# Presentation: 442591

## 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.**

##                 Cross Section & Mass Diffusion:                 Comparison of TVD and FronTier

**TVD**

**FronTier**

## Simulation of two immersed bubbles

**Simulation of two immersed bubbles**

## 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**

**Bubble velocity = single mode velocity + envelope velocity**

**Advanced **
**bubble**

**Retarded**
**bubble**

**Advanced **
**bubble**

##                   Analytic Models

**Statistical Models of Interacting Bubbles**
**Bubble Merger Models**

**Bubble velocity = single mode velocity + envelope velocity**

**Advanced **
**bubble**

**Retarded**
**bubble**

**Advanced **
**bubble**

##             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)**
**h****s**** = penetration of heavy fluid into light**
**h****s ****= ******** ****s**** Agt****2****    (RT)**
**COM ********  ******** ****s**** / ******** ****b**** = solution of quadratic equation**
**                        ******** ****s**** =   ******** ****s**** (******** ****b****)   **
**B. Cheng, J. Glimm, D. Saltz, D. Sharp**

##             Mixing Zone Edge Models

**Z****b,s**** (t)****	****= ****	****mixing zone edge****	**** **
**	****	****	****	****=****	****	****h****b,s**** in RT case**
**	****	****	****	****=****	****	********bs****  Agt****2**** ****in RT case**
**Buoyancy Drag equation for Z****b,s**** (t):**
**     **
**Determine C****b,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**

##    Multiphase  Averaged Equations

**Microphysics:**
**Macrophysics:**
**Closure Problem:  Determine F****ren**

##              Ensemble Averages

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

##                            Closure

**Assume:  *****v****** depends on *****v*****1 ****and *****v*****2 ****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 Z****k****(t), and velocities V****k****(t),**
**for incompressible flow.  Boundary accelerations             must be**
**must be supplied externally to this model.  **
**                                                       ****Drag + buoyancy**

##   Analytic Solution:  Incompressible Case

- Let

- Then

## Pressure versus distance

**Pressure versus distance**

## Asymptotic Expansion in Powers ofM = Mach Number

**0****th**** order ****	****	****	****	****=****	****incompressible v, ******
**1****st**** order****	****	****	****	****=****	****correction v, ******
**2****nd ****order****	****	****	****	****=****	****incompressible p****1****, p****2**** **
**                                     ****+ v, ********	****  correction**
**2****nd ****order p****1****, p****2****	****=****	****incompressible p****1****, p****2**** **
**              ******** ****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

## Illustration:Posterior reduces choices and uncertainty

## 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

## Simple Reservoir Description

- in unit square

- constant

## Ensemble

- 100 random permeability fields for each correlation length
-   ln*K*  gaussian, 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 References

- 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)

## Design of Study

- 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)

##  Bayes Theorem

- Permeability = geology
- Observation = past oil cut

- posterior;

- prior

## Errors and Discrepancies

- usually

- but

- implies

- geology

- geology

- Fine

- Coarse

##    Example: 

- Fig. 1  Typical errors (lower, solid curves) and discrepancies
- (upper, dashed curves), plotted vs. PVI.  The two families of
- curves are clearly distinguishable.

## 	     Mean error 

-      Sample covariance
- 	Precision Matrix
-      Gaussian error model:  has covariance C, mean

## In Bayes Theorem, assume           is exact.

- Then,     is an error, and probability

## Model Reduction:

- Limited data on solution errors
- Don’t over fit data
- Replace         by finite matrix

## Three Prediction Methods

- 	(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.

## Comparing Prediction Methods

- 	but not practical
- 	-uses fine grid solutions for 			 complete ensemble
- Prior prediction is worst
- 	- makes no use of production 			  data.

## Error Reduction

- per cent of prior prediction
- choose present time to be oil cut of 0.6

## Error Reduction

- 	(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

## Confidence Intervals

-      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