The source-type solution in the problem of unsteady filtration in a medium with random nonuniformity

Steady flows driven by sources of random strength in heterogeneous aquifers with application to partially penetrating wells

“…Note that this conclusion (similarly to the case d = 2) is in agreement with previous studies valid for flows driven by a source of deterministic strength (Shvidler 1966;Dagan 1982;Indelman and Abramovich 1994;Indelman 2001), since far Fig. 1 The r 2 order correction w d (x) to the mean Green function as function of the normalized distance x/I from the source enough from the singularity the Dirichlet boundary condition has no influence.…”

“…Therefore, hydraulics of partially penetrating wells (a wide exposition may be found in Cassiani and Kabala 1998) in heterogeneous aquifers is of definite practical interest (Streltsova 1988;Shad and Teutsch 1994). Most of steady state solutions for well flows in weakly heterogeneous formations are available for fullypenetrating wells in unbounded domains (e.g., Shvidler 1966;Naff 1991;Desbarats 1992;Oliver 1993;Desbarats 1994;Indelman et al 1996;Sanchez-Vila 1997;Fiori et al 1998;Indelman and Dagan 2004). …”

“…Only a few studies (Shvidler 1966;Matheron 1967;Neuman and Orr 1993;Desbarats 1994;Tartakovsky and Neuman 1998;Indelman 2001;Guadagnini et al 2003) considered three-dimensional media. In their fundamental paper Indelman and Abramovich (1994) have developed a general mathematical model for steady state flows driven by sources of deterministic strength, and have shown that the effective conductivity tensor K eff is a nonlocal medium property.…”

Transient well‐type flows in heterogenous formations

“…[6] The stochastic approach to modeling flow in random porous media has been developed during the last four decades [e.g., Shvidler, 1966Shvidler, , 1985Dagan, 1989;Gelhar, 1993]. The case investigated mostly is that of steady flow that is uniform in the average.…”

“…The first one is to derive the mean flow variables by solving the local flow equations in heterogeneous formation and subsequent averaging the solution. In his pioneering works, Shvidler [1966Shvidler [ , 1985 has developed a perturbation approach to solve a source-type flow and calculated the large time asymptotics of the mean head in isotropic and stratified media. Analytical perturbation methods and Monte-Carlo simulations have been applied to analyze the steady state well flow, mostly in two-dimensional media of random transmissivity [e.g., Oliver, 1990;Naff, 1991;Neuman and Orr, 1993;SanchezVila, 1997;Riva et al, 2001].…”