Non-Intrusive Stochastic Uncertainty Quantification Methods Don Zhang
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Non-Intrusive Stochastic Uncertainty Quantification Methods
Don ZhangUniversity of Southern California
Uncertainty Quantification WorkshopTucson, AZ, April 25-26, 2008
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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.
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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.
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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
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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
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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
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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?
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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
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( , )( ) ( , ) ( , )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
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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
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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
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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
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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
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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
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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)
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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)
= 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)
= 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
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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
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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
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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.
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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
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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
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Application to Multi-Phase Flow
1. Governing Equation for multi-phase flow:
2. PCM equations:
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• 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
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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
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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
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Field oil production Field gas production
var=0.25
var=1.00
Results
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Results
Field water cut Field gas oil ratio
var=0.25
var=1.00
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Mean:
STD:
PCM: MC:
Oil Saturation (var=1.0, CV=134%)
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Mean:
STD:
PCM: MC:
Gas Saturation (var=1.0, CV=134%)
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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
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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
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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].
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Acknowledgment
Financial Supports: NSF; ACS; DOE; Industrial Consortium “OU-CEM”
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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.