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) (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

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 Injector Producer

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)

Local Upscaling to Calculate k* or Local Extended Local Solve (kp)=0 over local region for coarse scale k * or T * Global domain Local BCs assumed: constant pressure difference Insufficient for capturing large-scale connectivity in highly heterogeneous reservoirs

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 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 Adaptive Local-Global Upscaling

Adaptive Local-Global Upscaling (ALG) Well-driven global coarse flow Local fine scale calculation Interpolated pressure gives Local BCs Coarse pressure Local fine scale calculation Interpolated pressure gives local BCs Coarse pressure y Coarse scale properties k* or T * and upscaled well index x Thresholding: Local calculations only in high-flow regions (computational efficiency)

Thresholding in ALG |q c| Identify high-flow region, >  (  0.1) Regions for Local calculations Permeability Streamlines Coarse blocks |q c| |q c|max Identify high-flow region, >  (  0.1) Avoids nonphysical coarse scale properties (T *=q c/p c) Coarse scale properties efficiently adapted to a given flow scenario

Multiscale Modeling Solve flow on coarse scale, reconstruct fine scale v, solve transport on fine scale 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)

Reconstruction of Fine Scale Velocity Partition coarse flux to fine scale Solve local fine scale (kp)=0 Upscaling, global coarse scale flow Reconstructed fine scale v (downscaling) Readily performed in upscaling framework

Results: Performance of ALG Averaged fine Pressure Distribution Coarse: extended local Coarse: Adaptive local-global Channelized layer (59) from 10th SPE CSP Upscaling 220  60  22  6 Q (Fine scale) = 20.86 ALG, Error: 4% Extended local, Error: 67% Flow rate for specified pressure Fine scale: Q = 20.86 Extended T *: Q = 7.17 ALG upscaling: Q = 20.01

Results: Multiple Channelized Layers 10th SPE CSP Extended local T * Adaptive local-global T *

Another Channelized System 100 realizations 120  120  24  24 k * only T * + NWSU ALG T *

Results: Multiple Realizations Fine scale mean 90% conf. int. 100 realizations BHP (PSIA) 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

Results: Multiple Realizations Coarse: Purely local upscaling Coarse: Adaptive local-global Time (days) BHP (PSIA) Time (days) BHP (PSIA) Mean (coarse scale) Mean (fine scale) 90% conf. int. (coarse scale) 90% conf. int. (fine scale)

Results (Fo): Channelized System Oil cut from reconstruction 220  60  22  6 ALG T * Flow rates Fine scale: Q = 6.30 Extended T *: Q = 1.17 ALG upscaling: Q = 6.26 Extended local T * Fine scale

Results (Sw): Channelized System Fine scale streamlines 1.0 0.5 0.0 Fine scale Sw (220  60) Reconstructed Sw from ALG T * (22  6) Reconstructed Sw from extended local T * (22  6)

Results for 3D Systems (SPE 10) Typical layers 50 channelized layers, 3 wells pinj=1, pprod=0 Upscale from 6022050  124410 using different methods

Results for Well Flow Rates - 3D Average errors k* only: 43% Extended T* + NWSU: 27% Adaptive local-global: 3.5%

Results for Transport (Multiscale) - 3D fine scale ALG T * local T * w/nw standard k* Producer 1 Fo PVI Producer 2 Quality of transport calculation depends on the accuracy of the upscaling

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

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) *  upscaled function c  coarse scale p, S

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 - Shares some similarities with earlier stochastic approaches of Lenormand & coworkers (1998, 1999)

Generalized Convection-Diffusion Model Coarse scale saturation equation: (modified convection m and diffusion D terms) “primitive” term GCD term Coarse scale pressure equation: (modified form for total mobility, Sc dependence)

Calculation of GCD Functions D and W2 computed over purely local domain: p = 1 S = 1 p = 0 (D and W2 account for local subgrid effects) m and W1 computed using extended local domain: (m and W1 - subgrid effects due to longer range interactions) target coarse block

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

Oil Cuts for M =1 Simulations 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) Oil Cut PVI GCD and pseudo models agree closely with fine scale (pseudoization technique selected on this basis)

Results for Two-Point Geostatistics Diffusive effects only x =0.05,  y = 0.01, logk = 2.0 10 5 100x100  10x10, Side Flow

Results for Two-Point Geostatistics (Cont’d) Permeability with longer correlation length x =0.5,  y = 0.05, logk = 2.0 10 5 100x100  10x10, Side Flow

Effect of Varying Global BCs (M =1) lx = 0.25, lz= 0.01, s =2  100 x 100  10 x 10 (GCD)  10 x 10 (primitive)  10 x 10 (pseudo) p = 1 S = 1 p = 0 lx = 0.25, lz= 0.01, s =2 0  t  0.8 PVI Oil Cut p = 0 p = 1 S = 1 t > 0.8 PVI PVI

Corner to Corner Flow (M = 5) lx = 0.2, lz= 0.02, s =1.5  100 x 100  10 x 10 (GCD)  10 x 10 (pseudo) Oil Cut PVI Total Rate Pseudo model shows considerable error, GCD model provides comparable agreement as in side to side flow

Effect of Varying Global BCs (M = 5) lx = 0.2, lz= 0.02, s =1.5  100 x 100  10 x 10 (GCD)  10 x 10 (pseudo) Oil Cut PVI Total Rate Pseudo model overpredicts oil recovery, GCD model in close agreement

Effect of Varying Global BCs (M = 5) lx = 0.5, lz= 0.02, s =1.5  100 x 100  10 x 10 (GCD)  10 x 10 (pseudo) Oil Cut PVI Total Rate GCD model underpredicts peak in oil cut, otherwise tracks fine grid solution

Combine GCD with ALG T* Upscaling Coarse scale flow: Pseudo functions: GCD model: T * from ALG, dependent on global flow *, m(S c) and D(S c) Consistency between T * and * important for highly heterogeneous systems

ALG + Subgrid Model for Transport (GCD) Stanford V model (layer 1) Upscaling: 100130  1013 Transport solved on coarse scale t < 0.6 PVI t  0.6 PVI flow rate oil cut

Unstructured Modeling - Workflow fine model coarse model upscaling gridding info. maps Gocad interface flow simulation flow simulation diagnostic

Numerical Discretization Technique j k Primal and dual grids CVFE method: Locally conservative; flux on a face expressed as linear combination of pressures Multiple point and two point flux approximations Different upscaling techniques for MPFA and TPFA qij = a pi + b pj + c pk + ... or qij = Tij ( pi - pj )

3D Transmissibility Upscaling (TPFA) Dual cells Primal grid connection p=1 fitted extended regions p=0 <qij> Tij*= - <pj> <pi> - cell j cell i

Grid Generation: Parameters min max 1 property cumulative frequency a b Pa Pb Sa Sb resolution constraint Specify flow-diagnostic Grid aspect ratio Grid resolution constraint: Information map (flow rate, tb) Pa and Pb , sa and sb N (number of nodes)

Unstructured Gridding and Upscaling velocity grid density Upscaled k* (from Prevost, 2003)

Flow-Based Upscaling: Layered System Layered system: 200 x 100 x 50 cells Upscale permeability and transmissibility Run k*-MPFA and T*-TPFA for M=1 Compute errors in Q/Dp and L1 norm of Fw p=0 p=1 1 0. 5 0.25

Flow-Based Upscaling: Results 8 x 8 x 18 = 1152 nodes 6 x 6 x 13 = 468 nodes 1 1 Reference (fine) 0.8 TPFA 0.8 Fw MPFA Fw 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 PVI PVI (from M. Prevost, 2003)

Layered Reservoir: Flow Rate Adaptation log |V| grid size sb sa Grid density from flow rate Flow results 0.2 0.4 0.6 0.8 1 PVI F w reference uniform coarse (N=21x11x11=2541) flow-rate adapted (N=1394) Qc=0.82 Qc=0.99 Fw (Qf = 1.0) (from Prevost, 2003)

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

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