Microporous and Mesoporous Materials 178 (2013) 64–68

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Microporous and Mesoporous Materials

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Magnetic resonance imaging of reaction-driven viscous fingering in a packed bed

Heather E.L. Rose, Melanie M. Britton ⇑School of Chemistry, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

a r t i c l e i n f o a b s t r a c t

Article history:Available online 14 March 2013

Keywords:Magnetic resonance imagingMRIViscous fingeringPacked bedReactive flow

1387-1811/$ - see front matter � 2013 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.micromeso.2013.02.039

⇑ Corresponding author. Tel.: +44 121 4144391.E-mail address: m.m.britton@bham.ac.uk (M.M. Br

The formation of 3-dimensional viscous fingers has been investigated in a packed bed using magnetic res-onance imaging. Fingering patterns are produced as a result of the formation of a highly viscous wormlikemicelle solution formed at the interface between solutions of cetyltrimethylammonium bromide (CTAB)and sodium salicylate (NaSal), which have similar viscosities, but that are significantly lower to thewormlike micelle solution. In this system, fingers are driven by an interfacial instability, rather than aSaffman–Taylor instability. The structure of the fingering patterns, which were found to be sensitive toflow rate, where analysed using histogram plots characterizing the distribution of pores containing theCTAB solution. This system shares similar characteristics to the growth of biofilms.

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1. Introduction

Viscous fingering, a hydrodynamic instability also commonlyknown as the Saffman–Taylor instability [1], forms when a moreviscous fluid is displaced by a less viscous fluid in a porous med-ium. This phenomenon is observed across a variety of disciplinesincluding oil recovery [2], hydrology [3,4] and separation science[5,6]. Of the many experimental [2,6–8] and theoretical [3,9] inves-tigations of this phenomenon, most have been focussed on non-reactive systems. However, in recent years, there has been increas-ing interest concerning the role reaction plays on the formationand development of these hydrodynamic instabilities [10–15]. Ithas been found that reaction can destabilise the otherwise stabledisplacement of a less viscous fluid by a more viscous fluid [16].It has also been observed that reaction can produce fingering whena rheological change occurs at the interface between two fluids ofidentical viscosity [15]. In this later situation, fingering has beenproduced when solutions of the cationic surfactant cetyltrimethyl-ammonium bromide (CTAB) and the organic salt sodium salicylate(NaSal) were brought into contact, producing highly viscoelasticwormlike micelles at the interface. These ordinarily miscible, aque-ous surfactant and co-surfactant solutions are of the same viscos-ity, and so should not normally produce viscous fingers. However,when mixed, CTAB and NaSal, which acts as a co-surfactant, orga-nise to form long wormlike micelles, which become entangled anddramatically increase the viscosity of the solution. The interfacebetween the two solutions thickens over time, as surfactant andco-surfactant molecules diffuse and aggregate. As only the regioncontaining wormlike micelles has increased viscosity, the

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itton).

phenomenon is an interfacial instability and occurs irrespectiveof which fluid displaces which. Thus it is not a Saffman–Taylorinstability. The fingering produced in this system, which sharesmany of the characteristics of a biofilm [15], has been investigatedas a function of solution concentration, flow rate and which solu-tion is injected or displaced [15]. A variety of fingering patternshave been observed, and characterized by their shape, such as ten-tacle, fan and mushroom fingers. The fingers develop from an ini-tially stable interface into these structures as a function of timeand flow rate. Belmonte and co-workers [15] only found slight dif-ferences in the patterns depending on which solution was dis-placed by which, and explained this by differences in thediffusivity between the different molecules and micelles [15].

Much of the experimental research investigating fingeringinstabilities has employed Hele–Shaw cells, which are composedof parallel transparent glass plates, separated by a narrow gap (typ-ically <1 mm) [2]. The cell is filled with one fluid, with the secondinjected into the first. Fluids are frequently distinguished visuallywith the aid of a dye. The advantages with Hele–Shaw cells are thatthey are simple to set up and fluids can be easily observed. The dis-advantages with this system are that it is only, at best, a 2-dimen-sional system and a rather crude model for a real porous medium.To date only experiments of fingering in reactive systems havebeen performed in Hele–Shaw cells. However, with the use of mag-netic resonance imaging (MRI) it is possible to investigate thesesystems in a 3-dimensional porous medium. This technique hasbeen successfully employed by Fernandez and co-workers to inves-tigate non-reactive viscous fingering in chromatography columns[5,17]. In these experiments a glycerol solution was displaced bysaline solution and the fingers were observed by doping the glyc-erol solution with Gd3+ and using T2 relaxation time contrast toobserve only the saline solution phase.

Fig. 1. Schematic diagram showing the orientation of images with respect to thetube, magnet and flow direction. The 50 mM CTAB solution (grey) is positioned atthe bottom of the tube and the 100 mM sodium salicylate solution (white) above.

H.E.L. Rose, M.M. Britton / Microporous and Mesoporous Materials 178 (2013) 64–68 65

In this paper, we report the first experimental investigation offingering instabilities in a reactive system in a 3-dimensional(3D) porous medium. Fingers were visualised in packed beds usingMRI, exploiting the inherent variation in T2 magnetic resonancerelaxation time of water in the different solutions as the meansof producing image contrast. One challenge with studying thesefingering patterns in a 3D packed bed is associated with how thesestructures are characterised. Hele–Shaw cells enable visualdescription of fingers, as patterns are formed in a plane. In a 3Dporous medium, the tortuosity of the pore space allows fingers todevelop with greater freedom to meander. We, therefore, appliedan alternative method for characterizing fingering patterns inpacked beds, using statistical analysis of the imaging data.

2. Experimental

2.1. Materials

Cetyltrimethylammonium bromide (CTAB) (Sigma Aldrich,P98%) and sodium salicylate (NaSal) (Sigma Aldrich, P99.5%)were used without further purification. Solutions of 50 mM CTABand 100 mM NaSal were prepared using distilled water. Thesesolutions were used to fill a packed bed comprising a cylindricalglass tube of length 30 cm and internal diameter 11 mm, packedwith 1 mm diameter borosilicate glass beads (Sigma). The packedbed was held vertically in a 7T magnet with the NaSal solutionabove the CTAB solution and the interface between the two solu-tions positioned at the bottom of the homogeneous region of a25 mm birdcage radiofrequency resonator. The CTAB solutionwas pumped from below using a syringe pump (Harvard pump22) at flow rates of 1–4 ml h�1.

2.2. Magnetic resonance experiments

All magnetic resonance (MR) experiments were performedusing a Bruker DMX 300 spectrometer, operating at a 1H resonancefrequency of 300.13 MHz, at a temperature of 294.0 ± 0.3 K. Relax-ation experiments [18] were performed to measure the spin-lat-tice, T1, and spin–spin, T2, MR relaxation times of the 50 mMCTAB, 100 mM NaSal and 50 mM/100 mM CTAB/NaSal wormlikemicelle solutions. Inversion-recovery (IR) experiments were per-formed to measure the T1 times for solutions, using 16 experi-ments with IR delays logarithmically spaced between 5 ls and15 s. The T1 relaxation times for the solutions were 2.3 ± 0.1 s(50 mM CTAB), 2.4 ± 0.1 s (100 mM NaSal) and 2.3 ± 0.1 s(50 mM/100 mM CTAB/NaSal). Carr–Purcell–Meiboom–Gill(CPMG) experiments were performed to measure the T2 for thesolutions, acquiring a maximum of 512 echoes with an echo spac-ing of 20 ms. The T2 relaxation times for the solutions were1660 ± 10 ms (50 mM CTAB), 240 ± 12 ms (100 mM NaSal) and664 ± 8 ms (50 mM/100 mM CTAB/NaSal).

Three-dimensional images were acquired using the fast spin-echo imaging technique RARE [19], with a field of view of 40mm � 20 mm � 12 mm and a pixel array of 256 (z) � 128(x) � 16 (y), respectively. A RARE factor of 128 was used, givingan effective echo time of Teff = 575 ms resulting in high signalintensity (SI) for the water in the CTAB solution and low SI forthe NaSal and micelle solutions. The repetition time was TR = 1 s,with a time resolution between images of 120 or 60 s for flow ratesof 1 ml h�1 or 4 ml h�1, respectively. The orientation of imageswith respect to the tube and magnet is shown in Fig. 1.

2.3. Image analysis

All MR data was analysed using Prospa [20]. Images were Fou-rier transformed and compressed from 256 pixels to 32 along the z

direction, so that each pixel shown was integrated over 8 pixelsalong this direction. The displacement of the CTAB and wormlikemicelle solutions into the NaSal solution was detected by monitor-ing the position of pixels above a threshold signal intensity (SI),determined by the average of the SI of the CTAB (high SI) and NaSal(low SI) solutions. As the relaxation time for the wormlike micellesolution was in between that of the CTAB and NaSal solutions, itwas, in principle, possible to distinguish between all three regions.However, as the Teff was comparable with the relaxation time ofthe wormlike micelle solution, it was not possible to distinguishbetween the wormlike micelle and NaSal solutions in the packedbed, hence, the NaSal and micelle solutions are considered to-gether. For each uncompressed 3D data set, the distribution ofthe high SI fluid was characterized using three histogram plots,giving the number of pixels, N(r), above the SI threshold, at a dis-tance r along the x, y and z axes from a root mean square positionfor the high SI fluid. The root mean square position for pixels filledwith the high SI fluid, which defines their centre, was determinedusing a plug-flow model for the filling of the packed bed. In theplug-flow model, the distribution of high SI fluid was allowed todevelop as the packed bed filled, as a plug, at a rate dependenton the volume flow rate and porosity of the packed bed (deter-mined from a spin density image of the packed bed).

The histogram plots for each image were then compared withthose associated with four simple models, where a spin-densityimage of the packed bed was allowed to fill with high SI fluid asa plug, single central-finger, single side-finger or pair of side-fin-gers. Each model system was created by selecting pores in certainregions to be filled with the displacing fluid. The distribution of thefilled pores were restricted to one of the four shapes used – plug,side finger, central finger or double finger. The cross-sectional areaof each finger was 33% of the width of the tube. The length of thefinger in each model system was allowed to increase as a functionof time and flow rate and was controlled by the volume flow rateand the pore space volume found in the experimental packedbed. By using these factors and the porosity of the packed bed, itwas possible to determine the number and distribution of the

Fig. 2. (a) A time-series of magnetic resonance images for a packed bed with CTABsolution (high SI) displacing sodium salicylate solution (low SI) at a flow rate of1 ml h�1. Images at the bottom are closest to the inlet and slices are shown at1.6 mm intervals, with an effective thickness of 1.25 mm. Images are shown at4 min intervals (left to right). (b) The corresponding histogram plots for the imagesin (a), showing the number of high SI pixels at a distance r from the centre of thehigh SI fluid. A set of larger images are available in the supplementary information(Fig. S1).

Fig. 3. (a) A time-series of magnetic resonance images for a packed bed with CTABsolution (high SI) displacing sodium salicylate solution (low SI) at a flow rate of4 ml h�1. (b) The corresponding histogram plots for the images in (a), showing thenumber of high SI pixels at a distance r from the centre of the high SI fluid. Imageand histogram plot parameters are the same as in Fig. 2. A set of larger images areavailable in the supplementary information (Fig. S2).

66 H.E.L. Rose, M.M. Britton / Microporous and Mesoporous Materials 178 (2013) 64–68

pixels filled with the high SI fluid. The filling of pores with the highSI fluid is primarily associated with the displacement of the NaSalsolution by the CTAB solution and not through the growth of thewormlike micelle region, as this solution cannot be detected inthe images. However, as the growth of the micelle region is diffu-sion-limited, it is also not expected that this region will grow sub-stantially during the time-scale of the experiments, as they wereundertaken over a short time-scale, compared to the diffusion ofsurfactant and co-surfactant molecules.

3. Results and discussion

The inherent T2 relaxation times of water in the CTAB and NaSalsolutions are sufficiently different, that the displacement of NaSalsolution by CTAB solution could be monitored directly from theT2-weighted RARE images. The reduced relaxation time for the

water in the NaSal solution is believed to be associated with chem-ical exchange between protons in the water and the hydroxidegroup of the salicylate ion. This resonance is typically not detectedin NMR spectra where water is used as the solvent. However, indeuterated dimethyl sulfoxide the chemical shift for this resonancefor lithium salicylate is extremely high at 15.7 ppm [21] and as it issufficiently different from water, it is expected that proton ex-change between these two resonances will reduce the T2 relaxationtime of the solvent.

Figs. 2a and 3a show a set of slices from the 3D images for thedisplacement of NaSal solution by CTAB solution at different flowrates. Figs. 1b and 2b show the associated histogram plots charac-terising the number, N(r), of high SI pixels (associated with theCTAB solution) at a distance r from a root mean square position.In both experiments there is a general advancement of the highSI fluid as the packed bed is filled from the bottom by the CTABsolution. However, for the images at 1 ml h�1, a finger can be seento form, which then propagates ahead of the bulk movement of the

Fig. 4. Histogram plots for model systems, taken at a single time-point, where highSI fluid fills a packed bed as plug flow (a), a single central-finger (b), a single side-finger (c) and a pair of side fingers (d).

Fig. 5. Histogram plots for experimental data at (a) 1 ml h�1 and (c) 4 ml h�1, withmodels for (b) a side finger and (d) plug flow.

H.E.L. Rose, M.M. Britton / Microporous and Mesoporous Materials 178 (2013) 64–68 67

CTAB solution. The shape of this finger is most closely related tothe tentacle fingers observed by Podgorski et al. in a Hele-Shaw cell[15]. Over time, the finger appears to extend and switch sides as itdevelops; first starting on the right side of the tube and thenswitching over to the left. The extension of the finger ahead ofthe bulk is best observed in the rz histogram plot, with its radialposition best observed in the ry plot. In addition to the fingerextending, it also appears to thicken with time. This can be

explained by considering how the enhanced viscosity of the worm-like micelle solution will restrict the flow where it forms, forcingthe CTAB solution to flow through regions where the wormlike mi-celle solution is less well developed. This is particularly an issue at

68 H.E.L. Rose, M.M. Britton / Microporous and Mesoporous Materials 178 (2013) 64–68

lower flow rates, where there tends to be sufficient time for a layerof wormlike micelles to form at the interface. Any weakness in thislayer is then exploited by the CTAB solution, which will flowthrough this region, resulting in a finger forming. As the wormlikemicelle layer at the tip and sides of the finger is weaker than else-where, the CTAB solution will preferentially travel through theseregions [15], causing the finger to get longer and wider. Thisbehaviour has been previously [2,15] described as spreading andis believed to be associated with the peak that develops at a dis-tance of around 10 mm in the rz histogram plot. In the images at4 ml h�1, there does not appear to be any significant finger forma-tion; rather the interface appears to move more as a plug. The ori-gins for this are believed to reside with the rate at which it takesfor the wormlike micelle layer to fully develop. At higher flowrates, the interface moves more quickly and there is not sufficienttime for the micelle layer to form substantially [15]. Hence, theinterface is able to move more uniformly up the packed bed. Asthe distance over which the interface is observed is relatively short,it is not possible to observe any distortion of the interface due thevariation in volume flow rate between the outer and inner regionsof the packed bed [22]. Nor is there the opportunity to allow themicelle layer to have had sufficient time to develop and affectthe shape of the interface.

The histogram plots for the experimental data were then com-pared with model systems. In Fig. 4, the histogram plots for thefour model systems considered are presented. The plots for plugflow (Fig. 4a) shows that the distribution of pixels is symmetricalalong the y and x axes, reflecting a cylindrical, randomly-packedbed. There is a sharp interface at high positive values along the zaxis, which tails away at negative values. The sharp cut-off at highpositive values reflects the sharp, artificial barrier created by ourmodel of plug flow. The slope at negative values is associated withthe pixels from the bottom of the images, where the tube is outsidethe homogeneous region of the rf coil and the signal drops off. Thejagged envelope to the profiles reflects the packing of the beads inthe tube. In Fig. 4b, the position of the central finger is seen mostclearly in the rz and rx histograms. The enhanced displacement isseen in the rz histogram, with the cross-sectional position givenby the rx plot. For the side finger model, the rz plot is similar to thatof the central finger, however, the rx plot is able to show the shift inposition to the side of the packed bed. The double finger model alsoshows a closer-packed advancement of fluid ahead of the bulk, andthe two fingers can be clearly seen in the rx plot.

The histogram plots for the model systems were compared withthose from the experimental data and are shown in Fig. 5. In thecase of the experiments at 1 ml h�1, a finger can be seen in the rz

histogram plot, advancing ahead of the bulk. Its position is shownin the ry histogram plot, which shows that the finger is a side fin-ger. In the model, the finger was centred in the y direction, but inthe experiment it is to the side in the y direction. In the 4 ml h�1

experiment, the histograms look closest to those of the plug-flowmodel. They are mostly symmetrical along all three axes.There isa more rounded edge to the front of the rz plot, indicating thatthe CTAB solution does not move with a sharp interface. This is

most likely because the advancing interface will move throughthe packing at a rate that is dependent on the tortuosity and pack-ing of the bed and so will not have a sharp interface, but rather onethat reflects the fluctuations in flow velocities between pores.

4. Conclusions

In this paper we report, for the first time, magnetic resonanceimages of viscous fingering in a 3D reactive system. These struc-tures are observed in a 3D packed bed using the difference in signalintensity between a surfactant (high SI) and co-surfactant (low SI)solution, which form wormlike micelles when they come into con-tact. This wormlike micelle solution has significantly higher viscos-ity than either of the other two solutions, which promotes theformation of fingering patterns, as the fluids are pumped througha packed bed. We show that at low flow rates, fingers form whichhave similar characteristics to the tentacle fingers previously ob-served in Hele-Shaw cells [15]. At higher flow rates, the interfacepropagates more like a plug. The fingering patterns were character-ised using statistical methods and compared with simple modelstructures.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.micromeso.2013.02.039.

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