Quantitative gas saturation estimation by frequencydependent

Transcription

1 Quanttatve gas saturaton estmaton by requencydependent AVO analyss Xaoyang Wu, * Mark Chapman, Xang-Yang L, and Patrck Boston Ednburgh Ansotropy Project, Brtsh Geologcal Survey, Murchson House, West Mans oad, Ednburgh EH9 3LA, U. School o Geoscences, Unversty o Ednburgh, The ng s Buldngs, West Mans oad, Ednburgh EH9 3JW, U. * Correspondng author, E-mal: xywu@bgs.ac.uk Abstract Sesmc ampltudes contan mportant normaton whch can be related to lud saturaton. The ampltude-versus-oset AVO analyss o sesmc data based on Gassmann s theory and approxmaton o the Zoepprtz equatons, has played an central role n reservor characterzaton. However, ths standard technque aces a long standng problem, ts nablty to dstngush between partal gas and zz-water wth lttle gas saturaton. In ths paper, we studed sesmc dsperson and attenuaton n partally saturated poroelastc meda by usng requency-dependent rock physcs model, through whch the requency-dependent AVO FAVO response s calculated as a uncton o porosty and water saturaton. We propose a crossplottng o two attrbutes derved rom FAVO response to derentate partal gas saturaton and zz-water saturaton. One o the attrbutes s a measure o low requency or Gassmann, relectvty, whle the other s a measure o the requency-dependence o relectvty. Ths s n contrast to standard AVO attrbutes, where there s typcally no such separaton. A pragmatc FAVO nverson or rock and lud propertes s also establshed based on Bayesan theorem. Synthetc study s perormed to explore the potental o the method to estmate gas saturaton and porosty varatons. An advantage o our work s that the method s n prncple predctve, openng the way to urther testng and calbraton wth

2 eld data. We beleve that such work should gude and augment more theoretcal studes o FAVO analyss.

3 eywords: AVO; requency-dependent; gas saturaton; Bayesan nverson; quanttatve; sesmc

4 Introducton eservor characterzaton and lud detecton are undamental goals o geophyscal technology. The relable estmaton o gas saturaton s a long standng problem Han and Batzle, 00, whch s mportant n both exploraton and nterpretaton o tme-lapse data. Despte some progress n the applcaton o Bayesan nverson concepts Bachrach, 006, relable estmaton o gas saturaton has remaned elusve. The root o the problem s the relatve nsenstvty o sesmc velocty to gas saturaton varatons. Introducton o a small amount o gas oten leads to a sudden drop n velocty, but only small urther varatons may be seen when the gas saturaton s ncreased urther. In prncple, consderaton o attenuaton may mprove the stuaton, snce the relatonshp between attenuaton and water saturaton typcally has a derent orm to that between velocty and water saturaton Murphy, 98, 984; Gst, 994; Cadoret et al., 998; Nakagawa et al., 03. Nevertheless, attenuaton-based technques ace obstacles n terms o the dculty o the measurements and the lack o approprate modellng and nterpretaton rameworks. Sgncant strdes have been made recently n the measurement o attenuaton rom sesmc relecton data Dasgupta and Clark, 998; Clark et al., 00; ene et al 009. The bass o the technques s measurng the cumulatve loss o energy durng transmsson through an attenuatng zone. The applcablty o the method s lmted by the thckness o the attenuatng zones, and some targets are too thn or successul applcaton o the method. An alternatve approach Chapman et al., 006; en et al., 009; Wlson et al., 009; Xu et al., 0, Zhang, 0; Innanen, 0, 0; Sun et al., 0; Wu et al., 03 ams to estmate sesmc attenuaton ndrectly through the requency-dependence o the relecton coecent rom attenuatve and dspersve meda.

5 The requency-dependence o sesmc veloctes and attenuatons n lud saturated rocks s the subject o ntensve current research rom both a theoretcal and expermental standpont orneev et al., 004; Müller and othert, 006; Quntal and Tsato, 03; Tsato and Quntal, 03. However, many mportant aspects reman poorly understood and controversal. It appears that no currently avalable theory can satsactorly descrbe and predct velocty dsperson and attenuaton. The case o partally saturaton appears to be partcularly dcult. In ths paper, we nvestgate the possblty o determnng gas saturaton rom the requencydependence o relectvty n the pre-stack doman usng smpled, but we beleve representatve, rock physcs theory. We show how exstng requency-dependent rock physcs theory can be used as a tool to match water saturaton normaton at the well to the output o requency-dependent ampltude-versus-oset analyss o sesmc relecton data. Our results are placed n a convenent Bayesan nverson scheme, whch allows the theoretcal detectablty o saturaton and porosty varatons to be assessed. Whle uture testng on eld data s requred to valdate the technque. We begn by brely revewng the current understandng o sesmc dsperson and attenuaton n lud saturated rocks. We then present a pragmatc approach to calculatng requency-dependent relectvty on the bass o parameters whch can be estmated rom well log data. The calculated data can then be used as a bass or our Bayesan nverson scheme targeted at recoverng porosty and saturaton varatons. A numercal example llustrates the potental power o the technque. Modellng attenuaton and dsperson n partally saturated rock The development o mathematcal theores to descrbe the dsperson and attenuaton o sesmc waves n lud saturated rock has been a ocus o research or over 50 years Bot, 956a, 956b, and the eld s stll actvely developng Müller et al., 00. The Bot theory o poroelastcty attempt to descrbe the behavour o global low n lud-saturated rocks, has

6 provded an elegant oundaton or many studes. The low requency lmtng veloctes predcted by Bot theory are consstent wth those predcted by the Gassmann equaton Gassmann, 95, whch s stll the cornerstone or most studes o lud substtuton n the petroleum ndustry. ecently, t has been suggested that the concept o squrt-low Mavko et al., 979 descrbng the mcroscopc wave nduced lud low between pores and cracks, provdes an mportant source o addtonal attenuaton, whch s not accounted or n the Bot theory eg. ng, 000. The rst mathematcal descrptons o squrt-low were produced n the 970 s O Connell & Budansky, 977. More recently, the development has proceeded down parallel paths separated by the preerred means o descrbng the pore space o the rock. One general approach Mavko & Jzba, 99; Dvorkn et al., 995; Prde et al., 004; Gurevch et al., 00; Gurevch, 03 avods an explct mathematcal descrpton o the pore space, and emphasses the use o averaged, bulk, rock propertes. Another approach Hudson et al., 996; Ponter et al., 000; Chapman et al., 00; Chapman, 003; Jakobsen et al., 003; Chapman, 009, Jakobsen and Chapman, 009 descrbes the squrt eect on the bass o ncluson models. Incluson models benet rom precse mathematcal descrptons o the stness o pores, but rely on dealzed geometres not encountered n nature. The utlty o ncluson models has been doubted by Avseth et al. 004 and others. In theory, the Bot theory assumes that the perod o the sesmc wave s sucently long to allow local lud pressure gradents to relax. Squrt-low theory, by contrast, consders that heterogenety n the pore structure gves rse to mcro-scale lud pressure gradents whch may not be equlbrated by lud low on the tme scale o the sesmc wave. When lud pressure gradents are un-relaxed, the rock s ster than n the relaxed case. Ths reasonng leads to a pronounced requency-dependence o sesmc veloctes, wth hgher requences beng assocated wth hgher veloctes.

7 Two major derences between the squrt low and Bot theores le n the magntude o the predcted dsperson and the behavour o the characterstc requency at whch the maxmum attenuaton occurs. Generally, squrt low s beleved to produce larger and more realstc attenuatons than the Bot theory. Both approaches suggest that the characterstc requency depends on the lud moblty dened as the rato o permeablty to lud vscosty, but the nature o that dependence s very derent. In the Bot theory, the characterstc requency moves hgher as the lud moblty s reduced, but the opposte s the case or the squrt theory. Batzle et al. 006 tentatvely suggest that laboratory measurements agree wth the squrt theory n terms o the dependence on lud moblty. When the rock s partally saturated, urther complcatons ensue. In a landmark study, Whte 975 gave a descrpton o how the exstence o gas patches gve rse to large attenuatons. Whte s model predcts no attenuaton or dsperson n the absence o a second lud. Dutta and Ode 979 extended Whte s approach wthn the ramework o Bot theory. Mavko and Mukerj 998 descrbe the eect o patchy saturaton, and show that ths concept gves rse to ncreased velocty dsperson. ecently, the role o the scale length o the heterogenetes has also emerged as an mportant concept, wth meso-scale structures smaller than the wavelength but larger than the gran scale beng assocated wth dsperson n the sesmc requency band. Ths phenomenon has been studed through numercal modellng ubno and Hollger, 0; Quntal and Tsato, 03. Laboratory measurements Murphy, 98; Murphy, 984; Cadoret et al. 984 suggest that attenuatons or small partal gas saturaton are hgher than or the ully saturated case. Gst 994 presents laboratory measurements on partally saturated rock and argues that ther nterpretaton requres a uned model wth both the squrt-low and gas pocket eects. Unortunately, no truly satsactory such model yet exsts. ecently, Amalokwu et al. 04 present measurements o attenuaton versus water saturaton n ractured and unractured

8 rocks whch appear to be a severe challenge to currently developed theory. The development o models or partally saturated rock s currently a ocus o ntensve research. Gven the uncertanty n the underlyng rock physcs, and the absence o a satsactory modellng approach, t s temptng to conclude that applcaton o the concepts to the nterpretaton o sesmc data s currently mpossble, and that progress must awat the development o more advanced theores. Ths paper explores the opposte pont o vew, presentng a pragmatc approach to the applcaton o exstng theory to ner varatons n porosty and saturaton. Our approach s based on three key assumptons. Frstly, we assume that the behavour o the rock s consstent wth the Gassmann-Wood predctons at low requency. We urther assume that partal gas saturaton generally gves hgher values o attenuaton than ull saturaton, wth maxmum values o attenuaton occurrng or low gas saturaton. Lastly, we assume that the dependence o the characterstc requency on lud moblty ollows the predctons o the squrt-low theory. We show below that t s possble usng exstng theory to provde a modellng scheme whch s broadly consstent wth these assumptons. We do not clam to be able to predct a pror values o attenuaton assocated wth a gven water saturaton. ather, we am to provde a modellng ramework whch s capable o beng calbrated when both sesmc data and water saturaton normaton rom log data are avalable. In ths way, we hope that our model may provde useul normaton even when the precse detals o the attenuaton mechansms may not be ully understood. Method or the calculaton o requency-dependent relectvty We base our calculatons on poroelastc rock physcs model presented by Chapman et al. 00. The model s prmarly a squrt low theory consderng the lud exchange between pores and cracks, as well as between cracks o derent orentatons due to wave propagaton.

9 Chapman et al. 00 dscussed the consstency o the model wth Gassmann s relatons, whch s an advantage n the current context. We propose to use the model n conjuncton wth an eectve sngle-phase lud model. Ths produces what may be a reasonable qualtatve relatonshp between attenuaton and water saturaton, but t should be borne n mnd that ths set up rules out many o the partal saturaton attenuaton mechansms whch have been dscussed n the lterature. Followng the theory o Chapman et al.00, the requency-dependent eectve bulk and shear modul e and µ e are expressed as ollows: B B ra A e c c e where ω s the angular requency,, µ, λ denote the bulk and shear modul and Lame parameter o the sold mneral matrx respectvely, whle the lud bulk modulus s, the total porosty s. r, a, ε are the aspect rato, unorm crack radus and crack densty o the cracks respectvely. A and B are: ' p c c p A 3 B c p c c p c ' 4 3 ' p 5

10 3 p 6 4 c p r c 8 where τ s the tmescale parameter that controls the requency regme over whch the dsperson occurs. When saturatng lud s an mxture o gas and water, densty o the partally saturated sandstone ρ sat used n the model was calculated as, sat 9 m S S 0 g g g w where ρ m, ρ, ρ g, ρ w s the denstes o rock matrx, eectve lud, gas and water respectvely. S g s gas saturaton. The bulk modulus o mxed lud s estmated by Wood s ormula, S / S / g g g w g and w are the bulk modul o gas and water respectvely. Vscosty o the eectve snglephase lud s averaged arthmetcally n term o volume percentage. Thus, vscosty and bulk modulus o the eectve lud change wth gas saturaton S g. Ths can urther changes attenuaton due to squrt low. Ater requency-dependent elastc modul has been derved, requency-dependent PP wave relectvty can be calculated usng the ansotropc Zoepprtz equaton generalzed by Schoenberg and Protazo 99, see Appendx I. Table dsplays the parameters o a two-layer model wth shales overlyng sandstone reservor. The lower layer s consdered to be dspersve. Fgure dsplays the attenuaton curves varyng wth arequency, bgas saturaton at derent porostes, and ccrack denstes, or the lower layer usng the theory o Chapman et al.00. As shown n Fgure

11 a and b, attenuaton peaks at about 50Hz, wth a correspondng Q value o 6. Fgure b shows that low attenuaton s predcted or ull saturaton by ether lud, wth a maxmum o attenuaton occurrng at an ntermedate saturaton and mnmum o attenuaton or ull gas saturaton. The locaton o the maxmum attenuaton s model dependent, but usually occurs or gas saturatons between 0% and 50%, wth a bas towards low gas saturatons. Fgure c shows the maxmum value o attenuaton s also senstve to the crack densty parameter. Ths parameter can be used to modulate the requency-dependence o relectvty and calbrate to the real data. Fgure dsplays the requency-dependent PP-wave relecton coecents when varyng water saturaton rom 00% to 0% every 0%. The nterace has hgh-to-low P-wave mpedance wth Class III AVO response. We can see that wth the ncreasng o gas saturaton, the value o relectvty become larger. An obvous characterstc s these plots llustrate nearly no dsperson at ull gas saturaton, slght dsperson at ull water saturaton a and consderable requency dependence at partal saturaton. Ths relectvty behavour s consstent wth the predcted attenuaton as shown n Fgure b. It s worth to menton that, or partal gas saturaton, the zero oset varaton o relecton coecent due to requency s 0.05, whch s sgncant n comparson to the varaton wth oset over the 40 range 0.05 when Sw=60% at 0Hz. Ths ndcates that the varaton o relecton coecents due to dsperson s sgncant n partally saturated sand. Spectral decomposton and balancng For real sesmc data, spectral decomposton technques may be used to calculate the spectral ampltude at derent requences. A varety o spectral decomposton technques, such as STFT Short Tme Fourer Trans orm, Partyka et al., 999, CWT Contnuous Wavelet Transorm, Snha et al, 005, MPD Matchng Pursut Decomposton, Wang et al., 007, WVD Wgner-Vlle Dstrbuton, Wu and Lu, 009 based methods, have been studed and

12 used or derent applcatons. Spectral decomposton transorms sesmc ampltudes, Dt, n, at tme t and recever n nto spectral ampltudes, St, n,, at requency such that, Dt, n St, n, Aterwards, spectral balancng should be perormed on the spectral ampltude to remove the overprnt eect o source wavelet Partyka et al., 999 the true spectral behavour o the geology and saturatng lud s desred. Varous methods such as spectral L-p norm scalng Marurt and rln, 00, spectral stablsaton Burnett et al., 003 and spectral balancng Odebeatu et al., 006; Wlson, 009 have been proposed. We remove the eect o the source wavelet by desgnng a sutable weght uncton wn,, Bt, n, =St, n, wn, 3 wn, s calculated rom a dened wndow wth k samplng ponts usng the rato o MS ampltudes at the chosen reerence requency 0 and other requences as ollowng, w, n k k S t, n, 0 S t, n, 4 Then the ampltudes at derent requences become comparable. We consder a two-layer Class III P-wave AVO model presented by Chapman et al. 005, where the top elastc layer had P- and S-wave veloctes o 743m/s and 394m/s. For the dspersve model, the lower layer s dened as a materal under water-saturaton then substtuted wth gas by changng the lud bulk modulus rom GPa to 0.GPa. For the elastc model, the P- and S-wave veloctes o lower layer were calculated rom elastc tensor or the dspersve model at low requency. Eleven traces or each model are generated usng 40Hz cker wavelet as the source. The trace space s 00m. Fgure 3 dsplays the synthetc gathers o the elastc and dspersve models at the nterace respectvely, both o whch the ampltudes ncrease wth the oset gradually. However, compared wth the elastc model, the ampltudes or the dspersve model have decreased.

13 The WVD based spectral decomposton method s perormed to calculate the spectral ampltudes at 5, 30, 40, 50, 60, 70 and 80Hz. A set o weghts are derved n the elastc model by matchng the peak ampltude o the sorequency trace to the 40Hz ampltude to remove the overprnt o source wavelet. These same weghts are appled to the dspersve model. Fgure 4 shows a comparson o sorequency sectons between the elastc and dspersve models at 5, 40, 60 and 80Hz. For the elastc model upper, smlar energy appears on each sorequency secton ater spectral balance, whle or the dspersve model lower, energy reduces at 5Hz as opposed to elastc model and decreases sgncantly wth the ncrease o requency. FAVO attrbutes AVO attrbutes such a s ntercept, gradent, lud actor have been derved rom standard AVO analyss and provded useul lud normaton. Analogously, we expect to derve attrbutes whch can delneate lud normaton rom FAVO analyss. Wlson et al. 009 s FAVO nverson produces an attrbute that measured the requency-dependence o relectvty. Clearly, t would be advantageous to be able to quanttatvely relate such attrbute to rock propertes o nterest. Our attenton s ocussed on the partcular rock property o nterest. We desgn a two-layer model wth elastc shale overlyng lud-saturated dspersve sandstone. Followng the theory o Boston 0, the relectvty at the nterace s calculated as a uncton o rock property or nstance, gas saturaton requency and oset. The relecton coecents o p derent values o gas saturaton X k k=,,,p, m angles o ncdence θ =,,,m, n requences j j=,,,n are calculated, and then splced together nto a large matrx o the ollowng orm:

14 p X X X n m m n m m n m m n n n n n n Followng the method o Causse et al. 007 and Varela et al. 009, the sngular value decomposton SVD s perormed on the matrx. Ths allows us to approxmate the relecton coecents as a seres o weghts C, whch are a uncton o rock property X and requency, and bass unctons h are a uncton o angle o ncdence:,, k j k j X h X C h X C j k 6 The dependence o the weght unctons on requency can be approxmated by a lnear relatonshp,, k j k j X dc CO X C 7 Ths gves rse to a orward modellng whch allows us to carry out ntercept-gradent analyss or the requency-dependent relectvty o each rock property. The ntercept attrbute s a measure o the low requency relectvty, whle the gradent attrbute s a measure o the requency-dependence o relectvty, whch s smlar to the attrbute by Wlson et al.009. Crossplottng the two attrbutes may help us derentate derent gas saturaton scenaros. We apply ths method to the two-layer model used by utherord and Wllams 989. The lower layer s consdered to be saturated wth eectve lud o gas and water, where dsperson and attenuaton are ntroduced. Gas saturaton s the reservor property o nterest. The materal parameters or the model are lsted n table. The relecton coecents matrx are calculated or 5 scenaros o gas saturaton, 5 cases rom % to 5% wth a step o %, and the other 0 cases rom 0% to 00% wth a step o 0%. Then the ntercept-gradent crossplot can be generated by the oregong method.

15 Fgure 5 dsplays the crossplot o the two attrbutes. It ndcates that the ntercept attrbute ncrease wth gas saturaton. Peak value o the attrbute measures requency-dependence o relectvty at 5% o gas saturaton. From ths crossplot, partal gas saturaton larger than 0% and smaller than 50% and zz water saturaton larger than % and smaller than 0% are clearly separated. Bayesan nverson usng FAVO analyss The Bayesan theorem n terms o probablty densty unctons pd can be wrtten as P d m P m P m d 8 P d where Pm d s the pd o model parameters m when gvng the observng data d. Pd m s the pd o d when gven m, whch s also known as lkelhood uncton. It measures the mst between the orward modellng data r derved rom m and the observng data d. Hgh mst between r and d wll gve small lkelhood. Pm s the pror normaton. Pd can be consdered as a constant snce the data d s observed. Here, we propose a model-based FAVO nverson technque to estmate porosty and gas saturaton rom analyses o well logs and pre-stack sesmc data, whch has the mert o beng readly applcable to eld data. The man steps are as ollows: Spectrally decomposng and balancng pre-stack data along the reservor poston to obtan the spectral ampltudes d varyng wth angle o ncdent θ at requences j, d = dθ, j. 9 Dervng a startng background model or both upper and lower layers rom well log data. Ths ncludes representatve values o V p, V s and densty, or dstrbutons o values, assocated wth gven petrophyscal parameters such as porosty φ and water saturaton S w. Calculatng the attenuaton and requency-dependent relecton coecent r or each realsaton o the petrophyscal parameters usng the parameters derved n step,

16 r = rθ, j. 0 v Comparng r wth d, and calculate the msts between them. Ths provdes a bass or an nverson o the data n terms o petrophyscal parameters. v Usng Bayesan theorem to estmate the probablty o each scenaro o petrophyscal parameters. For estmatng the posteror probablty o φ and S w when gven the observed spectral ampltudes d: P d, Sw P, Sw P, Sw d P d, The pror probablty Pφ, S w can be derved rom statstcal analyss o well log data. Pd φ, S w s the lkelhood uncton. We use an emprcal relaton to estmate Pd φ, S w : P d, Sw exp s E, where s s a constant, ΔE s the sum o absolute error between orward modellng relectvtes gφ, S w and d at each ncdent angle. E d g, Sw. 3 j Equaton llustrates that the posteror probablty depends on both pror normaton and lkelhood uncton. We use the parameters n Table, and assume 60% water saturaton n the pores to perorm a synthetc study. Under such case, the characterstc requency s 3Hz and.76 or correspondng Q value. Fgure 6 dsplays the requency-dependent relectvty and phase or the two-layer model. The hgh-to-low nterace gves rse to Class III AVO wth phase around 80 degree. The value o relectvty s hgher wth ncreasng o requency. Snce the spectral ampltude o a sesmc sgnal become postve value, we consder the ampltude value wthout phase eect. 0% Gaussan random nose was added nto the relectvty as the observng data. Then, we can study the potental resoluton o the method or determnng varatons n porosty and saturaton.

17 We then scan through derent combnatons o porosty φ and water saturaton S w, and compute mst between the theoretcal relectvty and the observng data. Ths analyss s repeated by replacng the requency-dependent modellng wth the sngle requency approach. The derved msts can be transormed to lkelhood uncton by usng equaton. Fgure 7 dsplays the lkelhood unctons, the well known result that t s hard to dstngush between low and hgh gas saturatons on the bass o Gassmann theory s recovered b, but the requency-dependent theory shows more promsng a. The reason or ths result s that whle hgh and ntermedate gas saturatons show the same relectvty or low requences, the requency-dependence o the relectvty s derent because o the behavour depcted n Fgure. Fgure 7a shows that the maxmum lkelhood occurs at 6% porosty and 55% water saturaton. The devaton o water saturaton rom true value 60% s due to the Gaussan random nose added nto the observng data. Three other local maxmum lkelhood values also occur. Ths ndcates that ncrease n porosty and water saturaton to a certan extent may have smlar sesmc response, whch leads to mult-solutons o petrophyscal parameters. Fgure 8 shows the ttng results at the maxmum lkelhood or a requency-dependent case and b Gassmann case. Fnally, normal dstrbutons o N0., 0.5 or porosty and N0.9, 0.5 or water saturaton are assumed respectvely as the pror normaton. Fgure 9 dsplays the pror probablty o porosty and water saturaton. Fgure 0 dsplays the posteror probablty, whch s calculated by multplyng the lkelhood unctons wth the pror normaton. The requency-dependent theory a provdes more accurate nverson o porosty and saturaton than Gassmann theory b. The smlar procedure can be appled to sesmc data or derent locatons. Sesmc data s compared to the theoretcal response or derent combnatons o porosty and saturaton. As such, ths technque can be used to assess the lkely changes n porosty and saturaton between zones where sesmc data are avalable.

18 Conclusons In ths paper, we have dscussed the easblty o usng FAVO response or quanttatve estmaton o gas saturaton. The Frequency-dependent rock physcs model s combned wth relectvty method to calculate the relectvty that s ncdent angle, requency and reservor propertes porosty and saturaton dependent. A crossplot o two FAVO attrbutes s derved usng sngular value decomposton, one measured the relectvty at low requency and ntercept the other measured the requency-dependence o relectvty gradent. Numercal study demonstrates that crossplottng o the two attrbutes s able to derentate the three cases o ull gas saturaton, ull water saturaton and partal gas saturaton. A Bayesan nverson o porosty and saturaton s also ntroduced based on FAVO response. Through synthetc modellng studes, we have demonstrated that the Bayesan nverson o FAVO response s theoretcally capable o provdng more quanttatve estmaton o petrophyscal parameters than the standard AVO technques. In real data processng, the eld data can be decomposed and balanced nto a set o spectral ampltudes, whle the parameter crack densty can be used to modulate the requency-dependence o relectvty and calbrate the theoretcal relectvty to real spectral ampltudes. It s worth to menton that not only sutable or attenuaton due to squrt low, our methodology o estmaton gas saturaton also apples to cases where other dspersve mechansms such as patchy saturaton gve rse to velocty dsperson. Hence, we argue that t may be possble to estmate gas saturaton rom FAVO response wth sutable processng and analyss o sesmc data. Acknowledgements Ths work was supported by the sponsors o the Ednburgh Ansotropy Project EAP, and s presented wth the permsson o the Executve Drector o the Brtsh Geologcal Survey NEC.

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20 Clark,. A., A. J. Carter, P. C. Nevll, & P. M. Benson, 00, Attenuaton measurements rom surace sesmc data - Azmuthal varaton and tme-lapse case studes: 63rd Conerence and Techncal Exhbton, EAGE, Expanded Abstracts, L8. Dasgupta,. & Clark,., 998. Estmaton o Q rom surace sesmc relecton data, Geophyscs, 636, 0-8. Dutta, N. C., & Odé, H., 979a, Attenuaton and dsperson o compressonal waves n ludlled porous rocks wth partal gas saturaton Whte model - Part I: Bot theory, Geophyscs, 44, Dutta, N. C., & Odé, H., 979b. Attenuaton and dsperson o compressonal waves n ludlled porous rocks wth partal gas saturaton Whte model-part II: esults, Geophyscs, 44, Dvorkn, J., G. Mavko, & A. Nur, 995, Squrt low n ully saturated rocks: Geophyscs, 60, Endres, A.L. & nght,.j., 997. Incorporatng pore geometry and lud pressure communcaton nto modelng the elastc behavour o porous rocks, Geophyscs, 6, Gassmann, F., 95, Über de Elastztät poröser Meden: Verteljahrsschrt der Naturorschenden Gesellschat n Zürch, 96, -3. Gst, G. A., 994. Interpretng laboratory velocty measurements n partally gas-saturated rocks: Geophyscs, 59, Gurevch, B., Makarynska, D., de Paula, O., & Pervukhna, M., 00. A smple model or squrt-low dsperson and attenuaton n lud-saturated granular rocks. Geophyscs. 75 6: pp.n09-n0. Gurevch, B. 03, gorous bounds or sesmc attenuaton and dsperson n poroelastc ocks: EAGE Workshop on Sesmc Attenuaton, Sngapore, O03. Han, D.-H., & Batzle, M. 00, Fzz water and low gas-saturated reservors: The Leadng Edge,, Hudson, J. A., E. Lu, & S. Crampn, 996, The mechancal propertes o materals wth nterconnected cracks and pores: Geophyscal Journal Internatonal, 4, 05-. Innanen,.A., 0. Inverson o the sesmc AVF/AVA sgnatures o hghly attenuatve targets. Geophyscs, 76, -4. Innanen,.A., 0. Anelastc P-wave, S-wave and Converted-wave AVO Approxmatons: 74th EAGE Conerence & Exhbton, Extended Abstracts, P97. Jakobsen, M., Hudson, J. A., & Johansen, T. A., 003. T-matrx approach to shale acoustcs. Geophyscal Journal Internatonal, 54, Jakobsen, M., Chapman, M., 009, Uned theory o global and squrt low n cracked porous meda. Geophyscs, 74, WA65-WA76.

21 ng, M. S., J.. Marsden, & J. W. Denns, 000, Bot dsperson or P- and S-wave veloctes n partally and ully saturated sandstones: Geophyscal Prospectng, 48, Marurt,.J. and rln,.l., 00. Narrow-band spectral analyss and thn-bed tunng. Geophyscs, 66, Mavko, G., jartansson, E., & Wnkler,., 979. Sesmc wave attenuaton n rocks. ev. Geophyscs., 7,55-64 Mavko, G., and D. Jzba, 99, Estmatng gran-scale lud eects on velocty dsperson n rocks: Geophyscs, 56, Mavko, G., and T. Mukerj, 998, Bounds on low requency sesmc veloctes n partally saturated rocks: Geophyscs, 63, Müller T. M., & E. othert, 006. Sesmc attenuaton due to wave-nduced low: Why Q n random structures scales derently, Geophyscal esearch Letters, VOL. 33, L6305. Müller T. M., Gurevch B. & Lebedev M., 00. Sesmc wave attenuaton and dsperson resultng rom wave-nduced low n porous rocks A revew, Geophyscs, 75, A47 A64. Murphy, W.F., 98. Eects o partal water saturaton on attenuaton n masslon sandstone and vycor porous glass, Acoust. Soc. Am. J., 7, Murphy, W.F., 984. Acoustc measures o partal gas saturaton n tght sandstones, J. geophys. es., 89, Nakagawa, S., neasey, T. J., Daley, T. M., Freeld, B. M. & ees, E. V., 03. Laboratory sesmc montorng o supercrtcal CO loodng n sandstone cores usng the Splt Hopknson esonant Bar technque wth concurrent x-ray Computed Tomography magng: Geophyscal Prospectng, 6, O Connell,.J. and Budansky, B., 977. Vscoelastc propertes o lud-saturated cracked solds. J.Geophys.es., 79, Odebeatu, E., Zhang, J., Chapman, M., Lu, E. & L, X.Y., 006. Applcaton o spectral decomposton to detecton o dsperson anomales assocated wth gas saturaton. The Leadng Edge, 5, Partyka, G.A., Grdley, J.M., & Lopez, J., 999. Interpretatonal applcatons o spectral decomposton n reservor characterzaton. The Leadng Edge 8 3, Prde S.., Berryman J. G. & Harrs J. M., 004. Sesmc attenuaton due to wave nduced low, Journal o Geophyscal esearch, 09, B00. Ponter T., Lu E. and Hudson J.A Sesmc wave propagaton n cracked porous meda. Geophyscal Journal Internatonal 4, Quntal, B. and Tsato, N., 03. Modelng Sesmc Attenuaton Due to Wave-Induced Flud Flow n the Mesoscopc Scale to Interpret Laboratory Measurements. Fth Bot Conerence on Poromechancs, Venna, pp

22 ene, C., van der Baan, M. & Clark,., 009. The robustness o sesmc attenuaton measurements usng xed-and varable-wndow tme-requency transorms, Geophyscs, 74, WA3 - WA35. en H., Goloshubn G. & Hlterman F., 009. Poroelastc analyss o ampltude-versusrequency varatons, Geophyscs, 7, N4 N48. ubno, J. G., & Hollger,. 0, Sesmc attenuaton and velocty dsperson n heterogeneous partally saturated porous rocks: Geophyscal Journal Internatonal, 88, utherord, S.. & Wllams,.H., 989. Ampltude-versus-oset varatons n gas sands. Geophyscs, 54, Sun, S., Jang, S., Sun, X., Yang, H., Han, J., & L, Y., 0. Flud dentcaton usng requency-dependent AVO nverson n dssoluton caved carbonate reservor. 8nd SEG Techncal Program Expanded Abstracts 0, pp.-5. Schoenberg, M. and Protazo, J., 99. Zoepprtz ratonalzed and generalzed to ansotropy. Journal o Sesmc Exploraton,, Snha, S., P. S. outh, P. D. Anno, & J. P. Castagna, 005. Spectral decomposton o sesmc data wth contnuous-wavelet transorm. Geophyscs, 706, 9-5. Tsato, N and Quntal, B., 03. Measurements o sesmc attenuaton and transent lud pressure n partally saturated Berea sandstone: Evdence o lud low on the mesoscopc scale, Geophyscal Journal Internatonal, 95, Varela, I., Maultzsch, S., Chapman, M. & L, X-Y., 009. Fracture densty nverson rom a physcal geologcal model usng azmuthal AVO wth optmal bass unctons. 79th SEG Annual Meetng Expanded Abstracts, Wang, Y., 007, Sesmc tme-requency spectral decomposton by matchng pursut. Geophyscs, 7, 3-0. Whte, J. E., 975. Computed sesmc speeds and attenuaton n rocks wth partal gas saturaton, Geophyscs, 40, 4-3. Wlson, A., Chapman, M., & L, X-Y., 009. Frequency-dependent AVO nverson, 79th annual SEG meetng Expanded Abstracts, 8, Wu, X., & Lu, T., 009. Spectral decomposton o sesmc data wth reassgned smoothed pseudo Wgner-Vlle dstrbuton, Journal o Appled Geophyscs, 683, Wu, X., Chapman, M., L, X.-Y., Angerer, E. & Boston P., 03. Bayesan nverson or porosty and saturaton usng requency-dependent rock physcs models 83rd annual SEG meetng Expanded Abstracts, Xu, D., Wang, Y-H, Gan, Q. & Tang, J., 0. Frequency-dependent sesmc relecton coecent or dscrmnatng gas reservors. Journal o Geophyscs and Engneerng, 8,

23 Zhang, S., Yn, X-Y & Zhang, G-Z, 0. Dsperson-dependent attrbute and applcaton n hydrocarbon detecton. Journal o Geophyscs and Engneerng, 8, 498.

24 Appendx I Schoenberg and Protazo 99 gave an explct soluton to the plane -wave relecton and transmsson problem n terms o sub-matrces o the coecent matrx o the Zoepprtz equaton. Based on Schoenberg and Protazo 99, Chapman and Lu 003 derved the relectvty, whch s ncdent angle and azmuthal dependent. When xng a polar angle ψ and an azmuthal angle θ, the drecton o propagaton o the plane wave s n = snψcosθ, snψsnθ, cosψ. Assumng a homogeneous ncdent P-wave travellng n cosψ, the axs normal to the nterace s taken to be the drecton n. The velocty eld o the ncdent wave wll then be o the orm: v= 0 e p expws x-t where e p s polarzaton and 0 s the ampltude. The slowness ξ o ths wave s calculated as the value o / havng the smallest real part or χ s an egenvalue o the matrx C jkl n n k and ρ s the densty o the rock. We then have s=ξn. The polarzaton s gven by the egenvector correspondng to χ. We now calculate expressons or the transmtted and relected waves. Snell s law states that the horzontal slownesses o all waves nteractng at the nterace are equal. Snce we know s and s, the condton: C jkl s s k - ρδ jl =0 In both the upper and lower meda gves a bcubc equaton on the vertcal slownesses. Solvng or the admssble values o s 3, the polarzatons, e, satsy the equaton: C jkl s s k - ρδ jl e l =0 and are obtaned through sngular value decomposton. The requred contnuty o dsplacement and tracton o the calculated waves gve the condtons on relected and transmtted ampltudes. Schoenberg and Protazo99 arranged the nterace condtons nto a convenent matrx representaton. Speccally, the 3 3 relecton matrx, s gven by:

25 = X - X -Y - Y X - X +Y - Y - Where the matrces X, X, Y and Y are as dened n Schoenberg and Protazo 99, wth the slownesses and polarzatons taken rom the above calculaton. The PP relecton coecent correspondng to the angles ψ, θ s gven by.

26 5 4 Por=0% Por=5% Por=0% Por=5% 00/Q 3 00/Q a cd=0.05 cd=0.0 cd=0.5 cd= gas saturaton c gas saturaton b Fgure P-wave attenuaton 00/Q varyng wth a requency; bgas saturaton; c crack denstes, predcted by the theory o Chapman et al. 00. A crack densty o 0.5 and porosty o 0% are gven n a. A crack densty o 0.5 s gven n b. A porosty o 0% s gven n c. The values o attenuaton peak at low gas saturaton about 0% and go down to mnmum values at ull gas saturaton. Attenuaton also depends on crack densty, whch can be used or calbratng theoretcal relectvty to real data. a Sw=00%,Sg=0% b Sw=80%,Sg=0% c Sw=60%,Sg=40% d Sw=40%,Sg=60%

27 e Sw=0%,Sg=80% Sw=0%,Sg=00% Fgure. P-wave relecton coecents as a uncton o ncdent angle or a seres o requences when varyng water saturaton rom 00% to 0%. The porosty s 0%. The maxmum requency-dependence o relectvty occurs when gas saturaton s 40%. Hgh requency-dependence o relectvty s predcted or ull water saturaton whle nearly requency-ndependence or ull gas saturaton. 0.8 aelastc model 0.8 btau=5e-3s Tme second Tme second ecever # ecever # Fgure 3 Synthetc relecton gathers at the nterace or the two-layer model, wth elastc layer Let and dspersve layer ght as the lower layer. Ampltudes exhbt Class III AVO eature or the two gathers. However, ampltudes or the dspersve model have decreased due to requency-dependence o elastc mpedance.

28 Fgure 4 Isorequency sectons o upper the elastc gather and lower dspersve gather at 5Hz, 40Hz, 60Hz and 80Hz. Spectral balance based on the nput wavelet has been carred out wth 40Hz as reerence requency. For the elastc case, relecton energy conserved at derent requency. For the dspersve case, relecton energy decreases wth ncrease o requency. Fgure 5: Crossplot o low-requency-relectvty Intercept versus requency-dependence o relectvty Gradent. Values labelled near the pnk squares reer to gas saturaton. Large ntercept and requencydependence occur around 5% gas saturaton. Full water saturaton has low ntercept. Full gas saturaton has nearly no requency-dependence o relectvty.

29 a b Fgure 6. Frequency-dependent relectvty a and phase b at the n terace o the two-layer model, usng Chapman et al.00 theory. The relectvty does not contan phase normaton. The hgh-to-low nterace gves rse to Class III AVO wth phase around 80 degree. a Frequency-dependent case b Gassmann case Fgure 7. Lkelhood unctons transormed rom msts between the theoretcal relectvty and the observng data usng an exponental relaton. ed and whte colours represent hghly lkelhood o the porosty and water saturaton to the true value. The true value o porosty 6% and water saturaton 60% s labelled wth whte crcle.

30 a Frequency-dependent case b Gassmann case Fgure 8. The ttng results or the maxmum lkelhood o porosty and water saturaton. For requencydependent case a, φ=6%, S w =55% and or Gassmann case b, φ=7%, S w =85%. Symbols represent the observng data, curves represent the modelng relectvty. Fgure 9. The pror normaton or porosty and water saturaton. Both o the porosty and water saturaton are assumed to be normally dstrbuted, wth mean value μ = 0% and varance σ = 0.5 or porosty, and μ = 90% and σ = 0.5 or water saturaton. a Frequency-dependent case b Gassmann case Fgure 0 The posteror probablty o porosty and water saturaton or requency-dependent case a and

31 Gassmann case b. The true value o porosty 6% and water saturaton 60% s labelled wth whte crcle. The requency-dependent case a provdes more accurate estmaton o porosty and water saturaton. Table The materal parameters or the two-layer model wth shales overlyng sandstone reservor. Layers Vpkm/s Vskm/s Vp/Vs Deng/cm 3 Porosty Crack densty Water saturaton Upper Lower Table The materal parameters or the two-layer model by utherord and Wllams 989 wth shales overlyng sandstone reservor. Layers Vpm/s Vskm/s Posson rato Deng/cm 3 Porosty Upper Lower