Slug Tests in the Presence of Background Head Trends

Transcription

1 Methods Note/ Slug Tests in the Presence of Background Head Trends by David W. Ostendorf 1 and Don J. DeGroot 2 Abstract We extend Bouwer and Rice (1976) slug test theory to incorporate background head trends that may be important in incompressible material of low permeability k. The extension, which features a convolution integral of the background head, is closed form for linear trends. A sensitivity study suggests that a rising background head can diminish the head changes associated with a slug-out test and underestimate k if it is ignored, as does falling background trend with a slug-in test. A falling background head can reinforce slug-in test head change and, if ignored, can overestimate k, as does a rising background head with a slug-out test. The simple extension is verified by field tests in glacial till and stratified drift deposits in eastern Massachusetts. Introduction This Method Note deals with slug tests in low permeability, incompressible material, which typically last days or longer. Bouwer and Rice (1976) developed a simple model describing the over-damped response of the head in a partially penetrating monitoring well to an abrupt change of water level in the well, subject to incompressible aquifer flow with a constant background head. This change can be established by manual insertion ( slug-in ) or withdrawal ( slug-out ) of a slug of known volume (Butler 1998), or by pneumatic release of headspace pressure in a well constructed in more permeable material (Prosser 1981). The models of Cooper et al. (1967) and Hyder et al. (1994) explicitly address the effects of compressibility and incorporate a more rigorous representation of vertical flow, respectively, but also retain the assumption of a steady background head. The 1 Corresponding author: Civil and Environmental Engineering Department, University of Massachusetts, Amherst, MA 01003; (413) ; fax: (413) ; ostendorf@ecs.umass.edu 2 Civil and Environmental Engineering Department, University of Massachusetts, Amherst, MA Received January 2009, accepted March Copyright 2010 The Author(s) Journal compilation 2010National Ground WaterAssociation. doi: /j x lengthy durations of slug tests in low permeability material can make this latter assumption questionable. We will show here how background head trends may be simply incorporated into the Bouwer and Rice (1976) model. Theory Low permeability material yields a slow recovery of head in the monitoring well H W in response to a slug test, so that the usual assumption of a steady background head H far from the well must be reconsidered. Flow between the well and the equilibrium radius r E is quasi steady and radial with the neglect of material storage, however, in accordance with Bouwer and Rice (1976) hydraulics for an over-damped slug test r dh dr = Qν 2πkgL (1) H = H (t) (r = r E ) (2) H = p(gage) + z(above pretest monitoring ρg well free surface) (3) with disturbed head H in the material, radial distance r from the center of the monitoring well, volumetric discharge Q leaving the well, screen length L, groundwater NGWA.org Vol. 48, No. 4 GROUND WATER July-August 2010 (pages ) 609

2 kinematic viscosity ν, groundwater density ρ, material permeability k, gravitational acceleration g, and time t from imposition of the slug. The pressure p is gauge and the elevation z refers to the pretest-free-surface elevation in the monitoring well. Thus H is assumed zero within and beyond the monitoring well prior to imposition of the slug. Equations 1 to 3 are integrated from the sand pack radius r P, where the disturbed head is at the monitoring well value, to the equilibrium radius, where the disturbed head is at the background value, with the result H W = H + Qν 2πkgL ln ( re Bouwer and Rice (1976) conserve water mass in the monitoring well casing, of radius r C, and relate Q to H W on the assumption of hydrostatic pressure in the well r P ) (4) πrc 2 dh W = Q dt (5) H W = H WO (t = 0) (6) H = 0 (t = 0) (7) with initial head H WO established in the well casing by the slug test. This latter parameter is signed a slug-in test raises the water level and generates a positive H WO, followed by a falling water column in the well (fallinghead slug test). A slug-out test lowers the free surface, and is signified by a negative H WO and a subsequent influx of groundwater into the well (rising-head slug test). Equations 4 to 7 govern the slug test, which is cast as a nonhomogeneous, first-order differential equation in H W 1 dh W λ dt + H W = H (t) (8) 2kgL λ = ( ) (9) νrc 2 ln re rp with slug test decay constant λ. This latter varies inversely with the characteristic time (T O ) of the Bouwer and Rice (1976) model. The permeable deposits analyzed by Bouwer and Rice (1976) recover quickly, so that the background head remains constant, and an exponential decay is implied by Equations 7 and 8. Low permeability material, however, responds slower and experiences a background head trend for the duration of the extended slug test, so that the Laplace transform of Equations 6 and 8 becomes H W = H WO s + λ + λh s + λ H W = H WO exp( λt) + λ t 0 (10) H (τ) exp[ λ(t τ)]dτ (11) with transformed monitoring well head H W and transformed background head H. A linearly varying background head rising (or falling) at rate κ generates an analytical model H = κt (12) H W = exp( λt) + κ [λt + exp( λt) 1] (13) H WO λh WO The background rate, like H WO, is also signed. A rising background head trend, associated with recharge, has a positive κ, while a falling head, induced by evapotranspiration or consumptive use, has a negative κ. Sensitivity Figure 1a is a semilogarithmic plot that displays the extended slug test solution as an interaction of slug test and background head trends (Equation 13). The exponential decay predicted by Bouwer and Rice (1976) appears as a straight line (κ = 0) in the figure. Figure 1a suggests that, if present, background head trends influence, mask, then dominate the slug-induced disturbance of the free surface in the monitoring well as time passes through progressively stronger nonlinear behavior of κ/(λh WO ) contours. If most of the period of record lies in this latter domain, slug tests may not be accurate or even appropriate estimators of permeability. This happens when κ/(λh WO ) exceeds unity, and the contours of Figure 1a depart from linearity at relatively low values of λt. The positive κ/(λh WO ) contours in Figure 1a describe this progression for recharge and a slug-in test, or evapotranspiration and a slug-out test. In both cases, the background head trend cancels the slug-induced transience, and eventually changes the direction of the water velocity in the monitoring well. Recharge, in time, raises the water level that initially fell during a slug-in test; while evaporation, in time, lowers the water level that rose to begin with in response to a slug-out test. A conventional Bouwer and Rice (1976) calibration of these cancelled tendencies performed without our extension would underestimate permeability by mistakenly attributing the delayed recovery of head in the monitoring well to flow resistance in the material. The negative κ/(λh WO ) contours in Figure 1a demonstrate reinforced behavior of slug and a background head trend. Recharge and a slug-out test both induce flow into the well, and a conventional Bouwer and Rice (1976) calibration without our extension would erroneously assign a higher permeability to the accelerated recovery of the free surface. In similar fashion, evapotranspiration and a slug-in test both induce flow out of the well. We note that compressibility imparts a convex upward behavior to the disturbed head, when plotted semilogarithmically (Chirlin 1989), while vertical flow can straighten curvature so that an apparently incompressible material may simply deliver axisymmetric flow to the monitoring well (Hyder et al. 1994). Thus semilogarithmic behavior at low λt values can be interpreted variously. Neither compressibility nor vertical flow persists when the slug test dies out however, so the presence and sign of a 610 D.W. Ostendorf, D.J. DeGroot GROUND WATER 48, no. 4: NGWA.org

3 Figure 1. (a) Semilogarithmic plots displaying the interaction of slug test and background head trends: cancelled background trend [positive κ/(λh WO )] delays water level response, while reinforced background trend [negative κ/(λh WO )] accelerates it; (b) slug test conditions for 5% error between extended and classical Bouwer and Rice (1976) theory: background head trends are unimportant for initial (t < t 0.05 ) period of record and for relatively large test amplitudes. background head trend is unambiguously defined by the asymptotic behavior of head on a semilogarithmic plot. All the contours in Figure 1a spread out with time. Field Verification The extended theory is calibrated with slug tests conducted in stratified drift and glacial till deposits at two sites in glacial terrane of eastern Massachusetts. A slug-in test of 0.67 m amplitude (H WO ) was conducted in a m radius (r C ) well (BC) in Spring 2008 in a 10-m-thick stratified drift deposit in the Neponset River floodplain, using the pneumatic testing apparatus described by Ostendorf et al. (2005). The material serves as a protective aquitard for an underlying aquifer and municipal supply well in the floodplain, and a network of monitoring wells characterizes the hydraulics of the drift. Ostendorf and Kilbridge (2009) analyze steady head in the leaky aquifer with the linear recharge model of Jacob (1946) and estimate a site averaged value for the aquitard permeability k = m 2 (steady pumping data, drift) (14) The slug test extension offers an independent, local check of the site averaged, time averaged, Equation 14 in the drift aquitard, based on a 1.52 m long well screen (r E value of 0.68 m) set in the interior of the low permeability material. Figure 2a displays water levels in a monitoring well adjacent to BC during the Spring 2008 sampling period aquifer pumping lowered the head during the test period, and a regression leads to a κ estimate of m/s. Adjacent monitoring wells with similar screen elevations are used to estimate the background head trends, based on monthly manual determinations of water levels with an electronic water level detector (Slope Indicator Co., Mukilteao, Washington). The use of adjacent monitoring wells with comparable screen elevations to independently establish κ estimates is particularly important in compressible material, which attenuates water table transience with significant time lags over the deposit thickness (Chapuis 2009). These vertical distances are much larger than r E values, and differentiate global from local conditions at the field site. A nested Fibonacci search through λ minimizes the root mean square error δ between the observed head and that predicted by Equation 13 for the drift test. The calibrated λ of s 1 implies a permeability value at BC when an adopted ν value of m 2 /s is substituted into Equation 9 k = m 2 (slug test, drift well BC) (15) The 2.3-cm error suggests an excellent calibration accuracy, as displayed in Figure 2b. The BC test exhibited reinforced transience with a κ/(λh WO ) value of 1.3: seasonal head fluctuations influence the results, although the test durations did not document the entire recovery period, with λt values less than 0.6 in size. The local Equation 15 agrees with the independently obtained Equation 14, and is consistent with size distributions in split spoon samples from the screened interval, which have 0% gravel, 2% sand, 87% silt, and 11% clay sized grain fractions. A week-long slug test was conducted in 1999 in the unweathered gray till of a drumlin at a second site in eastern Massachusetts (Ostendorf et al. 2004). The drumlin features 5 to 10 m of weathered brown till overlying 20 to 25 m of unweathered gray till, which in turn rests on bedrock. The gray till has a nonuniform grain size distribution, characterized by 17% gravel, 39% sand, 33% silt, and 11% clay, and a porosity of A network of monitoring wells of cm radius (r C ) and 1.52 m screen length (L) was constructed in the late 1990s to characterize the unconfined aquifer hydraulics of the drumlin at steady, seasonal, and diurnal scales. Steady head data calibrated the gray till permeability in a site wide axisymmetric model of unconfined aquifer NGWA.org D.W. Ostendorf, D.J. DeGroot GROUND WATER 48, no. 4:

4 Figure 2. Slug test calibrations with and without background head trends. (a) Water levels in adjacent monitoring wells used to estimate κ (left and bottom axes, circles and solid line for drift well BC; right and top axes, triangles and dashed line for till well MA). Observed (symbols), calibration with (solid curve), and without (dashed line) background head trend for slug tests in drift well BC (b) and till well MA (c). hydraulics with recharge, with the following result k = m 2 (steady axisymmetric data, gray till) (16) Manual slug tests were performed with 0.92 or 1.83 m long aluminum rods of m diameter, designed to generate nominal amplitudes of 0.5 and 1.0 m. Gauge water pressure was sensed by unvented, pressure transducers with dataloggers (Adara Systems; Richmond, British Columbia, Canada). The slug tests in the brown till were of hourly duration, while most of the slug tests in the gray till were diurnal. Figures 2a and 2c summarize the 1.37-m amplitude slug-in test in till well MA. The background head rose seasonally, and reflected the drumlin hydraulics, which amplify seasonal head variations near the outer boundary of the landform (Ostendorf et al. 2004). Although the ambient κ of m/s is large, it is still smaller than the 10 6 m/s scale of barometric fluctuations which dominate the diurnal scale, and preclude our ability is analyze the more permeable regions of the drumlin. Well MA was installed at the bedrock interface, with an r E of 2.48 m. The calibrated λ of s 1 for MA is an order of magnitude larger than its drift counterpart, and implies a higher permeability for gray till k = m 2 (slug test, till well MA) (17) Equation 17 is less than the site averaged Equation 16, as anticipated: future analysis of the more permeable, diurnal slug tests, perhaps with a barometric extension of the Bouwer and Rice (1976) model, would bring the average of the local determinations into agreement with the hydrologic estimate of Ostendorf et al. (2004). The κ/(λh WO ) value of for the till slug test is an order of magnitude less than the drift well ratio, with cancelled transience and delayed response to the slug test. Thus background head trend is less important in the drumlin, although the longer effective (λt as high as 6) duration shown in Figure 2c features the emergence of background or barometric unsteadiness in the latter phases of the till slug test. Practical Relevance The two sites included adjacent wells with screen intervals comparable to those tested, so that we were able to estimate κ independently. The practitioner, however, must often interpret slug test data from single wells with limited periods of record, precluding an independent estimate of background head trends. If background head trends are to be retained in the interpretation of single well data in this case, κ may join λ as a calibration parameter, or we may return after the completion of the test to measure a posttest head in the monitoring well. In the latter regard, the pre- and posttest heads, when divided by the sampling interval, yield an estimate of κ on the assumption of a linear background trend. 612 D.W. Ostendorf, D.J. DeGroot GROUND WATER 48, no. 4: NGWA.org

5 A third strategy limits our interpretation to those data relatively unaffected by the background head trend. Equation 13 may be rearranged to describe the departure of the slug test from the exponential behavior of Bouwer and Rice (1976) H W H WO exp( λt) H = κ WO λh [λt + exp( λt) 1] WO (18) We require that the Bouwer and Rice (1976) theory remain within 5% of the extended theory by setting the left hand side of Equation 18 equal to The 5% criterion, when substituted into Equation 18, establishes a relationship between κ/(λh WO ) and a time t 0.05, shown in Figure 1b. A truncated data set (t <t 0.05 ) may be analyzed without background trends, which progressively dominate the data for longer times. Higher amplitude slugs ensure that more data fall within this time period. Thus, typical κ ( m/s) and λ (10 6 s 1 ) values would require a 2-m-amplitude slug test in order to generate an appreciable period of record (λt 0.05 equals 1) that could accurately be described without the background trend theory. The stratified drift test yields a κ(λh WO ) value of 1.3, and Figure 1b accordingly suggests that λt 0.05 is 0.29 the first seconds of the data may be used with conventional Bouwer and Rice (1976) theory to estimate λ. This is shown at the dashed straight line in Figure 2b which corresponds to a calibrated λ value of s 1. This is 19% less than the estimate that includes background head trend (the solid line in Figure 2b): permeability (which varies directly with λ) computed without the reinforcing influence of the falling background head trend on the slug-in test underestimates the actual value at drift well BC. If we doubled the amplitude of the slug test (from 0.67 to 1.34 m), then κ(λh WO ) wouldfallto0.65andλt 0.05 would increase to Thus the first s of the test could be used with conventional Bouwer and Rice (1976) theory and since more data would be dominated by slug test transience, the agreement between the conventional and extended calibrations would improve. The glacial till test, on the other hand, implies a κ(λh WO ) value of Figure 1b then generates a λt 0.05 value of 3.0 the first sofslugtest data for MA may be fit with Bouwer and Rice (1976) theory to estimate λ, and the resulting calibration of s 1 is within 4% of the value implied by the extended theory (the dashed and solid lines coincide closely on Figure 2c). Far more of the period of record may be used and the permeability estimates with and without the background trend agree when κ(λh WO ) is small. Conclusions A simple extension of Bouwer and Rice (1976) slug test theory accommodates background head trends. Classical (unextended) slug test theory for cancelled transience exhibited by slug-out/falling or slug-in/rising background head underestimates material permeability by mistakenly attributing the delayed recovery of head in the monitoring well to losses in the material. Unextended theory for reinforced transience of slug-in/falling or slugout/rising background head overestimates material permeability by ignoring the contribution of the background trend to the recovery of head in the well. The difference between the classical and extended theories becomes more important for smaller amplitude tests and for the latter portion of the slug test. Acknowledgments The Massachusetts Department of Transportation Highway Division funded this research under Interagency Service Agreement The views, opinions, and findings contained in this Note are the Authors, and do not necessarily reflect the official view or policies of MassHighway. This Note does not constitute a standard, specification, or regulation. We acknowledge and appreciate reviews by Michael Cardiff, Robert Chapuis, and an anonymous reviewer. References Bouwer, H., and R.C. Rice A slug test for determining hydraulic conductivity of unconfined aquifers with completely or partially penetrating wells. Water Resources Research 12, no. 3: Butler, J.J The Design, Performance, and Analysis of Slug Tests. New York: Lewis Publishers. Chapuis, R.P Monitoring a well in a clay layer: revisiting the time lag problem. Bulletin of Engineering Geology and the Environment 68, no 10: Chirlin, G.R A critique of the Hvorslev method for slug test analysis: the fully penetrating well. Ground Water Monitoring and Remediation 9, no. 2: Cooper, H.H., J.D. Bredehoeft, and I.S. Papadopulos Response of a finite diameter well to an instantaneous charge of water. Water Resources Research 3, no. 1: Hyder, Z., J.J. Butler, C.D. McElwee, and W. Liu Slug tests in partially penetrating wells. Water Resources Research 30, no. 11: Jacob, C.E Radial flow to a leaky artesian aquifer. Transactions American Geophysical Union 27, no. 2: Ostendorf, D.W., and C.J. Kilbridge Roadside aquifer impact case study: A mass balance sampling network for the White Lodge Wellfield of the Dedham-Westwood Water District. Journal of the New England Water Works Association 123, no. 3: Ostendorf, D.W., D.J. DeGroot, P.J. Dunaj, and J. Jakubowski A closed form slug test theory for high permeability aquifers. Ground Water 43, no. 1: Ostendorf, D.W., D.J. DeGroot, W.M. Shelburne, and T.J. Mitchell Hydraulic head in a clayey sand till over multiple time scales. Canadian Geotechnical Journal 41, no. 1: Prosser, D.W A method of performing response tests on highly permeable aquifers. Ground Water 19, no. 6: NGWA.org D.W. Ostendorf, D.J. DeGroot GROUND WATER 48, no. 4: