Uncertainty quantification for porous media flows

Christie, Mike; Demyanov, Vasily; Erbas, Demet

doi:10.1016/j.jcp.2006.01.026

Cited by 91 publications

(50 citation statements)

References 21 publications

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“…Unlike a Cartesian parameter space, this space is not defined by any origin, dimension, or direction: It is only equipped with a distance. However, as shown in the next section, it is sufficient to implement a stochastic search, such as the neighborhood algorithm (NA) [15][16][17][18].…”

Section: Distance-based Model Parameterizationmentioning

confidence: 99%

“…The NA [15][16][17][18] is a stochastic optimization algorithm that explores a parameter space for multiple minima, partitioning the space into Voronoi cells as the function evaluation proceeds. Each function evaluation consists in running a flow simulation and calculating the mismatch with field data.…”

Section: Stochastic Search: the Neighborhood Algorithmmentioning

confidence: 99%

“…We denote this probability value as a "selection probability." Although the algorithm is originally proposed and applied for a traditional Cartesian parameter space [15][16][17][18], the method is easily reformulated to be applicable for the distance-based parameter space, as the definition of Voronoi cell requires only knowledge of a distance. An example of the step by step procedure of this approach is illustrated in Fig.…”

Section: Stochastic Search: the Neighborhood Algorithmmentioning

confidence: 99%

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Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization

2008

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This paper proposes a novel history-matching method where reservoir structure is inverted from dynamic fluid flow response. The proposed workflow consists of searching for models that match production history from a large set of prior structural model realizations. This prior set represents the reservoir structural uncertainty because of interpretation uncertainty on seismic sections. To make such a search effective, we introduce a parameter space defined with a "similarity distance" for accommodating this large set of realizations. The inverse solutions are found using a stochastic search method. Realistic reservoir examples are presented to prove the applicability of the proposed method.

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Section: Distance-based Model Parameterizationmentioning

confidence: 99%

Section: Stochastic Search: the Neighborhood Algorithmmentioning

confidence: 99%

Section: Stochastic Search: the Neighborhood Algorithmmentioning

confidence: 99%

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Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization

2008

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“…Uncertainty quantification is also a fundamental step towards validation and certification of numerical methods to be used for critical decisions. Fields of application of uncertainty quantification include but are not limited to turbulence, climatology [20], turbulent combustion [21], flow in porous media [8], fluid mixing [31] and computational electromagnetics [6].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of the Burgers’ equation in the presence of uncertainty

Pettersson

Iaccarino

Nordström

2009

Journal of Computational Physics

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The Burgers' equation with uncertain initial and boundary conditions is investigated using a Polynomial Chaos (PC) expansion approach where the solution is represented as a truncated series of stochastic, orthogonal polynomials.The analysis of well posedness for the system resulting after Galerkin projection is presented and follows the pattern of the corresponding deterministic Burgers equation. The numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. Similarly to the deterministic case, the explicit time step for the hyperbolic stochastic problem is proportional to the inverse of the largest eigenvalue of the system matrix. The time step naturally decreases compared to the deterministic case since the spectral radius of the continuous problem grows with the number of polynomial chaos * Corresponding author. Tel: 16508628295.Email addresses: massperp@stanford.edu (Per Pettersson), jops@stanford.edu (Gianluca Iaccarino), jan.nordstrom@foi.se (Jan Nordström) Preprint submitted to Journal of Computational PhysicsNovember 8, 2012coefficients. An estimate of the eigenvalues is provided.A characteristic analysis of the truncated PC system is presented and gives a qualitative description of the development of the system over time for different initial and boundary conditions. It is shown that a precise statistical characterization of the input uncertainty is required and partial information, e.g. the expected values and the variance, are not sufficient to obtain a solution. An analytical solution is derived and the coefficients of the infinite PC expansion are shown to be smooth, while the corresponding coefficients of the truncated expansion are discontinuous.

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“…Fields of application of uncertainty quantification include but are not limited to turbulence, climatology [13], combustion [14], flow in porous media [4], fluid mixing [15] and computational electromagnetics [3].…”

Section: Introductionmentioning

confidence: 99%

Boundary procedures for the time-dependent Burgers' equation under uncertainty

Pettersson

Nordström

Iaccarino

2010

Acta Mathematica Scientia

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The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to influence the solution; we will discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.

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Uncertainty quantification for porous media flows

Cited by 91 publications

References 21 publications

Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization

Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization

Numerical analysis of the Burgers’ equation in the presence of uncertainty

Boundary procedures for the time-dependent Burgers' equation under uncertainty

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