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Stochastic analysis of transport and multicomponent competitive monovalent cation exchange in aquifers

¹ Universidad de La Coruña, Campus de Elviña s/n, 15192 La Coruña, Spain

Most stochastic analyses of reactive transport in physically and geochemically heterogeneous aquifers have focused on the analysis of a single reactive species. Here we conduct the stochastic analysis of multicomponent competitive monovalent cation exchange. Transport equations for dissolved cations are coupled with nonlinear cation exchange terms, which, for chemical equilibrium, are described by mass-action law expressions. These equations can be effectively decoupled by assuming that the weighted sum of cation concentrations is constant. The weight of each cation is equal to the reciprocal of its selectivity. Randomness of cation exchange capacity (CEC) leads to random retardation factors. Analytical expressions for effective retardation factors, longitudinal macrodispersivities, and concentration spatial moments are derived for a chemical system made of three monovalent cations (Na⁺, K⁺, and Cs⁺) using the stochastic analytical solution of Miralles-Wilhelm and Gelhar (1996). Our results indicate that effective retardation factor, R_C,i, spatial moments, and macrodispersivities of K⁺ are significantly different from those of Na⁺. Effective retardation factors asymptotically attain their mean values after a transient phase of cation-dependent duration. They strongly depend on the correlation between log-permeability (log K) and CEC. Pre-asymptotic effective retardation factor values for a negative correlation are smaller than the mean value, regardless of the value of the coefficient of variation of CEC (CV_CEC). The smaller (larger) the variance of log K, , the greater (smaller) the effective retardation factor for a negative (positive) correlation. Cation macrodispersivities for a positive correlation structure are smaller than that of a nonreactive species and increase with increasing . On the other hand, for a negative correlation structure they are larger than that of a nonreactive species. Macrodispersivity of Na⁺ is smaller (larger) than that of K⁺ for a negative (positive) correlation structure. The macrodispersivity of K⁺ increases with k₁ and decreases with the Na⁺/Cs⁺ selectivity, k₂, for a positive correlation. The first-order spatial moment of Na⁺ is greater than that of K⁺, but is smaller than that of a nonreactive species at the asymptotic phase. Second-order spatial moments of cations are smaller than that of a nonreactive species for a positive correlation structure, but are larger for a negative correlation.

Keywords: stochastic analysis • cation exchange • physical • geochemical heterogeneities • spatial moment • macrodispersivity • effective retardation factor