Geochemical mole-balance modeling with uncertain data

Geochemical mole-balance models are sets of chemical reactions that quantitatively account for changes in the chemical and isotopic composition of water along a flow path. A revised mole-balance formulation that includes an uncertainty term for each chemical and isotopic datum is derived. The revised formulation is comprised of mole-balance equations for each element or element redox state, alkalinity, electrons, solvent water, and each isotope; a charge-balance equation and an equation that relates the uncertainty terms for pH, alkalinity, and total dissolved inorganic carbon for each aqueous solution; inequality constraints on the size of the uncertainty terms; and inequality constraints on the sign of the mole transfer of reactants. The equations and inequality constraints are solved by a modification of the simplex algorithm combined with an exhaustive search for unique combinations of aqueous solutions and reactants for which the equations and inequality constraints can be solved and the uncertainty terms minimized. Additional algorithms find only the simplest mole-balance models and determine the ranges of mixing fractions for each solution and mole transfers for each reactant that are consistent with specified limits on the uncertainty terms. The revised formulation produces simpler and more robust mole-balance models and allows the significance of mixing fractions and mole transfers to be evaluated. In an example from the central Oklahoma aquifer, inclusion of up to 5% uncertainty in the chemical data can reduce the number of reactants in mole-balance models from seven or more to as few as three, these being cation exchange, dolomite dissolution, and silica precipitation. In another example from the Madison aquifer, inclusion of the charge-balance constraint requires significant increases in the mole transfers of calcite, dolomite, and organic matter, which reduce the estimated maximum carbon 14 age of the sample by about 10,000 years, from 22,700 years to 12,600 years.