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Critical behavior in porous media flow
Marcel Moura, Knut Måløy, Renaud Toussaint
To cite this version:
Marcel Moura, Knut Måløy, Renaud Toussaint. Critical behavior in porous media flow. EPL -
Euro-physics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing,
2017, 118 (1), �10.1209/0295-5075/118/14004�. �hal-01628931�
Critical behavior in porous media flow
M. Moura1, K. J. M˚aløy1 and R. Toussaint2
1 PoreLab, Department of Physics, University of Oslo - PO Box 1048, Blindern, N-0316, Oslo, Norway.
2 Universit´e de Strasbourg, CNRS, IPGS UMR 7516 - 5 rue Ren´e Descartes, 67000, Strasbourg, France.
PACS 47.55.-t – Multiphase and stratified flows
PACS 47.15.gp – Hele-Shaw flows
PACS 64.60.Ht – Dynamic critical phenomena
Abstract –The intermittent burst dynamics during the slow drainage of a porous medium is studied experimentally. We have shown that this system satisfies a set of conditions known to be true for critical systems, such as intermittent activity with bursts extending over several time and length scales, self-similar macroscopic fractal structure and 1/fαpower spectrum. Additionally, we
have verified a theoretically predicted scaling for the burst size distribution, previously assessed via numerical simulations. The observation of 1/fαpower spectra is new for porous media flows and,
for specific boundary conditions, we notice the occurrence of a transition from 1/f to 1/f2scaling. An analytically integrable mathematical framework was employed to explain this behavior.
Introduction. – The topic of fluid motion inside a
1
porous network has deservedly been subjected to a
consid-2
erable number of studies over the past decades. Scientists
3
have studied the morphology and dynamics of the flow
4
[1–12] and proposed a set of numerical schemes able to
5
reproduce the observed macroscopic patterns [13–17] and
6
relevant pore-scale mechanisms [18–26]. The topic is also
7
of central importance for the study of groundwater flows
8
and soil contaminants treatment [27,28] and has direct
ap-9
plications in the energy sector, for example, in
hydrocar-10
bon recovery methods [29]. One particularly interesting
11
aspect of multiphase flow in porous media is its
intermit-12
tent dynamics [3, 4, 18], with long intervals of stagnation
13
followed by short intervals of strong activity. This kind of
14
general behavior [30–32] appears in many physical,
biolog-15
ical and economical systems, such as the stick-slip motion
16
of a block on an inclined plane [33], the propagation of a
17
fracture front in a disordered material [34–36], the number
18
of mutations in models of biological evolution [37], acoustic
19
emissions from fracturing [38, 39], variations in stock
mar-20
kets [40], and the rate of energy transfer between scales in
21
fully developed turbulence [41, 42]. Intermittent
phenom-22
ena arise irrespective of the (certainly different) specific
23
details of each system. In the particular case of porous
24
media flows, it is caused by the interplay between an
ex-25
ternal load (for example, an imposed pressure difference
26
across the system) and the internal random resistance due
27
to the broader or narrower pore-throats.
28
In the present work we show experimental results on
29
the burst dynamics during drainage in artificial porous 30
media and investigate the question of how the pressure 31
fluctuations (due to the burst activity) can encode useful 32
information about the system. The flows studied are slow 33
enough to be in the capillary regime, in which capillary 34
forces are typically much stronger than viscous ones [3,43]. 35
We have employed synthetic quasi-2D systems driven by 36
a controlled imposed pressure (CIP) boundary condition. 37
This boundary condition differs from the controlled with- 38
drawal rate (CWR), more commonly used [3, 9]. The dy- 39
namics is characterized both via direct imaging of the 40
flow and by local pressure measurements. We present 41
results related to the statistics of bursts, their morphol- 42
ogy and orientation within the medium, and the power 43
spectral density (PSD) associated with the fluctuations 44
in the measured pressure signal. In particular, we show 45
that for systems driven by the CIP boundary condition, 46
the PSD presents a 1/f scaling regime. The presence of 47
1/fαpower spectra is a widespread feature occurring in a 48
myriad of contexts [44–46], commonly signaling the collec- 49
tive dynamics of critical systems. Some examples are the 50
early measurements of flicker noise in vacuum tubes [47], 51
fluctuations in neuronal activity in the brain [48], quan- 52
tum dots fluorescence [49], loudness in music and speech 53
[50, 51] and fluctuations in the interplanetary magnetic 54
field [52]. Although 1/fα power spectra have also been
55
observed in some fluid systems, such as simulations and 56
experiments on hydrodynamic and magnetohydrodynamic 57
M. Moura et al.
netically forced flows [55], to the best of our knowledge the
59
results reported here provide the first experimental
obser-60
vations of 1/fαpower spectra in porous media flows.
61
Methodology. – Fig. 1 shows a schematic
representa-62
tion of the setup employed (additional details in Ref. [56]).
63
The quasi-2D porous network is formed by a modified
64
Hele-Shaw cell filled with a monolayer of glass beads
hav-65
ing diameters a in the range 1.0mm < a < 1.2mm. The
66
beads are kept in place by a pressurized cushion placed on
67
the bottom plate of the cell. A spongeous filter with pores
68
much smaller than those in the medium is placed between
69
the porous network and the outlet of the model. This filter
70
allows the dynamics to continue inside the medium even
71
after breakthrough [56]. Pressure measurements are taken
72
at the outlet with an electronic pressure sensor (Honeywell
73
26PCAFG6G) that records the difference between the air
74
pressure (non-wetting phase) and the liquid pressure
(wet-75
ting phase) at the outlet, i.e., pm= pnw− poutw . Since the
76
inlet is open to the atmosphere, pnw = p0 in all
experi-77
ments, where p0is the atmospheric pressure. The porous
78
matrix was initially filled with a mixture of glycerol (80%
79
in weight) and water (20% in weight) having kinematic
80
viscosity ν = 4.25 10−5m2/s, density ρ = 1.205 g/cm3
81
and surface tension γ = 0.064 N.m−1. We have
per-82
formed experiments on 4 different porous media with
di-83
mensions: (1) 27.3cm x 11.0cm, (2) 14.0cm x 11.5cm,
84
(3) 32.8cm x 14.6cm and (4) 32.0cm x 4.5cm, where the
85
first number corresponds to the length (inlet–outlet
di-86
rection) and the second to the width. The outlet of the
87
model is connected to an external reservoir. The height
88
difference h between the surface of the liquid in this
reser-89
voir and the model is used to control the imposed pressure
90
via an adaptive feedback mechanism (CIP boundary
con-91
dition). This mechanism guarantees that the pressure is
92
only increased when the system is in a quasi-equilibrium
93
situation (details in [56]). By slowly increasing the
im-94
posed pressure (via small steps in the height of the
reser-95
voir dh = 10µm =⇒ dp = ρgdh = 0.12 P a, where g
96
is the acceleration of gravity), new pore-throats may
be-97
come available to invasion. The value of dh was chosen to
98
satisfy the accuracy condition that the height would
typ-99
ically have to be increased several times before new pores
100
are invaded. As long as this condition is satisfied, the
101
results obtained should be independent of the particular
102
value of dh.
103
Burst size distribution. – We begin by analyzing
the size distribution of invasion bursts in a CIP experi-ment. A burst is understood as any connected set of pores
invaded in the interval Θ = t2−t1between two consecutive
time instants, t1and t2, at which the imposed pressure was
increased (i.e., the imposed pressure is constant during the
interval Θ, being changed only at its extremes t1 and t2).
Fig. 2 shows the individual bursts for experiment CIP-1 (the number identifies the model), colored according to their area (top) and randomly (bottom), the latter be-ing done to aid the visualization of separate bursts. Only
Fig. 1: (color online) Diagram of the experimental setup and boundary conditions (CIP or CWR). The numbers (1), (2) and (3) denote the porous medium, filter and external tubing.
bursts having their centroids in the mid 90% of the model’s length are considered, to avoid possible boundary effects [56]. A great deal of information can be obtained from
this image. Initially, one can observe the homogeneity
and isotropicality of the dynamics: the bursts don’t seem to follow a well defined size gradient (the top image does not seem to transition from blue to red following a spe-cific direction), nor have they a clear preferred orientation (they are not particularly elongated in any direction). It is hard, if not impossible, to say from this image in which di-rection the invasion takes place (it is from left to right). A
reflection (vertical or horizontal) or a 180◦rotation would
also not be clearly identified. The box counting fractal di-mension [57,58] of the invading cluster was measured to be D = 1.76 ± 0.05. Fig. 3 shows the burst size distribution N (n) for 3 separate experiments (the number of pores n being measured by normalizing the burst area by a
typi-cal pore area ≈ 0.3mm2). The system exhibits the scaling
N (n) ∝ n−τ, with τ = 1.37 ± 0.08, over at least two
decades. The burst dynamics is therefore spatially self-similar, a feature commonly associated with systems close to a critical transition [46, 57]. The exponent τ has been calculated via maximum likelihood estimation (MLE) [59] using the data from Fig. 2 for burst sizes in the interval 1 pore < n < 150 pores. MLE was used in order to avoid possible biases from data binning (MLE is a binning free method), see also [60]. The scaling is shown in Fig. 3 on top of the logarithmically binned histogram of the data for the sake of visualization. Experiment CIP-4 was left out of the analysis because boundary effects rendered the results unreliable (model 4 is too narrow). The measured expo-nent is consistent with the value τ = 1.30 ± 0.05 predicted by numerical simulations and percolation theory [21, 58]. Martys et al. [21] derived the analytical form
τ = 1 +De− 1/ν
0
D , (1)
where D and De are respectively the fractal dimensions 104
of the growing cluster and its external perimeter and ν0= 105
Burst size (number of por es) Bursts c ol ored r andomly
Fig. 2: (color online) Individual bursts for experiment CIP-1. The flow is from left to right, during ≈ 82h. Bursts color coded by their size normalized by a typical pore area (top) and randomly (bottom). The vast blue areas in the top image contain many smaller bursts (detail).
correlation length [57, 58]. Using the values D = 1.76
107
and De= 4/3 [14], we obtain τ = 1.33, very close to the
108
measured value τ = 1.37 ± 0.08 shown in Fig. 3. Our
109
measurements provide a direct experimental verification
110
of Eq. (1), proposed in Ref. [21].
111
Crandall et al. [61] performed measurements in a CWR
112
system finding the exponent τ = 1.53, which is compared
113
to the theoretical prediction of τ = 1.527 from Roux
114
and Guyon [62]. Nevertheless, Maslov [63] pointed out
115
an inconsistency in this theoretical prediction, the
cor-116
rect expression being given in Eq. (1). Modified invasion
117
percolation simulations and pressure measurements [3, 4]
118
have shown that, in a CWR, system very large bursts are
119
split into smaller ones. A burst size distribution was
ob-120
served, with exponent τ = 1.3 ± 0.05 for the simulations
121
and τ = 1.45 ± 0.10 for the experiments (consistent with
122
Eq. (1)), followed by an exponential cutoff [3, 4]. In the
123
CIP case large bursts can happen because the displaced
124
liquid can freely flow out of the model but in the CWR
125
case this is not possible since the available volume for the
126
displaced liquid is bounded by the outlet syringe volume.
127
Burst time distribution. – Let us now focus on the
128
distribution G(Θ) of time intervals Θ between two
succes-129
sive increments in the imposed pressure during which
inva-130
sion bursts have occurred. Fig. 4 shows this distribution,
131
produced for all bursts with Θ > 120s, a cutoff related
132
to the minimum time difference for proceeding the image
133
analysis used in the feedback mechanism [56]. It scales as
134
G(Θ) ∝ Θ−γwith γ = 2.04±0.15 (exponent was also
com-135
puted via MLE [59]). In the inset we show the distribution
136
of inverse intervals g(1/Θ), which is nearly uniform, since
137
it is related to G(Θ) by g(1/Θ) = G(Θ)Θ2 ∝ Θ2−γ. The
138
uniformity of g(1/Θ) will play an important role further
139
Fig. 3: (color online) Burst size distribution N (n). The line shows the scaling N (n) ∝ n−τ, with τ = 1.37 ± 0.08, which is consistent with the theoretical value τ = 1.30 ± 0.05 predicted by numerical simulations and percolation theory [21, 58]. The data has been shifted vertically to aid visualization.
on in the modeling of the pressure fluctuations PSD. 140
Connection between the burst size and time dis- 141
tributions. – We consider now the link between the 142
burst size distribution N (n) shown in Fig. 3 and the burst 143
time distribution G(Θ) in Fig. 4. Let ˙A = s/Θ denote 144
the average growth rate of a burst of area s during the 145
time interval Θ. This corresponds to an external perimeter 146
growth [57, 58], therefore ˙A ∝ ule where u is a character- 147
istic front speed (set by the Darcy law and the character- 148
istic capillary pressure) and le is the external perimeter, 149
related to the linear size across a cluster l as le ∝ lDe. 150
Since s ∝ lD, we have
151
s/Θ = ˙A ∝ ule∝ sDe/D =⇒ Θ ∝ sβ, (2)
with β = 1 − De/D. The distributions of s and Θ are 152
linked by |G(Θ)dΘ| = |p(s)ds| and since the area s of a 153
burst is proportional to its number of pores n (see Fig. 3), 154
it follows that p(s) ∝ s−τ. Therefore, 155
G(Θ) ∝ s−τds/dΘ =⇒ G(Θ) ∝ Θ−γ, (3)
with γ = (τ − 1 + β) /β = (τ − De/D) / (1 − De/D). Us- 156
ing the literature values τ = 1.3 [21, 58], De = 1.33 157
and D = 1.82 [58], we find γ = 2.11, quite close to the 158
measured value γ = 2.04 seen in Fig. 4. As an imme- 159
diate consequence of Eq. (3), the distribution of inverse 160
intervals scales as g(1/Θ) ∝ Θ−η, with η = γ − 2 = 161
(τ − 2 + De/D) / (1 − De/D). Using the literature values 162
above we find η = 0.11, which is in agreement with the 163
experimentally observed value η = γ − 2 = 0.04 ± 0.15 164
– i.e. these theoretical considerations explain the nearly 165
uniform distribution observed in the inset of Fig. 4. 166
Fluctuations in the measured pressure signal. – Next, we analyze the fluctuations in the pressure signal, following the pore invasion events. In Fig. 5, we show the typical pressure signature in a CIP experiment. The observed pressure pulses present a characteristic exponen-tial relaxation. We also observe that a pulse can trigger
M. Moura et al. 103 104 10-2 10-1 100 10-4 10-3 102 104
Fig. 4: (color online) Burst time distribution G(Θ). The scaling (red line) corresponds to G(Θ) ∝ Θ−γ with γ = 2.04 ± 0.15. In the inset we show the nearly uniform distribution g(1/Θ).
others and even give rise to large avalanches with the in-vasion of several pores. A pulse can be divided into two
phases: an initial fast drop in the capillary pressure pcand
a slower exponential relaxation back to the pressure level
ρgh set externally (see Fig. 1). The fast drop in pcoccurs
as the liquid is displaced (following the invasion of one or more pores) and subsequently redistributed to the sur-rounding menisci, causing a back-contraction of the inter-face [3,4,18]. The relaxation phase occurs as the liquid-air interface readjusts itself inside the available pore-throats and the liquid volume displaced from the pores flows out of the model. The fluid motion sets in viscous pressure drops which are reflected in the measured pressure, as seen in Fig. 5. These drops occur (see Fig. 1): 1) in the porous medium itself, 2) in the filter at the model’s outlet and 3) in the external tubing (the numbers are in correspondence with Fig. 1). The height difference h between the surface of the liquid in the reservoir and the model level accounts for a hydrostatic component ρgh. Adding these contribu-tions and assuming that the flow is governed by Darcy’s equation, we have pw− u µL1 k1 − uµL2 k2 − uS1µL3 S3k3 + ρgh = p0, (4)
where pwis the pressure in the wetting phase (liquid) just
167
after the liquid-air interface, u is the average Darcy
ve-168
locity of the flow in the porous network, µ = ρν is the
169
liquid’s dynamic viscosity, Li and ki with i = {1, 2, 3}
170
are the length and permeability respectively of the porous
171
network, filter and the tubing and S1 and S3 are the
172
respective cross sections of the model and the tubing.
173
Since the capillary pressure across the liquid-air interface
174 is pc= pnw− pw= p0− pw, Eq. (4) becomes 175 pc+ uR − ρgh = 0 , (5) where 176 R = R1+ R2+ R3 =⇒ R = µL1 k1 +µL2 k2 +S1µL3 S3k3 , (6) Pressure [Pa] 6.35 6.4 6.45 6.5 6.55 6.6 340 360 380 400 420
exponential signature of CIP pressure pulses
100s Time [s] ×104 ×104 Pressure [Pa] 100s 420 7 7.05 7.1 400 380 Time [s] multiple pulses superposition of pulses effectively increases
the decay time
Fig. 5: (color online) Typical exponential relaxation signature of pressure pulses. A pulse can trigger others and even give rise to a large avalanche (shown in the inset).
is equivalent to an effective resistance to the flow. The 177
volumetric flux in a pore is dV /dt = ua2/φ, where a is 178
a characteristic pore length scale (for example the bead 179
diameter) and φ is the porosity of the model. By intro- 180
ducing the concept of a capacitive volume κ = dV /dpc 181
(used first in Ref. [3]), where dV is the liquid volume dis- 182
placed from a pore throat in response to a change dpc in 183
capillary pressure, we have 184
dV dt = ua2 φ =⇒ u = κφ a2 dpc dt . (7)
Plugging this equation into Eq. (5), 185
κφR
a2
dpc
dt + pc− ρgh = 0 =⇒ pc(t) = ρgh + Ce
−t/tc, (8)
thus producing the exponential behavior seen in Fig. 5.
C = pc(0) − ρgh < 0 is a constant associated to how much
the capillary pressure decreases during the invasion of a set of pores before it starts to rise again. The characteristic time scale of the exponential decay is
tc=
κφR
a2 . (9)
The invasion of one pore quite frequently triggers the 186
invasion of others, in such a manner that before an ex- 187
ponential pulse decays completely, another one is seen 188
in the pressure signal, see Fig. 5. This mechanism de- 189
lays the complete relaxation of the pressure, effectively 190
increasing the decay time from tc to t∗ ≥ tc. Indeed, if 191
this relaxation-delaying mechanism was absent, the burst 192
time distribution G(Θ) shown in Fig. 4 should be peaked 193
around the value Θ = tc. Since we have shown that 194
G(Θ) ∝ Θ−γ we expect the effective exponential decay 195
time t∗ to follow the same distribution and, in particu- 196
lar, the effective decay rate λ = 1/t∗ should be uniformly
197
distributed in an interval [λmin, λmax] following the same 198
distribution as 1/Θ (see inset of Fig. 4). λmaxis related to 199
Fig. 6: (color online) Power spectral density comparison for CIP experiments (model’s numbers in the legend). Guide-to-eye lines are shown for the scaling S(f ) ∝ f−α, with α = 1 for lower frequencies and α = 2 for intermediate frequencies.
consider λmin= 0 for convenience. Later on we will show
201
that the distribution of decay rates has a crucial impact
202
on the power spectrum of the pressure signal.
203
Pressure signal PSD. – Next we analyze the power
204
spectral density (PSD) associated to the pressure signal
205
for the CIP experiments. The PSD S = S(f ) was
com-206
puted for all experiments using the Welch method [64].
207
We have noticed the existence of a 1/f scaling regime
208
(flicker/pink noise) for lower frequencies, followed by a
209
crossover and a 1/f2 scaling regime (brown noise) for
in-210
termediate frequencies. For higher frequencies, another
211
crossover follows and a region independent of f is seen
212
(white noise associated with fluctuations in the pressure
213
sensor and unimportant to our analysis). We see from
214
Fig. 6 that the scaling properties of the power spectrum,
215
in particular the occurrence of 1/f noise, seem to remain
216
unchanged despite the changes in both sample dimensions
217
and pore-size distribution (the samples were rebuilt before
218
each experiment, thus changing the pore-size distribution
219
[56]). The 1/f regime is associated with events having
fre-220
quency f < 10−2Hz, or alternatively, periods T > 100s.
221
From Fig. 5, we see that this corresponds to the
charac-222
teristic time intervals between the pressure pulses, thus
223
indicating that they are associated with the presence of
224
the 1/f scaling in the PSD.
225
Analytical modeling of the pressure signal and
226
PSD scaling explanation. – The non-trivial scaling
227
of the CIP power spectral density can be explained by the
228
following mathematical framework, which is an adaptation
229
of an argument proposed in [65] to explain a similar 1/f
230
to 1/f2transition in the very first reported observation of
231
1/f noise [47] (see also [66] and [67]). Apart from a nearly
232
constant offset, the pressure signal can be modeled as a
233
train of exponentially decaying pulses located at randomly
234
distributed discrete times tj,
235
pλ(t) =
X
j
AH(t − tj)e−λ(t−tj), (10)
where λ > 0 and A < 0 are initially taken to be constants 236
(the characteristic decay rate and amplitude of the pulses) 237
and H(t − tj) is the Heaviside step function, i.e., H(t − 238
tj) = 0 if t < tj and H(t − tj) = 1 if t ≥ tj. Let Pλ(f ) be 239
the Fourier transform of pλ(t). The PSD Sλ(f ) is 240
Sλ(f ) = lim T →∞ 1 T D |Pλ(f )| 2E = A 2r λ2+ 4π2f2, (11)
where r is the average rate of occurrence of pulses and the 241
brackets are the expected value operator (since in practice 242
one does not have access to an ensemble of measurements, 243
we have employed Welch’s method [64] to estimate the 244
PSD, which is based on the concept of a periodogram [68]). 245
The PSD shown in Eq. (11) is a Lorentzian curve which is 246
approximately constant for lower frequencies (f λ/2π) 247
and decays as 1/f2 for higher frequencies (f λ/2π).
248
A model with a single constant decay rate λ cannot 249
incorporate the 1/f region but, as previously argued, we 250
expect λ to follow the uniform distribution ξ(λ) = 1/λmax 251
in the interval [0, λmax]. Taking this distribution into ac- 252
count and writing λmax= 2 π ft, we have 253
S(f ) = Z λt 0 Sλ(f )ξ(λ)dλ = A2r 4π2f tf arctan ft f . (12) Eq. (12) has the asymptotic behavior
S(f ) = (A2r 8πft 1 f if f ft A2r 4π2 1 f2 if f ft , (13)
thus presenting the 1/f to 1/f2transition observed in the
254
experiments. The transition frequency ft in experiment 255
CIP-1 is roughly ft = 1.5 10−2Hz (see Fig. 6). By us- 256
ing the constant A2r as a fitting parameter we can com- 257
pare the measured PSD with the theoretical prediction in 258
Eq. (12). Fig. 7 shows the resulting comparison produced 259
using A2r = 1.5 P a2/s. The dashed red vertical line marks 260
the transition frequency ft. The analytical result repro- 261
duces the experimental findings very well, scaling as 1/f 262
for f ftand as 1/f2for f ft. Indeed, this theory not 263
only captures the 1/f and 1/f2 domains but also fits the
264
data well for the crossover region between these domains. 265
The transition frequency ft can be estimated using 266
Eq. (9) and the resistance R from Eq. (6). As a first 267
order approximation, let us consider only the contribu- 268
tion to R from the term R1 relative to the resistance in 269
the porous medium itself. Using µ = ρν = 5.1 10−2P a.s, 270
L1 = 0.27m, a = 10−3m, κ = 1.1 10−12m3/P a (from 271
Ref. [3]), k1 = 1.6 10−9m2 and φ = 0.63 (both measured 272
in a similar model in Ref. [5]), we find 273
ft= 1 2πtc = k1a 2 2πκφµL1 =⇒ ft≈ 2.6 10−2Hz , (14)
M. Moura et al.
not far from the transition frequency ft = 1.5 10−2Hz
274
shown in Fig. 7. The overestimation comes from the terms
275
R2 and R3 in Eq. (6), ignored in the calculation above.
276
Finally, notice also the existence of a single isolated
277
point in the very low frequency part of the PSD, falling far
278
from the scaling region (extreme left for all experiments
279
in Fig. 6). This point is not an outlier in the data: its
280
existence signals the very slow positive drift of the
pres-281
sure signal, which occurs since the capillary pressure has
282
to increase to allow the invasion of narrower pores [3, 56].
283
Comparison with a system driven under a CWR
284
boundary condition. – In order to test the effect of
285
the boundary conditions in the PSD, we have run a
con-286
trolled withdrawal rate (CWR) experiment using model
287
(1). The resulting PSD is shown in the inset of Fig. 7.
288
The PSD still presents an interesting scaling, but with
dif-289
ferent scaling regimes: 1/f1.5, for lower frequencies, and
290
1/f3.5, for intermediate frequencies. The 1/f region is
291
only observed for systems driven under the CIP boundary
292
condition. The fact that the exponents for CWR differ
293
from CIP is not surprising, since the pressure relaxation
294
in that case no longer exponential, but linear, see Ref. [3].
295
Connection between the measured pressure and
296
the capillary pressure. – The pressure sensor
mea-297
sures the difference between the pressure in the air and the
298
liquid at the outlet, i.e., pm= pnw− poutw . The measured
299
signal is not exactly the capillary pressure pc = pnw− pw
300
across the liquid-air interface, since pw 6= poutw given that
301
viscous losses occur between the liquid-air interface and
302
the outlet, thus generally making pw> poutw . Those losses
303
occur in the porous medium itself and in the filter at the
304
outlet of the model (numbers 1 and 2 in Fig. 1). The
305
connection between pm and pc is pm = pc+ u (R1+ R2),
306
where R1and R2are the resistance terms from the porous
307
network and the filter. Using Eqs. (7) and (8), we have
308 pm= ρgh + C 1 − R1+ R2 R1+ R2+ R3 e−t/tc. (15)
Therefore, by comparing Eqs. (8) and (15), we see that
309
pmdiffers from pconly in the amplitude of the pulses, but
310
not in their characteristic exponential decay. Since our
311
analysis depended only on the distribution of the decay
312
rates, the differences between pmand pc are not crucial.
313
Further generalizations of the PSD analytical
314
framework. – One possible generalization of the model
315
would be to consider a system with a distribution of
am-316
plitudes A instead of a single value (as we might expect
317
from Fig. 5). In this case the scaling properties of the
318
PSD would still be left unchanged but the constant A2in
319
Eq. (12) and (13) would be replaced by the expected value
320
of A2. Another possibility would be to consider a
distri-321
bution for λ of the form ξ(λ) ∝ λ−δ. Here the 1/f2region
322
is still left unchanged but the 1/f scaling is changed to
323 10-3 10-2 10-1 100 102 104 CIP-1 (experiment) CIP-1 (theory) 10-3 10-2 10-1 100 105 CWR-1 (experiment) α= 1 α= 2 α= 1.5 α= 3.5
Fig. 7: (color online) Comparison between theoretical predic-tion and experiments. The analytical result (thin blue line) is given by Eq. (12), where ft= 1.5 10−2Hz (vertical dashed red
line) and A2r = 1.5P a2/s. The analytical prediction match the experimental measurements (green crosses, experiment CIP-1) well. On the inset we show the PSD for experiment CWR-1.
1/f(1+δ)[69]. As previously noted, the distribution of
de-324
caying rates λ is the crucial figure behind the 1/f scaling. 325
Conclusions. – We have analyzed the burst dynam- 326
ics from slow drainage experiments in porous media. We 327
showed that this dynamics presents many features com- 328
monly associated to critical systems. Intermittent bursts 329
of activity were observed over many time and length scales 330
and a theoretical expression for their size distribution scal- 331
ing, Eq. (1), was verified experimentally. The pressure 332
signal of the invasion presented an interesting PSD scal- 333
ing, with a 1/f scaling region which further transitions to 334
1/f2in the case of the CIP boundary condition. We have
335
employed an analytical framework [65] which satisfactorily 336
reproduces the scaling properties of the PSD. The deriva- 337
tion of closed expressions relating the pressure signal PSD 338
to properties of the porous medium and the fluids can lead 339
to new techniques for indirectly probing such systems. For 340
example, if one has access to the PSD only and not to the 341
full pressure signal, the transition frequency ftcan still be 342
measured and information on the ratio k1/φ between the 343
permeability and the porosity of the medium can be found 344
via Eq. (14). If the PSD and ftare known, Eq. (12) can be 345
fitted to measure the product A2r between the amplitudes 346
and rate of occurrence of bursts. 347
∗ ∗ ∗
We acknowledge the support from the University of 348
Oslo, University of Strasbourg, the Research Council of 349
Norway through its Centre of Excellence funding scheme 350
with project number 262644, the CNRS-INSU ALEAS 351
program and the EU Marie Curie ITN FLOWTRANS net- 352
REFERENCES
354
[1] M˚aløy K. J., Feder J. and Jøssang T., Phys. Rev.
355
Lett., 55 (1985) 2688.
356
[2] Lenormand R., Proc. R. Soc. Lond. A, 423 (1989) 159.
357
[3] M˚aløy K. J., Furuberg L., Feder J. and Jøssang T.,
358
Phys. Rev. Lett., 68 (1992) 2161.
359
[4] Furuberg L., M˚aløy K. J. and Feder J., Phys. Rev.
360
E, 53 (1996) 966.
361
[5] Løvoll G., M´eheust Y., Toussaint R., Schmittbuhl
362
J. and M˚aløy K. J., Phys. Rev. E, 70 (2004) 026301.
363
[6] Toussaint R., Løvoll G., M´eheust Y., M˚aløy K. J.
364
and Schmittbuhl J., Europhys. Lett., 71 (2005) 583.
365
[7] Or D., Adv. Water Resour., 31 (2008) 1129 .
366
[8] Sandnes B., Flekkøy E., Knudsen H., M˚aløy K. and
367
See H., Nat. Commun., 2 (2011) 288.
368
[9] Moebius F. and Or D., J. Colloid Interface Sci., 377
369
(2012) 406 .
370
[10] de Anna P. et al., Phys. Rev. Lett., 110 (2013) 184502.
371
[11] MacMinn C. W., Dufresne E. R. and Wettlaufer
372
J. S., Phys. Rev. X, 5 (2015) 011020.
373
[12] Bultreys T. et al., Water Resour. Res., 51 (2015) 8668.
374
[13] Wilkinson D. and Willemsen J. F., J. Phys. A: Math.
375
Gen., 16 (1983) 3365.
376
[14] Furuberg L., Feder J., Aharony A. and Jøssang T.,
377
Phys. Rev. Lett., 61 (1988) 2117.
378
[15] Rothman D. H., J. Geophys. Res., 95 (1990) 8663.
379
[16] Misztal M. et al., Frontiers in Physics, 3 (2015) .
380
[17] Ferrari A., Jimenez-Martinez J., Borgne T. L.,
381
M´eheust Y. and Lunati I., Water Resour. Res., 51
382
(2015) 1381.
383
[18] Haines W. B., The Journal of Agricultural Science, 20
384
(1930) 97.
385
[19] Morrow N. R., Industrial & Engineering Chemistry, 62
386
(1970) 32.
387
[20] Lenormand R., Zarcone C. and Sarr A., Journal of
388
Fluid Mechanics, 135 (1983) 337.
389
[21] Martys N., Robbins M. O. and Cieplak M., Phys.
390
Rev. B, 44 (1991) 12294.
391
[22] Løvoll G., M´eheust Y., M˚aløy K. J., Aker E. and
392
Schmittbuhl J., Energy, 30 (2005) 861.
393
[23] Berg S. et al., Proc. Natl. Acad. Sci. U.S.A., 110 (2013)
394
3755.
395
[24] Holtzman R. and Segre E., Phys. Rev. Lett., 115
396
(2015) 164501.
397
[25] Trojer M., Szulczewski M. L. and Juanes R., Phys.
398
Rev. Applied, 3 (2015) 054008.
399
[26] Schlter S. et al., Water Resour. Res., 52 (2016) 2194.
400
[27] Guymon G., Unsaturated zone hydrology (Prentice Hall,
401
Englewood Cliffs, N.J) 1994.
402
[28] Jellali S., Muntzer P., Razakarisoa O. and
403
Sch¨afer G., Transport in Porous Media, 44 (2001) 145.
404
[29] Tweheyo M., Holt T. and Torsæter O., J. Pet. Sci.
405
Eng., 24 (1999) 179 .
406
[30] Pomeau Y. and Manneville P., Commun. Math. Phys.,
407
74 (1980) 189.
408
[31] Manneville, P., J. Phys. France, 41 (1980) 1235.
409
[32] Hirsch J. E., Huberman B. A. and Scalapino D. J.,
410
Phys. Rev. A, 25 (1982) 519.
411
[33] Gomes M. A. F., Souza F. A. O. and Brito V. P., J.
412
Phys. D: Appl. Phys., 31 (1998) 3223.
413
[34] M˚aløy K. J., Santucci S., Schmittbuhl J. and
Tou-414
ssaint R., Phys. Rev. Lett., 96 (2006) 045501. 415
[35] Grob M. et al., Pure Appl. Geophys., 166 (2009) 777. 416
[36] Tallakstad K. T., Toussaint R., Santucci S., 417
Schmittbuhl J. and M˚aløy K. J., Phys. Rev. E, 83 418
(2011) 046108. 419
[37] Sneppen K., Bak P., Flyvbjerg H. and Jensen M. H., 420
Proc. Natl. Acad. Sci. U.S.A., 92 (1995) 5209. 421
[38] Sethna J. P., Dahmen K. A. and Myers C. R., Nature, 422
410 (2001) 242. 423
[39] Stojanova M., Santucci S., Vanel L. and Ramos O., 424
Phys. Rev. Lett., 112 (2014) 115502. 425
[40] Liu Y. et al., Phys. Rev. E, 60 (1999) 1390. 426
[41] Kolmogorov A. N., J. Fluid Mech., 13 (1962) 82. 427
[42] Salazar D. S. P. and Vasconcelos G. L., Phys. Rev. 428
E, 82 (2010) 047301. 429
[43] Lenormand R. and Zarcone C., Transport in Porous 430
Media, 4 (1989) 599. 431
[44] Press W. H., Comments on Astrophysics, 7 (1978) 103. 432
[45] Mandelbrot B., The fractal geometry of nature (W.H. 433
Freeman, San Francisco) 1982. 434
[46] Schroeder M. R., Fractals, chaos, power laws: min- 435
utes from an infinite paradise (W.H. Freeman, New York) 436
1991. 437
[47] Johnson J. B., Phys. Rev., 26 (1925) 71. 438
[48] Novikov E., Novikov A., Shannahoff-Khalsa D., 439
Schwartz B. and Wright J., Phys. Rev. E, 56 (1997) 440
R2387. 441
[49] Pelton M., Smith G., Scherer N. F. and Marcus 442
R. A., Proc. Natl. Acad. Sci. U.S.A., 104 (2007) 14249. 443
[50] Voss R. and Clarke J., Nature, 258 (1975) 317. 444
[51] Gardner M., Scientific American, 238 (1978) 16. 445
[52] Matthaeus W. H. and Goldstein M. L., Phys. Rev. 446
Lett., 57 (1986) 495. 447
[53] Bourgoin M. et al., Phys. Fluids, 14 (2002) 3046. 448
[54] Dmitruk P. and Matthaeus W. H., Phys. Rev. E, 76 449
(2007) 036305. 450
[55] Herault J., P´etr´elis F. and Fauve S., EPL, 111 451
(2015) 44002. 452
[56] Moura M., Fiorentino E.-A., M˚aløy K. J., Sch¨afer 453
G. and Toussaint R., Water Resour. Res., 51 (2015) 454
8900. 455
[57] Feder J., Fractals (Plenum Press, New York) 1988. 456
[58] Stauffer D., Introduction to percolation theory (Taylor 457
& Francis, London Bristol, PA) 1994. 458
[59] Clauset A., Shalizi C. R. and Newman M. E. J., 459
SIAM Review, 51 (2009) 661. 460
[60] Iglauer S. and W¨ulling W., Geophys. Res. Lett., 43 461
(2016) 11,253 2016GL071298. 462
[61] Crandall D., Ahmadi G., Ferer M. and Smith D. H., 463
PHYSICA A, 388 (2009) 574 . 464
[62] Roux S. and Guyon E., J. Phys. A, 22 (1989) 3693. 465
[63] Maslov S., Phys. Rev. Lett., 74 (1995) 562. 466
[64] Welch P., IEEE Transactions on Audio and Electroa- 467
coustics, 15 (1967) 70. 468
[65] Ziel A. V. D., Physica, 16 (1950) 359 . 469
[66] Bernamont J., Ann. Phys. (Leipzig), 7 (1937) 71. 470
[67] Milotti E., ArXiv Physics e-prints, (2002) . 471
[68] Press W., Numerical recipes in FORTRAN: the art of 472
scientific computing (Cambridge University Press, Cam- 473
bridge England New York, NY, USA) 1992. 474
![Fig. 3: (color online) Burst size distribution N (n). The line shows the scaling N(n) ∝ n −τ , with τ = 1.37 ± 0.08, which is consistent with the theoretical value τ = 1.30 ± 0.05 predicted by numerical simulations and percolation theory [21, 58]](https://thumb-eu.123doks.com/thumbv2/123doknet/14777659.594675/4.892.56.419.132.405/distribution-scaling-consistent-theoretical-predicted-numerical-simulations-percolation.webp)
