High-resolution numerical methods for compressible multi-phase flow in hierarchical porous media. Final report, November 1992--August 1996

The objectives of this project were to develop computationally efficient numerical methods for modeling surfactant flooding in enhanced oil recovery and aquifer remediation. Surfactants have been considered by several oil companies to reduce the large residual oil saturations, and are being seriously considered for cleanup of dense contaminants in aquifers, particularly chlorinated hydrocarbons. The authors employed second-order Godunov methods for the discretization of the conservation laws, and lowest-order mixed finite element methods for the discretization of the pressure equation. They also used dynamically adaptive mesh refinement to concentrate the computational work. The development of the second-order Godunov method required a mathematical analysis of the hyperbolic wave structure; this analysis discovered undesirable features f the model that lead to infinite characteristic speeds. Minor modifications of the model to remove the infinite characteristic speeds improved the stability of the model considerably. The use of adaptive mesh refinement required the development of several techniques for upscaling various physical quantities, and a multigrid iteration for the pressure equation on an adaptively refined grid. Numerical simulations showed that the second-order Godunov method is reasonably effective in preserving sharp fluid fronts, but is too computationally expensive in so complex a fluid model. On the other hand, the same simulations showed that adaptive mesh refinement is very effective in reducing CPU time: computational time for adaptive simulations scale proportional to the total number of grid cells, while uniform grid calculations have computational time that scales with the number of cells times the number of timesteps

High-resolution numerical methods for compressible multi-phase flow in hierarchical porous media. Progress report, September 1993--September 1994

This is the second year in the proposed three-year effort to develop high-resolution numerical methods for multi-phase flow in hierarchical porous media. The issues being addressed in this research are: Computational efficiency: Field-scale simulation of enhanced oil recovery, whether for energy production or aquifer remediation, is typically highly under-resolved. This is because rock transport properties vary on many scales, and because current numerical methods have low resolution. Effective media properties: Since porous media are formed through complex geologic processes, they involve significant uncertainty and scale-dependence. Given this uncertainty, knowledge of ensemble averages of flow in porous media can be preferable to knowledge of flow in specific realizations of the reservoir. However, current models of effective properties do not represent the observed behavior very well. Relative permeability models present a good example of this problem. In practice, these models seldom provide realistic representations of hysteresis, interfacial tension effects or three-phase flow; there are no models that represent well all three effects simultaneously

Associations between Delta-8 THC and Four Loko retail availability in Fort Worth, Texas

Alcohol and cannabis are two of the most widely used substances among young people, and availability and price are two of the most significant determinants of use. Four Loko products contain up to 5.5 standard alcoholic drinks in a single can, are one of the least expensive ready-to-drink alcohol products on the market and are commonly consumed by underage drinkers. Delta-8 THC is a psychoactive substance with no federal regulations regarding minimum purchase age, ingredients and synthesis, marketing, and testing for potency or contaminants. Delta-8 THC products can be inexpensively synthesized and are sold for low prices. Given that young people often use both products, and use of these products can result in negative consequences, it is important to understand whether these products are being sold in the same stores, which would indicate the presence of niche stores marketing high-risk, youth-oriented substances. This study included 360 locations with off-premise beer or beer/wine licenses in Fort Worth, Texas. Locations were called and asked whether they sold Delta-8 THC. Four Loko’s availability was determined using the manufacturer’s website. A logistic regression model examined associations between the availability of Delta-8 THC and Four Loko. Of the 360 locations, 38% sold Four Loko and 9% sold Delta-8 THC. Delta-8 THC availability was significantly associated with higher odds of Four Loko availability (OR=2.15,95%CI=1.05,4.43). Given the associations between the retail availability of Delta-8 THC and Four Loko, policies that limit access to such products, including near schools and in stores that youth patronize, may be warranted

Macroscopic Equations of Motion for Two Phase Flow in Porous Media

The established macroscopic equations of motion for two phase immiscible displacement in porous media are known to be physically incomplete because they do not contain the surface tension and surface areas governing capillary phenomena. Therefore a more general system of macroscopic equations is derived here which incorporates the spatiotemporal variation of interfacial energies. These equations are based on the theory of mixtures in macroscopic continuum mechanics. They include wetting phenomena through surface tensions instead of the traditional use of capillary pressure functions. Relative permeabilities can be identified in this approach which exhibit a complex dependence on the state variables. A capillary pressure function can be identified in equilibrium which shows the qualitative saturation dependence known from experiment. In addition the new equations allow to describe the spatiotemporal changes of residual saturations during immiscible displacement.Comment: 15 pages, Phys. Rev. E (1998), in prin

Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

This paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory

On the Global Convergence of Broyden's Method

We consider Broyden's 1965 method for solving nonlinear equations. If the mapping is linear, then a simple modification of this method guarantees global and Q-superlinear convergence. For nonlinear mappings it is shown that the hybrid strategy for nonlinear equations due to Powell leads to R-superlinear convergence provided the search directions from a uniformly linearly indepenent sequence. We then explore this last concept and its connection with Broyden's method. Finally, we point out how the above results extend to Powell's symmetric version of Broyden's method