FluidsintheEarth’sCrustSeminar Homework1 Name_____________________________ Assigned:August30,2012Due:September13,2012Problem1.StartingwithStokesequationforsteady‐,uniform‐,laminar‐flowandusingyournotesinclassasaguide(derivingDarcy’slawforacapillarytube),determinetheparabolicformofthevelocitydistributionacrossauniformfracture.Alsodeterminetheaveragevelocityforthefracture.Showthatthepermeability(k)forthisfractureisequaltob2/12(seeFigure1).Stokesequationisgivenby:

€

µ∂ 2V∂z2

= ρg ∂h∂x

Subjecttothefollowingboundaryconditions:

€

V(z =b2) = 0

∂V z = 0( )∂z

= 0

Showallsteps.Hints:Youhavetointegratebothsidesoftheequationtwicewithrespecttozandevaluatetheconstantsofintegrationatz=0andz=b/2.Whenfindingtheaveragevelocity,usethelimitsofintegrationfromz=0andb/2andmultiplytheintegrandby2/b.Assumethatthecenterofthefractureisz=0.

Figure1.Schematicdiagramillustratingvelocitydistributionandpressuredropacrossauniformfracture.

Problem 2. A geothermal test well was drilled just west of the NM Tech campus (NMT-2GT; Figure 2). The slim-hole well (less than 7-inch in diameter) was drilled to a depth of about 1101 feet with rotary air equipment. The borehole allows for an assessment of the potential and feasibility of the Socorro Peak geothermal reservoir as a source for geothermal fluids to operate a district geothermal heating system on the NM Tech campus.

Figure 2. Location map of NMT-2GT drill site. During drilling in 2009-2010, water samples and fluid temperatures were collected from the blooie line of the drill rig for complete chemical analyses. These are reported in Table 1. Table 1. Major Ion Concentrations of Geothermal Well NMT-2GT. Concentrations reported in mg/l.

Na+ K+ Mg2+ Ca2+

SO4 2-

HCO3-

Cl-

δ18O‰ (SMOW)

δD‰ (SMOW)

625 26 16 105 32.1 305 935 -10.71 -74.98

Themaximumtemperaturesampledfromthiswellwas41oC.Estimatethemaximumreservoirtemperaturethatthesefluidswereexposedtousingbothsilica(Fournier,1977)andNa‐K‐Ca(FournierandTruesdall,1973)geothermometers.

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T(oC) =1309

5.19 − log10(SiO2)[ ]− 273.15 Fournier(1977)

T(oC) =1647

log10(Na /K) +1.33log10CaNa

+ 2.24

− 273.15 Fournier −Truesdall(1973)

whereSiO2isthedissolvedsilicaconcentrationinmg/l,Caisthedissolvedcalciumionicconcentrationinmoles/l,Kisthedissolvedionicconcentrationofpotassiuminmoles/l,andNaisthedissolvedionicconcentrationofsodiuminmoles/l.Thesegeochemicalgeothermometersassumethatgroundwaterhasequilibratedwithatypicalvolcanicrockassemblageatelevatedtemperaturesandthatduringupwellingandcooling,thegeothermalfluidsmaintainedtheirequilibriumconcentrations(slowchemicalkineticrates).YouwillhavetoconvertNa,K,andCaconcentrationfrommg/ltomoles/l(divideby1000anddividebythegramatomicweight).Required:Usethesetwogeothermometerstoestimatethemaximumtemperaturethesefluidswereexposedto.SilicaGeothermometertemperature_____________Na‐Ca‐KGeothermometertemperature______________Estimatethedepthofcirculationassumingageothermalgradientof40oC/km.ThemeanannualtemperatureatSocorroisabout15oC(temperatureatagivendepthisestimatedtobeequaltothelandsurfacetemperature+temperaturegradient*depth).Rangeofdepthofcirculation__________kmPlottheoxygenisotopiccompositionofthegeothermalfluidontheoxygen‐hydrogendiagraminFigure3.Isthereanyevidenceoffluid‐rockisotopicexchange(whichwouldbeindicativeofahightemperaturesystembetween200‐300oC)basedonthisisotopiccomposition?Arethegeothermometertemperaturesconsistentwiththeisotopicdata?

Figure3.Meteoricwaterlineforprecipitation.Problem3.AMATLABprogramhasbeenwrittenthatcalculatewaterdensityandviscosityofwateratelevatedtemperatures,pressures,andconcentrationsbasedontheequationsofstatepresentedinBatzelandWang(1992).Assumingasandstonepermeability(k)of10‐12m2(1000mD)andusingtheviscosity(viso,visf)anddensity(rhoo,rhof)valuesfromtheMatlabprogram,plothydraulicconductivity(inm/yr)versedepthfora500mthickoilreservoiratadepthbetween2000‐2500m(elevationof1000mto500m).RecallthathydraulicconductivityisKf=kρfg/µfandoilconductivityisKoil=kρoilg/µoilwhereρfiswaterdensity,µfiswaterviscosity,ρoilisoildensity,µoilisoilviscosity,KoilisoilconductivityandKfishydraulicconductivity,kispermeabity(mustuseconsistentunits),andgisthegravityconstant.

Required:Plotoilconductivityandhydraulicconductivityverseelevationassumingthefollowingconditions:

a. Hydrostaticpressuregradient,30oC/km,freshwaterb. Hydrostaticpressuregradient,30oC/km,saltwaterc. Lithostaticpressuregradient,30oC/km,saltwaterd. Hydrostaticpressuregradient,50oC/km,saltwater

Commentonwhethertheseresultsmakesense.Tocalculatepressure,recallthathydraulicheadisgivenby:

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h =Pρg

+ z

Snow(1968)providesamethodforestimatingpermeabilityforfracturesofconstantaperture(aswederivedabove)andspacing(N):

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k =Nb3

12

Assumingataperturespacingof1cm(N=0.01m‐1),whatfractureaperture(b)doesthispermeability(1000mD)correspondto?

b___________________(mm)

References:Batzle,M.andZ.Wang,1992Seismicpropertiesofporefluids,Geophysics,v57(11)

p.1396‐1408.Fournier.R.O,1977,Chemicalgeothermometersandmixingmodelsforgeothermal

systems,Geothermics,v.5:p.41‐50.Fournier,R.O,andTruesdell,A.H.1973.AnempiricalNa‐K‐Cageothermometerfor

naturalwaters:GeochemicaetCosmochimicaActa,v.37,p.1255‐1275.SnowDT(1969)Anisotripicpermeabilityoffracturedmedia.WaterResources

Research,5,86–95.