Time constant of hydraulic-head response in aquifers subjected to sudden recharge change: application to large basins

Abstract

Analytical formulae are proposed to describe the first-order temporal evolution of the head in large groundwater systems (such as those found in North Africa or eastern Australia) that are subjected to drastic modifications of their recharge conditions (such as those in Pleistocene and Holocene times). The mathematical model is based on the hydrodynamics of a mixed-aquifer system composed of a confined aquifer connected to an unconfined one with a large storage capacity. The transient behaviour of the head following a sudden change of recharge conditions is computed with Laplace transforms for linear one-dimensional and cylindrical geometries. This transient evolution closely follows an exponential trend exp(−t/τ). The time constant τ is expressed analytically as a function of the various parameters characterizing the system. In many commonly occurring situations, τ depends on only four parameters: the width a _c of the main confined aquifer, its transmissivity T _c, the integrated storage situated upstream in the unconfined aquifer M = S _u a _u, and a curvature parameter accounting for convergence/divergence effects. This model is applied to the natural decay of large aquifer basins of the Sahara and Australia following the end of the mid-Holocene humid period. The observed persistence of the resource is discussed on the basis of the time constant estimated with the system parameters. This comparison confirms the role of the upstream water reserve, which is modelled as an unconfined aquifer, and highlights the significant increase of the time constant in case of converging flow.

Résumé

Des formules analytiques sont proposées pour décrire le premier ordre de l’évolution temporelle de la charge hydraulique dans les grands systèmes d’eau souterraine (tels que ceux trouvés en Afrique du Nord ou dans l’Est de l’Australie) qui sont soumis à des modifications drastiques de leurs conditions de recharge (comme ceux du Pléistocène et Holocène). Le modèle mathématique est basé sur l’hydrodynamique d’un système aquifère mixte composé d’un aquifère captif relié à un aquifère libre caractérisé par une grande capacité de stockage. Le comportement transitoire de la charge hydraulique suite à un changement soudain des conditions de recharge est calculé avec des transformées de Laplace pour des géométries unidimensionnels et cylindriques linéaires. Cette évolution transitoire suit de près une tendance exponentielle exp(−t/τ). La constante de temps τ est exprimée analytiquement en fonction des différents paramètres qui caractérisent le système. Dans de nombreuses situations qui se produisent souvent, τ dépend de quatre paramètres: la largeur a _c de l’aquifère principal captif, sa transmissivité T _c, le stockage intégré situé en amont de l’aquifère libre M = S _u a _u, et un paramètre de courbure prenant en compte les effets de convergence/divergence. Ce modèle est appliqué à la désintégration naturelle des grands bassins aquifères du Sahara et de l’Australie après la fin de la période humide de l’Holocène moyen. La persistance observée de la ressource est examinée sur la base de la constante de temps estimée à l’aide des paramètres du système. Cette comparaison confirme le rôle de la réserve d’eau en amont, qui est modélisée en tant qu’aquifère libre, et met en évidence l’augmentation significative de la constante de temps dans le case d’un écoulement convergent.

Resumen

Se proponen formulas analíticas para describir la evolución temporal de primer orden de la carga hidráulica en grandes sistemas de agua subterránea (como las que se encuentran en el norte de África o este de Australia) que están sujetas a modificaciones bruscas de sus condiciones de recarga (como los del Pleistoceno u Holoceno). El modelo matemático se basa en la hidrodinámica de un sistema acuífero mixto compuesto de un acuífero confinado conectado a uno no confinado con una gran capacidad de almacenamiento. El comportamiento transitorio de la carga hidráulica luego de un cambio brusco en la condiciones de recarga se calcula con las transformadas de Laplace para geometrías unidimensionales y cilíndricas lineares. Esta evolución transitoria sigue estrechamente la tendencia exponencial exp(−t/τ). La constante de tiempo τ está analíticamente expresada como una función de varios parámetros que caracterizan el sistema. En muchos casos que ocurren comúnmente, τ depende solo de cuatro parámetros: el ancho a _c del principal acuífero confinado, su transmisividad T _c, el almacenamiento integrado situado agua arriba en el acuífero no confinado M = S _u a _u, y un parámetro de curvatura que tiene en cuenta los efectos de convergencia / divergencia. Este modelo se aplica al decaimiento en grandes cuencas acuíferas del Sahara y Australia a partir del final del período húmedo del Holoceno medio. Se discute la persistencia observada en el recurso sobre la base de la constante de tiempo estimada con los parámetros del sistema. Esta comparación confirma el papel de las reservas de agua aguas arriba, la cual se modela como un acuífero no confinado, y pone de manifiesto el significativo incremento de la constante de tiempo en el caso de la convergencia de flujo.

摘要

提出了描述经历过补给条件剧变(诸如更新世和全新世时期的剧变)大型地下水系统(如北非或澳大利亚东部发现的大型地下水系统)中水头的一级时间演化的解析公式。数学模型基于一个混合含水层系统的水动力学,这个混合含水层系统为一个承压含水层连接着另一个储量巨大的非承压含水层。通过线性一维和圆柱几何Laplace变换计算了补给条件突然变化的水头瞬时特性。这个瞬时演变紧紧遵循着指数趋势经验值(-t/τ)。时间常数τ 解析表达为描述系统的各种参数的函数。在许多经常出现的情况下,τ 只取决于四个参数:主要承压含水层的宽度ac 、承压含水层的导水系数Tc 、非承压含水层上游的综合储量M = Suau 以及表示聚合/散发效应的曲率参数。这个模型应用在了中全新世潮湿阶段结束后撒哈拉和澳大利亚大型含水层盆地的自然衰退中。根据利用系统参数估算的时间常数论述了观测到的资源持久性。这个对比确认了模拟为非承压含水层的上游水储量的作用,强调了汇聚水流情况下时间常数的显著增大。

Resumo

Fórmulas analíticas são propostas para descrever a evolução temporal de primeira ordem da carga em grandes sistemas de águas subterrâneas (como aqueles encontrados no Norte de África ou no leste da Austrália) que são sujeitos a modificações drásticas das suas condições de recarga (como as dos tempos Pleistoceno e Holoceno). O modelo matemático é baseado na hidrodinâmica de um sistema aquífero misto, composto por um aquífero confinado conectado a um aquífero livre com grande capacidade de armazenamento. O comportamento transiente da carga hidráulica, seguido de uma súbita alteração das condições de recarga, é computado com a transformada de Laplace para geometrias lineares unidimensionais e cilíndricas. Essa evolução transiente segue de perto uma tendência exponencial exp(−t/τ). A constante temporal τ é analiticamente expressa como uma função dos vários parâmetros que caracterizam o sistema. Em muitas ocasiões que ocorrem comumente, τ depende apenas de quarto parâmetros: da extensão a _c do aquífero confinado principal, da sua transmissividade T _c, do armazenamento integrado situado a montante no aquífero livre M = S _u a _u, e de um parâmetro de curvatura considerando efeitos de convergência/divergência. Este modelo é aplicado ao decaimento natural das grandes bacias aquíferas do Saara e da Austrália após o final do período húmido do Holocénico médio. A persistência observada do recurso é discutida com base na constante temporal estimada com os parâmetros do sistema. Esta comparação confirma o papel da reserva a montante, que é modelada como um aquífero livre, e salienta o incremento significativo da constante temporal no caso de fluxo convergente.

Appendices

Appendices

Appendix 1: Laplace transform solution for the 1-D mixed aquifer system

The auxiliary function g(x,t) defined in the text is satisfying:

$$ {S}_{\mathrm{u}}\frac{\partial g}{\partial t}={T}_{\mathrm{u}}\frac{\partial^2g}{\partial {x}^2} + B\kern1.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0<x<{a}_{\mathrm{u}} $$

(53)

$$ {S}_{\mathrm{c}}\frac{\partial g}{\partial t}={T}_{\mathrm{c}}\frac{\partial^2g}{\partial {x}^2}\kern1.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}}=a $$

(54)

Its initial condition is that g(x,0) = 0 for any x in the range (0, a = a _u + a _c). Its boundary conditions are:

$$ \begin{array}{l}{\displaystyle {T}_{\mathrm{c}}}\frac{\partial g}{\partial x}\left(0,t\right)=0\hfill \\ {}g\left(a,t\right)=0\hfill \\ {}{g}^{-}\left({a}_{\mathrm{u}},t\right)={g}^{+}\left({a}_{\mathrm{u}},t\right)\hfill \\ {}{\displaystyle {T}_{\mathrm{u}}}\frac{\partial {g}^{-}}{\partial x}\left({a}_{\mathrm{u}},t\right)={T}_{\mathrm{u}}\frac{\partial {g}^{+}}{\partial x}\left({a}_{\mathrm{u}},t\right)\hfill \end{array} $$

(55)

Let γ(x,p) be the Laplace transform of g(x,t); γ(x,p) satisfies:

$$ {S}_{\mathrm{u}}p\gamma ={T}_{\mathrm{u}}\frac{\partial^2\gamma }{\partial {x}^2} + \frac{B}{p}\kern1.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0<x<{a}_{\mathrm{u}} $$

(56)

$$ {S}_{\mathrm{c}}p\gamma ={T}_{\mathrm{c}}\frac{\partial^2\gamma }{\partial {x}^2}\kern1.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<\mathrm{x}<{a}_{\mathrm{u}}+a $$

(57)

with boundary conditions similar to those of g(x,t) as Eq. (55). When introducing the parameter q defined by:

$$ q=\sqrt{\frac{p{S}_{\mathrm{c}}}{T_{\mathrm{c}}}}=\sqrt{\frac{p}{D_{\mathrm{c}}}} $$

(58)

the general solution of Eq. (56) can be written as:

$$ \gamma \left(x,p\right)=\frac{B}{p^2{S}_{\mathrm{u}}}+\alpha \cosh \left(\frac{qx}{f}\right)+\delta \sinh \left(\frac{qx}{f}\right)\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0<x<{a}_{\mathrm{u}} $$

(59)

and that of Eq. (57) as:

$$ \gamma \left(x,p\right)=\lambda \kern0.2em \cosh (qx)+\xi \sinh (qx)\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{\mathrm{a}}_{\mathrm{c}} $$

(60)

The four constants a, δ, λ, ξ are obtained by the four boundary conditions (Eq. 55) and the solution is:

$$ \gamma \left(x,p\right)=\frac{B}{p^2{S}_{\mathrm{u}}}\left[1-r\kern0.2em \frac{ \cosh \left({a}_{\mathrm{c}}q\right) \cosh \left(qx/f\right)}{\Delta}\right]\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0<x<{a}_{\mathrm{u}} $$

(61)

$$ \gamma \left(x,p\right)=\frac{B}{p^2{S}_{\mathrm{u}}}\left[\frac{ \sinh \left({a}_{\mathrm{u}}q/f\right)\kern0.2em \cosh \left[q\left(a-x\right)\right]}{\Delta}\right]\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<a={a}_{\mathrm{u}}+{a}_{\mathrm{c}} $$

(62)

where Δ is the determinant defined by:

$$ \Delta =r \cosh \left({a}_{\mathrm{c}}q\right) \cosh \left({a}_{\mathbf{u}}q/f\right)+ \sinh \left({a}_{\mathrm{c}}q\right) \sinh \left({a}_{\mathrm{u}}q/f\right) $$

(63)

For taking the inverse Laplace transform, it is necessary to identify the singularities of γ(x, p) in the complex p plane. The function γ(x, p) is single valued. It has a simple pole at p = 0 and an infinite series of simple poles p _n on the negative part of the real axis.

The pole at p = 0 corresponds to the asymptotic trend of the solution for t → + ∞. Its expression is obtained as the limit of Eqs. (61)–(62) for p → 0:

$$ \gamma \left(x,p\to 0\right)\approx \frac{B}{p}\left(\frac{a_{\mathrm{u}}^2-{x}^2}{2{T}_{\mathrm{u}}}+\frac{a_{\mathrm{u}}{a}_{\mathrm{c}}}{T_{\mathrm{c}}}\right)\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.75em 0<x<{a}_{\mathrm{u}} $$

(64)

$$ \gamma \left(x,p\to 0\right)\approx \frac{B}{p}{a}_{\mathrm{u}}\frac{a-x}{T_{\mathrm{c}}}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}} $$

(65)

This asymptotic behaviour corresponds to the steady-state solution g(x,t → + ∞). From the definition of g(x, t) = h ⁰(x) − h(x, t), g(x,+ ∞) is also equal to the assumed initial value h ⁰(x), the expression of which (Eqs. 9–10) can be recognized in Eqs. (64)–(65) to within a 1/p multiplicative factor.

The other poles correspond to the p _n values where ∆(p _n ) = 0. Using Eq. (58), these poles in p correspond to purely imaginary values of ${q}_n=\sqrt{p_n/{D}_{\mathrm{c}}}=i{\beta}_n$ where β _n is real. Using a trigonometric transformation of Eq. (63), the poles are also the zeros in β of the function ∆β:

$$ \Delta \left(\beta \right)=r \cos \left({a}_{\mathrm{c}}\beta \right) \cos \left({a}_{\mathrm{u}}\beta /f\right)- \sin \left({a}_{\mathrm{c}}\beta \right) \sin \left({a}_{\mathrm{u}}\beta /f\right)=0 $$

(66)

The first and smallest zero of Eq. (66) can be approximated from a first-order expansion of ∆β) in the vicinity of β = 0:

$$ \Delta \left(\beta \right)\approx r-{\beta}^2\left[\frac{r}{2}\left(\frac{a_{\mathrm{u}}^2}{f}+{a}_{\mathrm{c}}^2\right)+\frac{a_{\mathrm{u}}{a}_{\mathrm{c}}}{f}\right] $$

(67)

This first root β ₁ of ∆β = 0 can be approximated by:

$$ {\beta}_1\approx \sqrt{\frac{r}{a_{\mathrm{u}}{a}_{\mathrm{c}}/f+r\left({a}_{\mathrm{u}}^2/f+{a}_{\mathrm{c}}^2\right)/2}}=\frac{1}{\sqrt{a_{\mathrm{u}}{a}_{\mathrm{c}}\frac{S_{\mathrm{u}}}{S_{\mathrm{c}}}+{a}_{\mathrm{u}}^2\frac{T_{\mathrm{c}}{S}_{\mathrm{u}}}{2{T}_{\mathrm{u}}{S}_{\mathrm{c}}}+\frac{a_{\mathrm{c}}^2}{2}}} $$

(68)

The inverse transform is obtained through integration in the complex p plane as (Carslaw and Jaeger 1959):

$$ g\left(x,t\right)=\frac{1}{2i\pi}\underset{c}{\int}\gamma \left(x,p\right)\kern0.2em \exp (pt)dp $$

(69)

The contour C of the integral lies parallel to the imaginary axis, from –∞ to + ∞, and is closed in such a way that all singularities of the function (p) are located in the left part of the complex p-plane. It is closed on a large circle in the left part of the p-plane, which gives a vanishing contribution. This contour integral allows the use of the residuals theorem and each pole gives a contribution to the integral Eq. (69) according to the behaviour of γ(x,p). The pole at p = 0 contributes to give the asymptotic function h ⁰(x). The other poles p _n associated with the β _n (n = 1,2…) solutions of ∆β = 0 contribute to give a function R _n (x)exp(−tD _c/β ² _n ). Here R _n (x) denotes the residual, i.e. the linear trend of the function γ(x,p) in the vicinity of the pole p _n  = − β ² _n D _c. This yields:

$$ g\left(x,t\right)=B\left(\frac{a_{\mathrm{u}}^2-{x}^2}{2{T}_{\mathrm{u}}}+\frac{a_{\mathrm{u}}{a}_{\mathrm{c}}}{T_{\mathrm{c}}}\right) - \frac{2B}{S_{\mathrm{u}}{D}_{\mathrm{c}}}{\displaystyle \sum_{n=1}^{\infty}\frac{r \sin \left({\beta}_n{a}_{\mathrm{u}}/f\right) \cos \left({\beta}_nx/f\right) \exp \left(-{\beta}_n^2{D}_{\mathrm{c}}t\right)}{\beta_n^3\left[{a}_{\mathrm{c}}+{a}_{\mathrm{u}}r/f+{a}_{\mathrm{c}}\left({r}^2-1\right){ \cos}^2\left({\beta}_n{a}_{\mathrm{u}}/f\right)\right]}} $$

(70)

for 0 < x < a _u and

$$ g\left(x,t\right)=B{a}_{\mathrm{u}}\frac{a-x}{T_{\mathrm{c}}}-\frac{2B}{S_{\mathrm{u}}{D}_{\mathrm{c}}}{\displaystyle \sum_{n=1}^{\infty}\frac{ \cos \left({\beta}_n{a}_{\mathrm{u}}\right) \sin \left({\beta}_n\left(a-x\right)\right) \exp \left(-{\beta}_n^2{D}_{\mathrm{c}}t\right)}{\beta_n^3\left[{a}_{\mathrm{c}}+{a}_{\mathrm{u}}/rf+{a}_{\mathrm{u}}\left({r}^2-1\right){ \cos}^2\left({\beta}_n{a}_{\mathrm{c}}\right)/rf\right]}} $$

(71)

for a _u < x < a _u + a _c.

Using Eq. (15), the expression of h(x,t) is found directly from that of g(x,t) as h(x,t) = h ⁰(x) – g(x,t) yielding the results (Eqs. 17–18).

In practice, as explained in the main text, the summation over n (1… ∞) can generally be restricted to the first term n = 1 since the other terms (n = 2,3…) give a negligible contribution. The time variation closely follows an exponential decrease h(x, t) ∝ exp(−β ²₁ D _c t) or h(x, t) ∝ exp(−t/τ) where τ is the time constant. From Eq. (68), this time constant may be approximated by:

$$ \tau \approx \frac{1}{\beta_1^2{D}_{\mathrm{c}}}\approx {a}_{\mathrm{u}}{a}_{\mathrm{c}}\frac{S_{\mathrm{u}}}{D_{\mathrm{c}}{S}_{\mathrm{c}}}+{a}_{\mathrm{u}}^2\frac{T_{\mathrm{c}}{S}_{\mathrm{u}}}{2{D}_{\mathrm{c}}{T}_{\mathrm{u}}{S}_{\mathrm{c}}}+{a}_{\mathrm{c}}^2\frac{1}{2{D}_{\mathrm{c}}}={a}_{\mathrm{u}}{a}_{\mathrm{c}}\frac{S_{\mathrm{u}}}{T_{\mathrm{c}}}+{a}_u^2\frac{S_{\mathrm{u}}}{2{T}_{\mathrm{u}}}+{a}_c^2\frac{S_{\mathrm{c}}}{2{T}_{\mathrm{c}}} $$

(72)

The relative importance of the three terms in Eq. (72) can be assessed as shown in Table 3 which gives in percent the relative contributions by these three terms for a wide range of aquifer parameters. Three parameter ratios (of storage coefficients, horizontal widths and transmissivities) have a range of variation inspired by the considered aquifer basins:

• The storage coefficient S _c/S _u ratio is in the range 10⁻⁴ – 10⁻¹ for characterizing the difference in storage between confined and unconfined aquifers.

• The horizontal width a _u/a _c ratio between the two types of aquifers is assumed to vary between 0.1 (localized recharge zone) and 0.5 (wide recharge zone).

• The transmissivity T _c/T _u ratio is largely arbitrary. It is assumed to vary in the range 10⁻¹ – 10¹.

Table 3 Relative value in % of the three terms of Eq. (72) vs. values of ratios S _c/S _u T _c/T _u, a _u/a _c

It is clear that, for S _c/S _u in the range 10⁻⁴– 10⁻², the last term of Eq. (72) is relatively negligible (less than 5 % of the sum) for any T _c/T _u or a _u/a _c and that when T _c ≤ T _u, the first term of Eq. (72) is clearly dominant (above 80 % of the sum), which results in further simplification of Eq. (72) as discussed in the main text.

Appendix 2: Laplace transform solution for the 1-D reservoir–aquifer system

The head h(x,t) satisfies the diffusion equation:

$$ {S}_{\mathrm{c}}\frac{\partial h}{\partial t}={T}_{\mathrm{c}}\frac{\partial^2h}{\partial {x}^2}\kern1.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}}=a $$

(73)

with the two boundary conditions:

$$ h\left({a}_{\mathrm{u}}+{a}_{\mathrm{c}}=a,t\right)=0 $$

(74)

$$ M\frac{\partial h\left({a}_{\mathrm{u}},t\right)}{\partial t}={\displaystyle {T}_{\mathrm{c}}}{\left.\frac{\partial h\left(x,t\right)}{\partial x}\right|}_{x={a}_{\mathrm{u}}} $$

(75)

The first boundary condition is that the head is 0 at the outlet x = a. The second boundary condition means that the aquifer is connected to a reservoir which has the same head h(a _u,t) at x = a _u as the aquifer. This reservoir is characterized by its integrated storage capacity M = a _u S _u. The reservoir–aquifer connection implies that any time variation of the reservoir volume induces a diffusive flux toward (or from) the aquifer at x = a _u.

The initial condition is that, at t = 0, the system is at equilibrium between a discharge at x = a and a recharge of the reservoir at a rate B per unit coordinate for 0 < x < a _u. This yields the following linear solution of the equilibrium equation in the aquifer:

$$ h\left(x,0\right)={h}_1\frac{a-x}{a_{\mathrm{c}}}\kern1.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<a $$

(76)

where h ₁ is such that the flux of water T _c h ₁/a _c entering in the aquifer compensates the integrated recharge Ba _u. Therefore:

$$ {h}_1=B\frac{a_{\mathrm{c}}{a}_{\mathrm{u}}}{T_{\mathrm{c}}} $$

(77)

Let η(x,p) be the Laplace transform of the function h(x,t); the associated equation for η(x,p) is:

$$ {S}_{\mathrm{c}}\left[p\eta -{h}_1\frac{a-x}{a_{\mathrm{c}}}\right]={T}_{\mathrm{c}}\frac{\partial^2\eta }{\partial {x}^2}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}} $$

(78)

Introducing $q=\sqrt{p/{D}_{\mathrm{c}}}$ the general solution of Eq. (78) is:

$$ \eta \left(x,p\right)=\frac{h_1}{p}\frac{a-x}{a_{\mathrm{c}}}+\alpha \cosh (qx)+\delta \sinh (qx) $$

(79)

with two constants α and δ which are determined to satisfy Eqs. (74)–(75). This gives:

$$ \eta \left(x,p\right)=\frac{h_1}{p}\left\{\frac{a-x}{a_{\mathrm{c}}}-\frac{T_{\mathrm{c}} \sinh \left(q\left(a-x\right)\right)}{q{a}_{\mathrm{c}}\left[{T}_{\mathrm{c}} \cosh \left(q{a}_{\mathrm{c}}\right)+M{D}_{\mathrm{c}}q \sinh \left(q{a}_{\mathrm{c}}\right)\right]}\right\} $$

(80)

Again, the function η(x,p) is single valued and has an infinite series of simple poles p _n on the negative part of the real axis. The value p = 0 is not a pole, as can be verified with a limited development of Eq. (80) for small p. The poles are the zeros of the expression at the denominator of Eq. (80) and can also be expressed as functions of the real variable $\beta =-iq=\sqrt{-p/{D}_{\mathrm{c}}}$. The corresponding β values are positive solutions of:

$$ \Delta \left(\beta \right)={T}_{\mathrm{c}} \cosh \left(q{a}_{\mathrm{c}}\right)+M{D}_{\mathrm{c}}q \sinh \left(q{a}_{\mathrm{c}}\right)={T}_{\mathrm{c}} \cos \left(\beta {a}_{\mathrm{c}}\right)-M{D}_{\mathrm{c}}\beta \sin \left(\beta {a}_{\mathrm{c}}\right)=0 $$

(81)

The Laplace inversion proceeds as for Appendix 1. The poles p _n corresponding to the positive β _n (n = 1,2…) solutions of Δ(β) = 0 contribute to the contour integral (as defined in Eq. 69). This contribution is a function R _n (x)exp(−tD _c/β ² _n ) where R _n (x) denotes the residual, i.e. the linear trend of the function η(x,p) in the vicinity of the pole ${p}_n=-{\beta}_n^2/{D}_{\begin{array}{l}\mathrm{c}\\ {}\end{array}}$. The expression for the residual is:

$$ {R}_n(x)=2{h}_1\frac{ \sin \left[{\beta}_n\left(a-x\right)\right]}{a_{\mathrm{c}}^2{\beta}_n^2 \sin \left({\beta}_n{a}_{\mathrm{c}}\right)\left[1+M/{S}_{\mathrm{c}}{a}_{\mathrm{c}}+{\left(M{\beta}_n/{S}_{\mathrm{c}}\right)}^2\right]} $$

(82)

and the required solution is an infinite series:

$$ h\left(x,t\right)=2{h}_1\sum_{n=1}^{\infty }\ \frac{ \sin \left[{\beta}_n\left(a-x\right)\right] \exp \left(-t{D}_c/{\beta}_n^2\right)}{\kern0.5em {a}_c^2{\beta}_n^2 \sin \left({\beta}_n{a}_c\right)\left[1+M/{S}_c{a}_c+{\left(M{\beta}_n/{S}_c\right)}^2\right]} $$

(83)

In practice, the summation over n (1… ∞) is restricted to the first term n = 1 since the other terms (n = 2,3…) give a negligible contribution. The time variation closely follows an exponential decrease h(x, t) ∝ exp(−β ²₁ D _c t) or h(x, t) ∝ exp(−t/τ) where τ is the time constant τ = 1/β ²₁ D _c. An approximate value of β ₁ is obtained from a first-order expansion of Δ(β) given by Eq. (82) in the vicinity of β = 0:

$$ {\beta}_1\cong \frac{1}{a_{\mathrm{c}}\sqrt{\left[1/2+M/\left({S}_{\mathrm{c}}{a}_{\mathrm{c}}\right)\right]}} $$

(84)

Table 4 allows a comparison between the approximate Eq. (84) and the exact value of β ₁ obtained by a numerical solution of Eq. (81). The approximation (Eq. 84) appears to be satisfactory to within 5 % as long as M/(S _c a _c) > 1.

Table 4 Comparison of exact numerical values of (β ₁ a _c) and approximation  (Eq. 84)

Therefore the time constant τ = 1/β ²₁ D _c may be approximated by:

$$ \tau =\frac{1}{D_{\mathrm{c}}{\beta}_1^2}\cong \frac{a_{\mathrm{c}}^2}{D_{\mathrm{c}}}\left[\frac{1}{2}+\frac{M}{S_{\mathrm{c}}{a}_{\mathrm{c}}}\right]=\frac{a_{\mathrm{c}}^2}{2{D}_{\mathrm{c}}}+\frac{a_{\mathrm{c}}{a}_u{S}_u}{T_{\mathrm{c}}} $$

(85)

Appendix 3: Laplace transform solution for the cylindrical reservoir–aquifer system with divergent or convergent flow

In radial coordinates, the head is assumed to depend only on the radius r and to satisfy the corresponding radial diffusion equation for h(r,t):

$$ \frac{T_{\mathrm{c}}}{r}\frac{\partial }{\partial r}\left(r\frac{\partial h}{\partial r}\right)={S}_{\mathrm{c}}\frac{\partial h}{\partial t} $$

(86)

For the case of diverging flow, the domain of the aquifer is R < r < R + a _c. A boundary condition is prescribed at the outlet r = R + a _c as h(R + a _c) = 0. The other condition at r = R is a connection with a reservoir with integrated storage given by M = a _u S _u. This implies a mixed boundary condition on the value of h at r = R:

$$ M\frac{\partial h\left(R,t\right)}{\partial t}={T}_{\mathrm{c}}{\left.\frac{\partial h\left(r,t\right)}{\partial r}\right|}_{\begin{array}{l}r=R\\ {}\end{array}} $$

(87)

The general solution of the steady-state equation which satisfies h(R + a _c) = 0 is:

$$ h(r)={h}_1\frac{ \ln \left(R+{a}_{\mathrm{c}}\right)- \ln (r)}{ \ln \left(R+{a}_{\mathrm{c}}\right)- \ln (R)} $$

(88)

h ₁ can be expressed as a function of the initial recharge rate B. For t ≤ 0, the water budget requirement is that the flow entering the aquifer is equal to the integrated recharge. Therefore:

$$ {h}_1=B\frac{a_{\mathrm{u}}R}{T_{\mathrm{c}}} \ln \left(\frac{R+{a}_{\mathrm{c}}}{R}\right)=B\frac{MR}{T_{\mathrm{c}}{S}_{\mathrm{u}}} \ln \left(\frac{R+{a}_{\mathrm{c}}}{R}\right) $$

(89)

Let η(r,p) be the Laplace transform of the function h(r,t); its associated equation is:

$$ {S}_{\mathrm{c}}\left[p\eta -{h}_1\frac{ \ln \left[\left(R+{a}_{\mathrm{c}}\right)/r\right]}{ \ln \left[\left(R+{a}_{\mathrm{c}}\right)/R\right]}\right]=\frac{T_{\mathrm{c}}}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \eta }{\partial r}\right)\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em R<r<R+{a}_{\mathrm{c}} $$

(90)

Using $q=\sqrt{p/{D}_{\mathrm{c}}}$, the general solution of Eq. (90) is obtained as a function of the modified Bessel functions I₀ and K₀:

$$ \eta \left(x,p\right)=\frac{h_1}{p}\frac{ \ln \left[\left(R+{a}_{\mathrm{c}}\right)/r\right]}{ \ln \left[\left(R+{a}_{\mathrm{c}}\right)/R\right]}+\alpha {\mathrm{I}}_0(qr)+\delta {\mathrm{K}}_0(qr) $$

(91)

α and δ are defined to satisfy the boundary conditions. This leads to:

$$ \begin{array}{l}\eta \left(x,p\right)=\frac{h_1}{pR \ln \left[\left(R+{a}_{\mathrm{c}}\right)/R\right]}\left\{R \ln \left[\frac{R+{a}_{\mathrm{c}}}{r}\right]\right.+\hfill \\ {}\left.\frac{\kappa \left[-{\operatorname{I}}_0(qr)\mathrm{K}{}_0\left[q\left(R+{a}_{\mathrm{c}}\right)\right]+\operatorname{I}{}_0\left[q\left(R+{a}_{\mathrm{c}}\right)\right]{\mathrm{K}}_0(qr)\right]}{p\left[{\mathrm{I}}_0(qR)\mathrm{K}{}_0\left[q\left(R+{a}_c\right)\right]-\mathrm{I}{}_0\left[q\left(R+{a}_c\right)\right]{\mathrm{K}}_0(qR)\right]-\kappa q\left[{\mathrm{I}}_1(qR)\mathrm{K}{}_0\left[q\left(R+{a}_{\mathrm{c}}\right)\right]+\mathrm{I}{}_0\left(q\left[R+{a}_{\mathrm{c}}\right]\right){\mathrm{K}}_1(qR)\right]}\right\}\hfill \end{array} $$

(92)

where the variable κ = T _c/M is introduced. In the complex p-plane, the function η(r,p) is regular at p = 0, as can be shown by an expansion of η for small p. This is expected since the limit p → 0 corresponds to the ultimate steady state which is h(r, + ∞) = 0. η(r,p) has a series of single poles on the negative real axis of p which correspond to the p values which nullify the denominator Δ of Eq. (92). The zeros can be expressed as functions of $\beta =\sqrt{-p/{D}_{\mathrm{c}}}$ using well known relations for Bessel functions of imaginary argument:

$$ \begin{array}{l}\Delta \left(\beta \right)=\left\{{\mathrm{J}}_0\left[\beta R\right]{\mathrm{Y}}_0\left[\beta \left(R+{a}_{\mathrm{c}}\right)\right]-{\mathrm{J}}_0\left[\beta \left(R+{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_0\left[\beta R\right]\right\}{D}_{\mathrm{c}}{\beta}^2+\hfill \\ {}\left\{{\mathrm{J}}_0\left[\beta \left(R+{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_1\left[\beta R\right]-{\mathrm{J}}_1\left[\beta R\right]{\mathrm{Y}}_0\left[\beta \left(R+{a}_{\mathrm{c}}\right)\right]\right\}\kappa \beta =0\hfill \end{array} $$

(93)

As previously, the poles p _n corresponding to positive solutions of Δβ = 0 contribute to the contour integral (as defined in Eq. 69) in order to give a function k _n (r)exp(−tD _c/β ² _n ). Here, k _n (r) denotes the residual, i.e. the linear trend of the function η(r,p) in the vicinity of the pole ${p}_n=-{\beta}_n^2/{D}_{\begin{array}{l}\mathrm{c}\\ {}\end{array}}$. When taking into account the fact that Δβ _n  = as well as classical relations for Bessel functions (see Carslaw and Jaeger 1959, p. 333), the expression for the residual can be written after some algebra as:

$$ {k}_n(r)=\frac{h_1\kappa \pi }{2R \ln \left(\frac{R+{a}_{\mathrm{c}}}{R}\right)}\frac{{\mathrm{J}}_0\left[{\beta}_n\left(R+{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_0\left[{\beta}_nr\right]-{\mathrm{Y}}_0\left[{\beta}_n\left(R+{a}_{\mathrm{c}}\right)\right]{\mathrm{J}}_0\left[{\beta}_nr\right]}{\frac{\rho_n}{2}-\frac{\beta_n^2}{2{\rho}_n}\kern0.2em \left({\kappa}^2+{\beta_n}^2{D}_{\mathrm{c}}^2-\frac{2\kappa {D}_{\mathrm{c}}}{R}\right)} $$

(94)

where the coefficients ρ _n are defined by:

$$ {\rho}_n=\frac{-{\beta}_n^2{D}_{\mathrm{c}}{\mathrm{J}}_0\left({\beta}_nR\right)+\kappa {\beta}_n{\mathrm{J}}_1\left({\beta}_nR\right)}{{\mathrm{J}}_0\left[{\beta}_n\left(R+{a}_{\mathrm{c}}\right)\right]} $$

(95)

The required solution is an infinite series h(x, t) = ∑ ^∞ _n = 1 k _n (r)exp(−tD _c/β ² _n ). Only the first major term is retained so that the approximate solution is:

$$ h\left(r,t\right)\cong {k}_1(r) \exp \left(-t{D}_{\mathrm{c}}/{\beta}_1^2\right)\kern0.5em \mathrm{or}\kern0.5em h\left(r,t\right)\cong {k}_1(r) \exp \left(-t/\tau \right) $$

(96)

β ₁ is obtained through an expansion of ∆β in the vicinity of β = 0. This requires the expansion of Bessel functions for small values of their argument u which are (here γ is Euler’s constant, 0.5772…):

$$ {\mathrm{J}}_0(u)\approx 1-{\left(u/2\right)}^2 $$

(97)

$$ {\mathrm{J}}_1(u)\approx u/2-{\left(u/2\right)}^3/3 $$

(98)

$$ {\mathrm{Y}}_0(u)\approx 2/\pi \left[ \ln \left(u/2\right)+\gamma -{\left(u/2\right)}^2 \ln \left(u/2\right)\right] $$

(99)

$$ {\mathrm{Y}}_1(u)\approx 2/\pi \left[-1/u+\left(u/2\right) \ln \left(u/2\right)+\left(u/2\right)\left(\gamma -1/2\right)\right] $$

(100)

Using this development for Δ, the evaluation of the smallest solution of Δβ = 0 is then obtained as:

$$ {\beta}_1\cong \frac{1}{\sqrt{R^2\left[\left(\frac{M}{S_{\mathrm{c}}R}-\frac{1}{2}\right) \ln \left(\frac{R+{a}_{\mathrm{c}}}{R}\right)+\frac{{\left(R+{a}_{\mathrm{c}}\right)}^2-{R}^2}{4{R}^2}\right]}} $$

(101)

For the case of converging flow, the aquifer domain is R – a _c < r < R. A boundary condition is prescribed at the outlet r = R – a _c as h(R – a _c) = 0. As previously, the other condition at r = R is a connection to a reservoir with integrated storage M = a _u S _u. The function h(r,t) satisfies the same Eq. (86) but, since the flow is now convergent, the mixed boundary condition at r = R is:

$$ M\frac{\partial h\left(R,t\right)}{\partial t}=-{T}_{\mathrm{c}}{\left.\frac{\partial h\left(r,t\right)}{\partial r}\right|}_{\begin{array}{l}r=R\\ {}\end{array}} $$

(102)

The general solution of the steady-state equation which satisfies h(R + a _c) = 0 is:

$$ h(r)={h}_1\frac{ \ln \left[r/\left(R-{a}_{\mathrm{c}}\right)\right]}{ \ln \left[R/\left(R-{a}_{\mathrm{c}}\right)\right]} $$

(103)

where h ₁ is expressed as a function of the initial recharge rate B:

$$ {h}_1=B\frac{a_{\mathrm{u}}R}{T_{\mathrm{c}}} \ln \left(\frac{R}{R-{a}_{\mathrm{c}}}\right) $$

(104)

The Laplace transform η(r,p) of the function h(r,t) yields the associated equation:

$$ {S}_{\mathrm{c}}\left[p\eta -{h}_1\frac{ \ln \left[r/\left(R-{a}_{\mathrm{c}}\right)\right]}{ \ln \left[R/\left(R-{a}_{\mathrm{c}}\right)\right]}\right]=\frac{T_{\mathrm{c}}}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \eta }{\partial r}\right)\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em R - {a}_{\mathrm{c}}<r<R $$

(105)

The general solution of Eq. (105) is expressed in terms of modified Bessel functions I₀ and K₀:

$$ \eta \left(x,p\right)=\frac{h_1}{p}\frac{ \ln \left[r/\left(R-{a}_{\mathrm{c}}\right)\right]}{ \ln \left[R/\left(R-{a}_{\mathrm{c}}\right)\right]}+\alpha {\mathrm{I}}_0(qr)+\delta {\mathrm{K}}_0(qr) $$

(106)

where α and δ are defined to satisfy the boundary conditions:

$$ \begin{array}{l}\eta \left(x,p\right)=\frac{h_1}{pR \ln \left[R/\left(R-{a}_{\mathrm{c}}\right)\right]}\left\{R \ln \left[\frac{r}{R-{a}_{\mathrm{c}}}\right]\right.+\hfill \\ {}\left.\frac{\kappa \left[{\mathrm{I}}_0(qr)\mathrm{K}{}_0\left[q\left(R-{a}_{\mathrm{c}}\right)\right]-\mathrm{I}{}_0\left[q\left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{K}}_0(qr)\right]}{p\left[-{\mathrm{I}}_0(qR)\mathrm{K}{}_0\left[q\left(R-{a}_{\mathrm{c}}\right)\right]+\mathrm{I}{}_0\left[q\left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{K}}_0(qR)\right]-\kappa q\left[{\mathrm{I}}_1(qR)\mathrm{K}{}_0\left[q\left(R-{a}_{\mathrm{c}}\right)\right]+\mathrm{I}{}_0\left[q\left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{K}}_1(qR)\right]}\right\}\hfill \end{array} $$

(107)

As in the diverging case, the function η(r,p) is regular at p = 0 and has a series of single poles on the negative real axis of p . These poles are the zeros of the denominator of the last member of Eq. (107). Using standard relations for Bessel functions with an imaginary argument, these zeros are obtained for the values of $\beta =\sqrt{-p/{D}_{\mathrm{c}}}$ satisfying:

$$ \begin{array}{l}\Delta \left(\beta \right)=\left\{-{\mathrm{J}}_0\left[\beta R\right]{\mathrm{Y}}_0\left[\beta \left(R-{a}_{\mathrm{c}}\right)\right]+{\mathrm{J}}_0\left[\beta \left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_0\left[\beta R\right]\right\}{D}_{\mathrm{c}}{\beta}^2+\hfill \\ {}\left\{{\mathrm{J}}_0\left[\beta \left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_1\left[\beta R\right]-{\mathrm{J}}_1\left[\beta R\right]{\mathrm{Y}}_0\left[\beta \left(R-{a}_{\mathrm{c}}\right)\right]\right\}\kappa \beta =0\hfill \end{array} $$

(108)

As previously, the poles p _n corresponding to the positive solutions β _n (n = 1,2…) of Δ(β) = 0 contribute to the contour integral (as defined in Eq. 69) via a function kk _n (r)exp(−tD _c/β ² _n ). Here kk _n (r) denotes the residual i.e. the linear trend of the function η(r,p) in the vicinity of the pole ${p}_n=-{\beta}_n^2/{D}_{\begin{array}{l}\mathrm{c}\\ {}\end{array}}$. After some algebra its expression is obtained:

$$ k{k}_n(r)=\frac{h_1\kappa \pi }{2R \ln \left(\frac{R}{R-{a}_{\mathrm{c}}}\right)}\frac{{\mathrm{J}}_0\left[{\beta}_n\left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_0\left[{\beta}_nr\right]-{\mathrm{Y}}_0\left[{\beta}_n\left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{J}}_0\left[{\beta}_nr\right]}{\frac{\rho_n}{2}-\frac{\beta_n^2}{2{\rho}_n}\left({\kappa}^2+{\beta_n}^2{D}_{\mathrm{c}}^2+\frac{2\kappa {D}_{\mathrm{c}}}{R}\right)} $$

(109)

where the ρ _n are defined by:

$$ {\rho}_n=\frac{-{\beta}_n^2{D}_{\mathrm{c}}{\mathrm{J}}_0\left({\beta}_nR\right)-\kappa {\beta}_n{\mathrm{J}}_1\left({\beta}_nR\right)}{{\mathrm{J}}_0\left[{\beta}_n\left(R-{a}_{\mathrm{c}}\right)\right]} $$

(110)

and the required solution is an infinite series h(r, t) = ∑ ^∞ _n = 1 kk _n (r)exp(−tD _c/β ² _n ). Only the first major term is retained so that the approximate solution is: h(r, t) = kk ₁(r)exp(−tD _c/β ²₁ ) or h(r, t) = kk ₁(r)exp(−t/τ).

β ₁ is obtained through an expansion of Δβ in the vicinity of β = 0 making use of the previous expansion (Eqs. 97–100) of Bessel functions. One obtains the first positive solution of Δβ = 0 as:

$$ {\beta}_1\cong \frac{1}{\sqrt{R^2\left[\left(\frac{M}{S_{\mathrm{c}}R}+\frac{1}{2}\right) \ln \left(\frac{R}{R-{a}_{\mathrm{c}}}\right)-\frac{R^2-{\left(R-{a}_{\mathrm{c}}\right)}^2}{4{R}^2}\right]}} $$

(111)