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Journal of Hydrology 428–429 (2012) 182–190

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Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol

Effects of free convection and friction on heat-pulse flowmeter measurement Tsai-Ping Lee a, Yeeping Chia a,⇑, Jiun-Szu Chen a, Hongey Chen a, Chen-Wuing Liu b a b

Department of Geosciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, ROC Department of Bioenvironmental Systems Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 28 June 2011 Received in revised form 4 January 2012 Accepted 1 February 2012 Available online 9 February 2012 This manuscript was handled by Philippe Baveye, Editor-in-Chief, with the assistance of Renduo Zhang, Associate Editor Keywords: Flowmeter Calibration Hydraulic conductivity Flow velocity Diverter

s u m m a r y Heat-pulse flowmeter can be used to measure low flow velocities in a borehole; however, bias in the results due to measurement error is often encountered. A carefully designed water circulation system was established in the laboratory to evaluate the accuracy and precision of flow velocity measured by heat-pulse flowmeter in various conditions. Test results indicated that the coefficient of variation for repeated measurements, ranging from 0.4% to 5.8%, tends to increase with flow velocity. The measurement error increases from 4.6% to 94.4% as the average flow velocity decreases from 1.37 cm/s to 0.18 cm/s. We found that the error resulted primarily from free convection and frictional loss. Free convection plays an important role in heat transport at low flow velocities. Frictional effect varies with the position of measurement and geometric shape of the inlet and flow-through cell of the flowmeter. Based on the laboratory test data, a calibration equation for the measured flow velocity was derived by the least-squares regression analysis. When the flowmeter is used with a diverter, the range of measured flow velocity can be extended, but the measurement error and the coefficient of variation due to friction increase significantly. At higher velocities under turbulent flow conditions, the measurement error is greater than 100%. Our laboratory experimental results suggested that, to avoid a large error, the heatpulse flowmeter measurement is better conducted in laminar flow and the effect of free convection should be eliminated at any flow velocities. Field measurement of the vertical flow velocity using the heat-pulse flowmeter was tested in a monitoring well. The calibration of measured velocities not only improved the contrast in hydraulic conductivity between permeable and less permeable layers, but also corrected the inconsistency between the pumping rate and the measured flow rate. We identified two highly permeable sections where the horizontal hydraulic conductivity is 3.7–6.4 times of the equivalent hydraulic conductivity obtained from the pumping test. The field test results indicated that, with a proper calibration, the flowmeter measurement is capable of characterizing the vertical distribution of preferential flow or hydraulic conductivity. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The hydraulic conductivity of an aquifer plays an important role in groundwater flow and contaminant transport. An equivalent hydraulic conductivity of the aquifer is commonly obtained by the conventional hydraulic testing. Nevertheless, most aquifers are somewhat heterogeneous in nature due to the spatial variation of deposition, weathering and deformation. Hydraulic conductivity may vary in both vertical and lateral directions (Rubin, 1982). Field techniques for estimating hydraulic conductivity at various depths have been studied by Molz et al. (1990, 1994), Taylor et al. (1990), Kabala (1993), Young et al. (1998), and Paillet (2000). Doublepacker tests can be used to measure the hydraulic conductivity interval by interval in an open-hole (Braester and Thunvik, 1984), but the operation is time-consuming (Borgne et al., 2006). An ⇑ Corresponding author. Tel./fax: +886 (02) 33665873. E-mail address: [email protected] (Y. Chia). 0022-1694/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2012.02.001

impeller flowmeter can be used to measure the borehole flow continuously (Molz et al., 1989; Hanson and Nishikawa, 1996), but the instrument only for rapid flow conditions, and requires frequent maintenance and calibration (Molz and Young, 1993). Electromagnetic flowmeter was developed to improve the resolution of borehole flow measurement in medium to high velocity conditions (Molz et al., 1994; Dinwiddie et al., 1998; Ruud et al., 1999; Arnold and Molz, 2000; Crisman et al., 2001). Heat-pulse flowmeter is a promising tool that can be used to measure the flow velocity at a stationary position in the borehole under low to medium flow velocity conditions. By integrating the velocities measured at various depths by the flowmeter and the equivalent hydraulic conductivity estimated by the hydraulic test, one can then establish the vertical profile of horizontal hydraulic conductivity in a heterogeneous aquifer (Molz et al., 1989; Kabala, 1994). However, the accuracy of calculated hydraulic conductivity could be affected by flowmeter measurement. Negative values of hydraulic conductivity were often reported in field applications


T.-P. Lee et al. / Journal of Hydrology 428–429 (2012) 182–190

(Paillet, 2000), but the mechanism behind the measurement error of the heat-pulse flowmeter remains unclear. Hess (1986) presented a calibration curve for heat-pulse flowmeter measurements, but did not provide detailed data at low velocities. Correction processes under various borehole conditions have been suggested for the heat-pulse and electromagnetic flowmeter measurement based on field practices by Paillet (2004). However, the lack of rigorous laboratory experimental test data and analysis of the measurement error in the previous studies may lead to improper use of the heat-pulse flowmeter or difficulties in data analysis and interpretation in the field. Here we conduct a laboratory test of flow through a carefully designed water circulation system for evaluating the error and precision of measured flow velocity using the heat-pulse flowmeter. The physical mechanism of the measurement error is discussed and an empirical formula for calibrating the measured flow velocity was developed. A field test was then conducted to demonstrate the application of the flowmeter measurement to characterization of the vertical distribution of hydraulic conductivity in an alluvial aquifer. Calibration against the effect of free convection and friction loss was performed to improve the accuracy in hydraulic conductivity of individual sections and to correct the inconsistency between the pumping rate and the measured flow rate. 2. Laboratory measurement A circulation system is established in the laboratory for measuring water flow velocity by heat-pulse flowmeter. The system is designed to minimize the influence of environmental factors and provides a stable condition for analyzing measurement data. 2.1. Heat-pulse flowmeter The heat-pulse flowmeter adopted in this study is manufactured by the Robertson Geologging Ltd. The instrument, approximately 2.24-m long and 5.0-cm in diameter, is designed for measuring vertical flow velocity at a stationary position in a borehole. The flowthrough cell is located near the bottom of the heat-pulse flowmeter. It contains a heating wire-grid and two thermistors located 5 cm above and below the grid (Fig. 1a). As water flows through the grid, a pulse of heat is produced by controlled electric current, heating water in the vicinity of the grid. The heated water is then transported 5 cm in distance toward the thermistor where it is detected.

(a)

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The data logger measures the elapsed time from heat injection to detection of the temperature difference by the thermistor. The plot of differential temperature between the two thermistors versus elapsed-time for the heat-pulse flowmeter measurement is shown in Fig. 1b. Time resolution is 0.06 s. The flow direction can be identified by the upward or downward direction of heat-pulse as implied by the temperature change. Based on the direction of heat pulse and the elapsed time of the warm fluid front detected by thermistors, the vertical flow velocity can be estimated. 2.2. Experimental setting The laboratory testing system is designed to accommodate flowmeter measurement at various inflow rates and to minimize the error due to artificial effects. The system consists of three components: heat-pulse flowmeter, electronic equipment, and circulation components (Fig. 2). The heat-pulse flowmeter measures water flow velocity. The electronic equipment includes data logger, computer, and plotter to process the heat command and to convert flowmeter signals to elapsed time readings. A 94-mm diameter transparent acrylic pipe was chosen to simulate a well, and a 20-cm thick layer of pebbles was placed at the bottom of the test pipe to reduce the turbulent instabilities near the inlet. The flow velocity is anticipated to be interfered by the placement of the instrument. The inlet of flow-through cell blocks about 4% of the cross section area of the pipe. It is possible to slightly reduce the flow velocity at the central position of the pipe (Fig. 1a). In order to minimize the instability of inflow rate caused by the pump, the inflow rate in our experimental setting was driven by the difference in elevation head and was controlled by a throttling pressure valve at the bottom of the pipe. The system allows water overflow to maintain a constant hydraulic gradient, and we collect overflowed water to measure the discharge rate. The flow system contains a pump and two vessels. The upper vessel receives the pumped water and provides hydraulic gradient to drive water flow to the overflow gate, and the lower vessel receives overflowed water from the acrylic pipe and keeps pumping water to the upper vessel to maintain a stable circulation system. The inflow rate ranges from 750 ml/min to 5700 ml/min. Furthermore, we designed a diverter to increase the velocity of the flow passing through the flow-through cell by reducing the flux area. The diverter assembly is a collar consisting of a sponge cylinder sandwiched between two rubber gasket rings that encircled the flowmeter as shown in Fig. 3. The rubber gasket is completely sealed between the flowmeter and the pipe. The lower limit of inflow rate can thus be extended to 210 ml/min. This setting with a diverter is then used to evaluate the performance of velocity measurement under high flow rate conditions and the tests with packers in the field. 2.3. Measurement error and precision

(b)

Fig. 4 illustrates the average flow velocity, measured flow velocity, Coefficient of variation (CV), and measurement error of heatpulse flowmeter measurements at different inflow rates. These tests results were obtained when the flowmeter was positioned at the center of pipe. The average flow velocity (Va) in the pipe is defined as the inflow rate per unit area which can be obtained by

V a ¼ Q =A ¼ 4Q=pD2

Fig. 1. (a) Schematic illustration of inner components of the heat-pulse flowmeter probe; (b) typical heat-pulse flowmeter response curves.

ð1Þ

where Q is the inflow rate, A is the cross-sectional area of the pipe, and D is the pipe diameter. The inflow rate is supposed to be equal to the measured discharge rate. According to the inflow rates in Fig. 4, the calculated average flow velocity ranges from 0.18 to 1.37 cm/s. The traveling time of the warm water front from the


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Fig. 2. Schematic diagram of laboratory test system for heat-pulse flowmeter measurement.

Fig. 3. Schematic diagram of the diverter for heat-pulse flowmeter measurement.

heating grid to the thermistor is recorded by the heat-pulse flowmeter, and the measured flow velocity (Vm) in the pipe can be computed by

V m ¼ 5=t

ð2Þ

where t is the elapsed time in seconds, and 5 cm is the distance between the heating grid and the thermistor. Eight repetitive heatpulse flowmeter measurements were performed for every specified inflow rate. The precision of measurement in terms of standard deviation varies from 0.05 s to 0.51 s. The coefficient of variation, defined as the ratio of the standard deviation to the mean elapsed time, varies from 0.4% to 5.8%. Because the time resolution of the instrument is 0.06 s, the coefficient of variation becomes large if the elapsed time is short or the flow velocity is high. Thus, the precision of repeated measurements is associated with flow velocity, as shown in Fig. 4. Fig. 4 shows a calibration curve obtained by a linear least squares fit to the upward flow velocities measured by the flowmeter. Every plotted flow velocity represents the mean value of eight repetitive test results at a specified inflow rate. The measured flow

velocity, which increases linearly with the average flow velocity, is generally faster than the average flow velocity. The measurement error is defined as the ratio of the difference between the measured velocity and the average flow velocity to the average flow velocity, i.e. (Vm Va)/Va. It is noted that as the average flow velocity decreases from 1.37 cm/s to 0.18 cm/s, the measurement error increases rapidly from 4.6% to 94.4%. We also measured flow velocity when the position of flowmeter was shifted from the center to the wall of the pipe. Apparently the measured flow velocity near the pipe wall is slower than that at the pipe center (Fig. 5). The measurement error increases from 2% to 50% when the average flow velocity decreases from 0.77 cm/s to 0.18 cm/s. Although the error is smaller than that at the pipe center, the slope of the regression line for the measured velocity near the pipe wall (1.15) is apparently larger than that for the measured velocity at the pipe center (1.05). The flow velocities measured by heat-pulse flowmeter with a diverter are shown in Fig. 6. As the inflow rate is constant, the average flow velocity can be accelerated due to the reduced flux area. Thus, the range of average flow velocity near the heating grid can be extended to 0.23–4.6 cm/s. The coefficient of variation varies from 0.46% to 5.5% when the average flow velocity is below 10 cm/s. Nevertheless, it increases to greater than 10% when the velocity exceeds 10 cm/s. As shown in Fig. 6, the measured flow velocity increases near-linearly with the average flow velocity in the laminar flow conditions (below 2.36 cm/s) and the measurement error varies between 50% and 75%. However, in the turbulent flow conditions (the average flow velocity is greater than 2.36 cm/ s), the error increases to beyond 100%. Such a large measurement error and CV may lead to a high fluctuation of flow velocities. 3. Calibration The increasing measurement error with decreasing flow velocity is a major concern for the application of the heat-pulse flowmeter. Prior to the development of an empirical formula for calibrating the measured flow velocity, it is essential to understand the effects of fluid dynamics and heat transfer on the measurement error.



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Fig. 4. Plot of the laboratory measured flow velocity against the average flow velocity, percentage error and the coefficient of variation (CV). Each flow velocity was obtained by the mean of eight flowmeter measurements. The broken line is a reference line where the measured velocity is the same as the average velocity. The solid line is the best-fit regression curve for the plot.

Fig. 5. Plot of the measured flow velocity against the average flow velocity. The solid squares denote the measured flow velocities when flowmeter is shifted to the pipe wall. The open circles denote the measured flow velocities when flowmeter is positioned right in the middle of the pipe.

3.1. Effects of fluid dynamics There are many factors in the water flow system that may influence the accuracy of heat-pulse flowmeter measurement in the laboratory, including turbulence, frictional loss related to the measurement, position in the pipe and the shape around the flowthrough cell. Whether the water flow in the pipe is laminar or turbulent is typically determined by the Reynolds number (Re) which is defined as the ratio of the inertial effects to the viscous effects

Re ¼ qVD=l

ð3Þ

where q is the fluid density, V is the flow velocity, l is the fluid dynamic viscosity, and D is the pipe diameter. Generally water flow

is laminar if the Reynolds number is less than 2000 (Bird et al., 1960). In our experimental system, the estimated value of Reynolds number based on the measured flow velocities ranges from 326 to 1397. Therefore, we can consider the water flow in the test cell as laminar flow. When flow velocity is measured in the system with a diverter, the Reynolds number ranges from 162 to 6967. As shown in Fig. 6, the relation between the measured velocity and the average velocity is near-linear in the laminar flow when the average flow velocity is below 2.36 cm/s or the Reynolds number is smaller than 1792. However, the relation becomes non-linear in the turbulent flow when the average flow velocity is above 2.63 cm/s or the Reynolds number is larger than 2386 (Fig. 6). It is anticipated that the water flow becomes turbulent as the Reynolds number increases from 1792 to 2386.


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Fig. 6. Plot of the laboratory measured flow velocity against the average flow velocity, percentage error and the coefficient of variation (CV). The solid squares denote the measured flow velocities when flowmeter is conducted with diverter under laminar flow. The open squares denote the measured flow velocities when flowmeter is conducted with diverter under turbulent flow.

When the Re of the pipe flow is small, the flow velocity will develop a parabolic profile after a certain distance. Due to the friction of the pipe wall, the flow is slower in the vicinity of the pipe wall and faster in the center. This phenomenon is called frictional loss effect. Frictional loss in the pipe is related to the entrance length (Le) that is defined as the distance from the inlet of water flow to the position where the flow velocity profile is fully developed. Entrance length can be expressed as

Le ¼ El D

ð4Þ

where El is the dimensionless entrance length number and D is the pipe diameter. The entrance length number can be correlated with the Reynolds number for the laminar flow

El ¼ 0:06Re

ð5Þ

The estimated entrance length in our experimental system varies from 184 to 788 cm. In our laboratory test, the flowmeter was placed approximately 20 cm above the pebble layer to reduce the influence of frictional loss on the flow velocity. The position of heat-pulse flowmeter in the pipe may also cause a significant error in the measurement of flow velocity. When the position of flowmeter is shifted from the center to the wall of the pipe, the measured flow velocity will be reduced because of frictional loss near the pipe wall. The winding flow path in the flowthrough cell of the heat-pulse flowmeter is another factor that causes measurement error. Although the geometric shape of flow-through cell varies with the design of flowmeters, it is likely to reduce the measured flow velocity to a certain extent due to the friction of the grid and the cell inlet. 3.2. Effects of heat transfer The mechanism of heat-pulse flowmeter measurement is based on heat transfer, including conduction, convection and radiation. The effect of thermal radiation could be neglected because the differential temperature is fairly small and the heat transfer is in the water. The Peclet number, which characterizes the strength of heat convection to that of heat conduction, can be expressed as

Pe ¼ LV=a ¼ Re Pr

ð6Þ

where L is characteristic length, V is velocity, a is thermal diffusivity, and Pr is the Prandtl number which is approximately 7 for water at 20 °C. Based on the inflow rate in our experimental system, the Peclet number is greater than 1981. The large Peclet number suggests that the dominant driving force for the transport of heat released by the grid is convection, rather than conduction, when water is flowing vertically through the flowmeter. The relative strength of heat transfer between free convection and forced convection can be described by the Archimedes number (Ar) which is the ratio of Grashof number (Gr) and the square of Reynolds number

Ar ¼ Gr=Re2

ð7Þ

The Grashof number, which is a ratio of buoyancy force and viscous force, can be expressed as

Gr ¼

gbðT s T 1 ÞL3

v2

ð8Þ

where g is the gravitational constant, b is the volumetric thermal expansion coefficient, Ts is the source temperature, T1 is the quiescent temperature, L is the characteristic length, and m is the kinematic viscosity. When Ar is far smaller than 1, heat transfer is controlled primarily by forced convection. When Ar is far greater than 1, heat transfer is controlled primarily by free convection. According to the Reynolds number calculated from measured flow velocity, the estimated value of Ar in our tests ranges from 2 104 to 9 103 assuming that the temperature difference between the heating grid and the thermistor is 0.1 °C (Hess, 1986). Thus, forced convection is the dominant heat transfer mechanism in our flowmeter measurements. Free convection, however, may have significant impact on the measurement results at low inflow rates. When the inflow rate or Re approaches 0, Ar becomes infinity according to Eq. (7). In other words, the process of heat transfer in stagnant water is dominated by free convection. 3.3. Empirical formula Hess (1986) proposed that the upward flow measured by the heat-pulse flowmeter under no-flow conditions was caused by



pulse buoyancy. Without rigorous tests in the laboratory, nevertheless, the effects of pulse buoyancy on the measured flow velocity remain poorly understood. In order to characterize the water flow induced by free convection, we conducted flowmeter measurements without any inflow in the laboratory. Results of the duplicate tests without flow from the inlet showed that a heat pulse was detected by the upper thermistor approximately 35 s after the heat was released from the grid. The same result was also obtained by measuring with a diverter under no-flow condition. The signal is anticipated to be generated by buoyancy-induced water flow. The elapsed time indicates that upward flow velocity due to free convection is approximately 0.143 cm/s. The test results suggest that a significant measurement error may result from free convection, particularly at low flow velocity. In addition to errors induced by free convection, in the flow– through cell of the heat-pulse flowmeter, the flow velocity could be changed due to the reduction of cross-sectional area and the friction of the inner wall and the heating grid of the flowmeter. We have conducted measurements of flow velocity in the pipe by the heat-pulse flowmeter for average flow velocities ranging from 0.18 cm/s to 1.37 cm/s. A linear relation is found in the plot of the average flow velocity against the measured flow velocity at 25 °C (Fig. 4). The general form of the linear regression curve can be expressed as

Va ¼ a Vm þ b

ð9Þ

where Va is the average flow velocity, Vm is the measured flow velocity, a is the slope and b is the intercept of the linear curve. Under our laboratory test conditions, a is 1.05 and b is 0.144 cm/s. The estimated slope of the calibration curve in the plot of Fig. 4, or the coefficient a, is 1.05. The slope implies that, without the effect of free convection, the measured velocity would be 5% slower than the average velocity when the flowmeter is centralized in the pipe and the frictional loss is minimized. The small deviation may have been caused by the shape factor associated with the irregular flowthrough cell of the heat-pulse flowmeter. When the measurement position of flowmeter is shifted away from the center of the pipe, the estimated slope of the calibration curve or the coefficient a, increases to 1.15 (Fig. 5). The results suggested a further decrease of measured velocity near the pipe wall due to frictional loss. A near-linear relation was also found when we conducted experiments with diverter within the range of laminar flow (Fig. 6). When the heat-pulse flowmeter measurement is conducted with a diverter, the frictional effect of geometric shape becomes more significant. However, the geometry of the diverter system is like a sudden contraction tube. The flow velocity measurement with a diverter is more complicated and it is difficult to discriminate the error sources. The reduced cross-section not only causes a faster velocity but also changes the flow pattern around the flowmeter inlet. We noticed that the measured flow velocity is approximately 50–70% faster than the average flow velocity under laminar flow conditions. The estimated slope, or the coefficient a, is approximately 0.60 suggesting an increase of measured velocity due to the reduced flux area. The intercept of the linear fit curve on the abscissa in Fig. 4 indicates that the measured velocity is estimated to be 0.144 cm/s when the average velocity is zero. The intercept in the plot of Fig. 5 is 0.138 cm/s. The values of the intercept in the plots of both Figs. 4 and 5 are close to the measured flow velocity, 0.143 cm/s, under no flow conditions. The coincidence suggests that the intercept is likely to be the buoyancy-induced flow velocity caused by free convection. It is noted that the inflow in our experiment is upward, and thus the pulse buoyancy tends to increase the measured flow velocity. In case the inflow is downward in the borehole, the buoyancy-induced flow velocity should be added to the measured flow velocity and the coefficient b in Eq. (9) becomes positive. The

187

coefficient a may vary with the geometry of the inlet and inflow pattern. In addition, the buoyancy effect is temperature dependent because the water density varies with temperature. As water density is the largest at 4 °C, heat-pulse would have negative buoyancy effect if the water temperature is below 4 °C. Hess (1986) estimated that the buoyancy-induced velocity would be less than 0.1 cm/s at 25 °C. Paillet (2004) suggested that measured flow rates lower than 0.05 l/min should be considered zero flow velocity due to buoyancy effect. According to the results of our study, free convection occurs at any inflow rate. As the heat pulse generated from the heating grid will change water density, all of the measured flow velocities are expected to be affected by pulse buoyancy. When the flow velocity is high, the effect of free convection is relatively smaller than that of forced convection on heat transfer. According to Eq. (9), there is only a minor correction for measured velocities. Under low flow velocity conditions, however, free convection becomes a dominant form of heat transfer. Therefore, any measured flow velocity must be properly calibrated to avoid overestimating the average flow velocity. Flowmeter measurements with a diverter reveal a non-linear relation between the measured flow velocity and average flow velocity in our set-up when the flow-rate is higher than 2400 ml/ min (Reynolds number is larger than 2386). In the field application, it is quite usual to encounter a high flow-rate with packers or a diverter. In these cases, the measured flow velocities could be overestimated and the errors would be too large to calibrate. Moreover, in order to minimize the measurement error and enhance the precision, the flowmeter is better conducted without packers or a diverter. 4. Field test The Laboratory experiment was followed by the field measurement at a monitoring well in the farm of the National Taiwan University. The test environment is simple and well controlled in conditions similar to the laboratory. The purpose is to verify the effects of free convection and frictional loss on measured velocities. The measurement and the distribution of hydraulic conductivities are calibrated by the empirical formula. 4.1. Site hydrogeology The test well is 23.6 m deep. Based on the drilling log of the well, as shown in Fig. 7, the aquifer is primarily composed of poorly-sorted gravelly sand from 2.5 m to 10 m in depth and fine sand from 10 m to 16 m. The overlying confining layer, ranging from the ground surface to a depth of 2.5 m, consists of silty clay. Below a depth of 16 m, a thick silt layer constitutes the underlying confining layer. The 97-mm diameter well casing is screened from 5 to 17 m. The water level of the aquifer is about 2.11 m below the ground surface. Prior to the heat-pulse flowmeter measurement, borehole geophysical logging was conducted to provide additional information about the depth and thickness of the aquifer. The natural gamma log indicates a relatively low content of clay in the depth from 2.5 m to 14 m (Fig. 7). Neutron log shows a relatively high count rate or a permeable zone from 2.5 m to 10 m in depth (Fig. 7). Integrated interpretation of well logs suggests that the permeable zone ranges in depth approximately from 2.5 m to 10 m, and possibly extends down to 14 m. 4.2. Test procedure The heat-pulse flowmeter measurement can be used to delineate the vertical distribution of horizontal hydraulic conductivity


188


Fig. 7. The vertical distribution of flow velocity measured by the heat-pulse flowmeter and the estimated hydraulic conductivity along the screen of the monitoring well. The stratigraphic column, well screen position, nature gamma log, and neutron log profiles on the left side show the subsurface information of the test well.

in the aquifer. Prior to the heat-pulse flowmeter measurement, the three-arm caliper logging and casing collar locator were used to provide a continuous record of casing diameter and positions of well screen. The results indicated that the inner diameter of both casing and screen remains 97 mm throughout the well, and thus no calibration is needed for the vertical distribution of flow rate. After pumping at a constant rate for an hour, we conducted heatpulse flowmeter measurements at various stationary positions without pumping. In order to ensure the precision of the measurement for data analysis, a repeated stationary measurement approach was applied at an interval of 25 cm or 50 cm along the well screen. Ambient flow was not detected in the well, implying that the vertical hydraulic gradient in the aquifer is very small and can be ignored. Following the ambient test, a small submersible pump was placed in the well approximately 2 m below the water level. During pumping, the water level was monitored by a pressure transducer and the discharge in excess of a constant rate, 4520 ml/ min, was returned to the well to maintain a constant pumping rate. After the water level reached a quasi-steady state, the heat-pulse flowmeter was lowered and placed at the top of the well screen. We took three measurements of flow rate at a stationary position before the probe was lowered to take another set of three measurements. This process was repeated until the probe was positioned at the bottom of the well screen. It was noted that the measured flow rate at the position above the screen is 4950 ml/min, which is 430 ml/min or 9.5% higher than the constant pumping rate. The measurement error can be minimized by calibrating measured flow velocities with Eq. (9). 4.3. Analysis of measured data Javandel and Witherspoon (1969) developed a theoretical approach to calculate the horizontal hydraulic conductivity along the vertical profile in a heterogeneous aquifer. The conceptual model of water flow between the aquifer and the wellbore under a

Fig. 8. Conceptual model of ground water flow in a heterogeneous aquifer and vertical distribution of flow rate in the wellbore under pumping conditions.

constant pumping rate is illustrated in Fig. 8. The aquifer is assumed to consist of several horizontal permeable and less permeable layers, and the water flow into the well is assumed to be horizontal even when the permeability contrast between the adjacent layers is large. The horizontal flow in each layer is treated as if it was from an aquifer of infinite horizontal extent and thickness. When pumping at a constant rate, the flow rate above point A represents the total discharge rate or the pumping rate, the flow rate at point B represents the total discharge from the open wellbore below point B, and at point C, the discharge rate should be close to zero. Based on these assumptions, the flow rate contributed by each individual layer is proportional to its transmissivity (Ti), or the thickness times the hydraulic conductivity, of that layer and can be expressed as



qi ¼ a T i ¼ a mi K i

ð10Þ

where a is the proportionality constant, Ki is the horizontal hydraulic conductivity of the i-th layer, qi is the discharge rate from the i-th layer, mi is the i-th layer thickness. In practice, continuous pumping is required until the flow rate from each thin layer is proportional to its transmissivity. Such a condition occurs when t > 100 r2w Ss =K, where t is the pumping time, Ss is the specific storage of aquifer, rw is well radius, and K is the average horizontal hydraulic conductivity of aquifer. The relation between the Ki and the equivalent K can be calculated as

K i m qi ¼ mi Q K

ð11Þ

where K is the equivalent hydraulic conductivity, Q is the total discharge rate, and m is the thickness of the aquifer (Molz et al., 1989). The equivalent hydraulic conductivity of the aquifer can be obtained from the hydraulic testing. The thickness of aquifer or thin layers can be estimated from core log or well log. Therefore, the hydraulic conductivity of an individual thin layer can be calculated using Eq. (11) if the discharge rate from each individual layer is obtained by means of heat-pulse flowmeter measurements. Fig. 7 shows the measured flow velocity and the calibrated flow velocity at discrete depths of the test well. The discharge rate from an individual layer can then be calculated by the difference of flow rates between adjacent layers at different depths. The transmissivity of the aquifer estimated from the pumping test is 0.84 m2/min. These data were then analyzed for hydraulic conductivity of discrete layers. The thickness of aquifer, m, is 13.5 m based on the drilling log. Thus, the equivalent hydraulic conductivity of the aquifer, K, is approximately 6.2 102 m/min, or 1.0 103 m/s. The thickness of individual layer is 25 cm or 50 cm. The total discharge rate or the pumping rate is 4520 ml/min. The ratio of the hydraulic conductivity of the individual layer to that of the whole aquifer, Ki/K, is then calculated using Eq. (11). The horizontal hydraulic conductivities of the discrete layers along the well screen, Ki, obtained by multiplying Ki/K value by the equivalent hydraulic conductivity of the aquifer are shown in Fig. 7. Analysis of heat-pulse flowmeter measurement data along the well screen interval indicates that the horizontal hydraulic conductivity fluctuates significantly with depth. A 6.25 m-thick permeable zone, ranging in depth from 6.25 m to 12.50 m, was delineated. The combined discharge rates from the permeable zone are nearly equivalent to the total discharge rate. The estimated aquifer thickness is about 13.5 m based on the drilling log and 7.5–11.5 m based on the well logs. Obviously the direct measurement by the heatpulse flowmeter provides a more precise estimate of permeable zone in comparison with drilling log and well logs. Two highly permeable sections located at depths of 6.25– 7.50 m and 10.00–10.75 m were identified. The estimated horizontal hydraulic conductivity is about 0.23–0.37 m/min between the depth of 6.25 m and 7.50 m, and 0.29–0.40 m/min between the depth of 10.00 m and 10.75 m. The combined discharge rates of these two highly permeable sections approach 65% of the total discharge rate. In other words, most of the groundwater discharge comes from these two sections that have a combined thickness of 2 m. The ratio of the hydraulic conductivity of these two sections to the equivalent hydraulic conductivity of the whole aquifer, (Ki/K), ranges from 3.7 to 6.4. This implies that the contaminant transport rate in an aquifer could be underestimated due to the presence of some highly permeable thin sections. Fig. 7 also shows the difference in the vertical distribution of horizontal hydraulic conductivity in the aquifer before and after the calibration of measured flow velocities with the empirical formula. The estimated Ki for highly permeable sections increases slightly after calibration. The calibrated Ki for less permeable

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sections, however, reduces considerably because the measured flow velocities are significantly enhanced by free convection. The estimated Ki for the less permeable sections below the depth of 12 m after the calibration is approximately 40% of that before the calibration. Therefore, the calibration of measured flow velocities can improve the contrast in hydraulic conductivity between more permeable and less permeable sections in the aquifer. 5. Conclusions Results of our laboratory measurements of flow velocity using heat-pulse flowmeter indicated that significant errors may arise due to free convection and frictional loss. At low flow velocities, a large measurement error could result primarily from pulse buoyancy due to free convection. At higher flow velocities, the error was caused mainly by frictional loss associated with the geometric shape around the inlet of flow-through cell and the position setting of the flowmeter in the pipe. The use of the flowmeter with a diverter may enlarge the error of measured flow velocities, particularly in turbulent flow. The precision of measurement is possibly caused by the time resolution of the flowmeter and the turbulence of flow. A large measurement error and coefficient of variation may result in a fluctuation of measured velocities in adjacent sections, possibly leading to a negative hydraulic conductivity. Therefore, it is preferred to conduct heat-pulse flowmeter measurement in laminar flow. The results of our field test indicated that flowmeter measurement can yield high resolution vertical distribution of hydraulic distribution in an aquifer. The calibration of measured flow velocity with the empirical formula developed in the laboratory could significantly reduce the large error of low-velocity measurement previously encountered. Moreover, it can minimize the inconsistency between the pumping rate and the measured flow rate. By eliminating the effect of free convection, it is possible to identify the boundary of impermeable sections and to enhance the contrast in horizontal hydraulic conductivity between permeable and less permeable sections. Our findings provide a fundamental understanding concerning the physical processes of heat-pulse flowmeter measurement error due to free convection and friction loss. As more complicated factors are often encountered in practice, the coefficients in the empirical formula may change with various field parameters. For instance, the calibration coefficient a may change with borehole size, roughness of borehole wall, geometric shape of the flowthrough cell or the packers used with the flowmeter. The coefficient b may change with different flow direction and temperature of the water. Therefore, further studies are necessary for the calibration of measured flow velocities in different field conditions. Acknowledgements The authors would like to acknowledge the support of the National Science Council of Taiwan (NSC-100-3113-E-002-008). References Arnold, K.B., Molz, F.J., 2000. In-well hydraulics of the electromagnetic borehole flowmeter: further studies. Ground Water Monit. Remediation Winter 2000, 52–55. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport phenomena. John Wiley. Borgne, T.Le., Bour, O., Paillet, F.L., Caudal, J.-P., 2006. Assessment of preferential flow path connectivity and hydraulic properties at single-borehole and crossborehole scales in a fractured aquifer. J. Hydrol. 328, 347–359. Braester, C., Thunvik, R., 1984. Determination of formation permeability by doublepacker tests. J. Hydrol. 72, 375–389. Crisman, S.A., Molz, F.J., Dunn, D.L., Sappington, F.C., 2001. Application procedures for the electromagnetic borehole flowmeter in shallow unconfined aquifers. Ground Water Monit. Remediation 21, 96–100.


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