(Received ?; revised ?; accepted ?. -To be entered by editorial office)During improved oil recovery, gas may be introduced into a porous reservoir filled with surfactant solution in order to form foam. A model for the evolution of the resulting foam front known as 'pressure-driven growth' is analysed. An asymptotic solution of this model for long times is derived that shows that foam can propagate indefinitely into the reservoir without gravity override. Moreover 'pressure-driven growth' is shown to correspond to a special case of the more general 'viscous froth' model. In particular, it is a singular limit of the viscous froth, corresponding to the elimination of a surface tension term, permitting sharp corners and kinks in the predicted shape of the front. Sharp corners tend to develop from concave regions of the front. The principal solution of interest has a convex front, however, so that although this solution itself has no sharp corners (except for some kinks that develop spuriously owing to errors in a numerical scheme), it is found nevertheless to exhibit milder singularities in front curvature, as the long-time asymptotic analytical solution makes clear. Numerical schemes for the evolving front shape which perform robustly (avoiding the development of spurious kinks) are also developed. Generalisations of this solution to geologically heterogeneous reservoirs should exhibit concavities and/or sharp corner singularities as an inherent part of their evolution: propagation of fronts containing such 'inherent' singularities can be readily incorporated into these numerical schemes.
Care is needed with algorithms for computer simulations of the Brownian motion of complex systems, such as colloidal and macromolecular systems which have internal degrees of freedom describing changes in configuration. Problems can arise when the diffusivity or the inertia changes with the configuration of the system. There are some problems in replacing very stiff bonds by rigid constraints. These problems and their resolution are illustrated by some artificial models; firstly in one dimension, then in the neighbourhood of an ellipse in two dimensions and finally for the trimer polymer molecule.
Microscale models of foam structure traditionally incorporate a balance between bubble pressures and surface tension forces associated with curvature of bubble films. In particular, models for flowing foam microrheology have assumed this balance is maintained under the action of some externally imposed motion. Recently, however, a dynamic model for foam structure has been proposed, the viscous froth model, which balances the net effect of bubble pressures and surface tension to viscous dissipation forces: this permits the description of fast-flowing foam. This contribution examines the behavior of the viscous froth model when applied to a paradigm problem with a particularly simple geometry: namely, a two-dimensional bubble "lens." The lens consists of a channel partly filled by a bubble (known as the "lens bubble") which contacts one channel wall. An additional film (known as the "spanning film") connects to this bubble spanning the distance from the opposite channel wall. This simple structure can be set in motion and deformed out of equilibrium by applying a pressure across the spanning film: a rich dynamical behavior results. Solutions for the lens structure steadily propagating along the channel can be computed by the viscous froth model. Perturbation solutions are obtained in the limit of a lens structure with weak applied pressures, while numerical solutions are available for higher pressures. These steadily propagating solutions suggest that small lenses move faster than large ones, while both small and large lens bubbles are quite resistant to deformation, at least for weak applied back pressures. As the applied back pressure grows, the structure with the small lens bubble remains relatively stiff, while that with the large lens bubble becomes much more compliant. However, with even further increases in the applied back pressure, a critical pressure appears to exist for which the steady-state structure loses stability and unsteady-state numerical simulations show it breaks up by route of a topological transformation.
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