Asymptotics of Solute Dispersion in Periodic Porous Media

The concentration C( x,t ) of a solute in a saturated porous medium is governed by a second-order parabolic equation ∂ C / ∂ t = U₀ b · ∇ C + 1/2 ∑ Dᵢⱼ ∂ ² C /∂ xᵢ ∂ xⱼ. In the case that b is periodic and divergence free, and Dᵢⱼ are constants and( ( Dᵢⱼ ) ) positive definite, the concentration is asymptotically Gaussian for large times. This article analyzes the dependence of the dispersion matrix k of the limiting Gaussian distribution on the velocity parameter U₀ and the period “a” It is shown that each coefficient Kᵢᵢ is asymptotically quadratic in aU₀if bᵢ - bⱼ. has a nonzero component in the null space of b · ∇, and asymptotically constant in aU₀if bᵢ - bⱼ. belongs to the range of b· ∇. It is shown in a more general context that k depends only on aU₀. An asymptotic expansion of the Cramer–Edgeworth type is derived for concentration refining the Gaussian approximation.