684 (ChineseJGeophys) 55 leastsquaresmethod(lsqr)andtruncatedsingularvaluedecomposition(tsvd)forverifying thehybridinversion methodaseriesofspinechotr

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1 CHINESE JOURNAL OF GEOPHYSICS Vol55No2 Feb2012 (T2D) (2): doi: /jissn Tan MJZouY LAhybridinversion methodof(t2d)2d NMRloggingandobservationparametersefectsChineseJ Geophys(inChinese)201255(2): doi:103969/jissn (T2D) ( )) (1D NMR) (2D (T2) (D) 2D NMR (LSQR) (TSVD) (T2) (D) T2 -D T2 -D 2D NMR 2D NMR doi:103969/jissn P Ahybridinversionmethodof D)2DNMRlogging andobservationparametersefects TAN Mao-Jin 12 ZOU You-Long 2 1Key Laboratoryof Geo-detection (China Universityof Geosciences)Ministryof EducationBeijing100083China 2Coledgeof Geophysicsand Information Technologyof China Universityof GeosciencesBeijing100083China Abstract Onedimensionalnuclearmagneticresonance(1D NMR)loggingtechnologyislimited forfluidtypingtwodimensionalnuclear magneticresonance (2D NMR)loggingcanprovide moreparametersincludingtransverserelaxationtime(t2)anddifusioncoeficient(d)inmulti- poremediaandthefluidtypingisrealizedsuccessfulyaccordingtothefluidpropertiesofnmr So2D NMR loggingis moreadvantageousthan1d NMR loggingpointtotherelaxation mechanismof2d NMR atgradientfieldtheechosimulationandinversionof2d NMR are discussedindetailandahybridinversionalgorithmisproposedwhichisbasedonadamping ( ) ( ) (2010ZY28) (2011ZX ) tanmj@cugbeducn

(T2) (D) T2 -D T2 -D 2D NMR 2D NMR doi:103969/jissn0001-5733201202032 P631 2011-06-132011-12-20 Ahybridinversionmethodof D)2DNMRlogging andobservationparametersefects TAN Mao-Jin 12 ZOU You-Long 2

2 684 (ChineseJGeophys) 55 leastsquaresmethod(lsqr)andtruncatedsingularvaluedecomposition(tsvd)forverifying thehybridinversion methodaseriesofspinechotrainsinanidealfluid modelarefirstly simulatedwithmultipleechospacing (TE)activationingradientfieldThenthesesynthesized echotrainsareinvertedbyhybridinversionmethodtheinversionresultsincludingt2anddare welconsistent withtheideal modelpresupposedandthe hybridinversion methodis more accurateand eficientthan singleinversion algorithm from the error analysis and runtime comparisonwhichindicatesthatthehybrid methodisefectiveandsuitstotheinversionof2d NMRloggingFurthermoretheLSQR-TSVDhybridmethodisappliedintotheinversiontestsof oil-waterand gas-water models with diferentobservation parameterssuchas magmaticfield gradientandechospacinggroupfromtheinversionresultsanalysisthereasonableobservation parametersareselectedandoptimizedforfluidtypingof(t2d)2d NMRLogging Keywords Two-dimension NMR (2D NMR)loggingLSQR-TSVD hybridinversiondifuse coeficient(d)transverserelaxationtime(t2)fluidtyping 1 Sun Dunn [12-13] Schlumberger Dol Hurlimann Venkatarama [14-15] 2 CPMG [19-20] (SVD) (NMR) [12] 2D NMR [3-8] NMR [21-22] NMR D) -T 1 ) (1D NMR) T 2 [23] T 2 (T W ) (LSQR) (2DNMR) [9-12] T 1 T 2 2D NMR 2D (T 1 ) (D) (G) [8] NMR Schlumberger MR-Scanner BakerAtlas MR-Explorer 2D NMR 2002 Chevron Boqin NMR 2D NMR LSQR (TSVD) 2D NMR - D) 2D NMR LSQR TSVD [18] 2D NMR - G)

modelpresupposedandthe hybridinversion methodis more accurateand eficientthan singleinversion algorithm from the error analysis and runtime comparisonwhichindicatesthatthehybrid

3 2 :(T2D) (T2D) U k ( 烄 β 1e 1 )=b AV k =U k+1b k (5) 烅 NMR 烆 A T U k+1 =V kb T k +αk+1v k+1e T k+ 1 e T k+1 n k+1 T 2 αk =V kykr k =b-aαkt k+1 =β (6) T W T E 1e1 -Bkyk D T W CPMG r k =b-ax k =U k+1 ( β 1e 1 )-AV kyk =U k+1 ( β 1e 1 )-U kb kyk =u k+1t k+1 (7) p m A ik = flje γ2 g 2 T 2 E D l t k ie -t i /T 2j +εik (1) l=1 j=1 A ik T W T Ek i εik p min β 1e1 -Bkyk LSQR m 32 TSVD flj D l T 2j (Singular Value Decomposition (2) : SVD) A ik = E ik ljflj +εik (2) E iklj = e γ2 g 2 2 T Ek Dl t i e -t i /T 2ii = 1 N Ek k =1 ql=1 pj=1 mn Ek k 3 Δx Ax -b 2 x Δx x 31 LSQR Δx x LSQR Paige Sanders 1982 Δx Δb A Lanczos A Ax =b min Ax -b 2 U k = [u 1 u 2 u k ] V k = [v 1 v 2 v k ] B k (k+1) k : α1 熿燄 (D) ) B k = β 2α2 αk (3) 燀 β k 燅 1 A : (T W ) β 1ui =b αiv i =A T u i 烌 T/cm βi+1 =Avi -αiui 烍 i=12 (4) 6 (T E ) αi+1v i+1 =A T β i+1vi+ T E [ 烎 αi =0 β i =0 u i v i =1 r k U k+1 yk t k +1 Ax -b 2 SVD (TruncatedSingularValueDecompositionTSVD) A D l T 2j flj x 0 b 0 = Ax 0 D) :A(x-x 0 )=b-b 0 AΔx = Δb AΔx-Δb 2 Δx x 0 + [24]

T 2j (Singular Value Decomposition (2) : SVD) A ik = E ik ljflj +εik (2) E iklj = e -1 12 γ2 g 2 2 T Ek Dl t i e -t i /T 2ii = 1 N Ek k =1 ql=1 pj=1 mn Ek k 3 Δx Ax -b 2 x Δx x 31 LSQR Δx x LSQR

4 686 (ChineseJGeophys) 55 1 TSVD 2 3 Table1 Gas-watermodelandfluidproperties LSQR T 2 D/(cm 2 s -1 ) T2/ms /% LSQR TSVD 96192]ms INT LSQR-TSVD (1440/T E )( INT ) 3a T 2 A s :x error = x-xmodel x model LSQR TSVD LSQR 2 TSVD LSQR ( 2a) 0558 TSVD T 2 T 2 (1~100 ms) TSVD T 2 (100~10000 ms) ( LSQR LSQR 2b ); ( 2b 7800s; TSVD ) 5608s LSQR-TSVD LSQR TSVD 3 TSVD s; ( 1) 200 ( 3a) s T 2 ( D ) T 2 T 2 (100~ 10000ms) ( 3b ) T 2 (1~100ms) ; Table2 Oil-watermodelandfluidproperties LSQR ( 3b ) LSQR s;TSVD s TSVD LSQR 33 LSQR TSVD TSVD T 2 TSVD LSQR LSQR x 0 TSVD s LSQR-TSVD TSVD 2 D/(cm 2 s -1 ) T2/ms /%

TSVD T 2 (100~10000 ms) ( LSQR LSQR 2b ); ( 2b 7800s; TSVD ) 5608s LSQR-TSVD LSQR TSVD 3 TSVD 400 01612 6752s; ( 1) 200 ( 3a) 01563 5536s T 2 ( D ) T 2 T 2 (100~ 10000ms) ( 3b ) T 2 (1~100ms) ;

5 2 :(T2D) (a) ;(b) Fig1 Fluidmodelandnumericalsimulationresults (a)fluidmodel;(b)syntheticechotrains 2 LSQR (a) ;(b) Fig2 InversionresultsusingLSQR method (a)inversionresults;(b)inversionresultscomparisonwithmodelon1d 3 TSVD (a) ;(b) Fig3 InversionresultsusingTSVD method (a)inversionresults;(b)inversionresultscomparisonwithmodelon1d

method (a)inversionresults;(b)inversionresultscomparisonwithmodelon1d 3 TSVD (a)

6 688 (ChineseJGeophys) 55 4 LSQR-TSVD (a) ;(b) Fig4 InversionresultsusingLSQR-TSVD method (a)inversionresults;(b)inversionresultscomparisonwithmodelon1d (A ) T/cm (b) (G) (T E ) TE 41 (G) [ ]ms [ ]ms [ ] ms [ T/cm T/cm T/cm T/cm T/cm T 2 D ( 5c) 7 ( 5ab) T/cm T E 10-3 T/cm T/cm T E T E 96192]ms INT(1440/T E )( INT ) 8 8 8c T a 8b (c) T T/cm T/cm 01258

96192]ms [061224366096]ms [102108120132156192] ms [06122448 20 10-3 T/cm 30 10-3 T/cm 4 0 10-3 T/cm 40 10-3 T/cm 5 40 10-3 T/cm T 2 D ( 5c) 7

7 2 :(T2D) (a)g= T/cm;(b)G= T/cm;(c)G= T/cm Fig5 Inversionresultsofoil-watermodelunderdiferentmagneticfieldgradients T E D) LSQR TSVD LSQR-TSVD D) 5 : (1)LSQR-TSVD

8 690 (ChineseJGeophys) 55 6 (a)g= T/cm;(b)G= T/cm;(c)G= T/cm Fig6 Inversionresultsofgas-watermodelunderdiferentmagneticfieldgradient LQSR TSVD (4) D)2D NMR (5) (2) T E D) (3) D)

9 2 :(T2D) (a)te=[ ]ms;(b)te=[ ]ms; (c)te=[ ]ms Fig7 Inversionresultsofoil-watermodelunderdiferentTEgroups ApplicationsinChinaWelLogging Technology (inchinese) (5): (References) [3] (3): [1] Zhao WJTan M JReviewandprospectonNMRlogging : 1998 applicationinshenglioilefieldprogressin Geophysics (in Xiao L Z NMR Imaging Logging Principle and NMR Chinese)200823(3): Petrophysics Experiment (in Chinese) Beijing: Science [4] Press (5): [2] WangZDWangHXiangT DetalIntegrationofNMR (5): DTWlogsandDTElogstoimprovetheoilbedinterpretation Xiao L Z Some Important Issues for NMR Logging accuracywlt (inchinese)200125(5):

(in Xiao L Z NMR Imaging Logging Principle and NMR Chinese)200823(3):814-821 Petrophysics Experiment (in Chinese) Beijing: Science [4] Press1998 200125(5):365-368 [2]

10 692 (ChineseJGeophys) 55 [5] TW setswithdiferentmeasurementerrorsjournalof Magnetic (4):38-40 Resonance (1): Tan MJZhao WJFanY RIdentificationoffluidproperty withnmrdual-tw welloggingdatanaturalgasindustry (inchinese)200626(4):38-40 [6] (in Chinese)Shandong: Press of China University of (4):96-98 Petroleum2010 Yun H YTan MJObservationwayandanalyticalapproach ofdualwaitingtimebynuclearmagneticresonancelogging Petroleum Geology and Recovery (in Chinese) (4):96-98 [7] (5): (1): ChengXZFanY RZhouCCIdentificationtechnologyof lowresistivitypaysinfresh waterreservoirearth Science Frontiers (inchinese)200815(1): [8] (3): (6): Shao W Z On efect of NMR diferential and shifted spectrum methodinlaboratorywlt (inchinese) (6): [9] SunB QDunn KJA globalinversion methodformulti- dimensionalnmrloggingjournalof Magnetic Resonance (1): [10] XieR HXiaoLZLu D W(T2T1)two-dimensional 20074(4): NMR methodforfluidtypingwellogging Technology (in BiLRLatestadvancesinNMRloggingtechnologyChinese Chinese)200933(1):10-20 Journalof Engineering Geophysics (inchinese)20074 [22] (T2D) (4): (9): [11] XieR HXiaoLZThe(T2D)NMRloggingmethodfor 20063(4): fluidscharacterizationchinesejgeophys(in Chinese) Lin Z ZPan H P The progress of nuclear magnetic resonanceloggingchinesejournalof Engineering Geophysics (inchinese)20063(4): [12] SunB QDunn K JMulti-dimensional NMRloggingJ Magnetic Resonance2005: [13] DunnKJBergmanDJLatorracaG ANuclearMagnetic ResonancePetrophysicalandLoggingApplicationsOxford: PergamonPress2002 [14] MoodyJBXiaYAnalysisofmulti-exponentialrelaxation datawithveryshortcomponentsusinglinearregularization Journalof Magnetic Resonance (1):36-41 [15] Hurlimann M D Venkataramanan LFlaum Cet al Difusion-editing:New NMR MeasurementofSaturationand PoreGeometryPaperFFFin43rd AnnualSymposium of SPWLAOisoJapanJune [16] DunnKJLatorracaG ATheinversionofNMRlogdata [17] : 2010 DunnKJXieR HNMRLoggingTheoryandApplications [18] (5): XieR HXiaoLZDunnKJetalTwo-dimensionalNMR loggingwellogging Technology (inchinese) [19] (3): GuZ BLiu WTheinversion oftwo-dimensional NMR mapchinesejournalof Magnetic Resonance (inchinese) [20] 2D NMR (4): GuZBLiu WSun D QetalApplicationof2D NMR techniquesinpetroleumloggingchinesejournalof Magnetic Resonance (inchinese)200926(4): [21] (T2T1) (1): (9): [23] TW (5): Tan MJShiY LZhao WJetalJointinversionmethod fornmrdual-twloggingdataandfluidtypingchinesej Geophys(inChinese)200851(5): [24] LSQR (2): LiuJSLiuF TLiuJetalParalelLSQRalgorithms usedinseismictomographygeophysics(inchinese) (2): [25] LSQR (2): Yang WLiu S XFeng Y Q A study ofthe LSQR algorithm for cross-hole tomography Geophysical and GeochemicalExploration (inchinese)200832(2): ( )

ofdualwaitingtimebynuclearmagneticresonancelogging Petroleum Geology and Recovery (in Chinese)200613 (4):96-98 [7] (5):430-434 200815(1):146-153 ChengXZFanY RZhouCCIdentificationtechnologyof

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