Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline...
Transcript of Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline...
Adaptive Multiscale Streamline Simulation andInversion for High-Resolution Geomodels
K.–A. Lie† V. R. Stenerud‡ A. F. Rasmussen†
† SINTEF ICT, Dept. Applied Mathematics
‡ StatoilHydro
ECMOR XI, Bergen, September 8–11, 2008
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Introduction: History matching
History matching is the procedure of modifying the reservoirdescription to match measured reservoir responses.
Initial: Matched: Reference:
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Introduction: History-matching loop
Evaluate misfit
No
Current reservoir parameters
Flow simulation
(observed - calculated)
Is misfit small enough
HM method/Inversion
Yes
E =∑
(dobs−dcal)2, dcal = g(m)
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1Time−shift residual
iteration
Rel
ativ
e re
sidu
al
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Challenges in history-matching loop
Evaluate misfit
No
Current reservoir parameters
Flow simulation
(observed - calculated)
Is misfit small enough
HM method/Inversion
Yes
Problems:
highly under-determinedproblem → non-uniqueness
errors in model, data, andmethods
nonlinear forward model
non-convex misfit functions
forward simulations arecomputationally demanding
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Challenge I: Non-convex misfit function
Inversion method: Generalized Travel-Time (GTT) inversion withanalytic sensitivities [Vasco et al. (1999), He et al. (2002)]
The generalized travel time is defined as the ’optimal’ time–shift thatmaximizes
R2(∆t) = 1−∑
[yobs(ti + ∆t)− ycal(ti)]2∑
[yobs(ti)− yobs(ti)]2.
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Travel-time inversion
Basic underlying principles for the history–matching algorithm
Minimize travel-time misfit for water–cut by iterative least-squareminimization algorithm.
Preserve geologic realism by keeping changes to prior geologicmodel minimal (if possible).
Only allow smooth large-scale changes. Production data have lowresolution and cannot be used to infer small-scale variations.
Minimization of functional:
∆t : Travel–time shiftS : Sensitivity matrixm : Reservoirparameters
‖∆t− SδR‖+
Regularization︷ ︸︸ ︷β1‖δR‖︸ ︷︷ ︸
norm
+β2‖LδR‖︸ ︷︷ ︸smoothing
S computed analytically along streamlines from a single flow simulation
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Streamline-based history matching
Features of streamlines
Very well suited for modeling largeheterogeneous multiwell systemsdominated by convection
Generally fast flow simulation
Delineate flow pattern(injector-producer pairs)
Enables analytic sensitivities Source: www.techplot.com
Classes of streamline-based history-matching methods
Assisted history matching
(Generalized) travel-time inversion methods
Streamline-effective property methods
Miscellaneous
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Example: Uncertainty quantification
Simple two-phase model (end-point mobility M = 0.5) on a 2Dhorizontal reservoir, 25× 25 cells with lognormal permeability. Resultafter eight iterations:
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days
fw
Pre data integration: P2
obsinit
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daysfw
Post data integration: P2
obsmatched
Statistical analysis of mean and standard deviation
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Challenge II: Long runtime for forward simulations
Evaluate misfit
No
Current reservoir parameters
Flow simulation
(observed - calculated)
Is misfit small enough
HM method/Inversion
Yes
Streamline simulation much fasterthan conventional FD-methods.
Still, room for improvement.
Observations:
pressure solver most expensivepart of flow simulation
parameters change very littlefrom one flow simulation tothe next
Idea: should reuse computationsin areas with minor changes−→ multiscale methods
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Challenge II: Long runtime for forward simulations
Evaluate misfit
No
Current reservoir parameters
Flow simulation
(observed - calculated)
Is misfit small enough
HM method/Inversion
Yes
Streamline simulation much fasterthan conventional FD-methods.
Still, room for improvement.
Observations:
pressure solver most expensivepart of flow simulation
parameters change very littlefrom one flow simulation tothe next
Idea: should reuse computationsin areas with minor changes−→ multiscale methods
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Multiscale pressure solver
Upscaling and downscaling in one step. Runtime like coarse-scalesolver, resolution like fine-scale solver.
Fine grid: 75× 30. Coarse grid: 15× 6Basis functions for each pairof coarse blocks Ti ∪ Tj :
Ψij = −λK∇Φij
∇ ·Ψij =
{wi(x), x ∈ Ti
−wj(x), x ∈ Tj
Global linear systemwith 249 unknowns:
∇·v = q, v = −λK∇p
Coarse grid: pressure and fluxes. Fine grid: fluxes
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Further computational savings
Can also reuse basis functions from previous forward simulation.General idea: use sensitivities to steer updating
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History matching on geological modelsGeneralized travel-time inversion on million-cell model
1 million cells, 32 injectors, and 69 producers2475 days ≈ 7 years of water-cut data
Analytical sensitivities along streamlines + travel-timeinversion (quasi-linearization of misfit functional)
Misfit CPU-time (wall clock)Solver T A ∆ ln k Total Pres. Transp.Initial 100.0 100.0 0.821 — — —TPFA 8.9 53.5 0.806 64 min 33 min 28 minMultiscale 11.2 47.3 0.812 43 min 7 min 32 minMultiscale 10.4 45.4 0.828 17 min 7 min 6 min
Time-shift misfit: ‖∆t‖2
Amplitude misfit: [∑
k
∑j(d
obsw − dcal
w )2]1/2
Permeability discrepancy:∑N
i=1 | ln krefi − ln kmatch
i |/N
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Extension to unstructured gridsPrimary concern: how to define smoothing stencil
Generalized Laplacian stencil
Li m = −wiimi +∑
j∈N (i) wjimj ,
wji = wnorm · ρ(ζ(i, j);R), wii =∑
j∈N (i) wji.
Neighborhood: N (i) Distance: ζ(i, j) Correlation: ρ(ζ; R, . . . )
Cell i
1 1
1
2 2
2 2
2 2
3 3 3
3 3
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3 3
Cell i
1 1
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2 2
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3 3 3
3 3
3 3
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1
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Cor
rela
tion
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1.0
Properties of smoothing: independent of grid density, reduce toLaplacian on Cartesian grids, decay with distance ζ(i, j), be zerooutside finite range, be bounded as ζ → 0.
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Extension to unstructured gridsSecond concern: sensitivities
Sensitivities are spatially additive.
Small/large cells −→ small/large sensitivities
Thus, grid effects are to be expected
sensitivity sensitivity density equisized grid
Rescaling of sensitivities:
Permeability modification ∆Ki scales with sensitivity Gi.Splitting cell in two =⇒ ∆K →∼ 1
2∆K in each subcell
Therefore: scale sensitivity by relative volume, Gi = Gi(V /Vi)
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Example:1200 days observed (5% noise added), 800 matched, 400 predicted
reference sensitivity rescaled sensitivity
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wat
er−c
ut
obsinitmatched
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obsinitmatched
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daysw
ater
−cut
obsinitmatched
worst match best match least data
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Example:Grid effects for models with thin layers
Synthetic test case:
21× 21× 7 tensor-product grid with layers of varying thickness.
Layer 1 2 3 4 5 6 7Thickness 1.0 0.089 0.164 0.212 0.251 0.285 1.0
Initial model: two different constant valuesTrue model: homogeneous
K from original Gij K obtained from scaled Gij
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Example:Corner-point grid with two non-sealing strike-slip faults
Infill drilling case:
Production data from 3000 days, 2500 used in history match.At 900 days:
new producer P5 drilled
producer P4 converted to injector
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Example:Corner-point grid with two non-sealing strike-slip faults, cont’d
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Producer P1
observedinitialmatched
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Producer P2
observedinitialmatched
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Producer P3
observedinitialmatched
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Producer P4
observedinitialmatched
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Producer P5
observedinitialmatched
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Extensions and similar ideas
Direct continuation of previous work:
Unstructured grids (done for inversion algorithm)
Corner-point grids (testing remains to be done on real models)
Other types of data / more general flow
Other possible directions using streamlines:
Closed-loop reservoir management
History-matching seismic data
Use of sensitivities for other optimization workflows
. . .
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