On Perturbative Expansions to the Stochastic Flow Problem

Stochastic computational modelling of highly heterogeneous poroelastic media with long‐range correlations

“…The white noise Gaussian field Z is taken to be piecewise constant on the mesh blocks of this computational grid. We distinguish two discretization errors arising in the numerical evaluation of (34). The first are local errors due to the finite size of the mesh blocks in the computational grid.…”

“…Given the above form for the deterministic kernel f one may discretize the convolution integral in (34) to obtain # Y Y numerically. To this end we approximate the required integrations by Riemann sums over a computational grid.…”

“…of (21) and (22) which, for a given order of accuracy, can be computed from the lower order solutions in the perturbation. By proceeding recursively one may establishing the following hierarchy: The above parametrized poroelastic equations represent a self-consistent recursive closure scheme which, up to any arbitrary order of approximation Oðs N Y Þ; can be solved for the stochastic displacement and pore pressure variables fu ðnÞ ; p ðnÞ g using for instance a representation based on Green's function techniques which lead to a straightforward derivation of effective equations for the statistical moments (see References [18,19,34] for flow in rigid porous media). An essential feature underlying in (23) is that the coefficients in the l.h.s.…”

“…Up to an arbitrary order of the scale parameter, the proposed perturbation scheme furnishes an hierarchical recursive system of poroelastic equations which can be solved for the stochastic unknowns by invoking a particular form of the closure problem (see e.g. References [21,23,34]). The proposed recursive scheme shows that heterogeneity acts to enhance spreading of the liquid drainage and rock consolidation, resulting in anomalous forms of Darcy's law and of the constitutive behaviour of the hydro-mechanical coupling (see e.g.…”

Dynamics of Coupled Contaminant and Microbial Transport in Heterogeneous Porous Media: Purdue Component

“…The bacteria modeled in this study, Desulfomonile tiedjei, is far from spherical and therefore it is better to use a drag force model for non-spherical particles, as provided by Haider and Levenspiel (1989). In this case C~=~(l+b Re~')+ b~Re bd+ Re (5) where the coefficients bl, b2, and b3 (Equations given in Appendix A) are fimctions of the shape factor @= s/S' which is the ratio of the surface areas of a sphere having the same volume as the particle, s, to the actual stiace area of the particle, S. If we consider Desulfomonile tiedjei to be of cylindrical shape, of height five times the diameter of the circular face, then @= 0.7.…”