Sesmc reflecton stuy n flu-saturate reservor usng asymptotc Bot s theory Yangun (Kevn) Lu* an Gennay Goloshubn Unversty of Houston Dmtry Sln Lawrence Bereley Natonal Laboratory Summary It s well nown that ampltue varatons of sesmc reflecte waves an energy loss are usually assocate wth flu-saturate porous roc. However to quantfy such effects an prect the sesmc sgnature generate by flu saturaton s a challenge. Ths paper presents some stuy on evaluatng sesmc response from flu-saturate porous permeable roc ue to the converson from Fast P wave to Slow P wave usng asymptotc Bot s soluton. Snce flusaturate layer also shows hgh sperson frequency epenency s also taen nto account. The fference between Hz calculaton an 1 Hz calculaton represents the fference between the Gassmann s theory an Bot s theory. It can be seen that for homogeneous reservor there s no bg fference between them whle for heterogeneous reservor the fference can be as hgh as 1% of the average ampltue. The heterogenety may be wth respect to porosty permeablty an flu phase between reservor layers. Introucton There are many publcatons for numercal calculatons of sesmc energy absorpton base on Bot s theory. In partcular Carcone et al. (3) clearly emonstrate remarable nfluence of heterogenety of porous flusaturate rocs to attenuaton. Lu et al. (9) utlze the asymptotc soluton on Bot s theory base on propagator matrx metho to calculate reflectvty from mult-layere meum nclung both Fast P wave an Slow P wave. It was emonstrate that for layer thcness hgher than one meter Slow P wave s har to etect. However the energy converte to Slow P wave stll has a sgnfcant effect on attenuaton. An we coul assume that all the energy converte to Slow P wave mnshes wthn the roc layer therefore by re-arrangng the propagator matrx to only ncorporate Fast P wave we coul calculate the sesmc response for relatvely thc poroelastc rocs. Snce the reflecton coeffcents are calculate usng the asymptotc Bot s soluton energy transferre to Slow P wave has alreay been taen nto account. Therefore the equaton of R+T1 wll no longer be satsfe here R s the reflecton coeffcent an T s the transmsson coeffcent. Inee R+T<1 snce the energy goes nto Slow P wave s equvalent to sesmc absorpton an we observe t to be a maor part of the attenuaton n flu-saturate rocs. Dutta & Oe (1983) calculate the sesmc reflectons as a functon of frequency for a gas-water bounary n san reservor usng Bot s moel (1956). Sln an Goloshubn (8 1) erve a low frequency asymptotc soluton of Bot s poroelastcty. For comparson the asymptotc soluton reuces the complexty of the calculatons an proves a smlar result at sesmc frequences relatve to exact Bot s soluton. Fgure 1 shows the attenuaton coeffcents for Fast P wave an Slow P wave as compute both by asymptotc formulas an Dutta an Oe s computaton n solvng exact Bot s moel. It can be seen that n sesmc frequency range the results are very smlar. Attenuaton (1/m).5 x 1-3 Fast P wave Gas Water Dutta & Oe Gas Dutta & Oe Water 1.5 1.5 1 1 1 1 3 Frequency (Hz) 1 1 1 1 3 Frequency (Hz) Fgure 1: Attenuaton coeffcent of Fast an Slow P wave calculate from asymptotc formulas (sol lnes) compare to the calculaton by Dutta an Oe (1983) (ots). 5 4 3 1 Slow P wave Gas Water Dutta & Oe Gas Dutta & Oe Water Table 1 summarzes all necessary nput parameters that are use n asymptotc soluton of Bot s theory. It can be seen that the nput parameters are as smple as the ones routnely use n flu substtuton technque base on Gassmann s equaton. Two atonal parameters (roc permeablty κ an flu vscosty η f ) are utlze n asymptotc formulas. Thus t allows beses realzaton of the flu substtuton technque to prove an nvestgaton K g ρ g K ry μ ry φ κ K f ρ f η f GPa g/cm 3 GPa GPa - Darcy GPa g/cm 3 cp Table 1. Input parameters an unts for asymptotc soluton. Here K f s the bul moulus of flu K g s the bul moulus of sol gran K ry s the ry roc bul moulus μ ry s the ry roc shear moulus φ s porosty ρ f s the flu ensty an ρ g s the gran ensty. of the nfluence of the permeablty (flu moblty) to sesmc response. All nput parameters can be acqure from log ata an laboratory measurements. 1 SEG SEG Denver 1 Annual eetng 715
Sesmc reflecton stuy usng asymptotc Bot s theory The rest of ths paper s presente as followng. At the begnnng we brefly escrbe the algorthm of asymptotc soluton of Bot s theory followe by ntroucton of the propagator matrx algorthm use n ths stuy. The matrx elements are also presente explctly. Fnally we show examples of calculatons on two reservor moels. Asymptotc calculaton reflecton an transmsson coeffcents. Note that across each bounary the Slow P wave s not taen nto account snce we assume that layer thcness s too thc (>1 meter) for Slow P wave to propagate an communcate wth Fast P wave therefore converson to Slow P wave s counte as sesmc absorptons. The reflecton an transmsson coeffcents are all calculate usng asymptotc soluton of Bot s theory thus the porton of energy converts to Slow P wave has alreay been taen away through each layer. The reflecton an transmsson coeffcents n asymptotc soluton are expresse as power seres of the square root of a mensonless parameter (Sln & Goloshubn 8 1): ρ f κω ε η where ρ f s flu ensty κ s permeablty η s flu vscosty ω s angular frequency an s the magnary unt. The reflecton an transmsson coeffcents from Fast P wave to Fast P wave are enote as R FF an T FF ; the reflecton an transmsson coeffcents from Fast P wave to Slow P wave are enote as R FS an T FS. They have the followng asymptotc forms for normal ncence: FF FF 1+ R R + R1 FF FF 1+ T 1+ R + T1 FS FS 1+ R R1 FS FS 1+ T T1 where R s the zero-orer classcal reflecton coeffcent an R FF 1 an T FF 1 are the frst-orer reflecton an transmsson coeffcents. Reaers are referre to ether Lu et al. (9) or Sln an Goloshubn (8 1) for a etale escrpton of asymptotc formulas. Propagator matrx metho A smplfcaton of the propagator matrx metho use n Lu et al. (9) has been one n ths paper. Accorng to the bounary conton n Fgure we can obtan the relatonshps between waveforms at nterface wth the corresponng reflecton an transmsson coeffcents. rff s use to represent the reflecton coeffcent of Fast P wave to Fast P wave whle the ncent wave s owngong an rffup s for the reflecton coeffcent of Fast P wave to Fast P wave whle the ncent wave s upgong. Thus rff u (t+1)/ (t-1) an rffup +1 (t)/u +1 (t). Smlar enotatons are use for other The bounary conton can be wrtten as: A + 1 where (t) u (t) tff A A ( t + 1) rff A α ( t ) + rffup + 1 ( t ) + tffup + 1 α h e s the attenuaton through layer s the attenuaton coeffcent (m -1 ) an thcness (m). Also efne D (z) as the z-transform of (t).e. n h s the layer t D ( z) z t n s the total number of samples n the tme seres of (t). An smlarly efne U (z) as the z-transform of u (t). Then we can obtan for any nterface : D owngong Fast P wave n layer upgong Fast P wave n layer Fgure : Schematc plot of wave propagaton through layer at normal ncence. The horzontal splacement correspons to tme elay. (s) z tffup D + 1 [ ] + 1 where [ ] s the x propagator matrx that communcates the waveforms between layer an +1. Each matrx 1 SEG SEG Denver 1 Annual eetng 716
Sesmc reflecton stuy usng asymptotc Bot s theory element of [ ] s also n z-transform.e. polynomals of z. Thus for wave propagaton n mult-layere mea (Fgure 3) we can obtan: D + 1 + 1 where (s) (s) z C Source D 1 D 1 D tffup D D +1 z tffup D ( ) C N s U U 1 U U Layer Layer 1 Layer Layer U +1 Layer +1 D ( ) s an an the multplcaton between any two matrx elements n [ ] s a convoluton of ther polynomal coeffcents. Fgure 3: Schematc plot of wave propagaton through multlayere mea at normal ncence. D s owngong Fast P wave n layer ; U s upgong Fast P wave n layer. We o not conser Slow P wave n ths stuy an assume t completely attenuates before reachng the next layer. Here we present our results of the matrx elements of. For any layer other than the frst layer (layer ) the two by two matrx can be expresse as: ( 11) ( tff tffup rff rffup ) A z (1) rffup A ( 1) rff A z () 1 A. An the matrx elements for can be expresse as: ( 11 ) tff tffup rff rffup ( 1) rffup ( 1) rff N () 1. Fnally by settng D (z) 1 an U +1 (z) we can obtan U (z) as the reflectvty seres of an mpulse Fast P wave travelng through the mult-layere mea wth mo converson to Slow P wave taen away as sesmc absorptons. Examples Homogeneous vs. heterogeneous reservors In ths example we calculate the response from two flusaturate layere reservor moels. The orgnal moel (oel 1) contents porous permeable sanstones wth total thcness beng meter an each layer thcness equal to 1 meter. Sanstones are saturate by gas an water. An there s varaton n such roc propertes as K ry u ry porosty an permeablty. The parameters are erve from log ata an laboratory measurements. oel 1 correspons to a heterogeneous reservor. oel s a result of averagng oel 1 an t represents a homogeneous moel of the reservor. The reflectvty seres from these two moels are plotte for 1 Hz an Hz as well as the fference between these two frequences as a functon of the total stance wave passe n Fgure 4. Therefore the reflecton at 44 m (two way propagaton) n the fgure correspons to the reflecton from the reservor bottom. It can be seen that for oel 1 the fference between 1 Hz an Hz s substantal. The evaton can be as hgh as 1% of the average ampltue. However such fference n oel s much smaller. The evaton s only about.1% of the average ampltue thus s neglgble. Ths experment tells us that for homogeneous reservor Gassmann s equaton can satsfy the nee n prectng the relatonshp between flu an reflecton response however for flu-saturate heterogeneous reservor Gassmann s equaton can lea to about 1% error n computng flu effect on sesmc response. An ths error shoul be taen nto account an try to reuce through Bot s theory. Asymptotc soluton of Bot s theory s a goo canate n ong such a computaton snce t has smple form an all parameters can be estmate from log ata an laboratory analyss. Quantfcaton of the qualty factor Q It can be estmate base on reflecton ampltue from the reservor zone bottom (A B at 44 meter n Fgure 4) at fferent frequences. We have estmate the Q factor for moels 1 an usng the followng equaton: 1 SEG SEG Denver 1 Annual eetng 717
Sesmc reflecton stuy usng asymptotc Bot s theory 1 ln A ln A π Δt f B B 1 Q where Δt s the total travel tme an f s frequency that equals to 1 Hz. So nee the Q factor calculate here s the Q factor at 1 Hz for moels 1 an. The results are summarze n Table. A B (1Hz) A B ( Hz) Q factor oel 1.15471.1478 33 oel.133654.133679 16797 Table. Reflecton ampltues an compute Q factors. The Q factor for moel 1 s realstc (Q33) an t s much smaller than the Q factor for the homogeneous moel. Conclusons We have analyze the reservor response from two moels usng both Gassmann s theory ( Hz) an Bot s theory (1 Hz). In case of heterogeneous reservor the Gassmann s theory can lea to about 1% error n computng flu effect on sesmc response. Ths error can be reuce by usng asymptotc soluton of Bot s theory. It proves smpler form of solutons whle stll has reasonable accuracy compare wth exact Bot s theory at sesmc frequences. Therefore t may be sutable for replacng Gassmann s calculaton wthout ncreasng computatonal effort. Sesmc absorpton by flu oscllaton has been evaluate an quantfe by estmatng ther Q factor. For heterogeneous reservor the Q factor s realstc (Q33) an t s much smaller than the Q factor for the homogeneous reservor. It can be conclue that sesmc absorpton ue to flu flow s not neglgble unless everythng can be treate as homogeneous. Acnowlegement The wor has been performe at the Unversty of Houston an Lawrence Bereley Natonal Laboratory. It has been partally supporte by CAGE consortum at the Unversty of Houston an DOE Grant No. DE-FC6-4NT1553..3. oel 1 1 Hz.3. oel 1 Hz.1.5 oel 1 fference.1.1 -.5 -.1 -.1 -.1 -.15 -. 4 6 8 -. 4 6 8 -. 4 6 8.3. oel 1 Hz.3. oel Hz 1 x 1-4 oel fference.1.1 -.1 -.1 -. 4 6 8 -. 4 6 8-1 4 6 8 Fgure 4: Reflectvty response of moel 1 (heterogeneous reservor) an moel (homogeneous reservor) n terms of permeablty an flu saturaton at 1 Hz an Hz. The fgures on the most rght ses show the fference between 1 Hz response an Hz response. It can be seen that for heterogeneous reservor the fference can be as hgh as 1% of the average ampltue however for homogeneous reservor t s very small an can be consere as neglgble. 1 SEG SEG Denver 1 Annual eetng 718
EDITED REFERENCES Note: Ths reference lst s a copy-ete verson of the reference lst submtte by the author. Reference lsts for the 1 SEG Techncal Program Expane Abstracts have been copy ete so that references prove wth the onlne metaata for each paper wll acheve a hgh egree of lnng to cte sources that appear on the Web. REFERENCES Bot. A. 1956a Theory of propagaton of elastc waves n a flu-saturate porous sol. 1. Lowfrequency range: The Journal of the Acoustcal Socety of Amerca 8 no. 168 178 o:1.111/1.19839. Carcone J.. H. B. Helle an N. H. Pham 3 Whte s moel for wave propagaton n partally saturate rocs: Comparson wth poroe lastc numercal experments: Geophyscs 68 1389 1398 o:1.119/1.159813. Dutta N. C. an H. Oe 1983 Sesmc reflectons from a gas-water contact: Geophyscs 48 148 16 o:1.119/1.1441454. Sln D. an G. Goloshubn 8 Sesmc wave reflecton from a permeable layer: Low-frequency asymptotc analyss: Proceengs of the Internatonal echancal Engneerng Congress an Exposton. Sln D. an Gennay Goloshubn 1 An asymptotc moel of sesmc reflecton from a permeable layer: Transport n Porous ea on-lne Sprgerln.com 1 SEG SEG Denver 1 Annual eetng 719