An upscaled rate law for mineral dissolution in heterogeneous media: The role of time and length scales

Abstract

An upscaled rate law is developed for mineral dissolution in heterogeneous media under variable residence time (flow velocity) and length scale conditions, expanding the previous rate law under one flow and length condition (Wen and Li, 2017). A total of 640 Monte-Carlo numerical experiments were carried out with magnesite dissolution within quartz matrix in spatially heterogeneous media characterized by permeability variance (σ2lnκ = 6.0) and correlation length (λ = 2–50 cm) under a range of domain length (1.0–5.0 m) and flow velocity (2.7 × 10−4 to 27.0 m/day) conditions. The upscaled rate law Rht=kAT1-exp-τeqτa1-exp-Lτaτad,rα consists of two parts. The first is the rate law in equivalent homogeneous media Rhm=kAT1-exp-τeqτa, where rates depend on rate constants (k), total mineral surface area (AT), and the relative timescales of reactions (τeq) and advection (τa)Da=τaτeq. The second term χ=1-exp-Lτaτad,rα is the heterogeneity factor that quantifies the deviation of heterogeneous systems from their homogeneous counterparts. The term includes the relative transport time ratio τaτad,r, domain length (L), and geostatistical characteristics of spatial heterogeneity (permeability variance in α=5σlnκ2. The ratio τaτad,r quantifies the relative timescales of transport at the domain scale (τa approximates τad) versus transport in the reactive zones (τad,r). Under low flow velocity and/or long domain length conditions where τa >τad,r, homogenization occurs so that χ is close to 1 and heterogeneity effects are negligible. In contrast, χ deviates from 1 by up to 2 orders of magnitude under short length and fast flow conditions where τa <τad,r and Lτaτad,r is small. Under such conditions, transport rates in and out of the reactive zones limits the overall dissolution at the domain scale. The rate law predicts that 1) heterogeneity effects are governed by the relative magnitude of mineral-water contact in reactive zones versus in the whole domain; 2) dissolution rates in heterogeneous media ultimately approach asymptotic values in homogeneous media at “sufficiently” long lengths. It takes longer distance (and time) for more heterogeneous systems with longer τad,r values to reach the asymptotic values. This is supported by carbonate dissolution rates calculated from river water chemistry data. Although the rate law is derived based on magnesite dissolution, it is expected to be applicable for the dissolution of other minerals, as long as the dissolution characteristics of other minerals (e.g., k, AT, and Keq) are incorporated in the rate law.