Eric S. Carlson

Devonian Shale Gas Production: Mechanisms and Simple Models

This paper shows that, even without consideration of their special storage and flow properties, Devonian shales are special cases of dual porosity. The authors show that wile neglecting these properties in the short term is appropriate, such neglect in the long term will result in an under-estimation of shale gas production.

White matter fiber tractography via anisotropic diffusion simulation in the human brain

A novel approach to noninvasively tracing brain white matter fiber tracts is presented using diffusion tensor magnetic resonance imaging (DT-MRI). This technique is based on successive anisotropic diffusion simulations over the human brain, which are utilized to construct three dimensional diffusion fronts. The fiber pathways are determined by evaluating the distance and orientation from the fronts to their corresponding diffusion seeds. Synthetic and real DT-MRI data are employed to demonstrate the tracking scheme. It is shown that the synthetic tracts are accurately replicated, and several major white matter fiber pathways can be reproduced noninvasively, with the tract branching being allowed. Since simulating the diffusion process, which is truly a physical phenomenon reflecting the underlying architecture of cerebral tissues, makes full use of the diffusion tensor data, including both the magnitude and orientation information, the proposed approach is expected to enhance robustness and reliability in white matter fiber reconstruction.

A fourth-order compact difference scheme on face centered cubic grids with multigrid method for solving 2D convection diffusion equation

We present a fourth-order compact finite difference scheme on the face centered cubic (FCC) grids for the numerical solution of the two-dimensional convection diffusion equation. The seven-point formula is defined on a regular hexagon, where the strategy of directional derivative is employed to make the derivation procedure straightforward, efficient, and concise. A corresponding multigrid method is developed to solve the resulting sparse linear system. Numerical experiments are conducted to verify the fourth-order convergence rate of the derived discretization scheme and to show that the fourth-order compact difference scheme is computationally more efficient than the standard second-order central difference scheme.

Performance of ILU preconditioning techniques in simulating anisotropic diffusion in the human brain

We conduct simulations for the unsteady state anisotropic diffusion process in the human brain by discretizing the governing diffusion equation on a face-centered cubic grid and adopting a high performance differential-algebraic equation solver, IDA, to deal with the resulting large-scale system of DAEs. Incomplete LU preconditioning techniques are used with the GMRES method to accelerate the convergence rate of the iterative solution. We then investigate and compare the efficiency and effectiveness of a number of ILU preconditioners, and find out that the ILUT with a dual dropping strategy gives the best overall performance when it is provided with the optimum choices of the fill-in parameter and the threshold dropping tolerance.

OpenMG: A New Multigrid Implementation in Python

In many large-scale computations, systems of equations arise in the form Au= b, where A is a linear operation to be performed on the unknown data u, producing the known right-hand side, b, which represents some constraint of known or assumed behavior of the system being modeled. Since such systems can be very large, solving them directly can be too slow. In contrast, a multigrid solver solves partially at full resolution, and then solves directly only at low resolution. This creates a correction vector, which is then interpolated to full resolution, where it corrects the partial solution. This project aims to create an open-source multigrid solver called OpenMG, written only in Python. The exist- ing PyAMG multigrid implementation is a highly versatile, configurable, black- box solver, but is difficult to read and modify due to its C core. Our proposed OpenMG is a pure Python experimentation environment for testing multigrid concepts, not a production solver. By making the code simple and modular, we make the algorithmic details clear. We thereby create an opportunity for education and experimentation with the partial solver (Jacobi, Gauss Seidel, SOR, etc.), the restriction mechanism, the prolongation mechanism, and the direct solver, or the use of GPGPUs, multiple CPUs, MPI, or grid computing. The resulting solver is tested on an implicit pressure reservoir simulation problem with satisfactory results.

Fiber tracking by simulating diffusion process with diffusion kernels in human brain with DT-MRI data

A novel approach for noninvasively tracing brain white matter fiber tracts is presented using diffusion tensor magnetic resonance imaging (DT-MRI) data. This technique is based on performing anisotropic diffusion simulations over a series of overlapping three dimensional diffusion kernels that cover only a small portion of the human brain volume and are geometrically centered upon selected starting voxels where a seed is placed. The simulations conducted over diffusion kernels are initiated from those starting voxels and are utilized to construct diffusion fronts. The fiber pathways are determined by evaluating the distance and orientation from fronts to their corresponding diffusion seed voxels. Synthetic and real DT-MRI data are employed to demonstrate the tracking scheme. It is shown that the synthetic tracts can be accurately replicated, while several major white matter fiber pathways in the human brain can be reproduced noninvasively as well. Since the diffusion simulation makes use of the entire diffusion tensor data, including both the magnitude and orientation information, the proposed approach enhances robustness and reliability in DT-MRI based fiber reconstruction.

Multidimensional Finite Differencing (MDFD) with Hypersphere-Close-Pack Grids

ABSTRACT This article concerns the use of hypersphere-close-pack (HCP) grids and quadratic multivariate Taylor polynomial interpolation functions with several additional higher-order terms for multidimensional finite differencing (MDFD). MDFD is a general methodology for the numerical solution of partial differential Equations (PDE) defined on irregular domains. MDFD uses domain embedding, which means that expensive body-fitted gridding is not required, but the methodology is flexible enough to allow any grid. The HCP grid allows efficient, stable, and robust implementation of MDFD. In this article, we discuss MDFD and related methods, summarize the HCP grids in two and three dimensions, outline the algorithms used, and present the results of a number of diagnostic examples. The examples concern numerical solution of the unsteady parabolic heat equation defined for a number of domains of different shapes. The examples suggest, among other things, that the algorithms work with domains that have no similarity to the grid geometry and that the solutions appear to have global O(h 2) convergence.

Numerical Analysis of Laminar Mixed Convection in a Lid Driven Square Cavity With an Isothermally Heated Square Internal Blockage

Laminar mixed convection characteristics in a square cavity with an isothermally heated square blockage inside have been investigated numerically using the finite volume method of the ANSYS FLUENT commercial CFD code. Various different blockage sizes and concentric and eccentric placement of the blockage inside the cavity have been considered. The blockage is maintained at a hot temperature, Th, and four surfaces of the cavity (including the lid) are maintained at a cold temperature, Tc, under all circumstances. The physical problem is represented mathematically by sets of governing conservation equations of mass, momentum, and energy. The geometrical and flow parameters for the problem are the blockage ratio (B), the blockage placement eccentricities (ex and ey), the Reynolds number (Re), the Grashof number (Gr), and the Richardson number (Ri). The flow and heat transfer behavior in the cavity for a range of Richardson number (0.01–100) at a fixed Reynolds number (100) and Prandtl number (0.71) is examined comprehensively. The variations of the average and local Nusselt number at the blockage surface at various Richardson numbers for different blockage sizes and placement eccentricities are presented. From the analysis of the mixed convection process, it is found that for any size of the blockage placed anywhere in the cavity, the average Nusselt number does not change significantly with increasing Richardson number until it approaches the value of the order of 1 beyond which the average Nusselt number increases rapidly with the Richardson number. For the central placement of the blockage at any fixed Richardson number, the average Nusselt number decreases with increasing blockage ratio and reaches a minimum at around a blockage ratio of slightly larger than 1/2. For further increase of the blockage ratio, the average Nusselt number increases again and becomes independent of the Richardson number. The most preferable heat transfer (based on the average Nusselt number) is obtained when the blockage is placed around the top left and the bottom right corners of the cavity.Copyright

Reconstructing Brain White Matter Pathways with Diffusion Tensor MRI Using Kernel-Based Diffusion Simulations

A novel algorithm for approximating anatomical brain connectivity in vivo is presented using diffusion tensor magnetic resonance imaging (DT-MRI). This technique relies on simulating diffusion process within a series of overlapping three dimensional diffusion kernels that cover only a small portion of the human brain volume. The shape of the anisotropic diffusion represented by diffusion fronts is used to estimate the directional organization of the underlying white matter fiber tracts. The proposed algorithm is tested on both simulated and real DT-MRI data. The demonstration shows that the synthetic tracts are accurately replicated, while various examples of white matter fiber pathways can be reconstructed as well, with assigned connectivity indices showing uncertainty. Several features of the algorithm are elucidated by the tracking experiments, including its capability of handling fiber branching and crossing, and robustness to noise. Impact of thresholding settings and the kernel size on performance of the algorithm is also analyzed.

Calculating immobile gas saturations

We present a simple, fast, analytical expression which can be used to estimate the reservoir pressure as a function of gas saturation, for gas saturations between zero and the critical gas saturation. Use of the relation also makes it extremely easy to assess the local recovery factor as a function of pressure from the bubble point down to the pressure at which the critical gas saturation is reached.