Avalanches in strong imbibition

Slow injection of non-wetting fluids (drainage) and strongly wetting fluids (strong imbibition) into porous media are two contrasting processes in many respects: the former must be forced into the pore space, while the latter imbibe spontaneously; the former occupy pore bodies, while the latter coat crevices and corners. These two processes also produce distinctly different displacement patterns. However, both processes evolve via a series of avalanche-like invasion events punctuated by quiescent periods. Here, we show that, despite their mechanistic differences, avalanches in strong imbibition exhibit all the features of self-organized criticality previously documented for drainage, including the correlation scaling describing the space-time statistics of invasion at the pore scale. Drainage and imbibition in porous media are two opposite fluid displacement scenarios with remarkably different underlying mechanisms. Here, the authors demonstrate that strong imbibition in porous media shows features of self-organized criticality previously observed only in drainage

T he complexity of the world around us-in rivers, climate patterns, landslides, and earthquakes-is often attributed to the prevalence of self-organized criticality (SOC), where systems naturally evolve toward a state in which small perturbations have scale-free consequences 1 . Slow injection of a nonwetting fluid (i.e. slow drainage, Fig. 1a-c) into a porous medium is arguably one of the most accessible examples of SOC [2][3][4][5][6][7] . It can be studied in great detail with benchtop experiments and simple pore-network models [6][7][8] , and the universality of observed trends can be tested by changing the properties of the fluids and the porous medium 9 .
In drainage, constant-rate displacement is achieved by forcing the invading fluid into the porous medium. In slow drainage, when viscous forces are negligible, the invading fluid advances into clusters of pore bodies via intermittent avalanches 10,11 , with waiting times between events and sizes of invasion clusters exhibiting scale-free behavior 6 . These scale-free features are the hallmarks of SOC.
Wetting conditions have a pronounced influence on invasion mechanisms and patterns in porous media 10,[12][13][14] . During the injection of a strongly wetting fluid (i.e. strong imbibition, Fig. 1d-f), the invading fluid advances by coating crevices and corners within the pore space. Slow, constant-rate displacement is achieved by resisting spontaneous imbibition, producing invasion patterns distinct from drainage 13 . However, much like slow drainage, slow strong imbibition evolves via intermittent avalanches 13 . Given the disparity in both pore-scale mechanisms and macroscopic patterns, it is not obvious that the scale-free features associated with drainage would translate to strong imbibition.
In this paper, we use experiments and simulations to show that slow strong imbibition in porous media exhibits all of the same scale-free features of SOC documented for drainage. In particular, we demonstrate that strong imbibition joins drainage as a second known example to follow the correlation scaling of Furuberg et al. 8 describing the space-time statistics of invasion at the pore scale.

Results and discussion
Approach. Experimentally, we examine fluid-fluid displacement in a micromodel, where cylindrical posts are placed at nodes of an irregular triangular mesh and confined between two transparent disks. Both node locations and post sizes are disordered. The micromodel is fabricated via soft lithography from a photocurable polymer (NOA81, Norland Optical Adhesives) 15,16 , which allows precise control of wettability conditions. We refer to this micromodel design as the benchmark geometry; micromodel characteristics and fabrication details can be found in Zhao et al. 13 . We use fixed contact angles of 150 ∘ ± 5 ∘ for drainage and 7 ∘ ± 3 ∘ for strong imbibition (Fig. 1), as measured through the invading phase. In both cases, we inject the invading fluid from the center of the micromodel at 0.4 μL min −1 while maintaining a constant pressure at the outer perimeter. This slow injection rate provides capillary-dominated flow with negligible viscous effects (Ca = 5.8 × 10 −3 , see Supplementary Methods 1 for details).
Slow fluid-fluid displacement in micromodels can be modeled quasi-statically by incorporating the pore-invasion events of Cieplak and Robbins 17,18 and corner-flow events, as described in detail in Primkulov et al. 19 , and compared against experiments in ref. 20 . In the model, pore-invasion pressures p are determined from local post geometry and wettability: (i) in slow drainage (Fig. 1b), invasion fills pore bodies in sequence according to the widest available pore throats; (ii) in slow strong imbibition (Fig. 1e), invasion proceeds by sequential post coating. We ran this model in strong imbibition on the benchmark geometry (see Supplementary Methods 2).
Pore-scale physics and intermittency. In both drainage and strong imbibition, intermittency emerges from capillarydominated interactions of the fluid-fluid interfaces with the quenched disorder of the pore geometry. In slow drainage, the invading fluid advances by progressively occupying new pore bodies. Posts act as local pinning sites of the displacement front ( Fig. 1). Since viscous forces are negligible, the pressure difference between the two fluids across the interfaces (the Laplace pressure) must be the same for all menisci 19 . As the invading fluid is injected, Laplace pressure builds uniformly across the micromodel until it matches the lowest capillary-entry pressure at the displacement front, at which point the associated meniscus becomes unstable. This meniscus then rapidly advances while all other menisci retract, a process known as a burst event 17,18 or Haines jump 21,22 . These events often occur in rapid successions or avalanches, the sizes of which are scale-free 6 . The repetition of this process generates the marked intermittency of slow drainage, where rapid invasion events are punctuated by periods of apparent inactivity. Since clusters of defending fluid occasionally get surrounded and disconnected (i.e., trapped) during the displacement, the quasi-static invasion process is analogous to invasion percolation with trapping (see Fig. 1 and Table 1).
We find that intermittent pore-scale invasion persists in slow strong imbibition, despite the substantial differences in the porescale displacement mechanisms (Fig. 1). In strong imbibition, the invading fluid advances by preferentially coating the corners f In the quasi-static limit, this process is equivalent to invasion percolation on a triangular lattice (Z = 6).
between posts and top/bottom plates 13,19 . After a particular postcoating event, the corner menisci swell as the Laplace pressure increases uniformly across the micromodel until the displacement front touches an uncoated post, triggering rapid coating of the new post 23 . The repetition of this process generates the marked intermittency apparent in experiments (see Supplementary Movie 1). As in drainage, the spatiotemporal evolution of the invasion front displays irregular changes in flow direction and the formation of invasion clusters-a signature of capillarydominated flow in disordered porous media 24 . Clusters of defending fluid occasionally get surrounded by chains of coated posts, but it remains unclear whether these clusters become disconnected and trapped. Our experimental observations are ambiguous but suggestive of trapping [Supplementary Movie 1], but our simulations do not include trapping because our model does not allow for trapping in strong imbibition 19 . We, therefore, assume that the quasi-static invasion process is analogous to invasion percolation without trapping ( Fig. 1 and Table 1). In the Supplementary Note 1, we use a simplified model to show that the presence or absence of trapping does not have significant influence on the invasion statistics.
SOC signature in waiting time. Scale-free waiting times between events are a signature of SOC. As a result, the timing of poreinvasion events in drainage is fundamentally unpredictable. To show that post-coating events in strong imbibition also behave in this way, we measure the waiting times between consecutive postcoating events in our experiments. We find that the histogram of waiting times follows a power-law scaling, similar to drainage 6 . The slope of the power-law fit to the experimental data is smaller than 2 (Fig. 2a), confirming that the distribution is scale-free.
SOC signature in avalanche size. An avalanche in strong imbibition is a cluster of consecutive post-coating events that originate from the same reference post (e.g., the sequence 88-98 in Fig. 2b). Avalanche size θ can be characterized by counting the number of events before encountering a post-coating event disconnected from the cluster (e.g., event 99 in Fig. 2b). For quasi-static invasion, a nearly equivalent definition of θ relies on the pressure signal 7,25,26 . Given the lowest capillary entry pressure p 0 at the displacement front at some reference time, θ can be defined as the number of pore-invasion events that occur before p 0 is next exceeded (Fig. 2c and ref. 25 ). As in drainage 6 , avalanches in strong imbibition retrieved by the counting method detailed in Fig. 2b reveal power-law distributions in both experiments and simulations (Fig. 2d, e). We obtain a similar power-law distribution from the quasi-static pressure signal, a portion of which is illustrated in Fig. 2c. Again, the slope of these power-law distributions is less than 2, indicating that avalanche sizes are scale free. For both drainage and strong imbibition, the scale-free distribution of avalanches is responsible for the fractal nature of the displacement patterns, which are aggregates of scale-free avalanche clusters. Therefore, slow strong imbibition, despite its distinct invasion mechanism, exhibits all the characteristics of SOC-scale-free waiting times between events, scale-free avalanches, and fractal displacement patterns-that have been documented in slow drainage 6 .
Furuberg scaling. The above findings raise the question of whether the correlation scaling of Furuberg et al. 8 , originally proposed for drainage, also holds in strong imbibition. The correlation function N(r, n) measures the probability of pore invasion at distance r in space and a number of events n in time away from a reference event. Here, we measure r as the Euclidean distance normalized by the characteristic distance between posts, and n as consecutive pore-scale event number. Furuberg et al. 8 found that in drainage, where D is the fractal dimension of the displacement pattern. For slow drainage, Furuberg et al. 8 obtained exponents a ≈ 1.4 and b ≈ 0.6 by fitting the results of an invasion-percolation model. Roux and Guyon 25 argued that a and b are linked to exponents of ordinary percolation theory, and Maslov 26 subsequently showed that, under that ansatz, where D e is the fractal dimension of the fluid-fluid interface and ν is the correlation-length-divergence exponent from ordinary percolation. Roux and Guyon 25 argued that a ≥ 1, while Moura et al. 7 showed that a = 1 + D e /D when n ≫ 1, so we should expect Expected values of exponents a and b from Eqs. (2) and (3) are reported in Table 1. Equation (1) was only recently verified experimentally in drainage by Moura et al. 7 .  Furuberg scaling in strong imbibition. Both model and experiment allow us to test the validity of Eq. (1) in strong imbibition (Fig. 3a). For a fixed value of n, N(r, n) is the histogram of distances r between pore-scale events separated in time by n events (Fig. 3b). The data from our model and experiments resemble those reported for strong drainage 7,8 , with peaks in N(r, n) moving to larger r as n increases. These data collapse for both model and experiment when we plot rN(r, n) against r D /n (Fig. 3c), with a peak near r D /n = 1 and power-law behavior on either side of the peak signifying the validity of Eq. (1) in strong imbibition.
We confirm the robustness of this collapse by running a set of additional simulations for both drainage and strong imbibition in which we place posts on a regular triangular lattice and draw post radii from a uniform distribution; the resulting data also collapses, as in Fig. 3, with values of exponents a and b reported in Table 2.
Roux and Guyon 25 demonstrated that the robustness of the collapse suggested by Eq. (1) relies on the power-law distributions of avalanches (P n ðθÞ $ θ Àτ b ) and distances between active pores at the displacement front (Q θ (r)~r α ). These conditions are satisfied in both drainage and strong imbibition, where fluid displacement exhibits features of SOC. However, the predictions of Eqs. (2) and (3) for the slopes a and b appear not to hold in strong imbibition, with the expected value of b (Table 1) significantly different from the value in the experiment (Fig. 3c) and almost two standard deviations away from the mean value in simulations (Table 2). In fact, Biswas et al. 9 showed that τ b is sensitive to the details of the pore structure and the presence of spatial correlation, something that the relatively wide standard deviation across our simulations also shows. This dependence is absent from Eq. (2), where b (which is a function of τ b 25 ) depends only on ν and fractal dimensions.

Conclusions.
We have demonstrated that slow fluid-fluid displacement in strong imbibition exhibits features of SOC previously documented for drainage. The invading fluid advances intermittently, and scale-free distributions emerge for waiting times and avalanche sizes. Both our model and our experiment also show that avalanches in strong imbibition robustly follow the pore-scale event correlation of Furuberg et al. 8 . Slow drainage and strong imbibition are thus the only two known examples of SOC that follow a definitive correlation of events in space and time [Eq. (1)]. It is surprising that these two phenomena-governed by entirely distinct pore-level mechanisms-both exhibit collapse in the correlated nature of invasion percolation.
Furthermore, we anticipate that the correlation scaling of Furuberg et al. 8 Table 2 Result of data collapse analogous to Fig. 3c for simulations on the benchmark geometry, and also on a set of regular lattices with post radii assigned from a uniform distribution. or experiments would be an intriguing next step. Additionally, we found that increasingly strong spatial correlation in wettability or pore-sizes eventually breaks the collapse of the correlation data (Fig. 3c). This is especially true in weak imbibition, where the invasion dynamics on a highly correlated regular lattice resembles the growth of a crystal 27 . While this discussion falls outside the scope of this letter, it is another great direction for follow-up studies. Scientific understanding of SOC is still at a relatively early stage 1,28,29 , and much of the progress in the field is made by studying one example of SOC and trying to extrapolate to others 30 . Many other natural examples of SOC, like landslides, snow avalanches, and earthquakes, share features similar to avalanches in drainage and strong imbibition 1 . For instance, in earthquakes: (i) the sliding of geologic faults occurs by means of intermittent stick-slip motion; (ii) magnitudes and waiting times between consecutive earthquakes are scale-free 1 , and (iii) earthquake locations within slip planes have been speculated to form fractal patterns 4 . An intriguing follow-up to this work would be to investigate whether the scaling in Eq. (1) holds for earthquakes, given the density and precision of modern earthquake catalogs 31,32 .

Data availability
All relevant data are available upon reasonable request from the corresponding author.

Code availability
A detailed description of the quasi-static model used in this study has been published in Primkulov et al. 19 . All the algorithmic details needed to reproduce the results of the numerical simulations are provided in that publication.
Received: 3 October 2021; Accepted: 4 February 2022; Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.