Scale dependence of reaction rates in porous media

doi:10.1016/j.advwatres.2005.05.007

Advances in Water Resources

Volume 29, Issue 1, January 2006, Pages 62-71

https://doi.org/10.1016/j.advwatres.2005.05.007 Get rights and content

Abstract

Elemental turnover in porous media depends on substrate concentrations at the pore-scale. In this study, the effect of small scale variability in concentration fields on reaction rate estimates and the validity of the continuum approximation in reactive transport models are investigated via a pore-scale numerical model. Artificial porous media are generated using an identical overlapping sphere algorithm. By comparison between explicit pore-scale simulations and macroscopic continuum approximations, it is shown that inhomogeneous solute distribution within the pores can affect estimates of elemental turnover rates. The error associated with large scale rate estimates depends on the type of reaction, pore geometry, reaction kinetics and macroscopic concentration gradient. A correction term that involves a phenomenological parameter which can be evaluated numerically and macroscopic concentration gradients is introduced to improve the accuracy of upscaled homogeneous reaction rates. Implications for macroscopic descriptions of surface processes and surface attached microbial populations are discussed and it is shown that pore-scale heterogeneity can substantially affect estimates of heterogeneous reactions, while for homogeneous reactions, the error amounts to only a couple of percents.

Introduction

In natural porous media, such as sediments or aquifers, complex reaction networks couple biogeochemical cycling of many elements. Prediction of the response to changes in environmental conditions requires an understanding of how individual processes contribute to the observed distribution patterns of chemical constituents. Reactive transport modeling permits a quantitative analysis of the interaction between transport and reaction processes and provides a framework to integrate insight gained both through laboratory experiments and field observations. A major obstacle to achieving prognostic power with simulations lies in the challenge to incorporate the effect of spatial heterogeneity on the system behavior. For example, at the field scale, modeled mineral weathering rates based on laboratory rates overpredicted actual field rates by 1–3 orders of magnitude [63], [57], [34]. Apart from differences of physico-chemical conditions in the laboratory compared to the field (e.g., temperature, pH), this has been attributed to a large extent to hydrological controls such as preferential flow patterns which reduce the mineral surfaces in contact with the percolating fluid.

At the pore-scale, limited exchange between macro- and micro-pores can lead to differences in the chemical environment. Modeling studies have shown the distribution of chemicals at the pore level to be important under advective flows (e.g., [13], [30], [50], [58]). For fast bimolecular kinetics the effect of pore-scale reactant distribution on estimates of reaction rates has been studied in advection dominated laboratory systems. A distinct impact of reactant segregation, with roughly 20–50% difference between measured rates and those calculated based on average concentrations was observed [21], [45]. In diffusion dominated settings, a theoretical analysis suggested only minor effect for reactions taking place in the pore fluid [39]. However, sub-millimeter scale heterogeneity has been observed for O₂ and trace metals in sediments and microbial mats (e.g., [20], [16]; see also [12]). This suggests that in these settings, diffusion may be sufficiently slow such that the distribution of chemicals at the pore level can be heterogeneous.

Reactive transport models commonly employ a continuum description and rely on volume averages. In porous media, averages are taken over scales larger than typical grain sizes (e.g., [3], [44], [69]). Hence, spatial heterogeneity in concentration fields below the scale of volume averaging is not resolved explicitly. The effect of features which are not accounted for explicitly is typically represented by phenomenological parameters and a closure model [5], [4]. Modeling pore-scale processes and subsequent upscaling to a macroscopic scale allows one to rigorously calculate macroscopic properties of complex porous media in terms of the statistical properties of the solution obtained at the pore-scale. To date, the majority of pore-scale modeling studies have focused on the basic transport properties of porous media including effective diffusion, conductivity, permeability and elasticity (among many, one may note [1], [29], [37], [49], [35], [40]).

In this communication, we investigate the effect of small-scale heterogeneity of reactants on estimates of reaction rates. We focus on settings where reactants are transported via diffusion (i.e., advective flow is negligible) which may include muddy sediments or soil aggregates (e.g., [24], [62]). We use high resolution computational models that explicitly resolve small-scale heterogeneity in porous media and upscale the pore-scale simulation results. To that purpose, first an artificial porous medium is generated. Then, numerical simulations of concentration fields at the pore-scale are performed and effective diffusion coefficients are evaluated. We calculate the volume averaged reaction rates using the concentration field obtained with the pore-scale simulations and compare them with reaction rates based on average concentrations. We discuss the consequences for biogeochemically relevant reaction kinetics and introduce a macroscopic rate correction term for homogeneous reactions.

Section snippets

Porous media

Pore connectivity is a key parameter for solute transport and the geometrical arrangement of pores and solid entities is a central issue in the computational study of small-scale processes. Several methods have been applied in the computational reconstruction of porous media, including pore networks consisting of pores and connecting throats, random networks where connectivity is based on statistical image analysis image analyses, or sedimentation reconstruction methods (e.g., [7], [9], [68],

Transport properties

From a macroscopic perspective, the distribution of a non-reactive tracer in a diffusive regime at steady state is governed by $\overset{⇀}{\nabla} \cdot ({\underset{̲}{\underset{̲}{D}}}^{eff} \overset{⇀}{\nabla} \bar{C}) = 0$ , where ${\underset{̲}{\underset{̲}{D}}}^{eff}$ is the diffusion tensor (see [23], [67] for the existence and uniqueness of the effective diffusion tensor in porous media). In an isotropic medium, it reduces to an effective diffusion coefficient, D^∗, chosen such that the volume average of the local flux is equal to the macroscopic flux. As the simulation domain is oriented such that

Reaction rates

In the case of reactive substances, the macroscopic governing equation at steady state takes the form $\overset{⇀}{\nabla} \cdot (D^{*} \overset{⇀}{\nabla} \bar{C}) - R^{*} = 0,$ where R^∗ indicates the larger scale reaction rate. R^∗ depends on volume-averaged state variables (ignoring small-scale heterogeneity in the reactant concentrations) and is chosen such that the left hand side of Eq. (3) is equivalent to the spatial average of the pore-scale expression, $〈 \overset{⇀}{\nabla} \cdot (D \overset{⇀}{\nabla} C) - R (C) 〉$ . Note that the use of Eq. (3) implies that D^∗/D determined from a numerical

Conclusions

In this study, we have investigated the upscaling of non-linear reaction rates by solving the reaction–diffusion equation at the pore-scale, employing the overlapping sphere algorithm for the generation of porous media. At reaction fronts exhibiting high concentration gradients, we observe a difference between the macroscopic rate estimate and the volume averaged reaction rate. Our numerical results show that the preservation of the pore-scale reaction function and the evaluation of the

Acknowledgement

We thank three anonymous reviewers whose comments helped improve the manuscript. This study was supported in part by the Shared University Research Grants from IBM Inc., to Indiana University, the IBM Life Sciences Institutes of Innovations and the Office of Science, the United States Department of Energy.

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Scale dependence of reaction rates in porous media

Abstract

Introduction

Section snippets

Porous media

Transport properties

Reaction rates

Conclusions

Acknowledgement

Adv Water Resour

FEMS Microbiol Rev

Current Opin Colloid Interface Sci

Mar Chem

Earth Planet Sci Lett

Geochim Cosmochim Acta

Chem Eng Sci

Geochim Cosmochim Acta

J Colloid Int Sci

Adv Water Resour

Geochim Cosmochim Acta

Chem Geol

J Hydrol

Geoderma

Chem Geol

Geochim Cosmochim Acta

Porous media: fluid transport and pore structure

Iterative solution methods

Introduction to modeling phenomena of transport in porous media

Measurement scale, network sampling scale, and groundwater model parameters

Water Resour Res

The universal structure of the groundwater flow equations

Water Resour Res

Hydraulic radius and transport in reconstructed model three-dimensional porous media

Transport Porous Media

Percolation theory and network modeling. Applications in soil physics

Surveys Geophys

Measurement of spatial correlation functions using image processing techniques

J Appl Phys

A theoretical investigation of the organic carbon-microbial biomass relation in muddy sediments

Aquat Microbial Ecol

Diagenetic models and their implementation

The role of Mg2+ as an impurity in calcite growth

Science

Dissolved metals in surface sediment and a microbial mat at 100-μM resolution

Nature

Surface control vs. diffusion control during calcite dissolution: dependence of step-edge velocity upon solution pH

Amer Mineral

Ferrous oxidation chemistry in passive abiotic systems for the treatment of mine drainage

Geochem: Explor Environ Analy

A new variational approach to the diffusion and the flow problem in porous media

J Phys Soc Jpn

Heterogeneity of oxygen production and consumption in a photosynthetic microbial mat as studied by planar optodes

J Phycol

Reactive transport in porous media: a comparison of model prediction with laboratory visualization

Environ Sci Technol

Homogenization and porous media

The role of Mg²⁺ as an impurity in calcite growth