Bruce Hunt

Solutions and verification of a scale-dependent dispersion model.

In this paper, analytical solutions are derived for a one-dimensional scale-dependent dispersion model (SDM), considering linear equilibrium sorption and first-order degradation for continuous and pulse contaminant sources, with a constant input concentration in a semi-infinite uniform porous medium. In the SDM model, dispersivity alpha(x) is replaced with a constant epsilon multiplied by the transport distance x. The solution for a pulse source is verified experimentally in the analysis of tritium data obtained from an 8-m-long homogenous pea-gravel column with multiple sampling locations, and the results are compared with those analysed by a commonly used solution of a constant dispersion model (CDM). The SDM predicts concentrations satisfactorily at all sampling locations, while the CDM fits the experimental data well for only one location. Both models are then calibrated for each individual concentration breakthrough curve, using local values for either epsilon in the SDM or alpha(x) in the CDM. Both models give equally good fits for appropriate choices of individual epsilon and alpha(x) values, and both indicate a linear increase in alpha(x) with distance. The epsilon values tend to change little as x increases and are expected to approach a constant at relatively large distances downstream. Hence, predictions from the SDM should become more accurate as x increases.

Stream Depletion in a Two-Layer Leaky Aquifer System

A new semianalytical stream-depletion solution is obtained for pumping from a well beside a stream in a two-layer leaky aquifer system. The well abstracts water from an upper unconfined aquifer underlain by an aquitard and a second semiconfined aquifer. The governing equations reduce to the equations that describe flow in a Hantush-Jacob leaky aquifer, in which the bottom aquifer has zero drawdown for all time, when the storativity of the bottom aquifer becomes infinite. This allows a Hantush-Jacob leaky aquifer solution to be compared with the solution for a bottom aquifer with finite storativity, and this comparison shows that the Hantush-Jacob approximation significantly underestimates stream depletion at larger values of time. In addition, comparison of the two-aquifer solution with a stream-depletion solution for a single aquifer shows that a single-aquifer solution closely approximates the two-aquifer solution when the single-aquifer transmissivity is replaced with the sum of transmissivities for the two-layer system. This suggests that a similar approximation might be used in a previously obtained solution to obtain a very general stream depletion model for multiaquifer systems.

Calculation of the leaky aquifer function

Abstract Two infinite series representations are obtained for the leaky aquifer function of Hantush and Jacob. These infinite series are absolutely convergent and can be used to calculate values of the leaky aquifer function for all possible values of the arguments. Some example calculations are carried out.

An analysis of the groundwater resources of Tongatapu Island, Kingdom of Tonga

Abstract An analysis is made of the groundwater resources of Tongatapu Island. The Ghyben-Herzberg approximation is used to estimate thicknesses of a fresh-water lens floating on seawater. Finite-difference calculations are used to estimate rainfall recharge rates, and calculations are made to investigate the dispersion of chloride ions across the fresh-water-salt-water interface. These calculations suggest that artificial recharge might be a useful device to control chloride concentrations in the fresh-water aquifer.

Water waves generated by distant landslides

Waves generated in a reservoir by a landslide are modeled by injecting an instantaneous point source of fluid through the bottom of a reservoir of infinite extent. This assumes that times are large relative to the duration time of the landslide and that distances are large relative to a characteristic horizontal dimension of the volume of water displaced by the slide material. Solutions are obtained for two-dimensional and axisymmetric problems, and the characteristics of each are discussed. Since the solutions assume a constant reservoir depth, and since maximum wave amplitudes always occur along shorelines of sloping beaches, it is suggested that these solutions may be of most use for providing boundary conditions for models of wave runup on sloping beaches. A comparison with experimental results is made, and agreement between the calculated and measured results is found to be good.

A simple approximation for a moving interface in a coastal aquifer

An approximate analytical solution for a moving interface in a confined coastal aquifer is found by solving a nonlinear first-order partial differential equation. The solution, which assumes that flow approaching the interface changes discontinuously with time from one constant value to another, coincides with exact steady-flow solutions obtained from the Dupuit approximation at both t = 0 and t = ∞. A comparison with numerical solutions suggests that maximum errors of about ten percent occur in the toe location for intermediate values of time and that the approximate solution tends to overestimate both the rate of advance and retreat of the toe. The approximation has a very simple form and, in contrast to numerical solutions, is easy to evaluate. Additional comparisons with numerical solutions also suggest that the analytical approximation can be applied with confidence to an unconfined flow only when the sudden change in approaching flow rate occurs at distances from the interface toe that are not too great. This is because free-surface storage effects slow the movement of the flow rate change before its effects are eventually felt downstream at the interface toe.

Dispersion calculations in nonuniform seepage

Abstract A singular perturbation approximation is derived and applied to some examples in groundwater dispersion. This approximation is general enough to be used for nonuniform, steady and unsteady seepage through heterogeneous aquifers, and it complements numerical methods since it becomes most accurate in regions where accurate numerical solutions are almost impossible to obtain.

Diffusion in laminar pipe flow

Abstract A combination of analytical reasoning and experimental observation is used to investigate the spreading of a solute that has been injected in fully-developed, laminar pipe flow. The results indicate that diffusion in laminar flow depends very much upon the magnitude of a dimensionless parameter, e, that is analogous to the reciprocal of the Peclet number of heat transfer. Most of the previous work in this area appears to apply for relatively large values of e, and perturbation methods are used to obtain some solutions for relatively small values of e.

Multiple-valued and non-convergent solutions in kinematic cascade models

Abstract The Lax-Wendroff finite-difference solution of the equations of motion with the kinematic flow approximation is now widely used in hydrology in cascade models of overland flow. The physical relevance of kinematic shocks and the stability and convergence of the finite-difference solution are problems that are often undetected in contemporary applications due to the complexity of model inputs. Past criteria, developed for discerning either the presence of a shock or the adequacy of finite-difference solutions, are inadequate for complex cascade models. A general method is devised for locating the point along a cascade segment where the solution first becomes multiple-valued, for the lateral inflow situation. An example comparison with an exact solution reveals that stable, finite-difference solutions apparently converge to single-valued hydrographs and water surface profiles with spurious peaks that can go undetected in complex models. Finally, shock propagation and the potential for error growth in cascade models are discussed.

A perturbation solution of the flood-routing problem

Singular perturbation techniques are used to obtain an approximate solution of the flood routing problem. The first-order outer solution is given by the kinematic-wave approximation. Singularities in this outer solution then lead to two inner solutions: one that follows a kinematic shock as it moves downstream and one either at the downstream boundary in subcritical flow or at the upstream boundary in supercritical flow. Solutions calculated in each of these subregions allow conclusions to be drawn about the relative importance of various terms in the momentum equation. The inner and outer solutions are combined to obtain composite solutions for flow depths and flow rates, and a numerical example is worked.