WATER FLOW AND SOLUTE MASS FLUX IN HETEROGENEOUS POROUS FORMATIONS WITH SPATIALLY RANDOM POROSITY

Abstract

The impact of the spatial variability of porosity on the solution of the flow equation is investigated using standard first-order perturbation analysis. We use existing theories of flow in heterogeneous media and incorporate spatially varying porosity that may or may not be correlated to the hydraulic conductivity. The first-order analysis shows that longitudinal velocity covariance and its cross covariance with the log-conductivity and head fields are significantly affected by porosity variability with increase or decrease in the magnitude of the covariance if there is a correlation between porosity and conductivity. In the absence of any correlation, only longitudinal velocity covariance is affected, but to a much lesser extent and with an increase in the velocity variance. In some cases where porosity variability is only one order of magnitude lower than that of the conductivity, the impact is so strong that it reverses the sign of the longitudinal velocity-head cross covariance. Transverse velocity covariance and its cross covariance with other flow quantities are not affected by porosity variability. The analytical results are confirmed by Monte Carlo simulations in terms of general patterns and porosity effects. The consequences of these results for transport processes are explored using the solute flux approach. The solute discharge and mass flux across an arbitrary control plane are strongly affected by porosity variability only when it is correlated to conductivity. The mean solute discharge is equally affected in both absolute and relative dispersion frameworks. The solute discharge variance computed within the absolute dispersion framework is more influenced by porosity variability than that computed within the relative dispersion framework. Accounting for the two-particle correlation of travel time and transverse displacement renders the effect of porosity variability less significant than when this correlation is dropped from the formulation of the point solute flux variance.