SOIL GEOMETRY AND SOIL-WATER EQUILIBRIA

BY E. C. CHILDS AND N. C. GEORGE

Received 9th January, 1948

Physically a soil may be regarded as completely specified by the geometry of the interface between the solid component and the void space together with that of the air-water interface within the voids. In the nature of the case it is most unlikely that a complete geometrical specification will ever be accomplished and one must be satisfied with less. The determination of the size distribution of the ultimate particles (mechanical analysis) and of aggregates stable to arbitrary degrees of violence, which constitute factors of the soil geometry, have long been weapons in the armoury of the soil scientist, but from this information alone one can only infer qualitatively the nature of the void space. In recent years the direct determination of pore-size distribution has become increasingly recognised as a complement to mechanical analysis1 ; and on its behalf it may be said that it is the water-occupied void space which, together with the solid surface, is the seat of physico-chemical activity and which largely determines the gross physical properties of the soil. The complete geometrical specification which the soil scientist seeks under the name " soil structure " may indeed be a will-o'-the-wisp, but it is reasonable to suppose, and we seek to show, that certain measurable physical properties of the soil may be related to measurable factors in soil geometry.

It is not our present function to discuss the physical chemistry of the soil solution, but some qualitative description must be included in the discussion of the determination of pore-size distribution. The total force acting on a volume element of water in the void space may be regarded as made up of four components, each making a greater or less contribution according to such circumstances as the nature of the solid surfaces, the moisture content, the content of soluble salts and the location of the element considered. The four components are : (a) gravitational attraction, always acting vertically downward, (b) hydrostatic pressure difference, (c) osmotic pressure differences due to differences of content both of soluble salts and of anions and cations in Gouy double-layers associated with the solid surfaces, and (a) adhesion of water to the solid surfaces. It simplifies matters if we speak in terms of the scalar potential rather than the vector force, defining the potential dif3erence between points A and B as the work done against the above-described force field in translating unit volume of water from the point A to the point B, and, as usual, adopting the potential at one convenient point as our arbitrary zero potential. The potential a t any point will then be the scalar sum of four components corresponding to the four groups (a) to (d) . The water is in hydrostatic equilibrium when the total force is everywhere zero, that is, when the total potential is everywhere the same, notwithstanding that single components may vary widely from point to point.

Childs, Soil Sci., 1940, 50, 239. Childs, Soil Sci., 1942, 53, 79. Donat, Trans. 6th Comm. Int. SOC. Soil Sci. B, 1937, p. 423. Haines, J. Agric. Sci., 1930. 20, 97.

Schofield, Trans. 1st Comm. Int. SOC. Soil Sci. A , 1938, p. 38. ti Learner and Lutz, Soil Sci., 1940, 49, 347.

' Swanson and Peterson, Soil Sci., 1942, 53, 173.

78

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online / Journal Homepage / Table of Contents for this issue

E. C. CHILDS AND N. C. GEORGE 79

The gravity component needs no explanation. The work done in lifting unit volume of water (density p) to a height h above an arbitrary datum level, and therefore the potential at height h, is given by gph. The hydro- static pressure term (b) is also familiar. If S is the surface tension, and the air-water interface in which it acts has radii of curvature rl and r2, then the pressure on the water side of the surface is less than that in the air by the amount AP = S(r/r1 + 1/r2), where rl and r 2 are regarded as positive for curvatures which are concave to the air. It is convenient to take atmospheric pressure as our datum; then the pressure P just inside an interface between soil water and air which is continuous with the outside air is given by P = -S(r/r1 + 1/r2). P may, of course, take other values at points remote from the interface. It is common to ignore the negative sign and to speak of pressure deficiency or suction pressure. The work done in taking unit volume of water from a point of zero pressure to one of pressure P, and consequently the potential at the latter point, is given by P. According to Gouy’s well-known concept,B a surface dissociating, say, cations, itself having a negative charge, tends to maintain increased cation and decreased anion concentration in its vicinity as compared with the equal concentrations in the more remote solution. From this it is a simple consequence of the Donnan equilibrium that the total ion concentration, and therefore the osmotic pressure, increases as the solid-liquid interface is approached ; the equilibrium of the double-layer is a balance of the concentrating electric and adsorbtive forces as opposed to the diluting osmotic pressure. Corre- sponding to this equilibrium there is an osmotic pressure gradient which is a function of distance from the solid surface. The osmotic pressure fi at any point corresponds to a hydraulic potential contribution -9, since it must be regarded as a hydrostatic pressure deficiency. This contribution to potential might commonly be appreciable for distances up to the order of 50 A. from the dissociating surface, but there seems to be no experimental confirmation of this osmotic potential distribution. Even less is known about the adhesion potential due to the attraction between the solid surface and the water dipolar molecules, but from the nature of dipole attraction these adhesion forces, whilst they may attain large values at close distances of approach, must be of very short range. Since the force is an attraction, the potential contribution wil l be negative relative to a point in the more remote solution; let us call it --n where n is positive.

The osmotic pressure term is less well defined.

The total potential, 9, is the sum 93 = gph + (P - + - n) - (1)

The total potential being everywhere the same at equilibrium, the hydro- static pressure at a given height will vary as @ and z vary, in such a way as to keep the term in brackets constant. In particular, there is no reason why P should not assume positive values in the immediate vicinity of solid surfaces where fi and rc may become large.

The total potential may be measured by allowing the soil system to come into equilibrium with a system of known potential components. Such a system may commonly be a manometer, the only potential components being directly measurable gravitational potential and hydrostatic pressure : equilibrium may be attained either by adjusting the manometer column or by altering the soil air-pressure, thus changing the arbitraxy pressure datum, as in the pressure-plate method of Richard~.~ Alternatively the soil system may be brought into equilibrium, via the vapour phase, with a hypothetical water surface at a hydrostatic pressure calculable from the

Gouy, J . Physique, 1910, (4) 9, 457. Richards and Fireman, Soil Sci., 1943, 56, 39;.

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online

80 SOIL GEOMETRY AND SOIL-WATER EQUILIBRIA

vapour pressure. Potential measurements by means of freezing point depression seem to be ambiguous of interpretation, since the results depend upon whether one assumes the ice formed to be released or not released from the pressure deficiency,lo l1 but fortunately the potential range for which this method is particularly suited is readily covered by the pressure- plate method. The total potential being known, the sum (P - $ : n), which we may call y, is clearly known, but we cannot at present precisely assess the separate contributions. We may write (I) in the form, q=gph+ty, and y is referred to as the capillary potential, without intentional stressing of the surface tension component which the term connotes. The capillary potential is commonly expressed as a pressure deficiency, from its method of measurement, and is not to be understood as referring necessarily to the hydrostatic pressure within the soil solution.

Pressure deficiency, dynes x I O ~ .

FIG. 1.-Moisture characteristic and shrinkage curve for a kaolin clay block.

The capillary potential components, and therefore y itself, will determine the moisture content of a given soil. The void space is continuous but not of uniform cross-section. It may be regarded as a series of cells connected to others via channels of smaller cross-section. For a given pressure deficiency a cell will be empty of water if its largest interconnecting channel is too large to accommodate an air-water interface of sufficiently sharp curvature to maintain that pressure defi~iency.~ As the suction is increased, cells will empty in succession, giving a suction-moisture-content curve characteristic of the soil; such curves have been called moisture charac- teristics.l If we make the assumption that large interconnecting channels are associated with large cells, and if we are justified in ignoring the contri- butions of fl and n to y (as in the case of non-shrinking, light, sandy soils at all but the lowest moisture contents), the moisture characteristic may clearly be interpreted as indicating the pore-size distribution in the soil. The moisture characteristic of a structured clay soil at low pressure deficiency can also be interpreted as the size distribution of the inter-crumb pores,l which are large enough for the contributions of $ and n to be negligible in this range of y.

Edlefsen and Anderson, Hzlgavdia, 1943, 15, 31. 11 Schofield and Botelho da Costa, J . Agvic. Sci., 1938, 28, 6 4 4

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online

E. C. CHILDS AND N. C. GEORGE 81

By contrast, Fig. I a and I b shows the moisture characteristic and the shrinkage curve (following Haines) respectively of a kaolin clay block initially wetted beyond the “ sticky point.” The range AB of Fig. I a corresponds to the range A’B’ of I b, and it will be seen that practically the whole loss of water is accounted for by a decrease of total volume, the initial and final air contents differing by only I CC./IOO g . dry clay. The mechanism of water removal is clearly not that which enables us to interpret the moisture characteristic as a reflection of pore-size distribution. The suction removes water by drawing the solid particles closer together, whether by orientation of the kaolin mineral plates into lower potential energy positions or by interpenetration of double-layers.12

The pressure required to allow water to re-enter an empty cell is higher (the suction less) than that at which the cell emptied, and consequently a moisture characteristic of the first-mentioned type is not reversible but shows hysteresis4 The difference between the emptying- and filling- pressure will depend upon the relative sizes of the cell and connecting channels, and most characteristics cannot be explained without assuming a proportion of cells which show little or no hysteresis.

If moisture characteristics of the second type involve a reorientation of mineral particles, they also might be expected to be irreversible, for one cycle at least of moisture change, so that the presence of hysteresis is not necessarily an indication of a particular mechanism of drying, nor a sure guide to the interpretation of the characteristic. Re-wetting of the kaolin sample already referred to could not readily be carried out in our pressure- plate apparatus, but an almost uncontrolled re-wetting to the “sloppy point ” brought the moisture content and volume back to the initial values ; we cannot therefore say there was no hysteresis shown by this sample, but there was no non-recoverable shrinkage. We propose in the near future to compare results of this kind with moisture and volume changes of the same materials under direct applied pressure.

Perhaps the soil property most obviously governed by pore-size distri- bution is the permeability to fluids. Formulae have been proposed inde- pendently by various workers,13 l4 l5 relating permeability to porosity and specific surface, and bearing a close resemblance to one another. In general it may be said that too small a range of experimentally available porosity prevents a satisfactory test as regards this factor, but, as regards specific surface, there is f a i r agreement between experiment and theory for such materials as have independently calculable specific surface. Such theories may be criticised on the grounds that, whilst porosity and specific surface are uniquely determined by the void-space geometry, which also determines permeability, the converse is not true, and it is possible to conceive of different pore-size distributions, with presumably diflerent permeabilities, yet having the same porosity and specific surface. Further, the void-space geometry undoubtedly accounts for the anisotropy which most soils exhibit to greater or less degree, since, for example, a structured soil with cracks lying predominantly in equipotential planes has lower permeability than that same soil with the equipotential planes perpendicular to the cracks. The specific surface would, of course, be the same in both cases, and might, moreover, be very little affected by the presence of the cracks, to which appreciable permeability is primarily due. Empirical approaches to the relationship between pore-size distribution and permeability have been

la Schofield, Trans. 3rd Int. Congr. Soil Sci., 1935, I , 30. l3 Fair and Hatch, J . Amer. Waterworks Assoc., 1933, 25, 1551. 1‘ Kozeny, Ber. Wien Akad., 1927, I-, 271. l5 Zunker, Trans. 6th Comm. Int. Soc. Soil Sci. B, 1933, p.18.

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online

82 SOIL GEOMETRY AND SOIL-WATER EQUILIBRIA

made by attaching varying weight to the contributions of pores lying in various size-groups.ls l7 l8

We have sought a theoretical treatment which brings out more clearly than hitherto the mechanism involved, and have tested it by measurements on a coarse sand fraction in which the pore-size distribution is certainly to be inferred from the moisture characteristic. The test permits the study of a great range of effective porosities, and also results in a measurement of direct interest in soil science, namely, the variation of soil permeability with moisture content .

the exposed ends, A and B, will exhibit pore-size distributions f(r) and F(r) where f(r)dr is the area of A devoted to pores of size between r and r + dr, whilst F(r) is similarly defined with respect to B. F ( r ) and f(r) are revealed by the moisture characteristic alone if the sand is isotropic, but may need the assistance of direct microscopic examination in various planes if the medium is anisotropic. When A and B are brought together to constitute a continuous column, the total area devoted to the sequence of a pore of size o followed or preceded by one of size p will be proportional to f(o)dr. F(p)Gr + f(p)dr . F(o)dr. If both columns consist of similar soil, so that the whole is a uniform-conducting column, then f(r) is identical with F(r) , and the area devoted to the given sequence is proportional to f(a)Sr . f(p)dr. We shall make two assumptions which involve small errors in opposite senses; first, that cells make no contribution to permeability other than by the direct sequence of the above type, and second that the permeability contribution is limited by the smaller cell in the sequence. The first assumption, by ignoring leakage via more devious paths, underestimates Permeability, the second, by ignoring the potential-drop in the larger cell, overestimates it. Suppose o<p ; then the number of flow-limiting cells in the sequence cross-section considered is proportional to f(o)Sr . f(p)6r/02. By Poiseuille’s equation, the contribution to flow of each individual cell-sequence is proportional to 04, hence the con- tribution, dK, of all the cells of this sequence to the permeability is dKa f(o)dr . f(p)dr . oz and the total permeability, K , is given by

Consider two soil columns of unit cross-section ;

R R

U==o p=o K a C C f(o)dr. f(p)dr . o2 * (2)

In (2) the final factor changes to p2 when p< a. The summation is carried out numerically from the moisture characteristic, and by stopping a t a chosen value R one can compute the permeability of a soil at any desired moisture content, i.e., a t a desired effective porosity.

The experimental determination of permeability at varying moisture content is beset with difficulties. The hydraulic potential gradient which produces the flow also plays its part in determining the moisture content, which therefore varies along the flow-path. Determinations both of the potential gradient and of moisture content are not straightforward. How- ever, if one can employ a sufficiently long column, it has been shown ls that the moisture content and pressure deficiency are sensibly constant over a considerable length. The potential gradient in a vertical column is therefore the gravitational gradient only and is known without measurement. Con- stancy of moisture content enables one to use indirect electrical (apparent capacitance) methods of measurement of moisture content without the use of troublesome guard-rings in alternating current circuits. Manometers

l6 Baver, Proc. Amer. Soil Sci. SOC., 1938, 3, 52. l7 Nelson and Baver, Proc. Amer. Soil Sci. SOC., 1940, 5, 69. l8 Smith, Browning and Pohlson, Soil Sci., 1944, 57, 197. lS Childs, Soil Sci., 1945, 59, 405.

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online

E. C. CHILDS AND N. C. GEORGE 83

were used only to indicate when conditions were steady after a change of rate of flow.

Fig. 3 shows the computed permeability curve for a sand fraction (moisture characteristic as in Fig. 2) separated between the I mm. and 0.5 mm. sieves, together with the experimentally observed curve, With which the computed

F I G . 2.-Moisture characteristic of a sand fraction used in obtaining subsequent figures.

- observe

- - - - - - - compuhd (Kozmy qpe formulo).

FIG. 3.-Permeability of the sand fraction as a function of moisture content, compared with computed curves, obtained by present method and by a Kozeny-type expression.

curve was matched at an arbitrary point. A second set of points was obtained for the condition of increasing moisture content, which was occa- sionally accidentally achieved by a momentary blockage of the inlet syphon feeding the flow-tube followed by a resumption of full flow. We have not

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online

84 SOIL GEOMETRY AND SOIL-WATER EQUILIBRIA

reproduced this curve since we have not yet enough points, but it seems that the permeability at a given moisture content is greater when that content is decreasing than when it is increasing, which argues a different distribution of water. A curve computed from a Kozeny-type formula, matched at the Same point, is also given.

The dependence of permeability upon moisture content has long been recognised in the definition of the so-called “ lento-capillary point,” the moisture content of soil at which the permeability is no longer appreciable. The “ definition” is, of course, vague and no more than a qualitative appreciation of an observed phenomenon, and we can see that there really is no such recognisable point. Another ‘‘ moisture constant ” illuminated by this discussion is the “ field capacity,” usually untenably defined as the moisture content of soil when freely draining under gravity. The field capacity is a property which is obtrusive mainly in dry regions where irrigation

FIG. 4.-Diffusion coefficient of the sand fraction as a function of moisture content.

is practised. The observed facts are that water moves down the soil profile relatively rapidly during the irrigation and for a few days subsequently, the added water redistributing itself to wet a certain depth to a certain moisture content. Thereafter the water moves relatively slowly, moisture content changes being due mainly to removal by the crop. There is no suggestion of complete attainment of equilibrium, but only of a sudden rapid decrease of movement after an initial free-moving stage. The moisture content a t the assessed “ end ” of this initial stage is the field capacity. Redistri- butions of water of this kind are reminiscent of diffusion.20 Darcy’s law, for a vertical potential gradient, is

dq/dt = -Kdp/dh . . . (3)

20 Childs, J . Agric. Sci. 1936, 26, 527.

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online

E. C. CHILDS AND N. C. GEORGE 85

where dq/dt is the apparent flow velocity, h is height above datum and c is the moisture content. Eqn. (4) is the diffusion equation with diffusion coefficient AC given by

The term gp . &/dc will often be negligible. Fig. 4 shows Kdy/dc as a function of c. for the sand described above. Up to about 25 yo water,

K = K(gp . dh/dc + dy/dc). . - (5)

the two factors change in opposite senses, with IC at a low value and changing relatively slightly, but above 25 yo water both factors increase together resulting in a rapid growth of diffusion co- efficient. Rough ‘‘ irrigation ” experiments (see Fig. 5) with initially dry sand in sectioned glass tubes qualitatively confirmed this conclusion, the slowly moving and steep water-front reflecting slow diffusion for moisture con- tents up to about 25 %. This figure also justifies the neglect of the gravity term excepting where the diffusion coefficient is in any case high. We do not wish to dismiss hysteresis as an agent in maintaining a long-lived moisture gradient,21 but to emphasise the existence of another factor which does not involve recognising such a gradient as a case of static equilibrium.

I

Depth. mm. FIG. 5.-Moisture distribution in a sand

column at various times after “ irriga- tion.” The initial water intake, mth water front advancing to about 5 cm., took place in only 7 sec.

Others of the moisture ‘* constants ” (e.g., wilting point) seem to fall either in the group which includes field capacity, being characterised by inherent importance, explicability in terms of more-or-less understood factors, but impossibility of precise definition ; or into a group characterised by precise. definition in arbitrary terms, and including, for example, the hygroscopic capacity, the “9, 4-2 point,” the moisture equivalent and the recently introduced 15 atm. and Q atm. points. These latter “ constants ” often owe their existence to the fact that, whilst capable of reproducible laboratory determination, they approximate to certain significant but indefinable constants of the first group.

81 Schofield, Trans. 3rd Int. Congr. Soil Sci., 1935, 2, 37.

Publ

ishe

d on

01

Janu

ary

1948

. Dow

nloa

ded

by G

eorg

ia I

nstit

ute

of T

echn

olog

y on

31/

10/2

014

14:3

7:06

. View Article Online