Analytical solutions of advection-dispersion-reaction equation with first decay under constant and time-dependent boundary conditions: Mass transfer shape factor effects

To describe the combined dispersion and convection mechanisms of solute transport through porous media, one may typically refer to the advection-dispersion equation (ADE). In this study, a generalized analytical solution to the ADE combined with first order decay term is derived for the problem of solute transport through porous media in one-dimensional finite region considering transient boundary conditions. First, the addressed convection-dispersion-reaction equation is simplified into a diffusion equation. Subsequently, an analytical solution is generated for a general fractured porous medium for a case where a fracture is viewed as the transient boundary condition. To verify the proposed analytical solution, experimental data acquired from literature as well as numerical techniques of finite difference and element are utilized. Previous studies have utilized fixed values for mass transfer shaper factor which is deployed in dual porosity modeling approach of fractured reservoirs, while in reality this parameter changes over time. In this regard, an attempt is made to attain mass transfer shape factor while considering the impacts of important parameters such as Péclet number, and decay coefficient during both pseudo steady state (PSS) and transient periods. Moreover, for two circumstances of linearly ascending and constant concentration boundary conditions, a correlation is developed to compute PSS mass transfer shape factor as a function of Péclet number, dimensionless time, and decay coefficient. The results illustrated the dependency of PSS shape factor on the decay coefficient, Péclet number, and time while transient shape factor solely depended on time. In addition, the findings showed that the larger the Péclet number, the larger the PSS matrix-fracture mass transfer shape factor. The obtained analytical solution is helpful when validating the numerical simulation results and for performing fast sensitivity analysis. The developed mass transfer shape factor can be applied in the dual porosity modeling part of the commercial reservoir simulation software packages for more accurately modeling the solute transport in the fractured reservoirs.