Builder Tutorial Geostatistical and Scripting Tools

1. Builder TutorialGeostatistical and Scripting Tools Gilles Bourgault Calgary September 2006

2. Aims and Goals • Build maps and a simulation grid. • Populate the simulation grid with spatial properties. • Familiarize yourself with Geostatistical and Scripting tools available in Builder.

3. Pre-Requisites • Familiar with Builder. • Concepts of spatial interpolation and mapping. • Concepts of geostatistics (variogram, kriging). • Concepts of conditional simulation. Suggested Geostatistical Text Books: An Introduction to Applied Geostatistics, Isaaks and Srivastava, Oxford Applied Geostatistics for Reservoir Characterization, Kelkar and Perez, SPE Geostatistics and Petroleum Geology, Hohn, Kluwer Geostatistical Reservoir Modeling, Deutsch, Oxford

4. Deterministic or Probabilistic? • Oil reservoir (Mineral deposit) are physical systems that are perfectly deterministic. • Geological systems are created by many processes at different scales. • We have few measurements. We still lack a great deal of information to fully determine those systems. • We cannot write the deterministic equations that will calculate the property value we observe at any given location in the system (e.g. porosity in the reservoir). We do not know the behavior of the system in terms of mathematical equations. Based on sampling, we can still describe its behavior in terms of statistics and probabilities.

5. Probabilities for Deterministic Objects Physical domain Probability space Z Spatial Random Functions D A spatial random functionZ(u,Ω) can be defined as a collection of regionalized randomvariables Z(u) over a spatial domain D. Space Coordinate u: Z(u) denotes a random variable at the location u Ω is the set of all possible realizations (outcomes) of the random function Values at each location u are described by a random variable. The set of all values in D represents 1 realization of the random function. Probability Space D Z Attributes Physical Space Z(uα) Z(u,W) Gold Porosity Permeability Mineral Deposit Oil Reservoir uα

6. uα Realization of Random Functions zω(u) denotes an outcome value or realization ω at the location u Probability space W is filled with multiple realizations D Probability Density Function at location uα Z(uα) z1(uα) D z2(uα) Z(uα) D z3(uα) Z(uα) is defined over multiple realizations w All realizations are equivalent The actual reservoir is one such realization

7. Random Functions Discretization of the Spatial Domain D Trivariate Bivariate Univariate D D D N-variate D N = NX*NY*NZ Random Function on a Grid => N-variate Distribution

8. Probabilities for Deterministic Objects Example of 1 realization of a Random Function in 1D z1(uα) D z1(ub) ub uα 1D line is discretized in 50 locations => 50-variate Random Function

9. uα Probabilities for Deterministic Objects Example of multiple realizations of a Random Function in 1D R.F. D 100 realizations (for short) uα n=100 Local pdf at location ua σ zw(uα) m Distribution of 100 outcomes at location ua R.V. at ua is characterized by a mean m and a standard deviation s

10. Probabilities for Deterministic Objects Stationary Random Function All random variables have same expected value and same variance n=100 (100 realizations) Stationary distribution (pdf) m and σ do not depend on the location u => No Trend local pdf is everywhere the same Stationarity => region with homogeneous statistics

11. Probabilities for Deterministic Objects Stationary Random Function Stationary LogNormal distribution (pdf) m and σ do not depend on the location u => No Trend local pdf is everywhere the same Stationarity => region with homogeneous statistics

12. Probabilities for Deterministic Objects Characterizing the Random Function with Correlations (100 realizations) h=49 h=1 closer random variables have higher correlation This is often observed in nature Coefficient of correlation Pairing locations n=100 n=100 lag distance h=49 lag distance h=1

13. Probabilities for Deterministic Objects Characterizing the Random Function with Moments of Inertia (100 realizations) h=49 h=1 closer random variables have a smaller moment of inertia in their x-plot Moment of inertia about first bisector dαβ n=100 n=100 lag distance h=49 lag distance h=1

14. Characterizing Random Functions (100 realizations) Correlogram = correlation as a function of lag distance Variogram = moment of inertia as a function of lag distance Correlation length Correlation vanishes when moment of inertia reach the maximum value h=lag distance between any two random variables not to scale Maximum correlation at h=0 corresponds to the variance of Z(u)

15. Probabilities for Deterministic Objects Ergodicity: Each realization reproduces the variogram and correlogram if observed over a Domain of infinite dimension (large enough) outcomes (actual values) random variables Example of a realization of Random Function in 1D Lag size z(ub) D z(uα) ub uα random variables outcomes (actual values)

16. Probabilities for Deterministic Objects Fluctuations in the Variograms when computed for each realization over a finite Domain 100 realizations 100 variograms Ergodic fluctuations The averaged variogram identifies the variogram of the random function. not to scale Ergodic fluctuations increases with the lag distance

17. Probabilities for Deterministic Objects Few observations (measurements) of a realization of a Random Function in 1D D The practice of computing variograms The actual reservoir (or deposit) is locally known at few data locations

18. Stationary Histogram (pdf) = Data Histogram Stationary distribution (pdf) Multiple realizations The actual reservoir (1 realization) is locally known at few data locations Assuming ERGODICITY => Stationary Histogram = Data Histogram Data histogram

19. Computing Variograms = Data Pairing 20 data locations, regular sampling => N(h=1) = 19 h=1 => N(h=2) = 18 h=2 => N(h=3) = 17 h=3 => N(h=19) = 1 h=19 regular sampling => regular lag distances

20. Computing Variograms = Data Pairing 11 data locations, irregular sampling Lag tolerance Lag size Lag1: N(h1)=7 pairs 0 0 1 1 0 1 1 0 1 2 0 1 Lag2: N(h2)=5 pairs 0 0 0 1 1 1 0 1

21. Computing Variograms = Data Pairing 2D sampling Example of 2D sampling geometry Angular and distance tolerances are used to get enough pairs of values in any given direction. Attribute Z Bandwidth 4 pairs Lag tolerance hi Lag size Direction angle hi hi = lag distance vector Angle tolerance (3D: vertical variogram is along wells or across grid layers)

22. Fitting Variograms Experimental g Variogram (h) Each experimental point may involve a different number of data pairs N(h). Rule of thumb >=30 pairs Semi-Variance Lag distance h (direction q) Theoretical

23. Fitting Variogram Anisotropy Fitting anisotropies => finding directions of major and minor axis Major axis = direction of maximum range Anisotropy Ellipse Minor axis = direction of minimum range

24. Variogram Map g(h) N h h h h Sill at 10 Top View. g(h) Vertical Cross-section along x-axis. Variogram is symmetric in h Would be a variogram cube in 3D h

25. Variogram and Spatial Heterogeneity N(0,1) N(0,1) N(0,1) g g g (h) (h) (h) . Nugget Model h . Spherical Model h Gaussian Model . h

26. Isotropic-Anisotropic N(0,1) N(0,1) g g (h) (h) Direction 120 degrees Direction 30 degrees All Directions h h Isotropic Anisotropic

27. Variogram and Trend Variogram does not reach a sill, instead it keeps increasing above the data variance level East-West Direction Northing Sill = Data Variance North-South Direction Easting If possible, remove Trend before computing variogram. Compute variogram in a direction where Trend is not present. Use only the beginning of the variogram curve. (Search radius <=5)

28. Reconciling Data and Random Function Stationary versus conditional distributions (pdf) Stationary Conditional probability density function at location u40 Conditional Stationary Considering only realizations with zw(u39) = 1 Stationary z(u39) = 1

29. Conditional Random Function Considering only realizations with z(u39) = 1 Conditional Stationary Conditional Stationary or Unconditional Local pdf at u40 is a conditional pdf

30. Stationary and Conditional Random Function Stationary Random Function before Observing Data pdf does not depend on the location Stationarity is lost when Random Function is Conditioned to Data pdf does depend on the location => conditional pdf Away from the data, Stationarity remains (n) Set of conditioning data, in this example n=1

31. Data and Random Function Unconditional Stationary Random Function Unconditional Random Variables Conditional Stationary Random Function (n) Conditional Random Variables Conditioned by (n) Data Uncertainty is reduced in the vicinity of the data

32. Kriging in a nutshell Kriging is a linear estimator searching neighborhood Linear combination of data ? u Random Function Model Spatial location u = (x,y) Linear combination of random variables Z(u) = property value at location u n(u) = # of data around location u Estimation Error: is also a R.V. As a R.V., the mean and variance of e(u) can be computed None-bias condition

33. Kriging in a nutshell Computing variance of estimation error Constraint Linear estimator Variance of estimation error Estimation error Find weights at min error variance under constraint Lagrange Multiplier n+1 equations n+1 unknowns n equations 1 equation Ordinary Kriging Minimum Variance

34. Kriging with 2 Random Variables Error Variance Estimator at the minimum error variance Error Variance at uo l2 l1 l1 l2 Ordinary Kriging with 2 Random Variables

35. Kriging in a nutshell g Variogram searching neighborhood (h) ? Semi-Variance u Lag distance h Kriging is a linear estimator Kriging=>Minimizing Error Variance Kriged Map Solve a linear system of equations written in terms of weights and variograms

36. Kriging in a nutshell Kriged Map

37. Kriging and Anisotropy Inverse distance Kriging Omni-directional Kriging Anisotropy 0 Degree Kriging Anisotropy 90 Degree Azimuth 90 Azimuth 0

38. Kriging Inverse distance Azimuth 90 Azimuth 0 Kriging Anisotropy 90 Degree Kriging Anisotropy 0 Degree

39. 0.99 0.01 u 0 2 1 Kriging Variance Variance = 1 range = 10 l2 l1 l1 l2 0.5 0.5 ? 0.5 0.5 ? u 1 1 u 5 5 0 2 1 Not to scale -4 6 1 l2 l1 Screening effect 0.668 0.332 ? ? u 1 1 1 1 1 3 0 2 1 l2 l1

40. Kriging Variance Spherical Gaussian Not to scale range = 10 range = 10 0.5 0.5 ? u 1 1 0 2 1 0.5 0.5 ? u 5 5 -4 6 1 0.5 0.5 ? u 10 10 -9 11 1 Less uncertainty with a slow ramping variogram

41. Estimation-Simulation 69 + 47 79 + + ? 60 + 27 + 78 64 + + Use the minimized error variance to characterize uncertainty around kriged value This uncertainty is often assumed to follow a Gaussian distribution=>Multi-Gaussian Dataset Linear combination of Gaussian distributions is also Gaussian u_0 Simulation => Draw a Z(u) value at random = Kriging variance = Error variance Lagrange Multiplier

42. Multi-Gaussian Distributions Trivariate Gaussian (ellipsoid) Bivariate Gaussian (ellipse) Univariate Gaussian (bell shape) N-variate Gaussian N = NX*NY*NZ (hyper-ellipsoid in N dimensions) Gaussian Random Function on a Grid => N-variate Gaussian Distribution => Gaussian Stationary Histogram

43. Making a distribution Gaussian Associating kriging mean and kriging variance with Gaussian distribution: =>Multi-Gaussiannity => Data histogram should be Gaussian F=>Data cdf G=>Standard Gaussian cdf 1 p 0.9 0.8 cumulative histogram 0.7 Probability 0.6 cdf 0.5 0.4 0.3 0.2 0.1 0 -2.5 -2 -1.5 -1 - 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y1 z1 Normal Scores Y Data Z Transform Back-Transform

44. Simulation of Gaussian Values Simulation at 1 location Probability values can be generated with a random number generator 1 Monte Carlo simulation calibrated by Kriging Standard Gaussian cdf: p=G(y) 0 p 1 0 1 0.9 0.8 0.7 m and sfrom kriging 0.6 Probability 0.5 0.4 0.3 0.2 0.1 0 -2.5 -2 -1.5 -1 - 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Normal Scores Rescaling according to kriging

45. Sequential Gaussian Simulation Multi-Gaussian Distribution as Multiplications of Univariate Gaussian Distributions Bivariate Gaussian Univariate Gaussian Z2 1 2 Pr{Z1} Pr{Z2|Z1} Z1 0 Z1 Z2 = z Z1 = z Z1 = z 1 z and z are correlated values Pr{Z1} Pr{Z1,Z2} = Pr{Z1}*Pr{Z2|Z1}

46. Sequential Gaussian Simulation 3 Z3 Z2 Z2 2 Z1 Multi-Gaussian Distribution as Multiplications of Univariate Gaussian Distributions Pr{Z1,Z2,Z3} = Pr{Z1}*Pr{Z2|Z1}*Pr{Z3|Z2,Z1} Trivariate Gaussian Z1 = z Z2 = z Z3 = z Z1 Pr{Z1} Pr{Z2,Z3|Z1} Z2 1 Z3 Pr{Z3 |Z2,Z1} z, z, and z are correlated values Pr{Z2|Z1}

47. Sequential Gaussian Simulation 1. Do SK on u1 using datum at u0. zsim1 zsim2 2. Monte-Carlo simulation of standard Gaussian value. px N(0,1) z 3. Rescale z using 4. Do SK on u2 using data at u0 and u1. 5. Repeat steps 2, 3, and 4. 1 datum, 2 blank grid nodes z0 ? ? u 0 2 1

48. Estimation-Simulation Multiple Geostatistical Realizations Kriged curve Kriged standard deviation Data Kriged curve = Average of all simulated curves

49. Geostatistical Simulation Data Kriging Simulation Kriging Estimation Kriged map = Average of all simulated maps

50. Geostatistical Simulation Variogram Reproduction Simulations Data estimation