Conditional Stochastic Simulations of Flow and Transport with Karhunen-Loève Expansions, Stochastic Collocation, and Sequential Gaussian Simulation

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.

On the Correspondence between Gaussian Processes and Geometric Harmonics

“…In this section we establish a link between diffusion maps [22] and Gaussian process regression [2] using the conditional Karhunen-Loève expansion [1,23,24]. This approach enables the study of the following topics from a different viewpoint than the usual one in which their approached:…”

Gaussian Process Regression and Conditional Polynomial Chaos for Parameter Estimation

“…The observations {Y (x (i) ) = ln κ(x (i) )} Nm i=1 are used to compute the (unconditional) stationary covariance describing an unconditional random field Y through the variagram analysis or by maximizing the likelihood function [5]. In addition, the observations can be used to model a conditional random field using GPR as presented in [25]. In the following, we will use x * to denote the set of N m observation locations {x (i) } Nm i=1 of the random field and Y (x * ) to denote the column vector of N m observations {Y (x (i) )} Nm i=1 .…”

“…The easyto-evaluate conditional gPC surrogate significantly accelerates the MCMC sampling and the parameter estimation and quarantees that the estimated parameters exactly match the parameter measurements. The conditional gPC is based on the data-driven conditional KL representation of unknown space-dependent parameters [25,14,15] that reduces the dimensionality of the parameter space and allows parameter estimation with relatively few measurements. There are two levels of the dimension reduction in the conditional KL and gPC methods.…”

Conditioning by Projection for the Sampling from Prior Gaussian Distributions

“…More recently a truncated Karhunen-Loève expansion (KLE) [29,16,22] has been applied both for dimensional reduction of the (computationally infeasible) large stochastic dimensions of fine grid discretizations of subsurface flow problems as well as for conditioning. In [37] one can find a discussion of the advantages of the KLE over the SGS. There are two lines of work for the conditioning of samples using the KLE.…”

“…This approach can be seen in [36] where a search through a set of matrices of size determined by the number of measurements is performed, and they take the one with the best condition number. In [37] the authors use the expressions for conditional means and covariances of Gaussian random variables [46] that require the inversion of a data covariance matrix.…”