J. David Logan

Solute transport in porous media with scale-dependent dispersion and periodic boundary conditions

Abstract An analytic solution is derived for the one-dimensional equations governing the transport of a tracer in a heterogeneous porous medium subject to rate-limited adsorption with a linear equilibrium isotherm, and also subject to decay. The kinetics of solute adsorption is given by a first-order rate law. The medium is semi-infinite, and periodic boundary conditions are imposed at the inlet boundary. Both Dirichlet-type and flux-type boundary conditions are examined. The heterogeneity arises from the assumption of a scale-dependent dispersion coefficient that increases exponentially as a function of distance up to some constant limiting value. Solutions are given in terms of hypergeometric functions, and comparisons are made with the corresponding problem in a homogeneous medium with a constant dispersion. The one-dimensional model gives information about how adsorption and decay effects can interact with heterogeneities in the medium. For example, when both adsorption and decay are present, the amplitude of waves in a homogeneous medium can exceed that in a heterogeneous medium, even though the latter has smaller dispersivity. In addition, because the input is periodic, breakthrough curves measured downstream have both an amplitude and phase shift; measurement of the latter may give additional information for determining medium characteristics.

Dimensional Analysis and the Pi Theorem

A new version of the Buckingham pi theorem is presented which reveals the underlying mathematical structure of that classical result. In this context it becomes a theorem in linear algebra, and it is formulated without reference to physical quantities, units, dimensions, and so on. Also, the classical approach of Birkhoff is reviewed and some points in his proof are expanded.

Similarity Solutions for Reactive Shock Hydrodynamics

It is shown how group-theoretic methods, or similarity methods, can be applied to determine the class of self-similar solutions for a one-dimensional, time-dependent problem in shock hydrodynamics,with a chemical reaction taking place behind the shock. The functional form of all reaction rates depending on the state variables is characterized under which the general system of differential equations and boundaryconditions admit self-similar solutions, It is shown that a subclass of these solutions can model certain processes in detonation physics.

Boundary Conditions for Convergent Radial Tracer Tests and Effect of Well Bore Mixing Volume

Convergent radial flow tracer tests have a complex spatial nonaxial transport structure caused by the flow in the vicinity of the injection well and its finite mixing volume. The formulation of the boundary value problem, and especially the treatment of the boundary conditions at the injection well, is nontrivial. Hodgkinson and Lever [1983],Moench [1989, 1991], and Welty and Gelhar [1994] have developed different models and methods for the analysis of breakthrough curves in the extraction well. To extend interpretation techniques to breakthrough curves in the zone between injection and extraction wells, an analysis of conventional transport models is given, and improved boundary conditions are formulated for a convergent radial tracer test problem. The formulation of the boundary conditions is based upon a more detailed analysis of the kinematic flow structure and tracer mass balance in the neighborhood of the injection well. Two practical applications of revised boundary conditions for field data analysis are given. First, the note explains anomalous high well bore mixing volumes of injection wells found by Cady et al. [1993] and allows one to establish the role of mixing versus other processes (retardation, matrix diffusion, etc.). Second, it is shown that the improper use of Moenchs [1989] model can produce bias in the characteristics of breakthrough curves in the extraction well under conditions that involve a significant mixing factor in the injection well. A numerical example indicates an error in peak concentrations on a breakthrough curve by as much as 70% and in peak arrival time by 10% for Peclet numbers Pe=102. The effect becomes slightly less significant for Pe=1.

Natural History of Mass-action in Predator-prey Models: A Case Study from Wolf Spiders and Grasshoppers

Abstract Mass-action models of predator-prey interactions assume that predators encounter prey according to their relative densities as scaled by functional responses, although models seldom specify critical natural history and behavioral mechanisms that ensure that encounters actually occur. As a case study of this assumption, we assess the hypothesis that daily and seasonal activity and microhabitat use by wandering wolf spiders (Lycosidae: Schizocosa) searching for four common grasshopper species (Orthoptera: Acrididae) are coincident under natural conditions. There was great overlap in seasonal phenology and use of microhabitats between spiders and grasshoppers. Grasshoppers that were suitably sized (10–20 mm in length) as prey for spiders were relatively abundant from late spring through summer in this grassland. Three of the four common grasshopper species used microhabitat in a similar way, but differed from a fourth common species, Phoetaliotes nebrascensis. However, when they were active, spiders were about equally distributed between open microhabitats on the ground and up in the vegetation so that all grasshopper species were at risk. In response to temperature, spiders were active for only a portion of the day during which grasshoppers were also active so that the actual daily “window-of-opportunity” for capture each day was much smaller than expected. Spiders were more likely to be active during the early morning and evening, while grasshoppers were active during all daylight hours, most likely because of differences in thermal preferences. Schizocosa and their grasshopper prey are largely coincident in time and space except for overlap in daily activity which, presumably, reflects differences in thermal preferences. Consequently, overlap in daily time budgets that ensure actual encounter was reduced about 50%. The significance of this difference to the inclusion of simple mass-action dynamics in predator-prey models requires further consideration, but may be important.

Chemical reactor models of optimal digestion efficiency with constant foraging costs

Abstract We develop quantitative optimization criteria for transient digestion processes in simple animal tracts that can be modeled by a semi-batch reactor or plug flow reactor. Specifically, we determine the residence time that optimizes the average net energy intake over the total residence time. The net energy is measured by the total energy intake, less the cost of foraging and digestion. Precise values for optimal residence times are presented for different chemical kinetics of substrate breakdown and of absorption. Both first-order kinetics and Michaelis–Menten kinetics are examined and compared, and it is determined how these residence times vary with foraging costs.

Macroparasite population dynamics among geographical localities and host life cycle stages: eugregarines in Ischnura verticalis.

Abstract: Populations of several species of gregarine parasites within a single host species, the damselfly Ischnura verticalis, were examined over the course of 1 season at 4 geographic localities separated by a maximum distance of 9.7 km. Gregarines, having a life cycle with both exogenous and endogenous stages, are subject to a wide variety of selective pressures that may drive adaptation. Gregarine species showed some specificity for host life cycle stage, i.e., Steganorhynchus dunwoodyi and Hoplorhynchus acanthatholius were most prevalent in larval hosts while Steganorhynchus dunwoodyi, Actinocephalus carrilynnae, and Nubenocephalus nebraskensis were most prevalent in adult hosts. Species prevalence and abundance differed by geographic locality. Gregarine prevalence was significantly higher in adult female damselflies than in males at 2 localities; sex differences in prevalence were insignificant for larval damselflies at all 4 localities. In larval hosts, gregarine abundance was independent of age (size). The present study, therefore, shows that pond characteristics, host life cycle stage, and adult host sex are the main factors that influence the prevalence and abundance of gregarine populations.

Time-periodic transport in heterogeneous porous media

Analytical solutions for the one-dimensional convection-diffusion equation in a semi-infinite, heterogeneous domain are obtained for periodic boundary conditions. The kinematic dispersion of the porous medium is taken to be either a linear function of distance, or an exponential function of distance that asymptotically becomes constant. The analytic solutions are given in terms of Kelvin functions and hypergeometric functions, respectively, which are computed using Mathematica. The solutions are compared with each other and with the constant dispersivity case, and the amplitudes and phase shifts are studied with respect to the frequency of the input wave at the boundary. All three models respond differently to different single frequency inputs, showing different decay rates at lower frequencies and more pronounced phase shifts at higher frequencies. The use of single frequency, periodic inputs appears to give more distinguishable solutions than traditional pulse-type data which contains multiple spectral frequencies, and the resulting solutions can be used as a verification for numerical schemes for convection-diffusion equations.

Type II functional response for continuous, physiologically structured models

The goal of this work is to formulate a general Holling-type functional, or behavioral, response for continuous physiologically structured populations, where both the predator and the prey have physiological densities and certain rules apply to their interactions. The physiological variable can be, for example, a development stage, weight, age, or a characteristic length. The model leads to a Fredholm integral equation for the functional response, and, when inserted into population balance laws, it produces a coupled system of partial differential-integral equations for the two species, with a nonlocal integral term that arises from rules of interaction in the functional response. The general model is, typically, analytically intractable, but specialization to a structured prey-unstructured predator model leads to some analytic results that reveal interesting and unexpected dynamics caused by the presence of size-dependent handling times in the functional response. In this case, steady-states are shown to exist over long times, similar to the stable age-structure solutions for the McKendick-von Foerster model with exponential growth rates determined by the Euler-Lotka equation. But, for type II responses, there are early transient oscillations in the number of births that bifurcate in a few generations into either the decaying or growing steady-state. The bifurcation parameter is the initial level of prey. This special case is applied to a problem of the biological control of a structured pest population (e.g., aphids) by a predator (e.g., lady beetles).

Nonlocal advection equations

Nonlocal problems are largely ignored in graduate and undergraduate texts on partial differential equations. Yet, nonlocal advection equations are important in many applications, and their solution provides an excellent illustration of the method of characteristics and how careful one must be to apply it.Nonlocal problems are largely ignored in graduate and undergraduate texts on partial differential equations. Yet, nonlocal advection equations are important in many applications, and their solution provides an excellent illustration of the method of characteristics and how careful one must be to apply it.