Exact solution for 1D deep bed filtration with particle capture by advection and dispersion
We consider a one-dimensional (1D) non-linear 2x2 hyperbolic problem of suspension-colloidal-nano transport
in porous media, so-called deep bed filtration. The particle capture rate is proportional to the overall
particle flux, which includes dispersion. The captured-particle concentration dependency on the proportionality
coefficient, which is typical for large captured concentrations, causes the non-linearity of the problem.
However, the 1D initial–boundary value problem allows for an exact solution. The introduction of the Riemann
invariant converts the 2x2 system into a scalar quasi-linear hyperbolic equation, which is implicitly solved
using the method of characteristics. The non-linear equation for particle capture rate yields a limited travelling
wave that approximates the exact solution for the initial–boundary problem.