Prediction of Tight Sand Reservoir with Multi-Wavelet Decomposition and Reconstructing Method

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1 Internatonal Journal of Geoscences, 2016, 7, Publshed Onlne Aprl 2016 n ScRes. Predcton of Tght Sand Reservor wth Mult-Wavelet Decomposton and Reconstructng Method Lfang Cheng, Yanchun Wang, Zhguo L, Fuxu Gong School of Geophyscs and Informaton Technology, Chna Unversty of Geoscences, Bejng, Chna Receved 20 February 2016; accepted 22 Aprl 2016; publshed 25 Aprl 2016 Copyrght 2016 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract Specal reservor or flud has an abnormal response to some certan frequences, so that sesmc decomposton and reconstructon are used to hghlght the sesmc reflecton at certan frequences useful to dentfy specal geologcal bodes. Because sesmc wavelets are tme-varyng and spatal-varable n the propagaton, synthetc traces based on sngle wavelet make some weak but useful nformaton lost, and make artfacts form. However, Morlet wavelet aggregaton wth mathematcal analytcal expresson s able to fully and correctly reflect the varatons of wavelet n the propagaton of underground medum. The matchng pursut algorthm on the bass of Morlet wavelet mproves the calculatng effcency n decomposton and reconstructon greatly. Ths method s appled to the actual study area to do conjont analyss of sngle well and well-te multwavelet decomposton. It s found that frequences senstve to nterest reservors range from 8 to 34 Hz. Reconstructng the wavelets at those specal frequences and analyzng the reconstructed sesmc data, t s ponted out that nterest reservors have abnormal characterstcs wth respectvely strong RMS ampltude n the reconstructed data. Crossplot of gamma value at wells and reconstructed RMS ampltude suggests that anomales caused by nterest reservors are well separated from the background anomales when the reconstructed RMS ampltude s greater than Quanttatve predcton results of nterest reservors dstrbuton n the study area reveal that nterest reservors of western and northern study area are dstrbuted annularly and bandedly, whle most contguous sandstone n eastern regons appears sporadcally. Keywords Morlet Wavelet, Matchng Pursut, Decomposton and Reconstructon, Tght Sandstone, Reservor Predcton How to cte ths paper: Cheng, L.F., Wang, Y.C., L, Z.G. and Gong, F.X. (2016) Predcton of Tght Sand Reservor wth Mult-Wavelet Decomposton and Reconstructng Method. Internatonal Journal of Geoscences, 7,

2 1. Introducton Wth the objects of sesmc exploraton and development transtng form the conventonal structural reservors to the subtle reservors ncludng low relef structure, thn nterbedded reservors, upward dppng formaton and lthologcal pnch out, the conventonal post-stack sesmc data cannot metculously descrbe varous geologcal anomales [1]. Especally the thckness of thn reservor s a few or decade meters whch s far less than a quarter of sesmc wavelength, so the reflected waves of thn sandstone nterbedded wth thn mudstone are nterfered wth each other, and t makes sesmc dentfcaton more dffcult. Therefore, t s necessary to use some specal frequency nformaton of sesmc sgnals to hghlght the certan geologcal anomales. Mult-wavelet sesmc decomposton and reconstructon s proposed to decompose the sesmc traces to multple wavelet wth lnear superposton frstly, and then determne the nterest frequences accordng to geologcal anomales havng responses at some certan frequences by means of tme-frequency transform method, and lnearly reconstruct the wavelets at those gven frequences fnally to get new sesmc data meetng the research purposes [1]. Mult-wavelet concept was frst proposed by Geronmo, Hardn and Massopust, et al. [2] n 1994, then Llly and Park [3] developed a new technology called mult-wavelet transform n Lots of researchers mproved the mult-wavelet decomposton technology based on ths prncple afterward. Nowadays there are many methods to acheve mult-wavelet decomposton and reconstructon, and one of them s Matchng Pursut (MP) algorthm frst proposed by Mallat and Zhang et al. [4] n It s on the bass of Gabor wavelet dctonary, but s characterzed by greed reducng the computaton effcency. Later Lu J. (2004) and Marfurt (2005) et al. [5] [6] successvely proposed new matchng pursut algorthm based on Rcker wavelet and Morlet wavelet to decrease the scale of wavelet dctonary. On top of that, these two researchers put forward the dynamc matchng pursut algorthm [7] to mprove the calculatng speed greatly. Many other researchers at home such as Song Weq (2007), Chen Ln (2008), Feng Le (2009), and Zhang Xanwen (2010) et al. mproved MP algorthm to one degree or another and obtaned good applcaton effects [8]-[11]. However, t s stll very rare that applyng the matchng pursut algorthm to the development stage to predct the nterest sandstone assocatons and to gude the deployment of development wells. Snce that the reservor n the research area s tght thn sandstone wth low porosty and low permeablty and that the sandstones vary quckly n lateral drecton, the general reservor predcton method cannot dentfy the dstrbuton of nterest reservors. Therefore, ths paper ntends to use matchng pursut decomposton based on Morlet wavelet to decompose the sesmc data volume nto data at sngle frequences, and pck out the frequences at whch the nterest reservors have an obvous abnormalty combned wth well data analyss. The data at nterest frequences are planned to be reconstructed to obtan a new sesmc data volume at whch the sesmc attrbutes and well data analyss are appled to predct the dstrbuton of nterest reservors. 2. Method Prncple 2.1. Mult-Wavelet Convoluton Model The convoluton model of common sesmc nterpretaton has an assumpton that the wavelet s unque and constant [12] [13]. The convoluton formula s shown n Formula (1). s( t) = W ( t) R( t) + N( t) (1) where s( t ) s synthetc record, W ( t ) s the sesmc wavelet, R( t ) s formaton reflecton coeffcent, and N( t ) s nose. However, n fact, because of the exstng factors such as energy dffuson, earth flterng, multple reflecton, nterference and other aspects, the wavelet morphology s tme-varyng and spatal-varable n the propagaton. Above assumpton dffers sgnfcantly from the actual stuaton, whch may lead to the loss of some weak effectve nformaton and generatng several artfacts. Thus, the wavelets wth dfferent frequences and ampltudes are used n convoluton respectvely and supermposed fnally, and the convoluton results are much closer to the real sesmc data [12] [14]. The mult-wavelet convoluton formula s ponted out n Formula (2). M (2) = 1 ( ) = ( ) ( ) + ( ) s t W t R t N t 530

3 where W ( t) ( 1, 2,, M) R ( t)( 1, 2,, M) = represents a wavelet aggregaton ncludng dfferent spectral characterstcs, and = s non-zero reflecton coeffcent seres. Sesmc wavelet morphology s arbtrary from the theoretcal model analyss, whle t has a defnte morphology after exctng the source wavelet n fact. Although varous factors n the propagaton make the wavelet morphology change, the performance of ths change s mostly a relatvely hgh frequency component attenuaton ncreasng and a relatvely low frequency attenuaton decreasng. The basc shape of those wavelets wth dfferent morphologes are smlar, so that t s consdered to use a mathematcal analytcal expresson to descrbe those wavelets aggregaton. For example, Morlet wavelet functon has a clear mathematcal analytcal form n tme and frequency domans, and t s able to characterze the energy attenuaton and velocty dsperson of sesmc waves n the propagaton of underground medum [15] [16]. Its mathematcal expressons n tme and frequency domans respectve are and 2 2 ( ) exp ( ln 2 ) exp ( 2π ) M t = t f k f t (3) m m ( f f ) 2 2 π ln 2 π M ( f ) = exp k fm ln 2 f where f s frequency, f m s mean or man frequency, and k s a constant actng on the Gauss part of wavelet functons to modulate the wavelet morphology. The hgher k value s, the more serous wave compresson s, the less sde lobe gets, and the narrower wavelets becomes Wavelet Decomposton and Reconstructon The basc prncple of mult-wavelet decompostonand reconstructon s decomposng the sesmc traces nto varous wavelets wth dfferent frequences and ampltudes n mathematcal methods, whch s shown n Fgure 1(a). Snce the decomposton s lnear, those wavelets are supermposed agan to get back nto the orgnal sesmc traces [17]-[20]. By analyzng the varatons of decomposed sngle frequency data n the nterest formatons, the useful wavelets are pcked out and supermposed agan wth mantanng ts poston unchanged to form new sesmc traces, whch s called sesmc reconstructon shown s Fgure 1(b). In new sesmc data after decomposton and reconstructon, the nterference waves are suppressed as much as possble and the useful nformaton as hghlghted Matchng Pursut Durng varous mathematcal algorthms of wavelet decomposton and reconstructon such as shot-term Fourer transform (STFT), wavelet transform (WT), matchng pursut (MP) and so on [12], the last one are adapted n ths study. Its basc thnkng s decomposng the sesmc sgnals va teratve algorthm nto a seres of Gabor wavelet functon aggregaton matchng wth tme-frequency characterstcs of sesmc sgnals. But ths method only consders the optmal matchng level locally, and makes the calculaton greedy [12] [21]. Fortunately, the matchng pursut based on Morlet wavelet uses the nstantaneous attrbute nformaton of complex sesmc traces to decrease the wavelet dctonary as well as tme of searchng for the best wavelet and mprove the computas t s band-lmted and tonal effcency greatly [6]. The prncple of ths technology s that sesmc sgnal ( ) can be expressed a s a lnear combnaton of M Morlet wavelets, as n m 2 m (4) Record 10 Hz 15 Hz 20 Hz 25 Hz 30 Hz 35 Hz 40 Hz 10 Hz 15 Hz 20 Hz 25 Hz 30 Hz 35 Hz 40 Hz Record Decomposton Reconstructon (a) Fgure 1. Schematc dagram of mult-wavelet decomposton (a) and reconstructon (b). (b) 531

4 where and M 1 ( M ). (5) = 1 ( ) (,, ϕ ) s t = AM t t f + R f A s the correspondng ampltude of decomposed th wavelet expressed n ϕ are tme delay, man frequency and phase of wavelet respectvely, and M, parameters t, f ( M ) R f s the resduals of reconstructon whch s also nose term. Sgnal s( t ) of complex sesmc traces can be expressed as * ( ) ( ) ( ) * where s ( t ) s the Hlbert transform of s( t ), and ( ) S t = s t + s t, (6) S t s just the complex sesmc trace [22] [23]. Instantaneous ampltude (envelope), nstantaneous phase and nstantaneous frequency of complex sesmc trace are and and θ 2 *2 ( ) ( ) ( ) At = s t + s t (7) ( t) = tan 1 s 2 ( t) + s *2 ( t) ( ) f t (8) ( t) 1 dθ = (9) 2π dt respectvely. Wheren nstantaneous frequency s the dervatve of nstantaneous phase versus tme. 3. Examples 3.1. Geologcal Background The H area s located on a faulted antclne structure of Turpan-Ham basn, Chna. It s almost 10 km 2 (Fgure 2), and the man gas-bearng reservor s the second member of Sanjanfang formaton, Jurassc system, whose lthology prmarly conssts of fne sandstone, sltstone and argllaceous sandstone. The thckness of sandstone s mostly less than 5 meters, so that the reservor s thn nterbed. There are totally 36 wells (Fgure 2) and varous well data n the study area. By dong statstcs of sandstone layers at wells n J 2 s 2 member, the lthologcal assocatons of H area are classfed nto three categores whch s shown n Fgure 3. The frst knd s sngle thn sandstone, the second type s thn nterbedded sandstone, and the last class s large thck sandstone. The nterest assocatons are both of the second and last class, whch s to say the assocatons wth hgh sand-to-formaton rato s the nterest reservor. Fgure 2. Structural map wth well dstrbutons n research area. 532

5 Fgure 3. Lthologcal correlaton of J2s2 member between H1, H2, H3, H4 and H5 well n H area. In the legend, 1 s yellow fne sandstone, 2 s yellow sltstone, 3 s yellow argllaceous sltstone, 4 s gray slty mudstone, 5 s gray mudstone, 6 s green mudstone, 7 s red-brown mudstone, 8 s varegated mudstone, 9 s black carbonaceous mudstone, and 10 s black coal seam. The colors of 1, 2 and 3 are changed to yellow from gray to hghlght ther lthology.h1 and H6 wells are classfed as Class I accordng to the lthologcal assocaton, and H3 and H5 wells are classfed as Class II as well as both H2 and H4 wells beng classfed as Class III Applcaton Analyss Usng dfferent lthology and flud havng varous responses to dfferent frequences, mult-wavelet decomposton and reconstructon are appled to J 2 s 2 member of the study area on reservor predcton. Analyss are from pont to lne and then to surface. Fgure 4 shows the decomposed wavelet dctonary of sesmc trace near H1 well and the correspondng ampltude spectrum. The fgure ponts out that ths sesmc trace s decomposed nto lots of Morlet wavelet, and ponts out that the man frequency ranges from 15 to 30 Hz. Transformng mult-wavelet decomposton results of sesmc trace near wells nto the ampltude spectrum shown n Fgure 5, t s found that the spectrum characterstcs of coal bed s strong energy, hgh frequency and broad bandwdth, and that of both of thn nterbedded sandstone and large thck sandstone s medum-to-strong energy and medum-to-hgh frequency as well as that of sngle thn sandstone beng characterzed by weak energy, low frequency and narrow bandwdth. The man frequency of energy group the s the correspondng frequency at the strongest energy n the tme-frequency spectrum, and oblateness s the rato of bandwdth of energy group to tme thckness. Results of statstcs and analyzng on tme-frequency spectrum of all wells n the study area reveals that the man frequency has a fttng relaton of power functon wth oblateness, whch s shown n Fgure 6. The fttng formula s as follows, O = f R 2 = (10) where O and f s the oblateness and man frequency of energy group respectvely, and R represents the correlaton coeffcent between fttng formula and scatter data. The closer to 1 the R value s gettng, the hgher the 2 fttng degree s. It ponts out that the shape of energy groups s gettng more oblate wth the man frequency ncreasng. Ths s because that the hgher man frequency s, the shorter tme-delay of wavelets, and the broader bandwdth n frequency doman, and the larger oblateness s, and vce versa. In addton, concentrated regularty of statstcal samples shows man frequency of the nterest reservor manly concentrates n three frequency bands. They are low band rangng from 8 to 16 Hz, medum band rangng from 22 to 34 Hz, and hgh band rangng from 40 to 50 Hz. The prmary frequency band s from 8 to 34 Hz. The orgnal sesmc records are decomposed nto plenty of data at sngle frequency rangng from 0 to 90 Hz va mult-wavelet matchng pursut decomposton method. Sesmc well-te sessons of Fgure 7(a) and Fgure 7(b) are orgnal data and sngle frequency data at 26 Hz respectvely. In the former fgure, the ampltude dfference among dfferent lthologcal assocatons s not very obvous and unable to be dstngushed vsually, whle n the latter one, the ampltude dfference of each wells are amplfed, wheren ampltude of both H2 and H4 wells s relatvely strong, whle that of H3 and H5 wells s relatvelyweak, and that of H1 and H6 wells cen- 533

6 (a) Fgure 4. Decomposton wavelet dctonary (a) and ts spectrum (b) of sesmc trace near H1 well. (b) Fgure 5. Mult-wavelet tme-frequency spectrum of sesmc traces near H1 well. Fgure 6. Crossplot of man frequency and flattenng of sesmc traces near wells n H area. The blue lne s the trend lne, and the black boxes represent the man frequency nterval. 534

7 (a) (b) Fgure 7. Energy dfference of dfferent lthologc assocaton between orgnal sesmc data (a) and sngle frequency data at 26 Hz (b). tered. Thus, responses of dfferent lthologcal assocatons to partcular frequences vares greatly. Observng the characterstc responses of dfferent lthologcal assocatons at frequences rangng from lower frequency of 10 Hz to hgher one of 50 Hz, t s found that sandstone n the second class s characterzed by enhanced energy at low frequences whch s less than 20 Hz, and that sandstone of the thrd class performs enhanced energy at mddle frequences from 20 to 34 Hz, and that sandstone of the frst class has stronger energy at hgh frequences more than 35 Hz. From the above analyss, t shows that frequences of energy anomales reflected by the nterest second and thrd classes s both low and medum frequency bands whch s less than 35 Hz. Consderng the analyss results of sngle well and well-te mult-wavelet decomposton, t s suggested that the nterest frequences of the study area n nterest reservors range from 8 to 34 Hz. Then wavelets of those frequences are lnearly supermposed by mult-wavelet matchng pursut reconstructon to obtan new reconstructed sesmc data whch s shown n Fgure 8. H2, H3, H4 and H5 wells wth nterest lthologcal assocatons have respectvely strong ampltude n the reconstructed data. Durng all 36 wells n the study area, there are only a quarter havng lthologcal data. For ths, comparng the gamma value of wells wthout lthology data n nterest reservors wth gamma value of wells wth lthology data, and combnng wth gamma characterstcs of dfferent lthologcal assocatons, the wells wthout lthology nformaton are classfed accordng to lthologcal assocatons. The crossplot of mean gamma values n nterest reservors and RMS ampltude at wells extractng from reconstructon data s shown n Fgure 9. When the RMS ampltude s greater than 3650, 63% wells wth nterest lthologcal assocatons are dstrbuted wthn ths range, whle 78% wells beyond nterest assocatons are elmnated. Therefore, the dstrbuton wth RMS ampltude greater than 3650 s the potental area of nterest reservors, and the predcton s shown n Fgure 10. The predcton results suggest that the nterest sandstone n nterest reservors are manly n the west of the study area. Wheren southwestern sandstone s characterzed by annular hgh ampltude anomales, and sandstone of western and northern areas demonstrate large-scale and banded dstrbuton, and contguous sandstone n eastern regons appears sporadcally. 535

8 Fgure 8. Reconstructon data at frequency from 8 to 34 Hz of J 2 s 2 member n study area. Fgure 9. Crossplot of both mean GR values and RMS ampltude of reconstructon data n J 2 s 2 member among all wells of study area. Fgure 10. Predcton of nterest reservor dstrbuton of J 2 s 2 member n H area. 536

9 4. Concluson In summary, mult-wavelet decomposton and reconstructon technology break through the lmtatons caused by the sngle wavelet assumpton n conventonal sesmc convoluton, and Morlet wavelet functon greatly descrbes frequency attenuaton and velocty dsperson of sesmc wavelet n the propagaton. Matchng pursut decomposton and reconstructon technology based on Morlet wavelets decomposes the orgnal sesmc traces nto multple Morlet wavelet lnearly supermposng. By analyzng the senstve frequences where geologcal bodes generate anomales, wavelets at those frequences are pcked out and lnearly reconstructed to obtan new sesmc data where reservor predctons are well done. Applcaton of ths method to H area n Turpan-Ham basn shows that mult-wavelet decomposton and reconstructon technology greatly descrbes the lateral dscontnuty of reservors, provdes strong support for reservor geology and geophyscal nterpretaton. References [1] Wang, Z. (2012) The Applcaton and Study of Spectral Decomposton Technology on Channel Sand Body Descrpton. Master Thess, Chna Unversty of Petroleum (East Chna), Qngdao. (In Chnese) [2] Geronmo, J.S., Hardn, D.P. and Massopust, P.R. (1994) Fractal Functons and Wavelet Expansons Based on Several Scalng Functons. Journal of Approxmaton Theory, 78, [3] Llly, J.M. and Park. J. (1995) Multwavelet Spectral and Polarzaton Analyses of Sesmc Records. Geophyscal Journal Internatonal, 122, [4] Mallat, S.G. and Zhang, Z.F. (1993) Matchng Pursut wth Tme-Frequency Dctonares. IEEE Transactons on Sgnal Processng, 41, [5] Lu, J.L., Wu, Y.F., Han, D.H. and L, Z.G. (2004) Tme-Frequency Decomposton Based on Rcker Wavelet. SEG Techncal Program Expanded Abstracts of the 74th SEG Annual Internatonal Meetng, Denver, October 2004, [6] Lu, J.L. and Marfurt, K.J. (2005) Matchng Pursut Decomposton Usng Morlet Wavelets. SEG Techncal Program Expanded Abstracts of the 75th SEG Annual Internatonal Meetng, Houston, 6-11 November 2005, [7] Lu, J.L. and Marfurt, K.J. (2007) Instantaneous Spectral Attrbutes to Detect Channels. Geophyscs, 72, [8] Song, W.Q., Zhu, W.X. and Sun, Y.J. (2007) Identfy Bed Layer Sandbody by Complex Wavelet Matchng Algorthm. Progress n Geophyscs, 22, (In Chnese) [9] Chen, L. and Song, H.B. (2008) Extracton of Sesmc Tme-Frequency Attrbute Based on Morlet Wavelet Match Tracng Algorthm. Ol Geophyscal Prospectng, 43, (In Chnese) [10] Fen, L. and Jang, Z.X. (2009) Spectral Decomposton Based on Matchng Pursut and ts Applcaton. Progress n Exploraton Geophyscs, 32, (In Chnese) [11] Zhang, X.W., Han, L.G., Wang, Y. and Shan, G.Y. (2010) Sesmc Spectral Decomposton Fast Matchng Pursut Algorthm and Its Applcaton. Geophyscal Prospectng for Petroleum, 49, 1-6. (In Chnese) [12] Qu, N. (2012) Research on Sesmc Wavelet Decomposton and Reconstructon Technology. Master Thess, Ocean Unversty of Chna, Qngdao. (In Chnese) [13] She, G., Zhou, X.Y. and Wang, J.B. (2013) Predcton of Sand Reservor wth Mult-Wavelet Sesmc Trace Decomposton and Reconstructon. Journal of Southwest Petroleum Unversty (Scence & Technology Edton), 35, (In Chnese) [14] Lu, X., Chen, C., Zhao, Y.T. and Wang, X. (2015) Mult-Wavelet Decomposton and Reconstructon Based on Matchng Pursut Algorthm Fast Optmzed by Partcle Swarm. Journals of Jln Unversty (Earth Scence Edton), 45, (In Chnese) [15] Morlet, J., Arens, G, Fourgeau, E. and Gard, D. (1982) Wave Propagaton and Samplng Theory-Part II: Samplng Theory and Complex Waves. Geophyscs, 47, [16] Xu, T.J., Shen, Z.M. and Wen, X.K. (2010) Research and Applcaton of Mult-Wavelet Decomposton and Reconstructon Technology. Journal of Chengdu Unversty of Technology (Scence & Technology Edton), 37, (In Chnese) [17] Da, S.H., Chen, Z.G., Yu, J.B., et al. (2011) Applcaton of Mult-Wavelet Decomposton and Reconstructon Technque n Reservor Characterzaton n TKT-NGS Olfeld, Algera. Ol Geophyscal Prospectng, 46, (In Chnese) [18] Huang, Y., Xu, D. and Wen, X.K. (2013) Optmzaton of Wavelets n Mult-Wavelet Decomposton and Reconstruc- 537

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