# Experimental Analysis of the Mechanisms of Reactive Mixing at the Seawater-Freshwater Interface

Kevin de Vriendt, Tanguy Le Borgne, Joris Heyman, Maria Pool, Francesco

Gomez, Yves Méheust, Marco Dentz

► To cite this version:

Kevin de Vriendt, Tanguy Le Borgne, Joris Heyman, Maria Pool, Francesco Gomez, et al. Experimental Analysis of the Mechanisms of Reactive Mixing at the Seawater-Freshwater Interface. Water Resources Research, 2025, 61 (8), pp.e2024WR038947. 10.1029/2024wr038947. insu-05240157

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Water Resources Research®

RESEARCH ARTICLE

10.1029/2024WR038947

**Key Points:**

* Chemiluminescence reaction used to quantify mixing-induced reactions at the salt-freshwater interface in sand-tanks

* The reaction rate is largely enhanced at the interface and varies both locally and globally as a function of fluid flow

* These reaction patterns and rates are captured by an analytical model, providing a new framework for mixing-induced reaction in coastal aquifers

**Correspondence to:**

M. Dentz,
marco.dentz@csic.es

**Citation:**

De Vriendt, K., Le Borgne, T., Heyman, J., Pool, M., Gomez, F., Méheust, Y., & Dentz, M. (2025). Experimental analysis of the mechanisms of reactive mixing at the seawater-freshwater interface. *Water Resources Research, 61*, e2024WR038947. https://doi.org/10.1029/2024WR038947

Received 18 SEP 2024
Accepted 16 JUN 2025

© 2025. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.

<figure><img src="image_2.png" /></figure>

Experimental Analysis of the Mechanisms of Reactive
Mixing at the Seawater-Freshwater Interface

Kevin De Vriendt¹ ID, Tanguy Le Borgne², Joris Heyman² ID, Maria Pool³ ID, Francesco Gomez², Yves Méheust²,⁴ ID, and Marco Dentz⁵ ID

¹Federal Institute for Geosciences and Natural Resources (BGR), Hannover, Germany, ²University of Rennes, CNRS, Géosciences Rennes – UMR 6118, Rennes, France, ³Amphos 21 Consulting S.L, Barcelona, Spain, ⁴Institut Universitaire de France (IUF), Paris, France, ⁵Spanish National Research Council (IDAEA-CSIC), Barcelona, Spain

**Abstract** The salt-freshwater interface (SFI) in coastal aquifers, where water bodies of differing chemical compositions converge, represents as a hotspot for chemical disequilibrium and biogeochemical reactions. The interplay between variable density flow, mixing and reaction in porous media leads to complex and highly localized reaction patterns across the SFI, which are difficult to visualize, understand and model. In this study, we present a novel experimental setup to quantify mixing-driven reactions at the SFI in a quasi two-dimensional laboratory-scale sand tank. We use a fast irreversible chemiluminescence reaction to obtain high resolution images of reaction intensity at the interface. We show that the reaction rate varies spatially along the interface, with up to a factor of four between minimum and maximum. We present a mechanistic model quantifying the evolution of the reaction rate along the interface by describing the coupled effects of solute dispersion, fluid compression and the change of the interface geometry under different flow rates. This modeling framework captures the spatial distribution of reaction along the SFI observed experimentally. It also explains and predicts the enhancement of the effective reaction rate across the interface when increasing freshwater flow. These findings provide a novel experimental technique and a new modeling framework to image and predict mixing-driven chemical reactions in coastal aquifers.

**1. Introduction**

The salt-freshwater interface (SFI) plays a central role in biogeochemical processes in coastal aquifers (Charette & Sholkovitz, 2002; Slomp & Cappellen, 2004; Spiteri et al., 2006; Werner et al., 2012). Since freshwater and saltwater have different densities and chemical compositions, fluid mixing at the SFI triggers a range of chemical and biological reactions (Rolle & Le Borgne, 2019; Valocchi et al., 2018), including precipitation/dissolution processes and contaminant degradation (Charette & Sholkovitz, 2002; Rezaei et al., 2005; Sanford & Konikow, 1989; Spiteri et al., 2006). The interplay between variable density flow and local scale transport leads to complex mixing and reaction patterns along the SFI (De Vriendt, Le Borgne et al., 2020; De Vriendt, Pool, & Dentz, 2020; Geng et al., 2020; Pool & Dentz, 2018; Rezaei et al., 2005; Sanford & Konikow, 1989). For instance, mixing-induced dissolution preferentially occurs at the toe and head of the mixing interface (Rezaei et al., 2005; Sanford & Konikow, 1989), which is thought to be a key mechanism for the formation of coastal caves and karst features (Back et al., 1986). Additionally, pH gradients across the SFI seen in Waquiot Bay have been shown to control iron oxide precipitation which can subsequently act to attenuate nutrients such as phosphate discharging into near-shore marine environments (Charette & Sholkovitz, 2002; Slomp & Cappellen, 2004; Spiteri et al., 2006). Uncovering the mechanisms controlling the location and intensity of reaction hotspots is a critical step to understand and predict biogeochemical processes at SFIs in the subsurface.

Over the past decade, a series of studies have used experimental sand tanks to study the SFI across a range of topics including the influence of engineered structures (Houben et al., 2018; Luyun, Momii, & Nakagawa, 2009), seasonal recharge (Chang & Clement, 2012), artificial recharge (Luyun, Momii, Nakagawa, & Takahashi, 2009), heterogeneity (Castro-Alcalá, 2019; Lu et al., 2013), tides (Kuan et al., 2012), as well as transport processes within and outside the saltwater wedge (Chang & Clement, 2013; Liu et al., 2014; Oz et al., 2015; Singh et al., 2006). The majority of these studies used conservative colored dyes to observe the movement of the salt-water body and the bulk movement of the interface. A few studies have quantified the extent of the mixing zone but it has been challenging so far to obtain a direct quantification of reaction rates along the interface. Abarca and Clement (2009) developed a colorimetric method for delineating the mixing zone based on the reaction between
alkaline freshwater and acidic saltwater. This allowed for the determination of dispersivity coefficients and a comparison with numerical models of conservative transport. Image processing techniques, such as pixel-wise regression demonstrated by (Robinson et al., 2016) and a two-step method by (Castro-Alcalá, 2019) that combines weighted kernel density estimation with Steering Kernel Regression, have been used to resolve mixing across dynamically moving SFI (e.g., seawater intrusion and recession). However, conservative scalar concentration images generally suffer from noise and are limited to a spatial macroscopic description of the extent of the interface and do not characterize local concentration gradients. This makes it difficult to precisely quantify mixing rates and infer reaction rates.

Imaging of reactive processes in porous media has classically been achieved with colorimetric reactions (Gramling et al., 2002; Oates & Harvey, 2006). In such reactions, two initially colorless reagents react and form a colored product which can be quantitatively measured in transparent porous media (Oates & Harvey, 2006). In the study by (Gramling et al., 2002), a copper sulfate-EDTA reaction was employed, which produces a dark blue product. The amount of light absorbance can then be related to the concentration of CuEDTA²⁻ using a pre-determined calibration curve, similar, but not exactly identical, to the logarithmic inverse Beer-Lambert law (Gramling et al., 2002). A disadvantage of these type of colorimetric methods is the non-linear and close-to-logarithmic relationship between the concentration and the transmitted light. For small light intensities, that is, small concentrations, a small uncertainty on the measured intensity result in a much larger intensity on the inferred concentration, and consequently the inferred concentration gradients are relatively noisy. To overcome this issue, (De Anna et al., 2013) used a peroxyoxalate chemiluminescent reaction (Jonsson & Irgum, 1999) to characterize reactive mixing by the flow through a two-dimensional porous medium. Unlike the aforementioned colorimetric technique, this method allows for direct visualization of local reactions rates. The emitted light detected from this reaction is directly proportional to the local reactive rate at the position from where the light is emitted (Blum, 1997; Mendenhall, 1993). The chemiluminescent reaction employed by (De Anna et al., 2013) is limited by the use of acetonitrile, whose fluid properties (ρ and μ of 787 kg/m³ and 3.4 × 10⁻⁴ kg/(m·s)) preclude its use in density driven flows at the Darcy scale. As an alternative luminol chemiluminescence was used by (Izumoto et al., 2023) in a Hele-shaw cell and a porous micromodel to investigate mixing and reactions in converging flows.

Here, we use luminol chemiluminescence to probe reactive mixing across the SFI. We thus obtain a direct quantification of mixing-induced reactions rates in a Darcy-scale experimental setup with fluid density gradients. We consider a reaction that occurs nearly instantaneously relative to relevant mass transfer time scales. Once reactants come into contact, they rapidly deplete, and the reaction ceases. Further reactions can only occur when additional mixing brings new reactants together (De Anna et al., 2013), the reaction is mixing-limited or mixing-induced. The considered chemiluminescence reaction is comparable to a fast irreversible bimolecular reaction of type A + B → C.

This paper is organized as follows. After presenting the experimental methods, we image reaction rates for several flow rates and highlight the influence of non-uniform flow on the spatial distribution and rates of mixing-induced reactions. Finally we present a mechanistic model capturing the location and intensity of the resulting reactive hot spots along the interface and the control of fluid flow on the effective reaction rate across the interface.

## 2. Materials and Methods

### 2.1. Sand Tank

Experiments were conducted in a rectangular tank (dimensions 0.38 m × 0.15 m × 0.025 m) in order to simulate seawater intrusion at laboratory scale. The tank consists of three main compartments, the central porous chamber and the reservoir chambers on each side (see Figure 1). The tank walls are made of 10 mm thick glass plates in order to optimize optical clarity while the top and bottom consist of acrylic glass (PMMA). The central porous chamber is bound by two rigid screens covered by a 1 mm nylon mesh, each of which define the boundary to one of the two reservoir chambers (inlet or outlet). The porous medium is composed of fluorinated ethylene propylene grains (FEP 100 X, Chemours®). This material has a similar refractive index ($n = 1.34$) to water ($n = 1.33$) while being fairly translucid. Emitted light can thus be transmitted without being refracted too much by the porous medium, though it absorbs light significantly. FEP grains have rectangular shapes of dimensions of approximately
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Schematic of the experimental setup and its components. (a) Front view of the sand tank. (b) Back view of the experimental setup showing the placement of the variable flow meter, pressure controllers and the camera.</figcaption></figure>

2 mm × 1 mm × 2 mm (L × W × H). Additional information regarding the transparency of FEP used in tank-scale mixing experiments can be found in (Rousseau et al., 2023).

## 2.2. Characterization of Local Reactions Across the Interface

### 2.2.1. Chemiluminescent Reaction

To visualize the local reaction rate occurring across the salt-fresh water interface, we used of a fast (flash-type) chemiluminescent reaction. In this reaction, blue light is emitted during the chemical reaction when an intermediate species in an excited state returns to the ground state. As the reactants mix, the amount of light generated is proportional to the reaction rate. A similar technique was previously used by (De Anna et al., 2013), who studied mixing and reaction kinetics in a two-dimensional porous medium study using a peroxyoxalate chemiluminescent reaction described by ref. (Jonsson & Irgum, 1999). Due to the specific nature of our flow configuration, we choose instead a luminol chemiluminescent reaction under the catalytic influence of cobalt. This allowed us to use water as our main solvent while maintaining an optimal reaction speed and light intensity. This method was similarly applied by (Izumoto et al., 2023) in a two-dimensional porous medium to study mixing-induced reaction in converging flows, in the absence of density contrasts. This reaction not only provides flexibility in adjusting the fluid density, but is also significantly more affordable while also limiting the amount of toxic chemicals that need to be handled during experiments. According to ref. (Burdo & Seitz, 1975), the reaction mechanism involves the formation of a cobalt-peroxide complex that acts as the rate-limiting reagent in the rate-limiting step. The generalized reaction can be expressed by,
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) Example of a batch test performed with the optimized reagent concentrations using the salt solution. Light gray lines denote four repeated tests and the dashed black line denotes the average. Light emissions have been normalized by the maximum light intensity and the peak is shifted to *t* = 0. The average reactive time scale, τr, for the batch tests was found to be 0.11 s. (b) Image of batch test set-up with a beaker placed upon a magnetic stirrer and the camera fixed above.</figcaption></figure>

$$ \text{Luminol} + \text{H}_2\text{O}_2 \xrightarrow{\text{CO}^{2+}} \text{Aminophthalatedianion} + \text{H}_2 + \text{N}_2 + \text{Hz} \quad (1) $$

We used this reaction to create an irreversible bi-molecular reaction A + B → C at the mixing interface. The seawater solution A contained 35 g/L salt (sodium chloride, NaCl), 10⁻³ M of the chemiluminescent agent (luminol, 5-amino-2,3-dihydrophthalazine-1,4-dione) and 5 × 10⁻⁴ M of the catalyst (cobalt chloride hexahydrate, CoCl₂·6H₂O). The pH of the solution A was adjusted to 12. Solution B contained 1 × 10⁻⁴ M of the oxidant hydrogen peroxide (H₂O₂). When the two solutions are put in contact, a chemiluminescent reaction ensues, resulting in a blue glow (of wave length λ = 420–460 nm). Luminol chemiluminescence is a notoriously sensitive reaction that is strongly influenced by very low concentrations of various organic and inorganic compounds (Rose & Waite, 2001). Therefore, all dilutions were carefully prepared using Milli-Q water prior to each experiment. Although pH is a parameter sensitive to the intensity of luminol reaction (Burdo & Seitz, 1975), it did not vary significantly during the experiments and therefore pH buffers were not used.

### 2.2.2. Reaction Kinetics

To characterize the kinetics of luminol chemiluminescence, we first performed a series of batch tests in well-mixed conditions. These tests were carried out to ensure that the reaction kinetics were optimized such that light emissions were largest and the reaction was sufficiently fast. We recorded the emitted light after mixing the reactants within a beaker with a magnetic stirrer using a sCMOS camera (Hamamatsu ORCA flash 4.0). Solution B, was first placed in a beaker with the magnetic stirrer. A small volume of solution A was then quickly added manually using a syringe while recording the light emission.

Following (De Anna et al., 2013), the characteristic reaction time τr was defined as the time at which the emitted light is reduced by a factor 10 in a well-mixed volume. Using the aforementioned reactant concentrations, we find τr = 0.4s. This is five times faster than obtained with other reactants by (De Anna et al., 2013), see Figure 2. To compare the advection and reaction time scales across the SFI, we define the Damköler number, Da, as the ratio of advection to reaction time scales, Da = τa/τr, where τa = l_c/v_c, with l_c defined as the characteristic grain size and v_c the characteristic velocity. For our experiment, v_c can be approximated using Darcy's law, v_c = KΔH/φL_D, where K is the hydraulic conductivity, ΔH is the hydraulic gradient, φ is the porosity and L_D is the horizontal length over which the hydraulic gradient is imposed. The resulting Da varies between 6 and 11.5 over the range of imposed ΔH. Although Da is not very large, the values satisfy our general requirement of fast chemistry where mixing processes are limiting (Da ≫ 1), rather than the chemical kinetics.
<table><caption>**Table 1**
Hydraulic Head Gradients Imposed for Each Steady State Experiment</caption><thead><tr><td>Position</td><td>ΔH (mm)</td></tr></thead><tbody><tr><td>P<sub>1</sub></td><td>11.5</td></tr><tr><td>P<sub>2</sub></td><td>10.5</td></tr><tr><td>P<sub>3</sub></td><td>9</td></tr><tr><td>P<sub>4</sub></td><td>7.5</td></tr><tr><td>P<sub>5</sub></td><td>6</td></tr></tbody></table>

Note. The experiment consists of five steady state positions during intrusion.

### 2.2.3. Experimental Methodology

Solution A and B were prepared on the same day as the experiment. Due to the poor solubility of luminol, solution A was prepared several hours in advance to mix sufficiently and left in a dark area to limit any possibility of photo-oxidation (Rose & Waite, 2001). Since the FEP grains are hydrophobic the tank was initially flushed with CO₂ gas, then flushed several times with water to reach neutral pH conditions. Three liters of solution B (approximately five pore volumes) was then flushed through the tank from the freshwater inlet, the excess of which was discharged from the saltwater reservoir outlet. Prior to starting the experiment, four bottles of solution B (6 L) and two bottles of solution A (3 L) were connected to the pressure controller and the variable

area flow meter (VAF) which feeds into the bottom of the tank (see Figure 1b). After each experiment, the tank was flushed and the grains removed to eliminate gas bubbles formed during experiments. Since luminol chemiluminescence is sensitive to low concentrations of various organic and inorganic compounds (Johnson & Sakamoto-Arnold, 1987; Kim et al., 2013; Rose & Waite, 2001; Zhao et al., 2007), the tank and grains were carefully rinsed using purified water and left to air dry. For each imposed freshwater head, the system was allowed to evolve until steady state was achieved. Steady state was found when three consecutive images showed no distinct changes in the distribution of light and position of the toe of the SFI. The Freshwater heads imposed for each successive steady state interface are provided in Table 1.

## 2.3. Image Acquisition and Processing

### 2.3.1. Steady State Images

Image acquisition was carried out using a high-sensitivity full-frame DSLR camera (Sony Alpha 7s, f-50 mm). The camera was placed 54 cm away from the tank in order to capture the entire central porous chamber. Under this configuration the pixel size (8.40µm) corresponds to 0.09 mm on the tank. An ISO of 32,000 was found to be an optimal compromise to capture relatively low light intensities while maintaining low readout noise. Lens focus was ensured prior to filling the tank by focusing on a ruler placed in the vertical mid-plane of the tank normal to the camera's optical axis. Several steady-state wedge configurations were established by changing the flow rate on the variable area flow meter. At each new position, three images (over the duration of 3 τ<sub>exp</sub>) were captured and averaged during image processing, where τ<sub>exp</sub> is the single shot exposure time. The reaction rate, given by local pixel light intensities, P evaluated after averaging was calculated as,

$$R_{i,j} = \frac{1}{N} \sum_{k=1}^{N} \frac{P_{i,j}^{k}}{\tau_{\text{exp}}} \quad (2)$$

where N is the number of steady state images for the given ΔH under consideration. To filter pore-scale variations in the emitted light, images were smoothed using a five pixels (~0.5 mm) moving average filter, a size corresponding to the mean grain diameter. Pixel intensities were normalized by the camera bit depth (2<sup>14</sup>).

### 2.3.2. Channel Color

Color images of the chemiluminescent reaction in FEP porous media showed that there were substantial differences in the signals given by the blue, red and green channels. The blue channel images were found to be brighter, but more blurry than the green and red channels. This is explained by the fact that the transparency of FEP varies with the wavelength of emitted light, with shorter wavelengths (i.e., blue) being absorbed and scattered more than green and red (Dever et al., 2001). In contrast, the red channel was very dark, because the chemiluminescence emission spectrum for this particular reaction is centered on blue wavelengths. Considering these two results, we only used the green channel to assess reaction rates.

### 2.3.3. Characterization of Conservative Transport

Concentration and concentration gradient for a steady state saltwater intrusion experiment were obtained using a conservative dye. The movement of the saltwater wedge is directly imaged using a light transmission technique.
In order to produce a uniform light distribution across the domain, a 100 W LED backlight panel was placed directly behind the sand tank. As tracer we use the natural food grade dye E 163. The denomination E 163 stands for are group of plant-based water soluble pigments, which are referred to as anthocyanins. We found that the dye was readily flushed off the FEP grains, and that horizontal and transverse concentration profiles from a slug injection test were Gaussian, which indicates that the dye is not sorbing on the FEP grains. A dye concentration of 0.1 g/L was mixed in a salt solution containing 35 g/L of NaCl was used for all conservative seawater intrusion test. Image acquisition was also carried out using the Sony Alpha 7s f-500 mm camera.

We follow Robinson et al. (2015) in order to best capture the mixing zone across the domain. For the calibration eight dye concentrations (0%, 5%, 10%, 25%, 50%, 75%, 90%, 100%) were used and flushed across the porous media. For each concentration approximately four pore volumes were used to flush the system, moving from the lowest concentration to the highest. This was deemed sufficient by ensuring light intensity across the entire domain was no longer fluctuating significantly for each concentration. For each concentration 10 images were captured and averaged. To relate the pixel light intensity to the concentration, use a pixel-wise regression method where we determine the power law regression coefficients for every pixel in the image. The concentration, C is therefore related to the light intensity, I following,

$$C = at^b - c, \quad (3)$$

where a, b and c are coefficients determined from the regression analysis. This method accounts for spatial variations in lighting due to bead heterogeneities and non-uniform back panel light (Robinson et al., 2015). Note that the calibration is specific to the domain being tested. Accordingly, the conservative seawater intrusion experiment was performed directly after the calibration was finished. Prior to doing so, approximately eight pore volumes of deionized water were flushed across the domain to ensure that all the dye was removed from the tank.

# 3. Results and Discussion

## 3.1. Mixing-Induced Reaction Along the Interface

The general characteristics of the chemiluminescent reaction across the interface at steady state are presented in Figure 3. The blue light emitted by the reaction is observed along the interface with the emitted light being most intense near the toe and head of the mixing zone.

In Figures 4a and 4b, we show the post-processed results of a conservative tracer experiment. The mixing zone appears relatively thin, consistent with observations from tank-scale experiments (Abarca & Clement, 2009; Robinson et al., 2015). However, we note that at very dilute concentrations, the mixing front extends further than expected. This effect is attributed to actual mixing processes rather than optical artifacts such as light diffraction. As expected, the concentration gradients from the conservative experiments are noisy and difficult to use to quantify mixing rates (Figure 4b). Furthermore, at Darcy scale, velocity dependent hydrodynamic dispersion must be considered in addition to concentration gradients to adequately describe mixing rates (Abarca & Clement, 2009; Pool & Dentz, 2018). These challenges in the quantification of mixing rates from conservative tracers motivate the use of chemiluminescent tracers, which provide a direct measurement of mixing-driven reactions.

Post-processed images of the reactive experiment are shown in Figures 4c and 4d. Although most of the reaction is localized at the interface, there is some background noise due to light produced by reaction occurring at the top of the salt water reservoir (Figures 3 and 4c). This is attributed to the large mixing occurring at the top of the discharge zone where fresh and mixed hydrogen peroxide flow across the top of the salt water reservoir. The light from the reaction is then scattered back into the top right region of the main tank due imperfect index matching of the grains with the resident fluid. There is also some light scattering occurring around the interface, due to the slight opacity of FEP for green wavelengths (Figures 3c–3g). To minimize the effect of this noise, we focus on the local maximum light intensity measured along the SFI, $R_{Max}(x)$. Because of the relative FEP opacity, part of the light emission was absorbed by the grains. However, since the SFI is two-dimensional, we expect the measure to be representative of the depth-average local reaction rate. Large reaction rates concentrate close to the SFI, where mixing occurs (Figure 4c), with an average intensity about five to eight times larger than the typical background intensity and maximum localized intensities ranging between 30 and 80 times larger than the background
<figure><img src="image_5.png" /><figcaption>**Figure 3.** (a) Raw image of chemiluminescent reaction across the mixing zone. Arrows depict general flow pattern within the flow cell at steady state. (b) Zoomed in image of salt water reservoir near discharge zone. (c–f) Show raw images of the reaction interface during the beginning stages of its advance prior to achieving steady state.</figcaption></figure>

intensity. The two local reaction rates maximum at the toe and at the head of the SFI (Figure 4d) have a reaction rate up to 2–4 times larger than in the central part of the SFI. Future studies aiming to mitigate light scattering should focus on identifying or designing porous media that are perfectly index-matched with the resident fluids,

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Image map of (a) the concentration field shown with the predicted interface position using the Glover solution (b), the normalized concentration gradient and (c) the normalized reaction rate for steady-state position $P_3$ and (d) Reaction maximum, $R_{max}$ along the interface normalized by the Reaction maximum at the toe $R_{max,t}$. The dark blue line denotes the average of the three steady reaction maximum profiles shown in light blue. Note position of the reaction hotspots which correspond to the toe and head of the salt-freshwater interface in (e). The toe length ($L_t$) is also shown.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Transient behavior observed throughout the experiment. (a) Top: Volumetric flow rate of solution B injected at the left boundary and bottom: toe length ($L_t$) over the duration of the experiment. (b) Local $R_{Max}$ values $R_{Max,l}$ at the toe, min and head ($R_{Max,t}$, $R_{Max,m}$ and $R_{Max,h}$ respectively) normalized by the lowest flow $R_{Max,m}$ value. (c) Integrated global reaction rate, $R_g$ normalized by the global reaction rate at the lowest flow, $R_{g,min}$ value. Vertical gray bars indicate steady state positions.</figcaption></figure>

without compromising the chemiluminescent reaction or disrupting variable density flow. However, to the authors' knowledge, materials that meet all these criteria are not readily feasible or readily available.

### 3.1.1. Evolution of the Reactive Front Under Different Flow Rates

We recorded the transient evolution of the SFI between successive steady state positions by a sudden drop of the freshwater head, leading to a forward movement of the SFI (Figure 5, Table 1). The flow rate stabilized quickly after each head drop (Figure 5a) and varied over a factor 3.1 in total between the beginning and end of the experiment. As the head gradient was reduced, the time required for the interface to reach quasi-equilibrium conditions became longer. Over the experiment, the interface length increases by a factor of 3.4 in total and the length at each steady state was approximately inversely proportional to the flow rate (Figure 7a), as expected from the Glover approximation (Glover, 1959).

During the advance of the front, the total reaction rate varied with each decrease in $\Delta H$ (Figure 5b) and reached a steady value over time scales similar to the toe length, $L_t$ (Figure 5a). The total reaction rate generally decreased when decreasing the flow rate and was approximately five times smaller for the lowest flow rate compared to the highest flow rate (Figure 5b). The maximum reaction rates at the toe and head (Figure 5c) were close to each other for each flow rate, and increased with the flow rate, following a similar trend as the average reaction rate. The
minimum reaction rate in the central part of the interface was two to four times lower than at the toe and head for all flow rates.

## 3.2. Reactive Mixing Model

To understand and quantify the evolution of the reaction rate along the interface, we build on the mixing model presented by De Vriendt et al. (2022a) that quantifies conservative mixing in SFIs. Specifically, the model accounts for the impact of interface deformation due to buoyancy effect on solute mixing, and identifies mixing and reaction hotspots along the deformed interface. Variable density flow plays a key role for the understanding of reaction patterns in the salt-freshwater mixing zone.

We consider a steady reactive mixing interface producing an irreversible bimolecular reaction $A + B \rightarrow C$ due to transverse mixing at the front. Since the experiment is characterized by a large Damköhler number, we assume that the reaction occurs instantaneously upon mixing across the interface. The local reaction rate at horizontal position $x$ is thus equal to the transverse dispersive flux of reactant A across the interface (Bandopadhyay et al., 2017; De Anna et al., 2013; Izumoto et al., 2023),

$$R_c(x) = -D_t \frac{\partial c_A}{\partial n} \approx \frac{D_t c_0}{s(x)}, \qquad (4)$$

where $c_A(x)$ is the local concentration of species A, $n$ is the coordinate in the direction perpendicular to the interface, $s(x)$ is the mixing width characterizing the transverse size of the interface, $D_t$ is the transverse dispersivity coefficient and $x = 0$ is the position of the toe. Note that we do not resolve here mixing at the pore scale and assume that mixing is mostly driven by Darcy scale concentration gradients. Under this hypothesis, Equation 4 shows that the local reaction rate results from the interplay between the dispersion coefficient $D_t$, which depends on the local velocity, and the mixing width $s(x)$, which determines the local concentration gradient. The mixing width evolves along the interface under the action of dispersion and diffusion, which tend to increase it, and compression by fluid flow (De Vriendt et al., 2022; Dentz et al., 2023; Rousseau et al., 2023), which reduces it thus enhancing concentration gradients.

We distinguish two flow regimes characterizing the competition between dispersive expansion and compression of the interface width as discussed by (De Vriendt et al., 2022). From the toe up to a characteristic distance $x_c$, the interface width expands under the action of dispersion, as the fluid velocity is small and its effect on the compression of the interface is negligible. For $x > x_c$, the fluid velocity accelerates with $x$ as the space available for the freshwater to flow reduces. Fluid incompressibility leads to a compression of the interface in the direction perpendicular to the acceleration, which dominates over the dispersive expansion. As shown by De Vriendt et al. (2022a), the transition between the two regimes occurs at a distance $x_c = Ng'/3$, where $Ng'$ is the modified gravity number defined as $Ng' = K\epsilon/q_f$, with $\epsilon = (\rho_s - \rho_f)/\rho_f[1 - (\alpha_t/b)^{1/4}]$, $\rho_s$ the saltwater density, $\rho_f$ the freshwater density, $q_f$ the freshwater velocity, $b$ the domain thickness and $\alpha_t$ the transverse dispersivity. This modified gravity number $Ng'$ quantifies the impact of dispersion on the interface position (Lu et al., 2013; Pool & Carrera, 2011), see Appendix A.

For $x < x_c$, the mixing width $s(x)$ hence increases with $x$, following the characteristic dispersive expansion (De Vriendt et al., 2022),

$$s(x) = \sqrt{\alpha_t x}. \qquad (5)$$

Note that this expression is obtained by setting $D_t = \alpha_t v_f$, where $v_f = q_f/\phi$. During this regime, $v_f$ is assumed to be approximately constant as compression rates are still sufficiently small and the interface continues to grow dispersively. Inserting Equation 5 into Equation 4, we obtain the total reaction rate in the region $x < x_c$,

$$R_c(x) \approx c_0 v_f \sqrt{\frac{\alpha_t}{x}}, \quad x < x_c. \qquad (6)$$

For $x > x_c$, flow acceleration is strong enough to dominate the interface dynamics. The balance between dispersion and compression leads to the mixing width (De Vriendt et al., 2022; Rousseau et al., 2023),
$$s(x) \approx \sqrt{\frac{D_t}{\gamma}}, \gamma = \frac{dv}{dz} \qquad (7)$$

the compression rate $\gamma$ is given by the derivative of the tangential flow velocity along the interface, and $D_t(x) = \alpha_t v(x)$ is the transverse dispersion coefficient. The average velocity $v(x)$ at a position $x$, $v(x)$, can approximated using the Glover equation for the interface depth (De Vriendt et al., 2022):

$$v(x) = \frac{v_f b}{\xi(x)}, \xi(x) = \sqrt{b^2 - \frac{2b}{\text{Ng}'}x}, \qquad (8)$$

where $\xi(x)$ denotes the interface depth, transformed such that $x = 0$ coincides with the toe of the interface. Thus, we obtain for the compression rate in terms of $v(x)$

$$\gamma = \frac{1}{\sigma(x)} \frac{dv}{dx}, \qquad (9)$$

where $\sigma(x) = dz(x)/dx$ is the variation of interface length with distance. In De Vriendt et al. (2022a), $\sigma(x)$ was set constant, which is a good approximation for long straight interfaces. However, in the reported experiments, the interfaces are short and sloped. Thus, we explicitly account for the variable slope of the interface, which gives for $s(x)$,

$$s(x) = \xi(x) \sqrt{\frac{\alpha_t}{b} \left[ \text{Ng}^2 + \frac{b^2}{\xi(x)^2} \right]^{1/4}}, \qquad (10)$$

as detailed in Appendix A. Using (Equations 8 and 10) in expression (Equation 4), we obtain for the maximum reaction rate

$$R_c(x) \approx \frac{c_0 \sqrt{\alpha_t b v_f b}}{\xi(x)^2 \left[ \text{Ng}^2 + \frac{b^2}{\xi(x)^2} \right]^{1/4}}, x > x_c. \qquad (11)$$

### 3.3. Modeling Reactive Mixing Along the Salt-Freshwater Interface

In order to compare the reactive mixing model to the experimental data, we identify the reaction rate $R_c(x)$ determined in the previous section with the maximum reaction rate $R_{Max}(x)$ measured in the experiments.

The model given by Equation 6 for $x < x_c$ and Equation 11 for $x > x_c$ is fitted to the data in Figure 6a. Note that similar to De Vriendt et al. (2022a), we also assume the interface extends vertically toward the top of the saltwater chamber, as opposed to truncating it at the shoreline as characterized by Glover (1959). Furthermore, we adjust the interface such that it extends 8 mm into the saltwater chamber in order to best-fit the observed experimental interface, see Appendix A. While the Glover solution fits the observed interface using the volumetric flux imposed in the experiment for the lowest flow conditions, for the two high flow steady state interfaces $Q_f$ was adjusted to match the location of the toe. This required an adjustment factor of 1.17 and 1.06 for steady state positions $P_1$ and $P_2$ respectively. Despite this, a relatively good agreement is found with the same parameters for all flow rates (Figures 6a–6e). Here we use a transverse dispersivity of 3 mm which is on the same order as the size of the FEP grains. The model identifies the location of the two reactive hot spots at the toe and head in addition to the location of the minimum reaction rate at the transition between the two regimes.

The presented model therefore captures the mixing-controlled non-monotonic reaction along the interface, resulting from the interplay between the evolution of the mixing width and of dispersion along the interface (Equation 4). The deep reactive hot spot at the toe is induced by the initial sharp gradient that results from the first contact between the reactive fluid. This gradient decays under the action of dispersion until it reaches a minimum at $x = x_c$. The acceleration of flow due to the reduction of the space available for freshwater flow then compresses the interface and enhances concentration gradients leading to the second hot spot at the head.

Note that the proposed model makes a few approximations. The velocity model assumes that the fluid can be discharged through the whole cross-sectional area between the top of the SFI and the top of the saltwater
<figure><img src="image_8.png" /><figcaption>**Figure 6.** (a–e) $R_{Max}(x)$ for each steady state position. Red and blue lines represent fitted curves using analytical predictions given by Equations 5 and 11 respectively. All curves have been scaled by $R_{Max,m}$ (see c). Dashed segment of curve represents solution after (red) and prior (blue) to the cross over distance. Note that analytical values above the local experimental maxima at the toe and head have been removed. (f) Collapse of $R_{Max}(x)$ from experiments after rescaling by $R_{Max,m}$ and $L_t$.</figcaption></figure>

boundary. In reality, flow is redirected up along saltwater boundary before discharging at the outlet. Furthermore, the model assumes that the mixing dynamics are the same for all steady state positions. It is however likely that a number of additional factors may influence the mixing and reaction across the interface such as the interface size, angle and freshwater flux. This is apparent from the images in Figure 3, where we see that for very small interfaces, the reaction rate is fairly uniform across the entire interface, it is only once the interface lengthens sufficiently for localized enhanced reaction hotspots to become apparent at the toe. This observation may also explain the better fitting of our model for intermediate steady state positions and poorer fitting to the first and last steady state positions.

Additionally, the chemical reaction is simplified. Even though there are numerous chemical species involved in the reaction, we do not account for individual species activities which may play a role on the overall reactivity across the interface. Despite these simplifications, the model captures the salient features of the reaction patterns along the interface. Specifically, it correctly predicts the non-monotonicity of reactivity along the SFI.

### 3.4. Control of Flow on the Global Reaction Rate

When rescaling the horizontal position $x$ by the interface length $L$ and the maximum reaction rate $R_{Max}(x)$ along the interface by its minimum value $R_{Max,m}$, the whole data set approximately collapses on a single trend (Figure 6f). This suggests that the reaction rate follows the scaling form,
<figure><img src="image_9.png" /><figcaption>Figure 7. Interface properties and reaction metrics as a function of $q_f$: (a) the interface toe length, $L_t$ (b) $R_{max,m}$ scaled by the highest $q_f$ value, $R_{max,m0}$ and (c) the integrated maximum reaction rates, $R_f$.</figcaption></figure>

$$R_{\text{Max}}(x) \sim R_{\text{Max},m} f\left(\frac{x}{L_t}\right). \qquad (12)$$

This scaling form is consistent with the mixing model of De Vriendt et al. (2022a). Equation 12 implies that the integrated reaction rate along the interface follows,

$$R_I = \int_0^L dx R_{\text{Max}}(x) = R_{\text{Max},m} L_t \int_0^1 dx f(x). \qquad (13)$$

The Glover solution for the interface (Equation 8) implies that its length scales as,

$$L_t \sim \text{Ng}' \sim q_f^{-1}. \qquad (14)$$

The minimum reaction rate is obtained by inserting $x_c = \text{Ng}'b/3$ into Equation 5:

$$R_{\text{Max},m} = R_{\text{Max}}(x_c) \sim v_f \sqrt{\frac{3\alpha_t}{\text{Ng}'b}} \sim q_f^{3/2}. \qquad (15)$$

These scaling laws are in good agreement with experimental measurements (Figures 7a and 7b). This leads to an expected growth of the integrated reaction rate as,

$$R_I \sim q_f^{1/2}, \qquad (16)$$

capturing relatively well the observed trend (Figure 7c). As described by Equation 13, this enhancement of reactive mixing by flow results from the balance between the reduction of interface length, which tends to reduce the effective reaction across the mixing front, and the enhancement of dispersion and concentration gradients, which tend to enhance reaction locally. The net effect of these two competing processes is thus the increase of mixing and reaction with the freshwater flow rate (Equation 16).

While experiment and our proposed model capture the effects of fast mixing-dependent reactions in a homogeneous system, natural systems that exhibit spatial heterogeneity may lead to a different spatial distribution of reaction patterns compared to the distributions observed in our study. The influence of heterogeneity in coastal aquifers for example, have been shown to have an influence over the localization of enhanced reactions across the SFI (De Vriendt, Le Borgne et al., 2020; De Vriendt, Pool, & Dentz, 2020; Sanford & Konikow, 1989). Understanding how these processes scale from controlled laboratory conditions to natural environments is critical for interpreting reactive transport dynamics in complex settings. Macroscopic heterogeneity for example, have been shown to lead to increased tortuosity of flow paths that lead to areas of flow focusing and enhanced transverse mixing (Werth et al., 2006). Complex three-dimensional flows owing to natural variations in permeability and
anisotropy have also shown propensity to increase transverse dispersion and reaction enhancement (Cirpka et al., 2015; Ye et al., 2016). However, consistent with the present findings, numerical simulations of calcite dissolution in coastal aquifers employing large-scale dispersion coefficients have shown spatial localization of reaction zones across the SFI (De Vriendt, Le Borgne et al., 2020; De Vriendt, Pool, & Dentz, 2020; Rezaei et al., 2005; Sanford & Konikow, 1989). This suggests that the qualitative behavior observed in the tank-scale experiments can be extrapolated to larger-scale coastal settings where reactions are fast, reinforcing the applicability of our findings beyond the laboratory setting.

While FEP media was used in this study to enhance the visualization of reactive processes, future research should also investigate the effects of more realistic pore geometries and heterogeneous porous media on localized reaction rates, particularly as new materials become available. Numerical simulations (Acharya et al., 2007), for instance, suggest that complex grain geometries can enhance transverse dispersion and reaction. This indicates that the observed results may be strongly influenced by pore-scale effects, driven by the non-homogeneous velocity structure induced by anisotropic grains. Consequently, Darcy-scale descriptions of reactive transport may be insufficient for accurately predicting local reaction rates in this setup, which can be further confounded in high Peclet number flows (Tartakovsky et al., 2009).

Finally, it should be acknowledged that natural systems are also influenced by a variety of dynamic stresses such as pumping, tides, flooding and more, which will impact mixing across the SFI. Transient fluctuations, for example, have been associated with an increase in transverse dispersion (Cirpka, 2005; Goode & Konikow, 1990; Rolle et al., 2009). In the context of seawater intrusion, it has been found flow is faster during a retreat than during an advance of the interface (Lu & Werner, 2013; Robinson et al., 2016). Faster flow velocities during a retreat may further enhance dispersion, which in turn plays a role in mixing and reaction dynamics. Transient dynamics should be considered in future studies to enhance the understanding of mixing and reaction mechanisms in dynamic coastal systems. Some further discussion on the impact of transient flow conditions can be found in Appendix B.

## 4. Conclusions

We have developed a new experimental technique, based on chemiluminescence, to visualize the spatial distribution of mixing-driven reaction rates across the salt-fresh water interface (SFI) in porous media. We show that the reaction rate varies locally along the interface, with two hotspots developing at the toe and head of the SFI. This is consistent with the locations of mixing hot spots observed in numerical simulations of conservative transport in SFIs (De Vriendt et al., 2022). These enhanced mixing zones also coincide with more complex reactions such as locations of enhanced calcite dissolution driven by mixing of freshwater and saltwater (Rezaei et al., 2005; Sanford & Konikow, 1989).

We have used these experimental insights to develop a mechanistic model quantifying the effect of dispersion and compression on mixing and reaction along the interface. The model captures the evolution of mixing-induced reaction along the SFI and explains the location of mixing and reaction hot spots at the toe and head. This framework allows understanding and predicting the effect of fluid flow on the effective reaction rate across the SFI. While an increase in freshwater flow reduces the length of the interface, we show that it also enhances dispersion and concentration gradients, leading to an increase of the reaction rate. These findings hence describe on how changes in hydrological processes inland may affect mixing and reaction at the interface with sea water.

These results offer new insights into the interaction between fluid flow, transport, and reactions at SFIs, which are crucial for biogeochemical processes at the interface of continental and oceanic environments (Charette & Sholkovitz, 2002; De Vriendt, Le Borgne et al., 2020; De Vriendt, Pool, & Dentz, 2020; Kroeger & Charette, 2008; Rezaei et al., 2005). This understanding is particularly relevant for assessing the chemical composition of submarine groundwater discharge (Moore, 1999) or to understand the formation of coastal caves and karst features (Back et al., 1986). These findings provide a foundation for investigating mixing and reaction in more complex aquifer conditions characterized by heterogeneous lithologies.
# Appendix A: Analytical Modeling

## A1. Glover Solution

For the convenience of the reader, we summarize here the Glover solution (Glover, 1959), which is used to approximate the freshwater velocity $v(x)$ along the interface. The depth to the interface $\xi(x)$ is given by De Vriendt et al. (2022).

$$ \xi(x) = \sqrt{b^2 - \frac{2b}{\text{Ng}'}x} \qquad (A1) $$

where $\text{Ng}' = K\epsilon/q_f$ is the modified gravity number, with $\epsilon = (\rho_s - \rho_f)/\rho_f[1 - (\alpha_f/b)^{1/4}]$ and $b$ the domain thickness. Note that the toe position is at $x = 0$, that is, $\xi(x = 0) = b$.

In Figure A1 we show the Glover solution super-imposed over the experimental images used to calculate $v(x)$. Note that to align the toe positions with the Glover solution in the experimental interface, we manually adjusted $q_f$.

<figure><img src="image_10.png" /><figcaption>Figure A1. Glover interface fitted to experimental salt-freshwater interfaces.</figcaption></figure>

## A2. Sloping Interface

In order to account for the sloping interface, we need to determine $\sigma(x) = dz(x)/dx$. The differential $dz$ can be written as

$$ dz = \sqrt{d\xi^2 + dx^2} = dx\sqrt{1 + \left(\frac{d\xi}{dx}\right)^2}. \qquad (A2) $$

Using Equation A1 for $\xi(x)$ gives

$$ \frac{d\xi}{dx} = -\frac{b}{\text{Ng}\xi(x)} \qquad (A3) $$

so that we obtain for $\sigma(x)$
$$ \sigma(x) = \sqrt{1 + \frac{b^2}{\text{Ng}^2 \xi(x)^2}} \qquad (A4) $$

The expression for the interface width is given by

$$ s(x) = \sqrt{\alpha_t v \left[ \frac{dv}{dz} \right]^{-1}} = \sqrt{\alpha_t v \left[ \frac{dv}{dx} \right]^{-1} \sigma(x)}, \qquad (A5) $$

and thus

$$ s(x) = \sqrt{-\alpha_t \left[ \frac{d \ln \xi(x)}{dx} \right]^{-1} \sigma(x)}. \qquad (A6) $$

Again, by using Equation A1 gives

$$ -\left[ \frac{d \ln \xi(x)}{dx} \right]^{-1} = \frac{\text{Ng}}{b} \left( b^2 - \frac{2b}{\text{Ng}} x \right) = \frac{\text{Ng}}{b} \xi(x)^2 \qquad (A7) $$

so that we obtain for the interface width

$$ s(x) = \xi(x) \sqrt{\frac{\alpha_t}{b} \left[ \text{Ng}^2 + \frac{b^2}{\xi(x)^2} \right]}^{1/4} \qquad (A8) $$

For $x > x_c$, the maximum reaction rate is

$$ R_{\text{Max}}(x) \approx \frac{\alpha_t v(x)}{s(x)}. \qquad (A9) $$

Recall that the flow velocity along the interface can be approximated by

$$ v(x) = \frac{v_f b}{\xi(x)}. \qquad (A10) $$

such that we obtain for $R_{\text{Max}}$

$$ R_{\text{Max}}(x) \approx \frac{\sqrt{\alpha_t b v_f b}}{\xi(x)^2 \left[ \text{Ng}^2 + \frac{b^2}{\xi(x)^2} \right]^{1/4}}. \qquad (A11) $$

## Appendix B: Transient Observations

Differences in reactive mixing behavior during an, advance, retreat and quasi-steady state are demonstrated in a single experimental test in Figure B1. Qualitatively, there is not a significant difference between an advancing wedge and quasi-steady state wedge. In contrast, during a retreat, the reactive interface becomes visibly wider, namely near the toe and discharge zone. These general observations also follow conservative mixing sand tank experiments of Abdoulhalik and Ahmed (2018), who found that during an intrusion there was little change in the average conservative mixing width, however during a retreat, there is a sizable increase in the mixing width emphasizing the asymmetry between intrusion and retreat. While a detailed investigation of the underlying mechanisms is beyond the scope of this study, our findings suggest that coastal aquifers subject to strong hydraulic perturbations, such as tidal fluctuations and seasonal recharge oscillations, may exhibit complex reactive responses due to changes in the velocity field.
<figure><img src="image_11.png" /></figure>

**Figure B1.** Image of reactive interface at comparable toe lengths during an active advance and an active retreat. Colormaps are normalized based on the maximum reaction rates from all images. Blue profiles above each image captures the local $R_{max}$ along the interface.

## Data Availability Statement

All the data used in this study is publicly available at (De Vriendt, Le Borgne et al., 2020).

## Acknowledgments

This research has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant 722028 ITN ENIGMA.

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