Effective hydraulic conductivity of bounded, strongly heterogeneous porous media

Abstract

This dissertation develops analytical expressions for the effective hydraulic conductivity Kₑ of a three-dimensional porous medium bounded by two parallel planes of infinite extent separated by a distance 2a. Head varies randomly along each boundary about a uniform mean value. The log hydraulic conductivity Y forms a homogeneous, statistically anisotropic random field having a variance σᵧ² and principal integral scales λ₁, λ₂, λ₃. Flow is uniform in the mean parallel to the principal coordinate χ₁. A solution is first derived for mildly nonuniform media with σᵧ² ≪ 1 via an approximate form of the 1993 residual flux theory by Neuman and Orr. It is then extended to strongly nonuniform media with arbitrarily large σᵧ² by invoking the Landau-Lifshitz conjecture as Kₑ = KG exp {σᵧ² [1/2 — (D + S)]} . Here, K(G) is the geometric mean of hydraulic conductivities and D and S are domain and surface integrals, respectively. Based on a rigorous limiting analysis we show that when the length scale ratio p = a / λ₁ → 0, Kₑ is equal to the arithmetic mean hydraulic conductivity K(A). This supports the theoretical finding of Neuman and Orr and the numerical result by Desbarats. When ρ → ∞ we obtain expressions for Kₑ that have been previously derived in the stochastic literature for infinite flow domains. For strongly anisotropic media with integral scale ratios ε₂ = λ₂ / λ₁ and ε₃ = λ₃ / λ₁ equal to each other and tending to zero or infinity ( ) i 0) we obtain the closed form solution Kₑ = K(G) exp {σᵧ²[exp(—p) — 0 .5]} . The latter reduces to K(A) when ρ → 0 and tends to the harmonic mean K(H) as ρ → ∞. One can think of the case ε₂ = ε₃ = 0 as mean flow along parallel channels having mutually uncorrelated hydraulic conductivities, and of the case ε₂ = ε₃ → ∞ as mean flow normal to layers having uniform hydraulic conductivities. For statistically isotropic media we show numerically that Kₑ equals K(A) when ρ = 0.01; when ρ ≥ 4, Kₑ = K(G) exp(σᵧ²/6) the three-dimensional infinite domain solution. Our results support the analytical finding of Rubin and Dagan, and predict and explain all related bounded domain numerical results. Finally, contrary to Dagan's assertion, we show that for small ρ boundary effects are extremely important; the absolute value of the surface integral S equals the value of the domain integral D.