A1. Assumptions and Definitions
 Consider an aquifer (along the horizontal direction x, infinitely wide in the orthogonal horizontal direction y) composed of two contiguous compartments indexed by c and u (confined and unconfined) (Figure 2). Assume that the slope and thickness are sufficiently small in both compartments so that, in both sections, the hydraulic potential satisfies the 1-D diffusion equation. In the compartment c, with length Lc, the hydraulic potential ϕ = ϕc follows the standard diffusion equation with a homogeneous transmissivity T, a storage coefficient Sc, and, therefore, a diffusivity Dc = T/Sc. In the neighboring compartment u, with length Lu, the transmissivity T is assumed to be the same, but the storage coefficient Su is different (much larger than Sc) as is the diffusivity Du = T/Su. In the latter compartment, the hydraulic potential ϕ = ϕc also follows the diffusion equation with diffusivity .
 Mathematically, the potential ϕ obeys the equation:
with D = Dc in the confined compartment and D = Du in the unconfined one whereas, at the interface x = Lc, the potential ϕ is continuous, as well as the hydraulic flow , so that:
 The other boundary conditions are that
 Since it is assumed that there is no producing source in the system, the boundary conditions impose that the steady-state potential equals h0 for any x. Now let us impose an initial condition, for time t = 0, corresponding to a large-scale disturbance of the system: at t = 0, the potential in both compartments is assumed to vary linearly with x: .
 When t increases, the potential ϕ(x,t) relaxes toward the steady state . For large times, by analogy with the solution of the confined aquifer presented in section 2.2, the relaxation behavior is expected to be characterized by an exponential decay. The problem is to define the coefficient attached to the time variable.
 Nondimensional variables are defined by:
 The difference between the two compartments is characterized by a fraction f: . Since is very small.
 In the two compartments u and c, the diffusion equations become:
 The boundary conditions are that
A3. The Inverse Laplace Transform
 The exact LT−1 of equations (A10) and (A11) is based on the inversion theorem [Carslaw and Jaeger, 1959] which requires the evaluation of the integral of ψ along a contour in the complex p-plane:
 The contour c of the integral ∫c is parallel to the imaginary axis p, running from to being a real value such that all singularities of the function ψ (p) of the complex variable p are located in the left part of the complex p-plane. It can be verified by limited development that the function ψ (p) is single valued and has no branch point associated with the dependence in qu or qc (it can be shown that functions such as , and are regular). The only singularities of ψ (p) are poles which make the denominator of equations (A10) and (A11) null, namely two type of poles:
 1. the pole p = 0, called p0
 2. an infinite series of poles corresponding to the zeros of Δ, i.e., solutions of:
 The function Δ(p) becomes null when is pure imaginary as, i.e., , (therefore, ). The zeros in p are then real negative or null and it can be shown [Carslaw and Jaeger, 1959, p. 325] that these are the only zeros of equation (A13).
 Hyperbolic trigonometry allows replacing equation (A13) by:
 The poles in p are real: p0 = 0 on the one hand and on the other hand.
 The contour of the integral equation (A12) can be closed on a large circle in the left part of the p-plane (Real (p) < 0); the contribution to the integral of this latter part of the contour vanishes when the circle radius tends to ∞. Since the integral contour includes all the poles of the function ψ, the application of the theorem of residues (Cauchy's theorem) provides the final result. The LT−1 is written as
where the summation ∑ is taken on all indices n = 0, 1,…, ∞ of ψ poles and where the Rn are the residues of these poles (i.e., the behavior of the ψ (p) in the vicinity of p = pn). One obtains
 The potential is the sum of a constant (in fact h0) and a series of exponential terms with negative time arguments, i.e., decreasing with reduced time t* according to a reduced time constant . Returning to actual variables, the time constant associated with index n is:
 The calculation of the residues is straightforward:
(A18) (A19) (A20)
 The final solution is a sum ∑ for :
 It may be verified that for ξ = ξc, the two expressions ϕc and ϕu and their space derivatives become identical (and therefore continuous) when taking into account equation (A13).
A4. Practical Use of the Series Development
 In the previous development, the time variation is ruled by the β's, the values of which require the solution of equation (A14). The only case where equation (A14) has an obvious solution is when f = 1; equation (A14) reduces to cos (β) = 0 the solutions of which are for n = 1, 2,…
 In the general case, β is a positive root of equation (A14). Let us call β1 the smallest positive root of equation (A14), also written as:
 It is convenient to begin with the case where is small which enables approximation of β1. For small values of the argument β1, the tangent function may be approached by its argument and equation (A23) is replaced by , so that
 Starting from this initial guess, the numerical solution of equation (A23) is easily obtained by Newton's tangent method. In a few iterations, the algorithm converges to the final value of β1. The value of the next roots of equation (A14), i.e., βn for can also be computed numerically using another algorithm based on the change of sign of Δ(β) while increasing β. These roots are of increasing magnitude and provide decreasing contributions to the series equations (A21) and (A22) as t increases, leaving the first exponential of the series development, , as the rapidly dominant term. Therefore, the first term of the series development gives a convenient approximation of the full solution.
 An illustration of this last statement is shown in Figure 6, using f = 0.01 and ξc = 0.7, which compares the result of a computation based on the full solution, either when retaining the first term (black curves), or using the first five (gray curves). As soon as t* > 0, the solution when only the first term is retained is not discernible from the more complete one, which justifies the fact that only the first exponential term of the series is retained. In supporting information, it is shown that this justification holds for a wide range of values of f and ξc.
 The relaxation of the initial disturbance toward the steady state is therefore characterized by an exponential decay according to , or in actual variables, with .
A5. Practical Estimate of β1: Limits and Further Development
 It remains to propose a practical approximation of β1. For low f values, the first development of Δ(p) for small p values yields a convenient approximation of the root β1, defined by the label β10:
and the corresponding time constant is evaluated by:
 Formula equation (A26) is interesting because of its simplicity and because it depends only on the diffusivity of the unconfined aquifer. However, it is clear that it breaks down when the length of the unconfined aquifer Lu tends to 0.
 More refined approximations can be obtained through a series expansion of Δ(β) about β = 0 through polynomial expansion of Δ(β) as a third degree polynomial in :
(A28) (A29) (A30)
 By truncating the polynomial of equation (A27) sequentially at orders 1, 2, 3, and equating the reduced polynomials to zero, respective β11, β12, β13 approximations β1 of the true β1 can be found by analytical solution formulae.
 The estimates of β and τm corresponding to the first-order approximation are given by
with the corresponding time constant:
 The estimates corresponding to the second-order approximation are
 Contrary to equation (A25), these last estimates of β1 are defined for any value of the unconfined aquifer length, even for ξu = 0. However, taking into account the value of f corresponding to actual situations (on the order of 10−3), it is possible to compare the accuracy of these various estimates of β1 as β10, β11, β12, and β13 for a large range of the ξu and f parameters. This is illustrated in Table 4, where values are given for the three approximations as well as the accurate value for f = 0.01, 0.001, and 0.0001. The ξc parameter varies from 0.5 to 0.995 in order to focus on the vicinity of ξu = 0. As expected, there is an improvement in accuracy with increasing order: β11 is closer to β1 than β10, β12 is even closer, and β13 is the closest.
Table 4. Values of β1 and Their Approximations as a Function Lc/L = ξc (i.e., ξu = 1−ξc)a
|f = 0.01|
|f = 0.001|
|f = 0.0001|
 The discrepancy in the vicinity of ξu = 0 is larger for the smaller values of f. However it appears that, to within about 5%, the simpler formula equation (A26) gives acceptable results for values as small as 0.05 (the unconfined aquifer then represents 5% of the total length) for f = 0.01; for f = 0.001, the approximation is correct when ξu is down to 0.005 and even smaller for f = 0.0001.
 Errors in the approximations of β1 translate into errors in the series equations (A21) and (A22). These will affect not only the target value of a 95% reduction of the initial perturbation at t = 3τm in the exponential term but also the coefficients of this term in the two series. These coefficients cause nonlinear behavior in ϕ across the aquifer as soon as t is greater than 0, as evidenced in Figure 6. This implies that the 95% reduction, or 5% residual, relative to the initial linear perturbation cannot be uniform across the aquifer. Consequently, an averaged target value is used. Typically, the range of residuals across the aquifer is between 4% and 7% with an average just over 5% when using β1. In supporting information, averaged residuals are calculated for a range of values of f and ξu for the two approximate roots β10 and β12 to determine the constraints on their usage. Setting an allowable average residual target value of 6.5% it is found that β10 can be used when the ratio f/ξu is smaller than 0.3. Returning to original physical quantities, this ratio has a specific meaning since, for ξu near 0, . Quantity as ScLc can be interpreted as the horizontally integrated storativity in the confined aquifer. Therefore, since f/ξu can be seen as the ratio of the integrated storativity of the confined aquifer to that of the unconfined one, only when the ratio is larger than 0.3, i.e., for a very small integrated storativity of the unconfined aquifer, would it be necessary to use other approximations such as β11 or β12 (which can be used for all f and ξu). Since the possibility that the unconfined aquifer is reduced to such a small relative length has no practical importance, it is clear that the approximation given by formula equation (A26) is relevant to the problem under study. With the range of f and ξu values for actual aquifers investigated in this paper formula (A26) can be used in all cases.
 A further question is that of initial and boundary conditions. If initial conditions are different from those used here, the effect is to change the right-hand sides of the Laplace transformed differential equations. But this will not change the determinant, Δ, given by equation (A13), and consequently the same τm prevails. The spatial variations of ϕ will be altered along with point wise values of percentage residuals at t = 3τm. Although not validated, the changes in percentage residuals are expected to be slight. In Carslaw and Jaeger [1959, chap. 3], there are worked examples for the single type of aquifer, which show that the τm values are unchanged for different initial conditions. The most likely variation in end boundary conditions is to assume that at the unconfined end, gradient conditions are no longer zero, i.e., at x = L. Again, this will not change Δ and τm.