We consider a model for precipitation and dissolution in a porous medium, where ions transported by a fluid through the pores can precipitate at the pore walls and form mineral. Also, the mineral can dissolve and become part of the fluid as ions. These processes lead to changes in the flow domain, which are not known a priori but depend on the concentration of the ions dissolved in the fluid. Such a system can be formulated through conservation equations for mass, momentum, and solute in a domain that evolves in time. In this case the fluid and mineral phases are separated by a sharp interface, which also evolves. We consider an alternative approach by introducing a phase field variable, which has a smooth, diffuse transition of nonzero width between the fluid and mineral phases. The evolution of the phase field variable is determined through the Allen--Cahn equation. We show that as the width of the diffuse transition zone approaches zero, the sharp-interface formulation is recovered. When we consider a periodically perforated domain mimicking a porous medium, the phase field formulation is upscaled to Darcy scale by homogenization. Then, the average of the phase field variable represents the porosity. Through cell problems, the effective diffusion and permeability matrices are dependent on the phase field variable. We consider numerical examples to show the behavior of the phase field formulation. We show the effect of flow on the mineral dissolution, and we address the effect of the width of the diffuse interface in the cell problems for both a perforated porous medium and a thin strip.

We consider the incompressible flow of two immiscible fluids in the presence of a solid phase that undergoes changes in time due to precipitation and dissolution effects. Based on a seminal sharp interface model a phase-field approach is suggested that couples the Navier–Stokes equations and the solid’s ion concentration transport equation with the Cahn–Hilliard evolution for the phase fields. The model is shown to preserve the fundamental conservation constraints and to obey the second law of thermodynamics for a novel free energy formulation. An extended analysis for vanishing interfacial width reveals that in this limit the sharp interface model is recovered, including all relevant transmission conditions. Notably, the new phase-field model is able to realize Navier-slip conditions for solid–fluid interfaces in the limit.

Enzymatically induced calcite precipitation (EICP) is an engineering technology that allows for targeted reduction of porosity in a porous medium by precipitation of calcium carbonates. This might be employed for reducing permeability in order to seal flow paths or for soil stabilization. This study investigates the growth of calcium-carbonate crystals in a micro-fluidic EICP setup and relies on experimental results of precipitation observed over time and under flow-through conditions in a setup of four pore bodies connected by pore throats. A phase-field approach to model the growth of crystal aggregates is presented, and the corresponding simulation results are compared to the available experimental observations. We discuss the model’s capability to reproduce the direction and volume of crystal growth. The mechanisms that dominate crystal growth are complex depending on the local flow field as well as on concentrations of solutes. We have good agreement between experimental data and model results. In particular, we observe that crystal aggregates prefer to grow in upstream flow direction and toward the center of the flow channels, where the volume growth rate is also higher due to better supply.

No abstract

We consider a phase-field model for the incompressible flow of two immiscible fluids. This model extends widespread models for two fluid phases by including a third, solid phase, which can evolve due to e.g. precipitation and dissolution. We consider a simple, two-dimensional geometry of a thin strip, which can still be seen as the representation of a single pore throat in a porous medium. Under moderate assumptions on the Péclet number and the capillary number, we investigate the limit case when the ratio between the width and the length of the strip goes to zero. In this way, and employing transversal averaging, we derive an upscaled model. The result is a multi-scale model consisting of the upscaled equations for the total flux and the ion transport, while the phase-field equation has to be solved in cell problems at the pore scale to determine the position of interfaces. We also investigate the sharp-interface limit of the multi-scale model, in which the phase-field parameter approaches 0. The resulting sharp-interface model consists only of Darcy-scale equations, as the cell problems can be solved explicitly. Notably, we find asymptotic consistency, that is, the upscaling process and the sharp-interface limit commute. We use numerical results to investigate the validity of the upscaling when discontinuities are formed in the upscaled model.