A quasi‐linear theory of non‐Fickian and Fickian subsurface dispersion: 1. Theoretical analysis with application to isotropic media

Abstract

A theory is presented which accounts for 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. Existing linear theories predict that, in the absence of local dispersion, transverse dispersivities tend asymptotically to zero as Fickian conditions are reached. According to our new quasi‐linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero Fickian asymptotes which are proportional to σY4 when the log hydraulic conductivity variance σY2 is much less than 1. All existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to σY2 when σY2 < 1, and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied. However, the quasi‐linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting σY2. It predicts that when σY ≫ 1 in isotropic media, both the longitudinal and transverse dispersivities ascend monotonically toward Fickian asymptotes proportional to σY.