A STREAM TUBE MODEL FOR MISCIBLE FLOW .1. MACRODISPERSION IN RANDOM POROUS-MEDIA

A simple theoretical model is described for deriving a 1-dimensional e quation for the spreading of a tracer in a steady flow at the field sc ale. The originality of the model is to use a stochastic appoach not i n the 3-dimensional space but in the 1-D space of the stream tubes. Th e simplicity of calculation comes from the local relationship between permeability and velocity in a 1-D flow. The spreading of a tracer fro nt is due to local variations in the cross-sectional area of the strea m tubes, which induces randomness in travel time. The derived transpor t equation is averaged in the main flow direction. It differs from the standard dispersion equation. The roles of time and space variables a re exchanged. This result can be explained by using the statistical th eory of Continuous Time Random Walk instead of a standard Random Walk. However, the two equations are very close, since their solutions have the same first and second moments. Dispersivity is found to be equal to the product of the correlation length by the variance of the logari thm of permeability, a result similar to Gelhar's macrodispersion.