains and the sorption of the compounds on the grain surfaces. However, natural porous media indeed has irregular shapes and physical and chemical sorption potentials. Such characteristics of the natural porous media have additional effects on solute dilution and mixing. In future laboratory experiments, we suggest more applications of the porous medium taken from the natural field. To simplify the experimental conditions, we could pre-treat the porous medium. For instance, we could sieve and classify the grains if the target of our research interest is the influence of the grain sizes and shapes on dilution and reactive mixing.
- The experimental studies have shown the importance of single or several high-permeability inclusions and macroscopic anisotropic lenses to dilution and reactive mixing. As to the larger scale, Di Dato et al. [106] performed numerical simulations to calculate the influence of multiple heterogeneous microstructures on macrodispersion and showed a high relevance between macrodispersion and heterogeneous microstructures. However, no experiments have ever been performed to test the different spatial distributions of multiple inclusions and lenses on large-scale dilution and mixing. In future laboratory experiments, we suggest properly enlarging the scale of the flow-through domain and including more heterogeneous lenses to resemble spatial features observed in the natural aquifers.
- So far, most experiments were performed involving conservative transport or very simple reactions. However biogeochemical reactive transport is the common phenomenon in natural aquifer systems. Even though complex reactive transport is rather difficult to operate in laboratory experiments, it should be tested to help better understand reactive mixing.
- Hydrodynamic dispersion, particularly mechanical dispersion, has an impact on the structure of fingers in density-driven nature convection. Recent experimental studies found that hydrodynamic dispersion enhances the merging of fingers and thus reduces the finger numbers [107,108,109]. Such a process indicates an effect of hydrodynamic dispersion on dilution and reactive mixing of solute fingers in density-driven natural convection. Therefore, density effects may also be considered and included in the future investigation of dilution and reactive mixing laboratory experiments.
- By complicating the experimental conditions, more advanced experimental operation and measurement techniques are required. Particularly in the three-dimensional systems, development of an easy and cheap method for observation and quantification of plume distributions in the inner porous media would help dramatically advance our understanding of dilution and reactive mixing from an experimental perspective. Therefore, during future experimental studies, development of better experimental methods and techniques could also be a key point to break through experimental studies of dilution and reactive mixing.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Pore-scale experimental setup with details of porous media structures: Solution A and Solution B are injected through two inlet ports; waste is extracted through outlet port; constant-rate injection and extraction are realized by three pumps.

**Figure 2.**Two-dimensional Darcy-scale flow-through experimental setup: solutions are injected through inlet ports and extracted through outlet ports; constant-rate injection and extraction are realized by two high-precision pumps; samples are collected at the outlet channels.

**Figure 3.**(

**a**) Heterogeneous isotropic porous media setup: three-dimensional overview, front view and side view. The orange cube represents the high-permeability inclusion with coarse glass beads, and the other porous media consist of fine glass beads (modified from [81]). (

**b**) Macroscopic heterogeneous anisotropic porous media setup: three-dimensional overview and top view of Layer 1 and Layer 2. Layer 1 and Layer 2 are structured as simplified herringbone cross-stratification with alternated fine and coarse glass beads, Layer 3 and the unsaturated zone consist of fine glass beads (modified from [80,82]).

**Figure 4.**Flux-related dilution indices at the inlet and outlet for different experimental setups: the black line represents no dilution between outlet and inlet (i.e., E

_{Q,outlet}= E

_{Q,inlet}); 2D and 3D indicate two-dimensional and three-dimensional flow-through systems respectively; v1, v3 and v5 represent flow velocities of 1 m/day, 3 m/day and 5 m/day respectively; Flu and Oxy represent a tracer solute of fluorescein and oxygen in conservative experiments respectively; Hom and Het represent homogenous and heterogeneous porous media.

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**MDPI and ACS Style**

Xu, T.; Ye, Y.; Zhang, Y.; Xie, Y.
Recent Advances in Experimental Studies of Steady-State Dilution and Reactive Mixing in Saturated Porous Media. *Water* **2019**, *11*, 3.
https://doi.org/10.3390/w11010003

**AMA Style**

Xu T, Ye Y, Zhang Y, Xie Y.
Recent Advances in Experimental Studies of Steady-State Dilution and Reactive Mixing in Saturated Porous Media. *Water*. 2019; 11(1):3.
https://doi.org/10.3390/w11010003

**Chicago/Turabian Style**

Xu, Tiantian, Yu Ye, Yu Zhang, and Yifan Xie.
2019. "Recent Advances in Experimental Studies of Steady-State Dilution and Reactive Mixing in Saturated Porous Media" *Water* 11, no. 1: 3.
https://doi.org/10.3390/w11010003