Average steady nonuniform flow in stratified formations

Abstract

Abstract. A mathematical model of average nonuniform flow in heterogeneous stratified media is developed. The averaged Darcy's law is obtained by deriving the analytical expression of the effective conductivity tensor for media with small heterogeneities. The fundamental solution of the average flow equation (mean Green function) corresponding to the flow toward a single point source of deterministic discharge is derived. The mean head distribution is calculated for a fully penetrating well, for a finite length well in an infinite domain, and for a dipole well. The concept of equivalent conductivity is defined for use in identifying the formation conductivity properties. The procedure for identifying the mean conductivity, log conductivity variance, and vertical integral scale is suggested for the case of dipole flow.

Modeling groundwater velocity uncertainty in nonstationary composite porous media

“…In most cases, these evaluations can be quite difficult. Most prior research focuses on some very special cases for which (10) can be reduced to a form that allows exact, closed-form integration [3,4,36,26]. In this paper, we evaluate (10) approximately under more general conditions.…”

“…This assumption may seem quite crude but proves to be highly effective and makes general, approximate variance modeling in nonstationary media possible. Previous studies investigating the effects of trending are all based on full integration of (9) and (10) or full solution of (1) that is intractable unless the trends are assumed to be of special forms [9]. These highly restrictive assumptions severely limit the practical utilities of the results.…”

“…Loaiciga et al [16], Gelhar [4], Li and McLaughlin [14], Rubin [25], Indelman and Rubin [9], Indelman and Zlotnik [10], and Ni and Li [21] later investigated flows in heterogeneous trending media and presented explicit, special analytical solutions in illustration of the effect of nonstationarity. More recently, Winter and Tartakovsky [32], Lu and Zhang [17], Rubin [26], and Guadagnini et al [7] derived closed-form solution for flow in special composite media that consist of multiple zones with nonstationary fluctuation statistics.…”

Recent advances in modeling of well hydraulics

“…The total responses are the linear superposition throughout the system; hence, the total solution can be obtained by adding the responses of influences in all parts of the domain [92]. Indelman and Zlotnik [93] used the mean Green function to develop a mathematical model of average nonuniform flow in heterogeneous stratified media. Using perturbation methods coupled with Green's function techniques, Axness and Carrera [94] presented a second-order analytical solution for the steady hydraulic head flow in a two-dimensional (2-D) confined heterogeneous aquifer.…”

Influence of aquifer heterogeneity and return flow on pumping test data interpretation

“…At the same time, investigations of the influence of heterogeneity on interpretation of time-drawdown data for unconfined aquifers are scarce (see Schad, 1997). Matheron (1967) introduced the concept of effective conductivity of a fictitious homogeneous medium that produces a well discharge equal to the mean discharge in the heterogeneous aquifer under the same test geometry (e.g., Indelman and Zlotnik, 1997). We use the term 'effective parameters' in the following sense: pumping a well at a constant rate in an unconfined heterogeneous aquifer induces drawdown in a finite number of piezometers that can be approximated by drawdown induced by a test of identical pumping rate and geometry in a fictitious homogeneous aquifer with effective hydraulic parameters.…”