Stochastic analysis of flow and transport in porous media Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/h415pd307

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  • Random fields are frequently used in computational simulations of real-life processes. In particular, in this work they are used in modeling of flow and transport in porous media. Porous media as they arise in geological formations are intrinsically deterministic but there is significant uncertainty involved in determination of their properties such as permeability, porosity and diffusivity. In many situations description of properties of the porous media is aided by a limited number of observations at fixed points. These observations constrain the randomness of the field and lead to conditional simulations. In this work we propose a method of simulating the random fields which respect the observed data. An advantage of our method is that in the case that additional data becomes available it can be easily incorporated into subsequent representations. The proposed method is based on infinite series representations of random fields. We provide truncation error estimates which bound the discrepancy between the truncated series and the random field. We additionally provide the expansions for some processes that have not yet appeared in the literature. There are several approaches to efficient numerical computations for partial differential equations with random parameters. In this work we compare the solutions of flow and transport equations obtained by conditional simulations with Monte Carlo (MC) and stochastic collocation (SC) methods. Due to its simplicity MC method is one of the most popular methods used for the solution of stochastic equations. However, it is computationally expensive. The SC method is functionally similar to the MC method but it provides the faster convergence of the statistical moments of the solutions through the use of the carefully chosen collocation points at which the flow and transport equations are solved. We show that for both methods the conditioning on measurements helps to reduce the uncertainty of the solutions of the flow and transport equations. This especially holds in the neighborhood of the conditioning points. Conditioning reduces the variances of solutions helping to quantify the uncertainty in the output of the flow and transport equations.
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