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.A curve is plotted giving the relation between a dimensionless parameter, tr*, which is a measure of the dispersion in a given medium, and a dimensionless parameter, H*, which is a measure of fluid velocity and tracer-element diffusivity in the same medium.It is predicted that, within the range of Darcy's law, as velocity increases dispersion decreases.A series of laboratory experiments designed to measure transverse dispersion in a relatively uniform medium, and performed in the range of Reynolds number from 0.04 (approx) to 1.0 (approx), showed that as velocity increased dispersion decreased.The measured standard deviation of the dispersion ranged from 1.57 cm to 0.89 cm (with some scatter) in a bulkflow distance of 119 cm, and the individual concentration distributions were found to be normal except near the end points.Experimental data were reduced to dimensionless form by the assigning of an arbitrary but reasonable value to a dimensionless constant, and in this way the measured dispersion is found to be roughly double that predicted by theory.By analysis of other possible mechanisms of dispersion one concludes that the observed dispersion is best explained by assuming that a velocitydependent dispersion (as predicted by theory) is superposed on a constant dispersion, independent of velocity, which results from wandering of stream paths (that is, nonzero circuity).Finally, it is shown that at very low fluid velocity (R<0.005 for the experimental porous medium) the effect of molecular diffusivity of the tracer elements overshadows the effects of other possible mechanisms causing dispersion.However, it is believed that dispersion caused by nonzero circuity increases with nonuniformity of the medium.The experimental medium was relatively uniform; hence, the relation between nonzero circuity and nonuniformity of the medium (as it occurs in most natural aquifers) remains to be investigated.1 Beran, M. J., 1955, Dispersion of soluble matter in slowly moving fluids: doctoral dissertation, Harvard Univ., p. 4:12.ing Department of the university kindly permitted the use of his laboratory for the measurement of molecular diffusivity of the tracer solution and helped in other ways.