Active Learning and Transparency in Teaching Gateway Mathematics Courses

The need: we concentrate our attention on the gateway courses (for example, PreCalculus ) which have had traditionally low passing scores and retention. Objectives: in our teaching practice we aim to be consistent assuring: Active participation in the class work based on understanding the concepts and ability to implement them; Clear understanding of the requirements and the grading system (what should be done, when it needs to be done, and how to do that?)https://digitalscholarship.unlv.edu/btp_expo/1075/thumbnail.jp

Exploring Parameter Sensitivity Analysis in Mathematical Modeling with Ordinary Differential Equations

This paper presents an exploration into parameter sensitivity analysis in mathematical modeling using ordinary differential equations (ODEs). Taking the first steps in understanding local sensitivity analysis through the direct differential method and global sensitivity analysis using metrics like Pearson, Spearman, PRCC, and Sobol’, we provide readers with a basic understanding of parameter sensitivity analysis for mathematical modeling using ODEs. As an illustrative application, the system of differential equations modeling population dynamics of several fish species with harvest considerations is utilized. The results of employing local and global sensitivity analysis are compared, shedding light on the strengths and limitations of each approach. The paper serves as a starting point for readers interested in exploring parameter sensitivity in their mathematical models

Multi-scale Asymptotic Analysis of Gas Transport in Shale Matrix

Organic-rich resource shales play an important role in global natural gas production. However, many uncertainties exist in an engineering analysis of gas transport and production such as the reservoir-scale flow simulation, history-matching, and optimization. In this work, we introduce a new set of governing equations to describe the characteristic features of porous structures of the organic-rich resource shale. We apply multi-scale analysis to mass balance equations, the equation of state (for free gas), and an adsorption isotherm. Using the macroscopic model, we study gas transport in shales, consisting of nanoporous organic material (kerogen) and the inorganic material. We conclude that both gas in-place and gas production rate depend on the amount of kerogen in the shale matrix. Adsorbed-phase transport by the organic pore walls is responsible for the increase in production rate. We investigate both Henry and Langmuir adsorption as well as different values of length scale ratio and diffusion coefficients

Elastic waves in fractured rocks under periodic compression

Abstract Background One of the current problems in studying the mechanical properties and behavior of structurally inhomogeneous media with cracks is the characterization of acoustic wave propagation. This is especially important in Geomechanics and prognosis of earthquakes. Methods In this work, the authors propose an approach that could simplify characterization of wave propagation in medium with cracks. It is based on homogenization procedure performed at a set of equations characterizing acoustic wave propagation in media weakened by fractures under condition of external distributed loading. Such kind of loading in most cases is close to the real one in case of consideration of Geomechanics problems. Results On the basis of the proposed homogenization technique, we performed characterization of elastic properties and plane acoustic waves propagation in a pre-loaded linear elastic medium weakened by a large amount of cracks. We have investigated two special cases of loading: uniaxial compression and complex compression. We have also studied how the wavespeeds depend on averaged concentration and distribution of craks. Conclusions Effective elastic properties were theoretically characterized for fractured media under external loading. The results revealed high dependency of the longitudinal wave propagation speed on the relation between stresses reasoned by an external loading

Multiscale Modeling of Gas Flow Through Organic-Rich Shale Matrix

In this work, we study gas flow through shale matrix consisting of a microporous inorganic material and nanoporous organic material (kerogen). We apply a multiscale analysis to mass balance equations taking into account such processes as desorption of gas from nanopores in kerogen, diffusion, and filtration. As a result, we get a macroscopic initial boundary-value problem with effective coefficients. The effective coefficients are determined from the solution of a problem on representative elementary volume (REV or periodicity cell). They are influenced by the structure of shale matrix. The values of the effective coefficients depend on permeability and porosity, diffusivity of adsorbed and free gas, adsorption/desorption mechanism, and the concentration of kerogen. We study the free gas amount across the shale matrix as a function of coordinates and time. We conclude that due to the adsorbed-phase transport by the organic pore walls, the amount of gas in-place and gas production rate increase with the concentration of kerogen. We investigate both the Henry and Langmuir adsorption, and also the effect of nonlinearity caused by the dependence of matrix permeability on pressure