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 [L] is the hydraulic head distribution in the flow domain. A general solution of equation (2) is given by Zaslavsky and Kirkham  as
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.
 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
where (dimensionless). By substituting equation (27) into equation (1) a general expression of F is obtained.
 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.