Capturing the Dynamic Processes of Porosity Clogging

Understanding mineral precipitation induced porosity clogging and being able to quantify its non‐linear feedback on transport properties is fundamental for predicting the long‐term evolution of energy‐related subsurface systems. Commonly applied porosity‐diffusivity relations used in numerical simulations on the continuum‐scale predict the case of clogging as a final state. However, recent experiments and pore‐scale modeling investigations suggest dissolution‐recrystallization processes causing a non‐negligible inherent diffusivity of newly formed precipitates. To verify these processes, we present a novel microfluidic reactor design that combines time‐lapse optical microscopy and confocal Raman spectroscopy, providing real‐time insights of mineral precipitation induced porosity clogging under purely diffusive transport conditions. Based on 2D optical images, the effective diffusivity was determined as a function of the evolving porous media, using pore‐scale modeling. At the clogged state, Raman isotopic tracer experiments were conducted to visualize the transport of deuterium through the evolving microporosity of the precipitates, demonstrating the non‐final state of clogging. The evolution of the porosity‐diffusivity relationship in response to precipitation reactions shows a behavior deviating from Archie's law. The application of an extended power law improved the description of the evolving porosity‐diffusivity, but still neglected post‐clogging features. Our innovative combination of microfluidic experiments and pore‐scale modeling opens new possibilities to validate and identify relevant pore‐scale processes, providing data for upscaling approaches to derive key relationships for continuum‐scale reactive transport simulations.

Reactive transport modeling (RTM) has become a versatile tool to assess the long-term evolution of such subsurface systems, coupling geochemical reactions and transport phenomena on different time and length scales.Classical RTM operates at the macroscopic (continuum)-scale, using the concept of the representative elementary volume (REV) (Bear, 1972;Noiriel & Soulaine, 2021;Scheibe et al., 2015).The REV defines the minimum size of a domain over which a property (e.g., porosity) is averaged over a certain length-scale (Bear, 1972).Clogging as a highly localized pore-scale (sub grid) heterogeneity challenges, thus, the predictive capability of continuum-scale RTM using averaged continuous (homogenized) parameters (Chagneau et al., 2015;Deng et al., 2021;Marty et al., 2009;Rajyaguru, 2018;Xie et al., 2015).Because of the inability to describe sub grid changes sufficiently, numerical approaches reveal discretization-dependent results.For instance, Marty et al. (2009) demonstrated how an increasing grid size results in an increase of the predicted time necessary to clog the porosity at a concrete/ clay interface.In order to capture effects of clogging at the sub-grid scale, continuum-scale modeling studies employed high-resolution grids, with sizes of the reaction zone smaller than the size of the single grains constituting the porous medium (Katz et al., 2011;Tartakovsky et al., 2008;Xie et al., 2022).Another approach to overcome mesh-discretization dependency at the continuum-scale is employing a hybrid continuum-and pore-scale model, resolving the process of precipitation at the pore-scale (Tartakovsky et al., 2008).Such hybrid (multiscale) models, however, still need an increase in the efficiency of coupling strategies, intending to combine the accuracy of pore-scale models and the computational efficiency of continuum-scale modeling (Battiato et al., 2011;Prasianakis et al., 2020;Scheibe et al., 2015;Tang et al., 2015).This used to be achieved by machine learning techniques with respect to decreasing computational time and interfacing various codes that operate at different length and time scales (Asadi & Beckingham, 2022;D'Elia et al., 2022;Deng et al., 2022;Menke et al., 2021;Prasianakis et al., 2020).
An alternative approach to quantify effective transport properties at the continuum-scale is the development of constitutive equations based on pore-scale modeling results (upscaling) (Deng et al., 2018;Hommel et al., 2018;Menke et al., 2021;Noiriel et al., 2012;Poonoosamy et al., 2020Poonoosamy et al., , 2022;;Prasianakis et al., 2020;Scheibe et al., 2015;Seigneur et al., 2019).Traditionally, empirical power-law functions are used to parameterize the effective transport properties as a function of porosity, for example, Archie's law (Archie, 1942).Modified versions of these laws also exist considering, in particular, critical minima to capture experimental observations and the expected behavior at the macroscopic scale (Cochepin et al., 2008;Deng et al., 2021;Hommel et al., 2018).There is ample evidence, however, that these empirical relationships are limited in predicting effective transport properties in response to a precipitation induced porosity reduction and pore space changes (Chagneau et al., 2015;Deng et al., 2021;Poonoosamy et al., 2020;Rajyaguru, 2018;Sabo & Beckingham, 2021).With increasing use, the combination of microstructural imaging (e.g., X-ray micro-CT scanning, 3D Raman imaging, positron emission tomography), pore network and pore-scale modeling provide valuable insights for the derivation of extended REV-scale constitutive equations (Agrawal et al., 2021;Kulenkampff et al., 2018;Menke et al., 2021;Noiriel & Soulaine, 2021;Poonoosamy et al., 2022;Roman et al., 2020;Soulaine et al., 2016).However, these extended laws have not been considered in recent modeling approaches, assessing the evolution of barrier interfaces of nuclear waste repositories (Xie et al., 2022) or geothermal systems (Tranter et al., 2021).One possible reason for this might be related to the unknown physical meaning of the added parameters and the difficulties in reliably defining them.
For the specific case of clogging and its impact on the porosity-diffusivity relation, Chagneau et al. (2015) observed a constant non-zero solute flux in response to a celestine precipitation induced porosity reduction and localized clogging.The 1D continuum-scale model based on Archie's law given by D e = D m a ϕ m , where D m is the molecular diffusion coefficient in free water (m 2 s −1 ) and a and m are empirical fitting parameters either overestimated the amount of precipitates or the diffusive tracer transport, highlighting the need for extensions to classical Archie's law to consider possibly complex precipitation kinetics.Based on a 2D micro-continuum model, Deng et al. (2021) evaluated the effect of different precipitation kinetics and initial pore geometries on the evolving porosity-diffusivity relationship, extending Archie's law by a critical porosity (ϕ c ) and critical effective diffusivity (D ceff ) to account for the effect of pore clogging.The predictive capability of the model was improved, but still existing discrepancies between the numerical results and experimental observations assumed yet undefined pore-scale processes during precipitation induced clogging.As demonstrated by previous column-scale experiments (Rajyaguru, 2018;Tartakovsky et al., 2008) and pore-scale modeling approaches, the complex nature of precipitation processes depends on several factors such as the initial pore geometry (Deng et al., 2021;Noiriel et al., 2016), the reaction rate as a function of supersaturation (Kim et al., 2020;Rajyaguru, 2018;Tartakovsky et al., 2008;Zhang et al., 2010), the probabilistic nature of nucleation (heterogeneous vs. homogeneous) and crystal growth (Deng et al., 2021;Kim et al., 2020;Nooraiepour et al., 2021).The evolving distribution patterns are affected, moreover, by the interplay of local hydrodynamics and reaction kinetics, quantified by the dimensionless Péclet (Pe) and Damköhler (Da) numbers (Battiato et al., 2011;Poonoosamy et al., 2019;Seigneur et al., 2019;Tartakovsky et al., 2008).The Pe is defined locally as: Pe = uL * /D m , where u is the fluid velocity (m s −1 ), and L * is a characteristic length (m), at the pore-scale, typically the average grain size (Seigneur et al., 2019;Soulaine et al., 2017;Steefel et al., 2013).The Da for diffusive systems is defined as Da II = (L *2 /D e )/(c/r), where D e is the effective diffusion coefficient in free water (m 2 s −1 ), c is the equilibrium concentration (  mol kg water −1 ) and r is the reaction rate (  mol kg water −1 s −1 ) (Steefel et al., 2013).In addition, the Reynolds number Re = uL*/ν, with ν as the kinematic viscosity (m 2 s −1 ) is used to describe the ratio between inertial and viscous forces (Steefel et al., 2013).However, all these aforementioned constraints challenge recent pore-scale modeling approaches that aim to predict precipitation processes, highlighting the need for further experimental studies to overcome limitations in the spatial and temporal resolution of the applied 2D/3D imaging techniques (Noiriel & Soulaine, 2021).
In this study, we provide new insights into mineral precipitation induced porosity clogging and its effect on the porosity-diffusivity relationship, combining innovative microfluidic experiments with pore-scale modeling.
The experimental setup follows our previous concept proposed by Poonoosamy et al. (2019Poonoosamy et al. ( , 2021)).The design of the microfluidic reactor allowed a real-time monitoring of precipitation-dissolution reactions under purely diffusive transport conditions in a quasi two-dimensional porous medium.The chemical system as well as the geometry of the microfluidic reactor were chosen in consideration of the work of Chagneau et al. (2015) and Deng et al. (2021).Based on the time-lapse optical images, simulations were performed to determine the effective diffusivity as a function of the evolving porous media, using pore-scale modeling.The diffusive transport of stable isotopes through the evolving porous medium was traced by Raman spectroscopy.The combination of time-lapse optical microscopy and confocal Raman spectroscopy opens up new possibilities to decipher pore-scale processes without the need of disturbing reaction and flow field conditions, overcoming limitations in the spatial and temporal resolution of typical flow cell or column-scale experiments (Chagneau et al., 2015;Katz et al., 2011;A. Tartakovsky et al., 2008) and providing data for upscaling parameters to derive key relationships for continuum-scale RTM.

Experimental Setup and Microreactor Design
The experimental setup combines a microfluidic device with high-resolution time-lapse optical microscopy and confocal Raman spectroscopy.The microfluidic reactor consists of a porous reaction chamber with dimensions of 100 × 66 × 1 μm³, referred to as quasi two-dimensional region of interest (ROI).The uniform distribution of cylindrical pillars (grains) of 10 μm in diameter distanced by 1 μm create a fully connected pore network with an initial porosity, ϕ i , of 0.36 (Figures 1b and 1c).The porous reaction chamber is connected to 10-μm deep supply channels via small inlets of 20 × 3 × 1 μm 3 on each side (Figure 1c).
The microreactor was mounted on an automatically moving x-y-z stage of the inverted microscope and connected to a syringe pump (NeMeSYS, Cetoni GmbH, Korbußen, Germany) (Figure 1a).The single-use polydimethylsiloxane (PDMS)-glass reactor was fabricated following the procedure from Poonoosamy et al. (2019) and references therein.Before the experiment, the reactor was purged with MilliQ water for 1 hr.The solutions of 0.05 M strontium chloride (SrCl 2 ) and sodium sulfate (Na 2 SO 4 ) were filtered with sterile Millex-GP syringe filter units (0.22 μm pore size) to remove solid particles and bacteria that may be present.The solutions were, then, injected with a constant flow rate of 1,000 nL min −1 into the microreactor to trigger a diffusion-driven precipitation of celestine (SrSO 4 ) inside the porous reaction chamber as follows: 10.1029/2023WR034722 4 of 14 The experiments were conducted at room temperature (22°C).Post-porosity clogging, a deuterated solution of SrCl 2 was injected to monitor the transport of D 2 O by Raman imaging.Deuterium oxide was chosen as a stable isotope tracer, here, as a hydrogen isotopic Raman tracer.The experiment was duplicated (Exp1 and Exp2).

Time-Lapse Optical Microscopy and Confocal Raman Spectroscopy
The microfluidic experiments were conducted with a Witec alpha300 Ri Inverted Confocal Raman Microscope, equipped with 100× oil immersion objective with a numerical aperture (NA) of 1.4, a working distance of 0.16 mm and a cover glass correction for 0.17 mm.The resolution of the microscope was estimated to be 0.6 × 0.6 × 1 μm 3 ; (voxel resolution -0.3 μm).Bright field images with a resolution of 6.85 pixel μm −1 were taken every 15 and 10 min, respectively, to monitor the evolving pore space geometry at regular time intervals.The amount of celestine was measured by segmenting and thresholding the optical images using the open-source software Fiji (Schindelin et al., 2012), allowing the derivation of the crystal growth rate (mol s −1 ), precipitation rate (mol m 2 s −1 ) and porosity (−) as a function of time.The 2D images were segmented into three phases (PDMS, aqueous phase, celestine), serving as input files for the pore-scale simulations by using the TauFactor Matlab application created by Cooper et al. (2016).
All Raman spectroscopic measurements were performed with a Nd:YAG laser (λ = 532 nm).The scattered light was dispersed by a grating of 600 grooves mm −1 of the Ultra-High-Throughput Spectrometer UHTS300 and detected by a thermoelectrically cooled CCD Camera.The usage of a 100× oil immersion objective yielded a theoretical diffraction-limited lateral and axial resolution of 464 nm (   = 1.4  ) and 1,629 nm (   = 4  2 ), considering a refraction index (n) of 1.5 for the glass cover lid of the reactor (Everall, 2010).The spectral resolution was ∼3.5 cm −1 .To avoid a potential heating of the microfluidic reactor, the laser power was set to 40 mW and an integration time of 0.25 s was chosen, resulting in a maximum output power of 0.5 mW.The output laser power was controlled by a handheld Laser Power Meter from Coherent before the experiments.The transport of D 2 O as a chemically "inert" and Raman active tracer was visualized by large area scans (115 × 75 μm 2 ) including the entire ROI (100 × 66 μm 2 ) with a step size of 1 μm.At each step, a Raman spectrum was recorded in the wavenumber range between 200 and 3,950 cm −1 to detect all representative Raman modes of celestine, deuterium and water.Further details about Raman mode assignment and data 10.1029/2023WR034722 5 of 14 treatment are given in Supporting Information S1.As the spatial resolution of the optical and Raman spectroscopic microscopy is above the depth of the porous reaction chamber, the region of interest displays a quasi two-dimensional system.

Modeling of Initial Velocity and Concentration Field
The initial transport velocity and solute concentration fields were modeled across the 3D microfluidic reactor at steady state using the software COMSOL Multiphysics 6.0 (COMSOL AB, Stockholm, Sweden).The conditions of mass transport were assessed in terms of the dimensionless Reynolds (Re) and Pe numbers (Equations S4 and S5 in Supporting Information S1).The Re was calculated for the supply channels using a characteristic length of 3 × 10 −4 m corresponding to the total lengths of the supply channels.The Pe was calculated for the ROI with a characteristic length of 1 × 10 −5 m corresponding to the diameter of the cylindrical pillars referred to as the average grain size.The diffusion coefficients were set to 1.23 × 10 −9 m 2 s −1 for Na 2 SO 4 , and 1.34 × 10 −9 m 2 s −1 for SrCl 2 , representing the diffusion coefficients of the respective salt solution, D salt (Equation S3 in Supporting Information S1; Lasaga, 1979).For the derivation of the theoretical maximum saturation index (SI), the species activities at the theoretical maximum concentration of SrCl 2 and Na 2 SO 4 were calculated with GEM-Selektor V3.42.Further details can be found in Text S2 and S4 in Supporting Information S1.

Simulation of Tracer Diffusion by Pore-Scale Modeling
The effective diffusivities as a function of the evolving pore geometry were calculated by using the MatLab application TauFactor (Cooper et al., 2016) based on the 2D segmented images.The diffusion equation in the porous medium domains for non-charged species is given by: with C p as the local concentration of the inert tracer at the pore-scale and D 0 as the local diffusion coefficient of the tracer in pure water.The constant concentrations (C in and C out ) at inlet and outlet are employed as the boundary conditions.At steady state, Equation 1 reduces to the Laplace equation.The effective diffusion coefficient, D e (m 2 s −1 ), was then calculated with the total mass flux J obtained at the steady state for the entire porous reaction chamber: (2) with L = 100 μm as the respective length of the ROI and S = 66 × 1 μm 2 as the total cross-sectional area.
For the clogged state, the concentration evolution with time was numerically solved by including a diffusivity of D 2 O in the precipitate, also named as the critical effective diffusivity, D ceff (m 2 s −1 ).D ceff (m 2 s −1 ) was determined by fitting with the experimentally determined time of diffusion, t diff , considering a minimum and maximum microporosity of 0.01 and 0.05, respectively, of the clogging precipitates.

Modeling of Initial Velocity and Concentration Fields
The transport velocity and the distribution of the SrCl 2 and Na 2 SO 4 concentrations for the non-reactive (initial) steady state of the 3D microfluidic reactor (i.e., before celestine precipitation occurred) were simulated, using COMSOL Multiphysics.The averaged velocity of 2.67 × 10 −2 m s −1 along the supply channels with a characteristic length of 3.00 × 10 −4 m yields a Re number of 8.02, ensuring a laminar flow and advective transport of reactants at a constant rate inside the supply channels.From the laminar flow regime inside the channels through the 1 μm-deep inlets toward the center of the porous reaction chamber, the average velocity significantly decreases by six orders of magnitudes.Inside the chamber, the simulation yields a maximum velocity of 8.87 × 10 −6 m s −1 , leading to a Pe number significantly below <0.1 and thus, indicating a purely diffusive transport of reactants within the porous reaction chamber.The distribution of Na 2 SO 4 and SrCl 2 concentrations within the supply channels shows a constant behavior, according to the laminar flow field (Figure 2a).Based on the concentration profiles of Na 2 SO 4 and SrCl 2 , a diffusive mixing of the reactants is expected to occur at a theoretical concentration of 25 mM, yielding a saturation index for celestine of 2.14 at the center of the reaction chamber (Figure 2b).

Experimental Evaluation of Mineral Reactions and Porosity
Figure 3 shows the evolving porous media in response to celestine precipitation induced clogging for Exp1 and Exp2.The processes can be subdivided into five stages: (a) induction time, preceding the appearance of the first crystallite, (b) crystal growth, (c) clogging, (d) dissolution, and (e) precipitation (Figure 4).Both experiments showed similar induction times and growth rates (2.0 × 10 −15 , 2.4 × 10 −15 mol s −1 ), describing a sigmoidal (logistic) growth function.The precipitation of celestine started within the center of the reactor (y = 50 μm), where the supersaturation with respect to celestine reaches a theoretical maximum of 2.14.However, the different locations in x-direction of the first occurring crystallites can be linked to the probabilistic nature of nucleation (Agarwal & Peters, 2014;Prieto, 2014), and initial defects on the surface of the chip.The flat euhedral celestine crystallites show a preferred direction of growth toward the incoming  SO4 2− .The growth rates were normalized to the reactive surface area, yielding initial precipitation rates of 8.90 × 10 −5 and 8.21 × 10 −6 mol m 2 s −1 that decreased to 2.8 × 10 −5 and to 2.4 × 10 −5 mol m 2 s −1 , respectively, until the porous reaction chambers became clogged, stopping further precipitation reactions.The precipitation rates averaged over time until clogging yielded an estimated Damköhler number of ∼22 (cf.Equation S6 in Supporting Information S1).At the clogged state, however, the initial porosity of 0.36 decreased to a minimum porosity of 0.29 and 0.26, respectively, referred to as the critical porosity, ϕ c .For Exp1, first signs of dissolution were noted after 18 ± 7 min of clogging, showing a relatively fast dissolution of the crystal surface facing the  SO4 2− inlet (red arrow in Figure 3a).The dissolving crystal surface partially reopened the disconnected porosity, providing further sulfate from the dissolving phase itself and via the  SO4 2− inlet.The post-clogging crystal growth decreased the critical porosity to 0.27, reaching the same value as observed for Exp2.In the case of Exp2, however, the state of clogging lasted significantly longer, showing first signs of dissolution after 115 ± 5 min of clogging along both sides of the precipitation front.In addition, the formation of a non-uniform microporosity of celestine was observed, indicating the evolution of a diffusivity of the clogging precipitates (blue arrows in Figure 3b).In the diffusion-controlled reaction chamber (low Pe), local precipitation along the dissolving crystal surfaces presumably lead to a second state of clogging.However, limitations in the spatial resolution did not allow to further define post-clogging reaction mechanisms and rates (kinetics) and to resolve the temporal evolution of the celestine microporosity.

Experimental Evaluation of the Critical Effective Diffusivity
As the pore network became microscopically clogged, a deuterated 0.05 M SrCl 2 solution was injected in the respective supply channel to monitor the transport of D 2 O through the evolving porous media using 2D Raman imaging.Figure 5 shows Raman distribution images of deuterium and celestine for Exp1 and Exp2 at two different time steps, t 0 and t diff .The stacked Raman spectra were obtained from the pore solution averaged over different areas labeled as A, B and C. At t 0 , deuterium is diffusing into the reaction chamber via the SrCl 2 -inlet (blue rectangle, spectrum A in Figure 5), whereby the pore solution within the disconnected porosity showed no detectable D 2 O (green rectangle, spectrum B in Figure 5).For Exp1, the first Raman signals of D 2 O diffused trough the clogging precipitates were detected after ∼2.93 hr (red spectrum C in Figure 5a).In the case of Exp2, the transport of deuterium through the evolving microporosity of the clogging precipitate took about 17.3 hr (red spectrum C in Figure 5b).The observed significant difference in the time of diffusion is attributed to the non-uniformly developing microporosity of the precipitate (cf.Section 3.2, Figure 3b-t d ).Both tracer   experiments, however, capture the diffusive transport of molecular water through the evolving microporosity of the precipitates.The diffusivity of the precipitates itself is referred to as the critical effective diffusivity, D ceff .To estimate the D ceff , a sensitivity analysis was performed by assuming a minimum and maximum microporosity of 0.01 and 0.05 of the clogging celestine (cf.Section 2.4).The study yielded critical effective diffusivities in the range from 5.1 × 10 −14 to 2.6 × 10 −13 m 2 s −1 .

Numerical Evaluation of the Effective Diffusivity
The 2D optical images shown in Section 3.2 were segmented into three phases (i.e., solid phase (initial porous medium), aqueous phase, and celestine precipitates) and used as input files to calculate the effective diffusivity for every time step, using the Matlab application TauFactor (see Text S5 in Supporting Information S1). Figure 6 displays the effective diffusivities as a function of decreasing porosity.The initial effective diffusivity of 8.72 × 10 −11 decreases to 5.00 × 10 −11 m 2 s −1 and 2.36 × 10 −11 m 2 s −1 , respectively, in Exp1 and Exp2 before reaching critical porosity at the clogged state.The difference in the evolution of the porosity-diffusivity observed for Exp1 and Exp2 is related to the distinctly evolving pore space morphology, leading to different effective diffusivities by the same reduction of porosity.However, Archie's law fails to describe the porosity-diffusivity relationship in response to mineral precipitation reactions, increasingly deviating from the simulated diffusivities (3) whereby a and n represent empirical coefficients (Sahimi, 1994).D m is set to 1 × 10 −9 m 2 s −1 as molecular diffusion coefficient.The minimum porosity reached during clogging is defined as the critical porosity, ϕ c .D ceff represents the critical effective diffusivity of the clogged porous medium.In the pore-scale modeling approach from Deng et al. (2021), no direct measurement of this parameter was performed and two values (5 × 10 −12 , 1 × 10 −15 m 2 s −1 ) were tested to reproduce the experimental tracer flux at the continuum-scale provided by Chagneau et al. (2015).In this study, therefore, the D ceff was estimated based on the Raman tracer experiments, yielding 5.1 × 10 −14 and 2.6 × 10 −13 m 2 s −1 within the range given by Deng et al. ( 2021) (cf.Section 3.3).
The application of the extended law (ϕ c -D ceff ) improves the description of the evolving effective diffusivity in response to mineral precipitation reactions using the experimentally determined critical parameters (Figures 6c  and 6d).However, the extended version does not consider post-clogging features with respect to a further evolving critical effective diffusivity.

The Non-Final State of Clogging
The microfluidic experiments of mineral precipitation induced porosity-clogging demonstrated the non-final state of clogging, showing a diffusive transport of molecular water through the evolving microporosity of celestine and a non-linear feedback on post-clogging reaction rates and porosity.Under diffusive transport conditions (low Pe numbers), the probabilistic nature of nucleation as well as the different diffusion coefficients of the dissolved reactants governed the final distribution of the clogging precipitates.The pattern of a uniform precipitation front perpendicular to the direction of diffusion as observed in Exp2 has also been described for column-scale experiments (Chagneau et al., 2015;Rajyaguru, 2018) and captured by pore-scale models (Deng et al., 2021;Tartakovsky et al., 2008).However, common setups used for diffusion-driven precipitation experiments do not allow the in situ observation of post-clogging reactions and thus, only a few pore-scale modeling approaches assumed dissolution as a post-clogging process, maintaining quasi-equilibrium chemical conditions near the precipitation zone (Tartakovsky et al., 2008).The microfluidic experiments clearly showed dissolution reactions as a post-clogging feature, occurring along the outer sides of the precipitation front due to a limited transport of reactants and causing the development of a microporosity in the precipitates.Raman tracer experiments qualitatively confirmed the assumed diffusivity of the precipitated phase, providing one explanation for the non-zero flux and its increase over time as reported from Chagneau et al. (2015).At this point, however, it should be noted that the microporosity and its temporal evolution could not be resolved, raising the question how the initial nano-to micro-porosity of the clogging precipitates controls the (critical) effective diffusivity.On the one hand, it is most likely that further precipitation in the microporous crystalline phase is inhibited due to the pore-size controlled solubility (PCS) effect (Emmanuel & Ague, 2009) or kinetic effect (Churakov & Prasianakis, 2018;Poonoosamy et al., 2016).The PCS effect could explain, in fact, the diffusive transport of deuterium through the clogging precipitates and the post-clogging precipitation reactions along dissolving crystal surfaces as clearly demonstrated in this study, confirming Deng et al.'s (2021) assumptions.

Simulated Porosity-Diffusivity Relation in Response to Clogging
Archie's law became inadequate to predict the porosity-diffusivity evolution in response to a significant porosity decrease, showing a deviant behavior in comparison to the simulated effective diffusivities.Similar observations were reported for diffusion-driven precipitation experiments at the column-scale (Chagneau et al., 2015;Rajyaguru, 2018) and in microfluidic experiment (Poonoosamy et al., 2022), demonstrating how a simple power law function underestimates the effect of a localized porosity reduction (Chagneau et al., 2015;Deng et al., 2018;Hommel et al., 2018;Poonoosamy et al., 2022;Rajyaguru, 2018).The differences in the evolution of the porosity-diffusivity evolution observed in this study can be ascribed to different locations of nucleation and crystal growth, resulting in distinct changes in the flow pathways and thus, in the effective diffusivity at the same reduction of porosity.Tartakovsky et al. (2007) reported similar observations for a pore-scale reactive transport model, demonstrating that the flux decreases less if the precipitation is uniform than if precipitation is non-uniform.
The extended version of Archie's law improved the description of the evolving effective diffusivity toward porosity clogging, considering a critical porosity and a critical effective diffusivity.The time-lapse optical images allowed an easy determination of the critical porosity, ϕ c , without disturbing or interrupting ongoing reaction and transport processes.In this context, a further decrease in the ϕ c could be observed during Exp1.Due to the ongoing post-clogging reactions, a reliable determination of a critical effective diffusivity referring to the total respective porous medium turned out to be challenging.Nevertheless, the estimated results were within the range of the assumed values from Deng et al. (2021) and showed a significant decrease in the effective diffusivity by two to three orders of magnitudes for a relatively low porosity reduction of 25%.Interestingly, this behavior is also predicted by various porosity-permeability relationships as reviewed in the work from Hommel et al. (2018).
In recent continuum and pore-scale modeling approaches, however, porosity clogging is often predicted as a final state at which the molecular diffusion of water stops after reaching a critical porosity that goes to zero or is set to <1% for numerical reasons, neglecting experimentally observed non-zero fluxes (Chagneau et al., 2015;Deng et al., 2021;Marty et al., 2009;Wasch et al., 2013;Xu et al., 2004).For instance, Deng et al. ( 2021) performed numerical tracer experiments on a 2D computational domain of similar size as the microfluidic reaction chamber, showing no break-through of the tracer after 10 million seconds (115.7 days) of clogging and ignoring dissolution as a post-clogging phenomenon.In this study, the diffusive transport through the precipitates was qualitatively approved after approximately 18 min and 2 hr of clogging, respectively, for Exp1 and Exp2, revealing significant differences in the time of diffusion due to the non-uniform evolution of the microporosity of the clogging precipitates.Deng et al. (2021) suggested a broad applicability of the extended law for 3D systems.Based on the work of Hunt (2004), a larger fitting exponent, m, of traditional Archie's law for 3D systems compared to 2D systems is to be expected.Here, our aim was particularly to experimentally validate the extended law proposed by Deng et al. (2021) and to determine the critical parameters (D ceff , ϕ c ) for a well-defined system.For extension to full 3D systems, further studies are needed to reliably evaluate the proposed critical parameters due to the related increase in complexity, for example, in pore space geometry.However, it is presumed that the fundamental formulation of the extended power law remains unchanged.
Future work needs, thus, to focus on the effects of different distribution patterns on the evolving effective diffusivity toward clogging, considering different initial geometries, minerals (crystal habit) and Da numbers, and, second, the effect of nano-to micro-porosity of the precipitates on the post-clogging evolution of the reactive porous medium using more advanced analytical techniques with higher spatial resolution, for example, FIB-SEM or FIB tomography.Additionally, the authors propose to conduct systematically sensitivity analysis using the extended law (Equation 3) for modeling subsurface evolution.

Conclusion
The microfluidic experiments of mineral precipitation induced porosity-clogging demonstrated the non-final state of clogging, showing a diffusive transport of molecular water through the evolving microporosity of celestine and a non-linear feedback on post-clogging reaction rates and porosity changes.The post-clogging dissolution reactions provided one explanation for the experimentally observed non-zero fluxes and its increase over time.The non-uniform evolution of a microporosity of the clogging precipitates challenged the determination of a critical effective diffusivity.However, the estimated values predicted a significant decrease in the effective diffusivity by two to three orders of magnitude by a total porosity reduction of 25%.The diffusivity of the clogging precipitates resulted in localized precipitation, potentially reclogging the porous media.Archie's law failed to describe the porosity-diffusivity relationship in response to mineral precipitation reactions, increasingly deviating from the simulated diffusivities toward clogging.The extended version of Archie's law proposed by Deng et al. (2021) was tested to explicitly account for the case of clogging.The extended law considering a critical porosity and a critical effective diffusivity improved the prediction of the porosity-diffusivity evolution in response to mineral precipitation induced clogging.However, post-clogging features are still not considered.Therefore, future pore-scale modeling approaches and microfluidic experiments are required to systematically assess the effect of a microporosity on the effective diffusivity, providing further data for upscaling approaches to derive key relationships for continuum-scale reactive transport simulations.Future work foresees three-dimensional microfluidic experiments under more realistic conditions, assessing different initial pore geometries and ranges of porosities.

Figure 1 .
Figure 1.Microfluidic experimental setup for precipitation induced clogging experiments monitored by time-lapse optical microscopy and confocal Raman spectroscopy.(a) PDMS-glass chip mounted on the automatically moving x-y-z stage for time-lapse optical imaging and Raman imaging.(b) Top and (c) Cross-sectional view of the microfluidic reactor.

Figure 2 .
Figure 2. Simulated (a) concentration field of SrCl 2 at steady-state displayed as mid-plane at z = 0.5 μm of the microfluidic reactor.The dashed line marks the position of the concentration profiles shown in (b).

Figure 3 .
Figure 3. Time-lapse bright field images representative for the five process stages of (a) Exp1 and (b) Exp2 (cf.Movies S1 and S2).The red arrow points to the dissolving crystal surface at the  SO4 2− inlet.The light blue arrows mark the evolving high microporosity of celestine.The dark blue arrows point to low microporosity areas of the crystal.t i : induction time; t g : crystal growth; t c : clogging; t d : dissolution, t p : precipitation.

Figure 4 .
Figure 4. (a) Amount of celestine as a function of time for Exp1 (black squares) and Exp2 (blue squares).The given numbers represent the crystal growth rates (mol s −1 ) obtained from a linear fit over t g -t c , and t c -t p, .(b) Porosity as a function of time until reaching the critical porosity, ϕ c , at the clogged state (dashed line).

Figure 5 .
Figure 5. Raman images of the ν s (D 2 O) and ν 1 (SO 4 ) intensities during clogging at t 0 and the time of D 2 O diffusion (t diff ) for (a) Exp1 and (b) Exp2.The blue, green and red rectangles mark the location of the averaged Raman spectra (right side): A, incoming deuterated SrCl 2 solution at t 0 ; B, D 2 O-free pore solution at t 0 ; C, pore solution at t diff .The PDMS spectrum (light gray) serves as reference for intensity.