### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Mathematical Problem
- 3. Effect of Packer Length on the Shape Factor
- 4. Summary and Conclusions
- Acknowledgments
- References

[1] Local estimates of hydraulic conductivity are often obtained from constant-head double-packer tests. The tests yield ratios of flow against head difference. These are converted to values of hydraulic conductivity by dividing by a shape factor. Previously used shape factors have generally assumed that the packers are infinitely large. Such shape factors lead to overestimates of hydraulic conductivity as they ignore the effect of water flow around the packers. A semianalytical solution is derived for a constant-head double-packer permeameter test. The solution is compared with existing, approximate techniques, including the spheroidal equipotential solution, the narrow packer approximation of Rehbinder (1996), and the long-packer approximation of Rehbinder (2005). Generally, it is shown that ignoring the length of the packer and assuming a fully cased well lead to an overestimate of hydraulic conductivity. The significance of this reduces with increasing well-screen aspect ratio. Evaluation of the solution is nontrivial. For this reason, a polynomial relationship between shape factor, well-screen aspect ratio, and packer length (obtained by curve fitting with the semianalytical solution) is provided.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Mathematical Problem
- 3. Effect of Packer Length on the Shape Factor
- 4. Summary and Conclusions
- Acknowledgments
- References

[2] Double-packer permeameters are routinely used to determine in situ vertical distributions of hydraulic conductivity in heterogenous systems [*Price et al.*, 1982; *Price and Williams*, 1993; *Nativ et al.*, 2003; *Muldoon and Bradbury*, 2005; *Williams et al.*, 2006]. Generally there are two approaches, constant-head and time-dependent “slug” testing. According to *Hvorslev* [1951], for constant-head tests, the horizontal hydraulic conductivity, *K* [LT^{−1}] is found from

where *F* [L] is a shape factor, *Q* [L^{3}T^{−1}] is the abstraction rate and ϕ_{0} [L] is the steady state hydraulic head difference between the well screen and the unperturbed aquifer.

[4] This paper is solely concerned with the derivation of the shape factor, *F*. The most commonly used shape factor for constant-head packer testing is [*Hvorslev*, 1951, Figure 12, Case 8]

where *r*_{0} [L] is the radius of the well, *k*_{D} [LT^{−1}L^{−1}T] is the ratio of vertical and horizontal hydraulic conductivity and *z*_{0} [L] is the half-length of the well screen.

[7] Unhappy with sacrificing the problem geometry (Figure 1b) for a spheroidal equipotential, *Bouwer and Rice* [1976] proposed that the shape factor, *F* could be found from the Thiem equation with a radius of influence, *r*_{e} over which the perturbation at the well-screen is dissipated. Interestingly, for large well-screen aspect ratios, according to Hvorslev's formula, *r*_{e} = 2*z*_{0}/*k*_{D}^{1/2} (see equation (5)). *Bouwer and Rice* [1976] obtained values of *r*_{e} using a resistance network analog for the more realistic scheme depicted in Figure 1b. Improved polynomial expressions for *r*_{e} can also be found in work by *Butler* [1998].

[8] *Dagan* [1978] derived a semianalytical solution for the shape factor described in Figure 1b, by assuming an infinitesimal well radius allowing the fixed head boundary in the well screen to be represented as a discretized source distribution along the well axis. Since then, solutions have been obtained for a finite well radius using a uniform well-screen flux distribution [*Dougherty and Babu*, 1984; *Hyder et al.*, 1994; *Moench*, 1997], a continuous nonuniform well-screen flux distribution [*Rehbinder*, 1996b, 2005], finite differences [*Rudd and Kabala*, 1997], dual-integral equations [*Cassiani and Kabala*, 1998; *Cassiani et al.*, 1999] and a discrete nonuniform well-screen flux distribution [*Novakowski*, 1993; *Chang and Chen*, 2002, 2003; *Perina and Lee*, 2005, 2006].

[9] Note that the scheme described in Figure 1b is particularly problematic because of the mixed-type boundary condition [*Sneddon*, 1966] (a Dirichlet along the well screen and a Neumann along the well casing). *Barker* [1981] presented a solution for flow in fractured rock, which avoided the issues associated with mixed-type boundary conditions by assuming that all the water entering the well screen travels through a horizontal, planar fracture, which draws water from an adjacent three-dimensional rock matrix. However, such an approach is not appropriate for rock formations where the representative volume is less than the dimensions of the well screen.

[10] An important consideration with all of the above approaches, is that they assume that the borehole is cased from the well screen to the ground surface (Figure 1b). With packer tests in open/fully screened boreholes this is not the case, as the borehole above and below a double-packer permeameter behaves as a Dirichlet condition (Figure 1c), which implies that there is flow from the well above and below the packered off interval.

[11] The effects of packer length have been explored using finite difference models by *Bliss and Rushton* [1984] and *Butler et al.* [1994] and finite element models by *Braester and Thunvik* [1984]. *Hayashi et al.* [1987] extended the semianalytical approach of *Dagan* [1978] by superimposing discretized source distributions over the vertical extent of the aquifer, above and below the packers. All of these techniques are complicated to implement and evaluate. Therefore only a limited range of simulations have been explored. *Rehbinder* [1996a] obtained a simpler solution by assuming that the no-flow condition associated with the packers could be replaced by a head distribution function that varies linearly with depth. However, while Rehbinder's solution is attractive, it remains to be tested against a more rigorous approach.

### 2. Mathematical Problem

- Top of page
- Abstract
- 1. Introduction
- 2. Mathematical Problem
- 3. Effect of Packer Length on the Shape Factor
- 4. Summary and Conclusions
- Acknowledgments
- References

[13] The problem to be solved is that illustrated in Figure 1c. Assuming that an aquifer is infinite, homogenous and its principal axes of anisotropy are horizontal and vertical, the governing equations for hydraulic head, ϕ [L] is Laplace's equation in cylindrical coordinates [e.g., *Dagan*, 1978]:

subjected to the boundary conditions (adapted from *Rehbinder* [1996a]):

where *r* [L] is radial distance from the center of the borehole, *z* [L] is elevation from the center of the well screen, *k*_{D} [LT^{−1}L^{−1}T] is the ratio of vertical and horizontal hydraulic conductivity, ϕ_{0} [L] is the constant applied head difference at the well screen, *r*_{0} [L] is the radius of the well, *z*_{0} [L] and *z*_{p} [L] are the elevations of the bottom and top of the packer. The length of the well screen is 2*z*_{0} and the packer lengths are *z*_{p} − *z*_{0}.

[14] *Chang and Chen* [2003] found that the infinite condition in equation (8) is adequately approximated by

when *b* ≥ 100 *z*_{0}. The advantage of doing this is that it facilitates the use of the finite Fourier cosine transform as opposed to the complete transform. The inverse finite Fourier cosine transform is an infinite series [e.g., *Sneddon*, 1951] and therefore much easier to evaluate. (Further discussion on the error associated with this approximation is given in section 3.)

[17] The boundless condition (9) then gives [*Dougherty and Babu*, 1984]

where *K*_{0} is a zero-order Bessel function of the second kind and *G*_{n} is defined by the boundary conditions over the entire length of the well, *r* = *r*_{0} (i.e., equations (10) and (11)).

#### 2.1. Rehbinder's Narrow Packer Approximation

[18] The issue is that while the boundary conditions in (10) are Dirichlet, the condition in (11) is Neumann. *Rehbinder* [1996a] solves this by assuming that ϕ varies linearly along the packer such that equation (11) is swapped with the Dirichlet condition:

*Rehbinder* [1996a] also assumes *b* ∞. For the purpose of comparison with our new solution (described in section 2.3) we extend his solution to finite values of *b*.

[20] Multiplying both sides by cos(*a*_{m}) and integrating with respect to *z* from 0 to *b* then leads to

[21] By inspection, it can be seen that all the terms on the left-hand side are zero apart from when *n* = *m* [*Moench*, 1997]. It follows that

[22] Given that the total flow, *Q* from the well screen can be found from

[23] The shape factor, *F* can then be found from (recall equation (1))

[24] When the packer length is infinitesimal (*z*_{p} *z*_{0}) the hydraulic gradient is singular at *r* = *r*_{0} and *z* = *z*_{0}. *Rehbinder* [1996a] speculated that the magnitude of this singularity was sufficient to make the flow, *Q* infinite and avoided deriving the solution for this situation. In fact, *Q* remains finite and the solution for the infinitesimal packer length is found from

#### 2.2. Rehbinder's Long-Packer Approximation

[25] *Rehbinder* [1996b, 2005] considers the problem when the packer is long (*z*_{p} ∞, i.e., a fully cased well). The mixed-type boundary condition is dealt with by swapping the Dirichlet condition in (10) for a Neumann condition. *Rehbinder* [2005] does this by assuming the flux distribution in the well screen obeys the heuristic function:

where *ω*(*z*_{0}) is a normalization factor yet to be defined. *Rehbinder* [1996b] used *N* = 0 and *Rehbinder* [2005] used *N* = 2.

[26] Again, for the purpose of comparison with our new solution (described in section 2.3) we extend the solution of *Rehbinder* [2005] to finite values of *b*.

[27] The normalization factor can be obtained as follows. Applying the conditions in (11) and (23) to equation (15) in a similar way as in section 2.1, and assuming *N* = 2, yields

where *J*_{0} and *J*_{1} are zero- and first-order Bessel functions of the first kind.

#### 2.3. Using a Discrete Nonuniform Well-Screen Flux Distribution

[30] In this section we develop a more accurate solution for both narrow and long packers. Consider the two auxiliary functions, *q*_{0}(*z*) and *q*_{p}(*z*) such that when

the condition in (10) is still satisfied.

[32] For the reinstatement of equation (10), the well screen and the borehole above the packer are discretized into *M* segments defined by the bounds *z*_{i−} < *z* < *z*_{i+}, *i* = 1, 2, . . *M*; a similar discretization practice was also employed by *Lee and Damiata* [1995], *Chang and Chen* [2002, 2003] and *Perina and Lee* [2005, 2006]. Corresponding to the *M* segments, the unknown functions *q*_{0}(*z*) and *q*_{p}(*z*) are replaced by *q*_{i}, *i* = 1, 2, ..*M*. For segment *i*, *q*_{i} is constant over *z*_{i−} < *z* < *z*_{i+} such that

and the solution in equation (15) becomes

where

[33] The next stage is to find *q*_{i}. Some of the segments are below the packer, *M*_{1} and some are above, *M*_{2} and *M* = *M*_{1} + *M*_{2}. Reapplying the boundary condition in equation (10) leads to

where

Note that *z*_{1−} = 0, = *z*_{0}, = *z*_{p}, *z*_{M+} = *b* and *z*_{i} = (*z*_{i+} + *z*_{i−})/2.

[35] Equation (35) can be easily evaluated using the MLDIVIDE command in MATLAB. For improved computation times Δ*z*_{i} = *z*_{i+} − *z*_{i−} should be nonuniform with the finest spacing just above and just below the packer.

[36] The abstraction rate, *Q* can then be found from (recall equation (20))

[37] Application of equation (1) then yields the shape factor

### 3. Effect of Packer Length on the Shape Factor

- Top of page
- Abstract
- 1. Introduction
- 2. Mathematical Problem
- 3. Effect of Packer Length on the Shape Factor
- 4. Summary and Conclusions
- Acknowledgments
- References

[38] Figure 2 shows plots of normalized hydraulic conductivity (assuming equation (1)) for different well-screen aspect ratios and packer lengths as calculated using the semianalytical solution in equation (37). Δ*z*_{i} was set to logarithmically decrease with elevation for *z* ≤ *z*_{0} and logarithmically increase with elevation for *z* ≥ *z*_{p}. Consequently, good accuracy (compare the semianalytical solution when *z*_{p} = *z*_{0} with the exact solution for the infinitesimal packer length) was obtained with just 10 segments, varying in size by three orders of magnitude, for 0 ≤ *z* ≤ *z*_{0}, and 10 segments, varying in size by four orders of magnitude, for *z*_{p} ≤ *z* ≤ *b*. *b* was routinely set to 100 × *z*_{0} so as to adequately approximate the condition *b* ∞ [see *Chang and Chen*, 2003]. Ten thousand terms were used to approximate the infinite series.

[39] The condition *b* > 100 *z*_{0} originated from studies of fully cased wells [*Chang and Chen*, 2003]. This paper is concerned with packers of finite length. Therefore water supply is mainly derived from the borehole boundary above and below the packers, as opposed to the unperturbed aquifer. Consequently, we found that even when *b* = 5*z*_{p} the error is less than 1% (i.e., − < 0.01) for (*z*_{p} − *z*_{0})/*z*_{0} < 1.0 and *z*_{0}/(*k*_{D}^{1/2}*r*_{0}) > 1.0.

[40] It can be seen that normalized hydraulic conductivity increases with increasing packer length. This suggests that assuming a fully cased well will lead to an overestimate in hydraulic conductivity. However, the curves (generated by the semianalytical solution) are all bounded by those for the infinitesimal packer (*z*_{p} *z*_{0}) and the fully cased well (*z*_{p} *b*). Furthermore, the infinitesimal packer and fully cased well curves converge, suggesting that packer length becomes less important, with increasing well-screen aspect ratio.

[41] For comparison, Figure 2 also shows Rehbinder's narrow packer *Rehbinder* [1996a] approximation. As suspected by Rehbinder, his narrow packer approximation becomes worse with increasing packer length and does not ultimately converge onto the fully cased well curve. The reason is that Rehbinder's assumption, that the head linearly varies across the packer, is progressively unrealistic as the packer length becomes increasingly large (see Figure 3).

[42] Figure 4 compares the spheroidal equipotential with the semianalytical solution when *z*_{p} *b* (analogous to *Perina and Lee* [2005]). The spheroidal equipotential can be seen to underestimate *K* for *z*_{0}/(*k*_{D}^{1/2}*r*_{0}) < 1 and overestimate it for *z*_{0}/(*k*_{D}^{1/2}*r*_{0}) > 1. Also shown is Rehbinder's long-packer approximation [*Rehbinder*, 2005], which overestimates *K* for small aspect ratios (*z*_{0}/(*k*_{D}^{1/2}*r*_{0})) but becomes increasingly more accurate with large well-screen aspect ratios.

[43] In practice, many workers are unlikely to want to evaluate the semianalytical solution as given by equations (31)–(37). For this reason it is useful to consider the following polynomial expression obtained by curve fitting:

[44] A comparison of the polynomial approximation with the semianalytical solution is shown in Figure 5. Note that shape factors for *z*_{0} / (*k*_{D}^{1/2}*r*_{0}) < 1 have not been considered in the curve-fitting process as they are of little practical interest and require a more complicated relationship.

### 4. Summary and Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Mathematical Problem
- 3. Effect of Packer Length on the Shape Factor
- 4. Summary and Conclusions
- Acknowledgments
- References

[45] A semianalytical solution for a constant-head double-packer test has been developed and shape factors have been derived for the double-packer permeameter. Three levels of approximation have taken place. First, a number of simplifying assumptions were made to obtain the mixed-type boundary value problem described in equations (6)–(12). Secondly, the mixed-type boundary along the wellbore has been transformed into a single-type boundary by approximating the fixed head boundaries along the well screen as a discrete nonuniform well-screen flux distribution (see equations (27) and (28)). This lead to the shape factor formula given in equation (37). Finally, because evaluation of equation (37) is nontrivial, its functional response has been further approximated by a polynomial relationship obtained by curve fitting (equation (38)).

[46] The shape factors have been compared with existing approximate techniques including the spheroidal equipotential solution [*Hvorslev*, 1951; *Mathias and Butler*, 2006], the narrow packer approximation of *Rehbinder* [1996a] and the long-packer approximation of *Rehbinder* [2005]. Generally it has been shown that ignoring the length of the packer and assuming a fully cased well leads to an overestimate of hydraulic conductivity. The significance of this reduces with increasing well-screen aspect ratio.