Environ. Sci. Technol. 2007, 41, 8453–8458
Probe for Measuring Groundwater Velocity at the Centimeter Scale W . L A B A K Y , † J . F . D E V L I N , * ,‡ A N D R.W. GILLHAM§ Schlumberger Water Services, Waterloo, Ontario, Canada, Department of Geology, Lindley Hall, University of Kansas, 1475 Jayhawk Boulevard, Lawrence, Kansas 66049, and Deptartment of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada
Received June 29, 2007. Revised manuscript received September 20, 2007. Accepted September 21, 2007.
A novel method of measuring small-scale groundwater velocities in unconsolidated noncohesive media uses the travel time of a tracer pulse between an injection port and two detectors located on the surface of a cylindrical probe, called a point-velocity probe (PVP), as the basis for velocity estimation. The direction and magnitude of the water velocity vector were determined to within (9% of magnitude and (8° in direction, on average, in ten laboratory tank tests conducted with the PVP, when the velocities were between 5 and 98 cm/ day. Numerical simulations supported the accuracy of the underlying theory for interpretation of the PVP data and indicated that the technology is capable of measuring velocity at a very fine scale (0.5 cm around the circumference). The benchtop and modeling investigations indicated that the probe is moderately sensitive to the condition of the porous medium immediately next to the cylinder surface, suggesting that challenges exist for the deployment of the instrument in the field.
Introduction The measurement of groundwater velocity is fundamentally important for the assessment of risk associated with groundwater pollution. Typically, this quantity is estimated based on a 1-D Darcy’s Law calculation to obtain the specific discharge, q (L/T), corrected for the porosity, n (dimensionless) v)
q K ∆H ) n n ∆x
(1)
where v is the average linear groundwater velocity (L/T), K is the hydraulic conductivity (L/T), H is hydraulic head (L), and x is distance in the direction of flow (L) (all units are given in generalized form where L is length, T is time). The direction is usually obtained assuming flow is perpendicular to contoured groundwater levels. Since the introduction of permeable reactive barriers (PRBs), there has been growing interest and recognition that both chemical and biological processes can exert important effects on groundwater flow (1–8). Conversely, flow velocity is known to affect chemical processes in a PRB because it * Corresponding author phone: 785-864-4994; fax: 785-864-5276; e-mail:
[email protected]. † Schlumberger Water Services. ‡ University of Kansas. § University of Waterloo. 10.1021/es0716047 CCC: $37.00
Published on Web 11/15/2007
2007 American Chemical Society
directly relates to the residence time of fluids in the PRB (9). Furthermore, the flow in aquifers and PRBs is threedimensional and not constrained to single paths, so simple one-dimensional column tests will not reproduce the outcomes of the chemical, biological, and hydrogeological interrelationships accurately. Thus, the study of chemical or microbiological processes without the context of flow is limiting, and field methods to investigate these dependencies, in three-dimensions, are needed. With the considerations above in mind, it is noteworthy that the conventional Darcy’s Law approach for determination of groundwater velocity (eq 1) is not well suited to an accurate evaluation of flow conditions on the scale of a few meters or less, such as those in a PRB. There are two major sources of uncertainty, stemming from scale and heterogeneity, that come from the Darcy-based method of velocity estimation: (a) accuracy of estimated K, which has been researched extensively for engineered and natural porous media (10–13), and (b) accuracy of hydraulic gradient, ∆H/ ∆x. With regard to the latter, Darcy-based calculations have been shown to be subject to large errors when the maximum well spacings are small or when the hydraulic gradient is small, such as at small sites or sites with highly permeable aquifers (hence low hydraulic gradients) (14). Detailed measurements of K, employing methods such as highresolution slug testing (15, 16) or borehole flowmeters (17), do not solve the problem completely because uncertainties in the gradient remain large. An alternative to the use of eq 1 for estimation of groundwater velocities is to measure the velocities directly. This is perhaps most simply done by injecting a tracer into the subsurface and tracking its progress through the aquifer (18). These tests are time and labor intensive, and provide velocity estimates that are averages, usually from injection point to monitoring point, rather than the actual velocities at the monitoring points. Other methods for direct velocity measurement are based on more highly localized tracer movement. Tracers include heat, colloids, radioactive tracers, and saline or dye tracers. In most cases, such as in the point dilution method (19), the colloidal borescope (20), the Geoflowmeter (21), and the laser Doppler velocimeter (22), the instrument is installed in a well. The advantages of this are that many locations can be tested repeatedly with a small number of instruments and without additional drilling, assuming that there are available wells. The disadvantages include the need for empirical calibration steps to account for flow distortion near the well screens. These can contribute uncertainty to the measurements. Over- or underdevelopment of well screens may also introduce bias to the measurements. Direct velocity measurements can also be made without a well. The SPFS technique uses a dedicated probe, installed in direct contact with the porous medium, that measures the velocity using the temperature distribution around a heated cylinder (23). The technique can yield a velocity vector in three-dimensions without the use of any chemical tracers. However, the probes are about 0.75 m long and not suitable for point-scale measurements. The cost of the probes is also prohibitive for multiple installations in many cases. The purpose of this work was to develop and test a simple, inexpensive probe suitable for centimeter-scale in situ measurements of groundwater velocity in noncohesive porous media, such as sand, without the need for a calibration step. The introduction of such an instrument makes it possible to investigate dependencies of flow on chemical and microbiological processes, and vice versa, at a scale VOL. 41, NO. 24, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Replacement of ω by 2v∞ sin θ r and ωapp by vapp r and evaluation of the integrals in eq 4 leads to (27) v∞ )
νapp × γ 2(cos R – cos (R + γ))
(5)
Similarly, eq 3 can be evaluated to give (26) 0.5 × ln v∞ ) vapp × γ
[ ] ( R +2 γ ) R tan ( ) 2
tan
(6)
Equations 5 and 6 can also be used to estimate the flow direction. Since the velocity next to the cylindrical surface is different at neighboring points along an arc, a probe constructed with two detectors will yield different apparent velocities at each detector. Moreover, the difference between the apparent velocities is a function of the R angle (Figure 1b). This fact can be exploited to estimate R. Since v∞ is the same for both detectors, eq 5 (or 6) expressed with variables from detector 1, vapp1, γ, can be equated to eq 5 (or 6) with variable values from detector 2, vapp2,, γ2. In the case of equation 5, this leads to (27) FIGURE 1. Plan view schematics of the point-velocity probe. (a) The velocity at the probe surface is a function of the angle, θ, to the flow direction. (b) A PVP device consists of an injection port, i, and two detectors, d1 and d2, for the measurement of the velocity magnitude and direction. comparable to that of multilevel sampling, that is, the centimeter scale. This scale has been shown to be relevant for the investigation of chemical and biological processes in unconsolidated sand (24).
Theory Flow around a smooth, solid cylindrical surface in the absence of a porous medium occurs as depicted in Figure 1a with the velocity, vθ, everywhere on the cylinder surface obtainable from (25) vθ2 ) 4v∞2
2
sin (θ)
(2)
where v∞ is the average linear groundwater velocity unaffected by the presence of the cylinder and θ is the angle defined in Figure 1a. If a tracer is released at location i and is detected at location d1 (see Figure 1b), then the apparent velocity of the tracer over the path traveled can be calculated from (26) r
vapp ) r
∫
∫
R+γ
R+γ
(3)
dξ
√
R
4v∞2
2
sin ξ
where ξ is an integration variable, r is the radius of the cylinder (L), and vapp refers to an apparent velocity (L/T). If the velocity is expressed as an angular velocity, ω (1/T), defined as ω ) v/r, then eq 3 can be rewritten as an average, apparent angular velocity
∫ ) ∫
R+γ
ωapp
ω dξ
R
R
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(4)
R+γ
(7)
which can be rearranged for the quantity R
[
R ) tan–1
νapp1γ(cos γ2 - 1) + νapp2γ2(1 - cos γ) νapp1γ sin γ2 - νapp2γ2 sin γ
]
(8)
Equation 6 cannot be rearranged to solve directly for R, but it can be written in a form suitable for use with a nonlinear optimizer (28)
[ ] [
R+γ tan vapp1 2 ln γ R tan 2
(
()
)
(
R + γ2 tan vapp2 2 ) ln γ2 R tan 2
()
)
]
(9)
Equations 6, 9 and 7, 8 produce similar estimates of v∞ and R except in the vicinity of stagnation points that exist at θ ) 0° and θ ) 180° (see Figure 1a). Care should be taken not to position the probe in the flow system at these angles. Instruments equipped with multiple injection ports and detectors located around the cylinder surface could overcome this limitation.
Materials and Methods
dξ
R
νapp2 × γ2 νapp1 × γ ) (cos R - cos(R + γ)) (cos R - cos(R + γ2))
dξ
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 41, NO. 24, 2007
To assess the viability of a groundwater velocity probe based on the theory described above, numerical modeling was conducted to generate velocities on a cylindrical surface that could be analyzed with eqs 5-9. In addition, laboratory testing was undertaken to verify the probe performance under physical conditions resembling an aquifer. A prototype PVP was constructed from two 7.5 cm long, 3 cm outside diameter (o.d.) half-cylinders of stainless steel, held together with machine screws (Figure 2). Half-cylinders were used for ease of construction. One half-cylinder was hollow and contained the injection line and detection wires. The two parts were threaded on both ends to accept drill rods at the top end and a drive point at the lower end.
FIGURE 3. Schematic of the tank experiments for assessing PVP performance.
FIGURE 2. Schematic of the PVP instrument. A saline tracer was released through a stainless steel screen (0.055 cm mesh) welded onto a 0.6 cm outer diameter stainless steel nut that tightened from the outside of the cylinder against a single small orifice connected to the injection line on the inside of the cylinder. The effective diameter of the screen was 0.3 cm. The injection line consisted of an L-shaped stainless steel tube, 0.3 cm o.d. connected to a section of polyethylene tubing of the same diameter with a Swagelok connector. The top end of the injection line was connected to a 60 mL plastic syringe filled with salt tracer solution. A graduated roller clamp, or a 1 mL plastic syringe, was attached to the injection line to deliver a small volume of tracer (0.01–0.25 mL) at the onset of each experiment. The tracer injection volumes were small enough that they exerted a negligible effect on the groundwater flow (see modeling results in Supporting Information). The detector system was designed to sense conductance changes in the water. It consisted of 0.075 cm gauge insulated copper wires positioned in individual grooves that were 0.3 cm apart on the surface of the probe. The copper wires in the detector were stripped of insulation where the wires came into contact with the porous medium. The detector wires were connected to a conductivity meter that output the data as µΩ, which were recorded with a data logger. The γ angles between the injection port and each of two the detectors (Figure 1b) were fixed at 40 and 70°, respectively. Experimental Procedure. Laboratory tests of the probe were performed in a tank 22 cm wide, 25 cm long, and 30 cm deep with open-water compartments at each end (Figure 3). The dimensions of this tank were considered large enough that boundary effects on the PVP measurements would be negligible. The central compartment was filled with homogenized, medium-sized sand from the Canadian Forces Base (CFB), Borden, Ontario, packed to a porosity of 0.36 ( 0.01, based on the mass of sediment used, an assumed particle density, Fs, of 2.65 g/cm3, and the volume of the tank that was packed. Flow was induced by pumping water from one open water compartment to the other. The probe was placed in the center of the tank. Two methods of placement were used. In the first method, the probe was lowered into the tank after it was filled with water, and then the sand was filled in around it. This method ensured near-uniform packing of the porous medium throughout the tank and next to the probe surface. In the second method, the probe was driven into the prepacked, saturated sand
using a hammer. In this case the porous medium was expected to be altered in its packing next to the probe. A typical tracer injection consisted of a 0.01 mL pulse of a 600 mg/L solution of NaCl. Conductance of the water was recorded with a data logger at intervals between 0.5 and 5 min, depending on the velocity of flow in the tank. The average flow velocity in the tank was determined by dividing the pump discharge (Q, L3/T) by the saturated cross-sectional area (A, L2) and porosity (n) according to the equation v)
Q An
(10)
The apparent velocity values, vapp, were determined by fitting tracer breakthrough curves from each detector to a solution of the advection–dispersion equation (see refs 26, 27, and Supporting Information). Experiments were conducted using velocities ranging between approximately 5 and 100 cm/day. In another set of experiments, the flow velocity was maintained at about 35 cm/day, and R was varied from 5 to 110 °. Replicate measurements (four on average) of each test were made to determine the standard deviation of the results. The uncertainty in the experimentally imposed velocity, or expected velocity, was determined by calculation of the relative standard deviation of eq 10. The pump discharge, Q, can be estimated from M/tF, where M is the mass of water pumped over time t, and F (M/L3) is the density of water, which was assumed to be 1.00 g/mL. Consequently, the relative standard deviation of the expected velocity was expressed as sv ) v
sh2 h2
+
sw2 w2
+
st2 t2
+
sM2
sn2 + M2 n2
(11)
where Sv, Sh ) Sw ) (0.05 cm, St ) (0.05 s, SM ) (5 × 10-5 g, and Sn ) (0.01 are the standard deviations of the various parameters, including thickness (h ) 28 cm) and width (w ) 22 cm) of the flow section. The uncertainty in the density of water, F, was assumed to be negligible. The values of v, t, and M were determined before and after every experiment. The uncertainty in measured parameters was taken to be 0.5 times the smallest division of the measuring instrument. The uncertainty on porosity was calculated by propagating the uncertainty in the terms in the following equation Ms (hwl)t Fs Vt - Vs ) n) Vt (hwl)t VOL. 41, NO. 24, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
(12)
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FIGURE 5. Agreement between velocities input to the SALTFLOW 3D flow and transport model and the velocities estimated from eq 5 (see text). The slight negative bias is thought to be related to boundary effects in the numerical model.
FIGURE 4. Breakthrough curves at detectors 1 and 2 for simulated tracer (symbols) at various velocities. Lines represent the fitted solutions to the advection dispersion equation using the FORTRAN program PULSEPE (see Supporting Information). Note the low conductance measured in the v ) 1 cm/day experiment is likely caused, in part, by diffusional loss of tracer away from the probe surface during the long data collection time. where Vt is the total volume of the tank (L3), and Vs is the volume of sand used (L3), assuming ( 0.05 cm in each length dimension, h, w, and l, and ( 57 g in the total sand mass, Ms, and negligible uncertainty in Fs (2.65 g/cm3). On the basis of eq 11, the standard deviation values for expected bulk velocities ranged between (0.2 and 2.7 cm/ day for velocities of magnitude of 5.3–97.3 cm/day, respectively. The average experimental relative standard deviation on expected velocity was somewhat higher at (9%, possibly because of installation issues and nonidealities at the boundaries of the tank. Numerical Simulations. Equation 2 was developed for flow in the absence of a porous medium (25). However, the continuity equation for flow in a porous medium is of the same form as that used for open-channel flow (29), and on this basis, it would seem that eq 2 should be applicable for flow in porous media. To test this assumption, numerical modeling was undertaken in which a simulated tracer was released on a cylinder, and the detector responses were converted to velocities using eqs 5 and 7. SALTFLOW 3.0, a 3D finite element code capable of simulating densitydependent flow and transport, assuming a saturated, nondeforming, isothermal medium with incompressible fluids, was used for the simulations (30). A radial grid with 400 000 elements was used to represent the porous medium in the vicinity of half-a PVP instrument surface. Dispersivity values were selected to be 0.9 mm in the direction of flow and 0.1 mm in the other two principle directions, based on tracer behavior in laboratory tests.
Results and Discussion The evaluation of the velocity probe concept was accomplished by testing the theory with a numerical model, SALTFLOW, and with a series of laboratory tank experiments. The numerical simulations gave a flow regime with velocities immediately next to the probe surface that were in close agreement with those of eq 2 for a variety simulated porous media permeabilities (Figure 4) (see Supporting Information). Moreover, the application of eqs 5 and 6 to the interpretation of tracer breakthrough curves from the simulations (Figures 4 and 5) gave estimates of v∞ that were within 3% to 7% of the correct values. Slight negative biases were thought to be 8456
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TABLE 1. Summary of Simulation Input Angles and Angles Estimated from Equation 7 (see text) model input (deg)
PVP analysis (deg)
20 30 45 60 90
17 28 45 61 76
related to the numerical boundary conditions. Flow directions were similarly well estimated (Table 1). The effect of porous media disturbance next to the cylindrical surface because of installation procedures was assessed by numerically simulating such zones with either enhanced or diminished K values. The K-diminished zones, or skins, were fixed at 50, 25, and 15% of the background K (1 × 10-3 cm/s). Additional simulations of 50% reduced K were conducted with skin thicknesses of 0.6, 1.5, and 3 cm to assess the effects of skin thickness. It was found that estimated velocities declined in direct proportion to the decline in K of the skin. Estimated velocities were not very sensitive to skin thicknesses. Similar findings, pertaining to velocity overestimations, were obtained from the simulations involving K-enhanced skins (see Supporting Information).
FIGURE 6. Agreement between expected velocities in the tank experiment and those estimated using eq 5 (see text). Open circles represent data from experiments in which the PVP was positioned before the sand was packed. The broken line represents the trendline through the circles. Note the trendline has a slope very near 1, indicating good accuracy. Error bars on the circles represent one standard deviation estimated from replicate experiments. Open triangles represent data from additional experiments in which the PVP was installed by hammering. The hydraulic conductivity of the sand in the tank was apparently reduced next to the probe as a result of compaction during the hammering. The observed bias in the data is clear and corresponds to measured velocities averaging about one-third the expected value.
TABLE 2. Summary of PVP Assessment in Laboratory Tank Experimentsa magnitude
direction
error
r (deg)
no. of replicate pairs
expected (cm/day)
measured (cm/day)
expected (cm/day)
measured (cm/day)
magnitude (%)
direction (deg)
5 20 45 45 45 50 60 70 85 110
3 3 4 3 3 5 5 7 5 3
34.7 ( 1.0 32.1 ( 0.9 5.3 ( 0.2 14.2 ( 0.4 93.5 ( 2.6 35.2 ( 1.0 97.3 ( 2.7 33.6 ( 0.9 32.8 ( 0.9 31.7 ( 0.4
33 ( 2.3 38.4 ( 9.4 5.1 ( 1.3 13.6 ( 0.8 93.6 ( 3.4 35.2 ( 8.7 95.5 ( 2.6 37.9 ( 4.7 34.9 ( 2.6 43.4 ( 7.9
5(5 20 ( 5 45 ( 5 45 ( 5 45 ( 5 50 ( 5 60 ( 5 70 ( 5 85 ( 5 110 ( 5
4(3 23 ( 5 46 ( 18 57 ( 3 68 ( 7 58 ( 10 51 ( 16 68 ( 13 86 ( 2 129 ( 8
-4.7 19.6 -1.6 -4.1 0.1 -0.2 -1.8 13 6.5 37.2
-1 3 1 12 23 8 -9 -2 1 19
a Control refers to the values expected based on the tank set up and pumping rate. Measured refers to values estimated from eqs 5 and 8 (see text).
In summary, the modeling study indicated that equations 5–9 are suitable for application to the estimation of groundwater velocities in porous media over a wide range of velocities typical of those found in natural geologic deposits (1–320 cm/day), provided that a minimal disturbance of the porous medium can be assured during installation of the device. In the laboratory tests where the PVP was installed before packing, to minimize skin effects, and in which velocities ranged between 5 and 98 cm/day, the PVP instrument was able to provide estimates within 15% of the expected value in 8 out of 10 tests. In all tests, the agreement between the measured and expected values was within 50% (Figure 6 and Table 2), and the average error was only about (9%. In addition, the estimated flow directions were within 15° in eight out of ten tests and did not exceed the expected angle by more than 23 ° in any tests. The average angular error was about (8°. A second tank experiment, in which the probe was hammered into position after the sand was packed, was performed to evaluate the effect of a skin (Figure 6). In that case the velocity estimates were noticeably different from the expected values, with negative biases amounting to as much as 80% ((expected – measured) × 100/expected). This degree of error might be considered large because it means that a measured velocity could be smaller than the actual velocity by a factor of 5. Nevertheless, given that the uncertainties associated the estimations of K and velocities derived from Darcy’s Law at field sites are recognized to be as much as an order of magnitude (13), the biases noted here (resulting from a skin effect) are not so severe that the measurements should be considered meritless. In summary, the laboratory testing of the PVP device showed that accurate measurements of velocity magnitude and direction are possible with the instrument. To use the technology to best advantage in field applications, care must be taken to minimize skin effects during installation. On the basis of the consistency of results from the modeling and laboratory testing and the good agreement between the algorithms developed here and the aforementioned tests, the prognosis for the use of PVPs in aquifers is considered very good.
Acknowledgments The NSERC/Motorola/ETI Industrial Research Chair in Groundwater Remediation, NSERC, CRESTech, The National Council for Scientific Research of Lebanon (NCSR), OGSST, and NSF under Grant No. 0134545 are acknowledged for funding this work. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the
National Science Foundation. John Molson provided assistance with the numerical modeling.
Supporting Information Available Additional details of the numerical modeling, results of the tank experiments to investigate skin effects, and details of the curve fitting procedure. This material is available free of charge via the Internet at http://pubs.acs.org.
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ES0716047
