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Updating prior probabilities through kriging in metric space: application to history matching
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Problem statement Given: a set of unknown parameters π a set of measured data π , we wish to update the prior probability π π π π π =π( π 1 , π 2 , π 3 β¦, π π π Letβs illustrate the proposed approach with an example
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Simple example to illustrate the approach
Black-oil Undersaturated reservoir 3 producers 2 injectors Aquifer Producers Injectors Aquifer
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Simple example to illustrate the approach
Dip structure, 1 fault Sand/shale facies Spatially varying sand % East/west deposition Fault Sand Shale Depositional direction
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8 years of historical production data
Field Production GOR Beginning of water injection Qo Qw
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P3 Well data: BHP Rates GOR P2 P1 Injectors Aquifer
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Summary of uncertain parameters
Global reservoir properties ο Aquifer strength, fault trans. ο Porosity, Kh, NTG, Kv/Kh ο Swi per facies Geobody locations ο Spatial uncertainty ο Depositional angle Geological interpretation ο # of facies Fault Aquifer 2 facies 3 facies
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Field Production β 280 Runs
Created 280 runs sampled from prior distributions of uncertain parameters Qo Wcut GOR
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Well P1 β 280 Runs Qo Wcut GOR BHP
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Well P2 β 280 Runs Qo Wcut GOR BHP
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Well P3 β 280 Runs Qo Wcut GOR BHP
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None of the models match all the data
Match threshold From these runs, can we infer what parameter values give us better match?
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Can we infer what parameter values give us better history match?
Letβs construct a metric space using weighted root mean square of Qo, GOR, and BHP
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Spatial information in metric (MDS) space
MDS: 280 runs We know location of data, but we donβt know PVMULT Can we use this spatial information to estimate PVMULT at the location of data? This is a problem of spatial statistics! Runs colored by PVMULT value Historical data
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Letβs state the problem in terms of random variables and spatial statistics
π π = uncertain parameter i π’ β = location of data π π π’ β = unknown parameter values i, π’ π = location of runs, π=1β¦ π π
π§ π π’ π = known parameter values i, at location π’ π , π=1β¦ π π
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Problem statement for continuous variables
For π= continuous variable, we wish to obtain: π π π π’ β β€ π§ π β π§ π π’ π , π§ π π’ π π=1β¦ π π
, πβ π NR prior runs π’ β = location of data π’ π π π = uncertain parameter i π§ π β = threshold value
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Problem statement for discrete variables
For π= discrete variable, we wish to obtain: π π π π’ β = π§ π β π§ π π’ π , π§ π π’ π π=1β¦ π π
, πβ π NR prior runs π’ β = location of data π’ π π π = uncertain parameter i π§ π β = outcome value
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Solve probabilities using indicator kriging
π½=1 π π
π π½ Cov π’ πΌ β π’ π½ =Cov π’ β β π’ πΌ , πΌ=1β¦ π π
IK solves for probabilities directly (dropping π§ π π’ π ) π π π π’ β β€ π§ π c π§ π π’ π π=1β¦ π π
π π π π’ β = π§ π c π§ π π’ π π=1β¦ π π
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Solve probabilities using kriging
Distance Matrix Next task: calculate the experimental covariance and construct the covariance model Covariance Model Kriging system of equations
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Cov Model Experimental Prior CDF min max cutoff #1 cutoff #2
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Cov Model Experimental numFacies prior PDF
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Solve IK system and update probabilities given 280 sensitivity runs
Black = prior Red = updated numFacies PVMULT
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Not all parameters have informative spatial structure
PERMX_1
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Kriging: challenges, assumptions
Key assumption is stationarity Assumes that m, s, covariance model are spatially invariant Future challenge: relax assumption that covariance is isotropic (same in all directions)
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Subtle point: Kriging is exact interpolator, which is not consistent with data error
π’ β = location of data Data measurements have error Data point in metric space is really a data βregionβ Proposed solution: allow for data to move in metric space (Change RHS of kriging system)
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Is stationarity an issue?
Big assumption that covariance is stationary But, certain variants of kriging relax βconstraintsβ of stationarity Kriging with trend Trend could be calculated externally Ordinary kriging: calculates mean from local data Similar to distance-based weighting of local data
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Local neighborhood: 10% runs closest to history
Example: IK vs OIK PERMX_1 Kx of facies #1 Simple IK Ordinary IK Local neighborhood: 10% runs closest to history
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Local neighborhood: 10% runs closest to history
Example: IK vs OIK NTG_1 NTG of facies #1 Simple IK Ordinary IK Local neighborhood: 10% runs closest to history OIK with local search neighborhood can give significantly different results
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Schematic diagram of method
(Indicator) Variogram Metric Space 1 2 3 d11 d12 d13 d21 d22 d23 d31 d32 d33 1 2 3 C(d11) C(d12) C(d13) C(d21) C(d22) C(d23) C(d31) C(d32) C(d33) Covariance CDF Indicator Kriging Eqns. P(x<x1) X1 X2 X
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Schematic diagram of method
(Indicator) Variogram Metric Space 1 2 3 d11 d12 d13 d21 d22 d23 d31 d32 d33 1 2 3 C(d11) C(d12) C(d13) C(d21) C(d22) C(d23) C(d31) C(d32) C(d33) Covariance 1 CDF Indicator Kriging Eqns. P(x<x2) X1 X2 X
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Schematic diagram of method : discrete parameter
(Indicator) Variogram Metric Space 1 2 3 d11 d12 d13 d21 d22 d23 d31 d32 d33 1 2 3 C(d11) C(d12) C(d13) C(d21) C(d22) C(d23) C(d31) C(d32) C(d33) Covariance Indicator Kriging Eqns. P(TI) TI1 TI10 TI13 PDF
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Uninformative on PVMULT
What if we separate out GOR, Qo, BHP for all wells into different metric spaces? Uninformative on PVMULT P2 Qo IK Informative on PVMULT P2 GOR IK Given Nd data types and Nw wells, one can have up to Nd X Nw separate metric spaces
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Multiple distances improve understanding of impact of different data on uncertainty
P3 Qo P1, P3 BHP CDF P1, P2, P3 GOR PVMULT
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Resampling: add new runs to metric space and repeat workflow
(Indicator) Variogram Metric Space 1 2 3 d11 d12 d13 d21 d22 d23 d31 d32 d33 1 2 3 C(d11) C(d12) C(d13) C(d21) C(d22) C(d23) C(d31) C(d32) C(d33) Covariance 1 CDF Indicator Kriging Eqns. P(x<x2) X1 X2 X Metric Space Optimization (MSO)
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Results from resampling
Resampling acts similar to optimization method OF 14 iterations HM models Run
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Comparison of MSO with standard HM optimization procedure
DiffEvol PORO_1 52 HM models 44 HM models Results from 430 runs each of MSO and DiffEvol
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Comparison of MSO with standard HM optimization procedure
DiffEvol PORO_2 52 HM models 44 HM models
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Comparison of MSO with standard HM optimization procedure
DiffEvol depoAngle Wider range of parameter values for MSO compared to differential evolution
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π π π β
π π π’ β π π’ π , π’ β = π’ β π π=1β¦ π π
Conclusion IK is approximation of joint probability: π π π β
π π π’ β π π’ π , π’ β = π’ β π π=1β¦ π π
Requires stationarity assumption and calculation of covariance Advantages: No need for dimension in MDS to be small (like KS) Can take advantage of existing geostat knowledge Applications: Iterative resampling (MSO) Visualization of impact of data on uncertainty
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Accounting for secondary variable π§ π
π π π π’ β = π§ π β π§ π π’ π , π§ 1 π’ π ,β¦, π§ πβ1 π’ π , π§ 1 π’ β ,β¦, π§ πβ1 π’ β Sensitivity runs Previously sampled values at u* Simplified approach: Calculate sensitivity of interactions for π§ π | π§ 1 ,β¦, π§ πβ1 If sensitive: Use only runs with values in the sampled cutoffs/categories of π§ 1 π’ β ,β¦, π§ πβ1 π’ β Recalculate covariance model for π π
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P2 Qo P2 GOR
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