?
An averaged model for colloidal transport to exhibit hyper-exponential particle retention
Modelling of colloidal and nano-suspension transport in porous media has garnered significant attention due to the prevalence of these processes in many engineering applications. A number of experimental studies have reported retention profiles after coreflooding that are hyper-exponential, a feature that the traditional models for deep bed filtration are unable to capture. The aim of this work is the development of a model for binary particle transport which can account for hyper-exponential retention profiles (HERPs). Averaging of the multicomponent model results in a non-linear dependence of the capture rate on the suspended particle concentration. The averaged model demonstrates how nonlinear capture behaviour arises due to particle heterogeneity even when individual particle populations filter traditionally. The averaged model exhibits classical filtration behaviour at low concentrations as a result of selective particle filtration, with a more rapid degeneration to this behaviour when the difference between particle populations is larger. The model establishes that nonlinear capture behaviour, including the prediction of HERPs, occurs even for dilute particle suspensions given that particle heterogeneity is significant. An exact solution of the colloidal transport equations is presented for any non-linear suspension function and any step-wise-constant injected concentration. The binary suspension function cannot be written explicitly, and so we present asymptotic expansions for several limiting cases where the suspension function can be expressed directly. The asymptotic expansions show good agreement with the binary model, and correctly capture the HERPs calculated with the exact solution. Several laboratory tests exhibiting HERPs have been matched by the traditional, binary, and asymptotic expansion models. The traditional model significantly deviates from the laboratory data due to its inability to capture the hyper-exponential behaviour, while both the binary and asymptotic models capture this behaviour well.