1. Stochastic inverse modeling under realistic prior model constraints with multiple-point geostatistics Jef Caers Petroleum Engineering Department Stanford Center for Reservoir Forecasting Stanford, California, USA

2. Acknowledgements I would like to acknowledge the contributions of the SCRF team, in particular Andre Journel and all graduate students who contributed to this presentation

3. Quote “Theory should be as simple as possible, but not simpler as possible….” Albert EINSTEIN

4. Overview Multiple-point geostatistics Why do we need it ? How does it work ? How do we define prior models with it ? Data integration Integration of multiple types/scales of data Improvement on traditional Bayesian methods Solving general inverse problems Using prior models from mp geostatistics Application to history matching

5. Part I Multiple-point Geostatistics

6. Limitations of traditional geostatistics Variograms EW 0.4 0.8 1.2 102030400 0.4 0.8 1.2 102030400 3 1 2 Variograms NS 2-point correlation is not enough to characterize connectivity A prior geological interpretation is required and it is NOT multi-Gaussian 123

7. Stochastic sequential simulation Define a multi-variate (Gaussian) distribution over the random function Z(u) Decompose the distribution as follows Or in its conditional form

8. Practice of sequential simulation A B1 B2 B3 B={B1,B2,B3} P(A|B) = N(m,  ) m,  given by kriging, depend on autocorrelation (variogram) function

9. Multiple-point Geostatistics Reservoir = well data multiple-point data event P ( A | B ) ? Sequential simulation A B

10. Extended Normal Equations u uu

11. Single Normal Equation

12. The training image module Training image module = standardized analog model quantifying geo-patterns P ( A | B ) = 1 / 4 A = Mud SNESIM algorithm Recognizing P(A|B) for all possible A,B

13. The SNESIM algorithm Training image Data template (data search neighborhood) Search tree Construction requires scanning training image one single time Minimizes memory demand Allows retrieving all training cpdf’s for the template adopted!

14. Probabilities from a Search Tree u Search neighborhood Search tree Training image 5 4 3 2 j=1 i = 1 2 3 4 5 u u u u u u u u u u u u u u Level 0 (no CD) Level 1 (1 CD) Level 4 (4 CD)............ 1 2 3 4 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 u u 1 1 2 2 3 3 u 1 2 3 4

15. Example 400 sample data Realization True image Training image CPU 2 facies, 1 million cells = 4’ 30” On 1GHz PC

16. Where do we get a 3D TI ? Training image requires "stationarity" Only patterns = "repeated multipoint statistics" can be reproduced Valid training imageNot Valid

17. Modular training image Modular ? * no units * rotation-invariant * affinity-invariant Training ImageModels generated with snesim using the SAME training image

18. Properties of training image Required Stationarity: patterns by definition repeat Ergodicity: to reproduce long range feature => large image Limited to 4-5 categories Not required Univariate statistics need not be the same as actual field No conditioning to ANY data Affinity/rotation need not be the same

19. Part II Multiple-point Geostatistics and data integration

20. Simple question, difficult problem… A geologist believes based on geological data that there is 80% chance of having a channel at location X A geophysicist believes based on geophysical data that there is 75% chance of having a channel at location X A petroleum engineer believes based on engineering data that there is 85% chance of having a channel at location X What is the probability of having a channel at X ? The essential data integration problem… P(A|B) P(A|C) P(A|D) P(A|B,C,D)?

21. Combining sources of information

22. Conditional independence O = In practice not necessarily YES

23. Correcting conditional independence

24. Permanence of ratios hypothesis

25. Advantages of using ratios No term P(B,C), hence McMC is not required Work with P(A|B),P(A|C), more intuitive than P(B|A),P(C|A) Verifies all consistency conditions by definition It is still a form of independence, Yet dependence can be reintroduced reintroducing dependence

26. Simple problem… P(A|B) = 0.80 P(A|C) = 0.75 => P(A|BC) Suppose P(A) = 0.5 => P(A|BC) = 0.92 Suppose P(A) = 0.3 => P(A|BC) = 0.95 = compounding of events Lesson learned : if geologist and geophysicist agree for almost 80%, you can be even more certain that there is a channel !

27. Example reservoir P(A|C) Training image P(A|B) Single realization

28. P(A|C), A = single-point ! P(A|C) Realization When combing P(A|B) from geology and P(A|C) from seismic to P(A|BC), ‘A’ is still a single point event !  Certain patterns, such as local rotation will be ignored  Honor seismic only as a single- point probability ?

29. Concept of MODULAR training image Modular ? * Stationary patterns * rotation-invariant * affinity-invariant * no units Modular Training ImageModels generated with snesim using the SAME training image

30. Local rotation angle from seismic P(A|C)Local angle

31. Results 2 realizations with anglewithout angle

32. Constrain to local “channel features” P(A|C) Hard data from seismic Soft data from seismic

33. Part III Inverse modeling with multiple-point geostatistics Application to history matching

34. Production data does not inform geological heterogeneity a a a a a a a a a a a a a a A Petroleum Engineer Geologist 1 Geologist 2Disagreeing Geologist ?

35. Approach Methodology Define a non-stationary Markov chain that moves a realization to match data, two properties At each perturbation we maintain geological realism use term P(A|B) Construct a soft data set “P(A|D)” such that we move the current realization as fast as possible to match the data => Optimization of the Markov chain at each step

36. Methodology: two facies D = set of historic production data (pressures, flows) Some notation: Initial guess realization: Realization at iteration

37. Define a Markov chain Define a transition matrix:

38. Transition matrix 2 x 2 transition matrix describes the probability of changing facies at location u and we define it as follows

39. Parameter r D

40. Determine r D Use P(A|D) as a probability model in multiple-point geostatistics  Combine P(A|B) (from training image) with P(A|D) from production data D into P(A|B,D)  Allows generating iterations that are consistent with prior geological vision  Allows combining geological information with production data  Allows determining an optimal value for r D as follows…

41. r D determines a “perturbation” r D =0.01 Some initial model r D =0.1 r D =0.2 r D =0.5r D =1 Find r D that matches best the production data Find r D that matches best the production data = one-dimensional optimization

42. Construct a Training Image with the desired geological continuity constraint Use snesim (P(A|B)) to generate an initial guess Until adequate match to production data D Define a soft data P(A|D) as function of r D Perform snesim with P(A|D) to generate a new guess Find the value of r D that matches best the data D Complete algorithm

43. Examples Generate 10 reservoir models that 1. Honor the two hard data 2. Honor fractional flow 3. Have geological continuity similar as TI I P

44. Single model

45. r D values, single 1D optimization r D value Objective function

46. Different geology

47. More wells

48. Hierarchical matching * First choose fixed permeability per facies, perturb facies model * Then, for a fixed facies perturb the permeability within facies (using traditional methods, ssc, gradual deformation

49. Example I P

50. Results Klow = 50 Khigh = 500 Klow = 50 Khigh = 500 Klow = 12 Khigh = 729 Klow = 12 Khigh = 729 Klow = 11 Khigh = 694 Klow = 150 Khigh = 750

51. Results

52. More realistic Reference Initial model matched model

53. Conclusions What can multiple-point statistics provide Large flexibility of prior models, no need for math. def. A fast, robust sampling of the prior A more realistic data integration approach than traditional Bayesian methods A generic inverse solution method that honors prior information

54. More on conditional independence Q? why should  B and  C be independent unless they are homoscedastic, i.e. independent of A ? Q? Is it not a mere transfer of independence hypothesis to  B and  C