Effect of Heterogeneity on Advection-Reaction Problem in Porous Media Using a Stochastic Eulerian Flow Model

Abstract Flow and transport in a porous medium coupled with a reaction is influenced by the medium heterogeneity. The effective macroscopic equations for flow and reaction must incorporate such information. In this paper, we consider upscaling of the advection-reaction problem using spectral theory. The approach follows original work by Gelhar and Axness [2]. The reaction occurs between an injected chemical and a stationary mineral residing in the pore space. A one-step nonlinear bimolecular dissolution reaction takes place between the two chemical species. The heterogeneity is in flow permeability field, which is random and correlated in space. Initial distribution of the mineral surface area is considered to be linearly correlated with the permeability field. A stochastic analysis of flow, transport and reaction is performed using firstorder perturbations. The reaction front propagation in the heterogeneous medium is then analyzed and field-scale expressions are obtained for the coefficients in reaction rate, effective flow velocity and longitudinal macrodispersivity. The heterogeneity is shown to influence flow, macrodispersion and reaction in porous medium. Furthermore, its presence develops inter-dependency between the three existing mechanisms. Introduction Behavior of fluids is influenced by the porous medium heterogeneities. While an attempt is being made to develop a quantitative description of flow in porous media in a scale much larger than an average pore size, dealing with the realm of heterogeneity becomes a fundamental and challenging problem. Finding appropriate average parameters which can be applied to flow, transport and reaction in the scale of interest, and at the same time being able to incorporate the influence of intrinsic heterogeneity on the modeling and predictions, is desired during an investigation. Rapid developments in theoretical research of fluid flow in porous media in a probabilistic framework have been experienced during the last two decades. This progress was motivated by the fact that (1) the medium is always heterogeneous, i.e., it has a large degree of variability in grain size, and shows a complex spatial structure at larger scales; (2) lack of knowledge of the detailed local structure of these spatial variations and (3) difficulties in obtaining sufficient data related to spatial and temporal distributions of mass and momentum variables, dictated by these large-scale variations. When dealing with subsurface processes such as movement of ground water, transport of its contaminant, recovery of crude oil and natural gas, the heterogeneities are more pronounced due to extremely complex diagenesis processes of the sediments [1]. Heterogeneity of a porous medium could be represented in terms of random quantities which characterize its pore structure, e.g., its mineral concentrations and internal surface area, permeability and porosity etc. In practice, spatial variations in permeability are often used to describe the medium heterogeneity. These variations subsequently influence flow, transport and reaction variables. Therefore, the latter are also random quantities, more precisely random fields with spatial and temporal arguments. Then, these random fields appear in the stochastic partial differential equations describing the phenomenon of interest. The results obtained by a variety of tools are then represented in terms of their statistical moments.

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