Aquifer Testing, Mathematical Modeling, and Regulatory Risk

Mathematical modeling founded on a strong field data base can be a valuable tool for the analysis of ground-water contamination problems. The purposes of this paper are threefold: (1) we demonstrate the dilemma of a knowledgeable ground-water quality regulator whose regulatory decision-making process is confronted with the output of a mathematical model that is based on very limited field test data; (2) we demonstrate a method available to a knowledgeable regulator for assessing approximately a range of possible performances of a contaminated ground-water recovery well field using a range of input data derived from a very limited data base; and (3) we present a strong case for presenting mathematical model outputs as ranges of values rather than as unique solutions. A range is determined by an examination of the level of sophistication of the field data base. Our experience with 12 field sites wherein ground-water contamination has occurred has led us to conclude that field data are seldom, if ever, adequate to defend a unique solution from a mathematical model. Regulatory decisions generally can be reduced to minimization of risks based on the smallest range of model outputs that can be defended on the basis of the field data base. The more limited the field data base, the greater must be the range of defensible model outputs, and consequently, the greater the risk inherent in subsequent regulatory decisions. The knowledgeable regulator can assess the risks in the regulatory decision-making process only if he is able to assess the extent to which the field data base for the mathematical model output reflects state-of-the-art data collection and analysis technologies and methodologies. If an applicant for a permit or license submits a less than adequate data base and concomitantly a broad range of defensible model outputs, he inherently requests that the knowledgeable regulator accept a risk greater than that required if adequate aquifer testing techniques had been employed.     

Lattice Boltzmann Models for Flow and Transport in Saturated Karst

Flow and transport simulation in karst aquifers remains a significant challenge for the ground water modeling community. Darcy's law–based models cannot simulate the inertial flows characteristic of many karst aquifers. Eddies in these flows can strongly affect solute transport. The simple two-region conduit/matrix paradigm is inadequate for many purposes because it considers only a capacitance rather than a physical domain. Relatively new lattice Boltzmann methods (LBMs) are capable of solving inertial flows and associated solute transport in geometrically complex domains involving karst conduits and heterogeneous matrix rock. LBMs for flow and transport in heterogeneous porous media, which are needed to make the models applicable to large-scale problems, are still under development. Here we explore aspects of these future LBMs, present simple examples illustrating some of the processes that can be simulated, and compare the results with available analytical solutions. Simulations are contrived to mimic simple capacitance-based two-region models involving conduit (mobile) and matrix (immobile) regions and are compared against the analytical solution. There is a high correlation between LBM simulations and the analytical solution for two different mobile region fractions. In more realistic conduit/matrix simulation, the breakthrough curve showed classic features and the two-region model fit slightly better than the advection-dispersion equation (ADE). An LBM-based anisotropic dispersion solver is applied to simulate breakthrough curves from a heterogeneous porous medium, which fit the ADE solution. Finally, breakthrough from a karst-like system consisting of a conduit with inertial regime flow in a heterogeneous aquifer is compared with the advection-dispersion and two-region analytical solutions.     

TRANSIENT ELECTROMAGNETIC RESPONSE OF A SPHERE IN A LAYERED MEDIUM*

The Galerkin method of solving integral equations is well suited to the solution of the integral equations describing the transient response of a sphere embedded in a layered medium, which is excited by a large co-axial loop. The transient response is calculated by transforming the steady state solutions obtained in the frequency domain. The analysis shows that the scattering matrix is extremely diagonally dominant and the maximum number of modes required to obtain convergence does not rapidly increase with frequency. The number of modes required is about eight. This type of scattering matrix can be taken to be an expression of the principle of elementary superposition. This principle is reflected in the decay curves. These show that the early part of the decay curves asymptotically approach the decay curves to be expected for a layered structure without the sphere. The slope of the latter stages of the decay curve gives a decay constant that is the same as was obtained for spheres in free space excited by planar or dipolar sources. The point of departure in time of these curves from the layered ground curves is delayed either by placing the sphere at a greater depth or by placing a more conductive overburden above the sphere.     

INVERSE SOLUTIONS OF HANTUSH'S EQUATIONS FOR AXISYMMETRIC MOUNDING

Abstract. Hantush (1967) provided time-dependent equations for describing ground-water mounds resulting from axisymmetric recharge. Because these equations contain the well function, they cannot be solved explicitly for the several variables employed in terms of the known mound height. Newton's method provides a means to achieve solutions for specific yield, hydraulic conductivity, time, or radius. An additional routine is provided which searches for the second solution in the case in which the inverse solution for hydraulic conductivity is doublevalued. The iterations can be performed on an HP-41C handheld calculator with three memory modules. The code is provided and will run with or without the printer attached.     

The Hydrological Effects of Gravel Winning in an Area West of London, United Kingdom

The ground-water flow and ground-water quality of extensive gravel deposits in the Lower Colne Valley, near London, England are described. The great demand for aggregates in southeast England over the past 30 years has resulted in extensive exploitation of these gravels, the worked-out pits being either left water-filled for recreational purposes, or filled with a variety of waste materials. Both after-uses are shown to alter the ground-water situation either by locally lowering or raising ground-water levels, such effects having been described in both French and German literature, or by causing ground-water and surface-water pollution problems. If the scale of pit development or landfilling is large as in the Lower Colne Valley, then remedial measures have to be undertaken to minimize the pollution risk and to obviate problems caused by raised or lowered ground-water levels.     

Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method

We extend the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) for solution of the advection-dispersion equation to two dimensions. The method can conserve mass globally and is not limited by restrictions on the size of the grid Peclet or Courant number. Therefore, it is well suited for solution of advection-dominated ground-water solute transport problems. In test problem comparisons with standard finite differences, FVELLAM is able to attain accurate solutions on much coarser space and time grids. On fine grids, the accuracy of the two methods is comparable. A critical aspect of FVELLAM (and all other ELLAMs) is evaluation of the mass storage integral from the preceding time level. In FVELLAM this may be accomplished with either a forward or backtracking approach. The forward tracking approach conserves mass globally and is the preferred approach. The backtracking approach is less computationally intensive, but not globally mass conservative. Boundary terms are systematically represented as integrals in space and time which are evaluated by a common integration scheme in conjunction with forward tracking through time. Unlike the one-dimensional case, local mass conservation cannot be guaranteed, so slight oscillations in concentration can develop, particularly in the vicinity of inflow or outflow boundaries.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

Three-dimensional semi-analytical solution to groundwater flow in confined and unconfined wedge-shaped aquifers

The Laplace domain solutions have been obtained for three-dimensional groundwater flow to a well in confined and unconfined wedge-shaped aquifers. The solutions take into account partial penetration effects, instantaneous drainage or delayed yield, vertical anisotropy and the water table boundary condition. As a basis, the Laplace domain solutions for drawdown created by a point source in uniform, anisotropic confined and unconfined wedge-shaped aquifers are first derived. Then, by the principle of superposition the point source solutions are extended to the cases of partially and fully penetrating wells. Unlike the previous solution for the confined aquifer that contains improper integrals arising from the Hankel transform [Yeh HD, Chang YC. New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions. Adv Water Resour 2006;26:471–80], numerical evaluation of our solution is relatively easy using well known numerical Laplace inversion methods. The effects of wedge angle, pumping well location and observation point location on drawdown and the effects of partial penetration, screen location and delay index on the wedge boundary hydraulic gradient in unconfined aquifers have also been investigated. The results are presented in the form of dimensionless drawdown-time and boundary gradient-time type curves. The curves are useful for parameter identification, calculation of stream depletion rates and the assessment of water budgets in river basins.     

ESSAIS D'AMELIORATION DU TRAITEMENT ET DE L'INTERPRETATION DES ENREGISTREMENTS MAGNETOTELLURIQUES*

The MT method has proved to be a useful tool for the resolution of some geophysical problems not well adapted to seismic prospecting. To meet the accuracy needed by such cases, the original field and processing techniques has to be somewhat improved. The processing of the natural EM field (with two electrical and three magnetic components recorded) is made in the frequency domain, and the usual MT parameters are calculated. However, the dispersion of the results is given a great importance and the final plots (apparent resistivity vs. frequency, azimuths of the main directions vs. frequency, etc.) illustrate the distribution of the calculated values. Average curves can then be selected and studied by means of the known master curves. However, if the earth can be assumed to consist of parallel layers, a computer program derives from the apparent resistivity diagram an automatic solution consisting of a great number of layers with the same ratio of thickness to square root of resistivity. This comprehensive solution is then simplified through an auxiliary diagram called the MT Dar Zarrouk curve and a set of geologically consistent solutions can be found.     

The need for operational reasoning in data‐driven rating curve prediction of suspended sediment

The use of data‐driven modelling techniques to deliver improved suspended sediment rating curves has received considerable interest in recent years. Studies indicate an increased level of performance over traditional approaches when such techniques are adopted. However, closer scrutiny reveals that, unlike their traditional counterparts, data‐driven solutions commonly include lagged sediment data as model inputs, and this seriously limits their operational application. In this paper, we argue the need for a greater degree of operational reasoning underpinning data‐driven rating curve solutions and demonstrate how incorrect conclusions about the performance of a data‐driven modelling technique can be reached when the model solution is based upon operationally invalid input combinations. We exemplify the problem through the re‐analysis and augmentation of a recent and typical published study, which uses gene expression programming to model the rating curve. We compare and contrast the previously published solutions, whose inputs negate their operational application, with a range of newly developed and directly comparable traditional and data‐driven solutions, which do have operational value. Results clearly demonstrate that the performance benefits of the published gene expression programming solutions are dependent on the inclusion of operationally limiting, lagged data inputs. Indeed, when operationally inapplicable input combinations are discounted from the models and the analysis is repeated, gene expression programming fails to perform as well as many simpler, more standard multiple linear regression, piecewise linear regression and neural network counterparts. The potential for overstatement of the benefits of the data‐driven paradigm in rating curve studies is thus highlighted. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical solution of specific energy and specific force equations: Trapezoidal and triangular channels

The specific energy and specific force equations have many applications in open-channel flow problems. At present, these equations have analytical solution only for rectangular channels. Trial and error procedure also graphical solutions are the existing methods of solving these equations. No analytical solutions are available in the technical literature for these equations in trapezoidal and triangular channels because it is presumed that these equations are quintic equations. The inversion of such equations consists of finding the roots of quintic equations. In the current study for a given channel geometry and discharge, the subcritical (supercritical) depth is analytically found in terms of the other supercritical (subcritical) depth. For this purpose, by considering physically meaningful domains, a quintic equation has been reduced to a quartic equation. In the next step, this quartic equation has been converted to a resolvent cubic equation and two quadratic equations. This research shows these steps clearly to reach an acceptable physical analytic solution for water depth in trapezoidal and triangular channels.     

RISING GROUND WATER – PROBLEM OR RESOURCE?

Abstract. Ground water rising to within 6 m (20 feet) of average ground surface elevations in Louisville, Kentucky caused concern to municipal officials and building owners in the central urban area. An average rise of more than 11 m (35 feet) occurred between 1969 and 1980.
An evaluation of foundation conditions and structural configurations in central Louisville indicated rising ground water could create:
1. slight but significant possibilities of structural settlement problems;
2. high possibilities of damage to basement floors and walls; and
3. very high possibilities for disruption of utility conduits.
Efforts to determine the cause of this rise in ground-water level have focused on the historical relationships between ground-water levels, pumpage rates and precipitation values.
Historical data indicated that ground-water levels in a system undisturbed by man could reach ground surface elevations in central Louisville. Preliminary studies indicated a strong relation between average ground-water levels and changes in pumping rates and incident precipitation. A further detailed study showed extremely high correlation (R = 0.995) between average ground-water levels in 1966–1980 and cumulative departures in precipitation and pumping rates from 1950–1965 average precipitation and pumping rates.
A study of the feasibility of lowering ground-water levels while simultaneously storing energy in the aquifer system was begun but was interrupted by devastating explosions of hexane in the sewers beneath south-central Louisville on February 13, 1981. Although a dry year in 1980 and no change in pumping rate have slowed the rise in ground-water level temporarily, long-term solutions to this problem need to be developed.     

瑞雷波多阶模式频散曲线稀疏正则化反演方法研究

目前瑞雷波多阶模式频散曲线反演中仅考虑数据的拟合,缺乏对模型的约束,不能很好地刻画地层间断面的问题,针对此问题,研究了瑞雷波多阶模式频散曲线稀疏正则化反演方法.正演模拟基于广义反射-透射系数法,数值计算上采用一种快速求根方法,与二等分方法相比,能够在很短的时间内达到最优的收敛效果;反演建模时采用L1范数正则化方法对模型...     

測量速度与积分电路时間常数的乘积（υ×τ）对自然伽（亻马）測井曲线的影响

一、引言目前,在勘探和开发放射性矿床、石油和其它有用矿藏的工作中,已广泛地应用自然伽（亻馬）測井。一般地說,在这些工作中,不仅需要根据自然伽（亻馬）測井曲线确定目的层的厚度及深度,而且在确定矿层的放射性元素合量、貯油层的合泥量与渗透率等工作中,还必須根据自然伽（亻馬）測井曲綫确定代表地层放射性的真正曲线幅度值。     

TOC Determinations in Ground Water

Determinations of total organic carbon (TOC) can provide valuable diagnostic evidence of the extent of ground-water contamination by organic compounds. The usefulness of conventional TOC results in monitoring efforts is limited by the bias introduced during the purging of inorganic carbon prior to analysis. A modified TOC procedure has been evaluated to permit the quantitation of the volatile organic carbon (VOC) fraction in water samples. The methodology consists of trapping the VOC in a manner analogous to commercial purge and trap instruments which are used for specific organic compound separations. The method has been found to be sensitive, accurate and reasonably precise for TOC determinations of standard solutions as well as on ground-water samples. Volatile organic carbon levels can range from 9–50% of the TOC in both uncontaminated and contaminated ground waters. The reporting of the volatile and nonvolatile fractions of the TOC will enhance both monitoring and research efforts, since it permits more complete characterization of the organic carbon content of ground-water samples.     

A fundamental solution due to a periodic point force in the interior of an elastic half-space

The fundamental solution for a periodic point force in the interior of a three-dimensional, homogeneous, isotropic, elastic half-space is derived. The method of synthesis and superposition is employed to obtain the solution in the Laplace transform as well as the frequency domain. These correspond to the dynamic equivalent of Mindlin's static half-space point force solutions. It is reduced, for certain limiting conditions, to the dynamic equivalent of Boussinesq's and Cerruti's problems of a normal and tangential periodic point force respectively, on the boundary of a half-space. Also, static solutions of Mindlin, Boussinesq and Cerruti are recovered for small frequency parameters. Finally, results are presented and compared with other available solutions.     

Model Calibration Techniques for Use with the Analytic Element Method

The combination of the analytic element method and a nonlinear parameter estimation technique forges a computationally efficient, information-rich, and cost-effective solution to the inverse ground-water flow problem. The recommended model calibration method uses a nonlinear least-squares objective, as quantified by misfitting the measured and modeled heads, and a modified Levenberg-Marquardt algorithm. As detailed and demonstrated by a steady-state regional aquifer model of Bemidji, Minnesota, automated calibration techniques make ground-water modeling feasible for a wider variety of projects where tight budgets and a lack of tools may have previously made such modeling inappropriate.     

Analysis of solute transport in rivers with transient storage and lateral inflow: An analytical study

Solute transport in rivers with transient storage and lateral inflow has been studied by Transient Storage Model (TSM). Analytical solution of the TSM is obtained by means of Laplace transform. In order to illustrate the use of present analytical solution, physical transport parameters are estimated for the observational data of Uvas Creek tracer experiment. Analytical concentration-time breakthrough curves are found to be in good agreement with the observed concentration-time breakthrough curves. A sensitivity analysis has been performed in order to identify the most critical parameter for predicting concentration. It has been found that lateral inflow rate is the most sensitive and the ratio of cross-sectional areas is the least sensitive parameter. A hypothetical situation has been considered to study the effects of background concentrations, lateral inflow concentration and its rate. The analytical solutions show that the solute concentration gets diluted or concentrated due to lateral inflow. Physical mechanisms of the problems are well reproduced by the present analytical solutions and these results can be used for analysis of tracer experiments.     

A spreadsheet program for two-well tracer test data analysis

Two-well tracer tests are often conducted to investigate subsurface solute transport in the field. Analyzing breakthrough curves in extraction and monitoring wells using numerical methods is nontrivial due to highly nonuniform flow conditions. We extended approximate analytical solutions for the advection-dispersion equation for an injection-extraction well doublet in a homogeneous confined aquifer under steady-state flow conditions for equal injection and extraction rates with no transverse dispersion and negligible ambient flow, and implemented the solutions in Microsoft Excel using Visual Basic for Application (VBA). Functions were implemented to calculate concentrations in extraction and monitoring wells at any location due to a step or pulse injection. Type curves for a step injection were compared with those calculated by numerically integrating the solution for a pulse injection. The results from the two approaches are similar when the dispersivity is small. As the dispersivity increases, the latter was found to be more accurate but requires more computing time. The code was verified by comparing the results with published-type curves and applied to analyze data from the literature. The method can be used as a first approximation for two-well tracer test design and data analysis, and to check accuracy of numerical solutions. The code and example files are publicly available.     

Non-Darcian groundwater flow in leaky aquifers

Abstract

A simplified method has been developed for solving leaky aquifer non-Darcian flow hydraulics. The principle of volumetric approach is combined with the confined-aquifer, time-dependent drawdown equation in an observation well. The groundwater flow in the leaky aquifer is assumed to obey a non-Darcian flow law of exponential type. The results are obtained in the form of type-curve expressions from which the necessary bundles of curves are drawn for a set of selective non-Darcian flow aquifer parameters. Although application of the methodology appears as rather limited but it provides a scientific contribution and extension of leaky aquifer theory towards nonlinear flow conditions. The methodology developed herein is applied to some actual field data from the eastern sedimentary basin in the Kingdom of Saudi Arabia.