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Register a new userInteracting length scales in the reactive-infiltration instability
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The reactive-infiltration instability, which develops when a porous matrix is dissolved by a flowing fluid, contains two important length scales. Here we outline a linear stability analysis that simultaneously incorporates both scales. We show that the commonly used thin-front model is a limiting case of a more general theory, which also includes convection-dominated dissolution as another special case. The wavelength of the instability is bounded from below, and lies in the range 1mm to 1km for physically reasonable flow rates and reaction rates. We obtain a closed form for the growth rate when the change in porosity is small.
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Reactive infiltration instability (RII) drives the development of many natural and engineered flow systems. These are encountered e.g. in hydraulic fracturing, geologic carbon storage and well stimulation in enhanced oil recovery. The surface area of the rocks changes as the pore structure evolves. We combined a reactor network model with grey scale tomography to seek the morphological interpretation for differences among geometric, reactive and apparent surface areas of dissolving natural porous materials. The approach allowed us to delineate the experimentally convoluted variables and study independently the effects of initial geometry and macroscopic flowrate. Simulations based on North Sea chalk microstructure showed that geometric surface not only serves as the interface for water-rock interactions but also represents the regional transport heterogeneities that can be amplified indefinitely by dissolutive percolation. Hence, RII leads to channelization of the solid matrix, which results in fluid focusing and an increase in geometric surface area. Fluid focusing reduces the reactive surface area and the residence time of reactants, both of which amplify the differences in question, i.e. they are self-supporting. Our results also suggested that the growing and merging of microchannels near the fluid entrance leads to the macroscopic fast initial dissolution of chemically homogeneous materials.
The tendency of irreversible processes to generate entropy is the ultimate driving force for the evolution of nature. In engineering, entropy production is often used as a measure of usable energy losses. In this study we show that the analysis of the entropy production patterns can help understand the vastly diversified experimental observations of water-rock interactions in natural porous media. We first present a numerical scheme for the analysis of entropy production in dissolving porous media. Our scheme uses a greyscale digital model of natural chalk obtained by X-ray nanotomography. Greyscale models preserve structural heterogeneities with very high fidelity, which is essential for simulating a system dominated by infiltration instability. We focus on the coupling between two types of entropy production: the percolative entropy generated by dissipating the kinetic energy of fluid flow and the reactive entropy that originates from the consumption of chemical free energy. Their temporal patterns pinpoint three stages of microstructural evolution. We then show that the regional mixing deteriorates infiltration instability by reducing local variations in reactant distribution. In addition, we show that the microstructural evolution can be particularly sensitive to the initially present transport heterogeneities when the global flowrate is small. This dependence on flowrate indicates that the need to resolve the structural features of a porous system is greater when the residence time of the fluid is long.
A wealth of experimental data indicate that while capillarity controlled infiltration gives an infiltration length that varies with the square root of time, reactive infiltration is characterised by a linear relationship between the two magnitudes. In addition the infiltration rate in the latter is at least two orders of magnitude greater than in the former. This work is addressed to investigate imbibition of a non-wetting, albeit reactive, liquid into a capillary, within the framework of a simple model that includes capillarity effects, viscosity and gravity. The capillary radius is allowed to vary, due to reaction, with both position and time, according to either an interface or a diffusion law. The model allows for capillary closure when reaction kinetics dominates imbibition. At short times, and depending on whether infiltration is capillarity or gravity controlled, the infiltrated length varies either as the square root or linearly with time. This suggest the following track for reactive infiltration: i) In most cases, the contact angle is initially larger than $90^circ$, ii) after some time, reaction gradually replaces the interface liquid/preform by the liquid/reaction product interface and, concomitantly, the contact angle gets closer to $90^circ$, iii) beyond that time, gravity triggers infiltration (actually the contact angle does not need to be smaller than $90^circ$ for the initiation of infiltration due to the metallostatic pressure exerted by the liquid metal on top of the porous preform), iv) thereafter infiltration is controlled by viscosity and gravity, provided that, due to reaction, the contact angle remains close to that at which infiltration was initiated.
Multi-phase reactive transport processes are ubiquitous in igneous systems. A challenging aspect of modelling igneous phenomena is that they range from solid-dominated porous to liquid-dominated suspension flows and therefore entail a wide spectrum of rheological conditions, flow speeds, and length scales. Most previous models have been restricted to the two-phase limits of porous melt transport in deforming, partially molten rock and crystal settling in convecting magma bodies. The goal of this paper is to develop a framework that can capture igneous system from source to surface at all phase proportions including not only rock and melt but also an exsolved volatile phase. Here, we derive an n-phase reactive transport model building on the concepts of Mixture Theory, along with principles of Rational Thermodynamics and procedures of Non-equilibrium Thermodynamics. Our model operates at the macroscopic system scale and requires constitutive relations for fluxes within and transfers between phases, which are the processes that together give rise to reactive transport phenomena. We introduce a phase- and process-wise symmetrical formulation for fluxes and transfers of entropy, mass, momentum, and volume, and propose phenomenological coefficient closures that determine how fluxes and transfers respond to mechanical and thermodynamic forces. Finally, we demonstrate that the known limits of two-phase porous and suspension flow emerge as special cases of our general model and discuss some ramifications for modelling pertinent two- and three-phase flow problems in igneous systems.
It is generally accepted that melt extraction from the mantle at mid-ocean ridges (MORs) is concentrated in narrow regions of elevated melt fraction called channels. Two feedback mechanisms have been proposed to explain why these channels grow by linear instability: shear flow of partially molten mantle and reactive flow of the ascending magma. These two mechanisms have been studied extensively, in isolation from each other, through theory and laboratory experiments as well as field and geophysical observations. Here, we develop a consistent theory that accounts for both proposed mechanisms and allows us to weigh their relative contributions. We show that interaction of the two feedback mechanisms is insignificant and that the total linear growth rate of channels is well-approximated by summing their independent growth rates. Furthermore, we explain how their competition is governed by the orientation of channels with respect to gravity and mantle shear. By itself, analysis of the reaction-infiltration instability predicts the formation of tube-shaped channels. We show that with the addition of even a small amount of extension in the horizontal, the combined instability favours tabular channels, consistent with the observed morphology of dunite bodies in ophiolites. We apply the new theory to MORs by calculating the accumulated growth and rotation of channels along streamlines of the solid flow. We show that reactive flow is the dominant mechanism deep beneath the ridge axis, where the most unstable orientation of high-porosity channels is sub-vertical. Channels are then rotated by the solid flow away from the vertical. The contribution of the shear-driven instability is confined to the margins of the melting region. Within the limitations of our study, the shear-driven feedback is not responsible for significant melt focusing or for shallowly dipping seismic anisotropy [abridged].
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