Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels Vegard Røine Stenerud † and Knut–Andreas Lie ‡ † NTNU, Department of Mathematical Sciences ‡ SINTEF ICT, Dept. Applied Mathematics February 12, 2008 Applied Mathematics Feb 2008 1/18
Introduction: History matching History matching is the procedure of modifying the reservoir description to match measured reservoir responses. Initial: Matched: Reference: Applied Mathematics Feb 2008 2/18
Introduction: History-matching loop Current reservoir parameters � d cal = g ( m ) ( d obs − d cal ) 2 , E = Flow simulation Evaluate misfit (observed - calculated) Is misfit small enough Yes No HM method/Inversion Applied Mathematics Feb 2008 3/18
Challenges in history-matching loop Current reservoir parameters Problems: highly under-determined Flow simulation problem → non-uniqueness errors in model, data, and Evaluate misfit methods (observed - calculated) nonlinear forward model non-convex misfit functions Is misfit small enough forward simulations are Yes computationally demanding No HM method/Inversion Applied Mathematics Feb 2008 4/18
Challenge I: Non-convex misfit function Inversion method: Generalized Travel-Time Inversion (GTTI) with analytic sensitivities [Vasco et al. (1999), He et al. (2002)] The generalized travel time is defined as the ’optimal’ time–shift that maximizes � [ y obs ( t i + ∆ t ) − y cal ( t i )] 2 R 2 (∆ t ) = 1 − . � [ y obs ( t i ) − ¯ y obs ( t i )] 2 Applied Mathematics Feb 2008 5/18
Travel-time inversion Basic underlying principles for the history–matching algorithm Minimize travel-time misfit for water–cut by iterative least-square minimization algorithm. Preserve geologic realism by keeping changes to prior geologic model minimal (if possible). Only allow smooth large-scale changes. Production data have low resolution and cannot be used to infer small-scale variations. Minimization of functional: ∆ ˜ t : Travel–time shift S : Sensitivity matrix Regularization � �� � m : Reservoir � ∆ ˜ t − S δ R � + β 1 � δ R � + β 2 � L δ R � parameters � �� � � �� � norm smoothing S computed analytically along streamlines from a single flow simulation Applied Mathematics Feb 2008 6/18
Streamline-based history matching Features of streamlines Very well suited for modeling large heterogeneous multi-well systems dominated by convection Generally fast flow simulation Delineate flow pattern (injector-producer pairs) Enables analytic sensitivities Source: www.techplot.com Streamline-based history-matching methods Assisted history matching (Generalized) travel-time inversion methods Streamline-effective properties methods Miscellaneous Applied Mathematics Feb 2008 7/18
Example: Uncertainty quantification Simple two-phase model (end-point mobility M = 0 . 5) on a 2D horizontal reservoir, lognormal permeability Pre data integration: P2 Post data integration: P2 1 1 obs obs 0.9 0.9 init matched 0.8 0.8 0.7 0.7 0.6 0.6 fw fw 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 days days Statistical analysis of mean and standard deviation Applied Mathematics Feb 2008 8/18
Challenge II: long runtime for forward simulations Streamline simulation much Current reservoir parameters faster than conventional FD-methods. Still, room for improvement. Flow simulation Observations: pressure solver most Evaluate misfit expensive part of simulation (observed - calculated) data changes very little from one simulation to the next Is misfit small enough Yes No HM method/Inversion Applied Mathematics Feb 2008 9/18
Challenge II: long runtime for forward simulations Streamline simulation much Current reservoir parameters faster than conventional FD-methods. Still, room for improvement. Flow simulation Observations: pressure solver most Evaluate misfit expensive part of simulation (observed - calculated) data changes very little from one simulation to the next Is misfit small enough Yes No Reuse computations in areas with minor changes − → HM method/Inversion multiscale methods Applied Mathematics Feb 2008 9/18
Multiscale pressure solver Upscaling and downscaling in one step. Runtime like coarse-scale solver, resolution like fine-scale solver. Fine grid: 75 × 30. Coarse grid: 15 × 6 Basis functions for each pair of coarse blocks T i ∪ T j : Ψ ij = − λK ∇ Φ ij � w i ( x ) , x ∈ T i ∇ · Ψ ij = − w j ( x ) , x ∈ T j Global linear system with 249 unknowns: ∇· v = q, v = − λK ∇ p Coarse grid: pressure and fluxes. Fine grid: fluxes Applied Mathematics Feb 2008 10/18
Multiscale methods: efficiency vs accuracy Ex: q5-spot, SPE 10 (layer 85) 1 , 60 × 220 → 10 × 22 Water cuts obtained by never updating basis functions: Water−Cut Water−Cut 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 Reference Reference Multiscale Multiscale 0 0 0 0.5 1 1.5 0 0.5 1 1.5 PVI PVI favorable ( M = 0 . 1) unfavorable ( M = 10 . 0) 1: Fluvial permeability field, Applied Mathematics Feb 2008 11/18
Multiscale methods: efficiency vs accuracy Ex: q5-spot, SPE 10 (layer 85) 1 , 60 × 220 → 10 × 22 Improved accuracy by adaptive updating of basis functions: Water−Cut Water−Cut 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 Reference Reference Multiscale Multiscale 0 0 0.65 0.7 0.75 0.8 0.85 0.9 0.65 0.7 0.75 0.8 0.85 0.9 PVI PVI no updating adaptive updating 1: Fluvial permeability field, Applied Mathematics Feb 2008 12/18
Further computational savings Can also reuse basis functions from previous forward simulation. General idea: use sensitivity to steer updating Stacked sensitivities 50 0 45 −1 40 −2 35 −3 30 25 −4 20 −5 15 −6 10 −7 5 −8 0 0 10 20 30 40 50 Applied Mathematics Feb 2008 13/18
History matching on heological models Generalized travel-time inversion on million-cell model Assimilation of production data to calibrate model 1 million cells, 32 injectors, and 69 producers 2475 days ≈ 7 years of water-cut data Analytical sensitivities along streamlines + travel-time inversion (quasi-linearization of misfit functional) Amplitude-residual Time-residual Computation time: ∼ 17 min on a desktop PC (6 iterations). Applied Mathematics Feb 2008 14/18
History matching on geological models Residuals and timing results, Intel Core 2 Duo (2.4 GHz, 4Mb cache) Misfit CPU-time (wall clock) Solver O/M T A ∆ ln k Total Pres. Transp. Initial — 100.0 100.0 0.821 — — — Std. (7 pt.) O 8.9 53.5 0.806 64 min 33 min 28 min Std. (7 pt.) M 9.6 50.4 0.806 39 min 30 min 5 min Multiscale O 11.2 47.3 0.812 43 min 7 min 32 min Multiscale M 10.4 45.4 0.828 17 min 7 min 6 min Misfit: Time-shift misfit � ∆ t � 2 Amplitude misfit [ � � j ( f obs − f cal w ) 2 ] 1 / 2 w k Permeability discrepancy 1 /N � N i =1 | ln k ref − ln k match | i i Applied Mathematics Feb 2008 15/18
Robustness with respect to data reduction Uncertainty quantification, revisited Mean time−shift residual Mean amplitude residual Mean average discrepancy of log(K) 12 50 100 10 40 80 8 % of initial % of initial 30 % of initial 60 6 20 40 4 10 2 20 0 0 0 0 0 0 25 25 25 % Dynamic Update % Dynamic Update % Dynamic Update 50 TPFA 50 TPFA 50 TPFA 100 100 100 75 75 75 75 75 75 % Initial Update % Initial Update % Initial Update 100 50 100 50 100 50 TPFA 25 TPFA 25 TPFA 25 0 0 0 Std. time−shift residual Std. amplitude residual Std. average discrepancy of log(K) 8 20 10 8 6 15 % of initial % of initial % of initial 6 4 10 4 2 5 2 0 0 0 0 0 0 25 25 25 % % Dynamic Update % Dynamic Update D 50 TPFA 50 TPFA 50 TPFA y n a 100 100 100 m 75 75 75 i c 75 75 75 U % Initial Update % Initial Update % Initial Update p 100 100 100 d 50 50 50 a t e TPFA 25 TPFA 25 TPFA 25 0 0 0 Applied Mathematics Feb 2008 16/18
Robustness with respect to data reduction Million-cell model, revisited Reduction in residuals Time−Shift Residual Amplitude Residual) Average Discrepancy of log(K) 15 60 120 50 100 10 40 80 % of initial % of initial % of initial 30 60 5 20 40 10 20 0 0 0 0 0 0 0 0 0 % Dynamic Update 25 % Dynamic Update 25 % 25 D TPFA TPFA y 50 50 n 50 TPFA a 100 100 m 100 75 75 i 75 c 75 75 U 75 % Initial Update % Initial Update p % Initial Update 100 50 100 50 d 100 a 50 t e TPFA 25 TPFA 25 TPFA 25 0 0 0 10 8 Speed−up 6 4 2 0 0 0 % Dynamic Update 25 50 TPFA 100 75 % Initial Update 75 100 50 Corresponding speedup: TPFA 25 0 Applied Mathematics Feb 2008 17/18
Extentions Unstructured grids (done for inversion algorithm) Corner-point grids (testing to be done on Norne-model) Other types of data / more general flow Inclusion of seismics Use of sensitivities for other optimization workflows . . . Applied Mathematics Feb 2008 18/18
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