A benchmark comparison of numerical methods for infinite Prandtl number thermal convection in two-dimensional Cartesian geometry

Abstract

A comparison is made between seven different numerical methods for calculating two-dimensional thermal convection in an infinite Prandtl number fluid. Among the seven methods are finite difference and finite element techniques that have been used to model thermal convection in the Earth's mantle. We evaluate the performance of each method using a suite of four benchmark problems, ranging from steady-state convection to intrinsically time-dependent convection with recurring thermal boundary layer instabilities. These results can be used to determine the accuracy of other computational methods, and to assist in the development of new ones.     

Grid resolution study of ground water flow and transport

Three-dimensional grids representing a heterogeneous, ground water system are generated at 10 different resolutions in support of a site-scale flow and transport modeling effort. These grids represent hydrostratigraphy near Yucca Mountain, Nevada, consisting of 18 stratigraphic units with contrasting fluid flow and transport properties. The grid generation method allows the stratigraphy to be modeled by numerical grids of different resolution so that comparison studies can be performed to test for grid quality and determine the resolution required to resolve geologic structure and physical processes such as fluid flow and solute transport. The process of generating numerical grids with appropriate property distributions from geologic conceptual models is automated, thus making the entire process easy to implement with fewer user-induced errors. The series of grids of various resolutions are used to assess the level at which increasing resolution no longer influences the flow and solute transport results. Grid resolution is found to be a critical issue for ground water flow and solute transport. The resolution required in a particular instance is a function of the feature size of the model, the intrinsic properties of materials, the specific physics of the problem, and boundary conditions. The asymptotic nature of results related to flow and transport indicate that for a hydrologic model of the heterogeneous hydrostratigraphy under Yucca Mountain, a horizontal grid spacing of 600 m and vertical grid spacing of 40 m resolve the hydrostratigraphic model with sufficient precision to accurately model the hypothetical flow and solute transport to within 5% of the value that would be obtained with much higher resolution.     

Statistic inversion of multi-zone transition probability models for aquifer characterization in alluvial fans

Understanding the heterogeneity arising from the complex architecture of sedimentary sequences in alluvial fans is challenging. This paper develops a statistical inverse framework in a multi-zone transition probability approach for characterizing the heterogeneity in alluvial fans. An analytical solution of the transition probability matrix is used to define the statistical relationships among different hydrofacies and their mean lengths, integral scales, and volumetric proportions. A statistical inversion is conducted to identify the multi-zone transition probability models and estimate the optimal statistical parameters using the modified Gauss–Newton–Levenberg–Marquardt method. The Jacobian matrix is computed by the sensitivity equation method, which results in an accurate inverse solution with quantification of parameter uncertainty. We use the Chaobai River alluvial fan in the Beijing Plain, China, as an example for elucidating the methodology of alluvial fan characterization. The alluvial fan is divided into three sediment zones. In each zone, the explicit mathematical formulations of the transition probability models are constructed with optimized different integral scales and volumetric proportions. The hydrofacies distributions in the three zones are simulated sequentially by the multi-zone transition probability-based indicator simulations. The result of this study provides the heterogeneous structure of the alluvial fan for further study of flow and transport simulations.     

Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations

Mathematical Geosciences - We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic...     

Hydrogeologic Controls on Induced Seismicity in Crystalline Basement Rocks Due to Fluid Injection into Basal Reservoirs

A series of M_b 3.8–5.5 induced seismic events in the midcontinent region, United States, resulted from injection of fluid either into a basal sedimentary reservoir with no underlying confining unit or directly into the underlying crystalline basement complex. The earthquakes probably occurred along faults that were likely critically stressed within the crystalline basement. These faults were located at a considerable distance (up to 10 km) from the injection wells and head increases at the hypocenters were likely relatively small (～70–150 m). We present a suite of simulations that use a simple hydrogeologic‐geomechanical model to assess what hydrogeologic conditions promote or deter induced seismic events within the crystalline basement across the midcontinent. The presence of a confining unit beneath the injection reservoir horizon had the single largest effect in preventing induced seismicity within the underlying crystalline basement. For a crystalline basement having a permeability of 2 × 10^?17 m² and specific storage coefficient of 10^?7/m, injection at a rate of 5455 m³/d into the basal aquifer with no underlying basal seal over 10 years resulted in probable brittle failure to depths of about 0.6 km below the injection reservoir. Including a permeable (k_z = 10^?13 m²) Precambrian normal fault, located 20 m from the injection well, increased the depth of the failure region below the reservoir to 3 km. For a large permeability contrast between a Precambrian thrust fault (10^?12 m²) and the surrounding crystalline basement (10^?18 m²), the failure region can extend laterally 10 km away from the injection well.