Introduction

In many parts of the world, especially where rainfall is scarce, surface water reserves (rivers, lakes or near-surface water bodies) are insufficient to satisfy human needs. Then, fresh groundwater resources from deep aquifers may offer an alternative supply. When these resources are exploited at a rate larger than their current recharge, their reserves will decrease more or less rapidly (Custodio 2002). Therefore, the understanding and quantification of processes associated with the recharge and discharge of these deep aquifers are essential for predicting and planning their exploitation (Scanlon et al. 2006).

The intensive exploitation of deep groundwater reserves is relatively recent (at most 50–100 years). Prior to human intervention, the recharge of groundwater systems had been modified by climate changes that occurred during the early Holocene or Pleistocene (Sonntag et al. 1980; Edmunds 2009). These changes in recharge conditions were particularly strong in arid and semi-arid areas of North-West to North-East Africa and in Eastern Australia. In the Sahara, the so-called “African Humid Period” (AHP) at the beginning of the Holocene was characterized by greater rainfall than currently. This is confirmed by many biological and archaeological observations and by isotopic records (Gasse 2000; Taylor et al. 2009; Lézine et al. 2011). The present intense aridity of the Sahara is interpreted as the result of a rapid hydrological change which occurred some 4,000 years ago (Kröpelin et al. 2008; Krinner et al. 2012). The same phenomenon was detected in East Australia (Love et al. 1994) indicating a possible worldwide trend.

Large variations in average rainfall (and therefore of recharge) induce modifications in the behaviour of deep groundwater systems but with a time lag as a consequence of the internal transient behaviour of such systems in response to changes of boundary conditions (Edmunds 1999). The quantification of this transient behaviour is fundamental in explaining or predicting the time required by the hydraulic system to adjust to the new boundary conditions. The aim of this study is to evaluate quantitatively the time constant that characterizes the duration of this transient state.

To this end, the modification of the relevant boundary conditions is assumed to be instantaneous. Prior to this modification, the system is assumed to be at equilibrium. Once the modification has taken place, the hydrodynamic system evolves toward a new equilibrium. This evolution can generally be approximated by a functional dependence in time of the form exp(−t/τ) where the time constant τ characterizes the transient duration.

The time constant τ is a function of the various physical parameters of the aquifer system. The simplest case is that of a one-dimensional (1-D) confined homogeneous aquifer where one end has a prescribed input flux due to recharge, and the other end is at a prescribed head. Initially the head presents a linear equilibrium profile. If at t = 0, the prescribed flux becomes null, i.e. the recharge is suddenly set to 0, the system evolves toward a new equilibrium profile where everywhere the head equals the prescribed head at the output. The transient evolution of the hydraulic head may be computed analytically as a solution of the diffusion equation. It can be shown (Domenico and Schwartz 1998) that the time constant τ is given by:

$$ \tau =\frac{4{a}^2}{\pi^2D} $$
(1)

where a is the length of the aquifer and D its hydraulic diffusivity (ratio of its transmissivity T to its storage coefficient S).

This classical expression was generalized by Rousseau-Gueutin et al. (2013) to account for a more complex situation inspired by the structure of the Great Artesian Basin (GAB) in Australia. Their model is a one dimensional (1-D) mixed-aquifer system composed of two contiguous compartments. The upper compartment is an unconfined aquifer (subscript u), recharged by rainfall and with a relatively large storage coefficient. The discharge takes place at the end of the lower compartment which is a confined aquifer (subscript c) with the same transmissivity as the unconfined one. Starting from an assumed initial condition, the hydraulic head relaxes toward equilibrium in an (almost) exponential way. The time constant τ of this relaxation is a function of the geometrical properties (lengths a u and a c), storage coefficients (S u and S c) and transmissivity T (assumed to apply to both compartments). Moreover, for values of the storage parameters such that a u S u is larger than a c S c, the parameter S c has a negligible influence and the expression of the time constant reduces to:

$$ \tau \approx \frac{a_{\mathrm{u}}{a}_{\mathrm{c}}{S}_{\mathrm{u}}}{T} $$
(2)

This study aims to further discuss the conditions under which this formula or alternative ones can be used. The previous approach by Rousseau-Gueutin et al. (2013) is generalized by introducing alternative assumptions on the geometry (1-D along the horizontal direction, or cylindrical), or by using different values of hydraulic parameters and initial and boundary conditions.

Sonntag (1986) already developed similar arguments for interpreting the transient behaviour of the large Nubian Sandstone Aquifer (NSA) in North Africa. His study is based on the use of elementary harmonic solutions of the time-dependant diffusion equation for the head. Both linear 1-D cases and a cylindrical geometry (cylindrical mountain or depression) are considered. The present study differs from Sonntag’s by the precise definition of initial conditions, by the use of a more realistic geometry and by the assumption of abrupt changes in recharge/discharge conditions allowing the use of Laplace transforms (Carslaw and Jaeger 1959).

In the present study, a generic model sketching the major features of some large aquifers of arid or semi-arid basins in North Africa and Australia is proposed. It is based on the observation of three geographic areas where large-scale characterization of aquifer behaviour are available: the North Western Sahara Aquifer System (NWSAS; Ould Baba Sy 2005; OSS 2003), the Nubian Sandstone Aquifer (NSA; Heinl and Brinkmann 1989), and the Great Artesian Basin in Australia (GAB; Rousseau-Gueutin et al. 2013). Using the same type of mathematical development, a set of models corresponding to various assumptions is proposed to describe the time evolution of the head in aquifers subjected to a sudden recharge variation. The time evolution of the head is obtained as a series of exponential terms decreasing with time. The first term of the series is strongly dominant which justifies the definition of a single time constant, expressed analytically as a function of a minimum number of parameters.

Formulation of a generic model: geometric and hydraulic parameters

The simplified generic model is based on reference examples of large aquifer basins with horizontal dimensions in the range 500–1,500 km such as NWSAS, NSA and GAB. A simplification of the structures described in the available monographs makes it possible to distinguish three zones in these large aquifer basins:

  1. 1.

    A recharge zone associated with the relief which generally bounds part of the basin. This relief receives (and has received in the past) a great amount of rainfall and generates high hydraulic heads in contiguous aquifers (Fig. 1a).

  2. 2.

    A large deep aquifer layer assimilated to a homogeneous and confined aquifer whose geometry can be greatly simplified (1-D in the horizontal direction or with a cylindrical symmetry).

  3. 3.

    An outlet zone where discharge occurs. This outlet may be either the sea or a surface-water body (lake, river, swamp) or a humid zone where evapotranspiration maintains a constant level.

Fig. 1
figure 1

Simplified scheme of the assumed 1-D aquifer system. a Schematic cross-section of a mixed aquifer: its upper part with width a u is unconfined and receives the rainfall recharge, and its lower part with width a c is confined down to the outlet. b Detail of the upstream recharge zone in the case where it corresponds to the surface outcrop of the aquifer. c Case where the upstream aquifer has a relatively wide dimension

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In more detail:

  1. 1.

    The recharge zone of any of the referenced basins is actually part of their mountainous borders: the Atlas Mountain for the NWSAS, the Enedi for the NSA and the Great Dividing Range for the East GAB. In arid and semi-arid regions, these mountainous zones are considered as the water towers of neighbouring basins (Viviroli et al. 2003, 2007) due to orographic meteorological effects (Chavez et al. 1994; Wilson and Guam 2000).

    Infiltration of runoff water in the foothills, in particular during floods (Chavez et al. 1994), is the main recharge of deeper aquifers. The connexion may be local on the outcrops of the deep aquifers as illustrated in Fig. 1b where the outcrops of deep aquifers form several-km wide strips such as occurs in NWSAS on the southern side of the Atlas Mountain (Ould Baba Sy 2005). Alternatively, the deep aquifer can be recharged through a wide, superficial water-table aquifer with a large extent (50–300 km) beginning at the front of the mountain range. This case is illustrated in Fig. 1c and would apply to NSA and GAB.

    In any case, the recharge zone can be assimilated to an unconfined aquifer characterized by its geometry (width a u), its hydraulic parameters (storage parameter Su and transmissivity T u) and its recharge rate (in fact the variation of this rate). For such an unconfined aquifer, the specific yield S u (or effective porosity) is in the range of 10−2 – 10 −1 for most porous or fractured sedimentary rocks. Conversely, the transmissivity T u—defined as the product of the permeability by the saturated thickness—is poorly known and will be parameterized in the model.

  2. 2.

    The large deep aquifers which occupy most of North Africa and East Australia have a wide horizontal range (up to 1,500 km) and a considerable vertical thickness (several hundred m) ensuring their hydraulic continuity. In the NWSAS of Algeria, two such aquifers are well known: the “Continental Intercalaire confined aquifer (CI)” comprising formations from Middle Jurassic to Lower Cretaceous and the “Complexe Terminal (CT)” with Upper Cretaceous to Miocene formations (Ould Baba Sy 2005). Both aquifers are considered confined or semi-confined. In North East Africa, the aquifer system of NSA corresponds to Lower Cretaceous formations with a horizontal continuity over 500 km from the Ennedi and Tibesti mountains down to the Mediterranean Sea (Thorweilhe and Heinl 2002; Hesse et al. 1987) and it is confined in its northern part. In East Australia, the GAB, which occupies more than 20 % of the continent, is a multilayer aquifer where the flow runs mainly from the Great Dividing Range in a south-western direction through a more than 1,300-km wide area. It is confined over most of its extent (Rousseau-Gueutin et al. 2013).

    In the generic model, the deep aquifer is schematized by a confined aquifer with width a c on the order of 500–1,500 km and homogeneous hydraulic properties: transmissivity T c and storage coefficient S c. Estimates of these parameters are available for the three basins under study. They are summarized in Table 1 and present quite a good consistency for the order of magnitude of T c and S c.

  3. 3.

    The natural outlets where these aquifers discharge are situated either in the sea or on the continental crust as springs or recharge of wadis (e.g. Oued Rhir in the NWSAS), surface-water bodies or chotts (i.e. playas) and swampy areas with low topography and a high evaporation rate (oases and humid areas with vegetation such as NSA).

    The model describes the natural evolution of the system, characterized by its specific time constant, when its recharge is submitted to external variations; therefore, artificial withdrawals for irrigated agriculture and domestic uses are not specifically taken into account. Moreover, a single outlet is assumed and corresponds to a condition of prescribed head.

    This simple geometry and the parameters defined in the preceding are the generic elements for computing the time constant. The basic model developed in the next section corresponds strictly to Fig. 1a. It consists of a mixed-aquifer system: its upper part is the recharge zone and its lower part is driving the flow toward the outlet. Then it is shown that, under certain conditions, it is possible to replace the recharge zone of this 1-D model by a boundary condition. Furthermore, the possible importance of horizontally convergent or divergent flow is studied in a following section, which describes discharge of a reservoir–aquifer system with cylindrical axial symmetry.

Table 1 Parameters characterizing the major confined aquifers of three basins
Full size table

Time constant for the discharge of a 1-D mixed aquifer

The 1-D mixed aquifer model (Fig. 1a) has two components: the first one with width a u is the recharge zone and the second one with width a c is the deep aquifer driving the flow toward the outlet. Both aquifers are assumed homogeneous but they differ by the values of the hydraulic parameters (transmissivity, specific yield and storage coefficient). Before the initial time t = 0, the head results from a stable equilibrium between an infiltration in the recharge zone and a discharge at the extremity of the deep aquifer. At time t = 0, the recharge is modified (for example it reduces to 0) and a transient evolution toward a new equilibrium occurs. This is a generalization of the model developed by Rousseau-Gueutin et al. (2013) with the possibility of using different transmissivities and more realistic initial values.

Making use of the Dupuit’s approximation (Marsily 1986), and assuming that in the unconfined aquifer at the recharge zone, variations of the saturated thickness can be neglected so that its transmissivity is constant, the hydraulic head h(x,t) then satisfies a linear 1-D diffusion equation in both aquifers, with different hydraulic parameters. The previous study by Rousseau-Gueutin et al. (2013) using a numerical model to check the influence of a constant saturated thickness has shown that this approximation is satisfactory.

Here, contrary to Rousseau-Gueutin et al. (2013), the two components of the aquifer (0 < x < a u and a u < x < a u + a c = a) are characterized by their own transmissivities T u,T c; but, as proposed by these authors, they have their own storage coefficients S u, S c as well as their diffusivity coefficients D u = T u/S u, D c = T c/S c. S u > > S c and the differences between the two components can be characterized by different ratios such as:

$$ r=\sqrt{\frac{T_{\mathrm{c}}{S}_{\mathrm{c}}}{T_{\mathrm{u}}{S}_{\mathrm{u}}}} $$
(3)
$$ f=\sqrt{\frac{T_{\mathrm{u}}{S}_{\mathrm{c}}}{T_{\mathrm{c}}{S}_{\mathrm{u}}}}=\sqrt[]{\frac{D_{\mathrm{u}}}{D_{\mathrm{c}}}} $$
(4)

from which:

$$ rf=\frac{S_{\mathrm{c}}}{S_{\mathrm{u}}}\kern0.5em \mathrm{and}\kern0.5em \frac{r}{f}=\frac{T_{\mathrm{c}}}{T_{\mathrm{u}}} $$
(5)

The boundary conditions are that the head is null at the outlet and that upstream of the recharge zone, at x = 0, the flux is null (∂h/∂x = 0). Moreover, at the limit between the two aquifers (x = a u), the head is continuous as well as the hydraulic flux. Therefore:

$$ \begin{array}{l}{T}_{\mathrm{u}}\frac{\partial h}{\partial x}\left(0,t\right)=0\hfill \\ {}h\left(a,t\right)=0\hfill \\ {}{h}^{-}\left({a}_{\mathrm{u}},t\right)={h}^{+}\left({a}_{\mathrm{u}},t\right)\hfill \\ {}{T}_{\mathrm{u}}\frac{\partial {h}^{-}}{\partial x}\left({a}_{\mathrm{u}},t\right)={T}_{\mathrm{c}}\frac{\partial {h}^{+}}{\partial x}\left({a}_{\mathrm{u}},t\right)\hfill \end{array} $$
(6)

where the superscripted symbols −/+ are relative to aquifers u/c.

Initially (t ≤ 0), the head is at equilibrium under recharge to the upper aquifer and discharge occurs at x=a. Let B be the recharge rate for 0 <x<a u. The equilibrium equations for the head write:

$$ {T}_{\mathrm{u}}\frac{\partial^2h}{\partial {x}^2}+B=0\kern0.75em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.75em 0<x<{a}_{\mathrm{u}} $$
(7)
$$ {T}_{\mathrm{c}}\frac{\partial^2h}{\partial {x}^2}=0\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.75em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}} $$
(8)

The solution which satisfies the boundary conditions (Eq. 6) is:

$$ {h}^0(x)=B\left(\frac{a_u^2-{x}^2}{2{T}_{\mathrm{u}}}+\frac{a_u{a}_{\mathrm{c}}}{T_{\mathrm{c}}}\right)\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.75em 0<x<{a}_{\mathrm{u}} $$
(9)
$$ {h}^0(x)=B{a}_{\mathrm{u}}\frac{a-x}{T_{\mathrm{c}}}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.75em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}} $$
(10)

When at time t = 0, the recharge suddenly stops, the system tends toward a new equilibrium where h(x, + ∞) = 0. This evolution is described by a diffusion equation assuming B = 0 and the same boundary conditions (Eq. 6). The time-dependant hydraulic head then satisfies for t > 0:

$$ {S}_{\mathrm{u}}\frac{\partial h}{\partial t}={T}_{\mathrm{u}}\frac{\partial^2h}{\partial {x}^2}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0<x<{a}_{\mathrm{u}} $$
(11)
$$ {S}_{\mathrm{c}}\frac{\partial h}{\partial t}={T}_{\mathrm{c}}\frac{\partial^2h}{\partial {x}^2}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}} $$
(12)

with the boundary conditions (Eq. 6) and the initial conditions (Eqs. 9–10).

As proposed by Carslaw and Jaeger (1959), it is useful to replace this problem by a “complementary” one where the solution, called g(x, t), has an initial condition g(x, 0) = 0 everywhere. g(x, t) is assumed to obey the same boundary conditions (Eq. 6) but now, for t > 0, it is subjected to a recharge B for 0 < x < a u. The equations for this complementary problem write:

$$ {S}_{\mathrm{u}}\frac{\partial g}{\partial t}={T}_{\mathrm{u}}\frac{\partial^2g}{\partial {x}^2} + B\kern2em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0<x<{a}_{\mathrm{u}} $$
(13)
$$ {S}_{\mathrm{c}}\frac{\partial g}{\partial t}={T}_{\mathrm{c}}\frac{\partial^2g}{\partial {x}^2}\kern3em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<{a}_{\mathrm{u}}+{a}_{\mathrm{c}} $$
(14)

It is easy to show that the initial head h(x,t) is related to g(x,t) by:

$$ h\left(x,t\right)={h}^0(x)-g\left(x,t\right) $$
(15)

This stratagem based on linearity and homogeneity of both equations and boundary conditions, allows a much easier solution. Moreover, it can be used to compute the evolution of the potential for any sudden change of recharge conditions. Assume that at t = 0 the recharge suddenly changes from B to B’, then the transient solution for the hydraulic head h(x,t) can be expressed as a function of the solution g(x,t) of Eqs. (11–12) by:

$$ h\left(x,t\right)={h}^0(x)+g\left(x,t\right)\frac{B^{\hbox{'}}-B}{B} $$
(16)

The mathematical solution of the “complementary” g(x,t) is developed in Appendix 1. Taking into account Eq. (15), the full solution of the initial problem is expressed as:

$$ h\left(x,t\right)=\frac{2B}{S_{\mathrm{u}}{D}_{\mathrm{c}}}{\displaystyle \sum_{n=1}^{\infty}\frac{r \sin \left({\beta}_n{a}_{\mathrm{u}}/f\right) \cos \left({\beta}_nx/f\right) \exp \left(-{\beta}_n^2{D}_{\mathrm{c}}t\right)}{\beta_n^3\left[{a}_{\mathrm{c}}+{a}_{\mathrm{u}}r/f+{a}_c\left({r}^2-1\right){ \cos}^2\left({\beta}_n{a}_u/f\right)\right]}}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0<x<{a}_{\mathrm{u}} $$
(17)
$$ h\left(x,t\right)=\frac{2B}{S_{\mathrm{u}}{D}_{\mathrm{c}}}{\displaystyle \sum_{n=1}^{\infty}\frac{ \cos \left({\beta}_n{a}_{\mathrm{u}}\right) \sin \left({\beta}_n\left(a-x\right)\right) \exp \left(-{\beta}_n^2{D}_{\mathrm{c}}t\right)}{\beta_n^3\left[{a}_{\mathrm{c}}+{a}_u/rf+{a}_{\mathrm{u}}\left({r}^2-1\right){ \cos}^2\left({\beta}_n{a}_{\mathrm{c}}\right)/rf\right]}}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<a $$
(18)

In these expressions, β n is the nth real and positive root of the equation:

$$ \Delta \left(\beta \right)=r \cos \left({a}_{\mathrm{c}}\beta \right) \cos \left({a}_{\mathrm{u}}\beta /f\right)- \sin \left({a}_{\mathrm{c}}\beta \right) \sin \left({a}_{\mathrm{u}}\beta /f\right)=0 $$
(19)

Therefore the hydraulic head h(x,t) is expressed as a series of exponential functions of time such as ∑R n (x)exp(−t/τ n ) where τ n is given by τ n  = 1/(D c β 2 n ). For a given time t, the successive terms R n (x)exp(−t/τ n ) of this series rapidly decrease with n. In practice, the series can be reduced to its first term R 1(x)exp(−t/τ 1). This simplification is quantitatively justified as illustrated in Fig. 2a which compares, for several truncations, the profile of the head, once normalized to its initial value at x = a u, i.e. from Eqs. (9–10) normalized by a u a c B/T c. For several values of t (t = 0, t = τ 1, 2τ 1) and several ratios of the hydraulic parameters, the first term of the series is compared to more complete series (up to order n = 5). From Fig. 2a, it is justified to retain only the first term: this is verified for the case T u = T c and will be later confirmed for more general cases. Therefore the full solution h(x,t) given by Eqs. (17–18) can be replaced by:

Fig. 2
figure 2

Evolution of the hydraulic head for unconfined-confined aquifers with a u = 0.3, a c  = 1 (a c + a u = 1.3), a ratio of specific storage S u/S c = 102 and three ratios of transmissivity: a T u/T c =1, b T u/T c =10, and c T u/T c =0.1. The various curves correspond to three values of time (0,τ/2, τ) and to various approximations. The head is initially at equilibrium between its recharge for 0 < x < 0.3 a c and discharge at x = a c + a u. The three curves labelled hEqu, h1(t = 0) and h12345(t = 0) correspond to the true value and to two approximations for the initial head using either the first root β 1 or the five first ones. The curves labelled h1(tau/2) and h12345(tau/2) illustrate two approximations for h(x,τ/2), and those labelled h1(tau) and h12345(tau) the corresponding approximations for h(x,τ). In most cases, the various approximations are practically indistinguishable.

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$$ h\left(x,t\right)\approx \frac{2B}{S_{\mathrm{u}}{D}_{\mathrm{c}}}\frac{r \sin \left({\beta}_1{a}_{\mathrm{u}}/f\right) \cos \left({\beta}_1x/f\right) \exp \left(-{\beta}_1^2{D}_{\mathrm{c}}t\right)}{\beta_1^3\left[{a}_c+{a}_{\mathrm{u}}r/f+{a}_{\mathrm{c}}\left({r}^2-1\right){ \cos}^2\left({\beta}_1{a}_{\mathrm{u}}/f\right)\right]}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.75em 0<x<{a}_{\mathrm{u}} $$
(20)
$$ h\left(x,t\right)\approx \frac{2B}{S_u{D}_c}\frac{ \cos \left({\beta}_1{a}_{\mathrm{u}}\right) \sin \left({\beta}_1\left(a-x\right)\right) \exp \left(-{\beta}_1^2{D}_{\mathrm{c}}t\right)}{\beta_1^3\left[{a}_{\mathrm{c}}+{a}_{\mathrm{u}}/rf+{a}_{\mathrm{u}}\left({r}^2-1\right){ \cos}^2\left({\beta}_1{a}_{\mathrm{c}}\right)/rf\right]}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em {a}_{\mathrm{u}}<x<a $$
(21)

Within the limits of this simplification, Eqs. (20–21) describe the exponential decrease with time of the head as exp(−t/τ 1) with a time constant τ 1 = 1/D c. β 1 2. As stated in the previous, the value of β 1 is the first positive root of Eq. (19) but it can be given an approximate analytical expression based on a limited development of Δ(β) in the vicinity of β = 0. A second-order Taylor series development in β of Eq. (19) yields the first root of Δ(β) = 0 as:

$$ {\beta}_1^2\cong \frac{r}{\left[{a}_{\mathrm{u}}{a}_{\mathrm{c}}/f+r\left({a}_{\mathrm{u}}^2/{f}^2+{a}_{\mathrm{c}}^2\right)/2\right]}=\frac{1}{S_{\mathrm{u}}/{S}_{\mathrm{c}}\left[{a}_{\mathrm{u}}{a}_{\mathrm{c}}+\left({T}_{\mathrm{c}}/{T}_{\mathrm{u}}\right){a}_{\mathrm{u}}^2/2\right]+{a}_{\mathrm{c}}^2/2} $$
(22)

and the corresponding time constant:

$$ {\tau}_1=\frac{1}{D_{\mathrm{c}}{\beta}_1^2}\cong \frac{a_{\mathrm{c}}{a}_{\mathrm{u}}{S}_{\mathrm{u}}}{T_{\mathrm{c}}}+\frac{a_{\mathrm{u}}^2}{2{D}_{\mathrm{u}}}+\frac{a_{\mathrm{c}}^2}{2{D}_{\mathrm{c}}} $$
(23)

This expression is similar to that given by Rousseau-Gueutin et al. (2013) with differences related to the fact that here T u ≠ T c. The time constant τ 1 given by Eq. (23) is expressed as a sum of three terms which have different values depending on the geometric and hydraulic parameters. The two last terms of Eq. (23) reflect the contribution by the unconfined (subscripts u) and the confined part (subscript c), respectively. The first term of Eq. (23) is a cross-contribution by u and c and will be shown to be the dominant contribution for the cases of interest.

The relative importance of the three terms in Eq. (23) can be assessed as shown in Table 3 (Appendix 1) which compares the relative contributions by the three terms of Eq. (23) for a wide range of aquifer parameters. It appears that, for S c/ S u in the range 10−4 – 10−2, the last term of Eq. (23) is relatively negligible (less than 5 % of the sum) for any T c/ T u or a u/a c. Moreover, when T c ≤ T u, the first term of Eq. (23) is clearly dominant (above 80 % of the sum). It should be noted that for T c = T u and S c < < S u the time constant Eq. (23) reduces to τ 1 ~ (S u / T c) a u (a c  + a u/2) which is exactly the formula proposed by Rousseau-Gueutin et al. (2013); however, a new aspect demonstrated by the present work is that the conductivity ratio T c/T u appears as a relatively important parameter, as illustrated in Fig. 2b,c.

The effect of this transmissivity difference can be assessed as follows: when T c/T u < 1, the hydraulic head along the unconfined aquifer (0 < x < a u) as illustrated by Fig. 2b is almost constant. In other words, for T c < T u, the unconfined aquifer is playing the role of reservoir with a hydraulic head that is constant with x and which is feeding the neighbouring confined aquifer. Conversely, when T c/T u > 1, as illustrated in Fig. 2c, the hydraulic head in the unconfined aquifer presents, near the position x = a u, a strong gradient due to the low transmissivity of the neighbouring confined aquifer.

The unconfined aquifer is assumed to model the relatively elevated margins of the main confined aquifer and its actual hydraulic transmissivity is difficult to assess (Chavez et al. 1994; Viviroli et al. 2003, 2007). Nevertheless, it seems plausible that this recharge zone has a transmissivity at least equal to that of the deeper confined aquifer, i.e. that T u ≥ T c. If this is the case, the unconfined aquifer can be modelled as a simple reservoir as developed in the next section.

From the mixed 1-D model to the discharge of a reservoir–aquifer system

As illustrated in Fig. 3, this 1-D model is obtained from the previous one in the case of very large T u. Then, lateral variations of the head h(x,t) in the unconfined aquifer can be neglected and the upper aquifer is reduced to a reservoir, or tank, of width a u and storage capacity S u connected to the lower confined aquifer. A single parameter M = S u a u now characterises the integrated storage capacity of the reservoir. For t < 0, the recharge still occurs for 0 < x < a u so that the hydraulic head of the reservoir is the same as that of the neighbouring confined reservoir at x = a u. This is accounted for by the boundary condition at x = a u: the flux entering the confined aquifer is equal to the rate at which the water mass inside the tank decreases, i.e. Mh = S u a u h. Mathematically, this is a boundary condition of the third type (or Fourier condition) prescribed at x = a u as a linear relation between the time derivative of h(x,t) for t > 0 and its space derivative:

Fig. 3
figure 3

Reservoir/confined-aquifer system. a Schematic cross-section of the system. The confined aquifer with width a c stands for a u < x < a c + a u. The outlet is at x = a c  + a u and the limit x = a u is connected to a reservoir with width a u and storage S u . b Corresponding evolution of the hydraulic head in the reservoir and in the confined aquifer.

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$$ M\frac{\partial h\left({a}_{\mathrm{u}},t\right)}{\partial t}={\displaystyle {T}_{\mathrm{c}}}{\left.\frac{\partial h\left(x,t\right)}{\partial x}\right|}_{x={a}_{\mathrm{u}}} $$
(24)

Before the initial time t = 0, the assumed hydraulic head in the system results from an equilibrium between the recharge at rate B in the tank (for 0 < x < a u) and the discharge at the end of the confined aquifer, at x = a u + a c. In the confined aquifer, the initial head varies linearly from h = h 1 at x = a u to h = 0 at x = a u + a c and the value h(a u,0) = h 1 is related to the initial recharge rate B by:

$$ {h}_1=\frac{B{a}_{\mathrm{c}}{a}_{\mathrm{u}}}{T_{\mathrm{c}}} $$
(25)

At t = 0, the recharge suddenly stops and the tank feeds the confined aquifer until it is completely depleted. The evolution of the head toward a new equilibrium is obtained as the solution of the diffusion equation using Laplace transforms as explained in Appendix 2. This solution for h(x,t) is shown to be:

$$ h\left(x,t\right)=2{h}_1{\displaystyle {\sum}_{n=1}^{\infty}\frac{ \sin \left({\beta}_n\left(a-x\right)\right) \exp \left(-{D}_{\mathrm{c}}{\beta}_n^2t\right)}{a_{\mathrm{c}}^2{\beta}_n^2 \sin \left({\beta}_n{a}_{\mathrm{c}}\right)\left[1+M/{S}_{\mathrm{c}}{a}_{\mathrm{c}}+{\left(M{\beta}_n/{S}_{\mathrm{c}}\right)}^2\right]}} $$
(26)

where β n is the nth positive root of the equation in β:

$$ \Delta \left(\beta \right)= \cos \left(\beta {a}_{\mathrm{c}}\right)-\left(\beta {a}_{\mathrm{c}}\right)\frac{M}{S_{\mathrm{c}}{a}_{\mathrm{c}}} \sin \left(\beta {a}_{\mathrm{c}}\right)=0 $$
(27)

As in the previous case, the first term (n = 1) of the series Eq. (26) is highly dominant so that this series can be truncated to yield:

$$ h\left(x,t\right)\cong 2{h}_1\frac{ \sin \left[{\beta}_1\left(a-x\right)\right] \exp \left(-{D}_{\mathrm{c}}{\beta}_1^2t\right)}{a_{\mathrm{c}}^2{\beta}_1^2 \sin \left({\beta}_1{a}_{\mathrm{c}}\right)\left[1+M/{S}_{\mathrm{c}}{a}_{\mathrm{c}}+{\left(M{\beta}_1/{S}_{\mathrm{c}}\right)}^2\right]} $$
(28)

which describes the exponential decrease of h(x,t). Furthermore, a linear development of Δ(β) in the vicinity of β = 0 shows that the first root β 1of Eq. (27) can be approximated by:

$$ {\beta}_1\cong \frac{1}{a_c\sqrt{\left[1/2+M/\left({S}_{\mathrm{c}}{a}_{\mathrm{c}}\right)\right]}} $$
(29)

As shown in Table 4 of Appendix 2, this approximation appears to be satisfactory to within 5 % as long as M/(S c a c) > 1.

According to these results, for any x, the head h(t) decreases exponentially toward 0 as exp(−t/τ) where the time constant τ is:

$$ \tau =\frac{1}{D_{\mathrm{c}}{\beta}_1^2}\cong \frac{a_{\mathrm{c}}^2}{D_{\mathrm{c}}}\left[1/2+M/\left({S}_{\mathrm{c}}{a}_{\mathrm{c}}\right)\right]=\frac{a_{\mathrm{c}}^2}{2{D}_{\mathrm{c}}}+\frac{a_{\mathrm{c}}{a}_{\mathrm{u}}{S}_{\mathrm{u}}}{T_{\mathrm{c}}} $$
(30)

On the basis of this formula, it is interesting to discuss the implication of the various hydraulic parameters on the order of magnitude of the time constant τ. When the integrated storage M = S u a u of the upper unconfined aquifer is negligible compared to that of the confined one (S c a c) then τ ≈ a c 2/2D c. This is consistent to within 20 % with the exact solution for an isolated confined layer given in Eq. (1). When using parameters relevant to this confined aquifer (T c ~ 102 m2s−1, S c ~ 104 implying D c ~ 102 m2s−1 and a c ~500 km), one obtains τ ~ 40 years which is a very low value.

Compared to this basic configuration of an isolated confined aquifer, the storage capacity due to the unconfined aquifer drastically increases the estimated value of τ as implied by the second term of Eq. (30). In fact, the case where S u a u > > S c a c is quite realistic, since S u is on the order of 0.1 which greatly exceeds S c ~10−4. Assuming a u ~ 100 km and a c ~ 500 km results in M/S c a c ~200 and a value of τ on the order of 15,000 years. This is well above the estimate neglecting the unconfined storage. With such assumptions, Eq. (30) reduces to:

$$ \tau \cong \frac{a_{\mathrm{c}}M}{T_{\mathrm{c}}}=\frac{a_{\mathrm{c}}{a}_{\mathrm{u}}{S}_{\mathrm{u}}}{T_{\mathrm{c}}} $$
(31)

which is similar to Eq. (2).

2-D effects: discharge of a reservoir–aquifer system with cylindrical symmetry

In the two previous sections, a 1-D model was assumed, which implies that, on average, the velocity vector is everywhere parallel to x. The present section deals with two-dimensional (2-D) effects due to converging or diverging flow prescribed by the basin geometry. For this study, the reservoir–aquifer system of Fig. 3 is extended by rotation around a vertical axis. With respect to Fig. 3, this vertical axis is fixed either at x < 0 in order to study a divergent flow or at x > a c + a u for the convergent one. Such geometries are quite similar to those proposed by Sonntag (1986) for the study of NSA (his models of a cylindrical mountain or cylindrical depression).

Diverging flow: model of an “island” or “cylindrical mountain”

This model requires a radial coordinate r which measures the distance with respect to the vertical axis of symmetry (at r = 0). A topographic height (or “island”) surrounds the axis. The central area 0 < r < R, or part of it, serves as a reservoir where the recharge takes place by precipitation, seepage, etc. This central reservoir is loading a radially diverging confined aquifer which occupies the area: R < r < R + a c. The outlet of the aquifer is at r = R + a c. The model is represented on Fig. 4. Of course, the model does not need to be fully cylindrical; a “slice” of the cylinder, like a slice of pizza, is sufficient.

Fig. 4
figure 4

Schematic cross-section and 3-D view of the model used for a divergent flow. a The evolution of the head in the aquifer along a radial direction or from the initial condition when recharge was occurring in the reservoir for R – a u < r < R. b A perspective view of the symmetrical model with divergent flow (from r = 0). The outlet is at r = R + a c

Full size image

The unconfined aquifer, where rainfall contributes to the recharge, is located along a circular zone defined by R – a u < r < R. Its specific yield is S u and its width along the radial distance is a u with a u ranging from 0 to R. For a u = 0, the reservoir comprises the whole circle r < R and the surface-integrated water reserve is given by S uπ R 2. The specific case considered here assumes a u < <R so that the reservoir forms a ring of small width a u with an integrated water surface reserve given by S u2π Ra u or a volume reserve S u2π Rha u.

The reservoir is connected to the external confined aquifer beginning at r = R. The decrease rate of the reservoir volume compensates the input flux into the surrounding aquifer. The boundary condition at r = R in the confined aquifer is a linear relation between the time derivative of the head and its normal derivative at r = R:

$$ 2\uppi R{S}_{\mathrm{u}}{a}_{\mathrm{u}}\frac{\partial h\left(R,t\right)}{\partial t}=2\uppi R{T}_{\mathrm{c}}{\left.\frac{\partial h\left(r,t\right)}{\partial r}\right|}_{\begin{array}{l}r=R\\ {}\end{array}} $$

or, with M = S u a u:

$$ M\frac{\partial h\left(R,t\right)}{\partial t}={T}_{\mathrm{c}}{\left.\frac{\partial h\left(r,t\right)}{\partial r}\right|}_{\begin{array}{l}r=R\\ {}\end{array}} $$
(32)

(The case of a reservoir using the whole surface r < R may be dealt with by defining M as S u R/2 instead of S u a u). The other boundary condition corresponds to the outlet (fixed head) prescribed at r = R + a c; therefore, h(R + a c,t) = 0.

The confined aquifer in the area R < r < R + a c is characterized by T c (its transmissivity) and S c (its storage coefficient), whence D c =T c/S c is its diffusivity. Since the flow is radial, the hydraulic head h(r,t) satisfies the diffusion equation in cylindrical coordinates.

$$ \frac{T_{\mathrm{c}}}{r}\frac{\partial }{\partial r}\left(r\frac{\partial h}{\partial r}\right)={S}_{\mathrm{c}}\frac{\partial h}{\partial t} $$
(33)

The initial hydraulic head of the confined aquifer is the result of an equilibrium between the prescribed recharge condition Eq. (32) at r = R and its discharge at r = R + a c prescribing h = 0. The general solution of the steady-state cylindrical equation (i.e. corresponding to Eq. (33) with ∂h/∂t = 0) which satisfies h(R) = h 1 and h(R + a c) = 0 is now logarithmic in r and can be written:

$$ h(r)={h}_1\frac{ \ln \left(R+{a}_{\mathrm{c}}\right)- \ln (r)}{ \ln \left(R+{a}_{\mathrm{c}}\right)- \ln (R)} $$
(34)

The constant h 1 (the initial head) is a function of the rainfall-seepage rate B:

$$ {h}_1=B\frac{a_{\mathrm{u}}R}{T_{\mathrm{c}}} \ln \left(\frac{R+{a}_{\mathrm{c}}}{R}\right) $$
(35)

From this initial solution, the transient evolution can now be calculated as a solution of Eq. (33) using the same techniques. This solution is developed in Appendix 3 as:

$$ h\left(r,t\right)={h}_1{\displaystyle {\sum}_{n=1}^{\infty } \exp \Big(-{D}_{\mathrm{c}}}{\beta}_n^2t\Big){k}_n(r) $$
(36)

where k n (r) is a function depending on Bessel functions J0, J1,Y0, and Y1, and β n is the nth positive root of the equation in β:

$$ \begin{array}{l}\left\{{\mathrm{J}}_0\left[{\beta}_nR\right]{\mathrm{Y}}_0\left[{\beta}_n\left(R+{a}_{\mathrm{c}}\right)\right]-{\mathrm{J}}_0\left[{\beta}_n\left(R+{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_0\left[{\beta}_nR\right]\right\}+\hfill \\ {}\left\{{\mathrm{J}}_0\left[{\beta}_n\left(R+{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_1\left[{\beta}_{\mathrm{n}}R\right]-{\mathrm{J}}_1\left[{\beta}_nR\right]{\mathrm{Y}}_0\left[{\beta}_n\left(R+{a}_{\mathrm{c}}\right)\right]\right\}=0\hfill \end{array} $$
(37)

The first root β 1 of this equation is approximated by expanding it when β is in the vicinity of 0. This yields:

$$ {\beta}_1\cong \frac{1}{\sqrt{R^2\left[\left(\frac{M}{S_{\mathrm{c}}R}-\frac{1}{2}\right) \ln \left(\frac{R+{a}_{\mathrm{c}}}{R}\right)+\frac{{\left(R+{a}_{\mathrm{c}}\right)}^2-{R}^2}{4{R}^2}\right]}} $$
(38)

and the corresponding time constant:

$$ \tau =\frac{1}{D_{\mathrm{c}}{\beta}_1^2}\cong \frac{a_{\mathrm{c}}^2}{D_{\mathrm{c}}}\left\{\frac{1+2R/{a}_{\mathrm{c}}}{4}-{\left(R/{a}_{\mathrm{c}}\right)}^2\left(\frac{1}{2}-\frac{M}{S_{\mathrm{c}}R}\right) \ln \left(1+{a}_{\mathrm{c}}/R\right)\right\} $$
(39)

When a c < <R, the cylindrical geometry approaches that of a 1-D case. In this case, the expression of τ given by Eq. (39) may be expanded as a function of a c/R:

$$ \tau \to {\tau}_{\lim}\cong \frac{a_{\mathrm{c}}^2}{D_{\mathrm{c}}}\left[\frac{1-{a}_{\mathrm{c}}/3R}{2}+\frac{M}{S_{\mathrm{c}}{a}_{\mathrm{c}}}\left(1-{a}_{\mathrm{c}}/2R\right)\right] $$
(40)

This expression is quite similar to Eq. (30) which applies to the 1-D model; the difference is that the first term is multiplied by (1 – a c/3R) and the second one by (1 – a c/2R). Compared to the model of parallel flow in the previous section, the divergent nature of the flow results in a decrease of the time constant τ as can be expected intuitively.

The limiting case where M> > S c R deserves attention. When the upstream storage of water is dominant, Eq. (39) reduces to:

$$ \tau \cong \ln \left(\frac{\rho }{R}\right)\frac{S_{\mathrm{u}}{a}_{\mathrm{u}}R}{T{}_{\mathrm{c}}}+\frac{\rho^2-{R}^2}{4{D}_{\mathrm{c}}} $$
(41)

where ρ = R + a c is the radius of the outlet. Moreover, if ρ/R is large enough so that ln(ρ/R) is at least on the order of 1, Eq. (41) becomes:

$$ \tau \cong \ln \left(\frac{\rho }{R}\right)\frac{S_{\mathrm{u}}{a}_{\mathrm{u}}R}{T{}_{\mathrm{c}}}= \ln \left(\frac{R+{a}_{\mathrm{c}}}{R}\right)\frac{S_{\mathrm{u}}{a}_{\mathrm{u}}R}{T{}_{\mathrm{c}}} $$
(42)

The latter expression presents some analogy with the corresponding asymptotic one (Eq. 30) valid for the 1-D case: Eq. (42) involves R instead of a c for Eq. (31) but τ decreases when a c/R increases.

Converging flux: model of a “lake” or “cylindrical depression”

This is the convergent version of the axially symmetric model. In this case, the recharge margins with high topography are located at r = R along the external limits of the basin, whereas the outlet lies closer to its centre at r = ρ = R – a c, (with a c < R). Between these two structures, for R – a c < r < R, a convergent flow occurs in the main confined aquifer, with transmissivity T c and storage coefficient S c over a width a c as shown in Fig. 5. The reservoir associated with the recharge zone is a narrow (a u < <R) circular fringe (R < r < R + a u) which is hydraulically connected to the main confined aquifer. The integrated storage capacity of this reservoir is characterized by the parameter M = S u a u.

Fig. 5
figure 5

Schematic cross-section and three-dimensional (3-D) view of the model used for a convergent flow. a The evolution with time of the head in the aquifer along a radial direction, from the initial condition when recharge was occurring in the reservoir. b A perspective view of the symmetrical model of convergent flow (toward r = 0). The outlet is at r = R – a c and the reservoir lies within R < r < R + a u

Full size image

In the aquifer, the head satisfies a diffusion equation similar to Eq. (33), the difference being the boundary conditions. The prescribed conditions at the reservoir–aquifer limit arise from the assumptions of head continuity and conservation of mass, which yields, at r = R:

$$ M\frac{\partial h\left(R,t\right)}{\partial t}=-{T}_{\mathrm{c}}{\left.\frac{\partial h\left(r,t\right)}{\partial r}\right|}_{\begin{array}{l}r=R\\ {}\end{array}} $$
(43)

At the outlet, the boundary condition is h(R – a c,t) = 0. (Note that the assumption a c = R would be misleading because for such convergent flow, the hydraulic head at r = 0 would be undefined). The general solution of the steady-state cylindrical equation (i.e. corresponding to Eq. (33) with ∂h/∂t = 0) which satisfies h(R) = h 1 and h(R-a c) = 0, is:

$$ h(r)={h}_1(r)\frac{ \ln \left(R-{a}_{\mathrm{c}}\right)- \ln (r)}{ \ln \left(R-{a}_{\mathrm{c}}\right)- \ln (R)} $$
(44)

and h 1 can be expressed as a function of the recharge rate and hydraulic parameters:

$$ {h}_1=B\frac{a_{\mathrm{u}}R}{T_{\mathrm{c}}} \ln \left(\frac{R}{R-{a}_{\mathrm{c}}}\right) $$
(45)

Starting from this initial solution, the transient evolution is calculated as a solution of Eq. (33) as developed in Appendix 3 and yields:

$$ h\left(r,t\right)={h}_0{\displaystyle {\sum}_{n=1}^{\infty } \exp \Big(-{D}_{\mathrm{c}}}{\beta}_n^2t\Big){kk}_n(r) $$
(46)

where kk n (r) is also a function depending on Bessel functions J0, J1,Y0, and Y1, and β n is the nth positive root of the equation in β:

$$ \begin{array}{l}\Delta \left(\beta \right)=\left\{-{\mathrm{J}}_0\left[{\beta}_nR\right]{\mathrm{Y}}_0\left[{\beta}_n\left(R-{a}_c\right)\right]+{\mathrm{J}}_0\left[{\beta}_n\left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_0\left[{\beta}_nR\right]\right\}+\hfill \\ {}\left\{{\mathrm{J}}_0\left[{\beta}_n\left(R-{a}_{\mathrm{c}}\right)\right]{\mathrm{Y}}_1\left[{\beta}_nR\right]-{\mathrm{J}}_1\left[{\beta}_nR\right]{\mathrm{Y}}_0\left[{\beta}_n\left(R-{a}_{\mathrm{c}}\right)\right]\right\}=0\hfill \end{array} $$
(47)

As previously, the first root β 1 of this equation may be approximated by:

$$ {\beta}_1\cong \frac{1}{\sqrt{R^2\left[\left(\frac{M}{S_{\mathrm{c}}R}+\frac{1}{2}\right) \ln \left(\frac{R}{R-{a}_{\mathrm{c}}}\right)+\frac{{\left(R-{a}_{\mathrm{c}}\right)}^2-{R}^2}{4{R}^2}\right]}} $$
(48)

corresponding to a time constant:

$$ \tau =\frac{1}{D_{\mathrm{c}}{\beta}_1^2}\cong \frac{a_{\mathrm{c}}^2}{D_{\mathrm{c}}}\left\{\frac{1-2R/{a}_{\mathrm{c}}}{4}-{\left(R/{a}_{\mathrm{c}}\right)}^2\left(\frac{1}{2}+\frac{M}{S_{\mathrm{c}}R}\right) \ln \left(1-{a}_{\mathrm{c}}/R\right)\right\} $$
(49)

For a c < <R, the expression of τ obtained by expanding Eq. (49) as a function of a c/R approaches that of the 1-D case:

$$ \tau \to {\tau}_{\lim}\cong \frac{a_{\mathrm{c}}^2}{D_{\mathrm{c}}}\left[\frac{1+{a}_{\mathrm{c}}/3R}{2}+\frac{M}{S_{\mathrm{c}}{a}_{\mathrm{c}}}\left(1+{a}_{\mathrm{c}}/2R\right)\right] $$
(50)

With respect to the 1-D model, the first term of τ is multiplied by (1 + a c/3R) and the second term by (1 + a c/2R). Therefore, the convergent flow is associated with an increase of the time constant τ as has already been suggested.

In the case where M> > S c R, Eq. (49) reduces to:

$$ \tau \cong \ln \left(\frac{R}{\rho}\right)\frac{S_{\mathrm{u}}{a}_{\mathrm{u}}R}{T{}_{\mathrm{c}}}+\frac{\rho^2-{R}^2}{4{D}_{\mathrm{c}}} $$
(51)

where ρ = R – a c is the radius of the outlet (ρ < R). Moreover, if R/ρ is large enough so that ln(R/ρ) is at least of the order of 1, Eq. (51) becomes:

$$ \tau \cong \ln \left(\frac{R}{\rho}\right)\frac{S_{\mathrm{u}}{a}_{\mathrm{u}}R}{T{}_{\mathrm{c}}}= \ln \left(\frac{R}{R-{a}_{\mathrm{c}}}\right)\frac{S_{\mathrm{u}}{a}_{\mathrm{u}}R}{T{}_{\mathrm{c}}} $$
(52)

which is analogous to Eq. (42) in the case of divergent flow. However, in this case, the logarithmic factor becomes very large when a c tends toward R, i.e. when the radius of the outlet becomes negligible.

Conclusion on convergence–divergence effects

The whole set of results concerning the effects of divergence/convergence on the time constant is summarized in Fig. 6, where the time constants τ οbtained for various models of flow (diverging, parallel and converging) are presented. The ordinate is the time constant τ normalized to the basic parameter D c/a c 2 and the abscissa is the ratio a c/R which characterizes the curvature of the system. The various curves correspond to various values of the normalized storage capacity M/(S c a c). The abscissa a c/R = 0 corresponds to the 1-D model, with parallel flow, the abscissa a c/R > 0 to a diverging flow and, by convention, the abscissa a c/R < 0 to a converging flow. In the latter case, the parameter a c/R ranges from 0 (no convergence) to −1; the last value corresponds to the limit where a c → │R│ so that the head has a singular behaviour. Figure 6 confirms that the flow divergence induces a decrease of τ as a function of the curvature, which is unlimited since a c/R is not limited. However, the convergence results in a very significant increase of τ, which may even be very large when the size of the outlet tends toward 0. For a small curvature (a c/R close to 0), the slope of the normalized τ as a function of │a c/R│ follows the previously predicted trends with a slope of −1/3 for low M/S c a c and −1/2 for large M/S c a c.

Fig. 6
figure 6

Chart of the normalized correction applied to the time constant in the divergent and convergent cases. The curves show the reduced value of τ (normalized by a c 2 /D c) as a function of the curvature of the reservoir zone, characterized by a c /R: by convention a c/R < 0 is a convergent flux. The curves are parameterized by the normalized storage of the reservoir M/S c a c

Full size image

Discussion and application

The obtained formulas are now applied to actual transient phenomena occurring in the selected basins. Since the presented models are based on many simplifications, only first-order phenomena are considered, with particular attention to the hydraulic parameters and the large-scale geometry of the systems.

The large aquifer basins of North Africa and East Australia received, during the early Holocene, abundant rainfall which stopped around 4,000 years BP resulting in the present arid climate (Gasse 2000; Taylor et al. 2009; Lézine et al. 2011). This drastic decrease in rainfall and recharge for a period of 4,000 years and the recent withdrawals of groundwater for agricultural use are clearly responsible for the critical piezometric falls observed in some deep aquifers (e.g. Besbes and Horriche 2007 for the NWSAS). However, these observations are hardly significant at the regional scale. In fact, local drawdowns of piezometric levels observed in areas of intense withdrawal are clearly associated with local pumping effects: their apparent time constant τ (defined as τ = h/(∂h/∂t), i.e. assuming an exponential decay of h(t)) is on the order of 100 years. This is much smaller than the time constant associated with recharge variations.

If one assumes that no rainfall has recharged these deep aquifers for about 4,000 years, the fact that they are not completely depleted indicates that their time constant τ is on the order of several thousand years. Alternatively, it has been proposed that some modern rainfall-recharge is still occurring even though it is much weaker than it was at the beginning of the Holocene (Ould Baba Sy 2005; Gonçalvez et al. 2013). In any case, the time constant defined in the present study is still valid since it represents a sudden modification of the recharge rate, no matter if the final state tends toward a complete drought. In this case, τ would characterize the evolution from an initial “very rainy” situation to a “less rainy” one.

As stated in the definition of the generic model, the confined aquifers of the large basins NWSAS, NSA and GAB are characterized by the following parameters: T c ~ 103–102 m2s−1, S c ~ 104 and a c ~500 km. Assuming that these confined aquifers are not connected to any unconfined aquifer operating as a reservoir, then from Eq. (1), their time constant τ would be in the range 40–400 years, i.e. much less than the 4,000 years of the beginning of dry conditions. Therefore, without modern recharge, these aquifers would be depleted.

As noted previously, the value of τ drastically increases when the main confined aquifer is connected to an upstream unconfined one with a much larger storage capacity. According to Eq. (30), valid for T u ≥ T c, the presence of an unconfined layer characterized by S u a u = 5 S c a c is sufficient to multiply the previous τ value by a factor of about 11. This factor can easily be obtained: assuming S c ~ 104 and a c ~ 500 km, even a narrow reservoir with a u ~ 2.5 km and S u ~ 101 would be sufficient.

Therefore, the presence of an upstream unconfined aquifer coupled with the recharge zone plays a major role for maintaining the head in the whole basin during periods on the order of 1–10 ka. In the cases of NSA (Heinl and Brinkmann 1989) and GAB (Habermehl 1980), relatively large unconfined aquifers with widths in the range 50–200 km, connected to the main confined one, have been observed. For NWSAS, the CI and CT main aquifers present many outcrops at the periphery of the basin (such as the Atlas Mountain range) with widths on the order of a few tens of km (Ould Baba Sy 2005). These narrow outcrops may behave as potential reservoirs where recharge occurred in the past and still continues.

Another important geometrical factor resulting in a larger τ value is the convergence of hydraulic flows which affects many basins, as shown by present piezometric maps. In the CI aquifer of NWSAS groundwater flows converge on the Gulf of Gabes in South Tunisia (Besbes and Horriche 2007). In Australia, a concentration of groundwater flows toward Lake Eyre occurs in the endorheic GAB basin (Habermehl 1980). The confined aquifer of NSA also presents converging features toward the Gulf of Syrte (Heinl and Brinkmann 1989). Such convergent geometry can be modelled by the cylindrical model for converging flow (Fig. 5). Assume that the discharge zone which represents the outlet has a radius of 100 km; if the main aquifer has a width a c = 500 km, the radius where the recharge zone is active becomes R = 600 km. This leads to a ratio a c/R = − 0.8 to –0.9, with a minus sign according to the convention used in Fig. 6. For such a ratio, a convergence-induced multiplicative factor of around 3–4 applies to τ, as can be seen in Fig. 6. This significantly contributes to the durability of the system. Therefore, the evaluation of the time constant of large aquifer basins suggests that their geometry and structure explain why they are still discharging water when they have been subjected to arid conditions for more than 4,000 years. Moreover, this evaluation is not modified by the possible contribution of a small modern rainfall recharge.

The concept of time constant can also be applied to a previous transient phenomenon: the onset of the “green Sahara”. At the beginning of the Holocene, the currently arid Sahara was much wetter than today (Gasse 2000; Taylor et al. 2009; Lézine et al. 2011). From about 12,500 to 6,000 years BP, its lowlands contained many freshwater bodies now almost completely dried out (this wet phase was followed by the current dry period which occurred around 4,000 years BP). During this humid event, a time-lag on the order of 3,000 years has been shown to have existed between the onset of the wet period and the maximum development of freshwater bodies. The chronology is based on the interpretation of many palaeo-hydrology observations at the scale of the whole Sahara (Lézine et al. 2011) and of high-quality data in specific areas such as Lake Yoa in Northern Chad (Grenier et al. 2009). A likely explanation is that this time-lag reflects the delayed contribution of groundwater to the near-surface-water bodies, i.e. the time necessary for the recharge of deep aquifers (starting at the onset of the humid period) to reach the outlets. The time constant computed here for application to the discharge of large Saharan basins may also be applied to their recharge. A time-lag of several thousand years between the onset of the humid period and their maximum extent is indeed consistent with the estimated time constant.

This study can be used to give a first-order estimate of the natural decay time of the hydraulic head. In many instances, unconfined aquifers provide a major contribution to water reserves and can be assimilated to reservoirs. A quite robust Eq. (31) was obtained for the case where the unconfined reservoir-like aquifer has an integrated storage capacity M = S u a u much larger than S c a c: the time constant τ reduces to τ ~ M a c/T c = S u a u a c/T c, which does not depend any more on S c and T u. Realistic evaluations of a c, a u and T c are often available and S u is related to the local porosity (another possibility is to make use of existing remote gravity measurements that record variations of the water mass over time (e.g. GRACE), as proposed by Sun et al. (2010).

For the previous applications, relatively large values of τ are explained by the major role of unconfined aquifers with storage coefficients much larger than those of confined ones. This feature has already been emphasized by many authors (e.g. OSS 2005, which is a report for the NWSAS) and deserves further discussion. The value of S u can generally be estimated as the accessible porosity, which is generally on the order of 0.1 (Marsily 1986). By contrast, the low storage coefficient S c of confined aquifers is due to the compressibility of the water and of the pore volume and depends mainly on the mechanical deformation ability of rocks. The resulting S c value lies in the range 1/100–1/1000 of that of S u; however, the existence of such a great difference of storativity suggests several limitations. First, during a depletion event, the top of the confined aquifer may become unsaturated in some places so that its behaviour becomes that of an unconfined one. Second, possible vertical fluxes (leakage) between neighbouring aquifers have been neglected, whereas they may contribute to increase the available water storage. For instance, Bredehoeft et al. (1983) found that leakage due to fractures is important in explaining the head distribution in the confined Dakota Aquifer System (USA). In fact, Eq. (30) applies to a case where the confined aquifer is hydraulically isolated from its surroundings except through the connection with its upstream unconfined neighbour.

Equation (31) may also receive a very simple physical interpretation: the time constant τ is similar to that of a falling head permeameter for measuring the “hydraulic resistance” a c/T c of a porous medium limiting the discharge toward an outlet (Marsily 1986). This analogy was already used by Ould Baba Sy (2005) to reconstruct the “initial” hydraulic head of the NWSAS aquifers at the beginning of the dry period.

Conclusions

This work is a continuation of the previous study by Rousseau-Gueutin et al. (2013) of the transient depletion of a mixed (unconfined-confined) aquifer system when subjected to a sudden recharge change. The various models developed here present both a generalization of the previous study for more complex and more realistic cases (initial condition, 2-D effects, etc.), and some simplifications where the initial water storage is geometrically concentrated.

When such a mixed aquifer system is submitted to an abrupt change of recharge, the hydraulic head evolution from an initial equilibrium to a final one closely follows an exponential trend exp(−t/τ) characterized by the time constant τ. For simple geometries, τ is expressed analytically as a function of the characteristics of the system: the hydraulic parameters (transmissivities T u,T c and storage coefficients S u, S c) and the geometrical ones (width of the two components a u, a c and for the cylindrical case, radius R of the recharge zone). Table 2 summarizes the analytical formulae expressing τ as a function of these parameters as developed in the previous analytical sections. These formulae are labelled from Nos. 1 to 10; each of them corresponds to a specific assumption made on the values of the parameters as well as on the actual geometry of the flow pattern. Since these various formulae are to be used in practical situations, the following paragraphs are providing some guidance for discussing their relevance domain.

Table 2 Summary of analytical expressions obtained
Full size table

Formulae Nos. 1–4 apply to the linear 1-D case where the curvature of the flow lines can be neglected, whereas formulae Nos. 5–10 emphasize the role of cylindrical flow characterised by the curvature radius R of the upstream reservoir.

For the linear 1-D case, formula No. 1 applies to the general case of a mixed aquifer and does not need any specific assumption. However, in practice, several approximations are often justified since the upstream aquifer component of the basic model is identified as an unconfined aquifer with large storage capacity (S c < < S u). In fact, the relevant formula also depends on the relative values of transmissivities: the case where T c = T u leads to formula No. 2 (already developed by Rousseau-Gueutin et al. 2013); if T u > > T c, it is possible to replace the upstream component of width a u and large capacity S u by an equivalent reservoir of integrated storage a u S u resulting in formula No. 3. If, moreover, the horizontally integrated a u S u is much larger that a c S c, then formula No. 3 reduces to No. 4 where the parameters S c and T u are absent.

The effects of radial convergence or divergence are to be discussed on the basis of cylindrical models; only the case where T u > > T c is considered where the upstream component can be replaced by an equivalent reservoir with curvature radius R. Formulae Nos. 5 and 7 respectively give the expression of τ for the general case of a diverging flow or a converging one; these formulae are the counterparts of formula No. 3 in the case of cylindrical symmetry. The effect of a weak curvature (a c < <R) is assessed by formulae Nos. 6 and 9 through comparison with formula No. 3 based on the linear model: the marginal effect of divergence (No. 6) or convergence (No. 9) is characterized by the factor ± a c/R. When, moreover, the horizontally integrated a u S u is much larger than a c S c, the effect of an important curvature (i.e. a c on the order of R) results in formula No. 7 (for divergence) and No. 10 (for convergence).

It is important to notice that, in many commonly occurring conditions, the analytical expression of τ depends on only four parameters that characterize the system: the width a c of the main confined aquifer, its transmissivity T c, the integrated storage situated upstream M = S u a u and the curvature R of the reservoir describing the convergence/divergence of the flow. These ultra-simplified expressions are given in Table 2 as Nos. 3, 7 and 10.

These expressions were applied to several large aquifer basins of the Sahara and Australia. Following the sudden occurrence of the present aridity around 4,000 years ago, these deep confined aquifers have been subjected to a natural head decay with a large time constant (τ > 5,000 years) resulting from their structure and geometry. This large time constant τ can be explained by the existence of an upstream water reserve in the form of an unconfined aquifer. The convergence of groundwater flow also results in a significantly enhanced time constant τ. Both effects have contributed to the natural persistence of flow at the outlet of these aquifers for 4,000 years. The same mechanisms may also have been responsible for the apparent time delay observed in the Sahara between the onset of a climatic wet phase at the beginning of the Holocene and the maximum extent of the humid zones.