numerical modelling of gravitational sinking of anhydrite ......numerical modelling approach, two...

Bollettino di Geofisica Teorica ed Applicata Vol. 57, n. 3, pp. 233-246; September 2016

DOI 10.4430/bgta0178

233

Numerical modelling of gravitational sinking of anhydrite stringers in salt (at rest)

S. Li1 and J. Urai2

1 College of Petroleum Engineering, China University of Petroleum, Beijing, China2 Structural Geology, Tectonics and Geomechanics, RWTH Aachen University, Germany

(Received: April 8, 2015; accepted: June 15, 2016)

ABSTRACT Alargenumberofsaltbodiescontainlayersofanhydritematerialwhichisgenerallyreferred to as “stringers”. The movement and deformation of embedded anhydritebodiesareprocesseswhicharenotyetfullyunderstood.Itisobservedthatstringerstendtosinktowardsthebottomofsaltbodiesatvelocitieshighlydependentonthemechanicalpropertiesofbothsaltandanhydrites,withgivendensitycontrastbetweensaltanddenseranhydrites.Therheologicaldifferencesbetweensaltandtheembeddedanhydrites are a major issue, contributing to the complexity of the problem. On ageological timescale, the salt behaves as a Newtonian or a power-law fluid. The anhydritestringerspresentelasticorbrittlepropertiesundercertainconditions.FiniteElementModelling(FEM)hasbeenemployedinthisstudybyusingtheFEMpackageABAQUS(SIMULIA,DassaultSystems)inordertonumericallysimulatethesinkingofananhydritestringerembeddedin thesalt.Furthermore,numericalmodellingofisolatedanhydritestringersinsaltatrestiscomparedwithobservationsofstringersinseismicdata.FEMsimulationoftheanhydritestringersinkingandthegravitationalsinkingofanhydriteblocksembeddedinthesaltwillbestudiedanddemonstratedwithtwodifferentmethodsofrheology,respectively.Thestudyresultsindicatethatsinkingvelocityiscloselyrelatedtoseveralfactors,includingtheviscosity,thethicknessofthestringer,aswellasthedensitycontrastbetweenstringerandsaltforagivenviscosity.The results also prove that anhydrite stringer fragments do not sink significantly over thegeological timescale if thehalite isdeformedbynon-Newtonianviscosity.But,when Newtonian viscosity is dominant, the fragments are likely to sink hundredsofmetresthroughtheZechsteinsaltduringafewMa.Inconclusion,themodellingof thesinkingofanhydriteoranhydrite inclusionsprovidesan important scope forunderstandingthemovementanddeformationofembeddedstringers.

Key words: stringer,rheology,Newtonian,power-law,sinkingvelocity,FEM.

© 2016 – OGS

1. Introduction

Anhydritebodyinclusionsarecommoninsaltdiapirs.However,themovementanddeformationofthoseembeddedanhydriteoranhydritebodiesarenotyetfullyunderstood.Evaporitescontainingthicklayersofanhydrite(“stringers”)havebeenexploredbymanyresearchers(Weinberg,1993;vanderBogert, 1997,1998;ChemiaandKoyi,2008;Chemiaet al., 2008,2009).Thesestudies

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Boll. Geof. Teor. Appl., 57, 233-246 Li and Urai

demonstrate that, due to the density contrast between salt and denser anhydrite or anhydrite,the stringers tend tomovedownwards in the salt atvelocitieswhicharehighlydependentonthemechanicalpropertiesofthesaltandthegeometricalpropertiesoftheembeddedstringers.In contrast to the Newtonian or power-law fluid behavior of the halite, the stringers can exhibit elastic to brittle behavior or fold and flow at much higher viscosity than the surrounding salt. In thesestudies,theriseandfallofadenselayerinsaltdiapirsismodelledinanumericalway,andtheupwardtransportofinclusionsinNewtonianandpower-lawsaltdiapirsisalsoexplored.Somereportsfocusonthein situstressmodelofthestringerinsalt.Physicalmodelswereusedtostudythedeformationofintra-stringers,whichisrelatedtohalokinesisandinversiontectonics(Eleman,1997).Inaddition,someresearchonthedeformationofsaltdiapirshasbeenpublished.Examplesinclude the following: mechanics of active salt diapirism (Schultz-Ela and Jackson, 1993), anumericalmodelsetupfortheinitiationofsaltdiapirswithfrictionaloverburdens(Podladchikovet al.,1993;Poliakov et al., 1993), the modelling of the salt flow by overburden (Koyi, 1996), theshapingofthesaltdiapirsexploredbyKoyi(1998),thesaltminibasinsinvestigatedbyIngsandBeaumont(2010),theeffectiveviscosityofrocksaltdiscussedbyvanKekenet al.(1993),whoseresearchisalsoconcernedwiththeprogramofradioactivewastemanagement.Physicalmodels(Koyi,2001)andnumericalmodels(Chemiaet al.,2008,2009)wereappliedtostudythewholeprocessinwhichanhydriteblocksareentrainedbysaltandthendescendasawholewithinthestructure.Numericalmodelswereused toquantify thedescent rateofentrainedanhydriteblockswithinasaltdiapir(Koyi,2001).Otherrelatedissueswerealsosystematicallystudied,suchastheeffectsofviscosity[Newtonianandnonlinear:Chemiaet al.(2008)],positionoftheanhydritelayer(ChemiaandKoyi,2008),thedifferentrisingrateofsaltdiapirsinconnectionwiththeinclusionofanhydritelayers/blocksandtheirdescentwithinasaltstructure(Chemiaet al.,2009),theeffectofsize/aspectratioandorientationofdenserblocksonsinkingrateandmode(Burchardtet al.,2011,2012a,2012b).

Inthispaper,thesinkingofstringersorstringerfragmentsinsaltblocksordomesisdiscussed.The researchers investigate how the sinking velocity of stringers is influenced by material conditions andgeometrical properties throughnumerical simulation.For thevalidationof thenumericalmodellingapproach,twosetsofsimulationshavebeenapplied.FirstinLiet al.(2008)andLi(2013),relationsbetweenstrainrateandstressobtainedfromsimulationandexperimentalmeasurementwerecomparedwitheachother.Therelationbetweenstrainrateandstresscanbeobtainedbyeithersimulationorexperimentalmeasurement.Fortheformermethod,FiniteElementModelling(FEM)isusedtosimulatethecylindricalsaltbodyunderuniaxialcompression.Asforthelattermethod,therelationbetweenstrainrateandstressisobtainedbyvaryingandmeasuringrheologicalparameters(UraiandSpiers,2007).

Second, researchers tested a simulation of the flow of a power-law fluid between two flat plates. The numerical solution for the flow velocity is compared to the analytical solution of Turcotte and Schubert(1982).Finally,comparisonofsimulatedpower-lawcreepwithatheoreticalresultwascompletedinLi(2013).Inthestudy,asinkingvelocityformula[i.e.,Stokeslaw:Lamb(1994)]wasusedtoevaluatethenumericalresultsinthemodelinwhichthevelocityofaballsinkingina steady-state Newtonian fluid (n=1)wassimulated.

Inrecentyears,thedeformationmechanismsandrheologyofrocksalthavebeeninvestigatedinalargenumberofstudies,includinglaboratoryexperimentsandmicrostructuralinvestigations(Carter and Hansen, 1983; Heard and Ryerson, 1986; Urai et al., 1986; Wawersik and Zeuch,

Numerical modelling of gravitational sinking of anhydrite stringers in salt Boll. Geof. Teor. Appl., 57, 233-246

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1986; Aubertin et al.,1989;Sensenyet al.,1992;Carteret al., 1993; Peach and Spiers, 1996; Weidinger et al., 1997; Cristescu and Hunsche, 1998; Spiers and Carter, 1998; Hunsche andHampel,1999;Martinet al.,1999;TerHeegeet al.,2005a,2005b;UraiandSpiers,2007;Uraiet al.,2008). Duringdynamicrecrystallizationatgeologicalstrainrate,saltisinthetransitionregime where both dislocation creep, related to Newtonian flow, and solution-precipitation creep, related to non-Newtonian flow, operate (Urai et al.,2008).Ifthedislocationcreepandpressuresolution act in parallel, the steady-state flow of a rock salt can be expressed as the summation of thestrainratefromdislocationcreepprocessesandthestrainratefrompressuresolution(Uraiet al., 2008). In power-law creep, the strain rate is related to the flow stress using the equation:

(1)

where ε.isthestrainrate,Δσ=σ1-σ3 isthedifferentialstress,andA=A0 exp(-Q/RT)istheviscosity

ofthesalt.Withintheviscositydescribedabove,A0isamaterial-dependentparameter,Q isthespecific activation energy, while Risthegasconstant(R=8.314Jmol-1k-1)andTisthetemperature.Theothermechanismissolution-precipitationcreep:

. (2)

The strain rate is dependent on the strain size D, Δσ=σ1-σ3 is the differential stress, andB=B0 exp(-Q/RT)(1/TDm)istheviscosityofthesalt.Withintheviscositydescribedabove,B0isamaterial-dependentparameter,Q is the specific activation energy, while Risthegasconstant(R=8.314J/mol-1k-1),andTisthetemperature.Theorderm influences strain rate dependent on thegrainsize.

Fig. 1 summarizes low-temperature laboratory data for a wide range of halite based onexperiments by Sandia, BGR, Utrecht University, and other laboratories. The dashed linerepresentsanextrapolationofthedislocationcreepwithn=5(30-50°C).Thesolidlinesareroom-temperaturesolution-precipitationcreeplawsfordifferentgrainsizes.Duringactivesalttectonics(differentialstressintheorderofafewMPaasshownbysubgrainsizepiezometry),thetwokindsofrheologyhavesimilarstrainrates,andbothareinterpretedtobeimportantincoarse-graineddiapiricsalt(SchléderandUrai,2005).Inthisstudy,numericalexperimentsweredesignedtobesensitivetosaltrheology,geometry,anddensityofstringerorfragment.Thepurposeandinterestof the research is to investigate the sinking velocity of stringers as influenced by salt rheology, densitycontrast,andgeometricalproperties.Inrecentyears,Koyi(2001)studiedtheeffectofthicknessondescendingrate.TheexperimentalresultsindicatedthatthickerblockssinkfasterintheNewtoniansalt.Chemiaet al.(2008,2009)studiedtheeffectofdifferencesintherheologyofsalt.ThesamegeometryasthatoftheGorlebensaltdiapirandtheanhydritelayerswassimulatedto study the interactionbetween these two rockunits.These studies investigated theeffectofincludingdifferentsaltrheologywithinthesamemodelinordertosimulatethedifferentsubgroupsofZechsteinformation.Inourstudy,wechangedthevalueofthickness,densitycontrasts,andtherheologyofthesaltwithinwhichtheembeddedstringerdescends.AgenericmodelandacasefromZechsteinsaltinthenorthernNetherlandswerechosen.Casesobtainedfromseismicdatatend to produce results which later can be compared with observations in real fields. In this way, moreinsightsarelikelytobeprovidedforfurtherstudies.DifferentfromChemiaet al.(2008),

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thesalttectonicsinourmodelsisnotactiveandthestringersinksaftersalttectonics.ThestudyissimilartotheresearchofBurchardtet al.(2011,2012a,2012b),whousedextensivenumericalmodelling to study sinking anhydrite blockswithin aNewtonian salt body. Inourmodel, thestringer is brittle rather than viscous, and salt flow presents the property of both Newtonian and non-Newtonian fluid. In this way, the stringers tend to sink due to gravitational loading and the densitycontrastbetweenitselfandsurroundingsalt.Therelationbetweensinkingvelocityandmaterialorgeometricalpropertiescanbeclearlyobserved.

2. Methods and models

2.1. Gravitational sinking of a single stringer in salt block (a generic model)As for the simplest, genericmodel for the sinkingof an anhydrite stringer embedded in a

saltbody,wehavechosenasetupconsistingofasinglerectangularstringerinsidearectangularsaltblock(Fig.2).Thegeometryofthemodelisdescribedbythewidthandheightofthesaltbody,whicharerespectivelysettobe8and4kminallofthefollowingmodels;hstandsforthethicknessofthestringer,wisthewidth,andDistheinitialdepth,whichismeasuredfromthetopofthestringer.Themodelconsistsofasaltblock,thegeometryofwhichis8kmwideand4kmhigh.

For the case of Newtonian flow, the rheology is presented with A0=4.70×10-10Pa-1s-1,Q=24530J/mol,m=3,n=1,D=0.01m,R=8.314J/(mol·k)(Spierset al.,1990).Forthecaseofnon-Newtonianflow, the rheology is presented with A0=1.82×10-39Pa-5s-1,Q=32400J/mol,n=5,R=8.314J/(mol·k)

Fig. 1 - Differential stress-strain rate diagram for rocksalt (modified after Schléder andUrai,2005).


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(Schoenherret al.,2007).Thesurfacetemperatureissettobe50oCand,atthedepthof4km,thetemperatureissettobe140oC(i.e.,thelowerlimitofthemodel).2Dplanestrainconditionsareapplied.Usingthismodel,weobservedthatthesinkingvelocityofthecentralpointinthestringerdepends on the following factors: a) thickness and aspect ratio of the stringer, b) the densitycontrastbetweenthestringerandthesurroundingsalt.

2.2. Gravitational sinking of 30-to-80 m-thick Zechstein intra-salt stringersTheconceptisbasedonseismicobservationsfromtheZechsteinsalt(vanGentet al.,2011;

Strozyket al.,2012).Duringsalttectonics,manyZechsteinintra-saltstringerfragmentsappearduetobreakage,thethicknessofwhichrangesfrom30to80m.Thefragmentsarelocatedintheupperpartofthesaltsection.Thewidthofthefragmentsrangesfrom100to1000m(Fig.3).TheMesozoicisthemainphaseofsalttectonicsandtectonicsstoppedfromthePaleogene(Geluk,2000,2005).Inourstudy,adaptiveremeshingtechniquesinFEMwereapplied(Liet al.,2012a,2012b).Themodelscalculatethesinkingvelocityandtrajectoryoffragmentgeometriesforthetwodifferentkindsofsaltrheologyaftersalt tectonicsstopped.Thecalculationresultscanbeveryuseful.Ontheonehand,wecanevaluatetherelationbetweensinkingvelocityofstringerfragmentsandviscosityforthetwodifferentkindsofsaltrheology.Ontheotherhand,wecanalsoevaluatetherelationbetweensinkingvelocityandthegeometricalpropertiesofthestringerfragment.Besides,bycomparingthemodellingresultswiththeobservationsofseismicdata,saltrheologycanbefurtherstudiedtofacilitateotherresearch(Liet al.,2012b).

Modelling and calculation involve two major steps (Li et al., 2012b). First, a verticaldisplacement of the top salt boundary should be applied to simulate the down-building ofoverburdensedimentsand formationofasaltpillow(Liet al.,2012a).Second,gravitationalloadingisappliedtothesaltsectionafter tectonicsandthesinkingvelocityandtrajectoryofstringerfragmentsaresurveyed.

Fig.2-Sinkingofastringerembeddedinarectangularsaltblock.

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Twomodelswithdifferentdominantsalt rheologyareapplied:modelA isperformedwithpressure-solutioncreep(n=1),andmodelBisperformedwithdislocationcreep(n=5;Table1).In addition, the geometry and configuration of stringer fragments in the two models are also different.Wecomparedthesinkingvelocityoffragmentsintwomodelswhichhavedifferentsaltdeformationmechanisms.Table1showstherheologicalandmechanicalpropertiesofboththesaltlayerandthestringerfragmentsusedforthetwomodels.

Fig. 3 - 3D seismic data (left) and 2D seismic profile (right) with interpreted, physically isolated Zechstein 3 stringer fragmentsembeddedinthedeformedsaltsections,northernNetherlands(bycourtesyofNAM).

Fig.4-Modelwidthis18kmandinitialmodelthickness1km.A:initialmodelsetupbeforesalttectonicswithoutdifferentialstresses.

Table1 -Material parameters for anhydrite and rock salt [following the twodifferent deformationmechanisms at50°C(i.e.,lowerlimitofthemodel)].

Salt A: pressure- Salt B: dislocation creep Anhydrite blocks solution creep (Non-Newtonian flow) (Newtonian flow)

A0 4.70×10-10 Pa-1 s-1 1.802×10-39 Pa-5 s-1 -

Q [J/mol] 24530 53920 -

R [J/(mol·k)] 8.314 8.314 -

m,n 3/1 0/ 5 -

ρ/density [kg/m3] 2200 2200 2900

E/Youngs Modulus [GPa] 10 10 40

υ/ Poisson’s ratio 0.4 0.4 0.3

Reference Spiers et al., 1990 Hunsche and Hampel, 1999 Sayers, 2008


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A simplified model of the Zechstein salt section (Fig. 7) was applied. This model included sevenstringerfragmentswhichareelongated,broken,andisolated.Thewidthofeachfragmentis chosen from 100, 600, and 1000 m, while thickness is chosen from 30, 60, and 80 m. All fragmentsareembeddedinarectangularsaltsection1kmthickand18mlong,whichisaplanestrainmodelbasedonseismicobservationsfromvanGentet al.(2011).Theboundaryconditionforthemodelisdisplacementequalszerowhenitisperpendiculartothebottomandbothsidesof the model. The top of the salt surface is fixed in xandydirectionaftersalttectonicslastingfor 4 Ma. When salt tectonics stops, a model with a time span of 60 Ma is performed to simulate theZechsteinsaltatrestintheTertiary.Table2isasummarizationwhichpresentsthegeometricparametersofstringerfragmentsandsaltsectionaswellasthetimedurationofeachexperimentalphase.

Table2-Geometricalandtemporalmodelsetup.

Width of salt body 18000 m

Height of salt body 1000 m

Stringer fragment thicknesses 30, 60, 80 m

Stringer fragment widths 100, 600, 1000 m

Amplitude down-building 300 m

Duration salt tectonics 4 Ma

Duration salt at rest 60 Ma

3. Results

3.1. Gravitational sinking of a simple stringer in salt block3.1.1. Stringer thickness

First we investigate the influence of the variations in the thickness, and therefore mass, of the stringer.Inthispart,thewidthofthestringerw isaconstantvalueandtheheightofthestringerhisvarious.Theboundaryconditionsarezerodisplacementperpendiculartotheboundaryatthebottomandatthesides.ThematerialpropertiesofthestringerareYoung’smodulusE=40GPa,Poissonratioυ=0.3,anddensityρ=2400kg/m3(Sayers,2008).Thisgeostaticstateisusedastheinitialstressconditionforthenextstep.

Fig. 5 - Sinking velocity of the stringer vs. stringer thickness. The variation in stringer thickness (140, 160, 180, 200, 220, and 240 m) results in aspect ratios of 8.6, 7.5, 6.7, 6.0, 5.4, and 5.0 respectively due to the constant length of the stringer(1200m).a)ThesaltrheologyisNewtonian(n=1);b)thesaltrheologyisnon-Newtonian(n=5).

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Fig.5illustratestherelationbetweenthesinkingvelocityandthethicknessofthestringer,separatelyforn=1andn=5.Thestringerthicknessh is 140, 160, 180, 200, 220, and 240 m, and therefore the aspect ratio is 8.6, 7.5, 6.7, 6.0, 5.4, and 5.0, respectively. The variation of the size, and thereby the mass, of the stringer influences the differential stress Δσ linearly.Thesinkingvelocityofthestringeraredeterminedbythestrainrateonthedifferentialstresswhenn=1and5thorderrelationwhenn=5.Thedisplacementandvelocityofthestringerhavethesametrendasthestrainrate.Moreover,fromtheresult,wecanknowthatthestringerwiththesamesizeinbothn=1andn=5rheologicalsalthassimilardifferentialstressΔσdistributionarounditbecauseofthesamemass,andthepeakdifferentialstressaroundthestringerchangesfrom100to150kPawithvariousstringersize.

3.1.2. Density contrastThe second parameter influencing the sinking velocity we discuss is the density contrast

between the salt and the stringer. A single stringer 1200 m wide and 160 m thick is located in the salt.ThematerialpropertiesofthestringerareYoung’smodulusE=40GPa,Poissonratioυ=0.3,anddensityρ=2400kg/m3.Thesaltdensityis2040kg/m3,andthestringerdensityis2200,2300,2400, 2500, 2600, 2700, and 2800 kg/m3,respectively.Theinitiallocationofthestringerisinthecentreofthesaltblock.

The results of the simulations confirm the expectations. The numerical solution shows that for Newtonian fluid flow (n=1),thesinkingvelocityofthestringerinthesaltfollowsalinearrelationdepending on the density contrast (Fig. 6a). For non-Newtonian fluid flow (n=5), the sinkingvelocityofthestringerfollowanonlinearrelationwith5th power order (Fig. 6b). The variation of the density influences the differential stress Δσinthesaltlinearly.Thestrainratevariesfromthedensitycontrastduetotherelationέ=A(Δσ)nandέ=B(Δσ).

3.2. Gravitational sinking of 30-to-80 m-thick Zechstein 2 intra-salt stringersFigs.7aand8ashowtheinitialmodelsetupwiththestringerfragmentsNos.1-7insaltsection.

Figs.7band8bshowthatthestringerfragmentsrotateduringsalttectonicswhenNewtonianlaw

Fig. 6 - Sinking velocity of the stringer vs. stringer density. The variation of stringer density is 2200, 2300, 2400, 2500, 2600, 2700, and 2800 kg/m3.a)ThesaltrheologyisNewtonian(n=1);b)thesaltrheologyisnon-Newtonian(n=5).


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Fig. 7 - Results of Model Ausing salt rheology of n=1.Modelwidthis18kmandini-tial model thickness 1 km: a)initial model setup before salttectonics without differentialstresses;b)ModelAduringsalttectonics (-62 Ma); c-g) Model A after salt tectonics (-60, -40, -20Ma,andtoday).

Fig. 8 - Results of Model B(right)usingsalt rheologiesofn=5.Modelwidthis18kmandinitialmodelthickness1km:a)initial model setup before salttectonics without differentialstresses;b)ModelBduringsalttectonics (-62 Ma); c-g) Model B after salt tectonics (-60, -40, -20Ma,andtoday).

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dominates,whilethefragmentsinthesaltwithn=5(ModelB;Figs.8band8c)almostdonotsink.Furthermore,wedetectaveragesinkingvelocityanddifferentialstressesaroundthestringerfragmentsinbothModelAandModelB,dependingstronglyonthegeometryandlocationofthefragmentsinthesaltbody(Table3).Finally,thebendingdeformationofstringerfragmentscanbeclearlyseeninModelAduringthesinkingprocessandthebendingdeformationalsoincreasestheinternalstressofthestringerfragment.

From Table 3, we can figure out the relation between the sinking velocity and the thickness ofthestringer,bothforModelA(n=1)andModelB(n=5).Hereaveragesinkingvelocityinthecentralpointofastringerfragmentisinvestigatedbecauseofrotationordeformation.Andthesinkingvelocityofastringerfragmentwilldecreasewhenitgetsclosetothebottomofthesaltsection.

The variation of the size and mass of the stringer linearly influences the differential stress, andthevelocityofthestringerisdeterminedbythestrainrate,dependingonthethicknessofthestringerinthesaltwhichislinearlydependentonthedifferentialstresswhenn=1andhasnonlinearrelationwhenn=5.Thedifferentialstressofastringerfragmentkeepsstableanddoesnotchangewithsaltrheology.ThestringerfragmentsNos.3,4,and5sharethesamewidthof1000m,buthave different thicknesses of 30, 60, and 80 m, respectively. For Model A, the average sinking velocitiesofthethreestringerfragmentsare28.4,49.8,and101.7m/Maseparately,whilethedifferential stresses around stringers are 133, 146 and 154 kPa. For Model B, the average sinking velocitiesare0.45,0.72,and0.90m/Ma,whilethedifferentialstressesaroundthestringersare131, 146, and 154 kPa. Moreover, the stringer fragments Nos. 1, 4, and 6 have the same width of 80mbutdifferentthicknessesof100,500,and1000m,respectively.ForModelA,theaveragesinkingvelocitiesare40.5,75.8,and101.7m/Ma,whilethedifferentialstressesaroundstringersare 136, 148, and 154 kPa. In Model B, the average sinking velocities are 0.63, 0.84, and 0.90 m/Ma,whilethedifferentialstressesaroundstringersare132,147,and154kPa.

Furthermore, the stringer fragments Nos. 2, 6, and 7 have the same width of 600 m but different thicknesses of 30, 60, and 80 m, respectively. For Model A, the average sinking velocities are 27.4, 49.8, and 75.8 m/Ma, while the differential stresses around stringers are 134, 138, and

Table3-Resultsofaveragesinkingvelocity,sinkingdisplacementanddifferentialstressofthestringerfragments1-7inModelAwiththerheologyn=1(pressure-solutioncreep)andModelBwiththerheologyn=5(dislocationcreep).

Model A Model B (n=1) (n=5)

Stringer Fragment Average Sinking Max. differential Average Sinking Max. differential fragment width/ sinking displacement stress around sinking displacement stress around No. thickness velocity in 10 Ma stringer velocity in 60 Ma stringer in salt (see Fig. 3a) [m] [m/M] [m] [×103 Pa] [m/M] [m] [×103Pa]

1 100/80 40.5 405 136.0 0.63 37.8 132.0

2 600/30 27.4 274 134.0 0.44 26.4 130.0

3 1000/30 28.4 284 133.0 0.45 27.0 131.0

4 1000/80 101.7 1017 154.0 0.90 54.0 154.0

5 1000/60 49.8 498 146.0 0.72 43.2 146.0

6 600/80 75.8 758 148.0 0.84 50.4 147.0

7 600/60 49.8 49.8 138.0 0.63 37.8 136.0


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148 kPa. For Model B, the average sinking velocities are 0.44, 0.63, and 0.84 m/Ma, while the differential stresses around stringers are 130, 136, and 147 kPa. Here one thing we should pay attentiontoisthatthelinearrelationisnotobviousinthisexampleforModelA(n=1),becausethemovementofstringerfragmentswithlargewidthandthicknessissometimesrestrictedbytheboundaryeffect.

4. Discussion

Simulationisusedtoevaluatetheeffectofmodelgeometryandmaterialparametersonthemovementofanhydritestringerswhichareembeddedinsalt.WhetherusingthegenericmodelorthemodelbasedontheZechsteinsalt,resultsindicatethatthesinkingvelocityofthestringervarieslinearlywithparameterAandparameterB[A=A0 exp(-Q/RT),B=B0 exp(-Q/RT)(1/TDm)],both of which are derived from the flow law έ=A(Δσ)n and έ=B(Δσ).However,anychangeinthe differential stress Δσ leads to either linear change in the sinking velocity for Newtonianrheology(n=1)ornonlinearchangeforpowerlawrheology(n>1)(thedifferentialstressΔσiscausedbyvaryingthesizeofthestringerorthedensitycontrastbetweenthestringerandthesalt).In thiscase, thenon-linearchangeof thesinkingvelocityfollowsapower lawwith thesameexponentn.DensityandgeometrychangeofthestringerisrelatedtoitsowngravitythataffectsthedifferentialstressΔσ.

Thecalculationresultofthesinglestringerinthegenericmodelshowsthatthevelocityofthesinkinganhydritestringerisgreaterthan1000m/Ma,whichindicatesthatifthesaltfollowstheruleofNewtonianrheologyandtheviscosityBis10-20Pa·s,thestringerwillgetclosetothebottomofthesaltbodyinatimerangeofabout10Ma.ThevelocityofthesinkinganhydritestringerissimilartothatoftheanhydriteblockswhichareembeddedinNewtoniansalt(Burchardtet al.,2011).Besides,thevelocityofstringerfragmentsvariesfrom15to100m/MaifanhydriteblocksareembeddedinZechsteinsaltatrest(effectivestressdecreasesafterrelaxation)whichfollowstheNewtonianlawandhasasaltviscosityofaround10-19Pa·s.AllofthestringerfragmentswouldgetclosetothebaseofthesaltsectionwithinatimespanoftheentireTertiary.However,itisobservedinseismicdata(vanGent et al.,2011;Strozyket al.,2012)thatsomestringersarestilllocated close to the top or in the upper part of the salt body after a longer time (~ 60 Ma). This discrepancysuggeststhatthelong-termrheologicalbehaviorofthesaltis,atleastinsomecases,quitedifferent fromwhatweexpected from the rheologicalparameterswhichareobtained inthelaboratory.Whenthesaltfollowspower-lawrheology,thesinkingvelocitiesareratherslow.Theexplanation for this situation is thatdue to thegravitational sinkingstringer in long-termrheologicalsalt,thepeakdifferentialstressaroundthestringeris100-150kPa,whichleadstoaratherlowstrainrateofaround10-20-10-19s-1. It is clear that stringers do not significantly sink over geologictimes.

Otherissuesdiscussedhereincludethattheinitiallocation(depth)ofthestringeralsohasabig impact on the salt flow and its own descent (Chemia et al.,2008).Inourstudy,thisfactorseemstobenotasimportantassaltrheology,densitycontrast,andgeometricalpropertiesofthestringer,becausethemethodemployedinthisstudyisgravitationalsinkingstringerinsaltatrest(after tectonics)sincewefocuson therelationbetweensinkingvelocityandfactors includingsalt rheology, density contrast, and geometrical properties of the stringer due to gravitational

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loading.Salt tectonicprocessor themovementof salt diapir canhaveadirect impacton theriseandfallofstringersorstringerfragments.Thecomplexmodelwhichincludessalttectonicmovementandgravitationalloadingcouldbetakenintoaccountinfurtherresearchofmovementanddeformationofembeddedstringers.

5. Conclusion

Thisstudyhasdemonstratedthatsaltrheologyandmodelgeometryofstringers(fragments)arethedominantfactorsaffectingthesinkingvelocityofanhydritestringersinthegenericmodeloranhydriteblocksinZechsteinsaltatrest(thestressinZechsteinsaltisreleasedaftertectonicmovement).ThestudyhasalsoindicatedthatthesinkingvelocityofthestringerlinearlychangeswiththerheologicalparametersA,B,andn.Moreover,asisshownintheresultsofthegenericmodel,thefactorsaffectingsinkingvelocityatagivenrheologyarestringerthicknessanddensitycontrast.ThicknessandwidthofstringerfragmentarealsofactorsthatcontrolsinkingvelocityinZechsteinsalt.Thevelocitychangesnonlinearlyforpower-lawrheology(n>1)andchangeslinearlyforNewtonianrheology whenthevalueofstressΔσchanges.Inthiscase,thenonlinearchangeofthesinkingvelocityfollowsapowerlawwiththesameexponentn.

Incontrast to theNewtonianrheology leading tohighsinkingspeed(>15~100m/Ma), thepower-law rheology results in stringers presents no significant sinking over the geologic timescale, whichisconsistentwiththeseismicdataobservations.Theselargedifferencesinsinkingvelocityareduetotheverylowdifferentialstressaroundthestringers,ontheorderof0.1MPa,aftersalttectonicsstopped.Finally,wecanconcludethatthesimulationofgravitationalsinkingofasimpleanhydritestringerinthegenericmodelandanhydriteblocksinrocksaltbasedonobservationsfromnaturalsettingsenablesaprofoundunderstandingofmovementanddeformationofembeddedstringersorfragments.

Acknowledgements. We thank the Ministry of Oil and Gas of the Sultanate of Oman and PetroleumDevelopmentOmanLLC(PDO)forgrantingpermissiontopublishtheresultsofthisstudy;NederlandsAardolieMaatschappojBVforprovidingtheseismicdata;andS.Abe,H.vanGent,F.Strozyk,andL.Reuningfortheirtechnicalassistanceonourresearchofsinkingstringers.Theresearchisalsofundedbythe startup project of China University of Petroleum, Beijing (n. 2462014YJRCOM) and supported by the Science Foundation of China University of Petroleum, Beijing (n. C20/60).

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Corresponding author: Shiyuan Li College of Petroleum Engineering, China University of Petroleum 18 Fuxue Road, Changping, Beijing 102249, China Phone:; fax:; e-mail: [email protected]

numerical modelling of gravitational sinking of anhydrite ......numerical modelling approach, two...

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