New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

High resolution schemes for hyperbolic conservation laws

Let X(t) be the number of tumor cells at time t, and Pr{X(t) = n} = pn(t) is the density of X. A “birth”, i.e., an increase of one of the total population of cancer cells, can occur either by mutation of a normal cell caused by the action of the carcinogen, consisting of randomly (Poisson) distributed hits, or by reproduction of existing cancer cells. A death of a tumor cell occurs as an additive combination of non-immunological and immunological elements. Once a tumor is initiated by carcinogenic action, it undergoes a birth and death process with infinitesimal birth rate linear and infinitesimal death rate composed of a linear and a nonlinear term, the former due to non-immunological deaths, the latter to immunological feedback. The death rate per tumor cell due to immunological response is proportional to the total number of antigen-producing (tumor) cells; thus, the total death rate is quadratic. Although this assumes a very simple mechanism for the action of immunological feedback, it is nevertheless a first step.

The Mathematical Model

Water chemistry has been shown experimentally to affect the stability of water films and the sorption of organic oil components on mineral surfaces. When oil is displaced by water, water chemistry has been shown to impact oil recovery. At least two mechanisms could account for these effects, the water chemistry could change the charge on the rock surface and affect the rock wettability, and/or changes in the water chemistry could dissolve rock minerals and affect the rock wettability. The explanations need not be the same for oil displacement of water as for water imbibition and displacement of oil. This article investigates how water chemistry affects surface charge and rock dissolution in a pure calcium carbonate rock similar to the Stevns Klint chalk by constructing and applying a chemical model that couples bulk aqueous and surface chemistry and also addresses mineral precipitation and dissolution. We perform calculations for seawater and formation water for temperatures between 70 and 130°C. The model we construct accurately predicts the surface potential of calcite and the adsorption of sulfate ions from the pore water. The surface potential changes are not able to explain the observed changes in oil recovery caused by changes in pore water chemistry or temperature. On the other hand, chemical dissolution of calcite has the experimentally observed chemical and temperature dependence and could account for the experimental recovery systematics. Based on this preliminary analysis, we conclude that although surface potential may explain some aspects of the existing spontaneous imbibitions data set, mineral dissolution appears to be the controlling factor.

/pdf/the-impact-of-pore-water-chemistry-on-carbonate-surface-18pa07prw0.pdf

The Impact of Pore Water Chemistry on Carbonate Surface Charge and Oil Wettability

Dean of Undergraduates John Hutchinson, the General Council and President David Leebron assembled the Working Group on University Responses to Federal Initiatives on Sexual Assault in June, according to Associate Vice Provost Matthew Taylor. The group formed in order to address federal measures as well as to continue changes already in progress. The federal government began releasing guidelines on how universities and colleges should address sexual assault beginning in April 2014. The White House task force report, Not Alone, features certain recommendations regarding training, prevention and adjudication of sexual assault. “The working group is looking at Rice’s policies, processes and communications about sexual violence and sexual harassment, in light of what the government is either recommending or requiring to make sure that we have a clear sense of the areas in which,” Taylor said. “We either exceed the recommendations or there are things that we need to do to meet the recommendations.” The working group consists of both undergraduate and graduate representatives, as well as leadership from the Student Wellbeing Office, Student Judicial Programs, the Rice University Police Department, the Rice Counseling Center, Athletics and Graduate and Post-Doctoral Studies, among others. There are approximately an equal number of males and females in the group. “I wouldn’t say [we have had] findings yet,” Taylor said. “We have tried to educate ourselves on the government’s recommendations.” According to Taylor, the working group focused on immediate shortterm issues for the summer. Many of the federal recommendations were already being met by Rice, and several aspects were already undergoing changes before the formation of the working group. Before the working group was formed, Associate Dean of Undergraduates Donald Ostdiek had already decided to hire a Title IX Resource Navigator, who will help students navigate the investigation if they choose to come forward and report a sexual assault. The main efforts of the working group are directed towards meeting federal guidelines that state that universities should designate employees to whom students can report sexual assault in confidence. Taylor said although Rice already has designated individuals, the policy’s language did not clearly outline whether those individuals are Title IX officers, responsible employees or non-disclosing employees. He said the new language encourages individuals to report cases. “The Title IX officers, in most cases, don’t have to report a student’s name, but we make it equally fair that, in some cases, the university would have to take action based on the law,” Taylor said. “Some of the people that we know students are most likely to go to for advice after a sexual assault are college masters, resident [associates] and college coordinators. We make it clear that they are responsible employees [who are] required to notify one of the Title IX coordinators if they are alerted by a student to a case of sexual assault or sexual violence.” Director of the Center for Student Wellbeing Kate Noonan and Wellbeing Advisor Saralyn Hernandez also contributed to the working group. Hernandez and Noonan said the Wellbeing Advisors are committed to providing trauma-informed support to students who have experienced sexual misconduct. “For survivors who prefer offcampus resources, Rice contracts with the Houston Area Women’s Center, [which] can provide support to Rice students outside the hedges,” Noonan and Hernandez said. Currently, new students undergo training during Orientation Week through Project Sexual Assault-Free Environment, presented by the Houston Area Women’s Resource Center and the Student Wellbeing Office. Although the working group did not have any influence on this year’s presentation of Project SAFE, the presentation already met federal recommendations by focusing on bystander prevention. The presentation discussed non-disclosing and responsible employees, although it did not talk about consequences for potential perpetrators. Colin Losey, a Martel College sophomore who transferred from the University of Chicago, said although he thought the presentation itself was satisfactory, he did not feel that all of the options for victims of sexual assault were covered, especially the option to report to police outside the university. “I’m a transfer student from a university that was recently hit with a major Title IX investigation, and a lot of what has come out from that is that the university’s adjudication process was used in a way to keep things internal,” Losey said. “Given what has happened at other college campuses across the country, students should be clear that this is criminal behavior, and they can press criminal charges.You don’t need to contact Rice first, and Rice need not be involved.” However, Taylor said offering Project SAFE only during O-Week is unsatisfactory and that there must Honor Council removes undergraduates from graduate cases, violates constitution

A new generation.

We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and, in general, are not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin [ Northeastern Math J., 5 (1989), pp. 395--422] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase flow equation, and hyperbolic conservation laws. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present context is to show that the (strongly degenerate) discrete diffusion term is sufficiently smooth. We provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes. Finally, we make some concluding remarks about monotone difference schemes for multidimensional equations.

Monotone Difference Approximations Of BV Solutions To Degenerate Convection-Diffusion Equations

Hybrid flux-splitting schemes for a common two-fluid model

Hybrid flux-splitting schemes for a two-phase flow model

https://www.math.ntnu.no/conservation/2007/025.pdf

Global existence of weak solutions for a viscous two-phase model

Effect of the aqueous chemistry on the mechanical strength of chalk has extensively been studied during the last decade. At high temperatures (~130°C), chalk exposed to seawater is significantly weaker compared to chalk exposed to distilled water when considering the hydrostatic yield strength and the following creep phase. The explanation of these experimental results must be of a chemical nature, as the density and viscosity of the aqueous phase vary little among these different brines. We present the results from simplified aqueous chemistry using MgCl2 brines, and compare these results with seawater. Previous studies show that different ions, e.g. Ca2+, Mg2+, $${{\rm SO}{_{4}}^{2-}}$$
 in the injected brine, as well as the chalk mineralogy have an impact on the stability of the rock. We performed mechanical tests on chalk cores from Liege and Stevns Klint; it was found that these two outcrop chalks exhibit an unexpected difference in their mechanical responses when comparing cores flooded with NaCl and MgCl2 at 130°C. The results of this study show that the effects of magnesium seem to be governed not only by the differences in mineralogy, but also a time dependency on chalk deformation is additionally observed. Independent of the chalk type tested, the chemical analyses performed show that when MgCl2 is flooded through the core, a significant loss of magnesium and a considerable additional amount of calcium are detected in the effluent. The experimental observations fit very well with the time-dependent chemical changes gained from the mathematical model of this study that accounts for transport effects (convection and molecular diffusion) as well as chemical processes such as precipitation/dissolution. Based on the calculations and chemical analyses, we argue that the loss of magnesium and the production of calcium cannot solely be a consequence of a substitution process. The calculations rather indicate that magnesium is precipitated forming new mineral phases and in this process not only calcite, but also silicates are dissolved. The amount of dissolved calcium and silicon from the rock matrix is significant and could thus cause an additional deformation to take place. Both the retention of magnesium in the chalk core and the formation of newly precipitated magnesium-bearing carbonates and/or magnesium-bearing clay-like minerals after flooding with MgCl2 brine were demonstrated using scanning electron microscopic methods. In addition, precipitation of anhydrite as a result of flooding with seawater-like brine was proven. The water-induced strain not only depends on the ion composition of the injected brine; moreover, the presence of non-carbonate minerals will most likely also have a significant influence on the mechanical behaviour of chalk.

Steinar Evje

Papers

Monotone Difference Approximations Of BV Solutions To Degenerate Convection-Diffusion Equations

Hybrid flux-splitting schemes for a common two-fluid model

Hybrid flux-splitting schemes for a two-phase flow model

Global existence of weak solutions for a viscous two-phase model

Chemical Alterations Induced by Rock–Fluid Interactions When Injecting Brines in High Porosity Chalks