Multiscale Wavelet and Upscaling-Downscaling for Reservoir Simulation

The unfortunate case of hydrocarbon reservoirs being often too large and filled with uncertain details in a large range of scales has been the main reason for developments of upscaling methods to overcome computational expenses. In this field lots of approaches have been suggested, amongst which the wavelets application has come to our attention. The wavelets have a mathematically multiscalar nature which is a desirable property for the reservoir upscaling purposes. While such a property has been previously used in permeability upscaling, a more recent approach uses the wavelets in an operator-coarsening- based upscaling approach. We are interested in enhancing the efficiency in implementation of the second approach. the performance of an wavelet-based operator coarsening is compared with several other upscaling methods such as the group renormalization, the pressure solver and local-global upscaling methods. An issue with upscaling, indifferent to the choice of the method, is encountered while the saturation is obtained at coarse scale. Due to the scale discrepancy the saturation profiles are too much averaged out, leading to unreliable production curves. An idea is to downscale the results of upscaling (that is to keep the computational benefit of the pressure equation upscaling) and solve the saturation at the original un-upscaled scale. For the saturation efficient solution on this scale, streamline method can then be used. Our contribution here is to develop a computationally advantageous downscaling procedure that saves considerable time compared to the original proposed scheme in the literature. This is achieved by designing basis functions similar to multiscale methods used to obtain a velocity distribution. Application of our upscaling-downscaling method on EOR processes and also comparing it with non-uniform quadtree gridding will be further subjects of this study

Converted wave imaging and velocity analysis using elastic reverse-time migration

Master's thesis in petroleum geosciences engineeringAlong the continuous evolution of exploration seismology, the main objective has been producing better subsurface seismic images that lead to lower risk exploration and enhanced production. The unique characteristics of converted (P-S) waves enable retrieving more accurate subsurface information, which made it play a complementary role in hydrocarbon seismic exploration, where the primary method of conventional compressional wave (P-P) data has limited capabilities. Conventional processing techniques of P-S data are based on approximations that do not respect the elastic nature of the subsurface and the vector nature of the recorded wave-fields, which urge the need for accurate modeling of subsurface velocity fields, and elastic imaging algorithm that can overcome the shortcomings following the conventional approximations. In this thesis we presented a novel workflow for accurate depth imaging and velocity analysis for multicomponent data. The workflow is based on elastic reverse-time migration as a robust migration algorithm, and automatic wave equation migration velocity analysis techniques. We practically tested novel imaging conditions for elastic reverse-time migration in order to overcome the polarity reversal problem and investigated the cross-talking between wave-modes. For velocity analysis we applied stack-power maximization to produce improved velocity fields that enhance the image coherency, then we applied co-depthing technique based on novel Born modeling/demigration method and target image fitting procedure in order to produce the shear-wave velocity model that result in depth consistent P-S and P-P images. We successfully implemented the workflow on synthetic and field datasets. The results obtained show the robustness and practicality of the workflow to produce enhanced velocity models and accurate subsurface elastic images

The Multiplicative Zak Transform, Dimension Reduction, and Wavelet Analysis of LIDAR Data

This thesis broadly introduces several techniques within the context of timescale analysis. The representation, compression and reconstruction of DEM and LIDAR data types is studied with directional wavelet methods and the wedgelet decomposition. The optimality of the contourlet transform, and then the wedgelet transform is evaluated with a valuable new structural similarity index. Dimension reduction for material classification is conducted with a frame-based kernel pipeline and a spectral-spatial method using wavelet packets. It is shown that these techniques can improve on baseline material classification methods while significantly reducing the amount of data. Finally, the multiplicative Zak transform is modified to allow the study and partial characterization of wavelet frames

Development of an adaptive multi-resolution method to study the near wall behavior of two-dimensional vortical flows

In the present investigation, a space-time adaptive multiresolution method is developed to solve evolutionary PDEs, typically encountered in fluid mechanics. The new method is based on a multiresolution analysis which allows to reduce the number of active grid points significantly by refining the grid automatically in regions of steep gradients, while in regions where the solution is smooth coarse grids are used. The method is applied to the one-dimensional Burgers equation as a classical example of nonlinear advection-diffusion problems and then extended to the incompressible two-dimensional Navier-Stokes equations. To study the near wall behavior of two-dimensional vortical flows a recently revived, dipole collision with a straight wall is considered as a benchmark. After that an extension to interactions with curved walls of concave or convex shape is done using the volume penalization method. The space discretization is based on a second order central finite difference method with symmetric stencil over an adaptive grid. The grid adaptation strategy exploits the local regularity of the solution estimated via the wavelet coefficients at a given time step. Nonlinear thresholding of the wavelet coefficients in a one-to-one correspondence with the grid allows to reduce the number of grid points significantly. Then the grid for the next time step is extended by adding a safety zone in wavelet coefficient space around the retained coefficients in space and scale. With the use of Harten's point value multiresolution framework, general boundary conditions can be applied to the equations. For time integration explicit Runge-Kutta methods of different order are implemented, either with fixed or adaptive time stepping. The obtained results show that the CPU time of the adaptive simulations can be significantly reduced with respect to simulations on a regular grid. Nevertheless the accuracy order of the underlying numerical scheme is preserved

Parallel MR Image Reconstruction Using Augmented Lagrangian Methods

Magnetic resonance image (MRI) reconstruction using SENSitivity Encoding (SENSE) requires regularization to suppress noise and aliasing effects. Edge-preserving and sparsity-based regularization criteria can improve image quality, but they demand computation-intensive nonlinear optimization. In this paper, we present novel methods for regularized MRI reconstruction from undersampled sensitivity encoded data-SENSE-reconstruction-using the augmented Lagrangian (AL) framework for solving large-scale constrained optimization problems. We first formulate regularized SENSE-reconstruction as an unconstrained optimization task and then convert it to a set of (equivalent) constrained problems using variable splitting. We then attack these constrained versions in an AL framework using an alternating minimization method, leading to algorithms that can be implemented easily. The proposed methods are applicable to a general class of regularizers that includes popular edge-preserving (e.g., total-variation) and sparsity-promoting (e.g., -norm of wavelet coefficients) criteria and combinations thereof. Numerical experiments with synthetic and in vivo human data illustrate that the proposed AL algorithms converge faster than both general-purpose optimization algorithms such as nonlinear conjugate gradient (NCG) and state-of-the-art MFISTA.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85846/1/Fessler4.pd