Transient well-type flows in heterogenous formations

Abstract

The problem of averaging transient flows by sources of a given head boundary condition in heterogeneous formations of random conductivity is investigated. The study generalizes the recently developed mathematical model of average transient nonuniform flow [Indelman, 1996; Tartakovsky and Neuman, 1998a]. The latter allows calculating the mean head for sources of flux boundary condition and, as such, is not applicable to modeling common well flows. To account for the head condition, the source term in the local flow model is modeled by a random function proportional to the hydraulic conductivity. The local flow equations are further averaged in order to determine the mean flow variables. It is shown that the effective conductivity cannot be defined for arbitrary initial head distributions. This precludes deriving the mathematical model of an average transient flow with the head boundary condition. However, the average flow equations are derived for the important particular case of a uniformly distributed initial head. The effective conductivity tensor is defined for an arbitrary heterogeneous formation and analyzed in detail for isotropic media. It is shown that at the initial stage of the transient flow, the effective conductivity is larger than the conductivity arithmetic mean. The fundamental solution of the average flow equations (mean Green function), corresponding to the mean head distribution due to the instantaneous injection of the unit mean amount of water through a point source, is derived at first order in the conductivity variance and is compared with the mean head for the flux boundary condition.