Analytical Solution for Conservative Solute Transport in One-Dimensional Homogeneous Porous Formations with Time-Dependent Velocity

The space-time variation in contaminant concentration in unsteady flow in a homogeneous finite aquifer subjected to point source contamination is analytically derived under two conditions: (1) the flow velocity in the aquifer is of sinusoidal form; and (2) the flow velocity is an exponentially decreasing function. The analytical solution is illustrated using an example. Analytical solutions are perhaps most useful for benchmarking numerical codes and solutions.     

Analytical Solution for Tidal Propagation in a Leaky Aquifer Extending Finite Distance under the Sea

This paper focuses on groundwater dynamics in response to the tidal fluctuation in a coastal aquifer system. An analytical solution is derived to describe groundwater level fluctuation in a leaky aquifer extending finite distance under the sea. Based on this solution, the joint effects of various parameters, such as the dynamic effect of water table fluctuation and the leakages of the inland and offshore, on the behavior of the groundwater level fluctuations in the inland part of the leaky confined aquifer can be thoroughly analyzed. When the roof length is increased, the dynamic effect of the water table fluctuation on the dimensionless groundwater amplitude, intrusion distance, and fixed phase shift in the unconfined aquifer become more important and the water table fluctuation approaches constant values when the roof length is greater than a threshold value. However, given the same values of dimensionless leakage and roof length, the dimensionless groundwater amplitude, intrusion distance, and fixed phase shift in the leaky aquifer with considering the dynamic effects are always larger than those of neglecting such effects.     

Analytical Solutions for Constant-Flux and Constant-Head Tests at a Finite-Diameter Well in a Wedge-Shaped Aquifer

Neglecting the effect of well radius may lead to a significant error in the predicted drawdown distribution near the pumping well area. New analytical solutions describing aquifer responses to a constant pumping or a constant head maintained at a finite-diameter well in a wedge-shaped aquifer are derived based on the image-well method and applicable to an arbitrarily located well in the system. The solutions are useful for quantifying groundwater exploitation from a wedge-shaped aquifer and for determining the hydrogeological parameters of a wedge-shaped aquifer in inverse problems.     

Effects of Primary Substrate Concentration on NDMA Transport during Simulated Aquifer Recharge

N-nitrosodimethylamine (NDMA) is known to be highly carcinogenic and is present in drinking water, wastewater, and a variety of foods. Because of its presence in chloraminated water at nanogram per liter concentrations, NDMA has become an emerging issue for reclaimed water which may be used for aquifer recharge or irrigation. This research investigated the fate of NDMA in two soil column systems used to simulate subsurface transport. One column system was operated under aerobic conditions with increasing primary substrate concentration where the biodegradable organic carbon (BDOC) in reclaimed water was used as the primary substrate. The reclaimed water content in the influent was increased from 0 to 25% in the column to increase the BDOC concentration. Negligible NDMA removal was observed at 0% reclaimed water and increasing the primary substrate in the influent resulted in NDMA removal suggesting that biodegradation of NDMA might be a cometabolic process. The effects of redox conditions on NDMA fate was studied by operating a second column system with 100% reclaimed water under anoxic conditions and then changing the conditions to aerobic. It was observed that NDMA removal was similar under both aerobic and anoxic condition, however, much lower effluent concentrations were observed under aerobic conditions. Under anoxic condition, a normalized mass removal rate of 254 ng NDMA/mg DOC was observed which increased to 273-ng NDMA/mg DOC under aerobic conditions. The majority of NDMA and substrate removal occurred in the first of three columns in series column under both aerobic and anoxic conditions. Normalized mass removal rates of NDMA after the first, second, and third columns were 372, 30, and 20 ng NDMA/mg DOC, respectively. Since the majority of dissolved organic carbon was also removed in the first column, NDMA biodegradation was consistent with cometabolic activity. Batch tests verified the biodegradation removal potential of NDMA. Addition of a methylotrophic substrate, methanol and an aromatic substrate, toluene, did not increase NDMA removal.     

Simulation of Radial Collector Well in Shallow Alluvial Riverbed Aquifer Using Analytic Element Method

To withdraw large quantities of groundwater from the alluvial aquifers for various uses near riverbeds, radial collector (RC) wells are often preferable to the installation of several small diameter tube wells. In regions where rivers are not perennial or have low flow conditions during most part of the year, the RC wells are placed in the riverbed to obtain uninterrupted supply of naturally filtered groundwater through highly permeable saturated riverbed aquifers. Due to the complexities of flow, no exact analytical solution exists to provide steady state discharge drawdown relationship for RC well. Numerical model construction using finite difference or finite element method is quite cumbersome because of the radial orientation of laterals. To overcome these difficulties, in this study a steady state simulation model based on analytic element method (AEM) is developed to simulate the discharge-drawdown relation for RC well in an unconfined riverbed aquifer. In the model, line-sink elements are used to represent stream as well as radial laterals with specified conductance. The model is used to study the effects of different lateral configurations, hydraulic conductivity of riverbed aquifer, radius of influence and conductance of laterals on the well discharge, and consequent drawdown. Further based on the results of simulations using the AEM model, an approximate empirical equation is developed to obtain the discharge of RC well readily for design purpose. Further, the developed model has been applied to a field study and results are obtained for different plausible configurations of radial wells. The proposed methodology based on the application of AEM modeling tool, has been found to be efficient in constructing riverbed aquifer model with RC well. The proposed model is recommended in designing new collector well by providing minimum length of laterals for the sustained yield.     

Water Table Fluctuation in the Presence of a Time-Varying Exponential Recharge and Depth-Dependent ET in a Two-Dimensional Aquifer System with an Inclined Base

An analytical solution is presented for water table fluctuation between ditch drains in presence of exponential recharge and depth-dependent evapotranspiration (ET) from groundwater table in a two-dimensional gently sloping aquifer. The groundwater head above the drain is small compared to the saturated thickness of the aquifer. A sound mathematical transformation is devised to transform the two-dimensional groundwater flow equation into a simple form, which makes possible to obtain an analytical solution. The transient midpoint water table variations from the proposed solution compare well with the already existing solutions for horizontal aquifer. A numerical example is used to illustrate the combined effect of depth-dependent ET coupled with a time-varying exponential recharge on the water table fluctuation. The inclusion of a depth-dependent ET in the solution results in water table decline at a faster rate as compared to the case when ET is not considered. With an increase in slope of the aquifer base, water table profiles become asymmetric and the water table divide shifts towards the lower drain. The height of the water table profiles increases on moving away from the boundary of the aquifer and the highest level of the ground water table is obtained in the central portion of the aquifer basin due to the presence of drainage ditches on the aquifer boundary. When the effect of ET is incorporated in combination with recharge, the analytical solution results in accurate and reliable estimates of water table fluctuations under situations subjected to a number of controlling factors. This study will be useful for alleviation of drainage problems of the aquifers receiving surface recharge and surrounded by streams.     

Ramp Kernels for Aquifer Responses to Arbitrary Stream Stage

Analytical expressions for ramp kernels (new kernels) for an improved convolution for obtaining aquifer responses, viz, groundwater head, rate, and cumulative volume of groundwater flow, to an arbitrary stage, are obtained. The use of the ramp kernels gives accurate aquifer responses and is superior to the conventional convolution in which numerical integration or pulse kernels are used. The extent of improvement in the results with the use of the ramp kernels is discussed and quantified for three examples, where the results are compared to analytical solutions. For the comparisons, the analytical solutions for linear and sinusoidal stream stages are derived. The use of the ramp kernels reproduces accurately the analytical solutions. The concept of ramp kernels can also be used for obtaining an accurate solution of convolution integrals observed in other fields.     

Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains

Generalized analytical solutions for groundwater head in horizontal aquifers in the presence of parallel subsurface drains are obtained considering a transient rate of recharge as a power series (polynomial) function and depth-dependent rate of evapotranspiration. A function, new to analytical drainage studies, is proposed for correctly representing the depth-dependent rate of evapotranspiration. The solutions are obtained considering the practical situation of drains placed at a shallow depth in a considerable depth of aquifer. Two conditions of large and small saturated thicknesses in comparison to the changes in groundwater head are considered. A mathematical criterion is proposed to distinguish between large and small saturated thicknesses.     

Comparing Three Models for Treatment of Stagnant Zones in Riverine Transport

The modified advection dispersion equation (MADE) model is compared to the transient storage model (TSM) and fractional advection dispersion equation (FRADE) model for modeling riverine transport. The comparisons are illustrated using field examples. The parameter F of the FRADE model is found to be inconsistent to its physical meaning and is unrelated to dead zone fraction. Also, the estimated parameters of TSM are found to be inconsistent to their physical meaning. On the contrary, the parameters of MADE are consistent and related to the dead zone fraction. It is observed that the MADE model is easy to use and yields reliable and consistent results.     

Hybrid-Cells-in-Series Model for Solute Transport in a River

A hybrid-cells-in-series (HCIS) model has been conceptualized to simulate transport of a conservative solute in a river. The concentration graph of the effluent from the first hybrid unit follows a skewed concentration-time profile more close to reality. When the linear size Δx of the hybrid unit is more than 4?DL/u, where DL = longitudinal dispersion coefficient, u = mean flow velocity, the concentration graphs at nΔx, n = 1,2,3,… are approximately equal to that predicted by Ogata and Banks’ model. The model parameters α, T1, and T2, which are the times of residence of solute in the plug flow zone, and in the first and second thoroughly mixed reservoir respectively, can be estimated from a time-concentration graph using: (1) partial moments, (2) three characteristics of the graph, i.e., time to peak, peak concentration and the partial first moment, and (3) least square optimization. The performance of the HCIS model has been verified using the data of field tests conducted in rivers. The model parameters estimated using one concentration graph simulate the concentration graphs observed at other location downstream with reasonable accuracy.     

Generalized Analytical Solutions for Groundwater Head in Inclined Aquifers in the Presence of Subsurface Drains

Analytical solutions for groundwater head in the presence of subsurface drains are important in assessing the effectiveness of an existing drainage system under a probable extreme variation in the rate of recharge and designing a new drainage system. Generalized analytical solutions for groundwater head in inclined aquifers in the presence of parallel subsurface drains are obtained considering the transient rate of recharge as a power series (polynomial) function and depth-dependent rate of evapotranspiration. An appropriate function, new to analytical drainage studies, is used for correctly representing the depth-dependent rate of evapotranspiration. The solutions are obtained considering the practical situation of drains placed at shallow depth in a considerable depth of aquifer. Two conditions of large and small saturated thicknesses in comparison to the increase in groundwater head are considered. A mathematical criterion is proposed to distinguish between large and small saturated thicknesses. The analytical equations for discharge to drains for different cases considered are also obtained. The discharge equations used by prior investigators are found inappropriate.     

Poroviscoelastic Two-Dimensional Anisotropic Solution with Application to Articular Cartilage Testing

The transverse anisotropic poromechanics solution for the two-dimensional Mandel-type problem geometry is extended in this paper to account for the orthotropic nature of the porous media, thus mimicking the response of articular cartilage samples when subjected to load perturbation. The anisotropic solution presented takes into account the viscoelastic and anisotropic nature of the fluid-saturated cartilage specimen sandwiched between two impermeable rigid plates and subjected to quasi-static step loading conditions; thus simulating the unconfined compressive test responses of cartilage samples in biomechanics laboratory setups. The solution addresses the stress, fluid pressure, and displacement results due to load application through exact modeling of the intrinsic nature of the orthotropic viscoelastic matrix structure as well as the compressible interstitial fluid flow responses. Poromechanical parameter characterization and modeling of biological tissues, such as cartilage, will find this analytical solution to the two-dimensional anisotropic poroviscoelastic geometry very useful. This problem will not only serve as a benchmark for validating numerical schemes and simulations but also assist in calibrating laboratory results on biological tissues, including cyclic loadings.     

Unsteady Solution for Well Recharge in a Low Diffusive Aquifer

Finite aquifer solution exists for the constant head in a fully penetrating well. Their use for well recharge is limited, as they do not permit simultaneous computation of unsteady wellhead pressure and variable recharge rate. In the present paper semianalytical solutions are presented for well recharge under variable head boundary condition. These solutions were developed using the method of separation of variables and Duhamel’s convolution theorem. The solution developed in the paper was verified with the Jacob-Lohman solution and subsequently validated using field data pertinent to constant-head boundary conditions. Subsequently for variable head boundary condition such an appropriate background was found missing in the literature.     

Model for Consolidation-Induced Solute Transport with Nonlinear and Nonequilibrium Sorption

A numerical model, called CST2, is presented for coupled large strain consolidation and solute transport in saturated porous media. The consolidation and solute transport algorithms include the capabilities of a previous model, CST1, with the addition of a variable effective diffusion coefficient during consolidation and nonlinear nonequilibrium sorption. The model is based on a dual-Lagrangian framework that tracks separately the motions of fluid and solid phases. Verification checks of CST2 show excellent agreement with analytical and numerical solutions for solute transport in rigid porous media. A parametric study illustrates that, for the test cases considered, variation of effective diffusion coefficient during consolidation has an important effect on solute transport, whereas nonlinearity of the sorption isotherm has a less important effect. Additional simulations show that nonequilibrium sorption can have a strong effect on consolidation-induced solute transport and that this effect becomes more important as the rate of consolidation increases. The simulations also corroborate previous findings that consolidation can have a lasting effect on solute migration because transient advective flows change the distribution of solute mass which then becomes the initial condition for subsequent transport processes.     

Analytical Solutions for Unsteady Transport Dispersion of Nonconservative Pollutant with Time-Dependent Periodic Waste Discharge Concentration

Analytical solutions have been obtained by the Fourier transform method for the case of unsteady transport dispersion of nonconservative pollutant/biochemical oxygen demand with first-order decay under each of the sine and cosine variation of waste discharge concentration at upstream boundary and nonzero initial condition throughout the river. The solutions give correct results along the whole length of the river for all times in contrast to those available in the literature which yield sensible results under quasisteady-state assumption and for large times only. Appropriate expressions for memory length and memory time have been derived so as to include the effect of decay rate of the pollutant in terms of the Thomann number.     

Analytical Solution of Direct Dynamics Problem in Cylindrical Coordinates

In this study, the analytical solution of the direct dynamics problem that mostly requires some numerical techniques is developed for two important cases. In the first case, two coupled differential equations of the motion of a particle acted upon by the constant radial force Fr and tangential force Fθ are solved. The analytical solution for the case of variable radial force Fr = ?kr and the constant tangential force Fθ is considered as a second stage. The present method is applied to some practical engineering problems.     

Diagnostic Curves for Identifying Leaky Aquifer Parameters with or without Aquitard Storage

Two sets of unimodal diagnostic curves, one set assumes no aquitard storage and the other set assumes aquitard storage, are developed for identifying the parameter of leaky aquifers from early drawdowns, which yields accurate estimates of the parameters and lessens the subjectivity due to personal errors. The proposed diagnostic curve method is simple, easy to apply, and is based on matching of the diagnostically plotted observed drawdowns to an appropriate diagnostic curve. The new method is simple, easy to apply, does not require either the initial guess for the parameter values or repetitive evaluation of the leaky aquifer well function, and outperforms the conventional curve-matching, optimization, extended Kalman filter, and artificial neural network methods. The proposed set of diagnostic curves has a good diagnostic property and is able to easily identify nonideal conditions. The new method suggests a shorter duration pumping test, which would save time, money, and water. It is hoped that the proposed method would be useful to the field engineers and practitioners.     

Analytical Models for the Design of Iron-Based Permeable Reactive Barriers

The preliminary design of iron-based permeable reactive barriers is often accomplished using analytic expressions for one-dimensional groundwater flow and contaminant transport. Typically, one or more of the governing processes is simplified or neglected to facilitate development of a tractable solution. This paper presents a set of improved design equations that include the effects of dispersion, finite domain boundary, sequential decay, and production processes, and increased flow through high conductivity barriers. When applied to realistic example problems, application of the expanded design equations typically results in the specification of a larger permeable reactive barrier thickness than obtained using conventional approaches.     

Analytical Solution for Diffusion of VOCs through Composite Landfill Liners

Analytical solutions are presented for analyzing volatile organic compound (VOC) diffusion through intact composite landfill liners for two scenarios with boundary conditions at the base of either a VOC concentration of zero or a VOC mass flux of zero. A time-dependent concentration top boundary condition is included in the presented analytical solutions to model typical variations of VOC concentration in the leachate over time. The presented solutions are verified against alternative numerical solutions and applied to analyze dichloromethane diffusion through a composite liner. The analytical solutions are found to provide useful predictions of VOC concentration and mass flux for the design of composite liners. VOC concentrations and fluxes at the base of the composite liner at 30?years predicted by consideration of representative transient variation in leachate concentration, for an example problem, are nearly half of those when a constant leachate concentration assumed.     

Solution for Flow Rates across the Wellbore in a Two-Zone Confined Aquifer

A closed-form solution for transient flow rates across the wellbore in a confined aquifer is derived from a two-zone radial ground-water flow equation subject to the boundary condition of keeping a constant head at the well radius. An aquifer may be considered as a two-zone system if the formation properties near the wellbore are significantly changed due to the well construction and/or well development. An efficient numerical approach is used to evaluate this newly derived solution. Values of the transient flow rate are provided in a tabular form and compared with those obtained by numerical inversion for the Laplace-domain solution. The results show that the two solutions are in good agreement. This newly derived solution can be used not only for predicting the transient flow rate across the wellbore but also for identifying the effects of a skin with a finite thickness on the estimation of transient flow rates in a ground-water system with two different formation properties.