Evaluation of oscillatory integrals for analytical groundwater flow and mass transport models

Modeling of transient dynamics of an interface between fluids of identical density and viscosity, but different otherwise, is of great interest in aquifer hydraulic, and advective contaminant transport, and has broad application. Closed-form solutions are often available for problems with simple, practically important geometry, but the integrals that appear in such solutions often have integrands with two or more oscillatory factors. Such integrals pose difficulties for numerical evaluation because the positive and negative contributions of the integrand largely cancel and the integrands decay very slowly in the integration domain. Some problems with integrands with a single oscillatory factor were tackled in the past with an integration/summation/extrapolation (ISE) method: breaking the integrand at consecutive zeros to obtain an alternating series and then using the Shanks algorithm to accelerate convergence of the series. However, this technique is ineffective for problems with multiple oscillatory factors. We present a comprehensive strategy for evaluation of such integrals that includes a better ISE method, an interval truncation method, and long-time asymptotics; this strategy is applicable to a large class of integrals with either single or multiple oscillatory factors that arise in modeling of groundwater flow and transport. The effectiveness of this methodology is illustrated by examples of integrals used in well hydraulics, groundwater recharge design, and particle tracking.