On the appropriate “equivalent aperture” for the description of solute transport in single fractures: Laboratory-scale experiments

Abstract

[1] Three distinct definitions of “equivalent aperture” have been used in the literature to describe variable-aperture fractures; however, significant inconsistencies exist in the literature as to which “equivalent aperture” is appropriate for simulating solute transport. In this work, a systematic series of hydraulic and tracer tests was conducted on three laboratory-scale fracture replicas, and the cubic law, mass balance, and frictional loss apertures were calculated. The analytical solution of the one-dimensional advection-dispersion equation was fit to the experimental breakthrough curves. Additionally, one of the experimental aperture fields was measured directly using a light transmission technique. The results clearly demonstrate that the mass balance aperture is the only appropriate “equivalent aperture” for describing solute transport in a single variable aperture fracture and that the mass balance aperture is an excellent approximation of the arithmetic mean aperture. Previously, these conclusions have been reached based only on theoretical and numerical analyses.

1. Introduction

[2] Prior to the 1980s, researchers in the field of fracture flow and transport conceptualized a single fracture, the basic unit of fractured media, as a pair of parallel plates with a constant aperture. It is now well recognized in the literature that single fractures are rough-walled conduits with variable apertures and points of contact [e.g., Durham and Bonner, 1994; Novakowski and Lapcevic, 1994; Brown et al., 1998]. Fracture apertures can be described by normal [e.g., Lee et al., 2003], lognormal [e.g., Keller, 1998; Keller et al., 1999], or gamma distributions [e.g., Tsang and Tsang, 1987], or a self-affine scale invariance [e.g., Plouraboue et al., 1995]. The natural fracture wall surface roughness can also be well described by a self-affine scale invariance [Brown and Scholz, 1985; Schmittbuhl et al., 1995; Bouchaud, 1997]. Although the combination of surface roughness and aperture field variability renders the parallel plate description of fractures inadequate for the majority of flow and transport problems [e.g., Durham and Bonner, 1994; Brown, 1995; Keller et al., 1999], researchers often find it convenient to represent aperture fields in terms of an “equivalent aperture” in both theoretical studies and experimental investigations. Various definitions of “equivalent apertures” appearing in the literature caused some confusion until Tsang [1992] reviewed the literature and categorized three distinct “equivalent apertures.” Specifically, Tsang [1992] described the mass balance aperture (μ_m), the cubic law aperture (μ_c), and the frictional loss aperture (μ_l), which typically have the following relationship: μ_m ≥ μ_c ≥ μ_l [Tsang, 1992] with some exceptions [Silliman, 1989]. The mass balance and cubic law aperture correspond to the arithmetic and geometric mean of the aperture field, respectively [Tsang, 1992]. Smith and Freeze [1979] employed numerical simulations to verify that the geometric mean permeability gives the correct volumetric flow rate for linear flow in a two-dimensional random field of permeabilities. Gelhar [1993] and Tsang [1992] concluded from their theoretical analyses that the geometric mean aperture is the appropriate “equivalent aperture” for use in hydraulic calculations, while the mass balance aperture is more appropriate for evaluating the mean displacement of a solute. Moreno et al. [1988] confirmed by numerical simulations that the mean tracer residence time based on the mass balance aperture agreed very well with that derived from the time moment of the tracer breakthrough curve obtained from particle-tracking simulations.

[3] However, these conclusions lack support from experimental observations in well-characterized fractures. Moreover, in practice, researchers do not always distinguish between the above-mentioned three equivalent apertures and therefore do not always use the mass balance aperture in their interpretation of solute tracer tests [e.g., Hinsby et al., 1996; Kosakowski, 2004]. Therefore the conventional procedure of fitting one-dimensional analytical solutions to experimental breakthrough curve data will lead to erroneous estimates of transport parameters. Furthermore, transport in fractured media is often affected by diffusion into the porous matrix and infill material among other factors. The discrepancy in breakthrough curve fit caused by using the hydraulic or frictional loss aperture instead of the mass balance aperture can be compensated for by adjusting parameters related to these factors, which may lead to additional errors when estimating solute transport parameters. The goal of this research is to demonstrate and confirm that the mass balance aperture is the only equivalent aperture appropriate for simulating solute tracer transport in single fractures in which the effects of both matrix diffusion and infill material are negligible. This goal was accomplished through conducting a series of systematic solute transport experiments at the laboratory scale, in well-characterized fractures with isolated parameters, and fitting the measured breakthrough curves to mathematical models.

2. Experimental Design

2.1. Materials and Methods

[4] A series of hydraulic and solute tracer transport tests were conducted in three transparent fracture replicas, F1, F2, and F3 (Table 1). These tests were designed as a 3² factorial experiment, with three specific discharges (based on μ_c) applied in each of the three synthetic fractures and all other variables held constant. The fabrication of the fracture replicas was based on the method presented by Dickson [2001] and employed an impervious epoxy thereby negating any matrix effects on flow and transport in the fracture.

Table 1. Summary of Hydraulic and Solute Tracer Test Results^a

Fracture ID	Dimension, mm	Manifold Volume^b		μc,^c mm	vc, mm/s	μm,^c mm	vm, mm/s	μl,^c mm	vl, mm/s	Estimated Fracture Aperture Volume,^d mL			Peclet Number^e	Reynolds Number^f
		Upstream, mL	Downstream, mL							Based on μc	Based on μm	Based on μl
F1	241.5 × 149.0	19	17	0.40 ± 0.07	0.55	0.79 ± 0.07	0.28	0.29 ± 0.05	0.77	14	28	10	248	0.25
				0.40	1.09	0.93		0.27		14				0.49
				0.40 ± 0.03	1.67	1.08 ± 0.03	0.63	0.24 ± 0.09	2.72	14	39	9	750	0.74
F2	350.0 × 214.5	45	43	0.54 ± 0.03	0.55	1.53 ± 0.02	0.21	0.33 ± 0.10	0.97	41	115	25	352	0.36
				0.54 ± 0.02	1.09	1.58 ± 0.01	0.38	0.33 ± 0.12	1.83	41	119	25	666	0.71
				0.54 ± 0.01	1.67	1.71 ± 0.01	0.53	0.32 ± 0.16	2.84	41	128	24	1001	1.10
F3	349.0 × 230.0	49	52	0.60 ± 0.03	0.55	1.49 ± 0.02	0.22	0.41 ± 0.10	0.80	48	120	33	369	0.41
				0.60 ± 0.01	1.09	1.54 ± 0.01	0.39	0.39 ± 0.12	1.55	48	124	31	670	0.81
				0.40 ± 0.07	1.67	1.93 ± 0.01	0.46	0.34 ± 0.18	2.60	48	155	27	989	1.24

• a All experiments were conducted at 25 ± 2°C, and the value of the parameters used in these calculations corresponds those at 25°C (i.e., ρ = 997 kg/m³, η = 0.00089 N.s/m²).
• b Manifold volume includes the volume of the manifold, sampling, and recirculation systems.
• c Plus or minus one standard deviation.
• d Estimated fracture aperture volume = L×W×μ, where μ represents μ_c, μ_m, or μ_l.
• e Peclet Number = v_m × μ_m/D_m.
• f Reynolds Number = ρ × v_c × μ_c/η.

[5] The experimental setup, illustrated in Figure 1, involved sealing the two opposite long edges of each fracture replica to serve as no-flow boundaries and operating the remaining two edges as constant-pressure boundaries. The fractures were oriented horizontally in all of the hydraulic and tracer tests. For the no-flow boundaries along each side of the fracture, a 0.2 cm thick gasket and a 1.1 cm thick piece of plexiglass were cut to the same dimensions as the side surface of the fracture block. The gasket was placed over the fracture trace along the edge of the fracture block, and the plexiglass was placed over the gasket. A steel frame was built (Figures 1b and 1c) in which each fracture replica was mounted. Bolt holes were drilled every 6.0 cm along each side of the frame so that the plexiglass could be tightened to compress the gasket thus sealing the fracture (Figures 1b and 1c). The bolts, as shown in Figures 1b and 1c, were tightened, and silicone sealant was then used to coat the seam between the plexiglass, the gasket, and the edge of the fracture to prevent any leaks.

Figure 1

Schematic diagram of the experimental apparatus for the hydraulic and tracer tests showing (a) the experimental setup, (b) a plan view of the fracture mounted in the steel frame, and (c) a cross section of the experimental fracture x-x′ mounted in the steel frame.

[6] The upstream and downstream ends of each fracture were sealed in a manner similar to the sides, except that a channel was cut out of the center of a 0.5 cm thick gasket, which served as a constant-pressure flow manifold. Additionally, four to six holes were drilled into the plexiglass and tapped to serve as flow ports, as shown in Figure 1a. The plexiglass pieces U1, U2, D1, and D2 (Figures 1b and 1c) were installed for two purposes. First, the plexiglass served as a support when the apparatus was suspended over a light box to facilitate photography. Second, it was found that when the aperture is too large, the pressure difference between the upstream and downstream manifolds is too small to be measured reliably. Therefore pressure was applied to the fracture by tightening the bolts through U2 and D2, and U1 and D1 served to distribute the applied pressure evenly over the fracture. This procedure effectively reduced the fracture aperture.

[7] The recirculation system for the upstream and downstream flow manifolds is illustrated in Figure 1a and consists of Teflon tubing strung through two bolt holes, one located at each the highest and lowest points of the manifold. The Teflon tube was perforated approximately every 5 mm, and each end of the tube was attached to pump tubing (MasterFlex, L/S 16) to form a closed-loop system. Throughout the duration of each experiment, the solution in the flow manifold was continuously recirculated using a peristaltic pump (MasterFlex, L/S 7543-20).

2.2. Hydraulic Tests

[8] Hydraulic tests were performed on each fracture plane to determine the equivalent hydraulic aperture as defined by Tsang [1992]. A peristaltic pump (MasterFlex, L/S 7523-70) was employed to inject water through the fracture plane, and an inclined piezometer was attached to the upstream and downstream manifolds to measure the head loss across the fracture. For F1 a pulsation dampener was used to eliminate pressure pulses in the inlet manifold caused by the pump. For F2 and F3 a multichannel peristaltic pump was used to minimize pulsations. Water exited the system through a constant head port on the downstream end of the fracture. A tipping bucket rain gauge (Davis, Rain Collector II) was installed directly below the constant-head outlet to measure the volumetric flow rate. The rain gauge was connected to a data logger (Lakewood Systems Ltd., UL16 GC), which recorded the number of tips at set time intervals. The rain gauge was calibrated regularly throughout the course of these experiments.

[9] The water employed in these experiments was prepared by degassing distilled water and equilibrating it to room temperature. This procedure prevented any gaseous phase from developing in the fracture plane during the experiments. Each fracture plane was saturated prior to commencing any experiments by first conducting a carbon dioxide flush to remove any air present in the system and then injecting the prepared water.

[10] Once the fracture plane was completely saturated, hydraulic tests were conducted by injecting water through the fracture at a constant flow rate and measuring the resulting head difference across the fracture plane. Hydraulic tests were repeated at various flow rates on each fracture. Each time a fracture plane was resaturated, hydraulic tests were performed to ensure that the equivalent hydraulic aperture remained unchanged.

2.3. Solute Tracer Tests

[11] The solute tracer tests were designed to determine the mass balance and frictional loss apertures under a range of flow rates. Tracer tests were performed using a pulse input of solute. Bromide was selected as the conservative tracer and was prepared as a 0.1 M solution using distilled water. The solution was then degassed and equilibrated to room temperature.

[12] The procedure for conducting the solute tracer experiments involved first saturating the fracture plane with deionized water as described above. Next, the tubing was clamped at C2 and C3 (Figure 1a) to prevent the loss of solute mass to the fluid in the manometer and at C1 (Figure 1a) to ensure that the solute mass entered the recirculation system rather than migrating upstream. A 5 mL disposable plastic syringe (BD 5 mL syringe) was used to inject 5 mL of 0.1 M bromide solution into the tubing immediately upstream of the recirculation system, and recirculation pump 2 was turned on for 20 s to mix the bromide solution evenly throughout the upstream manifold. A dye test was conducted prior to these experiments to determine the 20 s mixing requirement for this phase. Finally, the clamp was removed at C1 (Figure 1a), and feed pump 1 and recirculation pumps 2 and 3 were turned on simultaneously to begin the tracer test. Samples were collected at E2 (Figure 1a) every 1 min until the effluent bromide concentration was below the detection limit of the high-performance liquid chromatograph (HPLC) (Varian, ProStar).

[13] The samples collected from the tracer tests were analyzed using an HPLC (Varian ProStar) equipped with an autosampler (Varian, 410), a solvent delivery module (Varian, 230), and a conductivity detector (Dionex, CD25) at flow rate of 1.5 mL/min. A 4 × 200 mm column (Dionex, AS12A) was employed with a chemical suppressor (Dionex, AMMS III 4mm). The eluent solution was comprised of 0.3 mM NaHCO₃ and 2.7 mM Na₂CO₃. The regenerant for the chemical suppressor was a 12.5 mM H₂SO₄ solution.

2.4. Direct Aperture Field Measurement

[14] One of the fracture replicas, F2, was measured directly using the light transmission technique described by Renshaw et al. [2000]. The technique is based on measuring the transmitted light intensity of an aperture field, filled with Milli-Q water and subsequently dyed water, at each pixel on a digital image. A scientific grade charge-coupled device (CCD) camera (Photometrics, CoolSNAP_ES) was employed to shoot 100 images (50 Milli-Q-filled and 50 dye-filled), which were then averaged. The intensity at each pixel was then related to the aperture at that pixel through the Beer-Lambert Law. The reader is referred to Renshaw et al. [2000] for a discussion of the light transmission technique and Detwiler et al. [2000] for a discussion of the associated error.

3. Results and Discussion

[15] Hydraulic tests were conducted to determine the range of flow rates under which laminar flow dominates in F1, F2, and F3. Three “equivalent apertures,” specifically the cubic law (μ_c), mass balance (μ_m), and frictional loss (μ_l) apertures, were calculated based on the hydraulic and conservative solute tracer tests as follows [Tsang, 1992]:

with [Fahim and Wakao, 1982]

where Q (L³ · T⁻¹) is the volumetric flow rate, L (L) represents the length of the fracture in the direction of flow, W (L) is the fracture width (perpendicular to the direction of flow), ΔH (L) is the hydraulic head loss across the length of the fracture, η (M · L⁻¹ · T⁻¹) is the dynamic viscosity of the fluid, γ (M · L⁻² · T⁻²) is the specific weight of the fluid, t_m (T) is the mean residence time of tracer transport, and C_in(t) and C_out(t) represent the tracer concentration at time t in the upstream and downstream manifolds, respectively.

[16] The corresponding specific discharges, v_c (L · T⁻¹), v_m (L · T⁻¹), and v_l (L · T⁻¹), which are based on the cubic law, mass balance, and frictional loss apertures, respectively, are given by

Table 1 lists the equivalent apertures, specific discharges, estimated fracture volumes, and Peclet and Reynolds numbers calculated from these hydraulic and conservative tracer experiments.

[17] Figure 2 demonstrates a linear relationship between the specific discharge and hydraulic gradient across the fracture within the range of specific discharges employed in these experiments. The Reynolds number, calculated based on the hydraulic aperture and the corresponding specific discharge, is near to or less than one at each specific discharge employed in these experiments (Table 1). These data indicate that the flow is laminar within the range of specific discharges applied in F1, F2, and F3, and therefore the assumption of laminar flow made in the derivation of the cubic law is satisfied.

Figure 2

Hydraulic test results for F1 (diamonds indicate data and line indicate regression line (R² ≈ 1)), F2 (squares indicate data and line indicates regression line (R² ≈ 1)), and F3 (triangles indicate data and line indicates regression line (R² ≈ 1)).

[18] Assuming that the experimental fractures can be represented by parallel plates with an “equivalent aperture,” the following one-dimensional transport equation was used to model the conservative tracer transport [van Genuchten, 1981]:

where x (L) is the distance along the x-axis in the direction of flow, t (T) is the time from the beginning of the solute pulse release, C(x, t) (M · L⁻³) is the solute concentration at distance x and time t, v (L · T⁻¹) is the specific discharge, C₀ (M · L⁻³) is the concentration at x = 0 and t = 0, D (L² · T⁻¹) is the hydrodynamic dispersion coefficient, and λ (T⁻¹) is a decay constant. The solution to (8) is [van Genuchten, 1981]

with

and [Bear, 1972]:

where α_L (L) is the dispersivity and D_m (L² · T⁻¹) is the molecular diffusion coefficient in water.

[19] Owing to the significant volume of the flow manifolds at the upstream and downstream ends of each experimental fracture, the measured concentration was adjusted to represent the actual concentration of the solute entering and exiting the fracture. The upstream manifold was modeled using a mass balance on a completely mixed continuous flow reactor:

where Q (L³ · T⁻¹) represents the flow rate through the system, C_in(t) (M · L⁻³) represents the concentration of the solute entering the manifold at time t, C_out(t) (M · L⁻³) represents the concentration of the solute exiting the flow manifold at time t (i.e., entering the fracture), V_ol (L³) represents the combined volume of the manifold and recirculation system, and M (M) is the mass of the tracer injected. Equation (12b) assumes that the Br⁻ concentration at t = 0 is uniform in the upstream manifold. The solution to equations (12a) and (12b) is

Equation (13) indicates that the solute concentration in the upstream manifold decreased exponentially over the course of each tracer experiment, as shown in Figure 3. Therefore the constants C₀ and λ in equations (8), (9), and (10) are given by:

Also shown in Figure 3 is the measured concentration profile in the upstream manifold of F2 when the flow rate is 7.38 mL/min (v_c = 1.09 mm/s). The measured concentration profile is well approximated by the theoretical equation (13). The slight difference between the measured and theoretical concentration profiles can be explained by the existence of a dead volume or bypassing in the upstream manifold.

Figure 3

The concentration profile in the upstream manifold of F2 when v_c = 1.09 mm/s. The line shows the theoretical exponentially decreasing concentration profile and squares show the measured concentration profile.

[20] The concentration entering the downstream flow manifold from the fracture can also be described by equation (12a). Rearranging (12a) into difference form gives:

where C_eff-frac (M · L⁻³) represents the solute concentration exiting the fracture at time t, C_eff (M · L⁻³) represents the concentration measured in the effluent manifold at time t, and Δt (T) is the time between sampling events. The measured concentration of each sample taken from the downstream manifold was adjusted using equation (15) to reflect the concentration exiting the fracture. Additionally, all effluent samples were corrected for time due to the tubing volume, V_E1-E2, (see Figure 1a) between the downstream flow manifold and the sampling point as follows:

where t_corr (T) represents the corrected time of each sampling event and t_mea (T) represents the time from the start of the experiment at which the sample was actually taken.

[21] The adjusted experimental breakthrough curves were fit to equation (9) using PEST (Watermark Numerical Computing, Version 9.0). The dispersivity (α_L), both alone and in combination with the specific discharge (v), was employed as the fitting parameter(s). It is noteworthy that an “effective volume,” rather than the actual volume, of the downstream recirculation system was employed in equation (15) as there was likely a dead volume present in the downstream flow manifolds of F2 and F3. The actual volumes of each fracture's downstream recirculation system are listed in Table 1. When the effluent concentration profiles from F1 were adjusted using the actual manifold volume of 17 mL in equation (15), the analytical and experimental breakthrough curves matched well. However, for F2 and F3 the simulated breakthrough curves fit the experimental observations best when the values of V_ol in equation (15) were reduced to 17 mL. Since the pump capacity and tubing lengths of the downstream recirculation systems are identical for F1, F2, and F3, it is reasonable that the “effective volume” of the downstream manifolds, which represent the volumes completely mixed by the recirculation system, are also identical.

[22] Two approaches were employed to fit equation (9) to the measured experimental breakthrough curves using PEST. The first approach, hereinafter referred to as the single-parameter approach, employed the specific discharges calculated from the measured experimental breakthrough curves using equations (5), (6), and (7), and dispersivity was the only fitting parameter. This approach generated three distinct breakthrough curves for fitting purposes, one based on each of the three equivalent apertures calculated from the experimental breakthrough curves. The second approach, hereinafter referred to as the two-parameter approach, employed both dispersivity and specific discharge as fitting parameters. Figure 4 shows the experimental breakthrough curve for the solute tracer test through F2 with v_c = 1.09 mm/s, together with the three analytical breakthrough curve fits based on the single-parameter approach and the analytical breakthrough curve fit based on the two-parameter approach (Figure S1). The fracture dispersivities obtained from the first fitting approach, together with the specific discharges and dispersivities obtained from the second fitting approach, are listed in Table 2. The magnitude of the fit dispersivities relative to that of the fracture length is within the range of that reported by others [e.g., Thompson and Brown, 1991; Keller et al., 1995]. Additionally, the fit specific discharge, v_f, was very similar to the specific discharge based on the mass balance aperture, v_m, obtained from the experimental breakthrough curves. Therefore the breakthrough curve fits appear reasonable upon examination of the fitting parameter(s).

Figure 4

Breakthrough curves for the solute tracer test in F2 with v_c = 1.09 mm/s. Diamonds indicate experimental measurement (error bars represent one standard deviation), dashed line indicates simulation based on μ_c from physical experiment and α_c as the fitting parameter, dash-dotted line indicates simulation based on μ_m from physical experiment and α_m as the fitting parameter, dash-double dotted line indicates simulation based on μ_l from the physical experiment and α_l as the fitting parameter, and solid line indicates simulation employing both v_f and α_f as fitting parameters.

Table 2. Dispersivity and Specific Discharge Values Obtained From Fitting the Experimental Breakthrough Curves to the One-Dimensional Advection Dispersion Equation^a

Fracture ID	vc, mm	αc,^a mm (95% Confidence Limit)	αm,^a mm (95% Confidence Limit)	αl,^a mm (95% Confidence Limit)	With 95% Confidence Limit^b
					αf,^b mm/s	vf,^b mm/s
F1	0.55	36 (24, 48)	114 (99, 128)	46 (18, 73)	85 (72, 99)	0.33 (0.31, 0.35)
	1.67	49 (31, 68)	186 (160, 212)	63 (6,121)	149 (117, 182)	0.72 (0.64, 0.80)
F2	0.55	82 (45, 118)	135 (128, 142)	173 (47, 300)	149 (137, 160)	0.20 (0.19, 0.20)
	1.09	79 (44, 115)	171 (162, 180)	152 (45, 259)	156 (144, 167)	0.40 (0.39, 0.42)
	1.67	81 (41, 121)	214 (199, 229)	141 (168, 265)	173 (157, 190)	0.60 (0.57, 0.63)
F3	0.55	65 (41, 90)	123 (114, 133)	122 (53, 191)	124 (112, 137)	0.22 (0.21, 0.23)
	1.09	67 (40, 93)	185 (164, 207)	120 (43, 197)	223 (181, 265)	0.35 (0.31, 0.38)
	1.67	74 (40, 108)	241 (211, 271)	145 (28, 263)	275 (211, 340)	0.42 (0.36, 0.48)

• a Here α_c, α_m, and α_l represent the dispersivities calculated through fitting equation (8) to the experimental data using the single-parameter fitting approach with specific discharges v_c, v_m, and v_l, respectively (calculated from the experimental breakthrough curves).
• b Here α_f and v_f represent the dispersivity and specific discharge from the two-parameter fitting approach.

[23] The analytical curve fits were very sensitive to the specific discharge. Figure 4 clearly shows that when the specific discharge based on the mass balance aperture was employed in equation (9), very good agreement was obtained between the simulated and experimental breakthrough curves. The slight difference between the experimental curve and the simulated single-parameter mass balance and two-parameter fit curves can be attributed to the existence of dead volumes in both the upstream and downstream manifolds. These dead volumes resulted in slight discrepancies between the measured and theoretical input functions (Figure 3) and the adjusted and actual breakthrough curves. Figure 4 also shows that when the specific discharges based on the cubic law and frictional loss apertures were employed in equation (9), the resulting simulated fits to the experimental data were very poor. The fits between the analytical and experimental breakthrough curves shown on Figure 4 are typical of those obtained in all eight solute transport experiments. The observed breakthrough curves together with the analytical curve fits from the remaining seven experiments are provided as accompanying material online. It should be reiterated here that these experiments were conducted under flow conditions in which inertial forces were negligible.

[24] Table 3 shows the aperture field statistics of F2 obtained from the direct aperture field measurement. The mean experimental mass balance aperture (1.61 mm) is within 2% of the arithmetic mean aperture 〈b_a〉, which confirms Tsang's [1992] theoretical conclusion that the mass balance aperture is a good approximation of the arithmetic mean aperture. Additionally, the actual fracture volume obtained from the direct aperture field measurement (Table 3) lies within the 99.9% confidence interval of the mean volume of F2 calculated based on the mass balance aperture (Table 1) (120 mL ± 38 mL). The root mean square errors (RMSE) presented in Table 3 appear very low; however, each measurement presented in Table 3 is based on 50 million data points. The large number of data points significantly reduces the error associated with the standard deviation of 〈b_a〉. It is also noteworthy that the RMSEs presented in Table 3 are within the range of those presented by Detwiler et al. [2000].

Table 3. F2 Aperture Statistics Derived From the Direct Aperture Field Measurement

Fracture ID	F2
Arithmetic mean aperture (〈ba〉), mm	1.57 (RMSE = 0.3%)^a
Standard deviation (σ), mm	1.12
Coefficient of variation(= σ/〈ba〉)	0.71
Geometric mean aperture (〈bg〉), mm	0.90
Harmonic mean aperture (〈bh〉), mm	0.13
Fracture volume, mL	102 (RMSE = 0.3%)^a

• a RMSE = , where x represents 〈b_a〉 or fracture volume.

[25] Figure 5 shows that the specific discharge based on the mass balance aperture, v_m, agrees very well with that obtained through the two-parameter fitting approach, v_f. The fitting results shown in Figures 4 and 5 clearly demonstrate the theoretical conclusion proposed by Gelhar [1993], and the simulation results presented by Moreno et al. [1988], that the mass balance aperture is the only “equivalent aperture” appropriate for describing solute transport in single variable aperture fractures under laminar flow conditions.

Figure 5

Comparison of discharge rates based on the different equivalent apertures, where triangles indicate v_m, crosses indicate v_f, squares indicate v_c, and circles indicate v_l.

[26] The data presented in Table 1 reveal two additional noteworthy observations. First, the relative magnitude of the three equivalent apertures follows the relationship μ_m ≥ μ_c ≥ μ_l, which is consistent with the argument presented by Tsang [1992]. Second, although the hydraulic aperture remains relatively stable under the range of specific discharges tested in these experiments, the mass balance aperture increases and the frictional loss aperture decreases (to a lesser extent) with increasing specific discharge. This is due to the fact that smaller specific discharges provide the tracer with access to fewer flow paths than larger specific discharges. Therefore in equation (2), when Q increases by a factor of n, t_m decreases by a factor of 1/m where m < n. Therefore μ_m increases with increasing specific discharge. Since μ_l is inversely proportional to t_m^1/2, it is less sensitive to specific discharge than μ_m, although it does decrease with increasing specific discharge. These observations are supported by those of Moreno et al. [1990], who also found that the mass balance aperture was sensitive to flow rate in their numerical experiments.

[27] Table 2 shows that the dispersivity was not constant in any of the experimental fractures; it increased with the specific discharge. This is consistent with the experimental data presented by Dronfield and Silliman [1993] and Detwiler et al. [2000], which also demonstrate an increase in dispersivity with the specific discharge if a linear relationship between the hydrodynamic dispersion coefficient and the specific discharge is assumed (equation (11)). However, rather than assuming a linear relationship, Dronfield and Silliman [1993] derived a unique dispersivity for each aperture field in their research by assuming that the hydrodynamic dispersion coefficient is a function of the velocity raised to a power n (D = αvⁿ), and demonstrated that n took on values ranging from 1.3 to 2.0 depending on the fracture roughness. Detwiler et al. [2000] agreed that the hydrodynamic dispersion coefficient is nonlinearly dependent on velocity; however, they presented a different expression to describe this nonlinear dependence:

where τ(-) represents the tortuosity factor for molecular diffusion along the fracture, α_Taylor and α_macro are nondimensional coefficients representing the contributions of Taylor dispersion and macrodispersion, respectively, Pe(-) is the Peclet number, v (L · T⁻¹) is the mean solute velocity (equivalent v_m in this research), and 〈b_a〉 (L) is the arithmetic mean aperture. Currently, there is no agreement in the literature regarding the relationship between dispersivity and velocity in fractures, and therefore this subject merits further investigation.

4. Summary

[28] Laboratory-scale conservative solute tracer experiments were conducted at three different specific discharges through three distinct single fracture replicas. An analytical solution to the one-dimensional advection-dispersion equation was fit to each experimental breakthrough curve three times, each time applying v based on one of the three “equivalent apertures” derived from the tracer experiments, and employing α as the only fitting parameter, and a fourth time employing both v_f and α_f as fitting parameters. Additionally, one of the experimental fractures (F2) was measured directly. The excellent agreement between the experimental breakthrough curves and the simulated curves based on the single-parameter curve fit applying the mass balance aperture clearly demonstrates that the mass balance aperture is the only equivalent aperture appropriate for describing solute transport in single variable-aperture fractures. Additionally, the concurrence between the experimental mass balance aperture and the arithmetic mean aperture from the direct aperture measurement in F2 supports Tsang's [1992] arguments, which lend additional weight to this conclusion. This is the first set of laboratory data to explicitly demonstrate this fact, which is an important verification of the theoretical conclusions presented by Tsang [1992] and Gelhar [1993] and those based on numerical simulations presented by Moreno et al. [1988].

Acknowledgments

[29] Funding for this work was provided by the NSERC Discovery Grant Program.