Application of temporal moments to interpret solute transport with time-dependent dispersion

Numerical solution of the Burgers equation associated with the phenomena of longitudinal dispersion depending on time

“…Therefore, a time-dependent dispersion model can be used to provide a rough description of the transport scale in our study by exploiting the Burger equation. In this study, a linear and asymptotic form of the time-dependent dispersion coefficient is integrated into the nonlinear PDE equation (7) to study the phenomena of longitudinal dispersion of solutes in the underground environment; this has been mentioned in various works ( Guleria and Swami, 2018 ; Guleria et al, 2020 ). This type of dispersion will make it possible to better represent the results for a transport of solutes in a given porous medium.…”

“…The time-dependent linear dispersion increases with time without boundary conditions in the porous medium ( Gao et al, 2010 ; Guleria et al, 2020 ). The linear formula for the time dependent dispersion expression in equation (8) is given by equation (9) below: …”

“…The time-dependent asymptotic distance initially increases with time and eventually approaches an asymptotic value in the porous medium ( Guleria et al, 2020 ). The formula for time-dependent asymptotic dispersion expression in equation (8) is given by equation (10) below: Where is the maximum dispersion coefficient for the asymptotic time-dependent and uniform dispersion function for the linear time-dependent dispersion coefficient; is the effective diffusion coefficient of time; is the time-dependent asymptotic coefficient equivalent to the mean distance travelled by the pollutants in the aquifer; is the linear coefficient depending on time.…”

“…The values of the parameters of the time-dependent dispersion coefficient were determined by a sensitivity analysis of Barry and Garrison (1989) . The same values of the parameters of the linear and asymptotic time-dependent dispersion coefficient exploited in the work of Guleria et al (2020) which are also exploited in this work, are presented in the Table 1 .…”

“…An unconditionally stable Crank-Nicolson finite difference scheme was used by Ravi et al (2014) to analyze the constant and longitudinal dispersion profile of contaminants. In his studies, Ravi et al (2014) did not take into account the time-dependent dispersions coefficients, the decay rate constant and the zero-order production rate coefficient of solute in the liquid phase as done by Das et al (2017) , Guleria et al (2020) with a linear dispersion advection equation model. Our objective is to exploit the more stable fourth-order Runge-Kutta method (RK4) to evaluate the profile of C (X, T) contaminants through a porous medium and to conduct a comparative study between the profiles of the contaminants obtained from an asymptotic and linear dispersion coefficient, and on the other hand, from the initial and boundary conditions used in the work of Das et al (2017) .…”