Previous studies of fluid convection in porous media have considered the onset of convection in isotropic systems and the steady convection in anisotropic systems. This paper bridges between these and develops new results for the onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. These results are relevant to sedimentary formations where the average vertical permeability is some fraction γ of the average horizontal permeability. Linear and global stability analyses are used to define the critical time tc at which the instability occurs as a function of γ and the dimensionless Rayleigh-Darcy number Ra* for both thermal and solute-driven convection in an infinite horizontal slab. Numerical results and approximate analytical solutions are obtained for both a slab of finite thickness and the limit of large slab thickness. For a thick slab, the increase in tc as γ decreases is approximately given by (1+γ)4∕(16γ2). One important application is to the geological storage of carbon dioxide where it is shown that the use of an effective vertical permeability in estimating the critical time is only valid for low permeabilities. The time scale for the onset of convection in geological storage can range from less than a year (for high-permeability formations) to decades or centuries (for low-permeability ones).
We propose an (essentially combinatorial) approach to the correlation functions of the domain wall six vertex model.We reproduce the boundary 1-point function determinant expression of Bogoliubov, Pronko and Zvonarev, then use that as a building block to obtain analogous expressions for boundary 2-point functions.The latter can be used, at least in principle, to express more general boundary (and bulk) correlation functions as sums over (products of) determinants. IntroductionComputing off critical correlation functions 1 is probably the most challenging open problem currently under investigation in exactly solvable lattice models [1]. The six vertex model, with domain wall boundary conditions (dwbc's) is an ideal testing ground of possible approaches to such computations, particularly if one is interested in computations on a finite lattice 2 .The model was first introduced by Korepin [2], who also formulated recursion relations that uniquely determine the partition function. Korepin's recursion relations were solved by Izergin [3], who obtained a determinant representation for the partition function. * Supported by the Australian Research Council (ARC).1 The literature on off critical correlation functions is extensive. We refer the reader to [4] for references to the literature up to the early 90's and to a search of http://arXiv.org for more recent literature. 2 The previous footnote applies verbatim to the literature on the domain wall six vertex model.
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