On the tensor form of dispersion in porous media

Abstract

The variance of the bivariate normal distribution, which approximately defines the concentration distribution resulting from a tracer point injection into a uniform field of flow in a porous medium, is a second-rank tensor. When a point injection is subjected to a sequence of uniform movements in various directions, the final concentration distribution can be obtained by a summation of the tensors corresponding to the various movements. The concentration distribution across a transition zone, which develops when an abrupt interface between two miscible fluids is subjected to a sequence of uniform movements, can be determined by integrating the result for a single point injection over the entire tracer region. The property of isotropic porous media to disperse a tracer fluid is defined by the constant of dispersion which is shown to be a fourth-rank tensor. If the displacement is defined as a second-rank tensor, the variance of the distribution is obtained by the product of twice the constant of dispersion and this displacement tensor.