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

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 σ_Y⁴ when the log hydraulic conductivity variance σ_Y² is much less than 1. All existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to σ_Y² when σ_Y² < 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 σ_Y². It predicts that when σ_Y ≫ 1 in isotropic media, both the longitudinal and transverse dispersivities ascend monotonically toward Fickian asymptotes proportional to σ_Y.