Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture

Transcription

1 WATER RESOURCES RESEARCH, VOL. 40,, doi: /2003wr002356, 2004 Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture Julie S. Konzuk 1 and Bernard H. Kueper Department of Civil Engineering, Queen s University, Kingston, Ontario, Canada Received 25 May 2003; revised 14 October 2003; accepted 3 December 2003; published 6 February [1] This study provides an evaluation of various modifications of the cubic law, expanding upon previous work as follows: (1) Experimentally measured flow rates and apertures are the basis for the evaluation; (2) a rock fracture is used rather than an analog or numerically simulated fracture; (3) the fracture is not disturbed at any point during the testing; and (4) detailed measurements of the apertures and the top and bottom fracture surface profiles (931,988 measurements in total) are obtained, enabling assessment of the impact of fracture surface undulation and model discretization on the simulated flow rates. The cubic law calculated with either the geometric mean aperture or incorporating surface roughness factors provided reasonable (±10%) estimates of the observed flow rates for Re < 1. The cubic law applied locally (LCL) over-predicted the observed flow rates by at least 1.9 times. Modifying the LCL to incorporate a solution for tapered plates and correcting for surface undulation reduced the over-prediction to at least 1.75 times the measured flow rates. The primary conclusions that we can draw from this work are as follows: (1) There appears to be merit to conducting further studies of the cubic law applied at the single-fracture scale to determine whether similar results are achievable in all fracture types; and (2) the current understanding of when the LCL will provide an adequate representation of the true flow behavior is not entirely correct; more investigation into the effect of fracture surface undulation and other causes of abrupt aperture change (e.g., rock debris trapped within the fracture plane) is required. INDEX TERMS: 1829 Hydrology: Groundwater hydrology; 1832 Hydrology: Groundwater transport; 5104 Physical Properties of Rocks: Fracture and flow; KEYWORDS: fracture, flow, cubic law, modeling, Reynolds Citation: Konzuk, J. S., and B. H. Kueper (2004), Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture, Water Resour. Res., 40,, doi: /2003wr Introduction [2] Understanding the flow behavior of single-phase fluids through rock fractures is essential to understanding groundwater flow and solute transport in the context of water supply, containing and remediating hazardous dissolved contaminants, and designing for long-term nuclear waste storage in the subsurface. While fractures in rock are typically present in interconnected networks, the behavior of the fluid in a single fracture must first be understood before more complicated field-scale fracture networks can be addressed. Accurate prediction of the single-phase flow behavior is particularly important, as models of the flow behavior are often relied upon to engineer safe and sustainable solutions to meet our needs. [3] Numerous conceptual models of flow through single fractures have been proposed. For many years it was believed that the large-scale flow through a rough-walled fracture was similar to that through two smooth, parallel plates, for which a simple solution of the Navier-Stokes equations, called the cubic law (CL), has the form [Bear, 1972] 1 Now at GeoSyntec Consultants, Inc., Guelph, Ontario, Canada. Copyright 2004 by the American Geophysical Union /04/2003WR Q ¼ W rge3 h 12m L (see notation list for parameter definitions). Accurate predictions of laminar flow rates can be achieved using equation (1) when the apparent aperture e in the CL is set equal to a fitting parameter called the hydraulic aperture e h, which is calculated from measured flow rates and hydraulic gradients [Witherspoon et al., 1980]. However, attempts to calculate flow rates using measured apertures in the CL have been less successful [e.g., Lomize, 1951; Louis, 1969; Iwai, 1976; Witherspoon et al., 1980; Neretnieks, 1987; Moreno et al., 1988; Nicholl et al., 1999] due to the effect that fracture surface roughness and aperture variation has on the observed flow behavior, behavior that is not accounted for in the CL. Attempts have been made to improve the match between flow rates simulated using measured apertures in the CL and observed flow rates either through the use of different definitions of the CL aperture [e.g., Neuzil and Tracy, 1981; Tsang and Witherspoon, 1981; Smith and Freeze, 1979; Brown, 1987; Hakami and Barton, 1990; Tsang and Tsang, 1990; Zimmerman et al., 1991; Unger and Mase, 1993; Renshaw, 1995; Nicholl et al., 1999; Méheust and Schmittbuhl, 2001] or by applying a modifying factor to incorporate other information about the fracture [e.g., Lomize, 1951; Louis, 1969; Patir and Cheng, ð1þ 1of17

2 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS 1978; Witherspoon et al., 1980; Walsh, 1981; Walsh and Brace, 1984; Zimmerman et al., 1991; Gutfraind and Hansen, 1995]. Attempts to model the dispersion of contaminants in fractures and to more accurately account for flow tortuosity caused by fracture roughness led to the development of pore-scale approaches such as channel models or the local cubic law (LCL) method, which better model the local flow behavior by accounting for the aperture variation and, more recently, the fracture surface undulation [Nicholl et al., 1999; Brush and Thomson, 2003]. [4] Although our understanding of single-phase flow behavior in rough-walled fractures has improved a great deal over the past 2 decades, many of the conclusions stated in past research may be questionable. For example, comparisons of single-fracture scale flow to LCL simulated flow, which has often been assumed to represent the true flow behavior [e.g., Brown, 1987; Tsang and Tsang, 1990], may be erroneous due to the potential over-prediction of the true flow rates by the LCL [Hakami and Larsson, 1996; Nicholl et al., 1999; Renshaw et al., 2000]. Other errors are introduced through the use of simplified models of a natural, rough-walled fracture [e.g., Lomize, 1951; Louis, 1969; Glass et al., 1991; Brown et al., 1995; Gutfraind and Hansen, 1995; Mourzenko et al., 1995; Oron and Berkowitz, 1998; Yeo et al., 1998; Nicholl et al., 1999],or due to the use of a disturbed fracture [e.g., Reimus et al., 1993; Yeo et al., 1998; Renshaw et al., 2000]. Another source of error is the aperture definition. For example, apertures have been defined as the surface separation distance oriented in one direction globally (referred to here as a vertical aperture), or as the surface separation distance oriented perpendicular to the local fracture direction (referred to here as a perpendicular aperture), which varies with surface roughness [e.g., Mourzenko et al., 1995; Ge, 1997; Oron and Berkowitz, 1998]. [5] There has also been growing evidence that inertial forces become significant and the CL no longer adequately models the flow behavior as the fracture surface roughness and/or the Reynolds number (Re) increases [e.g., Louis, 1969; Brown, 1987; Glass et al., 1991; Zimmerman et al., 1991; Reimus et al., 1993; Mourzenko et al., 1995; Brown et al., 1995; Hakami and Larsson, 1996; Ge, 1997; Yeo et al., 1998; Oron and Berkowitz, 1998; Nicholl et al., 1999; Renshaw et al., 2000; Brush and Thomson, 2003]. Previous research has illustrated that aperture heterogeneity and surface roughness create tortuous flow paths through larger aperture regions and around fracture surface contact points [Walsh, 1981; Neuzil and Tracy, 1981; Tsang, 1984; Tsang and Tsang, 1987; Walsh and Brace, 1984; Brown et al., 1995; Nicholl et al., 1999; Brush and Thomson, 2003]. The velocity gradient perpendicular to the fracture plane near a rough fracture surface may be less steep than the parabolic profile predicted by the cubic law [Schlichting, 1979; Brush and Thomson, 2003]. Deviations in the velocity profile from the ideal parabolic shape have also been found to occur near diverging or converging fracture surfaces, or where differential acceleration of the fluid across the fracture is created by changes in the fracture direction [Brown et al., 1995; Brush and Thomson, 2003]. Flow separation happens in areas of the fracture where the angle of the fracture surface is significant, resulting in the formation of whorls and stagnant zones [Raven et al., 1988; Brown et al., 1995; Gutfraind and Hansen, 1995; Brush and Thomson, 2003]. Stagnant zones will also form near areas of contact between the two fracture surfaces [Oron and Berkowitz, 1998]. All of these factors are artifacts of fracture surface roughness and lead to an increase in the energy dissipation over that predicted by the cubic law ( pore and single-fracture scales). Lower flow rates are thus seen in rough-walled fractures than one would observe in fractures with smooth, parallel edges. [6] Several criteria have been proposed for estimating when the inertial forces begin to impact the flow behavior, based on (1) Navier-Stokes solutions using sinusoidal fractures [Zimmerman and Bodvarsson, 1996; Brush and Thomson, 2003]; (2) an order of magnitude analysis (Oron and Berkowitz [1998]; Zimmerman and Yeo [2000] as referenced by Brush and Thomson [2003]); and (3) comparison between Navier-Stokes and Stokes solutions of simulated rough-walled fractures [Brush and Thomson, 2003]. The Navier-Stokes solution includes the effect of inertial forces (e.g., flow separation), while the Stokes solution is a subset of the Navier-Stokes equations in which the inertial force terms have been neglected. The Stokes solution results in a parabolic velocity profile perpendicular to the local fracture midline direction and no flow separation. The proposed criteria include Re < 1[Schrauf and Evans, 1986; Zimmerman and Bodvarsson, 1996; Brush and Thomson, 2003], Re s e /e AM <1[Brush and Thomson, 2003], and Re e AM /l <1[Brush and Thomson, 2003]. [7] It is unclear, however, whether these criteria apply to all rough-walled fractures. For example, Nicholl et al. [1999] found that even for < Re < 4.3, < Re s e /e AM < 1.2, and < Re e AM /l < 1.2, the LCL overestimated the measured flow rates through analog fractures by 22 47%. These criteria have also yet to be tested to data obtained from a natural rock fracture, which may have different roughness characteristics than the simulated fractures upon which the criteria were developed. [8] The purpose of this paper is to present an evaluation of the ability of various cubic law based single-fracturescale and pore-scale LCL and channel model simulation approaches to accurately simulate the flow behavior through a natural, rough-walled rock fracture. This work expands upon previous work done in this area in the following ways: (1) Experimentally measured flow rates and apertures are the basis for the evaluation; (2) a natural rock fracture is used rather than an analog or numerically simulated fracture, which allows for a better evaluation of the model performance for field applications; (3) the fracture is not disturbed at any point during the testing; and (4) detailed measurements of the apertures and the top and bottom fracture surface profiles are obtained, enabling assessment of the impact of fracture surface undulation and model discretization on the simulated flow rates. [9] The remainder of this paper presents the methodology and results of the laboratory flow measurements (section 2) and aperture measurements (section 3), the results of the single-fracture-scale (section 4.1) and pore-scale (section 4.2) cubic law evaluation, and a summary of the findings of this work (section 5). Appendix A provides details on the geostatistical analysis and transformation of 2of17

3 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS 2. Laboratory Flow Experiment [10] The single-phase flow rates used to evaluate the performance of the various modeling approaches were obtained in a laboratory setting using a rock sample that had a single fracture induced in a manner intended to mimic natural fracture formation. A summary of the method of fracture generation, as well as a description of the flow measurement methodology and results are included below Fracture Generation [11] The rock sample was obtained from a massive dolomitic limestone outcrop in the Kingston, Ontario, Canada, area. The block was approximately 250 mm (width) by 203 mm (height) by 114 mm (depth perpendicular to the fracture plane) in size. The pores of the matrix were on the submicron scale, and the porosity of the limestone matrix was estimated as 0.8% using API RP40, indicating that significant wetting phase flow through the matrix is unlikely to occur. [12] A single tensile fracture with minimal shearing (i.e., mode I) was induced in the block under a low normal stress. In order to closely mimic natural conditions, the fracture was induced along a preexisting plane of weakness and was maintained in an undisturbed state by binding the rock prior to inducing the fracture. At no time was the stress released or the fracture taken apart. This method of fracture formation is similar to that which occurs in the field during near-surface erosional unloading in isotropic rocks or to differential volume changes in a heterogeneous formation caused by either decompression or cooling due to differences in compressibility or thermal contraction coefficients [Hobbs et al., 1976]. To maintain the undisturbed state of the fracture during the flow experiment and to prevent leakage out the sides of the fracture, the block was encased in a rigid epoxy mould (Devcon 1 Plastic Steel Liquid B) on four sides (excluding the top and bottom) Laboratory Flow Methodology and Results [13] The flow experimental apparatus consisted of the limestone block clamped with the fracture oriented vertically between a top and bottom reservoir (see Figure 1). The pressure of the water at the top and bottom of the fracture was kept constant through the use of a constant-head tank and a water overflow port. The flow rate (Q) of the water through the fracture was measured volumetrically for several hydraulic gradients, corresponding to a range in Re (see Figure 2). The Re was calculated following the procedure of Louis [1969] as outlined by de Marsily [1986]: Re ¼ rqd h Am ð2þ Figure 1. Apparatus used to measure the flow rates. All measurements are in millimeters. the fracture surface and aperture measurements into a fracture model suitable for the LCL numerical simulations. where the hydraulic diameter D h (D h =4A/p, where A is the cross-sectional area and p is the perimeter of the fracture perpendicular to the direction of flow) and area A were determined from detailed measurements of the apertures throughout the fracture plane (see section 3 below). The area A was calculated by (1) averaging apertures measured along a plane perpendicular to the direction of flow; (2) multiplying the average aperture by the average length of both fracture surfaces, assuming straight line segments between measurements; and (3) calculating an average area for the entire fracture by averaging the 38 individual plane areas. The perimeter of the fracture was calculated in a similar manner using the Figure 2. The measured flow rates and their corresponding Reynolds numbers. The horizontal and vertical error bars correspond to the uncertainty in the measured hydraulic gradients and flow rates, respectively. Also shown are linear and nonlinear fits to the data. The similarity between linear and nonlinear fits to the flow rates for Re < 5 demonstrates that minimal error should be introduced by extrapolating below Re <1. 3of17

4 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS lengths of the upper and bottom fracture surface traces and the average aperture of each plane. [14] The measured flow rates are shown in Figure 2 and appear to have a quadratic relationship with the hydraulic gradient over the range in Re (2.8 Re 14.3), indicating nonlinear flow occurred over at least part of this range. Since the cubic law is not intended to apply to the nonlinear regime, we fit an equation of the form rh ¼ 2: Q þ 6: Q 2 using the Levenberg-Marquardt method to the measured flow data and used this relationship to extrapolate expected flow rates for Re < 1 (see Figure 2). The data point for Re = 14.3 was not included during the parameter fitting or the subsequent evaluation of the cubic law since it appears to be an outlier in comparison with the other data. A linear fit using the hydraulic aperture calculated from the lowest Re data point is also shown in Figure 2 as a comparison. [15] For Re < 1, the fracture studied here meets the criteria suggested by Brush and Thomson [2003] as a requirement for adequate simulation of the flow behavior using the local cubic law (i.e., Re <1,Re s e /e AM < 1, and Re e AM /l < 1); the extrapolated data for Re < 1 was thus used for evaluation of the cubic law approach. The close agreement between the two fits for Re < 5 indicates that minimal error should be introduced if we extrapolate to Re < 1 using the nonlinear fitting approach and that flow was likely linear within this range. [16] The hydraulic aperture calculated at each of the eight measured flowrates using equation (1) ranged from 299 mm to 318 mm with an arithmetic mean of 307 mm. In comparison, the hydraulic aperture calculated for Re approaching zero was 332 mm. 3. Fracture Aperture Measurements [17] Detailed fracture aperture measurements were necessary for use as input for the evaluation of the flow models described below. An overview of the aperture measurement process and results are included below, as well as a statistical analysis of the measured apertures. More details are given by Konzuk [2001] Fracture Aperture Measurement Method [18] The aperture measurements were obtained using a method similar to that of Gale [1987] and Hakami and Larsson [1996] in which an image analysis system is used to trace and record the fracture surface coordinates at known locations within the fracture plane. The image analysis method was chosen over other faster, less labor intensive methods such as light attenuation [e.g., Detwiler et al., 1999; Brown et al., 1998; Nicholl et al., 1999; Renshaw et al., 2000], profilometry [e.g., Gentier, 1986; Brown and Scholz, 1985a, 1985b; Reimus et al., 1993], X-ray computed tomography [e.g., Johns et al., 1993; Keller, 1998; Bertels et al., 2001], and nuclear magnetic resonance imaging [e.g., Renshaw et al., 2000; Kumar et al., 1995], because of its accuracy, the detail that can be recorded, the ability to measure surface profiles simultaneously with apertures, and because the fracture is undisturbed during the measurement process. The accuracy of the measurement technique was estimated at 2 mm (220 magnification) and 4 mm (110 ð3þ magnification) for a well-defined edge [Konzuk, 2001]. The precision for a rougher edge, such as that seen on a roughwalled fracture, was estimated at 11 mm (220) and 22 mm (110). [19] The apparatus consisted of a stereo microscope providing 10 to 220 magnification, with a chargecoupled device (CCD) camera mounted directly onto the microscope and connected to a computer containing image analysis software. Upon formation of the fracture and prior to the flow experiments, the fracture sample was encased in epoxy mold on four faces of the sample in order to stabilize the fracture and ensure no change in the aperture. Once the flow measurements were completed, a resin was injected into the fracture to ensure that the fracture aperture was not changed due to potential damage occurring during the measurement process. The resin was not a necessary requirement to maintain fracture rigidity because of the presence of the epoxy mold; it provided added protection to eliminate local-scale damage along the fracture trace during grinding of the rock surface. A red dye was added to the resin to enhance the ability to distinguish between the rock and the fracture during the imaging process. The fracture sample was then mounted permanently onto a steel plate using plastic steel epoxy and locked into place on the table of a refurbished milling machine, which provided controlled movement in three perpendicular directions (x, horizontal along the fracture trace; y, horizontal perpendicular to the fracture trace; and z, vertical). [20] A magnification factor of 160 was typically used to capture images of the fracture, but was decreased in areas of large fracture aperture down to a minimum of 60. Measurements of the location of each fracture surface were taken in the horizontal plane along lines oriented in the y direction, with an average spacing between the lines of 20 mm. Between horizontal planes, the rock was ground down and polished using a wet grind process so that fracture surface data could be collected throughout the entire volume. The average vertical spacing between horizontal planes was approximately 5 mm and ranged from 2 to 8 mm. In total, 38 horizontal planes were analyzed, each containing an average of 25,000 data points, resulting in a total of 931,988 measurements of each fracture surface. [21] Many different methods have been proposed for estimating the fracture aperture including a vertical aperture, which is the separation distance between two surfaces according to some preestablished Cartesian coordinate system; a perpendicular aperture, which is the separation distance oriented perpendicular to the local trend of the fracture plane [Bear, 1993; Mourzenko et al., 1995; Ge, 1997]; and a segment aperture (see Figure 3 for an illustration), which is the perpendicular separation distance between the average position of each fracture surface over a segment [Oron and Berkowitz, 1998]. The roughness of our fracture precludes the use of the segment aperture in some locations within the fracture. The perpendicular aperture was calculated using the method described by Bear [1993] and Mourzenko et al. [1995] in which the aperture is assumed equal to the diameter of a circle (or sphere in three dimensions) that is drawn around the local centerline of the fracture and increased in size until the circle touches both walls of the fracture. The vertical aperture was calculated as the distance in the y direction between the two fracture surfaces. 4of17

5 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Figure 3. Illustration of the various aperture definitions as defined in the text. [22] The impact of the aperture definition on simulated flow behavior has yet to be evaluated using an actual roughwalled fracture; however, studies using either simulated or simplified fractures have illustrated that there may be a significant impact, particularly for undulating fractures [Mourzenko et al., 1995; Ge, 1997; Oron and Berkowitz, 1998]. We performed simulations using both vertical and perpendicular apertures to evaluate the magnitude of the impact of the aperture definition on the simulated flow rates (see section 4) Fracture Aperture Measurements [23] A typical horizontal plane through the sample showing a trace of the fracture surfaces is shown in Figure 4. Figure 5 contains the corresponding measured perpendicular apertures. These figures illustrate the undulating and rough-walled appearance of the fracture (Figure 4), as well as the large variation in perpendicular aperture size across the fracture (Figure 5). The aperture sizes range from 0 to mm within the entire fracture plane. Features of interest that are shown in Figure 4 are debris in the form of rock fragments that are trapped within the fracture, as well as secondary branching of the fracture (i.e., where the fracture has split into two or more branches that are nearly parallel and then rejoin further along the fracture). These features are important to the flow behavior since they create aperture constrictions and abrupt changes in aperture over a short distance. We estimate from visual inspection of the images that approximately 10 percent of the fracture contains one or more of these features. [24] Figure 6 shows plots of the entire measured vertical and perpendicular aperture fields. On these plots, pockets of large and small apertures can be seen, sometimes very close to one another. There is also an apparent trend in the horizontal direction from generally larger apertures on the right to smaller apertures on the left. A linear feature of small apertures can be seen on the lower right side of both plots; this is due to the fracture splitting into two smaller branches that extend nearly one third the width of the fracture. The top and bottom edge of the fracture have very large aperture regions (i.e., e > 3 mm) caused by pitting that occurred during sample preparation. There are, however, areas within the interior of the fracture that are unaffected by pitting but still have apertures greater than 2 mm. The percentage of closure within the primary fracture (i.e., excluding the secondary branches) was estimated as <1%, 5of17 which is in agreement with the assessment of Iwai [1976] and Hakami and Larsson [1996] for fractures under low normal stress. [25] The results of a statistical analysis of a randomly sampled subset (5000 data points) of the vertical and perpendicular aperture data sets are listed in Table 1. Random subsets were used for all statistical analyses conducted in the course of this study in order to minimize bias introduced due to spatial correlation between the apertures [Russo and Jury, 1987a; Woodbury and Sudicky, 1991]. A comparison between the vertical and perpendicular apertures illustrated differences between the two, including size differentials greater than 100 mm in some locations ( perpendicular apertures being smaller), and a perpendicular aperture histogram with a similar shape to the vertical apertures, but shifted toward the smaller apertures and having a mean that was 36 mm smaller (see Table 1). 4. Evaluation of the Cubic Law Based Simulation Approaches [26] The performance of the various cubic law based simulation approaches discussed in section 1 was evaluated through comparisons between the measured and extrapolated flow rates shown in Figure 2 and the simulated flow rates. The results are presented below in two sections addressing the single-fracture scale and the pore-scale simulation approaches Single-Fracture Scale Flow Simulation Methods and Results [27] Many different modifications of the CL (equation (1)) have been proposed at the single-fracture scale to account for the deviation from the ideal CL flow behavior caused by rough, nonparallel surfaces. The most popular approach has been to modify the definition of the aperture used in equation (1) [e.g., Smith and Freeze, 1979; Tsang and Witherspoon, 1981; Neuzil and Tracy, 1981; Tsang, 1984; Brown, 1987; Hakami and Barton, 1990; Tsang and Tsang, 1990; Zimmerman et al., 1991; Unger and Mase, 1993; Renshaw, 1995]. The remainder of the proposed methods incorporate other information about the fracture, such as the contact area [Walsh, 1981], the tortuosity of the Figure 4. A horizontal plane of the fracture showing typical traces of the fracture surfaces.

6 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Figure 5. The perpendicular aperture variation corresponding to the fracture surface traces of the horizontal plane shown in Figure 4. flow path [Walsh and Brace, 1984], or losses caused by surface roughness [Lomize, 1951; Louis, 1969; Patir and Cheng, 1978; Witherspoon et al., 1980; Zimmerman et al., 1991; Gutfraind and Hansen, 1995]. [28] The flow rates predicted using the various modifications to the CL were compared with the measured flow rates to assess their performance in predicting the observed flow behavior. The following proposed CL aperture definitions were evaluated: (1) the arithmetic mean (AM) [Brown, 1987]; (2) the geometric mean (GM) [Smith and Freeze, 1979; Tsang and Tsang, 1990; Unger and Mase, 1993; Renshaw, 1995]; (3) the harmonic mean (HM) [Unger and Mase, 1993]; (4) the volume averaged mean (VAM) [Schrauf and Evans, 1986; Hakami and Barton, 1990]; and means recommended by (5) Tsang and Witherspoon [1981] (TWM) and (6) Zimmerman et al. [1991] (ZM). The TWM and ZM means were developed to account for the variation of the apertures in a direction parallel and perpendicular to the direction of flow. Of the different modifications to the CL that employ modifying factors to incorporate other information about the fracture, the following were evaluated: (1) a contact ratio factor (CRF) [Walsh, 1981], and roughness factors proposed by (2) Louis [1969] (RF), (3) Zimmerman et al. [1991] (ZRF), (4) Patir and Cheng [1978] (PC), and (5) Gutfraind and Hansen [1995] (GH). Details of the equations used to calculate each parameter can be found in the original references. Table 2 contains a listing of the resulting parameter values. For the RF method, the roughness coefficient e m was calculated by determining the mean asperity height for 1 mm segments of the fracture surface, then averaging the mean heights to obtain e m =60mm. [29] Figure 7 provides a visual comparison between the measured and extrapolated Q and those simulated using the vertical (Figure 7a) and perpendicular (Figure 7b) apertures in the modified cubic law relationships listed in Table 2. These plots are formatted such that the vertical axis is in the form of a ratio between the simulated and measured flow rates (Q sim /Q obs ); therefore a horizontal line through Q sim / Q obs = 1 represents the best fit obtainable. A vertical line shows where Re becomes greater than 1, a point at which Figure 6. Contour plots of the entire measured (a) vertical and (b) perpendicular aperture fields. 6of17

7 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Table 1. The Results of Statistical Analyses of the Vertical and Perpendicular Aperture Data Sets Statistic Vertical Apertures Perpendicular Apertures Mean aperture, mm Aperture variance, mm Skewness, mm Minimum aperture, mm 0 0 Maximum aperture, mm nonlinear flow behavior is thought to become more prominent [Zimmerman and Bodvarsson, 1996; Brush and Thomson, 2003]. [30] A comparison between Figures 7a and 7b illustrates that the flow rates predicted using the vertical apertures are higher and, in general, provide a poorer match to Q obs than those predicted using the perpendicular apertures (the exception is the TWM model). This result is in agreement with the conclusions of Mourzenko et al. [1995], Ge [1997], and Oron and Berkowitz [1998] and also agrees with the fact that according to Darcy s law, the flow rate should be calculated using the cross-sectional area perpendicular to the local flow direction (i.e., not oriented according to a Cartesian coordinate system applied globally). Therefore only the results for the flow rates simulated using perpendicular apertures are presented herein. [31] Also seen in Figures 7a and 7b is the fact that the deviation between the simulated and observed flow rates increases as the Re increases above 1. This behavior is expected for nonlinear flow and substantiates the need to Figure 7. Results of the single-fracture-scale simulations using (a) vertical apertures and (b) perpendicular apertures. 7of17

8 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Table 2. The Resulting Parameter Values Used in the Simulations of the Modified CL Relationships Cubic Law Modification a Parameter Vertical Apertures Perpendicular Apertures AM, 1 e 417 mm 380 mm GM, 2 e 368 mm 335 mm HM, 3 e 200 mm 270 mm VAM, 4 e 379 mm 379 mm TWM, 5 e 21 mm 34mm ZM, 6 e 406 mm 378 mm CRF, 7 a RF, 8 R r ZRF, 9 f PC, 10 f GH, 11 f a Cubic law modifications proposed by 1, Brown [1987]; 2, Smith and Freeze [1979], Tsang and Tsang [1990], Unger and Mase [1993], and Renshaw [1995]; 3, Unger and Mase [1993]; 4, Schrauf and Evans [1986] and Hakami and Barton [1990]; 5, Tsang and Witherspoon [1981]; 6, Zimmerman et al. [1991]; 7, Walsh [1981]; 8, Louis [1969]; 9, Zimmerman et al. [1991]; 10, Patir and Cheng [1978]; and 11, Gutfraind and Hansen [1995]. evaluate the cubic law performance to the flow rates extrapolated to Re < 1. [32] For Re < 1, the GM is the only mean aperture definition that provides a reasonable fit to the extrapolated flow rates. The remainder of the mean apertures overpredict Q obs except for HM and TWM, which underpredict Q obs (see Figure 7b). In comparison, Brown [1987] found that the AM provided a reasonable match to the LCL Q. Nicholl et al. [1999] found that the AM and GM over-predicted flow rates measured in a textured glass analog fracture in the range of 40 76% (AM) and 32 58% (GM), in comparison to the 50% (AM) and 3% (GM) over-prediction found here. The aperture distribution of Nicholl et al. [1999] was skewed with the tail toward the smaller apertures, which is opposite to the lognormal-like distribution of this fracture. Smith and Freeze [1979], Tsang and Tsang [1990], and Renshaw [1995] found that the GM best predicted the flow rates through a lognormally distributed aperture field. Unger and Mase [1993] found that for their normally distributed aperture field, the GM performed best for smaller contact areas, but the HM was better for large contact areas. [33] Of the various modifying factors, the flow rates calculated using the roughness factors RF, ZRF, and PC are generally within ±10% of the observed rates for Re <1 (see Figure 7b). The similarity between the ZRF and the GM is not surprising given that the application of this roughness factor was intended to result in flow rates that are similar to those calculated using the geometric mean aperture [Zimmerman et al., 1991]. The RF method would not be expected to perform as well for a fracture with more closure as the roughness factor is calculated based on the asperity heights, which will not change beyond the point of initial fracture surface contact since the asperity height decreases at the same rate as the aperture [Gale, 1990]. This has not been experimentally demonstrated in other studies but can be seen from a study of the mathematics. The addition of the CRF has little effect on Q sim due to the low contact area. The GH results are not shown in Figure 7b since Q sim was negative due to a negative modifying factor, which was the result of the maximum asperity height (y max ) being larger than twice the arithmetic mean aperture (e AM ), indicating that the GH method is inappropriate for fractures having the y max > e AM. [34] Our results seem to concur with previous studies done using fractures with lognormally distributed apertures, similar to the fracture studies here. The use of the GM is supported by stochastic theory [Dagan, 1979] for fractures with lognormally distributed apertures displaying isotropic correlation structure. Our results and those of previous studies suggest that for a lognormally distributed aperture field with small contact areas, the GM and/or the ZRF and PC roughness factors combined with the arithmetic mean aperture may provide a reasonable estimate of the hydraulic aperture. However, more work is required to evaluate the appropriateness of using either the GM or the roughness factor approaches for other fracture types Pore-Scale Flow Simulations [35] Four pore-scale simulation approaches were evaluated, including two channel models, the classical LCL method, and a modified form of the LCL proposed by Brush and Thomson [2003] to account for fracture surface undulation. The LCL method, which is described below in more detail, requires detailed knowledge of the fracture apertures at specific locations that correspond to the nodes of a finite difference grid. Since the apertures were not sampled on a regular grid corresponding to the nodal locations, it was necessary to estimate the apertures between the sampling points using kriging. Section and Appendix A provide more details on the generation of a fracture model from the measured aperture data that is suitable for the numerical simulation Generation of the Fracture Model [36] The measured aperture data were used to estimate apertures at the model nodal locations using a geostatistical approach. Appendix A provides the details of the geostatistical analysis, which was conducted following the approach recommended by Armstrong [1984] and Woodbury and Sudicky [1991] consisting of (1) collecting and analyzing the data (i.e., determining the probability distribution, mean and higher moments, and identifying outliers), (2) estimating the sample semivariograms, (3) fitting a theoretical semivariogram model to the experimental data including trend identification, and (4) estimation (i.e., kriging). [37] The perpendicular aperture distribution is shown in Figure 8a, along with the best fit (using the Levenburg- Marquardt method) lognormal, truncated normal, and gamma probability distribution models. None of these probability distribution models fits the data significantly better than the other two. Several indicators alluded to the 8of17

9 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Figure 8. A comparison between the histograms of the perpendicular aperture data with (a) the secondary branches removed and (b) the trend removed. Best fit probability distribution model fits are also shown in both panels. presence of a trend in the aperture data in the horizontal direction, including a general trend in the aperture size seen in the plot of the aperture field (Figure 6b), anisotropy in the semivariograms (see Figure 9), and a horizontal semivariogram that increases with the lag distance h l without limit (Figure 9) [Neuman and Jacobson, 1984]. Plots of the moving mean apertures in the horizontal and vertical directions illustrated that the cause of the trend is due to a linear increase in the local mean apertures in the horizontal direction (see Figure 10). [38] The model aperture data were generated accounting for the trend in the measured data by choosing a trend model based on an understanding of the cause of the trend, subtracting this trend from the observed sample values to obtain the residuals, calculating the semivariogram of the 9of17 residuals, and using a simple kriging approach to estimate the apertures. The final step was to add the trend back into the kriged data. Where a large data set exists (similar to that available here), such that the search neighborhood for the kriging can be limited to a small area, the potential bias introduced in estimating the trend separately becomes negligible [Russo and Jury, 1987b]. [39] The estimated horizontal and vertical semivariograms of the residuals are shown in Figure 11. The residuals semivariogram in the vertical direction is similar in shape, thus indicating isotropy and stationary statistics of the residual apertures. The most appropriate semivariogram model (see Table 3 and Figure 11) for the kriging was found to be a combination of two exponential models, as chosen through the use of various criteria (see Appendix A).

10 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Figure 9. Semivariograms of the perpendicular aperture data including the trend estimated in the principal directions of anisotropy (aligned horizontally and vertically). The resulting best fit small- and large-scale correlation lengths in the horizontal direction were 3 mm and 69 mm, respectively. For the vertical direction, only the small-scale correlation length could be estimated with any confidence due to a lack of information, with a best fit value of 8 mm. Cross validation was used to evaluate the accuracy of the estimation process (see Appendix A for details). [40] Upon validation of the kriging approach, simple kriging of the residual apertures was performed. The residual apertures were estimated at locations corresponding to the nodal locations of a finite difference grid used in the pore-scale simulations for several grid sizes, with nodal spacings ranging from 250 mm, 500 mm, 1 mm, 5 mm, 10 mm, to 25 mm. The final aperture estimates were calculated from the estimated residual apertures by adding the horizontal trend in the mean apertures to the residual apertures. Figure 12 shows contour plots of the resulting estimated aperture fields Pore-Scale Simulation Methods and Results [41] Four pore-scale simulation approaches were evaluated, including two channel models, the classical LCL method, and a modified form of the LCL proposed by Brush and Thomson [2003] that accounts for fracture surface undulation. The two channel models consisted of (1) a set of constant aperture channels with the apertures varying only in a direction perpendicular to flow [Neuzil and Tracy, 1981]: X! e 3 Wrg n Q sim ¼ 12m h L and (2) the variable aperture channel model of Tsang and Tsang [1987]: ð4þ [42] Figure 13 shows a comparison between the simulated and observed Q. The constant-aperture channel model is seen to significantly over-predict the observed Q, which is not surprising given that the effect of aperture constrictions along flow paths is not accounted for using this approach. The excellent fit between Q sim and Q obs for the variable aperture channel model is not necessarily a reflection of the appropriateness of the model since a lack of a priori knowledge of the number of channels required that we treat this parameter as a fitting factor. Forty-six channels with widths equal to the aperture correlation length were required to obtain this fit, which corresponds to approximately 74% of the fracture plane. The large number of channels required to provide a good fit to the observed flow data suggests either that distinct channels do not exist in this fracture and that a channel model is thus not an appropriate choice, or that the number of channels had to be artificially increased in order to compensate for an under-prediction of the flow rate through each channel. [43] The classical [e.g., Tsang, 1984; Brown, 1987; Tsang and Tsang, 1989; Mourzenko et al., 1995; Nicholl et al., 1999] and modified [Brush and Thomson, 2003] LCL approaches were the third and fourth pore-scale simulation approaches evaluated. The classical and modified LCL methods are similar except in the manner in which the transmissivities are calculated and how the fracture geometry is represented. For the classical LCL, the fracture was modeled as a two-dimensional, variable-aperture medium having local parallel-plate segments, with negligible fracture surface undulation in the third dimension. The cubic law is assumed to apply locally, with the internode transmissivities calculated from the harmonic mean of the apertures as follows: e 3 i1=2;j ¼ 2e3 i;j e3 i1;j e 3 i;j þ e3 i1;j By substituting perpendicular apertures in equation (6) in place of vertical apertures, we can incorporate some information about the surface undulations. ð6þ Q i ¼ rg m X nc j¼1! 1 12x e 3 ah ð5þ ij The parameters rgh were added to equation (5) in place of the P w proposed by Tsang and Tsang [1987] to account for the effect of gravity in a nonhorizontal fracture. Both models assume channels with aperture distributions that are related to the probability distribution of the entire aperture field. 10 of 17 Figure 10. Moving means of the perpendicular aperture data in the horizontal and vertical directions. The trend lines were fit to the data using linear regression.

11 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Table 3. Parameters Describing the Best Fit Model Semivariograms in the Horizontal and Vertical Directions for the Perpendicular Apertures, and the Results of the SSQ, AIC, and c 2 Goodness-of-Fit Analyses Data Set Parameter Exponential Model Spherical Model Exponential Combination Model Horizontal direction a 1, mm c 1,mm a 2,mm 68.9 c 2,mm SSQ AIC c 2a c 2 88,0.025 b c 2 88,0.975 b Vertical direction a 1, mm c 1,mm a 2,mm c c 2,mm c SSQ AIC c 2a c 2 70,0.025 b c 2 70,0.975 b a Calculated statistic. b Theoretical statistic. c Large confidence intervals were estimated for a 1 and c 2, indicating an inability to estimate the parameters due to lack of information. [44] The modified LCL is formulated to provide a more realistic representation of the fracture surface by modeling the fracture as a variable-aperture medium with undulating surfaces and consisting of local tapered plates as proposed by Brush and Thomson [2003]. The analytical solution of the Stokes equation for two-dimensional flow through tapered plates was used to calculate the internodal transmissivities in the manner of Brush and Thomson [2003] using a weighted harmonic mean of the transmissivities as follows: e 3 iþ1=2;j x ¼ " x 2 1 b i;iþ1=2 e 3 i;iþ1=2 þ x 2 # 1 1 b iþ1=2;iþ1 e 3 ð7þ iþ1=2;iþ1 The internodal transmissivities e 3 i,i+1/2 and e 3 i+1/2,i+1 are calculated as outlined by Nicholl et al. [1999] but modified as proposed by Brush and Thomson [2003] to include terms to correct the aperture and nodal separation distance values to reflect fracture surface undulation (see original references for details). [45] Both LCL simulations were performed using a mass conservation approach, which for a nondeformable medium and an incompressible fluid under steady state conditions, is governed by the relationship i k ij i i ¼ 0 Equation (8) was solved using a block-centered finite difference scheme, which is fully implicit with a secondorder accurate spatial derivative. The resulting set of matrix equations was solved using a direct Gauss-Jordan elimination procedure. [46] Both numerical models were verified to three analytical solutions; the first using a parallel-plate fracture, the second employing a fracture with apertures varying in the direction parallel to flow, and the third with the apertures varying in the direction perpendicular to flow. The modified LCL was also verified to an analytical solution for a fracture ð8þ with apertures varying in the direction parallel to flow, but with the fracture surface inclined at a 45 degree angle in the third dimension. [47] The results in Figure 13 represent those simulated using a nodal spacing of 1 mm (see Figure 12c). The classical LCL Q sim range from approximately 1.9 to 2.5 times Q obs, even for the data extrapolated to Re <1. The classical LCL approach has over the years often been assumed to provide a better representation of the flow rates [e.g., Brown, 1987] than seen here, although our results are similar to those in a more recent study using a natural rock fracture and vertical apertures in which the simulated flow rates were 2.4 times the measured rates [Hakami and Larsson, 1996]. Nicholl et al. [1999] also found that the classical LCL over-predicted the measured flow rates by a factor of for analog fractures. Figure 11. A plot of the semivariograms of the perpendicular aperture data with the trend removed aligned in the horizontal and vertical directions. Also included are the best fit semivariogram models. 11 of 17

12 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Figure 12. Contour plots showing the estimated perpendicular apertures for nodal spacings of (a) 250 mm, (b) 500 mm, (c) 1 mm, (d) 5 mm, (e) 10 mm, and (f ) 25 mm. 12 of 17

13 KONZUK AND KUEPER: EVALUATION OF CUBIC LAW BASED MODELS Figure 13. Results of the pore-scale simulations. [48] The modified LCL provide a slightly improved fit to the measured flow rates; however, the simulated flow rates are still at least 1.7 times the measured rates, with the discrepancy increasing as the Re increases. In comparison, Nicholl et al. [1999] had simulated flow rates using this approach (not including corrections for surface undulation) that were 1.36 times the flow rates measured in an analog fracture. [49] The discrepancy between Q obs and Q sim has been suggested to be at least partially due to experimental error [Hakami and Larsson, 1996], the use of vertical as opposed to perpendicular apertures [Mourzenko et al., 1995; Ge, 1997; Oron and Berkowitz, 1998; Brush and Thomson, 2003], poor model spatial resolution [Glass et al., 1991; Nicholl et al., 1999], and not accounting for out-of-plane (i.e., perpendicular to the fracture plane) tortuosity caused by surface undulations [Brush and Thomson, 2003]. Errors may also be introduced through the removal of smaller secondary branches; however, the removal of the branching during the fracture model generation process should result in an under-prediction of the flow rates, rather than the overprediction seen here. [50] We conducted sensitivity analyses for these various factors and found, similar to Nicholl et al. [1999] and Yeo et al. [1998], that although these factors may be minor sources of error, they are not the principal causes of the discrepancy between Q obs and Q sim. For example, aperture measurement errors of 100 mm (over 4 times the estimated error) would be required throughout the entire fracture to reproduce the observed Q; errors of this magnitude are highly unlikely. The use of vertical apertures, since they are larger than the corresponding perpendicular apertures, would result in an even larger discrepancy between Q obs and Q sim. Including out-of-plane tortuosity, which occurs in a rough-walled fracture due both to the small-scale roughness of the fracture surfaces and to the large-scale undulations of the fracture [Brown et al., 1995; Ge, 1997; Oron and Berkowitz, 1998; Nicholl et al., 1999; Brush and Thomson, 2003], in the form of both a modifying factor following the procedure of Brown et al. [1995] and the modified LCL resulted in reductions in Q sim of only 18% and 10%, respectively. [51] Finally, we conducted a sensitivity study for nodal spacing resolution; the results of which are shown in Figure 14. These results demonstrate that varying the spatial resolution of the simulation between 250 and 25 mm (2 orders of magnitude) can result in flow estimates varying between 1.5 and 2.4 times the measured flow rates. If we compare the fracture field plots in Figure 12, we can see that as more detail is incorporated in the fracture apertures, more of the smaller flow paths will be represented. It appears that the simulated flow rates converge for spatial resolutions below the small-scale correlation length of approximately 3 mm, which agrees with Reimus et al. [1993] and Nicholl et al. [1999], who found that the spatial resolution of the simulation must be less than the correlation length to minimize the simulation errors. [52] According to Brush and Thomson [2003], the modified LCL should approximate the Stokes solution where s e /e AM < 1 and e AM /l < 0.2, and the Stokes solution should approximate the Navier-Stokes solution where Re <1,Re s e /e AM < 1 and Re e AM /l < 1. Both this study (for Re < 1 data) and the Nicholl et al. [1999] study meet these criteria, yet the flow rates are times the measured rates. While it is possible that extrapolating the flow rates to Re < 1 may introduce some error, the Brush and Thomson [2003] results indicate that even for Re = 2.8, the simulated flow rates should still be within 10% of the measured rates. Brush and Thomson [2003, Figure 10] also demonstrate that as the roughness of the fracture (as measured by the ratio s e /e AM and the magnitude of the fracture surface undulation in the third dimension, s y ) and the hydraulic gradient increases, the deviation between the Navier-Stokes and Stokes solutions increases. While the fracture studied here fell within the range of surface roughness tested by Brush and Thomson [2003], the surface undulation of the fracture was over 4 times higher (i.e., s y = 4.2 mm). Therefore it is possible that the inertial forces may have had a larger impact on the flow rates than those simulated by Brush and Thomson [2003]. The presence of debris (rock chunks) trapped within the fracture, creating abrupt changes in aperture, likely also added to the magnitude of Figure 14. Effect of model nodal spacing on the flow rates simulated using the classical LCL and perpendicular apertures. 13 of 17