A STREAM TUBE MODEL FOR MISCIBLE FLOW .2. MACRODISPERSION IN POROUS-MEDIA WITH LONG-RANGE CORRELATIONS

Large-scale dispersion in heterogeneous porous media is studied by using a simple model based on stochastic calculation of convective flow in a bundle of stream tubes. The advantage of this approach is that there is a local relationship between velocity and permeability in the 1-dimensional space of the stream tubes. Dispersion is due to the variation in stream tube cross-section, related to the permeability field. First, the arrival times of the tracer in the stream tubes are related to the stochastic properties of the permeability field (variance and covariance). Then, transport equations are derived from the moments of the arrival times. The results agree with more complicated studies. For a permeability field with long-range correlation, the transport equation is not unique. It depends on the assumptions involving moments higher than two. Assuming a Gaussian shape for the tracer flux leads to equations similar to the ones obtained in previous studies of time-dependent dispersivity. Without this approximation, the equation is non-local (integrodifferential) and leads to a memory effect. In the last part of this paper, the general results are illustrated with several correlation functions for the permeability field: purely random, exponential and power law covariance, and perfectly layered media.