Physics of swelling and cracking soils

Physics of Swelling and Cracking Sails* E D W A R D E. M I L L E R Departments of Physics and Soil Science, University of Wisconsin, Madison, Wisconsin 53706 Received July 30, 1974; accepted March 10, 1975 The environment of a layer of swelling, unsaturated soil is described by two quantities : The most direct choices are an overburden pressure P and a tensiometer pressure pw. Their difference, P -- p~ is the swelling pressure, P,. The water volume fraction 0 of rigid unsaturated soils depends only on p~. The porosity f of saturated swelling soils depends only on P,. Thus, it is convenient to regard the characteristics of general unsaturated swelling soils as dependent on p~ and P,. Simplifying to monotonic histories, the characteristics can be represented by two functions, o(p~, P,) and f(p~, P,). Introducing Darcy's law and the continuity condition leads to solutions for steady or unsteady state profiles subject to a "unidirectional" rheological constraint (zero horizontal strain). The advantage of this approach is that all the quantities employed can be measured physically and in situ. This contrasts to the use by Philip, Bolt, and others of an "overburden potential" described by an integral over a series of states of the properties of the soil. The motivation for this simpler approach was an interest in the nature of the irreversible or hysteretic characteristic of swelling soils. Experimentally, the advantage of a system based on directly measurable quantities is obvious. Surface cracking of swelling soils is discussed in the same terms but under a different (isotropic) theological constraint. INTRODUCTION Philip has stimulated a new interest in the theory of swelling soils, p a r t l y because of an interesting approach (5), and p a r t l y because of an interesting oversight t h a t provoked argum e n t (10). I n his thesis work, R a a t s successfully applied sophisticated concepts of cont i n u u m mechanics to general porous m e d i a (7, 8). The colloid chemistry of clay systems has been studied for m a n y years, in the course of which the general effects of applying external mechanical pressure to clay slurries were also investigated (1). I n contemplating an experimental approach to hysteresis of swelling media, I found myself dissatisfied with the concept of overburden potential ~2, even as refined b y Philip in collaboration with Bolt (2, 6). The purpose of this p a p e r is to outline an alternative approach to swelling media in which all the quantities employed can be effectively measured in situ, and * Presented at the 48th National Colloid Symposium.

in which the general effect of ~ will be transferred to the straightforward and experimentally measurable overburden pressure P . (This does not m e a n t h a t P can simply be written down in place of ~, of course; it is used in a basically different way.) T h e present approach m a y eventually lead to a somewhat more elaborate point of view on cracked surface layers than has been implied previously b y Philip. SWELLING SOIL RHEOLOGY Whenever a soil swells or shrinks it undergoes a rheological deformation. T h e n a t u r e of this deformation depends on the mechanical constraints to which the elements of soil are subjected. Conceptually simplest is a pure change of s c a l e - - a n isotropic swelling or shrinking of a soil (which we assume is itself internally isotropic) so t h a t there are no preferred directions either in the m e d i u m or in its external

434 Journal of Colloid and Interface Science. Vol. 52. No. 3. September1975

Copyright ~ 1975 by AcademicPress. Inc. All rights of reproductionin any form reserved.

SWELLING AND CRACKING SOILS constraints. In practice, however, swelling or shrinking is caused by transfer of moisture into or out of the sample, a transfer that is associated with gradients of moisture content. Gradients of moisture content imply corresponding gradients of swelling through the sample. But swelling gradients violate the condition of uniform change of scale. Hence, isotropic changes can occur only in the limit of infinite slowness, a limit that can best be approached in practice by the use of very small samples. The soil in a large flat field, if not cracked, is constrained horizontally to a constant size, i.e., to zero strain. Swelling and shrinking are then confined to one dimension, the vertical. Such vertical swelling and shrinking, constrained to horizontal strain, will be termed unidirectional. In an individual column of a cracked soil, on the other hand, the horizontal constraint must be more nearly one of zero stress rather than of zero strain, although it is clear that there will be more complicated stresses in the transition region near the base of the column, and also near the cracks whenever significant moisture gradients perpendicular to the crack surfaces begin to develop. Leaving such complications for the future, it will be useful to agree on a distinct name for swelling and shrinking deformations under this condition of zero horizontal stress. In elasticity, such a constraint, e.g., that existing in a taut wire, is called an axial stress. (The unidirectional constraint described above could logically have been called axial strain, but two terms so similar would be confusing.) Note that isotropic deformation constitutes the limit of the axial stress system of constraints when the vertical load goes to zero at the top of a column of cracked soil. I t is apparent that moisture gradients must exist in cracked soil columns and that the idea of these columns exhibiting pure axial stresses i s only an approximation, useful for practical simplification. D I R E C T L Y M E A S U R A B L E STRESSES

As an aid to visualization, let us imagine a very thin layer of soil contained between two

435

." .~- Air FIG. 1. L e a k y piston m o d e I - - a unidirectional system thin enough that gravity effects are negligible. P is total, or overburden, pressure; p~ is tensiometer pressure; P , = P -- p~ is swelling pressure. P and P , are positive; p~ m a y be positive or negative.

fiat leaky pistons sliding freely in a surrounding cylinder, as shown in Fig. 1. The lower piston A is assumed to be permeable only to air and the upper piston W only to water. Over the upper piston lies a layer of pure water which is covered in turn by an impermeable piston N. In practice, we would naturally assume that the pistons and cylinders were rigid structures so that (except for edge effects that can be made as small as desired by making the soil layer thin enough) the soil constraint is unidirectional, i.e., zero strain in horizontal directions. (However, for generality, we could at least imagine a similar arrangement for an axial stress type of constraint, an arrangement in which the pistons and cylinder could somehow expand or contract freely to permit a horizontally stress-free constraint. The force balance argument which follows, leading to Eq. rl-], would be equally valid for either type of constraint.) This entire assembly is imagined to be contained in a tank of air at pressure pa. The force on the bottom piston (expressed as force per unit area to put it in units of stress or pressure) will simply be the overburden pressure P, as used by Philip and many others. An opaque piston N separates the air and water; the pressure p~ on this piston is therefore exactly the easily measured tensiometer pressure, which will, of course, be negative in sign for unsaturated soils. The net force exerted on the soil by the water-permeable piston W will be called the "swelling pressure" 1~ of the soil. Since the soil slab is assumed to be made as thin as we wish, we can neglect gravity and

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436

EDWARD E. MILLER

write the force balance equation,

no effect on water content--excluding overburdens so high that irreversible crushing of P = p~ + Po. [1] the particles occurs. Thus, the wetness of rigid From the model, both P and pw are refer- soils is controlled by one parameter pw in the enced to the air pressure p~. We know how to range where it is negative. measure both p~ and P physically: p~ with a For saturated soils the bulk density can vary tensiometer; P by adding up the wet bulk only for the swelling types of soil. In saturated densities of soil above the profile level in swelling soils, if P, is held constant, a char_ge question. If we measure p~ and P, Eq. [-1] in pw can have no significant effect on bulk allows us to regard P, also as an experimentally density because of the large compressional measured quantity. stiffness of the condensed phases of matter No other mechanical stresses or pressures (liquids and solids). However, if P, (which by will be required or used in this discussion, Fig. 1 pushes only on the solid, not on the except for the air pressure pa which is essen- water) is increased, the particles are pushed tially inconsequential. Referring to Fig. 1, if closer to each other, against the spring-like we hold fixed the forces on both of the upper forces of the electrical double-layers that hold pistons, N and W (thus maintaining both pw them apart. Water is thereby squeezed out of and Ps), while changing the air pressure pa the soil much like a sponge being squeezed (indefinitely slowly so that air has plenty of under water. Thus, in the case of saturated time, not only to flow through the air channels swelling soils, the bulk density is controlled by within the soil, but even to diffuse into or out the single parameter P, in the range where it of entrapped air bubbles) the state of the is positive. soil--its wetness or swelling--will be unGENERAL CASE; CHOICE OF GOVERNING affected. This contention neglects such acaSTRESSES demic trivialities as the slight compressibility In the general case of swelling soils that are of solid or liquid phases, or the slight change of surface tension with changing concentration unsaturated, a single parameter is not sufficient to determine the state. This was the point of dissolved air. The chemical/osmotic character of the soil made by ¥oungs and Towner (10), and consolution is important to the behavior of swell- ceded by Philip (6) in his "Reply." Of the three ing soils, but will be taken to be beyond the possible choices of two parameters from the scope of the present discussion. We will concern three that are connected by Eq. [1], the two ourselves only with physical concepts, recogniz- foregoing special cases, requiring p~ for one ing that this is a practical oversimplification and P, for the other, make it most natural to that must certainly be dealt with in some later select these two, i.e., p,o and P,, for characterizing the external (mechanical) influences that discussion. shape the swelling and wetness of a general SPECIAL CASES: (1) RIGID MEDIA soil. (Of course, the force balance, Eq. [1], (2) SATURATED MEDIA permits us to refer to P as well, whenever it Rigid soils (sand, glass beads) have been suits an immediate purpose.) Although a major purpose of this paper is to studied extensively. Their challenge to analysis lies in the unsaturated range, where water exhibit a simpler approach, intended to be content is controlled by the curvature of air- more suitable for subsequent investigations of water interfaces, which is generated by the the nature of irreversible or hysteretic effects pressure p~ of the water relative to air [e.g., in swelling soils, it will nevertheless be con(4)-1. If there is an overburden pressure P, it venient to limit most of the discussion here to will be carried by solid-solid forces with no monotonic stress histories so that the largely effect on pore geometry and accordingly with unknown complications of irreversibility need Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

S W E L L I N G A N D C R A C K I N G SOILS

not be taken explicitly into account at this stage. MEASURES OF WETNESS AND SWELLING; INTERRELATIONS For a general unsaturated system, two independent descriptions of the state of wetness and swelling are needed. Half a dozen of these descriptors have been widely used, any of which is easily related to (almost) any two of the rest. Under these circumstances, it is convenient simply to admit all of them and write down a sufficient number of the interrelations to permit the use of whatever two choices may be most convenient for any particular application. The porosity f (volume of void/total volume), and the water volume fraction 0 (volume of water/total volume) are the most familiar to soil physicists, the dimensionless saturation S = O/f being a familiar alternative in petroleum literature. Wet bulk density pb (total [-mass or weight-I/total volume) is also in common use. To avoid writing g, I have chosen units of weight~volumefor all densities, with p denoting the density of water, om the density of solid matter, and R their dimensionless ratio, R = p/pro. Philip also uses the "wet specific gravity" of the soil ~, which is just oh/p, as well as the particle specific gravity ~,~ which is 1/R.) Referencing both void volume and water volume to the volume of solid has advantages in the analysis of swelling soil profiles, and for these I will conform to the notation used by Philip: the void ratio e (volume of void/volume of solid) ; and the moisture ratio 0 (volume of water/volume of solid). Convenient interrelations between these quantities (other than those already mentioned) are obvious from the basic definitions, for example, (1--f)-~=l+e;

S=O/e; pb/p=

O=O(1--f); (l--f)/R

C2]

Oq-

SOIL C H A R A C T E R I S T I C S ; D E P E N D E N C E OF W E T N E S S - S W E L L I N G u P o n T W O STRESSES

For a given general soil the state of wetnessswelling will be governed by the history of the

437

stresses pw and P, to which the soil has been exposed. From the interrelations [-2-] it is apparent that we can select any two independent quantities from the set of wetnessswelling descriptors. (Note that since f and e serve exactly the same purpose, they are not independent.) We can then measure the experimental dependences of our two choices on the history of (pw, P,). However, by limiting our present discussion to monotonic histories of pw, P,, we have only to assume in addition that these relations are time-scale invariant to reduce the two dependences to two ordinary functions of two variables. In our temporarily restricted context, these two functions will describe the complete characteristics of the medium. To give an example, we might select f(pw, P~) and O(p~,P~) as a convenient pair for exhibiting the characteristics of the medium. Note, however, that two measures of the wetness-swelling type are here represented as dependent on two measures of stress. In contrast Philip (6), for example, treats 0, P - - a wetness and a stress--as the independent variables which govern e and f~--a swelling and a stress. MATERIAL COORDINATES FOR SWELLING PROFILES

Before we consider the Darcian flow of water past the particles of a swelling profile (particles which may themselves be moving up or down en masse as the various layers of the soil profile are swelling or shrinking) we must define a reference system that is designed to keep track of the particle layers as they move. Such reference systems, called "material coordinates," are now standard for analysis of swelling profiles [-e.g., (5, 7-9)-]. Following Smiles, we will use m to denote the solid depth, i.e., the total volume of solid (per unit cross-sectional area) that lies between the surface and a point of the profile. With this definition, regardless of the swelling and shrinking that may be occurring, if we look at a given material depth m we will always find there the same layer of particular particles of soil.

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EDWARD E. MILLER

An increment dz of spatial depth is related to the corresponding increment, din, of solid depth by

dz = dm/(1 -- f) = din(1 + e).

[-3-]

Whenever we integrate [-3"] over a profile region, (1 -- f)-i = (1 + e) will normally be a function of depth. DARCY FLOW ASSUMPTION IN MATERIAL COORDINATES Denoting by v the flux (volume rate per unit area) of water past a fixed material depth m, the classical Darcian assumption of linearity for viscous flow of water past a layer of particles becomes = Kip

-

Op~/Oz]

[4]

where O represents the gravity portion, and --Op~/Oz represents the pressure portion of the driving force. Both z and v are taken positive downward. In [-4"], K must be a function of the state of wetness-swelling of the soil, hence equally a function of (pw, P.), so it will be written K(p~,P,). It is also useful to convert the spatial increment dz in [-4-] into dm by [-3-] to give v =

K(p,~,

P.)

X {o -- [-1 -- f(p,o, P,)]ap~/am}.

[5]

For rigid media the dependence of K on p~ is highly hysteretic, while from experiment the corresponding K (0) dependence has been found to exhibit very little hysteresis--for many practical purposes the hysteresis of K(O) is negligible. When more is known about the irreversible properties of K for general swelling media, it may be found similarly that some form of representation other than K(p,o, P~) will prove convenient. CONTINUITY CONDITION IN MATERIAL COORDINATES For flow systems subjected to unsteady flow conditions, if swelling or shrinking occurs as water is accumulated or depleted, conservation of matter requires the existence of flow rate divergences within the profile. The time

rate-of-change of water volume contained between two depths m and m + dm must match the difference of flow rates across levels m and m + din. Since the water volume content of a unit-area layer is dm and the difference of flow rate between m and m + dm is (Ov/Om) din, the continuity condition is (o/ot)a (p~, P~) = - Or~Ore

[6-]

Eqs. [-5"] and [-6-], along with a stipulation of boundary conditions, permit the analysis (by contemporary computer methods) of unsteadystate one-dimensional problems for profiles of general soils (swelling or not), temporarily subject to our assumed tactical limitation to monotonic histories. When more is known about the hysteresis of swelling systems, generalization will hopefully require only some type of hysteretic expression of the soil characteristics that appear in the Darcy/continuity equations, ]-5] and [-6"], i.e., in the dependences K(p~, P,), f(p~, P,), and v~(Pw, P,). APPLICATION TO STEADY-STATE PROFILES Steady-state profiles for which v is constant, independent of depth, could of course be analyzed by brute force using computer methods. However, in steady state, the water flux v is the same at all depths so that the continuity equation I-6"] is not needed, which simplifies matters somewhat. The essential aim of what follows is to telescope many possible profiles, with different boundary conditions but the same v, into a compact and simple representation. Since for steady states all layers of particles are stationary in height, we can use the spatial version [-4"] of Darcy's Law, rather than the material version [-5-]. When restricted to monotonic histories, the wetness-swelling characteristics of a soil will depend only on the values of p~, P,. This suggests transferring the basic analysis of profiles to a plot of p~ against P,. To this end, we can remove the dz from Darcy's Law by noting that the gravity-load increment across a thin layer is simply the total weight p d z of the layer and is also dP or dP~-Jr" dp,o for that layer.

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SWELLING AND CRACKING SOILS

Solving for dP~/dp~ in terms of the constant vertical velocity v and the soil characteristics pb(p~, P,) and K(p~, P,), we get for all steadystate vertical flows,

dP~/dp~ = 0

--

pdP~, P.) -v/K(p~, P~)

1.

r,,

",,,

"

[-7]

This is just an ordinary differential equation governing the track across a plot of p~ against P, that is followed by a profile having some steady-state water flux v. In practice, since pb and K are experimental characteristics we would start at some point p~, P~, compute the slope by [7-], and proceed in small steps to trace out, with the aid of a computer, the solution associated with that starting point. One might begin with a specified boundary condition p~, P at the top of the profile, convert this to p~, P, by [-1-1, and then compute out the track down into the profile. A more comprehensive approach would be to compute, for a given v, a family of closely-spaced profile tracks, eventually filling the p~, P~ plot with such curves to represent, by interpolation, the tracks of all possible profiles associated with the given flux, v. The advantage of this strategy over computing individual solutions as the need arises is that each individual curve also telescopes at a glance a single-parameter family of top-boundary loads--"piles of bricks." A family of such profile tracks is illustrated in Fig. 2, on which we have also plotted 45 ° lines of constant P for reference. In general, there will be an air-entry value curve, P~v(P~), on such a plot. Above this curve the soil is unsaturated, requiring the use of Eq. [-7-] for solution, as just described. In the saturated region below this curve, pb and K are functions only of P~, which permits a considerably simpler method of solution. This suggests starting the computation of each profile track in the middle, with an arbitrary point p~1, P,1 located on the air entry curve, then propagate the track first upward from that point across the unsaturated region as described, using F7-], and second extend the track downward across

FIG. 2. Steady-state profiles for fixed value of flux v. Illustrates telescoping of families of solutions on a p~, P , diagram. Profiles have top boundary conditions representing: (1) " L ak e bottom"; (2) "wet marsh"; (3) suction-but-saturated; (4) desaturated; (5) top desaturated, under toad P~.

the saturated region by the simpler method, explained below. For the saturated region, disappearance of the p~ dependence of ob and of K permits us to rearrange [-7~, segregating the terms in p~ and in P~ onto the two sides of the equation in order to integrate,

p~ -- pwl = Fo(P~) -- f ~(P~) where

f v(P~) =--fo P"

[-8-]

dP, Pb(P,)/EP -- v/K(e~)-]- 1

The subscript v associates the function F with a specific water flux v. Because of the experimental functions pb(P,) and K(P,) which appear, we must still use a computer to evaluate the integral [8-] for F o(P,), but only once. From Eq. [-8-] the extension into the saturated region from any other choices of P~I, P,1 is obtained by merely translating vertically the same curve of Fv(P~) until it passes through the desired crossing point p~l, P~I. In Fig. 2, five separate profile tracks have been illustrated to show the telescoping of all possible boundary conditions for a given flux v into this single family of curves. The top of Profile 1 is at a positive p~ = P, which would

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440

E D W A R D E. M I L L E R

be the proper boundary condition at the soil surface at the bottom of a lake. The top of Profile 2 is placed at pw = P = 0 which approximates the top boundary condition for any wet marsh. Profiles 3 and 4 also have no overburden, i.e., they have P = 0, at the top. Furthermore, both are under suction (p~ negative) at the top, but the top of 3 is saturated, while that of 4 is unsaturated. Profile 5 is also unsaturated at the top, but by terminating the track short of P = 0 we have illustrated the effect of a passive load imposed on the soil surface. Before we can claim to have analyzed any particular profile fully, we must associate with each point p~, P, on the profile track a corresponding value of the physical depth z. Let some quite arbitrary datum level Zxbe assigned to the air-entry curve crossing-point p ~ , pal. Taking first the saturated zone of the profile, we return to [7], and replace the weight of each layer, dP or dp~ + dP~, by p~(P~)dz; separate the variables; and then integrate to obtain

z -- z~ = G~(P~) -- G,(P,~) where

P"

dP.

[-9]

G~(P,) = j o

p~(P~) - p + ~ / K ( P ~ )

Eqs. [-8] and [9] thus yield two functions of P~ from which we can determine for the saturated portion of any profile transmitting a steady flux v the dependence on depth z of both P~ and p~. To associate z values with the unsaturated portions of the p~, P~ track for which the stepby-step computer calculation from [-7] was described earlier, let us first designate that computed track through the unsaturated region that ends on the air-entry curve at p~l, P,1 by the function H~i(p~) = P~, where of course H,i(p~i) must be just P,i. As before, we must restore z to [-7] by replacing the layer weight ob(Pw, P,)dz by dP, i.e., by dp,~ + dP~. However, since we are now working in the unsaturated region, we can solve for either p~ or P~. We will choose p~ this time because its

differential remains after that of P~ disappears, and we can replace the P, dependence of K by Ps = Hsi(pw), giving Zi - - Z = J d ( p w )

where J d(Pw) =

[-10]

f p~l dp~ ~0 p -- v/K[pw, H~l(pw)]

In general, the unsaturated track P~ = H,i (pw) must be calculated separately for each crossing point pwi, P,i. Because of the appearance of H,i(pw) in [-10], this function also must be computed separately for each new choice of air-entry crossing point pwi, P,1. The end result of all this is a set of profile tracks across both the unsaturated and saturated regions, as shown in Fig. 2, and for each track the associated values of z (relative to the arbitrary Zx). I t is, of course, trivial to shift the z reference at the end to give any other desired reference, for example, always to make z zero at the top of any profile. The equilibrium case, v = 0, is especially simple since from r l 0 ] J01(pw) is just ( p ~ l - pw)/p, giving z - zi, for the unsaturated zone as well as the saturated. Finding values of P, to associate with each p~ value is just a little easier than it was nonzero fluxes, since the v/K term disappears from Eqs. [-7] and [-8]. A P P L I C A B I L I T Y OF G R A V I T Y - F R E E INFILTRATION (BOLTZMANN PROBLEM)

Although the Darcy and continuity equations in material coordinates, F5] and [-6], form the basis for computer solutions of general transient problems, it will be useful to show that the simple system described here with no dependence on thermodynamic arguments leads directly to the classical Boltzmann solution for gravity-free infiltration into an infinite uniform medium when a step-function in pw is applied to the face at time zero. Though gravity is assumed absent, we will permit any desired constant envelope pressure P0 to be present, so from [-17, P~ becomes Po -- pw. Thus, even

Journal of Colloid and Interface Science, Vol. 52, No. 3, September 1975

SWELLING AND CRACKING SOILS for unsaturated systems, p~ becomes the only independent variable occurring in the Darcy and continuity equations, i.e.,

v = -- K(p,~, Po -- p,,,) x [1 - / ( p ~ , Po -

p~)]

X (Op~/Om) av/am

= -

(a/aOo[p~,

Po -

[113

p~).

The soil characteristic function 0 = 0 ( p ~ , P 0 - p~) can be inverted into a fcuntion pw = p,o(O, Po), whereupon K and f can be regarded as functions of the variable O, since P0 is only a constant parameter. Combining and expanding Opw/Om by the chain rule gives

0

--O

0{

= --

441

Subject to the boundary conditions U(01) = 0 ;

1/U'(Oo) = 0.

[-171

Note that while this cartesian coordinate solution for swelling media is formally analogous to the cartesian coordinate solution for rigid media given in (4, Appendix II), or in the author's contributed Appendix 2 in (3), the extensions of the solution in these two references to cylindrical and to spherical onedimensional infiltration (with a pole at the origin) are not applicable to swelling media because of the impossible rheological problem posed by swelling of the soil around the source. T H E P E N E T R A T I O N OF CRACKS C H A N G E S T H E RHEOLOGICAL CONSTRAINT

K ( a , Po)

When a field of swelling soil is allowed to dry on top, in due time a system of surface cracks X [1 -- f(O, eo)] \--/PoO0 am!--'" [-12] will appear that are characterized by some average spacing distance
Ot

am

1

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EDWARD E. MILLER

very small specimens and then change their conditions. This is a somewhat special arguenvironment, i.e., change p~, quite slowly. ment; it remains to be seen whether the Suppose that we have measured experimentally simpler assumption of Philip and others--that the characteristics, O(P,) say, of a saturated cracked layers can be treated macroscopically swelling soil under an isotropic, zero-load as simply unsaturated l a y e r s l c a n be a useful constraint. For a cracked field of this soil-- approximation in practice. It must be admitted assuming no air-entry into the body of the that field drying after wetting does tend to free-standing columns--the overburden pres- follow nearly the same history-track time after sure will be close to zero so that P, ~ - - pw. If time. we also know the tensiometer pressure p~ at Nevertheless, even if this should turn out various depths along these columns, we can to be a practical approximation, it would be deduce O(P~) = 0 ( - p w ) at these points. As useful to achieve a better understanding of the the columns shrink further, measured by de- processes of cracking, per se. For example, why crease of 0, water must flow away, the flow is the crack spacing (S) an order-of-magnitude being governed by the Darcy and continuity larger in some cases than in others? Philip has equations [-5~ and [-6~. If the bulk of this flow suggested (in discussions) that the reciprocal is upward or downward, the solution can be of characteristic length should be governed by calculated with reasonable accuracy for com- the surface gradient of the logarithm of almost parison with field observations of a cracking any measure of swelling; for example, the crack soil. (Raats, in an unpublished manuscript, has spacing (S) might be some constant times attacked one-dimensional flow along rods in a [-(de/dz)/e'] -1. This suggestion seems to me way that might well be extended to this type consistent with scaling concepts. For example, of analysis.) If horizontal flow to the crack if the stresses and wetness-swelling of two surfaces and out through the moving air inside profiles are made exact scale models of each the cracks becomes dominant, the calculations other, it seems to me that a rheological scaling become more difficult, but perhaps not entirely argument could be expected to show that the impractical. crack spacings in the two cases would conform Philip, Smiles, and others have tended to to the scaling ratio. regard the cracked surface layers of fields as Most gardeners have observed that wellsimply the normal unsaturated state of the structured swelling soils, soils that are easily soil. For purposes of computing overburdens desaturated at the microscopic level, somefor the deeper layers, there is no question that times show surface cracking as well. So long the actual weight of the cracked surface layers as P is nearly zero - p w ~---Ps so that there is is required. (Their argument, as I understand essentially one controlling stress-variable in it, is that if O is determined by the cracking this situation. Accordingly, there is no obvious process, in effect this plays the same role as O barrier to extending the foregoing suggestions determined by microscopic air entry at the to the cracking of microscopically unsaturated scale of pores.) However, for a fine-clay layer, soils. cracking can occur for very moderate suctions, At significant depths within field profiles, P provided the gradient of swelling is quite con- will not be negligible. An important point is siderable near the surface. On the other hand, that for any given layer of particles the variaif suction is applied from the bottom, then tion of P with cycles of wetting and drying is quite large values of suction can be reached an order of magnitude smaller than the value without the appearance of surface cracking or of P itself. For any given material depth mi, it any other form of air entry. In other words, air should therefore be a fairly good approximation will not enter at a given suction under the latter to treat P(mi) as a constant, so that conditions, whereas air-filled cracks will enter, P~ ~ P ( m ~ ) - p~. This applies not only to even at much lower suctions, under the former the lower parts of cracked-soil columns, but to Journal of Colloid and Interface Science, Vol..52, No. 3, September 1975

SWELLING AND CRACKING SOILS the general treatment of field profiles of swelling soils at all depths. Barring the works of man, including massive erosion or siltation, the overburden stress P for each layer of all soil profiles has been nearly constant, with very minor variations due to rainy and dry periods, for scores of centuries. It strikes me that the simplifying value of this layer-by-layer constancy of P has not yet been sufficiently appreciated. SUMMARY Although for simplicity and for lack of knowledge of the character of hysteretic (or perhaps history-function) properties of swelling soils, we have dealt here with monotonic processes, it has been shown that useful previous results of swelling soil analyses can be obtained in terms of directly-measurable physical parameters alone--quantities that therefore can be retained for later studies of more realistic reversing and cyclic processes. This approach avoids difficulties inherent in analyses that are made in terms of such quantities as Philip's overburden potential ~2, which are defined through arguments of reversible thermodynamics and in terms of experimentally-difficult integration processes. In addition, the view that cracked surface layers of swelling soils should be regarded as simply the unsaturated state of that soil has

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been questioned and a different approach with rheological aspects has been suggested. ACKNOWLEDGMENTS The author acknowledges useful discussions with J. R. Philip, P. A. C. Raats, D. E. Smiles, E. G. Youngs, and R. D. Miller, and beyond this is grateful to J. R. Philip for his active encouragement in attacking this problem. The Division of Environmental Mechanics, C.S.I.R.O. (Australia), and the Wisconsin Alumnae Research Foundation contributed to the support of the work. REFERENCES 1. BLACKMORE,A. V., AND MILLER, R. D., Soil Sci; Soc. Amer. Proc. 25, 169 (1961). 2. GROENEVELT, P. H., AND BOLT, G. H., Soil Sci.

113, 238 (1972). 3. HILLEL,D., "Soil and Water: Physical Principles and Processes," Academic Press, New York, 1971. 4. MILLER,E. E., ANDMILLER,R. D., J. Appl. Phys. 27, 324 (1956). 5. PmLIP, J. R., Water Resources Res. 5, 1070 (1969). 6. PI-IILIP,J. R., Water Resources Res. 6, 1248 (1970). 7. RAATS, P. A. C., AND KLUTE, A., Soil Sci. Soc. Amer. Proc. 32, 452 (1968). 8. RAATS,P. A. C., Afro KLUTE, A., Soil Sci. 107, 329 (1969). 9. StoLES, D. E., AND ROSENTHAL,M. J., Austral. J. Soil Res. 6, 257 (1968). 10. YOUNGS, E. G., AND TOWNER, G. E., Water Resources Res. 6, 1246 (1970).

Journal of Colloid and Interface Science, Vol. 52, No, 3, September 1975