Exact solution for 1D deep bed filtration with particle capture by advection and dispersion

Abstract

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.