Extending Darcy's Concept 

of Ground-Water Motion 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-F 

Prepared in cooperation with the 

U.S. Atomic Energy Commission, 

Division of Isotope Development and 

Division of Reactor Development 
Extending Darcy's Concept 

of Ground-Water Motion 

By H. E. SKIBITZKE 

FLUID MOVEMENT IN EARTH MATERIALS 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-F 

Prepared in cooperation with the 

U.S. Atomic Energy Commission, 

Division of Isotope Development and 

Division of Reactor Development 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964 
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary 

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director 

The U.S. Geological Survey Library has cataloged this publication as follows: 

Skibitzke, Herbert Ernest, 1921-

Extending Darcy's concept of ground-water motion. 

Washington, U.S. Govt. Print. Off., 1964. 

v, 6 p. illus., diagrs. 30 em. (U.S. Geological Survey. Professional paper 411-F) 

Fluid movement in earth materials. 

Prepared in cooperation with the U.S. Atomic Energy Commission, 

Division of Isotope Development and Division of Reactor Development. 

Bibliography : p. 6. 

1. Water, Underground. 2. Dispersion. I. U.S. Atomic Energy 

Commission. II. Title. (Series) 

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

Washington, D.C. 20402 - Price 20 cents 
PREFACE 

The ground-water hydrologist seeks to identify and use the physical parameters controlling 

ground-water flow. Scientists are seldom required to define a more heterogeneous or complex 

region than that involved in an underground flow system. In this respect, the hydrologist's 

problem is different from that of, say, the electrical engineer or the physicist, who normally 

analyzes regions constructed with homogeneous characteristics and definable geometries. 

The heterogeneity of the flow region involved in ground-water studies is so great that it is 

difficult to find physical characteristics adequate to describe the region. In effect, then, the 

region must be described by more complex parameters than are generally used in other scientific 

fields. 

Two such parameters are particularly significant. The first is the factor of heterogeneity, 

which somehow must be identified. But the nature of the heterogeneous region can hardly 

be described through reference to the individual geometric discontinuities. Such a description 

would require an endless compendium of individual descriptions, a device so obviously impractical that it renders the region not amenable to description by measurement of any of the 

characteristics visible or accessible from the surface of the region. The second region embraces 

the flow relations at any given point and these relations often reflect an angular discrepancy 

between potential gradient and direction of flow. Thus, tensor description is required because 

any scalar factor is inadequate. 

The measurement of the two factors-heterogeneity and flow relations-has been one of 

the principal difficulties in accomplishing an engineering analysis of ground-water flow. 

Laboratory experiments and study have illuminated some of the basic principles involved in 

the measurement of these factors. The effects of heterogeneity have been studied by hydraulic 

experiments on two types of laboratory models, one type constructed of heterogeneous materials 

and the other of homogeneous materials. A comparison of the experimental results for flow 

through the two model types shows that for heterogeneous porous material a dissolved constituent is dispersed widely; conversely, for material that is simply homogeneous a dissolved 

constituent is not dispersed in the moving fluid. This paper discusses results of experiments 

on the former type material. The paper was prepared and should be read as a companion to 

USGS Professional Paper 411-G, by Akio Ogata, in which experiments on homogeneous materials are discussed. 

Because the effects of heterogeneity cannot be expressed quantitatively, experimental 

procedures must entail the use of a more suitable concept. The observation of the flow is 

generally accomplished by following a tracer element in its progress downgradient. However, 

as tracer elements flow downstream they are dispersed to a degree depending on the conditions 

met in the porous medium-assuming the flow and fluid properties are unchanged. If it were 

possible to delineate a given series of streamlines, as they would be defined by Darcy's law, 

the need for describing the flow might be circumvented provided some measure of heterogeneity 

could be established. Generally, however, establishment of such measure is impossible because 

of physical limitations in experimental techniques. 

Because it is physically impossible to trace out any given streamline, the description of the 

dispersive mechanism may be of importance. Dispersion in porous media has been studied 

to some extent, and a general description of the spreading of the flowlines can be predicted for 

isotropic media. The paper by Ogata describes, for such a medimn, an experiment on dispersion transverse to the direction of flow. Good correlation was obtained between analytical 

and experimental results. The distribution of concentration about the centerline thread of 

flow was almost symmetrical. 

The tracer elements illustrate the history of events as the fluid progresses downstream. 

This seems to imply that if the dispersive effects can be separated the flow system can be comIII 
IV PREFACE 

pletely defined within the region. Although the tracer elements show the historical progress 

of each fluid element and even if it were possible to describe geometrically any flowline, the 

process of integrating this description throughout a region is extremely formidable-more 

likely impossible; hence, there is the need to establish a new outlook in describing the flow 

system, as discussed in some detail in this paper. 

Further research is required to determine the utility of a macroscopic concept of the flow 

system to replace the prevailing microscopic concept. The use of the microscopic analysis 

has led to the conclusion that the Navier-Stokes equation is valid; however, the integration of 

the point function throughout the region becomes impossible. In other words, each streamline needs to be described as a separate entity. Further, the microscopic concept has led to a 

statistical analysis of flow employing an ordered porous body and a random flow system, which 

is in direct contradiction to the conclusion that the N a vier-Stokes equation is valid. In 

essence, then, the microscopic analysis has produced a method of describing a point, whereas 

the description of a large-scale aquifer is the goal of the hydrologist. 

The implications of studies already accomplished are that further experiments should be 

tried in larger regions and that a few small-scale supporting laboratory experiments should be 

added. Because in many of the tests made to date only dissolved constituents were observed, 

some different and more suitable type of tracer must he used to reveal the components of the 

dispersion process. Radiotracers have proven especially well suited for this purpose in most 

laboratory experiments. 

The analysis of dispersion processes will necessarily require the use of some tracer techniques, simply because dispersion is the product of the motion of the dissolved component. 

In a macroscopic sense, of the order involved in ground-water field studies, the dispersion of 

contaminant does not depend upon any measurable hydraulic factor. Therefore, part of the 

experimental technique must involve addition of dissolved components to the fluid. Because 

the dispersion is also dependent upon the location and geometry of the injection site, no naturally occurring tracer element will suffice. Furthermore, once a tracer is decided upon, it will, 

in general, need to he a radiotracer because the problem of remote sensing is involved in groundwater studies. It seems necessary, then, that techniques he developed for extending the 

laboratory use of radiotracers into the field. 
CONTENTS 

Page 

~eface-------------------------~------------------ III 

Abstract___________________________________________ Fl 

The permeability concept____________________________ 1 

The ground-water flow regime________________________ 3 

Dispersion phenomena ___________ -------------------- Limitations of the calculus __________________________ _ Eteferences ________________________________________ _ 

ILLUSTRATIONS 

Page 

F5 

6 

6 

FIGURES 1-3. Diagrams of flow regime in region- Page 

1. Of constant permeability ________________________________________________________ ---_________ F3 

2. Where scalar permeability varies continuously _____________________________________ --__________ 3 

3. Where nonscalar permeability is continuously distributed _____________________ ---_----- _________ - 3 

4. Diagram showing-theoretical flowline refraction at a boundarY------------------------------------------ 4 

5. Laboratory photograph of flowline refraction_________________________________________________________ 4 

6. Diagram of the flow regime in region where tensor permeability is discontinuous __ ~- ____ --_- 8- w- _--- __ - -- - 4 

7. Diagram showing tracer dispersion in heterogeneous aquifer---------------------------------------- --~ 5 

8. Diagram showing the mean line of dispersion in heterogeneous aquifer __________ - _- _- _- _------ _-- ____ --- 6 

v 
FLUID MOVEMENT IN EARTH MATERIALS 

EXTENDING DARCY'S CONCEPT OF GROUND-WATER MOTION 

By H. E. SKIBITZKE 

ABSTRACT 

The tensor nature of permeability and its effect in a given 

ground-water flow regime have been acknowledged by various 

investigators. The effect on the spread of a given dissolved 

substance can be qualitatively discussed on the basis of the 

tensor characteristic of the permeability coefficient. The correlation between the dispersion coefficient and the tensor characteristic is one possible means of defining the flow regime within 

a given aquifer. Because of the large-scale changes in magnitude and direction of the tensor components of permeability, the 

c?ncept of a mean line of dispersion may produce a more sigmficant statement on the characteristics of flow than does the 

present concept of hydraulic potential and the streamline. 

THE PERMEABILITY CONCEPT 

The analysis of a region harboring ground-water flow 

requires the description of the permeability function in 

a quantitative sense. In studying most ground-water 

flow problems, the permeability of the porous earth 

material through which the flow occurs has been assumed a constant. At best the flow analysis may sometimes acknowledge the permeabilit ;- as a scalar, variable 

in space. This practice is illustrated by studies where 

permeabilities are determined by discharging-well tests 

of an aquifer and then are plotted on a regional map 

and used to develop contours of permeability. This 

simple scalar relationship is one wherein the permeability is constant at a given point-that is, it does not 

change in the different directions. The permeability 

can thus be defined by a single number at that point. 

If another point in the region is considered, a new number may represent the permeability at that point, but 

again the permeability is the same in any direction. 

Thus the scalar coefficient-permeability-of the Darcy 

equation may vary in space, but it will be constant for 

all directions at any point. 

Now suppose-that the permeability differs in various 

directions at any point; that is, if the permeability were 

measured in the east-west direction, it might differ from 

that measured in the north-south direction, which in 

turn might differ from the vertical direction. The manner in which th!3 permeability may vary can no longer 

be described mathematically by a simple coefficient such 

as the scalar permeability previously used. An adequate 

description will necessarily lead to a tensor concept for 

the vermeability relationship. 

The permeability concept can be extended further by 

considering spatial relationships as well as directional 

characteristics. Assume first that the permeability 

can be described by a function that would account for 

variations in direction-that is, by a tensor. It may 

then be assumed that permeability can vary in any 

direction specified by the Cartesian coordinate system 

and, further, that these variations change as the position of measurement is moved laterally or vertically 

within the aquifer. Measurements made at various 

points within an aquifer will then define a function 

varying with position. If these changes are continuous 

- that is, vary smoothly in a mathematical sense-the 

change of permeability can be described as a continuous 

function. This limitation of continuity is unnecessary, 

however, and its removal leads to the final enlargement 

of the permeability concept into something that is a 

discontinuous function. 

Imagine that the permeability of aquifer material is 

describable by a tensor relationship and that the tensor 

then can properly account for permeability variations in 

direction and magnitude. The tensor may vary not 

only in space but also in a discontinuous manner. 

Previously the permeability function was described as 

varying in space- both in the lateral and the vertical 

directions-but in a smooth manner, so that a continuous function accomplished its description. For discontinuous variation, permeability measurements at 

points adjacent to one another would reveal abrupt 

changes in the permeability tensor. This discontinuity 

condition leads to the most generalized concept of 

permeability necessary for describing the porous material constituting an aquifer. Thus an adequate 

Fl 
F2 FLUID MOVEMENT IN EARTH MATERIALS 

concept of the permeability function leads to the 

definition of a variable that changes with position in 

space as well as with direction at a point, the variations 

being discontinuous or continuous or both. 

Often in aquifer materials the variation in permeability is continuous over a small region, at the boundary 

of which it abruptly changes in a discontinuous manner. 

An analysis of the permeability factor requires determination of the implications involved in its variations. 

First, Darcy's equation must be revised. 

The Darcy equation has the general form 

~ ----+ 

Q=P grad h. (1) 

To develop a more elaborate description than is generally used, the scalar permeability P must be replaced 

in equation 1 by the tensor permeability P t' which represents a coefficient that would vary in space continuously or discontinuously-and generally both. 

Some comprehension of the tensor relationship may 

be extracted from a definition given by Margenau and 

Murphy (1959, p. 161), a part of which is quoted as 

follows: 

In many physical problems the notion of a vector is too re- ~ 

stricted. For example, in an isotropic medium, stress S and ~ 

strain X are related by the vector equation ~ ~ 

S=kX (2) ~ ~ 

X and S having the same direction. If the medium is not ~ ~ 

isotropic, S and X are not in general in the same direction; it 

is then necessary to replace the scalar k by a more general ~ 

mathematical construct capable, when acting on the vector X, 

of changing its direction as well as its magnitude. Such a construct is a "tensor." A similar generalization has to be made 

in the vector equations 

(3) ~ ~ 

where P and E represent electric polarization and field strength, 

and 

(4) ~ ~ 

where I and H represent intensity of magnetization and field 

strength; for anisotropic media, the susceptibilities E and p. must 

be replaced by tensors. 

Equally complex conditions confront the analysis of 

ground-water motion. If the permeability varies with 

direction, there is an angular difference between the 

hydraulic gradient and flow direction. Because of this 

difference, the vector descriptions require application 

of a tensor concept to the permeability coefficient. In 

the simplest situation the permeability function is a 

scalar, which in turn is the most degenerate form of 

tensor. More generally the tensor is a function that 

may be used to describe mathematically the relations 

between the hydraulic gradient and the direction and 

rate of ground-water flow, regardless of the directional , 

characteristics of the rock permeability. This tensor 

can vary both continuously and discontinuously in 

space. 

The tensor relationship changes the continuity 

conditions for which flow equations can be specified. 

The Laplace equation, written with a simple scalar 

permeability varying as a continuous function, describes the continuity condition of the liquid in a 

moving state. In this form the Laplace equation 

appears different from the form usually given, and one 

might argue that it is no longer properly termed 

Laplace's equation. As it is commonly written, 

Laplace's equation has the form 

v%=0 W 

where h is the hydraulic head. When the scalar 

permeability P varies in space, the equation must 

necessarily contain P=P(x,y,z) as a variable. Thus 

equation 5 would be rewritten 

~ 

V·PVh=O. (6) 

Upon completing the indicated mathematical operations, equation 6 becomes 

~ ~ 

Pv2h+vh·vP=O. (7) 

That equation 7 is reducible to equation 5 is readily 

shown. For the simplest concept of permeabilitythat is constant or unvarying with respect to space and ' ~ 

direction-the permeability gradient VP is obviously 

zero, and the second term in equation 7 is therefore 

zero. The remaining term can be divided by the constant permeability P, and the simple Laplacian form of 

equation 5 is thus left. However, the simple Laplacian 

form is not applicable to the ground-water conditions 

commonly found in nature. · 

The continuity equation for ground-water motion 

becomes even more complex when the Laplace equation 

is written in terms of a tensor relationship. A formal 

derivation and a discussion of its significance are not 

included here. These will be covered in a separate 

chapter of this Professional Paper. Sufficient for the 

present is an indication of the complexity in the form 

of Laplace's differential equation when written to 

show, say, the distribution of velocity potentials V in 

terms of a completely general coordinate system and 

the fundamental metric tensor Yt1· In tensor notation 

~ 

the Laplacian relation V2V =0 has the form 

giiVu=O (8) 

or 

_ _!_ ()( jggti()V j()xt) =0 .fiJ ()xt . (9) 

(See James and James, 194-9, p. 206.) 
EXTENDING DARCEY'S CONCEPT OF GROUND-WATER MOTION F3 

Finding solutions for a relationship similar to equation 

9 is much more difficult than for the simple Laplace 

equation. Actually, it would be virtually impossible 

to find any significant solutions for the tensor relationship. However, some flow features leading up to and 

involving the relationship might be briefly examined. 

THE GROUND-WATER FLOW REGIME 

It is helpful first to have clearly in mind, as a reference base or point of departure, the ground-water 

flow regime as it is commonly visualized and depictedpermeability in every place and direction being constant 

for the porous medium. These simple conditions are 

presented as case 1 in figure 1. Observe that the flowlines are coincident with the lines denoting hydraulic 

gradient, and the flow rate is labeled as inversely proportional to the distance between the equipotential 

contours. 

Now consider the matter of scalar variations in permeability in space-that is, the simple situation in 

which the permeability is always a scalar but varies 

continuously in space. The pertinent conditions are 

described as case 2, shown in figure 2; the changing 

permeability produces a flow pattern distorted in 

i'that the flow rate is not proportional to the gradient 

vector grad h. The flow is in general parallel to the 

gradient vector, but the rate is not proportional to the 

hydraulic gradients indicated by the spacing of contours denoting equipotential surfaces. This set of conditions, case 2, represents the simplest conceivable 

departure from the common idealized conditions 

illustrated in case 1 (fig. 1). 

A more general departure from the idealizations of 

case 1 is represented by the conditions shown as case 3 

in figure 3. The permeability not only varies continuFIGURE I.-Case 1: \72h=O. Diagram of the flow regime in a region in which 

permeability Pis constant throughout. Lines along gradient are streamlinesthat is, total flow Q is constant between streamlines. Flow rate is inversely 

proportional to the distance between contours. Arrows indicate direction of 

flow. 

741-755 0 - 64 - 2 

FIGURE 2.-Case 2: \7.p grad h=O. Diagram of the flow regime in a. region in 

which permeability P is a scalar varying continuously in space. Flow rate is 

not inversely proportional to the distance between contours. Arrows indicate 

direction of flow. 

ously with spaae but also varies with direction, and it 

has the characteristics of a tensor. Thus the ground 

water flows in directions differing from those indicated 

by the hydraulic gradients. 

A corollary case deserving brief mention involves the 

discontinuous condition resulting from the abrupt 

change between the simple permeabilities of two adjacent parts of an aquifer. The permeability P is a 

scalar quantity in each part but abruptly changes at 

the interface between the parts. The flow regime in 

such an environment has been analyzed &.nd discussed 

by a number of technical writers but is particularly 

well covered in the works of Muskat (1946) and Hubbert 

(1940). The sketches drawn in figure 4 are from 

Hubbert's discussion; they show how flowlines are 

abruptly refracted at the permeability discontinuity. 

Clearly illustrating the same refraction phenomena is 

the picture shown in figure 5, taken in the Phoenix 

research laboratory of the U.S. Geological Survey during 

FIGURE 3.-Case 3: \7. (Pt grad h) =0. Diagram of the flow regime in a region in 

which permeability Pt is a nonscalar continuously distributed in space. Flow 

direction is not parallel to the gradient. Flow rate is not inversely proportional 

to the distance between contours. Arrows indicat.e direction of flow. 
F4 FLUID MOVEMENT IN EARTH MATERIALS 

FIGURE 4.-Diagram showing theoretical flowline refraction across layers of coarse and fine sand. After Hubbert (1940). 

a hydraulic experiment on an artificial-sandstone 

model. Uniform flow is from left to right and the 

refraction of the black-appearing dye stream is clearly 

evident as it passes through the tapered band of 

higher permeability sandstone in the center of the 

model. 

To be examined as a final departure from the idealizations of case 1 are the completely general conditions 

presented as case 4 in figure 6. The permeability is 

now a tensor function and can vary with direction; it 

also can vary discontinuously in space. The groundwater flow is not only in directions differing from those 

indicated by the hydraulic gradients but appears to 

have several directions at each point. 

The net effect upon the flow system of combining 

all the individual conditions and factors discussed in 

the preceding paragraphs is most difficult to illustrate. 

A possible approach might be to construct in model 

form a heterogeneous medium having all these factors 

inherent in the construction. The author has concluded, after many such experiments (see, for example, 

Skibitzke and Robinson, 1963), that the most generalFIGURE 5.-Laboratory photograph showing refraction of flowlines in artificialsandstone model. 

FIGURE 6.-Case 4: \1 . (P t grad h) =0. Diagram of the flow regime in a region in 

whiJh permeability P t is a tensor function (either scalar or directional) il.nd is 

discontinuous in space. Flow is dispersed, appearing to have sevel'al directions 

at a point. Flow is not inversely proportional to the distance between contours. 

Arrows indicate direction of flow. 

ized state of the heterogeneous medium would lead to 

flowlines and related fea.tures as shown in figures 6 ·and 

7. The details in figure 7 are explained in the following 

paragraphs. Observe, however, that if the path of a 

water particle is traced, starting from any point, the 

flowline begins by briefly indicating laminar motion, 

according to the classical thinking represented by 

Darcy's law. As distance from the starting point 

increases and the flow pattern becomes more widely 

dispersed, however, the flowline quickly departs from 

the classical concepts. The manner in which the flow 

disperses downstream from a reference point is described by Ogata (1964) in a companion chapter to the 

present report. Presented and discussed therein are 

the experimental data collected in a series of experiments designed specifically to show the effects of dispersion. The important conclusion drawn by Ogata 

is that the spread of a tracer or contaminant migrating 

from a deposit in a heterogeneous aquifer is completely 

the result of the heterogeneity, insofar as can be 

observed. 
EXTENDING DARCEY'S CONCEPT OF GROUND-WATER MOTION F5 

Mean line of dispersion 

(apparent Darcy streamline) 

FlGVRE 7.-Diagram showlng dispersion of a tracer injected into a heterogeneous 

aquifer. Permeability P has scalar characteristics. 

DISPERSION PHENOMENA 

The preceding condusion leads to an important concept in describing the nature of flow in a ground-water 

reservoir. Two significant regions in the flow field can 

be visualized. In the first region the flow environment 

is so heterogeneous that the travel paths of a tracer or 

contaminant follow streamlines winding tortuously 

through the porous medium. Therefore, within this 

region, if a tracer-concentration profile were drawn for 

a section normal to the general direction of flow, it 

would be highly irregular in form (fig. 7). Further 

down the path of motion the effects of the intertwining 

of flowlines tend to blur out the apparent discontinuous 

features of tracer concentration. The region in which 

this blurring occurs can be regarded as merely a 

transition region (fig. 7). Ultimately, the flow reaches 

the !!!econd region to be visualized, which is at some 

di$tance downstream from the tracer or contaminant 

origin. In this second region (fig. 7), the porous 

medium appears to be more amenable to description 

by dispersive processes in a mathematical sense. If a 

coordinate system moving with the liquid is postulated, 

the mathematical description would be the Fickian 

diffusion equation, having the form 

(10) 

In equation 10, however, the magnitude of D is many 

powers of ten greater than it is in the ordinary process 

of molecular diffusion. 

Until region 2 is reached (fig. 7) and "complete" (in 

the mathematical sense) dispersion occurs, the discontinuities of flow virtually defy description. Their 

identification would require a complete kiwwledge of 

the permeability of the aquifer material in microscopic 

detail; in other words, the nature of about each cubic 

foot of the aquifer would have to be studied. Knowledge of the hydraulic gradient would also be require<;! 

in the same detail. Obviously these data, required 

primarily in region 1 (fig. 7), could never be obtained. 

To summarize, therefore, it must be concluded that 

around the tracer or contaminant source and proceeding downstream from it there exists first a region of 

indeterminate flow where the tracer is mixed discontinuously. This kind of mixing is illustrated schematically by the upper inset cross section of figure 7. This 

first region grades gradually through a transition region, 

where the flow and dispersion discontinuities "!:>egin to 

amooth out, into a second region where the flow can be 

characterized as simple dispersed flow. The effects 

are again schematically illustrated in the lower inset 

cross section of figure 7. However, it may take many 

times the length of any normal aquifer to arrive at the 

second region. Conversely, there may well be some 

situations in which the second region is reached in a 

matter of a few feet. Even in the simplest and most 

uniform of aquifers there will still be some intertwining 

of flowlines within the pore spaces themselves. The 

flow-regime features sketched in figure 7 typify the 

nature of fl0w and dispersion in a heterogeneous aquifer 

and show that a more generalized description of the 

flow pattern is needed. 

If the flow-regime idealizations based upon Darcy's 

law are compared with the features shown in figure 7, 

it may be concluded that the commonly used formula ~ 

Q=P grad h in reality describes only a generalized 

average direction of flow. The formula does not 

describe actual stream paths of tracer or contaminant 

elements, but describes instead what might be termed 

a mean path of dispersion. As a beginning step, 

therefore, in a more realistic analysis of flow through a 

heterogeneous aquifer, it would appear most desirable 

to promote this concept, or new definition, that the 
F6 FLUID MOVEMENT IN EARTH MATERIALS 

- central streamline, as it is identified classically by use 

of Darcy'f? equation, is more nearly the central line 

about which the dispersed tracer travels. That this 

is ···only a beg.inning step in a more realistic analysis 

can be inferred from the features shown in figure 6, 

characterizing the flow regime when the permeability 

is a tensor function. In this completely general 

_ situa.tioli the mean line of dispersion will depart from the 

lines representing the conventionally determined hydraulic gradient. Although the manner of departure 

is yet to be determined, it is schematically represented 

in figure 8. 

FIGURE 8.-Diagram showing the mean line of dispersion in a heterogeneous aquifer 

when permeability P t has directional characteristics. 

LIMITATIONS· OF THE CAL<:;ULUS 

One additional point merits brief discussion, and 

it concerns the nature of infinitesimals when working 

with geologic bodies the size of- regional a·quifers. 

In mathematical_ ~nalysis the · irifinitesimal_· generally 

must be considered as ever reducible to yet a smaller 

one without losing any of its inherent physical charac-.. 

teristics. If, for example, observation wells penetrate 

a regional aquifer on a ~-mile-square grid pattern,_ it 

must be assumed in any mathematical analysis, of data 

obtained from such wells that all aquifer ·elements 

smaller than the ~-mile-square segments will be 

identical in nature. If these "infinitesimals" remained 

the same in character as the elemental areas approached 

zero, a fairly representative mathematical picture of 

the aquifer would be obtained. If on the other hand 

the successively smaller infinitesimal elements-considered within the ~-mile-square segments-changed 

in their physical characteristics as the areas tended 

toward zero, the mathematical operations involvi:Qg the 

calculus would not be valid. Commonly the heterogeneity of porous media is such that the permeability 

function will contain changes in its nature through 

all magnitudes of areal size that may be chosen throughout the regional aquifer. It is not -unreasonable to 

expect, therefore, that flow patterns different from 

those described by simple solutions of Laplace's 

equation will be observed. In field practice the 

smallest infinitesimals that can be studied and described are those whose areas are delimited by the 

spacing of wells in the observation-well network. Thus 

any analytical conclusions dependent upon mathematical 

integration and differentiation processes will be invalid. 

For these reasons, employing the concept of the mean 

line of dispersion seems simpler and more promising 

than attempting to find the direct flow path. In 

heterogeneous acquifers the latter approach obviously 

requires a mode of description potentially involving an 

infinite number of paths. 

REFERENCES 

Hubbert, M. K., 1940, The theory of ground-water motion: 

Jour. Geology, v. 48, no. 8, pt. 1, p. 785-944. ,.. 

James, G., and James, R. C., 1949, Mathematics dictionary: 

New York, D. Van Nostrand Co., 432 p. 

Margenau, H., and Murphy, G. M., 1959, The mathematics of 

physics and chemistry: 2d ed., Ne:w York, D. Van Norstrand Co., 604 p. 

Muskat, M., 1946, The flow of homogeneous fluids through 

porous media: Ann Arbor, Mich., J. W. Edwards, 763 p. 

Ogata, A., 1964, The spread of a dye stream in an isotropic 

granular medium: U.S. Geol. Survey Prof. Paper 411-G 

(in press). 

Skibitzke, H. E., arid Robinson, G. M., 1963, Dispersion in 

ground water flowing through heterogeneous materials: u:s. 

Geol. Survey Prof. Paper 386-B, 3 p., 1 pl., 2 figs. -

U. S. GOVERNMENT PRINTING OFFICE : 1964 0- 741 - 755