physical modelling of sand injectites

Tectonophysics 474 (2009) 610–632

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Tectonophysics

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Physical modelling of sand injectites

N. Rodrigues a, P.R. Cobbold a,⁎, H. Løseth b

a Géosciences-Rennes (UMR6118), CNRS et Université de Rennes 1, 35042 Rennes Cedex, Franceb StatoilHydro Research Centre, Trondheim, Norway

⁎ Corresponding author.E-mail address: [email protected] (P.R.

0040-1951/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.tecto.2009.04.032

a b s t r a c t
a r t i c l e i n f o
Article history:Received 21 October 2008Received in revised form 20 March 2009Accepted 28 April 2009Available online 5 May 2009

Keywords:IntrusionInjectiteSandPhysical modellingOverpressureFluidization

Sand injectites are structures that result from intrusion of fluidized sand into fractures. We have studiedthem in the Tampen Spur area of the North Sea, and have reproduced them experimentally, by drivingcompressed air through layers of sand, glass microspheres, and silica powder. The silica powder was cohesiveand capable of hydraulic fracturing, whereas the sand and glass microspheres were almost non-cohesive andtherefore able to fluidize. The models were dynamically similar to their natural counterparts, for as long asequilibrium was static. When the processes became dynamic, so that inertial forces were significant, thescaling was approximate and the corresponding Reynolds numbers differed. The experimental apparatus wasa square box, 1 m×1 m wide, resting on a grid of fluid diffusers. During the experiments, the fluid pressureincreased, until it attained and surpassed the weight of overburden. Flat-lying hydraulic fractures, containingair, formed within cohesive and least permeable layers. Heterogeneities in material properties and layerthicknesses were responsible for localizing fracture networks. When any one network broke through to thesurface, rapid flow of air through the fractures fluidized the underlying mobile materials and even depletedsome of the layers. Some of the fluidized material extruded at the surface through vents, forming volcanoesand sheets. The remainder lodged at depth, forming sand injectites or laccoliths. Conical sand injectitesformed preferentially, where layers had high resistance to bending. Laccoliths formed nearer the surface,where overlying layers had low resistance to bending. The experimental sand injectites were broadly similarto those in the Tampen Spur area of the North Sea, as well as other areas.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Sand injectites are intrusive bodies, which result from the remo-bilization and injection of sand into fractures. Typically, such fracturesare in sedimentary strata. Early descriptions of sand injectites at outcropdate from the 19th century (Murchison, 1827). In the last few decades,many examples of sand injectites have come to attention, as the numberand quality of seismic surveys have increased (Huuse et al., 2007). Theinjectites have a worldwide distribution (Hurst and Cartwright, 2007)and occur in sediments of all ages, from Neoproterozoic (Williams,2001) to Holocene (Obermeier, 1989). Good examples occur at outcropin California (BoehmandMoore, 2002; Schwartz et al., 2003; Thompsonet al., 2007; Vigorito et al., 2008), France (Parize and Friès, 2003; Parizeet al., 2007), Greenland (Surlyk and Noe-Nygaard, 2001; Surlyk et al.,2007) and Chile (Winslow, 1983; Hubbard et al., 2007). In contrast, thebest seismic examples are probably those from theNorth Sea (Jenssen etal.,1993; Dixon et al.,1995;Molyneux et al., 2002; Lonergan et al., 2000;Duranti et al., 2002; Løseth et al., 2003; Huuse et al., 2004; Huuse andMickelson, 2004; Shoulders et al., 2007). From 2D and 3D seismicsurveys, the shapes of the intrusive bodies and their relationships withtheir host rocks are becoming clearer (Hurst and Cartwright, 2007;

Cobbold).

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Huuseet al., 2007). Like igneous intrusive bodies, sand injectites occur asdykes, sills or laccoliths. In the North Sea, conical bodies and laccolithsare common (Cosgrove and Hillier, 2000; Løseth et al., 2003; Huuseet al., 2007). The apical zone of a conical injectite is typically concordantto bedding (Cartwright et al., 2008) andmay be narrow (apical) orwide(flat-based). In contrast, the flanks are discordant to bedding. Apicalcones may be 0.5 to 2 kmwide, 50 to 300 m high, and 10 to 50 m thick(Molyneux et al., 2002; Huuse et al., 2007; Cartwright et al., 2008). Flat-based cones tend to be larger, i.e. 0.8 to 2.2 kmwide, and 130 to 290 mhigh (Cartwright et al., 2008). Laccoliths are typicallyelliptical to circularinplanview, 0.5 to 2 kmwide, and75 to 400mhigh (Hansen et al., 2005;Frey-Martínez et al., 2007). All these injectites are remarkably similar inshape to volcanic igneous bodies. For example, flat-based sand injectitesare similar to saucer-shaped magmatic sills (Polteau et al., 2008).

The sand injectites of the North Sea and Faeroe-Shetland basinshave intruded hemipelagic smectite-richmudstones of the Palaeoceneto Miocene Hordaland Group (Thyberg et al., 2000). In both basins,sand injectites are most common in Eocene mudstones (Molyneuxet al., 2002; Løseth et al., 2003; Huuse andMickelson, 2004; Shoulderset al., 2007). Other examples on the Norwegian continental margin arein Upper Cretaceous mudstones (Jackson, 2007). All these mudstonesare of very low permeability and form efficient seals (Wensaas et al.,1998; Jones et al., 2003). Today, the mudstones tend to be slightlyoverpressured (Teige et al., 1999).

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611N. Rodrigues et al. / Tectonophysics 474 (2009) 610–632

At first glance, sand injection and magmatic intrusion are broadlysimilar processes. At a large scale of observation, dykes, sills and lacco-liths result from hydraulic fracturing by overpressured fluids (Phillips,1972; Pollard and Johnson, 1973; Cosgrove, 2001; Jolly and Lonergan,2002). To fail in tension, the host rocks must be cohesive. Tensilehydraulic fractures form perpendicular to the least compressive stress(Hubbert and Willis, 1957; Secor, 1965). However, at a smaller scale ofobservation, sand injection and magmatic intrusion may be somewhatdifferent. Whatever its viscosity, magma can migrate no more than ashort distance through pore space, before it freezes. In contrast, aqueousfluids, which are responsible for sand injectites (Jonk et al., 2003, 2005),can migrate over much longer distances. Also, fluid flow through porespace, in response to pressure gradients, imparts seepage forces to thesolid framework (Mandl and Crans, 1981; Dahlen, 1990; Mourgues andCobbold, 2003; Cobbold and Rodrigues, 2007). Thus in a homogenouselastic material, under lithostatic conditions, fluid flowing verticallyupward through pore spacemay result in horizontal hydraulic fractures(Cobbold and Rodrigues, 2007).

Shallow domes are common above conical sand injectites, as aresult of bending (Shoulders and Cartwright, 2004; Shoulders et al.,2007). Such forced folds are similar to the ones above igneouslaccoliths (Pollard and Johnson, 1973). On a smaller scale, dykes andsills may be common in the roof of a laccolith, forming an intrusivehalo (Huuse et al., 2005, 2007; de Boer et al., 2007). For this to happen,stretching of the roof may be necessary (Huuse et al., 2007).

Pre-existing fractures, if suitably oriented, may reactivate as fluidoverpressure increases (Phillips, 1972; Jolly and Sanderson, 1997).Examples are polygonal faults in Eocene mudstones of the North Sea.These are extensional faults of modest throw that intersect, forming apolygonal network in map view (Cartwright, 1996; Cartwright andLonergan, 1996; Cartwright et al., 2003; Cartwright and Dewhurst,

Fig. 1.Map of North Viking Graben, North Sea, showingmainMesozoic rift faults (after Zanellasee Fig. 2. (For interpretation of the references to colour in this figure legend, the reader is r

1998). Lonergan and Cartwright (1999) identified some sand injectitesalong polygonal faults. However, Huuse et al. (2004) argued thatpolygonal faults are rarely connected and do not form conical structures.Shoulders et al. (2007) showed that polygonal faults are much steeperthan the flanks of conical intrusions and that sand injectites commonlycrosscut the polygonal faults.

The sand thatwas the source for the injectitesmusthavefluidizedandremobilized (Nichols et al., 1994; Nichols, 1995; Jolly and Lonergan,2002). A moving fluid entrains sand grains, when the viscous dragexceeds the effective weight of the grains; as well as the cohesive orfrictional stresses that keep them together. The fluid velocity at whichthis occurs is known as the minimum fluidisation velocity (Richardson,1971; Nichols et al., 1994; Nichols, 1995). To our knowledge, nobody hasyet described evidence for depletion of source layers. Even proving asource can be a major difficulty. In theMagellan Basin of southern Chile,Winslow (1983) inferred from fossil evidence that sand in dykes hadmigratedvertically through several kilometres. Shoulders andCartwright(2004) inferred a connection between source and injected sand in theFaeroe-Shetland basin. Rosales-Domínguez et al. (2005) foundOligoceneplanktonic foraminifera in sand dykes intruding Miocene sliciclasticrocks and estimated that the sand had moved upwards some 900 m.

In theNorth Sea, there is some evidence from seismic data that sandshave extruded at the sea bottom (Huuse et al., 2004, 2005; Shouldersand Cartwright, 2004; Hurst et al., 2005) The sand layers thin and dipaway from vents, typically forming low-angle laminae or thin beds(Hurst et al., 2006).

We are not aware of any previous attempts at producing sandinjectites experimentally. To do so has been a technical challenge for us.In this paper,wedescribe someof the difficulties of proper scaling,whenfluid flow is turbulent within open fractures. We explain what criteriaguided us in the choice of model materials. Thenwe describe some new

and Coward, 2003). Study area (red box) is on Tampen Spur. For seismic section (A′–B′),eferred to the web version of this article.)

Fig. 2. Sand injectites in Hordaland Group, Tampen Spur. For approximate location of seismic section, see Fig. 1. Vertical scale on seismic section is in milliseconds (two-way travel time). Velocity in Hordaland Group is about 2000 m/s (1 msrepresenting 1 m). Two of three wells (A, B and C) penetrated cemented sand (red traces are gamma-ray logs). Line drawing (top) shows interpreted bedding reflections, mobile sand (yellow) and mounds above laccoliths. Notice verticaloffsets of beds across V-shaped reflectors (V-brights). Injectites appear to terminate in upper part of Hordaland Group. Sand laccoliths in upper part of Hordaland Group may have fed extruded sand. For details, see text. (For interpretation ofthe references to colour in this figure legend, the reader is referred to the web version of this article.)

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experimental apparatus, which avoids unwanted boundary effects.During upward flow of compressed air through layers of quartz sand,glass microspheres and silica powder, we obtained sand injectites,which seemed to have realistic shapes and distributions. In the lastsection, we compare these experimental structures with possiblenatural counterparts from the Tampen Spur area of the North Sea basin.

2. Sand injectites in the Tampen Spur area

In this section, we describe some examples of sand injectites fromthe Tampen Spur area of the North Sea basin, west of the North VikingGraben (Fig. 1). Our aim is to summarize some of the salient features ofthese sand injectites, in two or three dimensions, and their relationshipto sedimentary strata. These structures served as templates for ourexperimental models.

The Tampen Spur area is a prolific oil province, comprising severalrotated Jurassic fault blocks (Fig. 1). Above the tilted and eroded crestof one such block, the Balder Fmconsists of (1) approximately 700mofCretaceous to Palaeocene clays, which are relatively flat lying, partly

Fig. 3. Structures at top of Balder Fm and distribution of high-amplitude cones. Divisions o(yellows) strikes NNE. Deeper areas (blues) are to east. Apices of cones are either just abovSegmented line (A′–B′) is of seismic section (Fig. 2). (For interpretation of the references to

consolidated and contain minor stringers of silt, sand and carbonate,and (2) Late Palaeocene tuff-rich muds. The overlying Eocene toOligocene Hordaland Group (Deegan and Scull, 1977; Isaksen andTonstad,1989) comprises 500–600m of smectite-richmuds (Wensaaset al., 1998; Thyberg et al., 2000) and minor sand intervals (Grid andSkade formations). The sands were originally thought to be openmarine deposits (Isaksen and Tonstad, 1989), but lately it has becomeclear that many are injectites (Løseth et al., 2003; Huuse andMickelson, 2004). Between the Oligocene strata of the HordalandGroup and the overlying Utsira Fm is a marked unconformity (Deeganand Scull, 1977; Isaksen and Tonstad, 1989). Upper Miocene to LowerPliocene glauconitic sands partly overly the unconformity in the studyarea (Eidvin and Rundberg, 2001). The sands may correlate with theUtsira Fm (Eidvin and Rundberg, 2001; Gregersen and Johannessen,2007). Seismic and well data reveal that the Utsira Fm consists ofseveral sub-units of sands and clays, which have had a complexgeological history (Gregersen and Johannessen, 2007). Overlying theUtsira Formation are approximately 600 m of prograding Pleistoceneglaciomarine strata.

n horizontal scale bars are every kilometre. North is at top. Ridge at top of Balder Fme top of Balder Fm (red clusters), or higher within Hordaland Group (violet clusters).colour in this figure legend, the reader is referred to the web version of this article.)

614 N. Rodrigues et al. / Tectonophysics 474 (2009) 610–632

According to basin modelling, the deeper parts of an upper Jurassicsource rock (Draupne Formation) entered the oil window during theLate Cretaceous (Johannesen et al., 2002). As a result of furthersubsidence during the Cenozoic, the deeper part of the North VikingGraben is now in thegaswindow. This gas formationmaybe responsiblefor some of the deep overpressure in the area.

Huuse and Mickelson (2004) and Huuse (2008) described andmapped high-amplitude discordant seismic anomalies in the TampenSpur area and interpreted them as conical sand injectites. They also(1) reported that ‘non-bright discordant anomalies of this type arerare, although their existence cannot be ruled out’, (2) suggested thatmounds in the upper part of the Hordaland Group were of mobilizedmud, and (3) speculated that large earthquakes, possibly assisted bypre-existing fluid overpressure, triggered sand intrusion.

In the westernmost part of the area, the flat-lying Hordaland Groupshows up exceptionally well on 3D seismic P-wave data. Explorationwells have encountered sandy intervals throughout the HordalandGroup. The sands are either non-cemented or partly cemented bycarbonate. Those that are partly cemented have high acoustic imped-ance, relative to the surroundingmuds. Where a cemented sand layer isthicker than the tuning thickness, it yields two reflections of highamplitude, a peak at the top of layer (blue or black on the seismic data),and a trough at the base (red or yellow on the seismic data). In contrast,non-cemented sandy intervals, although visible in thewells, yield weakseismic reflections. In places, vertical offsets of stratigraphic reflectionsprovide indirect evidence for discordant sands.

We have observed three types of sand-related seismic anomalieswithin the Hordaland Group, high-amplitude cones, low-amplitudecones, and mounds (Fig. 2).

2.1. High-amplitude cones

Several wells have penetrated seismic anomalies, which are cone-shaped (V-shaped in section, Fig. 2). For example,Well A penetrated thebase of one cone, whereas Well B penetrated the flank of another.

Fig. 4. Relief at top of Hordaland Fm. Vertical scale is inmilliseconds, two-way time. Divisionsrings above sand laccoliths. Erosional edge at top of Hordaland Fm (between yellows and gre(For interpretation of the references to colour in this figure legend, the reader is referred to

According to their gamma-ray logs, the corresponding intervals arecarbonate-cemented sands. In a neighbouring area, Huuse and Mick-elson (2004) studied the distribution of such high-amplitude cones. Thelargest ones have their apices just above the top of the Balder Fm (Fig. 3)and continue 400 m upwards into the Hordaland Group, where theybecome 2 km wide, or more. On a map, they seem to cluster along astructural high in the Balder Fm, next to a rift fault. The flanks of thecones appear to offset the patterns of stratigraphic reflections, in themanner of reverse faults (Fig. 2). The offsets tend to decrease upwardsand the upper parts producenohigh-amplitude reflections. Someapicesof high-amplitude cones continue downwards as short faults, havingsmall offsets at the base of the Balder Fm. Other apices overlie nearlycircular zones of low amplitude, whichmay contain fractures. Similar V-shaped anomalies, of high amplitude but small vertical extent (between100 and 200 m), are common about 100 m above the top of the BalderFm. Wells through their flanks and bases prove that these anomaliesconsist of sands, partly cemented by carbonates.

2.2. Low-amplitude cones

Within theHordalandGroup, some stratigraphic reflectionpatternsare vertically offset, but the zones of offset are not responsible for high-amplitude reflections. Theyappear as reverse faults in seismic sections,but produce low-amplitude reflections. In three dimensions, thezones of offset are conical. In some examples they extend upward, asfar as the unconformity above the Hordaland Group. This unconfor-mity truncates the conical injectites (Fig. 2). Some of the sands pene-trated bywells are not responsible for seismic reflections or discordantoffsets. Whether such sands are parallel to bedding or discordant isdifficult to interpret.

On theflanks of the conical intrusions, the dips are in the same rangeas for other areas. For the dips of 53 conical intrusions in Eocene andOligocene strata of the North Sea basin, Cartwright et al. (2008)measured a range of 7° to 33° and a mean of 22°. They also observed aslight flattening in dip with increasing depth. For the dips of 267 conical

on horizontal scale bars are every kilometre. North is at top right. Mounds (orange) formens) predates formation of mounds. Segmented line (A′–B′) is of seismic section (Fig. 2).the web version of this article.)


intrusions in the Faeroe-Shetland basin, Shoulders et al. (2007)measured a range of 6° to 57° and a mean of 26°.

2.3. Mounds

Well A penetrated a layer of sand, 65 m thick, in the upper part ofthe Hordaland Group (Fig. 2). Neither the top nor the base of the sandproduces good seismic reflections. However, some segments, severalhundred meters long, have high-amplitude reflections and variousdips and polarities. Their upper envelope describes mounds (Fig. 4),whereas their lower envelope is concordant with underlying strati-graphic reflections. The mounds are similar in shape to igneouslaccoliths. In some places, the longer discordant high-amplitudereflections are continuous, from the tops of mounds within theHordaland Group, to the flanks of mounds at the top of the HordalandGroup. In other places, the discordant reflections tie into amplitudeanomalies above the top of the Hordaland Group and these also aredue to sands. Both on amap and in section, themounds tend to overliethe upper edges of the conical injectites, indicating a degree ofcorrelation, if not interconnection, between mounds and cones.

2.4. Summary

The sand injectites in the Tampen Spur area are of several styles.Sills are common at depth and they tend to form conical networks,which have uplifted the overlying strata by an amount equivalent tothe thickness of sand. Nearer the surface, mounds (laccoliths) are

Table 1Mechanical properties of granular materials for tectonic modelling.

Material Grain size(μm)

Bulkdensity(g/cm3)

Coefficient ofinternal friction

Angle ofinternalfriction (°)

Cohesio(Pa)

Quartz sand 200–300 1.30 – 30 –

Quartz sand b500 1.53 0.58 (1.17) – 300 (42Quartz sand (Ottawa sand) 220 – – 47.6–70 –

Quartz sand 200–315 1.58 0.57 – 85

Glass microspheres (GMII) 90–180 1.61 0.65 33 160Sand (SII) 90–180 1.67 0.88 (1.60) 41.3 230 (66Quartz sand 100–300 1.44 0.65 33 –

Glass microspheres 300–400 1.60 0.44 23.9 –

Quartz sand (Fontainebleausand 1)

315–400 1.60 0.45 24.2 160


200–315 1.60 0.57 29.7 85


125–200 1.60 0.51 27 0

Loess 30–40 1.30 0.76 200Tapioca pearls 2000 – 0.74 36.5 39Quartz sand 315–400 1.60 0.42 – 11

Quartz sand b200 1.60 – – –

Quartz sand 200–315 1.60 0.35 – 28

Quartz sand b500 1.60 0.29 – 25

Hollow glass microspheres(SI-CEL)

25 0.15 0.44 23.9 1.5

Hollow aluminiummicrospheres (Microballs)

40 0.39 0.46 24.7 6

Silica powder (SI-CRYSTAL) 0–30 1.33 0.84 – 288Glass microspheres(SI-SPHERE)

0–45 1.56 – – –

Quartz sand (HN38) 0–350 1.52 – – –

Glass microspheres 100–200 1.51 0.94 43 0Glass microspheres 70–150 1.58 – – –

Glass microspheres 50–105 1.59 – – –

PVC 60–200 0.64 – 1.05 300Silica powder 0–200 1.55 – – –

Values in brackets take into account sidewall friction in testing machine (Mourgues and Co

common. Their roofs have bent, forming domes. The mounds andconical networks occur together and seem to interconnect. On thisbasis, it would seem that the bending resistance of the layers is animportant factor. Other factors of importance may be the permeabilityof the host rock and its tensile strength.

3. Physical modelling of tectonic processes usinggranular materials

The aim of this section is to describe the techniques of physicalmodelling, especially those that are relevant to sand injectites.Physical modelling of tectonic processes has a long history, goingback to Hall (1815). However, it has come of age only in recentdecades. There have been several reviews of the subject, mostlyfocusing on geological applications (see Koyi, 1997; Cobbold andCastro, 1999). Here we focus instead on the technological challengesand developments, which are relevant to the modelling of sandinjectites. Our approach is descriptive. The mathematically inclinedreader will find a more formal approach in Appendix A.

3.1. Scaling in theory and practice

Physical modelling in tectonics became quantitative, only afterHubbert (1937) reviewed and established the principles of dimen-sional analysis and scaling. This limited the range of investigation, butput it on a firmer basis. Indeed, we owe to Hubbert much of thesubsequent development of the technique.

n Tensilestrength(Pa)

Permeability(m2 Pa−1 s−1)

Intrinsicpermeability(m2)

Intrinsicpermeability(Darcy)

References

– – – – Vendeville et al. (1987)0) – – – – Krantz (1991)

– – – – Sture et al. (1998)– 1.71E−06 3.42E−11 35.258 Cobbold and Castro

(1999)– – – – Schellart (2000)

) – – – – Schellart (2000)– – – – Turrini et al. (2001)– – – – Turrini et al. (2001)– 5.00E−06 1.00E−10 101.327 Cobbold et al. (2001)

– 1.70E−06 3.40E−11 34.451 Cobbold et al. (2001)

– 1.40E−06 2.80E−11 28.372 Cobbold et al. (2001)

– 5.00E−11 1.00E−15 0.001 Cobbold et al. (2001)– – – – Grotenhuis et al. (2002)– – – – Mourgues and

Cobbold (2003)– – – – Mourgues and



Cobbold (2003)– – – – Rossi and Storti (2003)

– – – – Rossi and Storti (2003)

88 – – – Galland et al. (2006)– – – – Galland et al. (2006)

– 5.99E−09 5.80E−12 6.000 Graveleau (2008)– 1.47E−08 1.40E−11 14.500 Graveleau (2008)– 4.14E−09 4.20E−12 4.000 Graveleau (2008)– 3.45E−09 3.50E−12 3.500 Graveleau (2008)– 7.37E−09 7.20E−12 7.000 Graveleau (2008)– 4.60E−12 5.00E−14 0.050 Graveleau (2008)

bbold, 2003).

Table 2Theoretical values of average fluid velocity and Reynolds number for Couette flow of air or water in flat parallel-sided channels of various half-widths.

Half-width (m) Half-width (m)

1.00E−04 1.00E−03 1.00E−02 1.00E−01 1.00E−04 1.00E−03 1.00E−02 1.00E−01

Gradient (Pa/m) Density (kg/m3) Viscosity (Pa s) Velocity (m/s) Re (dimensionless)

Air in nature 20,000 1.20E+00 1.80E−05 3.70E+00 3.70E+02 3.70E+04 3.70E+06 4.94E+01 4.94E+04 4.94E+07 4.94E+10Water in nature 10,000 1.00E+03 1.00E−03 3.33E−02 3.33E−00 3.33E+02 3.33E+04 6.67E−00 6.67E+03 6.67E+06 6.67E+09Air in model 14,000 1.20E+00 1.80E−05 2.59E+00 2.59E+02 2.59E+04 2.59E+06 3.46E+01 3.46E+04 3.46E+07 3.46E+10Water in model 4000 1.00E+03 1.00E−03 1.33E−02 1.33E+00 1.33E+02 1.33E+04 2.67E+00 2.67E+03 2.67E+06 2.67E+09


For a physical model to be successful, it should be geometrically andkinematically similar to its natural prototype (Hubbert, 1937; Mandel,1962; Ramberg,1967). In practice, this is unlikely to occur, unlessmodeland prototype are dynamically similar. Geometrical similarity meansthat corresponding lengths in model and nature are proportional,according to themodel ratio of length. Kinematical similaritymeans thatcorresponding time intervals are proportional, according to the modelratio of time. In practice, the experimenter would like to choose thesetwo ratios independently, so as to have more freedom in designing anexperiment. However, in theory, this is not possible. Dynamicalsimilarity requires that forces at corresponding points should have thesame directions and proportional magnitudes. This applies to inertialforces, as well as to other kinds of forces. Hence, if both model andprototype are in the same gravitational field at the Earth's surface, theaccelerations at corresponding points must be identical. This in turnmeans that themodel ratios of length and timeare not independent (seeAppendix A). The restriction is strong—so strong, that proper scaling ofmodels is very difficult, if not impossible, under most circumstances(Hubbert, 1937). More generally, for complete dynamic similarity, thedimensionless ratios between forces of various origins should beidentical, in nature and experiment. The three most common kindsare gravitational forces, surface forces, and inertial forces. Their ratiosare the Ramberg number (of use for describing tectonic deformation),the Reynolds number (for turbulence), and the Froude number (forhydrodynamics).

Luckily for the experimenter, inertial forces are negligible ifprocesses are slow. This means that in practice the model ratios oflength and time are independent. Moreover, the only dimensionlessratio of consequence is the Ramberg number. To obtain identicalvalues of this ratio in nature and experiment is relatively easy. If modeland prototype are in the same field of gravity, and share the samedensity, any quantity having dimensions of stress should scale downin the sameway as the linear dimensions. Hubbert (1937) was the firstto realize this, and it was a major breakthrough, although it stillappears to be a source of confusion (Wickham, 2007).

3.2. Sand: a suitable material for brittle deformation

Hubbert (1951) was also the first to state that dynamical scaling ofbrittle models is especially simple, if inertial forces are negligible.Assuming that brittle rock and its model equivalent fail in compressionaccording to a Mohr–Coulomb criterion, two parameters are enough to

Table 3Mechanical properties of materials in our experiments.

Material Variety Roughness Grain size(μm)

d10(μm)

d50(μm)

d90(μm)

Silica powder Millisil C10 Angular 0–350 63.3 22.7 4.3Silica powder Millisil C4 Angular 0–150 177 64 8.8Diatomite powder DICS Angular 0–350 200 35–50 10Glass microspheres CVp Round 0–45 – 21 –

Quartz sand GA39 Sub-round 0–160 106 90 75

Measurements of cohesion are scarce, so we have ranked relative values according to heightshighest (5) being silica powder (from Galland et al., 2006).

describe their material properties: the cohesion, and the coefficient ofinternal friction. The cohesionhas thedimensionsof stress and thereforescales down as the linear dimensions. The coefficient of friction isdimensionless and therefore shouldbe identical inmodel andprototype.Such behaviour is not time-dependent and therefore it does not set anymodel ratio of time.

It so happens that dry sandmakes a goodmodelmaterial, because itscohesion is small and its coefficient of friction is similar to that of brittlerock. That is why Hubbert (1951) introduced sandbox modelling intotectonics. Although it was a few years before this very practicaltechnique became established (Horsfield, 1977; Davis et al., 1983;Malavieille, 1984), it then found applications to many problems intectonics.

More generally, the mechanical properties of a dry granularmaterialdepend on the shape, roundness, roughness, and packing of the grains(Table 1). Also, the cohesion tends to increase as the grain sizediminishes, in part because of electrostatic forces.

3.3. Use of pore fluids

Pore fluids may modify the effective stresses in a solid framework.Hubbert introduced the concept of fluid overpressure into tectonics, soas to explain large overthrusts (Hubbert and Rubey, 1959). However,there is no record thatheeverusedporefluids in sandboxmodels. Toourknowledge, the first to do so were Cobbold and Castro (1999) andCobbold et al. (2001).

The concept of effective stress came from soil mechanics (VonTerzaghi, 1923). A common rendering of it is that the effective stress isthe total stressminus the pore fluid pressure. However, this definition istoo simple. Instead, it is necessary to consider fluid migration throughpore space. According to Darcy's law, themacroscopic discharge velocity(Darcy velocity) of a pore fluid is directly proportional to the intrinsicpermeability of the porous medium, and to the overpressure gradient,and inversely proportional to the dynamic viscosity of the fluid. This lawholds well for granular materials, for values of fluid pressure as high asthe weight of overburden (Cobbold and Castro, 1999). Fluid flowingthrough the pores modifies the balance of forces, applying a seepageforce to each element of the solid framework, and this effect is readilyvisible in sandbox models, where it influences fault orientations(Mourgues and Cobbold, 2003, 2006a,b).

The effective stresses and fluid overpressure scale down in the sameway as the total stresses, if inertial forces are negligible. However, the

Particledensity(g/cm3)

Bulk density(g/cm3)

Rank forcohesion

Permeability(m2 Pa−1 s−1)

Intrinsicpermeability(m2)

Intrinsicpermeability(Darcy)

2.65 1.15 5 7.00E−08 1.400E−12 1.4192.65 1.34 4 8.00E−08 1.600E−12 1.6212.65 0.29 5 7.50E−07 1.500E−11 15.1992.46 1.39 2 1.00E−07 2.000E−12 2.0272.64 1.43 1 7.00E−07 1.400E−11 14.186

of free vertical faces, lowest (1) being quartz sand (fromMourgues and Cobbold, 2003),

Fig. 5. Experimental apparatus for modelling sand injectites. Longitudinal section (A) and three-dimensional view (B) show Plexiglas box for housing models (1), flow diffusercontaining sand (2), metallic mesh (3), reservoir for compressed air (4), pressure regulator (5) and tilting table (6). Pressure probes (narrow aluminium tubes) protrude horizontallythrough sidewalls of box (7). Their outer ends connect to U-tubes, containing water (8).

Table 4Reference values and scaling of models for steady flow (before fluidization).

Parameter Units Hordaland Group Experiment Model ratio

Length m 1.00E+03 0.10 1.00E−04Bulk density kg/m3 2.10E+03 1.20E+03 0.57Effective stress Pa 2.06E+07 1.18E+03 5.71E−05Permeability m2 Pa−1 s−1 2.66E−15 7.00E−08 2.63E+07Darcy velocity m/s 5.48E−11 8.24E−04 1.50E+07Time s 1.82E+13 1.21E+02 6.65E−12

Estimates for Hordaland group are from Wensaas et al. (1998).


intrinsic permeability and fluid viscosity set a time scale for steady flow.In practice, the experimenter may have some difficulties in reconcilingthis time scale with other time scales, coming from independentprocesses. The intrinsic permeability depends primarily on the sizes ofthe narrowest connections (throats) between the pores (Table 1)(Menéndez et al., 2001). These in turn depend on themeangrain size, asalso on the sorting and shapes of the grains.

3.4. Tensile failure, hydraulic fracturing and magmatic intrusion

Brittle rocksmay fail in tension.Under these conditions, theydevelopopen fractures, instead of shear fractures, and the Mohr–Coulombcriterionno longerholds. TheGriffith criterion is a commonreplacement(see Appendix A).

Hydraulic fracturing is a process whereby an overpressured fluid inan open fracture causes an increase in the tensile stress at the fracturetip, so that the fracture opens and propagatesmore readily. It is useful todistinguish between hydraulic fracturing of internal or external origins(Mandl and Harkness, 1987). In external hydraulic fracturing, the fluidcomes fromoutside andmigrates along the fracture,whereas, in internalhydraulic fracturing, the fluidmigrates through pore space, generating afracture at an internal weakness. Hubbert was the first to modelhydraulic fracturing (Hubbert and Willis, 1957). However his modelswere of gelatine, which is too strong at the scale of a sedimentary basin.To model tensile and hydraulic fracturing using granular materials ispossible, if they are truly cohesive. Luckily for the experimenter, suchmaterials exist. Examples are fine-grained powders, such as crushedsand or diatomite, in which the grains are irregular or even jagged. Thechallenge is to obtain a material having the right amount of cohesion(Galland et al., 2003). Such a material forms open fractures at shallowdepths, but shear fractures at greater depths (Galland et al., 2006).

For modelling magmatic intrusion, an additional challenge is tofind a model magma that solidifies at room temperature and does notpermeate the host powder significantly (Galland et al., 2003). In thisway, it is possible to model potentially complex interactions betweenmagmatic intrusion and faulting, in various tectonic contexts (Gallandet al., 2006, 2007; Mathieu et al., 2008).

3.5. Hydraulic fracturing during fluid migration

Here the challenge is to find materials that fail in tension, yet aresufficiently permeable to allow pervasive fluidmigration through porespace. Cobbold and Rodrigues (2007) did some preliminary experi-ments, in which the pore fluid was compressed air, migrating upwardthrough a pack of silica powder. They obtained horizontal fractures

and accounted for them in terms of seepage forces. However, the lowpermeability of the material made it difficult to measure and controlthe pore pressure within the powder.

3.6. Fluidization

Stokes's law describes the gravitational settling of an isolated spherethrough a viscous incompressible fluid (Appendix A). This is of interestfor our experiments, because a rising fluid will lift a grain of sand, if thefluid velocity is greater than the settling velocity. The settling velocity isdirectly proportional to the square of the radius and to the difference indensity between sphere and fluid, but inversely proportional to theviscosityof thefluid. Thus the settlingvelocityof a grainof sand is almosta hundred times greater in air, than it is in water. This is one of thedisadvantages of using air, instead of water, as a pore fluid. Anotherpotential problem is the grain size. A larger grain,1mmwide, settles 100times faster than a smaller grain, 0.1mmwide. Also, for grains of sand inwater, the flow around the particle is slow enough to be laminar,whereas for the same grains in air, the flow may be turbulent.

For a multitude of interacting grains within a sand pack, the flow ismore complex around the grains and within the pores, but Darcy's lawprovides a goodapproximation atmacroscopic scale, if inertial forces arenegligible. The law holds for increasing Darcy velocities, almost to thethreshold at which the overpressure gradient is equal to the density ofthe sand. At this point, the sandfluidizes. The settling velocity, accordingto Stokes law, provides a good approximation to the critical Darcyvelocity, otherwise known as the fluidization velocity.

We conclude that for studying fluidization, it is best to use smallgrains in water. However, experiments with water are more cumber-some and take longer than experiments with air.

Nichols et al. (1994) studied thefluidization of layered sediments, bypassing water through various granular materials. Horizontal fracturesand sand volcanoes are visible on their photographs, but there are noaccompanying pressure measurements.


3.7. Sand injectites

A challenge in modelling sand injectites is how to generate orintroduce an overpressured fluid, so that some layers undergohydraulic fracturing, while others fluidize. Fracturing may occur atdynamic equilibrium, whereas fluidization implies rapid flow, duringwhich inertial forces may be significant. Our response to thesechallenges has been pragmatic. First, we have tried to obtain realisticresults. Then we have checked on the correctness of the scaling, so asto improve the conditions of the next experiment.

We find that if a vertical fracture opens, the pore fluid tends to flowtowards and then along it, rather than through the less permeable

Fig. 6. Time-lapse photographs of top surface and plot of air pressure, Experiment 4. Blackvalue. For serial cross-sections (within white square), see Fig. 8.

matrix. If the fracture breaks through to the surface, the flowimmediately becomes much faster. We think that the reason for thisis a sudden drop in pressure at the leading end of the fracture. Forideal laminar flow of a fluid within a channel, the velocity isproportional to the overpressure gradient, the square of the channelwidth, and the viscosity of the fluid (Appendix A). Therefore, if thepressure drops to atmospheric values at the leading end of thefracture, the fluid velocity should also drop—and this is what weobserved.

It is possible to estimate a critical Reynolds number for thetransition from laminar to turbulent flow. The critical value dependson the shape of the channel (Appendix A). It is about 2300 for flow in a

glass microspheres erupted at upper surface when air pressure approached lithostatic


pipe, but 500,000 for flow in a parallel-sided channel. Pipe flow servesas an approximation for flow in a porous material, whereas channelflow serves as an approximation for flow in a fracture. Thus flow in afracture should be inherently more stable than flow in pore space or ina cylindrical vent.

In a fracture, 2 cmwide, containing air under a lithostatic pressuregradient, the fluid velocity is about 37 km/s and the Reynolds numberis about 4.9×107 (Table 2). For water, the corresponding velocity is333 m/s and the Reynolds number is about 6.7×106. Thus in either

Fig. 7. Time-lapse photographs of top surface and plot of air pressure, Experiment 5. Glassapproached lithostatic value (t1). For serial cross-sections (withinwhite square), see Fig. 10.to the web version of this article.)

fluid, flow is likely to be turbulent. However, for a channel width of2 mm or less, flow in either fluid is likely to be laminar. For pipe flow,flow is also likely to be laminar for widths of up to 1 mm. This showsthat it is reasonable to use air as a pore fluid in the experiments, forflow velocities as fast as the fluidization velocity. In fractures that arewider than 2 cm, the flow is likely to be turbulent, in nature as inexperiment. To obtain exactly the same Reynolds number is probablyimpossible—and we admit it. Hence we do not attach muchsignificance to the time scale, once fluidization is in action.

microspheres and blue sand erupted simultaneously at top surface when air pressure(For interpretation of the references to colour in this figure legend, the reader is referred

Fig. 8. Serial cross-sections (photographs and line drawings) from one small area of Experiment 4 (white square, Fig. 6). Black glass microspheres (red in line drawings) fill flat-lying fractures within layer of silica powder. Chains of fractures areV-shaped. Vents formed in final stages of experiment. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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4. Model materials

Thematerials in our experimentswere (1) fine quartz sand, (2) glassmicrospheres, (3) silica powder and(4)diatomitepowder. Silicapowderand diatomite powder are cohesive materials, and so they fracture intension, whereas quartz sand and glass microspheres have very smallvalues of cohesion and therefore fluidize more readily (Table 3).

We used two varieties of silica power, available from SIFRACO(Compiègne, France), under the trademarks, Millisil C10 and MillisilC4. The grain size was about 4.3–63.3 μm for Millisil C10 and 8.8–177 μm for Millisil C4. For both materials, the bulk density was 1.0–1.4 g/cm3. The diatomite powder was available from CECA (St Bauzile,France), under the trademark, Clarcel DIC S. The grain size was 10–200 μm and the bulk density was around 0.4 g/cm3. The quartz sandwas available from SIFRACO (Bourron-Marlotte, France), under thetrademark GA39. The grains were well rounded and well sorted. Themodal grain size was 90 μm, and 89% were between 75 and 106 μm.The bulk density was around 1.5 g/cm3. Finally, the glassmicrosphereswere available fromCVp (Linselles, France). Theywere similar to thoseof Galland et al. (2006). The grain size was in the range 0–45 μm, themode being about 30 μm. The bulk density was around 1.4 g/cm3.

Mechanically, the silica powder (Millisil C4 and C10) and diatomitepowder were cohesive enough, that a free vertical face, up to 15 cmhigh, did not collapse under its own weight. This height is similar toone reported by Galland et al. (2006), so we can infer similar values ofcohesion for the two materials. The microscopic reasons for suchcohesion are not yet clear, although some possibilities are (1) tiling ofplate-like grains, (2) interlocking of rough surfaces, and (3) electro-static forces between small grains.

The quartz sand (GA39) and glass microspheres had very smallvalues of cohesion, so that samples collapsed under their own weights,forming coneswith critical surface slopes. For colouring thesematerials,we used synthetic dyes available from Sika (France), under thetrademark SIKACIM COLOR. The coloured granular materials collapsedequally well under their ownweights, so we infer that the dyes did notsignificantly increase the cohesion.

To measure the permeability of a porous material, we (1) placed asample in a cylinder of Plexiglas, 11 cm in internal diameter and 10 cmhigh, (2) injected air upwards through it at a controlled rate, and(3) measured the vertical gradient of pore pressure. The cylinder had areservoir at its base. A metallic mesh prevented the granular materialfrom falling into the reservoir. A flowmeter served to control the rate ofinflow of compressed air. To measure the pressure internally, weinserted hypodermic needles, 5 cm long and 1 mm thick, through holesat various heights on the side of the cylinder. Theirs outer endsconnected to U-tubes, containing water. We poured the granularmaterial from a height of 15 cm into the cylinder and around the

Fig. 9. Three-dimensional reconstruction of injectite from serial sections, Experiment 4 (Fig(For interpretation of the references to colour in this figure legend, the reader is referred to

needles, and then scraped the upper surface level. To compact thematerial, we tapped the base and side of the cylinder several times.Where necessary, we added more material to make up the requiredheight. We then measured values of internal pressure for various flowrates. For air pressures exceeding the weight of overburden, the sand orglass microspheres fluidized, whereas the silica or diatomite powdersfractured at the base. The calculated values of intrinsic permeability(Table 3) are of the same order of magnitude as those of Cobbold et al.(2001), but one order of magnitude higher (for sand) and two orders ofmagnitude higher (for silica powder) than the values of Graveleau(2008), who used water as a pore fluid.

5. New experimental apparatus

Along the lines of earlier attempts (Cobbold et al., 2001; MourguesandCobbold, 2003; Cobbold et al., 2004),we built somenewapparatusfor studying the effects of migrating pore fluids in granular materials.

In 2006 we built a simple rectangular box, 20 cmwide, 30 cm longand10 cmhigh. As inprevious apparatus, the sourceof overpressurewasbeneath themodel, not within it. As a result, the overpressure increasedlinearly with depth and Darcy flow occurred within the model. Wesucceeded in obtaining some realistic injectites, but most of themformed near the sidewalls of the box.

So as to avoid such perturbations at lateral boundaries, we built awider box (Fig. 5). Instead of being rectangular, this box is square and1 mwide. The model is 18 cm thick, at most. These dimensions reducesidewall friction andother boundaryeffects during theexperiments. Thesidewalls of the new apparatus are of transparent Plexiglas. Underlyingthe box is a flow diffuser, consisting of 400 contiguous and verticalaluminium tubes, 30 cm long and 5 cm square. A highly permeablemetallic mesh separates the tubes from an underlying reservoir, whichprovides fluid at uniform pressure. Sand in the tubes acts as a buffer,regulating fluid flow between reservoir and model. The metallic meshprevents the sand from falling into the reservoir. The tubes channel theflow vertically and the columns of sand render the flow rate moreuniform. The box can take either compressed air or water as a pore fluid.

For our first set of experiments on sand injectites, we usedcompressed air as a pore fluid. To make pressure measurements insidethe model, we used narrow probes, which were round tubes ofaluminium, 40 cm long and 2 mm in diameter. The probes protrudedhorizontally through the sidewalls of the box. Their outer endsconnected to U-tubes, containing water. Rather than risk deformingthe model, we chose to put the probes in place first, and then built themodel around them. To prevent sand in the model from blocking theprobes, we covered their inner endswith pieces of finemetallic mesh. Aflow regulator and a pressure gauge served to control the flow ofcompressed air from an external source into the reservoir.

. 6). Bowl-shaped upper part (blue) and undulating lower part (orange) merge at back.the web version of this article.)


6. Experimental procedure

To test the suitability of the apparatus and materials for physicalmodelling of sand injectites,wedid a set of 8 experiments. In Experiments4, 5 and 8, themodels were thick (13, 8 and 13 cm, respectively) and theirinternal structures had good resolution. The other models were thinner(between5.5 and7.5 cm)andso their internal structureswere smaller andless visible. Nevertheless, all the experiments yielded consistent results.

Fig. 10. Full-length serial cross-sections of entire model, Experiment 5. Black glass microsphpowder and ponding at interface between upper silica powder and white sand. Sections 6experiment to facilitate wetting of model and to protect surface during cutting. Emptyinterpretation of the references to colour in this figure legend, the reader is referred to the

Each model consisted of a series of horizontal layers of variousgranular materials. We poured each material in turn, from a height of5 cm, forming a layer of uniform thickness. So as to obtain a horizontalupper surface, we scraped off the excess material. Once all the layerswere in place, we compacted the model, by tapping all around the boxwith a rubber hammer.

Themodelmaterials were (1) silica powder (Millisil C10), (2) quartzsand (GA39), and (3) glass microspheres. Depending on the method of

eres and blue quartz sand have migrated to upper levels, intruding layers of white silicaand 7 show zone of strong eruptions. Uppermost layer of sand was added at end of

vertical fractures within silica powder resulted from shrinkage during wetting. (Forweb version of this article.)

Fig. 10 (continued).


preparation, the density of the material was between 1.0 and 1.3 g/cm3

for the silica powder, and between 1.2 and 1.4 g/cm3 for the glassmicrospheres or quartz sand. By comparison, Krantz (1991) measured adensity of 1.78 g/cm3 for sprinkled sand,1.75 g/cm3 for sifted sand, and1.53 g/cm3 for poured sand, whereas Galland et al. (2006) measured adensity of 1.0 g/cm3 for non-compacted silica powder and 1.3 g/cm3 forcompacted silica powder.

In Experiment 4, from top to bottom, the model consisted of 3layers: (1) silica powder (10 cm thick), (2) glass microspheres (1 cm),and (3) quartz sand (2 cm). In Experiment 5, the model consisted of 7

layers: (1) silica powder (1.5 cm thick), (2) quartz sand (0.5 cm),(3) silica powder (1.5 cm), (4) quartz sand (0.5 cm), (5) silica powder(1.5 cm), (6) glass microspheres (0.5 cm), and (7) quartz sand (2 cm).Thus three of the layers were of silica powder.

In both experiments, the initial state of stress was lithostatic, thesidewalls of the box being stationary. We increased the air pressuresteadily, until it approached the weight of overburden. At this point,someof thematerialfluidized and reached the surface through vents. Atthe end of each experiment, we cut serial sections, for observation ofinternal structures. To do so, we first covered the model with a further

Fig. 11. Enlargements (photographs and line drawings) of serial cross-sections from small area of Experiment 5 (white square, Fig. 7). Black glass microspheres (red in line drawings) and blue quartz sand (blue in line drawings) fill fractureswithin layer of silica powder and at interface between upper silica powder and white sand. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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layer of sand, 2 cm thick. Then we sprinkled the surface with water,rendering the materials more cohesive. During this wetting, the entiremodel shrank, presumably as a result of capillary forces. The silicapowder shrank the most, by about 20%. The horizontal component ofshrinkage was enough to produce some open vertical fractures in thesilica powder. Cuttingwith a knife caused somehorizontal smearing, butdid not obscure the internal structures.

In these experiments, the intrinsic permeability and fluid viscosityset the time scale for steady Darcy flow, during which inertial forceswere negligible (Table 4). For turbulent flow, leading to fluidization ofweakly cohesive material, we were no longer able to constrain thetime scale with any confidence.

7. Experimental results

For Experiments 4 and 5, which had the best resolution, wedescribe surface observations and internal structures.

7.1. Surface observations

Experiment 4 lasted for 9 min. As the air pressure increased, thesurface initially remained stable (Fig. 6). After 6 min, a circular ventformed in a central position, and glass microspheres erupted throughit. At that stage, within the range of experimental error, the airpressure was equal to the weight of overburden. As the pressureincreased further, the rate of eruption increased. The microspheressettled around the vent, forming a volcano. At the same time, themodel expanded, the surface rising about 1 mm, except against thesidewalls. After a period of rapid extrusion, the air pressure dropped atthe base of the model, but remained high within the layer of silicapowder.

Experiment 5 lasted for 10 min. As the air pressure increased, thesurface initially remained stable (Fig. 7). After 5 min, vents appearedin several places. Two kinds of material, black glass microspheres andblue quartz sand, erupted through them. At that stage, within therange of experimental error, the air pressure was slightly smaller thanthe weight of overburden. As the pressure increased further, the rateof eruption increased. The particles settled around the vents, formingvolcanoes. At the same time, the model expanded, the surface risingabout 1 mm, except against the sidewalls. At that stage, the airpressure exceeded the weight of overburden within the lowermostlayer of silica powder. Both kinds of particles continued to extrude forthe entire duration of the experiment.

Fig. 12. Three-dimensional reconstruction of injectite from serial sections of small a

7.2. Internal structures

For Experiment 4, serial cross-sections of the model at the end ofthe experiment revealed a network of hydraulic fractures in the silicapowder. The fractures contained glass microspheres, which had risenfrom the underlying layer (Fig. 8). The fractures were mostly less than1 mmwide and flat lying. In detail, they were mostly en-echelon andformed open vs. Restoration of the layer of silica powder yielded arange of initial dips for the flanks of the vs. (3° to 58°, most beingbetween 15° and 30°). In three dimensions, the fracture networkconsisted of intersecting upper and lower segments (Fig. 9). The upperpart was somewhat conical. On the central sections, some widerfractures were visible beneath the surface vents.

For Experiment 5, cross-sections of the model revealed a morecomplex arrangement of structures (Figs. 10–13).

1. Two kinds of mobile material (black glass microspheres and bluequartz sand) reached the surface, next to vents.

2. Beneath these vents, the corresponding source layers were thinner,as a result of depletion.

3. As in Experiment 4, hydraulic fractures were visible within thesilica powder. However, in Experiment 5 there were three suchlayers, and all of them had fractured.

4. Sills and laccoliths of remobilizedmaterial were visible beneath thelayers of silica powder, especially the uppermost one.

5. In the roofs of laccoliths, the uppermost layer of silica powderformed domes, as a result of bending.

6. In areas of violent eruption, fractures were more numerous in thesilica powder and the mobile materials (glass microspheres andsand) had intermixed.

7. In areas where the layer of glass microspheres had lost volume, theoverlying layer of silica powder had bent downward.

8. In some places the sand layer had become thinner, due to depletion,whereas in others it had become thicker.

9. Sills of remobilized material had wings at their tip, especially in theuppermost layer of silica powder.

8. Discussion

Various kinds of sand injectites (sills, laccoliths or conical injectites)formed in our experiments. The main controlling parameters appear tohave been the permeability, tensile strength, and flexural resistance ofthemore cohesive layers, as well as themobility of the fluidizing layers.

rea in Experiment 5 (Fig. 7). Main intrusive body (laccolith) has lateral wings.

Fig. 13. Enlarged cross-sections (photographs and line drawings) from Experiments 4 and 5, showing structural details. Notice small horizontal sills of glass microspheres (a),laccolith of remobilized black glass microspheres and blue quartz sand (b), wings at terminations of some laccoliths and sills (c), deposits of glass microspheres extruded at uppersurface of model (d), depleted layers of black glass microspheres and blue sand (e), and transported block of silica powder (f). (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)


Hydraulic fractures formed in the silica powder, when the fluidoverpressure reached values approximately equal to the weight ofoverburden, plus the tensile strength of the material. The fracturesweremainly horizontal or flat lying and at first they contained air. Thatis good evidence for the action of seepage forces during upward flowof air through the pores (Cobbold and Rodrigues, 2007). Later, whenthe fluid pressure decreased at the end of the experiment, most ofthese fractures closed again, becoming almost invisible.

Granular materials moved up through some of the fractures, wherethese had broken through to the surface, forming vents. At thatinstant, the flow rate of air increased greatly, fluidizing and entrainingthe granular materials. Although some materials extruded at thesurface, more remained at depth, forming injectites of various kinds.These injectites were not distributed uniformly throughout the silicapowder, but concentrated in some areas. There they formed networks,which were somewhat conical. Thus the main fractures may have


resulted from concentrated loading at specific points, rather than fromuniform loading.

Because of involuntary errors in constructing a physical model,heterogeneity was the norm, rather than the exception. Our materialswere inevitably heterogeneous, in terms of packing, density, strength,andpermeability. Also the layers had variations in thickness, of the orderof 1 mm. Thus one would have expected the model to fracture morereadily at some points, than at others—and this is what happened.

In Experiment 5, laccoliths formed where the uppermost layer ofsilica powder bent upward toward the free surface, forming domes.Similarly, depletion of source layers at depth caused downward bending(downwarping) of the overlying layers. Thus bendingwas an importantcomponent of the deformation field. Similar bending occurs aboveigneous intrusions, especially laccoliths. In exact solutions for bendingoflinear elastic materials, vertical displacement is proportional to theelastic rigidity of the material and to the cube of the layer thickness(Pollard and Johnson, 1973). Although thematerials in our experimentswere not elastic, we would expect a strong dependence on layerthickness. This explains why laccoliths formed readily beneath the thinuppermost layer of silica powder, rather than deeper in the model.

Some conical injectites in the experiments appeared to beanalogous to magmatic cone sheets in nature. Similarly, Cartwright

Fig. 14. Theoretical model for progressive development of injectites, as fluid pressure increasshow effective shear stress in solid framework, as function of effective normal stress. Circles r(Cobbold and Rodrigues, 2007). Blue curve represents failure envelope. At first stage, fluidsecond stage, effective stress (red circle) becomes tensile enough to cause failure beneathbecomes tensile enough to produce conical network of barren hydraulic fractures within lowfluidize lowermost sand (yellow), which fills fractures and laccolith, and extrudes at surfac

et al. (2008) noticed analogies between conical sand injections in theNorth Sea Basin and magmatic cone sheets. A popular explanation formagmatic cone sheets is that they are tensile-shear fractures, resultingfrom fluid pressure of magma in a chamber (Phillips, 1974, 1986). Thecone sheets propagate out from the chamber, when an increase inmagma pressure causes the overburden to bend slightly and fracture.The irregular injectite in Experiment 4 probably formed in a similarway. The upper layer probably had a higher flexural resistance than inExperiment 5 and so it did not allow laccoliths to form.

More generally, the sand injectites in our experiments were broadlysimilar to those in nature, and especially to those from the Tampen Spurarea of theNorth Sea. The similarity is not only geometrical. In nature, asin experiment, there can be little doubt that the structures formedsequentially, from bottom to top (Shoulders and Cartwright, 2004;Shoulders et al., 2007; Cartwright et al., 2008), and that they involvedoverpressured fluids. In this view, hydraulic fracturing of sealing layerswas the main mechanism at depth, whereas doming of thinner layerswas themainmechanismnearer the surface. In the experiments, as soonas the overpressure exceeded the weight of overburden, the sandfluidized, intruded the overburden, and extruded at the free surface.Correspondingly, we predict that in nature similar processes should becommon, in areas of highfluid overpressure (Cosgrove, 2001;Hillier and

es. Permeability and tensile strength vary, from one layer to next. Mohr diagrams (right)epresent decreasing principal values of effective elastic stress, as fluid pressure increasespressure is high enough to drive vertical flow (arrows), but not to cause failure (A). Atuppermost sealing layer, forming barren laccolith (B). At third stage, effective stress

ermost sealing layer (C). At fourth stage, overpressure gradient becomes high enough toe (D).


Cosgrove, 2002). This of course does not answer the question as to whythe overpressure arose in the first place.

In summary, from our experimental observations, the naturalexamples, and previous theoretical considerations (Cobbold andRodrigues, 2007),we draw the following inferences about the processesthat may operate (Fig. 14). Our analysis refers to four stages ofdevelopment in a simple 4-layer model, through which fluid flowsvertically, as a result of an overpressure gradient. The permeability andcohesion vary, from one layer to the next, whereas the density isinvariant. Assuming a uniform vertical flow rate and Darcy's law, thepressure profile varies from layer to layer, according to the permeability.

1. For low values of basal fluid pressure, the fluid pressure is every-where smaller than the vertical stress (A, Fig.14). Hence the effectivestress does not reach the critical values necessary for failure (theMohr circle does not touch the failure envelope).

2. At a faster flow rate, the least principal effective stress becomestensile across the base of the uppermost sealing layer (B, Fig.14). Thelayer detaches and forms a dome. The underlying space fills withfluid, forming a laccolith.

3. At an even faster flow rate, the fluid pressure equals the overburdenweight plus the tensile strength of the material, within a lowersealing layer (C). Hydrofractures therefore form within this layer.The fractures are mainly flat lying or horizontal, but their dips andpositions reflect the shear stresses acting on the flanks of localdomes. The fractures interconnect, forming a conical network.

4. Finally, the fractures at different depths become interconnected andreach the surface (D). The pressure gradients within the fracturesincrease and so therefore does the rate of flow. This leads tofluidization of mobile materials, which fill the fractures or extrude atthe free surface.

9. Conclusions

1. We have succeeded in generating sand injectites experimentally,by driving compressed air through layers of sand, glass micro-spheres, and silica powder.

2. The silica powder was cohesive and capable of hydraulic fracturing,whereas the sandandglassmicrosphereswere almostnon-cohesiveand therefore able to fluidize.

3. Themodelsweredynamically similar to their natural counterparts,for as long as equilibriumwas static. When the processes becamedynamic, so that inertial forces were significant, the scaling wasapproximate and the corresponding Reynolds numbers differed.

4. The experimental apparatus was a square box, 1 m×1 m wide,resting on a grid of fluid diffusers.

5. During the experiments, the fluid pressure increased, until itattained and surpassed the weight of overburden.

6. Flat-lying hydraulic fractures, containing air, were first to formwithin cohesive and least permeable layers.

7. Heterogeneities in material properties and layer thicknesses wereresponsible for localizing fracture networks.

8. When any one network broke through to the surface, rapid flow ofair through the fracturesfluidized the underlyingmobilematerialsand even depleted some of the layers.

9. Some of the fluidized material extruded at the surface throughvents, forming volcanoes and sheets. The remainder lodged atdepth, forming sand injectites or laccoliths.

10. Conical sand injectites formed preferentially where layers hadhigh resistance to bending.

11. Laccoliths formed where overlying layers had low resistance tobending. This occurred preferentially near the surface.

12. The experimental sand injectites were broadly similar to those inthe Tampen Spur area of the North Sea, as well as other areas.Presumably they all formed by similar mechanisms. However, we

are aware that other choices of materials and boundary conditionsmay result in more faithful models in the future.

Acknowledgements

We are grateful to StatoilHydro for funding this project onexperimentalmodelling of sand injectites. Jean-Pierre Caudal, Ingénieurd'Etudes au CNRS, helped in designing the apparatus, Sylvan Rigauld,technician at the Université de Rennes 1, helped in its construction, andJean-Jacques Kermarrec, Ingénieur d'Etudes au CNRS, and PascalRolland, technician at the Université de Rennes 1, helped in makingimprovements. Nuno Rodrigues is grateful to the “Fundação para aCiência e Tecnologia”, Portugal, for a post-graduate studentship (No.SFRH/BD/12499/2003). Two anonymous reviewers made usefulcomments.

Appendix A. Scaling of the experiments

A.1. Similarity and model ratios

For a physical model to be successful, it should be geometrically,kinematically, and dynamically similar to its natural prototype(Hubbert, 1937; Mandel, 1962; Ramberg, 1967).

Geometrical similarity means that corresponding lengths in modeland nature are proportional:

γL =Lmod

LnatðA1Þ

Here γL is model ratio of length (L), and the suffixes “mod” and“nat” indicate model and nature.

Kinematical similaritymeans that corresponding time intervals areproportional:

γt =tmod

tnatðA2Þ

Here γt is the model ratio of time (t). From Eqs. (A1) and (A2), it isclear that corresponding velocities and accelerations will also beproportional.

Dynamical similarity requires that corresponding masses beproportional:

γm =mmod

mnatðA3Þ

Here γm is themodel ratio of mass (m). Also for dynamic similarity,forces at corresponding points should have the same directions andproportional magnitudes:

γF =Fgmod

Fgnat=

Fimod

Finat=

Fsmod

Fsnat= γmγLγ

−2t = γmγa ðA4Þ

Here γF is the model ratio of force (F) and the subscripts (g, i, s anda) denote gravity, inertia, surface and acceleration. If both model andprototype are at the Earth's surface, so that they are in the samegravitational field:

γa = γg = γLγ−2t = 1 ðA5Þ

Under these conditions, only twomodel ratios are independent, forexample, γm and γL, or γm and γt. The model ratios of length and timeare mutually dependent. This very strong restriction means thatproper scaling of models is very difficult, if not impossible, undermostpractical conditions (Hubbert, 1937).


Also for complete dynamic similarity, the following dimensionlessratios must each be equal, in model and prototype.

Ra =FgFs

Re =FiFv

Fr =FiFg

ðA6Þ

Here Ra is the Ramberg number, Re is the Reynolds number(Reynolds, 1883), and Fr is the Froude number.

A.2. Stresses at equilibrium

Luckily for the experimenter, if inertial forces are negligible, scalemodelling becomes feasible under a much wider range of conditions(Hubbert, 1937; Ramberg, 1967). For slowly moving tectonic systems,where stresses are effectively at equilibrium:

Aσij

Axj= ρgi ðA7Þ

Here σ is the stress tensor, x is the position vector, ρ is the density,the suffixes, i, j, refer to Cartesian tensor components, and we use thesummation convention for repeated suffixes. Each parameter in Eq. (A7)can be written as a product of a dimensionless quantity (denoted by anasterisk) and its reference value (denoted by the subscript zero). Theequation then becomes:

Aσ ij⁎

Axj⁎= Ra ρ⁎gi⁎ð Þ ðA8Þ

Here the Ramberg number is:

Ra = ρ0g0x0 = σ0 ðA9Þ

This has a value of unity, if the reference value of stress is thevertical component, due to a column of constant density.

A.3. Rheological properties of brittle rock

For the upper crust, we assume that rock is brittle, and that theMohr–Coulomb criterion adequately describes its behaviour at yield,when stresses are compressive (Byerlee, 1978):

τ = μσ Vn + c ðA10Þ

Here μ is the coefficient of internal friction, c is the cohesion, τ isthe shear stress acting on a plane of failure, and σ ′n is the effectivenormal stress acting on that plane.

When stresses are tensile, the Griffith criterion is more adequate(Secor, 1965):

τ2 + 4Tσ Vn − 4T = 0 ðA11Þ

Here T is the tensile strength of the material.

A.4. Fluid migration through porous rock

The law of Darcy (1856) for fluid flow through a porousmedium is:

qi = − kμAPAxjjnh

ðA12Þ

Here qi is the macroscopic discharge velocity (Darcy velocity) ofthe pore fluid, μ is its dynamic viscosity, k is the intrinsic permeabilityof the porous medium, P is the fluid pressure, and nh indicates its non-hydrostatic part, in other words, the overpressure.

Fluid flowing through the pores modifies the balance of forcesacting on each element of the solid framework. Von Terzaghi (1923)defined an effective stress as the difference between the applied stressand the pore fluid pressure. Although this principle would appear tobe correct, it is dangerous to calculate the effective stress at a point, byfirst considering the total stress without regard for fluid flow, and thensubtracting a fluid pressure (Mourgues and Cobbold, 2003). Instead, ifthere is an overpressure gradient, causing fluid flow, it will impart aseepage force per unit volume to the solid framework, so modifyingthe effective stresses:

AσVij

Axj= ρbgi −

APAxi

ðA13Þ

Here σ′ is the macroscopic effective stress tensor for the solidframework, P is the macroscopic fluid pressure, x is the macroscopicposition vector, ρb=(1−ϕ) ρs+ϕρf is the bulk density, ϕ is theporosity, and ρs and ρf are the average densities of the solid particlesalone and the fluid, respectively.

The effective stresses and fluid overpressure scale in the same wayas the total stresses, if inertial forces are negligible. The permeabilitysets the time scale for fluid flow.

A.5. Settling of a sphere

Stokes law describes the viscous drag on an isolated sphere as itsettles through an incompressible fluid, under its own weight:

Fd = 6πμRU ðA14Þ

Here Fd is a drag force, μ is the dynamic viscosity of the fluid, R isthe radius of the sphere, and U is its settling velocity. This law is valid ifRe is smaller than unity. The settling velocity of the sphere is:

U =2 ρs − ρfð ÞgR2

9μðA15Þ

Here rs and rf and are the densities of the fluid and the sphere.Conversely, if a fluid wells up, it will lift a sphere, if the fluid velocity isgreater than the settling velocity of the sphere. Thus Eq. (A15) providesan estimate of theminimum fluidization velocity of a granular material.

A.6. Laminar flow in a channel

Channels are of great importance in the study of fluid flow throughporousor fracturedmaterials. Dependingon the shape of the channel, sothe internal flow patternwill be different. Exact solutions exist for slowlaminar motions of Newtonian fluids in channels of simple shapes,where the fluid adheres to the walls.

For flow in a flat parallel-sided channel (Couette flow) the velocityprofile is parabolic:

v = AP = Azð Þ X2 − x2� �

= 2μ ðA16Þ

Here v is the vertical velocity of the fluid, P is thefluid overpressure, zis the distance along the channel, x is the distance across the channel, Xis its half-width, and μ is the dynamic viscosity of the fluid. Byintegration, the average vertical velocity is:

v = AP = Azð ÞX2= 3μ ðA17Þ


Similarly, for flow in a cylindrical pipe of circular cross section(Poiseuille flow), the velocity profile is also parabolic:

v = AP = Azð Þ r2 − R2� �

= 4m ðA18Þ

Here r is the radial distance across the pipe and R is its inner radius.By integration, the average vertical velocity is:

v = AP = Azð ÞX2= 8μ ðA19Þ

A.7. Turbulent flow in a channel

Reynolds (1883) argued that flow in a channel becomes turbulentat a critical value of the dimensionless (Reynolds) number:

Re = ρvL = μ ðA20Þ

Here L is the half-width of the channel. The critical value of Redepends on the shape of the channel. For example, it is about 2300 forPoiseuille flow, but as much as 500,000 for Couette flow. Poiseuilleflow serves as an approximation for flow in a porous material,whereas Couette flow serves as an approximation for flow in afracture. Thus flow in a fracture is inherently more stable than flow inpore space or in a cylindrical vent. In a fracture, 2 cmwide, containingair under a lithostatic pressure gradient, the fluid velocity is about37 km/s and the Reynolds number is about 4.9×107 (Table 2). Forwater, the corresponding velocity is 333 m/s and the Reynoldsnumber is about 6.7×106. Thus in either fluid, flow is likely to beturbulent. For a channel width of 2 mm or less, flow in either fluid islikely to be laminar.

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