Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline...

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Adaptive Multiscale Streamline Simulation and Inversion 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 Applied Mathematics Sept 2008 1/19

Transcript of Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline...

Page 1: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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

Applied Mathematics Sept 2008 1/19

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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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Page 6: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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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Page 7: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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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Page 8: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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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1

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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Page 9: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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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Page 10: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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

Applied Mathematics Sept 2008 9/19

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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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50Stacked sensitivities

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−1

0

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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

3 3

3 3

Cell i

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tion

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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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er−c

ut

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wat

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ut

obsinitmatched

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1P4

daysw

ater

−cut

obsinitmatched

worst match best match least data

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Page 17: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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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Page 18: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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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Page 19: Adaptive Multiscale Streamline Simulation and Inversion ... · Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels K.–A. Lie† V. R. Stenerud‡

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

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Producer P3

observedinitialmatched

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Producer P4

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Producer P5

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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

. . .

Applied Mathematics Sept 2008 19/19