Experimental analysis on the effects of variable apertures on tracer transport

WATER RESOURCES RESEARCH, VOL. 39, NO. 1, 1015, doi:10.1029/2001wr001246, 2003 Experimental analysis on the effects of variable apertures on tracer transport Jaehyoung Lee Research Institute of Energy and Resources, Seoul National University, Seoul, South Korea Joe M. Kang and Jonggeun Choe School of Civil, Urban and Geosystems Engineering, Seoul National University, Seoul, South Korea Received 11 February 2002; revised 19 September 2002; accepted 19 September 2002; published 21 January 2003. [1] Aperture distributions have been measured using replicas of apertures to investigate the effect of variable aperture on solute transport in five transparent fracture samples. The coefficient of variation (C v ), correlation length (l), and anisotropy ratio (AR) were used to quantify the aperture distributions. Step-type tracer tests showed that dispersion is significantly influenced by both l and C v. Dispersivity (a) increases with l and C v. The magnitude of a is proportional to a polynomial function of natural logarithm of C 2 v and is linearly proportional to l and the total sill measured from the semivariogram. The effect of AR is insignificant to those of the other parameters on a. Despite assumptions of the small perturbation theory developed by Gelhar [1993], the prediction of a from the theory is similar to a measured in this study. If the small perturbation theory is valid and reliable data for the tailing in a breakthrough curve are obtained, the analysis of the curve by the moment method gives a more reliable result than by the conventional fitting of a onedimensional analytical solution to the curve. INDEX TERMS: 1832 Hydrology: Groundwater transport; 5104 Physical Properties of Rocks: Fracture and flow; 5139 Physical Properties of Rocks: Transport properties; KEYWORDS: variable-apertures, aperture distribution, fracture, dispersivity Citation: Lee, J., J. M. Kang, and J. Choe, Experimental analysis on the effects of variable apertures on tracer transport, Water Resour. Res., 39(1), 1015, doi:10.1029/2001wr001246, 2003. 1. Introduction [2] Variable aperture models have lead to considerable progress in the study of fluid flow through a single fracture. To represent aperture distribution of a fracture, many researchers have used the lognormal distribution [Gentier, 1986; Gale, 1987] and Gaussian distribution [Brown, 1995; Hakami and Larssen, 1996; Lapcevic et al., 1999]. In the numerical study by Moreno et al. [1988], they showed preferential flows took place through a few dominant paths in the fracture plane. Severe deviations of the flow pattern from a parallel plate model have been found in reactive and nonreactive solute transport through real fractures [Tsang et al., 1988; Johns and Roberts, 1991; Haldeman et al., 1991; Dronfield and Silliman, 1993; Brown, 1995; Keller et al., 1999]. While the importance of apertures on fluid flow and transport has been recognized, understanding of what controls the aperture distribution of a given fracture or the extent of channeling within a given formation remains limited [Renshaw et al., 2000]. [3] The previous works on solute transport through a single fracture lack experimental observations and represent the variable aperture characteristics insufficiently. Methods for measuring an aperture can be categorized into destructive and nondestructive methods. The destructive methods include the surface topography method [Gentier, 1986; Iwano and Einstein, 1993], the casting method [Hakami Copyright 2003 by the American Geophysical Union. 0043-1397/03/2001WR001246 SBH 7-1 and Barton, 1990; Yeo et al., 1998], and the injection method [Tsang and Tsang, 1990; Hakami and Stephansson, 1993]. The nondestructive methods use differences of reflection intensity of magnetic resonance [Kumar et al., 1995; Brown and Bruhn, 1998; Dijk et al., 1999; Renshaw et al., 2000] or absorbance intensity of x-ray [Johns et al., 1993; Keller et al., 1995; Keller, 1998; Keller et al., 1999] and light [Persoff and Pruess, 1995; Detwiler et al., 1999; Renshaw et al., 2000]. [4] Generally, the destructive methods are cost-effective, but their data can be less reliable due to inconsistency in assembling a fracture sample, while the nondestructive methods are easy to perform, but facilities are relatively expensive and conversion of intensity to aperture size is difficult. These drawbacks have made little experiment including accurate and sufficient aperture variability observed, despite an important need for experiments with known variable aperture fractures to verify transport theories. [5] Numerical studies by Tsang and Tsang [1990] suggested that the mean, standard deviation, and correlation length of the apertures could be used to represent flow patterns through a fracture. A recent experimental study with a variable aperture fracture demonstrated that geostatistical parameters of the fracture aperture could be used to predict the approximate breakthrough curve for solute transport in a fracture [Keller et al., 1999]. It is crucial to understand the effects of geostatistical parameters of the aperture distribution on the characteristics of solute transport for accurate numerical modeling and for predicting the breakthrough curve from the aperture distribution or vice

SBH 7-2 LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT versa. Numerical models for solute transport typically assume that the cubic law is valid locally using the Reynolds equation. However, it remains uncertain whether the local cubic law is valid for aperture distributions typical of natural fractures [Nicholl et al., 1999; Renshaw et al., 2000]. In many of the past experimental research, only one or two fracture samples have been used to study tracer transport through a fracture. A few experimental studies include the effects of parameters of the aperture distribution on the transport characteristics. Besides the above parameters, the anisotropic nature of the fracture controls the direction of preferential flow, and the assumption of isotropy may lead to errors in the analysis of transport characteristics. Not many studies have addressed the effect of anisotropy in a fracture on solute transport [Thompson and Brown, 1991; Cady et al., 1993; Yeo et al., 1998]. [6] This experimental study aims (1) to test the small perturbation theory [Gelhar, 1993] against our experimental results, and (2) to quantify the effects of three geostatistical parameters of an aperture distribution (the coefficient of variation C v ; correlation length l; anisotropy ratio (AR) on solute transport. To accomplish these objectives, the geostatistical parameters were calculated from measured aperture distributions in five different transparent fracture models. The applied tracer tests are step-type injections of inert dyed water. Breakthrough curves from the tracer tests were analyzed with a one-dimensional analytical solution and a moment method of breakthrough curve analysis. 2. Experimental Procedures 2.1. Preparation of Fracture Models [7] Five epoxy fracture samples were molded from silicone rubber castings of artificially fractured sandstones. Each transparent fracture replica was used to monitor flow images of the dyed water by a charge-coupled device (CCD) camera. Berea sandstone cores were artificially fractured in the length direction by a wedge under a uniaxial loading machine. Aperture variations of the artificial fractures were further generated by shear displacement. Information on the construction of fracture replicas is given by Hakami and Barton [1990]. All fracture models were mated and carefully machined to approximately 3.75 cm in thickness, 5 cm in width, and 29 cm in length. The sides of the fracture samples were sealed with silicone sealant. [8] Since the aperture distribution of a fracture is very sensitive to the confining pressure, it is critical to maintain a constant confining pressure during the flow test. We made a transparent cylindrical sample holder to accommodate a fracture model for a stable confining pressure up to 1 MPa. Two endpieces of the holder, which were carefully machined to reduce dead volume, were installed at the ends of the sample with silicone sealant. The total dead volume of the two endpieces and pipes is 0.9 cm 3. A leaking test was performed to check the sealing before the flow tests. Table 1 shows the dimensions of the fracture models. 2.2. Tracer Tests and Analysis [9] Figure 1 shows a schematic diagram of the experimental apparatus used in this study. After placing a fracture replica into the core holder, we filled the annulus between the holder and the fracture sample with water for a better view. A constant confining pressure (345 kpa) of gas was Table 1. Statistical Parameters of Aperture Distribution and Dimension of the Fracture Samples Sample F1 F2 F3 F4 F5 Length (L, cm) 30.2 29.1 29.1 28.4 29.6 Width (W, cm) 4.5 4.5 4.5 4.8 4.8 Fracture volume (V f,cm 3 ) 4.40 5.02 4.59 3.53 5.45 Mean aperture (hbi, mm) 323 383 351 259 384 Standard deviation (s, mm) 264 253 246 226 153 Coefficient of variation (C v ) 0.82 0.66 0.70 0.87 0.40 applied to the sample through the fluid accumulator. Two separate pumps were used for dyed and clear water to achieve a constant concentration of the dyed water. The fracture model was evacuated for 24 hours and clear water imbibed into the model. The temperature was set at 20 C. The constant rate of pure water injection produced a steady state flow field in the fracture. By switching the Rheodyne valve from clear water to dyed water, we performed a tracer test, captured images of tracer movement with the CCD camera, and measured effluent concentrations with an inline spectrophotometer at a fixed time interval. [10] To obtain the quantitative characteristics of solute transport in the tracer test, we fit experimental breakthrough curves with the one-dimensional analytical solution of the advection-dispersion equation (ADE) [Kreft and Zuber, 1978]: C R ¼ 1 x vt erfc pffiffiffiffiffiffiffi þ 1 vx exp erfc x þ vt pffiffiffiffiffiffiffi ð1þ 2 4Dt 2 D 4Dt Initial and boundary conditions of equation (1) are C R (x,0)= 0 for x > 0, and C R (0, t) =1fort 0 and lim x!1 C R(x, t) =0. In this application, x is the longitudinal dimension in the direction of flow [L]; C R is the concentration relative to the inlet concentration (C/C 0 ); D is the longitudinal dispersion coefficient [L 2 T 1 ]; v is the average velocity [LT 1 ], and t is the time [T]. It is assumed that the dispersion coefficient (D) is proportional to the average velocity (v) and is defined as: D ¼ av where a is the dispersivity [L]. Fitting of an observed breakthrough curve to equation (1) yields two transport parameters, v and D. Since equation (1) is a solution for a one-dimensional transport model in parallel plates, the tailing phenomena may not be fit well using equation (1). [11] To compensate for this, we adopted the moment method of breakthrough curve analysis [Yu et al., 1999]. The velocity and the dispersion coefficient were directly calculated from the following expressions: m 2 ¼ M 1 ¼ v ¼ L M 1 D ¼ m 2v 3 2L Z 1 0 Z 1 0 tdc R ð2þ ð3þ ð4þ ð5þ ðt M 1 Þ 2 dc R ð6þ

LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT SBH 7-3 Figure 1. A schematic diagram of tracer tests in this study. where M 1 and m 2 are the normalized first moment and the normalized second central moment of the breakthrough curve. It is one of the advantages of the moment method that underlying physical model of a fracture is not required [Yu et al., 1999]. However, it is still difficult to exactly monitor the long tail of the breakthrough curve, which affects the magnitude of D in this method. [12] For both cases, dead volume (V DV ) was considered for time correction and no dispersion was assumed in the sections of dead volume. The delayed time (t DV ) due to the dead volume in the setup was calculated and subtracted from the monitored time (t m ) to obtain corrected time (t corr ), as follows. t DV ¼ V DV =Q t corr ¼ t m t DV where Q is the injection rate [L 3 /T]. After time was corrected, the measured effluent concentration was normalized and plotted against the corrected time. Using equations (1) and (2), we obtained dispersivity from the breakthrough curve fitting. In the case of the moment method, M 1 and m 2 values were calculated by integrating the normalized breakthrough curve according to equations (5) and (6). Then, a was calculated using equation (2) and parameters v and D, which were determined according to equations (3) and (4). 2.3. Measurement of Aperture Size [13] We used a casting method to measure the aperture distribution of the fracture samples. An aperture replica of the fracture was made using silicone rubber. The detailed procedure for the casting method is given by Yeo et al. [1998]. It is crucial that the replica retains the void geometry of the fracture sample during the flow tests. The side of the sealant connection on the fracture was cut in order to maintain the same relative position of the upper and lower surfaces of the fracture sample. We took apart the fracture and poured silicone rubber on both surfaces of the fracture while preserving the other side of the sealant connection. The fracture sample was then placed under a loading-equipment ð7þ ð8þ at 345 kpa to maintain the same confining condition for the tracer tests. Excess silicone rubber flowed out of the fracture sample through the open-ended inlet and outlet faces prior to 3 hours of solidification. After 24 hours of solidification, each aperture replica was sliced at 1 mm intervals in the transverse direction. Apertures were measured along these sections by the CCD camera equipped with a micro-lens. The pixel values of the captured image of the aperture replica section were converted to aperture sizes. We used a scale reference to convert pixel value into length, whereby 1 mm corresponds to 213 pixels (1 pixel = 4.7 mm). [14] A light transmission technique [Detwiler et al., 1999] was adopted to validate the casting method. The contours of the aperture from the light was compared with those from the casting method. Image processing was applied to obtain the final picture of the fracture. After placing a fracture sample into the sample holder, we took pictures of the fracture filled with clear and dyed water. The latter image was subtracted from the former one to correct the material and light source effects. The resulting image was transformed into a negative form for the pixel values to increase with aperture size. To convert light intensity into aperture size, we constructed a cell using two parallel plates whose length and width are 30 cm and 5 cm, respectively. Two parallel plates were joined at one end, along the width, and the other end was separated by inserting a 3-mm-thick metal sheet. Therefore we assumed that aperture sizes between the two parallel plates vary linearly from 0 to 3 mm. Both sides in the length direction were sealed with silicone sealant. For the cell placed in the sample holder, the same image processing procedure was applied. By comparing pixel values of the final image with the gap size of the cell, we obtained the conversion relation of b = 53.36 e pixel/67.48, where b is aperture size in mm. 2.4. Statistical Parameters for Aperture Distribution [15] For characterization of the fracture morphology, three parameters were calculated: C v, correlation length (l), and AR. The C v, defined as the ratio of standard deviation (s) to mean aperture (hbi), is regarded as the extent of the fracture deviation from a parallel plate. Dispersion in a fracture mainly results from heterogeneity of

SBH 7-4 LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT Figure 2. technique. Aperture contour maps of the fracture samples: (a) casting method and (b) light transmission the flow field through the fracture, which is closely related to C v. Moreover, the spatial characteristics of the aperture distribution can cause flow patterns to be very different even when the fractures have statistically similar distributions [Lapcevic et al., 1999]. [16] Effects of C v and l on dispersion can be explained theoretically by the small perturbation theory, and the dispersivity (a) of a variable aperture fracture can be derived [Gelhar, 1993]. a ¼ 4 þ 0:205s 2 b þ 0:16s4 b þ 0:45s6 bþ0:0115s 8 b s 2 b l for 0 > s 2 b < 5 ð9þ where s b 2 is the variance of the natural log of aperture distribution. If an aperture distribution is lognormal, s b 2 is calculated from equation [2.1.3] of Gelhar [1993]. s 2 b ¼ ln C2 v þ 1 ð10þ Equation (9) was derived from the assumptions that the aperture distribution is lognormal and isotropic, the traversed length is greater than 10 times of the correlation length, and the mean concentration varies slowly for local stationary of concentration. This approach was first attempted experimentally by Keller et al. [1995, 1999]. In their studies, the estimated a of two fractures were 33% and 81% larger than measured ones. [17] It is critically important to evaluate the correlated structure of an aperture distribution for predicting a of a fracture using equation (9). A semivariogram has been fit using a linear combination of models. It is necessary to use a combination of more than three models to precisely capture the semivariogram. The correlation length of each direction was obtained by fitting equation (11) to the semivariogram, gðhþ ¼ nugget þ A 1 1 exp 3h "!# þ A 2 1 exp 3h2 l e l 2 g ð11þ where A 1 and A 2 are the normalized sill values for the exponential and Gaussian models, respectively; h is the lag distance; l e and l g are the range for exponential and Gaussian models, respectively. The anisotropy ratio from the semivariogram (AR l ) is defined as the ratio of longitudinal correlation length (l L ) to transverse correlation length (l T ). As mentioned by Yeo et al. [1998], fracture apertures have directional properties, which are the main cause of anisotropy in their hydraulic properties. [18] Equation (9) was based on that the normalized sill from the semivariogram is equal to one, but the normalized sill is not always equal to one in semivariogram analysis. To reflect this, s 2 b in equation (9) was replaced by the modified 2 variance (s b,mod ) as follows. s 2 b;mod ¼ s2 b A ð12þ

LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT SBH 7-5 Figure 3. Aperture histograms and cumulative distribution curves of the fracture samples. where A is the normalized total sill determined from the semivariogram. Therefore a from equation (9) is proportional to the normalized total sill in addition to C v and l. 3. Results and Discussion 3.1. Morphology of the Fracture Samples and Geostatistical Parameters of Their Aperture Distributions [19] In this section, we present the results of the aperture distribution measurement and semivariogram analysis. Measured aperture contours from the casting method were verified by comparing the images from the light transmission technique. Aperture size in mm is expressed as the gray level in Figure 2. Due to the limited resolution of the captured image, which is composed of 256 gray colors, the large aperture regions in Figure 2b are larger than those in Figure 2a. The aperture distributions are similar to each other despite a slight difference in the sizes of the large aperture regions. We can confirm the reliability of the aperture measurements of the casting method indirectly from this resemblance, and these data will be used for further discussion. [20] As shown in Figure 3, aperture distributions of fracture 5 (F5) appears normal, but the others lognormal. Table 1 shows calculated statistical parameters of aperture distribution for each sample, such as hbi, s, C v, fracture width (W ), fracture length (L), and fracture volume (V f ). Fracture volume was calculated as L W hbi. [21] The results from the semivariogram analyses are presented in Figure 4 and Table 2. The nugget and the sill were normalized to the variance of the apertures. For fracture F1, the total sill (A) is the sum of the nugget and the sill of exponential model (A 1 ), and the correlation length (l) is the range of exponential model (l e ), since we could fit the semivariogram by an exponential model only. For fractures F2 and F3, A is the sum of the nugget and A 1, and l is l e. Because there is intermediate plateau in semivariance, and the semivariance grows continuously after the intermediate plateau, the sill value and the range

SBH 7-6 LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT Table 2. Correlation Structure of Aperture Distribution in the Fracture Samples Sample F1 F2 F3 F4 F5 Isotropic case nugget 0.000 0.252 0.344 0.062 0.194 A 1 1.056 0.525 0.456 0.671 0.509 l e (cm) 1.820 2.827 3.327 0.352 0.587 A 2 0 * * 0.303 0.372 l g (cm) 0 * * 3.755 6.053 A 1.056 0.777 0.800 1.036 1.075 l (cm) 1.820 2.827 3.327 3.755 6.053 Anisotropic transverse direction Anisotropic longitudinal direction Anisotropy ratio from correlation length (AR l ) nugget 0.000 0.308 0.310 0.196 0.050 A 1 1.041 0.429 0.313 0.548 0.628 l Te (cm) 1.838 2.285 0.670 0.363 0.374 A 2 0 * * 0.268 0.436 l Tg (cm) 0 * * 2.760 4.100 l T (cm) 1.838 2.285 0.670 2.760 4.100 nugget 0.000 0.268 0.183 0.298 0.243 A 1 1.049 0.460 0.390 0.475 0.389 l Le (cm) 1.726 4.130 1.322 0.745 0.629 A 2 0 * * 0.307 0.449 l Lg (cm) 0 * * 18.81 5.994 l L (cm) 1.726 4.130 1.322 18.81 5.994 0.93 1.81 1.97 6.81 1.46 of Gaussian model (A 2 and l g ) do not exist, as can be seen in Figure 4. For fractures F4 and F5, there are fitted ranges and sills for both the exponential and Gaussian models, respectively. A is the sum of the nugget, A 1, and A 2, and l is chosen as the larger one between l e and l g. These determinations of A and l are reasonable in the sense that total sill is a value at the point where semivariance is converged and correlation length is maximum distance between two correlated values. [22] Without performing a tracer test, we can predict the transport characteristics of the fracture from the measured parameters of the aperture distribution. Since flow becomes more tortuous as C v increases, we can expect that dispersivity would increase with increasing C v. The degree of dispersivity differs depending on the correlated structure of apertures, such as l and AR. The effect of C v on increasing dispersivity will become more significant for the case of well-correlated apertures, and that will be reduced for the case of randomly distributed apertures. Sharp increase of dispersivity occurs regardless of the pattern of the aperture distributions when channeling takes place. A larger C v will increase the possibility of the channeling. The value of AR will determine the direction of the channel and its length. The channel will be meandering with a smaller AR; the length of the channel is relatively long compared to that of the fracture. 3.2. Validity of the Small Perturbation Theory and the Moment Method of the Breakthrough Curve [23] There was almost no difference among tracer breakthrough curves of various flow rates for the same fracture model, if time was converted to injection volume. This Figure 4. (opposite) Curve fittings of semivariograms to equation (11): (a) isotropic case, (b) anisotropic case in longitudinal direction, and (c) anisotropic case in transverse direction.

LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT SBH 7-7 result indicates that molecular diffusion is not an important mechanism of transport through the fractures. Dispersion is linearly proportional to the velocity in the range of water velocities in the study, and we can neglect the effect of velocity on dispersivity. [24] Figure 5 shows the results of the tracer test analysis using equation (1). Measured points are marked as black circles and fitted curves are drawn as lines. The normalized dye concentration was plotted against the normalized time. The normalized time (PV) was calculated as the cumulative volume of injected water divided by the measured total pore volume of the fracture (V f ). Equation (1) does not adequately explain the breakthrough curves at later time of tailing for fractures F2, F3, and F5. The inadequate model for tailing can underestimate dispersivity. For fracture F4, equation (1) shows a lower concentration near tailing than the measured one, causing overestimation of dispersivity. We calculated a and v using the moment method of the breakthrough curves in Figure 5 for compensation of the model inadequacy. [25] From equation (9), we calculated the theoretical dispersivity (a p ) of the fracture samples by isotropic results. We used a measured s b 2 instead of calculated s b 2 from equation (10), because the measured aperture distributions were not exactly lognormal. Table 3 summarizes the results from equation (1), equation (9), and the moment method. Figure 6 shows the comparisons between a p versus a f from equation (1) and a p versus a m from the moment method. Differences between a f and a p are 2% +134% and their average is 46%. Large differences between a p and a f for fractures F2 (74%) and F5 (134%) are due to the fact that discrepancy exists between equation (1) and the measured breakthrough curves at later time. Differences between a m and a p are 32% +31% and their average is 19%. In both the conventional fitting process and the moment method, estimation of a from the small perturbation theory may be acceptable, because equation (9) was derived under the assumption of isotropy, lognormal distribution of apertures, local stationary in concentration, and L > 10l. [26] The predicted dispersivities (a p ) show closer agreement with a m than a f except for fractures F3 ( 32%) and F4 (31%). It may be a problem which dispersivity between a m and a f is closer to the true dispersivity of the fracture sample. Errors in the estimation of a may be attributed to the deviations from the assumptions of the theory. In particular, the channeling phenomenon may be a main cause for larger difference of a p from a m than that of a p from a f. Figure 7 shows the flow images obtained during the tracer test. The darker the gray scale is, the earlier the time. The time interval of imaging was not constant, but was selected to improve the contrast between the flow images. In Figures 7c and 7d, a channel is formed at the upper edges of F3 and F4. Abrupt change in aperture size resulted in these highly narrow channels, producing relatively large errors of a p. [27] The signs of errors are opposite, even though the geostatistics of the aperture distributions are similar for fractures F3 and F4. The negative sign in F3 means that the measured a is larger than a p, and therefore solutes underwent larger dispersion than expected. This can be attributable to the second channel at the lower edge of F3. The second channel grew later than the first channel. The Figure 5. Fitted curve of the one-dimensional analytical solution to measured breakthrough curve.

SBH 7-8 LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT Table 3. Calculated Dispersivity From Aperture Distribution Using the Small Perturbation Theory, Fitted Parameters of Breakthrough Curves to the Equation (1), and Statistical Parameters Calculated From Moment Method of Breakthrough Curve Fitted Parameters From Equation (1) a p From Equation (9), cm D L,cm 2 /s v, cm/s a f,cm Moment Method of BTC p f f D L,cm 2 /s v, cm/s a m,cm p m m F1 5.90 0.522 0.104 5.02 0.18 0.624 0.1003 6.22 0.05 F2 4.41 0.241 0.095 2.54 0.74 0.365 0.0915 3.99 0.10 F3 5.12 0.582 0.111 5.20 0.02 0.803 0.1066 7.53 0.32 F4 12.7 1.311 0.108 12.1 0.05 1.078 0.1112 9.69 0.31 F5 4.21 0.161 0.090 1.80 1.34 0.308 0.0851 3.62 0.16 difference in developing time of the two channels caused a large dispersion through the fracture. On the other hand, the measured a is smaller than a p in F4, and solutes experienced smaller dispersion than expected. This may be partly due to the large AR of 6.81. The traverse length of solute becomes shorter as AR increases with constant C v and the omni directional l. [28] In summary, a m may be closer than a f to the true dispersivity of the fracture sample and the estimation of a using equation (9) is acceptable within approximately 30% except for two cases from the conventional ADE fitting. This is plausible considering several assumptions in the small perturbation theory. Accurate results were obtained when the breakthrough curves were analyzed using the moment method if equation (1) did not fit the tailing phenomena in the breakthrough curves. 3.3. Effects of the Geostatistical Parameters on Solute Transport Through Fracture [29] The geostatistical parameters of the fracture samples are discussed in this section. A large C v indicated strong possibility of channeling for fracture F1. From the results of the tracer test, we obtained a slightly larger dispersivity than those of the other fracture samples. A low AR for fracture F1 caused the channel to be meander along the transverse direction as seen in Figure 7a. We expected little and moderate channelings in F2 and F3, which resulted in smaller and moderate dispersivity among those of the fracture samples respectively. Figure 7d shows that the most severe channeling occurred in F4. The large value of AR for fracture F4 caused a straight channel from the inlet to the outlet. The most stable waterfront was observed in F5 due to the smallest C v and the closest AR to isotropy in Figure 7e. [30] We can quantify the effects of the geostatistical parameters on dispersivity based on the small perturbation theory for a variable aperture fracture. We can infer the effects of C v and l on a directly from equation (9). In Figure 8, we plot a m as a black circle and a p as a surface constructed from C v and l. The thick line shows the distance between a m and a p. a p is calculated from equation (9) under the assumption that the sill (A) in equation (12) is 1. a m follows the tendency of a p relatively well as C v and l change. We can deduce from equation (9) that a has a linear relationship with l and the polynomial function of natural logarithm of C 2 v, and the effect of C v on a increases with l. Because the measured A is not always equal to 1, a is proportional to the measured sill value (A) of the semivariogram. In all samples, the differences between a p and a m are about 32% at most regardless of the wide variation of AR. The predicted error of a in F4 is 31%, even though large value of AR (6.81) significantly deviates from the assumption of the small perturbation theory. The results indicate that the effect of AR on a is insignificant compared with those of C v and l, and AR reduces a slightly when AR is greater than 1. 4. Summary and Conclusions [31] Experimental analyses were conducted to examine the effect of aperture variability on the major transport parameter (a) in a single fracture by three statistical parameters, C v, l, and AR. It was demonstrated that the dispersivity from the moment method (a m ) was in a good agreement with that from the small perturbation theory (a p ) in a variable aperture fracture by Gelhar [1993] within 32%. The theory can be acceptable considering the complex phenomena of solute transport through a fracture. On the whole, the predicted a p is closer to a m than to a f from the conventional fitting process using a one-dimensional analytical solution. Figure 6. Comparison of estimated dispersivity (a p ) with dispersivity (a m ) from the moment method and dispersivity (a f ) from conventional fitting process.

LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT SBH 7-9 Figure 7. Transient images of flowing front.

SBH 7-10 LEE ET AL.: EFFECTS OF APERTURES ON TRACER TRANSPORT Figure 8. Effects of coefficient of variation (C v ) and correlation length (l) on dispersivity (a) of fracture. Poor agreements (±32%) of a p with a m for fractures F3 and F4 are caused by channeling phenomena and high value of AR. [32] It is very difficult to conclude which method is superior as far as numerical modeling of a fracture is concerned. However, we can say that it would be appropriate to estimate a using the moment method rather than the conventional fitting process when accurate tailing data of a breakthrough curve are employed. If the accuracy of the measured tail in the breakthrough curve is not guaranteed, the parameters from the conventional fitting process could be used. [33] The effects of C v and l on a are quantitatively analyzed on the basis of the validity of the small perturbation theory. As can be seen from Gelhar s derivation, the magnitude of a is proportional to the polynomial function of natural log of C 2 v. The value of a shows a linear relationship with l at constant C v. The prediction of a can be improved if the measured total sill from the semivariogram is used for the calculation of a. The influence of AR on dispersivity seems to be insignificant compared with those of C v and l. If errors are tolerated within approximately 30%, the effects of the channeling and AR are negligible when estimating a from the small perturbation theory. It is concluded that the small perturbation theory is applicable even for the channeling effect and large variation in AR. [34] Acknowledgment. The authors acknowledge the financial support of the Korea Ministry of Science and Technology under the National Research Laboratory Program, contract M10104000042-01J000001700. References Brown, S. R., Simple mathematical model of a rough fracture, J. Geophys. 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