Transverse dispersion in liquid flow through porous media

Abstract

It is assumed that a line source of tracer elements exists in a porous medium through which an incompressible fluid is flowing. Each tracer element is assumed (a) to be completely miscible in the fluid, (b) to be unaffected by the solid medium or by any other tracer element, and (c) to have a path dependent both on the motion of the fluid and on its own molecular diffusivity. As the tracer elements move away from the source they will disperse. A mathematical model, describing the dispersion transverse to the direction of bulk flow, is formulated under the following assumptions: (a) each path has an average direction parallel to the direction of bulk flow, herein termed "zero circuity"; and (b) tracer elements may transfer from one flow path to another by virtue of the molecular diffusivity of the elements. It is then found that the standard deviation of the dispersion is proportional to the square root of the product of grain size, length of flow, and probability that a tracer element will transfer from one flow line to another in a given unit distance. An estimate of the probability is obtained by equating it to that fraction of a large number of tracer elements that would diffuse across the centerline of an idealized pore space. The fraction will depend on (a) size of pore space; (b) velocity of fluid; and (c) coefficient of diffusivity of tracer elements.