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### Chaos, Fluid Mixing, Uncertainty

James Glimm

Department of Applied Mathematics & Statistics

State University of New York

Stony Brook, NY 11794-3600

Center for Data Intensive Computing

Brookhaven National Laboratory

Upton, NY 11973-5000

- with -

D. Sharp, B. Cheng, X.L. Li, H. Jin, D. Saltz, Z. Xu, F. Tangerman,

M. Laforest, A. Marchese, E. George, Y. Zhang, S. Dutta, H. J. Kim, Y. Lee,

K. Ye, S. Hou

Chaos and Prediction

Chaos: Sensitive dependence on data, for example, initial conditions

Result: Predictability lost

Cure: Predict averages, statistics, probabilities

Result: Predictability regained

Examples

1. Fluid Mixing

- acceleration of density contrasting layers

- unknown and sensitive initial data

2. Flow in Porous Media

- unknown heterogeneous geology

Prediction for Multiscale Chaos

- Fine scale features influence macroscopic flow

- Multiscale defines frontier problems in many areas of science

Basic Methods

I. Fine Scale Science

- Study single unstable mode

- Study mode coupling and interactions

- Analytic methods

- DNS = direct numerical simulation

Basic Methods

II. Micro-Macro Coupling

- Statistical models of mode interactions

- Statistical analysis of DNS

III. Macro Theories

- Averaged equations

- Validated by comparison to I, II

- Validated by comparison to experiment

- Mathematical analysis of averaged equations

Basic Methods

IV. Prediction and Uncertainty

- Errors in micro-macro approximation

- Errors in numerics and experiment

- Observational/exp. data: reduce uncertainty

- Statistical inference: quantify uncertainty

Fluid Mixing Simulation

Early time FronTier simulation of Late time FronTier simulation of a

3D RT mixing layer. 3D RT mixing layer.

Comparison to Laboratory Experiments

penetration distance of light fluid into the heavy fluid (bubbles)

FronTier and TVD Simulations Compared with Experiment

b = 0.06-0.07 FronTier (above)

b = 0.03-0.04 TVD (below)

b = 0.05 Experiment (Dimonte)

b = 0.06 - 0.07 (Youngs)

Comparison of Simulations

FT nondiffusive across interface

TVD diffusive across interface

Mass diffusion appears to cause

~ 50% error in mixing rate.

Simulation in Spherical Geometry

Cross-sectional view of the growth of instability in a randomly perturbed axisymmetric sphere driven by an imploding shock wave in air. The

shock Mack number M is 1.2 and the Atwood number A is 2/3.

Spherical Mix, Later Time

Later time in the instability evolution.

Analytic Models

Statistical Models of Interacting Bubbles

Bubble Merger Models

Advanced

bubble

Retarded

bubble

Advanced

bubble

Bubble velocity = single mode velocity + envelope velocity

Analytic Models

Statistical Models of Interacting Bubbles

Bubble Merger Models

Advanced

bubble

Retarded

bubble

Advanced

bubble

Bubble velocity = single mode velocity + envelope velocity

Bubble Merger Criterion

Envelope velocity > 0 advanced bubble

Envelope velocity < 0 retarded bubble

Remove bubble from ensemble where velocity = 0:

single mode velocity = envelope velocity

Statistical Physics Model of Mixing Rate

Solve statistical bubble model at renormalization group fixed point

B. Cheng, J. Glimm, D. Sharp

b 0.05 - 0.06

COM Hypothesis

Stationary Center of Mass (COM)

(approx. valid unless A 1)

hs = penetration of heavy fluid into light

hs = s Agt2 (RT)

COM s / b = solution of quadratic equation

s = s (b)

B. Cheng, J. Glimm, D. Saltz, D. Sharp

Mixing Zone Edge Models

Zb,s (t) = mixing zone edge

= hb,s in RT case

= bs Agt2in RT case

Buoyancy Drag equation for Zb,s (t):

Determine Cb,s from FT theory above

ODE valid for arbitrary acceleration

Chunk Mix Model

• Complete fluid variables for each fluid

-- Mathematically stable equations

• Improved physics model for mix

-- Pressure difference forces ~ drag

• Thermodynamics is process independent

• New closure proposed and tested

-- Zero parameters (incompressible flow)

• Analytic solution for incompressible case

Ensemble Averages

Assume two fluids, labeled k=1 (light) and k=2 (heavy). Define

Closure

Assume: v* depends on v1 and v2 and spatially dimensionless quantities only.

Assume: regularity of v*.

Theorem:

(convex combination) and related relations for

p* and (pv)*

Assume: all ’s depend on k and t only.

Explicit Model: Zero Parameters (incompressible) One Parameter (compressible)

Assume are fractional linear in k. Then

with k’ denoting the other fluid index and

With the mixing zone boundaries Zk(t), and velocities Vk(t),

for incompressible flow. Boundary accelerations must be

must be supplied externally to this model.

Drag + buoyancy

Asymptotic Expansion in Powers ofM = Mach Number

0th order = incompressible v,

1st order = correction v,

2nd order = incompressible p1, p2

+ v, correction

2nd order p1, p2 = incompressible p1, p2

constraint:

“missing” incompressible pressure equation

Summary: Multiscale Science

Multiple Methods to Solve Chaotic Mix Problem

-- Analytic Methods

-- Microscopic Simulation (DNS)

-- Edge Motion Models

-- Averaged Solutions

Closed Form Solutions

Asymptotics

Numerical Solutions

Summary: Multiscale Science

Scientific Understanding:

Consistent theory, experiment, simulation

Still Needed:

Other mix/chaos problems

Transients, shock waves

Validation, comparison to other closures

FronTier Lite

Tech transfer to other codes

Prediction and Uncertainty for Chaotic Flow

- Prediction of Oil Reservoir Production
- Confidence Intervals
- Allows Evaluation of Risk in Decision Making
- Reservoir Development Choices
- Sizing of Production Equipment
- Location of New Wells

Basic Idea: I

- Match geology to past oil production datai with probability of error
- Start with geostatistical probability model for geology (permeability, etc).
- Observe production rates, etc.

Basic Idea: II

- Multiple simulations from ensemble
- (Re)Assign probabilities based on data, degree of mismatch of simulation to history

Basic Idea: III

Redefine probabilities and ensemble to be consistent with:

(a) data

(b) probable errors in simulation and data

Basic Idea: IV

New ensemble of geologies = Posterior

Prediction = sample from posterior

Confidence intervals come from

- posterior probabilities

- errors in forward simulation

New Result:

Predict outcomes and risk

Risk is predicted quantitatively

Risk prediction is based on

- formal probabilities of errors

in data and simulation

- methods for simulation error analysis

- Rapid simulation (upscale) allowing

exploration of many scenarios

Problem Formulation

Simulation study:

Line drive, 2D reservoir

Random permeability field

log normal, random correlation length

Upscaling

Solution from fine grid

100 x 100 grid

Solution by upscaling

20 x 20, 10 x 10, 5 x 5

Upscaled grids

Upscaling by

Wallstrom, Hou, Christie, Durlofsky, Sharp

1. Computational Geoscience 3:69-87 (1999)

2. SPE 51939

3. Transport in Porous Media (submitted)

Examples of Upscaled, Exact Oil Cut Curves

Scale-up: Black (fine grid) ,

Red (20x20), Blue (10x10), Green (5x5)

Select one geology as exact.

Observe production for

Assign revised probabilities to all

500 geologies in ensemble based on:

(a) coarse grid upscaled solutions

(b) probabilities for coarse grid errors.

Compared to data (from “exact” geology)

Example:

Fig. 1 Typical errors (lower, solid curves) and discrepancies

(upper, dashed curves), plotted vs. PVI. The two families of

curves are clearly distinguishable.

In Bayes Theorem, assume is exact.

Then, is an error, and probability

Prediction based on

(a) Geostatistics only, no history match (prior).

Average over full ensemble

(b) History match with upscaled solutions (posterior). Bayesian weighted average over ensemble.

(c) Window: select all fine grid solutions “close” to exact over past history.

Average over restricted ensemble.

Window prediction is best,

but not practical

-uses fine grid solutions for complete ensemble

Prior prediction is worst

- makes no use of production data.

Prediction error reduction, as

per cent of prior prediction

choose present time to be oil cut of 0.6

Window based error reduction: 50%

(fine grid: 100 x 100)

Upscaled error reduction:

5 x 5 23%

10 x 10 32%

20 x 20 36%

Confidence Intervals

5% - 95% interval in future oil production

Excludes extreme high-low values with 5%

probability of occurrence

Expressed as a per cent of predicted

production

Predicted future production confidence interval

prediction + 13%

- 28%

Prediction depends on reservoir

% depends on reservoir

% averaged over choice of reservoir

Summary and Conclusions

- New method to assess risk in prediction of future oil production
- New methods to assess errors in simulations as probabilities
- New upscaling allows consideration of ensemble of geology scenarios
- Bayesian framework provides formal probabilities for risk and uncertainty

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