Scale dependence of mineral dissolution rates within single pores and fractures
Introduction
The dissolution of minerals plays a major role in various physical, chemical, and biological processes in nature. Mineral dissolution affects the formation of soils, influences the degradation of radioactive waste and its containers (Spycher et al., 2003) and the subsequent migration of heavy metals and radionuclides to the biosphere (Lovley, 1993, Lovley and Coates, 1997, Chorover et al., 2003), and at larger space and time scales, it regulates the atmospheric concentrations of CO2 (Berner, 1995, Berner and Berner, 1997). Laboratory-measured dissolution rates of many minerals have been consistently found to be several orders of magnitude faster than those observed in the field (White and Brantley, 2003, Maher et al., 2004), although it is not clear in all cases that this represents a true discrepancy in the rate constants as opposed to a failure to take into account the intrinsic differences in chemical and/or physical conditions between laboratory and field settings (Steefel et al., 2005). Such rate discrepancies need to be resolved, however, since they seriously hinder the application of laboratory-measured dissolution rates to natural systems.
A variety of factors have been proposed to contribute to the discrepancies between laboratory and field rates, including the differences in reactive surface area of the fresh and weathered minerals (White and Peterson, 1990, Anbeek, 1993, White, 1995), the effect of reaction affinity (White, 1995, White and Brantley, 2003, Maher et al., 2006), the precipitation rate of secondary clay minerals (Steefel and Van Cappellen, 1990, Alekseyev et al., 1997, Zhu et al., 2004, Maher et al., 2006), and the age of the reacting material (White and Brantley, 2003, Maher et al., 2004). Recent studies have also shown that physical and chemical heterogeneities in soils and aquifers where subsurface flow occurs may contribute to a scale dependence to mineral dissolution rates, and thus potentially to discrepancies between laboratory and field rates (Malmstrom et al., 2000, Malmstrom et al., 2004, Li et al., 2006, Meile and Tuncay, 2006). For example, Li and coworkers (Li et al., 2006, Li et al., 2007) found that variations in the spatial distributions of minerals with differing reactivity can result in the development of concentration gradients of chemical species involved in dissolution reactions, thus leading to erroneous predictions of reaction rates when the scale dependence is not accounted for. More generally, any chemical, physical, or microbiological heterogeneity that results in the formation of concentration gradients in the subsurface can lead to a scale dependence of the rates, and thus potentially to discrepancies between laboratory and field rates.
While the discrepancies between laboratory and field rates cannot be attributed entirely to the effect of physical and chemical heterogeneities, as indicated by studies of weathering rates in physically and chemically homogeneous media (Maher et al., 2006), it is clear that a comparison of lab and field rates requires careful consideration of the inherent differences between the laboratory and natural systems as suggested above. The laboratory measurement of dissolution rates usually employs well-mixed batch or flow through reactors. In these experimental systems, the aqueous phase is stirred rapidly enough that the aqueous phase becomes well-mixed, thus eliminating the effect of transport. In such cases, mineral dissolution is surface-controlled and depends only on the uniform chemistry of the aqueous solution. In natural systems, however, reactions are inevitably subject to the influence of transport via advection, molecular diffusion, and/or dispersion. As such, the mineral dissolution rates are an outcome of coupling between the reaction and transport processes.
In addition to these differences, natural porous or fractured media are also very different from laboratory settings in terms of their structure and heterogeneous nature. For example, natural porous media typically possess hierarchical structures and spatial scales that range from the pore scale, to the continuum scale, and finally to the field scale. The pore scale focuses on individual pores at the spatial scale of tens to thousands of microns, while the continuum scale, often at the scale of millimeters to centimeters, contains a sufficient number of pores that allows the definition of statistically averaged properties of porous media, including porosity and permeability (Bear, 1972). The field scale, often from meters to kilometers, is the scale at which we examine specific processes in natural field settings. Heterogeneities in the physical and chemical properties of porous media exist at all spatial scales. While at the continuum and field scales porous media are often represented by properties and processes “averaged” over a large number of pores, the pore scale is where processes such as flow, transport, and reactions actually take place. As such, pores are the fundamental building blocks of natural porous media and are an important starting point to examine the scaling issue with regard to reaction rates.
The effect of transport, especially diffusion, on mineral reaction rates in natural systems has been discussed previously in the context of transport-controlled versus surface-controlled reaction kinetics and the validity of the local equilibrium assumption (Berner, 1978, Dibble and Tiller, 1981, Casey, 1987, Murphy et al., 1989). These studies did not address the detailed flow patterns developed in single pores and fractures and focused primarily on single component systems so that analytical solutions could be derived for the time evolution of the aqueous concentrations. In this work, we examine the effects of advective and diffusive transport on mineral dissolution that involve multiple species in single pores and fractures, where flow can no longer be described completely by an average Darcy velocity—gradients in flow velocity inevitably exist within individual pores as a result of the fundamental physics. Such flow gradients, as well as the limited rates of diffusive transport, can present conditions where reactions are limited by flow and transport, which are very different from conditions in well-mixed batch or flow through reactors in laboratory settings. We aim to understand the mechanisms that contribute to the rate discrepancies between laboratory and field settings at the single pore and fracture scale and to identify the conditions under which such rate discrepancies become significant.
In this study, laboratory-measured reaction kinetics are applied at the scale of the mineral–water interface where they occur, and then coupled with a rigorous treatment of flow and transport. Although the model for a single pore is highly idealized in terms of its geometry, first-order effects in terms of the coupling of flow, transport, and reaction kinetics are captured. The numerical model used here for flow and transport is verified with an analytical solution. In addition, the coupled representation of flow, transport, and reaction by the model is validated with a microfluidic reactive flow experiment carried out at a spatial scale of hundreds to thousands of microns. We then compare rates within the single pores and fractures obtained from these fully coupled reactive transport simulations to rates obtained from a model that assumes complete mixing within the pore, while maintaining the same reactive surface area, average flow rate, geometry, and multicomponent chemistry.
For single fractures, the effect of transverse mixing is evaluated by comparing the rates from the fully coupled 2D reactive transport simulations to those calculated from a 1D plug flow reactor model that assumes complete transverse mixing within the fracture. The comparisons were made under various flow, pore size, and fracture length conditions to identify and quantify the conditions under which gradients in concentration (and thus, in rates) may develop.
Section snippets
Reactions and rate laws
In this work, we focus on the dissolution of three important subsurface mineral phases with a range of reaction rates: calcite, plagioclase, and iron hydroxide. Calcite dissolution is one of the most important and rapid mineral reactions in the subsurface (Morse and Arvidson, 2002), while plagioclase dissolution is one of the slowest, with its rate about 5–6 orders of magnitude less than that of calcite (Blum and Stillings, 1995, White and Brantley, 1995). Dissimilatory iron reduction is an
Model for a single pore
To analyze the scale dependence of mineral dissolution rates at the pore scale, we developed two models: (1) a Poiseuille Flow model that applies laboratory-measured dissolution kinetics at the pore or fracture wall and couples this to a rigorous treatment of both advective and diffusive transport, and (2) a Well-Mixed Reactor model that assumes complete mixing within the pore, while maintaining the same reactive surface area, average flow rate, geometry, and multicomponent chemistry as the
Verification of transport for a cylindrical pore
The numerical model implemented in CrunchFlow is verified against an analytical solution for Taylor dispersion (Taylor, 1953). Using Eq. (12) for the flow field within a cylindrical pore, Taylor derived an analytical expression for hydrodynamic dispersion (Taylor, 1953)where Dh is the dispersion coefficient, and D is the molecular diffusion coefficient. Note that the molecular diffusion coefficient appears both in the first term on the right hand side of Eq. (17), where it contributes
Development of concentration gradients at the pore scale
To illustrate the effects of coupling between flow, transport, and reaction on the scale dependence of reaction rates (and thus to discrepancies between lab and field rates that may be due to processes operating at the pore scale), we show a steady-state concentration field that develops within a calcite pore of 100 μm in length at a flow velocity of 0.1 cm/s. The pore is infiltrated by a dilute pH 5 solution with 0.01 mM dissolved NaCl, equilibrated with atmospheric CO2 gas. Under these slightly
Discussion
The results presented here demonstrate that a scale dependence to mineral dissolution rates, leading potentially to a discrepancy between laboratory and field rates, arises when large concentration gradients develop. Where such conditions arise, the situation is inherently different from the well mixed laboratory conditions where most mineral reaction rates are measured. This implies that it is only in transport regimes where homogeneous concentration fields form that laboratory-measured
Conclusions
In summary, we examined the effects of flow and transport processes on mineral dissolution kinetics in single pores and fractures. Laboratory-measured reaction kinetics were coupled with Poiseuille flow and advective and diffusive transport to unravel some of mechanisms that might contribute to a scale dependence of the dissolution rates, and thus to a discrepancy between laboratory and field rates. The simulations were intended to identify the conditions under which such a rate discrepancy
Acknowledgments
Funding was provided to the Center for Environmental Kinetics Analysis (CEKA) by the U.S. Department of Energy’s Environmental Remediation Science Program as part of a joint NSF-DOE Environmental Molecular Science Institute at Pennsylvania State University. Additional funding was provided by the Laboratory-Directed Research and Development program at Lawrence Berkeley National Laboratory. We are grateful for the review of an early version of the manuscript provided by Dr. Donald DePaolo. We
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