Chaos, Fluid Mixing, Uncertainty

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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

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Chaos and Prediction

• Chaos Sensitive dependence on data, for example,
  initial conditions
• Result Predictability lost
• Cure Predict averages, statistics,
  probabilities
• Result Predictability regained

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Examples

• 1. Fluid Mixing
• - acceleration of density contrasting layers
• - unknown and sensitive initial data
• 2. Flow in Porous Media
• - unknown heterogeneous geology

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Prediction for Multiscale Chaos

• - Fine scale features influence macroscopic
  flow
• - Multiscale defines frontier problems in many
  areas of science

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Basic Methods

• I. Fine Scale Science
• - Study single unstable mode
• - Study mode coupling and interactions
• - Analytic methods
• - DNS direct numerical simulation

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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

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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

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Fluid Mixing Simulation
Early time FronTier simulation of Late time
FronTier simulation of a 3D RT mixing
layer. 3D RT mixing layer.
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Comparison to Laboratory Experiments

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

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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)
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Comparison of Simulations

• FT nondiffusive across interface
• TVD diffusive across interface
• Mass diffusion appears to cause
• 50 error in mixing
  rate.

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Cross Section Mass Diffusion
Comparison of TVD and FronTier
TVD
FronTier
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Simulation of two immersed bubbles
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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.
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Spherical Mix, Later Time
Later time in the instability evolution.
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Analytic Models

• Statistical Models of Interacting Bubbles
• Bubble Merger Models

Advanced bubble
Retarded bubble
Advanced bubble
Bubble velocity single mode velocity envelope
velocity
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Analytic Models

• Statistical Models of Interacting Bubbles
• Bubble Merger Models

Advanced bubble
Retarded bubble
Advanced bubble
Bubble velocity single mode velocity envelope
velocity
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Bubble Merger Criterion

• Envelope velocity gt 0 advanced bubble
• Envelope velocity lt 0 retarded bubble
• Remove bubble from ensemble where velocity 0
• ?single mode velocity? ?envelope velocity ?

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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

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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

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Mixing Zone Edge Models

• Zb,s (t) mixing zone edge
• hb,s in RT case
• ?bs Agt2 in RT case
• Buoyancy Drag equation for Zb,s (t)
• Determine Cb,s from FT theory above
• ODE valid for arbitrary acceleration

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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

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Multiphase Averaged Equations

• Microphysics
• Macrophysics
• Closure Problem Determine Fren

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Ensemble Averages

• Assume two fluids, labeled k1 (light) and k2
  (heavy). Define

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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.

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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

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Analytic Solution Incompressible Case
Let
Then
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Pressure versus distance
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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

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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

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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

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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

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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.

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Basic Idea II

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

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Basic Idea III
Redefine probabilities and ensemble to be
consistent with (a) data (b) probable
errors in simulation and data
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Basic Idea IV
New ensemble of geologies Posterior Predicti
on sample from posterior Confidence
intervals come from - posterior
probabilities - errors in forward simulation
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IllustrationPosterior reduces choices and
uncertainty
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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
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Problem Formulation
Simulation study Line drive, 2D reservoir Random
permeability field log normal, random
correlation length
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Simple Reservoir Description
in unit square
constant
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Ensemble
100 random permeability fields for each
correlation length lnK gaussian,
correlation length
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Upscaling
Solution from fine grid 100 x 100 grid Solution
by upscaling 20 x 20, 10 x 10, 5 x 5 Upscaled
grids
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Upscaling References
Upscaling by Wallstrom, Hou, Christie,
Durlofsky, Sharp 1. Computational Geoscience
369-87 (1999) 2. SPE 51939 3. Transport in
Porous Media (submitted)
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Examples of Upscaled, Exact Oil Cut Curves
Scale-up Black (fine grid) , Red (20x20),
Blue (10x10), Green (5x5)
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Design of Study
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)
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Bayes Theorem
Permeability geology Observation past oil cut
prior
posterior
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Errors and Discrepancies
Fine
Coarse
usually
but
implies
geology
geology
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Example
Fig. 1 Typical errors (lower, solid curves) and
discrepancies (upper, dashed curves), plotted vs.
PVI. The two families of curves are clearly
distinguishable.
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Mean error Sample covariance Pre
cision Matrix Gaussian error model has
covariance C, mean
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In Bayes Theorem, assume is
exact. Then, is an error, and probability
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Model Reduction
Limited data on solution errors Dont over fit
data Replace by finite matrix
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Three Prediction Methods
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.
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Comparing Prediction Methods
Window prediction is best, but not
practical -uses fine grid solutions for
complete ensemble Prior prediction is worst -
makes no use of production data.
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Error Reduction
Prediction error reduction, as per cent of
prior prediction choose present time to be oil
cut of 0.6
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Error Reduction
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
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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
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Confidence Intervals
Predicted future production confidence interval
prediction 13 -
28 Prediction depends on reservoir ? depends
on reservoir ? averaged over choice of reservoir
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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