(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.

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Abstract. The pressure-driven growth model is considered, describing the motion of a foam front through an oil reservoir during foam improved oil recovery, foam being formed as gas advances into an initially liquid-filled reservoir. In the model, the foam front is represented by a set of so-called "material points" that track the advance of gas into the liquid-filled region. According to the model, the shape of the foam front is prone to develop concave sharply curved concavities, where the orientation of the front changes rapidly over a small spatial distance: these are referred to as "concave corners". These concave corners need to be propagated differently from the material points on the foam front itself. Typically the corner must move faster than those material points, otherwise spurious numerical artifacts develop in the computed shape of the front. A propagation rule or "speed up" rule is derived for the concave corners, which is shown to be sensitive to the level of anisotropy in the permeability of the reservoir and also sensitive to the orientation of the corners themselves. In particular if a corner in an anisotropic reservoir were to be propagated according to an isotropic speed up rule, this might not be sufficient to suppress spurious numerical artifacts, at least for certain orientations of the corner. On the other hand, systems that are both heterogeneous and anisotropic tend to be well behaved numerically, regardless of whether one uses the isotropic or anisotropic speed up rule for corners. This comes about because, in the heterogeneous and anisotropic case, the orientation of the corner is such that the "correct" anisotropic speed is just very slightly less than the "incorrect" isotropic one. The anisotropic rule does however manage to keep the corner very slightly sharper than the isotropic rule does.

This version is available at https://strathprints.strath.ac.uk/52032/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the AbstractFoam is often used in improved oil recovery processes to displace oil from an underground reservoir. During the process, the reservoir is flooded with surfactant, and then gas is injected to produce foam in situ, with the foam front advancing through the reservoir. Here the effect of surfactant slumping (downward movement of surfactant in relation to a lighter phase) upon the advance of a foam front is presented. Slumping which can be associated with foam drainage, coarsening and collapse, causes a rise in mobility of the foam front specifically near the top of the front. The description of a foam front displacement for an initially homogeneous foam mobility is therefore modified to account for slumping-induced inhomogeneities. Numerical solution for the front shape shows that, although slumping transiently produces a localised concave region on the otherwise convex front, this concavity has little effect on the long term front evolution. In fact in the long-time limit, a convex kink develops on the front: an analytical solution describing the convex kink agrees very well with the numerics.

This work focuses on the mathematical analysis of the controlled release of a standardized extract of A. chica from chitosan/alginate (C/A) membranes, which can be used for the treatment of skin lesions. Four different types of C/A membranes were tested: a dense membrane (CA), a dense and flexible membrane (CAS), a porous membrane (CAP) and a porous and flexible membrane (CAPS). The Arrabidae chica extract release profiles were obtained experimentally in vitro using PBS at 37 °C and pH 7. Experimental data of release kinetics were analyzed using five classical models from the literature: Zero Order, First Order, Higuchi, Korsmeyer–Peppas and Weibull functions. Results for the Korsmeyer–Peppas model showed that the release of A. chica extract from four membrane formulations was by a diffusion through a partially swollen matrix and through a water filled network mesh; however, the Weibull model suggested that non-porous membranes (CA and CAS) had fractal geometry and that porous membranes (CAP and CAPS) have highly disorganized structures. Nevertheless, by applying an explicit optimization method that employs a cost function to determine the model parameters that best fit to experimental data, the results indicated that the Weibull model showed the best simulation for the release profiles from the four membranes: CA, CAS and CAP presented Fickian diffusion through a polymeric matrix of fractal geometry, and only the CAPS membrane showed a highly disordered matrix. The use of this cost function optimization had the significant advantage of higher fitting sensitivity.