scholarly journals Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods

2007 ◽  
Vol 43 (9) ◽  
Author(s):  
Heng Li ◽  
Dongxiao Zhang
Keyword(s):  
Porous Media ◽  
Collocation Method ◽  
Flow In Porous Media ◽  
Stochastic Methods ◽  
Probabilistic Collocation Method ◽  
Probabilistic Collocation

Conditional Simulation of Flow in Heterogeneous Porous Media with the Probabilistic Collocation Method

2014 ◽  
Vol 16 (4) ◽  
pp. 1010-1030 ◽  
Author(s):  
Heng Li
Keyword(s):  
Porous Media ◽  
Hydraulic Conductivity ◽  
Collocation Method ◽  
Heterogeneous Media ◽  
Conditional Simulation ◽  
Stochastic Approach ◽  
Heterogeneous Porous Media ◽  
Propagation Of Uncertainty ◽  
Probabilistic Collocation Method ◽  
Probabilistic Collocation

AbstractA stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

Efficient and Accurate Global Sensitivity Analysis for Reservoir Simulations By Use of the Probabilistic Collocation Method

SPE Journal ◽  
2013 ◽  
Vol 19 (04) ◽  
pp. 621-635 ◽  
Author(s):  
Cheng Dai ◽  
Heng Li ◽  
Dongxiao Zhang
Keyword(s):  
Sensitivity Analysis ◽  
Collocation Method ◽  
History Matching ◽  
Global Sensitivity Analysis ◽  
Computational Cost ◽  
Global Sensitivity ◽  
Reservoir Simulations ◽  
Probabilistic Collocation Method ◽  
The Impact ◽  
Probabilistic Collocation

Summary Reservoir simulations involve a large number of formation and fluid parameters, many of which are subject to uncertainties owing to the combination of spatial heterogeneity and insufficient measurements. Accurately quantifying the impact of varying parameters on simulation models can reveal the importance of the parameters, which helps in designing field-characterization strategies and determining parameterization for history matching. Compared with the commonly used local sensitivity analysis (SA), global SA considers the whole variation range of the parameters and can thus provide more-complete information. However, the traditional global sensitivity analysis that is derived from Monte Carlo simulation (MCS) is computationally too demanding for reservoir simulations. In this study, we propose an alternative approach that is both accurate and efficient. In the proposed approach, the model outputs such as pressure and reservoir production quantities are expressed by polynomial chaos expansions (PCEs). The probabilistic collocation method is used to determine the coefficients of the polynomial expansions by solving outputs at different sets of collocation points by means of the original partial-differential equations. Then, a proxy is constructed with such coefficients. Accurate statistical sensitivity indices of the uncertainty parameters can be obtained by running the proxy. We validate the approach with 2D examples by comparing with the MCS-based global SA. It is found that with only a small fraction of the computational cost required by the MCS approach, the new approach gives accurate global sensitivity for each parameter. The proposed approach is also demonstrated on a large-scale 3D black-oil model, for which the MCS-based global SA is found to be computationally infeasible. It is found that the developed approach possesses the following key advantages: It requires a much smaller number of reservoir simulations for accurate global SA; it is nonintrusive and can be implemented with existing codes or simulators; and it can accommodate arbitrary distributions of parameters encountered in realistic geological situations.

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