11-Geostatistical Methods for Seismic Inversion. Amílcar Soares CERENA-IST

Transcription

1 11-Geostatistical Methods for Seismic Inversion Amílcar Soares CERENA-IST

2 01 - Introduction Seismic and Log Scale Seismic Data

3 Recap: basic concepts Acoustic Impedance Velocity X Density = AI

4 Recap: basic concepts Acoustic Impedance = Velocity X Density Incident wave Reflected wave Layer 1 impedance = Velocity(1) x Density(1) = Z1 Transmitted wave Layer 2 impedance = Velocity(2) x Density(2) = Z2 Since reflections are caused by changes in velocity and density, these two parameters are combined into a parameter called impedance. This is the product of velocity and density

5 Recap: basic concepts Reflection coefficient Incident wave Reflected wave R = Reflected wavelet amplitude Incident wavelet amplitude R = Z2 - Z1 Z2 + Z1 R = (V2 x D2) - (V1 x D1) (V2 x D2) + (V1 x D1) Transmitted wave The ratio of the incident amplitude to the reflected amplitude is called the Reflection Coefficient. Reflection coefficient can be seen a measure of the impedance contrast at the interface.

6 Recap: basic concepts Reflection coefficient Layered earth Impedance Reflection Coefficients

7 Recap: basic concepts Wavelet Marine air gun Land dynamite Time A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. C - 2

8 Recap: basic concepts Wavelet Minimum phase Time (Sec.) Zero phase Time origin

9 Recap: basic concepts Wavelet Lithology Impedance Minimum phase Zero phase Low velocity density High velocity density

10 Recap: basic concepts Wavelet Lithology Impedance Zero phase wavelets High velocity density Low velocity density High velocity density

11 02 Seismic Inversion Convolution Impedance = Velocity X Density Incident wave Reflected wave Layer 1 impedance = Velocity(1) x Density(1) = Z1 Transmitted wave Layer 2 impedance = Velocity(2) x Density(2) = Z2

12 02 Seismic Inversion Convolution Reflection coefficient Incident wave Reflected wave R = Reflected wavelet amplitude Incident wavelet amplitude R = Z2 - Z1 Z2 + Z1 R = (V2 x D2) - (V1 x D1) (V2 x D2) + (V1 x D1) Transmitted wave

13 Layered earth Convolution Impedance Reflection Coefficients

14 Principle of Seismic Inversion Convolving the reflectivity coefficients c(x) with a given wavelet w, one obtain the synthetic seismic amplitudes a*(x)= c(x) * w

15 Earth Convolution - Forward exercise

16 Convolution - Forward exercise Earth Impedance

17 Convolution - Forward exercise Earth Impedance Reflection Coefficients

18 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet

19 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition

20 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition

21 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition

22 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition

23 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition

24 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition

25 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition Recorded Trace

26 Convolution - Forward exercise Earth Impedance Reflection Coefficients Wavelet Wavelet Superposition Recorded Trace Seismic Section

27 Seismic Section Convolution - Inverse Exercise

28 Convolution - Inverse Exercise Seismic Section Recorded Trace

29 Convolution - Inverse Exercise Seismic Section Recorded Trace Wavelet

30 Convolution - Inverse Exercise Seismic Section Recorded Trace Wavelet Reflection Coefficients

31 Convolution - Inverse Exercise Seismic Section Recorded Trace Wavelet Reflection Coefficients Reflection Coefficients

32 Convolution - Inverse Exercise Seismic Section Recorded Trace Wavelet Reflection Coefficients Reflection Coefficients

33 Low Frequency Model

34 amplitude Inverse Modeling is based on the physical relation: Convolving the reflectivity coefficients c(x) with a given wavelet w, one obtain the synthetic seismic amplitudes a*(x)= c(x) * w * = ms Typical Inverse Problem: one whish to know the acoustic impedances which give rise to the known real seismic.

35 Typical Inverse Problem: one wish calculate the parameters ( high resolution grid of acoustic impedance) that give rise to the solution we know (the real seismic) Outline of the iterative method Space of the Parameters Change the set of parameters in order to make the process convergent Solution for the set of parameters Compare with the known real solution Is the match satisfactory? N In this problem there is not a unique solution. One whish to find the set of solutions that accomplish the spatial requisites of the acoustic impedance grid: spatial continuity pattern, global CDfs,...

36 Geostatistical Seismic Inversion The aim of geostatistical inversion of seismic is to produce high resolution of numerical models that have two properties: The numerical model honors a physical relationship (convolution model) with the actual data. The numerical model reflects the spatial continuity and the global distribution functions.

37 amplitude Geostatistical Seismic (Trace-by-Trace) Inversion (Bertolli et al, 1993): it is an iterative process based on the sequential simulation of trace values of acoustic impedances Choose randomly a trace to be generated. Simulation of N realizations of AI of that trace * = ms 2- Convolution with a known wavelet N Sinthetic trace realizations Optimization algorithm 3-Compare with the real seismic, choose and retain the best realization 4- return until all traces are simulated

38 GSI Global Stochastic Inversion Geostatistical Inversion With Global Perturbation Method Part I - Theory

39 GSI Global Stochastic Inversion The approach of Global Stochastic Inversion is based on two key ideas: the use of the sequential direct cosimulation as the method of transforming 3D images, in a iterative process and to follow the sequential procedure of the genetic algorithms optimization to converge the transformed images towards an objective function

40 amplitude 1 Simulation of Acoustic Impedance 2- Convolution of transformed Simulated Acoustic Impedance * ms

41 3 Comparing the synthetic amplitudes a*(x) with the real seismic a(x) obtaining local correlation coefficients cc(x) 4 From the N realizations, retain the traces with best matches and compose a best image of AI 5 Return to step one to obtain a new generation of AI images until a given objective function is reached.

42 An iterative inversion methodology is proposed based on a direct sequential simulation and co-simulation approaches: Several realizations of the entire 3D cube of acoustic impedances are simulated in a first step, instead individual traces or cells; After the convolution local areas of best fit of the different images are selected and merged into a secondary image of a direct co-simulation in the next iteration; The iterative and convergent process continues until a given match with objective function is reached. Spatial dispersion and patterns of acoustic impedances (as revealed by histograms and variograms) are reproduced at the final acoustic impedance cube. In a last step, porosity images are derived from the seismic impedances and the uncertainty derived from the seismic quality is assessed based on the quality of match between synthetic seismogram obtained by seismic inversion and real seismic.

43 The use of Direct Sequential Co-Simulation for global transformation of images. Let us consider that one wish to obtain a transformed image Z t (x), based on a set of Ni images Z 1 (x), Z 2 (x), Z Ni (x), with the same spatial dispersion statistics, e.g. variogram and global histogram: C (h), (h), F (z) Direct co-simulation of Z t (x), having Z 1 (x), Z 2 (x), Z Ni (x) as auxiliary variables, can be applied (Soares, 2001). The collocated cokriging estimator of Z t (x) becomes: Z t x Z ( x ) m ( x ) x Z ( x ) m ( ) ( x0 ) * mt ( x0 ) 0 t t i 0 i 0 i x0 Ni i 1 Colocated data of N i secondary images

44 Variable Z 1 (x) 3 realizations from variable Z 2 (x)

45 Markov-type approximation: The crossed correlograms 12 (h) are calibrated by the correlation coefficient between variables Z 1 (x) and Z 2 (x). 12* (0): * 12( h) 12(0). 1( h) * 12(0) 12( h) 12 ( h) * global (0) 12 global

46 Simulation of variable Z 2 (x) Variable Z 1 (x) =.95 =.80 =.60

47 Since the models i (h), i=1, Ni, and t (h) are the same, the following approximation is, in this case, quite appropriated: t, i h t, i 0 t the corregionalization models are totally defined with the correlation coefficients t,i (0) between Z t (x) and Z i (x). t h 0 Remarks: The affinity of the transformed image Z t (x) with the multiple images Z i (x) are determined by the correlation coefficients t,i (0). Hence, one can select the images which characteristics we wish to preserve in the transformed image Z t (x)

48 Local Screening Effect Approximation Assumption: to estimate Z t (x 0 ) the collocated value Z i (x 0 ) of a specific image Z i (x), with the highest correlation coefficient t,i (0), screens out the influence of the effect of remaining collocated values Z j (x0), j i. Hence, colocated co-kriging can be written with just one auxiliary variable : the best at location x 0 : Z t x Z ( x ) m ( x ) x Z ( x ) m ( ) ( x0 ) * mt ( x0 ) 0 t t i 0 i 0 i x0 The best colocated data at x 0.

49 ) ( ) ( ) ( ) ( ) ( ) * ( x m x Z x x m x Z x x m x Z i i Ni i i t t t t... ) ( ) ( ) ( ) ( ) ( )* ( x m x Z x x m x Z x x m x Z i i i t t t t The best colocated data at x 0: highest Correlation Coeffificient t,i (0).

50 Outline of the proposed methodology GSI Global Stochastic Inversion i- Generate a set of initial images of acoustic impedances by using direct sequential simulation. ii- Create the synthetic seismogram of amplitudes, by convolving the reflectivity, derived from acoustic impedances, with a known wavelet. iii- Evaluate the match of the synthetic seismograms, of entire 3D image, and the real seismic by computing, for example local correlation coefficients.

51 iv - Ranking the best images based on the match (e.g. the average value or a percentile of correlation coefficients for the entire image). From them, one select the best parts- the columns or the horizons with the best correlation coefficient of each image. Compose one auxiliary image with the selected best parts, for the next simulation step. v- Generate a new set of images, by direct co-simulation, and return to step ii) until a given threshold of the objective function is reached.

52 03 Algorithm Description Algorithm Description N stochastic simulations of AI based upon well data and variograms. Calculation of Coefficients of Reflection (CR) Calculation of the N Synthetic cubes: convolution of CR cubes with a wavelet. Wavelet Calculation of Correlation Coefficient (CC) between the synthetics and the seismic cubes. 3D seismic cube n iterations A new CC map (Best Correlation Map, BCM) and the corresponding AI secondary image (Best AI, BAI) are created: The highest CC of the N CC maps is allocated to each x 0 location. The corresponding AI values are used to build the BAI cube to be used as secondary data set. AI from wells N stochastic co-simulations (DSco-S) of AI based upon well data and conditioned to BCM.

53 Algorithm Description AI from wells Variograms from wells 1 DSS 2 CR & SY 3 CC Direct Sequential Simulation 4 BCM & BAI 5 DSco-S AI N Simulated cubes of AI

54 Algorithm Description 1 DSS AI N 2 CR & SY Cr( t) Ai( t 1) Ai( t) Ai( t 1) Ai( t) 3 CC CR N Coefficient of Reflection cubes 4 BCM & BAI Convolution Sy( t) Cr( t) wave( z) 5 DSco-S SY N Synthetic cubes Wavelet

55 Algorithm Description SY N 1 DSS 2 CR & SY 3 CC x, y Cov( X, Y) x y 4 BCM & BAI 5 DSco-S CC cube Real seismic cube CC N Correlation cubes

56 1 DSS AI CC Algorithm Description N & & & & & & N 2 CR & SY 3 CC 4 BCM & BAI 5 DSco-S N N BCM BAI

57 Algorithm Description AI from wells Variograms from wells 1 DSS BCM BAI 2 CR & SY 3 CC Direct Sequential co-simulation 4 BCM & BAI 5 DSco-S AI N Simulated cubes of AI

58 Algorithm Description N stochastic simulations of AI based upon well data and variograms. Calculation of Coefficients of Reflection (CR) Calculation of the N Synthetic cubes: convolution of CR cubes with a wavelet. Wavelet Calculation of Correlation Coefficient (CC) between the synthetics and the seismic cubes. 3D seismic cube n iterations A new CC map (Best Correlation Map, BCM) and the corresponding AI secondary image (Best AI, BAI) are created: The highest CC of the N CC maps is allocated to each x 0 location. The corresponding AI values are used to build the BAI cube to be used as secondary data set. AI from wells N stochastic co-simulations (DSco-S) of AI based upon well data and conditioned to BCM.

59 04 Results Seismic Data Set Data extracted from a reservoir Interpreted Horizons to quality control

60 Variograms

61 Histogram, basic statistics and Wavelet

62 Wells From 19 only 2 had Velocity log

63 04 Results Wells

64 04 Results Wells

65 04 Results Wells Histogram and Basic Statistics Acoustic Impedance

66 04 Results Results from iteration 0 - Unconditional AI from Simulation 1 AI from Simulation 15

67 04 Results Results from iteration 0 - Unconditional SY from Simulation 1 SY from Simulation 15

68 04 Results Results from iteration 0 - Unconditional CC from Simulation 1 CC from Simulation 15

69 04 Results Results from iteration 0 - Unconditional Average from Simulations Standard Deviation from Simulations

70 04 Results Results from iteration 0 - Unconditional Best Acoustic Impedance cube Best Correlation Cube

71 04 Results Correlation Results from Process Iterations

72 04 Results Results from iteration 5 AI from Simulation 3 AI from Simulation 28

73 04 Results Results from iteration 5 SY from Simulation 3 SY from Simulation 28

74 04 Results Results from iteration 5 CC from Simulation 3 CC from Simulation 28

75 04 Results Results from iteration 5 Average from Simulations Standard Deviation from Simulations

76 Good match with the horizons in the final AI cube

77 04 Results Synthetic Seismic Real Seismic

78 Practice VII- Seismic Inversion Practice with GSI (Student) Global Stochastic Inversion Practice with S-GeMS Stanford Geostatistical Modelling Software