Non-Intrusive Stochastic Uncertainty Quantification Methods Don Zhang University of Southern California donzhang@usc.edu

1. Non-Intrusive Stochastic Uncertainty Quantification Methods Don Zhang University of Southern California donzhang@usc.edu Uncertainty Quantification Workshop Tucson, AZ, April 25-26, 2008

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

3. 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.

4. Intrusive vs. Non-Intrusive Approaches • Moment equation methods are intrusive • New governing equations • Existing deterministic simulators cannot be employed directly • Monte Carlo is non-intrusive: • Direct sampling • Same governing equations • Not efficient • More efficient non-intrusive stochastic approaches are desirable

5. Stochastic Formulation • SPDE: which has a finite (random) dimensionality. • Weak form solution: where

6. 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]:

7. 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?

8. Karhunen-Loeve Expansion: Eigenvalues & Eigenfunctions For CY(x,y) = exp(-|x1-x2|/1-|y1-y2|/2)

9. 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.

10. Polynomial Chaos Expansion (PCE) • Express a random variable as:

11. 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

12. 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:

13. Stochastic Collocation Methods (SCM) • Leading to a set of independent equations evaluated at given sets of interpolation nodes: • Statistics can be obtained as follows:

14. 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

15. MCS vs. PCM/SCM • MCS: • Random sampling of (realizations) • Equal weights for hj (realizations) • PCM/SCM: • Structured sampling (collocation points) • Non-equal weights for hj (representations)

16. PDF of Pressure Pressure head at position x = 4 Pressure head at position x = 6

17. Error Studies • 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

18. 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

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

20. Energy Retained • The approximate random dimensionality Nc versus the retained energy for the same energy for the same error for the same error

21. 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

22. Application to Multi-Phase Flow 1. Governing Equation for multi-phase flow: 2. PCM equations:

23. Application: The 9th SPE Model • 3D dipping reservoir • (7200x7500x360 ft) • Grid: 24x25x15 • 3 phase model • Heterogeneous Initial oil Saturation

24. 3D Random Permeability Field • Kx = Ky, Kz = 0.01 Kx A realization of ln Kx field: Kx: 3.32--1132 md

25. MC: 1000 realizations PCM: 231 representations (N = 20, d = 2), shown right, constructed with leading modes (below) MC vs. PCM Representation of random perm field

26. Results Field gas production Field oil production var=0.25 var=1.00

27. Results Field gas oil ratio Field water cut var=0.25 var=1.00

28. Oil Saturation (var=1.0, CV=134%) MC: PCM: Mean: STD:

29. Gas Saturation (var=1.0, CV=134%) MC: PCM: Mean: STD:

30. 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

31. 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

32. 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].

33. AcknowledgmentFinancial Supports: NSF; ACS; DOE; Industrial Consortium “OU-CEM”

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