Coupled Adsorption/Precipitation Modelling of Phosphonate Scale Inhibitors in a Batch Reactive System

Scale inhibitor squeeze treatments are one of the most common ways to prevent scale deposition. The mineral scale will be inhibited if the concentration of the scale inhibitor (SI) in the produced water is above a certain threshold, known as the Minimum Inhibitor Concentration (MIC), which is controlled by scale inhibitor retention. Therefore, accurate modelling of the SI retention through adsorption (Γ) and precipitation (Π) is critical to the successful design and implementation of squeeze treatments. In this study, an equilibrium model has been developed to simulate the coupled adsorption-precipitation (Γ/Π) of phosphonate scale inhibitors in reactive formations, such as carbonates, in the presence of calcium and magnesium cations. In this approach, the scale inhibitor (SI) was considered as a poly weak acid that may be protonated (H_nA), resulting in the complexation with Ca/Mg ions, leading to the precipitation of SI_Ca/Mg complexes. All these reactions occur in an integrated system where carbonate system reactions and adsorption of the soluble species are occurring in parallel. In the adsorption process, all the SI derivatives remaining in the solution, including free and complex species, are considered to participate in the adsorption process, described by an an adsorption isotherm model (e.g., Freundlich). For the precipitation part, the model considers the following reactions: (i) the carbonate system, (ii) SI speciation, considered as weak polyacid, H_nA, (iii) the SI-metal (Ca and Mg) binding complexes, and (iv) subsequent precipitation of the SI-Ca/Mg complex. The system charge balance and the mass balances for calcium, magnesium, carbon, and SI are considered, to numerically equilibrate the system (excluding the adsorbed species), by solving a determined set of non-linear equations numerically. Following the algebraic reduction of the equations, the system is reduced to three non-linear equations that may be solved by the Newton-Raphson method. The precipitation of the SI-Ca/Mg is modelled in the equilibrium model based on the solubility of SI in the solution, determined from the lab experiments. The reliability of the proposed model was established by comparison with experimental results from a previous study (Kalantari Meybodi et al., 2023) on the interactions of DETPMP in a Calcite/brine (containing free Ca/Mg) system, where the final concentration of SI, Ca²⁺, Mg²⁺, CO₂ and pH were compared. The modelling showed good general agreement with the experimental results, and a further sensitivity analysis was performed to examine the behaviour of some uncertain parameters, such as the stability constant of complexes.