Forced imbibition in porous media: a fourfold scenario

TL;DR

This paper provides a detailed analysis of four distinct pattern formation scenarios during forced imbibition in porous media, linking pore-scale instabilities to large-scale morphologies through experiments and microscopy.

Contribution

It introduces a comprehensive framework connecting pore-scale instabilities to large-scale patterns in forced imbibition, based on experimental observations and microscopy.

Findings

Identification of four distinct imbibition scenarios

Linking pore-scale transitions to large-scale patterns

Observation of long-time pattern coarsening

Abstract

We establish a comprehensive description of the patterns formed when a wetting liquid displaces a viscous fluid confined in a porous medium. Building on model microfluidic experiments, we evidence four imbibition scenarios all yielding different large-scale morphologies. Combining high-resolution imaging and confocal microscopy, we show that they originate from two liquid-entrainment transitions and a Rayleigh-Plateau instability at the pore scale. Finally, we demonstrate and explain the long-time coarsening of the resulting patterns.

Full text

About a hundred years ago field engineers noticed that water does not homogeneously displace oil confined in porous media Buckley and Leverett (1942). After a century of intense research, our understanding of driven liquid interfaces remains surprisingly unbalanced. The case of drainage where the defending fluid wets the confining solid, is fairly well understood, and so is the spontaneous imbibition of a wetting liquid through an empty porous structure, such as the rise of a coffee drop in a sugar cube Lenormand et al. (1988); Lenormand (1990); Sandnes et al. (2011); Sahimi (1993); Alava et al. (2004). In stark contrast, much less attention has been paid to the situation where imbibition is used to mobilize a more viscous liquid Stokes et al. (1986); Weitz et al. (1987); Lenormand (1990). For more than twenty years, and despite its relevance to a number of industrial processes, our knowledge about forced imbibition has been mostly limited to a seminal phase diagram constructed by Lenormand from heuristic arguments  Lenormand (1990). However, recent microfluidic experiments Levaché and Bartolo (2014); Wexler et al. (2015); Zhao et al. (2016); Trojer et al. (2015), and simplified 2D simulations Holtzman and Segre (2015) have revisited this framework, and established the dramatic impact of wetting properties on liquid-displacement patterns. Nonetheless, we still lack a comprehensive picture of forced-imbibition patterns, and of the underlying interfacial dynamics at the pore scale.

In this letter, we contribute to resolving this situation. Building on model microfluidic experiments, we evidence four classes of imbibition regimes yielding non-monotonic variations of the recovery rate with the driving strength. We show that these patterns stem from four different microscopic dynamics separated by two film-entrainment transitions and one interfacial instability. Finally, we demonstrate and explain the long-time coarsening of two classes of imbibition patterns.

Our experiment consists in injecting an aqueous liquid in a hydrophilic porous medium filled with viscous oil. Details about the materials and methods are provided in a Supplementary Document supp . In brief, we use a simplified 2D geometry consisting of a square lattice of posts forming interconnected channels of dimensions $200\,\rm\mu m\times 80\,\mu m\times 70\,\mu m,$ see in Figs. 1a. The patterns are imprinted on microfluidic stickers bonded to quartz slides, and made hydrophilic by in situ exposure to deep-UV light Bartolo et al. (2008); Levache et al. (2012). We first fill the channels with a mixture of silicon oils, which we then displace by injecting an aqueous solution of dye and SDS ($1\,\rm wt\%$). The surface tension between the two liquids does not depend on the oil molecular weight: $\gamma=13\pm 2\,\rm mN/m$, and we find advancing contact angles $\theta_{\rm A}=30\pm 2^{\circ}$, and $\theta_{\rm A}=20\pm 2^{\circ}$, respectively on the treated quartz and sticker surface. The water solution is injected at constant flow rate $Q$, and the large-scale morphology of the imbibition patterns is observed with a 14-bit, CCD-camera with a spatial resolution of $11\,\rm\mu m/pixel$. We gain information about the interfacial morphology in the third dimension by locally converting the transmitted-light intensity into the $z$-averaged thickness of the water patterns with a relative precision of $1\,\rm\mu m$ Levaché and Bartolo (2014). In addition, 3D reconstructions of the liquid interfaces at the pore scale are achieved in separate experiments using confocal microscopy.

Let us first discuss the impact of flow rate on the pattern morphologies, while keeping constant both the wetting contact angles and the viscosity ratio, here $M\equiv\eta_{\rm oil}/\eta_{\rm water}=560$. Without any a priori knowledge of the microscopic dynamics, a natural dimensionless control parameter is given by the capillary number ${Ca}=\eta_{\rm water}Q/(\gamma S)$, where $S$ is the cross-section area of the microchannels separating the posts. Increasing $Ca$, we observe four different patterns exemplified in Fig. 1a:

– Regime A: When $Ca<2.3\times 10^{-5}$, the patterns are typical of capillary imbibition Lenormand (1990). A single macroscopic finger proceeds at constant speed, and its faceted edges reflect the square geometry of the underlying periodic lattice Courbin et al. (2007); Cubaud and Fermigier (2001).

– Regime B: Increasing the flow rate, $2.3\times 10^{-5}<Ca<9.1\times 10^{-5}$, we still observe the growth of a single yet narrower finger. However, the dynamics of the water-oil interface does not reduce to the motion of a stable meniscus. Instead, the spatial fluctuations of the transmitted light indicate that thin water films form and propagate throughout the lattice.

– Regime C: Further increasing the flow rate, $9.1\times 10^{-5}<Ca<1.1\times 10^{-3}$, the patterns evolve into even thinner films forming river networks with multiple narrow branches and reconnections. We emphasize that the existence of closed loops distinguishes this geometry from typical viscous fingering motifs, and from all instances of Laplacian growth patterns Sahimi (1993).

– Regime D: Finally when $Ca>1.1\times 10^{-3}$, water films self-organize into non-intersecting branched patterns. They are reminiscent of dendritic growth and drainage dynamics Chen and Wilkinson (1985); Ben-Jacob et al. (1985); Budek et al. (2015). Pushing the flow rate to even higher values yields the fragmentation of the oil-water interfaces, and to the formation of both direct and inverse emulsions.

These four dynamical patterns have a clear signature on four global observables. In Fig. 1b we show the variations of the water saturation $S_{\rm w}$ defined as the volume fraction of water, or equivalently as the fraction of oil extracted from the porous medium, when the water front reaches the end of the device. $S_{\rm w}$ undergoes non-monotonic variations with $Ca$ and reaches a minimum exactly at the transition between regimes B and C. Surprisingly, at small flow rates pushing faster hinders oil extraction. Ignoring the heterogeneities of the water-film thickness, the area fraction of the porous medium in contact with water, $A_{\rm w}$, also varies non-monotonically with $Ca$. $A_{\rm w}$ reaches a minimum when a river network emerges (regime C). Above this transition, the increase of $A_{\rm w}$ echoes the widening of the rivers.

The transitions toward the two other regimes is clearly visible when inspecting the distribution of the water-film thickness, $P(h)$, and the non-monotonic variations of $\langle h\rangle$ in Figs. 1d and 1e. At low $Ca$ (regime A), the distribution is bimodal: the global maximum is located at $h=H$, the mean height of the channels. A smaller peak also exists at $h=0$ and trivially corresponds to the boundaries of the imbibition pattern. The transition from regime A to regime B coincides with the inversion of the magnitude of the peaks at $h=0$ and $h=H$ in Fig. 1d. The ratio between the typical and the average film thickness drops down to zero at this transition, Fig. 1e. The second transition from regimes B to C is also visible on $P(h)$. When the river network forms, the thickness distribution is peaked at $h=0$ and plateaus at intermediate thicknesses. Finally, the increase of the mean thickness $\langle h\rangle$ in Fig. 1e signals the onset of regime D.

Three comments are in order. Firstly, the abrupt changes in these macroscopic observables confirm that the four imbibition regimes correspond to four distinct physical processes. Secondly, as demonstrated in a Supplementary Document supp , the same transitions from regimes A to B, and B to C are also observed keeping $Ca$ constant and increasing the viscosity ratio $M$. Consistently, these transitions have the same signature on the film-thickness distribution. Finally, this classification in four regimes extends that proposed in Zhao et al. (2016) where disorder and the radial variations of the local flow rate are likely to blur the transitions between regimes A, B, and C.

We now single out the origin of these four regimes by investigating the imbibition dynamics at the pore scale illustrated in Fig. 2a.

– Regime A: The corresponding snapshots provide a prototypical illustration of capillary imbibition. The menisci proceed uniformly through the channels, merge at the vertices and grow until touching the next post. Then, they split into two interfaces invading the adjacent channels.

– Regime B and C: The corresponding snapshot sequences reveal a marked change in the microscopic dynamics. In both regimes, as the contact line reaches a corner formed by a square post and a confining wall, the perimeter of the post is coated leaving a significant fraction of oil immobile in the channels. The coating layer then growths, contacts adjacent posts, and therefore discretely progresses through the lattice. This behavior is consistent with the observations of Zhao et al. (2016). Confocal imaging is however necessary to complete this scenario. The confocal pictures in Fig. 2b are taken $5\,\rm\mu m$ above the bottom surface of the channel (sticker surface). The progression of the water-oil interface in this plane is very similar to the capillary-imbibition sequences illustrating regime A in Fig. 2a. Even though the contact line accelerates when reaching a corner, it also freely advances away from the posts along the direction of the microchannels. The advancing of the water film does not require any additional Laplace-pressure pumping from the negatively curved gutters at the base of the wetted posts Concus and Finn (1969). A possible explanation could be the existence of a rough structure further driving the imbibition process at the sub-optical scales. This hypothesis is ruled out by Atomic Force Microscopy imaging, see Supplementary Informations. Both the quartz and sticker surfaces are smooth down to the $10\,\rm nm$ scale supp . Therefore the only possible reason for the formation of a thin film in regimes B and C is a dynamical wetting transition Snoeijer and Andreotti (2013); Levaché and Bartolo (2014). The confocal image in the $xz$ plane indeed shows that water films are entrained only on the bottom surface and fully cover it as they grow, Fig. 2c. Increasing the capillary number, the contact line on the sticker surface entrains the aqueous liquid at higher speed than the average liquid front. The localization of the entrainment on the sticker surface is very likely to stem from the lower advancing contact angles value. Liquid entrainment prevents the steady motion of a stable meniscus in agreement with observations in simpler Hele-Shaw geometries Levaché and Bartolo (2014).

The dynamics of the water front are identical in regimes B and C. However an interfacial instability distinguishes these two regimes, characterized by different film thickness distributions in Fig. 1d. In both cases, the incomplete displacement of the non-wetting fluid results in the formation of hemicylindrical interfaces, Fig. 2c. In regime B, these interfaces are unstable to the Rayleigh-Plateau instability which breaks the oil cylinders into droplets receding to the vertices of the lattice, Fig. 2a and  2d. Conversely, increasing the flow rate to regime C, the Rayleigh-Plateau instability is suppressed. In principle the stabilization of the oil hemicylinders could be either due to local pinning of the contact line Gau et al. (1999); Speth and Lauga (2009), or by the convection of the interface fluctuations as in quickly stretched liquid filaments Tomotika (1936), and in coflowing liquid streams Eggers and Villermaux (2008). In order to single out the origin of stabilization, we performed additional experiments deep in regime C ($Ca=2.2\times 10^{-4}$) followed by an abrupt stop of the water flow illustrated in fig. 2e. As the flow stops, the interfaces become unstable. The snap-off events and the resulting droplet pattern typical of regime B are recovered. This observation unambiguously confirms that the oil interface is dynamically stabilized by the flows in the water films.

– Regime D: At the highest capillary numbers, the local dynamics undergoes a strong qualitative change. Given the speed of the water front, confocal imaging is not possible anymore. However, it is already clear from the high-resolution snapshots in Fig. 2a that the apparent contact angle between the aqueous liquid and the solid walls is reversed: the porous medium becomes effectively hydrophobic. We explain this reversal by another wetting transition first reported for colloidal liquids with ultra-low surface tensions Setu et al. (2013). In this high-$Ca$ regime the destabilization of the interface occurs at the center of the channels where a finger of the aqueous liquid grows and leaves an immobile oil layer on the solid surfaces. In turn, the lubricated motion of the water fingers drastically changes the exploration of the post lattice. The imbibition patterns become analogous to dendritic growth as observed in conventional anisotropic drainage experiments Chen and Wilkinson (1985); Ben-Jacob et al. (1985); Budek et al. (2015).

In line with earlier studies, until now we have limited our discussion to pattern formation prior to the percolation of the driving fluid through the lattice. This limitation is unimportant when liquid mobilization is complete at the pore scale (i.e. in drainage experiments and in regime A). The driving fluid then merely flows through the percolating pattern which does not age Lenormand (1990). In contrast, we now demonstrate that the thin-film patterns undergo significant structural changes over long time scales. Fig. 3a show the evolution of the film-thickness distribution over hours of continuous injection in regime C ($Ca=5\times 10^{-4}$). From now on we measure time in term of the volume of injected water normalized by the overall volume of the porous medium $V_{\rm inj}=Qt/V_{\rm Total}$. In the example of Fig. 3a, $V_{\rm inj}=0.27$ at percolation, and the pattern forms a river network. Further increasing the injected volume, the oil-water interfaces are not destabilized by the Rayleigh-Plateau instability. Instead the wetting pattern slowly evolves into compact islands of pores fully filled with water connected by a sea of thin films, Fig. 3a. The islands grow at the expense of the thin films and eventually coalesce into a percolating cluster of pores where oil has been fully displaced. More quantitatively, see Figs. 3b and 3b inset, the water-thickness distribution evolves from a decreasing function with a thickness plateau comprised between 10 and 50 microns, to a bimodal distribution peaked both at $h=0$ (pattern edges and residual thin films) and $h=H=70\,\rm\mu m$ (filled channels). This slow dynamics has a clear microscopic origin. As sketched in Fig. 3c, the channels where the water films are thin correspond to smaller curvature of the oil-water interface, and hence to smaller Laplace pressure. Therefore Laplace pressure gradients cause the oil to flow out of regions where the water-films are thicker until total water saturation. This coarsening process relies on the capillary drive of the more viscous liquids which de facto results in a slower dynamics than the forced propagation of the water films. The collective relaxation of the initial distribution of $h$ further increases the volume of injected water needed to reach a stationary pattern. The typical water volume, $V_{\rm inj}^{\star}$ required to reach the asymptotic steady state is measured from the variations of the the water saturation $S_{\rm w}$, Fig. 3d inset. We find in Fig. 3d that $V_{\rm inj}^{\star}$ monotonically increases with $Ca$: the higher the flow rate the higher the water volume required to achieve maximal saturation and form stationary patterns. However, unlike its value at percolation, the asymptotic value of $S_{\rm w}^{\star}$ monotonically increases with $Ca$: pushing harder mobilizes more oil, see Fig. 3e.

Together the variations of $V_{\rm inj}^{\star}$, and $S_{\rm w}^{\star}$ are useful guidelines to design optimal oil-recovery strategies. Decontamination processes requiring the mobilization of the maximal amount of oil, at any cost, would be optimized at the highest possible injection rate. Conversely, commercial oil extraction would require a trade-off, e.g. maximizing the $S_{\rm w}^{\star}/V_{\rm inj}^{\star}$ ratio, which would correspond to minimizing the water flow rate, Fig. 3e (light circles). From a more fundamental perspective, we hope our findings will stimulate experimental and theoretical investigation to quantitatively elucidate the interfacial instabilities underlying the fourfold dynamical scenarios established in this letter.