Geofluids

Volume 2018, Article ID 9260603, 8 pages

https://doi.org/10.1155/2018/9260603

## Upscaling Strategies of Porosity-Permeability Correlations in Reacting Environments from Pore-Scale Simulations

^{1}Laboratory for Waste Management LES, Nuclear Energy and Safety Department, Paul Scherrer Institute, 5232 Villigen, Switzerland^{2}Institute of Geological Sciences, University of Bern, 3012 Bern, Switzerland

Correspondence should be addressed to Nikolaos I. Prasianakis; hc.isp@sikanaisarp.soalokin

Received 2 February 2018; Accepted 29 April 2018; Published 19 June 2018

Academic Editor: Ilenia Battiato

Copyright © 2018 Nikolaos I. Prasianakis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In geochemically reacting environments, the mineral dissolution and precipitation alters the structural and transport properties of the media of interest. The chemical and structural heterogeneities of the porous media affect the temporal evolution of the permeability with respect to porosity. Such correlations follow a nonlinear trend, which is difficult to estimate a priori and without knowledge of the microstructure itself, especially under the presence of strong chemical gradients. Macroscopic field-scale codes require such an input, and in the absence of exact descriptions, simplified correlations are used. After highlighting the diversity of microstructural evolution paths, due to dissolution, we discuss possible upscaling strategies.

#### 1. Introduction

Precipitation and dissolution reactions in porous media dominate and control a large number of geochemical processes and industrial applications. The precipitation and dissolution of minerals from aqueous solutions alters the pore space and its connectivity. This has a strong effect on the mass convection and diffusion through the porous medium. When a mineral precipitates/dissolves at the reactive porous surface, the overall porosity decreases/increases, leading to a subsequent decrease/increase in permeability and effective diffusivity. At the same time, the connectivity of the pores can also change in a way to block or to facilitate the mass diffusion processes.

Reactive transport modelling at the field scale is usually based on a macroscopic finite element or a finite volume discretization scheme [1]. In such descriptions, the computational domain is divided into small elements/volumes, the so-called voxels. The voxels are typically several orders of magnitude larger than the typical pore diameter, and as a consequence, all chemical and transport properties within such volumes are homogenized and smoothed out. The pore space and its transport properties are therefore represented using macroscopic parameters, such as the porosity, the tortuosity, the diffusivity, and the permeability. In such a description, the small scale geometrical characteristics and the heterogeneities of the porous materials are neglected. Such an assumption allows making accurate numerical predictions in the case of relatively mild chemical gradients, as well as in the case where chemical reactions can be approximated by equilibrium values. However, when strong chemical gradients are present with simultaneous dissolution and precipitation reactions, the pure macroscopic reactive transport codes fail to make accurate predictions of the evolution of the system.

Dissolution and precipitation reactions change the pore space and the resulting material properties, in a nonlinear way, and therefore have a strong feedback in the transport properties of the porous medium. The lack of explicit pore structure description and of appropriate material-specific correlations is responsible, for example, for the numerical artefact, relevant to the dependency of the resulting clogging times on the spatial grid discretization [1–5]. An improvement to the numerical predictions can be achieved (a) by coupling pore-level solvers to the macroscopic ones in a multiscale simulator [6–8] or (b) by providing the necessary microscopic feedback in terms of appropriate correlations or tabulated values, which can be defined from pore-level simulations (upscaling of results). Pore-level methods allow the simulation of the advection-diffusion-reaction processes at the pore space, where surface charges and reactive surface areas can be explicitly considered. Representative structures can be generated via computer models or can be obtained via X-ray or other microtomography techniques. When combined with appropriate kinetic and thermodynamic solvers that act at the submicrometer level, it is possible to reproduce accurately the experimental observations.

Depending on the level of abstraction, different pore-level methodologies exist. The more detailed ones solve the relevant flow equations or some approximation depending on the flow regime and the flow physics that are involved, in realistic geometries. Such methods are the lattice Boltzmann method [9–13], the methods based on particle hydrodynamics [14, 15], as well as the standard finite volume methods when applied in complex geometries with moving boundaries [16–18]. Lattice Boltzmann models can resolve transport processes in realistic complex geometries, involving complex interactions between species and phases, but are more computational intensive compared to pore network models. A significant advantage of the lattice Boltzmann methodology is the minimum effort to discretize the realistic computational domain, as well as the efficient continuous solid structure update per time step. Such an example is the evolution of a system when simultaneous dissolution and precipitation processes are competing and drive the evolution of the system [11, 19]. Efficient parallelization though allows running simulations with many billion degrees of freedom in relatively small computer clusters, especially when GPGPUs are used [20, 21]. We note that the numerical extraction of microscopic properties using realistic geometries has the potential to provide useful input to the effective medium theories, which are used to upscale porous medium flows [22]

In this paper, we construct pore geometries with target porosity and initial permeability following a methodology similar to [23, 24]. The target permeability is selected in a way to represent limestone rock samples commonly found in hydrocarbon reservoirs or in geothermal fields. The implemented chemical reaction is representing the calcite dissolution under the presence of acid, a common process used in the field stimulation, in order to enhance the permeability of the formation. The evolution of selected geometries is examined using the lattice Boltzmann framework, and permeability-porosity correlations are numerically extracted. Upscaling strategies that allow passing information to the macroscopic solvers and therefore bridge pore level and macroscopic scales, are discussed based on the output of the simulations.

#### 2. Reactive Transport Modelling

##### 2.1. Pore-Level Modelling

For the simulation presented in the next sections, the lattice Boltzmann methodology is implemented. This method is a special discretization of the Boltzmann equation. The elementary variables are the so-called populations or velocity probability distribution functions . At every distribution function corresponds to a discrete velocity vector [25–27]. Different discretization schemes lead to different numbers of discrete velocities, which results in several lattice models [28, 29]. For two-dimensional simulations, the standard D2Q9 square lattice with 9 discrete velocities is usually implemented due to its simplicity and robustness in complex geometry domains (see Figure 1).