Capillary equilibrium of bubbles in porous media

Significance

We show how pore geometry modulates the capillary equilibrium states of a trapped bubble inside a porous medium using a simple conceptual model. The model explains recently published data on bubbles trapped in porous rock samples. We explain the counterintuitive observation that bubbles with large surface areas in the subsurface can remain thermodynamically stable for geologically long times. The stability is the result of a modified relationship between a bubble’s surface free energy and volume due to geometric confinement. The implications are relevant to applications ranging from petroleum recovery, to CO₂ sequestration, to groundwater oxygen supply, and to fuel-cell water management.

Abstract

In geologic, biologic, and engineering porous media, bubbles (or droplets, ganglia) emerge in the aftermath of flow, phase change, or chemical reactions, where capillary equilibrium of bubbles significantly impacts the hydraulic, transport, and reactive processes. There has previously been great progress in general understanding of capillarity in porous media, but specific investigation into bubbles is lacking. Here, we propose a conceptual model of a bubble’s capillary equilibrium associated with free energy inside a porous medium. We quantify the multistability and hysteretic behaviors of a bubble induced by multiple state variables and study the impacts of pore geometry and wettability. Surprisingly, our model provides a compact explanation of counterintuitive observations that bubble populations within porous media can be thermodynamically stable despite their large specific area by analyzing the relationship between free energy and bubble volume. This work provides a perspective for understanding dispersed fluids in porous media that is relevant to CO₂ sequestration, petroleum recovery, and fuel cells, among other applications.

Bubbles are generated, trapped, and mobilized within porous media as a consequence of incomplete fluid–fluid displacements (1, 2), phase changes (3, 4), chemical and biochemical reactions (5, 6), or injection of emulsified fluids and foams (7, 8). Compared to continuously connected phases, the behavior of dispersed bubbles, or ganglia, are far less understood. In particular, the thermodynamic stability of bubbles, despite their large specific surface area, remains a puzzle. The difficulty comes from the fact that each bubble can attain a volume (V), topology, and capillary pressure (P_c) that is distinct from other bubbles in the medium (9). The variability poses challenges to understanding the transport and trapping mechanisms of bubbles in geologic CO₂ sequestration (10, 11), hydrocarbon recovery (12, 13), fuel cell water management (14, 15), and vadose zone oxygen supply (16, 17).

The dominant factor controlling a bubble’s behavior in a porous medium is capillarity, which is typically much larger than either viscous, gravitational, or inertial forces (18, 19). Capillary pressure, P_c, allows a closure relationship for two-phase Darcy Eqs. (20–22) and influences thermodynamic properties like phase partition (23). Capillary pressure is derived from the Young–Laplace equation P_c = γκ, where γ is the interfacial tension and κ is the surface curvature. In an open space without obstacles, a bubble spontaneously evolves into a sphere to minimize its total interfacial energy. Thus, P_c is a continuous and monotonically decreasing function of V (Fig. 1A). However, in a porous medium, bubble’s P_c–V relation is more complicated due to the geometric confinement imposed by the porous structure and topological evolution (24). A bubble can no longer remain spherical as it grows in size but must conform to the geometry of the pore(s) it occupies. Therefore, a bubble’s P_c is a function of not only its volume and interfacial tension but also its topology as dictated by the confining porous medium, as confirmed by recent laboratory experiments and numerical simulations (25–29). The mere presence of confinement therefore engenders a host of phenomena that would otherwise be absent, such as capillary trapping (30, 31), anticoarsening of bubble populations (32, 33), and complex ganglion dynamics (11, 18). Furthermore, theoretical studies in mathematical topology (28, 34, 35) prove that immiscible fluids can be fully characterized by d+1 Minkowski functionals, where d is the problem dimension. Such characterizations remove the path-dependent (or hysteretic) behavior common to these systems (34, 35).

Fig. 1.

(A) Spherical bubbles inside a bulk fluid. (B) Micromodel observations show that bubbles are nonspherical in porous media and may occupy multiple pores. This image is from SI Appendix, Movie S1. (C) A 2D porous medium comprised of an ordered array of identical circular grains. A bubble occupying multiple pores including a zoom-in to a portion of it. (D) Illustration of the full state. (E) Illustration of the critical state. (F) Decomposition of a bubble into four distinct parts: minor arc menisci shown by dark blue cap-shaped regions, throats shown by light blue diamond-shaped regions, inner bulk bodies shown by red star-shaped regions, and major arc menisci shown by dark green cap-shaped regions.

Recent developments in microfluidics and micro computed tomography imaging allow detailed pore-scale visualizations of fluids inside porous media, including the morphology of bubbles and ganglia (25, 36–39). Garing et al. (25) experimentally measured the equilibrium capillary pressure of trapped air bubbles inside sandstone and bead-pack samples. They found that, unlike bubbles within a bulk fluid, the P_c of trapped bubbles shows no clear dependence on V and seems to fall within a bounded interval, except for vanishingly small V. Xu et al. (40) proposed an empirical correlation for the P_c trapped bubbles based on microfluidic observations. In this correlation, as V increases, P_c decreases until a minimum is reached and then increases linearly. In the first stage, the bubble is unconfined, whereas in the second, it is reshaped by the surrounding solid walls. The proposed correlation, however, is only valid for bubbles in a single pore and not bubbles that span multiple pores. The latter seems to be rather common in nature as evidenced by recent direct observations (Fig. 1B) (2, 25).

Here, we propose a simple conceptual model to describe the equilibrium states of a bubble with arbitrary size trapped inside a porous medium. The model accounts for the bubble’s morphology, the geometry of the solid matrix, and the wettability between the two. We derive all metastable configurations of the bubble analytically and highlight the thermodynamic states the bubble assumes when it is static, growing, or shrinking. We also show that the relationship between surface free energy (F) and volume (V) of large bubbles is approximately linear, which explains the previously counterintuitive observation that such bubbles are thermodynamically stable despite having large surface areas. Our work provides a step toward understanding the capillary state, stability, and evolution of dispersed immiscible fluids in porous media.

Conceptual Model

Consider a static bubble inside a two-dimensional (2D) porous medium consisting of a rectangular array of identical circular grains (Fig. 1C). The bubble is assumed to be perfectly nonwetting (i.e., contact angle is $\theta \hspace{0.17em}=\hspace{0.17em}0$). The impact of noncircular grain shapes and nonzero contact angles will be discussed later. We call the void space enclosed between four adjacent grains a pore and the narrow constriction that connects two neighboring pores a throat. The bubble can reside either within one pore or multiple pores. The number of pores occupied by the bubble is referred to as its pore occupancy, n. When n > 1, the bubble may have multiple branches that meander and occupy pores in different directions. Moreover, the branches can self-intersect, leaving isolated islands of the wetting phase stranded in the middle (Fig. 1F). To quantify the topology of the bubble, we use the Euler characteristic, $\chi$ (30, 41). The Euler characteristic is defined as χ = β₀ − β₁ + β₂, where β_i are called Betti numbers (26, 28). β₀ denotes the number of disconnected objects, β₁ the number of redundant loops, and β₂ the number of cavities in the object. Because we only analyze one bubble at a time in this work, β₀ = 1, and because we focus on 2D, β₂ = 0. The only Betti number of relevance to us is therefore β₁, which is proportional to the number of self-intersections of the bubble’s branches (i.e., each intersection creates a new hole). In SI Appendix, SI.3.2, we provide a detailed analysis that shows χ and n are constrained by the following interval: $2\sqrt{n}-n\hspace{0.17em}\le \hspace{0.17em}\chi \hspace{0.17em}\le \hspace{0.17em}1$.

In a 2D porous medium, three functionals are needed to be specified in order to constrain the thermodynamic state of a bubble (28, 35). Recent studies (29) show the hysteresis of immiscible two-fluid systems to be an artifact of omitting one of these functionals. Here, we choose V, n, and χ as the three parameters to describe equilibrium states of a bubble, which are related (as we shall see) to the aforementioned Minkowski functionals.

Geometric parameters relevant to our study are annotated in Fig. 1C. We denote the radius of the grains by R₁, the half-distance between the centers of two adjacent grains by R₀ and the half-width of the throats by H. The volume (i.e., area in a 2D system) of a pore is denoted by V_pore and its area (i.e., or perimeter in 2D) by A_pore. We assume a free surface is constrained by two neighboring solid grains for simplicity.

We assume that the bubble is in static equilibrium and that no external fields are imposed. The 2D bubble’s free surface (not touching the grain surfaces) has therefore a uniform interfacial curvature denoted by κ = 1/r, where r is the radius of curvature. Since snap-off events can’t happen in our 2D (42, 43), the bubble is assumed to remain as one connected piece as it evolves in size, which simplifies our analysis. To analyze and determine the thermodynamic state of the bubble, we divide it into four distinct element types as shown by Fig. 1F. Calculations related to each element type are provided in Methods.

We next describe the capillary pressure (P_c) and surface (or Helmholtz) free energy (F) of the bubble for all metastable states it can assume. The mathematical details for calculating P_c and F are provided in Methods. Here, we simplify the discussion by introducing the following terminology that correspond to two important bubble states:

• •The full state refers to when the bubble attains the maximum volume it can sustain at a given pore occupancy n and Euler characteristic χ (Fig. 1D). The capillary pressure associated with the full state is the capillary entry pressure of the throats, P_{c, throat}.
• •The critical state refers to when the bubble attains the minimum capillary pressure it can sustain at a given n and χ (Fig. 1E). The capillary pressure associated with the critical state is denoted by P_{c, min}, which corresponds to the capillary pressure of the maximum inscribed sphere of a pore.

In the following, we repeatedly refer to the static bubble as “trapped” even though its size may grow or shrink due to ripening or other mass transfer processes. The term “trapped” here means “static” and “in capillary equilibrium.” We shall use the two terms interchangeably.

Results and Discussion

In this section, we first analyze the equilibrium states of a trapped bubble through a series of demonstrative examples. We then focus on the bubble’s surface free energy and show how it depends on the bubble’s volume and morphology. Implications for the stability of dispersed bubble populations are highlighted. We finally generalize our discussion by considering different grain shapes, pore-throat aspect ratios, and contact angles. In all subsequent figures, $R_0/H=6$, which is representative of previous micromodel experiments (33).

Bubble Multistability.

In SI Appendix, SI.1, we derive a closed-form equation for P_c (and F) as a function of V, n, and $\chi$. The model shows that for a bubble of fixed V, multiple metastable states corresponding to different (n, χ) pairs exist. Since n and $\chi$ are integers, the P_c equation depicted by Fig. 2A is a piecewise continuous function that consists of many disconnected segments. Each segment corresponds to a different (n, χ) pair, for which $P_c=\partial F/\partial V$ holds. The discontinuous and highly oscillatory dependence of P_c on V, n, and $\chi$ highlights the need for a different approach to describing the capillary equilibrium of trapped bubbles than the one provided by existing Darcy-scale theories of two-phase flow, which treat nonwetting fluids as continuously connected phases.

Fig. 2.

Relationship between P_c, V, A, n, and $\chi$ for $R_0/H=6$ in an ordered array of circular grains. (A) A schematic of the relation P_c = f(V, n, χ). Each segment corresponds to a different (n, χ) pair, stretching along the P_c-V plane. Larger V entails larger n and thus a higher possibility of self-intersection leading to more negative χ. (B) A projection of P_c = f(V, n, χ = 1) on the P_c-V plane and V axis is in log scale. The Inset is a zoom-in at large V. The solid black curve corresponds to n = 24, and the solid gray curves correspond to other values of n. (C) A projection of P_c = f(V, n = 24, χ) on the P_c-V plane. More negative χ corresponds to steeper P_c-V curve segments. In C, D, and E, the solid black curve indicates the maximum $\chi =1$ and the solid red curve the minimum $\chi =\left\{2\sqrt{n}\hspace{0.17em}-n\right\}$. (D) Dimensionless specific interfacial area (Γ_d) versus dimensionless bubble volume. The solid blue curve corresponds to a spherical bubble in open space. The Inset is a zoom-in at large V. (E) Relation between dimensionless total surface area and dimensionless bubble volume. At large V, the area A is proportional to V, while at small V, A is proportional to V^1−1/d. d is the problem dimension (=2 here). The dotted blue lines correspond to the slope 1−1/d and 1.

For n = 1, it is easy to verify that χ = 1. This corresponds to the first segment of P_c-V in Fig. 2A, which decreases from infinity at very small V to a minimum P_{c, min}, followed by an increase toward P_{c, throat}. For n > 1, all P_c-V segments are bounded within the interval [P_{c, min}, P_{c, throat}] regardless of n or χ. The result is consistent with previous laboratory experiments in homogeneous bead packs (25). There, the authors found that the P_c of trapped bubbles fall within a narrow interval, where the correlation to V is weak except at very small V.

Fig. 2B shows the P_c-V segments associated with χ = 1 but different values of n. Fig. 2C shows the P_c-V segments associated with n = 24 and all possible values of χ. Here, P_c changes more rapidly with V for smaller values of χ (i.e., corresponding to more self-intersections). We also note that the number of metastable states increases with V, as shown by Fig. 2 A and B.

Surface Free Energy and Bubble Population Stability.

Here, we analyze the thermodynamic stability of a population of trapped bubbles by considering their surface free energy (F) under isothermal conditions and no external fields (9). Much like P_c, F is also a function of V, n, and $\chi$ in 2D porous media. We choose the reference energy as that of a fully saturated wetting phase without any bubbles, to which we assign the value zero.

The stability of a population of isolated bubbles is governed by the specific relation between F and V. Assuming $F\propto V^m$ holds for each bubble, let us define the following ratio:

$$\begin{array}{c}\frac{F_{connnected}}{F_{isolated}}\hspace{0.17em}=\hspace{0.17em}\frac{{\left({\displaystyle \sum }\hspace{0.17em}{V}_i\right)}^m}{\left({\displaystyle \sum }\hspace{0.17em}{V}_i^m\right)}\hspace{0.17em}\{\begin{array}{c}<1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}m<1\\ =1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}m=1\\ >1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}m>1\end{array}\end{array},$$ [1]

where V_i is the volume of each isolated bubble. F_isolated denotes the collective free energy of all the isolated bubbles, whereas F_connected denotes the free energy of a hypothetical bubble that has a volume equal to the sum of the volumes of all the isolated bubbles. We see that when the exponent m <1, it is energetically favorable for the bubbles to coalesce into one because the free energy is then lower. This is the case with spherical bubbles in the absence of confinement, where m = 1−1/d such that d is the problem dimension. Such a sublinear F-V scaling is indeed the driving force behind classical Ostwald ripening in bulk fluids that coarsens the bubble population (44).

However, the validity of the above argument for a bubble confined inside a porous medium was heretofore unknown. To address this gap, we examine the dependence of F upon V, n, and $\chi$. Notice first that $F=\gamma A=\gamma \mathrm{\Gamma }V$, where A is the total area of the bubble and Γ = A/V is its specific surface area. In SI Appendix, SI.1.8, we derive closed-form expressions for F and Γ as functions of V, n, and $\chi$. In Fig. 2D, we plot Γ_d = R₀A/V (dimensionless Γ) versus V/V_pore (dimensionless volume) for all possible values of n. Only graphs corresponding to the maximum (black) and minimum (red) values of χ, at each V and n, are shown. The graphs correspond to the black and red lines in Fig. 2C. We see that for very small V, Γ_d is identical to that of an unconfined spherical bubble (blue line). But as V increases further, Γ_d starts to fluctuate within a bounded interval. This interval is analytically derived and given below:

$$\begin{array}{c}{\mathrm{\Gamma }}_d\in \hspace{0.17em}2\pi \cdot \left[\frac{1}{4R_0/R_1-\pi R_1/R_0},\hspace{0.17em}\frac{1}{4+\pi -2\pi R_1/R_0}\right]\end{array}.$$ [2]

The upper bound of this interval corresponds to the full state of a bubble at maximal χ =1, whereas the lower bound corresponds to a state with dΓ/dV = 0 (not the same as critical state) at minimal χ = $2\sqrt{n}-n$. The interval shrinks for smaller values of porosity ($\phi =1-\pi R_1^2/4R_0^2$).

In geologic porous media, R₀/R₁ < 2 (corresponding to φ<0.8) typically holds (45–47). As a result, the interval in Eq. 2 for Γ_d is $2\pi \cdot \left[0.16,\hspace{0.17em}\hspace{0.17em}0.25\right]$. This implies a nearly constant specific surface area at large bubble volumes, which is in agreement with previous experimental observations in homogenous rock samples (Fig. 14 in ref. 39 and Fig. 6 in ref. 48). Since Γ = A/V is almost constant, the relation between A = ΓV and V (for n > 1) must be approximately linear as shown by Fig. 2E. Therefore, m ∼1 in Eq. 1, also in agreement with published data (2, 49–51). Notice that n and $\chi$ play a relatively minor role in the calculated value of Γ and thus F. This is similar to the findings of ref. 29, where some of the geometric state variables of two-fluid systems can be dependent on one another provided a minor error can be tolerated.

The linearity of F-V implies that the coalescence of confined bubbles is energetically less favorable (m ∼ 1) than free bubbles in a bulk fluid (m < 1). Coalescence, therefore, does not necessarily reduce F except at vanishingly small and spherical bubble volumes ($F\propto V^{1-1/d}$). This explains why trapped bubble populations are frequently found within geologic porous media. Despite their large Γ, the bubbles have no affinity to merge and can remain stable over geologic time scales. Now the kinetics of bubble coarsening, or the time needed to reach equilibrium, is a separate but important issue that the authors we will explore in the future.

Capillary Hysteresis during Bubble Growth–Shrinkage.

Bubble growth and shrinkage often accompany physical and chemical processes such as phase change, degas/dissolution, gas-generating/consuming chemical/biochemical reactions, and volumetric expansion due to pressure and temperature changes. Here, we track the changes in bubble morphology, P_c, and F during a growth–shrinkage cycle (Fig. 3A). We show that the process is hysteretic because of the multistability of trapped bubbles in porous media.

Fig. 3.

(A) A growth and shrinkage loop of a bubble. Blue arrows show a bubble growing from small V with n = 1, while red arrows show a bubble shrinking from large V. (B and C) The nonmonotonic and discontinuous P_c-V and F-V paths of growth and shrinkage. Blue corresponds to growth and red to shrinkage. $R_0/H$ = 6 and the domain is a circular disk pack.

Theoretical tools in mathematical topology, such as Minkowski functionals, provide an elegant framework with which to analyze and understand hysteresis of general immiscible multiphase fluid systems (35). Here, we use conceptually simpler tools to examine hysteresis during growth–shrinkage cycles so as to appeal to the physical intuition of the process. For simplicity, we take χ = 1, although a similar analysis holds for other values of χ.

Consider a bubble that grows from an initial volume V = 0. With reference to the blue arrows in Fig. 3A, the following stages govern growth:

• •Free growth (FG) is the period when the bubble is too small to be constricted by the pore geometry. P_c gradually decreases with V in a manner identical to a spherical bubble free from any solid constraints.
• •Constricted growth (CG) starts when the bubble has grown sufficiently large to have touched the four confining grains, as the solid blue bubble in Fig. 1E illustrates. Further growth in V leads to the distortion of the bubble surface to conform to the pore geometry. In CG, P_c increases gradually until the full state is reached, where P_c = P_c,throat.
• •Breakthrough event (BT) occurs right after the bubble reaches the full state. Any further increase in V causes the penetration of one of the end-point menisci through a throat. The entire configuration of the bubble becomes unstable, and a sudden redistribution of mass ensues. BT is manifested by a sudden jump in P_c -V from one curve segment to the next. During a BT, P_c drops from P_{c, throat} to a lower value, and n increases by one (Fig. 1E).
• •Further increase in V leads to a repetition of CG and BT stages. As a result, the P_c-V appears to fluctuate up and down.

We next track a bubble that shrinks from an initially large V and n. With reference to the green and red arrows in Fig. 3A, the following stages govern shrinkage:

• •Constricted shrinkage (CS) is the period when the bubble shrinks without detaching from any of the surrounding grains, as the solid blue bubble in Fig. 1F illustrates. If the bubble is initially at the full state, P_c gradually decreases from P_{c, throat} with all the menisci retracting simultaneously and maintaining a uniform curvature.
• •Flinch event (FC) occurs right after the bubble reaches the critical state. Any further decrease in V leads to an unstable configuration that cannot be sustained by the current n. Hence, the bubble retracts inward to rearrange its shape. The result of FC is a sudden surge in P_c and a decrease in n by one (Fig. 1F).
• •Further decrease in V leads to a repetition of CS and FC stages. As a result, the P_c-V relation appears to fluctuate up and down while bounded below by P_{c, min}.
• •Free shrinkage (FS) is the period when the bubble has reduced sufficiently in size to occupy a single pore without touching any of the surrounding grains. The P_c-V relation overlaps with that of free growth (FG).

Fig. 3B shows that the above growth–shrinkage cycle exhibits by a sawtooth P_c-V path. The growth route traces the maximum possible P_c at any given V, while the shrinkage path the minimum possible P_c at any given V. The two paths, therefore, are completely different and show significant hysteresis. The F-V paths corresponding to the above growth–shrinkage cycle are also shown in Fig. 3C, which similarly exhibit hysteresis.

The FG, CG, CS, and FS periods are all reversible. The only sources of irreversibility, and thus hysteresis, are BT and FC events that dissipate energy by rapid reconfiguration. The two events are very similar to the classical pore-scale irreversible processes that govern fluid–fluid displacements in porous media: Haines jumps (rheons) and Melrose events (9, 52, 53). BT and FC events lead to the adjustment of the bubble interfaces in every occupied pore. For example, a BT results in not only the interface advancing in the newly invaded pore but also simultaneous interface recoiling in all previously occupied pores as compensation to keep bubble volume conservative. SI Appendix, Movie S1 visualizes BT events in a micromodel experiment where a bubble grows due to slow depressurization. We note that recent works in mathematical topology provide a promising means of characterizing the observed hysteresis herein with a unique state function (see Discussions under the “ink bottle problem” therein) (27). This pore-scale picture of bubble capillary hysteresis is aligned with classical pictures for fluid–fluid displacement hysteresis at pore scale (9, 24) and mesoscale (54).

As shown by Fig. 3B, the oscillations in P_c attenuate as V increases and disappear altogether at the limit $V\to \infty$. At large V, the growth path asymptotes to P_{c, throat} and the shrinkage path to P_{c, min}. More importantly, discontinuities in both paths gradually disappear, and the P_c-V curves become continuous. Since an infinitely large bubble is essentially a continuous phase, we see indications that both P_c-V paths seem to approximate drainage/imbibition P_c-S curves (where S is saturation) used in classical Darcy-scale theories of fluid displacement. An exact characterization of these limits, however, requires additional tools outside the scope of this paper. Recent advances in mathematical topology (29) provide a promising avenue.

Impacts of Geometry and Wettability.

To determine the generality of the results discussed thus far, we examine the qualitative impacts of grain shape, throat-to-grain ratio, and contact angle (<π/2) on the equilibrium states of a trapped bubble.

In SI Appendix, SI.4, we show that none of the geometric factors or wettability qualitatively alter our previous conclusions about the multistability of bubbles, linear scaling between free energy and volume, and hysteresis during growth–shrinkage cycles, at least for the parameter ranges considered. They do, however, have an important quantitative impact on, for example, the amplitude and wavelength of P_c-V and F-V oscillations. More specifically, larger throat-to-grain ratios result in weaker fluctuations of P_c and F; different grain shapes change each P_c-V segment quantitatively; and nonzero contact angles sometimes lead to negative P_{c, min}.

Implications and Limitations.

Our conceptual model is distinct from classical capillary pressure models at the Darcy scale. The latter assume that each phase is hydrodynamically connected throughout the sample (13, 19) and that fluid–fluid interfaces are always at capillary equilibrium. However, bubbles are disconnected entities with possibly different thermodynamic states.

Our conceptual model is admittedly limited by several assumptions. These include the following: 1) the roles of heterogeneity and polydispersity in pore sizes are neglected that will likely have significant impact on bubble growth, shrinkage, and hysteresis (54, 55); 2) All external fields such as gravity, background flow, and concentration gradients are neglected. The last item can induce Marangoni effects at the bubble surface driving the system away from the metastable states discussed herein and toward new ones (56); 3) The model does not apply to very tight packings of bubbles, like foams, as that requires keeping track of the separating lamella that delineate the boundary of each bubble; and 4) Perhaps most importantly, our model is 2D. In three-dimensional (3D), interfaces consist of two principle curvatures, with possibly opposite signs (i.e., be saddle points) (57) that may lead to new metastable states. The reason for this assumption was that closed-form expressions seemed possible, at least to us, only in 2D but immensely difficult in 3D. While no conclusive claims can yet be made about capillary equilibria of 3D bubbles, we suspect the qualitative observations made herein remain intact. A quantitatively rigorous analysis in 3D is possible with computational techniques such as level set (58, 59) and lattice Boltzmann (60–62) methods, which pose an obvious next step for future analysis.

Conclusion

We developed a 2D conceptual model that describes the equilibrium capillary pressure (P_c) and surface free energy (F) of a static bubble trapped inside a porous medium. Closed-form equations are derived for both quantities as functions of bubble volume V, pore occupancy n, and Euler characteristic χ. The conceptual model revealed that bubbles have fundamentally different capillary properties than those in a bulk fluid.

For a 2D ordered disk pack, a typical P_c-V curve consists of many piecewise continuous segments, each of which corresponding to a different (n, χ) pair. For n > 1, all segments fall within the interval [P_{c, min}, P_{c, throat}], which itself depends on the pore geometry. The free energy (or specific surface area) of the bubble falls, surprisingly, within a similar but narrower interval. The implication is an approximately linear relationship between F and V. The linearity means that the merger of isolated bubbles is not necessarily favorable energetically, which explains the thermodynamic stability of large ganglia observed in a geologic porous media. This model explains the observations from many previous experimental observations (25–28, 33, 39, 48)

We also compute P_c and F during a growth–shrinkage cycle of a trapped bubble and find that both quantities are highly oscillatory and discontinuous functions of V. Moreover, the growth and shrinkage paths do not overlap. The hysteresis is attributed to energy dissipations during sudden and thus irreversible breakthrough and flinch events following changes in pore occupancy. The oscillations in both P_c and F disappear as V approaches infinity, indicating some sort of “convergence” toward a macroscopic state.

Though our model is simple, we believe that it serves as a useful entry point toward predictive macroscopic theories of bubble mobilization, trapping, and ripening in porous media. Such theories are needed to answer important questions related to hydrocarbon migration, gas-hydrate formation, CO₂ sequestration, and fuel-cell design.

Methods

Correlate P_c to V, n, and χ.

We divide the bubble into four distinct element types as shown in Fig. 1F: (a) minor arc menisci of volume V_minor, (b) throats of volume V_throat, (c) inner regions of the pore-body of volume V_body, and (d) major arc menisci of volume V_major. The V_minor, V_throat, V_body, and V_major are all functions of the free surface curvature radius, r, for a given matrix geometry and wettability. The total volume of the bubble can thus be written as

$$\begin{array}{c}V=n_1\cdot V_{minor}\left(r\right)\hspace{0.17em}+\hspace{0.17em}{n}_2\cdot V_{throat}\left(r\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{n}_3\cdot V_{body}\left(r\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{n}_4\cdot V_{major}\left(r\right)\end{array},$$ [3]

where n_i denotes the total number of each element in the bubble. To obtain n_i, we solve the following system of equations, formulated by imposing a set of morphological constraints:

$$\begin{array}{c}\{\begin{array}{c}\begin{array}{c}4n_3=n_1+2n_2-n_4\\ n_3+n_4=n\\ n_3+n_4-n_2=\chi \end{array}\end{array}\end{array}.$$ [4]

Expressions for n_i, V_i, and A_i are derived in SI Appendix, SI.1. The first row of Eq. 4 can be interpreted as an accounting of all the pores’ open faces occupied by the bubble. The second row of Eq. 4 defines the pore occupancy, n. The third row defines the Euler characteristic, $\chi$ (30, 41). We note that, as long as $0\le n_4\le \min \left\{\text{n},2\left(\text{n}+\text{χ}\right)/3\right\}$, there could be multiple solutions to Eq. 4. While all solutions ensure the bubble is at equilibrium (i.e., all interfaces share the same curvature), they do not necessarily represent metastable states. At a metastable state, the morphology of a bubble should spontaneously recover from an infinitely small perturbation of the bubble’s position. By contrast, a bubble that is not at a metastable state will amplify the tinniest of perturbations until the bubble finds a metastable state. See Fig. 4, for example. After a small perturbation, the solid red bubble will change to the dashed black bubble and become unstable. We therefore need more constraints to exclude unstable configurations.

Fig. 4.

An example of a bubble with an unstable morphology. The solid red line is the bubble at equilibrium before perturbation, while the black dashed line is the bubble after perturbation. Th perturbation amplifies while V remains unchanged.

Details of the analysis of metastable bubble morphology are given in SI Appendix, SI.2. The conclusion is that n₄ can only be 0 or 1. Specifically, when $n_4=0$, the bubble is always stable and $\partial V/\partial r<0$ holds, but when $n_4=1$, the bubble is stable only when $\partial V/\partial r>0$. This criterion is derived from general physical principles and is valid regardless of grain shape, contact angle, and dimension. In short, the constraint to ensure metastability is as follows:

$$\begin{array}{c}{n}_4=0\hspace{0.17em}{{\displaystyle \cup }}^{\text{}}\hspace{0.17em}\left\{1\hspace{0.17em}{{\displaystyle \cap }}^{\text{}}\hspace{0.17em}\partial V/\partial r>0\right\}\end{array}.$$ [5]

In (SI Appendix, SI.4.2), we derive explicit expressions for V_minor(r), V_throat(r), V_body(r), and V_major(r) for different grain shapes and wettability. Combined with Eqs. 3–5 and the mathematical derivation in SI Appendix, SI.3.1, we are able to obtain a closed-form equation for $r=\hspace{0.17em}f\left(V,\hspace{0.17em}n,\hspace{0.17em}\chi \right)$. P_c is then readily obtained via P_c = γ/r. Because r is a function of V, n, and χ, so is P_c.

F versus V, n, and χ.

For a given n and χ, dF/dV = P_c (9). Since P_c is a function of V, n, and χ obtained from the previous section, F can also be written as a function of V, n, and χ. We set the reference free energy as that of a fully saturated liquid with no bubbles.

To determine F of a static equilibrated bubble, we also look into the four distinct element types as shown in Fig. 1F and denote the surface area of 1) minor arc menisci of surface area A_minor, 2) throats of surface area A_throat, 3) inner regions of the pore-body of surface area A_body, and 4) major arc menisci of surface area A_major. The A_minor, A_throat, A_body, and A_major are all functions of r for a given matrix geometry and wettability. The total surface area of the bubble can thus be written as the following:

$$\begin{array}{c}F=n_1\cdot {\gamma }_1{A}_{minor}\left(r\right)\hspace{0.17em}+\hspace{0.17em}{n}_2\cdot {\gamma }_2{A}_{throat}\left(r\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{n}_3\cdot {\gamma }_2{A}_{body}\left(r\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{n}_4\cdot {\gamma }_1{A}_{major}\left(r\right)\end{array},$$ [6]

where γ₁ is the interfacial tension at the free surface and γ₂ is the interfacial tension at a constrained surface. For a completely nonwetting bubble, ${\gamma }_1={\gamma }_2={\gamma }_{gl}$; for a nonzero contact angle (=$\theta$) case, ${\gamma }_1={\gamma }_{gl}$, and ${\gamma }_2={\gamma }_{gs}-{\gamma }_{ls}={\gamma }_{gl}\cos \theta$ is the difference between solid–gas interfacial tension and solid–liquid interfacial tension. In SI Appendix, SI.4.2, we derive explicit expressions for A_minor(r), A_throat(r), A_body(r), and A_major(r) for different grain shapes and wettability. We then obtain a functional form of $F=\hspace{0.17em}g\left(V,\hspace{0.17em}n,\hspace{0.17em}\chi \right)$.

Data Availability

All study data are included in the article and/or supporting information.

Change History

May 03, 2021: Equation 1 has been updated.