Stochastic analysis of macrodispersion in heterogeneous bimodal variably saturated formations: The effect of water saturation

The effect of water saturation on macrodispersion in bimodal, heterogeneous, variably saturated formations under steady state, gravity‐dominated flow conditions was analyzed here. The present study was motivated by the inconsistency between the results of the first‐order analysis of Russo [2002] and those of the simulations of Russo et al. [2001]. The latter results suggested that near saturation, increasing mean pressure head H (i.e., decreasing water saturation) may decrease log conductivity variance, σ y 2 , and concurrently, may decrease macrodispersion, while the former results suggested the opposite. Results of the present first‐order analysis suggest that this inconsistency stems from the fact that σ y 2 derived by Russo [2002] is valid only for H = 0. The σ y 2 derived in the present study, valid for H ≥ 0, is a concave function of H, which exhibits a minimum at H = H m , (H m > 0), and may exceed its counterpart in saturated flow when H > H c (H c > H m ). The latter σ y 2 leads to a time‐dependent macrodispersion tensor whose principal components decrease with increasing H when 0 < H < H m′ (H m′ < H m ) and increase with increasing H when H > H m′ and may exceed their counterparts in saturated flow when H > H c′ , (H c′ < H c ), consistent with the results of the simulations of Russo et al. [2001]. Results of the present analysis suggest that for a formation of given statistics and for relatively small inclusions volume fraction P* when the formation is relatively wet, i.e., when 0 < H < H c′ , the macrodispersion is larger in formations in which the texture of the embedded soil is finer than that of the background soil, while the converse is true when the formation becomes less saturated, i.e., when H > H c′ . When P* is relatively large, however, the reverse situation occurs.