This research improves field based estimates of aquitard compressibility and permeability. A semianalytical model of partially penetrating, overdamped slug tests achieves this objective. The short term solution is an existing fully penetrating model, the long term solution is the polar residue of an inverse Laplace transform, and an exponential spline function patches the solutions together. Large amplitude slug test data from ten pairs of partially penetrating monitoring wells installed in an unweathered till at Scituate Hill in eastern Massachusetts calibrate the model. The deposit is bound by weathered till and the Dedham Granite fracture zone, and both are far more permeable than the unweathered till. The calibrated till permeability of 8.4 × 10^{–16} m^{2} is about 25% less than existing model calibrations that include boundary recharge in permeability values. The calibrated till compressibility of 5.1 × 10^{–10} Pa^{–1} reflects the proper inclusion of recharge as a long term source of groundwater, rather than the unrealistically large compressibility calibrations required by fully penetrating models.

## NOMENCLATURE

*A*_{J}*J*th Fourier sine series coefficient (m-s)*a*_{0}simple pole (1/s)

*B*_{J}integrating factor of

*J*th sine series coefficient*b*unweathered till thickness (m)

*c*_{J}*J*th integration constant (m-s)*D*hydraulic diffusivity (m

^{2}/s)*F*_{Short}transform of short term solution (m-s)

*f*(*t*):*F*(*p*)Laplace transform pair

*g*gravitational acceleration (m/s

^{2})*H*_{O}amplitude of slug test (m)

*h*disturbed hydraulic head (m)

*h*_{Long}long term solution for disturbed water level in well (m)

*h*_{Res}residue of simple pole (m)

*h*_{S}disturbed water level in well (m)

*h*_{Short}short term solution for disturbed water level in well (m)

*h**transformed hydraulic head (m-s)

transformed hydraulic head in well (m-s)

*J*_{N}Bessel function of the first kind, of order

*N**K*_{N}modified Bessel function of the second kind, of order

*N**k*permeability of unweathered till (m

^{2})*k*_{BR}Bouwer & Rice (1976) estimate of unweathered till permeability (m

^{2})*L*length of sandpack (m)

*L*_{Match}matching length (m)

*m*_{J}*J*th radial eigenfunction (m-s)*p*Laplace transform variable (1/s)

*p*_{Gage}gage pressure (kg/m-s

^{2})*q*_{J}*J*th vertical eigenfunction*R*distance along negative real axis of Bromwich contour (1/s)

*R*_{J}*J*th branch point (1/s)*r*radial distance from center of well (m)

*r*_{C}radius of casing (m)

*r*_{e}effective radius in Bouwer & Rice (1976) model (m)

*r*_{S}radius of sandpack (m)

*S*unweathered till storativity

*T*unweathered till transmissivity (m

^{2}/s)*t*time from start of slug test (s)

*u*changed variable of integration in Cooper

*et al.*(1967) model*Y*_{N}Bessel function of the second kind, of order

*N**z*elevation above lower recharge boundary (m)

*z*_{S}elevation of the base of the sandpack (m)

*z*_{T}elevation of transducer (m)

*α*unweathered till compressibility (m-s

^{2}/kg)*β*sum of integrating factors

*ρ*density of groundwater (kg/m

^{3})*ν*kinematic viscosity of groundwater (m

^{2}/s)

## INTRODUCTION

We derive and calibrate a closed form model of partially penetrating slug tests to improve field based estimates of aquitard compressibility and permeability. Aquitards are important because they protect underlying aquifers from surface contamination (van der Kamp 2001), an attribute of particular significance in stratified drift deposits (Klinger 1996) and till mantled bedrock (Lukas *et al.* 2015) of New England. Aquitard leakage affects confined aquifer hydraulics (Wang *et al.* 2015), and an improved understanding of the confining layer properties distinguishes recharge from aquifer compressibility in the calibration of transient pump test data (Neuman & Witherspoon 1969).

Slug tests estimate hydraulic properties of soil near monitoring wells, as documented by an established (Butler 1997) and continuing (Sakata *et al.* 2015) literature. Impermeable soil responds monotonically to an abrupt change of head in the well, and Cooper *et al.* (1967) use Laplace transforms to derive an analytical model for a fully penetrating sandpack in a confined aquifer. The overdamped slug test theory balances radial flow and storage in the soil with an integrated conservation of mass for the monitoring well imposed as a boundary condition at the interface of the sandpack and soil. Water level observations in the well calibrate the aquifer transmissivity and storativity, hence the radial permeability and compressibility of the soil. Partially penetrating wells, packers, or unconfined aquifers add a vertical component to the hydraulics of overdamped slug tests (Zlotnik 1994). Bouwer & Rice (1976) neglect soil storage, and compare cylindrical finite difference solutions with a radial closed form solution for overdamped slug tests in an unconfined aquifer to determine an equivalent radius in the soil. The latter determines the exponential decay constant for the water level in the monitoring well, along with the calibrated soil permeability. Hvorslev (1951) adopts a similar approach, but uses a steady cylindrical solution and shape factors to infer soil permeability from an exponential decay constant. Hyder *et al.* (1994) include soil storage in their cylindrical model of overdamped slug test hydraulics and derive a closed form Laplace transform solution for partially penetrating wells in a confined aquifer. Their transformed solution is inverted numerically, and features in many recent investigations. Paradis & Lefebvre (2013) use it to interpret slug test data in observation and slugged intervals in a single monitoring well equipped with multiple packers. Brauchler *et al.* (2010) use the Hyder *et al.* (1994) model to interpret slug test data in separate observation wells. Yeh *et al.* (2008) add a skin zone to the cylindrical model of a partially penetrating slug test in a confined aquifer and, like Hyder *et al.* (1994), invert the analytical Laplace transform solution numerically. Multiple wells and more complicated groundwater settings require numerical models, and Yang *et al.* (2015) summarize a recent MODFLOW application in aquitards. The challenge of heterogeneous property estimation from an array of observation and partially penetrating slugged wells is considered as an inverse tomographic exercise by Paradis *et al.* (2015).

Our study was motivated by the documented response of unweathered till monitoring wells to the introduction of seasonal pumping from underlying fractured bedrock more than a decade after the initial site characterization experiments. The latter followed van der Kamp's (2001) strategy of aquitard analysis across steady, seasonal, and diurnal scales; and included Hvorslev (1951) permeability calibration of slug tests and downward attenuation of storm scale transience for compressibility estimation (Ostendorf *et al.* 2004). Bedrock pumping introduces seasonal transience from below, and reveals the fracture zone as a leaky aquifer (Hantush 1960). Discrete (Ostendorf *et al.* 2015a) and continuous (Lukas *et al.* 2015) records of drawdown and recovery in the monitoring wells calibrate site averaged values for permeability and compressibility of the till. Ostendorf *et al.* (2015b) confirm the corresponding bedrock properties with a novel theory of slug tests and geophysical logs in uncased boreholes through the fracture zone. The present study completes the cycle in the till, with an improved model of partially penetrating slug tests that includes compressibility and recharge boundaries.

## SITE DESCRIPTION

### Scituate Hill

The theory is used to calibrate unweathered till properties at a well characterized field site in eastern Massachusetts (Figure 1). The site is a Massachusetts Department of Transportation highway deicing agent storage facility built in the 1960s on Scituate Hill, a glacial lodgment till drumlin in Cohasset (Ostendorf *et al.* 2004). The ground surface is about 55 m above mean sea level (mmsl), the water table is about 50 mmsl and occurs in a weathered till about 8 m thick, which overlies the unweathered till at about 45 mmsl. Soil moisture and grain size distributions from split spoon samples obtained during monitoring well construction suggest that the till is well graded and uniformly dense with an average porosity of 22%. Typical average particle size fractions are 17% gravel, 39% sand, 33% silt and 11% clay. Nearby excavations suggest cobbles and boulders are also present. The unweathered till in turn overlies the fracture zone of the Dedham Granite (Goldsmith 1991) with a contact surface at about 15 mmsl in the study area. Thus the unweathered till thickness *b* is set equal to 30 m, as cited in Table 1, which summarizes adopted and calibrated parameter values site wide.

Parameter | Value | Source |
---|---|---|

Unweathered till thickness (b) | 30 m | Soil borings |

Groundwater kinematic viscosity (ν) | 1.3 × 10^{–6} m^{2}/s | White (2011) |

Groundwater density (ρ) | 1.0 × 10^{3} m^{3}/s | White (2011) |

Weathered till permeability | 10^{–13} m^{2} | Ostendorf et al. (2004) |

Leaky bedrock aquifer transmissivity | 4 × 10^{–5} m^{2}/s | Ostendorf et al. (2015a) |

Casing radius (r) _{C} | 2.5 cm | Ostendorf et al. (2004) |

Unweathered till permeability (k) | 8.4 × 10^{–16} m^{2} | Geometric calibration average |

Unweathered till compressibility (α) | 5.1 × 10^{–10} Pa^{–1} | Arithmetic calibration average |

Matching length (L) _{Match} | 6.3 m | Arithmetic calibration average |

Parameter | Value | Source |
---|---|---|

Unweathered till thickness (b) | 30 m | Soil borings |

Groundwater kinematic viscosity (ν) | 1.3 × 10^{–6} m^{2}/s | White (2011) |

Groundwater density (ρ) | 1.0 × 10^{3} m^{3}/s | White (2011) |

Weathered till permeability | 10^{–13} m^{2} | Ostendorf et al. (2004) |

Leaky bedrock aquifer transmissivity | 4 × 10^{–5} m^{2}/s | Ostendorf et al. (2015a) |

Casing radius (r) _{C} | 2.5 cm | Ostendorf et al. (2004) |

Unweathered till permeability (k) | 8.4 × 10^{–16} m^{2} | Geometric calibration average |

Unweathered till compressibility (α) | 5.1 × 10^{–10} Pa^{–1} | Arithmetic calibration average |

Matching length (L) _{Match} | 6.3 m | Arithmetic calibration average |

Prior testing, modeling, and analysis have established that the weathered till and bedrock fracture zones are more than an order of magnitude more permeable than the unweathered till between them. Infiltration and unconfined aquifer hydraulics at steady and seasonal scales have established weathered till permeability of 10^{–13} m^{2} in radial (Ostendorf *et al.* 2004) and horizontal (Ostendorf *et al.* 2015a) flow fields along the toe and flank of the drumlin. Irrigation pumping from the Dedham Granite at an adjacent apartment complex completed in 2011 has revealed the fracture zone hydraulics to function as a confined aquifer of 4 × 10^{–5} m^{2}/s transmissivity that receives leakage from the unweathered till.

### Monitoring well details and slug test protocol

Twenty-two monitoring well clusters were constructed by the University of Massachusetts Amherst in the late 1990s and early 2000s to investigate the fate, transport, and remediation of deicing agent chloride contamination of the groundwater (Ostendorf *et al.* 2006). Ten of the clusters include pairs of 0.025 m radius (*r _{C}*) PVC monitoring wells, with 1.52 m screen sections with uniform sandpacks in closely spaced (<5 m) boreholes. One well was drilled to the contact surface with the Dedham Granite and a second was drilled 15–25 m higher up in the unweathered till. Table 2 lists attributes of the pairs, which feature sandpack radii

*r*varying from 0.049 to 0.108 m and sandpack lengths

_{S}*L*from 2.13 to 2.74 m in extent, and, in one instance, 5.34 m. The sandpacks were sealed with bentonite grout or chips, and each monitoring well was finished with a locked steel cover and concrete collar at the ground surface.

Well (date) | r, m _{S} | L, m | z, m _{S} | β | z, m _{T} | H, m _{O} |
---|---|---|---|---|---|---|

AB (1/29/13) | 0.108 | 2.59 | 0.0 | 0.086 | 1.1 | –21.0 |

AD (7/28/15) | 0.049 | 2.13 | 18.5 | 0.070 | 18.8 | –8.88 |

BB (7/28/15) | 0.108 | 1.89 | 16.9 | 0.062 | 17.5 | –7.17 |

BC (1/29/13) | 0.108 | 2.59 | 0.0 | 0.086 | 6.4 | –22.3 |

CB (1/29/13) | 0.108 | 2.44 | 0.0 | 0.082 | 1.8 | –23.0 |

CD (7/28/15) | 0.108 | 2.13 | 20.2 | 0.070 | 20.5 | –7.19 |

DB (1/29/13) | 0.108 | 2.13 | 0.0 | 0.070 | 3.7 | –25.3 |

DC (7/28/15) | 0.108 | 2.13 | 21.8 | 0.070 | 21.9 | –7.66 |

EB (8/4/15) | 0.108 | 2.21 | 0.0 | 0.072 | 0.3 | –15.6 |

ED (7/28/15) | 0.108 | 2.44 | 16.7 | 0.082 | 17.4 | –10.4 |

GB (1/29/13) | 0.049 | 5.34 | 0.0 | 0.18 | 1.0 | –18.6 |

GC (7/28/15) | 0.049 | 2.13 | 18.0 | 0.070 | 18.9 | –7.46 |

HB (1/29/13) | 0.049 | 2.44 | 0.0 | 0.082 | 0.7 | –19.6 |

HC (7/28/15) | 0.049 | 2.59 | 18.9 | 0.086 | 20.7 | –7.08 |

IB (1/29/13) | 0.108 | 2.13 | 0.0 | 0.070 | 0.1 | –24.0 |

ID (7/28/15) | 0.108 | 2.44 | 25.9 | 0.081 | 26.2 | –6.41 |

LA (1/29/13) | 0.108 | 2.13 | 0.0 | 0.070 | 14.5 | –26.5 |

LD (7/28/15) | 0.108 | 2.44 | 21.7 | 0.070 | 23.0 | –8.78 |

TA (8/4/15) | 0.049 | 2.74 | 0.0 | 0.090 | 1.0 | –19.3 |

TD (7/28/15) | 0.049 | 2.44 | 16.1 | 0.081 | 16.6 | –10.0 |

Well (date) | r, m _{S} | L, m | z, m _{S} | β | z, m _{T} | H, m _{O} |
---|---|---|---|---|---|---|

AB (1/29/13) | 0.108 | 2.59 | 0.0 | 0.086 | 1.1 | –21.0 |

AD (7/28/15) | 0.049 | 2.13 | 18.5 | 0.070 | 18.8 | –8.88 |

BB (7/28/15) | 0.108 | 1.89 | 16.9 | 0.062 | 17.5 | –7.17 |

BC (1/29/13) | 0.108 | 2.59 | 0.0 | 0.086 | 6.4 | –22.3 |

CB (1/29/13) | 0.108 | 2.44 | 0.0 | 0.082 | 1.8 | –23.0 |

CD (7/28/15) | 0.108 | 2.13 | 20.2 | 0.070 | 20.5 | –7.19 |

DB (1/29/13) | 0.108 | 2.13 | 0.0 | 0.070 | 3.7 | –25.3 |

DC (7/28/15) | 0.108 | 2.13 | 21.8 | 0.070 | 21.9 | –7.66 |

EB (8/4/15) | 0.108 | 2.21 | 0.0 | 0.072 | 0.3 | –15.6 |

ED (7/28/15) | 0.108 | 2.44 | 16.7 | 0.082 | 17.4 | –10.4 |

GB (1/29/13) | 0.049 | 5.34 | 0.0 | 0.18 | 1.0 | –18.6 |

GC (7/28/15) | 0.049 | 2.13 | 18.0 | 0.070 | 18.9 | –7.46 |

HB (1/29/13) | 0.049 | 2.44 | 0.0 | 0.082 | 0.7 | –19.6 |

HC (7/28/15) | 0.049 | 2.59 | 18.9 | 0.086 | 20.7 | –7.08 |

IB (1/29/13) | 0.108 | 2.13 | 0.0 | 0.070 | 0.1 | –24.0 |

ID (7/28/15) | 0.108 | 2.44 | 25.9 | 0.081 | 26.2 | –6.41 |

LA (1/29/13) | 0.108 | 2.13 | 0.0 | 0.070 | 14.5 | –26.5 |

LD (7/28/15) | 0.108 | 2.44 | 21.7 | 0.070 | 23.0 | –8.78 |

TA (8/4/15) | 0.049 | 2.74 | 0.0 | 0.090 | 1.0 | –19.3 |

TD (7/28/15) | 0.049 | 2.44 | 16.1 | 0.081 | 16.6 | –10.0 |

*z*of a piezoresistive pressure transducer (In Situ, Fort Collins, CO, USA) above the cluster interface with the Dedham Granite (Table 2). The transducer was removed briefly (<10 minutes) while a Waterra purge pump (New England Environmental, Waltham, MA, USA) evacuated the monitoring well in order to establish the amplitude of the slug test. The transducer was then reinserted into the monitoring well and the subsequent recovery of head in the well logged at 15 minute intervals. The gage pressure

_{T}*p*sensed by the vented transducer estimates the disturbed head measured in the well:

_{Gage}The groundwater density is *ρ* and gravitational acceleration is *g*.

### Modifications for the study site

Our study site featured large amplitude, partially penetrating slug tests in an aquitard with permeable boundaries, so the existing aquifer theory had to be modified to calibrate the results. Figure 2 shows disturbed water levels *h _{S}* observed in monitoring wells AB and HB. The wells were pumped dry at time

*t*equals zero, so that the slug test amplitude

*H*was negative and large, with absolute values in excess of 20 m at the start of the tests. Semilog plots of the first 150 hours suggest that AB was far less permeable than HB, as the levels decreased by an order of magnitude in 3 days and 5 hours, respectively (Figure 2(a) and 2(b)), followed by a slower decrease from 1 m to centimeter scale disturbances (Figure 2(c) and 2(d)). The asymptotic approach to straight lines in the semilogarithmic Figure 2(a) and 2(b) does imply exponential late term behavior, as is also predicted by Bouwer & Rice (1976). Figure 2(c) and 2(d) suggest that the slope of the regression is distorted by ambient transience at the centimeter scale, however. Figure 2(e) and 2(f) are linear plots of the last 150 hours of both tests. The monitoring wells recovered hydraulic heads above their starting levels at the end of the tests, implying a seasonal rise of head due to recharge and cessation of irrigation pumping adjacent to the site (Ostendorf

_{O}*et al.*2015a). Aperiodic storm scale transience appeared at the 230–260-hour intervals in both wells, and a semidiurnal earth tide was easily seen in well HB, due to its proximity to underlying permeable bedrock. Our use of large amplitude slug tests adds a second order of magnitude to the data before distortion by earth tides, barometric pressure fluctuations, recharge, and pumping disturbances.

The exact integration of the Bromwich contour needed to invert transformed cylindrical hydraulics is complicated, as demonstrated by Neuman & Witherspoon (1969) for pump tests in leaky aquifers. We follow Hantush (1960) instead, who interpolates between simple short and long term solutions in his analysis of pump tests in leaky aquifers. Our long term solution is exponential, with a pole and a residue that permit a semilogarithmic regression whose slope and intercept calibrate the compressibility and permeability of the unweathered till. The short term solution is similar to that of Cooper *et al.* (1967), with the inclusion of a factor dependent on partial well geometry. Our closed form model, though new, does rest on its predecessors.

## MATHEMATICAL MODEL

### Governing equation and assumed transformed solution for cylindrical flow

*h*in a confined, homogeneous, isotropic till of compressibility

*α*: with radial distance

*r*from the axis of a well and elevation

*z*above the lower recharge boundary, and elevation of the base of the sandpack

*z*(Figure 3). The permeability

_{S}*k*combines with

*ρ*and kinematic viscosity

*ν*of the groundwater to define a hydraulic diffusivity

*D*. These properties describe the propagation of disturbances caused by an overdamped slug test in a well that partially penetrates the till.

*h**of Equations (2a) and (2c) is thus the sum of vertical

*q*and radial

_{J}(z)*m*eigenfunctions, each governed by an ordinary, second order differential equation: The Laplace transform variable is

_{J}(r)*p.*Boundary conditions constrain the eigenfunctions by distinguishing their type and integration constants.

*N*order modified Bessel functions of the second kind

*K*are chosen for

_{N}*m*: Branch points

_{J}*R*are defined for each eigenfunction. The transformed solution is: The solution approaches a vertically uniform transformed head at the sandpack radius over the sandpack length, and vanishes elsewhere as the radius goes to zero:

_{J}*A*:

_{J}*B*reflect the partial penetration of the sandpack. In view of Equations (6) and (8), the integration constants

_{J}*c*depend on :

_{J}The Laplace transform of the slug test condition, Equations (2f) and (2g), is:

*r*and evaluated at the sandpack radius. The result is solved for : Equation (11b) defines the transmissivity T.

### Short term solution

*p*. The logic is applied to Equation (11a) with the limiting result: The series sum

*β*reflects the partial penetration of the sandpack, in view of Equation (8c).

*h*(

_{Short}*t*) reduces to an integral of the transformed function

*F*(

_{Short}*p*) along the negative real axis -

*R*of the Bromwich contour: Carslaw & Jaeger (1973) use Bessel function identities and a change of variables to invert Equation (13) for heat transfer:

*N*order Bessel functions of the first and second kind are

*J*and

_{N}*Y*. We substitute Equation (14d) into Equation (13a), take the complex conjugate, invoke Equation (14a), and so derive the short term solution for the present case: Equation (15b) defines the storativity

_{N}*S*. We use 64 point Gauss quadrature (Bathe & Wilson 1976) to estimate the integral in Equation (15a), with coefficients provided by Abramowitz & Stegun (1972).

If the sandpack extends over the entire till, *z _{S}* is zero,

*L*is

*b*,

*β*is one (Spiegel & Liu 1999), and Equation (15a) becomes identical to the Cooper

*et al.*(1967) model for a fully penetrating, overdamped slug test in a confined aquifer. The coincidence suggests that most of the initial recovery of the head in the well is due to radial flow of stored groundwater released by soil near the sandpack. Leakage and cylindrical flow happen later.

### A simple pole as the long term solution

*a*and branch points along the negative real axis of the Bromwich contour, as shown in Figure 4(b). The modified Bessel functions in Equation (16b) have real arguments to the right of their branch points and imaginary arguments to the left, and the first branch point is the lower limit of integration.

_{0}*et al.*(1974) use a derivative of the denominator of Equation (16b) to evaluate the residue, and Hildebrand (1976) provides derivatives for the Bessel functions necessary to evaluate the residue series terms. Equation (2b) simplifies the result: The simple pole

*a*lies between the origin and the first branch point and is defined by a zero in the denominator of Equation (16b). This establishes an implicit equation for the pole: The polar residue is the long term solution

_{0}*h*for the slug test, since the exponential term in the integrand of Equation (16a) eliminates the branch point integral from the solution as time increases:

_{Long}### Spline function between the short and long term solutions

*h*in the monitoring well requires complex contour integration of Equation (16a), which is difficult. A heuristic patching of long and short term solutions is adopted instead: The matching length

_{S}*L*is less than

_{Match}*b*, which characterizes the long term solution, and Equation (21) uses vertical diffusion across this length to interpolate between the asymptotic behavior. Figure 5 displays the patched solution for slug test AB, a –21 m slug test in a 2.5 cm radius monitoring well with an 11 cm radius sandpack 2.6 m long, resting on the lower boundary of a 30 m thick till layer of 0.00092 m

^{2}/s diffusivity. The 4.8 × 10

^{–16}m

^{2}permeability and 4.0 × 10

^{−10}Pa

^{–1}compressibility imply a –13 m polar residual. The matching length of 6.4 m shown in the figure creates an asymptotic departure from the short term solution at 1 hour and a long term approach about 3 days into the test.

### Calibration

*a*and

_{0}*h*. The former value of –1.0 × 10

_{Res}^{–5}s

^{–1}and latter value of –13.0 m are cited in Table 2. Equations (18) and (19) yield an implicit equation for the hydraulic diffusivity in terms of the calibrated polar values:

*–a*–

_{0}/(H_{O}/h_{Res}*1)*value of 1.6 × 10

^{–5}s

^{–1}, which converges to a

*D*of 0.00092 m

^{2}/s. The value is cited in Table 3. Equation (19a) yields the transmissivity (1.1 × 10

^{–7}m

^{2}/s) with

*D*and

*a*known:

_{0}Well | a, s_{0}^{–1} | h, m _{Res} | L, m _{Match} | D, m^{2}/s | S | T, m^{2}/s | k, m^{2} | α, Pa^{–1} |
---|---|---|---|---|---|---|---|---|

AB | –1.0 × 10^{–5} | –13 | 6.4 | 0.00092 | 0.00012 | 1.1 × 10^{–7} | 4.8 × 10^{–16} | 4.0 × 10^{–10} |

AD | –2.1 × 10^{–5} | –4.0 | 7.0 | 0.0019 | 0.00021 | 3.9 × 10^{–7} | 1.7 × 10^{–15} | 7.0 × 10^{–10} |

BB | –1.3 × 10^{–4} | –3.4 | 7.0 | 0.012 | 0.00018 | 2.1 × 10^{–6} | 9.3 × 10^{–15} | 6.1 × 10^{–10} |

BC | –1.1 × 10^{–5} | –3.6 | 4.1 | 0.00098 | 0.00012 | 1.2 × 10^{–7} | 5.2 × 10^{–16} | 4.0 × 10^{–10} |

CB | –8.3 × 10^{–6} | –3.5 | 4.1 | 0.00076 | 0.00012 | 9.4 × 10^{–8} | 4.1 × 10^{–16} | 4.2 × 10^{–10} |

CD | –3.2 × 10^{–6} | –4.4 | 6.1 | 0.00029 | 0.00016 | 4.6 × 10^{–8} | 2.0 × 10^{–16} | 5.3 × 10^{–10} |

DB | –3.1 × 10^{–5} | –3.8 | 3.7 | 0.0028 | 0.00014 | 3.8 × 10^{–7} | 1.7 × 10^{–15} | 4.6 × 10^{–10} |

DC | –2.1 × 10^{–5} | –2.3 | 4.1 | 0.0019 | 0.00016 | 3.0 × 10^{–7} | 1.3 × 10^{–15} | 5.3 × 10^{–10} |

EB | –1.9 × 10^{–6} | –14.0 | 5.3 | 0.00017 | 0.00013 | 2.3 × 10^{–8} | 1.0 × 10^{–16} | 4.5 × 10^{–10} |

ED | –7.1 × 10^{–6} | –6.0 | 7.8 | 0.00065 | 0.00014 | 9.3 × 10^{–8} | 4.1 × 10^{–16} | 4.8 × 10^{–10} |

GB | –3.8 × 10^{–5} | –0.90 | 2.4 | 0.0034 | 8.3 × 10^{−5} | 2.8 × 10^{–7} | 1.2 × 10^{–15} | 2.8 × 10^{–10} |

GC | –6.4 × 10^{–6} | –3.0 | 16 | 0.00059 | 0.00021 | 1.2 × 10^{–7} | 5.4 × 10^{–16} | 7.1 × 10^{–10} |

HB | –5.0 × 10^{–5} | –2.0 | 3.4 | 0.0045 | 0.00016 | 7.3 × 10^{–7} | 3.2 × 10^{–15} | 5.5 × 10^{–10} |

HC | –6.5 × 10^{–5} | –4.0 | 5.7 | 0.0060 | 0.00017 | 1.0 × 10^{–6} | 4.6 × 10^{–15} | 5.9 × 10^{–10} |

IB | –9.7 × 10^{–6} | –3.5 | 3.0 | 0.00086 | 0.00014 | 1.2 × 10^{–7} | 5.3 × 10^{–16} | 4.6 × 10^{–10} |

ID | –2.6 × 10^{–5} | –3.5 | 4.7 | 0.0024 | 0.00013 | 3.2 × 10^{–7} | 1.4 × 10^{–15} | 4.5 × 10^{–10} |

LA | –7.6 × 10^{–6} | –15 | 5.7 | 0.00069 | 0.00014 | 9.4 × 10^{–8} | 4.2 × 10^{–16} | 4.6 × 10^{–10} |

LD | –5.9 × 10^{–6} | –5.0 | 6.4 | 0.00054 | 0.00016 | 8.4 × 10^{–8} | 3.7 × 10^{–16} | 5.3 × 10^{–10} |

TA | –1.4 × 10^{–5} | –5.0 | 19.9 | 0.0013 | 0.00015 | 2.0 × 10^{–7} | 8.7 × 10^{–16} | 5.1 × 10^{–10} |

TD | –1.6 × 10^{–5} | –1.8 | 3.7 | 0.0015 | 0.00018 | 2.7 × 10^{–7} | 1.2 × 10^{–15} | 6.3 × 10^{–10} |

Well | a, s_{0}^{–1} | h, m _{Res} | L, m _{Match} | D, m^{2}/s | S | T, m^{2}/s | k, m^{2} | α, Pa^{–1} |
---|---|---|---|---|---|---|---|---|

AB | –1.0 × 10^{–5} | –13 | 6.4 | 0.00092 | 0.00012 | 1.1 × 10^{–7} | 4.8 × 10^{–16} | 4.0 × 10^{–10} |

AD | –2.1 × 10^{–5} | –4.0 | 7.0 | 0.0019 | 0.00021 | 3.9 × 10^{–7} | 1.7 × 10^{–15} | 7.0 × 10^{–10} |

BB | –1.3 × 10^{–4} | –3.4 | 7.0 | 0.012 | 0.00018 | 2.1 × 10^{–6} | 9.3 × 10^{–15} | 6.1 × 10^{–10} |

BC | –1.1 × 10^{–5} | –3.6 | 4.1 | 0.00098 | 0.00012 | 1.2 × 10^{–7} | 5.2 × 10^{–16} | 4.0 × 10^{–10} |

CB | –8.3 × 10^{–6} | –3.5 | 4.1 | 0.00076 | 0.00012 | 9.4 × 10^{–8} | 4.1 × 10^{–16} | 4.2 × 10^{–10} |

CD | –3.2 × 10^{–6} | –4.4 | 6.1 | 0.00029 | 0.00016 | 4.6 × 10^{–8} | 2.0 × 10^{–16} | 5.3 × 10^{–10} |

DB | –3.1 × 10^{–5} | –3.8 | 3.7 | 0.0028 | 0.00014 | 3.8 × 10^{–7} | 1.7 × 10^{–15} | 4.6 × 10^{–10} |

DC | –2.1 × 10^{–5} | –2.3 | 4.1 | 0.0019 | 0.00016 | 3.0 × 10^{–7} | 1.3 × 10^{–15} | 5.3 × 10^{–10} |

EB | –1.9 × 10^{–6} | –14.0 | 5.3 | 0.00017 | 0.00013 | 2.3 × 10^{–8} | 1.0 × 10^{–16} | 4.5 × 10^{–10} |

ED | –7.1 × 10^{–6} | –6.0 | 7.8 | 0.00065 | 0.00014 | 9.3 × 10^{–8} | 4.1 × 10^{–16} | 4.8 × 10^{–10} |

GB | –3.8 × 10^{–5} | –0.90 | 2.4 | 0.0034 | 8.3 × 10^{−5} | 2.8 × 10^{–7} | 1.2 × 10^{–15} | 2.8 × 10^{–10} |

GC | –6.4 × 10^{–6} | –3.0 | 16 | 0.00059 | 0.00021 | 1.2 × 10^{–7} | 5.4 × 10^{–16} | 7.1 × 10^{–10} |

HB | –5.0 × 10^{–5} | –2.0 | 3.4 | 0.0045 | 0.00016 | 7.3 × 10^{–7} | 3.2 × 10^{–15} | 5.5 × 10^{–10} |

HC | –6.5 × 10^{–5} | –4.0 | 5.7 | 0.0060 | 0.00017 | 1.0 × 10^{–6} | 4.6 × 10^{–15} | 5.9 × 10^{–10} |

IB | –9.7 × 10^{–6} | –3.5 | 3.0 | 0.00086 | 0.00014 | 1.2 × 10^{–7} | 5.3 × 10^{–16} | 4.6 × 10^{–10} |

ID | –2.6 × 10^{–5} | –3.5 | 4.7 | 0.0024 | 0.00013 | 3.2 × 10^{–7} | 1.4 × 10^{–15} | 4.5 × 10^{–10} |

LA | –7.6 × 10^{–6} | –15 | 5.7 | 0.00069 | 0.00014 | 9.4 × 10^{–8} | 4.2 × 10^{–16} | 4.6 × 10^{–10} |

LD | –5.9 × 10^{–6} | –5.0 | 6.4 | 0.00054 | 0.00016 | 8.4 × 10^{–8} | 3.7 × 10^{–16} | 5.3 × 10^{–10} |

TA | –1.4 × 10^{–5} | –5.0 | 19.9 | 0.0013 | 0.00015 | 2.0 × 10^{–7} | 8.7 × 10^{–16} | 5.1 × 10^{–10} |

TD | –1.6 × 10^{–5} | –1.8 | 3.7 | 0.0015 | 0.00018 | 2.7 × 10^{–7} | 1.2 × 10^{–15} | 6.3 × 10^{–10} |

The storativity (0.00012) follows from Equation (19b), then the permeability (4.8 × 10^{–16} m^{2}) and compressibility (4.0 × 10^{–10} Pa^{–1}) from Equations (11b) and (15b), respectively. The long term behavior alone estimates intrinsic properties of the till, by graphical calibration.

The splined approximation Equation (21) of the complete solution then drives a Fibonacci search (Knuth 1973) for the matching length that minimizes the root mean square of the error defined by Equations (1) and (21). A value of 6.4 m yields an rms error of 2.7 cm, and the calibration plots as the line in Figure 6(b). The AB calibration is also plotted in Figure 5.

## RESULTS

### Calibrated parameter values

This graphical calibration of the long term solution is carried out for all 20 slug tests with the individual results summarized in Table 3 and the site averages in Table 1. The polar residues are from 5 to 90% of the slug amplitudes; thus an exponential decay constant does not describe the first two orders of magnitude of the recovery of head from the slug. The matching lengths vary from 2.4 to 20 m, with a 6.3 m arithmetic average. Figures 6(b) and 7 suggest that the approximation accurately patches the observed long and short term behavior of the test.

The long term behavior generates polar decay constants that range from –1.9 × 10^{–6} to –1.3 × 10^{–4} s^{–1}, so that wells regain their ambient water levels within a few hours or a few days (Figure 2). The permeabilities vary from 1.0 × 10^{–16} m^{2} in monitoring well EB to 9.3 × 10^{–15} m^{2} in well BB with a (geometric) average value of 8.4 × 10^{–16} m^{2}. Table 1 confirms that the weathered till is over an order of magnitude more permeable than the unweathered till beneath it. The transmissivities of Table 3 range from 2.3 × 10^{–8} to 2.1 × 10^{–6} m^{2}/s, with a (geometric) average (1.9 × 10^{–7} m^{2}/s) that is two orders of magnitude less than the fractured bedrock zone transmissivity cited in Table 1. The unweathered till is far less permeable than the underlying Dedham Granite, and the recharge boundary assumptions of the closed form analysis are justified at the site.

The calibrated compressibilities range from 2.8 × 10^{–10} to 7.1 × 10^{–10} Pa^{–1}, with an arithmetic average of 5.1 × 10^{–10} Pa^{–1}. The values are an order of magnitude larger than the product of the 22% till porosity and the 4.4 × 10^{–10} Pa^{–1} compressibility of water (White 2011) and the 2 × 10^{–11} Pa^{–1} compressibility of unfractured Dedham Granite (Leet & Ewing 1932). Storage comes from the rearrangement of incompressible grains of the till in incompressible groundwater.

### Bouwer & Rice (1976) and Cooper *et al.* (1967) comparisons

The effective radius *r _{e}* depends on the well geometry and distance of the sandpack from the water table, and the impermeable lower boundary of the aquifer.

Figure 8 plots the permeability *k _{BR}* estimated by the Bouwer & Rice (1976) model against the Table 3 values for deep (circles) and shallow (squares) monitoring wells. The permeability of the deep slug tests is slightly overpredicted by the Bouwer & Rice (1976) model:

*k*is 12–30% higher than

_{BR}*k*with an average of 24%. The overprediction may be attributed in part to the assumption of an impermeable bottom boundary condition in the Bouwer & Rice (1976) analysis. Upward inflow from the Dedham Granite is responsible for some of the slug recovery as well as downward inflow from the till above the sandpack, and the former is not reflected in the effective radius curves relating to the numerical model of Bouwer & Rice (1976) to the simpler one dimensional approximation of the quasi-steady aquifer flow. By contrast, the permeability of the shallow tests is underpredicted by the Bouwer & Rice (1976) model:

*k*is 27–35% lower than

_{BR}*k*, with an average of 31%. The water table of Bouwer & Rice (1976) contributes more flow than the upper (zero disturbed head) recharge boundary of the present model.

Lukas *et al.* (2015) use Cooper *et al.* (1967) to calibrate the entire slug test for six of the deep till experiments. The flow field is purely radial in this approximation, and the confined aquifer thickness is equated to the sandpack length. The resulting compressibility estimates range from 2.6 × 10^{–10} to 2.4 × 10^{–4} Pa^{–1} for the slug tests under this assumption. Figures 5 and 9 help explain this behavior. The latter displays the data of experiment AB, plotted with the nondimensionalized, semilogarithmic format suggested by Cooper *et al.* (1967), but using the storativity and transmissivity calibrations of Table 3. Type curves from the Cooper *et al.* (1967) analysis are also shown in Figure 9. The progression from early to late term behavior, recovered by the heuristic spline curve of the cylindrically based Figure 5, implies a shift from a low compressibility Cooper *et al.* (1967) type curve to a much higher compressibility type curve. Our use of Cooper *et al.* (1967) as a short term asymptote preserves the low compressibility calibration of the parameter. Late term behavior introduces vertical flow as the source of groundwater, rather than an increased release of storage from more compressible soil. The spline function of Figure 5 shifts the short term (type curve) behavior to an exponential late term prediction, rather than a higher compressibility type curve.

## DISCUSSION

As a practical matter, the use of quasi-steady, radial slug test models to estimate permeability is confirmed by the cylindrical, closed form approach of the present analysis. Figure 8 establishes that Bouwer & Rice (1976) calibrations are well within an order of magnitude of the Table 3 permeabilities. Ostendorf *et al.* (2004) summarize Hvorslev (1951) calibrations of small amplitude slug tests at the site with an average permeability (1.4 × 10^{–15} m^{2}) that is close to the Table 1 value. Hyder *et al.* (1994) arrive at a similar conclusion through an extensive set of simulations. Our method does yield a tighter compressibility estimate than the radial models when the boundary conditions are met, however. The parameter is absent altogether from the Bouwer & Rice (1976) and Hvorslev (1951) models, while the Cooper *et al.* (1967) application precludes vertical flow and leakage. The radial models offer little insight into storage properties for partially penetrating slug tests in till.

Although the –20 m scale amplitude of the deep and –10 m scale amplitude of the shallow slug tests eliminate complicating effects of ambient transience on till property estimation, centimeter scale fluctuations can be used to advantage in simpler analyses. Ostendorf & DeGroot (2010) incorporate a linearly varying ambient head into a Bouwer & Rice (1976) analysis of a small amplitude slug test at a deep monitoring well (MA) at the Cohasset site; the 6.0 × 10^{–16} m^{2} permeability compares favorably with the Table 3 values. The downward attenuation of storm scale disturbances observed at the G cluster implies a hydraulic diffusivity of 0.00043 m^{2}/s in the unweathered till (Ostendorf *et al.* 2004). This is less than the 0.0013 m^{2}/s site-wide value implied by Table 1, though it is the same order of magnitude. A site average compressibility of 3 × 10^{–9} Pa^{–1} and a site average permeability of 5.9 × 10^{–16} m^{2} calibrate the response of the unweathered till to seasonal irrigation pumping of the Dedham Granite at an adjacent apartment complex (Ostendorf *et al.* 2015a).

## CONCLUSIONS

A new semianalytical cylindrical model of a slug test in a partially penetrating monitoring well calibrates hydraulic properties of till bound by zero head recharge layers. A heuristic spline function patches initial and long term solutions together. Ten pairs of large amplitude slug tests in deep and shallow monitoring wells calibrate permeability and compressibility values at a till drumlin in eastern Massachusetts. The 8.4 × 10^{–16} m^{2} permeability estimate is close to the value established by simpler radial slug test models at the site. The calibrated compressibilities of the 20 tests do not vary much from the site average value of 5.1 × 10^{–10} Pa^{–1} because the present analysis models vertical flow and recharge as a progressively more important groundwater source later in the experiment. Thus the low compressibility calibration accurately reflects the short term response to the test, when soil immediately near the sandpack flows radially into the well. Radial models rely on storage release for the groundwater throughout the test duration, and calibrate late term response with unrealistically high compressibilities as a consequence.

## ACKNOWLEDGEMENTS

The Massachusetts Department of Transportation Highway Division funded this research under Interagency Service Agreement No. 73140 with the University of Massachusetts Amherst. The views, opinions, and findings contained in this paper are those of the authors and do not reflect MassDOT official views or policies. This paper does not constitute a standard, specification, or regulation.