A natural gradient experiment on solute transport in a sand aquifer: 2. Spatial moments and the advection and dispersion of nonreactive tracers

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Abstract
The three‐dimensional movement of a tracer plume containing bromide and chloride is investigated using the data base from a large‐scale natural gradient field experiment on groundwater solute transport. The analysis focuses on the zeroth‐, first‐, and second‐order spatial moments of the concentration distribution. These moments define integrated measures of the dissolved mass, mean solute velocity, and dispersion of the plume. Moments are estimated from the point observations using quadrature approximations tailored to the density of the sampling network. The estimators appear to be robust, with acceptable sampling variability. Estimates of the mass in solution for both bromide and chloride demonstrate that the tracers behaved conservatively, as expected. Analysis of the first‐order moment estimates indicates that the experimental tracer plumes traveled along identical trajectories. The horizontal trajectory is linear and aligned with the hydraulic gradient. The vertical trajectory is curvilinear, concave upward. The total vertical displacement is small, however, so that the vertical component of the mean solute velocity vector is negligible. The estimated mean solute velocity is identical for both tracers (0.091 m/day) and is spatially and temporally uniform for the first 647 days of travel time. After 647 days of transport, the plume apparently encountered a relatively large‐scale heterogeneity in the velocity field, leading to a distinct vertical layering, and slowing the rate of advance of the center of mass of the plume as a whole. The estimated horizontal components of the covariance tensor evolve over time in a manner consistent with the qualitative shape changes observed from plots of the concentration data. The major principal axis, initially aligned roughly perpendicular to the hydraulic gradient, rotates smoothly over time until it is nearly aligned with the mean solute velocity vector, as the plume itself elongates and orients its long axis with the direction of movement. Plots of the components of the covariance tensor as functions of time show evidence of what is commonly called “scale‐dependent” dispersion: the rate of growth of the covariance over time is not linear. The theoretical results of G. Dagan (1984) calibrate well to the estimated covariance data for the first 647 days of transport. The calibrated values of the parameters of the hydrualic conductivity distribution closely match independently measured values from the site. The asymptotic longitudinal dispersivity obtained from the calibration is 0.49 m, although asymptotic conditions were apparently not reached. The estimated covariance terms for the last sampling session, 1038 days after injection, are inconsistent with the earlier data and with the Dagan model, particularly for the transverse and off‐diagonal components. This behavior is probably attributable to the observed large‐scale heterogeneity in the velocity field.