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

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This document discusses a new multiscale parameterization and history matching workflow for structured and unstructured grids. At the regional scale, it introduces a novel grid-connectivity-based parameterization that flexibly captures heterogeneity while reducing the number of parameters. At the local scale, it leverages established streamline-based methods to refine preferential flow paths. The workflow iterates between regional and local scales to history match production data in a computationally efficient manner.

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Multiscale Parameterization and History Matching in Structured and Unstructured Grids: Theory and Field Application E. W. Bhark, A. Rey, A. Datta-Gupta and B. Jafarpour
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
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
Why re-parameterization? • Reduce redundant model information  Preserve important heterogeneity ~5,000 Unknowns 100 Unknowns 50 25 Ex., high-resolution (3D) abs. permeability • Improves:  Solution non-uniqueness and stability, computational efficiency 4
Parameterization by linear transform = v1 + v2 + v3 + … + v50 + … + vN N-parameter high-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
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
Basis development Concept: Develop as generalization of discrete Fourier basis KEY: 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
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
Basis functions: Examples Corner-point Grid (Brugge) • Modal shape  modal frequency • Constant basis  Zero frequency • Discontinuities honored Basis vec. 1 Basis vec. 2 Basis vec. 3 Basis vec. 4 Basis vec. 5 Basis vec. 9 9
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 structure Multiple subdomains 10
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
Field application: Offshore reservoir Reservoir • > 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), SPE124950 Parameter • Permeability Data • Water cut 12
Conceptual heterogeneity model Prior model facies (5) Prior geo-model P2 I2 P1 P3 I1 Initial Kx: I3 Average of measurements at wells per facies (5) Facies ID P4 I4 Next objective: Use parameterization to assist in heterogeneity identification and updating 13
Workflow: Prior model & multiplier field F2 Prior geo-model Multiplier field F6 F5 F3 F1 14
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 15 v1 …+ v3 …+ v6 …+ v8 …+ v15 15
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 functions Multiscale inversion • End refinement when production data become insensitive to addition of basis functions 16
Multiscale update Number of leading basis Kx: Adaptive multiscale functions per facies 10 10 10 1 5 36 17
Comparison with previous calibration This study Rey et al. (2009) Tx multiplier Facies zonation Tx multiplier Adaptive multiscale Manual zonation 18
Data misfit: WCT Initial and multiscale P2 P3 P4 P1 19
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 functions Multiscale inversion Streamline-based inversion 20
Streamline-based update Final Kx match Kx change Kx (md) Kx (md) • Local updates • Minimal updates along prior preferential flow paths 21
Final Data misfit: WCT Multiscale and streamline P2 P3 P4 P1 22
Comparison of data misfit: WCT Multiscale/SL and Business Unit P2 P3 P4 P1 23
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) TMX mult. High perm (> upper limit near P3) Potential channel 24
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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Bhark, E.W., Structured History Matching Workflow using Parameterization and Streamline Methods

  • 1. Multiscale Parameterization and History Matching in Structured and Unstructured Grids: Theory and Field Application E. W. Bhark, A. Rey, A. Datta-Gupta and B. Jafarpour
  • 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. 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. Why re-parameterization? • Reduce redundant model information  Preserve important heterogeneity ~5,000 Unknowns 100 Unknowns 50 25 Ex., high-resolution (3D) abs. permeability • Improves:  Solution non-uniqueness and stability, computational efficiency 4
  • 5. Parameterization by linear transform = v1 + v2 + v3 + … + v50 + … + vN N-parameter high-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. 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. Basis development Concept: Develop as generalization of discrete Fourier basis KEY: 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. 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. Basis functions: Examples Corner-point Grid (Brugge) • Modal shape  modal frequency • Constant basis  Zero frequency • Discontinuities honored Basis vec. 1 Basis vec. 2 Basis vec. 3 Basis vec. 4 Basis vec. 5 Basis vec. 9 9
  • 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 structure Multiple subdomains 10
  • 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. Field application: Offshore reservoir Reservoir • > 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), SPE124950 Parameter • Permeability Data • Water cut 12
  • 13. Conceptual heterogeneity model Prior model facies (5) Prior geo-model P2 I2 P1 P3 I1 Initial Kx: I3 Average of measurements at wells per facies (5) Facies ID P4 I4 Next objective: Use parameterization to assist in heterogeneity identification and updating 13
  • 14. Workflow: Prior model & multiplier field F2 Prior geo-model Multiplier field F6 F5 F3 F1 14
  • 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 15 v1 …+ v3 …+ v6 …+ v8 …+ v15 15
  • 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 functions Multiscale inversion • End refinement when production data become insensitive to addition of basis functions 16
  • 17. Multiscale update Number of leading basis Kx: Adaptive multiscale functions per facies 10 10 10 1 5 36 17
  • 18. Comparison with previous calibration This study Rey et al. (2009) Tx multiplier Facies zonation Tx multiplier Adaptive multiscale Manual zonation 18
  • 19. Data misfit: WCT Initial and multiscale P2 P3 P4 P1 19
  • 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 functions Multiscale inversion Streamline-based inversion 20
  • 21. Streamline-based update Final Kx match Kx change Kx (md) Kx (md) • Local updates • Minimal updates along prior preferential flow paths 21
  • 22. Final Data misfit: WCT Multiscale and streamline P2 P3 P4 P1 22
  • 23. Comparison of data misfit: WCT Multiscale/SL and Business Unit P2 P3 P4 P1 23
  • 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) TMX mult. High perm (> upper limit near P3) Potential channel 24
  • 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