Abstract
We derive a new method of conditional Karhunen-Loève (KL) expansions for stochastic coefficients in models of flow and transport in the subsurface, and in particular for the heterogeneous random permeability field. Exact values of this field are never known, and thus one must evaluate uncertainty of solutions to the flow and transport models. This is typically done by constructing independent realizations of the permeability field followed by numerical simulations of flow and transport for each realization and assembling statistical estimates of moments of desired quantities of interest. We follow the well-known framework of KL expansions and derive a new method that incorporates known values of the permeability at given locations so that the realizations of the permeability field honor this data exactly. Our method relies on projections to an appropriate subspace of random weights applied to the eigenfunctions of the covariance operator. We use the permeability realizations constructed with our stochastic simulation method in simulations of flow and transport and compare the results to those obtained when realizations are constructed with sequential Gaussian simulation (SGS). We also compare efficiency and stochastic convergence with that of stochastic collocation.
Highlights
Computational modeling of flow and transport in the subsurface requires detailed knowledge of coefficients of partial differential equations (PDEs), in particular of permeabilities K and porosities Φ
We see that the three corresponding methods, respectively, sequential Gaussian simulation (SGS)-MC, KLMC, and KL with Stochastic Collocation (KL-stochastic collocation (SC)), require the knowledge of Q(⋅) evaluated for each realization ωk or at each collocation point yk
Different choices of the number Nm of conditioning points are used; we vary the number of terms N in the KL expansions (24) and the number of realizations M, as well as the polynomial order m in KL-SC
Summary
Computational modeling of flow and transport in the subsurface requires detailed knowledge of coefficients of partial differential equations (PDEs), in particular of permeabilities K and porosities Φ. These are heterogeneous; that is, K = K(x) where x ∈ D and D ⊂ Rd is the domain of flow. KL expansions account for spatial variability of K(x) through a sequence of eigenfunctions of C smooth in x Since these correspond to rapidly decreasing eigenvalues, one typically truncates the KL expansion to N terms in KN(x, ω) which capture the desired proportion of the variance of the field
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