11. Inversion_for_acoustic_and_elastic_impedances

11. Inversion_for_acoustic_and_elastic_impedances - GPHY...

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Unformatted text preview: GPHY 3423 Petroleum Geology and Geophysics Kurt J. Marfurt (University of Oklahoma) Inversion for Acoustic and Elastic Impedance 11-1 Inversion for Acoustic and Elastic Impedance After this section you will be able to: Use inversion to integrate a user-defined earth model and synthetic seismograms to `best fit' the measured seismic data, Differentiate and ideally choose between relative, bandlimited, model-driven, and geostatistical inversion algorithms Relate impedance inversion to deconvolution, and Use elastic inversion effectively as a lithologic indicator. 11-2 The Acoustic Impedance Attribute Matrix : Vm, m Porosity Fluid : Vf, f Acoustic impedance = P-Velocity x Density A.I. of a rock is a function of : - Matrix (lithology), - porosity, - the fluid content - Pore pressure - and maybe the shape of the pores! A.I. 1 A.I. 2 A.I. 3 A.I. 4 11-3 Model-driven AI inversion is performed in the time domain on post stack migrated data. (Veekan et al, 2002) Hierarchy of acoustic impedance attributes None (0) Few (1-2) Near/far angle stacks Normal impedance and scaled Poisson's ratio Fluid factor=RP-kRS. Increasing resolution Acoustic/elastic impedance inversion Lambda-mu-rho inversion Neural network well correlation Simultaneous inversion Geostatistical Inversion Regional Prospect Development Characterizatio n Number of wells More (5-10) Many ( > 10) 11-4 (Modified after Ross, 2002) Conversion of Reflectivity to Acoustic impedance Disadvantages of interpreting amplitudes 1. Advantages of using acoustic impedance 1. Represents a contrast between layers (relative measurement) Affected by thin-bed tuning. Upward-fining and upward coarsening produce subtle amplitude anomalies. Contaminated by noise Not directly measured at the wells Directly represents layer properties such as porosity Removes phase rotations due to tuning, upward fining, and upward coarsening Constrained inversion suppresses noise inconsistent with model Measured at well locations Excellent parameter for geobody visualization 2. 2. 3. 3. 4. 4. 5. Limited value in constructing geobodies 5. 11-5 (Modified from Duboz et al. , 1998) Inversion for acoustic impedance (wedge model) 0.3 0.4 0.5 0.6 0.3 0.4 0.5 sidelobes! Time (s) `true impedance model' colored seismic synthetic sidelobes! Time (s) variable area seismic synthetic 0.6 0.0 0.1 0.2 0.3 0.4 0.5 Thickness (s) 0.6 0.7 inverted acoustic impedance 0.0 0.1 0.2 0.3 0.4 0.5 Thickness (s) 0.6 0.7 11-6 (Latimer et al., 2000) Inversion for acoustic impedance (ramp model) 0.0 Z (g/cm3-ft/s) 20000 19000 0.1 18000 17000 16000 15000 Time (s) 0.2 14000 13000 -0.05 Time (s) 0.3 0.00 +0.05 0.4 11-7 (Latimer et al., 2000) Impact of low and high frequencies on resolution Better for calibration to wells! 10-80 Hz 10-500 Hz 0-80 Hz (Latimer et al., 2000) 11-8 Extending the seismic spectrum Amplitude Final inversion spectrum Model band Seismi c band 0 20 40 Frequency (Hz) 60 80 11-9 (Pendrel and van Riel,2000) Inversion: improvement of vertical resolution 0.95 0 Time (s) 1.00 0 c top i sm i Se Seis mi c bas e Seismic does not reveal reservoir thickening off the horst block Impedance inversion reveals both thickening and reservoir quality Increased resolution improves interpretation confidence 1.05 0 0.95 0 Time (s) 1.00 0 er Inv nt si o op 1.05 0 0. 00 0. 0 00 0.012 0. 4 31100 00 8 Inver s i on bas e Thickness (s) Seismic thickness Inversion thickness 31050 CDP no. 31000 11-10 (http://www.e-seis.com/Bulletin_PDF/e-pb300.pdf) Seismic Modeling and Inversion Subsurface acoustic impedance Forward Modeling Wavelet Seismic trace with Seismic trace Wavelet Subsurface acoustic impedance Inversion with 11-11 Non-uniqueness in impedance inversion Seismic data are band-limited Basically, any one of many 1-ft resolution impedance models having the same average (over the distance of a seismic wavelet) can fit the seismic data. The low frequency trend is not measured at all. The interpreter can add any information he or she wishes. Seismic noise can be incorrectly converted to seismic impedance changes This non-uniqueness results in alternative inversion strategies: Favor models that are blocky (sparse-spike inversion) Favor components of a user-generated background model (model-driven inversion) Examine all possible models that fit the data and user-defined constraints (geostatistical inversion) 11-12 Hierarchy of Seismic Inversion 1. Inversion of normal-incidence or post-stack seismic data - Bandlimited impedance inversion - Sparse-spike impedance inversion - Model-based inversion Cell-based parameterization Stratigraphic parameterization Global optimization - Geostatistical inversion 2. Inversion of prestack seismic data - Sequential common-angle ("elastic") inversion - Simultaneous prestack inversion AVO-based Wave-equation-based 3. Hybrid inversion 11-13 If the impedance profile is smoothly varying and the sampling is sufficiently fine, Ri = Z i +1 - Z i Z i +1 + Z i Impedance Inversion (trace integration implementation) can be approximated by Ri Z i 2Zi 1 d ln Z i 2 dt ( ) By integrating, we obtain: ln Z n = 2 R i =0 n i Finally, we exponentiate to obtain: n 2 Z n = exp Ri i =0 11-14 Red Fork horizon slices, Anadarko Basin, USA Band-limited acoustic impedance Seismic amplitude 11-15 (Suarez et al., 2008) Hierarchy of Seismic Inversion 1. Inversion of normal-incidence or post-stack seismic data - Bandlimited impedance inversion - Extending the bandwidth - Sparse-spike impedance inversion - Model-based inversion Cell-based parameterization Stratigraphic parameterization Global optimization - Geostatistical inversion 2. Inversion of prestack seismic data - Sequential common-angle ("elastic") inversion - Simultaneous prestack inversion AVO-based Wave-equation-based 3. Hybrid inversion 11-16 Flowchart for extending the band-width of seismic inversion Migrated seismic data Density logs x Sonic logs Impedance logs Trace integration Bandlimited inversion + Broadband inversion Low-pass filter Lowfrequency model 11-17 Band-limited Inversion 1.0 Addition of low frequency velocity trend from migrationdriven velocity analysis and well logs to the mid frequency component obtained by inversion 2.0 3.0 11-18 Images courtesy: ONGC, India Bandlimited Inversion Swan Hills carbonate, Alberta, Canada Porosity Synthethic sonic long (in transit time) generated using the `seislog' process. 11-19 (Lindseth, 1979) Hierarchy of Seismic Inversion 1. Inversion of normal-incidence or post-stack seismic data - Relative acoustic impedance - Bandlimited impedance inversion - Sparse-spike impedance inversion - Model-based inversion Cell-based parameterization Stratigraphic parameterization Global optimization - Geostatistical inversion 2. Inversion of prestack seismic data - Sequential common-angle ("elastic") inversion - Simultaneous prestack inversion AVO-based Wave-equation-based 3. Hybrid inversion 11-20 Model-Based Inversion 1. A geological (impedance) model is first built consistent with the seismic data. A synthetic trace is generated from the model and compared with the equivalent seismic trace. If the difference is small, the model trace is taken as the final solution. If the model trace is significantly different from the equivalent seismic trace, the impedance model is perturbed in terms of reflection time and impedance to yield lateral and temporal impedance changes. The process is repeated iteratively till the difference falls below a set threshold. 2. 3. 4. 11-21 Inverse modeling technique Initial model A.I Synthetic Seismic seismogram trace Residual trace Iteration 1 Iteration 2 Iteration 3 (Veekan et al, 2002) 11-22 Time Model-based inversion objective function Misfit between modeled data, d(m), and measured data, d0 Minimize the squared error E = [d(m) - d 0 ] + [m - m 0 ] + [m - 0] 11-23 2 T -1 Cd [d(m) - d 0 ] Misfit between new model and a priori model T -1 Cm [m - m 0 ] -1 Cm [m - 0] T Misfit between smoothness of model and a model that is `perfectly smooth' Classical inversion Start Define initial a priori model, m0, Using horizon picks and AI logs Model seismic, d(m) Calculate d=d(m)-d0 Does modeled seismic match measured data? no Output AI model, mi Output residuals d Generate objective function, E2(m) yes Perturb model to obtain E2/m Stop Solve for m Update mi=mi-1+ m 11-24 Local minimu mA 1 n 2 Objective fucntion, E2 Infeasible solutions Local minimum B Initial model Global minimu m Local minimum C Feasible solutions mj User constra int 11-25 User constra int Infeasible solutions Local minimum A 2 6 pdf 8 7 (high T)5 4 3 1 Local minimum C Local minimum B Objective fucntion, E2 Infeasible solutions Initial model Global minimum Infeasible solutions Feasible solutions mj User constraint User constraint 11-26 Local minimum A pdf (mid T) 37 1 5 8 4 6 2 Local minimum B Initial model Global minimum Infeasible solutions mj User constraint User constraint Local minimum C Objective fucntion, E2 Infeasible solutions Feasible solutions 11-27 Local minimum A pdf (low8T) 7 41 2 5 3 6 Local minimum B Initial model Global minimum Infeasible solutions mj User constraint User constraint Local minimum C Objective fucntion, E2 Infeasible solutions Feasible solutions 11-28 Start Define initial a prior model, m0, Using horizon picks and AI logs Decrease mutations or temperature Generate PDF Inversion using simulated annealing or genetic algorithms Choose a random value of mi Model seismic, d(m) Calculate d=d(m)-d0 Generate objective function, E2(m) More random models? no Output AI model, mi Output residuals d yes Does modeled seismic match measured data? yes Stop no 11-29 Model-Based inversion Amplitude D de ow lin nd ea ip tio n Up di High P-Impedance (ft/s-g/cm3) 28000 26000 24000 water 22000 gas 20000 p water de lin ea tio n water gas gas Low gas gas 18000 Amplitude P-wave impedance Model-based inversion discriminates between the gas-bearing zones and the non-gas zone. 11-30 (Russell et al., 2006) Case study Delineation of a chert deposit Central Basin Platform, west Texas 11-31 Impedance inversion applied to a chert deposit 10-32 11-32 (Ruppel, 2001) Thick-bedded chert Chert/carbonate Laminae Major Lithofacies (Good porosity) Brecciated Chert Carbonates 11-33 (Fu et al., 2006) Thick-bedded Chert, West Texas Spicultic Chert Sponge Spicules 11-34 (Ruppel and Barnaby, 2001) Type logs of WRE (Well "W") Gamma Porosity Thick-bedded Chert Density Vp Impedance Thirtyone Laminated Chert Laminated chert BEG-A Carbonates BEG-B Laminated Chert BEG-C Thick-bedded Chert Laminated Chert 11-35 (Fu et al., 2006) Porosity vs. Lithology 0.300 0.250 0.200 Laminated Chert 0.150 & Limestone 0.100 0.050 0.000 Thick-bedded Chert Samples 11-36 (Fu et al., 2006) Grain Density vs. Lithology 2.75 2.70 2.65 2.60 2.55 2.50 ) c / g ( y t s e D n i a r G Limestone Cherts 2.45 Lithofacies Samples 11-37 (Fu et al., 2006) P-wave Impedance vs. Lithology 20.00 18.00 P-wave Impedance (km*g/s*cc) 16.00 12.00 10.00 8.00 6.00 4.00 CB-S CB-S CB-S CB-P CB-S CB-S CB-BB CB-S C-B Thick-bedded Chert Laminated Chert CL-D LM Sample Dry sample Water satuated 11-38 (Fu et al., 2006) Limestone 14.00 P-wave Impedance vs. Porosity (Chert) Chert Impedance versus Porosity 17.00 15.00 Impedance (g*km/cc*s) 13.00 11.00 9.00 7.00 5.00 0% 10% y = -28.86x + 14.94 2 R = 0.9838 y = -27.851x + 15.088 2 R = 0.9826 y = -26.226x + 15.327 2 R = 0.9782 20% 30% Zpd Zpw Zpo Porosity 11-39 (Fu et al., 2006) Impedance vs. Porosity Log data & Core measurements 18 17 16 15 14 13 12 11 10 9 8 Impedance (km/s*g/cc) Log data Core data phi>0.1 phi<=0.1 0 0.05 0.1 0.15 Porosity (fraction) 0.2 0.25 11-40 (Fu et al., 2006) Typical Synthetic Seismogram (745er) Marker Gamma Sonic T D RC Wavelet Syn. Seismic Thirtyone BEG-C Frame 11-41 (Fu et al., 2006) Crossplot Gamma Neutron logs to discriminate lithologies Depth (ft) Shale Thick-bedded Chert Neutron Porisity (v/v) Gamma Ray (gAPI) 11-42 (Fu et al., 2006) Conventional Seismic Data X Y Z 300 m Thirtyone BEG-C Frame 11-43 (Fu et al., 2006) Initial Model for Impedance Inversion Impedance (km/s*g/cm3) 17 16 1100 W X Y 300 m 15 14 13 Time (ms) 1200 11 Thirtyone 10 9 8 1300 BEG-C High vertical resolution Low lateral resolution 11-44 Smooth to < 8Hz (Fu et al., 2006) Impedance Inversion Results W X Y 300 m Impedance (km/sec*g/cc) 17 16 1100 15 14 13 Time (ms) 1200 11 Thirtyone 10 9 1300 BEG-C Lower vertical resolution Higher lateral resolution 11-45 8 (Fu et al., 2006) Impedance Inversion Results X Y Z 300 m Impedance (km*g/s*cc) 17.8 16.6 15.4 Time (ms) 14.3 13.1 11.9 Thirtyone BEG-C Thirtyone 10.7 9.5 8.4 Frame 11-46 (Fu et al., 2006) Impedance Inversion Results A 300 m A' Impedance (km*g/s*cc) 17 16 15 14 13 Thirtyone 11 BEG-C 10 9 8 Extract RMS Average Impedance in a 10ms window below BEG-C 11-47 Frame (Fu et al., 2006) RMS Average Impedance - 10ms window below BEG-C High AI Low Artifact of faults Low AI High 11-48 (Fu et al., 2006) Most Negative Curvature 3 km Horizon slice along BEG-C 11-49 Time Slice t = 1222ms (Fu et al., 2006) Production vs. Curvature Time Slice Large fractures, poor wells! 11-50 (Fu et al., 2006) Acoustic impedance calibration P-impedance (g/cc*ft/s) 11-51 Gamma ray (gAPI) (Latimer et al., 2000) Detection of geobodies AI thresholds determined from calibration step 11-52 (Latimer et al., 2000) Detection of geobodies Geobody thresholds determined by volume of connectivity 11-53 (Latimer et al., 2000) Geostatistical inversion Seismic inversion combined with geostatistical data analysis and modeling is referred to as geostatistical inversion. Geostatistical analysis generates spatial and vertical statistics: (1) Vertical variograms are generated from well-bore measurements Horizontal variograms are generated from acoustic impedance values taken from the starting impedance model. Such a model could be the impedance model generated by recursive inversion. Steps (2) 1. 2. 3. 11-54 4. Extraction of wavelet by comparing well log reflectivities with seismic traces at well locations. Calculation of synthetic well traces by convolving reflectivities with wavelets. Tying of wells to surrounding seismic traces-correlation between synthetic and real traces are optimized (tolerance shift of 1 or 2 traces horizontally and 1 or 2 samples in time). Bring in statistical analysis in step 3. Classical Inversion (calculate expected value of AI) Geostatistical Conditional Simulation Geostatistical Inversion 11-55 (Dubrule, 2003) 11-56 (Dubrule, 2003) Start Geostatistical Inversion Pick an unoccupied grid point at random Use variograms from well data and all previously defined grid points to estimate mean and standard deviation Choose a random AI value that fits mean and standard deviation at current grid point. (local realization) Stop no yes More global realizations? Does seismic match measured data? yes Output AI global realization More grid points? Model seismic no no yes 11-57 Geostatistical Inversion 11-58 (Dubrule, 2003) Acoustic Impedance Seismic Seismic match improves as the number of local realizations increases! 11-59 (Dubrule, 2003) Extending the seismic spectrum using geostatistical inversion Background model Amplitude Seismic amplitude Geostatistical variogram model 0 50 100 150 Frequency (Hz) 200 250 11-60 (Dubrule et al., 2003; Pendrel and van Riel, 2000) Results of geostatistical inversion One realization Wells are honored by GI Mean Standard deviation Lateral variogram ensures lateral correlation between traces Standard deviation is zero at the wells 11-61 (Lamy et al., 1999; Dubrule, 2003) Hierarchy of Seismic Inversion 1. Inversion of normal-incidence or post-stack seismic data - Relative acoustic impedance - Bandlimited impedance inversion - Sparse-spike impedance inversion - Model-based inversion Cell-based parameterization Stratigraphic parameterization Global optimization - Geostatistical inversion 2. Inversion of prestack seismic data - Sequential common-angle ("elastic") inversion - Simultaneous prestack inversion AVO-based Wave-equation-based 3. Hybrid inversion 11-62 Hierarchy of Seismic Inversion 1. Inversion of normal-incidence or post-stack seismic data - Relative acoustic impedance - Bandlimited impedance inversion - Sparse-spike impedance inversion - Model-based inversion Cell-based parameterization Stratigraphic parameterization Global optimization - Geostatistical inversion 2. Inversion of prestack seismic data - Sequential common-angle ("elastic") inversion - Simultaneous prestack inversion AVO-based Wave-equation-based 3. Hybrid inversion 11-63 Sensitivity of finite-offset data to shear waves Boundary conditions: Continuous displacement, u Continuous stresses, n ^ 11-64 Offset (m) Common angle stacks Time (s) 11-65 (Barklay et al., 2008) Elastic Impedance Aki-Richards equation V 1 p 1 R() = + 2 V p 2 Vp VV sV s 2 s + -4 2 -2 2 V 2 p V Vp p sV 2 Vp 1 2 - sin 2 sin + tan 2 V p We define a function which has properties similar to acoustic impedance so that reflectivity can be derived from it. So 11-66 Normalized Elastic Impedance One shortcoming of the above EI function is that it dimensionality varies with , and so the values of EI varies appreciably with angle. Because of this, displaying of AI as well as EI logs becomes inconvenient. Whitcombe (2002) introduced constants (Vp0, Vs0 and 0) in the equation as shown below, where they represent averages of the Vp, Vs and density logs. The average value for the EI logs for the 204/24a-2 well, west of Shetland, plotted as a function of incidence angle. The shape of this curve depends on the units in which the data have been specified. In this example, velocities were measured in meters per second and density was measured in grams per cubic centimeter. 11-67 Normalized Elastic Impedance Whitcombe (2002) further shows that if the function is scaled by a factor Vp00, the dimensionality of EI and AI becomes the same. AI and EI (30 degrees) traces for the 204/24a-2 well, west of Shetland. (a) The Vpo, Vso, o values have been used appropriate to the start of the log. The AI and EI logs intersect at this point. (b) The Vpo, Vso, o values have been used appropriate to a time of 2150 ms, appropriate to the shales between the pay zones at approximately 2130 and 2180 ms. 11-68 Whitcombe (2002) Elastic Impedance EI(30) log curve scaled so that the shale baseline is approximately the same as AI curve shows behavior consistent with class III sands (Tertiary reservoirs, west of Shetlands) 11-69 Segment of EI(30) log curve overlain on the inverted 30 degree angle stack. (Connolly, 1999) Elastic impedance Gaussian curves showing distribution of AI and EI(30) values for three lithologies. Frequency (%) 10000 15000 20000 25000 30000 Frequency (%) 10000 15000 20000 25000 30000 Acoustic Impedance (ft/s-g/cm3) Normalized Elastic Impedance at 300 (ft/s-g/cm3) Normalized standard deviation AI Shale Brine sand 11-70 Oil sand 0.15 0.12 0.11 EI (30) 0.10 0.08 0.09 (Connolly, 1999) Elastic impedance 7.0*106 Elastic Impedance at 300 Acoustic Impedance 6.5*106 6.0*106 5.5*106 5.0*106 0.4 1.4*10 6 1.3*10 6 1.2*10 6 1.1*10 6 1.0*10 6 0. 4 0. 5 0. 0. 6 7 Oil Saturation 0. 8 0. 9 0.5 0.6 0.7 Oil Saturation 0.8 0.9 Relationship between oil saturation and AI (left) and EI(30) (right) from core sample measurements from the Foinaven Field (west of Shetlands). Evidently, the far offsets are more sensitive to changing saturation than the near ones. 11-71 (Connolly, 1999) Angle Impedance Concept of angle impedance is similar to elastic impedance, but reportedly angle impedance helps in taking the AVO inversion results to physical properties in the form of acoustic. Impedance, and Poisson's ratio and predict about lithology and fluids. Angle impedance is defined as Where Zp= acoustic impedance Zs= shear impedance = angle of incidence C = constantZp exp ([lnZp - 2lnZs + C] sin2) Z = 11-72 Resnick (1993) Angle Impedance In the example discussed here the goal was to separate sand from shale and determining fluids in the reservoir. Data from only one well are available. The reservoir is at approx. 2 sec. Steps: 1. Seismic data are processed as angle stacks where average angles of incidence for the three stacks are 10, 20 and 30 degrees. A separate wavelet is estimated for each stack. The three angle stacks are inverted for angle impedance and these are found to be better than seismic data as the wavelet is removed and noise is damped using horizontal continuity. Acoustic impedance, shear impedance and Poisson's ratio were calculated from these impedance volumes. 2. 3. 4. 11-73 Input Seismic Data (10 degrees angle stack) Well A Gamma ray log 11-74 (Espersen et al., 2000) Input Seismic Data (20 degrees angle stack) Well A Gamma ray log 11-75 (Espersen et al., 2000) Input Seismic Data (30 degrees angle stack) Well A Gamma ray log 11-76 (Espersen et al., 2000) Well Log Data Dipole sonic P-wave sonic Density Gamma ray A 0 2 4 2 3 11-77 (Espersen et al., 2000) 150 Well Inversion Results Z = Z P exp[( ln Z P - 2 ln Z S + C ) sin 2 ] (10 degrees angle stack) A 11-78 (Espersen et al., 2000) Well Inversion Results Z = Z P exp[ ( ln Z P - 2 ln Z S + C ) sin 2 ] (20 degrees angle stack) A 11-79 (Espersen et al., 2000) Well Inversion Results Z = Z P exp[ ( ln Z P - 2 ln Z S + C ) sin 2 ] (30 degrees angle stack) A A 11-80 (Espersen et al., 2000) Derivation of rock properties Impedance variation with angle log(v) slop / e~ Intercept AI ~(vP)/ vP 0.0 10 20 0.2 30 0.4 40 ( vP ) 2 R ( ) = cos + sin 2 vP 0.6 50 60 0.8 70 1.0 80 sin2 (deg) 11-81 (Espersen et al., 2000) Acoustic impedance Well Log data Inversion result 2.0 9.0 2.0 11-82 4.3 6.7 (106 kg/m2-s) 9.0 (Espersen et al., 2000) Poisson's ratio Well Log data Inversion result 0.00 0.50 0.00 0.16 0.33 0.50 11-83 (Espersen et al., 2000) Well log data crossplots to estimate lithology Acoustic impedance Gamma ray color bar Poisson's ratio 11-84 (Espersen et al., 2000) Sand Acoustic impedance Resisitivity color bar Shale Poisson's ratio 11-85 (Espersen et al., 2000) Well log data crossplots to estimate lithology Water sand Oil sand Acoustic impedance Gas sand Shale Probability density functions Poisson's ratio 11-86 (Espersen et al., 2000) Time slice 16 ms below top reservoir reservoir Time slice through the Lithology Cube shows the probabilitiy for lithology and fluids Probability 11-87 Oil sand Gas sand Shale Water sand (Espersen et al., 2000) Vertical slice through the Lithology Cube. Probability 11-88 Oil sand Gas sand Shale Water sand (Espersen et al., 2000) Inversion for Acoustic and Elastic Impedance In Summary: Modern acoustic impedance inversion algorithms integrate seismic data, well control, estimations of the source wavelet, and interpreter constraints on structure and lithology in a mathematically rigorous manner Acoustic impedance `deconvolves' the seismic data to remove the source wavelet shape For the above reasons, acoustic impedance has higher resolution than conventional seismic data Of all the attributes, acoustic impedance is one of the most closely tied to rock porosity Elastic impedance and extensions of classical AVO analysis, promise to provide high resolution estimates of lithology and fluid product Geostatistical inversion provides realistic high resolution images that fit both the well statistics, depositional model, and seismic data. 11-89 ...
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This note was uploaded on 02/04/2012 for the course GPHY 3423 taught by Professor Marfurt during the Fall '11 term at The University of Oklahoma.

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