; T1. SEG Houston 2009 International Exposition and Annual Meeting

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1 Asymptotc calculatons o the Bot s waes n porous layered lud-saturated meda Yanun Lu* and Gennady Goloshun nersty o Houston Dmtry ln Lawrence Bereley Natonal Laoratory ummary Bot s poroelastcty model predcts the exstence o a slow compressonal (duson) wae due to the relate low o pore lud wth respect to the sold roc. Ths phenomenon opens an opportunty or nestaton o lud propertes o the hydrocaron-saturated reserors n partcular lud molty rom sesmc ampltude. Low-requency asymptotc descrpton o the Bot s model prodes relately smple orm o relecton and transmsson coecents. In case o normal ncdent the coecents only depend on mpedance contrast and small dmensonless parameter that s a product o lud densty scosty and roc permealty. All parameters are measurale. The oal o the present study s to use asymptotc soluton and propaator matrx method or nestaton o the slow P wae eect on relectty o the porous layered lud-saturated meda. Introducton Dependence o sesmc relectons on requency rom a as-water oundary n a porous sand reseror was calculated y Dutta & Ode (1983) usn exact Bot s model (1956). It was also shown y Dutta & Ode (1979a ) that relate lud moement ecomes nelle at sesmc requences the porous materal s homoeneous whle a heteroeneous lud saturaton leads to a sustantal eect on attenuaton. Carcone et al. (3) also demonstrated a sncant sesmc attenuaton due to heteroenetes n ether permealty or lud saturaton. Thus t s well nown that stron attenuaton o the transmtted P-waes s usually assocated wth heteroenetes n permealty and lud saturaton. Howeer esdes the attenuaton o the transmtted P-waes the eect o oth lud saturaton and lud molty on sesmc reseror relectty has not een extensely studed. ln and Goloshun (8 9) carred out a low requency asymptotc analyss o Bot s poroelastcty. They used oth lud low and scattern mechansms to dere a requency dependent relecton. In ths case the relecton and transmsson coecent are expressed as power seres o the square root o a dmensonless parameter: where s lud densty s permealty s lud scosty s anular requency and s the manary unt. A re descrpton o asymptotc calculaton s summared n the next secton. Asymptotc calculaton In case o ast P ncdent wae the relecton and transmsson coecents rom ast P wae to ast P wae are denoted as R and T the relecton and transmsson coecents rom ast P wae to low P wae are denoted as R and T. They hae the ollown asymptotc orms or normal ncdent P wae: 1 R R 1 T 1 R T1 1 R 1 T T1 where R s the ero order classcal relecton coecent and the rst order relecton and transmsson coecents R 1 and T 1 hae the orm: Z ( T1 ) ( T1 ) T1. Z Z Here Z 1 and Z are the acoustc mpedance o medum 1 and medum respectely. The rst order relecton and transmsson coecents R 1 and T 1 hae the orm: A A 1 1 T1 D D 1 here suscrpt 1 and ndcates medum numer. Other nternal descrptons related to lud and sold are: 1 (1 ) 1 s 1 Z A 1 1 Z 1 Z D Z Z EG Houston 9 Internatonal Exposton and Annual eetn 717

2 Asymptotc calculatons o the Bot s waes where s 1 and 1. 1 Here s the ul modulus o lud s the ul modulus o sold ran s the dry roc ul modulus s porosty s the lud densty and s the ul densty. A summary o all necessary nput propertes s demonstrated n Tale 1. It can e seen that the nput parameters are ran ul modulus dry roc ul modulus dry dry roc shear modulus dry lud ul modulus ran densty and lud densty. These parameters are routnely used n lud susttuton technque ased on Gassmann s equaton. The asymptotc descrpton o the Bot s model ncludes two addtonal parameters (roc permealty and lud scosty ). Thus t allows esdes realaton o the lud susttuton technque to prode an nestaton o the nluence o the permealty (lud molty) to sesmc response. All nput parameters can e acqured rom lo data and laoratory measurements. Hence t maes the asymptotc descrpton o the Bot s model more practcal or applcaton. dry dry Tale 1. Input parameters and unts or asymptotc soluton. urthermore elocty o ast P wae and low P wae V and V attenuaton coecents o ast P wae and low P wae a and a (n unt o m -1 ) can e calculated rom: where a V V a 1 1 Gpa /cc Gpa Gpa \ darcy Gpa /cc cp 1 1. Asymptotc soluton also prodes the relecton and transmsson coecents n case o low P wae as ncdent wae where the relecton and transmsson coecents or conerted low to low P wae are denoted as R and T and the relecton and transmsson or conerted low to ast P wae are denoted as R and T. They hae the asymptotc orms elow: R T Z( 1 R T ) R Z ( 1 R T ) T Z where 1 1. Propaator matrx method ollown the descrpton o Ronson (1967) we demonstrate a normal ncdent ast P wae and low P wae propaaton throuh porous layered meda usn the Thomson (195) and Hasell (1953) propaator matrx method. Accordn to the oundary condton n ure 1 we can otan the relatonshps etween all waeorms at nterace wth the correspondn relecton and transmsson coecents. We use r to represent the relecton coecent o ast P wae to ast P wae whle the ncdent wae s downon and rup to represent the relecton coecent o ast P wae to ast P wae whle the ncdent wae s upon. Thus r = u (t+1)/d (t-1) and rup = d +1 (t)/u +1 (t). mlar denotatons are used or other relecton and transmsson coecents. The tme delay or ast P wae trael throuh any layer s taen to e 1 unt o tme. The tme delay or low P wae trael throuh any layer s taen to e unt o tme. EG Houston 9 Internatonal Exposton and Annual eetn 718

3 Asymptotc calculatons o the Bot s waes Thus depends on the relate elocty o ast P wae and low P wae.e. ast. slow Also dene D () as the -transorm o d (t).e. D n t d ( t) n s the total numer o samples n the tme seres o d (t). And smlarly dene D () () () as the -transorm o d (t) u (t) u (t) respectely. Then we can otan or any nterace : D D tup tssup tsup t tsup D D ( ) ( ) where the our y our matrx [ ] s the propaator matrx that communcates the waeorms etween layer and +1. Each matrx element o [ ] s also n -transorm.e. polynomals o. Thus or wae propaaton n multlayered meda (ure ) we can otan: It can e seen that or homoeneous lud saturated meda there s almost no sle low P wae eect whle or nhomoeneous lud saturated meda a sncant low P wae eect exsts. And ths low P wae eect ares or derent requences. A loner and stroner low P wae d (t) d (t) u (t) u (t) Layer d (t-1) d (t-) u (t+1) u (t+) u +1 (t) u +1 (t) d +1 (t) d +1 (t) Layer +1 d (t) d (t) u (t) u (t) downon ast P wae n layer downon low P wae n layer upon ast P wae n layer upon low P wae n layer ure 1: chematc plot o wae propaaton throuh layer at normal ncdence. The horontal dsplacement corresponds to tme delay. D D ( tup tssup tsup tsup ) D ( ) D and the multplcaton etween any two matrx elements n [ ] s a conoluton o ther polynomal coecents. By settn D () = 1 D () = +1 () = and +1 () = we can otan () as the relectty seres o an mpulse ast P wae traeln throuh the mult-layered meda wth mod conerson to low P wae and multples taen nto account. Example 1: Homoeneous s. nhomoeneous lud saturaton In ths example we calculate the response rom two types o lud saturated mult-layered meda. Both meda are porous permeale sandstone ully saturated y some lud. Howeer one s only saturated y water whch corresponds to a homoeneous lud saturaton the other one s alternately saturated y as and water (Tale ) whch leads to nhomoeneous lud saturaton. The relectty seres rom these two types o meda are plotted as a uncton o requency (ure 3 and ure 4). ure : chematc plot o wae propaaton throuh multlayered meda at normal ncdence. D and D are downon ast and low P wae n layer and are upon ast and low P wae n layer respectely. eect exsts or lower requency and shorter and weaer low P wae eect exsts or hher requency. Thus t can e expected n sesmc prole some enery wll appear elow an nhomoeneous lud saturated reseror or low requences ut dsappear or hher requences. uch phenomenon s smlar to the low requency shadows osered y nstantaneous spectral analyss technque demonstrated y Castana et al. (3). Hence we thn EG Houston 9 Internatonal Exposton and Annual eetn 719

4 Asymptotc calculatons o the Bot s waes that low P wae may e a maor cause or the low requency shadows. urthermore t s well nown that low requency shadows are always assocated wth as reseror. We thn the reason or ths assocaton s ecause that ree as n the as reseror wll nduce some deree o nhomoenety y the orm o as ules and consequently enhances the low P wae eect. Ths eect appears n a sesmc prole le shadows eneath the reseror one at lower requences. At hher requences snce low P wae attenuates qucly those shadows would dsappear. We must coness that t s not adequate to draw a concluson at ths stae and more edence towards ths postulate wll need to e carred out. Example : Permeale s. mpermeale meda Both models n ure 3 are porous permeale meda the permealty were taen to e darcy (Tale ). I the meda ecomes low permeale low P wae eect s sncantly weaened (ure 4). Conclusons Dynamc modeln on mult-layered meda was appled ased on asymptotc descrpton o Bot model and propaator matrx technque. The relectty seres as a uncton o requency or derent types o lud saturaton and permealty are otaned. A stron low P wae eect s osered or low requency hh permealty and nhomoeneous lud saturated meda. urthermore due to the smlarty etween low P wae phenomenon and low requency shadows osered y nstantaneous spectral analyss technque we thn that low P wae may e the maor cause or these shadows. ure 3: Relectty seres s. requency or an mpulse ast P wae trael throuh 7 layers o (a) homoeneous only water saturated meda (wth roc property chane alternately) () nhomoeneous as water alternately saturated meda (Tale ). Acnowledement The wor has een perormed at the nersty o Houston and Lawrence Bereley Natonal Laoratory. It has een partally supported y RQL consortum at the nersty o Houston and DOE Grant No. DE-C6-4NT1553. The authors are thanul to Dr. John Castana and Dr. Chrs Lner or remars and suestons. dry dry ure 4: ame as ure 3 () only the permealty s chaned to.5 darcy. [Gpa] [/cc] [Gpa] [Gpa] [darcy] [Gpa] [/cc] [cp] Tale : Input parameters or the porous permeale nhomoeneous as water alternately saturated meda. One-way trael tme or ast P wae n each layer s.1 ms. Relectty results or 7 layers are plotted n ure 3 (). EG Houston 9 Internatonal Exposton and Annual eetn 7

5 EDITED REERENCE Note: Ths reerence lst s a copy-edted erson o the reerence lst sumtted y the author. Reerence lsts or the 8 EG Techncal Proram Expanded Astracts hae een copy edted so that reerences proded wth the onlne metadata or each paper wll achee a hh deree o lnn to cted sources that appear on the We. REERENCE Bot. A. 1956a Theory o propaaton o elastc waes n a lud-saturated porous sold. 1. Low-requency rane: Journal o the Acoustcal ocety o Amerca Theory o propaaton o elastc waes n a lud-saturated porous sold.. Hher requency rane: Journal o the Acoustcal ocety o Amerca Carcone J.. H. B. Helle and N. H. Pham 3 Whte s model or wae propaaton n partally saturated rocs: Comparson wth poroelastc numercal experments: Geophyscs Castana J. P.. un and R. W. ered 3 Instantaneous spectral analyss: Detecton o low-requency shadows assocated wth hydrocarons: The Leadn Ede 1 7. Dutta N. C. and H. Ode 1979a Attenuaton and dsperson o compressonal waes n lud-lled porous rocs wth partal as saturaton (Whte model) Part 1: Bot theory: Geophyscs Attenuaton and dsperson o compressonal waes n lud-lled porous rocs wth partal as saturaton (Whte model) Part : Results: Geophyscs esmc relectons rom a as-water contact: Geophyscs Hasell N The dsperson o surace waes n multlayered meda: Bulletn o the esmolocal ocety o Amerca Ronson E. A ultchannel tme seres analyss wth dtal computer prorams: Holden-Day. ln D. and G. Goloshun 8 esmc wae relecton rom a permeale layer: Low-requency asymptotc analyss: Proceedns o the Internatonal echancal Enneern Conress and Exposton. 9 A low-requency asymptotc model o sesmc relecton rom a hh-permealty layer: Lawrence Bereley Natonal Laoratory Report. Thomson W. 195 Transmsson o elastc waes throuh a strated sold medum: Journal o Appled Physcs EG Houston 9 Internatonal Exposton and Annual eetn 71