1 / 34

Non-Intrusive Stochastic Uncertainty Quantification Methods Don Zhang

Non-Intrusive Stochastic Uncertainty Quantification Methods Don Zhang University of Southern California donzhang@usc.edu Uncertainty Quantification Workshop Tucson, AZ, April 25-26, 2008. East-West Cross Section at Yucca Mountain [Bodvarsson et al., 1999]. Distance (ft). Large Dimensions.

ariasb
Download Presentation

Non-Intrusive Stochastic Uncertainty Quantification Methods Don Zhang

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related