This report discusses three topics relating to errors of numerical methods and to improvements of numerical approximations. The introduction connects these topics to the secondary mathematics curriculum. The three chapters which follow develop the three selected topics: improving approximations of irrational numbers, error analysis of numerical integration methods, and discretization versus rounding error in Euler’s Method for solving ordinary differential equations. The conclusion describes specific national secondary mathematical standards and classroom activities relevant to numerical approximations and error analysis. / text

Macroscopic transport properties of porous media depend on textural rock parameters such as porosity, grain size and grain shape distributions, surface-to-volume ratios, and spatial distributions of cement. Although porosity is routinely measured in the laboratory, direct measurements of other textural rock properties can be tedious, time-consuming, or impossible to obtain without special methods such as X-ray microtomography and scanning electron microscopy. However, by using digital three-dimensional pore-scale rock models and physics-based algorithms researchers can calculate both geometrical and transport properties of porous media. Therefore, pore-scale modeling techniques provide a unique opportunity to explore explicit relationships between pore-scale geometry and fluid and electric flow properties. The primary objective of this dissertation is to investigate at the pore-scale level the effects of grain shapes and spatial cement distribution on macroscopic rock properties for improved understanding of various petrophysical correlations. Deposition and compaction of grains having arbitrary angular shapes and various sizes is modeled using novel sedimentation and cementation pore-scale algorithms. Additionally, the algorithms implement numerical quartz precipitation to describe preferential cement growth in pore-throats, pore-bodies, or uniform layers. Subsequently, petrophysical properties such as geometrical pore-size distribution, primary drainage capillary pressure, absolute permeability, streamline-based throat size distribution, and apparent electrical formation factor are calculated for several digital rock models to evaluate petrophysical correlations. Furthermore, two geometrical approximation methods are introduced to model irreducible (connate) water saturation at the pore scale. Consolidated grain packs having comparable porosities and grain size distributions but various grain shapes indicate that realistic angular grain shape distribution gives the best agreement of petrophysical properties with experimental measurements. Cement volume and its spatial distribution significantly affect pore-space geometry and connectivity, and subsequently, macroscopic petrophysical properties of the porous media. For example, low-porosity rocks having similar grain structure but different cement spatial distribution could differ in absolute permeability by two orders of magnitude and in capillary trapped water saturation by a factor of three. For clastic rocks with porosity much higher than percolation threshold porosity, pore-scale modeling results confirm that surface-to-volume ratio and porosity provide sufficient rock-structure character to describe absolute permeability correlations. In comparison to surface-to-volume ratio, capillary trapped (irreducible) water saturation exhibits better correlation with absolute permeability due to weak pore space connectivity in low-porosity samples near the percolation threshold. Furthermore, in grain packs with fine laminations and permeability anisotropy, pore-scale analysis reveals anisotropy in directional drainage capillary- pressure curves and corresponding amounts of capillary-trapped wetting fluid. Finally, results presented in this dissertation indicate that pore-scale modeling methods can competently capture the effects of porous media geometry on macroscopic rock properties. Pore-scale two- and three-phase transport calculations with fast computers can predict petrophysical properties and provide sensitivity analysis of petrophysical properties for accurate reservoir characterization and subsequent field development planning. / text

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The ability to discriminate relative and absolute number has been researched widely in both human and nonhuman species. However, the full extent of numerical ability in nonhuman animals, and the nature of the underlying numerical representation, on which discriminations are based, is still unclear. The aim of the current research was to examine the performance of pigeons and humans in tasks that require the discrimination of relative number (a bisection procedure), and absolute number (in a reproduction procedure). One of the main research questions was whether numerical control over responding could be obtained, above and beyond control by temporal cues in nonhuman animals, and if so, whether it was possible to quantify the relative influences of number and time on responding. Experiment 1 examines nonhuman performance in a numerical bisection task; subjects were presented with either 2 and 6, 4 and 12, or 8 and 24 keylight flashes across three different conditions, and were required to classify these flash sequences as either a “large” or “small” number, by pecking the blue or white key, respectively. Subjects were then tested with novel values within and 2 values higher and lower than the training values. Experiments 2-4 investigate responding in a novel numerical reproduction procedure, in which pigeons were trained to match the number of responses made during a production phase to the number of keylight flashes (2, 4, or 6) in a recently completed sample phase. Experiments 2 and 2A examined discrimination performance when the temporal variables, flash rate and sample phase duration, were perfectly correlated (Experiment 2) or only weakly correlated (Experiment 2a) with flash number. Acquisition of performance in the numerical reproduction procedure was investigated in Experiment 3. For Experiments 1-3, hierarchical regression analyses showed significant control by number over responding, after controlling for temporal cues. Additionally, positive transfer to novel values both within and outside the training range was obtained when the temporal organization of test sequences was similar to baseline training. Experiment 4 investigated the effects of increasing or decreasing the retention interval (RI) on performance in the reproduction procedure, and found this produced a response bias towards larger numbers, contrary to predictions based on previous RI research, and suggested responding was not affected by memorial decay processes. The structure of the representation of number developed by subjects in the bisection and reproduction procedures was investigated using analyses of responding and response variability in Chapters 2 and 6, respectively. Bisection points obtained in Experiment 1 were located at the arithmetic, not geometric mean of all three scales, and coefficients of variation (CVs) obtained in both the bisection and reproduction experiments tended to decrease as flash number increased. Additionally, analyses of the acquisition data found differences in average response number was better fit by a linear than logarithmic scale. These results show that responding did not conform to scalar variability and is largely inconsistent with previous nonhuman research. Together these results suggest responding appeared to be based on a linear scale of number with constant generalisation between values, similar to that associated with human verbal counting, rather than a logarithmic scale with constant generalisation or a linear scale with scalar generalisation between values. Experiment 5 compared pigeons’ and humans’ verbal and nonverbal discrimination performance with numbers 1-20 in analogous bisection, reproduction and report tasks. Human verbal and nonverbal performance in the three tasks was similar and resembled nonhuman performance, although verbal discriminations were more accurate and less variable. The main findings from Experiments 1 and 2A were replicated with humans; bisection points were located at the arithmetic mean, average response number increased linearly as sample number increased, though there was a tendency to underestimate sample number, and decreasing CVs were also obtained for values less than 8. An additional, interesting finding was that CVs showed scalar variability for values greater than 8, suggesting a less exact representation and discrimination process was being used for these values. Collectively, these five experiments provide new evidence for a nonverbal ability to discriminate relative and absolute number with increasing relative accuracy resembling human verbal counting in both human and nonhumans.