Non-Intrusive Stochastic Uncertainty Quantification Methods Don Zhang University of Southern California Uncertainty Quantification Workshop Tucson, 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 10 5 to 10 6 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 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

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

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]:

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 C Y (x,y) = exp(-|x 1 -x 2 |/  1 -|y 1 -y 2 |/  2 )

Flow Equations Consider first transient single phase flow satisfying subject to initial and boundary conditions Log permeability or log hydraulic conductivity Y=ln K s is assumed to be a random space function.

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

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

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:

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

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

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

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

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

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, N c

The critical random dimensionality (N c ) increases with the decrease of correlation length.

Energy Retained The approximate random dimensionality N c versus the retained energy for the same energy 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 N c1 N c2 E c1 E c

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

3D Random Permeability Field Kx = Ky, Kz = 0.01 Kx A realization of ln Kx field: Kx: 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

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, 2 nd order polynomial and N=6 –Number of coefficients: M=28 –Choosing 28 sets of points: –3 rd Hermite polynomials: –Roots in decreasing probability: –Choose 28 points out of The selected collocation points for each (N,d) can then be tabulated.