Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

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 \cite{ketelson2013}, and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.Comment: 29 pages, 6 figure

Multilevel Uncertainty Quantification Techniques Using Multiscale Methods

In this dissertation, we focus on the uncertainty quantification problems in sub-surface flow models which can be computationally demanding because of the large number of unknowns in forward simulations. First, 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 at different resolutions, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. By suitably choosing the number of samples at different levels, one can use less of expensive forward simulations on the fine grid, while more of inexpensive forward simulations on the coarse grid in Monte Carlo simulations. Further, we describe a Multilevel Markov Chain Monte Carlo (MLMCMC) method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on the fine grid, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale feature of the GMsFEM with the multilevel feature of the MLMC methods, and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates. We also propose a multiscale space-parameter separation model reduction method for handling uncertainties in forward problems. The method is based on the idea of separation of variables. This involves seeking the solution in terms of an expansion, where each term is a separable function of space and parameter variables. To find each term in the expansion, we solve a minimization problem associated with the forward problem. The minimization is performed successively for each term consisting of a separable function. In this proposed approach, we need to solve the PDE repeatedly, where we use GMsFEM to speed up the computation. We discuss how the GMsFEM can be used in this context and how the computational gain can be achieved. We present numerical results, which illustrate the efficiency and accuracy of our method. We also discuss efficient sampling techniques for uncertainty quantification in inverse problems. In particular, we consider Approximate Bayesian computation (ABC) and develop a Multilevel Approximate Bayesian computation (MLABC) by using a hierarchy of forward simulation models within the MLMC framework. This approach improves the MLMCMC approach. In this part of the dissertation, we develop a mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving parameter-dependent two-phase flow problems with transport model. A hierarchy of approximations at different resolutions can be provided by this mixed GMsFEM. ABC can be incorporated in different levels to reduce the computational cost, and to produce an approximate solution by ensembling at different levels

A Comprehensive Model for Real Gas Transport in Shale Formations with Complex Non-planar Fracture Networks

A complex fracture network is generally generated during the hydraulic fracturing treatment in shale gas reservoirs. Numerous efforts have been made to model the flow behavior of such fracture networks. However, it is still challenging to predict the impacts of various gas transport mechanisms on well performance with arbitrary fracture geometry in a computationally efficient manner. We develop a robust and comprehensive model for real gas transport in shales with complex non-planar fracture network. Contributions of gas transport mechanisms and fracture complexity to well productivity and rate transient behavior are systematically analyzed. The major findings are: simple planar fracture can overestimate gas production than non-planar fracture due to less fracture interference. A “hump” that occurs in the transition period and formation linear flow with a slope less than 1/2 can infer the appearance of natural fractures. The sharpness of the “hump” can indicate the complexity and irregularity of the fracture networks. Gas flow mechanisms can extend the transition flow period. The gas desorption could make the “hump” more profound. The Knudsen diffusion and slippage effect play a dominant role in the later production time. Maximizing the fracture complexity through generating large connected networks is an effective way to increase shale gas production

Fracture Detection and Numerical Modeling for Fractured Reservoirs

The subsurface fractures could impact the fluid mechanisms dramatically, which makes the modeling of the hydraulic and natural fractures an essential step for fractured reservoirs simulations. However, because of the complexities of fracture patterns and distributions, it is difficult to detect and quantify the fracture networks. In this study, line detection techniques are designed and applied to quantify the fracture segments from fracture figures. Using this fracture detection algorithm, the fracture segments could be located by detecting the endpoints and the intersections of fractures, thus that the fracture patterns could be accurately captured and characterized. The proposed method is applied to two previous well-known field cases and the pressure distribution results are consistent with the micro-seismic data profiles. These two field cases are simulated and computed by using a semianalytical model and Embedded Discrete Fracture Model (EDFM) respectively. The third case is constructed by the fracture outcrop figure and simulated by a numerical simulator with EDFM implemented. The simulation results are accurate and clearly illustrate the important role fractures play in unconventional reservoirs. The technology proposed in this study could be used to quantify the fracture input data for reservoir simulations and be easily expanded for fracture detection and characterization problems in other fields

Efficient Pre-Conditioned Descent Search Detector for Massive MU-MIMO

Massive multi-user (MU) multiple-input multiple-output (MIMO) is an enabling technology for the next-generation wireless systems as it provides significant improvements in terms of spectral efficiency and per-user peak data rates compared to existing, small-scale MIMO. The presence of increasing antennas at the infrastructure base-station results, however, in high computational complexity for data detection, which requires low-complexity algorithms and efficient hardware designs. Descent search (DS) algorithms have been proposed to reduce complexity of data detection and enable efficient hardware implementations. However, their error-rate performance is not always satisfactory for channels that exhibit correlation or for systems with small base-station to user antenna ratios. In this paper, we propose a linear inequality constraint quadratic programming (LICQP) model, which enables the design of a constrained DS (CDS) detector that outperforms existing DS algorithms. To further improve the performance of CDS in the presence of challenging channel conditions, we propose a universal pre-conditioner. Theoretical analysis and numerical results indicate that CDS with pre-conditioning reduces the complexity over the state-of-the-art (SOA) by 60%. We develop a reference Xilinx Virtex-7 FPGA design, which demonstrates that our implementation achieves 1.75x higher throughput (35 Mbps in a 128 antenna, 8 user system) at comparable hardware resource utilization.ISSN:0018-9545ISSN:1939-935

Asiatic acid improves high-fat-diet-induced osteoporosis in mice via regulating SIRT1/FOXO1 signaling and inhibiting oxidative stress

Asiatic acid can attenuate osteoporosis through suppressing adipogenic differentiation and osteoclastic differentiation. Oxidative stress enhances osteoclastic differentiation but represses osteogenic differentiation to promote osteoporosis. However, the role and mechanism of asiatic acid in osteoporosis have not been reported. Firstly, mice were fed with high-fatdiet (HFD) with or without asiatic acid for 16 weeks. Data from an automatic biochemical analyzer showed that HFD induced down-regulation of high-density lipoprotein (HDL) and an increase of serum levels of triglyceride (TG), total cholesterol (TC) and low-density lipoprotein (LDL). However, asiatic acid administration attenuated the decrease of HDL and increase of serum TG, TC and LDL in osteoporotic mice. Secondly, HFD induced high levels of malondialdehyde (MDA) and reactive oxygen species (ROS), low levels of superoxide dismutase (SOD) and glutathione peroxidase (GSH-Px) in osteoporotic mice. However, the levels of MDA, ROS, SOD and GSH-Px in osteoporotic mice were reversed by asiatic acid administration. (this section is unclear and requires revision) Asiatic acid administration reduced expression of c-telopeptide of type 1 collagen (CTX-1), enhanced alkaline phosphatase (ALP) and procollagen type 1 N-terminal propeptide (P1NP) in HFD-induced osteoporotic mice. (this section is unclear and requires revision) Thirdly, asiatic acid promoted calcium deposition in bone marrow cells and osteogenic differentiation in osteoporotic mice, but decreased ALP in bone marrow cells. Lastly, asiatic acid enhanced SIRT1 and nuclear FOXO1 (Nu-FOXO1) expression, while it reduced Acetyl FOXO1 (AcFOXO1) in osteoporotic mice. In conclusion, asiatic acid might inhibit oxidative stress and promote osteogenic differentiation through activating SIRT1/FOXO1 to attenuate HFD-induced osteoporosis in mice