Probability distribution methods for nonlinear transport in heterogeneous porous media
- Because geophysical data are inexorably sparse and incomplete, stochastic treatments of simulated responses are crucial to explore possible scenarios and assess risks in subsurface problems. In particular, nonlinear two-phase flow in porous media is essential, yet challenging, in reservoir simulation and hydrology. Adding highly heterogeneous and uncertain input, such as the permeability and porosity fields, transforms the estimation of the flow response into a tough stochastic problem for which computationally expensive Monte Carlo simulations remain the preferred option. In this thesis, we first propose an alternative approach to evaluate the probability distribution of the (water) saturation for nonlinear transport in strongly heterogeneous porous systems. We build a physics-based, computationally efficient and numerically accurate method to estimate the one-point probability density and cumulative distribution functions of the saturation. The distribution method draws inspiration from a Lagrangian approach of the stochastic transport problem and expresses the saturation probability density function and cumulative distribution function essentially in terms of a deterministic nonlinear mapping of scalar random fields. In a large class of applications these random fields are smooth and can be estimated at low computational costs (few Monte Carlo runs), thus making the distribution method attractive. Once the saturation distribution is determined, any one-point statistics thereof can be obtained, especially the saturation average and standard deviation. More importantly, the probability of rare events and saturation quantiles (e.g. P10, P50 and P90) can be efficiently derived from the distribution method. These statistics can then be used for risk assessment, as well as data assimilation and uncertainty reduction in the prior knowledge of geophysical input distributions. We provide various examples and comparisons with existing methods to illustrate the performance and applicability of the new developed method. In the second part of the thesis, we present a procedure to analytically obtain the multi-point cumulative distribution function of the saturation for the stochastic two-phase Buckley-Leverett model with random total-velocity field. The multi-point distribution function is determined by first deriving a partial differential equation for the saturation raw cumulative distribution function at each point, then combining these equations into a single partial differential equation for the multi-point raw cumulative distribution function. This latter stochastic partial differential equation, linear in the space-time variables, can be solved in a closed form and semi-analytically for spatial one dimensional problems or numerically for higher spatial dimensions. Finally, the ensemble average of its solution gives the saturation multi-point cumulative distribution function. We provide numerical results of distribution function profiles in one spatial dimension and for two points. Besides, we use the two-point distribution method to compute the saturation auto-covariance function, essential for data assimilation. We confirm the validity of the method by comparing covariance results obtained with the multipoint distribution method and Monte Carlo simulations.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Institute for Computational and Mathematical Engineering.
|Statement of responsibility
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis (Ph.D.)--Stanford University, 2016.
- © 2016 by Fayadhoi Ibrahima
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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