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Summary: The dissolution-induced finger or wormhole patterns in porous medium or fracture rock play a crucial role in a variety of scientific, industrial, and engineering practices. Although previous studies have extensively presented a number of numerical models which couples a system of nonlinear governing equations of porosity change due to mineral dissolution, the conservations of groundwater flow and transport of chemical species to investigate the morphological pattern of a chemical dissolution front within a fluid-saturated porous medium, whereas the mechanical dispersion effect has generally been neglected in the model development. This study addresses the effects of mechanical dispersion on the morphological evolution of a chemical dissolution front for a variety of cases. Mechanical dispersion processes is incorporated with the coupled nonlinear governing equation system so as to rebuild a newly numerical model. The results of numerical simulations demonstrate that mechanical dispersion has pronounced impacts on the morphological pattern of the chemical dissolution front. For single local non-uniformity case, mechanical dispersion reduces the finger length of an unstable single-fingering front or retains the shape of a stable planar front while speeding up the front advancement. In the case of two local non-uniformities, adding mechanical dispersion with different flow conditions can yield one of the following results: (1) the shape of the stable planar front is maintained but its advancement is accelerated; (2) the shape of the unstable single-fingering front is maintained but its length is reduced; (3) the unstable double-fingering front is merged into an unstable single-fingering front; and (4) the shape of the unstable double-fingering front is preserved but its fingering length is reduced. A comparison between the behavior diagrams of dissolution front morphology (with and without considering mechanical dispersion) shows that the double-fingering front occurs under condition where the upstream pressure gradient is higher and the non-uniformity spacing is larger while mechanical dispersion is taken into consideration. [Copyright &y& Elsevier]

Abstract: This study presents a novel mathematical model for analysis of non-axisymmetrical solute transport in a radially convergent flow field with scale-dependent dispersion. A two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates is derived based on assuming that the longitudinal and transverse dispersivities increase linearly with the distance of the solute transported from its injected source. The Laplace transform finite difference technique is applied to solve the two-dimensional, scale-dependent advection–dispersion equation with variable-dependent coefficients. Concentration contours for different times, breakthrough curves of average concentration over concentric circles with a fixed radial distance, and breakthrough curves of concentration at a fixed observation point obtained using the scale-dependent dispersivity model are compared with those from the constant dispersivity model. The salient features of scale-dependent dispersion are illustrated during the non-axisymmetrical transport from the injection well into extraction well in a convergent flow field. Numerical tests show that the scale-dependent dispersivity model predicts smaller spreading than the constant-dispersivity model near the source. The results also show that the constant dispersivity model can produce breakthrough curves of averaged concentration over concentric circles with the same shape as those from the proposed scale-dependent dispersivity model at observation point near the extraction well. Far from the extracting well, the two models predict concentration contours with significantly different shapes. The breakthrough curves at observation point near the injection well from constant dispersivity model always produce lesser overall transverse dispersion than those from scale-dependent dispersivity model. Erroneous dimensionless transverse/longitudinal dispersivity ratio may result from parametric techniques which assume a constant dispersivity if the dispersion process is characterized by a distance-dependent dispersivity relationship. A curve-fitting method with an example is proposed to evaluate longitudinal and transverse scale-proportional factors of a field with scale-dependent dispersion. [Copyright &y& Elsevier]

The authors discuss the article "A New Method for Laboratory Estimation of the Transverse Dispersion Coefficient," by Marco Massabò, Federico Catania and O. Paladino. The authors say that Massabò and colleagues recommended a technique for transverse dispersion coefficient determination. They state that the authors specified the Dirac delta function as the instantaneous slug input. They also suggested an approach for associated boundary and initial conditions.

Summary: This study presents exact analytical solutions to the two-dimensional advection–dispersion equation in cylindrical coordinates in finite domain subject to the first- and third-type inlet boundary conditions. The second kind finite Hankel transform and the generalized integral transform technique are adopted to solve the two-dimensional advection–dispersion equation in cylindrical coordinates and its associated initial and boundary conditions. The developed analytical solutions are compared with the solutions for semi-infinite domain subject to the first- and third-type inlet boundary conditions available in literature to illustrate the impacts of the exit boundary conditions. Results show that significant discrepancies between the breakthrough curves obtained from analytical solutions for the finite domain and infinite domain for small Peclet number. Numerical evaluations of the developed analytical solutions for finite domain are computationally intensive because that the convergences of the series progress slowly for medium Peclet number. The developed solutions should be especially useful for testing numerical model simulated solutions for the finite domain subject to first- and third-type inlet boundary conditions. [Copyright &y& Elsevier]

Abstract: Exact analytical solutions for two-dimensional advection–dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer. [Copyright &y& Elsevier]

Summary: The convergent flow tracer test is an efficient method for determining dispersivity in field, but the traditional curve-fitting method for the estimation of dispersivity from a convergent flow tracer test is quite time-consuming. In this study, we present a model to improve the evaluation of longitudinal and transverse dispersivities from a convergent flow tracer test which couples a back-propagation neural network (BPN) model with a two-dimensional convergent flow tracer transport model. The prediction errors for the training and validation data show that with the effective porosity fitting model, the scale-dependent longitudinal dispersivity fitting model, and the scale-dependent transverse dispersivity fitting model, we can obtain satisfactory prediction accuracy with much less computational time. The applicable ranges of parameters are: The Peclet number is between 0.5 and 100, the effective porosity is between 0.05 and 0.5 and the scale-dependent transverse dispersivity is between 0.01 and 10 m. One set of hypothetical data and one set of field data are used to demonstrate the robustness and accuracy of the back-propagation neural network fitting model (BPNFM). The results demonstrate that BPNFM has the advantage of significantly saving the computational time and giving more accurate transport parameter values. The developed BPNFM is an effective tool for fast and accurate evaluation of the longitudinal and transverse dispersivities for a field convergent flow tracer test. [ABSTRACT FROM AUTHOR]

Summary: This study presents a novel mathematical model for describing the transport of the remedial reagent in a vertical circulation flow field in an anisotropic aquifer. To develop the mathematical model, the radial and vertical components of the pore water velocity are calculated first by using an analytical solution for steady-state drawdown distribution near a vertical circulation well. Next, the obtained radial and vertical components of the pore water velocity are then incorporated into a three-dimensional axisymmetrical advection–dispersion equation in cylindrical coordinates from which to build the reagent transport equation. The Laplace transform finite difference technique is applied to solve the three-dimensional axisymmetrical advection–dispersion equation with spatial variable-dependent coefficients. The developed mathematical model is used to investigate the effects of various parameters such as hydraulic conductivity anisotropy, longitudinal and transverse dispersivities, the placement of the extraction and injection screened intervals of the vertical circulation well and the injection modes on the transport regime of the remedial reagent. Results show that those parameters have different degrees of impacts on the distribution of the remedial reagent. The mathematical model provides an effective tool for designing and operating an enhanced groundwater remediation in an anisotropic aquifer using the vertical circulation well technology. [ABSTRACT FROM AUTHOR]