Hydraulic conductivity is a fundamental hydrogeological parameter, whose in situ measurement at a local scale is principally performed through injection tests from screened probes or using impermeable packers in screened wells. The shape factor F [L] is a proportionality constant required to estimate conductivity from observed flow rate to injection head ratios, and it depends on the geometric properties of the flow field. Existing approaches for determination of F are either based on geometric or mathematical simplifications and are limited to particular assumptions about the flow domain's external boundaries. The present work presents a general semianalytical solution to steady state axisymmetric flow problems, where external boundaries may be nearby and of arbitrary combinations of impermeable and constant head type. The inner boundary along the probe or well may consist of an arbitrary number of impermeable and constant head intervals resulting in a mixed-type boundary value problem, for which a novel and direct solution method based on trigonometric interpolation is presented. The approach is applied to generate practical nondimensional charts of F for different field and laboratory situations. Results show that F is affected by less than 5% if a minimum distance of 10 probe or well diameters is kept between the injection screen and a nearby boundary. Similarly, minimum packer lengths of two well diameters are required to avoid increasing F by more than 10%. Furthermore, F is determined for laboratory barrel experiments giving guidelines for achieving equal shape factors as in field situations without nearby boundaries. F for the theoretical case of infinitely short packers is shown to be infinitely large.
 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 m3) 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.  demonstrate a “mini slug test” method, while Butler et al.  and Dietrich et al.  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 [L3/T] to injection heads [L] and K. Thus, knowing F and observing K can be estimated from
 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.
 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.
 In what follows we take advantage of a general solution given by Zaslavsky and Kirkham  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  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  previous analytical solution for infinitesimally short packers.
 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.
2. General Solution of the Flow Problem
 An example of an axisymmetric flow domain is given in Figure 1, where r [L] and z [L] are the radial and vertical coordinates, respectively, being delimited by a < r < b [L] and 0 < z < d [L]. To represent the radius of the probe or well, a is used, which is assumed to span the entire distance d between top and bottom boundaries, while b is the radial distance to a lateral boundary. Moreover, h1 through h4 [L] delimit different boundary type segments along the well or probe. The governing Laplace equation for steady state flow and isotropic hydraulic conductivity in axisymmetric cylindrical coordinates is
where ci for i = 1, 2, …, 12 and mi for i = 1, 2, 3 are arbitrary real constants (mi being positive) and I0, K0, J0, and Y0 are Bessel functions of order zero [Dwight, 1947]. By superimposing solutions of equation (3) with different sets of constants, specific boundary conditions, including arrangements of mixed-type boundary conditions (this becomes clearer in the subsequent discussion), can be met. This will first be done for external boundaries I, II, and III shown in Figure 1 and subsequently for internal boundary segments IV–VIII along the well or probe.
2.1. External Boundaries
 While the types of internal boundary conditions along the probe are defined by the injection test setup, the types of external boundaries may be different combinations of constant head and impermeable. Both top and bottom boundaries may be considered impermeable, for example, if an injection test is performed in a (thin) stratum between two confining layers. In the case of permeability injection testing of a sealing layer under a landfill, for example, both top and bottom boundaries may be well approximated by two constant head boundaries. A mix of constant head top and impermeable bottom boundary may well represent conditions in a (shallow) unconfined aquifer or beneath surface water bodies. If the distance between injection screen and one or both of top and bottom boundaries is sufficiently small, an impact of the nearby boundary (boundaries) onto the outcome of the injection test may be expected and accounted for by a respective adjustment in the shape factor F. Similarly, modeling an impermeable lateral boundary is of interest for laboratory barrel tests, where either top, bottom, or both boundaries are constant head, while the lateral barrel wall is impermeable. Thus, the external boundary conditions in Figure 1 may be represented as follows: I, constant head (Ic) or impermeable (Ii) lateral boundary; II, constant head (IIc) or impermeable (IIi) top boundary; III, constant head (IIIc) or impermeable (IIIi) bottom boundary. Mathematically, this can be expressed as
 1. Considering constant head top and bottom boundaries, i.e., for all a ≤ r ≤ b, of all the terms in z in equation (3), these conditions can be met by sin(m1z) with as well as n and N being arbitrary positive integers, such that after superposition
Bn are a new set of constants encompassing c1, c3, and c4, and f0(m1r) is a function to be defined involving terms of equation (3) containing K0(m1r) and I0(m1r).
 2. Considering impermeable top and bottom boundaries, i.e., at z = 0 and z = d for all a ≤ r ≤ b, of the terms in z in equation (3), these conditions can be met by cos(m1z) with as well as by ln(r/m3) with arbitrary m3 and by the final (arbitrary) constant, which may be incorporated into m3. Superposition of these solutions gives
Bn and f0 are analogous to equation (7), and B0 is an additional constant; c12 = 0 and m3 = b are chosen such that the leading term on the right-hand side becomes zero for r = b, as required for a constant zero head boundary at radial distance b. The constant term ln(b/a) in the denominator is added to simplify expressions in the sequel by taking the ratio to 1 for r = a and allowing for a particular interpretation of B0.
 3. Considering impermeable top and constant head bottom boundary, i.e., and at z = d for all a ≤ r ≤ b, of the terms in z in equation (3), the first (constant head) condition may be met by sin(m1z), sinh(m2z), zln(r/m3), and z, where m1, m2, and m3 may be arbitrary. Among these solutions, however, the second (no flow) condition may only be satisfied by sin(m1z) with such that superposition leads to
 It is observed that equations (7), (8), and (9) meet boundary conditions II and III independent of the choice of the coefficients B0 and Bn and the function f0. This allows using f0 to independently satisfy the type of lateral boundary condition I in equations (7), (8), and (9). From equation (3) and its terms in z retained in equations (7), (8), and (9), it is evident that f0 has to consist of linear combinations of K0(m1r) and I0(m1r) as follows.
 1. For constant head lateral boundary, i.e., at r = b for all 0 ≤ z ≤ d, this can be achieved by imposing c3I0(m1b) + c4K0(m1b) = 0 in equation (3) and results in
where additional constants are included to make f0c = 1 at r = a for later convenience. For later convenience and knowing that dK0(r)/dr = −K1(r) and dI0(r)/dr = I1(r), we also introduce
 2. For impermeable lateral boundary, i.e., at r = b for all 0 ≤ z ≤ d, this condition may be achieved by assuring f1i = −(1/m1)df0i/dr = 0 for r = b in equations (7) and (9). Since f0i has to be a linear combination of I0 and K0 and because of the respective derivatives given above, f1i has to be a linear combination of I1 and K1, such that imposing c3I1(m1b) + c4K1(m1b) = 0 leads to
where additional constants are used to make f1i = 1 for r = a. From this, f0i may be obtained by integration as
 In the case of equation (8), where top and bottom are already impermeable, an impermeable lateral boundary only makes physical sense if the same flow injected is again extracted by the well or probe (e.g., vertical recirculation well [Zlotnik and Ledder, 1996; Peursem et al., 1999]). If this is the case, then it can be shown that B0 = 0, and hence, for r = b is again met. The approximations given with equations (10)–(13) are for large values of b/a and become exact for b → ∞, i.e., laterally unbounded flow domains (both K0(m1b) and K1(m1b) approach zero in this case, while both I0(m1b) and I1(m1b) approach infinity).
2.2. Trigonometric Interpolation Approach to the Mixed-Type Internal Boundaries
 In equations (7), (8), and (9), the values of the coefficients B0 and Bn do not affect compliance with the external boundary conditions I, II, and III such that these coefficients can be used to independently meet the internal (mixed) boundary conditions along the device (i.e., for r = a). According to Figure 1, for a double-packer test these boundaries are constant head open screen interval at bottom (IV), impermeable bottom packer (V), constant head injection screen interval (VI), impermeable top packer (VII), and constant head open screen interval at top (VIII). Mathematically, this may be written as
where and [L] are the constant hydraulic heads in the bottom and top open screen intervals, respectively. In order to determine B0 and Bn, the respective “raw” solution of for a given set of external boundary conditions from equations (7), (8), or (9) is substituted into equations (14)–(18). Considering, for example, the case of impermeable top and bottom boundaries in combination with a constant head lateral boundary, equation (8) (with f0 = f0c and f1c from equations (10) and (11)) is used to obtain the following system of equations to impose the internal boundary conditions.
 To achieve an exact solution, N must be set to infinity, for which equations (7), (8), and (9) become Fourier series. Although Sneddon  discusses solutions to similar systems of equations, analytical solutions for the mixed-type boundary value problems are generally intractable.
 However, by limiting N to finite values (i.e., truncating the trigonometric series) and discretizing the device length 0 ≤ z ≤ d into a number NB (dimensionless) of equidistant intervals delimited by zi− and zi+ with i = 1, 2,…, NB, such that z1− = 0, zNB+ = d, zi+ = z(i + 1)−, zi+ − zi− = d/NB, and zi = (zi+ + zi−)/2, equations (19)–(23) may be rewritten in a discretized form by simply substituting zi for z everywhere. With this, equations (19)–(23) constitute a linear system of NB equations in N + 1 unknown coefficients. This system may be regarded in a curve-fitting context, where it is the goal to determine the unknown coefficients of equation (8) (for finite N) to best approximate the right-hand sides of equations (19)–(23) containing punctual information about ϕ and For NB > N + 1, this may be done in a linear regression (i.e., least squares) sense, while for NB = N + 1, curve fitting becomes exact and thus transitions into the field of trigonometric interpolation. For NB < N + 1, the system is underdetermined. In the present work, NB is set equal to the number of unknown coefficients, i.e., N + 1 or N, depending on whether B0 is present or not. Note that the discretized well flux approach presented previously by Mathias and Butler  represents a special case of the more general approach presented above.
 Before solving the resulting system it is noted that and/or in equations (19)–(23) are only known if the top and bottom open screen intervals are connected to constant head top and/or bottom boundaries. For impermeable top and bottom boundaries, and are initially unknown, but instead, two additional equations can be formulated imposing zero total (i.e., integrated over z) inflows or outflows QIV and QVIII [L3/T] from the open screen intervals below and above the packers, respectively:
 The result is an extended system of NB + 2 linear equations in NB + 2 unknowns, for which many standard methods are available for the solution. For example, equations (19)–(25) may be converted from summation into matrix form [A] × [B] = [C], giving
such that matrix division immediately results in the required vector [B] = [A]−1[C] of unknown coefficients B0 and Bn from equation (8) as well as and . Note that the index “Nx” with z in equation (26) stands for the number of discretization points zi between z = 0 and the top of the boundary segment denoted by “x” in Figure 1 (e.g., zNIV is the last discretization point at the top of boundary segment IV, and zNIV+1 is the first one in boundary segment V; also NVIII = N). Appendix A gives further details about the convergence behavior for increasing N and shows that resulting flow field parameters, including F, may be hyperbolically extrapolated to the exact solutions corresponding to N → ∞. Thus, the problem is solved in a novel and direct way without the need for previous conversion of the mixed-type into a single-type boundary value problem.
 For the simpler configuration of injection from a push-in probe (i.e., in the absence of open screen intervals above and below the packers), equations (19), (23), (24), and (25) become irrelevant. The system reduces to equations (20), (21), and (22) with h1 = 0 and h4 = d, which further corresponds to a respective reduction of equation (26). In the opposite case of infinitesimally short packers such that h1 = h2 and h3 = h4, all boundary segments along the probe are of the constant head type, and the problem reduces to a single-type boundary value problem. Under this scenario (and known and ) the present trigonometric interpolation approach becomes identical to performing the discrete Fourier (in particular sine or cosine) transform on . The discrete Fourier transform is again known to become identical to the (classic) Fourier decomposition of a continuous periodic function if NB → ∞. The latter is performed in Appendix B to obtain a fully analytical solution and discussion of the infinitesimally short packer problem.
2.3. Shape Factor F
 Integrating local radial fluxes leaving the probe (i.e., for r = a), the injection flow rate Q [L3/T] may be written as
 In the simpler case of injection through a single screen from a push-in probe (or packers extending to top and bottom boundaries) and in the case of using shorter packers between impermeable top and bottom boundaries the integration limits in equation (28) may be set from zero to d, for which the following simplified expressions are found.
 Constant head top and bottom boundaries
 Impermeable top and bottom boundaries
 Impermeable top and constant head bottom boundary
where and i.e., dimensionless coefficients for unit injection head. As above, f1 in equations (29) and (31) is chosen from equations (11) and (12) to honor a constant head or impermeable lateral boundary, respectively. Interesting to note is that equation (30) only depends on B0u, which by inspecting equation (8) with r = a in a Fourier series context, is seen to represent the mean head along the device (for unit ). In other words, B0 is the head required at a fully screened probe to inject an equal flow rate Q as from a partially screened probe using an injection head . Also, for a fully screened probe, Bn for n > 0 in equation (8) become zero, and the solution correctly collapses to radial flow toward a fully penetrating well in a confined aquifer with a constant head outer boundary. Equation (30) further reflects that b/a needs to be finite in order to achieve flow (i.e., F > 0) for a finite injection head if both top and bottom boundaries are impermeable.
3. Results and Validation
 For the example configuration of equations (19)–(25) (and NB = 1280), Figure 2 depicts resulting dimensionless head and dimensionless radial flux qpr/K distributions along a fully screened well with injection between two packers. Solid lines correspond to the case of both top and bottom boundaries impermeable, while dashed lines are for constant head top and bottom boundaries; the lateral boundary is constant head in both cases. Compliance with prescribed constant head and no flow boundary conditions over their respective intervals may be easily verified in both cases along with the fact that is constant but larger than zero in the open screen intervals beyond the packers if the adjacent boundaries are impermeable. Furthermore, Figure 2b suggests (in agreement with Figure A1) that local fluxes at the transitions between constant head and impermeable boundary segments along the probe or well become locally very large or infinite. Finally, it may be correctly observed that flow along the open screen intervals is negative (i.e., inward) if top and bottom boundaries are constant head (dashed line), while the direction of flow along the open screen intervals is inward and then outward for impermeable top and bottom boundaries (solid line) to meet the condition of zero total inflow and outflow over those segments as imposed by equations (24) and (25). Figure 3 further illustrates and validates solutions by showing contour plots of for different internal and external boundary conditions. Figures 3a and 3b correspond to the solid and dashed lines of Figure 2, respectively. Compliance with external boundary conditions is verified by observing constant head lines approaching impermeable boundaries perpendicularly and by constant head lines becoming parallel to constant head boundaries in their vicinity. Figures 3c and 3d represent two cases of injection from a probe (i.e., packers extend to top and bottom boundaries) where the lateral boundaries are impermeable (e.g., as in laboratory barrel experiments). In Figure 3c both top and bottom are constant head, while in Figure 3d, only the top is constant head and the bottom impermeable.
3.1. Injection From a Probe (Field Conditions)
 As shown in Figure A1 of Appendix A, the convergence behavior of F with increasing N is of the same hyperbolic type as with other flow field parameters and may, hence, be conveniently extrapolated to exact solutions for N → ∞. For an error margin of 1% in extrapolated F values, Figure 4 summarizes the resulting shape factors from equations (29), (30), and (31) for injection from a probe with h1 = 0 and h4 = d in Figure 1 as a function of s/a (where s = h3 – h2 [L] is the length of the injection screen vertically centered between boundaries), d/s, and different top and bottom boundary types (thick solid line, constant head top and bottom; thin solid line, impermeable top and bottom; dashed line, constant head top and impermeable bottom or vice versa). Typical for field conditions, the constant head lateral boundary is located at b/a = 1000, which is large enough to not affect the outcome if at least one of top and bottom boundaries is constant head. In case both top and bottom boundaries are impermeable, as stated above, a finite value of b/a is required for F > 0. As to be expected, for a given geometric configuration, i.e., s/a and d/s, F/a is always largest if both top and bottom are constant head and smallest if both are impermeable. For increasing values of d/s, however, it is seen that the types of top and bottom boundaries have a decreasing effect on F/a, i.e., the three curves for a given value of s/a converge to a shape factor, whose value increases with s/a and which corresponds to an infinite flow domain, where the types and distances of top and bottom boundaries do not matter anymore. On the other hand, for small values of d/s approaching 1 such that top and bottom boundaries approach the injection screen extremes, F/a approaches infinity if at least one of the boundaries is constant head (flow short-circuiting; see Appendix B). If both boundaries are impermeable, then F approaches a minimum equal to which corresponds to the case of a fully penetrating well in a confined aquifer.
 While Figure 4 is limited to injection screens vertically centered between top and bottom boundaries, Figure 5 displays F/a for different values of screen length s/a and distance h1/s to a single nearby boundary. In Figure 5, d/s = 50 is set to be large enough for the distant boundary to not have a significant impact on results (compare Figure 4), and the lateral boundary is located at infinity. Thick lines correspond to the injection screen approaching a constant head boundary, while thin lines are for nearby impermeable boundaries. It is seen that for large values of h1/s, values of F/a agree with those from Figure 4 for large values of d/s; as h1 decreases, the screen approaches one of the boundaries. As to be expected, if the approached boundary is constant head, the thick lines indicate that F/a increases (up to a theoretically infinite value for h1 = 0; see Appendix B), and if the boundary is impermeable, then F/a decreases to a minimum value. In general, it may be observed from Figure 5 that the proximity of the injection screen to an impermeable or constant head boundary does not significantly affect F by more than an absolute value of ±a as long as h1/s > 5. By multiplying the abscissa by s/a of each line, F/a is obtained as a function of h1/a. The double arrows in Figure 5 are located at h1/a = 20, which appears to be the limit, below which the relative impact on F because of a nearby boundary exceeds approximately 5%. As conductivity measurements may vary over several orders of magnitude, the latter interpretation in terms of h1/a seems to be more useful in practice.
 For the absence of any nearby boundaries, Figure 6 gives a graphical validation of the present approach against previous ones defined by Hvorslev  (spheroidal approximation),
by the American Society for Testing and Materials (ASTM) [Chapuis and Chenaf, 2003] (equal surface area spherical approximation),
 For Mathias and Butler's  full semianalytical approach as well as Rehbinder's  long packer approximation, no simple expressions are available. Figure 6 illustrates that the ASTM formula is in good agreement down to s/a ≈ 1, while Hvorslev's formula begins to significantly overestimate F/a below s/a ≈ 2. The latter is a known shortcoming and is discussed by Mathias and Butler , who offer an improvement for Hvorslev's formula for s/a < 2. Equation (33) is based on a spherical equal surface approximation of the screen and, hence, becomes increasingly inaccurate as s/a grows. While equations (32) and (33) are based on geometrical simplifications, the other approximations try to honor the exact geometry of the boundaries and rely on more complex mathematical approximations. The agreement of the results of Ratnam et al.  and Rehbinder  (thin black line) with the present results is seen to be good over the entire range depicted. Excellent agreement (with, in fact, indistinguishable lines in Figure 6; thick black line) is achieved with the results of Mathias and Butler . Yet another independent validation of the present method may be obtained by comparison with results from a large numerical model for a direct-push permeameter of Liu et al. , who report hydraulic heads at two locations along a probe for injection from a short screen. Agreement is good (within 5% of injection head) when assuming impermeable top and bottom boundaries and even better (within 0.5% of injection head) when assuming constant head top and bottom.
3.2. Injection From a Probe (Laboratory Conditions)
 Calibration of injection probes under field conditions is problematic, as true values of hydraulic conductivity are generally not available. Laboratory barrel experiments are convenient because the entire barrel packed with test material above a bottom gravel pack may be used as a column in a constant or falling head conductivity experiment for obtaining independent and relatively reliable measurements of K. Figure 7 depicts F/a for different screen and barrel geometries (s/a, d/a, and b/a) as well as boundary conditions at the barrel bottom (gravel pack or not). The top boundary is always constant head (i.e., water level above test material), while the barrel wall represents an impermeable boundary. The thick black line is the same as in Figure 6 and corresponds to a field situation where external boundaries do not have a significant impact. For the remaining lines, two lines with a particular color and pattern are given to correspond to the same boundary and barrel geometries (see legend). The top line of each pair of lines corresponds to a constant head bottom boundary, while the bottom one corresponds to an impermeable bottom. Obviously, the particular barrel geometry and type of bottom boundary condition significantly affect F, and hence, they need to be accounted for explicitly in the test interpretation. However, it is interesting to note that barrels of d/b = 1 with impermeable bottoms (bottom blue lines) as well as barrels of d/b = 2 with constant head bottoms (top green lines) lead to shape factors very close to field conditions in the absence of nearby boundaries (black line). For b/a > 25 this holds with high accuracy for s/a < 10 and up to an approximate error of 10% for s/a < 20. Thus, designing a barrel test of known Klab such that its shape factor is equal to F under field conditions allows for an extremely simple injection test interpretation as This avoids dealing with probe specific shape factors in laboratory and field practice.
3.3. Injection Using Single or Double Packers
 The solid lines in Figure 8 summarize shape factors F/a from the present solutions for cases of injection from a screen interval of length s delimited by two impermeable packers of finite (and equal) length p with open screen intervals above and below the packers. While F/a is a function of b/a even if b/a >> 1 when both top and bottom boundaries are impermeable, Figure 8 considers all boundaries constant head with b/a >> 1 and d/s = 50, i.e., far away from the screen. Shorter packers (smaller p/s), as expected, increase F/a in an approximately constant manner, which indicates that changes in packer length only affect the flow fields in the vicinity of the screen ends, thus being essentially independent of s/a unless s/a is very small. It is interesting to note for practice that above p/s ≈ 1, F is only slightly affected by the finite length of the packers; especially for s/a > 5, the difference in F to longer packers is consistently smaller than approximately a. By multiplying the values of p/s in Figure 8 with s/a from the abscissa, lines of F/a over s/a for constant values of p/a are obtained. The circles and squares represent points on such lines for p/a = 1 and 4, respectively. Similar to Figure 5, it may be observed that the influence of packer length on F may be more conveniently expressed in relative terms using p/a instead of the perhaps more intuitive first choice p/s. In particular, it can be inferred from the circles and squares in Figure 8 that p/a = 1 and 4, for example, correspond to increases in F by approximately 20% and 10%, respectively, with respect to p >> a. The influence of nearby top or bottom boundaries on double-packer injection tests depends on a series of parameters (s/a, p/a, h1/a, and type of boundary approached) and is best evaluated individually for a given test configuration (i.e., s/a and/or p/a). For injection between a packer and an impermeable bottom layer (single packer test), F/a may also be found from Figure 8 by entering the chart with p/(2s) instead of p/s and 2s/a instead of s/a and then halving the respective outcome for F/a. This is a consequence of the fact that the flow field for a single-packer injection test corresponds exactly to one half of the flow field of a double-packer injection test.
Figure 8 is also valuable for the situation of injection from a probe if p is taken to represent the impermeable probe tip length under the injection screen. For this case, a lower bound for F/a is found by assuming that the inner boundaries above and below these fictitious packers are impermeable, while an upper bound for F/a is found by assuming that the inner boundaries above and below the fictitious packers are constant head, i.e., infinitely conductive. In reality, the inner boundary above the top packer is impermeable (probe casing), and below the bottom packer (i.e., below the probe tip) is the transition to a cylinder of some finite conductivity. From this, it may be concluded that errors in F because of the conceptual assumption of a long probe tip beneath the injection screen become less than a for p/s ≥ 1 (given s/a ≥ 5) or less than 10% for p/a ≥ 4.
 The scenario of double-packer injection with distant boundaries is also considered by Mathias and Butler  and by Rehbinder  for a short-packer approach, and their results are used for additional validation of the present results in Figure 8. While Rehbinder's short-packer solution (shown as dashed lines for p/s = 0.05 and 1) presents moderate agreement, the results of Mathias and Butler  are again indistinguishable from the present ones. However, as shown in Appendix B, Mathias and Butler's  result for infinitesimally short packers contains an unsatisfactorily divergent infinite series. The 10,000 terms used to approximate the infinite series in their equations (21) and (31) are insufficient to achieve convergence for very small values of p/s. However, the regression relationship given in their equation (40) accurately approximates the case when p/s = 0.005 and should be assumed for p/s = 0.005 (as opposed to p/s = 0, which is what is currently stated in their paper). Fortunately, this does not impair the validity and relevance of Mathias and Butler's  results for all practical purposes.
3.4. Anisotropic Conductivity
 In practice it is common to encounter situations where hydraulic conductivity in the horizontal direction Kr [L/T] is significantly different (e.g., several orders of magnitude larger) than its vertical counterpart Kz [L/T]. Given the respective anisotropy ratio (dimensionless), a scaling of the horizontal coordinate to while maintaining z′ = z allows for treating the resulting flow domain as isotropic with conductivity Kr (such that continuity of flows after scaling is assured [Hvorslev, 1951]). Thus, the present approach remains generally valid if hydraulic conductivity is axially anisotropic with the principal anisotropy axes being horizontal and vertical. In particular, results for F/a reported above remain directly applicable if is used instead of a such that equation (1) estimates Kr to subsequently obtain However, is generally not known before hand and needs to be assumed or measured independently.
 Subsurface hydraulic conductivity K is a fundamental hydrogeological parameter whose in situ measurement is generally performed through injection tests from screened probes or well screen intervals delimited by impermeable packers. While K is directly proportional to an observed ratio of injection flow rate Q to injection head , it is also proportional to the shape factor F, which is determined by the geometry of the injection flow field and, hence, the geometry of the injection device (internal boundary conditions) as well as the flow domain (external boundary conditions). For the purpose of evaluating F over a wide range of scenarios, the present work presents general solutions to the axisymmetric steady state flow problem for arbitrary combinations of external boundary types and distances. The internal boundary may consist of an arbitrary number of impermeable and constant head intervals. The resulting mixed-type boundary value problem is solved directly in a novel and relatively simple way using a trigonometric interpolation approach. This avoids previous conversion into a single boundary type problem as required with recent alternative approaches and allows for a simple (hyperbolic) extrapolation from approximate to exact results. Through an adequate scaling of the radial coordinate the approach becomes generally valid for flow domains with anisotropic conductivity, where K is different between the vertical and horizontal directions.
 A series of dimensionless charts is given to allow for a quick determination of F for different injection screen geometries s/a under a range of scenarios: (1) injection from a probe (or using long double packers) centered between different combinations of impermeable and constant head top and bottom boundaries at different distances, (2) injection from a probe (or using long double packers) near a single impermeable or constant head boundary, (3) injection in laboratory barrel experiments with all boundaries nearby, the lateral being impermeable, and (4) injection between double packers of different lengths and distant external boundaries. Interesting findings include F being affected by less than 5% if a minimum distance of approximately 10 times the probe diameter is maintained to an impermeable or constant head horizontal boundary (Figure 5). Similarly, using a minimum packer length of twice the well diameter does not affect F by more than 10% compared to longer packers; the latter also applies to the length of impermeable tips of push-in probes below the screen. For injection tests in sand barrels (e.g., for probe calibration) it is found that shape factors within 10% of those of field situations (in the absence of nearby boundaries) may be created by using d/b = 1 for impermeable barrel bottoms and d/b = 2 if a gravel pack at the barrel bottom is deployed to establish a constant head boundary (with water freely draining out of barrel).
 Comparison and validation of results is performed against a number of existing approximate, semianalytical, and numerical approaches available for scenarios where the influence of external boundaries is negligible. Notably, respective shape factors of the present approach are identical to those obtained by the method of Mathias and Butler . However, for extreme cases of very short packers, Mathias and Butler's  solution was found to be incomplete. We present a respective correction in the form of a fully analytical solution for the limit of infinitesimally short packers (see Appendix B).
Appendix A:: Convergence Behavior
 For the wide range of boundary configurations investigated, it was observed that the Bn coefficients always converge to zero as n increases and that particular flow field parameters (e.g., F or local heads and fluxes) converge hyperbolically toward a stable value for NB → ∞ (such hyperbolic convergence is also observed by Boast and Kirkham ). Figure A1 represents examples of the latter for arbitrarily chosen h1/a = 4, h2/a = 5, h3/a = 7, and h4/a = 8, i.e., for injection from a screen of length 2a delimited by two packers of length a, which are asymmetrically located in a stratum between two confining layers. Depicted as functions of 1/NB are the relative heads (dashed lines) and relative fluxes q/K (dot-dashed lines) for z/a = 4.5, 6, and 8 (indicated in the indices), i.e., for the center of the bottom packer, the center of injection interval, and the top extreme of top packer, respectively. NB is increased from an initial value of 10 through consecutive multiplication by 2 until 1280 (circles). It is observed that all and q/K approach a relatively straight line toward the left when plotted over 1/NB. This allows for simple linear extrapolation of two consecutive data points onto 1/NB = 0, i.e., the exact solutions for NB → ∞, and a sufficiently large value of NB is reached when two consecutive extrapolated values are within a prescribed margin. An exception to this is q8/K, which provides some evidence that flow is singular (infinite flux) at the transitions between impermeable and constant head boundaries along the well or probe. As infinite fluxes are physically impossible, the governing equation (equation (3)) of the present approach appears to be invalid near the extremes of well casings and packers. However, F converges to a finite value, indicating that injection flow does remain finite (integrable). Successful validation of F against independent (e.g., numerical) methods, which do not produce these flow singularities, demonstrates that equation (3) is, indeed, physically valid everywhere else (i.e., at all nonsingular locations) in the flow domain. Although not a necessary requirement, maximum convergence with NB is achieved if the screen and packer limits coincide with discretization interval limits and if average values over discretization intervals are used for any location inside a respective interval. An attempt to lower required NB by enlarging interval spacing distant from the screen and packers (as done by Mathias and Butler ) leads into the context of trigonometric interpolation for unevenly spaced data and was not further pursued as computational time was found not to act as a limiting factor.
Appendix B:: Analytical Solution for Infinitesimally Short Packers
 If h1 = h2 and h3 = h4 in Figure 1, the impermeable packers become infinitesimally short, and the internal boundary conditions are all of the constant head type, which is a significant simplification with respect to the general mixed boundary problem. A fully analytical solution is found for this case and provides some interesting theoretical insight on whether total injection flow becomes infinite or not, a topic which Rehbinder  and Mathias and Butler  are in disagreement about. For the sake of concurrence with these previous studies, consider constant head top, bottom, and lateral boundaries, such that equations (7) and (10) apply. Equation (10) is identical to 1 for r = a (i.e., along the well surface), and if N → ∞, the coefficients Bn in equation (7) become the coefficients of a sine-Fourier series of the known head distribution along the well ( for 0 ≤ z ≤ h2 and h3 ≤ z ≤ d; for h2 ≤ z ≤ h3), which may be found as
such that the analytical solution of the problem is complete. By using equation (28) a general expression of F is obtained as
 From equation (11) it may be seen that f1c(m1a) converges to 1 as n increases and the squared term in brackets is always positive. This and the fact that is sufficient to prove that equation (B2) does not converge; that is, F and, hence, Q are infinitely large in the case of infinitesimally short packers (the only exception to this is the case of h2 = h3, i.e., when the injection screen interval itself becomes infinitesimally short). Clearly, infinitesimally short packers and infinite injection flows are beyond physical reality; however, the result bears some significance in that practitioners are warned from arbitrarily minimizing packer sizes as, depending on particular site conditions, the governing equation (3) based on Darcy flow may not hold over significant portions of the flow domain near the short packers. These conclusions are also applicable to injection screens immediately next to a top or bottom constant head boundary where the geometric distance (flow paths) between source (screen) and sink (boundary) is even shorter. Note, however, that flow singularities discussed in Appendix A do not share the property of causing F and Q to be infinite.
 In contrast, if both top and bottom boundaries are impermeable, equation (30) dictates that F has to be finite at steady state even for infinitesimally short packers since B0u was found to be the (naturally finite) mean head along the well. Also, the same flow entering the open screen intervals needs to leave them again and reenter the aquifer flow field until meeting the lateral constant head boundary. Using a similar procedure as in equations (B1) and (B2) for constant head top and bottom boundaries, it may be shown that the head difference between the injection screen and the open screen intervals above and below the infinitesimally short packers becomes infinitesimally small, such that the whole well behaves as if injection was uniformly applied along all of it (no packers present). For a given set of parameters a, b, and d the shape factor F is then a maximum, independent of s.
 The first author would like to thank the Austrian Science Fund (FWF) for an Erwin-Schrödinger Fellowship (project number J2677-N14) to perform this work. Further funding was provided by the Environmental Remediation Science Program (ERSP), the U.S. Department of Energy (grant number DE-FG02-08ER64585), and the U.S. Department of Defense (project number ER0831) under the Environmental Security Technology Compliance Program (ESTCP).