## 1. Introduction

[2] Hydraulic conductivity *K* [*L*/*T*] with its horizontal and vertical variability is a parameter of paramount importance for the modeling and management of a large number of natural and engineered processes, including infiltration, irrigation, drainage, groundwater extraction and injection, soil compaction, landfill impermeabilization, and contaminant transport [*Sedighi et al.*, 2006; *Sudicky*, 1986; *Hvorslev*, 1951]. Under saturated conditions, i.e., below the water table of an aquifer, classically, pump or slug tests with their well-known individual advantages and drawbacks are performed for investigations of *K* at different scales [*Weight and Sonderegger*, 2001]. Accordingly, these tests may be performed on an entire well or on various portions of a well screen by use of single- or double-packer systems [*Butler et al.*, 2009; *Price and Williams*, 1993]. Different types of small-diameter (i.e., lower centimeter range) drive point (also called push-in or direct-push) probes have also been proposed for quick and flexible investigation of *K* in unconsolidated media at highly local (i.e., <1 m^{3}) scales. Because of the small spatial scale the associated flow systems reach steady state rapidly and do not require a permanent installation of an injection well or piezometers around it. Using such push-in probes with short injection screen intervals near the probe tips, *Hinsby et al.* [1992] demonstrate a “mini slug test” method, while *Butler et al.* [2007] and *Dietrich et al.* [2008] apply a “direct-push permeameter” and a “direct-push injection logger,” respectively. The difference between the latter two direct-push methods is that the injection logger uses the variability in recorded ratios of injection pressures and flow rates as a function of depth to estimate variability in local *K* without, however, assigning absolute *K* values. The “push-in permeameter” uses two additional head observations along the probe to also quantify absolute values of *K*. Whenever the goal is to estimate such absolute values of *K*, a so-called shape factor (often denoted by *F* [*L*]) is required, which serves as the proportionality constant between ratios of observed injection flow rates *Q* [*L*^{3}/*T*] to injection heads [*L*] and *K*. Thus, knowing *F* and observing *K* can be estimated from

[3] More precisely, hereby is the excess hydraulic head at the screen because of pumping with respect to ambient (no pumping) conditions, and as used in the sequel, it is assumed to be constant both in time (steady state) as well as over the injection screen surface.

[4] Many variants of in situ measurement methods for *K* have been developed, and a correspondingly large number of theoretical models have been invoked for test interpretation, i.e., determination of *F*. However, a common feature of virtually all methods is that an axially (or rotationally) symmetric potential flow field is generated through injection or extraction of water from some cylindrical well or probe surface, which may or may not include the tip of a well or probe. An important and complicated issue is the presence of mixed-type boundary conditions, which arises because of the simultaneous presence of both no-flow and constant head segments along the inner boundary associated with the well or probe surface.

[5] Early models which persist until the present day apply geometric approximations of the cylindrical (constant head) injection surface by spheroids [*Mathias and Butler*, 2006; *Hvorslev*, 1951]. Other approximations use distributed point sources along a line [*Zlotnik and Ledder*, 1996] or over a cylindrical surface [*Peursem et al.*, 1999]. More recently, computationally intensive finite difference or finite element methods have been used to better reproduce geometric and hydraulic boundary conditions at the injection screen [*Liu et al.*, 2008; *Ratnam et al.*, 2001]. Another approach that has proved popular involves conversion of the mixed-type boundary problem along the well or probe into a single-type boundary problem by either assuming approximate flux distributions along constant head boundary segments [*Chang and Chen*, 2002, 2003; *Rehbinder*, 2005; *Perina and Lee*, 2006; *Mathias and Butler*, 2007] or assuming approximate head distributions along no flow boundary segments [*Rehbinder*, 1996]. While the analytical approaches of *Rehbinder* [1996, 2005] use predefined continuous functions for these approximations, *Chang and Chen* [2002, 2003], *Perina and Lee* [2006], and *Mathias and Butler* [2007] use adjustable functions by making them piecewise constant in a semianalytical approach.

[6] In what follows we take advantage of a general solution given by *Zaslavsky and Kirkham* [1964] to derive different forms of steady state solutions to the axisymmetric flow problem for all possible combinations of constant head and impermeable top, bottom, and lateral boundaries at arbitrary distances. We further present a novel, direct, and simple semianalytical method related to trigonometric interpolation to directly deal with the mixed boundary value problem along the injection well or probe (i.e., without requiring conversion into a single-type boundary value problem as done in previous work) and use the observed convergence behavior to extrapolate toward exact solutions. Results are applied to investigate effects of nearby boundaries on injection test results and to provide practical charts of shape factors *F* for different scenarios. Validation is achieved by comparison with equivalent results previously obtained by *Mathias and Butler* [2007] for sufficiently distant boundaries such that they can be ignored. A clarification is also made concerning the divergent series contained within *Mathias and Butler*'s [2007] previous analytical solution for infinitesimally short packers.

[7] Although injection tests from push-in probes or packered-off screen intervals may be limited to local scales not containing any external boundaries, situations may arise where proximity to a confining layer, a surface water body, or the water table has to be accounted for [*Lui et al.*, 2008]. In particular, injection near constant head boundaries may be strongly distorted because of flow short-circuiting between the screen and the boundary. Similarly, laboratory testing in sand barrels is a routine procedure for injection test calibration, and the effects of nearby impermeable barrel walls deserve particular investigation.