An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

Rupert, C. P.; Miller, Cass T.

doi:10.1016/j.jcp.2007.07.001

Cited by 29 publications

(14 citation statements)

References 70 publications

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“…The error measures the deviation of the solutions by collocation methods from the MC solutions. Following common sense, the result of the collocation methods is expected to converge to the MC reference, as the number of retained random dimensions in KLE increases and the target covariance function becomes more accurate [Rupert and Miller, 2007;Chang and Zhang, 2009]. However, in this strongly nonlinear case, the error from the PCM does not decline, but rather inclines ( Figure 12).…”

Section: One-dimensional Heterogeneous Casementioning

confidence: 93%

Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

Liao

Zhang

2013

Water Resour. Res.

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[1] In this work, we propose a new collocation method for uncertainty quantification in strongly nonlinear problems. Based on polynomial construction, the traditional probabilistic collocation method (PCM) approximates the model output response, which is a function of the random input parameter, from the Eulerian point of view in specific locations. In some cases, especially when the advection dominates, the model response has a strongly nonlinear profile with a discontinuous shock or large gradient. This nonlinearity in the space domain is then translated to nonlinearity in the random parametric domain, which causes nonphysical oscillation and inaccurate estimation using the traditional PCM. To address this issue, a new location-based transformed probabilistic collocation method (xTPCM) is developed in this study, inspired by the Lagrangian point of view, in which model response is represented by an alternative variable, i.e., the location of a particular response value, which is relatively linear to the random parameter with a smooth profile. The location is then approximated by polynomial construction, from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples and to estimate the statistical properties. The advantage of the xTPCM is demonstrated through applications to multiphase flow and solute transport in porous media, which shows that the xTPCM achieves higher statistical accuracy than does the PCM, and produces more reasonable realizations without oscillation, while computational effort is greatly reduced compared to the direct sampling Monte Carlo method.

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Section: One-dimensional Heterogeneous Casementioning

confidence: 93%

Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

Liao

Zhang

2013

Water Resour. Res.

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“…However, the particular form of this coefficient allows for a "log-transformed" reformulation of the problem as a convection-diffusion problem. This approach is mentioned, for example, in [50, section 1.4] and [40]. Multiplying both sides of (1.1) by exp(−a) and rearranging, we obtain the equation we have arrived at (1.2).…”

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confidence: 99%

Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems

Ullmann¹,

Elman²,

Ernst³

2012

SIAM J. Sci. Comput.

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Abstract. We consider the numerical solution of a steady-state diffusion problem where the diffusion coefficient is the exponent of a random field. The standard stochastic Galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it. We consider a reformulated version of this problem as a stochastic convection-diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated Gaussian random variables. The associated Galerkin matrix is nonsymmetric but sparse and allows for fast matrix-vector multiplications with optimal complexity. We construct and analyze two block-diagonal preconditioners for this Galerkin matrix for use with Krylov subspace methods such as the generalized minimal residual method. We test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection-diffusion formulations.

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“…[22] The heterogeneity in the porous medium is treated probabilistically by modeling the intrinsic permeability and porosity of the medium as stochastic processes [Ghanem and Dham, 1998;Rupert and Miller, 2007]. In this paper, two distinct models are used for representing the uncertain medium properties.…”

Section: Stochastic Reservoir Modelmentioning

confidence: 99%

Characterization of reservoir simulation models using a polynomial chaos‐based ensemble Kalman filter

Saad

Ghanem

2009

Water Resources Research

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[1] Model-based predictions of flow in porous media are critically dependent on assumptions and hypotheses that are not always based on first principles and that cannot necessarily be justified on the basis of known prevalent physics. Constitutive models, for instance, fall under this category. While these predictive tools are usually calibrated using observational data, the scatter in the resulting parameters has typically been ignored. In this paper, this scatter is used to construct stochastic process models of the parameters which are then used as the cornerstone in a novel model validation methodology useful for ascertaining the confidence in model-based predictions. The uncertainties are first quantified by representing the unknown model parameters via their polynomial chaos decompositions. These are descriptions of stochastic processes in terms of their coordinates with respect to an orthogonal basis. This is followed by a filtering step to update these representations with measurements as they become available. In order to account for the non-Gaussian nature of model parameters and model errors, an adaptation of the ensemble Kalman filter is developed. Instead of propagating an ensemble of model states forward in time as is suggested within the framework of the ensemble Kalman filter, the proposed approach allows the propagation of a stochastic representation of unknown variables using their respective polynomial chaos decompositions. The model is propagated forward in time by solving the system of partial differential equations using a stochastic projection method. Whenever measurements are available, the proposed data assimilation technique is used to update the stochastic parameters of the numerical model. The proposed method is applied to a black oil reservoir simulation model where measurements are used to stochastically characterize the flow medium and to verify the model validity with specified confidence bounds. The updated model can then be employed to forecast future flow behavior.Citation: Saad, G., and R. Ghanem (2009), Characterization of reservoir simulation models using a polynomial chaos-based ensemble Kalman filter, Water Resour. Res., 45, W04417,

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An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

Cited by 29 publications

References 70 publications

Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems

Characterization of reservoir simulation models using a polynomial chaos‐based ensemble Kalman filter

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