Search

Flow and transport take place in a heterogeneous medium made up from inclusions of conductivity K submerged in a matrix of conductivity K0. We consider two-dimensional isotropic media, with circular inclusions of uniform radii, that are placed at random and without overlap in the matrix. The system is completely characterized by the conductivity contrast κ=K/K0 and by the volume fraction n. The flow is uniform in the mean, of velocity U=const. The derivation of the velocity field is achieved by a numerical method of high accuracy, based on analytical elements. Approximate analytical solutions are derived by a few methods: composite elements, effective medium, dilute systems and first-order approximation in logconductivity variance. The latter was employed by Rubin (1995), while the dilute system approximation was used by Eames and Bush (1999) and Dagan and Lessoff (2001). Transport is solved in a Lagrangean framework, with trajectories determined numerically from the velocity field, by particle tracking. Results for the velocity variance and for the longitudinal macrodispersivity, for a few values of κ and n, are presented in Part 2. [ABSTRACT FROM AUTHOR]

In the present part the results of numerical simulations of flow and transport in media made up from circular inclusions of conductivity K that are submerged in a matrix of conductivity K0, subjected to uniform mean velocity, are presented. This is achieved for a few values of κ=K/K0 (0.01, 0.1 and 10) and of the volume fraction n (0.05, 0.1 and 0.2). The numerical simulations (NS) are compared with the analytical approximate models presented in Part 1: the composite elements (CEA), the effective medium (EMA), the dilute system (DSA) and the first-order in the logconductivity variance (FOA). The comparison is made for the longitudinal velocity variance and for the longitudinal macrodispersivity. This is carried out for n<0.2, for which the theoretical and simulation models represent the same structure of random and independent inclusions distribution. The main result is that transport is quite accurately modeled by the EMA and CEA for low κ, for which αL is large, whereas in the case of κ=10, the EMA matches the NS for n<0.1. The first-order approximation is quite far apart from the NS for the values of examined κ. [ABSTRACT FROM AUTHOR]

The theoretical results of part 1 [ Dagan and Fiori, 2003] for modeling time-dependent, advective transport of a conservative solute in porous formations of bimodal structure are applied to illustrate the behavior of a few trajectory statistical moments as function of time, of the permeability contrast κ, and of the inclusions volume density n. The computations are carried out for circular (2D) and spherical (3D) inclusions to represent isotropic media. Advective transport is solved by studying the distortion of a thin plume, linear (2D) or planar (3D), normal to the mean velocity U and moving through a single inclusion. The deformation of the plume is determined from the residual trajectories of solute particles that are derived numerically by a quadrature. The longitudinal macrodispersivity is defined by α L( t; n, κ) = (2 U)−1 dX11/ dt, where X11 is the trajectories second moment in the mean flow direction. The general behavior of the time-dependent longitudinal dispersivity α L( t; n, κ) and, in particular, its constant, large time limit are examined. The tendency of α L to the 'Fickian' limit with time depends strongly on the conductivity contrast; in particular, for low permeable inclusions (κ ≪ 1) it may be extremely slow. It is shown that the first-order approximation in the conductivity contrast κ is of limited validity (0.3 < κ < 2). The transverse moment X22 tends asymptotically to a constant value. The analysis of the trajectory high order moments shows that the probability density function (pdf) of the solute trajectories tends to normality at large time. Similar to the 'Fickian' limit the normal distribution may be reached at very large time in presence of low conductivity inclusions, with the pdf characterized by significant tailing for the trailing part of the pdf. [ABSTRACT FROM AUTHOR]

Explores the application of stochastic approaches in modeling groundwater flow. Subsurface hydrology; Factors contributing to the slow process in subsurface hydrology; Hydrogeologic modeling.