Upscaling solute transport in porous media in the presence of an irreversible bimolecular reaction

Abstract

We present a volume-averaging based theoretical analysis of upscaling of reactive transport processes involving fast bimolecular homogeneous irreversible reactions occurring within a porous medium. We start from the formulation driving the system dynamics at the pore scale and derive the governing equations at observation scales associated with laboratory-scale scenarios involving advection-dominated transport of two reactants and the resulting product in the presence of different relative strengths of Dahmköhler (Da) and Péclet (Pe) numbers, i.e., Da ≫ Pe and O(Da) ≈ O(Pe). We provide an original theoretical formulation describing the space–time propagation of advection-dominated conservative components. This formulation includes time dependent dispersive terms, consistently with previous experimental and numerical works. The (upscaled) system of reactive transport equations includes non-Fickian and time dependent dispersive terms which embed a direct link between pore scale dynamics and the chemistry of the problem. We then discuss the appropriateness of adopting effective reaction parameters of the type proposed in previous literature studies for the conditions analyzed. Our results provide a theoretical support to observations according to which (a) dispersion coefficients calibrated under conservative transport scenarios and (b) kinetic parameters measured in a batch reactor might not be appropriate to model the distribution of reactive species in the presence of fast homogeneous irreversible reactions without a proper scaling.

Introduction

Spatial distribution of solutes’ concentrations in porous media is driven by a variety of processes including, e.g., chemical reactions, molecular diffusion and transport by advection. Each of these processes acts at the pore level and its contribution to solutes’ mass conservation can be described in terms of quantitative models characterized by their own functional format and parameters, e.g., chemical reactions rates, diffusion coefficients and the velocity field. Even when an accurate knowledge of the solutes’ fate at the pore scale is possible, transport modeling taking into account all the system’s pore structure is still not affordable in laboratory and field scale applications. In this context, upscaled (continuum scale) models are usually formulated by averaging the equations governing the system at the pore level over characteristic observation scales. Amongst others, one of the typically adopted upscaled models describing reactive transport is the advection–dispersion–reaction equation (ADRE). This model is based on the assumptions that well mixed transport conditions occur at the pore scale and hydrodynamic dispersion can be described by a Fickian analogy. This formulation has been adopted to describe reactive transport processes (e.g., [1], [2] and references therein) and has been embedded in a variety of numerical codes designed to model multi-component reactive transport in porous media.

A number of works question the appropriateness of ADRE based continuum scale models to adequately include the key effects of pore scale processes and properly represent an upscaled transport depiction within the porous system. Experimental observations (e.g., [3], [4], [5], [6]), pore network or direct pore-scale numerical simulations of reactive transport in the presence of homogeneous and/or heterogeneous reactions within flow-through [5], [7], [8], [9] and purely diffusive systems [10], [11] suggest that microscale features which are not included in the standard ADRE model can significantly influence the evolution of reactive transport processes. Battiato et al. [11] and Battiato and Tartakovsky [12] consider transport of solutes in the presence of mixing controlled heterogeneous reactions and show theoretically that the adoption of the ADRE model is warranted under some sufficient conditions which insure that scale separation holds. Besides some geometrical length scale constraints involving the pore scale geometry, these conditions are directly linked to hydrodynamic and chemical transport mechanisms associated with the problem through the Péclet (Pe) and Dahmköhler numbers (Da).

Here, we focus on upscaling of reactive transport processes in the presence of fast irreversible homogeneous reactions. We consider the case where a bimolecular irreversible chemical reaction of the type A + B → C is induced in a porous system. Due to the homogeneous nature of the reaction, this setting allows to investigate mixing-driven reactions and has been considered in a series of recent literature works (e.g., [3], [4], [8], [9]), where experimental or numerical pore-scale models results are analyzed in terms of the standard homogenized ADRE. While the chemical problem is rather simple when compared against natural geo-chemical systems associated with field applications, it allows grasping the key points associated with the assumptions underlying homogenized (continuum-scale) ADRE-based formulations which are commonly applied.

Our theoretical derivations are motivated by recent modeling strategies based on continuum-scale models which have been proposed to interpret experimental observations of the kind presented in [3], [4]. In these experiments a laboratory scale column is initially filled with a background solution of a chemical species A. The chemical species B is then introduced in the column as a step input and C is formed in the mixing zone while A is displaced. These experiments are characterized by large values of Pe and Da. Gramling et al. [3] and Raje and Kapoor [4] compare their measurements against results obtained by classical (continuum scale) ADRE models. These modeling attempts were not conducive to correct estimates of the total produced amount of the reaction product C and the authors suggest that incomplete mixing of the reactants taking place in the pore space prevents the use of continuum scale formulations. This indicates that the evolution of the concentration field within the (disordered) porous system is strongly influenced by pore-scale features which are typically not included in the ADRE model, and highlights the multiscale nature of the processes observed.

Several modeling strategies have been presented to interpret these experiments. Edery et al. [13], [14] provide an interpretation of the experimental observations of [3] via a continuous time random walk particle tracking (CTRW PT) approach and adopting the concept of particles’ radius of interaction, the latter being a model calibration parameter. They show that particle tracking methods can deal with mixing problems involving homogeneous reactions, given a suitable statistical description of the particles’ transition and interaction. On the basis of the good agreement between their numerical results and the experimental published data the authors suggest that continuum models may not be suited to interpret the transport scenario analyzed in [3]. To model these experiments, Rubio et al. [15] and Sanchez-Vila et al. [16] propose the adoption of a continuum formulation which models the effect of incomplete mixing at the pore scale in terms of a time dependent effective kinetic reaction parameter, K_eff. While Sanchez-Vila et al. [16] propose modeling K_eff by a formulation which is conceptually equivalent to an upscaled two region mass transfer scheme, the model in [15] is inspired by the concept of reactant segregation intensity ([17], [18]). Rubio et al. [15] provide a comparison between their effective model and data presented in [3] and [4]. Sanchez-Vila et al. [16] show the robustness of their model with reference to a complete time dependent data set associated with one of the experiments in [3]. These authors note that, albeit the tails of the spatial distributions of concentrations are not well described by their continuum model, whereas they are properly reproduced by the CTRW PT approach, the agreement between the modeling results and the key experimental features observed is remarkable. The modeling strategies presented in [15], [16] are not supported by a theoretical upscaling analysis which formally includes the pore-scale description of the system dynamics. As such, Sanchez-Vila et al. [16] point out that the appropriateness of depicting an average system behavior in terms of a macro-scale (time-dependent) effective reaction rate is still an open question.

In this context, our key contribution is the theoretical derivation of the upscaled system of equations describing the above referenced reactive system at a continuum scale level for Pe ≫ 1 and Da ≫ 1, i.e., for an advection dominated transport setting in the presence of fast bi-molecular reaction. While it is shown that the sufficient conditions for the classical ADRE model to hold are not met in this case ([12]), the theoretical formulation of the upscaled system of equations is currently lacking. Our derivations allow exploring the main conceptual points of the continuum formulations adopted by Rubio et al. [15] and Sanchez-Vila et al. [16] to model the available experimental observations. Theoretical upscaling of the pore-scale governing equations is performed by following the classical volume averaging framework outlined in [19]. This methodology has been applied in a considerable number of studies focused on reactive transport in porous media (see [19] and references therein). The relative magnitudes of Pe and Da in the conditions we investigate require adopting some assumptions which affect the closure step.

The paper is organized as follows. Section 2 is devoted to the illustration of the problem formulation and the key equations describing mass balance at the pore scale. Volume averaging of the system is introduced in Section 3 and details of the closure of the volume averaged equations are discussed. Two different cases are distinguished according to the relative magnitude of Da with respect to Pe. The upscaled system is analyzed in Section 4, where terms representing scales interplay are highlighted and their role is discussed. The theoretical analysis is then discussed in light of published continuum-based interpretations of the experiments in [3], [4].