On the variances of second spatial moments of a nonergodic plume in heterogeneous aquifers

Abstract

The expressions for the variances of the second spatial moments of a solute plume, Var[Sij], were formulated and evaluated for a line source of finite length either normal or parallel to the uniform mean flow velocity, μ, in a two‐dimensional heterogeneous and statistically isotropic aquifer. The evaluations were performed by using the analytical solutions for the velocity covariance derived previously under the first‐order approximation to the particle displacement. The main finding is that for the special case of a line source either normal or parallel to the mean flow, not all the variances of the second spatial moments decrease but all the coefficients of variation of the second spatial moments, CV[Sii], decrease as the initial length of a line source increases. The first‐order longitudinal CV[S11] is independent of the variance of transmissivity, σY2, and the transverse CV[S22] is inversely proportional to σY in the case of a line source normal to μ. Comparison between the first‐order theoretical results obtained in this study with previous Monte Carlo simulations shows that the match between the theoretical and simulated CV[S11] improves as τ2 increases and in a less heterogeneous aquifer.