Imbibition in porous media: correlations of displacement events with pore-throat geometry and the identification of a new type of pore snap-off

1Institute of GeoEnergy Engineering, Heriot-Watt University, EH14 4AS Edinburgh, U.K. 2Qatar Carbonates and Carbon Storage Research Centre, Department of Earth Science and Engineering, Imperial College London, SW7 2AZ London, U.K. 3Department of Earth Science and Engineering, Imperial College London, SW7 2AZ London, U.K. 4Department of Geology, Ghent University, 9000 Ghent, Belgium *Corresponding author: Kamaljit Singh, k.singh@hw.ac.uk


Introduction
Multiphase flow, including drainage and imbibition, in porous media occur, for instance, during water infiltration in soils, geo-sequestration of supercritical CO 2 in deep saline aquifers or depleted oil and gas fields, subsurface non-aqueous phase liquid contaminant transport, and oil recovery from reservoir rocks (1 , 2 ). At low injection rates, typical in earth science applications, capillary forces tend to dominate drainage and imbibition, which consist of a sequence of several types of fluid displacement events (3 -8 ). The displacement of a wetting fluid by a non-wetting fluid from porous media is called drainage, whereas the displacement of a non-wetting fluid by a wetting fluid is called imbibition. Drainage is controlled by the narrowest regions, called throats, between the adjoining widest regions, called pores, and consists of Haines jumps (in which fluid-fluid interface makes a sudden and fast jump through the throat to the adjacent pores, halting at the next throat which has a higher threshold pressure) and Roof-snap-o↵ events (3 , 9 -13 ).
On the other hand, imbibition is controlled by both pores and throats (14 ). During secondary imbibition, when the wetting fluid is initially present in small pores and throats as well as in pore corners before imbibition, the two most important processes that compete are pore-filling and snap-o↵ (15 -17 ). These processes strongly influence the trapping of the non-wetting phase which is reflected in oil recovery, CO 2 storage and remediation eciency of contaminated land. The pioneering work of Lenormand and co-workers (4 , 16 ), using a twodimensional micro-model consisting of a square lattice with a co-ordination number of four, provided a detailed analysis of threshold capillary pressure required for pore-filling events under di↵erent fluid configurations, called I n events. Figure 1 a-c show schematics of three di↵erent fluid configurations. In I 1 displacement events, only one of the connecting throats is initially filled with a non-wetting fluid (Figure 1a). Similarly, in I 2 and I 3 events, two and three connecting throats are initially filled with a non-wetting fluid respectively (Figure 1 b-c). During waterflooding, pore-filling is more favorable for I 1 compared to I 2 for the same pore size, as I 1 can happen at higher capillary pressure that can be estimated using simple geometrical calculations and the Young-Laplace equation. A lower curvature and capillary pressure is needed to displace the non-wetting phase in I 2 (see the dashed oil-brine interfaces in Figure 1 a-b). Similarly, I 3 is the least favorable among these three displacement events.
Apart from these I n pore-filling mechanisms, a nonwetting fluid can be snapped-o↵ and expelled from a throat when wetting layers in throat corners swell and eventually touch, leading to an unstable interface (1 , 3 , 18 ). Figure 1 d-g show a schematic of such an event for an I 1 fluid configuration. In this case, the non-wetting fluid in the pore space can either be expelled in a piston-like displacement (Figure 1e), or it can be trapped due to wetting layer swelling in the neighbouring throat leading to the snap-o↵ of the non-wetting phase (Figure 1f-g). A snap-o↵ can also occur without trapping, as shown in (Figure 1h-i). In this case, after a snap-o↵ event, fluid configurations are reduced in order from I n to I n 1 (e.g., I 2 to I 1 in Figure 1 h-i), which can lead to the displacement of the non-wetting phase by a filling process which was not favorable previously.
The prevalence of snap-o↵ is a function of porethroat geometry and wettability (described by fluidfluid-solid contact angle). Among various geometrical parameters, aspect ratio (the ratio of pore radius to throat radius), angularity or shape and coordination number (the number of throats connected to a pore in consideration) have been investigated in previous two-dimensional and numerical simulation studies (1 , 14 , 19 -21 ). In a two-dimensional system, it has been reported that the minimum (threshold) value of aspect ratio for which snap-o↵ events occur increases from 1.5 to 1.75 for an increase in contact angle from 0 to 55 (14 , 20 ). The threshold aspect ratio becomes infinite for contact angles larger than 70 , and snap-o↵ cannot occur in throats. In general, snap-o↵ is theorized to be the preferred filling mechanism when the aspect ratio is larger than 2 and the contact angle is close to 0 (for strongly wetting or hydrophilic conditions) (1 ).  Figure 1: In pore-filling and snap-o↵ with and without trapping. (a-c) In pore-filling description in a square lattice with a co-ordination number of four (after Lenormand and Zarcone (12)). A displacement event is called I 1 if one of the throats is initially filled with a non-wetting fluid (a). Similarly, I 2 and I 3 pore-filling events are defined when two and three throats are initially filled with the non-wetting fluid respectively (b and c respectively). w and nw refer to wetting and non-wetting fluids respectively. Red arrows show the direction of fluid displacement. (d-g) An I 1 configuration experiencing either pore-filling (d and e) or wetting layer swelling (f) leading to snap-o↵ and trapping (g). (h-i) Snap-o↵ without trapping. Wetting layer swelling between two connected pores with an I 2 configuration (h) leading to snap-o↵ (i). In this case, the non-wetting fluid after snap-o↵ is still connected.
Secondly, snap-o↵ is favorable for more angular throats in which wetting layers can reside and swell with time (with decreasing capillary pressure or increasing water pressure). Angularity is often characterized by calculating a "shape factor", traditionally defined as G=A/P 2 , where A is the crosssectional area and P is the perimeter of a pore or throat (22 ). Lastly, snap-o↵ is less likely to occur in a well-connected pore space (described by the coordination number) (1 ). Most of the available studies on snap-o↵ and pore-filling have focused on two-dimensional systems; however, we still lack a clear understanding of these processes in natural three-dimensional porous media as these have relatively complex pore geometries compared to previously used model systems.
With recent advances in computational power and pore-scale imaging using synchrotron X-ray microtomography, the fluid displacement in porous media can be observed directly in three dimensions at the micrometer scale. This technique has been used to study the dynamics of burst instabilities after Haines jumps (10 , 11 , 23 ) and Roof snap-o↵ (3 , 13 ) during drainage. In general, the drainage process is fairly well understood; however, secondary imbibition has not been studied in such detail.
Only a handful studies have investigated the threedimensional dynamics of imbibition processes, e.g., (3 , 18 , 23 -28 ). In terms of morphological features, Herring et al. (29 ) and Andersson et al. (30 ) have reported a strong correlation of residual saturation of the non-wetting phase with morphological and persistent homology-based aspect ratios and a weaker correlation with the traditional pore-throat aspect ratio. On the other hand, Tanino and Blunt (31 ) reported a good correlation between residual saturation and the traditional aspect ratio. In most of the previous studies, an average aspect ratio (ratio of the pore radius to the mean of all bounding throat radii) was used, which does not directly reflect local trapping and pore-filling events. Moreover, a pore-by-pore analysis of various displacement events has not been explored yet, possibly due to lack of time-resolved tomographic data.
In this study, we have used synchrotron X-ray micro-tomography to capture pore-to-pore displacement during capillary-controlled secondary imbibition in a water-wet Ketton limestone sample. We have analyzed and correlated various pore-scale displacement and trapping events to pore and throat geometrical parameters extracted from the segmented pore space (32 ). We also report a new type of pore snap-o↵ mechanism leading to the trapping of the non-wetting fluid in a fraction of a single pore body. Note that this work is purely based on the analysis of experimentally observed displacement events. We have used pore-throat parameters obtained from a pore-network extraction code; however, the pore-network flow model is not used in this study. A comparison of fluid displacement observed in this experiment to quasi-static pore network flow model simulations are reported elsewhere (33 , 34 ).

Materials
The experiment was conducted on a 3.8 mm diameter and 10 mm long water-wet Ketton limestone rock sample from the Ketton quarry, Rutland, UK, which contains 99.1 % calcite with the remaining fraction being quartz (35 ). These samples were cleaned with methanol using Soxhlet extraction apparatus for 24 hours, followed by drying in a vacuum oven at 100 C for 24 hours. A solution of potassium iodide (KI) salt (puriss, 99.5 %, Sigma-Aldrich, UK) in deionized water with a salin-ity of 1.8 mol/L was used as the aqueous phase, which provided an e↵ective X-ray contrast between brine (wetting phase) and oil (non-wetting phase). Decane (ReagentPlus, 99 %, Sigma-Aldrich, UK) was used as the oil phase.

Experimental protocol
The rock sample was placed on a water-wet porous plate in a Viton sleeve. The assembly was loaded in a Hassler-type flow cell made of carbon fiber that is nearly transparent to X-rays. The end pieces that were fitted to either sides of the sleeve were connected to high-pressure pumps (the details of experimental methodology are presented in Singh et al. (18 , 36 )). First, the sample was saturated with brine. The system was then pressurized to 10 MPa with a confining pressure of 11.2 MPa. The oil was then injected from the top at a constant pressure drop of 50 kPa to start drainage. Once there was no further displacement of brine from the sample, the flow was reversed by injecting brine from the base of the sample at a constant pressure drop of 22 kPa, to start secondary imbibition. The capillary number N c =v µ/ (where v is the Darcy velocity of the invading fluid, µ is the viscosity of the invading fluid, and is the brine-oil interfacial tension) during brine flooding was 1.26 ⇥ 10 9 , representing a capillary-dominated flow regime. The sample was imaged continuously during drainage and secondary imbibition using synchrotron X-ray microtomography at the Diamond Light Source, with a voxel size of 3.28 µm and a time-step between consecutive images of 38 s. These tomographic datasets have been uploaded on a public repository (BGS), the details of which are provided in Singh et al. (36 ).

Image processing
The reconstructed tomographic images were filtered using an edge preserving non-local mean filter (37 ). The dry filtered image (without fluids) was segmented into two phases (pores and grains) using a seeded watershed algorithm. We then obtained parameters of pores and throats from the isolated pore space using a pore-throat network approach (32 ). This method uses a watershed algorithm on the distance map of the segmented pore space image to identify throats (local constrictions) as boundary surfaces between adjacent pores.
The time-series data containing oil and brine phases were subtracted from the initial image of the rock saturated with brine, to isolate the oil phase, which was then segmented using simple thresholding. The segmented oil was applied as a mask on the pore-throat network to identify the pores and throats that were occupied by oil and brine. By repeating this procedure for each time step (for 285 tomograms), we obtained information on the characteristics of pores and throats involved in each snap-o↵ and pore-filling event. Note that this analysis on time-resolved images was conducted manually for each time step for quality control. We discarded the events that occurred at the periphery of the sample and at the top and the base of the sample, as the pore-network extraction at these boundaries can result in inaccurate assignment of pore and throat radii.

Statistical analysis
The correlation of two variables was obtained by calculating Pearson's correlation coe cient (Pearson's r ).
wherex andȳ are the mean values of variables x and y respectively. x i and y i are individual data points of variables x and y respectively. The values of Pearson's r can range from 1 to +1, where values close to 1 and +1 indicate strong correlations between two variables, whereas a value of 0 shows no correlation.
Secondly, we have calculated p-value, which is a probability test that describes whether the null hypothesis is true or there is any relationship between two variables. The p-value is based on the t-test, described as: where r is Pearson's correlation coe cient, and n is the number of data points. The p-value can be determined from t-test and the degrees of freedom (f =n−2) using a t-distribution table. A p-value close to zero indicates that there is a strong relationship between two variables, whereas a high value close to 1 indicates either the lack of correlation (null hypothesis) or that the data are insu cient to prove a correlation.

Results and Discussion
A sequence of pore-filling and snap-o↵ processes during secondary imbibition is shown in Singh (38 ) (or doi.org/10.6084/m9.figshare.4235381.v1). During brine injection, the brine-oil interface moves from pore to pore in a piston-like displacement without oil trapping, until a pressure disequilibrium condition is reached, at which the brine layer in the throat corners (ahead of an oil-filled pore) starts to grow. The brine layer continues to grow until the interface is no longer stable, and results in snap-o↵ of the interface and trapping of oil. In our previous study (18 ), we investigated the mechanism of snap-o↵ by calculating local capillary pressures (obtained from liquid-liquid curvatures) and timescales of brine layer swelling. In this study, we have extended our analysis to obtain conditions that are favorable for pore-filling and snap-o↵ based on porethroat geometry.
Pore-throat geometry and fluid displacement events: A localized subset analysis Here in this section, we provide a pore-scale displacement analysis on a small subset (an example case as shown in Figure 2) of the scanned image, and then extend this analysis to the complete image in the next section. Figure 2a shows a network of pores (spheres) and throats (tubes) in the subset. The pores are the largest inscribed spheres in the separated pore space, which are colored according to size. Throats are the narrowest constriction along the medial axis connecting two adjacent pores. An illustration of a throat between two pores in two dimensions is shown in Figure 2b.
The oil phase from the time-resolved segmented data (at t = 25 min, 58 s) is superimposed on the pore-throat network as shown in Figure 2c. With continuous brine flooding, the interface jumped to adjacent pores (Figure 2d), which is indicated by yellow arrows for pore-filling event A and B. By subtracting consecutive images and superimposing them on the pore-throat network, we obtained information on the characteristics of pore and throats involved in these pore-filling events. Similarly, we performed this analysis on the complete image sequence in this subset containing nine events (the video is available in (39 )   Pore 1 and the radius of the throat connecting the two pores. We also calculated shape factors of pores and throats involved in each event. Here, the shape factor is defined as G = R 2 /(4A) where R is the radius of the inscribed sphere, and A is the crosssectional area of the pore or throat (32 , 40 ). Note that this description of the shape factor is di↵erent from the traditional definition.
From the analysis, we find a significant di↵erence in the aspect ratios and throat shape factors for pore-filling events compared to those for snap-o↵ events. The aspect ratio for pore-filling events is in the range of 1.2 to 1.6, except for the event I with a value of 2.1. An aspect ratio less than 1.6 appears to be favorable for pore-filling. Note that the event H and event I occurred during the same time step after the snap-o↵ event F. The interfacial jump after event F could create su cient pressure fluctuations in the adjacent pores, which could lead to the pore-filling event for an unfavorable aspect ratio of 2.1 in event I. The pore-scale fluctuations and inertial e↵ects after a snap-o↵ event have also been observed in a recent study by Rücker and coworkers (20). On the other hand, the aspect ratio for the snap-o↵ event F is 3, which is significantly larger than that for pore-filling events, therefore, resulting in unfavorable conditions for pore-filling.
The shape factor of throats involved in pore-filling events is in the range of 0.029 to 0.049, whereas for the snap-o↵ event F, the value is 0.026. The lower value of throat shape factor is favorable for snap-o↵, since this represents throats with a large surface area and smaller corner angles, which are likely to have more corners in which wetting layers can reside and swell with time (with decreasing capillary pressure or increasing water pressure) leading to snap-o↵. Although this example subset shows that a lower value of throat shape factor is favorable for snap-o↵, our analysis on the whole image of the complete time-series datasets (reported in the forthcoming section) shows that the throat shape factor is poorly correlated with displacement events.
Correlation analysis of event types (snap-o↵ and I n pore-filling) and porethroat geometry: Complete tomographic datasets We now extend the subset analysis in Figure 2 to the complete scanned image of the whole time-series (285 tomograms). Table 1 shows the frequency of di↵erent snap-o↵ and I n events observed during brine injection. The majority of fluid displacements were I 1 and I 2 events, which occurred more frequently than snap-o↵. Only three I 3 events were observed in the data analyzed. Dashed gray and black lines show minimum (threshold) values of aspect ratios for snap-o↵ resulting in oil trapping (for I 1 configurations) and snap-o↵ (without trapping) between connected oil in two pores respectively. (b) Throat shape factor distribution for I 1 , I 2 and I 3 events. Throat shape factors for snap-o↵ events are overlain on the data along the top x -axis. We first compare situations where I 1 events took place to those where throat snap-o↵ led to oil trapping, as illustrated in Figure 1 d-g. Here we consider what we call I 1 configurations where an oilfilled pore is surrounded by throats of which only one is oil-filled. If an I 1 event occurs, both the pore and the remaining throat fill with brine and there is no trapping (see Figure 1e); if snap-o↵ occurs in the oil-filled throat, oil is trapped in the pore (Figure 1 f-g). Figure 3a shows the distribution of aspect ratios of the pore-throat space where various I 1 events occurred. Most of the events occur for an aspect ratio less than 1.6. On the other hand, the smallest (threshold) value of the aspect ratio for which we observe a snap-o↵ followed by trapping is 2.15. This is in agreement with the previous subset analysis in Figure 2, and also with the theoretical value reported in the literature (1 ). We also observed a pore-filling I 1 event which occurred for an unfavorable aspect ratio of 2.49. This pore-filling occurred immediately after a snap-o↵ event in the adjacent pore, and may be related to inertial e↵ects as described for event I in Figure 2 and reported in a previous study (24 ).
Similarly, the snap-o↵ in a throat between two pores containing connected oil ( Figure 1 h-i) shows a good correlation with aspect ratio. A threshold value of 2.03 is found for snap-o↵ without trapping ( Figure  3a). In this case, the maximum value of aspect ratio between two pores that are connected to the throat undergoing snap-o↵ is considered.
Many I 2 pore-filling events occurred for an aspect ratio larger than 2 ( Figure 3a). In these events, it is likely that a snap-o↵ occurred in one of the throats leading to an I 1 piston-like displacement in the second throat. The time resolution of tomography is not su cient to confirm this hypothesis. This would need further investigation by conducting faster tomography or direct numerical simulations.
On the other hand, there is no clear correlation between the observed I n events and the throat shape factor (Figure 3b). Although a majority of snap-o↵ events happen in throats with lower throat shape factors, this still overlaps with the distribution of throat shape factors for I 1 , I 2 and I 3 events. This could be due to a limited voxel count in throats (due to image resolution) which does not allow for the accurate estimation of shape factors.
We have also calculated Pearson's correlations coefficient (Pearson's r, Eq. 1) between pore-throat geometric parameters and displacement events (snap-o↵, I 1 and I 2 ), as shown in Table 2. In this case, in Eq. 1, we have used x i as an indicator function which shows whether it is an I 1 event (value = 0) or a snap-o↵ (value = 1). The second variable y i is the pore-throat parameter studied (e.g., aspect ratio, pore radius, throat radius, throat shape factor and Table 2: Pearson's correlation coe cient, number of data points, t-distribution and null hypothesis probability for displacement events vs pore-throat geometry. Here, AR is the aspect ratio, Rp and Rt are pore and throat radii respectively, Gt is the shape factor for throats, and Zp is the pore coordination number. The counts (n) show the total number of events identified in the analysis. In the case of AR for snap-o↵/I 1 , a total number of 128 events were analyzed, in which 124 were I 1 and 4 snap-o↵ with trapping, see Table 1. In the case of AR for snap-o↵/I 2 , 40 I 2 configurations were considered, in which each I 2 has two throat connections. Three of the throats were not captured clearly and were not considered in the analysis. Therefore, a total of 85 events (77 throats and 8 snap-o↵ without trapping) were analyzed. Similarly, in the case of Rp for snap-o↵/I 1 , 128 events were considered. In the case of Rp for snap-o↵/I 2 , 48 events were analyzed, in which 40 were I 2 events and 8 snap-o↵ without trapping (refer to Table 1).  pore coordination number). A high absolute value of the Pearson's correlation coe cient indicates a strong correlation between the event type and geometric parameters. Table 2 also shows the probability of null hypothesis (p-value). A high p-value indicates either the lack of correlation (null hypothesis) or insu cient experimental data for proving the existence of the correlation.

Geometry and events
The analysis provided in Table 2 again shows that the aspect ratio is strongly correlated to the occurrence of snap-o↵ or I 1 and I 2 events, with Pearson's r in the range 0.60-0.67 and low p-values. This dependence on aspect ratio is not explicitly implemented in many pore network models to capture the competition between snap-o↵ and pore filling (41 , 42 ), and can be used to improve the accuracy of these models.
Similarly, pore radius (R p ) for snap-o↵/I 1 and snap-o↵/I 2 shows a good correlation with low p-values, indicating that pore filling is suppressed when the pore involved is large; however, throat radius (R t ) shows a poor correlation with high p-values. This shows that just having a small throat does not necessarily favor snap-o↵ over pore-filling: the key criterion is the throat radius in relation to the pore, captured by the aspect ratio. On the other hand, pore coordination number for snap-o↵/I 1 shows a good correlation with a low p-value. Snap-o↵ is more likely in the remaining throat in pores with a larger coordination number: here it is because the more connected pores tend to be larger with a higher aspect ratio. The coordination number analysis for snap-o↵/I 2 events shows a poor correlation, which could be possibly due to misidentification (due to limited time resolution) of I 2 events compared to snap-o↵ followed by an I 1 event.
The throat shape factor shows weaker correlations with higher probability of null hypothesis (Table  2), which confirms the findings reported in Figure  3b. This indicates that the implementation of the dependence of displacement events on the throat shape factor in pore-network models could produce biased results, and must be carefully evaluated.
The time at which these events happen gives an indication of the order in which an event is preferred and the resulting fluid distribution. Table  3 shows the correlation of the pore-throat geometry and the time when events took place. In this case, we have used the event time as x i variable and pore-throat parameters (e.g., pore radius) as y i variable in Eq. 1. For I 1 and I 2 events, only pore radius shows a good correlation and has low p-values. Other parameters, such as shape factor and coordination number show a reasonable correlation, however with high p-values.
Pore radius of displacement events as a function of their time of occurrence Figure 4 shows pore radii for various events as a function of time. Although larger pores appear to be filled at later times, which is expected as the capillary pressure decreases, we still observe many small pores experiencing I 1 events at later times. This could be due to transition of configurations from I 2 to I 1 with fluid displacement. With continuous brine injection, it is possible that one of the throats in an I 2 configuration is filled due to either a snap-o↵ in the throat or I 1 displacement in the adjacent pore. This leaves the pore in question with an I 1 configuration at late times. As described in the schematic in Figure 1 a-b, an I 1 event is more favorable than the original I 2 event for the same pore size and can happen at a higher capillary pressure. For this reason, small pores changing from an I 2 to an I 1 configuration at later stages of brine injection are potentially in an unstable configuration and will be rapidly filled with brine.
Similarly, invasion of an I 2 configuration for smaller pore radii at later times was observed. In I 2 configurations, snap-o↵ in one of the throats could have preceded the actual pore filling event, making the latter an I 1 rather than an I 2 invasion. Such events may have been misidentified in the images due to limited time resolution. This hypothesis is supported by the aspect ratio distribution (Figure 3a) in which we observe many I 2 events for aspect ratios more than 2, which tend to be favorable for snap-o↵. Therefore, the pore radii cross-plot with time for I 2 events (Figure 4) could be biased, and this is likely to a↵ect its correlation with time. An example of such an event transition can be seen in Figure 2 g-i, in which snap-o↵ event F changes the fluid configuration in the adjacent pore (marked by cyan circle in Figure 2i) from I 3 (before snap-o↵) to I 2 (after snap-o↵), thereby facilitating the displacement of oil through event G and H after snap-o↵. Note that the radius of this particular pore (marked by cyan circle in Figure 2i) is smaller compared to two other pores that were invaded through porefilling events in earlier times (events D and E ), and could be expected to fill first. However, this did not take place due to the I 3 configuration in this pore prior to snap-o↵, which impedes pore-filling. With its transition from I 3 to I 2 , a pore-filling becomes more favorable.
Unexpected I 2 pore-filling event We have observed an I 2 pore-filling event with unfavorable aspect ratios. Figure 5a shows an I 2 fluid configuration where two throats connected to a pore are occupied by oil. In the next time step, oil is displaced to the adjacent pores through both throats (Figure 5b). The aspect ratios involved in the displacement events A and B are 3.3 and 4.4 respectively, which are favorable for snap-o↵ and oil trapping; however, the snap-o↵ does not occur in any of the throats, rather the oil migrates to the adjacent pores without being trapped. It is also possible that a snap-o↵ event occurred in one of the throats forming an I 1 configuration in the second throat in which the displacement could occur due to inertial e↵ects after a snap-o↵ in the neighbouring throat.
To explain this type of displacement event, further research is needed. New type of pore snap-o↵ leading to oil trapping We have found a new type of displacement event leading to the formation of sub-singlet ganglia (occupying less than a pore body) in the pore space. Figure 6 shows the formation of such a ganglion, in which the oil in an I 1 fluid configuration (Figure 6a) is displaced to the adjacent pore (through throat 3 ) in the next time step (Figure 6b). The aspect ratio of the pore-throat configuration is 2.7, therefore a traditional snap-o↵ in throat 3 leading to trapping in the middle of the pore space was expected. However, neither a complete pore-filling nor a snap-o↵ (due to brine layer swelling) occurred in this case, rather the oil-brine interface jumped to the adjacent pore leaving a small fraction of oil which appears to be in a pore corner close to throat 2. The oil appears to be stuck to the pore wall crevice, and does not occupy the larger space in the center of the pore as in the case of trapping after a traditional snap-o↵ (18 ).
This type of snap-o↵ and trapping has similarities to the fragmentation of a non-wetting phase (oil) observed during desaturation at fast flow rates (43 ); however, it cannot be related to the same mechanisms as the flow rate in our study is slower by four orders of magnitude. It may be argued that the interfacial velocity during pore-filling in a slow capillary-dominated flow might be extremely high to result such a fragmentation; however, we hypothesize that the formation of such ganglia is likely to be due to interface pinning at rough edges and corners of the pore space.
The interface pinning at solid surfaces can be observed in Figure 6c, which results in an advancing contact angle more than 90 during displacement. Using direct numerical simulations (44 ), we show that this pinning holds the interface at the threephase contact points on the solid surfaces; however, with the increasing water pressure (or decreasing capillary pressure), the water continues to displace the oil in the centre of the pore body. This eventually empties the pore centre, leaving the oil stuck at the pore walls. Once the oil is disconnected, the interfaces rearrange to attain an equilibrium pressure condition (Figure 6d).
We have found four such sub-singlet ganglia formed in single pore bodies in our time-series data. The pore-throat aspect ratio in these events is the range of 1.6 to 2.7, which indicates that this type of pore snap-o↵ is independent of aspect ratio. Although these events result in trapping of small volumes of oil which may not be significant for oil recovery applications, it would be crucial for designing e↵ective remediation techniques for non-aqueous phase liquid contaminated soils.
I 2 pore-filling, bypassing and trapping Apart from snap-o↵, the second type of trapping is due to pore-filling and bypassing. This is illustrated in Figure 7. Here, an I 2 pore-filling occurs in either Pore 1 or Pore 2, leading to oil bypassing and trapping in two pore bodies. Aspect ratios of the pore and throats involved in this event are small (in the range 1.1 to 1.9, considering both throats connected to Pore 1 and Pore 2 ), therefore a snap-o↵ is not favorable. This type of bypassing and trapping, which can lead to the formation of large ganglia occupying several pore bodies, has been widely investigated, especially in a pore doublet geometry in micromodels (45 ); however, this is the first time that it has been demonstrated in three dimensions using a time-resolved study. Oil configuration in two consecutive tomographic time steps: (a) connected oil, and (b) trapped oil. Oil in the pore space is superimposed on pores (colored spheres) and throats (gray tubes).

Conclusions
We have used fast synchrotron X-ray microtomography to investigate various displacement events such as I n pore-filling, snap-o↵, bypassing and trapping during secondary imbibition in a Ketton limestone sample. By extracting a pore-throat network and superimposing it on the oil phase in consecutive time-series images, we have analyzed conditions that are favorable for each type of event.
The Pearson's correlation coe cient and probability of null hypothesis showed that the aspect ratio has a strong influence on whether a snap-o↵ or a pore-filling event occurred. Aspect ratios less than 2.03 were found to be favorable for pore-filling, except for two pore-filling events that occurred for values of 2.1 and 2.49. This unfavorable pore-filling may be related to inertial e↵ects, especially after a snap-o↵ event in the adjacent throat; however, further research is needed to support this hypothesis. On the other hand, throat shape factors were weakly correlated to snap-o↵ and I n events. This may have been influenced by the limited number of voxels in a throat, which did not allow us to resolve its shape accurately. Moreover, the pore radius in I 1 and I 2 events was found to be positively correlated to event time or, equivalently, to a decreasing capillary pressure with time.
We have also observed a new type of pore snap-o↵ leading to oil trapping in a fraction of a pore body, which is related to interface pinning at rough pore surfaces. Finally, we have also identified an unexpected I 2 pore-filling event for high aspect ratios that were favorable for snap-o↵, however a pistonlike displacement occurred through both throats. Future work can focus on faster synchrotron imaging or the visualization of flow in three-dimensional printed micromodels along with direct numerical simulations, to investigate these processes in detail.
Overall, the analysis and correlations provided in this study are expected to improve our general understanding of the physics of fluid flow in water-wet porous media, and consequently of pore-scale models of fluid displacement processes. Further experiments and similar analysis on porous media with altered wettability, with di↵erent initial conditions, or on other types of porous media, e.g., sand packs, sandstones and more complex carbonate rocks, can provide the generality of our approach.