Dispersion in Ground Water 

Flowing Through Heterogeneous 

Materials

GEOLOGICAL SURVEY PROFESSIONAL PAPER 386-B

Prepared on behalf of the U.S. Atomic 

Energy Commission
Dispersion in Ground Water 

Flo wing Through Heterogeneous 

Materials

By H. E. SKIBITZKE and G. M. ROBINSON

CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

GEOLOGICAL SURVEY PROFESSIONAL PAPER 386-B

Prepared on behalf of the U.S. Atomic 

Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For safe by the Superintendent of Documents, U.S. Government Printing Office

Washington 25, D,C,
CONTENTS

Page 

Abstract. ________________________________ Bl

Introduction ___..__ _____________________ 1

Dispersion phenomena-_________________ 1

Effects of heterogeneity______________-____ 2

References cited________________________ 3

ILLUSTEATIONS

Page

PLATE 1. Typical dispersion of dyes in model with heterogeneity.----- Facing B2

FIGUBE 1. Typical cross section of sandstone models__________--_---_----------- 2

2. Section showing theoretical flow lines near region of high premeability 3

m

668729 63

CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

DISPERSION IN GROUND WATER 

FLOWING THROUGH HETEROGENEOUS MATERIALS

By H. E. SKIBITZKE and G. M. KOBINSON

ABSTRACT

Laboratory studies of dispersion in water flowing through 

porous media show that the heterogeneity of the medium is the 

dominant dispersive factor. All other factors have insignificant 

effects on dispersion.

INTRODUCTION

Man has always discharged wastes on or in the 

ground, so contamination of ground water by wastes is 

not a new problem. The amount of wastes of all kinds, 

however, is increasing rapidly. This increase and the 

recent appearance of radioactive wastes, which are being 

disposed of directly or indirectly into ground-water 

systems, make it essential to learn in detail the nature 

of ground-water movement and the rates and directions 

of flow at specific locations.

The direction of ground-water flow, according to 

classical concepts, is perpendicular to the equipotential 

lines (lines connecting points of equal head) in an 

aquifer. This is literally correct, however, for only 

perfectly homogeneous granular, porous materials. In 

real aquifers, all of which contain inhomogeneities, the 

actual microscopic flow pattern is very complex in detail, and the directions of flow diverge from the hydraulic-potential gradients. This causes wide dispersion of 

water-soluble and -miscible liquids that enter an 

aquifer.

Determination of equipotential lines is relatively 

simple compared to the determination of real flow lines, 

because the principles by which equipotential values 

may be determined are much simpler than those which 

govern the direction of flow. Ground-water hydrologists generally have concerned themselves largely with 

lines of equal potential, but the advent of waste-disposal 

problems requires a change of emphasis.

DISPERSION PHENOMENA

Many investigators have used tracers to determine directions of ground-water flow. As early as 1899, Slichter (1899) reported experiments with tracers, but since

then very little knowledge has been gained about relations between direction of flow and hydraulic gradient.

Recently, many investigators have studied dispersion 

in ground water. They have observed the spread of 

waste liquids in the vicinity of waste-disposal wells, for 

example, and have found that the spread far exceeds 

that which would be predicted by classical hydrodynamics. One notion is that chemical diffusion, compounded by flow around sand grains in an aquifer, 

causes dispersion. Experiments by the writers, however, show that such is not the case.

In many of the recent laboratory experiments the dispersion was much too great to be accounted for by diffusion and flow around sand grains. This conclusion 

necessitates analysis of the precise meaning of the permeability coefficient in Darcy's law. Many writers have 

noted that the permeability coefficient actually is a tensor quantity, though it is generally applied as a scalar. 

Permeability being a tensor, rather than a scalar, would 

account for a difference between the direction of flow 

and the direction of the maximum hydraulic gradient. 

However, the explanation of extensive dispersion is 

more involved because streamlines do more than simply 

bend; they also diverge. Divergence of streamlines is 

the principal concern in this discussion.

In most laboratory tests reported in the literature, 

observed dispersion in homogeneous media can be 

attributed to disturbing unnatural factors, such as differences in the chemical composition and physical characteristics of the tracer and the liquid, or to the physical 

and physico-chemical properties of the permeable material used in the experiments. Tracers having greater 

viscosity than the liquid (such as colored glycerin) or 

tracers of greater density (such as an aqueous solution 

of dye) seem to introduce excessively disturbing conditions in the experiments. To overcome difficulties of 

that sort, we used a radioactive-tracer solution which 

had the same viscosity, density, and other properties as 

the liquid (Skibitzke and others, 1961).

Bl
B2 CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

In the experiment, a dilute solution of nonradioactive phosphoric acid was allowed to flow through a 

porous block of artificially cemented homogeneous sandstone. Flow was maintained for about 1 month to bring 

the solution into ion-exchange equilibrium with the 

solid block. A tracer solution then was introduced 

through a small hollow needle near the input end of the 

block. The tracer was phosphoric acid which contained 

radioactive phosphorous but was otherwise physically 

and chemically identical with the liquid flowing through 

the block. By isotropic exchange along the filament of 

tracer solution downstream from the point of injection, 

radioactive phosphorus replaced some of the stable 

phosphorus which had been adsorbed on the sand 

grains. After 6 weeks of continuous flow with the 

tracer present, the block was cut into several slices and 

the distribution of radiophosphorus was determined. 

Analysis with radiographic film and with radiation 

counters showed that the sand grains actually retard 

diffusion rather than accelerate it. That is, diffusion 

was less than would be expected in a body of free water. 

Thus, strong dispersion in porous material does not 

seem to be caused by diffusion.

EFFECTS OF HETEROGENEITY

In additional experiments, we observed dispersion in 

models of heterogeneous media. Several hydraulic 

models, having various sizes and width-to-length ratios, 

were built by cementing sand with epoxy resin to form 

an artificial sandstone. All the finished models were 

less than 4 feet long and 2 feet wide and were about 2 

inches thick. Heterogeneity was introduced before 

cementing the blocks by embedding bodies of coarse 

sand in a matrix of fine sand or by embedding bodies 

of fine sand in a matrix of coarse sand. Most of the embedments were sinuous channels, U-shaped in cross section, and with depths less than one-third the thickness of

the model (fig. 1). Glass plates, cemented along the top 

and bottom of the sandstone model, permitted direct observation of saturated flow. By cementing the glass to 

the sand, we avoided the boundary effects that have distorted the flow patterns in many experiments. Liquid 

was permitted to flow through the completed model, 

and small streams of dye were injected in the flow field 

at upstream points. Although physical properties of a 

dye may cause some dispersion, dye tracer permits 

visual comparison of phenomena in homogeneous and 

heterogeneous materials. Flow velocities approximating 20 feet per day were within the range of velocity of 

ground water under average natural conditions in 

similar materials.

A typical example of wide dispersion caused by heterogeneity is shown in plate 1. The model was about 

4: feet long and about 1 foot wide. In cross section, the 

shape of the embedments was much like those illustrated 

in figure 1, and they were much higher in permeability 

than the matrix. The two dye-injection points were 

near the inflow edge of the model. Eed dye was introduced at one point and green at the other. During the 

experiment, dye spread through nearly the whole width 

of the model by the time it reached the downstream end.

The process by which the dye spreads is analogous 

to the process by which an influent stream recharges an 

aquifer, or by which effluent ground water feeds a 

stream (fig. 2). The position of the streamline which 

separates recharge flow from through flow (represented 

by the heavy line in figure 2) depends on the rate of 

recharge or discharge in the region of high permeability 

compared to the rate of flow in the part of the matrix 

with through flow. The curve representing any streamline is asymptotic to a straight line parallel to the direction of through flow. Highly permeable channels in 

the model shown in figure 1 carried liquid at a much 

higher velocity than did the adjoining less permeable

Artificial sandstone (matrix) Contrasting permeability

Glass plate

Epoxy seal Glass plate

FIGURE 1. Typical cross section of sandstone models used for studying effects of heterogeneity on dispersion. Section transverse

to direction of fluid movement.
UNITED STATES DEPARTMENT OF THE INTERIOR 

GEOLOGICAL SURVEY

PROFESSIONAL PAPER 386-B 

PLATE 1

Direction 

of flow

Injection/ 

points

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INTERIOR GEOLOGICAL SURVEY. WASHINGTON. D. C. 62293

TYPICAL DISPERSION OF DYES IN A MODEL WITH HETEROGENEITY

DISPERSION IN GROUND WATER FLOWING THROUGH HETEROGENEOUS MATERIALS

Dividing streamline x , Region of high permeability

B3

FIGURE 2. Section showing theoretical flow lines near a region of high permeability. General direction is to the right if head is relatively

high or to the left if relatively low in the region of high permeability.

matrix. Liquid flowed from the model matrix into 

the channel and from the channel into the matrix. 

Thus, dyed liquid that had entered the channel recharged the less permeable matrix along much of the 

channel length, thereby spreading dye through most 

of the matrix.

Figure 1 shows dispersion created by long sinuous 

zones of high permeability which are alined with the 

general direction of flow. Further studies are in 

progress with models having highly permeable zones of 

shorter length, different shapes, and various orientations with respect to the direction of flow.

Flow lines are known to refract at heterogeneity 

boundaries; and in tests in which refraction alone was 

analyzed it was shown that a large angle of refraction 

at the boundary between materials of contrasting permeability accelerated spreading of the dye. In all our 

experiments the effect was always the same: once the

dye entered a region of contrasting permeability, dispersion was widespread.

The experiments are being continued, and the observations made to date are being analyzed to determine 

the quantitative relations among the degree of dispersion, the hydraulic characteristics of the matrix and of 

the imposed heterogeneities, and the rates of flow in the 

system. Quantitative knowledge is prerequisite to forecasting the spread of waste materials which enter the 

zone of saturation.

REFERENCES CITED

Skibitzke, H. E., Chapman, H. T., Robinson, G. M., and McCullough, R. A., 1961, Radiotracer techniques for the study of 

flow in saturated porous materials: Internat. Jour. Appl. 

Radiation and Isotopes, v. 10, no. 1, p. 38-^6.

Slichter, C. S., 1899, Theoretical investigation of the motion of 

ground waters: U.S. Geol. Survey 19th Ann. Rept., pt 2, 

p. 285-394.

O