Reply to Comments on Comparison of Observations from a Laboratory Model with Stochastic Theory: Initial Analysis of Hydraulic and Tracer Experiments,Transport in Porous Media 52, 111–115, 2003

Transport in Porous Media -     

Contents of Volume 42

Volume Contents

Contents of Volume 42     

Comparison of Observations from a Laboratory Model with Stochastic Theory: Initial Analysis of Hydraulic and Tracer Experiments

Hydraulic and tracer tests were conducted in a flow cell containing a mixture of sediments designed to mimic a two-dimensional, log-normally distributed, second-order stationary, exponentially correlated random conductivity field. With 60 integral scales in the direction of mean flow and 25 integral scales perpendicular to this direction, behavior of flow and transport in the interior of the flow cell can be compared directly with stochastic solutions for flow and transport. Using 144 piezometers and 361 platinum electrodes, the distribution of hydraulic head and the concentrations of an ionic tracer could be monitored in substantial detail. The present discussion presents the details of the experimental equipment. Results and initial analysis of hydraulic measurements and characterization of a two-dimensional tracer plume are also presented. Analysis using first-order hydraulic theory shows that the flow through the medium was consistent with an effective conductivity equal to the geometric mean of the conductivity distribution. Further, the semivariogram of head increments as observed in the experimental results was consistent with the semivariogram predicted by theory. The chemical transport experiments are here compared with the early solutions presented by Dagan (1984, 1987). The observed rate of longitudinal spread of two tracer plumes was slightly less than that predicted using this theory. Further, the spread in the transverse dimension was observed to decline from the initial plume dimensions and then remain constant or increase slightly, but at a rate lower than predicted by the theory. The difference between the hydraulic and transport results is believed to be related to the fact that the hydraulic results were averaged over a very large portion of the flow cell such that ergodic conditions could be assumed. In contrast, the initial geometry of the plume covered only approximately five integral scales in the transverse direction such that the validity of the assumption of ergodic conditions must be questioned in the analysis of results for the chemical transport.     

Complete controllability to a closed target set

The complete controllability to a closed target setG of nonlinear systems is studied. In the linear case, states completely controllable toG are characterized in closed form. Using a geometric growth condition, a necessary and sufficient condition for the complete controllability toG of the system =A(t)x+k(t, u) is given, making it possible to study the perturbed nonlinear system. Some examples are given.