| Steady flow through a heterogeneous porous medium in a bounded domain is investigated using a recursive perturbation scheme. The effect of boundary conditions for a two dimensional flow with arbitrary variation of the mean flow (allowing large gradients) is investigated using analytical expressions for the head and velocity covariance functions. Boundary conditions were decomposed into deterministic and stochastic components. When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, ln KG, is assumed to have finite variance, s2f . Historically, perturbation schemes have involved the assumption that s2f < 1. Here it is shown that sf may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, s21f , is a more appropriate choice. Bacterial motility is studied at the pore scale. The present work focuses on individual bacteria movement rather than biofilm growth. It is necessary to understand individual bacterial motion before an aggregated model for larger length scales and bacterial populations can be rationally defined. These results will be integrated at a later date in an aggregated model. Here we seek to incorporate attachment and detachment of bacteria to pore walls and mechanisms and behavior of bacterial (non flagellated) movement under a convective drift which is in addition to the tumble and run bacterial behavior observed, measured, and modeled by previous researchers. |
