Theory of Slug Testing

Submitted by Karl DeBisschop on Mon, 10 Dec 2007 - 20:21

A slug test involves inducing a rapid change in water level in a test well (see note: slug). By measuring and recording the rate of return to static conditions (recovery), one is able to estimate the local horizontal hydraulic conductivity of the material surrounding the well. Slug test data are generally analyzed using relatively standard analytical solutions to the equations which govern groundwater flow. Homogeneity and constant aquifer thickness are common assumptions for conditions within the area of influence of the test. In practice, these are usually met because the radius of influence of most slug tests is fairly small.

Two classes of solutions are generally used in evaluating slug test data. The class of solutions which includes the Bouwer and Rice, and Hvorslev Methods relies on the assumption that water and soil are incompressible--that is, aquifer storativity is zero. This assumption allows use of a modified Thiem equation to predict well response. The potential which drives flow into or out of the well is expressed as a difference between head in the well and at a so-called "radius of influence." Using the somewhat artificial concept of radius of influence eliminates the need to consider aquifer storage in this treatment. (see note: Thiem)

Alternate methods of analysis, including the Cooper-Bredehoeft-Papadopulos method, assume a non-zero storativity which must be calculated to obtain the correct solution. These methods yield solutions analogous to the Theis equation for radial flow to a well. Bedrock aquifers generally meet, in a rough sense, the assumptions of the analysis and may potentially be analyzed with this spreadsheet, though extreme caution should be taken in interpreting the results. For many applications, however, the zone of interest may be a single fracture or discrete fracture zone (e.g. multilevel monitoring wells and packer testing). It may therefore be misleading to present results as hydraulic conductivity without knowing the thickness of the fracture or fracture zone. In such cases, techniques such as CBP may be more appropriate.

Another point of concern lies in the fact that the upper boundary condition for the Thiem analysis is a no-flow boundary, whereas the saturated overburden is more likely to act as a constant head condition. Considering the anisotropy of the roughly horizontal fractures which a vertical well is likely to intersect, direct leakage from overburden will frequently serve simply as the means by which the condition of constant water level at infinite distance is maintained. The satisfaction of this condition, however, cannot generally be confirmed on the basis of the slug test data.

Basic Development

Figure 1 shows a schematic of a typical well installation.

To solve for drawdown in a well following the slug test, consider the Thiem equation:

Q = 2 π L_I K y / ln(R_E /R_W)

where

K: is the hydraulic conductivity of the aquifer
y: is the vertical distance between the water level in the well and the equilibrium water table in the aquifer
L_I: is the well intake length
R_E: is the effective radius over which drawdown y is dissipated
R_W: is the horizontal distance from the well center to the original aquifer material

The rate of water level change in the well is related to the rate of water level recovery by conservation of mass:

d_y/d_t = -Q / π R_C²

where R_C is the radius of the well casing. Combining these equations and integrating between the limits y_o at time = 0 and y_t at time = t, yields

K = (R_C² ln(y_o/y_t))/ Ft

where F is a shape factor which includes terms dependent on the well geometry.

Shape Factors

Although presented in different forms, the time lag procedure described in Hvorslev's original paper is equivalent to the slope of line procedure described by Bouwer and Rice. Therefore, the determination of shape factors is the only substantive difference between Hvorslev's method and the method of Bouwer and Rice.

Hvorslev

A series of analytical solutions is presented in Hvorslev (1951). A critical review of shape factors and their validity is presented in Chapuis (1989). Chapuis particularly identifies discrepencies in practice that affect wells with L_I/R_W less than 16 (for typical environmental wells drilled with an 8-inch auger, this means sand pack lengths less than 5 feet). It is also useful to note that many authors use approximations to the exact analytical shape factors, which are only valid for large length and/or anisotropy. The worksheet currently uses 5 shape factors based on the geometry of the well:

4 R for L_I=0, L_W=0 (a)
5.5 R for L_I=0, L_W>0 (b)
2 π L_I / sqrt(L_I/[(2R_W) + 0.25]) for L_I<16R_W (c)
2 π L_I / ln[mL_I/R_W + sqrt(1+[mL_I/R_W]²)] for L_I=L_W (d)
2 π L_I / ln[mL_I/2R_W + sqrt(1+[mL_I/2R_W]2)] for L_I (e)

where the transform factor, m, reflects aquifer anisotropy, and is the square-root of the quotient of horzontal and vertical hydraulic conductivity [(K_h/K_v)1/2]. These formulae are exact representations of the analytical solution, and do not use the approximation for large LI noted above.

Chapuis (1989) also suggests that many cases the contribution of well bottom flow (5.5 R) should be subtracted from the 4c, 4d, and 4e above. In practice, environmental monitoring wells often have a highly permable sand pack which defines the boundary of the well (rather than the actual slotted casing). In such cases, flow does occur through the bottom of the well bore, and this correction is not recommended. For wells in which the gravel pack is not significantly more permeable than the formation, this correction can be applied, but it must be asked in such cases if slug test data are measuring the aquifer response, or the response of the sand pack. For typical environmental applications, the value of this correction term is less than 5 percent. There is a logical parameter in the function definition for determination of the Hvorslev shape factor which, when changed to a non-zero value, causes this correction to be included.

Cedergren (1989) provides additional shape factors for short wells screened at the top of an aquifer and for fully-penetrating wells, but these are not typically used because 1) the appropriate range of applicability is poorly defined, 2) the transition between domains is not smooth, and 3) there is no provision for anisotropy. If well geometries enter this range, a reference to Cedergren's text is provided, and the shape factor formula can be modified on a case-by-case basis if the user wishes to incorporate these cases. Typical corrections due to this factor seem to be on the order of 20 percent. A parameter is provided in the Hvorslev shape factor definition to control inclusion of this factor.

Bouwer and Rice

Bouwer and Rice (1976) proposed a shape factor of the form

F = 2 p LI / ln(R_E /R_W)

Values for ln(R_E/R_W) were determined by electrical resistance network for various combinations of well length (L_W), well radius (R_W), aquifer thickness (H), and intake length (L_I). Results of the network simulation are given as curves of three parameters A, B, and C. As part of the development of this spreadsheet, an analytic expression for the values of these parameters was formulated by regression analysis. The resulting expressions are:

Function A(x)
  If x < 2.554422663 Then
    A = 1.638445671 + 0.166908063 * x + 0.000740459 * Exp(6.17105281 * x - 1.054747686 * x * x)
  Else
    A = 11.00393028 - 170.7752217 * Exp(-1.509639982 *x)
  End If
End Function

Function B(x)
  If x < 2.596774459 Then
    B = 0.174811819 + 0.060059188 * x + 0.007965502 * Exp(2.053376868 * x - 0.007790328 * x * x)
  Else
    B = 4.133124586 - 93.06136936 * Exp(-1.435370997 * x)
  End If
End Function

Function C(x)           (eq.6c)
  If x < 2.200426117 Then
    C = 0.074711376 + 1.083958569 * x + 0.00557352 * Exp(2.929493814 * x - 0.001028433 * x * x)
  Else
    C = 15.66887372 - 178.4329289 * Exp(-1.322779744 * x)
  End If
End Function

where x is log(L_I/R_W). Figure 2 shows a comparison of the values predicted by these equations with the published curves. The RMS error for these curve fits are 0.017 for A (64 data points), 0.0059 for B (88 data points), and 0.048 for C (88 data points). The greatest error is 3 percent, which occurs for small values of B. All other errors are 1 percent or less.

Double Straight-Line Effect

Bouwer (1989) discusses a phenomena he terms the "double straight-line effect" which can occur in wells with a sand pack intersecting the water table. In such situations, the drainage out of or into the sand pack contributes to the volumetric flow term of the governing equations for the slug test. An initial drainage curve is observed, followed by the aquifer response curve on which the analysis should be conducted. An intriguing suggestion in a letter following the 1989 update involves using the known volume of the slug to aid in estimating the porosity or radius of the sand pack for this calculation. In the response, details of this methodology are left for the reader. Two separate treatments of this issue are used for this spreadsheet implementation, either of which can be used based on the judgement of the user. These are discussed under the Casing Radius topic of the Required Data section of this paper.

Confined Aquifers

Another comment in the update indicates that the Bouwer and Rice method is applicable to confined aquifers. There are no intrinsic difficulties in the use of these methods in confined aquifers. Some care, however, is required to choose the correct values for the well geometry. The aquifer thickness is taken, as expected, to be the distance between the upper and lower boundaries of the physical aquifer of concern. The intake length, as described by Bouwer and Rice (1976) is the length of the well screen, or the length of the sand pack or developed zone if significantly more permeable than the aquifer itself. The length of the well is taken as the distance between the upper aquifer surface and the bottom of the well. Casual inspection of Figure 1 in Bouwer and Rice (1976) might suggest that the water table should be taken as the upper surface for these quantities, but it must be remembered that the original paper was written to address the unconfined case only.

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