Mritunjay Kumar Singh

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.

Solute Transport along Temporally and Spatially Dependent Flows through Horizontal Semi-Infinite Media: Dispersion Proportional to Square of Velocity

According to the hydrodynamic dispersion theories, the dispersion parameter is proportional to a power n of the velocity; the power ranges between 1 and 2. Based on the value n=1, analytical solutions of the dispersion problems along temporally dependent flow domains were obtained in previous works. In the present work, two dispersion problems are addressed for n=2. Using the Laplace transform technique, analytical solutions are obtained for two-dimensional advection-diffusion equations describing the dispersion of pulse-type point source along temporally and spatially dependent flow domains, respectively, through a semi-infinite horizontal isotropic medium. Point sources of a uniform and varying nature are considered. The inhomogeneity of the medium is demonstrated by the linearly interpolated velocity in the space variable. Introduction of new space variables enable one to reduce the advection-diffusion equation in both problems to a one-dimensional equation with constant coefficients. The solutions are...

Analytical Solution for Two-Dimensional Solute Transport in Finite Aquifer with Time-Dependent Source Concentration

Using the Hankel Transform Technique, an analytical solution is derived for two-dimensional solute transport in a homogeneous isotropic aquifer. The aquifer is subjected to time-dependent point source contamination. The solution is derived under two conditions: (1) the flow velocity in the aquifer is a sinusoidally varying function and (2) the flow velocity is an exponentially decreasing function. Initially the aquifer is assumed solute free. The analytical solution is illustrated using an example.

Contaminant Concentration Prediction Along Unsteady Groundwater Flow

One-dimensional model describing contaminant concentration pattern, governed by advective-diffusive process, is discussed in the present chapter. Solute mass dispersion originating from a pulse type time dependent source is used along a homogeneous semi-infinite aquifer, defined by the Heaviside unit step function. Linear, exponentially decreasing, and sigmoid forms of unsteady velocities are considered. Using suitable transformations, variable coefficients are reduced to constant coefficients. Laplace transformation is used to get the analytical solutions. The analytical solution is compared with the numerical solution of the same problem. To get numerical solution, the semi-infinite domain is converted into a finite domain. The unsteadiness of velocity is defined with the help of two parameters. Comparisons are made for a wide range of combinations of the different parameters; interesting results are observed.

Solution of One-Dimensional Time Fractional Advection Dispersion Equation by Homotopy Analysis Method

AbstractThis study develops a homotopy analysis method (HAM) for analytically solving a one-dimensional time-fractional advection-dispersion equation (FADE). The HAM is a powerful method for solvin...

One-dimensional uniform and time varying solute dispersion along transient groundwater flow in a semi-infinite aquifer

An analytical solution for the space-time variation of contaminant concentration in one-dimensional transient groundwater flow in a homogenous semi-infinite aquifer, subjected to time-dependent source contamination, is derived. The uniform and time varying dispersion along transient groundwater flow is investigated under two conditions. First, the flow velocity distribution in the aquifer is considered as a sinusoidally varying function, and second, the flow velocity distribution is treated as an exponentially increasing function of time. It is assumed that initially the aquifer is not solute free, so the initial background concentration is considered as an exponentially decreasing function of the space variable which is tending to zero at infinity. It is assumed that dispersion is directly proportional to the square of the velocity, noting that experimental observations indicate that dispersion is directly proportional to the velocity with a power ranging from 1 to 2. The analytical solution is illustrated using an example and may help benchmark numerical codes and solutions.

Pollutant’s Horizontal Dispersion Along and Against Sinusoidally Varying Velocity from a Pulse Type Point Source

An analytical solution of a two-dimensional advection diffusion equation with time dependent coefficients is obtained by using Laplace Integral Transformation Technique. The horizontal medium of solute transport is considered of semi-infinite extent along both the longitudinal and lateral directions. The input concentration is assumed at an intermediate position of the domain. It helps to evaluate concentration level along the flow as well as against the flow through one model only. The source of the input concentration is considered to be of pulse type. In the presence of the source, it is assumed to be decreasing very slowly with time, and just after the elimination of the source it is assumed to be zero. The dispersion coefficient and the advection parameter are considered directly proportional to each other. The analytical solution may be used to predict the solute concentration level with position and time in an open medium as well as in a porous medium. The effect of heterogeneity on the solute transport may also be predicted.

Mathematical modeling of groundwater contamination with varying velocity field

Abstract In this study, analytical models for predicting groundwater contamination in isotropic and homogeneous porous formations are derived. The impact of dispersion and diffusion coefficients is included in the solution of the advection-dispersion equation (ADE), subjected to transient (time-dependent) boundary conditions at the origin. A retardation factor and zero-order production terms are included in the ADE. Analytical solutions are obtained using the Laplace Integral Transform Technique (LITT) and the concept of linear isotherm. For illustration, analytical solutions for linearly space- and time-dependent hydrodynamic dispersion coefficients along with molecular diffusion coefficients are presented. Analytical solutions are explored for the Peclet number. Numerical solutions are obtained by explicit finite difference methods and are compared with analytical solutions. Numerical results are analysed for different types of geological porous formations i.e., aquifer and aquitard. The accuracy of results is evaluated by the root mean square error (RMSE).

Analytical modeling of solute transport in homogeneous porous media with Cauchy type boundary condition

Analytical solution is obtained to describe the nature of concentration of solute, transported in saturated porous media. The aquifer is assumed to be homogeneous and finite in nature in which groundwater flow is considered unsteady. Initially the aquifer is not supposed to be solute free which means that aquifer is not clean i.e. some initial background concentration exists in the aquifer system which is represented by uniform source of concentration. A Cauchy type input source concentration is considered in the intermediate portion of aquifer system. At the other end, the concentration gradient is supposed to be zero. Two types of temporally dependent velocities are considered. Laplace Transform Technique (LTT) is used to determine the analytical solution. The results are perhaps most useful for benchmarking numerical codes and solutions. It may also be used as a predictive tool of groundwater resource management.

Analytical Approach to Solute Dispersion along and against Transient Groundwater Flow in a Homogeneous Finite Aquifer: Pulse Type Boundary Conditions

A one-dimensional advective-dispersive equation is analytically solved to predict patterns of contaminant concentration distribution in a homogeneous and finite aquifer. The dispersion of solute along and against transient groundwater flow is considered. Initially the aquifer is assumed to be not clean, which means that some initial background concentration exists and it is represented by uniform concentration. A pulse-type exponentially decreasing temporally dependent source concentration is considered in the intermediate portion of the aquifer and at other end, the concentration gradient is supposed to be zero. The Laplace Transform Technique (LTT) is used to obtain the solution of the problem of contaminant distribution in two different domains. In fact, in one domain the distribution pattern is depicted along groundwater flow and in other domain, it is depicted against groundwater flow. This represents a realistic situation of the contaminant concentration distribution pattern in an aquifer in the presence or absence of temporally dependent source concentration. The time-varying velocity expressions are considered. The dispersion is directly proportional to the seepage velocity used in which the effect of molecular diffusion is not taken into account because the value of molecular diffusion does not vary significantly for different soil and contaminant behaviors. Results of the obtained analytical solution may form useful complements to benchmark numerical models or for their verification. The long-term vertical transport of solute during transient saturated groundwater flow can be described correctly with appropriate analytical models. It may also be used as a preliminary predictive tool for groundwater resource management to estimate transport parameters.