The interaction of bacteria in the fluid phase with pore walls of a porous material involves a wide range of effective reaction times which obey a diversity of substrate-bacteria adhesion conditions, and adhesive mechanisms. For a transported species, this heterogeneity in sorption conditions occurs both in time and space. Modern experimental methods allow one to measure adhesive reaction times of individual bacteria. This detailed information may be incorporated into nonequilibrium transport-sorption models that capture the heterogeneity in reaction times caused by varying chemical conditions. We have carried out particle (Brownian dynamic) simulations of adhesive, self-motile bacteria convected between two infinite plates as a model for a microflow cell. The adhesive heterogeneity is included by introducing adhesive reaction time (understood as time spent at a solid boundary once the particle collides against it) as a random variable that can be infinite (irreversible sorption) or vary over a wide range of values. This is made possible by treating this reaction time random variable as having an alpha-stable probability distribution whose properties (e.g., infinite moments and long tails) are distinctive from the standard exponential distribution commonly used to model reversible sorption. In addition, the alpha-stable distribution is renormalizable and hence upscalable to complex porous media. Simulations are performed in a pressure-driven microflow cell. Bacteria motility (driven by an effective Brownian force) acts as a dispersive component in the convective field. Upon collision with the pore wall, bacteria attachment or detachment occurs. The time bacteria spend at the wall varies over a wide range of time scales. This model has the advantage of being parsimonious, that is, involving very few parameters to model complex irreversible or reversible adhesion in heterogeneous environments. It is shown that, as in Taylor dispersion, the ratio of the channel half width b to the Brownian bacteria motility coefficient (D0 or dispersion coefficient) t(b)=b(2)/D(0) controls the different adhesion regimes along with the value of alpha. Universal scalings (with respect to dimensionless time t(*)=t/t(b)) for the mean position,
Abstract. Steady flow through a heterogeneous porous medium in a bounded domain is investigated using a recursive perturbation scheme. The effect of boundary conditions on 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. Two flow cases with the same zero-order and different first-order boundary conditions were analyzed. Boundary conditions are deterministic for the first case and random for the second. Significant differences between the two cases indicate random (or absence of randomness) processes must be modeled at boundaries. First-order solutions for the head and velocity covariance functions for a bounded rectangular domain are derived. The resulting integral kernels involve Greens functions, and they are evaluated by numerical integration. Boundary conditions for higher-order problems influence the absolute value and shape of the kernels (and therefore of the head and velocity variance and covariance functions). It is found that the validity of the perturbation scheme is dependent on the magnitude of the kernels and not only on the condition rrf << 1, where rrf is the variance of the log hydraulic conductivity, assumed Gaussian and weakly homogeneous in space. The type of boundary conditions affects the values of the kernels and therefore determine the convergence limits for the problem. Milder head gradients also allow larger values of rrf. Nonlocality is also contingent on boundary type. Weakly homogeneous log-fluctuating conductivity fields give rise to head and velocity covariances which are not weakly homogeneous. The inhomogeneous fields are obtained from a linear filter of the solution to the problem without stochasticity. Higher-gradient regions induce higher head and velocity variances. In the presence of space-varying gradients, nonlocal effects are most important away from the boundaries ("center of the domain"). Small local head gradients result in small head and velocity gradients, and therefore where observations are made in stagnation regions, data should be analyzed taking into account this effect. The results may be used to interpret experimental data for columns or data taken where conditions do not fit the average uniform flow assumption and when processes at the boundaries influence the flow and therefore the mixing of contaminants.
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