Allen F. Moench

Flow to a well of finite diameter in a homogeneous, anisotropic water table aquifer

A Laplace transform solution is presented for the problem of flow to a partially penetrating well of finite diameter in a slightly compressible water table aquifer. The solution, which allows for evaluation of both pumped well and observation piezometer data, accounts for effects of well bore storage and skin and allows for the noninstantaneous release of water from the unsaturated zone. For instantaneous release of water from the unsaturated zone the solution approaches the line source solution derived by Neuman as the diameter of the pumped well approaches zero. Delayed piezometer response, which is significant during times of rapidly changing hydraulic head, is included in the theoretical treatment and shown to be an important factor in accurate evaluation of specific storage. By means of a hypothetical field example it is demonstrated that evaluations of specific storage (Ss) using classical line source solutions may yield values of Ss that are overestimated by a factor of 100 or more, depending upon the location of the observation piezometers and whether effects of delayed piezometer response are included in the analysis. Theoretical responses obtained with the proposed model are used to suggest methods for evaluating specific storage.

Convergent Radial Dispersion in a Double-Porosity Aquifer with Fracture Skin: Analytical Solution and Application to a Field Experiment in Fractured Chalk

An exact Laplace transform solution to the problem of dispersion, advection, and adsorption of a tracer due to its injection in a steady, horizontal, radially convergent flow field in a densely fractured, porous formation (double-porosity aquifer) is presented. The porous blocks were assumed to be covered with a layer of material (fracture skin) of negligible volume and storage capacity that provides a resistance to diffusion in the rock matrix. Longitudinal dispersion, advection, and adsorption dominate transport of the tracer in the fractures, and diffusion and adsorption dominate movement of the tracer in the blocks. Dimensionless breakthrough curves are used to illustrate the influence of various aquifer and tracer properties. In support of the model a detailed analysis is performed of a published multitracer field test, conducted in a layer of densely fractured chalk in Bethune, France. Of the three tracers analyzed, two are nonsorptive but have widely different free water diffusion coefficients, and one is slightly sorptive. Analysis of measured breakthrough curves, matched by trial and error to theoretical responses, reveals that by allowing for fracture skin on block surfaces, one can obtain (1) pure-advection arrival times that are independent of the tracer used, (2) values of mass recovery consistent with measured values, and (3) relative values of effective diffusion coefficients that are consistent with known free water diffusion coefficients for the separate tracers. Reasonable estimates of longitudinal dispersivity and fracture porosity are also obtained.

Aquifer response to stream-stage and recharge variations. II. Convolution method and applications

Abstract In this second of two papers, analytical step-response functions, developed in the companion paper for several cases of transient hydraulic interaction between a fully penetrating stream and a confined, leaky, or water-table aquifer, are used in the convolution integral to calculate aquifer heads, streambank seepage rates, and bank storage that occur in response to stream-stage fluctuations and basinwide recharge or evapotranspiration. Two computer programs developed on the basis of these step-response functions and the convolution integral are applied to the analysis of hydraulic interaction of two alluvial stream–aquifer systems in the northeastern and central United States. These applications demonstrate the utility of the analytical functions and computer programs for estimating aquifer and streambank hydraulic properties, recharge rates, streambank seepage rates, and bank storage. Analysis of the water-table aquifer adjacent to the Blackstone River in Massachusetts suggests that the very shallow depth of water table and associated thin unsaturated zone at the site cause the aquifer to behave like a confined aquifer (negligible specific yield). This finding is consistent with previous studies that have shown that the effective specific yield of an unconfined aquifer approaches zero when the capillary fringe, where sediment pores are saturated by tension, extends to land surface. Under this condition, the aquifers response is determined by elastic storage only. Estimates of horizontal and vertical hydraulic conductivity, specific yield, specific storage, and recharge for a water-table aquifer adjacent to the Cedar River in eastern Iowa, determined by the use of analytical methods, are in close agreement with those estimated by use of a more complex, multilayer numerical model of the aquifer. Streambank leakance of the semipervious streambank materials also was estimated for the site. The streambank-leakance parameter may be considered to be a general (or lumped) parameter that accounts not only for the resistance of flow at the river–aquifer boundary, but also for the effects of partial penetration of the river and other near-stream flow phenomena not included in the theoretical development of the step-response functions.

Aquifer response to stream-stage and recharge variations. I. Analytical step-response functions

Abstract Laplace transform step-response functions are presented for various homogeneous confined and leaky aquifer types and for anisotropic, homogeneous unconfined aquifers interacting with perennial streams. Flow is one-dimensional, perpendicular to the stream in the confined and leaky aquifers, and two-dimensional in a plane perpendicular to the stream in the water-table aquifers. The stream is assumed to penetrate the full thickness of the aquifer. The aquifers may be semi-infinite or finite in width and may or may not be bounded at the stream by a semipervious streambank. The solutions are presented in a unified manner so that mathematical relations among the various aquifer configurations are clearly demonstrated. The Laplace transform solutions are inverted numerically to obtain the real-time step-response functions for use in the convolution (or superposition) integral. To maintain linearity in the case of unconfined aquifers, fluctuations in the elevation of the water table are assumed to be small relative to the saturated thickness, and vertical flow into or out of the zone above the water table is assumed to occur instantaneously. Effects of hysteresis in the moisture distribution above the water table are therefore neglected. Graphical comparisons of the new solutions are made with known closed-form solutions.

Convergent Radial Dispersion: A Note on Evaluation of the Laplace Transform Solution

A numerical inversion algorithm for Laplace transforms that is capable of handling rapid changes in the computed function is applied to the Laplace transform solution to the problem of convergent radial dispersion in a homogeneous aquifer. Prior attempts by the author to invert this solution were unsuccessful for highly advective systems where the Peclet number was relatively large. The algorithm used in this note allows for rapid and accurate inversion of the solution for all Peclet numbers of practical interest, and beyond. Dimensionless breakthrough curves are illustrated for tracer input in the form of a step function, a Dirac impulse, or a rectangular input.

Estimation of hectare-scale soil-moisture characteristics from aquifer-test data

Abstract Analysis of a 72-h, constant-rate aquifer test conducted in a coarse-grained and highly permeable, glacial outwash deposit on Cape Cod, Massachusetts revealed that drawdowns measured in 20 piezometers located at various depths below the water table and distances from the pumped well were significantly influenced by effects of drainage from the vadose zone. The influence was greatest in piezometers located close to the water table and diminished with increasing depth. The influence of the vadose zone was evident from a gap, in the intermediate-time zone, between measured drawdowns and drawdowns computed under the assumption that drainage from the vadose zone occurred instantaneously in response to a decline in the elevation of the water table. By means of an analytical model that was designed to account for time-varying drainage, simulated drawdowns could be closely fitted to measured drawdowns regardless of the piezometer locations. Because of the exceptional quality and quantity of the data and the relatively small aquifer heterogeneity, it was possible by inverse modeling to estimate all relevant aquifer parameters and a set of three empirical constants used in the upper-boundary condition to account for the dynamic drainage process. The empirical constants were used to define a one-dimensional (1D) drainage versus time curve that is assumed to be representative of the bulk material overlying the water table. The curve was inverted with a parameter estimation algorithm and a 1D numerical model for variably saturated flow to obtain soil-moisture retention curves and unsaturated hydraulic conductivity relationships defined by the Brooks and Corey equations. Direct analysis of the aquifer-test data using a parameter estimation algorithm and a two-dimensional, axisymmetric numerical model for variably saturated flow yielded similar soil-moisture characteristics. Results suggest that hectare-scale soil-moisture characteristics are different from core-scale predictions and even relatively small amounts of fine-grained material and heterogeneity can dominate the large-scale soil-moisture characteristics and aquifer response.

WTAQ - A computer program for aquifer-test analysis of confined and unconfined aquifers

Computer program WTAQ was developed to implement a Laplace-transform analytical solution for axial-symmetric flow to a partially penetrating, finite-diameter well in a homogeneous and anisotropic unconfined (water-table) aquifer. The solution accounts for well-bore storage and skin effects at the pumped well, delayed response at an observation well, and delayed or instantaneous drainage from the unsaturated zone. For the particular case of zero drainage from the unsaturated zone, the solution simplifies to that of axial-symmetric flow in a confined aquifer. WTAQ calculates theoretical time-drawdown curves for the pumped well and observation wells and piezometers. The theoretical curves are used with measured time-drawdown data to estimate hydraulic parameters of confined or unconfined aquifers by graphical type-curve methods or by automatic parameter-estimation methods. Parameters that can be estimated are horizontal and vertical hydraulic conductivity, specific storage, and specific yield. A sample application illustrates use of WTAQ for estimating hydraulic parameters of a hypothetical, unconfined aquifer by type-curve methods.