## 1. Introduction

[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.

[3] For slug tests *Hvorslev* [1951] suggests that

where *r*_{c} [L] is the radius of the test borehole where the head change takes place, *y*/*y*_{0} [LL^{−1}] is the proportional head change at time, *t* [T] time interval from the head perturbation. For discussions on the validity of equation (2) see *Hyder et al.* [1994], *Demir and Narasimhan* [1994], *Brown et al.* [1995], and *Beckie and Harvey* [2002].

[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.

[5] Hvorslev's formula was derived by assuming that the cylindrical well screen could be exchanged for a fixed spheroidal equipotential (see Figure 1a). In fact, equation (3) is incorrect [*Mathias and Butler*, 2006]. The exact formula for the fixed spheroidal equipotential is [*Moon and Spencer*, 1961, p. 242]

[6] Interestingly, for large *z*_{0}/(*k*_{D}^{1/2}*r*_{0}), equations (3) and (4) both reduce to [*Mathias and Butler*, 2006]

[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.

[12] In this paper a packer test shape factor, for the model geometry depicted in Figure 1c, is derived using the discrete nonuniform well-screen flux distribution previously used by *Chang and Chen* [2002, 2003] and *Perina and Lee* [2005, 2006] for fully cased wells. The results are compared with the approximate solutions of *Rehbinder* [1996a] and *Hvorslev* [1951, Figure 12, Case 8].