Glenn Ledder

Groundwater flow near vertical recirculatory wells: effect of skin on flowgeometry and travel times with implications for aquifer remediation

Abstract Flow structure around a recirculation well in a uniform anisotropic aquifer is investigated using potential theory and Stokes’ stream function techniques. A vertical recirculation well consists of two screened sections (chambers) separated by an impermeable casing: one section extracts water from the aquifer and the other section injects the water back into the aquifer. Analytical formulae are derived for the drawdown and the stream function for a model in which the extraction and injection chambers of the well are modeled as uniformly distributed cylindrical sinks and sources, respectively. Travel times are analyzed for water particles traveling from the injection chamber to the extraction chamber along streamlines containing various percentages of the flow. We consider a disturbed zone (skin) having properties which are different from those in the uniform anisotropic aquifer owing to disturbance of the area near the well during installation. The effects of this skin on the streamlines and travel times are analyzed for various lengths separating the chambers and for various skin conductivities. The method completely eliminates the use of numerical finite-difference or finite-element methods and can be used for optimization of technological parameters in a remediation system.

Groundwater flow in a compressible unconfined aquifer with uniform circular recharge

The distributions of the hydraulic head and velocity components of the transient groundwater flow in an unconfined compressible aquifer of finite thickness under constant uniform circular recharge are obtained from the linearized mathematical model by the use of integral transforms. The result generalizes Dagans (1967) solution which was derived by neglecting the compressibility. By treating the compressibility parameter as a small value, the formula for the hydraulic head is analyzed by asymptotic methods, resulting in approximations to the exact solutions for the head and velocities on small and large time scales. The hydraulic head and flow velocities can be accurately approximated by Dagans formula for large times; for small times, neglecting the compressibility gives a large relative error but small absolute error.

Groundwater Velocity in an Unconfined Aquifer With Rectangular Areal Recharge

Three-dimensional solutions for the hydraulic head and velocity components for transient groundwater flow in an unconfined compressible aquifer of finite thickness with a rectangular areal recharge source are analyzed using a linearized mathematical model. The solution generalizes Dagans (1967) results, which were obtained for an idealized aquifer of infinite thickness, and Hantushs (1967) results, which were derived using the Dupuit assumptions. Hydraulic head and velocity components are estimated by asymptotic methods for large times, which are most appropriate for contaminant transport problems. Numerical analysis of the solution demonstrates spatial features of the velocity components in general and a transient effect of upward flow in particular, similar to that found for circular sources (Zlotnik and Ledder, 1992). Obtained formulas show that applicability of the Dupuit assumption for computation of velocity components in an unconfined aquifer is limited to a far-field zone given approximately by |x| > 1.5X, |y| > 1.5Y, where X and Y are the half width and half-length, respectively, of the contaminant source.

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).

An Experimental Approach to Mathematical Modeling in Biology.

Abstract The simplest age-structured population models update a population vector via multiplication by a matrix. These linear models offer an opportunity to introduce mathematical modeling to students of limited mathematical sophistication and background. We begin with a detailed discussion of mathematical modeling, particularly in a biological context. We then describe Bugbox-population, a virtual insect laboratory that allows students to make observations and collect quantitative data easily, thereby learning mathematical modeling in the context of its use in scientific research. Creating a mathematical model for boxbugs involves the same intellectual work as creating a mathematical model for real insects, but without the difficulties involved in collecting real biological data. The analysis of the Bugbox-population data leads to the development of the eigenvalue problem for population projection matrices.

Mathematics for the life sciences

Mathematics for the life sciences : , Mathematics for the life sciences : , کتابخانه مرکزی دانشگاه علوم پزشکی ایران

Contamination and remediation waves in a filtration model

Abstract We propose a model for the filtration of suspended particles in porous media and we examine some of its mathematical properties. The model includes a variable porosity that depends on the volume of particles retained through filtration and a kinetics law that allows both a positive and negative rate of particle accretion. We characterize the properties of accretion rates that lead to contamination and remediation wave fronts in the model.

A SINGULAR PERTURBATION PROBLEM IN FRACTURED MEDIA WITH PARALLEL DIFFUSION

We study differential equations that model contaminant flow in a semi-infinite, fractured, porous medium consisting of a single fracture channel bounded by a porous matrix. Models in the literature usually do not incorporate diffusion in the porous matrix in the direction parallel to the fracture, and therefore they must omit a no-flux boundary condition at the edge, which, in some problems, may be unphysical. Herein we show that the problem usually treated in the literature is the outer problem for a correctly posed singular perturbation problem which includes diffusion in both directions as well as the no-flux boundary condition.

An integral equation for the planar ablation problem

A Laplace transform in the spatial variable is used to obtain a Volterra integral equation for the problem of determining the speed and position of the moving boundary of an evaporating surface. The integral equation is applicable for an arbitrary Stefan number and variable heat input. Asymptotic formulas for the short-time behavior are derived. The integral equation is solved numerically to illustrate several features of planar ablation. Of particular interest is the approach to a steady state, which is seen to be very slow compared to the preheat time or the characteristic diffusion time.

Scaling for Dynamical Systems in Biology

Asymptotic methods can greatly simplify the analysis of all but the simplest mathematical models and should therefore be commonplace in such biological areas as ecology and epidemiology. One essential difficulty that limits their use is that they can only be applied to a suitably scaled dimensionless version of the original dimensional model. Many books discuss nondimensionalization, but with little attention given to the problem of choosing the right scales and dimensionless parameters. In this paper, we illustrate the value of using asymptotics on a properly scaled dimensionless model, develop a set of guidelines that can be used to make good scaling choices, and offer advice for teaching these topics in differential equations or mathematical biology courses.