Phases saturation control on mixing driven reactions in 3D porous media
Ishaan Markale, Gabriele M. Cimmarusti, Melanie M. Britton, Joaquin Jimenez-Martinez
PPhases saturation control on mixing drivenreactions in 3D porous media
Ishaan Markale, † , † Gabriele M. Cimmarusti, ¶ Melanie M. Britton, ¶ and JoaquínJiménez-Martínez ∗ , ‡ , † † Eawag, Swiss Federal Institute of Aquatic Science and Technology, 8600 Dübendorf,Switzerland ‡ Department of Civil, Environmental and Geomatic Engineering, ETH Zürich,Stefano-Franscini-Platz 5, 8093 Zürich, Switzerland ¶ School of Chemistry, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
E-mail: [email protected]/[email protected]
Abstract
Transported chemical reactions in unsaturated porous media are relevant across arange of environmental and industrial applications. Continuum scale dispersive modelsare often based on equivalent parameters derived from analogy with saturated condi-tions, and cannot appropriately account for processes such as incomplete mixing. Itis also unclear how the third dimension controls mixing and reactions in unsaturatedconditions. We obtain 3 D experimental images of the phases distribution and of trans-ported chemical reaction by Magnetic Resonance Imaging (MRI) using an immisciblenon-wetting liquid as a second phase and a fast irreversible bimolecular reaction. Keep-ing the Péclet number (Pe) constant, we study the impact of phases saturation on the a r X i v : . [ phy s i c s . f l u - dyn ] F e b ynamics of mixing and the reaction front. By measuring the local concentration of thereaction product, we quantify temporally resolved effective reaction rate ( R ). We de-scribe the temporal evolution of R using the lamellar theory of mixing, which explainsfaster than Fickian ( t . ) rate of product formation by accounting for the deformationof mixing interface between the two reacting fluids. For a given Pe, although stretchingand folding of the reactive front are enhanced as saturation decreases, enhancing theproduct formation, this is larger as saturation increases, i.e., volume controlled. Afterbreakthrough, the extinction of the reaction takes longer as saturation decreases be-cause of the larger non-mixed volume behind the front. These results are the basis fora general model to better predict reactive transport in unsaturated porous media notachievable by the current continuum paradigm. The chemical and biological evolution of many natural, engineering and industrial systems isgoverned by reactive mixing interfaces.
Chemical reactions in fluid are driven by mixing,which is a process that brings together segregated substances. Mixing is primarily driven bystretching, due to the existence of fluid velocity gradients, and diffusion.
Porous media aretopologically complex environments with highly heterogeneous fluid flow dynamics. Underunsaturated conditions, i.e., in the presence within the pore space of another phase such asan immiscible liquid or gas, this is exacerbated by a much more complex spatial configurationof the phases. This affects the internal connectivity of the system, further increasing thefluid flow heterogeneity with the formation of preferential paths (high velocity zones) andstagnation zones (low velocity zones). Understanding mixing and reactions in unsaturatedflows in porous media is fundamental to predicting the dynamics of contaminants and toevaluating the role of the soils in controlling global carbon, i.e., soil respiration, nitro-gen, and trace element cycles. It is also key as it plays a major role in applicationssuch as catalytic reactors, contaminant (bio-)remediation, enhanced oil recovery and2uclear waste disposal. The classic modeling approaches, also called Fickian or dispersive-diffusive, for reactivetransport in unsaturated conditions are often based on equivalent parameters (e.g., disper-sion) derived from analogy with saturated conditions, with systematic ad hoc incorporationof saturation dependency (fraction of the pore volume occupied by one of the immisciblephases). Fundamentally, these models cannot predict accurately the kinetics of transportedreactions resulting from mixing, which intrinsically occur at pore scale.
Stretching andfolding of reactive fronts by the high flow heterogeneity enhance mixing and thus reactioncompared to diffusive mixing only.
Recently developed lamellar mixing models thatcouple stretching and diffusion to capture the pore-scale concentration fluctuations are apromising avenue to predict reactive processes in these highly complex systems.
Mixing and reactive transport at pore scale has been studied experimentally using 2 D milli- and microfluidic approaches in saturated and unsaturated conditions. Various nu-merical studies have further added insights under both conditions.
However, the impactof incomplete mixing processes and non-Fickian dispersion on reaction kinetics is still notcompletely understood, especially in 3 D , where the third dimension adds spatial hetero-geneities to the system and greater tortuosities. While some recent studies, both numericaland experimental, have addressed the complexity added by the third dimension on mix-ing and reactive processes, our current understanding of flow dynamics and associatedreactive processes in unsaturated porous media is very limited, owing to both the complexi-ties associated with the presence of multiple phases and the difficulty of experimentally (inparticular, optically) accessing these systems.The development, over the last few decades, of non-invasive and non-destructive 3 D imag-ing techniques at pore scale opens a wide spectrum of possibilities to address the complexityof natural media, in which accessibility is limited. To date, the use of techniques suchas laser tomography, confocal microscopy, X-ray absorption computed tomography or mag-netic resonance imaging (MRI) has been limited to imaging the distribution of phases and3ultiphase flow, fluid flow in saturated and unsaturated media, conservative transportin saturated conditions, and propagation of reactive waves. In this work, we employMRI to visualize the 3 D distribution of phases and reactants in unsaturated porous media.The main goals of the present work are to measure pore-scale dynamics of local concentra-tion to identify the mechanisms that control kinetics of transported reactions in unsaturatedporous media, and to systematically characterize the impact of phase saturation on the ef-fective reaction rate of the system, i.e., on the product formation. For this purpose, weuse an analogous porous medium consisting of glass beads and a catalyst/indicator reaction(mixing-limited bimolecular irreversible reaction). Two different glass bead sizes are used forthe analysis, and several phases saturation and flow rates are explored. This experimentalapproach provides a completely new perspective for reactive transport in unsaturated porousmedia, and the experimental results are used to develop a theoretical framework in 3 D basedon the lamella mixing and reaction models by providing scaling laws for the effective reac-tion rate as a first step in the linking of the pore-scale phenomena to the Darcy (continuum)scale. The transport at pore scale of two initially segregated reactants is governed by advection,dispersion and reaction. In the absence of inertia effects, Navier-Stokes equation can be usedas the governing equation: dc i dt + ∇ · ( v c i ) − D ∇ c i = r i (1)where c i is the concentration of the respective reactants, D is the molecular diffusioncoefficient, v is the velocity field (calculated from the solution of the flow problem), and4 i is the local reaction rate. The heterogeneous velocity field, with complex streamlinetopologies, leads to the deformation of the interface, i.e., mixing front, between reactants.The deformation of the mixing front can enhance mixing and reaction rates by increasingthe area available for diffusive mass transport. We consider an initially 3 D flat (non-deformed) interface involving a mixing driven ir-reversible reaction A + B → P in a porous medium. The lamella theory of mixing andreaction, a Lagrangian framework that links the distribution of stretching rates along mix-ing interfaces to mixing and reaction rates, assumes that the mixing front is a collectionof numerous stretched lamellae, also called sheets in 3 D . At pore scale, the deformation ofthe mixing interface of area ε , with initial area ε , is quantified by the elongation ρ = ε/ε and its transport as: dc i dt − γn dc i dt − ddn (cid:18) D dc i dn (cid:19) = r i (2)where γ is the stretching rate defined as γ = (1 /ρ )( dρ/dt ) and n is the coordinate perpendic-ular to the lamella. Since the concentration gradients along lamella are small, we assumethat diffusive mass transfer is dominant only perpendicular to the lamella, i.e., along n .The transport and reaction regimes are characterized by the dimensionless numbers Péclet(Pe) and Damköhler (Da), respectively. Péclet number, Pe = τ d /τ a = ¯ vξ/ D , represents theratio between the characteristic time of diffusion and the characteristic time of advection overa typical pore throat ξ , being ¯ v the mean pore water velocity. Damköhler number, Da = τ a /τ r = ξc k/ ¯ v , represents the ratio of transport to reaction timescales. The characteristicreaction time, under well mixed conditions, is calculated as τ r = 1 /c k , where k is the rateconstant of our chosen reaction and c is the initial concentration. When reaction time scaleis much smaller than the advection and diffusion time scale, i.e., Da >> , the reaction isdriven by mixing. 5 AB xyz fl o w shear s ( t ) Figure 1: 3 D reactive lamella inside a fully saturated ( S w = 1 ) packed bed of 8 mm glassbeads (only three beads in grey color are shown for visualization simplicity). Concentrationof the reactants within the mixing volume is shown in warm colors, in which the lightest colorindicates equal concentration of the invading ( A ) and resident ( B ) reactant. A is pumpedinto the porous medium at constant flow rate ( Q = 0 . mm /s) from bottom to top. Shearis indicated with velocity vectors of different magnitude. The width or transverse thicknessof the lamella is labeled as s ( t ) . Note that the lamella shown is cut by an x-z (vertical) planepassing through the middle of the domain; beads are not cut by the plane (see SI, MovieS1). 6 .2 Effective Reaction Rate: Mass of Product For a fast irreversible reaction, in which the mass of A that diffuses from the interface into B reacts to produce P , the global kinetics, i.e., effective reaction rate R , of mass of product M P over the interface Π can be written as: R = dM P dt = D Z Π |∇ c A | dε ≈ Dc S w ε (1 + γt ) s ( t ) (3)where s is the width or transverse thickness of the interface (see Figure 1). It is assumedthat |∇ c | ∼ c /s . In natural systems, and in unsaturated soils in particular (with pore sizes[1-10 − ] mm and pore flow velocities [10 − -10 − ] mm/s), a shear flow regime is expected,in which the mixing front deforms by the gradient of velocity in the direction transverseto the main flow. As detailed for 2 D flows, shear flows in 3 D can also lead to alinear increase of elongation ( ρ = ∇ vt ) and a linear increase of the mixing interface areaas ε = S w ε (1 + γt ) , where S w is the wetting saturation defined as ratio of the volumeoccupied by the wetting phase (i.e., phase in which reaction is taking place) to that ofthe porous space. We estimate γ as v/λ = v/ ( ξ/S w ) , where λ is the velocity correlationlength. λ increases as S w reduces as has been observed recently. The elongation of themixing front increases the area available for diffusive mass transport, but also enhances theconcentration gradients by compression normal to the surface. The competition betweencompression and molecular diffusion controls s as follow: s ( t ) = s s β − γt ) β (1 + γt ) (4)where β = s γ/D and s is the initial interface thickness. When compression balanceswith diffusion, concentration gradients decrease, and s grows diffusively as s ∼ t / . Thetime at which concentration gradients are maximum corresponds to the so-called mixingtime t mix = ∇ v − ( s ∇ v/D ) / . Note that the characteristic shear time, defined as τ s = ∇ v − , and the characteristic reaction time τ r have been used to define the transition between7ifferent temporal scaling laws for R . Analytical expressions for those scaling laws have beenderived for both weak (Pe’< Da’) and strong (Pe’ > Da’) stretching, with Pe’ = τ D /τ s andDa’ = τ D /τ r , and where τ D = s /D . The different temporal scaling laws for the reactionrate R controls the mass of product M P . The mass is computed as M P ( t ) = R V c P d x at theinstant t , where V is the total pore volume of the phase in which transport and reaction arehappening. The incomplete mixing behind the reactive front makes the reaction persists locally in spaceand longer in time. The existence of concentration gradients mainly transverse to the mainflow direction are responsible of the resilience in the system. The decay of the concentrationgradients and therefore of the reaction is mainly driven by the diffusion into the low velocityregions or stagnation zones.
For a given position, i.e., a plane transverse to the mainflow direction, after breakthrough of the reactive front, the reaction rate R drops and canbe scaled by a power law with an exponent, α . This scaling is expected to be controlledby saturation S w , but also by Pe and Da. In this work we consider a monodispersed 3 D porous medium consisting of spherical grains(glass beads) contained in a cylindrical column of internal diameter 16 mm. Two differentglass bead diameters (1 and 4 mm) are used, allowing to explore a wide range of experimentalconditions. The column consists of a conical section at the bottom, a diffusing section,and a cylindrical section at the top, which represents our porous medium. Through theconical section, the injected liquid transitions from the diameter of the connecting pipe tothe diameter of the cylindrical section. The height of the cone was designed with the help8f computational fluid dynamics to minimize the boundary effects for the experimental flowrates expected (see SI, Figure S1). The diffusing section, composed of a fritted glass filterof 2 mm thickness and pore size 500 µm, is used to hold the beads and to homogenize theflow of the invading reactant liquid (henceforth called A ). It also acts as the limit betweenthe invading and resident (henceforth called B ) reactant during the experiment set-up. Thecolumn is connected to a syringe pump (Harvard Apparatus PHD Ultra) to control theinjection flow rate ( Q ) of the injected reactant A . We control the syringe pump remotelyusing the FlowControl software (Figure 2).As immiscible non-wetting phase, a fluorinated hydrocarbon, Tetrafluorohexane ( C F ,henceforth called O ) is used for the unsaturated experiments. Tetrafluorohexane is a stableorganic compound which does not react with any of the chemicals used (see Section 3.2) andis not miscible with water. We chose a manganese reduction reaction to study the transport of reaction front. A residentsolution of Mn ions ( B ) in the pore space is displaced by an invading solution containingMn ( A ) and the reactant CHD. The reduction reaction consists of two steps (see Eq. 5) anda conversion from Mn to Mn . The reacting solution used is H SO (Sigma Aldrich),1,4-cyclohexanedione (CHD, Sigma Aldrich) and manganese(III) acetate (Sigma Aldrich): + CHD → + H Q + 2H + (5a) + H Q → + Q + 2H + (5b)where H Q is an intermediate organic molecule, 1,4-hydroquinone, and Q is quinone. Asolution of . × − M manganese(III) acetate initially filled the packed bed and a solutionof 0.1 M CHD and . × − M manganese(II) acetate was injected into the packed bed.9igure 2: Scheme of the experimental setup. Glass beads are held within a glass cylinder bya fritted glass filter located at the bottom of the column. A syringe pump is used to injectreactant A at constant flow rate into the column containing glass beads and reactant B .Flow sense is upward. The immiscible phase ( O ) for unsaturated experiments is allocatedwithin the porous medium accessing it from the upper part of the column. The porousmedium column is placed inside the MRI magnet used for imaging the reaction. The visiblefield of view is 32 ×
16 mm (vertical × horizontal).10his ensures a constant total concentration of Mn ions (Mn and Mn ), and that changesin signal intensity come from changes in the oxidative state of the Mn , rather than adepletion of manganese ions. All solutions were prepared in distilled, deionized water. Theseexperiments were performed in high sulfuric acid conditions which stabilizes Mn ions inaqueous solutions. We prepared fresh solutions before each experiment. The time scaleof the reaction, which is visible optically, is calculated by a batch experiment. Using a CMOScamera, we first measure the time scale of dilution (without one of the reactants), and secondthe time scale of dilution+reaction. The difference between both times provides the reactiontime scale, which is found to be τ r ∼ However, the Nuclear Magnetic Resonance (NMR) signal intensity ( I ) depends onconcentration of manganese ions and their oxidative state (Mn or Mn ). There are moreunpaired electrons in Mn ions than Mn , and as a result the relaxation time for moleculessurrounding Mn is shorter. This produces the necessary contrast with which to visualizereactive fronts inside the porous medium using MRI. Therefore, initially the non-wettingphase ( O ) and beads appear dark, and the resident Mn ( B ) appears as the lighter phase.As the reaction occurs, the oxidative state of manganese ions changes which in turn changesthe visible intensity. Note that the immiscible phase O does not contain protons and henceis visualized by afterwards. The relaxation times (transverse or spin-spin relaxation times, T ,m ) for water can be converted into concentrations given that the total concentration ofmanganese ions remains constant ([Mn] = [Mn ] + [Mn ]). T = k x (6) T = 1 T , Mn + 1 T , Mn + 1 T , beads (7)11 ) a) b) Figure 3: a) The natural logarithm of the signal intensity ( I ) is proportional to the relaxationrate ( /T ) (data from ). b) Natural logarithm of I against concentration of Mn in theexperiments.There is a simple linear relationship between the overall relaxation rate ( /T ) of a para-magnetic species ( m ) and its concentration (Eq. 6), however it is also affected by thediamagnetic relaxation of the solvent and proximity to the glass beads (see SI for details).The intensity observed is a combination of the total concentration of manganese ions [Mn]and the glass beads (Eq. 7). Calibration experiments were performed in order to quan-tify the contribution of glass beads by only measuring the relaxation time (and hence theintensity) for each of the individual species ( m ). The natural logarithm of intensity ( I ) isdirectly proportional to the relaxation rate ( /T ) (Figure 3 a). This is correlated to theconcentration of species as shown in Figure 3 b. For our calibration, the data follows a linearfit with a regression coefficient of R = 0 . . The monodispersed 3 D porous media were used to run saturated and unsaturated experi-ments at different flow rates Q . For the unsaturated experiments, the immiscible phase ( O )to ’desaturate’ the system is injected from the upper side of the column, which was open12Figure 2). A syringe to control the volume and a needle were used to allocate O randomly(and homogeneously in statistical sense) within the porous medium. Once the beads andresident chemicals (either B , or B and O for saturated and unsaturated experiments, re-spectively) were placed inside the column, the latter was then carefully installed inside themagnet before the injection of A (Figure 2). Subsequently, the continuous injection of A was started. While for the saturated experiments, and in both porous media, two differentflow rates ( Q ) were used, the flow rate in the unsaturated ones was modified in order toget same Pe and very similar Da as in one of the saturated experiments for comparison. Toensure that there is no flow behavior missed between two consecutive scans, the upper limitof the imposed flow rate is determined by the acquisition time (16 s for a full 3 D scan), i.e.,the acquisition time must be longer than the advective time ( τ a ). Owing to the capillaryforces and low flow rates used, the immiscible phase was immobile during the course of theexperiments, i.e., the magnitude of the viscous forces was smaller than the magnitude ofcapillary forces. The saturation degrees ( S w ), the flow rates imposed, the resulting meanpore water velocities ( v ), and the Péclet (Pe) and Damköhler (Da) numbers experimentedare summarized in Table 1.1H and 19F magnetic resonance 3 D images were acquired using a Bruker Avance III HDspectrometer which comprised a 7 T wide-bore superconducting magnet operating at a protonresonance frequency of 300.13 MHz. All images were acquired using a micro 2.5 imagingprobe equipped with a dual resonance 1H/19F 25 mm radio frequency (RF) birdcage coil.The temperature of the imaging probe was maintained at 293 ± 0.3 K by the temperatureof water-cooled gradient coils. Vertical (sagittal) 3 D images of the system were acquiredusing the fast spin-echo imaging sequence RARE.
1H 3 D images were recorded using a × × pixel matrix, with a field of view of × × pixels. A RARE factor of 32,echo time of 3.2 ms and repetition time of 500 ms were used. For each experiment, a range ofimages (from 20 to 100) were collected depending on the injection time. 3 D
19F MR imageswere acquired to visualize the non-wetting phase O at t = 0 (before the injection of A ) and13t the end of the injection. These 3 D images were acquired using the same parameters asthe 3 D
1H MRI images, except for the repetition time, which was 1 s. Scans, at each timestep, consisted of 16 slices (in the x direction), of 1 mm thickness, with a resolution of 0.25mm (in the y and z directions). The size of each voxel is . × . × mm, and thus thevoxel volume equal to 0.0625 mm . Signal intensity was stored in 16-bit gray scale images.All images were processed and then converted to images of concentration, averaging overthe voxel size, using the calibration described above (Figure 3). Artifacts were corrected(see SI, Figure S3), and segmentation was used to differentiate the glass beads from theliquid phase. The segmented liquid volume is compared to the actual volume hosted in theporous medium. This action was also performed for the volume of the immiscible phase.The acquisition was carried out until no further reaction was detected in the visible domain.We compute for every time step the mass of Mn injected and the excess mass visible inthe pore volume as a consequence of reaction. The difference between these two providesthe mass of Mn that has reacted to produce Mn in each voxel at every time step (see SI,Figure S4). We then calculate the total mass of product formed M P as the sum in all voxelsin the visible domain. The rate of change of M P provides the effective reaction rate R . Figure 4 shows the spatial distribution of the immiscible non-wetting phase O in the porespace of a porous medium built from 4 mm glass beads. Clusters of this phase connectingseveral pores and isolated drops can be recognized. The saturation of the wetting phasein this case is S w = 0 . . O was immobile during the reactive transport experiment. Forthe same experiment, snapshots at six equispaced times of the concentration of the reactantswithin the mixing volume are shown in Figure 5, for an injection flow rate Q = 7 . mm /s.Experimental conditions of the reactive experiments in both porous media (i.e., different glass14igure 4: Distribution of the resident reactant B (semi-transparent white color) and of theimmiscible non-wetting phase O (semi-transparent blue color) within the monodispersedporous medium of 4 mm glass beads (non visible). Saturation of the wetting phase is S w = 0 . . O is randomly distributed within the pore space.beads diameter) are summarized in Table 1. As the invading chemical A enters the domain, itpenetrates the channels created by the grains and O (see Figure 5). Initially the interface Π ishighly stretched and due to heterogeneity of the pore space a collection of lamellar topologicalstructures (or fingers) develops (Figure 5 a-c). As the reaction propagates through the porespace, these fingers merge by diffusion and a more homogeneous reaction front propagatesthrough the medium (Figure 5 d). After breakthrough, incomplete mixing makes the reactionpersist behind the front (Figure 5 e and f) (see SI, Movie S2).Table 1: Experimental conditions of the reactive experiments in the packed beds of 1 and 4mm glass beads for the different saturation degrees S w . Q is the imposed flow rate, ¯ v is themean pore water velocity, Pe is the Péclet number, and Da is the Damköhler number.1 mm grains 4 mm grains S w Q [mm /s] ¯ v [mm/s] Pe Da S w Q [mm /s] ¯ v [mm/s] Pe Da1.00 3.865 0.050 25 20 1.00 10.996 0.125 250 321.00 1.933 0.025 12.5 40 1.00 17.593 0.200 400 200.77 3.286 0.054 25 18 0.88 9.346 0.119 250 330.43 1.933 0.058 25 17 0.78 7.697 0.112 250 35As the chemicals react, we measure in time the effective reaction rate R and the mass15f reaction product M P for each experiment (symbols in Figure 6). For all our experiments, R initially increases. After breakthrough, R rapidly decreases and eventually the reactiondies out. For a given saturation, in this case S w = 1 , a faster increase with time of R asflow rate increases, i.e., as Pe increases, is observed, irrespective of the grain size. For agiven Pe, a lower magnitude of R as S w decreases is observed (Figure 6 a and b). Thetemporal evolution and magnitude of M P follows the patterns dictated by R , although somefeatures are better recognized as follows. In Figure 6 c and d, it is seen that M P for fullysaturated cases depends on the Pe. A higher Pe leads to a higher mass production. M P for unsaturated cases reduces as S w decreases. However, for 1 mm sized grains (Figure 6c), as S w reduces to 0.77, the mass produced at the very early times is larger than in fullysaturated conditions. This trend reverses when S w reduces to 0.43, i.e., the rate of increaseof M P reduces as compared to S w = 0 . . For low Pe (i.e., 1 mm sized grains), the rate ofproduction of M P decreases before the breakthrough (denoted by vertical lines in Figure 6c). On the contrary, for higher Pe (i.e., 4 mm sized grains), M P production does not decreasebefore breakthrough (Figure 6 d). After breakthrough, for both 1 and 4 mm sized grains,the rate of M P production decreases slower as saturation decreases (Figure 6 c and d). Notethat despite the differences between the measured porosities (0.3844 and 0.4375, for 1 and4 mm sized grains, respectively), the maximum mass produced for S w = 1 in 1 mm sizedgrains and Pe = 25 is M P = 43 . mg, whereas for the 4 mm sized grains and Pe = 250, it is M P = 44 . mg. Thus the Pe and pore size plays a key role in determining R and thus M P . We now compare the results obtained from the experiments with the theoretical modelpresented in Section 2. We use the lamella model to describe the mixing interface as acollection of stretched lamellae and to predict the global reaction rate R as function of S w .The model parameters ( s , ε , γ ) used in Equation 3 are given in Table 2. The solid lines16igure 5: Time series of the transported reaction as A is pumped at constant flow rate( Q = 7 . mm /s) into an unsaturated packed bed of 4 mm glass beads. Concentration ofthe reactants within the mixing volume is shown in warm colors, in which the lightest colorindicates equal concentration of A and B (i.e., equal concentration of Mn and Mn , mM).Non-wetting phase O is shown in a semi-transparent blue color and does not move duringthe experiment. Saturation of the wetting phase is S w = 0 . . Pe and Da numbers for thisexperiment are 250 and 35, respectively. The arrow denotes the mean flow direction. Imageswere acquired every 16 s (see SI, Movie S2).Table 2: Parameters used in the model prediction. s and ε are the initial interface thicknessand area, respectively. γ is the shear deformation rate. t modelmix is the mixing time computedfrom the lamella based model and compared with the one inferred from the experiments. t expmix is experimentally observed only for 1 mm sized grains since it is not reached in the 4mm sized grains experiments.1 mm grains 4 mm grains S w Pe ε [m ] s [m] γ [s − ] t expmix [s] t modelmix [s] S w Pe ε [m ] s [m] γ [s − ] ) d) a) b) Figure 6: a, b) Comparison of temporal evolution of the global reaction rate R for 1 and 4 mmsized grains, respectively, and different S w between the MRI experiment results (symbols)and the reactive lamella model (solid lines). Note that the same Pe is used for comparingdifferent S w except where specified. The stretching ( t . ) and the Fickian ( t . ) regime areindicated with the solid black lines. The mixing time ( t mix ) between both regimes is alsoshown. The model prediction is shown until only breakthrough (i.e., the reaction frontreaches the end of the observable domain). The dotted line in (a) depicts the speed α ofreaction extinction after breakthrough. c, d) Temporal evolution of M P obtained from MRIexperiments (symbols) for 1 and 4 mm sized beads, respectively, and different S w . Verticallines denote when breakthrough happens. 18n Figure 6 a and b show the results of the model (only until breakthrough) and how theycompare with the experimental observations (symbols).At early times and for all S w and Pe, R grows in time faster than Fickian, i.e., R ∼ t . .In the temporal evolution of R for 1 mm sized grains, a change in the scaling from t . to t . for all S w before breakthrough is observed (Figure 6 a). This change happens attime t mix when diffusion overcomes compression and concentration gradients are no longerenhanced. At later times, folding of the plume over itself promotes the lamellae interactiondue to diffusion and they coalesce into bundles. For 4 mm sized grains, the front breaksthrough before t mix and the coalescence regime, thus no change in the scaling of R is observed(Figure 6 b). After t mix , ε no longer grows linearly and rate of product formation slowsdown. At longer times and for fully saturated conditions, R is expected to decay in time asFickian (i.e., R ∼ t − . ). However, for unsaturated conditions, this decay is expected tobe slower than Fickian and it would reduce as S w decreases, as observed for 2 D flows. Comparing different S w (same Pe), t mix is reached earlier at S w = 0 . than at S w = 1 . ,but a further reduction of S w (i.e., . ) increases the mixing time (Figure 6 a and Table2). An analogy of this inversion in the trend can be found for conservative transport inunsaturated porous media, where a higher value of dispersivity and dispersion coefficientas saturation decreases is observed. However, further observations indicate that thisrelation is not monotonic, and the maximum dispersivity and dispersion coefficient occursat an intermediate saturation, called critical saturation . While this has been explainedby some authors for being the saturation where the tortuosity has its highest value, othersargue the strong channeling effects in both fully saturated and low-saturation cases, beingless significant at intermediate saturation values. The model overall gives a good agreement to our observations (until breakthrough).Hence the assumption of linear stretching holds for spherical grains and the range of Peused, based on the fact the shear is induced by the velocity gradients between the no-slipboundary condition at the grain walls and the maximum velocity at the pore center. We19 ) b) Figure 7: s ( t ) and ε (inset) evolution computed from the reactive lamella model ( (Eqs. 2-4)for the time experiment before breakthrough in a) 1 mm sized grains and b) 4 mm sizedgrains.hypothesize this assumption is also valid in unsaturated conditions as shown below, becausethe impact of the non-slip condition at the liquid-gas interfaces on transport processes hasbeen recently demonstrated for being negligible. For a given Pe, as S w decreases, globalreaction rate R can scale slightly higher than t . (Figure 6 a). This can be explained bythe temporal evolution of s ( t ) and ε before breakthrough (Figure 7). For 1 mm sized grains, s ( t ) decreases by compression until it reaches a minimum after which it grows diffusively. ε is highest for S w = 1 , but a slightly faster growth is observed for S w < due to a highershear rate γ (Figure 7 a). For the range of Pe and S w explored, a constant gradient ofvelocity results from shear deformation, which is characterized by the transient mixing frontstrictly increasing linearly in time even for low S w . According to Rolle and Le Borgne, astrong stretching regime (i.e., Pe’>Da’) for the range of Pe and Da studied here is alwaysexperienced by the mixing and reactive front (Figure 7). We are able to characterize all theresults (for different Pe and S w ) using linear stretching ( ρ ∼ t ). This also gives an insightinto the permeability field of the domain: between a moderate and strong heterogeneityfield. The reactive lamella model (Eq. 2) reduces to three parameters, s , ε and γ , whichcan be evaluated as ¯ v/ ( ξ/S w ) . Based on these estimations and without any other further20tting, the model provides a reasonable estimate of the global reaction rate in both fully andpartially saturated conditions over the range of Pe investigated by taking into account theincomplete mixing at the pore scale. Once breakthrough occurs, there is still reaction happening within the system due to incom-plete mixing.
As saturation decreases, it takes longer for the reaction to completelyextinguish (see Figure 6). The presence of non-wetting phase O increases the volume fractionof the wetting fluid with low velocities, i.e., stagnation zone. This promotes further reactionwith the creation of concentration gradients between the high velocity regions or preferentialpaths and the low velocity regions. In some cases, the consumption of the resident reactantcan take place only by diffusion. We define α as the rate of reaction extinction. Asdepicted in Figure 6, α is the slope of the reaction rate R tail calculated after breakthroughhas already taken place. Figure 8 compares α for both grain sizes and all Pe studied. α increases as S w decreases. While previous 2 D observations shown an enhancement of mixingand reaction as S w decreases, this is exacerbated in 3 D by the presence of helical flowcomponents and transverse mixing. For S w = 1 , increasing Pe (more than an order ofmagnitude, from 25 to 250) does not increase α by the same amount (Figure 8). When S w < , α increases by a similar order of magnitude independently of the Pe. Thus, a higherPe does not necessarily lead to an increase in resilience of the reaction, it is in fact S w whichis the main controller of how long a reaction lasts inside the pore space.We have demonstrated and quantified experimentally the incomplete mixing at pore scalein 3 D porous media and the impact of the presence of an immiscible phase on it. The mixingand reactive front have been depicted by a lamella like topology. The front is advected anddeformed by the heterogeneous velocity field, being subjected to stretching and folding andresulting in a competition between compression (creation of concentration gradients) anddiffusion (destruction of concentration gradients). Compared to the 2 D case, we get a more21igure 8: Impact of S w on reaction extinction after breakthrough for different grain sizesand P: 4 mm sized grains, Pe = 250 and 400, purple symbols; 1 mm sized grains, Pe = 25and 12, green symbols.stretched 3 D reaction front, however, the mechanisms which control the dynamics remainsame. Although the fringes of the plumes are considered reaction hotspots, subsurfaceenvironments are in general poorly mixed. For instance, the initial response (at earlytimes) can be crucial in understanding biogeochemical processes in unsaturated soils. Whilewe considered a fast reaction compared to the transport in a small domain, such reactions arerelatively common in natural environments. These findings have implications for effectivereactive transport modeling in a variety of applications, since the basic phenomena we studiedoccurs in a wide range of flows.
Acknowledgement
IM and JJM acknowledge the financial support from the Swiss National Science Foundation(SNF, grant Nr. 200021_178986). MMB and GC acknowledge the financial support fromthe University of Birmingham and the Engineering and Physical Science (EPSRC) ResearchCouncil, UK (EP/K039245/1). 22 upporting Information Available
The following files are available as Supporting Information (SI).• Movie_S1.mp4: Propagation of reaction front inside the pore space of 8 mm sizedglass beads.• Movie_S2.mp4: Propagation of reaction front (and incomplete mixing after break-through) in the pore space of an unsaturated 4 mm sized glass beads.• SI.pdf: Reaction time scale from batch experiment, CFD simulation of column designand details on calibration procedure.
References (1) De Simoni, M.; Carrera, J.; Sanchez-Vila, A.; Guadagnini, A. A procedure for thesolution of multicomponentreactive transport problems.
Water Resour. Res. , ,1–16.(2) Tartakovsky, A. M.; Redden, G.; Lichtner, P. C.; Scheibe, T. D.; Meakin, P. Mixing-induced precipitation: Experimental study and multiscale numerical analysis. WaterResour. Res. , .(3) Rezaei, M.; Sanz, E.; Raeisi, E.; Ayora, C.; Vázquez-Suñé, E.; Carrera, J. Reactivetransport modeling of calcite dissolution in the fresh-salt water mixing zone. J. Hydrol. , , 282–298.(4) Ottino, J. The kinematics of mixing: Stretching, chaos, and transport ; CambridgeUniversity Press, 1989.(5) Ranz, W. E. Applications of a stretch model to mixing, diffusion, and reaction inlaminar and turbulent flows.
AIChE J. , , 41–47.236) Villermaux, E.; Duplat, J. Mixing as an aggregation process. Phys. Rev. Lett. , , 184501.(7) Duplat, J.; Villermaux, E. Mixing by random stirring in confined mixtures. J. FluidMech. , , 51–86.(8) Jiménez-Martínez, J.; Negre, C. F. A. Eigenvector centrality for geometric and topo-logical characterization of porous media. Phys. Rev. E , , 013310.(9) Xu, L.; Baldocchi, D. D.; Tang, J. How soil moisture, rain pulses, and growth alter theresponse of ecosystem respiration to temperature. Global Biogeochem. Cycles. , .(10) Giardina, C. P.; Litton, C. M.; Crow, G. P., Susan E.and Asner Warming-relatedincreases in soil CO2 efflux are explained by increased below-ground carbon flux. Nat.Clim. Change. , , 822–847.(11) Ebrahimi, A.; Or, D. Dynamics of soil biogeochemical gas emissions shaped by remoldedaggregate sizes and carbon configurations under hydration cycles. Glob. Change Biol. , , e378–e392.(12) Sebilo, M.; Mayer, B.; Nicolardot, B.; Pinay, G.; Mariotti, A. Long-term fate of nitratefertilizer in agricultural soils. Proc. Natl. Acad. Sci. , , 18185–18189.(13) Helton, A. M.; Ardón, M.; Bernhardt, E. S. Thermodynamic constraints on the utilityof ecological stoichiometry for explaining global biogeochemical patterns. Ecol. Lett. , , 1049–1056.(14) Kravchenko, A. N.; Toosi, E. R.; Guber, A. K.; Ostrom, N. E.; Yu, J.; Azeem, K.;Rivers, M. L.; Robertson, G. P. Hotspots of soil N2O emission enhanced through waterabsorption by plant residue. Nat. Geosci. , .2415) Winkel, L.; Vriens, B.; Jones, G.; Schneider, L.; Pilon-Smits, E.; Bañuelos, G. Seleniumcycling across soil-plant-atmosphere interfaces: a critical review. Nutrients , ,4199–4239.(16) Duduković, M. P.; Larachi, F.; Mills, P. L. Multiphase catalytic reactors: a perspectiveon current knowledge and future trends. Catal. Rev. , , 123–246.(17) Rolle, M.; Eberhardt, C.; Chiogna, G.; Cirpka, O. A.; Grathwohl, P. Enhancement ofdilution and transverse reactive mixing in porous media: Experiments and model-basedinterpretation. J. Contam. Hydrol. , , 130–142.(18) Jiménez-Martínez, J.; Porter, M. L.; Hyman, J. D.; Carey, J. W.; Viswanathan, H. S.Mixing in a three-phase system: Enhanced production of oil-wet reservoirs by CO2injection. Geophys. Res. Lett. , , 196–205.(19) Winograd, I. Radioactive waste disposal in thick unsaturated zones. Science , ,1457–1464.(20) Šimůnek, J.; van Genuchten, M. T. Modeling nonequilibrium flow and transport pro-cesses using HYDRUS. Vadose Zone J. , , 782.(21) Williams, K. H.; Kemna, A.; Wilkins, M. J.; Druhan, J.; Arntzen, E.; N’Guessan, A. L.;Long, P. E.; Hubbard, S. S.; Banfield, J. F. Geophysical monitoring of coupled microbialand geochemical processes during stimulated subsurface bioremediation. Environ. Sci.Technol. , , 6717–6723, PMID: 19764240.(22) Heyman, J.; Lester, D. R.; Turuban, R.; Méheust, Y.; Le Borgne, T. Stretching andfolding sustain microscale chemical gradients in porous media. Proc. Natl. Acad. Sci. , , 13359–13365.(23) Wright, E. E.; Richter, D. H.; Bolster, D. Effects of incomplete mixing on reactive25ransport in flows through heterogeneous porous media. Phys. Rev. Fluids , ,114501.(24) Le Borgne, T.; Ginn, T. R.; Dentz, M. Impact of fluid deformation on mixing-inducedchemical reactions in heterogeneous flows. Geophys. Res. Lett. , , 7898–7906.(25) Le Borgne, T.; Dentz, M.; Villermaux, E. Stretching, coalescence, and mixing in porousmedia. Phys. Rev. Lett. , , 1–5.(26) Le Borgne, T.; Dentz, M.; Villermaux, E. The lamellar description of mixing in porousmedia. J. Fluid Mech. , , 458–498.(27) Lester, D. R.; Dentz, M.; Le Borgne, T. Chaotic mixing in three-dimensional porousmedia. J. Fluid Mech. , , 144–174.(28) de Anna, P.; Jiménez-Martínez, J.; Tabuteau, H.; Turuban, R.; Le Borgne, T.; Der-rien, M.; Méheust, Y. Mixing and reaction kinetics in porous media: An experimentalpore scale quantification. Environ. Sci. Technol. , , 508–516.(29) Jiménez-Martínez, J.; de Anna, P.; Tabuteau, H.; Turuban, R.; Le Borgne, T.;Méheust, Y. Pore-scale mechanisms for the enhancement of mixing in unsaturatedporous media and implications for chemical reactions. Geophys. Res. Lett. , ,5316–5324.(30) Jiménez-Martínez, J.; Le Borgne, T.; Tabuteau, H.; Méheust, Y. Impact of saturationon dispersion and mixing in porous media: Photobleaching pulse injection experimentsand shear-enhanced mixing model. Water Resour. Res. , , 1457–1472.(31) Karadimitriou, N. K.; Joekar-Niasar, V.; Babaei, M.; Shore, C. A. Critical role of theimmobile zone in non-Fickian two-phase transport: A new paradigm. Environ. Sci.Technol. , , 4384–4392. 2632) Willingham, T.; Werth, C.; Valocchi, A. Evaluation of the effects of porous mediastructure on mixing-controlled reactions using pore-scale modeling and micromodelexperiments. Environ. Sci. Technol. , , 3185–3193.(33) Li, P.; Berkowitz, B. Characterization of mixing and reaction between chemical speciesduring cycles of drainage and imbibition in porous media. Adv. Water. Resour. , , 113–128.(34) Jiménez-Martínez, J.; Alcolea, A.; Straubhaar, J. A.; Renard, P. Impact of phasesdistribution on mixing and reactions in unsaturated porous media. Adv. Water. Resour. , , 103697.(35) Dentz, M.; Le Borgne, T.; Englert, A.; B., B. Mixing, spreading and reaction in het-erogeneous media: A brief review. J. Contam. Hydrol. , , 1–17.(36) Ghanbarian, B.; Hunt, A. G.; Ewing, R. P.; Sahimi, M. Tortuosity in porous media: Acritical review. Soil. Sci. Soc. Am. J. , , 1461.(37) Comolli, A.; De Wit, A.; Brau, F. Dynamics of A + B → C reaction fronts under radialadvection in three dimensions. Phys. Rev. E , , 052213.(38) Britton, M.; Sederman, A.; Taylor, A.; Scott, S.; Gladden, L. Magnetic resonanceimaging of flow-distributed oscillations. J. Phys. Chem. A , , 8306–8313.(39) Wildenschild, D.; Vaz, C.; Rivers, M.; Rikard, D.; Christensen, B. Using X-ray com-puted tomography in hydrology: systems, resolutions, and limitations. J. Hydrol. , , 285 – 297.(40) Krummel, A. T.; Datta, S. S.; Münster, S.; Weitz, D. A. Visualizing multiphase flowand trapped fluid configurations in a model three-dimensional porous medium. AIChEJ. , , 1022–1029. 2741) Berg, S.; Ott, H.; Klapp, S. A.; Schwing, A.; Neiteler, R.; Brussee, N.; Makurat, A.;Leu, L.; Enzmann, F.; Schwarz, J.-O.; Kersten, M.; Irvine, S.; Stampanoni, M. Real-time 3D imaging of Haines jumps in porous media flow. Proc. Natl. Acad. Sci. , , 3755–3759.(42) Deurer, M.; Vogeler, I.; Khrapichev, A.; Scotter, D. Imaging of water flow in porousmedia by magnetic resonance imaging microscopy. J. Environ. Qual. , , 487–493.(43) Greiner, A.; Schreiber, W.; Brix, G.; Kinzelbach, W. Magnetic resonance imaging ofparamagnetic tracers in porous media: Quantification of flow and transport parameters. Water Resour. Res. , , 1461–1473.(44) Rose, H. E.; Britton, M. M. Magnetic resonance imaging of reaction-driven viscousfingering in a packed bed. Micropor. Mesopor. Mat. , , 64 – 68.(45) Martínez-Ruiz, D.; Meunier, P.; Favier, B.; Duchemin, L.; Villermaux, E. The diffusivesheet method for scalar mixing. J. Fluid Mech. , , 230–257.(46) Meunier, P.; Villermaux, E. The diffusive strip method for scalar mixing in two dimen-sions. J. Fluid Mech. , , 134–172.(47) Bandopadhyay, A.; Le Borgne, T.; Méheust, Y.; Dentz, M. Enhanced reaction kineticsand reactive mixing scale dynamics in mixing fronts under shear flow for arbitraryDamköhler numbers. Adv. Water. Resour. , , 78–95.(48) Bandopadhyay, A.; Davy, P.; Le Borgne, T. Shear flows accelerate mixing dynamics inhyporheic zones and hillslopes. Geophys. Res. Lett. , , 11659–11668.(49) Vanderborght, J.; Vereecken, H. Review of dispersivities for transport modeling in soils. Vadose Zone J. , , 29–52.(50) de Anna, P.; Dentz, M.; Tartakovsky, A.; Le Borgne, T. The filamentary structure of28ixing fronts and its control on reaction kinetics in porous media flows. Geophys. Res.Lett. , , 4586–4593.(51) Emmanuel, V. Mixing by porous media. Comptes Rendus Mécanique , , 933–943.(52) Velásquez-Parra, A.; Tomás, A.; Willmann, M.; Méheust, Y.; Le Borgne, T.; Jiménez-Martínez, J. Sharp transition to strongly anomalous transport in unsaturated porousmedia. In preparation ,(53) An, S.; Hasan, S.; Erfani, H.; Babaei, M.; Niasar, V. Unravelling effects of the pore-sizecorrelation length on the two-phase flow and solute transport properties: GPU-basedpore-network modeling.
Water Resour. Res. , .(54) Meunier, P.; Villermaux, E. How vortices mix. J. Fluid Mech. , , 213–222.(55) Dentz, M.; Icardi, M.; Hidalgo, J. J. Mechanisms of dispersion in a porous medium. J.Fluid Mech. , , 851–882.(56) Nissan, A.; Alcolombri, U.; de Schaetzen, F.; Berkowitz, B.; Jimenez-Martinez, J. Re-active transport with fluid–solid interactions in dual-porosity media. ACS ES&T Water , , A–J.(57) FlowControl. 2002; .(58) Britton, M. M. Measurement of the concentration of Mn and Mn in the Manganese-Catalyzed 1,4-Cyclohexanedione/Acid/Bromate reaction using redox-triggered Mag-netic Resonance Spectroscopy. J. Phys. Chem. A , , 13209–13214.(59) Cotton, F. A.; Wilkinson, G.; Murillo, C. A.; Bochmann, M. Advanced Inorganic Chem-istry , 6th ed.; Wiley, 1999. 2960) Kluh, I.; Doležal, J.; Zyká, J. Rasche reduktometrische Bestimmung von Mangan inLegierungen und Mineralien. , 14–20.(61) Britton, M. M. Spatial quantification of Mn and Mn concentrations in theMn-catalyzed 1,4-Cyclohexanedione/Acid/Bromate reaction using magnetic resonanceImaging. J. Phys. Chem. A , , 2579–2582.(62) Britton, M. M. MRI of chemical reactions and processes. Prog. Nucl. Magn. Reson.Spectrosc. , , 51–70.(63) Britton, M. M. Magnetic resonance imaging of chemistry. Chem. Soc. Rev. , ,4036–4043.(64) Ramskill, N.; Bush, I.; Sederman, A.; Mantle, M.; Benning, M.; Anger, B.; Appel, M.;Gladden, L. Fast imaging of laboratory core floods using 3D compressed sensing RAREMRI. J. Magn. Reson. , , 187 – 197.(65) Tang, J.; Smit, M.; Vincent-Bonnieu, S.; Rossen, W. R. New capillary number definitionfor micromodels: The impact of pore microstructure. Water Resour. Res. , ,1167–1178.(66) Hennig, J.; Nauerth, A.; Friedburg, H. RARE imaging: A fast imaging method forclinical MR. Magnetic resonance in medicine. Magn. Reson. Med. , 823–833.(67) Muller, K. A.; Ramsburg, C. A. Influence of nonwetting phase saturation on dispersivityin laboratory-scale sandy porous media.
Environ. Engin. Sci. ,(68) Matsubayashi, U.; Devkota, L. P.; Takagi, F. Characteristics of the dispersion coefficientin miscible displacement through a glass beads medium.
J. Hydrol. , , 51–64.(69) Toride, N.; Inoue, M.; Leij, F. J. Hydrodynamic dispersion in an unsaturated dunesand. Soil Sci. Soc. Am. J. , , 703–712.3070) Raoof, A.; Hassanizadeh, S. M. Saturation-dependent solute dispersivity in porous me-dia: Pore-scale processes. Water Resour. Res. , , 1943–1951.(71) Birkholzer, J.; Tsang, C.-F. Solute channeling in unsaturated heterogeneous porousmedia. Water Resour. Res. , , 2221–2238.(72) Rolle, M.; Le Borgne, T. Mixing and reactive fronts in the subsurface. Rev. Mineral.Geochem. , , 111–142.(73) Guédon, G. R.; Inzoli, F.; Riva, M.; Guadagnini, A. Pore-scale velocities in three-dimensional porous materials with trapped immiscible fluid. Phys. Rev. E , ,043101.(74) Triadis, D.; Jiang, F.; Bolster, D. Anomalous dispersion in pore-scale simulations oftwo-phase flow. Transport Porous Med. , , 337–353.(75) Le Borgne, T.; Dentz, M.; Davy, P.; Bolster, D.; Carrera, J.; de Dreuzy, J.-R.; Bour, O.Persistence of incomplete mixing: A key to anomalous transport. Phys. Rev. E , , 015301.(76) Valocchi, A.; Bolster, D.; Werth, C. Mixing-Limited Reactions in Porous Media. Trans-port Porous Med. , , 157–182.(77) Cirpka, O. A.; Chiogna, G.; Rolle, M.; Bellin, A. Transverse mixing in three-dimensionalnonstationary anisotropic heterogeneous porous media. Water Resour. Res. , ,241–260.(78) Ye, Y.; Chiogna, G.; Cirpka, O. A.; Grathwohl, P.; Rolle, M. Experimental investigationof transverse mixing in porous media under helical flow conditions. Phys. Rev. E , , 013113.(79) Ye, Y.; Chiogna, G.; Lu, C.; Rolle, M. Effect of anisotropy structure on plume entropyand reactive mixing in helical flows. Transport Porous Med. , , 315–322.3180) McClain, M. E.; Boyer, E. W.; Dent, C. L.; Gergel, S. E.; Grimm, N. B.; Groff-man, P. M.; Hart, S. C.; Harvey, J. W.; Johnston, C. A.; Mayorga, E.; McDowell, W. H.;Pinay, G. Biogeochemical hot spots and hot moments at the interface of terrestrial andaquatic ecosystems. Ecosystems , .(81) Stegen, J. C.; Fredrickson, J. K.; Wilkins, M. J.; Konopka, A. E.; Nelson, W. C.;Arntzen, E. V.; Chrisler, W. B.; Chu, R. K.; Danczak, R. E.; Fansler, S. J.;Kennedy, D. W.; Resch, C. T.; Tfaily, M. Groundwater–surface water mixing shiftsecological assembly processes and stimulates organic carbon turnover. Nat. Comm. , .(82) Pool, M.; Dentz, M. Effects of heterogeneity, connectivity, and density variations onmixing and chemical reactions under temporally fluctuating flow conditions and theformation of reaction patterns. Water Resour. Res. , , 186–204.(83) Kitanidis, P.; McCarty, P. Delivery and mixing in the subsurface: Processes and designprinciples for in situ remediation , 1st ed.; Springer, 2012.(84) Sparks, D. L.
Kinetics of soil chemical processes ; Academic press, 2013.32 raphical TOC Entry upporting Information:Phases saturation control on mixing drivenreactions in 3D porous media Ishaan Markale, † , † Gabriele M. Cimmarusti, ¶ Melanie M. Britton, ¶ and JoaquínJiménez-Martínez ∗ , ‡ , † † Eawag, Swiss Federal Institute of Aquatic Science and Technology, 8600 Dübendorf,Switzerland ‡ Department of Civil, Environmental and Geomatic Engineering, ETH Zürich,Stefano-Franscini-Platz 5, 8093 Zürich, Switzerland ¶ School of Chemistry, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
E-mail: [email protected]/[email protected]
Contents of this file • Column design• Reaction time scale calculation• Calibration 1 a r X i v : . [ phy s i c s . f l u - dyn ] F e b dditional Supporting Information (files uploaded sepa-rately) • Captions for movies S1 and S2 a) b) v [mm/s] fl o w xy xyz Fritted glass fi lter Figure S1: a) 2 D section of steady state velocity field after passing the fritted glass filterperpendicular to the flow direction for a constant flow rate Q =20 mm /s from bottom totop. b) Velocity field in the flow direction along a vertical cross-section in the middle of thedomain.A Darcy-Brinkman formulation is used to simulate the fluid flow velocity in the conicalshape of the vessel, in the fritted glass filter (2 mm thickness; ∼
500 µm pore size) and inthe porous medium (1 and 4 mm sized glass beads). While Navier-Stokes equations are used2n the first case, a Darcy (continuum) formulation is used in the two porous media. FigureS1a shows how the fritted glass filter homogenizes the flow before the porous medium ofinterest and reaction occurs. Figure S1b shows the velocity field in the flow direction alonga vertical cross-section. The velocity profile shows that for the given length of the cone,the flow remains attached to the sidewalls for a flowrate Q =20 mm /s (which is the highestflowrate used in the experiments). Figure S2: Exponential curve fitted to the reaction and dilution processes when Mn toMn reduction reaction occurs.A manganese catalysed 1,4 cylohexanedione (CHD) acid reaction is used to study thereaction kinetics (Eq. 5). Mn is first produced by adding manganese (III) acetate toa strong acid H SO (2.5 M). The reaction which consists of conversion between the twooxidative states of manganese from Mn to Mn is accompanied by a change in colour.We captured this colour change using a CMOS camera at 15 frames per second. The visible3rocess in the camera is that of reaction and dilution. The dilution was captured in anotherexperiment where there is no reaction occurring. To determine the rate constant of thereaction, the effect of dilution has to be removed (green curve in Figure S2). Fitting thereaction curve to an exponentially decaying function, a st order reaction with a rate constant k = 1 . is determined. Thus, the time scale of the reaction is /k ≈ s. Figure S3: Schematic showing MRI artifacts close to the beads. A representative glass beadis shown in green colour, the voxels with artifacts in gray and voxels without artifacts inpink.Before injecting any reactant, we measure the signal intensity in each voxel. Due to thedifference in magnetic susceptibility differences of the beads and liquid, there exist artifactsvery close to the glass beads. In Figure S3, as an example we show that gray voxels nextto the beads have these artifacts which gives an incorrect concentration map based on theintensity observed. For the pink colored voxels there exist no artifacts. At initial time, allthe liquid is unreacted solvent (and at final time all liquid is reacted solvent), hence thegray voxels must have the same intensity as the pink ones. For intermediate images, asthe reaction occurs, we correct the intensity values for the gray voxels. We observe thatthe temporal changes in intensities for all the pink voxels follow a normal distribution. We4se the mean value of this distribution to predict the intermediate intensity values in thegray voxels. Our method is corroborated by the fact that we see mass conservation in thedomain, i.e.. the mass produced (of Mn ) in the pore space is same as what is calculatedtheoretically. This is further explained by Figure S4.Figure S4: Mass of Mn plotted as a function of time.As an example, Figure S4 shows in green colour the resident mass of the reaction productwithin the observable domain, i.e., injected Mn plus the mass produced by the reaction,as a function of time for 1 mm sized glass beads with S w = 1 and Pe = 25. The mass ofreaction product is calculated as the difference between the resident mass and the injectedmass. The final resident mass can be calculated as following M Mn = c · M w · φ · V = 0 . g where c = 0.5 mM is the concentration, M w = 173 . g/mol is the molecular weightof Mn , φ = 0 . is the porosity of the system (1 mm beads) and V = 5286 mm is thevolume of the observed region. The final resident mass is used to validate the one inferredfrom MRI. 5 aption Movie S1 Transport of the reactive front inside a fully saturated ( S w = 1 ) packed bed of 8 mm glassbeads. Concentration of the reactants within the mixing volume is shown in warm colors, inwhich the lightest color indicates equal concentration of the invading ( A ) and resident ( B )reactant. A is pumped into the porous medium at constant flow rate ( Q = 0 . mm /s)from bottom to top. Note that the lamella shown is cut by an x-z (vertical) plane passingthrough the middle of the domain; beads (in grey color) are not cut by the plane. Only threebeads are shown for illustration purpose. Caption Movie S2
Transport of the reaction as A is pumped at constant flow rate ( Q = 7 . mm /s) into anunsaturated packed bed of 4 mm glass beads. The flow is from left to right. Concentrationof the reactants within the mixing volume is shown in warm colors, in which the lightestcolor indicates equal concentration of A and B (i.e., equal concentration of Mn and Mn ,mM). Non-wetting phase O is shown in a semi-transparent blue color and does not moveduring the experiment. Saturation of the wetting phase is S w = 0 .78