Bhark, E.W., Structured History Matching Workflow using Parameterization and Streamline Methods

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Structured multiscale history matching workflow, parameterization, streamlines, GTTI

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Bhark, E.W., Structured History Matching Workflow using Parameterization and Streamline Methods

  1. 1. Multiscale Parameterization andHistory Matching in Structured andUnstructured Grids:Theory and Field ApplicationE. W. Bhark, A. Rey, A. Datta-Gupta and B. Jafarpour
  2. 2. Motivation• Develop structured history matching workflow• Coarse (regional) scale  Novel grid-connectivity-based parameterization • Flexible, efficient application for large models, complex geology  Calibrate multiscale heterogeneity  Avoid traditional regional multipliers• Local (grid cell) scale  Established streamline-based method • Vasco et al. (1998); Datta-Gupta and King (2007)  Refine prior preferential flow paths 2
  3. 3. Outline of presentation• Parameterization in history matching  Methods of linear transformation  Grid-connectivity-based parameterization• Structured history matching workflow• Field application  Offshore reservoir model (Rey et al. [2009], SPE124950) 3
  4. 4. Why re-parameterization? • Reduce redundant model information  Preserve important heterogeneity ~5,000 Unknowns 100 Unknowns 50 25Ex., high-resolution(3D) abs. permeability • Improves:  Solution non-uniqueness and stability, computational efficiency 4
  5. 5. Parameterization by linear transform = v1 + v2 + v3 + … + v50 + … + vN N-parameterhigh-resolution model  u= v Φ M N u  v for M << N  u1   1  u   2   v1  • Required basis properties  2     v2         Compression power: most                  energy in fewest coefficients vi          M     v M     Amenable to efficient u   N application for very large grids 5
  6. 6. Highlights of new basis  u1  1 u     2 v  2    v     Grid-connectivity-based transform basis =                        M     v M  (1) Model (or prior) independent u   N  Can benefit from prior model information (2) Applicable to any grid geometry (e.g., CPG, irregular unstructured, NNCs, faults) (3) Efficient construction for very large grids (4) Strong, generic compression performance (5) Geologic spatial continuity 6
  7. 7. Basis developmentConcept: Develop as generalization of discrete Fourier basisKEY: Perform Fourier transform of function u by (scalar) projection on eigenvectors of grid Laplacian (2nd difference matrix) • Interior rows  Second difference  Periodic operator (circulant matrix) • Exterior rows  Boundary conditions control eigenvector behavior 7
  8. 8. Basis development CPG Unstructured Grid Laplacian 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 2-point connectivity (1/2/3-D)• Decompose L to construct basis functions (rows of )  Always symmetric, sparse  Efficient (partial) decomposition by restarted Lanczos method  Orthogonal basis functions; Φu  v  u  Φ1 v  ΦT v• In general (non-periodic) case  Eigen(Lanczos)vectors  vibrational modes of the model grid  Eigenvalues represent modal frequencies 8
  9. 9. Basis functions: Examples Corner-point Grid (Brugge) • Modal shape  modal frequency • Constant basis  Zero frequency • Discontinuities honoredBasis vec. 1 Basis vec. 2 Basis vec. 3 Basis vec. 4 Basis vec. 5 Basis vec. 9 9
  10. 10. Basis functions: Examples Unstructured grid Basis function 1 Basis function 3 Basis function 5 Basis function 8 Basis function 10 Unstructured grid (local refinement) Channel structureMultiple subdomains 10
  11. 11. Structured multiscale workflow(1) START: Prior model (2) Regional update (3) Local update Prior spatial hydraulic Parameterize property model Streamline-, multiplier field sensitivity-based inversion (GTTI) Update in transform domain Multiscale iterate Gradient-based iterate Back-transform Unit-multiplier field at multiplier field to grid cell resolution spatial domain Calibrated Model FINISH Flow and transport Add higher- simulation frequency modes to basis NO Data misfit tolerance? YES Additional YES spatial detail? NO 11
  12. 12. Field application: Offshore reservoirReservoir• > 300,000 cells• Mature waterflood• 8 years of production history• 4 producers and 4 water injectors• Complex depositional sequence of turbidite sand bodies / facies• Rey et al. (2009), SPE124950Parameter• PermeabilityData• Water cut 12
  13. 13. Conceptual heterogeneity model Prior model facies (5) Prior geo-model P2 I2 P1 P3 I1Initial Kx: I3Average of measurementsat wells per facies (5) Facies ID P4 I4Next objective:Use parameterization to assistin heterogeneity identificationand updating 13
  14. 14. Workflow: Prior model & multiplier field F2 Prior geo-model Multiplier field F6 F5 F3 F1 14
  15. 15. Facies basis functions Facies 5: Prior geo-model • Multiplier field is linear combination of basis functions Multiplier field Basis functions F5 multiplier field: u= 1 3 6 8 15v1 …+ v3 …+ v6 …+ v8 …+ v15 15
  16. 16. Adaptive multiscale inversion Prior geo-model • Sequentially refine within-facies heterogeneity  From coarse to finer scales  Adaptive inclusion of basis functions Multiplier field 1 5 10 Basis functionsMultiscale inversion • End refinement when production data become insensitive to addition of basis functions 16
  17. 17. Multiscale update Number of leading basisKx: Adaptive multiscale functions per facies 10 10 10 1 5 36 17
  18. 18. Comparison with previous calibrationThis study Rey et al. (2009)Tx multiplier Facies zonation Tx multiplier Adaptive multiscale Manual zonation 18
  19. 19. Data misfit: WCTInitial and multiscale P2 P3 P4 P1 19
  20. 20. Streamline-based inversion High-resolution Prior geo-model permeability model • Refine at grid-cell scale • Streamline paths determined by Multiplier field heterogeneity, well pattern Basis functionsMultiscale inversion Streamline-based inversion 20
  21. 21. Streamline-based update Final Kx match Kx changeKx (md) Kx (md) • Local updates • Minimal updates along prior preferential flow paths 21
  22. 22. Final Data misfit: WCTMultiscale and streamline P2 P3 P4 P1 22
  23. 23. Comparison of data misfit: WCTMultiscale/SL and Business Unit P2 P3 P4 P1 23
  24. 24. Comparison with previous calibration P3 This study • Regional I3 I4 SOURCE parameterization I2 more consistent with model constraints I3 I4 TMX: Rey et al. (2009) Figure 26: Rey et al. (2009)TMXmult. High perm (> upper limit near P3) Potential channel 24
  25. 25. Summary• Multiscale approach to history matching  Builds on well-established ‘structured’ workflow  Regional heterogeneity  Generalized grid-connectivity-based parameterization  Efficient, flexible application to any reservoir model geometry  Refine local heterogeneity  Prior preferential flow paths captured by streamlines• Field application  Demonstrates practical feasibility  Improvement upon heterogeneity characterization using standard zonation approaches 25

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