[8] 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, *h*_{1} through *h*_{4} [*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* [1964] as

where *c*_{i} for *i* = 1, 2, …, 12 and *m*_{i} for *i* = 1, 2, 3 are arbitrary real constants (*m*_{i} being positive) and *I*_{0}, *K*_{0}, *J*_{0}, and *Y*_{0} 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

[9] 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 (I_{c}) or impermeable (I_{i}) lateral boundary; II, constant head (II_{c}) or impermeable (II_{i}) top boundary; III, constant head (III_{c}) or impermeable (III_{i}) bottom boundary. Mathematically, this can be expressed as

[11] 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(*m*_{1}*z*) with as well as *n* and *N* being arbitrary positive integers, such that after superposition

[12] *B*_{n} are a new set of constants encompassing *c*_{1}, *c*_{3}, and *c*_{4}, and *f*_{0}(*m*_{1}*r*) is a function to be defined involving terms of equation (3) containing *K*_{0}(*m*_{1}*r*) and *I*_{0}(*m*_{1}*r*).

[13] 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(*m*_{1}*z*) with as well as by ln(*r*/*m*_{3}) with arbitrary *m*_{3} and by the final (arbitrary) constant, which may be incorporated into *m*_{3}. Superposition of these solutions gives

[14] *B*_{n} and *f*_{0} are analogous to equation (7), and *B*_{0} is an additional constant; *c*_{12} = 0 and *m*_{3} = *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 *B*_{0}.

[15] 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(*m*_{1}*z*), sinh(*m*_{2}*z*), *z*ln(*r*/*m*_{3}), and *z*, where *m*_{1}, *m*_{2}, and *m*_{3} may be arbitrary. Among these solutions, however, the second (no flow) condition may only be satisfied by sin(*m*_{1}*z*) with such that superposition leads to

[16] It is observed that equations (7), (8), and (9) meet boundary conditions II and III independent of the choice of the coefficients *B*_{0} and *B*_{n} and the function *f*_{0}. This allows using *f*_{0} 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 *f*_{0} has to consist of linear combinations of *K*_{0}(*m*_{1}*r*) and *I*_{0}(*m*_{1}*r*) as follows.

[17] 1. For constant head lateral boundary, i.e., at *r* = *b* for all 0 ≤ *z* ≤ *d*, this can be achieved by imposing *c*_{3}*I*_{0}(*m*_{1}*b*) + *c*_{4}K_{0}(*m*_{1}*b*) = 0 in equation (3) and results in

where additional constants are included to make *f*_{0c} = 1 at *r* = *a* for later convenience. For later convenience and knowing that d*K*_{0}(*r*)/d*r* = −*K*_{1}(*r*) and d*I*_{0}(*r*)/d*r* = *I*_{1}(*r*), we also introduce

[18] 2. For impermeable lateral boundary, i.e., at *r* = *b* for all 0 ≤ *z* ≤ *d*, this condition may be achieved by assuring *f*_{1i} = −(1/*m*_{1})d*f*_{0i}/d*r* = 0 for *r* = *b* in equations (7) and (9). Since *f*_{0i} has to be a linear combination of *I*_{0} and *K*_{0} and because of the respective derivatives given above, *f*_{1i} has to be a linear combination of *I*_{1} and *K*_{1}, such that imposing *c*_{3}*I*_{1}(*m*_{1}*b*) + *c*_{4}K_{1}(*m*_{1}*b*) = 0 leads to

where additional constants are used to make *f*_{1i} = 1 for *r* = *a*. From this, *f*_{0i} may be obtained by integration as

[19] 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 *B*_{0} = 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 *K*_{0}(*m*_{1}*b*) and *K*_{1}(*m*_{1}*b*) approach zero in this case, while both *I*_{0}(*m*_{1}*b*) and *I*_{1}(*m*_{1}*b*) approach infinity).

#### 2.2. Trigonometric Interpolation Approach to the Mixed-Type Internal Boundaries

[20] In equations (7), (8), and (9), the values of the coefficients *B*_{0} and *B*_{n} 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 *B*_{0} and *B*_{n}, 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 *f*_{0} = *f*_{0c} and *f*_{1c} from equations (10) and (11)) is used to obtain the following system of equations to impose the internal boundary conditions.

[21] To achieve an exact solution, *N* must be set to infinity, for which equations (7), (8), and (9) become Fourier series. Although *Sneddon* [1966] discusses solutions to similar systems of equations, analytical solutions for the mixed-type boundary value problems are generally intractable.

[22] However, by limiting *N* to finite values (i.e., truncating the trigonometric series) and discretizing the device length 0 ≤ *z* ≤ *d* into a number *N*_{B} (dimensionless) of equidistant intervals delimited by *z*_{i}_{−} and *z*_{i+} with *i* = 1, 2,…, *N*_{B}, such that *z*_{1−} = 0, *z*_{NB+} = *d*, *z*_{i+} = *z*_{(i + 1)−}, *z*_{i+} − *z*_{i}_{−} = *d*/*N*_{B}, and *z*_{i} = (*z*_{i+} + *z*_{i}_{−})/2, equations (19)–(23) may be rewritten in a discretized form by simply substituting *z*_{i} for *z* everywhere. With this, equations (19)–(23) constitute a linear system of *N*_{B} 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 *N*_{B} > *N* + 1, this may be done in a linear regression (i.e., least squares) sense, while for *N*_{B} = *N* + 1, curve fitting becomes exact and thus transitions into the field of trigonometric interpolation. For *N*_{B} < *N* + 1, the system is underdetermined. In the present work, *N*_{B} is set equal to the number of unknown coefficients, i.e., *N* + 1 or *N*, depending on whether *B*_{0} is present or not. Note that the discretized well flux approach presented previously by *Mathias and Butler* [2007] represents a special case of the more general approach presented above.

[24] The result is an extended system of *N*_{B} + 2 linear equations in *N*_{B} + 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 *B*_{0} and *B*_{n} from equation (8) as well as and . Note that the index “*N*_{x}” with *z* in equation (26) stands for the number of discretization points *z*_{i} between *z* = 0 and the top of the boundary segment denoted by “x” in Figure 1 (e.g., *z*_{N}_{IV} is the last discretization point at the top of boundary segment IV, and *z*_{N}_{IV+1} is the first one in boundary segment V; also *N*_{VIII} = *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.

[25] 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 *h*_{1} = 0 and *h*_{4} = *d*, which further corresponds to a respective reduction of equation (26). In the opposite case of infinitesimally short packers such that *h*_{1} = *h*_{2} and *h*_{3} = *h*_{4}, 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 *N*_{B} ∞. 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*

[26] Integrating local radial fluxes leaving the probe (i.e., for *r* = *a*), the injection flow rate *Q* [*L*^{3}/*T*] may be written as

where (dimensionless). By substituting equation (27) into equation (1) a general expression of *F* is obtained.

[27] 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.

[28] Constant head top and bottom boundaries

[29] Impermeable top and bottom boundaries

[30] Impermeable top and constant head bottom boundary

where and i.e., dimensionless coefficients for unit injection head. As above, *f*_{1} 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 *B*_{0u}, 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, *B*_{0} 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, *B*_{n} 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.