Simulation of porosity reduction in random structures
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We describe an algorithm for computing the motion of a solid-liquid interface in 2D, which is applicable to geological pressure solution or to pressure sintering. The backward motion (toward the solid) of the interface is due to dissolution of the solid, and the forward motion (away from the solid) is due to the inverse process of reprecipitation. The interface velocity is assumed proportional to the difference between the solubility of the solid and the concentration of the solution. The former is dependent upon stress (the phenomenon of "pressure solution"), so our algorithm must also keep track of the stress. We use a Lagrangian grid, with constant-stress periodic boundary conditions. The method has been applied to porosity reduction in sandstone. It is applicable to other interface-following problems, such as freezing, if the motion is slow enough that heat transport is not rate-limiting.
I. INTRODUCTION
In this paper we describe an algorithm for simulating the motion of a solid-liquid interface, whose motion is due to dissolution of the solid in the liquid or reprecipitation of the solid. The method was developed to study porosity reduction in sandstone, in which case the solid is silica and the liquid is an aqueous silica solution. An important phenomenon in determining the structure of sandstone is "pressure solution", that is, the dependence of the solubility of silica in water on the stress in the silica. This causes silica to dissolve away at grain contacts where the stress is high and to reprecipitate elsewhere. The result of this effect is to decrease the porosity, as the contact points dissolve away and the grains settle closer to one another. Thus we must calculate the stress field everywhere within the system, in order to account for its effect on the solubility and thereby on the interface velocity. A. Interface tracking
Various methods have been used previously to simulate interface-motion problems. Some of these methods keep track of an interface by solving a differential equation for the evolution of the interface (a parameterized arc, in 2D), a technique which is most useful for studying a single interface1 or the growth of a single grain.2'3 More realistic, topologically complex interfaces can be modeled by explicit interface-following or implicitly, by calculating an order parameter whose sudden change indicates the presence of an interface. Implicit interface-following has been done in continuum models of a liquid-liquid interface, by defining, necessarily somewhat arbitrarily, an interface position in a liquid of continuously-varying composition,4'5 in the context of analytic calculations of light scattering. However, microscopically detailed calculations have 2184
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J. Mater. Res., Vol. 5, No. 10, Oct 1990
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mostly used Potts-like lattice models (for example, to study grain growth during annealing6'7). Such discretevalued lattice models allow the simulation of arbitrarily shaped random structures, but because of their stochastic natur
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