Utilised subsurface systems are often affected by continuous changes in rock properties due to water-rock-interaction. There are applications, where mineral precipitation or dissolution induced rock alterations are intended, e.g., in reactive barriers for nuclear waste repositories . In other cases, they are an unwanted side effect, for example, barite scalings in geothermal systems or during oil and gas extraction, where they can induce a massive loss of injectivity or productivity . A comprehensive understanding of the reactive processes taking place is crucial, so they can be incorporated into prediction models that anticipate and quantify the behaviour of the system, paving the way for a successful utilisation. As opposed to commonly applied empirical formulations for describing rock property alterations, process-based models are more robust and flexible. In order to develop reactive transport models that are applicable to a broad range of boundary conditions and scenarios, it is necessary to identify, parametrise and calibrate the relevant processes with the aid of laboratory experiments.

A recent experimental study investigated the role of barite supersaturation on its precipitation mechanisms caused by concurrent celestite dissolution . To this aim, quasi one-dimensional flow-through column experiments were conducted, providing insights into pore-scale evolution during mineral exchange reactions. Three different orders of magnitudes of barite supersaturation were applied, where each caused different precipitation patterns. The authors identified barite nucleation as a key process that becomes increasingly relevant at higher supersaturations. Nuclei formation increases exponentially with supersaturation, and in turn creates reactive surface area for consecutive crystal growth . Thus, at high input concentrations, a passivation effect occurred due to complete or partial coverage of the celestite grains, preventing any further dissolution. At low input concentrations, nucleation played a lesser role, enabling the replacement reaction to take place. The authors tested the validity of conceptual models to describe precipitation induced reactive surface area development together with celestite dissolution kinetics and barite equilibrium reactions. They concluded that a single empirical relationship is insufficient, but rather two or more are needed to represent the observed responses at all input concentrations. However, it remains open which saturation threshold is to be used for switching instances, and how transition ranges should be treated.

In this study, we provide a comprehensive geochemical modelling approach to match the reported experimental responses. A digital rock representation of the granular celestite sample was applied. The derived rock properties were then used as initial conditions for one-dimensional reactive transport simulations. Next to bulk dissolution and precipitation kinetics, process-based heterogeneous nucleation applying classical nucleation theory and geometrical crystal growth were considered in the coupling. The modelled mineral phase volume fractions in the column and effluent chemistry were compared to the experimental results.

For matching the results of the reactive transport simulations with the experimental data, only the nucleation process was calibrated manually. By adjusting θ and Γ0 to 32∘ and 7.0×10-181/s, these were found to be the best matching values to reproduce the experimental data with respect to effluent chemistry and mineral substance amount in the column. The results of the simulations using these parameters are presented in the following.

Continuum scale reactive transport simulations were applied to match the experimental results. Barite precipitation likewise caused the reactive surface area of barite to increase and that of celestite to decrease, up to five orders of magnitude. These large variations justify to take dissolution kinetics of celestite and precipitation kinetics of barite into account . The precipitation mechanism of barite was identified to consist of two steps, heterogeneous nucleation on celestite substrate and subsequent growth of these nuclei to become larger crystals. Nucleation was treated deterministically with the classical theory . Crystal growth was implemented as the averaged geometrical growth of nuclei bodies, where the volume increase was taken from bulk precipitation rate.

The overgrowth of celestite with barite crystals had a passivation effect at high and medium input concentrations (Bain2+=100mM and Bain2+=10mM). This happened when a high enough number of nuclei formed during the initial surge of nucleation. The subsequent crystal growth covered all the celestite surface and prevented any further dissolution. At low supersaturations (i.e., for Bain2+=1mM), the passivation effect was not observed, since significantly fewer nuclei formed in the beginning. Thus, fewer barite crystals grew to larger sizes compared to the experiments with 100mM and 10mM input concentration, covering the celestite surface only in parts. Therefore, a complete mineral replacement took place.

The modelled distribution patterns of barite crystals match well with the SEM images of the laboratory experiments for all input concentrations . The experiment with high input concentration showed celestite grains overgrown uniformly with a thin barite rim (∼3 µm). The other two experiments showed distinct zonation patterns across the column with mineral phase substitution of different degrees. The medium input concentration mode exhibited a transition zone in the center with thicker rims (∼5 µm) and generally decreasing barite content on either end of the column. At low BaCl2 input concentration, a sharp reaction front at the upstream was observed, where the average thickness of overgrowth was about ∼7 µm. Simulated crystal sizes are slightly larger, corresponding to final rim thicknesses of 4, 8, and 12 µm for experiments with high, medium, and low input concentration, respectively.

Nucleation was parametrised assuming spherical cap shaped nuclei and a respective interfacial tension from the literature . Two parameters were fitted to match the laboratory experiments: Γ0 and θ. Γ0 is part of the pre-exponential factor Γ of the nucleation rate (Eq. ), which quantifies the diffusive attachment rate of monomers from solution to sub-critical clusters. Compared to the exponential term, where parameter uncertainties are much more significant, approximating the order of magnitude of Γ is usually sufficient. However, many of the parameters for calculating Γ are challenging to quantify. It is uncertain, how many monomers in the pore fluid actually play a role in the nucleation process, or if only monomers in the diffusive layer surrounding the substrate should be considered. Furthermore, the available nucleation sites can only be judged from the total substrate surface area and the approximate size of a nuclei. Calibrating Γ0 accounts for these uncertainties in the considered system. The contact angle θ of the nuclei and the substrate depends on the structural similarity between the substances. At 180∘ the contact is practically only one point, at 0∘ the “wets” the substrate. The fitted value of 32∘ accounts for the similarity between barite and celestite, both crystallise in the orthorhombic system. It also compares well to the value of 30∘ used by  for a similar system.

Modelling all three experiments with empirical relationships required at least two different models to account for the reactive surface area evolution . However, for the modeller it remains impossible to know, which empirical relationship to use a-priori. Furthermore, they seem insufficient to be used for the transitional case (Bain2+=10mM). In this study, the identified chemistry-based processes are taken into consideration explicitly in coupled models. The resulting transient reactive surface areas are used in both kinetic rates for barite and celestite, compared to only celestite kinetics and barite equilibrium reactions . After calibration of the here provided models, the effluent and column chemistry of laboratory experiments at medium (10mM) and low (1mM) barium input concentrations could be reproduced almost exactly, and at high (100mM) input concentration the match was good with slight deviations. The main benefit is that no knowledge of the supersaturation in the system has to be known in advance, which also solves the transitional case well (medium input concentration).

Calibration of the presented models may be improved by further refining the grid size and increasing the iteration steps of nucleation and crystal growth between transport steps, thus coupling them more tightly together with the kinetics solver. However, model run times on a regular desktop working machine (2.3 GHz Quad-Core Intel Core i5) were in the range (12–20) h for one experiment run on a single CPU with 30 grid elements. Increasing the grid size would make manual calibration unfeasible due to too long model run times. In future work, this could be solved by using approaches for chemistry speed-ups in reactive transport simulations . Furthermore, a more detailed crystal size distribution map using digital pore-scale models instead of mean values in each cell may improve determination of transient reactive surface areas. However, nucleation happens predominantly in the beginning, thus the comparably low amount of new nuclei later in the experiment do not change the mean crystal size of each cell significantly. The assumption of tracking only one mean size per cell appears sufficient as the models can describe the investigated system qualitatively well and moreover the data basis does not cover this in enough detail.

A geochemical modelling approach was presented to simulate barite formation in a celestite grain packed column. Celestite dissolution and barite precipitation kinetics, as well as barite nucleation and barite crystal growth were included explicitly as processes in the model coupling. After calibration of the nucleation process, of the three different precipitation patterns observed in the experiments, two were reproduced almost exactly and one was matched qualitatively well by only varying the input concentration. Compared to previous modelling approaches using various empirical relationships to take reactive surface area evolution into account, the provided models can be applied to systems with a broad range of input concentrations without a-priori knowledge of the prevailing barite supersaturations. This can be of great benefit for modelling the evolution of subsurface systems due to barite formation, where only the prevalent solute concentrations are known. This is foremost important in geothermal reservoirs or in reactive barriers near nuclear waste repositories, where it is crucial to predict the response of the system in advance, so it can be incorporated into the project design. In future work, it is planned to couple reactive transport and digital pore-scale models more tightly together. The aim is to track pore-scale alterations in detail and exploit the capabilities of digital rock physics for deriving rock properties: evolution of reactive surface areas and feedback of resulting geometrical and porosity changes on permeability evolution. Furthermore, the use of surrogate models to speed-up geochemical calculations will be a valuable improvement in the future, making higher grid discretisation and inverse modelling feasible for more accurate parameter determination.