On the pH-dependent export of anthropogenic alkalinity in pore water through soil: Implications for enhanced rock weathering

Enhanced rock weathering (ERW) is a highly scalable carbon dioxide removal (CDR) strategy. In ERW deployments, pulverized mineral feedstock is spread on agricultural or managed lands. Upon the dissolution or weathering of the feedstock while exposed to the elements, cations are released, altering the charge balance of the pore water in the soil. The introduction of the divalent cations calcium (Ca²⁺) and magnesium (Mg²⁺) catalyzes the conversion of CO₂ to dissolved inorganic carbon (DIC), principally bicarbonate (HCO₃⁻). The efficacy of ERW hinges on the maintenance and export of HCO₃⁻ through the soil. To explore the mechanistic underpinnings of ERW, we develop a theoretical framework for the pH-dependent adsorption and desorption of cation solutes onto charged mineral surfaces in a porous medium. We develop a theory for the solution of the Riemann problem for a one-dimensional, quasi-linear 2x2 system of conservation laws for the transport of Ca²⁺ and Mg²⁺. The system is strictly hyperbolic, with the full pH-dependent system being non-genuinely nonlinear due to the non-convex, competitive Langmuir-type isotherms governing adsorption of Ca²⁺ and Mg²⁺. We present the structure of the eight fundamental solutions to the Riemann problem which are comprised of elementary and compound solute waves. Where alkaline solutions infiltrate acidic solutions in the pore water, compound reaction waves may arise. Such waves are always composed of fast-moving solute fronts and a slower-moving shock where Ca²⁺ and Mg²⁺ compete with hydrogen ion (or proton, H⁺) for adsorption onto charged surface sites. The solutions to the Riemann problem presented here help interpret measurements from column experiments run to examine the efficiency of ERW and the general transport alkalinity in soils.