A quasilinear theory of time-dependent nonlocal dispersion in geologic media.

Abstract

A theory is presented which accounts for a particular aspect of nonlinearity caused by the deviation of plume "particles" from their mean trajectory in three-dimensional, statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities. Quasilinear expressions for the time-dependent nonlocal dispersivity and spatial covariance tensors of ensemble mean concentration are derived, as a function of time, variance σᵧ² of log hydraulic conductivity, degree of anisotropy, and flow direction. One important difference between existing linear theories and the new quasilinear theory is that in the former transverse nonlocal dispersivities tend asymptotically to zero whereas in the latter they tend to nonzero Fickian asymptotes. Another important difference is that while all existing theories are nominally limited to situations where σᵧ² is less than 1, the quasilinear theory is expected to be less prone to error when this restriction is violated because it deals with the above nonlinearity without formally limiting σᵧ². The theory predicts a significant drop in dimensionless longitudinal dispersivity when σᵧ² is large as compared to the case where σᵧ² is small. As a consequence of this drop the real asymptotic longitudinal dispersivity, which varies in proportion to σᵧ² when σᵧ² is small, is predicted to vary as σᵧ when σᵧ² is large. The dimensionless transverse dispersivity also drops significantly at early dimensionless time when σᵧ² is large. At late time this dispersivity attains a maximum near σᵧ² = 1, varies asymptotically at a rate proportional to σᵧ² when σᵧ² is small, and appears inversely proportional to σᵧ when σᵧ² is large. The actual asymptotic transverse dispersivity varies in proportion to σᵧ⁴ when σᵧ² is small and appears proportional to σᵧ when σᵧ² is large. One of the most interesting findings is that when the mean seepage velocity vector μ is at an angle to the principal axes of statistical anisotropy, the orientation of longitudinal spread is generally offset from μ toward the direction of largest log hydraulic conductivity correlation scale. When local dispersion is active, a plume starts elongating parallel to μ. With time the long axis of the plume rotates toward the direction of largest correlation scale, then rotates back toward μ, and finally stabilizes asymptotically at a relatively small angle of deflection. Application of the theory to depth-averaged concentration data from the recent tracer experiment at Borden, Ontario, yields a consistent and improved fit without any need for parameter adjustment.