Multi-scale Averaging and Analysis of Transport and Reaction Phenomena in Porous Media
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In the first part, a systematic procedure of multi-scale averaging, based on Lyapunov-Schmidt (L-S) technique of bifurcation theory, is presented where low-dimensional models are derived for two problems: dispersion of a non-reacting tracer in laminar flow in a tube (Taylor dispersion); and, diffusion, convection and reaction in a catalytic monolith with porous washcoat. The averaged model for Taylor dispersion developed by the L-S procedure is exact for general inlet/initial conditions including point sources. It predicts no centroid displacement or variance deficit as other models in the literature. Truncated hyperbolic models are also presented along with inlet/initial conditions to the same accuracy. The reduced order model developed for catalytic monoliths is presented in terms of three concentration modes and it is shown for time-varying inlet conditions, the interfacial flux depends on all three modes. In such cases, in contrast to the traditional two-phase model, the three-mode reduced order model retains the feature of the detailed model. In the second part, modeling and simulation of reactive dissolution of carbonates with in-situ gelling acids is presented. Stimulation of oil-wells in carbonate-reservoirs using an acidic-solution is a common practice to enhance oil production. However, due to heterogeneity, acid flows preferentially in high-permeability zones, which results into under stimulation of low-perm regions. Therefore, in-situ gelling acids are used that block the high-permeability region by forming a gel and diverts more acid to the low-permeability zones. Here, a rheological model for in-situ gelling acids is developed and combined with an extended two-scale-continuum model to describe the transport and reaction of in-situ gelling acids in carbonates. Three-dimensional simulations predict dissolution patterns in various flow regimes that are in accordance with experimental results. The effect of rheological parameters on flow diversion, optimum injection rates, wormhole diameter and gel front-width and speed is studied using scaling analysis. Finally, guidelines for optimal stimulation of carbonates with in-situ gelling acids are presented.
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