Roughening Dynamics of Radial Imbibition in a Porous Medium

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Roughening Dynamics of Radial Imbibition in a Porous Medium Yong-Jun Chen,*,†,‡ Shun Watanabe,‡ and Kenichi Yoshikawa*,‡ †

Department of Physics, Shaoxing University, Shaoxing, Zhejiang Province 312000, China Faculty of Life and Medical Sciences, Doshisha University, Kyotanabe, Kyoto 610-394, Japan

‡

ABSTRACT: We report radial imbibition of water in a porous medium in a Hele−Shaw cell, including forced imbibition and spontaneous imbibition. Washburn’s law is confirmed in our experiment. Radial imbibition follows scaling dynamics. For forced radial imbibition, anomalous roughening dynamics is found when the front invades the porous medium, and the roughening dynamics depend on the flow rate of the injected fluid. The growth exponents increase linearly with an increase in the flow rate while the roughness exponents decrease with an increase in the flow rate. For spontaneous imbibition, we found a growth exponent (β = 0.6) that was independent of the pressure applied at the liquid inlet, and the roughness exponent decreased with an increase in pressure. Thus, it has become evident that the roughening dynamics of radial imbibition is markedly different from one-dimensional imbibition with a planar interface window.

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INTRODUCTION Interface growth is found throughout nature.1 Propagation of an interface shows roughening dynamics.1 Roughening dynamics of the interface are characterized by sets of critical exponents. Different interface growths with the same critical exponents belong to the same universal class of roughening dynamics.2 The roughening process of a propagating front is often described in terms of the root-mean-square width W of the interface fluctuation. Family and Vicsek (FV) proposed a dynamic scaling hypothesis for a fixed-size interface:1,3 W(L) ∼ tβ for t < τ, W(L) ∼ Lα for t > τ, and τ ∼ Lz, where L is the lateral size of the interface, t is time, τ is the saturation time of roughness, β is a growth exponent, α is a roughness exponent, and z = α/β is a dynamic exponent. The scaling dynamics of the initially planar interface with a fixed-size window have been studied intensively.1 In addition, the role of geometric properties, for example, the size of the interface window, the shape of the interface, and non-Euclidean geometry, has attracted attention.4−6 As a matter of fact, many systems do not exhibit a fixed-size window during interface propagation, including both biological and physical interfaces. In these cases, the interface grows in all directions and forms not only overhangs but also non-Euclidean geometry. Among these, a traditional scaling analysis has been adopted to understand the universality class of tumor growth.7 The previous studies have shown that radial and linear growth of interface front obey the same universality class of the Kardar−Parisi−Zhang (KPZ) model, and the universality of KPZ class was enforced in the previous studies.8−13 It is under debate which point, initial origin point or the center of mass (MC), should be chosen as the origin point to calculate the roughness of circular interface.6,14 Here we solve this debate by a simple example. For example, if we calculate the roughness of a circle which has the roughness of zero, we will get nonzero value of roughness based on any initial point other than the center of the circle © 2015 American Chemical Society

(MC). The center of mass is the suitable choice for calculating the roughness of circular interface. Unfortunately, in most of past experimental studies7−11 including tumor evolution and theoretical studies,12,13 the roughness of circular interface was calculated based on origin point of the initial interface growth, which leads to misleading results. Thus, the relation of radial and linear growth of interface remains unclear yet. It is noted whether the universality class of roughening dynamics of interface is universal for different geometry of interface front is the key for the existence of universality class of roughening scaling dynamics. To revisit this fundamental problem, in this article, we observed radial imbibition in a porous medium confined within a Hele−Shaw cell, as a simple and representative model system for studying various growth phenomena. Imbibition, or the displacement of one fluid by another in a porous medium, exhibits a roughening transition with scaling dynamics during the motion of the invading front.15 The kinetic roughening process is governed by nonlocal dynamics. Because of its importance in industry, such as in oil recovery, printing, food industry, textile construction, and medicine transport, and to gain a fundamental understanding of a nonequilibrium process, imbibition has been studied intensively both theoretically and experimentally.16−26 For a fixed-size interface window, the anomalous roughening of imbibition has recently been observed.16−18 It is believed that roughening dynamics are caused by the disorder in the field of the porous medium, including capillary disorder, permeability disorder, pressure disorder, and so on.15 The local motion of the interface is related to the curvature of the local interface and is governed by Darcy’s law.27,28 Thus, it is reasonable to adopt a Received: April 1, 2015 Revised: May 12, 2015 Published: May 14, 2015 12508

DOI: 10.1021/acs.jpcc.5b03157 J. Phys. Chem. C 2015, 119, 12508−12513

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The Journal of Physical Chemistry C curvature-driven model to describe the motion of the imbibition front.29 However, recent studies have shown that geometric differences of propagation window can cause a striking difference in the scaling dynamics.5,6 Here, we observed radial fluid imbibition in a porous medium including forced imbibition and spontaneous imbibition. We found that radial imbibition follows scaling dynamics and the scaling behavior depends on the flow rate of the fluid. Anomalous roughening behavior was also found in radial forced imbibition. In comparison with one-dimensional imbibition with a fixed-size window, radial imbibition in two dimensions demonstrates new roughening dynamics.

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EXPERIMENTAL METHODS The experiment was performed using a Hele−Shaw cell constructed from two glass plates (30 cm × 30 cm) aligned parallel to each other with a gap size of 0.5 mm. The Hele− Shaw cell was filled with approximately two layers of glass beads with a diameter of 0.20 ± 0.05 mm. The glass beads were packed tightly by manual vibration during filling. Three of the four boundaries of the Hele−Shaw cell were sealed while one was left open so that air could come out. For forced imbibition, pure water was injected from the center of the cell at constant flow rates between 0.060 and 2.0 mL/min, using an auto syringe pump. For spontaneous imbibition, the pure water penetrated into the porous medium from the center of the Hele−Shaw cell, which was connected to a reservoir of water by a tube. The invasion under a fixed injection flow rate or a fixed pressure was observed repeatedly, and its reproducibility was confirmed. The invasion of the imbibition front was monitored using a digital CCD camera. Lateral optical illumination yielded a sharp contrast at the invading front. The experimental data were analyzed using image-analysis software. All experiments were performed under ambient conditions (20 °C).

Figure 1. Imbibition in close-packed glass beads. (a) A typical morphology of imbibition front. Scale bar: 50 mm. (b) Spatial− temporal evolution of the invasion front. The time interval between neighboring fronts is 16.67 s. The flow rate is 200 μL/min. (c) Plot of the squared mean radius r2̅ against time. The linear fit demonstrates Washburn’s law for the imbibition of fluid in a porous medium. The inset is a plot of the squared prefactor A2 in Washburn’s law against the flow rate.

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RESULTS Forced Radial Imbibition in the Hele−Shaw Cell. Figure 1a shows a typical radially propagating front of an air− water interface. The front roughens during invasion into the air in the porous medium as shown in Figure 1b. The average position r ̅ of the front, i.e., the radius to the center of mass of the front, evolves according to Washburn’s law, r ̅ = At1/2, where t is time and A is a constant, as shown in Figure 1c. Conservation of the liquid volume leads to slowing of the average position of front. The flow in the porous medium can be described using Darcy’s law, dr/dt ̅ ∼ ∇p ∼ f/r,̅ where p and f are the pressure and the injection flow rate. Consistent with Washburn’s law, A2 increases linearly with an increase in the flow rate, A2 ∼ f, as shown in the inset of Figure 1c. For a large flow rate, as f = 2000 μL/min, the increase in A2 decreases relative to the linearity (inset in Figure 1c). In our experiment, the front was digitized and analyzed numerically. Before calculating the roughness of the interface, we redistributed the original digitized points to uniform distribution on the front interface. The overhangs were included in our calculation. The roughness of the front is the root-mean-square fluctuation W = ⟨r 2 − r ̅ 2⟩1/2

Figure 2. A log−log plot of roughness against time at different flow rates. The guidelines of linear fit from up to down have slopes of 0.26, 0.28, 0.38, 0.56, and 0.71, respectively. To avoid crowding of the data, −1, −0.5, and −0.25 have been added to the logarithm of roughness for flow rates of 2000, 1000, and 400 μL/min, respectively. The right inset is a plot of the growth exponent β against the flow rate.

evolution of roughness exhibits scaling dynamics of interface propagation, W ∼ tβ. The fluctuation in the log−log plot of the scaling law is attributable to the overhang and coalescence of the front. We obtained growth exponents of 0.26, 0.28, 0.38, 0.56, and 0.71 for flow rates of 60, 200, 400, 1000, and 2000 μL/min, respectively. The growth exponent increases linearly with the flow rate, as shown in the inset of Figure 2. Roughness was not saturated in our experiments. For circular propagation, roughness should usually show saturation because the curvature decreases with an increasing in the mean radius of the front.6 This suggests that the correlation length of the interface grows slowly and is always smaller than the size of the invasion front. This is markedly different from the scaling behavior in the case of a fixed-size interface planar window.1,30 To obtain the roughness exponent, we calculated the structure function of the invading front, S(k,t) = |r(k,t)|2, where r(k,t) is a Fourier transformation of the radius r and k is

(1)

where r and r ̅ are the radius to the center of mass (CM) and its mean value, and ⟨ ⟩ represents the average over the space. Figure 2 is the log−log plot of roughness against time. The 12509

DOI: 10.1021/acs.jpcc.5b03157 J. Phys. Chem. C 2015, 119, 12508−12513

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The Journal of Physical Chemistry C the wavenumber. For a self-affine interface, the structure function has the scaling form2 S(k , t ) = k−(2α + 1)fs (kt 1/ z)

(2)

with ⎧ u 2(α − αs) if u ≫ 1 fs (u) ∼ ⎨ ⎩ u 2α + 1 if u ≪ 1 ⎪

⎪

where fs(u) is a spectral scaling function, α is a roughness exponent, and αs is a spectral roughness exponent. Figure 3 Figure 4. A log−log plot of interface roughness and structure function for different window sizes. (a) Local roughness of the invasion front as a function of time. The slopes of dashed guidelines from up to down (center angle from 2π to π/6) are 0.28, 0.27 0.25, 0.24, 0.24, 0.23, 0.21, 0.16, 0.16, 0.16, 0.16, and 0.16, respectively. We have βloc = 0.16. (b) Local structure function. The time is 86.67 s. The slopes of the dashed guidelines are −3.4, −3.3, −3.2, −3.1, −3.0, −3.0, and −3.0, respectively. We have αloc = 1.0. The flow rate is 200 μL/min. The data are artificially shifted vertically.

exponents βloc are 0.16 ± 0.05, 0.16 ± 0.05, 0.24 ± 0.05, 0.40 ± 0.05, and 0.50 ± 0.05 for the flow rates of 60, 200, 400, 1000, and 2000 μL/min, respectively. The local roughness exponents αloc are 1.0, 1.0, 1.0, 0.7, and 0.5 for flow rates of 60, 200, 400, 1000, and 2000 μL/min, respectively. If we collapse the data in Figure 3a (the inset of Figure 3a), we have αs = α. When α > 1.0 and αloc = 1.0, the interface is super-rough with flow rates of 60, 200, and 400 μL/min. Spontaneous Radial Imbibition in the Hele−Shaw Cell. We studies spontaneous imbibition with an increase in the water-column height (H), corresponding to an increase in the applied pressure difference at the fluid inlet. Figure 5 shows the

Figure 3. A log−log plot of the structure function S(k,t) against k. (a) Structure function against k for a flow rate of 200 μL/min at different time points. Inset: scaling function f(kt1/z) = S(k,t)k(2α+1) vs kt1/z (α = 1.2, z = β/α = 0.23). (b) Structure function against k for different flow rates (unit: μL/min). The data were artificially shifted vertically. The dashed lines are guidelines for the data, and the slope (−(2α + 1)) is indicated in the figure.

shows the log−log plot of the structure function against wavenumber. Figure 3a is the structure function at different time points with a flow rate of 200 μL/min. There are two regimes with different scaling behaviors, i.e., a small value of k (corresponding to a large size in real space) and a relatively large value of k (corresponding to a small size in real space). The roughness exponent does not change with time, as shown in Figure 3a. The inset in Figure 3a shows a log−log plot of S(k,t)k(2α+1) versus kt1/z with roughness exponent α = 1.2 when the flow rate is 200 μL/min. The spectral scaling function fs(kt1/z) collapses to the same scaling at a different time. From the collapse of the spectral scaling function, we found that α = αs. As shown in Figure 3b, the roughness exponent is 1.30 ± 0.1, 1.20 ± 0.1, 1.10 ± 0.1, 0.95 ± 0.1, and 0.90 ± 0.1 for flow rates of 60, 200, 400, 1000, and 2000 μL/min, respectively. From the surface geometry, it is easy to see the general trend that the front is rougher and thus has a greater roughness exponent when the flow rate is smaller. The local roughening dynamics of the invasion front were calculated. We divided the closed front into 12 equal parts using the center angle from the mass center of the front, as shown in Figure 4. The local roughness of the front and the local structure function are Wloc(θ,t) = ⟨r2 − r2̅ ⟩θ1/2 and Sloc(k,t) = |r(k,t)|θ2, respectively. Figure 4 shows the typical log−log plots of the local roughness against time (Figure 4a) and the local structure function against wavenumber (Figure 4b) for invasion fronts of different sizes (center angle from the mass center) when the flow rate is 200 μL/min. The roughness evolution and structure function depend on the center angle of the interface. According to our calculation, the local growth

Figure 5. Spatial−temporal propagation of the imbibition front at different pressure differences: (a) H = 0 cm, (b) H = 1.0 cm, (c) H = 4.0 cm, (d) H = 6.0 cm, (e) H = 12.0 cm.

morphology of the invasion front at various pressure differences (represented by the height of water column, H). The interface undergoes a roughening transition during imbibition into the porous medium of glass beads. The roughening process depends on the pressure difference at the inlet of fluid. Figure 6 shows Washburn’s law for the average position r ̅ of the invasion front, r ̅ = At1/2, where t is time and A is a constant. According to Darcy’s law, the value of A2 should depend 12510

DOI: 10.1021/acs.jpcc.5b03157 J. Phys. Chem. C 2015, 119, 12508−12513

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Figure 6. Plot of the squared mean radius against time. The linear fit shows Washburn’s law for the imbibition of water in the porous medium. The inset is a plot of the squared prefactor A2 in the Washburn equation against the height of the liquid reservoir.

Figure 8. Structure function. (a) Structure function of fronts at different time when H = 0 cm. The inset is the collapse of the log−log plot of S(k,t)k(2α+1) versus kt1/z (z = α/β = 2.08). (b) Structure function for invasion fronts at various pressure differences.

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linearly on the pressure difference. However, as shown in the inset of Figure 6, the value of A2 does not increase linearly as the height of the water column increases. The roughness W of the front was calculated according to eq 1. Figure 7 shows the log−log plot of roughness against time

DISCUSSION Let us compare the above results with those in flow imbibition with a fixed-size window in quenched disorder. For forced imbibition, as reported by Soriano et al.,30 the growth exponent with a fixed-size window is 0.5, which is independent of the experimental parameters, and the roughness exponent on a short length scale decreases with a decrease in the flow rate. For weak quenched disorder, the scaling behavior does not depend on the disorder configuration.30 However, for strong quenched disorder, forced imbibition demonstrates anomalous roughening behavior with β = 0.50, βloc = 0.25, α = 1.0, and αloc = 0.5 when the flow rate is smaller than a threshold flow rate.31 There is a clear difference between fixed-sized forced imbibition and our radial forced imbibition, as shown above. In our radial forced imbibition, due to the enlargement of the fluid front and the conservation of liquid, the local flow rate decreases in the radial direction. Such different scaling dynamics is related not only to the different disorder but also to the different geometry of the interface. For spontaneous imbibition, anomalous roughening dynamics have been reported for spontaneous imbibition with a planar interface window in a porous medium in a Hele−Shaw cell,17 and the roughening behavior depends on the pressure difference with a crossover from negative pressure to positive pressure.18 We examined the evolution of local roughness and local structure factor in the present study on spontaneous radial imbibition. There was no apparent anomalous scaling. The roughness process followed Family−Vicsek (FV) scaling, as shown in the collapse of data in the inset of Figure 8a using a dynamic exponent z = α/β = 2.08 for H = 0. This result is surprising. The invasion front should experience a similar roughening process due to the quenched disorder of the porous medium at a local point in both one-dimensional imbibition and two-dimensional radial imbibition. In addition, spontaneous radial imbibition has a growth exponent β = 0.60, independent of the pressure difference, while in one-dimensional imbibition, the value of growth exponent decreases when the pressure difference increases from negative to positive, as reported by Planet et al.18 At an intermediate pressure difference between a negative and positive pressure difference, Planet et al.18 suggested that there exists a crossover regime which lacks scaling behavior. It has been reported that the roughness exponent 0.81 for small length scales crosses over to quenched KPZ (QKPZ) roughness exponent 0.6 for spontaneous one-dimensional imbibition.32 We can see that the roughening dynamics of spontaneous radial imbibition are markedly different from those of spontaneous one-dimensional

Figure 7. A log−log plot of roughness against time for various pressure differences. The slopes of the dashed fitting lines are 0.60 for H = 0, 1.0, 4.0, and 6.0 cm. In contrast, the slope of the fitting line for H = 12.0 cm is 0.50.

for different heights of the water column. The evolution of roughness exhibits scaling dynamics for interface propagation, W ∼ tβ. The fluctuation in the log−log plot of the scaling law is attributable to the overhang and coalescence of the front. By fitting the data linearly, we obtain the growth exponent β = 0.60 ± 0.05. The growth exponent does not depend on the pressure difference, as shown in Figure 7. When the pressure difference is high (H = 12.0 cm), the data become somewhat scattered (Figure 7). To obtain the roughness exponent, we calculated the structure function of the invading front, S(k,t) = |r(k,t)|2, where r(k,t) is a Fourier transformation of the radius r and k is the wavenumber. Figure 8 shows a plot of structure functions. From Figure 8a, we can obtain the roughness exponent α = 1.25 when the pressure difference H = 0, indicating that the invasion front is super-rough.2 The collapse of the log−log plot of S(k,t)k(2α+1) versus kt1/z with roughness exponent α = 1.25 when H = 0 is shown in the inset of Figure 8a. From this collapse, we find that α = αs. Figure 8b shows the dependence of the roughness exponent on the pressure difference. A greater height H leads to a smoother front with a smaller value of the roughness exponent. We obtained roughness exponents of 1.10, 1.0, 0.93, and 0.75 for the height H values of 1.0, 4.0, 6.0, and 12.0 cm, respectively. Thus, there is a clear trend in the roughening process of the front, as shown in Figure 5. 12511

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definitions of capillary number.35 The microscopic condition for the imbibition determines the noise and penetration of liquid into the pores. In addition, the surface tension of interface affects the roughening dynamics as shown by previous simulation.6 For interface with low interfacial tension, for example, liquid−liquid interface and granular front,37 the interface will have more overhangs, and it becomes rougher. For interface with low tension, large instability of the front and anisotropy of interface propagation deviate the whole interface from circular shape. Coalesce of front will affect the roughness of the interface. And also the interface will develop into noncircular interface with large anisotropy for imbibition in media with large aspect ratio because liquid will always penetrate preferably in some specific direction. How to define the roughness of an anisotropic interface is not yet clear. We will study this problem in the future.

imbibition with a planar interface window. We should note that the pressure difference in our experiment is different from that used by Planet et al.18 In our experiment, the pressure difference at the inlet of the fluid ranged from 0 to over 10 cm, while in Planet’s experiment the largest pressure difference was 1.5 cm and, in fact, a negative pressure difference was possible. When H = 1.5 cm, the interface becomes smooth in onedimensional imbibition. Based on the roughness exponents in our experiment, the interface is still rough even when the pressure difference H is over 6.0 cm. These differences suggest that the mechanism of interface roughening should be different in the two cases. For imbibition with a fixed-size planar window, the finite size of the interface window and the boundary effect should affect the observed scaling behavior. As known in anomalous roughening, a different size for the interface should lead to a difference in the local roughness evolution and structure function.1 Especially, the boundary effects on local roughening dynamics are critical and yet still not clear.15 As shown in an experiment,15 the boundary will tilt the invading front, and we have to subtract such tilting caused by the boundary effect. However, this subtraction of the interface morphology must be performed very carefully according to the insight obtained in our present study. If we include boundary effects, local disorder, and so on, we obtain exponents with large variation.15 For radial imbibition, while there is no such disadvantage, we do observe enlargement of the interface. In addition, a previous consideration of the interface roughening mainly focused on one-dimensional propagation of the interface, while lateral “soliton-like” propagation was neglected.32 In our radial imbibition lateral propagation is nontrivial. For local roughening dynamics, interface growth should follow the same physical process for both fixed-size and radial imbibition. As shown by the simulation based on curvature-driven growth of interface,6 a variable interface window leads to striking difference in the roughening dynamics of the interface propagation. It is noted that the propagation of the interface is related to capillary number and pore-scale mechanism of fluid imbibition.33−36 The capillary number affects the roughness of the propagating front.33,34 As shown by previous studies, the capillary number reflecting imbibing mechanism can be defined as pore-scale capillary number Nc1, Newtonian-fluid capillary number Nc2, and apparent capillary number Nc3.33 As found in our experiment, displacement will show various behaviors under different capillary numbers, for example, second imbibition. The saturation of the porous medium is related to the capillary number. But the relation between the definition of capillary number and the behavior of interface roughening is rather complicated. Here we primarily discuss Newtonian-fluid capillary number. As shown by Washburn’s law in Figures 1c and 6, Darcy’s velocity υ of propagating interface decreases with time as υ = 1/2At−1/2. According to definition of Newtonian capillary, Newtonian capillary number decreases proportionally as Nc2 ∼ υ/(cos θ), where θ is contact angle. For different geometry of propagating front, the change of capillary number during propagation of the interface should be different due to the conservation of the liquid volume. Different evolution of capillary number will affect the roughening dynamics of the imbibition front as shown by the previous research.24 The porescale mechanisms, wettability, pore-body structure, and porethroat geometry have effects on the interface roughness, and they can be correlated with one another by using different

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CONCLUSION We observed the scaling dynamics of radial imbibition in a porous medium made from close-packed glass beads in a Hele− Shaw cell. We demonstrate the first confirmation of Washburn’s law in radial imbibition. Radial imbibition exhibits anomalous roughening dynamics. The growth and roughness exponents depend on the flow rate of the injected fluid. The local exponents are substantially different from the global exponents. The imbibition front is super-rough when the flow rate is small because the pinning effect dominates the roughening of the invading front. The radial imbibition in our porous medium is strikingly different from that with a fixed-size interface window. These findings provide the new horizon to understand the essential difference between the scaling behaviors of fluctuation in interface propagation with a fixed-size window and a variable window. As a final remark, we would like to point out the delicate problem on the influence of the geometric shape of interface on the roughening dynamics of the propagating interface. A naive imagination on the roughening dynamics of circular KPZ growth will show that growth exponent and roughness exponent are independent of the shape of interface and belong to the same KPZ universality class of onedimensional planar growth. The interface growing which belongs to the same universality class will have same mechanism of interface growth.1,2 As has been clarified in the present article, roughening dynamics of the radial propagation of interface is markedly different from one-dimensional roughening process and has different exponents. Thus, we should think more deeply whether the same class of interface growth with different interface shapes belongs to the same class of roughening dynamics and, more specifically, whether circular “KPZ growth” and one-dimensional “KPZ growth” belong to the same KPZ universality class (with different exponents). If they do not belong to the same class because of different exponents, what does the term “universality class” mean or does “universality class” really exist? Physically, change from onedimensional growth to circular growth should not lead to change of the general mechanism of the interface growing, which determines the exponents of interface roughening. Thus, we could conclude that the “universality class” for the interface growth does not exist. We have immersed ourselves in studies on the interface growth and the celebrated KPZ universality class for over two decades. Shall we continue to find universality class of the interface growth in nature? We hope that the above thought will stimulate new direction of studies on the interface growth. 12512

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(20) Rubio, M. A.; Edwards, C. A.; Dougherty, A.; Gollub, J. P. Selfaffine fractal interfaces from immiscible displacement in porous media. Phys. Rev. Lett. 1989, 63, 1685. (21) Horvath, V. K.; Family, F.; Vicsek, T. Comment on “self-affine fractal interfaces from immiscible displacement in porous media. Phys. Rev. Lett. 1990, 65, 1388. (22) Horvath, V. K.; Family, F.; Vicsek, T. Anomalous noise distribution of the interface in two-phase fluid flow. Phys. Rev. Lett. 1991, 67, 3207. (23) Martys, N.; Cieplak, M.; Robbins, M. O. Critical phenomena in fluid invasion of porous media. Phys. Rev. Lett. 1991, 66, 1058. (24) He, S.; Kahanda, G. L. M. K. S.; Wong, P. Z. Roughness of wetting fluid invasion fronts in porous media. Phys. Rev. Lett. 1992, 69, 3731. (25) Dubé, M.; Rost, M.; Elder, K. R.; Alava, M.; Majaniemi, S.; AlaNissila, T. Liquid conservation and nonlocal interface dynamics in imbibition. Phys. Rev. Lett. 1999, 83, 1628. (26) Miranda, A. M.; Menezes-Sobrinho, I. L.; Couto, M. S. Spontaneous imbibition experiment in newspaper sheets. Phys. Rev. Lett. 2010, 104, 086101. (27) Lee, J. H.; Kim, S. K.; Kim, J. M. Growth with surface curvature on quenched potentials. Phys. Rev. E 2000, 62, 3299. (28) Ganesan, V.; Brenner, H. Dynamics of two-phase fluid interfaces in random porous media. Phys. Rev. Lett. 1998, 81, 578. (29) Koplik, J.; Levine, H. Interface moving through a random background. Phys. Rev. B 1985, 32, 280. (30) Soriano, J.; Ortín, J.; Hernández-Machado, A. Experiments of interfacial roughening in Hele-Shaw flows with weak quenched disorder. Phys. Rev. E 2002, 66, 031603. (31) Soriano, J.; Ortín, J.; Hernández-Machado, A. Anomalous roughening in experiments of interfaces in Hele-Shaw flows with strong quenched disorder. Phys. Rev. E 2003, 67, 056308. (32) Geromichalos, D.; Mugele, F.; Herminghaus, S. Nonlocal dynamics of spontaneous imbibition fronts. Phys. Rev. Lett. 2002, 89, 104503. (33) Jamaloei, B. Y.; Asghari, K.; Kharrat, R. The investigation of suitability of capillary number definitions for flow behavior characterization of surfactant-based chemical flooding in heavy oil reservoirs. J. Pet. Sci. Eng. 2012, 90−91, 48−55. (34) Jamaloei, B. Y.; Ahmadloo, F.; Kharrat, R. The effect of pore throat size and injection flowrate on the determination and sensitivity of different capillary number values at high-capillary-number flow in porous media. Fluid Dyn. Res. 2010, 42, 055505. (35) Jamaloei, B. Y.; Asghari, K.; Kharrat, R.; Ahmadloo, F. Pore-scale two-phase filtration in imbibition process through porous media at high- and low-interfacial tension flow conditions. J. Pet. Sci. Eng. 2010, 72, 251−269. (36) Jamaloei, B. Y.; Kharrat, R.; Asghari, K.; Torabi, F. The influence of pore wettability on the microstructure of residual oil in surfactantenhanced water flooding in heavy oil reservoirs: Implications for porescale flow characterization. J. Pet. Sci. Eng. 2011, 77, 121−134. (37) Cheng, X.; Xu, L.; Patterson, A.; Jaeger, H. M.; Nagel, S. R. Towards the zero-surface-tension limit in granular fingering instability. Nat. Phys. 2008, 4, 234−237.

AUTHOR INFORMATION

Corresponding Authors

*Tel 86-15067509284; e-mail [email protected] (Y.J.C.). *Tel 81-774-65-6264; e-mail [email protected] (K.Y.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (No. 11204181), SRF for ROCS, SEM and Kakenhi (Nos. 23240044 and 25103012).

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REFERENCES

(1) Barabasi, A. L.; Stanley, H. E. Fractal Concepts in Surface Growth; Cambridge University Press: Cambridge, 1995. (2) Ramasco, J. J.; López, J. M.; Rodríguez, M. A. Generic dynamic scaling in kinetic roughening. Phys. Rev. Lett. 2000, 84, 2199. (3) Family, F.; Vicsek, T. Scaling of the active zone in the eden process on percolation networks and the ballistic deposition model. J. Phys. A: Math. Gen. 1985, 18, L75−L81. (4) Block, M.; Schöll, E.; Drasdo, D. Classifying the expansion kinetics and critical surface dynamics of growing cell populations. Phys. Rev. Lett. 2007, 99, 248101. (5) Escudero, C. Dynamic scaling of non-euclidean interfaces. Phys. Rev. Lett. 2008, 100, 116101. (6) Chen, Y. J.; Nagamine, Y.; Yamaguchi, T.; Yoshikawa, K. Anomalous roughening of curvature-driven growth with a variable interface window. Phys. Rev. E 2010, 82, 021604. (7) Brú, A.; Pastor, J. M.; Fernaud, I.; Brú, I.; Melle, S.; Berenguer, C. Super-rough dynamics on tumor growth. Phys. Rev. Lett. 1998, 81, 4008. (8) Takeuchi, K.; Sano, M. Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Phys. Rev. Lett. 2010, 104, 230601. (9) Takeuchi, K. Experimental approaches to universal out-of equilibrium scaling laws: turbulent liquid crystal and other developments. J. Stat. Mech. 2014, P01006. (10) Takeuchi, K.; Sano, M. Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquidcrystal turbulence. J. Stat. Phys. 2012, 147, 853−890. (11) Takeuchi, K.; Sano, M.; Sasamoto, T.; Spohn, H. Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep. 2011, 1, 34. (12) Takeuchi, K. Statistics of circular interface fluctuations in an offlattice Eden model. J. Stat. Mech.:Theor. Exp. 2012, P05007. (13) Alves, S. G.; Oliveira, T. J.; Ferreira, S. C. Universal fluctuations in radial growth models belonging to the KPZ universality class. EPL 2011, 96, 48003. (14) Ferreira, S. C.; Alves, S. G. Pitfalls in the determination of the universality class of radial clusters. J. Stat. Mech.: Theor. Exp. 2006, P11007. (15) Alava, M.; Dubé, M.; Rost, M. Imbibition in disordered media. Adv. Phys. 2004, 53, 83−175. (16) Soriano, J.; Ramasco, J. J.; Rodríguez, M. A.; HernándezMachado, A.; Ortín, J. Anomalous roughening of hele-shaw flows with quenched disorder. Phys. Rev. Lett. 2002, 89, 026102. (17) Soriano, J.; Mercier, A.; Planet, R.; Hernández-Machado, A.; Rodríguez, M. A.; Ortín, J. Anomalous roughening of viscous fluid fronts in spontaneous imbibition. Phys. Rev. Lett. 2005, 95, 104501. (18) Planet, R.; Pradas, M.; Hernández-Machado, A.; Ortín, J. Pressure-dependent scaling scenarios in experiments of spontaneous imbibition. Phys. Rev. E 2007, 76, 056312. (19) Cieplak, M.; Robbins, M. O. Dynamical transition in quasistatic fluid invasion in porous media. Phys. Rev. Lett. 1988, 60, 2042. 12513

DOI: 10.1021/acs.jpcc.5b03157 J. Phys. Chem. C 2015, 119, 12508−12513