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DIVISION S-1-SOIL PHYSICS

Using Surface Crack Spacing to Predict Crack Network Geometry in Swelling Soils

Faculty of Agricultural Engineering, Technion, Haifa 32000, Israel

agvictor{at}tx.technion.ac.il

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Prediction of the geometrical characteristics of steady crack networks in swelling soils is possible using a previously published model based on concepts of multiple cracking and fragmentation. Applications of the model require estimation of the model parameters: the thickness of the upper intensive-cracking layer, z_o, and maximum depth of the cracks, z_m. A technique of such an estimation is proposed based on the correlation of these parameters with the mean spacing between the shrinkage cracks at the soil surface. The data from three previously published papers are used for validation of the correlation. The data are related to 10 different soil profiles and different geographical areas. Estimates of standard deviations of comparable values show their overlap within the limits of measurement errors.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
SHRINKAGE CRACKS in swelling soils are a major factor determining their hydraulic properties. The development of a crack network is a complex process consisting of a number of stages. Chertkov and Ravina (1998) proposed a probabilistic approach to the description of the quasi-steady state of average cracking as a result of drying based on the concepts of multiple cracking and an intensive-cracking layer. They introduced two parameters: the maximum crack depth, z_m, and the thickness, z_o, of an upper layer (as a rule of a few tens of centimeters) of intensive cracking at the quasi-steady state. Parameters z_o and z_m, together with a depth-linear shrinkage curve (i.e., a vertical profile of linear shrinkage), enable us to estimate, at the stage close to steady-state cracking, the distributions of crack width, the surface area of a crack cross-section, and the crack volume as functions of soil depth and crack tip depth. However, measurement of z_o and z_m, based on their definitions (Chertkov and Ravina, 1998), as well as their independent estimation based on crack width and depth distributions, using the least-squares criterion (Chertkov and Ravina, 1998), are time-consuming. Estimation of z_m as the depth of the ground water level (and then z_o 0.1z_m) (Chertkov and Ravina, 1998) can be insufficiently accurate.

The objective of this work was to propose and validate, by available data, a simple method of estimating z_o and z_m based on their correlation with the measurable mean spacing, S, between shrinkage cracks at the soil surface at the steady state. The value of S is easily measured, using a sufficiently long tape, as the mean spacing between the crack intersections with the tape (Zein el Abedine and Robinson, 1971).

For verification of the technique to be proposed for estimating z_o and z_m values, we used data from Zein el Abedine and Robinson (1971), Yaalon and Kalmar (1984), and Dasog et al. (1988) on the mean crack spacing at the soil surface (see below). It is worth noting that these researchers published data of five types on soil cracking relating to different geographical areas. For validation of their model, Chertkov and Ravina (1998) used the data of four types from Zein el Abedine and Robinson (1971), Yaalon and Kalmar (1984), and Dasog et al. (1988).

1. Data on distributions of crack width at the surface and crack depth (Tables II and IV in Zein el Abedine and Robinson, 1971) were used for estimation of z_o and z_m for eight profiles (Table 6 in Chertkov and Ravina, 1998) using the least-squares criterion.
   
2. Data on the depth-linear shrinkage curves (Fig. 3 in Zein el Abedine and Robinson, 1971; Fig. 2 in Yaalon and Kalmar, 1984; and Fig. 3 in Dasog et al., 1988) were used for Chertkov and Ravina's (1998) prediction of crack volume in soil layers and the total crack volume.
   
3. Data on crack volume distribution (Table VI in Zein el Abedine and Robinson, 1971) and the total crack volume (Fig. 1 in Yaalon and Kalmar, 1984; Table 3 in Dasog et al., 1988) were used for comparison with Chertkov and Ravina's (1998) prediction.
   

Independent data on the mean crack spacing at the soil surface were not used by Chertkov and Ravina (1998); namely, these data are the experimental background of the present paper.

	Theory

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
First, let us summarize the major concepts and relationships of Chertkov and Ravina's (1998) basic model. Notation is summarized in Table 1 .

(1)

where d(z) is the mean crack spacing at depth z. Chertkov and Ravina (1998) also remarked that at the initial transitional stage of drying the concept of the intensive-cracking layer is not feasible. That is, for an insufficiently developed crack system the condition given by Eq. [1] cannot be satisfied, and for some time the layer simply does not exist. In sufficient development of the crack system, when the condition given by Eq. [1] has meaning, the thickness of the intensive-cracking layer, z_o, varies in the range (Eq. [24] in Chertkov and Ravina, 1998)

(2)

Such a crack system is referred to as quasi-steady. At this stage of development the velocities of cracks change smoothly, and the mean crack spacing at depth z, d(z), does not explicitly depend on time, but only through parameters z_o and z_m. z_m increases rather faster than z_o, and in the field of the quasi-steady state (Eq. [2]) z_m is equal to its highest value (Chertkov and Ravina, 1998).

Some Remarks Connected with Scaling
It should be emphasized that the concepts of multiple cracking and fragmentation underlying Chertkov and Ravina's (1998) model include presentation of any macrocrack as the result of a number of random sequential coalescences of increasingly larger cracks across a range of spatial scales, beginning from microcracks.

For instance, many very small cracks develop at the soil surface, but not all of them continue to increase. However a number of cracks with slightly larger dimensions develop by jumps with a sharp increase in velocity (Salganik and Chertkov, 1969). Such macrocracks are referred to as primary ones (Konrad and Ayad, 1997). Their depths can be in the development process from the first centimeters to z_m and, as with spacings, depend on physical soil properties and boundary conditions at the surface and at depth z_m.

It is also worth noting the difference between the theoretical mean crack spacing at the surface, d(0), and the measurable mean crack spacing at the surface, S. At the soil surface, the number of small cracks is very large, and d(0) = 0 (Chertkov and Ravina, 1998). The measurable mean crack spacing, S, accounts for cracks only with a width (at the surface) more than the diameter of flexible wire (1.5–3 mm) for crack depth measurement (Zein el Abedine and Robinson, 1971).

Correlation between Surface Crack Spacing (S) and Steady Thickness of Intensive-Cracking Layer (z_os)
According to Eq. [2], z_o changes from an initial value

(3)

to a steady one

(4)

Equation [3] corresponds with the initial (quasi-steady) state of intensive-cracking layer, which appears in drying with some delay after the appearance of primary shrinkage cracks at the surface. Equation [4] corresponds with the steady crack network. In further drying, crack width and volume may increase without changing the dependence d(z) (Chertkov and Ravina, 1998). Our aim here is to state a number of considerations that are possibly nonstrict, but heuristic, and permit us to formulate an assumption of the above-mentioned correlation that can be checked using available data.

Let us consider the simplest possible mechanism of transition from developing primary cracks to an initial quasi-steady state of an intensive-cracking layer (Fig. 1) . First, sufficiently small primary cracks develop as isolated. When their depth, h (Fig. 1), becomes comparable with the mean crack spacing between primary cracks, S_o (Fig. 1), the cracks begin to interact (Cherepanov, 1979). That is, they form a system and demonstrate an instability of development of the system of parallel interacting cracks (Kuznetsov, 1977) (when some of the cracks stop at a given depth) (Fig. 1). Thus, condition

(5)

determines the transition from developing isolated primary cracks to a qualitatively new state where the primary crack spacing (S_o) at a depth, h, coincides (by order of magnitude) with this depth, and, at the same time, there are cracks of rather more depth (Fig. 1). It is clear that this state of the crack system is identical to the state when an intensive-cracking layer exists, and Eq. [5] is identical to Eq. [1] if we take

(6)

View larger version (9K): [in this window] [in a new window]   Fig. 1 A system of developing primary cracks originating the initial state of an intensive-cracking layer. So is the mean spacing between primary cracks at the soil surface, and h is a depth of primary cracks, coinciding by order of magnitude with spacing So

(7)

According to Eq. [5] through [7]

(8)

Thus, we come to the assumption that spacing between primary shrinkage cracks at the surface (S_o) is approximately equal to the initial thickness of the intensive cracking layer (z_oi).

To pass from the mean spacing between primary cracks (at the surface), S_o, to the mean crack spacing at the steady state (at the surface), S, one can assume, based on Konrad and Ayad's (1997) and Yaalon and Kalmar's (1984) data on the mean crack spacing of a developing crack network, that the secondary cracks divide the spacing between the primary cracks at the soil surface, on the average, into two. An effect of the small superficial cracks of the following generations on the measured spacing S at the steady state at the surface is negligible because such cracks are not accounted for in Zein el Abedine and Robinson (1971) method (see remark at the end of the above section). Therefore, we can assume that

(9)

According to Eq. [3] and [4], the initial (z_oi) and steady (z_os) values of the intensive-cracking layer thickness are connected

(10)

Substituting to Eq. [8] for S_o and z_oi values from Eq. [9] and [10], we come to the assumption that the steady thickness of an intensive-cracking layer (z_os) is approximately equal to the surface crack spacing (S)

(11)

It should be noted that Chertkov and Ravina (1998) did not separately consider the boundaries of the range given by Eq. [2] and did not introduce notations z_oi and z_os. For the steady thickness of an intensive-cracking layer (z_os) and for the thickness of an intensive-cracking layer as a variable they used the same designation, z_o. Since we are interested hereafter only in z_os and S values, to simplify and unify notation we will also write z_o implying z_os. Then Eq. [11] is rewritten as

(12)

.

	Materials and methods

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
For validation of the relation defined in Eq. [12] we used the data of Zein el Abedine and Robinson (1971), Yaalon and Kalmar (1972, 1984), and Dasog et al. (1988) on the mean crack spacing at the soil surface (these data were not used in Chertkov and Ravina's analysis, 1998).

Data of Zein el Abedine and Robinson (1971)
These authors measured crack depths, widths, and spacing at the surface using steady crack networks in Vertisols of the eastern part of the central clay plain of the Sudan and calculated the corresponding crack volume. Data on crack depths and widths for two profiles, GTO3 and GTO4, in central Gezira (clay = 51–58%); two profiles, GTO5 and GTO6 in south Gezira (clay = 60–64%); two profiles, GHATO5 and GHBTO7, in the Gash delta area (clay = 55–67%); and two profiles, O4H and O4MH, in the Link canal area (clay = 71–76%) were analyzed in the frame of Chertkov and Ravina's (1998) model. Below we use data (not previously used) on the mean crack spacing (S) at the soil surface from Zein el Abedine and Robinson's (1971) Table V, and Chertkov and Ravina's (1998) estimates of the intensive-cracking-layer thickness (z_o) for the steady crack networks of the indicated profiles (Table 2) .

Data of Dasog et al. (1988)
These researchers measured crack widths, depths, and spacing of clay soils in Saskatchewan, Canada. We used the data of a site at the Agriculture Canada research farm at Saskatoon (Regina soil; clay = 64–80%). Data (not previously used) on the half mean spacing of primary cracks at the soil surface, S (Table 3 of Dasog et al., 1988), and the estimate of the thickness of the intensive-cracking layer at the steady state, z_o (Chertkov and Ravina, 1998), are presented in Table 2.

	Results and discussion

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
For a statistically convincing comparison of the measured values of S (Zein el Abedine and Robinson, 1971; Yaalon and Kalmar, 1984; Dasog et al., 1988) and model estimates of z_o (Chertkov and Ravina, 1998) (Table 2), we need the independent estimates of standard deviations of the mean values of S and of z_o for all 10 profiles (Table 2). We designate the standard deviations by _S and _zo, respectively.

Comparison between S and z_o
Based on the data of Zein el Abedine and Robinson (1971)
These authors divided the total range of spacing values for every profile into intervals and gave the fractions of the spacing numbers occurring in every interval. These data on the crack spacing distribution at the surface (Table V in Zein el Abedine and Robinson, 1971) allow us to estimate not only the mean value, S, but also the standard deviation, , of the distribution

(13)

where the sum is taken on all the intervals numbered by subscript i = 1, ...; S_i is the mean spacing value of the ith interval; and f_i is the fraction of the spacing number occurring in the ith interval. The standard deviation, _S, of the mean value S_i, is estimated by the value and the total number of measured crack spacings, n

(14)

The approximation given by Eq. [14] corresponds with n > 30 (Hamilton, 1964). The number n is estimated by the total length of measuring tape, L

(15)

In Zein el Abedine and Robinson's (1971) measurements, L = 20 m. Hence, for their eight profiles, the number n is in the range 31 n 51, justifying the approximation of Eq. [14]. The estimates of _S for the eight profiles are given in Table 2.

The relative standard deviation, _S/S, for the eight profiles changes from 0.07 to 0.09. Such small changes allow us to assume that this relative standard deviation is a random variable characterizing the measurement method by itself and does not depend on the soil type. Then from Table 2 we can estimate the average for the eight profiles by using

(16)

as a value characterizing the method.

To estimate the standard deviation, _zo, for the eight profiles we use Chertkov and Ravina's (1998)(Table 6) estimates of z_o and z_m (based on handling Zein el Abedine and Robinson's, 1971, data on crack width and depth using the least-squares method) and compare this estimate, z_o, with another possible estimate, z_o z_m/10.5 (Chertkov and Ravina, 1998)

(17)

The estimate of _zo can be made more accurate compared with that given by Eq. [17]. We assume that the relative standard deviation, _zo/z_o, also characterizes the method of estimation z_o, used by Chertkov and Ravina (1998) and does not depend on a profile. Then the average of the relative standard deviations, _zo/z_o, on all the eight profiles, calculated from Eq. [17] and Chertkov and Ravina's (1998) Table 6 is

(18)

The more accurate estimates, are given in Table 2.

Thus, for Zein el Abedine and Robinson's (1971) eight profiles Table 2 shows that in five cases (the profiles GTO3, GTO4, O4MH, GHATO5, and GHBTO7), there is an overlapping of the ranges S ± _S and z_o ± _zo. In three cases (the profiles GTO5, GTO6, and O4H), the ranges S ± 2_S and z_o ± 2_zo overlap.

Based on the data of Yaalon and Kalmar (1972, 1984) and Dasog et al. (1988)
These researchers used the same methods of measurement as Zein el Abedine and Robinson (1971). Hence, in estimating _S values (Table 2) by S values we can use Eq. [16]. Similarly, in estimating _zo values (Table 2) by z_o values we can use Eq. [18], because the method of estimation of z_o (Chertkov and Ravina, 1998) was the same as in the case of Zein el Abedine and Robinson's (1971) eight profiles.

For all 10 profiles in Table 2 the values of S and z_o coincide within the limits of measurement error because discrepancies between them do not surpass the sum of one or two standard deviations of both values. Hence, the available data do not contradict the proposed estimate of the thickness of the intensive-cracking layer of the steady crack network, z_o, by the spacing of the shrinkage cracks at the soil surface, S. This results in the simple method of estimating z_o (and z_m 10z_o) by easily available measurements of S. Thus, based on these simple measurements as well as data on the depth-linear shrinkage curve (or data on water content profile and shrinkage curve) and using Chertkov and Ravina's (1998) approach, we can estimate the characteristics of the crack network that develops under given conditions.

	Summary and conclusions

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Knowledge of the intensive-cracking-layer thickness, z_o, enables us to predict a number of geometrical characteristics of shrinkage crack networks in a steady state. Direct experimental estimation of this parameter is time-consuming. We propose a simple approach for estimating z_o by measuring the spacing (S) of shrinkage cracks at the soil surface. The correlation of z_o with S was validated by the data of three published works for 10 soil profiles from different geographical areas. It is shown that within the limits of two standard deviations of the estimates of z_o and S that these values coincide. This result opens the possibility of a practical prediction of the geometrical characteristics of quasi-steady crack networks under given conditions.

	ACKNOWLEDGMENTS

 
The research was supported in part by the Technion V.P.R. Fund - M. and C. Papo Research Fund.

Received for publication September 13, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 

• Cherepanov G.P. Mechanics of brittle fracture. New York: McGraw-Hill, 1979.
• Chertkov V.Y., Ravina I. Modeling the crack network of swelling clay soils. Soil Sci. Soc. Am. J. 1998;62:1162-1171.[Abstract/Free Full Text]
• Dasog G.S., Acton D.F., Mermut A.R., De Jong E. Shrink–swell potential and cracking in clay soils of Saskatchewan. Can. J. Soil Sci. 1988;68:251-260.
• Hamilton W.C. Statistics in physical science. New York: Ronald Press, 1964.
• Konrad J.-M., Ayad R. Desiccation of a sensitive clay: field experimental observations. Can. Geotech. J. 1997;34:929-942.
• Kuznetsov, V.M. 1977. Mathematical models of blasting. (In Russian.) Nauka, Siberian branch, Novosibirsk, Russia.
• Salganik R.L., Chertkov V.Y. Reduction of strength under the action of shrinkage stresses. Mech. Solids 1969;4(3):126-133.
• Yaalon D.H., Kalmar D. Vertical movement in an undisturbed soil: Continuous measurement of swelling and shrinkage with a sensitive apparatus. Geoderma 1972;8:231-240.
• Yaalon, D.H., and D. Kalmar. 1984. Extent and dynamics of cracking in a heavy clay soil with xeric moisture regime. In J. Bouma and P.A.C. Raats (ed.) Proc. ISSS Symp. on water and solute movement in heavy clay soils. ILRI, Wageningen, the Netherlands.
• Zein el Abedine A., Robinson G.H. A study on cracking in some vertisols of the Sudan. Geoderma 1971;5:229-241.

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