Stochatic inverion of eimic PP and PS data for reervoir parameter etimation Jinong Chen*, Lawrence Berkeley National Laboratory, and Michael E. Glinky, ION Geophyical Summary We develop a hierarchical Bayeian model to invert eimic PP and converted-wave (i.e., PS) data for reervoir parameter. The model i an extenion of the model-baed Bayeian method by Gunning and Glinky (004) with converted wave repone and PS time regitration a additional data and with two-way PS travel time and PS reflectivity a additional unknown. We implement the model by reviing their open-ource oftware, Delivery, written in Java. We demontrate the ue of the revied code by alying them to two ynthetic cae. One i a three-layer and wedge model with variable thickne, and the other i a ix-layer floating-grain model baed on actual field data. The cae tudy reult how that eimic PS data greatly enhance the reolution of eimic data for thin layer, and they provide more information on reervoir parameter than far-offet PP data. Introduction Multicomponent eimic urveying ha been ued for hydrocarbon exploration more than a decade becaue it can capture the eimic wave-field more completely than conventional ingle-element technique (Stewart et al., 00). Many type of energy converion may occur when eimic wave pa through the underlying earth. However, tranmitted or multiple converion generally have much lower amplitude than the P-down and S-up reflection (Rodriguez-Saurez, 000). Conequently, for many alication of multi-component eimic data, the ue of converted-wave or PS image receive much attention (Stewart et al., 00; Mahmoudian and Margrave, 004; Veire and Landro, 006). Several obtacle exit currently that make the ue of converted-wave data, a a routine practice, difficult. The firt one i the high acquiition cot of collecting multicomponent eimic data compared to conventional eimic urvey. The econd one i that multicomponent data proceing i till challenging. Compared with P-wave velocity analyi, identifying pure S-wave event in multicomponent eimic data i much harder (Veire and Landro, 006). Finally, we till do not have good method to jointly integrate multiple eimic data, epecially in the ene of joint inverion, becaue PP and PS data are recorded in different time domain. Thi tudy i an effort to combine eimic PP and converted-wave for etimating reervoir parameter uing a Bayeian hierarchical framework. We extend the model-baed Bayeian method developed by Gunning and Glinky (004) for inverting eimic AVO data, by reviing their open-ource Java code (i.e., Delivery ) to allow converted-wave repone and PS event time regitration a additional data. We ue the ame rock phyic model and Markov chain Monte Carlo (MCMC) ampling trategie a Delivery. Since thi tudy i built on the previou work, the ubequent decription will be focued on the new development and alication, and the detail of other part can be found in Gunning and Glinky (004). Method Hierarchical Bayeian model We ue the ame notation a Gunning and Glinky (004) and aume the uburface can be divided into n horizontal layer. We conider reervoir parameter at each layer a unknown, which include (1) poroity and net-to-gro (i.e., the ratio of permeable to impermeable rock by volume), () fluid aturation (e.g., brine, oil, and ga aturation), and (3) rock phyic attribute of permeable and impermeable rock (e.g., P-wave and S-wave velocity and denity). For eae of decription, we let vector α repreent all the parameter related to the rock phyic model. We conider effective eimic P-wave and S-wave velocity ( v and v ) and denity ( ρ ), and eimic PP and PS p reflectivity ( R and R ) a unknown. They are function p of rock phyic parameter through uitable rock phyic model. We conider PP traveltime ( t ) a a primary unknown, and both layer-thickne ( d ) and PS traveltime ( t ) can be derived from the PP traveltime and aociated p effective eimic attribute. The data ued for inverion include eimic PP and PS full-waveform ( S, and S ) p and PP and PS event regitration time ( T and T ). If p available, they include other information from nearby borehole, uch a thickne contraint ( D ). Figure 1 how all the unknown, available data, and their relationhip, and the dahed rectangle highlight our extenion to Delivery. Specifically, we add two unknown related to converted wave (i.e., t p and R ) and two new p data (i.e., T and p S ). Following the direct graphical p model, we have the following hierarchical Bayeian model: b
Stochatic inverion of PP and PS data f ( α, t, t, d, v, v, ρ, R, R S, S, T, T, D ) p p p p p b f( S t, R ) f( S t, R ) f( T t ) p p p f( T t ) f( D d) f( R v, v, ρ) p p b p f( R v, v, ρ) f( d t, v ) f( t t, v, v ) p p p p p f( v, v, ρ α) f( α) f( t ). p Equation 1 define a joint poterior probability ditribution function of all unknown parameter, up to a normalizing contant. The firt five term on the right ide of the equation are the likelihood function of available data, which link data to unknown, and all other term on the right ide are the prior probability ditribution, derived from other ource of information. We define all the likelihood function and prior ditribution in a imilar way to Delivery (ee Gunning and Glinky, 004). In the following, we only decribe the new development. PP and PS reflectivitie We ue the linearized Zoeritz aroximation (Aki and Richard, 1980) to obtain PP and PS reflectivity at an interface, which are given below: R R p 1 ΔVp Δρ = + Vp ρ 1 ΔV V Δρ ΔV + + + V p V p ρ V p 4 4 θ O( θ ), 1 Δρ V Δρ ΔV = + + + ρ V p ρ V 3 θ O( θ ). In equation and 3, Vp = ( Vp 1+ Vp)/, V = ( V 1+ V)/, ρ = ( ρ1+ ρ)/, Δ Vp = Vp V, Δ V p1 = V V1, and Δ ρ = ρ ρ1, where ( V p1, V 1, ρ 1 ) and ( V, V p, ρ ) are P- and S-wave velocity and denity in the layer above and below the interface. Symbol θ i the PP incident angle in the unit of radiu. The PP and PS reflectivitie have the fourth and third order of accuracy in term of the incident angle. Linkage between PP and PS data and event time regitration Traditional method for joint inverion of PP and PS data are primarily baed on maing of PS data to PP time (or domain converion), in which PS data are conidered a additional eimic tack. Although thi aroach i imple to implement, it uffer from difficultie, uch a wavelet ditortion (Banal and Matheney, 010), becaue the converion of PS time to PP time need interval eimic P-to- S velocity ratio, which are not known a priori. In thi tudy, we avoid the PP-to-PS domain converion and ue PS data directly in the PS time domain. We firt pick one or more PS event from PS eimogram that have good correpondence with the PP eimogram along the ame profile, and we refer them to a the mater PS horizon. In the (1) () (3) forward imulation, we calculate all the PS time relative to the mater horizon. The ue of event time regitration a data i one of main advantage of Delivery a well a the current extenion becaue PP event time i directly related to P-wave velocity and PS event time directly related to P-wave and S-wave velocity. They provide additional information to contrain the etimate of P-wave and S-wave velocity beyond the reflectivity baed PP and PS full-waveform. Example We ue two example to tet the revied code and to demontrate the benefit of including converted-wave data in reervoir parameter etimation. The firt example i a threelayer wedge model with variable thickne, which i imilar to the one by Gunning and Glinky (004) and Puryear and Catagna (008). In the example, we will how how converted-wave data help to improve parameter etimation in thin layer. The econd example i baed on actual field data given by Gunning and Glinky (007) with ix layer. We will how how converted-wave data help to etimate floatinggrain fraction, which wa demontrated by Gunning and Glinky (007) to be a difficult parameter to etimate under the field condition. In both example, we ue the PP and PS wavelet typical of thoe derived from field borehole log for pare-pike inverion uing the method of Gunning and Glinky (006). The PP and PS wavelet have the peak frequencie of 3 Hz and 13 Hz, repectively (ee Figure ). Three-layer and wedge model We ue the ame three-layer and wedge model given by Gunning and Glinky (004) to tet the developed method. In thi model, a and wedge i pinched out by the urrounding hale layer. Baed on the PP wavelet given in Figure and uing the formula from Chung and Lawton (1995), we get the PP tuning thickne of 8 m. For thi tudy, we let the and wedge thickne increae from 1/8 to 5/4 of the PP tuning thickne with an increment of 1/8 tuning thickne. Thi yield the thickne of 3.5, 7, 10.5, 14, 17.5, 1, 4.5, 8, 31.5, and 35 m. The true net-to-gro of the and wedge i 0.5. For inverion, we et the prior a the truncated normal ditribution with the true value a mean and 0.5 a the tandard deviation. The rock phyic model ued for thi tudy i given by Mavko et al. (1998, page 97) baed on data from Han (1986) for tight-ga andtone. We generate ynthetic PP and PS data by uing equation and 3 to calculate reflectivity. At the top interface of the and wedge, the reflectivitie of the near PP tack (incident angle = 6 degree), the far PP tack (incident angle = 30 degree), and the far PS tack (incident angle = 30 degree) are 0.0864, 0.0677, and -0.1003, repectively. At it bottom interface, the reflectivitie of PP near and far tack and PS far tack are - 0.0860, -0.060, and 0.1179. We convolved thoe reflectivitie with the given wavelet (ee Figure ) to get
Stochatic inverion of PP and PS data eimic full waveform. Figure 3 how the ynthetic eimic data, and we aume thoe data have Gauian random noie with the tandard deviation of 0.005. Inverion reult and analyi We invert the ynthetic data for ten different thicknee uing three different et of data in the inverion. They are: (1) only near-offet PP tack, () the near and far PP tack, and (3) the near PP tack and the far PS tack. Figure 4 how the median, 95% predictive interval, and the true value for (a) the net-to-gro (NG), (b) thickne, (c) P-wave velocity, and (d) S-wave velocity a a function of wedge thickne. The black quare and the dahed line in each figure how the true value and the prior bound. The red, green, and blue line are the etimate obtained uing the only the near PP tack, the near and far PP tack, and the near PP tack and the far PS tack. The olid line repreent the etimated median and the dahed line repreent 95% predictive interval. Comparing the etimated NG, and the P-wave and S-wave velocitie with their correponding true value, we found that when uing only the near PP tack data, the etimated median are very cloe to their true value for thickne greater than 17.5 m (i.e., 6.5% of the tuning thickne) but the etimate have ignificant uncertainty. By adding the far PP tack data, the etimated median have good agreement with the true value a long a the wedge thickne i greater than 10.5 m (i.e., 37.5% of the tuning thickne). By replacing the PP far tack with the PS far tack, the etimated median are unbiaed for thicknee greater than 7 m or even le. PS data i better at etimating thicknee and NG of thin layer. From thi example, we can ee that the ue of converted wave ignificantly improve the reolution of parameter etimation. The poible reaon may include (1) the PS data extending the frequency range of data, () the PS wavelet providing additional information, and (3) the PS event time regitration providing additional information. Six-layer floating-grain model We ue the example B in Gunning and Glinky (007) a the econd tet cae. The example i baed on actual field data and ha ix layer with the fourth and ixth layer being oil reervoir. A general layer equence i given by (1) marl, () ilt-marl tringer complex, (3) hale, (4) uer pay and, (5) hale, and (6) lower pay and. A pointed out by Gunning and Glinky (007), thi i a difficult cae tudy a the oil reervoir are caed by a complex draping tructure including thick, acoutically hard marl and thin, oft, ilty layer. each layer interface, calculated uing equation and 3. We can ee that the near PP reflectivity coefficient at the firt two interface dominate the reflection from the three deeper interface. Both PP and PS far tack provide much larger reflectivity at thoe deeper interface, with the PS reflectivity being lightly better than the PP reflectivity. Inverion reult and analyi We ue the ame PP and PS wavelet a for the firt example and the floating-grain model developed by DeMartini and Glinky (006) and reformulated by Gunning and Glinky (007) for Delivery. Our primary focu i on the etimate of floating-grain fraction, and we want to ee what we can gain by including converted-wave data in the etimation. Baed on the field data, we et the true floating-grain fraction to 3.5% and the net-to-gro ratio to 65+/-10%. Other parameter are et to be the mot plauible value at thi ite (ee Gunning and Glinky, 007 for detail). For inverion, we give a prior of truncated Gauian ditribution with mean of % and tandard deviation of 3% for the floating-grain fraction. Thi ditribution give ignificant prior probability to the zerofloat (i.e., the clean and cae). Figure 5 compare the etimated poterior probability ditribution of floating-grain fraction and effective S-wave velocity in the uer and lower pay layer (i.e., layer 4 and 6) under variou ituation with their correponding prior probability ditribution. The black curve are the prior probability ditribution, and the red, green, and blue curve are the etimated probability denitie obtained uing only the near PP tack, the near and far PP tack, and the near PP and far PS tack, repectively. The comparion between the reult uing the near and far PP tack i imilar to thoe by Gunning and Glinky (007). We can ee that the far PS tack ignificantly improve the etimate of effective S-wave velocity (ee column ) and provide more information than the far PP tack when etimating the floating-grain fraction (ee column 1, blue curve v. green curve). Concluion The extended Bayeian model i effective when jointly inverting PP and PS data. PS data can provide complementary information to PP data and thu have the potential of ignificantly improving parameter etimation reult. Acknowledgment We thank ION Geophyical for funding and for permiion to publih thi work. We thank Jame Gunning from CSIRO for providing help in undertanding the Delivery code. Again, we ue two-offet tack eimic data (i.e., near and far), which have the incident angle of 0 and 30 degree. Table 1 lit PP reflectivity coefficient of the near and far tack and the PS reflectivity coefficient of the far tack for 3
Stochatic inverion of PP and PS data Figure 1. Hierarchical tructure of unknown parameter and available data, where the arrow how conditional relationhip. Figure. PP and PS wavelet, derived from borehole log of a real field. They have the peak frequencie of 3 Hz and 13 Hz, repectively. Figure 3. Seimic near PP tack, far PP tack, and far PS tack for the three-layer wedge model. The PP data are in the PP time domain, while the PS data are in the PS time domain. Figure 4. Comparion of prior and poterior etimate for the threelayer wedge model. The black how the prior (dahe) and the true value (quare). The red, green, and blue how the median (olid) and the 95% bound (dahe) obtained from the near PP tack, the near and far PP tack, and the near PP and the far PS tack. Table 1: PP and PS reflectivity coefficient for the ix-layer floating-grain model Interface Near PP tack Far PP tack Far PS tack Marl/tringer -0.1586-0.0887 0.363 Stringer/hale 0.1007 0.0589-0.155 Shale/uer 0.0036-0.059-0.0571 pay and Uer pay 0.048 0.0394 0.015 and/hale Shale/lower pay and -0.0100-0.035-0.0386 Figure 5. Comparion between the prior and poterior etimate of floating-grain fraction ((a) and (c)) and S-wave velocity ((b) and (d)). The black curve and magenta vertical line repreent the prior and the true value. The red, green, and blue curve are the poterior probability denitie obtained from the near PP tack, the near and far PP tack, and the near PP and far PS tack. 4
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