Quantification of uncertainty in production injection forecasting is an important aspect of reservoir simulation studies. Conventional approaches include intrusive Galerkin-based methods (e.g., generalized polynomial chaos (gPC) and stochastic collocation (SC) methods) and non-intrusive Monte Carlo (MC) based methods. Nevertheless, the quantification is conducted in reformulations of the underlying stochastic PDEs with fixed well controls. If one wants to take various well control plans into account, expensive computations need to be repeated for each well design independently. In this project, we take advantages of the equation-free spirit of convolutional neural network (CNN) to overcome this challenge and thus achieve the flexibility of efficient uncertainty quantification with various well controls. We are interested in the development of surrogate models for uncertainty quantification and propagation in reservoir simulations using a deep convolutional encoder-decoder network as an analogue to the image-to-image regression tasks in computer science. First, a U-Net architecture is applied to replace conventional expensive deterministic PDE solver. Then we adopt the idea from shape-guided image generation using variational U-Net and design a new variational U-Net architecture for "control-guided" reservoir simulation. Backward propagation is learned in the network to extract the hidden physical quantities and then predict the future production by the learned forward propagation using the hidden variable with various well controls. Comparisons in computational efficiency are made between our proposed CNN approach and conventional MC approach. Significant improvements in computational speed with reasonable accuracy loss are observed in the numerical tests.
The subsurface is such a complex system involving interactive physics in multiple scales that there is no accurate deterministic model. Thus probabilistic models have been explored to account for the uncertainties from model errors, model parametrization, heterogeneity of the environment and various geometries of the boundary, giving rise to extensive research interests in uncertainty quantification in reservoir engineering. A dominant strategy for such problems is to solve the deterministic problem at a finite large number of realizations of the random inputs using Monte Carlo sampling @cite_5 . Variations of MC methods, including quasi-MC @cite_23 , Multi-Level-MC(MLMC) @cite_13 @cite_6 , stratified MC @cite_21 , and etc, are designed to make more efficient sampling. Intrusive methods, (e.g., moment equations @cite_19 , polynomial chaos @cite_7 , stochastic collocation @cite_14 , method of distributions @cite_4 ), as alternatives to MC simulations, have been widely studied in the past decades. Although efficient in some problems, intrusive methods, in general, require additional efforts in reformulation of the model and reconstruction of the deterministic solvers. In the meantime, it is well known that all intrusive methods suffer from curse of dimensionality (in random space). We refer to reviews on these topics in @cite_22 .
@cite_14 @cite_4 @cite_22 @cite_7 @cite_21 @cite_6 @cite_19 @cite_23 @cite_5 @cite_13
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Received 28 May 2002; revised 18 September 2002; accepted 24 September 2002; published 14 February 2003. [1] When an inverse problem is solved to estimate an unknown function such as the hydraulic conductivity in an aquifer or the contamination history at a site, one constraint is that the unknown function is known to be everywhere nonnegative. In this work, we develop a statistically rigorous method for enforcing function nonnegativity in Bayesian inverse problems. The proposed method behaves similarly to a Gaussian process with a linear variogram (i.e., unrestricted Brownian motion) for parameter values significantly greater than zero. The method uses the method of images to define a prior probability density function based on reflected Brownian motion that implicitly enforces nonnegativity. This work focuses on problems where the unknown is a function of a single variable (e.g., time). A Markov chain Monte Carlo (MCMC) method, specifically, a highly efficient Gibbs sampler, is implemented to generate conditional realizations of the unknown function. The new method is applied to the estimation of the trichloroethylene (TCE) and perchloroethylene (PCE) contamination history in an aquifer at Dover Air Force Base, Delaware, based on concentration profiles obtained from an underlying aquitard. INDEX TERMS: 1831 Hydrology: Groundwater quality; 1869 Hydrology: Stochastic processes; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: stochastic inverse modeling, contaminant source identification, inference under constraints, Markov chain Monte Carlo (MCMC), Gibbs sampling, Bayesian inference Quantitative descriptions of flow and transport in subsurface environmentsare often hampered by uncertainty in the input parameters. Treatingsuch parameters as random fields represents a useful tool for dealingwith uncertainty. We review the state of the art of stochasticdescription of hydrogeology with an emphasis on statisticallyinhomogeneous (nonstationary) models. Our focus is on composite mediamodels that allow one to estimate uncertainties both in geometricalstructure of geological media consisting of various materials and inphysical properties of these materials. In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in [26], and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates. In this paper we revisit the computation and visualization of equivalents to isocontours in uncertain scalar fields. We model uncertainty by discrete random fields and, in contrast to previous methods, also take arbitrary spatial correlations into account. Starting with joint distributions of the random variables associated to the sample locations, we compute level crossing probabilities for cells of the sample grid. This corresponds to computing the probabilities that the well-known symmetry-reduced marching cubes cases occur in random field realizations. For Gaussian random fields, only marginal density functions that correspond to the vertices of the considered cell need to be integrated. We compute the integrals for each cell in the sample grid using a Monte Carlo method. The probabilistic ansatz does not suffer from degenerate cases that usually require case distinctions and solutions of ill-conditioned problems. Applications in 2D and 3D, both to synthetic and real data from ensemble simulations in climate research, illustrate the influence of spatial correlations on the spatial distribution of uncertain isocontours.