Physical modelling of sand injectites
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Tectonophysics 474 (2009) 610632
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.e linjectites 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 Fris, 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 et
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-Martnez et al., 2007). All these injectites are remarkably similar inshape to volcanic igneous bodies. For example, at-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 Palaeoceneal.,1993; Dixon et al.,1995;Molyneux et al., 20Duranti et al., 2002; Lseth et al., 2003; HuusMickelson, 2004; Shoulders et al., 2007). Frsurveys, the shapes of the intrusive bodies andtheir host rocks are becoming clearer (Hurs
Corresponding author.E-mail address: email@example.com (P.R.
0040-1951/$ see front matter 2009 Elsevier B.V. Adoi:10.1016/j.tecto.2009.04.032. In the last few decades,attention, as the number(Huuse et al., 2007). The
(at-based). In contrast, the anks 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-manyexamples of sand injectites have come toand quality of seismic surveys have increased1. Introduction
Sand injectites are intrusive bodiebilization and injection of sand into fraare in sedimentary strata. Early descriptdate from the 19th century (Murchisonair, 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 ow of air through the fractures uidized the underlying mobile materials and even depletedsome of the layers. Some of the uidized 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.
h result from the remo-. Typically, such fracturessand injectites at outcrop
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; Lseth 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) orwideSandPhysical modelling
increased, until it attained and surpassed the weight of overburden. Flat-lying hydraulic fractures, containing
Injectite a square box, 1 m1 m wide, resting on a grid of uid diffusers. During the experiments, the uid pressureKeywords: equilibrium was static. Whscaling was approximate and the corresponding Reynolds numbers differed. The experimental apparatus wasPhysical modelling of sand injectites
N. Rodrigues a, P.R. Cobbold a,, H. Lseth b
a Gosciences-Rennes (UMR6118), CNRS et Universit de Rennes 1, 35042 Rennes Cedex, Frb StatoilHydro Research Centre, Trondheim, Norway
a b s t r a c ta 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
Sand injectites are structurthem in the Tampen Spurcompressed air through layeand capable of hydraulic fratherefore able to uidize. Th
j ourna l homepage: www02; Lonergan et al., 2000;e et al., 2004; Huuse andom 2D and 3D seismictheir relationships witht and Cartwright, 2007;
ll rights reserved.e
hat result from intrusion of uidized sand into fractures. We have studieda of the North Sea, and have reproduced them experimentally, by drivingf sand, glass microspheres, and silica powder. The silica powder was cohesiveing, whereas the sand and glass microspheres were almost non-cohesive andodels were dynamically similar to their natural counterparts, for as long as
the processes became dynamic, so that inertial forces were signicant, the
sev ie r.com/ locate / tectoto Miocene Hordaland Group (Thyberg et al., 2000). In both basins,sand injectites are most common in Eocene mudstones (Molyneuxet al., 2002; Lseth 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 efcient seals (Wensaas et al.,1998; Jones et al., 2003). Today, the mudstones tend to be slightlyoverpressured (Teige et al., 1999).
At rst 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 uids (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, aqueousuids, which are responsible for sand injectites (Jonk et al., 2003, 2005),can migrate over much longer distances. Also, uid ow 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, uid owing 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 uidoverpressure 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,
1998). Lonergan and Cartwright (1999) identied 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 anks of conical intrusions and that sand injectites commonlycrosscut the polygonal faults.
The sand thatwas the source for the injectitesmusthaveuidizedandremobilized (Nichols et al., 1994; Nichols, 1995; Jolly and Lonergan,2002). A moving uid 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 uid velocity at whichthis occurs is known as the minimum uidisation 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 difculty. 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-Domnguez 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 difculties of proper scaling,whenuid ow is turbulent within open fractures. We explain what criteriaguided us in the choice of model materials. Thenwe describe some new
611N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Fig. 1.Map of North Viking Graben, North Sea, showingmainMesozoic rift faults (after Za
see Fig. 2. (For interpretation of the references to colour in this gure legend, the reader is rand Coward, 2003). Study area (red box) is on Tampen Spur. For seismic section (AB),
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 reections, mobile sand (yellow) and mounds above laccoliths. Notice verticaloffsets of beds across V-shaped reectors (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 gure legend, the reader is referred to the web version of this article.)
experimental apparatus, which avoids unwanted boundary effects.During upward ow 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 prolic 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 at lying, partly
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 500600m 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 (Lseth 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.
613N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Fig. 3. Structures at top of Balder Fm and distribution of high-amplitude cones. Divisio(yellows) strikes NNE. Deeper areas (blues) are to east. Apices of cones are either just
Segmented line (AB) is of seismic section (Fig. 2). (For interpretation of the references ton 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 gure legend, the reader is referred to the web version of this article.)
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 uid overpressure, triggered sand intrusion.
In the westernmost part of the area, the at-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 reections 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 reections. In places, vertical offsets of stratigraphic reectionsprovide 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).
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 anks of thecones appear to offset the patterns of stratigraphic reections, in themanner of reverse faults (Fig. 2). The offsets tend to decrease upwardsand the upper parts producenohigh-amplitude reections. 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 anks and bases prove that these anomaliesconsist of sands, partly cemented by carbonates.
2.2. Low-amplitude cones
Within theHordalandGroup, some stratigraphic reectionpatternsare vertically offset, but the zones of offset are not responsible for high-amplitude reections. Theyappear as reverse faults in seismic sections,but produce low-amplitude reections. 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 reections or discordantoffsets. Whether such sands are parallel to bedding or discordant isdifcult to interpret.
614 N. Rodrigues et al. / Tectonophysics 474 (2009) 6106322.1. 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 ank of another.
Fig. 4. Relief at top of Hordaland Fm. Vertical scale is inmilliseconds, two-way time. Divisirings above sand laccoliths. Erosional edge at top of Hordaland Fm (between yellows and
(For interpretation of the references to colour in this gure legend, the reader is referred toOn theanks 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 attening 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 (AB) 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.
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 reections. However, some segments, severalhundred meters long, have high-amplitude reections and variousdips and polarities. Their upper envelope describes mounds (Fig. 4),whereas their lower envelope is concordant with underlying strati-graphic reections. The mounds are similar in shape to igneouslaccoliths. In some places, the longer discordant high-amplitudereections are continuous, from the tops of mounds within theHordaland Group, to the anks of mounds at the top of the HordalandGroup. In other places, the discordant reections 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.
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
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 nd 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 rmer basis. Indeed, we owe to Hubbert much of thesubsequent development of the technique.
Table 1Mechanical properties of granular materials for tectonic modelling.
Material Grain size(m)
Coefcient ofinternal friction
Angle ofinternalfriction ()
Permeability(m2 Pa1 s1)
Quartz sand 200300 1.30 30 Vendeville et al. (1987)Quartz sand b500 1.53 0.58 (1.17) 300 (420) Krantz (1991)Quartz sand (Ottawa sand) 220 47.670 Sture et al. (1998)Quartz sand 200315 1.58 0.57 85 1.71E06 3.42E11 35.258 Cobbold and Castro
615N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Glass microspheres (GMII) 90180 1.61 0.65 33 160Sand (SII) 90180 1.67 0.88 (1.60) 41.3 230Quartz sand 100300 1.44 0.65 33 Glass microspheres 300400 1.60 0.44 23.9 Quartz sand (Fontainebleausand 1)
315400 1.60 0.45 24.2 160
Quartz sand (Fontainebleausand 2)
200315 1.60 0.57 29.7 85
Quartz sand (Fontainebleausand 3)
125200 1.60 0.51 27 0
Loess 3040 1.30 0.76 200Tapioca pearls 2000 0.74 36.5 39Quartz sand 315400 1.60 0.42 11
Quartz sand b200 1.60
Quartz sand 200315 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) 030 1.33 0.84 288Glass microspheres(SI-SPHERE)
Quartz sand (HN38) 0350 1.52 Glass microspheres 100200 1.51 0.94 43 0Glass microspheres 70150 1.58 Glass microspheres 50105 1.59 PVC 60200 0.64 1.05 300Silica powder 0200 1.55 Values in brackets take into account sidewall friction in testing machine (Mourgues and Co Schellart (2000)) Schellart (2000)
Turrini et al. (2001) Turrini et al. (2001) 5.00E06 1.00E10 101.327 Cobbold et al. (2001)
1.70E06 3.40E11 34.451 Cobbold et al. (2001)
1.40E06 2.80E11 28.372 Cobbold et al. (2001)
5.00E11 1.00E15 0.001 Cobbold et al. (2001) Grotenhuis et al. (2002) Mourgues and
Cobbold (2003) Mourgues and
Cobbold (2003) 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)
5.99E09 5.80E12 6.000 Graveleau (2008) 1.47E08 1.40E11 14.500 Graveleau (2008) 4.14E09 4.20E12 4.000 Graveleau (2008) 3.45E09 3.50E12 3.500 Graveleau (2008) 7.37E09 7.20E12 7.000 Graveleau (2008) 4.60E12 5.00E14 0.050 Graveleau (2008)bbold, 2003).
For a physical model to be successful, it should be geometrically and describe their material properties: the cohesion, and the coefcient of
Table 2Theoretical values of average uid velocity and Reynolds number for Couette ow of air or water in at parallel-sided channels of various half-widths.
Half-width (m) Half-width (m)
1.00E04 1.00E03 1.00E02 1.00E01 1.00E04 1.00E03 1.00E02 1.00E01Gradient (Pa/m) Density (kg/m3) Viscosity (Pa s) Velocity (m/s) Re (dimensionless)
Air in nature 20,000 1.20E+00 1.80E05 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 1.00E03 3.33E02 3.33E00 3.33E+02 3.33E+04 6.67E00 6.67E+03 6.67E+06 6.67E+09Air in model 14,000 1.20E+00 1.80E05 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 1.00E03 1.33E02 1.33E+00 1.33E+02 1.33E+04 2.67E+00 2.67E+03 2.67E+06 2.67E+09
616 N. Rodrigues et al. / Tectonophysics 474 (2009) 610632kinematically 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 eld 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 strongso strong, that proper scaling ofmodels is very difcult, 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 eld 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 rstto 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 rst 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 MohrCoulomb criterion, two parameters are enough toTable 3Mechanical properties of materials in our experiments.
Material Variety Roughness Grain size(m)
Silica powder Millisil C10 Angular 0350 63.3 22.7 4.3Silica powder Millisil C4 Angular 0150 177 64 8.8Diatomite powder DICS Angular 0350 200 3550 10Glass microspheres CVp Round 045 21 Quartz sand GA39 Sub-round 0160 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).internal friction. The cohesionhas thedimensionsof stress and thereforescales down as the linear dimensions. The coefcient 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 coefcient 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 (Horseld, 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 uids
Pore uids may modify the effective stresses in a solid framework.Hubbert introduced the concept of uid overpressure into tectonics, soas to explain large overthrusts (Hubbert and Rubey, 1959). However,there is no record thatheeverusedporeuids in sandboxmodels. Toourknowledge, the rst 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 uid pressure. However, this denition istoo simple. Instead, it is necessary to consider uid migration throughpore space. According to Darcy's law, themacroscopic discharge velocity(Darcy velocity) of a pore uid is directly proportional to the intrinsicpermeability of the porous medium, and to the overpressure gradient,and inversely proportional to the dynamic viscosity of the uid. This lawholds well for granular materials, for values of uid pressure as high asthe weight of overburden (Cobbold and Castro, 1999). Fluid owingthrough the pores modies the balance of forces, applying a seepageforce to each element of the solid framework, and this effect is readilyvisible in sandbox models, where it inuences fault orientations(Mourgues and Cobbold, 2003, 2006a,b).
The effective stresses and uid overpressure scale down in the sameway as the total stresses, if inertial forces are negligible. However, theParticledensity(g/cm3)
Permeability(m2 Pa1 s1)
2.65 1.15 5 7.00E08 1.400E12 1.4192.65 1.34 4 8.00E08 1.600E12 1.6212.65 0.29 5 7.50E07 1.500E11 15.1992.46 1.39 2 1.00E07 2.000E12 2.0272.64 1.43 1 7.00E07 1.400E11 14.186
of free vertical faces, lowest (1) being quartz sand (fromMourgues and Cobbold, 2003),
617N. Rodrigues et al. / Tectonophysics 474 (2009) 610632intrinsic permeability and uid viscosity set a time scale for steady ow.In practice, the experimenter may have some difculties 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)(Menndez 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 MohrCoulombcriterionno longerholds. TheGrifth criterion is a commonreplacement(see Appendix A).
Hydraulic fracturing is a process whereby an overpressured uid 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 uidcomes fromoutside andmigrates along the fracture,whereas, in internalhydraulic fracturing, the uidmigrates through pore space, generating afracture at an internal weakness. Hubbert was the rst to modelhydraulic fracturing (Hubbert and Willis, 1957). However his modelswere of gelatine, which is too strong at the scale of a sedimentary basin.
Fig. 5. Experimental apparatus for modelling sand injectites. Longitudinal section (A)containing sand (2), metallic mesh (3), reservoir for compressed air (4), pressure regulatthrough sidewalls of box (7). Their outer ends connect to U-tubes, containing water (8To model tensile and hydraulic fracturing using granular materials ispossible, if they are truly cohesive. Luckily for the experimenter, suchmaterials exist. Examples are ne-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 tond a model magma that solidies at room temperature and does notpermeate the host powder signicantly (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 uid migration
Here the challenge is to nd materials that fail in tension, yet aresufciently permeable to allow pervasive uidmigration through porespace. Cobbold and Rodrigues (2007) did some preliminary experi-ments, in which the pore uid was compressed air, migrating upwardthrough a pack of silica powder. They obtained horizontal fracturesand accounted for them in terms of seepage forces. However, the lowpermeability of the material made it difcult to measure and controlthe pore pressure within the powder.
Stokes's law describes the gravitational settling of an isolated spherethrough a viscous incompressible uid (Appendix A). This is of interestfor our experiments, because a rising uid will lift a grain of sand, if theuid 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 uid, but inversely proportional to theviscosityof theuid. 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 uid. 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 ow around the particle is slow enough to be laminar,whereas for the same grains in air, the ow may be turbulent.
For a multitude of interacting grains within a sand pack, the ow 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 of
three-dimensional view (B) show Plexiglas box for housing models (1), ow diffuser) and tilting table (6). Pressure probes (narrow aluminium tubes) protrude horizontallythe sand. At this point, the sanduidizes. The settling velocity, accordingto Stokes law, provides a good approximation to the critical Darcyvelocity, otherwise known as the uidization velocity.
We conclude that for studying uidization, 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 theuidization of layered sediments, bypassing water through various granular materials. Horizontal fracturesand sand volcanoes are visible on their photographs, but there are noaccompanying pressure measurements.
Table 4Reference values and scaling of models for steady ow (before uidization).
Parameter Units Hordaland Group Experiment Model ratio
Length m 1.00E+03 0.10 1.00E04Bulk density kg/m3 2.10E+03 1.20E+03 0.57Effective stress Pa 2.06E+07 1.18E+03 5.71E05Permeability m2 Pa1 s1 2.66E15 7.00E08 2.63E+07Darcy velocity m/s 5.48E11 8.24E04 1.50E+07Time s 1.82E+13 1.21E+02 6.65E12
Estimates for Hordaland group are from Wensaas et al. (1998).
3.7. Sand injectites
A challenge in modelling sand injectites is how to generate orintroduce an overpressured uid, so that some layers undergohydraulic fracturing, while others uidize. Fracturing may occur atdynamic equilibrium, whereas uidization implies rapid ow, duringwhich inertial forces may be signicant. 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 nd that if a vertical fracture opens, the pore uid tends to owtowards and then along it, rather than through the less permeable
matrix. If the fracture breaks through to the surface, the owimmediately becomes much faster. We think that the reason for thisis a sudden drop in pressure at the leading end of the fracture. Forideal laminar ow of a uid within a channel, the velocity isproportional to the overpressure gradient, the square of the channelwidth, and the viscosity of the uid (Appendix A). Therefore, if thepressure drops to atmospheric values at the leading end of thefracture, the uid velocity should also dropand this is what weobserved.
It is possible to estimate a critical Reynolds number for thetransition from laminar to turbulent ow. The critical value dependson the shape of the channel (Appendix A). It is about 2300 for ow in a
618 N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Fig. 6. Time-lapse photographs of top surface and plot of air pressure, Experiment 4. Bl
value. For serial cross-sections (within white square), see Fig. 8.glass microspheres erupted at upper surface when air pressure approached lithostatic
pipe, but 500,000 for ow in a parallel-sided channel. Pipe ow servesas an approximation for ow in a porous material, whereas channelow serves as an approximation for ow in a fracture. Thus ow in afracture should be inherently more stable than ow in pore space or ina cylindrical vent.
In a fracture, 2 cmwide, containing air under a lithostatic pressuregradient, the uid velocity is about 37 km/s and the Reynolds numberis about 4.9107 (Table 2). For water, the corresponding velocity is333 m/s and the Reynolds number is about 6.7106. Thus in either
uid, ow is likely to be turbulent. However, for a channel width of2 mm or less, ow in either uid is likely to be laminar. For pipe ow,ow is also likely to be laminar for widths of up to 1 mm. This showsthat it is reasonable to use air as a pore uid in the experiments, forow velocities as fast as the uidization velocity. In fractures that arewider than 2 cm, the ow is likely to be turbulent, in nature as inexperiment. To obtain exactly the same Reynolds number is probablyimpossibleand we admit it. Hence we do not attach muchsignicance to the time scale, once uidization is in action.
619N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Fig. 7. Time-lapse photographs of top surface and plot of air pressure, Experiment 5. Gapproached lithostatic value (t1). For serial cross-sections (withinwhite square), see Fig.
to the web version of this article.)microspheres and blue sand erupted simultaneously at top surface when air pressure(For interpretation of the references to colour in this gure 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) ll at-lying fractures within layer of silica powder. Chains of fractures areV-shaped. Vents formed in nal stages of experiment. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of this article.)
4. Model materials
Thematerials in our experimentswere (1) ne 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 uidize more readily (Table 3).
We used two varieties of silica power, available from SIFRACO(Compigne, France), under the trademarks, Millisil C10 and MillisilC4. The grain size was about 4.363.3 m for Millisil C10 and 8.8177 m for Millisil C4. For both materials, the bulk density was 1.01.4 g/cm3. The diatomite powder was available from CECA (St Bauzile,France), under the trademark, Clarcel DIC S. The grain size was 10200 m 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 m, and 89% were between 75 and 106 m.The bulk density was around 1.5 g/cm3. Finally, the glassmicrospheres
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 owrates. For air pressures exceeding the weight of overburden, the sand orglass microspheres uidized, 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 uid.
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 uids in granular materials.
In 2006 we built a simple rectangular box, 20 cmwide, 30 cm long
621N. Rodrigues et al. / Tectonophysics 474 (2009) 610632were available fromCVp (Linselles, France). Theywere similar to thoseof Galland et al. (2006). The grain size was in the range 045 m, themode being about 30 m. 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 notsignicantly 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 owmeter served to control the rate ofinow 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
(For interpretation of the references to colour in this gure legend, the reader is referred toand10 cmhigh. As inprevious apparatus, the sourceof overpressurewasbeneath themodel, not within it. As a result, the overpressure increasedlinearly with depth and Darcy ow 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 ow 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 uid at uniform pressure. Sand in the tubes acts as a buffer,regulating uid ow between reservoir and model. The metallic meshprevents the sand from falling into the reservoir. The tubes channel theow vertically and the columns of sand render the ow rate moreuniform. The box can take either compressed air or water as a pore uid.
For our rst set of experiments on sand injectites, we usedcompressed air as a pore uid. 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 rst, and then built themodel around them. To prevent sand in the model from blocking theprobes, we covered their inner endswith pieces of nemetallic mesh. Aow regulator and a pressure gauge served to control the ow 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.
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
622 N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Fig. 10. Full-length serial cross-sections of entire model, Experiment 5. Black glass micropowder and ponding at interface between upper silica powder and white sand. Sectioexperiment to facilitate wetting of model and to protect surface during cutting. Emp
interpretation of the references to colour in this gure legend, the reader is referred to theeres 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. (For
web version of this article.)
623N. Rodrigues et al. / Tectonophysics 474 (2009) 610632preparation, 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
Fig. 10 (contlayers: (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 thematerialuidized and reached the surface through vents. Atthe end of each experiment, we cut serial sections, for observation ofinternal structures. To do so, we rst 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) ll fractureswithin layer of silica powder and at interface between upper silica powder and white sand. (For interpretation of the references to colour in this gure legend, the reader is referred to the web version of this article.)
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 uid viscosityset the time scale for steady Darcy ow, during which inertial forceswere negligible (Table 4). For turbulent ow, leading to uidization ofweakly cohesive material, we were no longer able to constrain thetime scale with any condence.
7. Experimental results
For Experiments 4 and 5, which had the best resolution, wedescribe surface observations and internal structures.
7.1. Surface observations
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 at 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 anks 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. 1013).
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 the
6. In areas of violent eruption, fractures were more numerous in thesilica powder and the mobile materials (glass microspheres and
625N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Experiment 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 asand) had intermixed.7. In areas where the layer of glass microspheres had lost volume, the
overlying 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 the
uppermost layer of silica powder.
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 exural resistance ofthemore cohesive layers, as well as themobility of the uidizing layers.silica 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.rea in Experiment 5 (Fig. 7). Main intrusive body (laccolith) has lateral wings.
626 N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Hydraulic fractures formed in the silica powder, when the uidoverpressure reached values approximately equal to the weight ofoverburden, plus the tensile strength of the material. The fracturesweremainly horizontal or at lying and at rst they contained air. Thatis good evidence for the action of seepage forces during upward owof air through the pores (Cobbold and Rodrigues, 2007). Later, whenthe uid pressure decreased at the end of the experiment, most ofthese fractures closed again, becoming almost invisible.
Fig. 13. Enlarged cross-sections (photographs and line drawings) from Experiments 4 andlaccolith of remobilized black glass microspheres and blue quartz sand (b), wings at terminsurface of model (d), depleted layers of black glass microspheres and blue sand (e), and trangure legend, the reader is referred to the web version of this article.)Granular materials moved up through some of the fractures, wherethese had broken through to the surface, forming vents. At thatinstant, the ow rate of air increased greatly, uidizing 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
5, showing structural details. Notice small horizontal sills of glass microspheres (a),ations of some laccoliths and sills (c), deposits of glass microspheres extruded at uppersported block of silica powder (f). (For interpretation of the references to colour in this
resulted from concentrated loading at specic 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 othersand 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 eld. 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
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 uid 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 exural 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 uids. 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 sanduidized, intruded the overburden, and extruded at the free surface.Correspondingly, we predict that in nature similar processes should becommon, in areas of highuid overpressure (Cosgrove, 2001;Hillier and
627N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Fig. 14. Theoretical model for progressive development of injectites, as uid pressure incshow effective shear stress in solid framework, as function of effective normal stress. Circ(Cobbold and Rodrigues, 2007). Blue curve represents failure envelope. At rst stage, second stage, effective stress (red circle) becomes tensile enough to cause failure benbecomes tensile enough to produce conical network of barren hydraulic fractures within
uidize lowermost sand (yellow), which lls fractures and laccolith, and extrudes at surfaces. Permeability and tensile strength vary, from one layer to next. Mohr diagrams (right)epresent decreasing principal values of effective elastic stress, as uid pressure increasespressure is high enough to drive vertical ow (arrows), but not to cause failure (A). Atuppermost sealing layer, forming barren laccolith (B). At third stage, effective stressermost sealing layer (C). At fourth stage, overpressure gradient becomes high enough to
628 N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Cosgrove, 2002). This of course does not answer the question as to whythe overpressure arose in the rst 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 uid owsvertically, 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 ow rate and Darcy's law, thepressure prole varies from layer to layer, according to the permeability.
1. For low values of basal uid pressure, the uid 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 ow 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 lls withuid, forming a laccolith.
3. At an even faster ow rate, the uid 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 at lying or horizontal, but their dips andpositions reect the shear stresses acting on the anks 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 ow. This leads touidization of mobile materials, which ll the fractures or extrude atthe free surface.
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 uidize.
3. Themodelsweredynamically similar to their natural counterparts,for as long as equilibriumwas static. When the processes becamedynamic, so that inertial forces were signicant, the scaling wasapproximate and the corresponding Reynolds numbers differed.
4. The experimental apparatus was a square box, 1 m1 m wide,resting on a grid of uid diffusers.
5. During the experiments, the uid pressure increased, until itattained and surpassed the weight of overburden.
6. Flat-lying hydraulic fractures, containing air, were rst 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 ow ofair through the fracturesuidized the underlyingmobilematerialsand even depleted some of the layers.
9. Some of the uidized 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, weare aware that other choices of materials and boundary conditionsmay result in more faithful models in the future.
We are grateful to StatoilHydro for funding this project onexperimentalmodelling of sand injectites. Jean-Pierre Caudal, Ingnieurd'Etudes au CNRS, helped in designing the apparatus, Sylvan Rigauld,technician at the Universit de Rennes 1, helped in its construction, andJean-Jacques Kermarrec, Ingnieur d'Etudes au CNRS, and PascalRolland, technician at the Universit de Rennes 1, helped in makingimprovements. Nuno Rodrigues is grateful to the Fundao para aCincia 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:
Here L is model ratio of length (L), and the sufxes mod andnat indicate model and nature.
Kinematical similaritymeans that corresponding time intervals areproportional:
Here t 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:
Here m is themodel ratio of mass (m). Also for dynamic similarity,forces at corresponding points should have the same directions andproportional magnitudes:
= mL2t = ma A4
Here F 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 eld:
a = g = L2t = 1 A5
Under these conditions, only twomodel ratios are independent, forexample, m and L, or m and t. The model ratios of length and timeare mutually dependent. This very strong restriction means thatproper scaling of models is very difcult, if not impossible, undermost
practical conditions (Hubbert, 1937).
629N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Also for complete dynamic similarity, the following dimensionlessratios must each be equal, in model and prototype.
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:
= gi A7
Here is the stress tensor, x is the position vector, is the density,the sufxes, i, j, refer to Cartesian tensor components, and we use thesummation convention for repeated sufxes. 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:
= Ra gi A8
Here the Ramberg number is:
Ra = 0g0x0 = 0 A9
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 theMohrCoulomb criterion adequately describes its behaviour at yield,when stresses are compressive (Byerlee, 1978):
= Vn + c A10
Here is the coefcient of internal friction, c is the cohesion, isthe shear stress acting on a plane of failure, and n is the effectivenormal stress acting on that plane.
When stresses are tensile, the Grifth criterion is more adequate(Secor, 1965):
2 + 4T Vn 4T = 0 A11
Here T is the tensile strength of the material.
A.4. Fluid migration through porous rock
The law of Darcy (1856) for uid ow through a porousmedium is:
qi = kAPAx j nh A12jHere qi is the macroscopic discharge velocity (Darcy velocity) ofthe pore uid, is its dynamic viscosity, k is the intrinsic permeabilityof the porous medium, P is the uid pressure, and nh indicates its non-hydrostatic part, in other words, the overpressure.
Fluid owing through the pores modies the balance of forcesacting on each element of the solid framework. Von Terzaghi (1923)dened an effective stress as the difference between the applied stressand the pore uid pressure. Although this principle would appear tobe correct, it is dangerous to calculate the effective stress at a point, byrst considering the total stress without regard for uid ow, and thensubtracting a uid pressure (Mourgues and Cobbold, 2003). Instead, ifthere is an overpressure gradient, causing uid ow, it will impart aseepage force per unit volume to the solid framework, so modifyingthe effective stresses:
= bgi APAxi
Here is the macroscopic effective stress tensor for the solidframework, P is the macroscopic uid pressure, x is the macroscopicposition vector, b=(1) s+f is the bulk density, is theporosity, and s and f are the average densities of the solid particlesalone and the uid, respectively.
The effective stresses and uid overpressure scale in the same wayas the total stresses, if inertial forces are negligible. The permeabilitysets the time scale for uid ow.
A.5. Settling of a sphere
Stokes law describes the viscous drag on an isolated sphere as itsettles through an incompressible uid, under its own weight:
Fd = 6RU A14
Here Fd is a drag force, is the dynamic viscosity of the uid, 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 s f gR2
Here rs and rf and are the densities of the uid and the sphere.Conversely, if a uid wells up, it will lift a sphere, if the uid velocity isgreater than the settling velocity of the sphere. Thus Eq. (A15) providesan estimate of theminimum uidization velocity of a granular material.
A.6. Laminar ow in a channel
Channels are of great importance in the study of uid ow throughporousor fracturedmaterials. Dependingon the shape of the channel, sothe internal ow patternwill be different. Exact solutions exist for slowlaminar motions of Newtonian uids in channels of simple shapes,where the uid adheres to the walls.
For ow in a at parallel-sided channel (Couette ow) the velocityprole is parabolic:
v = AP = Az X2 x2
= 2 A16
Here v is the vertical velocity of the uid, P is theuid overpressure, zis the distance along the channel, x is the distance across the channel, Xis its half-width, and is the dynamic viscosity of the uid. Byintegration, the average vertical velocity is:
v = AP = Az X2 = 3 A17
630 N. Rodrigues et al. / Tectonophysics 474 (2009) 610632Similarly, for ow in a cylindrical pipe of circular cross section(Poiseuille ow), the velocity prole is also parabolic:
v = AP = Az r2 R2
= 4m A18
Here r is the radial distance across the pipe and R is its inner radius.By integration, the average vertical velocity is:
v = AP = Az X2 = 8 A19
A.7. Turbulent ow in a channel
Reynolds (1883) argued that ow in a channel becomes turbulentat a critical value of the dimensionless (Reynolds) number:
Re = vL = A20
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 ow, but as much as 500,000 for Couette ow. Poiseuilleow serves as an approximation for ow in a porous material,whereas Couette ow serves as an approximation for ow in afracture. Thus ow in a fracture is inherently more stable than ow inpore space or in a cylindrical vent. In a fracture, 2 cmwide, containingair under a lithostatic pressure gradient, the uid velocity is about37 km/s and the Reynolds number is about 4.9107 (Table 2). Forwater, the corresponding velocity is 333 m/s and the Reynoldsnumber is about 6.7106. Thus in either uid, ow is likely to beturbulent. For a channel width of 2 mm or less, ow in either uid islikely to be laminar.
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Physical modelling of sand injectitesIntroductionSand injectites in the Tampen Spur areaHigh-amplitude conesLow-amplitude conesMoundsSummary
Physical modelling of tectonic processes using granular materialsScaling in theory and practiceSand: a suitable material for brittle deformationUse of pore fluidsTensile failure, hydraulic fracturing and magmatic intrusionHydraulic fracturing during fluid migrationFluidizationSand injectites
Model materialsNew experimental apparatusExperimental procedureExperimental resultsSurface observationsInternal structures
DiscussionConclusionsAcknowledgementsScaling of the experimentsSimilarity and model ratiosStresses at equilibriumRheological properties of brittle rockFluid migration through porous rockSettling of a sphereLaminar flow in a channelTurbulent flow in a channel