Constriction Size of Geotextile Filters

Transcription

1 Constriction Size of Geotextile Filters Ahmet H. Aydilek, M.ASCE 1 ; Seyfullah H. Oguz 2 ; and Tuncer B. Edil, M.ASCE 3 Abstract: Filtration performance of nonwoven geotextiles strongly depends on pore opening constriction size, i.e., the minimum opening size of flow channels across the geotextile. Currently available methods of pore opening characterizations do not provide accurate information about the constriction size. This paper presents the constriction size distributions of eleven nonwoven geotextiles commonly used as filters based on a probabilistic approach coupled with image analysis, named constriction size (CONS). The image analysis method was developed using mathematical morphology operations. Randomness in the structure of geotextile media was modeled using Markov chain processes. Two characteristic constriction sizes, C 95 and C 50, were determined and compared with the available results from laboratory tests, theoretical equations, and manufacturer s reported apparent opening sizes. The CONS-based C 95 constriction size was comparable to C 95 determined using the bubble point test. On the other hand, the CONS-based C 95 was consistently lower than the manufacturer s reported apparent opening size, which is typically determined on the basis of the dry sieving test. Similarly, the CONSbased C 95, was highly comparable with the value suggested by a theoretical equation. The observed differences in the constriction sizes determined by CONS versus the available physical tests may be reflective of the nature and indirectness of the physical tests rather than inaccuracies associated with CONS, which is based on direct observation of pore openings. DOI: /(ASCE) (2005)131:1(28) CE Database subject headings: Geosynthetics; Digital techniques; Imaging techniques; Pore size distribution; Markov chains. Introduction 1 Assistant Professor, Dept. of Civil and Environmental Engineering, Univ. of Maryland, 1163 Glenn Martin Hall, College Park, MD aydilek@eng.umd.edu 2 Staff Systems Engineer, Qualcomm, Inc., 5775 Morehouse Dr., AM-215F, San Diego, CA soguz@qualcomm.com 3 Professor & Chair, Dept. of Geological Engineering, and Professor, Dept. of Civil and Environmental Engineering, Univ. of Wisconsin- Madison, Madison, WI edil@engr.wisc.edu Note. Discussion open until June 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 12, 2003; approved on April 30, This paper is part of the Journal of Geotechnical and Geoenvironmental Engineering, Vol. 131, No. 1, January 1, ASCE, ISSN /2005/ /$ Filters are used in geotechnical engineering to prevent erosion of soils in contact with the filter without impeding the flow of seeping water through the soil. The two types of filters commonly used are granular filters composed of gravel or sand, and woven or nonwoven geotextiles. Nonwoven geotextiles are more frequently used as filters than woven geotextiles due to their high permittivity and nonuniform pore structure. The heterogeneous pore structure of nonwoven geotextiles unlike woven geotextiles is similar to that of a well-graded granular filter and this heterogeneous pore structure provides a better retention capability. Accurate information is needed about the shape and size distribution of the pore channels of nonwoven geotextiles to assess their filtration performance. Various investigations have examined the pore opening size distribution of granular filters and its effect on filter selection criteria (Wittmann 1979; Kenney et al. 1985; Giroud 1996; Bhatia et al. 1998; Indraratna and Locke 2000; Locke et al. 2001). A number of methods also have been developed to determine the pore opening size distribution (PSD) of nonwoven geotextiles (Prapaharan et al. 1989; Faure et al. 1990; Fischer et al. 1990; Bhatia et al. 1994, 1996; Giroud 1996). Recent studies (Fischer et al. 1990; Giroud 1996; Vermeersch and Mlynarek 1996; Giroud et al. 1998) indicated that constriction sizes of a filter, the minimum pore opening sizes of its flow channels, impact filtration performance and these sizes should be determined before applying filter selection criteria. Constriction size is different than pore opening size. It may be located at any depth inside the geotextile and not necessarily at the top surface. In their current status, a majority of the methods used for geotextiles cannot determine constriction size. Dry sieving test (ASTM D 4751) (ASTM 2003a) and bubble point test (ASTM F316) (ASTM 2003b) are the two commonly used methods for determining the pore opening size distribution of geotextiles. Dry sieving test measures the nearly largest pore opening size O 95 of a geotextile, called the apparent opening size (AOS); however, the smaller pore opening sizes determined by this method may not be accurate due to various problems in testing procedure (Van der Sluys and Dierickx 1991; Giroud 1996). Furthermore, this method does not determine the constriction size in flow channels. However, the method is widely used by the geosynthetic manufacturers because an American Society of Testing of Materials standard is available for it (ASTM D 4751). The bubble point test, on the other hand, determines constriction sizes indirectly by approximating them from the measured minimum constriction area. Recent studies (Bhatia and Smith 1995, 1996) indicated that wetting fluid, air pressure and equipment type plays a major role on the results and a new ASTM standard test procedure is currently being developed (Christopher, personal communication 2002). Image analysis method is perceived to provide a more direct and reliable way of determining constriction sizes of a nonwoven geotextile. However, it has not been extensively used due to the lack of a standard and proven procedure. Most existing image analysis methods use two-dimensional planar or cross-sectional 28 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005

2 Table 1. Properties of Nonwoven Geotextiles Used for Constriction Size Distribution Determination Geotextile Stucture, polymer type Mass/unit area g/m 2 Thickness (mm) Apparent opening size (AOS) (mm) Porosity (%) Permittivity s 1 J NW,NP,STF,PP I NW,NP,STF,PP P NW,NP,STF,PP L NW,NP,STF,PP K NW,NP,CF,PP C2 NW,NP,CF,PP M NW,NP,CF,PP C4 NW,NP,CF,PP C5 NW,NP,CF,PP D4 NW,NP,STF,PP/PET D1 NW,NP,STF,PP/PET N1 NW,CF,HB,PP N2 NW,CF,HB,PP N3 NW,CF,HB,PP N4 NW,CF,HB,PP Note: NW=nonwoven, NP=needle-punched, HB=heat-bonded, STF=staple fiber, CF=continuous filament, PP=polypropylene, PET=polyster. All properties are the manufactruer s reported values based on the ASTM (2003c) standards with the exception of the porosity values, which were determined using the method described by Wayne and Koerner (1993). The lower bound for the AOS values is reported based on a personal communication with the manufacturers. view images of a geotextile (Masounave et al. 1980; Smith 1993; Bhatia et al. 1993; Elsharief and Lovell 1996), and are usually operator-dependent. Moreover, most of the methods do not consider randomness in manufacturing of nonwoven geotextiles, and the measurements are image-specific rather than representing the entire geotextile layer. Thus, there is a need for a methodology that more accurately and consistently determines the constriction sizes in nonwoven geotextiles. In response to this need, a new method, constriction size (CONS), was developed. This method uses image analysis and a probabilistic description of constriction size. The procedure is also applicable to granular filters although this is not presented here. The method consists of three steps: specimen preparation, image analysis, and probabilistic analysis. It uses planar and cross-sectional views to capture the three-dimensional structure of a nonwoven geotextile and constructs a probabilistic description of pore channels using Markovian process. The characteristic constriction sizes, C 95 and C 50, were determined for a variety of geotextile specimens and compared with the values based on bubble point tests and theoretical equations. The same constriction sizes were also compared with manufacturer s reported AOS values to show the differences between pore opening and constriction sizes (i.e., O 95 versus C 95 ). Materials and Methodology Geotextiles Fifteen different types of nonwoven geotextiles from four different manufacturers covering a wide range of AOSs and permittivities were analyzed. Nine of the fifteen had been previously used by Smith (1993) to evaluate PSD determination methods. The physical and hydraulic properties of the geotextiles used in this study are given in Table 1. Specimen Preparation The three-dimensional structure of nonwoven geotextiles presents difficulties in capturing pore structures in contrast to more twodimensional structure of woven geotextiles which can be captured from two-dimensional images. Planar and cross-sectional thin sections are necessary to provide detailed three-dimensional (3D) information. The thin sectioning process involved planar sections as well as sections normal to the plane of the geotextile. The latter were termed cross sections. From each geotextile type, three sets of specimens were prepared, with each specimen yielding three cross-sectional and two planar sections (underlying planar sections) (Fig. 1). The thin sections of the geotextiles investigated in this research were prepared following the procedures generally used for preparing thin sections of soil and rock. This required a series of steps: epoxy-resin impregnation, cutting, grinding, lapping, and polishing. Aydilek et al. (2002) provide details of the specimen preparation technique. Image Analysis Image Capturing Pore structure images of nonwoven geotextiles were captured using an optical light microscope having a macro zoom lens, placed on a 300 mm 250 mm workstation platform, and coupled with an image-capturing software called Pixera (Pixera 2003). A digital camera was attached to the microscope, which sent 8-bit per pixel digital luminance images to the computer. Zoom ratios of 5 and 2.75, and resolutions of 1, pixels and pixels were used for planar and cross-sectional images, respectively. Magnification (objective) ratio was 0.5 for all specimens. The specimens were illuminated from the bottom as proposed by Jang et al. (1999). The light intensity was adjusted so that the background pixels had a maximum grayscale value of 255 (i.e., pure white) without saturation, providing the largest JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005 / 29

3 Fig. 1. Specimen preparation: (a) typical locations of sections in epoxy-resin impregnated block and (b) sections cut from block contrast. Figs. 2(a and b) shows planar and cross-sectional images, respectively, of geotextile P. Image Processing In image processing, thresholding is a particularly important process, which produces binary images by differentiating the background from the image objects of interest. To obtain threshold values for the nonwoven geotextile images, a computer program was developed following the approach of Chow and Kaneko (1971) and Kapur et al. (1985). The program determines optimal threshold values by fitting two Gaussian distributions to the bimodal curves of the pixel intensity value histograms, and selects the pixel intensity value at the intersection point of the two Gaussian distributions as the threshold gray tone. This thresholding algorithm was used for the images of all geotextiles. An exception was the analysis of heat-bonded geotextiles (geotextiles N1, N2, N3, and N4), which required a special algorithm, due to their relatively thin and translucent fibers (Aydilek et al. 2002). An edge-detection operation was necessary to clearly identify the fibers before thresholding. The commonly used Canny edge algorithm along with a nonmaximal suppression method was used to threshold the images (Matlab 2001). The binary images generated by thresholding were then morphologically filtered using structuring elements. In mathematical morphology, a structuring element is a template (square-shaped matrix) designed to detect or modify certain shape features of the signal under consideration (Haralick et al. 1987). Initial observations showed that the porosity and pore opening sizes based on planar section images were nearly the same throughout the entire thickness of a geotextile with no apparent directionality of fibers. Fig. 3 is given as an example of the process performed on Geotextile P. This observation supported the use of cross-sectional images to extract information about the three-dimensional pore structures. To investigate the threedimensional structure of pores, a specific slicing algorithm was developed and applied to the cross-sectional images (Fig. 4). This algorithm first generated a uniform slicing grid, i.e., mask, which Fig. 2. Images of geotextile P: (a) top (planar) image; (b) cross-sectional image; and (c) slicing vector multiplied by original cross-sectional image divided the underlying cross-sectional image into horizontal slices of equal thickness. The slice thickness was set equal to the mean fiber thickness of the geotextile under consideration. The second step involved aligning this grid with the horizontally oriented fiber segments observed in the cross-sectional image. For this purpose, horizontal fiber streaks were detected through the use of directional image filtering operations, and all other fiber segments with different orientations were temporarily removed from the cross-sectional image. The criterion of best alignment of the slicing in a cross-sectional image was defined as maximizing the total count of pixels representing the horizontal fiber streaks lying within the boundaries of slices. Once the best alignment was determined, the optimal slicing grid was used to slice the original (cross-sectional) input image [Fig. 2(c)], enabling the calculation of two-dimensional porosity values over the length of each slice. This porosity was named the longitudinal porosity. Aydilek et al. (2002) provides detailed descriptions of the thresholding, morphological de-noising, filtering, and slicing operations. Pore Structure from Cross-Sectional Images The longitudinal porosities obtained from the cross-sectional images were essentially uniform throughout the entire thickness of the geotextile (passed the Chi Square and Kolmogorov Smirnov tests at 95% confidence interval) and comparable to the porosity determined from the planar section images. Fig. 3(b) shows the 30 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005

4 Fig. 3. Geotextile P: (a) distribution of pore opening sizes of underlying planar sections and (b) longitudinal porosity calculations across cross-sectional image sample measurements for one of the geotextiles. This is consistent with the findings of Masounave et al. (1980) and further substantiates that the cross-sectional images could represent the pore structure of a geotextile medium reasonably well. After the identification of fibers and pores in the slices, first the size of pore openings in each individual cross-sectional image was determined using an algorithm developed in IMAQ, an image processing software program functioning under the instrumentation software called LabVIEW. Then, a probabilistic model based on Markov chain theory was used to determine the constriction sizes. Herein, the pore opening size term was reserved for the recognized pores in a cross-sectional image, and the constriction size for the smallest size in a pore channel which is formed of communicating pore openings [Fig. 6(b)]. In the IMAQ algorithm, a range of sizes, comparable to the bead sizes commonly used in a dry sieving test, was defined and each size was compared to the minor axis length of an ellipse having the same area and the perimeter as the pore opening (Aydilek 2000). The number of pixels given in the output was defined as the number of pore openings having a minor axis length greater than a given diameter, which simulated retaining percentages in the dry sieving test. The process was repeated for the crosssectional images of all samples of a particular geotextile, and an average distribution of the pore opening sizes based on these images was reported. Fig. 5(a) shows the histogram of the pore opening sizes determined by this method for the cross-sectional images of geotextile P. Markov Chain Process Pore channels in a geotextile are nonuniform across depth (thickness), and distribution of minimum pore size across pore channels Fig. 4. Flow chart of slicing algorithm is of primary interest for filtration. As mentioned before, this minimum size is generally referred to as constriction size, and also corresponds to the pore opening size determined in some physical tests, such as the bubble point method. Based on the principle of stereology, Serra (1982) showed that spatial information about the 3D structure of a material could be estimated from limited number of two-dimensional images of sections cut parallel. However, the sections should be uniformly distributed, and the objects of interest in each section should not have any preferred orientation (Kuo and Frost 1996). Masad et al. (2002) used the approach to estimate the volume of images from their area fractions. The described method of determining pore opening sizes of a geotextile through its cross-sectional images is a reasonable way of determining the constriction sizes, since no specific directional structure of fiber orientations was observed in the horizontal slices uniformly extracted from cross-sectional images. To simulate the random distribution of pores in a geotextile and thus to estimate constriction sizes, a Markov chain process was used. Markov chains may be continuous or discrete, but discrete processes are more commonly used, in which the process changes state at discrete times that the future of the process depends only on its present state, and is independent of its past (Kijima 1997). Preliminary observations of cross-sectional images of the geotextiles showed that a discrete-time Markov chain can be used for modeling pore structure, since the size of a pore opening in a pore channel is generally dependent on the size of its underlying pore opening. For instance, if a relatively large pore opening is encountered in a cross-sectional image, then it is frequently under- JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005 / 31

5 Fig. 6. (a) Discretely varying finite number of pore openings in cross-sectional image and (b) recognition of pores and fibers in hypothetical cross section of geotextile Fig. 5. (a) Pore opening size histogram of geotextile P and (b) dividing area under curve into 20 equal subareas lain by another large pore opening. Herein, the size of a particular pore opening and its underlying companion (through the thickness of the geotextile) would constitute the present and future states of the discrete-time Markov chain, respectively. Furthermore, the preliminary analysis conducted on the cross-sectional images based on horizontal slicing revealed that a discretely varying pore opening size structure prevails throughout the thickness of a geotextile, i.e., the pore channel sizes are not changing in a continuous manner, and the pore openings can be classified into a finite number of size ranges (Fig. 6). Specifying a discrete-time Markov chain requires the definition of a set of states and corresponding transition probability matrix. In order to define the states of the discrete-time Markov chain used in this study, correlation structures of pore openings and fiber segments identified in the horizontal slices of crosssectional images were investigated as a function of slice displacement. The pore opening sizes observed in these cross-sectional images were grouped into a number of subclasses. Then, the statistical relationship between each pair of pore opening sizes, i.e., the probability of finding a pore opening size O x under a pore opening size O y, was identified to form the transition probability matrix. Analysis of Correlation Structure The cross-sectional image of a nonwoven geotextile including pores and fibers [Fig. 6(b)] is treated as a two-dimensional random field. A stationary distribution is assumed to exist in this random field, since both the mean and the variance of the pore opening sizes observed in any particular cross-sectional image were almost constant (Aydilek et al. 2002). In order to analyze the correlation structure between uniformly extracted slices of this random field, pore opening size histograms of the cross-sectional images of a particular geotextile were generated and an average of these histograms from different cross-sectional images was calculated. A log-normal distribution fit well to the histogram data of the geotextiles in most cases [Fig. 5(a)]. The best fit to the data was assured by minimizing the mean square difference. Similar log-normal distributions were reported for the pore opening size histograms by previous researchers (Masounave et al. 1980; Glantz and Schuler 1996; Vermeersch et al. 1997). The area under the curve was divided into 20 equal areas (or classes) as shown in Fig. 5(b), to facilitate the calculations. The median diameter of each area defined the pore diameter assigned to that particular area. Therefore, the sliced cross-sectional images of each particular geotextile were simplified to consist of 20 different pore opening sizes. This implies that a discrete-time Markov chain with 20 states represented by a transition probability matrix is used. Preliminary results indicated that this choice for the number of states was sufficient for the sensitivity of the model (Aydilek 2000). The vertical correlation structure of pores in the sliced crosssectional images as a function of the discrete distance unit, defined as the slice thickness, was investigated along the entire depth of the geotextile. Correlation lags (separation distances) specified in terms of this discrete unit were consequently replaced by the equivalent number of pixels. Autocorrelation functions were calculated in the vertical direction along the depth of the geotextile according to the following equation using the information provided by the sliced cross-sectional images: 1 k O k i=1 i o O i+r o r = In Eq. (1) above, r = lag value= separation distance/slice thickness; r =autocorrelation function value at lag r; O i =size of a o / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005

6 Characteristics of Markov Chain Used in Model The discrete-time Markov chain used for modeling the random pore channel structure, consisted of a single recurrent class of states. This is due to the fact that the probability of finding another pore along the pore channel with a size in the same size class as the current pore (corresponding to the same discrete Markov chain state) is equal to one. Therefore, none of the states were transient as the random process represented by this Markov chain, leaves a state and returns back to it, implying an infinite number of revisits to all states. A single recurrent finite class of states generates an irreducible Markov chain in which every state is accessible from every other state and there are no absorbing states, i.e., a state impossible to leave once entered. The transition probability matrix was formed assuming that the underlying Markov chain had a stationary distribution described by the discrete probability density function of the pore opening sizes. During the simulations, the Markov chain model was triggered to start evolving from an initial location randomly defined by its stationary distribution, since the pore opening size distribution characteristics essentially did not depend on the depth parameter, i.e., starting point of the process was not important. Fig. 7. (a) Autocorrelation functions for three main pore openings sizes in geotextile P and (b) Markov chain process pore opening belonging to a particular size class (state); O i+r =size of the pore opening which is located vertically below and at a lag r from the pore opening with which O i is associated; o = mean value of the random variable representing the pore opening size; k=total number of data points associated with the pore opening size class and lag value under consideration; and o 2 = variance of the random variable representing the pore opening size. The autocorrelation function provided information about how far a pore belonging to a certain size class is correlated with other pores below it throughout the depth of a geotextile (as a function of the slice separation distance). Fig. 7(a) shows the calculations performed for geotextile P. The distribution of pore opening sizes along the entire geotextile thickness was assumed to be stationary. This assumption was supported by the fact that the autocovariance matrix of pore opening sizes was independent of an initial offset in the location of the slicing grid on a crosssectional image. Analysis conducted by Aydilek (2000) suggested that the autocorrelation functions data could be grouped into three fundamental classes of autocorrelations representing essentially large, medium, and small pores. Gaussian and exponential decays were fitted best to the autocorrelation data by least squares. The autocorrelation function associated with large pores usually had the heaviest tail, i.e., the slowest decay of similarity vertically along the depth of the geotextile. It was followed by medium and then small sized pores in terms of the persistence of vertically located pore opening size similarity as a function of slice separation. Pores belonging to each of the three fundamental size classes were assumed to be correlated with their vertically aligned neighboring pores in lower slice layers up to a distance referred to as the range throughout the depth of the geotextile. Beyond these ranges, the correlations ceased to exist. The ranges are marked as N S, N M, and N L for small, medium and large size pores, respectively, as shown in Fig. 6. Construction of Transition Probability Matrix The transition probability matrix defines the transition probabilities between each pair of different states as well as the transition probabilities of states onto themselves. A transition probability p ij is defined as the probability of making a transition to state j in the next evolution epoch given that the current state is i. After forming the transition probability matrix (as described next), the random process described by this Markov chain was simulated to generate random pore channel structures. Initially, the three different classes of pore opening sizes, i.e., small, medium, and large, were assigned as the states of a discrete Markov chain, as shown in Fig. 7(b). As stated above, the associated transition probability matrix included the transition probabilities of the states onto themselves, i.e., P SS, P MM, and P LL,as well as the transition probabilities between three pairs of adjacent states, i.e.,,, and [Fig. 7(b)]. To determine P SS, P MM, and P LL, a procedure originally developed by Kemeny and Snell (1960) and described by Harr (1977) was used. Along with the suggestions of Elsharieff (1992), the transition probabilities, P SS, P MM, and P LL, were derived from the ranges as follows: P SS = N S 1 N S P MM = N M 1 N M P LL = N L 1 N L The transition probabilities,, and, identifying the three remaining degrees of freedom in the definition of the transition probability matrix, were determined by finding the least-squares solution of the following system of linear equations known as balance equations required for a stationary distribution to exist: = P 3 In Eq. (3), P and =transition probability matrix and the stationary distribution vector, respectively, and are defined as 2 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005 / 33

7 and P = P SS 1 P SS 1 P MM P MM 4 1 P LL P LL = S M L 5 The elements S, M, and L of the stationary distribution vector =relative frequencies of the pores (normalized to a sum of 1) with opening sizes belonging to the small, medium, and large size classes, respectively, and they were determined from the pore opening size distribution histogram (Fig. 5). Since a stationary distribution existed as implied by empirical evidence, the transition probability matrix P should map the vector onto itself as required by the definition of a stationary (invariant) distribution. The discrete Markov chain model described above is based on an oversimplification of the real case, in which the three main pore opening size classes, i.e., small, medium, and large, contain multiple subclasses. To consider these subclasses in the model, the pore openings were divided into 20 size classes [as shown in Fig. 5(b)] and autocorrelation functions were calculated for each of these classes. Consequently, a discrete Markov chain with 20 states was designed and used in the simulations. Fig. 8. Simulated pore channels using Markov chain-based model Simulation of Random Process for Modeling Pore Channels Sample paths of the random process defined by the Markov chain model construct pore channel structures. Therefore, in order to construct a random pore channel structure, the evolution of this Markov chain needs to be simulated. The first step of this simulation is the generation of an initial state from which the Markov chain will start its evolution. Since the Markov chain used to model the random pore channels is assumed to be in its steadystate across the entire simulated geotextile thickness, the generation of an initial state has to be performed in a manner consistent with the stationary distribution of the chain. Hence, the probability mass function of pore opening sizes is used to randomly generate initial states. Consequently, the chains were generated through a Monte Carlo simulation using the transition matrix P. Fig. 8 shows sample images of the random pore channel structures generated using the Markov chain model described. A sensitivity analysis indicated that at least 10,000 sample path realizations (i.e., 10,000 pore channels) were necessary to provide consistent results. After this point (i.e., 10,000), an increase in the number of simulations did not have a significant effect on the constriction sizes. The constriction sizes of all channels, i.e., the smallest pore opening diameter in the channel, were determined and their distributions were then plotted. This distribution was called the constriction size distribution (CSD) herein. Results and Discussion Comparison of Predicted Constriction Sizes with Reported Sizes The image-based probabilistic model developed in this research program, named CONS, determines the constriction size of the pore channels of a nonwoven geotextile. Two commonly used characteristic constriction sizes, C 95 and C 50, obtained by using CONS are compared with the manufacturer s reported AOS O 95 values and the values obtained by other researchers in Fig. 9(a). As mentioned before, the manufacturer s reported AOS O 95 values are based on a crude testing technique, which does not deter- Fig. 9. Comparison of (a) constriction size-based C 95 constriction sizes with manufacturer s reported values and laboratory-based measurements and (b) constriction size-based C 50 constriction sizes with ones determined from bubble point tests 34 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005

8 mine constriction sizes. However, the method is commonly used by the geosynthetic manufacturers and comparisons are made herein to delineate the difference between constriction size and opening size (i.e., C 95 and O 95 ). Smith (1993) and Bhatia et al. (1996) conducted an extensive testing program to determine the pore opening size of nonwoven geotextiles with five different methods, including the dry sieving and bubble point tests. Bhatia et al. (1996) used a variety of geotextiles including the geotextiles C2, C4, C5, D1, D4, N1, N2, N3, and N4 of the current testing program. Fig. 9(a) shows that the constriction size predictions of CONS are consistent with the bubble point-based values for these geotextiles, but generally lower than the values obtained from dry sieving and manufacturer s reported AOS values. This is meaningful, because among various methods only the bubble point test can roughly estimate the constriction size. C 50 values obtained by Smith (1993) and Bhatia et al. (1996) using the bubble point test are compared with the predictions of CONS in Fig. 9(b). C 50 values predicted by CONS are generally comparable to the bubble-point based values. The differences may be attributed to the variability in geotextile samples used in two different studies due to manufacturing and random locations of the fibers. Comparison of Predicted Constriction Sizes with Theoretical Sizes Giroud (1996) proposed a theoretical equation for calculating the constriction size of nonwoven geotextiles. The equation is based on the porosity, thickness, and fiber diameter of a geotextile O f d f = 1 1 n 1+ nd f 1 n t where d f =fiber thickness; n=porosity; t=geotextile thickness; =dimensionless parameter; and O f =filtration opening size and is usually given by the nearly largest constriction size of a particular geotextile (e.g., C 95 ). Comparisons of with the data of 52 laboratory tests published by Rigo et al. (1990) indicated that the use of =10 is reasonable (Giroud 1996). The fiber thickness d f was measured from the captured images and an equation given by Wayne and Koerner (1993) was used to determine the porosity n n =1 M 7 t where =fiber density; and M and t=mass/unit area and thickness of the geotextile, respectively. Fig. 10(a) compares the CONS-based C 95 constriction sizes with those calculated by using the Giroud (1996) method. The values are highly comparable which gives further substantiation to the accuracy of the imagebased probabilistic approach. Faure et al. (1990) and Gourc and Faure (1990) also presented a theoretical technique for determining constriction size based on Poissonian polyhedra model. Detailed mathematical explanation of the Poissonian polyhedra model can be found in Matheron (1971) and Rollin et al. (1982). Faure et al. (1990) sliced the epoxy-impregnated specimens and modeled the nonwoven geotextile as a pile of elementary layers, randomly distributed in planar images of the geotextile. The cross-sectional images of the geotextiles were sliced at a thickness of fiber diameter d f and statistical distribution of pores were calculated by inscribing a circle into each polygon defined by the fibers similar to a technique described by Aydilek et al. (2002). Finally, the probability 6 Fig. 10. Comparison of (a) constriction size-based C 95 constriction sizes with ones determined from theoretical equations, and (b) constriction size distribution with pore size distribution determined from planar images of Geotextile C4 of passage of different spherical particles (similar to glass beads in a dry sieving test) through the layers forming the geotextile were determined using the following formula: P g d = n 2 d ++d 2N f N d 2+ d f exp 8 where P g d =probability of a particle with a diameter d passing through a pore channel in the geotextile; n=porosity of the geotextile; =total length of straight lines per unit area in a planar surface (also termed as specific length); and N=number of slices in a cross-sectional image. Giroud (1996) provides a clear and detailed explanation of the technique and summarizes the calculated constriction sizes of various geotextiles. Fig. 10(a) compares the CONS-based C 95 constriction sizes with those calculated by using the Faure et al. (1990) method. Overall the values are comparable; however, as expected the Faure et al. (1990) approach generally produces lower values due to an assumption that at relatively high geotextile thicknesses the constriction size tends to approach zero (Giroud 1996). The use of this method is not recommended for geotextiles with a porosity of 50% or less because of this assumption. Therefore, the comparisons with geotextiles N1 and N2 are not provided herein. JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005 / 35

9 opening size distribution methods, CONS provides the probabilistic shape of pore channels, which could further be used to quantify the clogging potential of geotextile and granular filters with different soils in contact with them. The CONS is embedded in the actual image and geometry of the pore opening structure and is expected to provide a truer reflection of pore size. It is expected that by providing a complete constriction size distribution, this method will stimulate development of probabilistic filter models as a basis for the development of rational and more general filter criteria for nonwoven geotextiles than has been possible based on empirical approaches. Summary and Conclusions Fig. 11. Comparison of C 95 and C 50 constriction sizes with pore opening sizes determined from planar images Comparison of Predicted Constriction Sizes with Pore Opening Sizes Determined from Planar Images The distribution of constriction sizes (CSD) by the CONS is based on the cross-sectional images of a geotextile. In contrast, the CSD in earlier image-based analyses was usually determined from the top planar image only. Constriction sizes are different than the pore opening sizes of the top planar image, since the largest or smallest pore in a channel is not necessarily located on the top planar surface. Fig. 10(b) shows that the CSD of a nonwoven geotextile is different than the pore opening size distribution of its planar image. For instance, C 95, is estimated as 0.08 mm, for geotextile C4, from the CSD curve, and is comparable to the value obtained from the bubble point test mm as determined by Bhatia et al. (1996). However, if only the planar images were analyzed, the size would have been overpredicted as 0.12 mm. The difference is usually larger for the smaller pore opening sizes. Two characteristic pore opening sizes, O 95 and O 50, determined from planar images are compared with the CONS-based values in Fig. 11. Almost in all cases, the CONS-based C 95 constriction sizes are smaller than the ones determined from the planar images. The difference is 1 26%. The CONS-based C 50 s are also 1 96% smaller than the O 50 s determined from the planar images. The comparisons herein indicated that CONS is effective in determining the constriction sizes. In contrast to the current pore Existing studies indicate that constriction sizes of a filter play a significant role in its filtration performance. Due to the shortcomings of existing methods used for determining the CSD, a new probabilistic method coupled with image analysis, named CONS, was developed and applied to a wide variety of nonwoven geotextiles (both needle punched and heat bonded) commonly used in filtration applications. Several image morphology algorithms were used to analyze the planar and cross-sectional images of the geotextiles. The randomness of fibers in the geotextile media was modeled using a Markov chain process, and the constriction size and shape of the pore channels of geotextiles were obtained. The two characteristic constriction sizes, C 95 and C 50, were determined using the CSDs obtained from the generated pore channel structures. The predictive capability of CONS was evaluated against the reported AOS values as well as values of O 95, C 95, and C 50 based on laboratory tests and theoretical equations. The model also has potential for determining the constriction size distribution of granular filters although not dealt with here. The following conclusions are advanced: 1. The image-based probabilistic model CONS provides a quasithree-dimensional CSD for a given geotextile. 2. Comparisons made between the pore opening size distributions based on the planar images and CSDs determined using CONS applied to cross-sectional images suggest that planar images may not closely model the nonuniform pore channels in a geotextile. Therefore, pore opening size measurements based solely on planar images should be avoided. 3. The C 95 constriction sizes of various geotextiles obtained by CONS are comparable to those determined from the bubble point tests and the values determined using a theoretical equation. However, the predicted C 95 constriction sizes are somewhat lower than the manufacturer s reported values (typically based on dry sieving) and the other reported values based on dry sieving, due to the nature and established shortcomings of the dry sieving test. The C 50 constriction sizes obtained by CONS are generally comparable to those obtained by the bubble point test. The reported differences may be due to material variability and randomness of fibers. 4. In contrast to the current pore opening size distribution methods, CONS provides the probabilistic shape of pore channels, which could further be used to quantify the clogging potential of geotextile and granular filters with different soils in contact with them. Acknowledgments The writers would like to express their appreciation to Mr. Brian Hess, manager of the Thin Section Laboratory of the Department 36 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JANUARY 2005

10 of Geology and Geophysics at the University of Wisconsin- Madison, for his assistance and guidance during the specimen preparation. Dr. Gregory B. Baecher and Dr. David J. Lovell of the University of Maryland are thanked for reviewing the initial draft of this paper. Notation The following symbols are used in this paper: d f fiber thickness; k total number of data points associated with the pore size class and lag value under consideration; M mass/unit area of geotextile; N number of slices in cross-sectional image; N X range that autocorrelation function for class X is correlated; n geotextile porosity; O f filtration opening size of geotextile; O i size of pore opening belonging to a particular pore class (state); O x geotextile pore opening/constriction size that x% of pores are smaller than that size; P transition probability matrix; P g probability of particle with diameter d passing through pore channel in geotextile; P XX transition probability of pore class X onto itself; r lag value=separation distance/slice thickness; t geotextile thickness; probability of transition from small to medium class; probability of transition from medium to large class; probability of transition from large to small class; total length of straight lines per unit area in planar surface (also termed a specific length); 0 mean value random variable representing pore opening size; dimensionless parameter; stationary distribution vector; X % frequencies of pore opening sizes belonging to class (state) X; fiber density; r autocorrelation function at lag r; and 2 0 variance of the random variable representing pore opening size. 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