Exact solutions for two-phase colloidal-suspension transport in porous media

Two-phase transport of colloids and suspensions occurs in numerous areas of chemical, environmental, geo-, and petroleum engineering. The main effects are particle capture by the rock and altering the flux by changing the suspended and retained concentrations. Multiple mechanisms of suspended particle capture are discussed. The mathematical model for m independent particle-capture mechanisms is considered, resulting in an (m + 2) × (m + 2) system of partial differential equations. Using the stream-function as an independent variable instead of time splits the system into an (m + 1) × (m + 1) auxiliary system, containing only concentrations and one lifting hydrodynamic equation for an unknown phase saturation. Introduction of the concentration potential linked with retention concentrations yields an exact solution of the auxiliary problem. The exact formulae allow for predicting the profiles and breakthrough histories for the suspended and retained concentrations, and phase saturations. The solution shows that for small retained concentrations, the suspended concentration is in a steady-state behind the concentration front, where all the retained concentrations are proportional to the mass of suspended particles that passed via a given reservoir cross-section. The maximum penetration depths for suspended and retained particles are the same and are equal to those for a single-phase flow.