Early-Time Recession Solution from a Steady-State Initial Condition for the Horizontal Unconfined Aquifer

Abstract

In this work, we present the semi-analytical solution for the early-time recession phase of the horizontal unconfined aquifer of finite length for steady-state initial conditions. This is a case where self-similarity arguments are not applicable. The solution is built as linear perturbations from the initial steady state. The solution is determined via a Sturm–Liouville eigenvalue problem, which should be solved numerically. On the other hand, the immediate response of the aquifer to the sudden switching off the recharge, i.e., in the earliest times, is obtained by deducing analytically the large eigenvalue asymptotic solutions of the problem. We find analytically that in this time regime, the outflow Q is given by Q = Q₀ − 1.4Q₀^5/3L^−4/3n^−2/3t^2/3, where Q₀ is the initial outflow rate, L is the length of the aquifer, n is the porosity of the formation and t is the time from the start of the recession. The stated result is very accurate for times t up to ~0.01 nL^3/2k^−1/2Q₀^−1/2, where k is the hydraulic conductivity of the formation. The analytical and quantitative relation of the presented solution with the classical recession phase asymptotic solutions derived in the past by Polubarinova (early-time solution) and Boussinesq (separable, late-time solution) is discussed in detail. The presented results can be used as a benchmark solution for modeling or numerical validation purposes.

1. Introduction

The Boussinesq equation is a groundwater flow dynamical equation that models one-dimensional water flows in homogeneous unconfined aquifers over horizontal or uniformly sloping beds [1,2]. The Boussinesq equation is an expression of local conservation of mass (the continuity equation) for saturated flow which follows from the Dupuit–Forchheimer assumptions [3], that the groundwater flows primarily along the flat bed, combined with Darcy’s law. The Dupuit–Forchheimer assumptions imply that the Darcy flux through any cross section of the flow is proportional to the flow depth. This implies that the Boussinesq equation is non-linear (quadratic) with respect to the water depth.

The assumptions about the geometric and hydraulic properties of the flow imply that the Boussinesq equation is a model that describes very approximately the groundwater flow of real aquifers. That means that, as is most usually the case with engineering applications, its primary usefulness rests on simple and clear results that are strongly based on the physical content of the model rather than on complex solutions of the corresponding equation. Such results usually take the form of scaling relations and asymptotic solutions.

For the problem at hand, the recession phase of the unconfined aquifer, there are two classic asymptotic solutions: one derived by Polubarinova for the semi-infinite aquifer with constant boundary depths [4] by imposing self-similarity, and another derived by [5] by imposing separability (which is a special form of self-similarity) on a finite length aquifer. These solutions are the early- and late-time asymptotic solutions of the problem, respectively. The Polubarinova solution also applies to finite aquifers with vanishing outlet depth and corresponds to a horizontal initial profile, see, e.g., [6,7], as we explicitly discuss in this work. Pertinent use of both asymptotic solutions has been made in application-oriented works such as [8].

Self-similar solutions for the recession phase have been constructed over the decades by many authors; see, for example, the discussion in [9]. Barenblatt in [10] derived a power series solution for a self-similar Boussinesq equation assuming a power law time-dependence of the water depth at one end of a semi-infinite horizontal aquifer. Both the Polubarinova and the Barenblatt boundary conditions still generate a certain amount of literature, see, e.g., [11,12,13,14,15,16,17,18,19,20,21,22]. It should be noted that, in contrast, the literature on the early-time solution of the build-up phase is limited; see [23,24] for the exact and approximate solutions to this problem, respectively.

When the initial state of a finite length aquifer is a smooth profile, for example, it may be the steady-state profile of a previous constant recharge period, the Polubarinova problem which leads to a singular outflow rate Q proportional to 1/√t, with t being the recession duration, is no longer the correct response to the sudden change in recharge rate. Boussinesq equation implies that, whenever the depth squared profile is finite at the outlet, there is no reason for Q to start with an (integrable) infinity; Q is perfectly continuous through the sudden change. On the other hand, although the general smooth initial condition is consistent with the outlet boundary condition at start of recession t = 0, it is not consistent dynamically. As we shall see, that requires a vanishing second derivative at the outlet for the depth squared profile. That means that, dQ/dt must exhibit an integrable singularity at t = 0 for the smooth initial conditions. The main objective of this paper is to deduce in the most detailed manner the law of the aquifer response in this problem. For definiteness we focus on the steady-state initial condition, although the formulation is easily generalizable to other initial profiles.

This paper is organized as follows. In Section 2, we set the notation and summarize certain facts necessary for the rest of the work. In Section 3, we develop the analysis of this work. We start by formulating the specific problem one needs to solve, i.e., the dynamics of the perturbations of the initial state, which is then solved both numerically, as well as analytically through asymptotic methods. In Section 4, we summarize our findings and present concluding remarks.

2. The Boussinesq Equation

We assume uniform and constant hydraulic conductivity k and porosity n throughout the aquifer. The flow will be considered saturated and effectively one-dimensional, and the recharge rate r wherever it appears is regarded as constant. The impermeable bottom of the aquifer is assumed flat and horizontal. The aquifer is assumed to be of finite length L.

The water table height h is governed by the Boussinesq equation.

$$n{\displaystyle \frac{\partial h}{\partial t}}-k{\displaystyle \frac{\partial }{\partial x}}\left[h{\displaystyle \frac{\partial h}{\partial x}}\right]=r$$

(1)

The boundary conditions that we will impose are:

$${\displaystyle \frac{\partial h}{\partial x}}(0,t)=0,\hspace{1em}h(L,t)=0$$

(2)

which expresses that there is no inflow at the point x = 0 and that the water table has zero depth at the outlet of the aquifer.

Let [x] and [h] be the scales of the extent and height of the water table, and [t] a time scale. For r = 0, Equation (1) implies that

$$k{\displaystyle \frac{{[h]}^2}{{[x]}^2}}=n{\displaystyle \frac{[h]}{[t]}}$$

(3)

We choose [x] = L and for any scales [h] and [t] consistent with (3) we define dimensionless variables X, T, H by

$$x=LX,\hspace{1em}h=[h]H,\hspace{1em}t=[t]T$$

(4)

Clearly, 0 ≤ X ≤ 1. Then, Equation (1), for zero recharge rate reads

$${\displaystyle \frac{\partial H}{\partial T}}=\frac{1}{2}{\displaystyle \frac{{\partial }^2{H}^2}{\partial X^2}}$$

(5)

while the boundary conditions take the form

$${\displaystyle \frac{\partial H}{\partial X}}(0,T)=0,\hspace{1em}H(1,T)=0$$

(6)

In these variables, the storage and outflow rate read, respectively

$$S={\displaystyle {\int }_0^{1}H(X,T)dX,\hspace{1em}Q=-\frac{1}{2}}(H^2{)}^{\prime }(1,T)$$

(7)

3. Early Time Asymptotic from an Initial Profile with a Continuous Slope

3.1. General Mathematical Analysis

Consider an initial profile H₀(X). We assume that this function has a continuous slope and obeys the boundary conditions (6). The slope may become minus infinity at the outlet X = 1 to be consistent with the boundary condition the zero-height boundary condition and a finite or infinite outflow rate Q, given by (7).

By Equation (7), Q is finite if the slope of the squared profile is finite at X = 1. The Polubarinova early-time recession solution [4], reviewed in some detail in Appendix A, is a solution consistent with a uniform initial profile for all X < 1. This means that the square profile has an effectively infinite slope for this solution. This is why Q starts with a singularity in time: Q is proportional to 1/√t.

If the initial profile is, for example, a steady state, or any profile with finite Q before a recession starts, there is no such initial singularity in the flow rate. On the other hand, Equations (5) and (6) say that the outlet boundary condition implies dynamically that the square profile must have a vanishing second derivative at X = 1. This is certainly not a feature of the steady state, or of any generic initial profile with a finite Q. That means that the Boussinesq equation will transform the initial profile into something with this property through a sudden but finite change in ∂H/∂T. That means the second, time derivative of the profile must be infinite for T = 0⁺. That in turn means that dQ/dT will be infinite for T = 0⁺. We show here that this is realized by a (smaller than 1) power law departure from the initial value of the outflow rate.

Consider small perturbations δH(Χ,Τ), that is,

$$S={\displaystyle {\int }_0^{1}H(X,T)dX,\hspace{1em}Q=-\frac{1}{2}}(H^2{)}^{\prime }(1,T)$$

(8)

Given that the initial profile obeys the boundary conditions, the perturbation must also obey them:

$${\displaystyle \frac{\partial \delta H}{\partial X}}(0,T)=0,\hspace{1em}\delta H(1,T)=0$$

(9)

with initial condition δH (Χ,0) = 0.

The perturbation is governed by the linear equation

$${\displaystyle \frac{\partial \delta H}{\partial T}}={\displaystyle \frac{{\partial }^2(H_0\delta H)}{\partial X^2}}+\frac{1}{2}{\displaystyle \frac{{\partial }^2{H}_0^2}{\partial X^2}}$$

(10)

which is obtained by substituting (8) in (5) and dropping the quadratic term. The second term on the right hand side does not depend on the field δH, and hence acts as a source/forcing term.

Aiming at an expansion in orthogonal functions, we look for solutions of the form

$$\delta H(X,T)=e^{-{\kappa }^{2}T}{\phi }_{\kappa }(X)$$

(11)

where κ is a constant, to be determined by the boundary conditions. Substituting into (10) without the source term, we obtain

$$(H_0(X){\phi }_{\kappa }(X){)}^{″}=-{\kappa }^2{\phi }_k(X)$$

(12)

This is an eigenvalue equation, where the eigenfunctions φ_κ(Χ) obey the boundary conditions

$${\phi }_{\kappa }^{\prime }(0)=0,\hspace{1em} {\phi }_{\kappa }(1)=0$$

(13)

Define the functions Φ_κ(Χ) = H₀(Χ) φ_κ(Χ). These functions clearly obey the boundary conditions (13) and the equation

$${\mathrm{\Phi }}_{\kappa }^{″}({\rm X})=-{\kappa }^2{\displaystyle \frac{1}{H_0(X)}}{\mathrm{\Phi }}_{\kappa }(X)$$

(14)

Due to the linearity of the eigenvalue Equation (14) the normalization of Φ_κ(Χ) is left unspecified. We impose the simple condition Φ_κ(0) = 1.

Equation (14) defines a Sturm–Liouville eigenvalue problem with weight 1/H₀(X): The solution of (14) with the given boundary conditions leads to a discrete set of functions Φ_κ(Χ), labeled by a discrete set of constant rates of decay (eigenvalues) κ², which form a complete and orthogonal set of functions in space of weighted square integrable functions, which obey the given boundary conditions. The orthogonality relation explicitly reads

$${\mathrm{\Phi }}_{\kappa }^{″}({\rm X})=-{\kappa }^2{\displaystyle \frac{1}{H_0(X)}}{\mathrm{\Phi }}_{\kappa }(X)$$

(15)

for any two values of eigenvalues κ and κ′. In terms of the original functions the orthogonality reads

$${\displaystyle {\int }_0^1{\phi }_{\kappa }(X)}\hspace{0.17em}{\phi }_{{\kappa }^{\prime }}(X)H_0(X)dX=N_{\kappa }{\delta }_{\kappa {\kappa }^{\prime }},\hspace{1em} N_{\kappa }={\displaystyle {\int }_0^1{\phi }_{\kappa }^2(X)}\hspace{0.17em}{H}_0(X)dX$$

(16)

Expand now the general solution of (10) in these orthogonal functions

$$\delta H(X,T)={\displaystyle \sum _{\kappa }{c}_{\kappa }(T){\phi }_{\kappa }(X)}$$

(17)

where the function c_κ(Τ) are determined by (10) and the initial condition δH(Χ,0) = 0. This sum is inverted by the orthogonality relation (16):

$$c_{\kappa }(T)=N_{\kappa }^{-1}{\displaystyle {\int }_0^1\delta H(X,T){\phi }_{\kappa }(X)H_0(X)dX}$$

(18)

Multiplying (10) with H₀(X)φ_κ(Χ) and integrating over the interval 0 < X < 1, using also the orthogonality relation (16) and Equation (18), we obtain a dynamical equation for c_κ(Τ):

$$c_{\kappa }^{\prime }(T)=-{\kappa }^2{c}_{\kappa }(T)-d_{\kappa }$$

(19)

where d_κ are given by

$$d_{\kappa }=-\frac{1}{2}{N}_{\kappa }^{-1}{\displaystyle {\int }_0^1{H}_0(H_0^2{)}^{″}{\phi }_{\kappa }(X)dX}=-\frac{1}{2}{N}_{\kappa }^{-1}{\displaystyle {\int }_0^1(H_0^2{)}^{″}{\mathrm{\Phi }}_{\kappa }(X)dX}=\mathrm{constant}$$

(20)

with the initial condition c_κ(0) = 0. We immediately obtain

$$c_{\kappa }(T)=-{\displaystyle \frac{d_{\kappa }}{{\kappa }^2}}(1-e^{-{\kappa }^{2}T})$$

(21)

Hence, as long as one is able to solve (14), the solution of the problem is

$$\delta H(X,T)=-{\displaystyle \sum _{\kappa }\frac{d_{\kappa }}{{\kappa }^2}(1-e^{-{\kappa }^{2}T}){\phi }_{\kappa }(X)}$$

(22)

The storage associated with this solution reads

$$\begin{array}{l}S(T)={\displaystyle {\int }_0^1{H}_0(X)dX+{\displaystyle {\int }_0^1\delta H(X,T)dX}}=\\ \hspace{1em}\hspace{1em}=\hspace{0.33em}{\displaystyle {\int }_0^1{H}_0(X)dX}-{\displaystyle \sum _{\kappa }\frac{d_{\kappa }}{{\kappa }^2}(1-e^{-{\kappa }^{2}T})e_{\kappa }},\\ e_{\kappa }={\displaystyle {\int }_0^1{\phi }_{\kappa }(X)dX}={\displaystyle {\int }_0^1{\mathrm{\Phi }}_{\kappa }(X)\frac{dX}{H_0(X)}}\end{array}$$

(23)

Therefore, the outflow rate reads

$$Q(T)=-{\displaystyle \frac{dS(T)}{dT}}={\displaystyle \sum _{\kappa }{d}_{\kappa }{e}_{\kappa }{e}^{-{\kappa }^{2}T}}$$

(24)

Let us now focus on a specific initial condition of this kind, i.e., the steady-state solution. This result is further analyzed below.

Regarding the range of times where the solution (35) is expected to be a reasonably good approximation, one may note that the solution holds well as long as the time-dependent exponentials are small. That is, it is expected to be a good approximation for times t << 1/κ_MIN² where κ_ΜΙΝ² is the smallest eigenvalue.

Without loss of generality, we may write

$$H_0(X)=\sqrt{1-X^2}$$

(25)

Indeed, the most general steady state is of the form h₀(X) = h_o√(1 − X²), for some constant h_o, in terms of the original (dimensionful variable). We may then set the height scale [h] = h_o, which fixes the scale of time by Equation (3):

$$[t]={\displaystyle \frac{n}{k}}{\displaystyle \frac{L^2}{h_o}}$$

(26)

That of course means that the scales of storage and outflow rate are also fixed

$$[S]=n{h}_{o}L,\hspace{1em}\hspace{1em}[Q]={\displaystyle \frac{[S]}{[t]}}=k{\displaystyle \frac{h_o^2}{L}}$$

(27)

3.2. Numerical Treatment of the Problem

It is useful first to solve numerically the eigenvalue problem (14) for large enough eigenvalues κ. (In the next section we shall calculate analytically the critical quantities). By Equation (24), the early-time behavior of the flow rate is dominated by the asymptotic behavior of the various quantities of large eigenvalues. The numerical solution of the eigenvalue Equation (14) is facilitated by the observation that the boundary condition Φ′_κ(0) = 0, and normalizing so that Φ_κ(0) = 1, the first two terms in a power series expansion of the eigenfunctions read

$${\mathrm{\Phi }}_{\kappa }(X)=1-\frac{1}{2}{\kappa }^2{X}^2$$

(28)

Therefore, Equation (14) can then be solved with initial conditions Φ_κ(ε) = 1 − (1/2)κ²ε² and Φ′κ(ε)= −κ²ε, with ε is a small number and the eigenvalues κ² are determined by the condition that Φ_κ(1) is zero (within tolerance). We use ε = 10⁻⁹ and |Φ_κ(1)| < 10⁻⁸. We obtain results for the (root) eigenvalues κ, and the constants d_κ and e_κ defined in Equations (20) and (23), respectively, for the first 100 modes.

We find that the smallest root eigenvalue is κ₁ = 1.50. That is, the longest mode decay time is 1/κ₁² = 0.44. That means that the solution should make sense for times T << 0.44. On the other hand, for a given time T, the series adequately represents the solution of (10) when we include modes such that κ²T >> 1, that is, when we include decay times such that 1/κ² << Τ. We have included modes up to decay times of order 10⁻⁵; hence, we may safely represent times nearly as small as 0.0001.

We may also utilize these results to deduce the behavior of the outflow rate given by Equation (24) for small times, looking for a more explicit expression than Equation (24). To this end, we shall also make explicit certain properties of the eigenvalue and eigenfunctions which define the solution to the problem at hand. First of all, at T = 0⁺, the flow rate reads

$$Q(0^{+})={\displaystyle \sum _{\kappa }{d}_{\kappa }{e}_{\kappa }}$$

(29)

Although the time derivative of the profile and hence of the storage is discontinuous at T = 0, the flow rate, defined by (24), is continuous: Q(0⁺)= Q(0⁻) = 1, using the fact that the steady state (25) produces unit flow rate from the results of the 100 modes we obtain [that is, the error appears to be of order O(100⁻²)]. In order to determine the behavior of (37) for small T, we need the asymptotic behavior of (square root) eigenvalues κ and of the product e_κd_κ. As we shall explicitly see below, the behavior of these quantities is needed to deduce the asymptotic early-time behavior of the outflow rate (25).

Indeed, we find that

$$\kappa (i)=2.622\hspace{0.17em}i-1.093$$

(30)

where i is the integer index of the eigenfunction corresponding to the number of its nodes. This is obtained by calculating the slope and the intercept of the line κ(i) calculated in a moving window of 10 eigenvalues. The results are plotted in Figure 1 where the slope of the line κ(i) is shown in a dotted line (primary axes) and its intercept is shown dashed line (secondary axes). As we shall see, what matters is the linearity of the relationship between the eigenvalue κ and the integer index i.

The behavior of the product eκdκ is best described by the logarithmic derivative with respect to κ as a function of the node number i, which illustrates a power law dependence of the product on κ. Figure 2 illustrates this dependence for up to node number 100. The result suggests that the product eκdκ exhibits a power law dependence on large κ:

$$e_{\kappa }{d}_{\kappa }=\mathrm{constant}\times {\kappa }^{-\nu },\hspace{1em}\hspace{1em}\nu =2.33\simeq 7/3$$

(31)

Such an estimate allows us to quantify analytically the early-time behavior of the flow rate Q in this problem. Indeed, the small time T behavior of Q given by Equation (24) is dominated by the large κ behavior of the summand. In this limit, the discrete index i, or equivalently via Equation (30), the index κ, is regarded as a continuous variable:

$$Q(T)=\mathrm{constant}{\displaystyle {\int }_{\alpha }^{\infty }{\kappa }^{-\nu }{e}^{-{\kappa }^{2}T}}d\kappa$$

(32)

where α denotes an (undeterminable) lower-end point of order 1. We have

$$Q(T)=\mathrm{constant}\left({\displaystyle \frac{{\alpha }^{1-\nu }}{v-1}}+{\displaystyle \frac{1}{2}}\mathrm{\Gamma }\left({\displaystyle \frac{1-\nu }{2}}\right)T^{{\displaystyle \frac{\nu -1}{2}}}\right)+O(T)$$

(33)

for the small times T. We know that the constant term must be 1. We also observe that the dominant term in the expansion of small times is a non-analytic power law. The reason is that the exponent ν ≈ 7/3 does not allow expanding the exponential to obtain even just one term (as the sum does not converge), or equivalently, the sum is not differentiable even once near T = 0, as we expected from our discussion previously. Explicitly, we have

$$Q(T)=1-\mathrm{c}\times T^{2/3}+O(T)$$

(34)

where the c is a constant of order 1, which is not determined analytically in this work. Equation (34) encodes explicitly the dynamical response of the aquifer to the initial state, which respects the boundary conditions but it is not consistent with its dynamical preservation.

It is clear that Equation (34) applies to even earlier times than the complete result in Equation (24). Upon comparing it with the result of Equation (24), we may determine the numerical constant c. Indeed, for c = 1.414, the result (34) fits very well with the earliest time behavior of (24). The resulting curves are shown in Figure 3. Indeed, for the smallest times of order 0.0001, the relative difference is of order 1:105 while at time 0.05 is 0.5%. It is also worth quoting the result (34) in dimensionful form, using the scales (26) and (27), and the fact that h₀ = √(r/k)L for the steady state with uniform and constant recharge rate r. We then have

$$(\mathrm{dimensionful})\hspace{1em}\hspace{1em}\hspace{1em}Q(t)=Q_0-1.4{\displaystyle \frac{{(Q_0)}^{5/3}}{L^{4/3}{n}^{2/3}}}{t}^{2/3},\hspace{1em}{Q}_0=rL$$

(35)

This solution holds well for dimensionful times t up to 0.01 nL^3/2k^−1/2Q₀^−1/2. For example, if we consider a porosity of 30%, aquifer length of 1 km, mean hydraulic conductivity 10⁻⁴ m/sec and discharge rate of 10⁻⁴ m³/sec, the solution holds very accurately for 11 days since the recharge switch-off.

3.3. Analytical Treatment of the Problem by Large Eigenvalue Asymptotics

We may complete our analysis by estimating analytically the important results (30) and (31). We start by determining the asymptotic behavior of the large (square root) asymptotic behavior of the solution of (14). This is a well-known type of problem, treated usually by standard asymptotic methods, such as the WKB approximation [25]. It is instructive to first obtain some insight into the problem by looking at the behavior of (14) near the boundary points. We first observe that for X~0, the eigenvalue Equation (14) takes the form

$${\mathrm{\Phi }}_{\kappa }^{″}({\rm X})=-{\kappa }^2{\mathrm{\Phi }}_{\kappa }(X)$$

(36)

applying also the boundary condition Φ′_κ(0) = 0 and normalization condition Φ_κ(0) = 1. Then, the solution of (36) is

$${\mathrm{\Phi }}_{\kappa }(X)=\cos (\kappa X)$$

(37)

Applying the WKB approximation in its first two orders (which are usually adequate) is essentially equivalent to looking for a solution in the form of the following ansatz

$${\mathrm{\Phi }}_{\kappa }(X)=A(X)\cos (\kappa \varphi (X))$$

(38)

for some amplitude function A(X) and phase function $\varphi (X)$ to be specified. Indeed, by substituting (38) into (14), we find (dividing also everything with κ²):

$$\begin{array}{l}A(X)\cos (\kappa \varphi (X))\left[-{({\varphi }^{\prime }(X))}^2+{\displaystyle \frac{1}{H_0(X)}}\right]+\\ {\displaystyle \frac{1}{\kappa }}\sin (\kappa \varphi (X))\left[-2A^{\prime }(X){\varphi }^{\prime }(X)-A(X){\varphi }^{″}(X)\right]+\\ {\displaystyle \frac{1}{{\kappa }^2}}\cos (\kappa \varphi (X))A^{″}(X)=0\end{array}$$

(39)

For large κ, the dominant term with respect to κ implies the following simple equation for the unknown function $\varphi (X)$:

$${({\varphi }^{\prime }(X))}^2={\displaystyle \frac{1}{H_0(X)}}$$

(40)

Hence, we obtain

$$\varphi (X)={\displaystyle {\int }_0^X\frac{d{X}^{\prime }}{\sqrt{H_0(X^{\prime })}}}$$

(41)

Thus, we have determined the wave part of the WKB approximation without yet determining the prefactor A(X). This solution allows us immediately to determine the (asymptotic) form of the eigenvalues. Indeed, applying the boundary condition Φ_κ(1) = 0 on the ansatz (38) means that it must be cos(κφ(1)) = 0. Hence, we find

$$\kappa (i)=(i+\frac{1}{2}){\displaystyle \frac{\pi }{\varphi (1)}}$$

(42)

We see that the eigenvalue is indeed a linear function of the integer counting index i. Moreover, let us apply this result to the case of steady-state initial condition (38): H₀(X) = √(1 − X²). Then

$$\varphi (1)={\displaystyle {\int }_0^1\frac{dX}{{(1-X^2)}^{1/4}}}=2\sqrt{\pi }{\displaystyle \frac{\mathrm{\Gamma }(\frac{3}{4})}{\mathrm{\Gamma }(\frac{1}{4})}}=1.198140$$

(43)

Hence, the slope of κ(i) is

$$\kappa (i+1)-\kappa (i)={\displaystyle \frac{\pi }{\varphi (1)}}=2.62206$$

(44)

which is the asymptotic slope we estimated numerically (Figure 1).

Impose now the first subdominant term, of order 1/κ, in (39). We obtain

$$-2A^{\prime }(X){\varphi }^{\prime }(X)-A(X){\varphi }^{″}(X)=0$$

(45)

This implies

$$A(X)={\displaystyle \frac{C}{\sqrt{{\varphi }^{\prime }(X)}}}=C H_0^{1/4}$$

(46)

Applying this equation specifically for the case of the steady-state initial condition and applying the normalization Φ_κ(0) = 1, we find

$$A(X)={(1-X^2)}^{1/8}$$

(47)

In all, the eigenfunction Φ_κ(X) under these approximations reads

$${\mathrm{\Phi }}_{\kappa }(X)= {(1-X^2)}^{1/8}\cos {(\kappa {\displaystyle {\int }_0^X(1-{\chi }^2})}^{-1/4}d\chi )$$

(48)

Note that, by Equation (40),

$$\varphi (1)-\varphi (X)={\displaystyle {\int }_X^1(1-{\chi }^2}{)}^{-1/4}d\chi$$

(49)

This allows us to rewrite (48) in the form

$${\mathrm{\Phi }}_{\kappa }(X)= {(1-X^2)}^{1/8}\sin {(\kappa {\displaystyle {\int }_X^1(1-{\chi }^2})}^{-1/4}d\chi )$$

(50)

where we have dropped a factor sin(κφ(1)) = ±1. Obtaining (50), we used the condition that defines the eigenvalues, i.e., cos(κφ(1)) = 0.

We may now come to the calculation of d_κe_κ. The quantity d_κ is defined as in Equation (20). It involves the quantity N_κ, which is defined in Equation (15). The quantity e_κ is defined in Equation (23). Working with the steady-state initial condition, we see that we should estimate the following three integrals for large κ:

$${\displaystyle {\int }_0^1{\mathrm{\Phi }}_{\kappa }(X)dX},\hspace{1em}{\displaystyle {\int }_0^1\frac{{\mathrm{\Phi }}_{\kappa }^2(X)}{\sqrt{1-X^2}}dX},\hspace{1em}{\displaystyle {\int }_0^1\frac{{\mathrm{\Phi }}_{\kappa }(X)}{\sqrt{1-X^2}}dX}$$

(51)

In particular, we need to estimate their scaling with κ.

There are quick but somewhat handwaving ways to determine this scaling based on Equation (50); hence, we shall refrain from writing them down. The robust way to obtain the desired results is to proceed by a pertinent general asymptotic procedure, such as the steepest descent method [25]. The cumbersome chain of details of this calculation does not add anything to our main argument and we present it in Appendix C. Indeed, we find that for large κ

$${\displaystyle {\int }_0^1{\mathrm{\Phi }}_{\kappa }(X)dX\propto {\kappa }^{-3/2}\hspace{1em}}$$

(52)

$${\displaystyle {\int }_0^1\frac{{\mathrm{\Phi }}_{\kappa }^2(X)}{\sqrt{1-X^2}}dX}=\mathrm{constant}$$

(53)

$${\displaystyle {\int }_0^1\frac{{\mathrm{\Phi }}_{\kappa }(X)}{\sqrt{1-X^2}}dX}\propto {\kappa }^{-5/6}$$

(54)

Therefore, using these results in Equations (20) and (23) we find

$$d_{\kappa }{e}_{\kappa }\propto {\kappa }^{-3/2}{\kappa }^{-5/6}={\kappa }^{-7/3}$$

(55)

which is what we estimated numerically above (Figure 2).

It is instructive to obtain the important result (55) in a less general but more direct way. This can be carried out by looking at the behavior of the eigenvalue Equation (14) near the other end of the interval, X~1. This is a non-trivial point, as the coefficient κ²/H₀(Χ) diverges there. Usually, such singularities conceal, and with a bit of effort give easily away, a lot of important information about the solution. Moreover, as we learned explicitly through the steepest descent analysis in Appendix C, the large κ asymptotics rely on the behavior of the solutions in the neighborhood of that singular point. This fact can be understood from the very form of the eigenvalue equation: the coefficient κ²/H₀(Χ) is large either by κ being large or by X being close to 1.

Indeed for X~1, and focusing specifically on the steady-state initial condition, we may approximate H₀(X) = √(1 − X²) = √2(1 − X). Then, Equation (14) reads

$${\mathrm{\Phi }}_{\kappa }^{″}({\rm X})+{\displaystyle \frac{{\kappa }^2}{\sqrt{2(1-X)}}}{\mathrm{\Phi }}_{\kappa }(X)=0$$

(56)

This equation is solved in terms of the Bessel function of the first kind (Arfken et al. [26]):

$${\mathrm{\Phi }}_{\kappa }(X)=C\hspace{0.33em}\sqrt{1-X}{J}_{2/3}(\frac{2}{3}{2}^{3/4}\kappa {(1-X)}^{3/4})$$

(57)

where C is an integration constant. There is also a solution in terms of the Bessel function of order –2/3. But this solution is not zero at X = 1; hence, it is not consistent with the boundary condition Φ_κ(1) = 0, and hence the associated integration constant should vanish. The constant (2/3)2^3/4 = 1.12 appearing in (57) is an approximation of φ(1) in Equation (57), as a result of linearization of H₀(X). We now apply the solution (57) over the whole interval imposing the normalization Φ_κ(0) = 1 to obtain

$${\mathrm{\Phi }}_{\kappa }(X)=\sqrt{1-X}{\displaystyle \frac{J_{2/3}(\frac{2}{3}{2}^{3/4}\kappa {(1-X)}^{3/4})}{J_{2/3}(\frac{2}{3}{2}^{3/4}\kappa )}}$$

(58)

We apply the boundary condition Φ′_κ(0) = 0. It can only be applied asymptotically for large κ. We obtain

$$\cos (\frac{2}{3}{2}^{3/4}\kappa -\frac{\pi }{12})=0$$

(59)

which produces again a linear relationship for κ with respect to the counting index. This linear relationship is an approximate form of the one given in (42).

We now have all the information we need to calculate the asymptotics of d_κe_κ. The three integrals (51) can be easily estimated for large κ. This is carried out in Appendix D, using the asymptotic forms of the Bessel functions and the condition (59). Hence, the result (55) is obtained through a different method. It is worth noting that the linearity of κ with respect to the counting index guaranteed also by (59) together with (55) is all we need to deduce the outflow rate early-time asymptotic in Equation (34). Hence, the near X~1 analysis is an alternative derivation of (34).

4. Early and Late-Time Asymptotics

In what follows, we present the (full) early-time solution, given by Equations (8) and (22) and the numerically obtained eigenfunctions, together with the late-time asymptotic solution introduced by [5]. Boussinesq looked for the simplest kind of self-similar solution, an exact separable solution, which is non-trivial due to the non-linearity of the Boussinesq equation. This important solution is presented in Appendix B. The simultaneous presentation of both asymptotics, together with the numerical solution of the Boussinesq equation, illustrates in a specific manner their extent of validity and the transition between them, expressed in various variables: water table profile, storage and outflow.

A direct test is provided by the time variation in the ratio S²/Q, which is constant for the separable solution. This is given in Figure 4.

The ratio S²/Q is a pure constant with a value of 0.693 for the separable solution, as shown in Appendix B. The numerical solution of the Boussinesq Equation (5) provides the storage by spatial integration, while the outflow rate is deduced by the mass balance, Q(T) = –S′(T). Presumably, these results show first that the separable solution is indeed a ‘late’-time asymptotic: its deviation from the Boussinesq equation solution is 1% at time 0.11, and 1‰ at time 0.29. The early-time asymptotic holds for times much earlier than 0.44 (as expected from our general considerations): the deviation from the solution of the Boussinesq equation appears to be 5% at time 0.05. On the other hand, we shall see below the ratio S²/Q provides a rather conservative criterion for convergence between solutions.

At the level of the time variation in the storage and outflow, the difference between the solutions is much less pronounced, as shown in Figure 5. Regarding the outflow rate, the two asymptotic solutions deviate visibly from the Boussinesq solution in a neighborhood of T = 0.1. In the case of the storage, the early-time solution eventually deviates from the Boussinesq solution, while the separable solution appears to provide a fairly good approximation of the Boussinesq solution at essentially all times.

It should be mentioned that, in order to produce the specific curves for the separable solution, one has to fix the integration constant T₀. There is no unique choice for this number, and it is possibly the only remnant of the initial condition. If T₀ is fixed too small, the convergence of the solution to the Boussinesq equation solution is delayed; if it is fixed too large, there is a larger disagreement between the solution and the Boussinesq solution at earlier times. To obtain a guiding guess, we may fix T₀ so that the separable solution at T = 0 contains the same volume as the initial (steady) state, i.e., π/4, although this is not physically necessary or even reasonable. This gives T₀ = (4/π)(4/9J³) = 0.882. A slightly different value T₀ = 0.887 appears to give better results and is the one adopted here.

In Figure 6, comparisons of the water table profile are given between asymptotic solutions and the solution of the Boussinesq equation. For clarity purposes, the comparisons are given separately, following the same color coding as in Figure 4 and Figure 5. All curves are drawn for times T = 0.002, 0.02, 0.05, 0.10, 0.15, 0.20, 0.30 and 0.40 (time increases downwards). As expected from our analysis above, the early-time solution deviates visibly after time 0.10, while the separable solution observes well the behavior of the Boussinesq equation after time 0.10.

5. Summary and Conclusions

In the present work, we deduce the early-time response solution of the horizontal, unconfined aquifer to a sudden switching-off of the previously constant recharge. The results are not only novel but in some sense complete the list of fundamental asymptotic solutions of the Boussinesq equation for the recession phase, in addition to the solutions obtained by Boussinesq and Polubarinova-Kochina in the past. The steady-state initial profile is chosen as a simple and naturally smooth initial condition, in contrast to the Polubarinova solution, which starts off with a uniform initial profile in the finite-length aquifer.

As opposed to these classic solutions, the problem at hand cannot be solved by self-similarity arguments. Indeed, the problem is analyzed by solving the dynamics of the depth profile perturbations off the initial condition. This essentially translates to solving an eigenvalue/eigenfunction problem. This problem cannot be solved in closed form. It is either solved numerically for all eigenvalues, or by analytical means for large eigenvalues. The latter takes the form of asymptotic analysis utilizing, e.g., the WKB approximation to solve the eigenfunction problem for large eigenvalues, which carry the behavior of the aquifer for the earliest of times. Indeed, we establish that the outflow Q drops by 2/3 power law of time at the very early stages of the sudden recession. This verifies our argument, given in the introduction, that dQ/dt forms an integrable singularity at the beginning of the recession. We also quantify explicitly the times through which this law holds. The numerical solution of the eigenfunction problem is used to illustrate and compare the perturbations solution against the numerical solution of the full non-linear Boussinesq equation as well as the late time asymptotic by Boussinesq.

The details of our derivations show that with moderate effort it may be possible to establish the early response of the aquifer to the sudden recession for generic initial conditions, or at least, more general than a steady state. This endeavor, which may lead to statements of greater hydrologic interest, that is, set such results against field and laboratory data, will be left for future work. Our present results can certainly be used as a benchmark solution for modeling or numerical validation purposes.