Builder TutorialGeostatistical and Scripting ToolsGilles BourgaultCalgary September 2006

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

Pre-RequisitesFamiliar 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

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.

Probabilities for Deterministic ObjectsSpatial Random FunctionsA spatial random function Z(u,) can be defined as a collectionof regionalized random variables Z(u) over a spatial domain D. Z(u) denotes a random variable at the location uDSpace Coordinateuu:Mineral DepositOil ReservoirValues at each location u are described by a random variable. The set of all values in D represents 1 realization of the randomfunction.Z(u)Z(u,W) is the set of all possible realizations (outcomes) of the random functionZGoldPorosityPermeabilityProbability SpacePhysical SpaceAttributesZProbability spacePhysical domainD

DDDRealization of Random FunctionsProbability space W is filled with multiple realizationsz1(u)z2(u)z3(u)ProbabilityDensityFunction atlocation uis definedover multiplerealizations wz(u) denotes an outcome value or realization at the location uZ(u)Z(u)Z(u)All realizations are equivalentThe actual reservoir is one such realization

UnivariateBivariateTrivariate N-variateN = NX*NY*NZ Random Function on a Grid => N-variate DistributionRandom FunctionsDDDDDiscretization of the Spatial Domain D

Probabilities for Deterministic ObjectsExample of 1 realization of a Random Function in 1DDuz1(u)ubz1(ub) 1D line is discretized in 50 locations => 50-variate Random Function

Probabilities for Deterministic ObjectsExample of multiple realizations of a Random Function in 1D100 realizationsuD(for short)mn=100zw(u)Distribution of 100 outcomes at location uaLocal pdf at location ua R.V. at ua is characterized by a mean m and a standard deviation sR.F.

Probabilities for Deterministic ObjectsStationary Random Functionm and do not depend on the location u(100 realizations)All random variables have same expected value and same varianceStationary distribution (pdf)n=100local pdf is everywhere the same=> No TrendStationarity => region with homogeneous statistics

Probabilities for Deterministic ObjectsStationary Random FunctionStationary LogNormal distribution (pdf)m and do not depend on the location ulocal pdf is everywhere the same=> No TrendStationarity => region with homogeneous statistics

Probabilities for Deterministic ObjectsCharacterizing the Random Function with Correlations(100 realizations)closer random variableshave higher correlationThis is often observedin natureh=49h=1lag distance h=49lag distance h=1n=100n=100Coefficient of correlationPairinglocations

Probabilities for Deterministic ObjectsCharacterizing the Random Function with Moments of Inertia(100 realizations)closer random variableshave a smaller momentof inertia in their x-plotdh=49h=1lag distance h=49lag distance h=1n=100n=100Moment of inertia about first bisector

h=lag distance between any two random variablesCharacterizing Random FunctionsCorrelogram = correlation as a function of lag distance(100 realizations)Variogram = moment of inertia as a function of lag distanceCorrelation lengthCorrelation vanisheswhen moment ofinertia reach themaximum valueMaximum correlation at h=0 corresponds to the variance of Z(u)not to scale

Probabilities for Deterministic ObjectsExample of a realization of Random Function in 1DDuubLag sizez(u)z(ub)Ergodicity: Each realization reproduces the variogram and correlogram if observed over a Domain of infinite dimension (large enough)outcomes (actual values)random variablesoutcomes (actual values)random variables

100 realizations100 variogramsProbabilities for Deterministic ObjectsFluctuations in the Variograms when computed for each realizationover a finite DomainThe averaged variogramidentifies the variogramof the random function.Ergodic fluctuationsErgodic fluctuations increases with the lag distancenot to scale

Probabilities for Deterministic ObjectsFew observations (measurements) of a realization of a Random Function in 1DDThe practice of computing variogramsThe actual reservoir (or deposit) is locally known at few data locations

Stationary Histogram (pdf) = Data HistogramThe actual reservoir (1 realization) is locally known at few data locationsStationary distribution (pdf)Multiple realizationsAssuming ERGODICITY => Stationary Histogram = Data HistogramData histogram

Computing Variograms = Data Pairing20 data locations, regular sampling => N(h=1) = 19 => N(h=2) = 18 => N(h=3) = 17 => N(h=19) = 1h=1h=2h=3h=19 regular sampling => regular lag distances

Computing Variograms = Data Pairing1111001020Lag sizeLag toleranceLag1: N(h1)=7 pairs1011100100Lag2: N(h2)=5 pairs11 data locations, irregular sampling

Computing Variograms = Data PairingAttribute Zhi4 pairshihi = lag distance vectorBandwidthDirection angleAngle toleranceAngular and distance tolerancesare used to get enough pairs of values in any given direction.Lag sizeLag toleranceExample of 2D sampling geometry(3D: vertical variogram is along wells or across grid layers)2D sampling

Fitting VariogramsVariogramLag distance hg(h)Semi-Variance(direction q)ExperimentalTheoreticalEach experimental point may involve a different number of data pairs N(h).Rule of thumb >=30 pairs

Fitting Variogram AnisotropyMajor axis = direction of maximum rangeMinor axis = direction of minimum rangeFitting anisotropies => finding directions of major and minor axisAnisotropy Ellipse

Variogram MapTop View.Vertical Cross-sectionalong x-axis.Sill at 10hhhhhg(h)g(h)N Would be a variogram cube in 3D Variogram is symmetric in h

Variogram and Spatial Heterogeneity.Spherical Modelh.Nugget Modelh.Gaussian Modelh

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Isotropic-AnisotropichIsotropichDirection 30 degreesDirection 120 degreesAnisotropicAll Directions

Variogram and TrendEast-West DirectionNorth-South DirectionSill = Data VarianceIf possible, remove Trend before computing variogram.Compute variogram in a direction where Trend is not present.Use only the beginning of the variogram curve.Variogram does not reach a sill, instead it keeps increasing above the data variance level(Search radius

Reconciling Data and Random FunctionConditional probability densityfunction at location u40Stationary versus conditional distributions (pdf)z(u39) = 1Considering only realizations with zw(u39) = 1StationaryStationaryStationaryConditional

Conditional Random FunctionStationary or UnconditionalConditionalConsidering only realizations with z(u39) = 1Local pdf at u40 is a conditional pdfStationaryConditional

Stationary Random Function before Observing DataStationarity is lost when Random Function is Conditioned to DataStationary and Conditional Random Functionpdf does notdepend on thelocationpdf does dependon the location=> conditional pdf(n) Set of conditioning data, in this example n=1Away from the data, Stationarity remains

Unconditional Stationary Random FunctionData and Random FunctionConditional Stationary Random FunctionUnconditionalRandom VariablesConditionalRandom VariablesConditioned by(n) DataUncertainty is reduced in the vicinity of the data(n)

Kriging in a nutshelluSpatial location u = (x,y)Z(u) = property value at location uKriging is a linear estimatorsearching neighborhoodLinear combination of dataLinear combination of random variablesRandom Function Modeln(u) = # of data around location uEstimation Error:is also a R.V.As a R.V., the mean and variance of e(u) can be computed? None-bias condition

Kriging in a nutshelln+1 equationsn+1 unknownsMinimum VarianceLinear estimatorConstraintEstimation errorFind weights at min error variance under constraintVariance of estimation errorComputing variance of estimation errorOrdinary KrigingLagrange Multipliern equations1 equation

l1l2Error Variance at uol1l2at the minimum error varianceOrdinary Kriging with 2 Random VariablesKriging with 2 Random VariablesEstimatorError Variance

Kriging in a nutshellVariogramLag distance hg(h)Semi-VarianceuKriging is a linear estimatorsearching neighborhoodKriged MapKriging=>Minimizing Error Variance Solve a linear system of equationswritten in terms of weights and variograms?

Kriging in a nutshellKriged Map

Kriging and AnisotropyKriging Omni-directionalInverse distanceKriging Anisotropy 90 DegreeKriging Anisotropy 0 DegreeAzimuth 90Azimuth 0

KrigingInverse distanceKriging Anisotropy 90 DegreeKriging Anisotropy 0 DegreeAzimuth 90Azimuth 0

Kriging Variance012u0.50.5range = 10Variance = 1l1l2Screening effectl1l2012u0.3320.6683l1l2-416u0.50.5l1l2111155111Not to scale????

Kriging Variance012u0.50.5-416u0.50.51155-9111u0.50.51010range = 10Sphericalrange = 10GaussianNot to scaleLess uncertainty with a slow ramping variogram???

Estimation-Simulationu_0= Kriging variance = Error varianceSimulation => Draw a Z(u) value at randomUse the minimized error variance to characterize uncertainty around kriged valueThis uncertainty is often assumed to follow a Gaussian distribution=>Multi-Gaussian DatasetLagrange MultiplierLinear combination of Gaussian distributions is also Gaussian

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

Making a distribution Gaussian10 1.5 0.5 2 1 2.5 4 3 4.5 3.5 5G=>Standard Gaussian cdfData ZProbability- 0.5-1-1.5 -2 -2.5Normal Scores Yz1y1F=>Data cdf0.90.80.70.60.50.30.20.100.4TransformBack-TransformAssociating kriging mean and kriging variance with Gaussian distribution: =>Multi-Gaussiannity => Data histogram should be Gaussianpcumulative histogramcdf

10 1.5 0.5 2 1 2.5 4 3 4.5 3.5 5Standard Gaussian cdf: p=G(y)Probability- 0.5-1-1.5 -2 -2.5Normal Scores0.90.80.70.60.50.30.20.100.41001Probability values can be generated with a random number generatorpSimulation of Gaussian ValuesMonte Carlo simulation calibrated by Krigingm and s from krigingSimulation at 1 locationRescaling according to kriging

Sequential Gaussian SimulationUnivariate Gaussian0112Bivariate GaussianPr{Z1}Z1Z2Z1Pr{Z1}Pr{Z2|Z1}Pr{Z1,Z2} = Pr{Z1}*Pr{Z2|Z1}Multi-Gaussian Distribution as Multiplications of Univariate Gaussian DistributionsZ1 = zZ1 = zZ2 = zz and z are correlatedvalues

Sequential Gaussian SimulationTrivariate GaussianPr{Z1}Z1Z2Z3Pr{Z2|Z1}Pr{Z3 |Z2,Z1}Pr{Z1,Z2,Z3} = Pr{Z1}*Pr{Z2|Z1}*Pr{Z3|Z2,Z1}1Z1 = zZ3 = zZ2 = zMulti-Gaussian Distribution as Multiplications of Univariate Gaussian DistributionsPr{Z2,Z3|Z1}z, z, and z are correlated values

012u1 datum, 2 blank grid nodesz0??Sequential Gaussian Simulation

Estimation-SimulationDataKriged curveMultiple Geostatistical RealizationsKriged curve = Average of all simulated curvesKriged standard deviation

Geostatistical SimulationKriging EstimationKriging SimulationDataKriged map = Average of all simulated maps

Geostatistical SimulationDataestimationSimulationsVariogram Reproduction

Geostatistical SimulationKrigingDataSimulationHistogram Reproduction

Geostatistical SimulationKriging EstimationKriging SimulationOne answerMultiple answersFlow SimulatorFlow SimulatorUncertainty Analysis

Co-Kriging in a nutshelluPrimary-HarduThis weight depends oncorrelation hard-softThese weights depend on variogramuSecondary-Soft??

Co-KrigingKrigingSoftColocated-Kriging cc=0.7Colocated-Kriging cc=0.9

Tutorial Main Steps: 1. Building GridLoad Well TrajectoriesLoad Top DataInterpolating Top with KrigingImporting Thickness DataInterpolating Thickness with KrigingCombining Top and ThicknessDefine the Grid Layers

Tutorial Main Steps: 2. Populating GridImport Well Logs for PorositySimulate Porosity Data with GeostatisticsImport and Calibrate Well Test DataSimulate Permeability with Geostatistics

Tutorial Final Steps: Automatic WorkflowCreate a Script that will generate multiple versions (cases) and save the datasetsRun and Save the Script

Getting StartedStart LauncherSet the data directoryClick and Drag the data file tutorial_start.dat on the Builder Icon

Open WORD document GeostatisticsTutorial.doc and follow the instructionsPlease, ask questions whenever needed.