Primary cementing of vertical wells: displacement and dispersion effects in narrow eccentric annuli

Laminar miscible displacement flows in narrow, vertical, eccentric annuli show a wide range of interesting physical phenomena, e.g. static layers and channels, dispersive spikes/fronts and various instabilities. These are of relevance to the primary cementing of wells, where gas leakage and greenhouse gas emissions can result from ineffective cementing. The current popular way to model this process is via a Hele-Shaw approach, reducing the Navier-Stokes equations via scaling arguments and then averaging across the annular gap: the two-dimensional (2-D) gap averaged (2DGA) model. This leads to a reduced model that is computationally efficient and represents some important features of cementing flows, but is also deficient in capturing the effects of dispersion. While three-dimensional (3-D) simulations are able to capture dispersion and most other physical phenomena, they suffer from excessive computational times due to the extreme aspect ratios of cementing geometries and non-Newtonian properties of the fluids. Here we extend the 2DGA approach by modelling the high Peclet number limit of miscible duct displacement flows, to take into account the leading-order effects of dispersion. Our modified dispersive 2DGA model is well able to approximate dispersive effects that are observed by gap-averaging the results of 3-D computations, including geometric effects of eccentricity and instabilities. This represents a marked improvement over the 2DGA approach, while keeping the computational advantages of a 2-D model.