Capillary jumps of ﬂuid-ﬂuid fronts across an elementary constriction in a model open fracture

We study experimentally the quasistatic displacement of an oil-air front across a localized constriction in a model open fracture. The front experiences capillary jumps at one end of the constriction in both imbibition and drainage, leading to a microscale pressure-saturation hysteresis cycle. At the other end the front is reversibly pinned. A condition of local mechanical equilibrium between the restoring elasticity of the front and the distortion produced by the local change in aperture captures all we measure quantitatively, in terms of material and geometrical properties only.


I. INTRODUCTION
Two-phase flows in porous and fractured media proceed by the fluid invasion or withdrawal of individual pores and openings [1], which ultimately control the large-scale behavior of environmentally relevant processes such as secondary oil recovery, soil irrigation, and CO 2 sequestration [2,3]. Observations in micromodels and natural porous media using state-of-the-art imaging techniques, with optimized temporal and spatial resolution [4][5][6][7][8][9][10][11][12], show that local bursts at individual pores may give rise cooperatively to spatially extended, rapid fluid invasions known as Haines jumps [13]. The complexity of porous and fractured materials, however, makes the analysis of these rapid events highly challenging. Thus, even though Haines jumps are acknowledged to be the leading contribution to the energy dissipated in immiscible fluid-fluid displacements [6,14,15], their impact on imbibition-drainage hysteresis is still poorly understood and remains a subject of active research [5,[16][17][18][19].
Immiscible fluid-fluid displacements through fractured and porous media are controlled at continuum scales by the relative importance of viscous to capillary forces and by the displacement direction. This leads to distinguishing drainage (the displacement direction in which the less wetting fluid displaces the most wetting fluid) from imbibition (the opposite) [1,2]. The viscosity contrast also plays a very important role: it makes the front unstable against viscous fingering when the displacing fluid is the less viscous, and stable in the opposite direction [20,21]. These two governing effects, combined, give rise to a very rich phenomenology of two-phase patterns and interfacial dynamics [22][23][24][25][26][27][28][29][30][31].
No less important, but less well understood, is the physics of local fluid motions driven by capillary forces at the scale of individual pores and throats [32][33][34][35][36]. Models based on capillaries of sharply varying cross section (ink bottles) show that fluid motion between consecutive stable interfacial configurations can be either smooth and reversible or abrupt and irreversible, depending on cross-section gradients and wettability contrast [5,13,19,37,38]. Here we extend these ideas to the motion of an oil-air interface in a Hele-Shaw cell with a localized constriction or expansion. In contrast to the ink-bottle models, the interface here is strongly confined in one lateral dimension and spatially extended in the other. The motion in this case mimics the displacement of a two-phase fluid front in an open fracture of variable aperture. The stability of the displacement is determined by the sign of the difference Bo − Ca, where Ca and Bo are the capillary and Bond numbers, i.e., the ratio of viscous to capillary forces and of gravitational to capillary forces, respectively [39]. By tilting the cell we introduce an effective gravitational field g e such that Bo > 0, and consider quasistatic displacements for which Ca Bo. In this way we prevent viscous fingering in drainage and in imbibition. We aim at understanding the basic physics behind the motion of an extended front induced by a localized constriction or expansion. We will see that a local pressure balance at the interface accurately reproduces all the configurations observed experimentally, without having to account in detail for the complex structure of the extended meniscus. This sets the basis to later understand how localized fluid front motions give rise to large cooperative displacements (Haines jumps) in two-phase fluid displacements in disordered media.

II. EXPERIMENTS
Our experiments were conducted in a Hele-Shaw cell consisting of two closely spaced parallel plates of size 190 × 500 mm 2 (x × y). The top one was a 10-mm-thick glass plate and the bottom one a fiberglass plate with a thin (0.060 ± 0.010 mm) copper layer, sitting on a sturdy steel platform 15 mm thick. The plates were separated by a PVC frame which imposed controlled gap spacings b 0 = 0.46, 0.56, and 0.66 ± 0.02 mm. The steel platform and the cell on top were inclined by angles α = 2 • 21 , 3 • 19 , 4 • 9 , and 5 • 3 ± 2 , so that an effective gravity g e = g sin α prevented the formation of viscous fingers in drainage. Mesa-shaped defects were produced by carefully etching the copper layer of the fiberglass plate using standard printed circuit techniques. Constrictions were thus rectangular copper patches of thickness δb = 0.060 ± 0.010 mm, and expansions were copperfree rectangular depressions of thickness δb = −0.060 ± 0.010 mm. In either case their length was = 60.00 ± 0.02 mm and their width w = 3.00, 9.00, and 27.00 ± 0.02 mm (Fig. 1). In a complete imbibition-drainage cycle the cell was initially filled with air. A silicone oil [surface tension γ = 20.7 mN/m, kinematic viscosity ν = 50 mm 2 /s, density ρ = 998 kg/m 3 , and equilibrium contact angle with all exposed surfaces (glass, copper, and fiberglass) θ 0 • at RT] was removed from or injected to the cell through its left end (inlet), and air entered or left the cell through its right end (outlet) which was open to the atmosphere. The lateral sides were sealed. Oil injection (imbibition) and withdrawal (drainage) were carried out quasistatically by changing the inlet pressure externally in very small increments and allowing the system to equilibrate after each step. Front motion was recorded with a MotionBLITZ cube6 CMOS camera with a spatial resolution of 0.24 mm/pixel. The camera was inclined by the same angle as the cell to avoid a parallax in the recorded images. Interface positions were extracted by image analysis, using classical edge detection techniques.
A sequence of front configurations recorded in an imbibition-drainage cycle across a constriction (δb > 0) is shown in Fig. 2 [movie S1 of the Supplemental Material (SM) [40]). In imbibition [panel (a)] we observed first a capillary instability right at the moment that the initially flat interface reached the left end of the constriction (y d = 0 in Fig. 1). The deformation at the centerline jumped abruptly from η(0) = 0 to a maximum value η c . The Reynolds number corresponding to the fast advancement observed in this abrupt jump was Re 0.4. Despite the large acceleration, inertial effects were negligible because of the fluid confinement and its high viscosity, as is typically the case for Hele-Shaw flows.
Following this forward jump in imbibition, the front moved with pistonlike motion and constant deformation η c until the tip reached the right end of the constriction. The sequence of front 044002-3 configurations that followed was strikingly different from the sequence observed at the left end. The interface progressively flattened, but the tip of the front remained pinned to the constriction until the whole front was flat, and only then did it leave the constriction.
In the reverse displacement [drainage, panel (b)] the front reached first the right end of the constriction. Its central point remained pinned there until the deformation was maximum, practically equal to the maximum deformation η c reached in imbibition. This maximally deformed front then moved unchanged in a pistonlike motion across the whole constriction, until the tip reached the left end. At this stage the front depinned abruptly from the constriction and recovered its original flatness, closing the cycle.
A plot of the centerline deformation η(0) against the distance to the left end of the mesa defect y d in the whole imbibition-drainage cycle [panel (c)] reveals that the two irreversible capillary events at the left end gave rise to a hysteresis cycle [panel (d)]. In contrast, front pinning at the right end was reversible, with identical deformation paths in the two directions [panel (e)]. Remarkably, analogous displacements across an expansion (δb < 0) instead of a constriction displayed identical behavior with left and right ends interchanged (movie S2 of the SM [40]).

III. THEORY
Our theoretical approach builds on the Joanny and de Gennes pioneering approach to the deformation of a contact line in the presence of wettability defects [41][42][43]. In the present case, however, the fluids are confined between the two close plates of a Hele-Shaw cell in one lateral dimension, so that instead of a contact line plus a free surface we have two contact lines tied together by a meniscus [44]. This changes the functional form of the propagator of the deformation, as shown below.
For a viscous fluid in a Hele-Shaw cell inertial forces are negligible, and the flow is governed by the Stokes equation. The gap-averaged, two-dimensional velocity field in each fluid satisfies Darcy's law, where κ is the hydraulic permeability of the cell (κ = b 2 0 /12 for uniform plate spacing b 0 ), μ i and ρ i are the dynamic viscosity and density of fluid i = 1, 2, and p i is the corresponding pressure field. Ordinary liquids are incompressible, ∇ · u i = 0, and the pressure field then obeys Laplace's equation ∇ 2 p i = 0. Gaseous phases have negligible viscosity, μ i 0, and for them the pressure field p i is constant.
In the base state the fluid-fluid interface is flat, h(x, t ) = h 0 , and the two fluids move with the same velocity u 0 i = V ∞ĵ . Taking the pressure origin at the interface (y = h 0 = V ∞ t) the pressure distribution takes the form where g e is the magnitude of the effective gravitational acceleration. If the interface is now slightly distorted from flatness ( Fig. 1) such that h(x, t ) = V ∞ t + η(x), pressure field and velocity field are modified correspondingly: Darcy's law and the condition of fluid incompressibility apply also to these perturbed fields. The local pressure jump across a curved interface is given by the Young-Laplace equation where γ is the interfacial tension between the two fluids and κ , κ ⊥ are in-plane and perpendicular curvatures of the meniscus, respectively. The strongest curvature of the meniscus in a Hele-Shaw cell is κ ⊥ , but in a cell of uniform gap spacing, b 0 , and for a constant (static) contact angle, θ , it can be approximated by the constant value 2 cos θ/b 0 . This approximation disregards the eventuality 044002-4 that the more wetting fluid may leave coating films on the cell plates when it is displaced [45]. Since we assume that |∂η/∂x| 1, we can approximate the in-plane curvature by κ ∂ 2 η/∂x 2 to first order in η. The extra pressure jump across the distorted interface due to the perturbation of the front shape is thus given by to linear order in η. The terms on the right-hand side of the equation account, respectively, for the capillary, viscous, and hydrostatic pressure contributions to the difference in pressure jump. The difference p d (x) may be restoring or destabilizing. When V ∞ < −κ ρg e / μ, where ρ ≡ ρ 1 − ρ 2 and μ ≡ μ 1 − μ 2 , i.e., when Bo − Ca < 0, p d (x) is destabilizing: it leads to the formation of incipient Saffman-Taylor fingers [20,21]. Very quickly, however, the linear approximation fails in this unstable case. Contrarily, for V ∞ > −κ ρg e / μ, i.e., Bo − Ca > 0, p d (x) is restoring: it tends to bring the interface back to the flat configuration η(x) = 0. In this case the interface between two immiscible fluids in a Hele-Shaw cell of constant gap thickness can be regarded as an elastic line, which opposes a local deformation η with a Hookean elastic force kη (Fig. 1). The elastic response originates from the combined action of interfacial tension, viscous damping, and effective gravity.
In order to determine the spring constant associated with the "elastic" restoring force of the interface in the stable case, Bo − Ca > 0, we write Eq. (5) as p d (x) −Lη(x), where L is a linear differential operator. When a local perturbation corresponding to a pressure field p a (x, y) produces an in-plane distortion of the interface y = h(x), the corresponding equilibrium condition reads Since L is a linear differential operator, where G(x) is the propagator of the elastic response of the interface to a localized disturbance. The explicit form of the propagator is obtained by solving LG(x) = δ(x), where δ(x) is Dirac's delta function. In our case this equation reads in Fourier space, where q is the conjugate variable of x andĜ(q) is the Green's function in Fourier space. The result of inverting this equation depends on the sign of 2 g ≡ κγ /( μV ∞ + κ ρg e ). Here 2 g is the (squared) lateral length scale in which capillary forces prevail. For unstable fluid displacements ( 2 g < 0, i.e., Bo − Ca < 0) the propagator in real space takes the form For marginally stable ones ( 2 g = ∞, i.e., Bo − Ca = 0) it takes the form Finally, for stable displacements ( 2 g > 0, i.e., Bo − Ca > 0, the case of interest here) it reads This novel propagator describes the linear "elasticity" of the interface between two immiscible fluids in a Hele-Shaw cell. Remarkably it is qualitatively different from the propagators of the restoring force of a contact line [41] or a fracture front [46,47], which depend logarithmically on 044002-5 the reciprocal of the distance. We write the distorting pressure in the form where p a is the strength of the defect and f (x) a dimensionless function 0 f (x) 1, symmetric around x = 0. Then a Hookean (linear) relation p a = kη(0) holds, where k = 1/ dx G(x ) f (x ) is the corresponding spring constant. For a mesa defect of width w the spring constant reads In the particular case Ca → 0, i.e., for quasistatic displacements, the term of viscous pressure damping in Eq. (5) vanishes. In this limit the elastic response of the interface is provided only by surface tension and gravity. In our experiment the distorting pressure is produced by a constriction (expansion), which locally reduces (increases) by δb the gap thickness of the Hele-Shaw cell. As a result, the curvature of the fluid-fluid meniscus across the gap of the cell changes abruptly from 2 cos θ/b 0 to 2 cos θ/(b 0 − δb) on the defect (Fig. 1). In quasistatic conditions θ can be assumed to take its static (equilibrium) value. This abrupt change in curvature produces a distorting pressure which is responsible for pulling the wetting phase towards the defect when δb > 0 (constriction) and for pushing it away when δb < 0 (expansion). The local change in gap thickness changes locally also the hydraulic permeability of the cell (which in turn would change the viscous pressure drop in the oil phase), but these changes are irrelevant for quasistatic displacements. Similarly, the condition of continuity is always satisfied in quasistatic displacements. From the Hookean relation p a = kη(0), the stationary deformation experienced by the front reads Figure 3 shows that this prediction is in excellent agreement with our experimental results; we emphasize that no fitting parameters are used in this comparison. Further physical insight can be gained from energy considerations [41]. Firstly, the fluid front behaves as an elastic spring whose elastic energy density is u el = kη 2 0 /2, where η 0 ≡ η(0). Secondly, the work done by the localized expansion or constriction in deforming the front is u W = −p a (η 0 − y d ). The energy density of the fluid-fluid interface can then be built as a piecewise function of its distance to the mesa defect, y d , which is controlled externally by the applied pressure: The shape of u(η 0 , y d ) for a constriction (δb > 0) is shown in Fig. 4 for different values of y d . In panel (a) we see that the two irreversible capillary events at the left end of the constriction, in the absence of external fluctuations, arise from the disappearance of the energy barrier separating the equilibrium states η 0 = 0 and η c = p a /k. These two nonequilibrium jumps involved in the drainageimbibition hysteresis cycle of Fig. 2 give rise therefore to large and abrupt dissipations of elastic energy. Indeed, the corresponding elastic waves have been measured as acoustic emissions in the passage of fluid fronts within assemblies of glass beads [48]. Panel (b), in contrast, shows the reason why the front pinning observed at the right end of the constriction is reversible: the stable minima at pinning are located now at the crossing points of the last two pieces of Eq. (15), which are the same in imbibition and drainage, and correspond to smoothly varying deformations of the front. An equivalent analysis in terms of pressures instead of energies is provided in the SM [40].

IV. DISCUSSION AND CONCLUSIONS
We have studied the interfacial dynamics of two-phase fluid displacements across a localized constriction or expansion in a Hele-Shaw cell. Our results highlight the rich phenomenology of the motion of interfaces over defects. Specifically, our problem lies between the ink bottle, where a completely confined interface moves through a tube of variable cross section, and the contact line, where a completely unconfined interface moves over a solid surface. In the present case the interface is extended in one lateral dimension and confined in the other, mimicking the passage of a fluid front  through an open fracture with a localized change in aperture. For stable displacements (Bo − Ca > 0), the lateral extension of the front in one dimension has the advantage that a differential equationthe linearized Saffman-Taylor equation of motion in a Hele-Shaw cell-governs the dynamics, in contrast with the ink-bottle problem. The confinement of the interface between the two plates of the cell-quantified by the local out-of-plane curvature of the meniscus-introduces, on the other hand, the nonlinearities responsible for the variety of equilibrium and nonequilibrium behaviors observed experimentally, even in the quasistatic limit.
Our approach identifies the consecutive equilibrium states of the fluid-fluid front and their relative stability. This reveals already the hysteretic nature of imbibition-drainage cycles at the scale of individual heterogeneities. However, as the pressure balance on which it is based is valid only in quasistatic conditions, the present approach cannot account for the front dynamics during abrupt changes of η(x). Not surprisingly, viscous effects become very important in those jumps, as shown by the arrows in Fig. 2(d). Departure from the quasistatic limit to account for finite flow rates leads to new research avenues that will be pursued in the future.
The model predicts also that the front should adopt the same stationary deformation, η c , in both imbibition and drainage. We observe, however, in Figs. 2(d) and 2(e), a small but systematic difference. We attribute the larger deformation in drainage to the influence of the coating layers of oil left on the cell plates [45], which formed in spite of the extremely low velocity of the front during pistonlike motion. These layers might have effectively reduced the cell gap thickness, and thus increased slightly the distorting pressure in drainage, revealing ultimately the importance of wetting layers in two-phase fluid displacements.
To conclude, an analysis of the local pressure balance at a fluid-fluid front confined in a Hele-Shaw cell explains both qualitatively and quantitatively the sequence of equilibrium configurations reached when the front is driven quasistatically across a localized heterogeneity in the form of a mesa defect. This approach accounts for all the features observed experimentally, including metastability, multivaluedness, hysteresis, and pinning in cyclic drainage-imbibition displacements. Its local character makes us think that it may be extended to more realistic disordered media, featuring multiple and randomly distributed constrictions and expansions, thus paving the way to understand the physics behind cooperative fluid displacements (Haines jumps) and their contribution to macroscale pressure-saturation hysteresis in imbibition-drainage cycles.