Adaptive Multiscale Streamline Simulation and Inversion for - - PowerPoint PPT Presentation

Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels

Vegard Røine Stenerud† and Knut–Andreas Lie‡

† NTNU, Department of Mathematical Sciences ‡ SINTEF ICT, Dept. Applied Mathematics

February 12, 2008

Applied Mathematics Feb 2008 1/18

Introduction: History matching

History matching is the procedure of modifying the reservoir description to match measured reservoir responses.

Initial: Matched: Reference:

Applied Mathematics Feb 2008 2/18

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)

Applied Mathematics Feb 2008 3/18

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-determined problem → non-uniqueness errors in model, data, and methods nonlinear forward model non-convex misfit functions forward simulations are computationally demanding

Applied Mathematics Feb 2008 4/18

Challenge I: Non-convex misfit function

Inversion method: Generalized Travel-Time Inversion (GTTI) with analytic sensitivities

[Vasco et al. (1999), He et al. (2002)]

The generalized travel time is defined as the ’optimal’ time–shift that maximizes R2(∆t) = 1 − [yobs(ti + ∆t) − ycal(ti)]2 [yobs(ti) − ¯ yobs(ti)]2 .

Applied Mathematics Feb 2008 5/18

Travel-time inversion

Basic underlying principles for the history–matching algorithm Minimize travel-time misfit for water–cut by iterative least-square minimization algorithm. Preserve geologic realism by keeping changes to prior geologic model minimal (if possible). Only allow smooth large-scale changes. Production data have low resolution and cannot be used to infer small-scale variations. Minimization of functional:

∆˜ t : Travel–time shift S : Sensitivity matrix m : Reservoir parameters

∆˜ t − SδR +

Regularization

• β1δR

norm

+ β2LδR

• smoothing

S computed analytically along streamlines from a single flow simulation

Applied Mathematics Feb 2008 6/18

Streamline-based history matching

Features of streamlines Very well suited for modeling large heterogeneous multi-well systems dominated by convection Generally fast flow simulation Delineate flow pattern (injector-producer pairs) Enables analytic sensitivities

Source: www.techplot.com

Streamline-based history-matching methods Assisted history matching (Generalized) travel-time inversion methods Streamline-effective properties methods Miscellaneous

Applied Mathematics Feb 2008 7/18

Example: Uncertainty quantification

Simple two-phase model (end-point mobility M = 0.5) on a 2D horizontal reservoir, lognormal permeability

500 1000 1500 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 days fw Pre data integration: P2

• bs

init 500 1000 1500 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 days fw Post data integration: P2

• bs

matched

Statistical analysis of mean and standard deviation

Applied Mathematics Feb 2008 8/18

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 faster than conventional FD-methods. Still, room for improvement. Observations: pressure solver most expensive part of simulation data changes very little from one simulation to the next

Applied Mathematics Feb 2008 9/18

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 faster than conventional FD-methods. Still, room for improvement. Observations: pressure solver most expensive part of simulation data changes very little from one simulation to the next Reuse computations in areas with minor changes − → multiscale methods

Applied Mathematics Feb 2008 9/18

Multiscale pressure solver

Upscaling and downscaling in one step. Runtime like coarse-scale solver, resolution like fine-scale solver.

Fine grid: 75 × 30. Coarse grid: 15 × 6 Basis functions for each pair

• f coarse blocks Ti ∪ Tj :

Ψij = −λK∇Φij ∇ · Ψij =

• wi(x),

x ∈ Ti −wj(x), x ∈ Tj Global linear system with 249 unknowns: ∇·v = q, v = −λK∇p Coarse grid: pressure and fluxes. Fine grid: fluxes Applied Mathematics Feb 2008 10/18

Multiscale methods: efficiency vs accuracy

Ex: q5-spot, SPE 10 (layer 85)1, 60 × 220 → 10 × 22

Water cuts obtained by never updating basis functions:

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PVI Water−Cut Reference Multiscale 0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PVI Water−Cut Reference Multiscale

favorable (M = 0.1) unfavorable (M = 10.0) 1: Fluvial permeability field,

Applied Mathematics Feb 2008 11/18

Multiscale methods: efficiency vs accuracy

Ex: q5-spot, SPE 10 (layer 85)1, 60 × 220 → 10 × 22

Improved accuracy by adaptive updating of basis functions:

0.65 0.7 0.75 0.8 0.85 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 PVI Water−Cut Reference Multiscale 0.65 0.7 0.75 0.8 0.85 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Water−Cut PVI Reference Multiscale

no updating adaptive updating 1: Fluvial permeability field,

Applied Mathematics Feb 2008 12/18

Further computational savings

Can also reuse basis functions from previous forward simulation. General idea: use sensitivity to steer updating

10 20 30 40 50 5 10 15 20 25 30 35 40 45 50 Stacked sensitivities −8 −7 −6 −5 −4 −3 −2 −1

Applied Mathematics Feb 2008 13/18

History matching on heological models

Generalized travel-time inversion on million-cell model

Assimilation of production data to calibrate model 1 million cells, 32 injectors, and 69 producers 2475 days ≈ 7 years of water-cut data

Analytical sensitivities along streamlines + travel-time inversion (quasi-linearization of misfit functional) Time-residual Amplitude-residual

Computation time: ∼ 17 min on a desktop PC (6 iterations).

Applied Mathematics Feb 2008 14/18

History matching on geological models

Residuals and timing results, Intel Core 2 Duo (2.4 GHz, 4Mb cache)

Misfit CPU-time (wall clock) Solver O/M T A ∆ ln k Total Pres. Transp. Initial — 100.0 100.0 0.821 — — —

• Std. (7 pt.)

O 8.9 53.5 0.806 64 min 33 min 28 min

• Std. (7 pt.)

M 9.6 50.4 0.806 39 min 30 min 5 min Multiscale O 11.2 47.3 0.812 43 min 7 min 32 min Multiscale M 10.4 45.4 0.828 17 min 7 min 6 min Misfit: Time-shift misfit ∆t2 Amplitude misfit [

k

• j(f obs

w

− f cal

w )2]1/2

Permeability discrepancy 1/N N

i=1 | ln kref i

− ln kmatch

i

|

Applied Mathematics Feb 2008 15/18

Robustness with respect to data reduction

Uncertainty quantification, revisited

25 50 75 100 TPFA 25 50 75 100 TPFA 2 4 6 8 10 12 % Initial Update Mean time−shift residual % Dynamic Update % of initial 25 50 75 100 TPFA 25 50 75 100 TPFA 10 20 30 40 50 % Initial Update Mean amplitude residual % Dynamic Update % of initial 25 50 75 100 TPFA 25 50 75 100 TPFA 20 40 60 80 100 % Initial Update Mean average discrepancy of log(K) % Dynamic Update % of initial 25 50 75 100 TPFA 25 50 75 100 TPFA 2 4 6 8 % Initial Update

• Std. time−shift residual

% D y n a m i c U p d a t e % of initial 25 50 75 100 TPFA 25 50 75 100 TPFA 5 10 15 20 % Initial Update

• Std. amplitude residual

% Dynamic Update % of initial 25 50 75 100 TPFA 25 50 75 100 TPFA 2 4 6 8 10 % Initial Update

• Std. average discrepancy of log(K)

% Dynamic Update % of initial

Applied Mathematics Feb 2008 16/18

Robustness with respect to data reduction

Million-cell model, revisited

Reduction in residuals

25 50 75 100 TPFA 25 50 75 100 TPFA 5 10 15 % Initial Update Time−Shift Residual % Dynamic Update % of initial 25 50 75 100 TPFA 25 50 75 100 TPFA 10 20 30 40 50 60 % Initial Update Amplitude Residual) % Dynamic Update % of initial 25 50 75 100 TPFA 25 50 75 100 TPFA 20 40 60 80 100 120 % Initial Update Average Discrepancy of log(K) % D y n a m i c U p d a t e % of initial

Corresponding speedup:

25 50 75 100 TPFA 25 50 75 100 TPFA 2 4 6 8 10 % Initial Update % Dynamic Update Speed−up

Applied Mathematics Feb 2008 17/18

Extentions

Unstructured grids (done for inversion algorithm) Corner-point grids (testing to be done on Norne-model) Other types of data / more general flow Inclusion of seismics Use of sensitivities for other optimization workflows . . .

Applied Mathematics Feb 2008 18/18