Temporal Moment‐Generating Equations: Modeling Transport and Mass Transfer in Heterogeneous Aquifers

Abstract

We present an efficient method for determining temporal moments of concentration for a solute subject to first‐order and diffusive mass transfer in steady velocity fields. The differential equations for the moments of all orders have the same form as the steady state nonreactive transport equation. Thus temporal moments can be calculated by a solute transport code that was written to simulate nonreactive steady state transport, even though the actual transport system is reactive and transient. Higher‐order moments are found recursively from lower‐order moments. For many cases a small number of moments sufficiently describe the movement of a solute plume. The first four moments describe the accumulated mass, mean, spread, and skewness of the concentration histories at all locations. Actual concentration histories at any location can be approximated from the moments by applying the principle of maximum entropy, a constraint consistent with the physical process of dispersion. The forms of the moment‐generating equations for different mass transfer models provide insight into reactive transport through heterogeneous aquifers. For the mass transfer models we considered, the zeroth moment in a heterogeneous aquifer is independent of the mass transfer coefficients. Thus, if the velocity field is known, the mass transported past any point, or out any boundary, can be calculated without knowledge of the spatial pattern of mass transfer coefficients and, in fact, without knowledge of whether mass transfer is occurring. Also, for both first‐order and diffusive mass transfer models, the mean arrival time depends on the distribution coefficient but is independent of the values of the rate coefficients, regardless of the spatial variability of groundwater velocity and mass transfer coefficients.