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Reservoir Simulation:From Upscaling to Multiscale Methods
Knut–Andreas Lie
SINTEF ICT, Dept. Applied Mathematicshttp://www.math.sintef.no/GeoScale
Multiscale Computational Science and Engineering,September 19–21, Trondheim, Norway
Applied Mathematics 21/09/2007 1/47
Reservoir SimulationWhat and why?
Reservoir simulation is the means by which a numerical model ofthe petrophysical characteristics of a hydrocarbon reservoir is usedto analyze and predict fluid behavior in the reservoir over time.
Reservoir simulation is used as a basis for decisions regardingdevelopment of reservoirs and management during production. Tothis end, one needs to
predict reservoir performance from geological descriptions andconstraints,
fit geological descriptions to static and dynamic data,
assess uncertainty in predictions,
optimize production strategies,...
Applied Mathematics 21/09/2007 2/47
Reservoir SimulationWhat are the challenges today?
Reservoir modelling is a true multiscale discipline:
Measurements and models on a large number of scales
Large number of models
Complex grids with a large number of parameters
High degree of uncertainty...
There is always a need for faster and more accurate simulators thatuse all available geological information
Applied Mathematics 21/09/2007 3/47
Physical Scales in Porous Media FlowOne cannot resolve them all at once
The scales that impact fluid flow in oil reservoirs range from
the micrometer scale of pores and pore channels
via dm–m scale of well bores and laminae sediments
to sedimentary structures that stretch across entire reservoirs.
−→
Applied Mathematics 21/09/2007 4/47
Physical Scales in Porous Media FlowMicroscopic: the scale of individual sand grains
Flow in individual pores between sand grains
Applied Mathematics 21/09/2007 5/47
Physical Scales in Porous Media FlowGeological: the meter scale of layers, depositional beds, etc
Porous sandstones often have repetitive layered structures, butfaults and fractures caused by stresses in the rock disrupt flowpatterns
Applied Mathematics 21/09/2007 6/47
Physical Scales in Porous Media FlowReservoir: the kilometer scale of sedimentary structures
Applied Mathematics 21/09/2007 7/47
Physical Scales in Porous Media FlowChoosing a scale for modelling
Applied Mathematics 21/09/2007 8/47
Geological ModelsThe knowledge database in the oil company
Geomodels:
are articulations of the experts’perception of the reservoir
describe the reservoir geometry(horizons, faults, etc)
give rock parameters (e.g.,permeability K and porosity φ)that determine the flow
In the following: the term “geomodel” will designate a grid modelwhere rock properties have been assigned to each cell
Applied Mathematics 21/09/2007 9/47
Flow SimulationModel problem: incompressible, single phase
Consider the following model problem
Darcy’s law: v = −K (∇p− ρg∇D) ,
Mass balance: ∇ · v = q in Ω,
Boundary conditions: v · n = 0 on ∂Ω.
The multiscale structure of porous media enters the equationsthrough the absolute permeability K, which is a symmetric andpositive definite tensor with uniform upper and lower bounds.
We will refer to p as pressure and v as velocity.
Applied Mathematics 21/09/2007 10/47
Flow SimulationThe impact of rock properties
Rock properties are used as parametersin flow models
Permeability K spans many lengthscales and have multiscale structure
maxK/minK ∼ 103–1010
Details on all scales impact flow
Ex: Brent sequence
Tarbert Upper Ness
Challenges:
How much details should one use?
Need for good linear solvers, preconditioners, etc.
Applied Mathematics 21/09/2007 11/47
Flow SimulationGap in resolution and model sizes
Gap in resolution:
High-resolution geomodels may have 106 − 1010 cells
Conventional simulators are capable of about 105 − 106 cells
Traditional solution: upscaling of parameters
Assume that u satisfies the elliptic PDE:
−∇(
K(x)∇u)
= f.
Upscaling amounts to finding a newfield K∗(x) on a coarser grid such that
−∇(
K∗(x)∇u∗)
= f ,
u∗ ∼ u, q∗ ∼ q .
⇓
Applied Mathematics 21/09/2007 12/47
Upscaling Geological ModelsIndustry-standard methods
How do we represent fine-scale heterogeneities on a coarse scale?
Combinations of arithmetic, geometric, harmonic averaging
Power averaging(
1|V |
∫
Va(x)p dx
)1/p
Equivalent permeabilities ( a∗xx = −QxLx/∆Px )
V’
p=1 p=0
u=0
u=0
V
p=1
p=0
u=0 u=0V
V’
Lx
Ly
Applied Mathematics 21/09/2007 13/47
Upscaling Geological ModelsIs it necessary and does one want to do it?
There are many difficulties associated with upscaling
Bottleneck in the workflow
Loss of details
Lack of robustness
Need for resampling for complexgrid models
Not obvious how to extend theideas to 3-phase flows
10 20 30 40 50 60
20
40
60
80
100
120
140
160
180
200
220
2 4 6 8 10
2
4
6
8
10
12
14
16
18
20
22
Need for fine-scale computations?
In the future: need for multiphysics on multiple scales?
Applied Mathematics 21/09/2007 14/47
Fluid Simulations Directly on Geomodels
Research vision:
Direct simulation of complex grid models of highly heterogeneousand fractured porous media - a technology that bypasses the needfor upscaling.
Applications:
Huge models, multiple realizations, prescreening, validation,optimization, data integration, ..
To this end, we seek a methodology that
incorporates small-scale effects into coarse-scale system;
gives a detailed image of the flow pattern on the fine scale,without having to solve the full fine-scale system;
is robust, conservative, accurate, and efficient.
Applied Mathematics 21/09/2007 15/47
Multiscale Pressure SolversEfficient flow solution on complex grids – without upscaling
Basic idea:
Upscaling and downscaling in one step
Pressure varies smoothly and can be resolved on coarse grid
Velocity with subgrid resolution
Applied Mathematics 21/09/2007 16/47
From Upscaling to Multiscale Methods
Standard upscaling:
⇓
Coarse grid blocks:
⇓
Flow problems:
Multiscale method:
⇓
Coarse grid blocks:
⇓
Flow problems:
Applied Mathematics 21/09/2007 17/47
From Upscaling to Multiscale Methods
Standard upscaling:
⇓ ⇑
Coarse grid blocks:
⇓ ⇑
Flow problems:
Multiscale method:
⇓ ⇑
Coarse grid blocks:
⇓ ⇑
Flow problems:
Applied Mathematics 21/09/2007 18/47
The Multiscale Mixed Finite-Element Method
Standard finite-element method (FEM):
Piecewise polynomial approximation to pressure,∫
l∇K∇p dx =∫
lq dx
Mixed finite-element methods (MFEM):
Piecewise polynomial approximations to pressure and velocity
∫
Ω
k−1v · u dx−
∫
Ω
p ∇ · u dx =
∫
Ω
k−1ρg∇D · u dx ∀u ∈ U,
∫
Ω
l ∇ · v dx =
∫
Ω
ql dx ∀l ∈ V.
Multiscale mixed finite-element method (MsMFEM):
Velocity approximated in a (low-dimensional) space V ms designed toembody the impact of fine-scale structures.
Applied Mathematics 21/09/2007 19/47
Multiscale Mixed Finite ElementsGrids and basis functions
Assume we are given a fine grid with permeability and porosityattached to each fine-grid block:
Ti
Tj
We construct a coarse grid, and choose the discretisation spaces Uand V ms such that:
For each coarse block Ti, there is a basis function φi ∈ U .
For each coarse edge Γij , there is a basis function ψij ∈ V ms.
Applied Mathematics 21/09/2007 20/47
(Multiscale) Mixed Finite ElementsDiscretisation matrices (without hybridization)
Saddle-point problem:
(
B CCT 0
)(
vp
)
=
(
fg
)
,
bij =
∫
Ωψik
−1ψj dx,
cij =
∫
Ωφj∇ · ψi dx
Basis φj for pressure: equal one in cell j, zero otherwise
Basis ψi for velocity:
1.order Raviart–Thomas: Multiscale:
Applied Mathematics 21/09/2007 21/47
Multiscale Mixed Finite ElementsBasis for the velocity field
Velocity basis function ψij : unit flowthrough Γij defined as
∇ · ψij =
wi(x), for x ∈ Ti,
−wj(x), for x ∈ Tj ,
and no flow ψij · n = 0 on ∂(Ti ∪ Tj).
Global velocity:
v =∑
ij vijψij , where vij are (coarse-scale) coefficients.
Applied Mathematics 21/09/2007 22/47
Multiscale Simulation versus Upscaling10th SPE Comparative Solution Project
Producer A
Producer B
Producer C
Producer D
Injector
Tarbert
UpperNess
Geomodel: 60 × 220 × 85 ≈ 1, 1 million grid cells,maxKx/minKx ≈ 107, maxKz/minKz ≈ 1011
Simulation: 2000 days of production (2-phase flow)
Commercial (finite-difference) solvers: incapable of running the whole model
Applied Mathematics 21/09/2007 23/47
Multiscale Simulation versus Upscaling10th SPE Comparative Solution Project
Upscaling results reported by industry
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (days)
Wate
rcut
Fine GridTotalFinaElfGeoquestStreamsimRoxarChevron
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (days)
Wate
rcut
Fine GridLandmarkPhillipsCoats 10x20x10
single-phase upscaling two-phase upscaling
Applied Mathematics 21/09/2007 24/47
Multiscale Simulation versus Upscaling10th SPE Comparative Solution Project
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wate
rcut
Producer A
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wate
rcut
Producer B
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wate
rcut
Producer C
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wate
rcut
Producer D
ReferenceMsMFEMNested Gridding
ReferenceMsMFEMNested Gridding
ReferenceMsMFEMNested Gridding
ReferenceMsMFEMNested Gridding
upscaling/downscaling, MsMFEM/streamlines, fine grid
Runtime: 2 min 22 sec on 2.4 GHz desktop PC
Applied Mathematics 21/09/2007 25/47
RobustnessSPE10, Layer 85 (60 × 220 Grid)
Applied Mathematics 21/09/2007 26/47
Comparison of Multiscale and Upscaling Methods
1 Local-global upscaling (Durlofsky et al)
global boundary conditions, iterative improvement (bootstrap)reconstruction of fine-grid velocities
2 Multiscale mixed finite elements (Chen & Hou, . . . )
multiscale basis functions for velocitycoarse-scale pressure
3 Multiscale finite-volume method (Jenny, Tchelepi, Lee,. . . )
multiscale basis functions for pressurereconstruction of velocity on fine grid
4 Numerical subgrid upscaling (Arbogast, . . . )
direct decomposition of the solution, V = Vc ⊕ Vf
RT0 on fine scale, BDM1 on coarse
Applied Mathematics 21/09/2007 27/47
Comparison of Multiscale and Upscaling MethodsSPE 10, individual layers
Saturation errors at 0.3 PVI on 15 × 55 coarse grid
0 10 20 30 40 50 60 70 80 90
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Layer #
δ(S
)
MsMFEM
MsFVM
ALGUNG
NSUM
X
Applied Mathematics 21/09/2007 28/47
Comparison of Multiscale and Upscaling MethodsVelocity errors for Layer 85
MsMFEM: MsFVM:
1120
2244
55110
5
6
10
12
15
20
30
0
0.5
1
1.5
2
1120
2244
55110
5
6
10
12
15
20
30
0
10
20
30
40
δ(v) = 0.80 δ(v) = 4.93
ALGUNG: NSUM:
1120
2244
55110
5
6
10
12
15
20
30
0
0.5
1
1.5
2
2.5
1120
2244
55110
5
6
10
12
15
20
30
0
1
2
3
4
δ(v) = 1.16 δ(v) = 1.49
Applied Mathematics 21/09/2007 29/47
Comparison of Multiscale and Upscaling MethodsAverage saturation errors on Upper Næss formation (Layers 36–85)
Cartesian coarse grids:
Multiscale methods give enhanced accuracy when subgridinformation is exploited.
5x11 10x22 15x55 30x1100.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
MsMFEM
MsFVM
ALGUNG
NSUM
PUPNG
HANG
5x11 10x22 15x55 30x1100.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
MsMFEM
MsFVM
ALGUNG
NSUM
PUPNG
HANG
Fluid transport: coarse grid Fluid transport: fine grid
Applied Mathematics 21/09/2007 30/47
Comparison of Multiscale and Upscaling MethodsMsMFEM versus upscaling on complex coarse grids
Complex coarse grid-block geometries:
MsMFEM is more accurate than upscaling, alsofor coarse-grid simulation.
3 x 3 x 3 5 x 5 x 5 10 x 10 x 10 15 x 15 x 15 30 x 30 x 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
MsMFEM
A−UP
G−UP
H−UP
3 x 3 x 3 5 x 5 x 5 10 x 10 x 10 15 x 15 x 15 30 x 30 x 300
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
MsMFEM
A−UP
G−UP
H−UP
Coarse-grid velocity errors Coarse-grid saturation errors
Up-gridded 30 × 30 × 333 corner-point grid with layered log-normal permeability
Applied Mathematics 21/09/2007 31/47
Computational ComplexityOrder-of-magnitude argument
Assume:
Grid model with N = Ns ∗Nc cells:
Nc number of coarse cellsNs number of fine cells in each coarse cell
Linear solver of complexity O(mα) for m×m system
Negligible work for determining local b.c., numericalquadrature, and assembly (can be important, especially for NSUM)
Direct solution
Nα operations for a two-point finite volume method
MsMFEM
Computing basis functions: D ·Nc · (2Ns)α operations
Solving coarse-scale system: (D ·Nc)α operations
Applied Mathematics 21/09/2007 32/47
Computational ComplexityExample: 128 × 128 × 128 fine grid
0
0.5
1
1.5
2
2.5
3
x 108
MsMFEM
NSUM
MsFVM
ALGUNG
MsMFEM
NSUM
MsFVM
ALGUNG
MsMFEM
NSUM
MsFVM
ALGUNG
MsMFEM
NSUM
MsFVM
ALGUNG
Nc = 8
3Nc = 16
3Nc = 32
3Nc = 64
3
Local work
Global work
Fine scale solution
Comparison with algebraic multigrid (AMG), α = 1.2
Applied Mathematics 21/09/2007 33/47
Computational ComplexityExample: 128 × 128 × 128 fine grid
0
1
2
3
4
5
x 109
MsMFEM
NSUM
MsFVM
ALGUNG
MsMFEM
NSUM
MsFVM
ALGUNG
MsMFEM
NSUM
MsFVM
ALGUNG
MsMFEM
NSUM
MsFVM
ALGUNG
Nc = 8
3Nc = 16
3Nc = 32
3Nc = 64
3
Fine scale solution
Local work
Global work
Comparison with less efficient solver, α = 1.5
Applied Mathematics 21/09/2007 34/47
Multiphase FlowTime-dependent problems: ∇(K(x)λ(S)∇p) = q(S)
Direct solution may be more efficient, so why bother with multiscale?
Full simulation: O(102) timesteps.
Basis functions need notalways be recomputed
Also:
Possible to solve very largeproblems
Easy parallelization8x8x8 16x16x16 32x32x32 64x64x64
0
1
2
3
4
5
6
7
8x 10
7
Computation of basis functions
Solution of global system
Fine scale solution
Applied Mathematics 21/09/2007 35/47
Two-Phase FlowExample: quarter five-spot, Layer 85 from SPE 10, coarse grid: 10 × 22
Water cuts obtained by never updating basis functions:
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PVI
Water−Cut
Reference
MsMFEM (δ(ω) = 0.1551)
MsFVM (δ(ω) = 0.0482)
Ms−NSUM (δ(ω) = 0.1044)
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PVI
Water−Cut
Reference
MsMFEM (δ(ω) = 0.0077)
MsFVM (δ(ω) = 0.0146)
Ms−NSUM (δ(ω) = 0.0068)
favorable (M = 0.1) unfavorable (M = 10.0)
Applied Mathematics 21/09/2007 36/47
Two-Phase FlowExample: quarter five-spot, Layer 85 from SPE 10, coarse grid: 10 × 22
Improved accuracy by adaptive updating of basis functions:
0.65 0.7 0.75 0.8 0.85 0.90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
PVI
Water−Cut
Reference
MsMFEM (δ(ω) = 0.1551)
MsFVM (δ(ω) = 0.0482)
Ms−NSUM (δ(ω) = 0.1044)
0.65 0.7 0.75 0.8 0.85 0.90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Water−Cut
PVI
Reference
MsMFEM (δ(ω) = 0.031)
Ms−NSUM (δ(ω) = 0.036)
no updating adaptive updating
Applied Mathematics 21/09/2007 37/47
Application: History Matching on Geological Models
Assimilation of production data to calibrate model
1 million cells, 32 injectors, and 69producers
2475 days ≈ 7 years of water-cut data
6 iterations in data integration method
7 forward simulations, 15 pressureupdates each
Computation time (on desktop PC):
Original method: ∼ 40 min (pressure solver: 30 min)
Multiscale method: ∼ 17 min (pressure solver: 7 min)
Applied Mathematics 21/09/2007 38/47
Geological Models as Direct Input to Simulation‘Medium-fitted’ grids to model complex reservoir geometries
Another challenge:
Industry-standard grids are often nonconforming and containskewed and degenerate cells
There is a trend towards unstructured grids
Standard discretization methods produce wrong results onskewed and rough cells
Corner point: Tetrahedral: PEBI:
Applied Mathematics 21/09/2007 39/47
Corner-Point GridsIndustry standard for modelling complex reservoir geology
Specified in terms of:
areal 2D mesh of vertical orinclined pillars
each volumetric cell is restriced byfour pillars
each cell is defined by eight cornerpoints, two on each pillar
Applied Mathematics 21/09/2007 40/47
Discretisation on Corner-Point GridsExotic cell geometries from a simulation point-of-view
Skew and deformed gridblocks:
Non-matching cells:
Can use standard MFEM provided that one has mappings andreference elements
Can subdivide corner-point cells into tetrahedra
We use mimetic finite differences (recent work by Brezzi,Lipnikov, Shashkov, Simoncini)
Applied Mathematics 21/09/2007 41/47
Discretisation on Corner-Point GridsMimetic finite differences, hybrid of MFEM and multipoint FVM
Let u, v be piecewise linear vector functions and u, v be thecorresponding vectors of discrete velocities over faces in the grid,i.e.,
vk =1
|ek|
∫
ek
v(s) · nds
Then the block B in the mixed system satisfies
∫
ΩvTK−1u = v
TBu
(
=∑
E∈Ω
vTEBEuE
)
The matrices BE define discrete inner products
Mimetic idea:
Replace BE with a matrix ME that mimics some properties of thecontinuous inner product (SPD, globally bounded, Gauss-Green forlinear pressure)
Applied Mathematics 21/09/2007 42/47
Mimetic Finite Difference MethodsGeneral method applicable to general polyhedral cells
Standard method + skew grids = grid-orientation effects
K: homogeneous and isotropic,symmetric well pattern−→ symmteric flow
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Water−cut curves for two−point FVM
PVI0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Water−cut curves for mimetic FDM
PVI
Streamlines with standard method Streamlines with mimetic method
Applied Mathematics 21/09/2007 43/47
Multiscale Mixed Finite ElementsAn automated alternative to upscaling?
Coase grid = union of cells from fine grid
MsMFEMs allow fully automated coarse gridding strategies: gridblocks need to be connected, but can have arbitrary shapes.
Uniform up-gridding: grid blocks are shoe-boxes in index space.
Model is courtesy of Alf B. Rustad, Statoil
Applied Mathematics 21/09/2007 44/47
Multiscale Mixed Finite ElementsExamples of exotic grids
Applied Mathematics 21/09/2007 45/47
Multiscale Mixed Finite ElementsIdeal for coupling with well models
Fine grid to annulus, one coarse block for each well segment =⇒no well model needed.
Applied Mathematics 21/09/2007 46/47
SummaryAdvantages of multiscale mixed/mimetic pressure solvers
Ability to handle industry-standard grids
highly skewed and degenerate cells
non-matching cells and unstructured connectivities
Compatible with current solvers
can be built on top of commercial/inhouse solvers
can utilize existing linear solvers
More efficient than standard solvers
faster and requires less memory than fine-grid solvers
automated generation of coarse simulation grids
easy to parallelize
Applied Mathematics 21/09/2007 47/47