1-DIMENSIONAL, 2-DIMENSIONAL, AND 3-DIMENSIONAL EFFECTIVE CONDUCTIVITY OF AQUIFERS

Starting with a stochastic differential equation with random coefficie nts describing steady-state flow, the effective hydraulic conductivity of 1-, 2-, and 3-dimensional aquifers is derived. The natural logarit hm of hydraulic conductivity (lnK) is assumed to be heterogeneous, wit h a spatial trend, and isotropic. The effective conductivity relates t he mean specific discharge in an aquifer to the mean hydraulic gradien t, thus its importance in predicting Darcian discharge when field data represent mean or average values of conductivity or hydraulic head. E ffective conductivity results are presented in exact form in terms of elementary functions after the introduction of special sets of coordin ate transformations in two and three dimensions. Ir was determined tha t in one, two, and three dimensions, for the type of aquifer heterogen eity considered, the effective hydraulic conductivity depends on: (i) the angle between the gradient of the trend of lnK and the mean hydrau lic gradient (which is zero in the one-dimensional situation); (2) (in versely) on the product of the magnitude of the trend gradient of lnK, b, and the correlation scale of lnK, lambda; and (3) (proportionally) on the variance of lnK, sigma(f)(2). The product b lambda plays a cen tral role in the stability of the results for effective hydraulic cond uctivity.