Radial transport in a porous medium with Dirichlet, Neumann and Robin-type inhomogeneous boundary values and general initial data: analytical solution and evaluation

Abstract

The analytical solution is presented to the convection-diffusion equation describing the concentration of solutes in a radial velocity field due to extracting groundwater from or injecting water into an aquifer with arbitrary initial concentration data F(r ), with r the radial distance, and an inhomogeneous mixed boundary condition G(t), with t the time, at the well radius r = r 0 . The analytical solution is obtained with a generalized Hankel transformation or with a Laplace transformation. The Hankel transformation turns out to be easier for G = 0, F = 0, while the Laplace transformation is easier for F = 0, G = 0. Both techniques can, however, deal with the full problem. The representation found by the generalized Hankel transform can also be found by the Laplace transform, through modification of the contour through the complex plane in the Bromwich integral for the inverse Laplace transform to the real axis. In practice, the numerical evaluation of the integral representation is difficult, due to the oscillating behavior of the integrands. A more appropriate numerical inversion procedure is also suggested, which circumvents the integration of the oscillating integrands, by an alternative modification of the contour in the Bromwich integral such that the new contour follows the steepest descent path starting from a saddle point at the real axis.

A general model of radial dispersion with wellbore mixing and skin effects

“…Because of the benefits mentioned above, significant efforts have been put forward over many decades in developing advanced analytical models of radial dispersion. Some examples include the works of Hoopes and Harleman (1967), Gelhar and Collins (1971), Tang and Babu (1979), Moench and Ogata (1981), Chen (1985Chen ( , 1986Chen ( , 1991, Hsieh (1986), Tang and Peaceman (1987), Yates (1988), Falade and Brigham (1989), Novakowski (1992), Philip (1994), Veling (2001Veling ( , 2011, Huang and Goltz (2006), Chen et al (2007Chen et al ( , 2011Chen et al ( , 2012, Gao et al (2009a), Cihan and Tyner (2011), Wang and Zhan (2013), Hsieh and Yeh (2014), Zhou et al (2017), Wang et al (2018Wang et al ( , 2020, Huang et al (2019), and Li et al (2020). A general trend of such developments is to provide more robust models that can better represent physical reality.…”

A General Model of Radial Dispersion with Wellbore Mixing and Skin Effect

“…Firstly, the injected solute goes through a mixing process with native (or pre-injection) water in the wellbore at the early stage of injection, which is called mixing effect. Probably due to the small radius of the well, the mixing effect has been overlooked by almost all the analytical solutions mentioned above except Novakowski (1992) and Wang et al (2018), e.g., either by assuming that the well radius was infinitesimal, or assuming that the solute concentration in the wellbore was the same as the concentration of the injected solution (Hoopes and Harleman, 1967;Veling, 2011;Zhou et al, 2017). Consequently, the solutions developed without considering the wellbore mixing effect may overestimate concentration values in both the wellbore and the aquifer (Novakowski, 1992;Wang et al, 2018).…”

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

“…Convection-diffusion equation (CDE) is one of the most challenging problems and frequently used in various branches of engineering and applied science, especially in radial transport in a porous medium [1], heat transfer in a nanofluid filled [2], heat transfer in a draining film [3], and water transport in soil [4]. Also, the applications of CDE can be found in [5]- [8].…”