J Geophys Res-Sol Ea 2010 Nermoen

Experimental and analytic modeling of piercement structures

A. Nermoen,1 O. Galland,1 E. Jettestuen,1,2 K. Fristad,1 Y. Podladchikov,1 H. Svensen,1

and A. Malthe‐Sørenssen1

Received 23 March 2010; revised 1 June 2010; accepted 24 June 2010; published 8 October 2010.

[1] Piercement structures such as mud volcanoes, hydrothermal vent complexes,pockmarks and kimberlite pipes, form during the release of pressurized fluids. The goalof this work is to predict under which conditions piercement structures form from theinsights gained by sand box experiments injecting compressed air through an inlet of widthw at the base of a bed of glass beads of height h. At an imposed critical velocity vf, afluidized zone consisting of a diverging cone‐like structure formed with morphologicalsimilarities to those observed in nature. Dimensional analysis showed that vf is correlatedto the ratio of h over w. In addition, we derived an analytical model for vf which iscompared to the experimental data. The model consists of a force balance between theweight and the seepage forces imparted to the bed by the flowing gas. The analytic modelreproduces the observed correlation between vf and h/w, although a slight underestimatewas obtained. The results suggest that the gas‐particle seepage force is the main triggeringfactor for fluidization and that the commonly used proxy, which the fluid pressure mustequal or exceed the lithostatic weight, needs to be reconsidered. By combining theexperiments and the model, we derived critical pressure estimates which were employed to avariety of geological environments. Comparing the estimated and measured pressuresprior to the Lusi mud volcano shows that the presented model overestimates the criticalpressures. The model paves the way for further investigations of the critical conditions forfluidization in Earth systems.

Citation: Nermoen, A., O. Galland, E. Jettestuen, K. Fristad, Y. Podladchikov, H. Svensen, and A. Malthe‐Sørenssen (2010),Experimental and analytic modeling of piercement structures, J. Geophys. Res., 115, B10202, doi:10.1029/2010JB007583.

1. Introduction

[2] Piercement structures, the structures forming whenfluids are forced through sediments and rocks, are commonin nature and they manifest themselves in a variety casesranging from pockmarks, mud volcanoes, hydrothermal ventcomplexes to kimberlite pipes. Despite their diverse origin,they show structural similarities consisting of circular pipes(and pipes within pipes) containing intensely deformedrocks, often brecciated, with a lack of internal structure [e.g.,Hovland et al., 1985; Planke et al., 2003; Jamtveit et al.,2004; Svensen et al., 2006; Sparks et al., 2006; Walters et al.,2006].[3] On seismic data, piercement structures are inferred

from pipe‐like domains marked by discontinuities of theseismic reflectors, with large height to width ratios [e.g.,Lancelot and Embley, 1977; Planke et al., 2005; Hansen,2006; Moss and Cartwright, 2010]. Field data shows thatthe pipes contain brecciated elements with lithologies dif-fering from the surrounding rocks. Moreover, mud volcanoesand hydrothermal vent complexes are normally surrounded

by inward dipping strata suggesting a convective movementwithin the pipes, where rocks are transported upward in thecenter and with a downward collapse of the adjacent margin(see Figure 1) [e.g., Planke et al., 2003; Svensen et al., 2006;Walters et al., 2006; Svensen et al., 2007b].[4] In some hydrothermal vent complexes (HTVC) (e.g.,

the Witkop II and III complexes in the Karoo Basin (SouthAfrica)), pipes containing fine‐grained fluidized sand areobserved and an occurrence of zeolites indicate that hydro-thermal fluids are involved in their formation [Svensen et al.,2007b]. In offshore pockmarks, fluids are sometimes observedseeping out of the craters [e.g., Hovland et al., 1987; Karpenet al., 2006]. The observations listed above shows some ofthe evidences indicating the close association between pipesand localized zones of elevated fluid pressures, although thecause and effect at the formation of piercement structures isnot obvious; that is, how can circular pipes form whenconsolidated sediments are brecciated by pressurized fluids?[5] Themechanism for fluid pressure buildup varies between

the different geological cases:[6] 1. The origin of the pressures for the formation of

seafloor pockmarks has been explained by several hypotheses[Hovland et al., 1985] such as the rapid expulsion of variousgases, especially methane, originating from depth. Alterna-tive hypotheses suggest that pockmarks form in response tothe pressure buildup due to rapid pore collapse of water

1Physics of Geological Processes, University of Oslo, Oslo, Norway.2International Research Institute of Stavanger, Stavanger, Norway.

Copyright 2010 by the American Geophysical Union.0148‐0227/10/2010JB007583

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, B10202, doi:10.1029/2010JB007583, 2010

B10202 1 of 15

http://dx.doi.org/10.1029/2010JB007583

bearing quick clay [Harrington, 1985] or that they formwhen groundwater is expelled onto the seafloor [e.g., Jensenet al., 2002; Karpen et al., 2006].[7] 2. Mud volcanoes are regularly observed in settings

with organic rich, water‐bearing sediments with a highsedimentation rate and low geothermal gradient causing thebuildup of fluid pressure at depth [e.g., Jakubov et al., 1971;Dimitrov, 2002; Planke et al., 2003; Svensen et al., 2007a;Mazzini et al., 2007].[8] 3. Hydrothermal vent complexes form in volcanic

basins during magmatic sill emplacement into sedimentaryrocks [e.g., Jamtveit et al., 2004; Svensen et al., 2004;Planke et al., 2005]. The heat transported into the sedimentsresults in rapid maturation of the organic compounds andboiling of the pore water, leading to rapid buildup of thefluid overpressure.[9] 4. Kimberlite pipes form when ascending volatile‐rich

kimberlite magma is decompressed, causing exsolution offluids out of the magma leading to magmatophreatic erup-tions which result in deeply rooted explosions [e.g.,Woolseyet al., 1975; Clement, 1975; Lorenz, 1975; Walters et al.,2006; Lorenz and Kurszlaukis, 2007].[10] Despite the fact that the geological examples are of

very different scales and origins, this study aims at explainingthem within a unique framework. Whatever the details of

the pressure buildup mechanism at depth, the overlying rocksdeform, and the fluids are released through eruptions at fluidpressures exceeding a critical pressure. In this study we aimat understanding under which physical conditions, with afocus on the critical pressures at which, eruptions occur andpiercement structures may form.

1.1. Mechanical Effects of Elevated Pore FluidPressure

[11] It is well known that the effective strength of rockis reduced by elevated fluid pressures [Terzaghi, 1943]. Inaddition, fluid pressure gradients within a porous materialinduces a fluid flow affecting the stress state in the solidmatrix [e.g., Paterson and Wong, 2006]. Consequently, thiseffect changes the Mohr circle causing yielding in situationsthat would otherwise be stable [e.g., Sibson, 2003; Cox,2005]. Mourgues and Cobbold [2003] demonstrate howseepage forces induced by gradients in the pressure changesthe total stress state in a simple sand box model. Cobbold andRodrigues [2007] use this concept to show how bedding‐parallel veins of fibrous calcite may form under lithostaticconditions.[12] If the local pore fluid pressure exceeds a critical

threshold, given, e.g., by the effective strength of the sub-jected material, fluid‐induced deformation occurs that

Figure 1. A schematic drawing of different piercement structures in nature. (a) Submarine pockmarks off-shore Norway [after Hovland et al., 1985]. (b) Witkop II hydrothermal vent complex, Karoo Basin, SouthAfrica [after Svensen et al., 2006]. (c) Mud volcano Azerbaijan [after Planke et al., 2003]. (d) Kimberlitepipe, Botswana [after Walters et al., 2006].

NERMOEN ET AL.: MODELING OF PIERCEMENT STRUCTURES B10202B10202

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effectively expel the pressurized fluids onto the surface.Possible deformation mechanisms range from focuseddeformation generating tensile hydraulic fractures, to adeformation of a more diffuse, less focused nature, when thematrix moves together with the flowing fluid [Lemaireet al., 1991; Hirata, 1998]. In addition to that, the locali-zation of overpressures may cause localized compactionthat generate porosity waves that can significant alter to theflow pattern in a solid matrix [Connolly and Podladchikov,1998; Chauveau and Kaminski, 2008].[13] The matrix deformation mechanism that realizes itself

in response to the injected fluids depends on the physicalproperties of both the infiltrated fluids, the properties of thesubjected material and the stress state of the solid. It isunclear how the different deformation mechanisms impedesor accelerate each other in the different cases. For example,the threshold between the different regimes can be related toa threshold permeability or porosity. In the case of a localreduction of porosity at the front of the porosity wave, thefluid pressure will increase such that fluidization may occur.On the contrary, however, fluidization may become easierat larger porosities.[14] Another example include the history dependence

within dynamically evolving systems. When the fluidpressure exceeds a critical limit [e.g., Rozhko et al., 2007],hydrofracturing and brecciation of consolidated materialsmay occur. If the pressure gradient is maintained, thebrecciated elements may fluidize during later stages of theemplacement process. In this case, it might be the fluid-ization process that one observes within in piercementstructures today (e.g., for the Venetia Kimberlite structure inSouth Africa [Walters et al., 2006]). Fluidization/liquefac-tion is defined as the solid‐liquid phase transition occurringwhen the matrix no longer can withstand any differentialstresses imparted to the system. In this case shear strengthdrops to zero and the matrix starts moving in response tothe interstitial flow of air [Kunii and Levenspiel, 1969;Gidaspow, 1994; Dudukovic et al., 1999; Wilhelm, 2000].See section 4.4.[15] The process of fluidization, when the previously

static solid‐like material changes to a flowing, liquid‐likestate, has been widely studied during the last decades.Jaeger et al. [1996] reviewed the phases and phase transi-tions of granular materials and showed how the solid‐liquidtransition can be induced by several mechanisms such asshaking, pouring and the interstitial flow of fluids [deGennes,1999].[16] The physical conditions determining the onset of

fluidization were studied by, e.g., Kunii and Levenspiel[1969], who first defined the concept of the minimalvelocity of fluidization. This concept is based on the quasistatic approximation obtained when the fluid‐particle vis-cous drag balances the buoyant weight of the matrix. Thisconcept has been used in several experimental studies onfluidization applied to the emplacement of Kimberlite vol-canoes [Woolsey et al., 1975; Freundt, 1998; Walters et al.,2006; Sparks et al., 2006]. However, a qualitative andquantitative application of fluidization to the formation ofpiercement structures in general has not yet been performed.In numerical continuum models, fluidization has been sim-ulated by the use of, e.g., Bingham rheologies where theeffective rheological material properties depend on the

subjected stress. In these models, fluidization is simulatedby abrupt drop in the viscosity [e.g.,Mazzini et al., 2009], orthe previously elastic material changes to a flowing viscousbehavior [e.g., Gittings et al., 2006], when the differentialstresses exceed a critical level.[17] In geology, several different mechanisms lead to

localized increased fluid pressures at depth. It is observed thatin apparently similar geological settings the fluid pressuresometimes does, and other times does not, form piercementstructures. For physical parameters below a critical limit,Darcian flow is sufficient to bring the influx of fluids (relatedto the rate of fluid production) to the surface. Analogously,above the critical limit, Darcian flow is insufficient as a fluidtransport mechanism leading to unstable situations where thefluid pressure builds up, and highly permeable pathwaysrepresented by the piercement structures form. When thespecific conditions are met, the piercement structures connectthe pressure anomaly to the free surface causing rapid releasesof the pressurized fluids. We use this scenario to explore thegoverning physical parameters and their internal quantitativerelationship that determine the onset of fluidization.[18] In this study we use the results from a well‐defined

experiment as a starting point to discuss the formation ofpiercement structures in some geological settings. Whendesigning the experiments, a vertically oriented Hele‐Shawcell, equivalent to the ones used by, e.g., Rigord et al.[2005] and Johnsen et al. [2008] was used. We will showthat the critical velocity increases with the ratio of the fillingheight by the inlet width as a power law with an exponentclose to one. Then, we derive an expression for the ventingnumber that enables us to estimate the critical fluid over-pressure for the formation of piercement structures in gen-eral. We apply our results to four specific geological caseswhere we estimate the critical pressure at formation. Thecases are the Lusi mud volcano (Indonesia), hydrothermalvent complexes in Karoo (South Africa), pockmarks (off-shore Norway) and kimberlites (Venetia Kimberlite struc-tures, South Africa). We compare the calculated pressureestimates with the pore pressure measurements in a nearbydrilling well prior to the Lusi mud volcano eruption [Mazziniet al., 2007].

2. Methods

2.1. Experimental Setup

[19] The experimental setup consisted of a verticallyoriented Hele‐Shaw cell (H‐S cell) consisting of two parallelglass plates 0.8 cm apart. (see Figure 2). The 59 cmwide glassplates are sealed at the bottom and at the two sides and open atthe top. A replaceable inlet was raised 7 cm into the cell toprevent the airflow to focus along the walls of the H‐S cell.Four inlet widths w were used (w = 3, 9, 25, and 59 mm),each with breadth b = 7 mm. Each experiment was preparedby slowly pouring glass beads, through a funnel, from thetop of the fixed H‐S cell to a desired filling height h,between 1 and 25 cm, measured from the upper tip of theinlet to the free surface. By using the funnel in the fillingprocedure we could control the filling height and make a bedwith a flat surface. This produced a similar initial packingfor each experiment, although some striped features in thetransmitted light intensity were observed. The gradients inthe transmitted light intensity is associated with local var-


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iations in bead size and porosity in the packing. This effectarise when differently sized beads segregate when pouredinto the cell [Lebron and Robinson, 2003]. The fillingprocedure, and thus the formation of the initial packing het-erogeneities, was not varied systematically in the experi-ments, but this did not, as seen later, affect the reproducibilityof the experiment. Therefore it is considered second order.[20] Overpressurized air was injected into the bead bed

through the inlet placed within the bed. The inlet air flux (Q)was controlled by slowly opening a valve nearby a OmegaFMA‐1610 flowmeter ∼1 m from the inlet. The air over-pressure was recorded in a T junction (with an inner diameterof 2.45mm) 10 cm from the inlet by a Omega PX139‐004DVpressure sensor (Figure 2). In the experimental setup wecontrolled the flux, while the pressure measurements variedas a proxy of the bulk permeability in the overlying bed.Digital gray scale images were captured at 20 frames persecond by a high‐resolution Jai CV‐M4 CCD camera with aNikon AF Nikkor 20 mm lens. A Labview (http://www.ni.com/labview/) program triggered the simultaneous loggingof the pressure, flow measurements and the images. Theinlet air flux was slowly increased until deformation of theoverlying bed occurred. At this point in time, the criticalvalues of flux and pressure were recorded.

2.2. The Material

[21] Beijer glass beads (http://www.beijer.no) was used inthe experiments. The glass beads had a close to sphericalshape (Figure 3a), with a diameter in the range of d = 280 ±50mm. The porosity (�) of the packing was estimated frommeasurements of the bed density rb = 1547 kgm−3 (Figure 3b)

and the density of silica glass rg = 2460 kg m−3, yielding abulk porosity of ∼37% [e.g., Bernal and Mason, 1960;Jaeger and Nagel, 1992].[22] The permeability k the packing of beads was esti-

mated from the ratio of the measured air filtration velocity(v) and pressure difference (P) using Darcy’s law,

k ¼ � v�h

P; ð1Þ

where h is the viscosity of the gas, and h is the fillingheight. For the k measurement we used a tube with adiameter of 3.5 cm filled with glass beads. Wall effects areexpected to be negligible within these measurements sincethe diameter of the tube exceeds more than 25 particlediameters. This procedure was repeated for eight differentfilling heights (h′ = 17, 24, 44, 57, 65, 76 and 78 cm).Figure 3c shows how the measurements of the velocityversus the pressure gradient (P/h′) collapses onto a commoncurve. The slope yields an estimate for the ratio of thepermeability by the air viscosity k/h. The linear fit of thedata gave k/h = 6.8 ± 0.4 × 10−6 m2 Pa−1 s−1. Given an airviscosity at 20°C, of 1.76 × 10−5 Pa s this yields a perme-ability of 1.2 ± 0.4 × 10−10 m2. These experiments wereperformed at very low fluxes to prevent nonlinear effects ofturbulence, dynamic pressure and dynamic permeabilityinduced by deformation of the matrix. The measured valueis comparable to the Carman‐Kozeny estimate [Carman,1937] of the permeability given by the measurements ofthe mean particle diameter and the porosity,

k ¼ hdi245

�3

ð1� �Þ2 ’ 2:2� 10�10 m2: ð2Þ

Figure 2. Sketch of the experimental setup used in this study consisting of a vertically oriented Hele‐Shaw cell with backlighting. Air was injected into the bed of glass beads through an inlet. Inlet area and/or filling height were varied for each experiment.


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[23] In order to characterize the mechanical properties ofthe packing we estimated the cohesion (C), the frictionalcoefficient (m) and internal angle of friction (g) using aclassical Hubbert‐type apparatus [Hubbert, 1951; Schellart,2000; Galland et al., 2007]. For each shear test experiment,the shear stress (t) required for failure under a given normalstress (s) was measured. The normal stresses ranged from700 Pa to 2500 Pa and the measured shear stresses areplotted in Figure 3d where the linear trend defines the Mohr‐Coloumb (M‐C) envelope. The slope of the M‐C envelopeis defined as the frictional coefficient, with the internal angleof friction defined as g = arctan(m). By extrapolating of thefailure envelope down to zero normal stress the cohesion ofthe glass beads could be estimated. The best linear fit of theM‐C envelope gave m = 0.37 ± 0.02 and g = 20 ± 1° with acohesion of C = 64 ± 34 Pa. The relatively large value of C isdiscussed in section 4.1. The physical parameters measured inthe experiment are listed in Table 1.

3. Results

[24] Each experiment started with a static granular bed inthe Hele‐Shaw cell as seen in Figures 4a and 5a. The imposedair flux was slowly increased by manually opening a valveto a chamber containing compressed air. The pressure andflux were increased slowly until fluid induced deformationwere observed in the matrix. At the initiation of deformation

the critical conditions for flow velocity and pressure wererecorded.[25] In the high filling experiments with h > 14 cm, a tiny

bubble formed on top of the inlet at critical values ofpressure Pb and air flux Qb (Figure 5b). The bubble did notevolve through time at a constant inlet flux, although byincreasing the inlet flux the size of the bubble increased.This observation, and the fact that we increased the fluxslowly, supports the idea that our experiment evolved viaequilibrium states. Observations show that the presence of

Figure 3. (a) Microscope image of the glass beads (d = 280 ± 50 mm) used in the experiment. The whitedot in the center of each bead is caused by an illumination effects as the light is transmitted through thebead. (b) Mass versus volume plot used to estimate the packing density (rb = 1547 kg m−3). (c) Measure-ments of air velocity versus air pressure gradient (∼P/h′) through a tube for varying filling heights h′ ofglass beads. The slope is defined as the k/h ratio. (d)Plot of shear stress at failure versus applied normalstresses for the glass beads. The slope of the trend gives the friction coefficient m of the packing of glassbeads, while the intercept gives the cohesion C.

Table 1. Experimentally Measured Variables

Variable Value

m 0.37C 64 Paa 70 ± 5°g 20 ± 1°� 37%h 1.76 × 10−5 Pa sk 1.2 × 10−10 m2

rs 2450 kg m−3

rf 1.3 kg m−3

d 280 mmb 0.7 cmbexp 12.4xexp 0.88bana 6.8xana 1.02


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the bubble induced both lateral compaction of the matrixand a gentle lift up in the center, generating two steeplydipping reverse shear bands (Figures 4b and 5b) originatingon each side of the inlet. The uplift shear zone produces acone‐like structure with an angle a ∼ 65 ± 6° to horizontal.Although there is scattering in the experiments, observationsshow that a decreases toward the top of the bed into adiverging trumpet‐like structure and that the mean angleis positively correlated with the filling height [Nermoen,2006].[26] When increasing the inlet flux and pressure to values

exceeding a second critical threshold (Pf and Qf), full flu-idization of the bed occurred. Full fluidization occurred forall experiments, although the system behaved differently upuntil fluidization for different filling heights. In the experi-ments with h < 14 cm the transition from diffusive flow tofull fluidization occurred directly, without the static bubblephase (such that Qb = Qf) (Figure 4). In contrast, in theexperiments with h > 14 cm (Figure 5) the static bubbleformed at Qb, and by increasing the imposed velocity furtherto Qf, the bubble rapidly rose to the surface initiating thefluidization. The fluidization morphology is characterizedby convective movement where the grains are transportedupward together with rapidly ascending gas bubbles in the

center, while flowing downward along the sides (Figures 5cand 5d). Consequently, the lateral sides of the fluidized zoneexhibited inward dipping beds (Figure 5d). The fluidizationwas mainly constrained in between the preexisting shearbands imposed by the bubble described above, althoughsome lateral growth of the fluidized zone was observed withtime. The ejected material formed a crater at the rim of thefluidized zone as the materials were spewed onto the sur-face. Due to the (close to) preservation of the dip angle ofthe shear bands, the diameter of the crater increased with h.[27] For all the experiments, we plotted P as a function of

Q. In experiments with h < 14 cm, the relation between theair flux and the pressure was nearly linear before fluidization(Figure 4d). Subsequently, at the onset of fluidization, thepressure suddenly dropped whereas the flux remained con-stant (Figure 4d). In experiments with h > 14 cm, the evo-lution of P versus Q was more complex (Figure 5e). At lowfluxes, the relation between the air flux and the pressure waslinear. As the flux increased and a static bubble formed, andthe pressure values became smaller than the values expectedfor a linear trend. The size of the bubble increased forincreasing air flux, which increased the bulk permeability,and further decreased the pressure measurements. Finally,similar to the low filling height experiments, at the onset of

Figure 4. Pictures of the experiment 11 at three stages (w = 0.9 cm and h = 7 cm) (Figures 4a–4c).(a) The initial configuration of the bed. (b and c) A bubble forms above the inlet that instantaneouslygrows to the surface triggering the onset of fluidization. Dilation can be seen as brighter region extendingupward at an angle from the inlet onto the surface in Figure 4b. A distinct crater formed on top in Figure 4c.(d) Plot of themeasurements of the pressureP versus volumetric fluxQ, with the pictures indicated along themeasurement. At fluidization, the pressure drops from ∼3400 to ∼1500 ± 300 Pa while Q remains constant.


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fluidization the pressure abruptly dropped while the fluxremained constant. At fluidization the pressure dropped.[28] By going through the recorded images and selecting

the time at which the onset of fluidization occurred, wecollected the values of vf = Qf /(wb) and Pf in Table 2 for allexperiments (see Figures 6 and 7). The experiments wereorganized into four series corresponding to the inlet widthsw. In general, the required critical fluidization velocityincreased for the narrower inlet widths and increasing h.Within each series of experiments the fluidization velocitiesvf exhibited almost a linear correlation with h. It is importantto note that the scaling in the fluidization velocity does notchange whether the static stable bubble (observed for h >14 cm) exists or not. This observation indicates that thestatic bubble phase does not affect the critical conditions forfluidization, which means that the dynamic permeabilityeffects induced by the presence of the bubble plays no major

role in the experiment. We will come back to the scaling ofvf in section 4.3. The pressure at fluidization Pf is positivelycorrelated to the filling height except in the case of thesmallest inlet w = 3.6 mm. We will discuss in detail the useof the pressure measurements in section 4.2.

4. Discussion

4.1. Cohesion Measurements

[29] The glass beads used in the experiment are cohe-sionless like dry sand. However, the shear test experimentsprovided an estimate of C ∼ 65 Pa, (Figure 3d). This value ishigh compared with recent cohesion estimates of dry sandwhere they show that the Mohr‐Coulomb envelope bendsdownward for normal stresses below 200–300 Pa [Schellart,2000; Mourgues and Cobbold, 2003]. In our shear tests, thesmallest value of normal stress was ∼700 Pa such that we

Figure 5. Pictures of an experiment at four different stages (w = 0.9 cm and h = 17 cm) (Figures 5a–5d).The passive markers are spray painted glass beads. (a) The starting configuration. (b) A tiny static bubbleforms above the inlet. (c) The bubble rapidly grows to the surface and a fluidized cone (trapezoid) devel-ops. (d) The air supply is switched off. A distinct crater has formed and the passive markers dip inwardalong the margin of the zone. (e) Plot of measurements of the pressure P versus the inlet flux Q during theevolution of the experiment, with the pictures indicated along the plot. The curved red line showsthe pressure corrected by the dynamic pressure described in section 4.2.1. At fluidization (Figure 5c)the pressure drops from ∼8000 to ∼4000 ± 1000 Pa while the inlet flux remains constant.


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might have missed this effect. The extrapolation of normalstresses from 700 down to 0 Pa may, therefore, have causedthe large errors and the overestimate of the cohesion value(similar to Krantz [1991]).

4.2. Pressure Measurements

[30] The evolution of the bulk bed permeability betweenthe inlet and the free surface was captured by the dynamics ofthe pressure and velocity measurements through equation (1).We notice that in Figure 4d, the P versus Q curve is linearuntil fluidization, whereas in Figure 5e the curve bendsdownward at higher fluxes. We can relate this difference totwo effects in the experiment: dynamic pressure and bulkpermeability changes caused by grain reorganization.4.2.1. Dynamic Pressure Effects[31] The pressure measurements were taken in the T

junction ∼10 cm from the inlet (Figure 2). At the T junction,the inner area of the pipe decreased to ∼5 mm2, while thearea of the injection was, e.g., ∼63 mm2 for the w = 9 mminlet. Due to flux conservation the flow velocity increasedby the ratio of the two areas. At these velocities, Bernoulli’slaw predicts the increasing importance of the dynamicpressure Pdyn = rvT

2/2. In Figure 5e the dynamic pressure issubtracted from the Darcian linear trend extrapolated fromsmall velocities. The resulting calculated pressures corre-spond with the measured pressures at higher imposedvelocities (Figure 5e). We can thus infer that for high fluidvelocities, the dynamic component of the pressure becomesincreasingly important, causing a potential underestimate ofthe actual gas pressure above the inlet.4.2.2. Bulk Permeability Effects[32] During some of the experiments, we often observed

abrupt jumps in the pressure measurements synchronouslywith reorganization of the grains above the inlet (Figure 4d).For a packing of spheres with porosity close to random close

Table 2. Measurements of the Critical Pressures Pf and CriticalFlow Velocities vf at the Onset of Fluidization for the Given FillingHeight h and Inlet Width w

Experiment w (mm) h (cm) vf (m s−1) Pf (Pa)

1 3.6 5 5.6 2442 3.6 7 6.7 20323 3.6 10 9.5 6334 3.6 12 9.3 6095 3.6 14 12.7 32466 3.6 15 12.8 26747 3.6 17 14.9 4318 3.6 19 16.2 5489 3.6 21 17.0 57310 9.0 1 0.9 18311 9.0 3 1.6 80512 9.0 7 3.0 339513 9.0 10 4.0 398114 9.0 11 4.3 441115 9.0 15 5.9 698116 9.0 20 6.6 802317 9.0 25 7.2 904718 24 5 0.7 34019 24 7 1.1 95120 24 9 1.5 332821 24 10 1.7 445122 24 12 2.0 476823 24 14 2.2 421624 24 14 2.5 868925 24 14 2.3 872026 24 15 2.7 987227 24 17 2.8 973528 24 19 3.2 1283129 24 21 3.5 1261030 59 5 0.4 192431 59 10 0.6 305632 59 15 0.8 540333 59 20 1.3 931234 59 25 1.3 17250

Figure 6. Plot of the measurements of Pf versus the fillingheight h for all experiments. Scattering is observed withineach series. In general, larger filling heights require largerPf, but there is no clear correlation between Pf and the inletsize w.

Figure 7. Plot of the critical fluidization velocity vf /� (inthe bed) versus the filling height h for all experiments. Apositive correlation is observed for increasing filling heightat constant inlet size w. Smaller inlets require larger vf andvice versa.


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packing any reorganization lead to both dilation along shearbands and also potential compaction, which will affect thebulk permeability of the bed.[33] The experiments with filling heights h > 14 cm, the

formation of the static bubble above the inlet rearrangesthe grains such that the bulk permeability between the inletand the free surface is affected. The bulk permeability isexpected to increase because (1) the distance between the tipof the bubble and the surface is reduced compared to theinitial distance from the inlet to the surface, (2) the perme-ability in the bubble (free space) is significantly higher thanthe permeability in the porous packing and (3) dilation ofthe packing occurs along two shear bands originating fromthe inlet onto the surface at an angle a. As explained insection 3, the geometry and size of the bubble depends onthe imposed flux. Increasing in flux increases the bubblesize, with a subsequent increase in the bulk permeabilitysuch that the measured pressure decreases further comparedto the expected linear trend.[34] To summarize, the nonlinearity of the P versus Q

curve is potentially explained by a combination of dynamicpressure effects in the T junction and the dynamic bulkpermeability k increase due to lateral compaction, dilationalong the shear bands and the reduced height to the surfacefrom the bubble tip. The measured fluid pressure is thus nota relevant proxy for constraining the physical processes inthe experiments. Any variation in k caused by the listedeffect resulted in variations in P while the inlet filtrationvelocity vin = Q/(wb) remained unaffected. Therefore, wewill use and discuss vin as the controlling parameter for the

onset of deformation, similar to what is done by, e.g., Kuniiand Levenspiel [1969], Freundt [1998], and Sparks et al.[2006].

4.3. Dynamics of Fluidization

[35] In all experiments, two steeply dipping reverse shearbands formed either during the bubble phase or at the onsetof fluidization. In Figure 5b, the shear bands are clearlyvisible due to the deformation of the passive markers. Forexperiments without the passive markers, shear bands canbe seen as brighter zones due to dilation in the packing(Figure 4b). The shear bands accommodated uplift of thebed above the inlet. The force resulting from the injection ofthe air into the bed exceeded both the weight and theapparent strength of the bed. Similar processes have beenobserved in other experiments such as, e.g., Walter andTroll [2001], Galland et al. [2007], and Galland et al.[2009], who simulated the inflation of shallow magmaintrusions in the upper crust.[36] One might expect that the formation of the static

stable bubble affect the dynamics of fluidization. As dis-cussed above, the formation of the bubble modifies the localpermeability, and thus the pore pressure distribution. How-ever, we noticed that the existence of the static stable bubbleabove the inlet at high sediment thickness does not affectthe scaling of the fluidization velocity vf (Figure 6). On thatbasis we infer that the static bubble forming on top of theinlet prior to fluidization plays a minor role for the criticalfluidization conditions, and we will not consider it in thefollowing analytic development.4.3.1. Governing Parameters[37] The relevant parameters in the system were assumed

to be the permeability of the bed k, the air viscosity h, thefilling height h, the inlet width w, the gravity g, the densityof the air rf and the glass beads rs. The cohesion of the bedis assumed negligible in the further discussion due to itsrelatively low value compared to the weight of the bed, C�((1 − �) (rs − rf)gh). Thus, we treat the experiments as anexample of fluid flow through a cohesionless density con-trolled deformable porous media. Dimensional analysis re-duces the number of parameters to two dimensionlessparameters, h/w and vf /(�vd), where vf is the inlet fluidvelocity at fluidization (and vf /� is approximately the gasvelocity within the bed) and vd is the intrinsic Darcy velocitygiven by the parameters listed above,

vd ¼ k

�ð1� �Þð�s � �f Þg ’ 0:11 m s�1: ð3Þ

[38] By plotting the scaled critical fluidization velocitiesvf /(�vd) versus the aspect ratio h/w, we obtain a data col-lapse of all our measurements onto a master curve (Figure 8).The curve seems to follow a power law of the form

vf = �vdð Þ ¼ �h

w

� ��

; ð4Þ

with the parameters b = 12.4 ± 0.3 and x = 0.88 ± 0.03. Nosystematic dependency is observed to the existence, or non-existence, of the bubble phase.

Figure 8. Phase diagram for all experiments showing thevelocity at fluidization vf /� (in black) and the bubbling veloc-ity vb/� (in green) normalized by the intrinsic Darcy velocityvd versus the aspect ratio h/w. The solidline represents thescaling relation fitted to the measurements of vf /(vd�) (withthe parameters b and x listed in Table 1).


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[39] A combination of equations (3) and (4) yielded asemiempirical expression for the critical velocity,

vf ¼ �k

�ð1� �Þð�s � �f Þg� h

w

� ��

; ð5Þ

where the first part on the right‐hand side, k/h, correspondsto the diffusivity of the grain packing, i.e., the ability ofthe fluid to flow through the porous bed. Increasing diffu-sivity increases the critical velocity at which fluid‐induceddeformation occurs. The second part of this equation, (1 −�) (rs − rf)g, represents the weight of the bed corrected bybuoyancy effects. Decreasing the difference in densitybetween the injected fluid and the bed lowers the criticalvelocity. The third part of the equation, the aspect ratio(h/w)x, quantifies the geometrical effect of the setting on thefluidization velocity. Increasing aspect ratio, (the ratio ofthe sediment thickness to the inlet width) increases thecritical fluidization velocity vf. Finally, b is a constant ofproportionality.

4.4. A Lower Bound for vf[40] In order to understand and predict the initiation of

fluidization and the experimental scaling of vf /(�vd) wepresent an analytical model inspired by the minimal velocityof fluidization Kunii and Levenspiel [1969]. We evaluate themodel predictions with the experimental results of b and x.The analytical model setup is shown in Figure 9 and consistsof a H‐S cell with lateral domain width 2lx, filled with agranular media with permeability k and sediment thicknessh. A fluid of viscosity h is injected through the bottom at avelocity vin through a central inlet of width w. Above theinlet, we define a truncated wedge that represents the flu-idized zone. The wedge is bounded by two shear planesoriginating from the inlet sides at an angle a with the hor-

izontal. In the proceeding analysis we make the approxi-mation that the angle a of the initial shear zone is constantthroughout the domain and for all filling heights.[41] Along the lines of Kunii and Levenspiel [1969] and

Gidaspow [1994] the minimum velocity of fluidization iscalculated from the quasi‐static approximation implying thatfluidization occurs when the seepage force Fd equals thebuoyant weight of the bed Fg integrated over the wholetruncated wedge (see Figure 9). In Appendix A we calculatethe pressure field analytically by solving the Laplaceequation (r2P = 0) with boundary conditions similar tothose in the experiment (see Figure 9),

Pðx; yÞ ¼ �

k

vin�

w

lxðh� yÞ þ

X1n¼1

2 sinðknw=2Þn�kn coshðknhÞ

"

� cosðknxÞ sinhðknðh� yÞÞ#; ð6Þ

where the origin of the coordinate system is placed at theinlet and kn = 2np/lx. A plot of the 2‐D pressure field isshown in Figure 9.[42] The weight of the truncated wedge is estimated byFg =

(1 − �) (rs − rf)gV with the volume V = h(w + 2 cot(a)). Thetotal seepage force transmitted from the fluid onto the solidmatrix can be found by integrating the vertical gradients inthe pressure over the total volume V of the truncated wedge,

Fdðvin; h;wÞ ¼Z

@Pðx; y; vin; h;wÞ@y

dV ; ð7Þ

where P(x, y) is given in equation (6). The critical inletvelocity was found by calculating the y derivative of thepressure field in equation (6) as a function of the inlet velocity,integrating up ∂P/∂y throughout the truncated wedge and

Figure 9. Calculated pressure field within the H‐S cell. The boundary conditions are along the top sur-face P = 0, along the two sides we allow no horizontal gradients in the pressure field (i.e., no flux bound-aries), and along the bottom we impose a constant influx only through the central inlet ∂P/∂y = hvin/k� =const (no vertical flux elsewhere). The plot shown here is of filling height h = 30 cm, inlet width w = 3 cmand inlet flow velocity vin = 1 m/s. The truncated wedge is denoted between the two solid lines originatingfrom the inlet onto the surface at an angle a.


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comparing the weight and the seepage force, and thenincreasing vin if the integrated value of Fd is lower (or higher)than Fg. At fluidization, we employ the quasi‐static approx-imation, that the total weight and the viscous drag balance out(Fd = Fg), such that vin = vf could be uniquely defined. Asystematic use of the procedure above for the experimentalvariations in h and w, generate the analytic phase diagramshown in red (gray) in Figure 10.4.4.1. Comparing the Experimental and AnalyticResults for vf /(fvd)[43] In Figure 10 the fluidization velocities obtained from

the experimental and the analytic models are plottedtogether. The results of the analytic model are consistentwith the experimental results: an identical trend is observedwith similar scaling factors (xexp = 0.88 versus xana = 1.02).However, the analytic model underestimates the experi-mental results since bana ’ 6.8 while bexp ’ 12.4 suggestingthat additional effect can be included in the full theoreticalanalysis. The consistency between the experimental andanalytic scaling indicate that the main triggering force forfluidization are the bulk seepage force (∂P/∂y) integratedover the entire truncated wedge, and not only the pressure atthe inlet.[44] This result, together with the consistency of the

scaling of vf, has implications for interpreting the influenceof the static bubble prior to fluidization occurring only in thehigh filling experiments. The effects of the local dynamicpermeability at the bubble are small compared to the forcesintegrated over the rest of the system. Consequently, thediscrepancy between the analytic and experimental models(in Figure 10) cannot be explained by dynamic permeabilityeffects.

[45] The underestimate in the theoretical prediction couldbe caused by frictional effects in the experiment causingadditional dissipation of the energy imparted to the systemby the imposed flow. Examples include friction along thesidewalls Janssen [1895] and shear resistance within thepacking. Gas‐induced fluidization is defined by the transi-tion from the static solid‐like behavior, where the packingcan sustain differential stresses, to a liquid‐like behaviorwhere any nonzero differential stresses induce flow. Inweight dominated and cohesionless systems the shearstrength of the material is given by the frictional coefficientand the normal stress onto a shear plane, ts = msn. Thenormal stress onto the plane directed in the a direction canbe calculated by sn = sv (� sin2 a + cos2 a), in plane stressconditions where � is the coefficient of side stresses relatedto the Poisson ratio by � = n/(1 − n). If now the verticalstress is reduced by the increasing seepage forces driven bythe vertical gradients in the pressure field,

vðyÞ ¼Z y

0�bg � @P

@y

� �dy; ð8Þ

we approach the condition for fluidization when sv → 0which leads to both sn and the shear strength ts → 0. If theshear strength is negligible, the frictional dissipation withinthe grain package drops to zero. This, however, is a meanfield argument, which does not exclude the existence oflocal frictional effects still being present within the system.An additional argument is that the planar area, which theshear force acts over at initiation, is small compared to thetotal system size.[46] In the experiment the gas is flushed into the cell

through the inlet raised 7 cm into the bed. This effect is nottaken into account in the analytic boundary conditionswhere the inlet is placed at the bottom of the cell. Thisdifference could presumably also lead to an underestimateof the analytic prediction of vf since the gas can also flowdownward at both sides of the inlet.

4.5. Predicting Critical Fluid Pressures

[47] In the experiment, we controlled the fluid velocitywhile the pressure measurements varied in response tovariations in the bulk permeability caused by deformation ofthe bed. By dimensional analysis we obtained a data col-lapse of all critical fluidization velocities onto a mastercurve. Such a master curve was not obtainable for the crit-ical pressures since the velocity, and not the pressure, wasthe controlling parameter in the experiments. The pressuredropped abruptly at fluidization such that an uniquely criticalpressure could not be defined. However, based on theory[e.g., Kunii and Levenspiel, 1969; Terzaghi, 1943; Rozhkoet al., 2007], experiments [e.g., Mourgues and Cobbold,2003; Walters et al., 2006; Freundt, 1998; Sparks et al.,2006] and field observations [e.g., Cobbold and Castro,1999; Jamtveit et al., 2004; Planke et al., 2005; Svensenet al., 2006], pore fluid pressure is expected to be the con-trolling parameter for the formation of piercement structuresin geological environments. We will therefore use the ana-lytic model for the pressure field in combination with boththe experimental and analytic scaling relation for vf to pre-dict the critical pressures. Combining the pressure equationin equation (6), the scaling relation in equation (5) and the

Figure 10. Plot of the analytic model of vf /(�vd) in red forvalues of h/w corresponding to those in the experiment(black). The black solid line represents the experimental dataof the critical velocity at fluidization plotted in Figure 8 withbexp = 12.4 and x = 0.88. In the analytic model the cor-responding values are bana = 6.8 and xana = 1.02.


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definition of the Darcian velocity the critical overpressureabove the inlet (y = x = 0) is written

Popc ¼ ð�b � �f Þg� h

w

� �� wh

lxþX1n¼1

2 sinðknwÞ tanhðknhÞn�kn

" #; ð9Þ

where b and x can be replaced by the experimental andanalytical values.[48] In a natural setting there is no lateral box‐size lx such

that by letting lx → ∞ the sum is replaced by an integral,

Popc ¼ ð�b � �f Þg� h

w

� ��2

�

Z 1

0

sinð~kwÞ tanhð~khÞ~k2

d~k: ð10Þ

This expression is used to estimate the critical pressure inselected geological settings (see section 4.6). equation (10)can be interpreted to be a quantification of how the locali-zation of the pressure anomaly affects the critical pressure atfluidization.[49] By normalizing the overpressure by the hydrostatic

pressure we obtain the definition of the venting numberproposed by Jamtveit et al. [2004],

Ve ¼ Popc

�f gh

¼ �b � �f�f

�h

w

� ��Z 1

0

2 sinð~kwÞ tanhð~khÞ~k2 d~k; ð11Þ

where the solution of the integral evaluated is evaluatednumerically. This relation can be used as a predictive toolto estimate the critical fluid pressures needed to form thepiercement structures in geological systems.

4.6. Geological Implications

[50] There are close morphological similarities betweenour experiments and the piercement structures observed innature (see section 1 and Figure 1. Some primary similaritiesare (1) the apparent conical shape observed in, e.g., shallowhydrothermal vents in the Eastern Karoo [Svensen et al.,2006] and kimberlite pipes [Walters et al., 2006] with adipping angle a ∼ 65°, (2) the bedding in the vicinity of thevents exhibit inward dipping structures [e.g., Walters et al.,2006; Svensen et al., 2006], and (3) the evidence of intensefluid‐assisted deformation within the vents, i.e., brecciationand fluidization. For these reasons we infer that the experi-ments simulate the key processes involved in the formation

of piercement structures in general. We use equation (10) toobtain crude estimates of critical fluid overpressures in somegeological cases.4.6.1. Lusi Mud Volcano[51] Mazzini et al. [2007] documented the prehistory and

birth of the Lusi mud volcano (East Java, Indonesia). Priorto the eruption, pore pressure data was recorded in a drillingsite ∼150 m away of the main eruption site. The origin of theejected fluids have been estimated by comparing the ejectedmud with drilling samples to a depth of h ∼ 1200 m. At thisdepth the fluid overpressure is ∼27 MPa [see Mazzini et al.,2007, Figure 2].[52] We can compare this value to the critical pressure

estimated from the analytic and experimental models(equation (10)). The missing parameter in the model is thewidth w of the pressure source. Seismic profiles [seeMazziniet al., 2009, Figures 6A and 6B] suggest that the Lusi mudvolcano is rooted at the top of a diapiric like structure. If weconsider the diapir as an effective “inlet,” the parameterw canbe taken as the width of the diapir. We assume in the fol-lowing that w ’ 600 m. By estimating the bulk bed densityof the country rock to rb = 2000 kg m−3 and the mud to rf =1200 kg m−3 we find critical fluid overpressures of 66 and113 MPa from the analytic and experimental models,respectively (see Table 3). This number is an overestimate ofthe measured fluid pressure.4.6.2. HTVC, Kimberlites, and Pockmarks[53] For the HTVC in the Karoo Basin, the fluids involved

in the process are likely to be aqueous fluids with a densityof rf ∼ 1000 kg m−3. The bed density of the country rock canbe approximated to rb ∼ 2100 kg m−3. Typical valuesextracted from the geological observation are of the orderof h ∼ 500 m and a roughly estimated width of the feederzone is w ∼50 m [Svensen et al., 2006]. These values pro-vide critical pressure estimates of 84 and 113 MPa from theanalytic and experimental models, respectively. Similarly,estimates of Pf for kimberlites and pockmarks with thegeometrical assumptions shown in Table 3 are 246 and1.2 MPa from the analytic model and 364 and 2.1 MPa fromthe experimental model, respectively. This is consistent withtheir respective geological settings, as pockmarks are shal-lower than Witkop II and III vents, which are shallower thanthe kimberlitic structures. The estimated values of Pf basedon both the experiment and analytic prediction are sum-marized in Table 3.[54] These values are much larger than the lithostatic

stresses at the depth of fluidization (Table 3). This suggests

Table 3. Example of Estimated Critical Pressures in Some Geological Cases Using Equation (10) and the Experimental and AnalyticValues of b and xa

Setting rf rb h w Pf,mod Pf,exp Plith Fluid

Lusi 1200 2100 1200 600 66.3 MPa 113 MPa 24 MPa Water+MudPockmarks 1000 1700 30 30 1.2 MPa 2.1 MPa 0.5 MPa WaterWitkop II and III 1000 2100 500 50 84 MPa 113 MPa 10 MPa WaterKimberlites 1500 2500 2000 400 246 MPa 364 MPa 49 MPa WaterExperiments 1.3 1540 0.1 0.024 17 kPa 26 kPa 1.5 kPa Air

aWe use data for the Lusi mud volcano from Mazzini et al. [2007]; pockmarks from Ø. Hammer (personal communication, 2008); Witkop II and IIIhydrothermal vent complexes from Svensen et al. [2006]; and kimberlites from Lorenz and Kurszlaukis [2007]. The input values are rough order ofmagnitude estimates, which will vary significantly from case to case. The estimate of Pf fits within order magnitude to the experiments (∼10 kPa in theexperiment) but overestimates the measured pore fluid pressure prior to eruption for the Lusi mud volcano (27 MPa, see Mazzini et al. [2007, Figure 2]).


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that the models overestimate the critical fluid pressure.Further experimentation and theoretical modeling needs tobe performed in order to better constrain the critical pressuresof fluidization.

5. Summary and Conclusion

[55] Piercement structures, such as pockmarks, mud vol-canoes, hydrothermal vent complexes and kimberlites, arecommon features on earth. These structures form in responseto the buildup of fluid pressures at depth. The goal of thiswork was to predict the conditions at which piercementstructures form in general.[56] We designed an experimental setup to study fluidiza-

tion processes. The setup consisted of a vertically orientedHele‐Shaw cell filled with glass beads, in which compressedair was injected. We varied the inlet width w and depth h. Ourmain conclusions from the experiments are as follows:[57] 1. At a critical air velocity, fluidization occurred

within a cone, with morphological characteristics similar tothose observed in nature.[58] 2. The inlet velocity is the main proxy for initiation of

fluidization in the presented experiments. Our results showthat vf /(�vd) almost scales linearly with h/w.[59] We proposed an analytic model for vf defined when

the weight is balanced by the seepage force integrated overthe fluidized zone. The main conclusions from the analyticmodel are the following:[60] 1. The model predicts the scaling observed in the

experiments between vf and h/w.[61] 2. The analytical model underestimates the experi-

mental results. This discrepancy could be related to the artifi-cial constraints imposed by the walls of the H‐S cell, localizedfrictional effects at fluidization, the position of the inlet, andturbulence effects leading to nonlinear seepage forces.[62] In geological systems the pore fluid pressure is com-

monly used as the primary proxy for fluid‐induced defor-mation, fluidization and the formation of the piercementstructures in general. By converting the critical velocitiesobtained from the models, we derived an analytical equationfor the critical pressure for fluidization. We employed thisexpression to a variety of geological settings such as pock-marks, mud volcanoes, hydrothermal vent complexes andkimberlites. The values obtained from the model seem tooverestimate the critical pressures, notably so at the Lusimud volcano (Indonesia) where the pore fluid pressureswere measured prior to the eruption.

Appendix A: Calculating P(x, y)

[63] In Appendix A we sketch the derivation of thepressure field used in equation (6) calculated from theLaplace equation r2P = 0 with the boundary conditionsnoted in Figure 9. First, assume that the pressure can bewritten as a product of two functions P = e(x) f (y), such thatby using the Laplace equation,

1

e

@2e

@x2¼ � 1

f

@2f

@y2¼ �k2; ðA1Þ

since e is only a function of x and f is only a function ofy implies that k is constant, such that

@2

@x2e ¼ �k2e ðA2Þ

@2

@y2f ¼ k2f ðA3Þ

which has the basic solutions e1 = sin(kx), e2 = cos(kx),f1 = exp(ky) and f2 = exp(−ky), where k > 0. The k = 0term is treated separately. Now since the left and right side-walls have zero flux conditions, i.e., ∂P/∂x = 0 at x = 0, fromthis we can omit the e1 solution.We find k by evaluating ∂e/∂x=−ksin(kx) = 0 at x = ~lx which is satisfied only if kn = np/~lx.[64] For k = 0 equation (A1) reads

@2e

@x2¼ 0 ¼ @2f

@y2ðA4Þ

which shows that e(x) and f(y) are linear functions in x and y,respectively. Inserting the right‐ and left‐hand boundaryconditions, we note that e(x) is constant.[65] The general solution of the Laplace equation with

zero flux at the left‐ and right‐hand boundaries is the linearcombination of the above derived basic solutions,

Pðx; yÞ ¼ a0yþ b0 þX1n¼1

cosðknxÞ an expðknyÞ þ bn expð�knyÞð Þ:

ðA5Þ

On the top boundary, y = h, we impose zero pressure. Sincethis is valid for all x, we get

an expðknyÞ þ bn expð�knyÞ ¼ 0; for all n � 1 ðA6Þ

a0yþ b0 ¼ 0; ðA7Þ

and we obtain the solution

Pðx; yÞ ¼ cðh� yÞ þX1n¼1

dn cosðknxÞ sinhðknðh� yÞÞ; ðA8Þ

where dn = −2an exp(kh).[66] Until now we have not used the bottom boundary

condition such that both c and dn are still unconstrained. Thefull solution of the pressure equation is obtained by con-sidering the bottom boundary condition g(x).[67] Consider the case

@

@yPðx; y ¼ 0Þ ¼ gðxÞ ¼

�

k

vin�

if 0 < x < ~w ¼ w=2

0 if ~w < x < ~lx ¼ lx=2

8<: ðA9Þ

when the origin of the coordinate system is placed on top ofthe inlet. We now want to express this condition as a cosineexpansion:

gðxÞ ¼X1n¼0

cn cosðknxÞ: ðA10Þ


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The coefficients in the cosine expansion are found by usingthe orthogonality of the basis functions,

Z 1

0cosðn�uÞ cosðm�uÞdu ¼ 1

2n;m; ðA11Þ

where dn,m is the Kronecker delta function. Equation (A10)is multiplied by cos(kmx) and integrated from x = 0 to x = ~lx,and by using that g(x) = �

kvin� up to ~w and zero elsewhere, we

obtain

�

k

vin�

Z ~w

0cosðkmxÞdx ¼

X1n¼0

cn

Z ~lx

0cosðknxÞ cosðkmxÞdx; ðA12Þ

so that

�

k

vin�

sin km~w

km¼ 1

2

X1n¼0

cn~lxn;m ¼ 1

2cm~lx: ðA13Þ

After inserting for the expression for ~lx and ~w we get

c0 ¼ �

k

vin�

w

lxðA14Þ

cn�1 ¼ 2�

k

vin�

sinðknw=2Þ�n

: ðA15Þ

Comparing the cosine expansion of g(x) and (∂P/∂y) eval-uated at (x, y = 0), we obtain that

�c ¼ c0 ðA16Þ

�dn coshðknhÞkn ¼ cn; for n � 1: ðA17Þ

Inserting the above coefficients into equation (A8), weobtain the analytic solution of the pressure,

Pðx; yÞ ¼ �

k

vin�

w

lxðy� hÞ �

X1n¼1

2 sinðknw=2Þn�kn coshðknhÞ

"

� cosðknxÞ sinhðknðh� yÞÞ#; ðA18Þ

where kn = np/~lx = 2np/lx.

Notation

b inlet breadth [m].C interparticle cohesion [Pa].d bead diameter [m].Fd seepage (drag) force [Pa m2].Fg buoyant weight [Pa m2].Fs integrated shear force [Pa m2].g constant of gravity [m s−2].h bed filling height [m].k bed permeability [m2].P fluid pressure [Pa].Pd dynamic pressure [Pa].

Pf, Pf pressure at fluidization/bubbling [Pa].Pop fluid overpressure [Pa].Phyd hydrostatic pressure [Pa].

Q inlet air flux [m3 s−1].Qf, Qb air flux (fluidization/bubbling) [m3 s−1].vf, vb critical inlet air velocity at fluidization and bubbling

[m s−1].vd Darcian velocity (in the bed) [m s−1].vin inlet air velocity from Q/(wb) [m s−1].v air filtration velocity within the bed, equal to vin /�.

Ve venting number [1].w inlet width [m].a angle of shear zone [°].

bana,exp constant of proportionality in equation (4) [1].h air viscosity [Pa s].g internal angle of friction [°].� porosity [1].� coefficient of side stresses [1].m frictional coefficient [1].rf air density [kg m−3].rb bed density [kg m−3].rg glass density [kg m−3].s normal stress [Pa].t shear stress [Pa].x power law exponent in equation (4) [1].

[68] Acknowledgments. The study was supported by grants from theNorwegian Research Council to B. Jamtveit, A. Malthe‐Sørenssen, andH. Svensen through a Centre of Excellence and PetroMaks grant to Physicsof Geological Processes (PGP). We have benefited from fruitful discussionsfrom our colleagues of PGP, especially Dag K. Dysthe, Simen Bræck, SeanHutton, Simon D. deVilliers, Øyvind Hammer, Galen Gisler, and AdrianoMazzini. Two anonymous reviewers are thanked for their insightful andvaluable input improving the quality of the article.

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