Advection-Dispersion-Reaction Equation

• Problem Description

  • The transport of dissolved solutes in groundwater is often modeled using the Advection-Dispersion-Reaction (ADR) equation. Advection refers to the bulk movement of solutes carried by flowing groundwater. Dispersion refers to the spreading of the contaminant plume from highly concentrated areas to less concentrated areas. Dispersion coefficients are calculated as the sum of molecular diffusion, mechanical dispersion, and macrodispersion. Reaction refers to changes in the mass of the solute within the system resulting from biotic and abiotic processes.

• Model Descriptions

  • The ADR equation in one spatial dimension is given by the following linear partial differential equation (PDE)

\[\begin{array}{rl} {d C \over d t} \: = \: & D {\partial^2 C \over \partial x^2} - \: v {d C \over d x} \: - \: R C \\ \\ & \mbox{where:} \\ & \ \ D \mbox{ is the hydrodynamic dispersion coefficient in the } x \mbox{ direction} \\ & \ \ v \mbox{ is the advective transport or seepage velocity in the } x \mbox{ direction} \\ & \ \ R \mbox{ is the reaction coefficient which takes into acount the linear} \\ & \ \ \ \ \mbox{equilibrium retardation factor and effective first order decay} \\ & \ \ \ \ \mbox{rate due to combined biotic and abiotic processes} \end{array}\]

• • A nonlinear response function is built to model the concentrations over the spatial and temporal domain, resulting from the ADR PDE

  • For each model, a spatial domain of \(0<x<10\) was used, with \(D = 0.5\), \(v = 2.0\), and the boundary condition \(C(0) = 1\).

• Results

  • Resulting \(C(t)\) curves are generated for each model, for \(x\) values of 2, 4, 6, and 8. Differential equation model values are shown as data points, and nonlinear response function model values are shown as lines.

  • A prediction curve is generated for \(x\) value of 5.

  • All of the resulting plots are shown here.

• Conclusions

  • The nonlinear response function show accurate fits to the data derived from solution of the partial differential equation.

  • The nonlinear response function is able to accurately predict the time curve for \(x = 5\).

  • The nonlinear response function provides an accurate model for time curves over the entire spatial domain, thereby extending its applicability in modeling nonlinear dynamics to spatially-variant nonlinear dynamics.