Non-Intrusive Stochastic Uncertainty Quantification Methods

Don ZhangUniversity of Southern California

donzhang@usc.edu

Uncertainty Quantification WorkshopTucson, AZ, April 25-26, 2008

Distance (ft)East-West Cross Section at Yucca Mountain [Bodvarsson et al., 1999]

Large Dimensions

• Large physical scale leads to a large number of gridblocks in numerical models

•105 to 106 nodes

• Parameter uncertainty adds to the problem additional dimensions in probability space.

Stochastic Approaches

• Two common approaches for quantifying uncertainties associated with subsurface flow simulations:

Monte Carlo simulation (MCS)

Statistical Moment Equation (SME): Moment equations; Green’s function; Adjoint state

• These two types of approaches are complementary.

Intrusive vs. Non-Intrusive Approaches

• Moment equation methods are intrusiveNew governing equationsExisting deterministic simulators cannot be employed

directly

• Monte Carlo is non-intrusive:Direct samplingSame governing equationsNot efficient

• More efficient non-intrusive stochastic approaches are desirable

Stochastic Formulation

• SPDE:

which has a finite (random) dimensionality.

• Weak form solution:

where

1 2where ( , ,..., )TN ξ

( ; , ) ( , ), , ,L u x g x P x D ξ ξ ξ

ˆ( ; , ) ( ) ( ) ( , ) ( ) ( )P P

L u x w p d g x w p d ˆ( , ) , where trial function spaceu x V V

( ) , where test (weighting) function spacew W W ( ) probability density function of ( )p ξ

Stochastic Methods• Galerkin polynomial chaos expansion (PCE) [e.g., Ghanem and Spanos, 1991]:

• Probabilistic collocation method (PCM) [Tatang et al., 1997; Sarma et al., 2005; Li and Zhang, 2007]:

• Stochastic collocation method (SCM) [Mathelin et al., 2005; Xiu and Hesthaven, 2005]:

1 1( ) , ( )

M M

i ii iV span W span

1 1( ) , ( )

M M

i ii iV span W span

1 1( ) , ( )

M M

i ii iV span L W span

1where { ( )} lagrange interpolation basisMi iL

1where ( ) orthogonal polynomials

M

i i

Key Components for Stochastic Methods

• Random dimensionality of underlying stochastic fields – How to effectively approximate the input random fields with finite dimensions– Karhunen-Loeve and other expansions may be used

• Trial function space– How to approximate the dependent random fields– Perturbation series, polynomial chaos expansion, or Lagrange interpolation basis

• Test (weighting) function space– How to evaluate the integration in random space?– Intrusive or non-intrusive schemes?

Karhunen-Loeve Expansion:Eigenvalues & Eigenfunctions

For CY(x,y) = exp(-|x1-x2|/1-|y1-y2|/2)

n

n

10 20 30 400.00

0.10

0.20

x1

x 2

0 2 4 6 8 100

2

4

6

8

101.510.50

-0.5-1-1.5

(c) n=10x1

x 2

0 2 4 6 8 100

2

4

6

8

101.510.50

-0.5-1-1.5

(b) n=4

x1

x 2

0 2 4 6 8 100

2

4

6

8

101.510.50

-0.5-1-1.5

(d) n=20x1

x 2

0 2 4 6 8 100

2

4

6

8

101.31.21.110.90.80.70.60.50.4

(a) n=1

1

( , ) ( ) ( )N

n n nn

Y f

x x

( , )( ) ( , ) ( , )s S

h tK h t G t S

t

x

x x x

Flow Equations

• Consider first transient single phase flow satisfying

subject to initial and boundary conditions

Log permeability or log hydraulic conductivity Y=ln Ks is assumed to be a random space function.

( ) ', oro

pp gz S G

t

k

Polynomial Chaos Expansion (PCE)

• Express a random variable as:

1 21

0 0 1 21 1 1

31 1 1

( , ; ) ( , ) ( ), ( ) ( , , , )

( ) ( , )

( , , )

-Multi-dimensional Hermite

MT

j j Nj

i

i i ij i ji i j

ji

ijk i j ki j k

d

h t c t

a a a

a

x x ξ ξ

orthogonal

polynomials of degree d

Other (generalized) orthogonal polynomials are also possible

PCM• Leading to M sets of deterministic (independent)

equations:

which has the same structure as the original equation

• The coefficients are computed from the linear system of M equations

1

( , )exp ( ) ( ) ( , ) ( , )

Nj

i i i j Si

h tY f h t g t S

t

xx x x x

1 2

1 2

[ , , , ]

=[ , , , ]

( )!

! !

TM

TM

h h h

c c c

N dM

N d

hZ C h

C

Post-Processing

• Probability density function: statistical sampling

– Much easier to sample from this expression than from the original equation (as done by MCS)

• Statistical moments:

1

2 2 2

2

( , ) ( , )

( , ) ( , )M

h j jj

h x t c x t

x t c x t

1

( , ) ( , ) ( )M

j jj

h t c t

x x ξ

Stochastic Collocation Methods (SCM) • Leading to a set of independent equations evaluated at

given sets of interpolation nodes:

• Statistics can be obtained as follows:

1 1( ) , ( )

M M

i ii iV span L W span

1

ˆ( , ) ( )( , ) ( , ) ( )M

i ii

u u u L

x x xI

( ( ); , ) ( , ) , ,i i iL u f D x x x x

0 0

( ) ( ) ( ) ( ) ( ) ( )M M

i i i iP Pi i

u u d u L d u c

2

2

0 0

( ( ))M M

i i i ii i

Var u u c u c

Choices of Collocation Points• Tensor product of one-dimensional nodal sets

• Smolyak sparse grid (level: k=q-N)

• Tensor product vs. level-2 sparse grid– N=2, 49 knots vs. 17 (shown right)– N=6, 117,649 knots vs. 97

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Each dimension: knots

dimension: N

m

N M m

For N>1, preserving interpolation

property of N=1 with a small number

of knots

0 1 2 3 4 5 6 7 8 9 105

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

2nd

order PCM: 28 representations, = 4.0, Y2 = 1.0

x

Hea

d, h

MCS vs. PCM/SCM

0 1 2 3 4 5 6 7 8 9 105

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

MC: 1000 realizations = 4.0, Y2 = 1.0

x

Hea

d, h

PCM/SCM: • Structured sampling (collocation points)• Non-equal weights for hj (representations)

MCS: • Random sampling of (realizations) • Equal weights for hj (realizations)

4 4.5 5 5.5 6 6.5 7 7.50

0.2

0.4

0.6

0.8

1

1.2

h(4)

PDF

= 4.0, Y2 = 1.0

PCM, 2nd order

KLME, 2nd orderMC (10,000)

4 4.5 5 5.5 6 6.5 7 7.5 80

0.2

0.4

0.6

0.8

1

1.2

h(6)

PDF

= 4.0, Y2 = 1.0

PCM, 2nd order

KLME, 2nd orderMC (10,000)

Pressure head at position x = 4

Pressure head at position x = 6

PDF of Pressure

1

( ) ( ) ( )M

j jj

h c

x x ξ

Error Studies

0 200 400 600 800 10000.000

0.001

0.002

0.003

0.004

err

or

M

PCM Smolyak

/L=0.4, 2

Y=1.0, N=6

d1=2d1=4

d1=6

d2=1

d2=2

d2=3

(a)

0 200 400 600 800 10000.000

0.004

0.008

0.012

0.016

err

or

M

PCM SmolyakL=0.4, 2

Y=2.0, N=6

d2=1

d2=2 d2=3

d1=2

d1=4 d1=6

(b)

• In general, the error reduces as either the order of polynomials or the level of sparse grid increases

• Second-order PCM and level-2 sparse grid methods are cost effective and accurate enough

( )!PCM:

! !

N dM

N d

2 3 4 5 6 7 8 9 10 11 12 13 140.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

I

N

L=0.4, 2

Y=1.0

L=0.4, 2

Y=2.0

L=0.4, 2

Y=3.0

L=0.4, 2

Y=4.0

Level-2 Smolyak

(b)

2 3 4 5 6 7 8 9 10 11 12 13 140.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

i

N

/L=0.4, 2

Y=1.0

L=0.4, 2

Y=2.0

L=0.4, 2

Y=3.0

L=0.4, 2

Y=4.0

2nd order PCM

(a)

Approximation of Random Dimensionality • For a correlated random field, the random dimensionality

is theoretically infinite• KL provides a way to order the leading modes

• How many is adequate? The critical dimension, Nc

2 3 4 5 6 7 8 9 10 11 12 13 140.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

i

N

L=0.7, 2

Y=1.0

L=0.4, 2

Y=1.0

L=0.1, 2

Y=1.0

2nd oder PCM

(c)

2 3 4 5 6 7 8 9 10 11 12 13 140.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

i

N

L=0.7, 2

Y=1.0

L=0.4, 2

Y=1.0

L=0.1, 2

Y=1.0

Level-2 Smolyak

(d)

• The critical random dimensionality (Nc) increases with the decrease of correlation length.

Energy Retained

•The approximate random dimensionality Nc versus the retained energy

1 12

1

c cN N

n nn nc

Ynn

ED

0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

30

35

40

45

Nc

/L

convergence criterion 90% energy criterion

(a)

2

Y=1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

convergence criterion

Ec

/L

2

Y=1.0

(b)

for the same

energy

for the same

error

for the same

error

Two Dimensions

•In 2D, the eigenvalues decay more slowly than in 1D •However, it does not require the same level of energy to achieve a given accuracy in 2D

•Reduced energy level•Moderate increase in random dimensionality

η/L Nc1 Nc2 Ec1 Ec2

0.7 5 15 0.94 0.87

0.4 6 20 0.91 0.80

0.1 9 30 0.77 0.44

Application to Multi-Phase Flow

1. Governing Equation for multi-phase flow:

2. PCM equations:

• 3D dipping reservoir

• (7200x7500x360 ft)

• Grid: 24x25x15

• 3 phase model

• Heterogeneous

Application: The 9th SPE Model

Initial oil Saturation

1

( , ) ( , ) ( )

being , ,...

M

j jj

i i

Q t c t

Q P S

x x ξ

3D Random Permeability Field

• Kx = Ky, Kz = 0.01 Kx

0.12 Y

21 2 1 1 2 2 1 2 3

( ) ln ( )

= exp(-|x -x |/ -|y -y |/ -|z -z |/ )Y Y

Y k

C x x

31 2 0.4Lx Ly Lz

A realization of ln Kx field:Kx: 3.32--1132 md

MC vs. PCM

• MC: 1000 realizations

• PCM: 231 representations (N = 20, d = 2), shown right, constructed with leading modes (below)

Representation of random perm field

Field oil production Field gas production

var=0.25

var=1.00

Results

Results

Field water cut Field gas oil ratio

var=0.25

var=1.00

Mean:

STD:

PCM: MC:

Oil Saturation (var=1.0, CV=134%)

Mean:

STD:

PCM: MC:

Gas Saturation (var=1.0, CV=134%)

Summary (1)

• The efficiency of stochastic methods depends on how the random (probability) space is approximated

– MCS: realizations– SME: covariance– KL: dominant modes

• The number of modes required is– Small when the correlation length/domain-size is large– Large when the correlation length/domain-size is small

• Homogenization, or low order perturbation, may be sufficient

Summary (2)

• The relative effectiveness of PCE and PCM/SCM depends on how their expansion coefficients are evaluated– PCE: Coupled equations– PCM & SCM: Independent equations with the same

structure as the original one

• PCM & SCM: Promising for large scale problems

Summary (3)

• The PCM or SCM has the same structure as does the original flow equation.

• PCM /SCM is the least intrusive !

• For this reason, similar to the Monte Carlo method, the PCM/SCM can be easily implemented with any of the existing simulators such as• CHEARS, CMG, ECLIPSE, IPARS, VIP

• MODFLOW, MT3D, FEHM, TOUGH2

• The expansions discussed also form a basis for efficiently assimilating dynamic data [e.g., Zhang et al., SPE J, 2007].

Acknowledgment

Financial Supports: NSF; ACS; DOE; Industrial Consortium “OU-CEM”

Selection of Collocation Points• Selection of collocation points: roots of (d+1)th order

orthogonal polynomials

• For example, 2nd order polynomial and N=6

– Number of coefficients: M=28

– Choosing 28 sets of points:

– 3rd Hermite polynomials:

– Roots in decreasing probability:

– Choose 28 points out of (0, 3, 3)

63 729

33( ) 3H

1 2 6( , , , ) j

The selected collocation points for each (N,d) can then be tabulated.