HOMOGENEOUS, ANISOTROPIC MEDIA The horizontal permeability of natural soil formations is

frequently greater than the vertical permeability on account of the w a y

in

which deposits are fist laid d o w n and then consolidated. Samside showed

in 1930

that a suitable geometrical transformation could transform the natural soil mass into an imaginary, isotropic for-

Fig. 6.

mation. T h e foregoing resdts concerning diffusion horns can then be applied to Samside’s imaginary soil mass and,

by the

inverse transformation, the diffusion horns for the real soil mass can be found.

I

Fig. 7.

The

statics and dynamics of underground water

REFRACTION

W h e n flow talres place from a homogeneous, isotropic medium of a certain permcability into another similar medium of a different permeability, the streamlines (mean trajectories within the horns) are refracted at the interface between the two media. T h e horns themselves are refracted

in

a similar way.

INLET CONDITIONS

U p

to this point w e have presumed that the upper end

of a diffusion horn always terminated

in

a point; in practice, however, some departure from this condition

may

occur.

If

rapid diffusion is sought for, injections can be made through a spreading device, but, in doing this, the flow under study is inevitably modified to s o m e extent.

Having seen h o w a jet of coloured

liquid

(or of elec- trolyte) diffuses out in the form of a diffusion horn, w e shall n o w see h o w injections can he used for velocity and discharge measurements.

M E A S U R E M E N T O F T H E V E L O C I T Y O F I N F I L T R A T I O N O F G R O U N D W A T E R

Our

first task is to define the different velocities which can be considered in ground-water flow.

T h e velocity of inatration at a given point will be defined

by

the equation:

ui = c!

S

where

Q

is the clischarge passing through the total cross-sectional area

S;

and

U i

is therefore the m e a n discharge per unit area y.

Darcy’s law can be written U,

=

q

=

KI

where

K =

coefficient of permeability,

I =

slope of the ground-water table at the point

in

question.

the medium is porous, the water is excluded from pace occupied

by

the grains

and

the velocity of inatration

U,

defined above is not the real velocity of the water.

If

it is supposed that all the water within the interstices is

in

motion, then the m e a n velocity

U,

can be defined as follows:

1 I

I

Fig. 8.

io2

where

p

is the porosity, i.e. the ratio of the volume of voids to the total volume of the granular medium.

However, only part of the intersticial water is really in motion;

if

this is taken account of, the effectiye m e a n velocity or effective velocity of the water m a y be defined as:

KI

p, p,

u

^{e -}

--=4

where p c is the effective porosity.

Direct measurement

by

means of salt or colourinfiltra- tion tests enables the latter velocity alone to be found. T h e determination of the m e a n velocity

U,

^.

necessitates permeability and porosity measurements

in

addition.

Very m a n y procedures are used for measuring effect- ive velocities;

in

the present paper w e shall confine ourselves to making certain remarks on the dissolved salt or dyestuff methods.

In

this case, the velocity of propagation of a batch of electrolyte or colouring matter is measured

by

timing between two well-points

in

the ground-water nappe. It

is

most important for the t w o well-points to be located on the same streamline, or at least within the same diffusion horn. The concen- tration of electrolyte or of colouring matter reaching the downstream well-point vanes with time; it increases from zero to a m a x i m u m and thereafter falls very slowly back to zero (see

Fig. 8).

T h e curve of concentration as a function of time resembles a Pearson probability curve. T h e passage of the batch of electrolyte or colouring matter lasts a considerable time and it is no easy matter to determine the m e a n duration of infiltration corresponding to the

T h e m e a s u r e m e n t of g r o u n d - w a t e r $ow

:I

m e a n velocity. T h e only sure deduction that can he m a d e is that the first appearance of the colouration gives the m a x i m u m velocity. Certain workers have taken the time corresponding

to

the peak concentration as a n estimate of

U,.

This

point can be illustrated

in

the following simple way. Let us replace the porous medium

by

a series of capillary tubcs, and let

us

examine the propagation of a certain quantity of coloured

liquid

along one of these tubes (see

Fig. 9).

It is well known that the velocity at any point

within

the tube depends on the distance of this point from the centre of the tube, as compared to r,, the radius of the tube.

T h e velocity at any point is given

by

the formula:

U

=

K (rz0

-

^rz)

where

K = - - - dP 1 ax 4p (p. is the coefficient of viscosity of water).

and is given

by Kr2,.

velocity within the tube is just half the maximum.

T h e m a x i m u m velocity of flow corresponds to r

= O

It is easy to show that the m e a n

G 2 0

and U,,,,

= -

₂

If

a granular medium can be likened to a series of capillary tubes of equal diameter, pointing

in

the direction of Bow, it would be easy to measure the m e a n velocity

by

colouration tests. The first appearancc of the dyestuff is ‘easily detected, and gives a velocity just twice the effective velocity

U,.

However, the complexity of a granular medium makes it improbable that such a schematization holds true

in

the general case. Thus, let us examine a series of capillary tubes of different. diameters Zr,, 2r2, 2r3, etc.

. . .;

22, is the biggest of these.

The m a x i m u m velocity for such a bundle of tubes is given

by U,,,,, = Kr12,

that is

by

the maximum velocity within the biggest tube.

T h e m e a n velocity

will

be given

by

the ratio:

1

Total discharge

= -

2 ^{M K (rI4}

+

rZ4

+

2 3

+ . . .)

Total cross-section M

(riz +

r 2 2

+

raz

+ . . .)

1 r14

+

rZ4

+ r 3 + . . .

2 r ? + r 2 + r 3 2 +

...

u , =

-IC-

O

I

r

x

I I

^I

o b C

PO

t O ..-.-.-

-C

-

Fig.

9.

b e assumed that the coloured water fills the space between two sections au and

bb,

at a distance e from each other, at time t

= O

(see

Fig. 9).

A t a later instant of time t, each surface of the coloured water

will

be formed

by

a paraboloid, the front paraboloid being similar to, and at a distance e from, the rear paraboloid.

Seen

in

plan, these paraboloids have the equations

a n d

x

=

K(rzo

-

r2) t x

=

I<(rz0

-

rz) t

+

e

Taking

the origin of the x axis as being the section ua of the initial layer, w e shall now calculate the volume contained in each paraboloid d o w n to a section ce for which x

= L.

With such a n arrangement the m a x i m u m velocity Krlz

will

be more than twice

U , .

W e can thus conclude that for flow

in

a porous medium the effective velocity is generally less than, or at most equal to, half the m a x i m u m speed corresponding to the appearance of the dyestuff.

W e shall n o w study the motion of a layer

of

coloured water within a capillary tube of diameter Zr,; it

w i l l

por

the front paraboloid, the volume is &en

by:

The

statics and dynamics of underground water

For the

hind

paraboloid: T h e point of intersection of the two curves corresponds to the point for

which

-

^{rol (L}

-

^e)

⁺ -

1 ^(L

-

^]')e

2 Kf

! L o

dt that is,

The instantaneous rates of flow are

t =

-

L

f l l

-

^{Kra4 La}

PI

----..-

df 2 2 Kfz K r o 2

and

Kr04 (L

-

^{e)S [2]} T o s u m

up,

before the paraboloid au reaches the sec-

at -

2 ^_I 2 Kt2 tion cc, the rate of discharge of colouring matter is

given by

the expressión

Equations [i] and

[Z]

are plotted

in

Fig.

10.

As

long as

the

paraboloid

bb

does not reach' section cc, equation

[2]

alone gives the discharge; that is, the discharge of coloured

liquid

equals

r04

- ^___

sequently,

when

the paraboloid au passes cc,

-

²

F... ^-

^K2$2 the rate of flow decreases accord@ to the law

Te (2 L

-

e)

1

. K

(L

-

^e)z

In

the next stage, for which both ql and q2 are positive

quantities, i.e. for K2t2

t >

the discharge is given

by

pective concentrations are

given by:

(L

-

^e)2

2 2 1-- Kz tZ rO4

x

KrO4 -

^'

dv, d v l

-

^{7~ e} (2

L -

^e)

---_-

dt dt 2

K

t2 ' and

(2 L

-

^e)

^.

^{7~ Kra*}

^-

^{e (2 L}

-

^e)

2 Kt' 2 K2 t2 rO4 z e

.-_--

T h e m a x i m u m degree of concentration is reached at time

t = - L

fio2

T h e lapse of time between the first appearance of colouring matter and the attainment of this m a x i m u m depends only on

the

thickness of the initial layer of colouring matter.

If

it is supposed that flow

in

a porous medium is similar, it can

be

concluded that the effective velocity cannot be calculated from

the

point of m a x i m u m concentration; this depends only on the quantity of dyestuff injected originally, or on

the

length of time taken for this injection.

In

the decreasing part of the curve,

the

concentration of colouring matter is roughly proportional to the thickness of the original layer, e

(if

e is supposed m u c h smaller than

L)

and inversely proportional to the square of the time.

Experimental tests have furnished essentially similar results. Thus,

if

the -granular soil mass

is

compared to a bundle of capillary tubes of

differing

diameters, placed parallel to each other, the discharge of colouring matter is

given by

the s u m of the rates of flow

in

each tube. According to what has already been said, the decreasing part of

the

curve is independent of the

dia-

i

L- e L T

kr,

eKr,

Fig.

10,

104

the starting point on this curve

is

all that distinguishes one tube from another. For the rising concentrations, however, each tube follows a separate curve (see

Fig. 11).

If

n o w these curves are summed, a curve resembling the experimental curves is obtained. Thus, just after the colouring matter first appears, the curve

may

be concave upwards, as is

found

to be the case experi- mentally. T h e summit, instead of being a sharp point quite near

in

time to the first appearance of the dyestuff, is rounded and further a w a y from the starting point.

The tail of the curve is however similar to that found

with

a single capillary tube. T o s u m

up,

the simple theoretical flow pattern just studied gives results similar to typical experimental results with granular materials.

It

is

plain

that ground-water flow, even

in

homoge- neous soils, is much more complicated and varied

and

that the theoretical analysis given above has

only

a qualitative value. T h e merit of this theory is to correlate certain aspects of ground-water flow with k n o w n facts concerning laminar flow

in

capillary tubes.

The theory seems particularly useful

in

carrying out basic research into the flow of ground water, especially when difficulties anse

in

comparing the results of different methods of ground-water prospection. Basic research of this nature could be carried out with very simple equipment, and

would

be concerned with the dispersion

in

time and space

of

injected liquid, with the effect of variations

in

the concentration and quantity of dyestuff injected and with the effect of rapid changes

in

head on ground-water flow. T h e latter effect is similar to that produced

by

injection of colouring matter at a well-point; a sudden rise in the ground- water table occurs

in

the well and the resulting dis- turbance is, according to some writers, propagated as a wave with a speed m u c h greater than the velocity of infiltration;

in

this way, a n error m a y be introduced into the results of

field

prospections.

T o resume, it seems that measurements m a d e with colouring matter or salt solutions give a m a x i m u m velocity, as determined

by

the &st appearance of the injected solution, at least double the effective velocity

U,.

Experimental evidence shows that,

in

fact, the m a x i m u m velocity is about two or three times the effective velocity.

If

the effective velocity is known, the flow of ground water

may

be calculated

by

the formula

g

=

U,P,

where p e

is

the effective porosity and can'be measured

by

saturation or drainage methods

by

field, as well as

by

laboratory, tests. T h e length of time taken for drain- age tests is a most important factor; thus, the Ameri- can writer

King

points out that

22.26

per cent of

It has been seen that

if

continuous injections of dyestuff or salt are m a d e at a constant discharge, the concen- tration measured at a point

on

the diffusion horn varies as the inverse square of the distance from the point of injection (for a diffusion horn

in

radial flow). More generally, the concentration varies inversely as the cross-section of the horn and, iii particular, will remain ,

constant in a flow converging at the same rate as the diffusion angle (about 60).

Crosswise, the curve of concentration is bell-shaped (i.e. like a normal probability curve) for a freely-diffus-

ing

dyestuff, tha? is, w h e n neither fixed boundaries \ nor free surfaces are\encountered.

On

the other hand,

if

the diffusion horniis confined between fixed boundaries as

in

a real pipe, the concentration varies inversely as the cross-section of the conduit so formed and soon ceases to vary across the conduit.

injection of pquid at a given point is considered, then the curve of concentration as a function of time

will

be like that which has just been studied.

In

this case the dissymmetrical, bell-shaped curve

giving

the con- centration at: a given point as a function of time

will

flatten out as w e

go

further

way

from the starting point.

Let us examine the relation existing between t w o concentration curves, one for a point

A

situated at distance

L

from the oiigin, and a second for point

B

at a distance,

KL.

T h e foregoing theoretical analysis shows than the scale of time willbe

multiplied by K

and the concentrations divided

by K; in

other words, the concentrations measured

during

the passage of a

I

i

If, instead' of a constant input of dyestuff, a brief f

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