Concentration variance and spatial covariance in second-order stationary heterogeneous conductivity fields

The concentration variance and spatial covariance resulting from convective-dispersive transport driven by a uniform mean flow in second-order stationary conductivity fields was derived in a Lagrangian framework based on a first-order approximation of solute particle trajectories. The approximation of the concentration (co)variances for large injection volumes is considerably simplified by defining the concentration at a certain location x and time t in terms of the "backward" solute trajectory probability distribution. This is the probability that the trajectory of a fictitious microscopic and indivisible solute particle, which is in a volume Deltax centered around x at time t(0) was at the time of solute injection, t(0), in the injection volume, V-0. The approximated concentration (co)variances were validated against concentration covariances derived from transport simulations in generated second-order stationary conductivity fields. The approximate solutions reproduced fairly well the effects of local scale dispersion and of the spatial variability of the hydraulic conductivity on the concentration (co)variance. The effect of the spatial structure of the hydraulic conductivity field on the spatial covariance of the concentrations was investigated in order to identify parameters that can be unequivocally determined from the structure of the concentration field. For a given spreading of the solute plume in the mean flow direction, X-11(t), a given spatial correlation length of the log, transformed hydraulic conductivity in the transverse to flow direction, I-f2, and a given local scale dispersion D-d the concentration covariance was nearly invariant and hardly influenced by the anisotropy of the covariance function, e, and the variance, sigma (2)(f), of the log, transformed conductivity. As a result, e and sigma (2)(f) cannot be unequivocally determined from the spatial structure of the concentration field. The concentration variance and spatial covariance in the transverse to mean flow direction are predominantly determined by the lateral component of the local scale dispersion, D-d22, and by the spatial correlation length of the log, transformed conductivity in the transverse to flow direction, I-f2. These two parameters might be unequivocally derived from the concentration (co)variance.