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Fast Simulation of Highly Heterogeneous andFractured Porous Media
Knut–Andreas Lie
SINTEF ICT, Dept. Applied Mathematics
Bergen, May 2006
Applied Mathematics May 2006 1/20
The GeoScale Project Portifolio
Vision:
Direct simulation of fluid flow on high-resolution geomodels ofhighly heterogeneous and fractured porous media in 3D.
Research keywords:
multiscale methods, upscaling/downscaling
robust discretisations of pressure equations
fast simulation of fluid transport
Contact:
http://www.math.sintef.no/geoscale/
+47 22 06 77 10 / +47 930 58 721
Applied Mathematics May 2006 2/20
GeoScale Portifolio – A Collaborative Effort
Partners:
SINTEF
Universities of Bergen, Oslo, and Trondheim
Schlumberger, Shell, Statoil
Education:
4-5 PhD grants (RCN, UoB, NTNU, Shell)
4 postdoc grants (RCN, EU, Schlumberger)
2 master students
Collaboration:
Stanford, Texas A&M, Ume̊a
Schlumberger Moscow Research, Statoil Research Centre
Applied Mathematics May 2006 3/20
Simulation on Geological Models
For various reasons, there is a need fordirect simulation on high-resolutiongeomodels. This is difficult:
K spans many length scales andhas multiscale structure
maxK/ minK ∼ 103–1010
Details on all scales impact flow
Gap between simulation models and geomodels:
High-resolution geomodels may have 107 − 109 cells
Conventional simulators are capable of about 105 − 106 cells
Applied Mathematics May 2006 4/20
Motivation
Applications for fast (and lightweight) simulators for :
direct simulation of large geomodels
multiple realisations
history-matching
. . .
Long-term collaboration with Schlumberger:
Petrel workflow tools
FrontSim streamline simulator
Applied Mathematics May 2006 5/20
Developing More Robust Discretisations
Accurate simulation on industry-standard grid models ischallenging!
Skew and deformed gridblocks:
Non-matching cells:
Our approach: finite elements and/or mimetic methods
Applied Mathematics May 2006 6/20
Developing an Alternative to Upscaling
We seek a methodology that:
gives a detailed image of the flow pattern on the fine scale,without having to solve the full fine-scale system
is robust and flexible with respect to the coarse grid
is robust and flexible with respect to the fine grid and thefine-grid solver
is accurate and conservative
is fast and easy to parallelise
Applied Mathematics May 2006 7/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓
⇑
Coarse grid blocks:⇓
⇑
Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓ ⇑Coarse grid blocks:
⇓ ⇑Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓ ⇑Coarse grid blocks:
⇓ ⇑Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓ ⇑Coarse grid blocks:
⇓ ⇑Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓ ⇑Coarse grid blocks:
⇓ ⇑Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓
⇑
Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓ ⇑Coarse grid blocks:
⇓ ⇑Flow problems:
Multiscale method:
⇓
⇑
Coarse grid blocks:
⇓ ⇑Flow problems:
Applied Mathematics May 2006 8/20
From Upscaling to Multiscale Methods
Standard upscaling:
⇓ ⇑Coarse grid blocks:
⇓ ⇑Flow problems:
Multiscale method:
⇓ ⇑Coarse grid blocks:
⇓ ⇑Flow problems:
Applied Mathematics May 2006 8/20
Advantage: Accuracy10th SPE Comparative Solution Project
Producer A
Producer B
Producer C
Producer D
Injector
Tarbert
UpperNess
Geomodel: 60× 220× 85 ≈ 1, 1 million grid cells
Simulation: 2000 days of production
Applied Mathematics May 2006 9/20
Advantage: AccuracySPE10 Benchmark (5× 11× 17 Coarse Grid)
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
tProducer A
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
t
Producer B
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
t
Producer C
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
t
Producer D
ReferenceMsMFEM Nested Gridding
ReferenceMsMFEM Nested Gridding
ReferenceMsMFEM Nested Gridding
ReferenceMsMFEM Nested Gridding
Nested gridding: upscaling + downscaling
Applied Mathematics May 2006 10/20
Computational Complexity
Direct solution may be more efficient, so why bother with multiscale?
Full simulation: O(102) timesteps.
Basis functions need not berecomputed
Also:
Possible to solve very largeproblems
Easy parallelization8x8x8 16x16x16 32x32x32 64x64x64
0
1
2
3
4
5
6
7
8x 107
Computation of basis functionsSolution of global system
Fine scale solution
Applied Mathematics May 2006 12/20
Flexibility
Multiscale mixed formulation:
coarse grid = union of cells in fine grid
Given a numerical method thatworks on the fine grid, theimplementation is straightforward.
One avoids resampling when goingfrom fine to coarse grid, and viceversa
Other formulations:
Finite-volume methods: based upon dual grid −→ special casesthat complicate the implementation in the presence of faults, localrefinements, etc.
Applied Mathematics May 2006 13/20
Flexibility wrt. GridsFracture Networks
1
1Grid model courtesy of M. Karimi-Fard, Stanford
Applied Mathematics May 2006 17/20
Fast Simulation of Fluid Transport
discontinuous Galerkin
reordering
large time-steps
multiphase transport
tracer flow
delineation of volumes
Applied Mathematics May 2006 18/20
Future Work
Multiscale methods
Well models (adaptive gridding, multilaterals)
More general grids (block-structured, PEBI, ..)
Compressibility, multiphase and multicomponent
Adaptivity
Fractures and faults
Applications:
Multiscale history matching
Carbonate reservoirs..?
CO2..?
Applied Mathematics May 2006 20/20