Upscaling of Geocellular Models for Flow Simulation Louis J. Durlofsky Department of Petroleum Engineering, Stanford University ChevronTexaco ETC, San Ramon, CA.
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Upscaling of Geocellular Models for Flow Simulation Louis J. Durlofsky Department of Petroleum Engineering, Stanford University ChevronTexaco ETC, San Ramon, CA Acknowledgments • Yuguang Chen (Stanford University) • Mathieu Prevost (now at Total) • Xian-Huan Wen (ChevronTexaco) • Yalchin Efendiev (Texas A&M) 2 (photo by Eric Flodin) Outline • Issues and existing techniques • Adaptive local-global upscaling • Velocity reconstruction and multiscale solution • Generalized convection-diffusion transport model • Upscaling and flow-based grids (3D unstructured) • Outstanding issues and summary 3 Requirements/Challenges for Upscaling • Accuracy & Robustness – Retain geological realism in flow simulation – Valid for different types of reservoir heterogeneity – Applicable for varying flow scenarios (well conditions) • Efficiency Injector Producer Producer Injector 4 Existing Upscaling Techniques • Single-phase upscaling: flow (Q /p) – Local and global techniques (k k* or T *) • Multiphase upscaling: transport (oil cut) – Pseudo relative permeability model (krj krj*) • “Multiscale” modeling – Upscaling of flow (pressure equation) – Fine scale solution of transport (saturation equation) 5 Local Upscaling to Calculate k* or Local Global domain Extended Local Solve (kp)=0 over local region for coarse scale k * or T * • Local BCs assumed: constant pressure difference • Insufficient for capturing large-scale connectivity in highly heterogeneous reservoirs 6 A New Approach • Standard local upscaling methods unsuitable for highly heterogeneous reservoirs • Global upscaling methods exist, but require global fine scale solutions (single-phase) and optimization Adaptive Local-Global Upscaling • New approach uses global coarse scale solutions to determine appropriate boundary conditions for local k* or T * calculations – Efficiently captures effects of large scale flow – Avoids global fine scale simulation 7 Adaptive Local-Global Upscaling (ALG) Well-driven global coarse flow y Local fine scale calculation Coarse pressure Interpolated pressure gives local LocalBCs BCs Coarse scale properties x k* or T * and upscaled well index • Thresholding: Local calculations only in high-flow regions (computational efficiency) 8 Thresholding in ALG Regions for Local calculations Permeability Streamlines • Identify high-flow region, • • Coarse blocks |q c| |q > ( 0.1) c| max Avoids nonphysical coarse scale properties (T *=q c/p c) Coarse scale properties efficiently adapted to a given flow scenario 9 Multiscale Modeling • Solve flow on coarse scale, reconstruct fine scale v, solve transport on fine scale S * c ( vS ) 0 k p 0 t • Active research area in reservoir simulation: – Dual mesh method (FD): Ramè & Killough (1991), Guérillot & Verdière (1995), Gautier et al. (1999) – Multiscale FEM: Hou & Wu (1997) – Multiscale FVM: Jenny, Lee & Tchelepi (2003, 2004) 10 Reconstruction of Fine Scale Velocity k p 0 * c Partition coarse S ( vS ) 0 t flux to fine scale Upscaling, global coarse scale flow Solve local fine scale (kp)=0 Reconstructed fine scale v (downscaling) • Readily performed in upscaling framework 11 Results: Performance of ALG Pressure Distribution Channelized layer (59) from 10th SPE CSP Averaged fine Upscaling 220 60 22 6 25.0 Q (Fine scale) = 20.86 Flow rate for specified pressure ALG, Error: 4% 20.0 Coarse: extended local • Fine scale: Q = 20.86 15.0 Q • Extended T *: Q = 7.17 10.0 Extended local, •5.0 ALG upscaling: Q = 20.01 Error: 67% 0.0 0 1 2 Iteration 3 4 Coarse: Adaptive local-global 12 Results: Multiple Channelized Layers 10th SPE CSP Extended local T * Adaptive local-global T * 13 Another Channelized System 100 realizations 120 120 24 24 k * only T * + NWSU ALG T * 14 Results: Multiple Realizations Fine scale BHP (PSIA) 100 realizations mean 90% conf. int. Time (days) • 100 realizations conditioned to seismic and well data • Oil-water flow, M=5 • Injector: injection rate constraint, Producer: bottom hole pressure constraint • Upscaling: 100 100 10 10 15 Results: Multiple Realizations Coarse: Adaptive local-global BHP (PSIA) BHP (PSIA) Coarse: Purely local upscaling Time (days) Time (days) Mean (coarse scale) Mean (fine scale) 90% conf. int. (coarse scale) 90% conf. int. (fine scale) 16 Results (Fo): Channelized System Oil cut from reconstruction Fractional Flow Curve 1.2 1 220 60 22 6 0.8 ALG T * Fo 0.6 Flow rates • Fine scale: Q = 6.30 • Extended T *: Q = 1.17 Extended local T * 0.4 0.2 Fine scale 0 0 0.2 0.4 PVI 0.6 0.8 1 • ALG upscaling: Q = 6.26 17 Results (Sw): Channelized System 1.0 0.5 0.0 Fine scale streamlines Fine scale Sw (220 60) Reconstructed Sw from Reconstructed Sw from extended local T * (22 6) ALG T * (22 6) 18 Results for 3D Systems (SPE 10) 50 channelized layers, 3 wells pinj=1, pprod=0 Typical layers P1 P2 I Upscale from 6022050 124410 using different methods 19 Results for Well Flow Rates - 3D 4000 Fine Standard k* T*+NWSU ALG 3500 Well Rate 3000 2500 2000 1500 1000 500 0 I P1 P2 Average errors • k* only: 43% • Extended T* + NWSU: 27% • Adaptive local-global: 3.5% 20 Results for Transport (Multiscale) - 3D Producer 2 Producer 1 fine scale Fo ALG T* local T * w/nw Fo standard k* standard k* local T * w/nw PVI • fine scale ALG T * PVI Quality of transport calculation depends on the accuracy of the upscaling 21 Velocity Reconstruction versus Subgrid Modeling • Multiscale methods carry fine and coarse grid information over the entire simulation • Subgrid modeling methods capture effects of fine grid velocity via upscaled transport functions: - Pseudoization techniques - Modeling of higher moments - Generalized convection-diffusion model 22 Pseudo Relative Permeability Models • Coarse scale pressure and saturation equations of same form as fine scale equations • Pseudo functions may vary in each block and may be directional (even for single set of krj in fine scale model) (x, S )k p 0 , * c * c S c F* ( x , S c ) 0 t * k rw * k * ( x, S c ) = ro , Fi* v ic f i * ( x, S c ) w μo ( k ) μw fi (S ) * ( k rw )i μw + ( k ro* )i μo * c * rw i * upscaled function c coarse scale p, S 23 Generalized Convection-Diffusion Subgrid Model for Two-Phase Flow • Pseudo relative permeability description is convenient but incomplete, additional functionality required in general • Generalized convection-diffusion model introduces new coarse scale terms - Form derives from volume averaging and homogenization procedures - Method is local, avoids need to approximate vi(x)vj ( y ) - Shares some similarities with earlier stochastic approaches of Lenormand & coworkers (1998, 1999) 24 Generalized Convection-Diffusion Model • Coarse scale saturation equation: S c G(x, S c ) D(x, S c ) S c t (modified convection m and diffusion D terms) G(x, S c ) v c f ( S c ) m(x, S c ) • Coarse scale pressure equation: * (x, S c )k * p c 0 “primitive” term GCD term * ( S c ) W1 (x, S c ) W2 (x, S c ) S c (modified form for total mobility, Sc dependence) 25 Calculation of GCD Functions • D and W2 computed over purely local domain: D( S ) S v f ( S ) vf ( S ) p=1 S=1 p=0 (D and W2 account for local subgrid effects) • m and W1 computed using extended local domain: m( S ) v f ( S ) vf ( S ) D( S ) S (m and W1 - subgrid effects due to longer range interactions) target coarse block 26 Solution Procedure • Generate fine model (100 100) of prescribed parameters • Form uniform coarse grid (10 10) and compute k* and directional GCD functions for each coarse block • Compute directional pseudo relative permeabilities via total mobility (Stone-type) method for each block • Solve saturation equation using second order TVD scheme, first order method for simulations with pseudo krj fine grid: lx lz Lx = Lz 27 Oil Cuts for M =1 Simulations Oil Cut lx = 0.25, lz= 0.01, s =2, side to side flow 100 x 100 10 x 10 (GCD) 10 x 10 (primitive) 10 x 10 (pseudo) PVI • GCD and pseudo models agree closely with fine scale (pseudoization technique selected on this basis) 28 Results for Two-Point Geostatistics • Diffusive effects only x =0.05, y = 0.01, slogk = 2.0 10 5 0 100x100 10x10, Side Flow 29 Results for Two-Point Geostatistics (Cont’d) • Permeability with longer correlation length x =0.5, y = 0.05, slogk = 2.0 10 5 0 100x100 10x10, Side Flow 30 Effect of Varying Global BCs (M =1) lx = 0.25, lz= 0.01, s =2 p=1 S=1 100 x 100 10 x 10 (GCD) lx = 0.25, lz= 0.01, s =2 10 x 10 (primitive) 10 x 10 (pseudo) p=0 Oil Cut 0 t 0.8 PVI p=0 p=1 S=1 t > 0.8 PVI PVI 31 Corner to Corner Flow (M = 5) Oil Cut Total Rate lx = 0.2, lz= 0.02, s =1.5 PVI 100 x 100 10 x 10 (GCD) 10 x 10 (pseudo) PVI • Pseudo model shows considerable error, GCD model provides comparable agreement as in side to side flow 32 Effect of Varying Global BCs (M = 5) Oil Cut Total Rate lx = 0.2, lz= 0.02, s =1.5 PVI 100 x 100 10 x 10 (GCD) 10 x 10 (pseudo) PVI • Pseudo model overpredicts oil recovery, GCD model in close agreement 33 Effect of Varying Global BCs (M = 5) Oil Cut Total Rate lx = 0.5, lz= 0.02, s =1.5 PVI 100 x 100 10 x 10 (GCD) 10 x 10 (pseudo) PVI • GCD model underpredicts peak in oil cut, otherwise tracks fine grid solution 34 Combine GCD with ALG T* Upscaling Coarse scale flow: Pseudo functions: GCD model: *k* p c 0 * ( S c ) * (S c , S c ) T * from ALG, dependent on global flow *, m(S c) and D(S c) • Consistency between T * and * important for highly heterogeneous systems 35 ALG + Subgrid Model for Transport (GCD) • Stanford V model (layer 1) • Upscaling: 100130 1013 • Transport solved on coarse scale t < 0.6 PVI t 0.6 PVI flow rate oil cut 36 Unstructured Modeling - Workflow fine model upscaling coarse model gridding info. maps Gocad interface flow simulation flow simulation diagnostic 37 Numerical Discretization Technique k Primal and dual grids i • CVFE method: j – Locally conservative; flux on a face expressed as linear combination of pressures – Multiple point and two point flux approximations qij = a pi + b pj + c pk + ... or qij = Tij ( pi - pj ) • Different upscaling techniques for MPFA and TPFA 38 3D Transmissibility Upscaling (TPFA) Primal grid connection Dual cells p=1 fitted extended regions Tij*= p=0 - <qij> <pj> - <pi> cell j cell i 39 Grid Generation: Parameters • Specify flow-diagnostic cumulative frequency 1 Pb • Grid aspect ratio • Grid resolution constraint: Pa – Information map (flow rate, tb) – Pa and Pb , sa and sb a min b property – N (number of nodes) max resolution constraint Sb Sa min a property b max 40 Unstructured Gridding and Upscaling velocity grid density Upscaled k* (from Prevost, 2003) 41 Flow-Based Upscaling: Layered System • Layered system: 200 x 100 x 50 cells p=1 p=0 1 0. 5 0.25 • Upscale permeability and transmissibility • Run k*-MPFA and T*-TPFA for M=1 • Compute errors in Q/p and L1 norm of Fw 42 Flow-Based Upscaling: Results 6 x 6 x 13 = 468 nodes 8 x 8 x 18 = 1152 nodes 1 1 Reference (fine) TPFA MPFA 0.8 Fw 0.8 Fw 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0 PVI 0.2 0.4 0.6 0.8 1 PVI error in Fw error in Q/p TPFA 7.6% -1.2% MPFA 17.9% -25.2% error in Fw error in Q/p TPFA 16.8% -5.9% MPFA 21.3% -31.7% 43 (from M. Prevost, 2003) Layered Reservoir: Flow Rate Adaptation sa • Grid density from flow rate log |V| grid size sb • Flow results 1 reference uniform coarse (N=21x11x11=2541) flow-rate adapted (N=1394) 0.8 0.6 Fw F w Qc=0.99 0.4 Qc=0.82 (Qf = 1.0) 0.2 0 0 (from Prevost, 2003) 0.2 0.4 PVI PVI 0.6 0.8 1 44 Summary • Upscaling is required to generate realistic coarse scale models for reservoir simulation • Described and applied a new adaptive local-global method for computing T * • Illustrated use of ALG upscaling in conjunction with multiscale modeling • Described GCD method for upscaling of transport • Presented approaches for flow-based gridding and upscaling for 3D unstructured systems 45 Future Directions • Hybridization of various upscaling techniques (e.g., flow-based gridding + ALG upscaling) • Further development for 3D unstructured systems • Linkage of single-phase gridding and upscaling approaches with two-phase upscaling methods • Dynamic updating of grid and coarse properties • Error modeling 46