How long does it take for aquifer recharge or aquifer discharge processes to reach steady state?

Groundwater ﬂow models are usually characterized as being either transient ﬂow models or steady state ﬂow models. Given that steady state groundwater ﬂow conditions arise as a long time asymptotic limit of a particular transient response, it is natural for us to seek a ﬁnite estimate of the amount of time required for a particular transient ﬂow problem to eﬀectively reach steady state. Here, we introduce the concept of mean action time (MAT) to address a fundamental question: How long does it take for a groundwater recharge process or discharge processes to eﬀectively reach steady state? This concept relies on identifying a cumulative distribution function, F ( t ; x ), which varies from F (0; x ) = 0 to F ( t ; x ) → 1 (cid:0) as t → ∞ , thereby providing us with a measurement of the progress of the system towards steady state. The MAT corresponds to the mean of the associated probability density function f ( t ; x ) = d F= d t , and we demonstrate that this framework provides useful analytical insight by explicitly showing how the MAT depends on the parameters in the model and the geometry of the problem. Additional theoretical results relating to the variance of f ( t ; x ), known as the variance of action time (VAT), are also presented. To test our theoretical predictions we include measurements from a laboratory–scale experiment describing ﬂow through a homogeneous porous medium. The laboratory data conﬁrms that the theoretical MAT predictions are in good agreement with measurements from the physical model.

t → ∞. This kind of scenario, where recharge is applied to an existing un-23 confined groundwater flow system, leads to an increase in the saturated depth 24 corresponding to an increase in the amount of water stored in the aquifer. 25 The details of how to design and operate such recharge systems have been de- numerical computation to answer these questions would be useful since it is 41 not obvious how, for example, changing the properties of the porous medium 42 or the geometry of the groundwater flow system would affect the time taken 43 for the rate of change of water stored in the aquifer to effectively reach zero. 44 Understanding this timescale may have several practical uses; for example, if 45 we were to design an artificial recharge program it would be of interest to 46 monitor the increase in storage in the aquifer with time and to have a criteria to indicate when the system would effectively reach steady state.
whereh is the average saturated thickness (Bear 1972;Bear 1979 where D = Kh/S y [L 2 /T] is the diffusivity and W = R/S y [L/T] is a zero 112 order constant source term which is used to model recharge (Bear, 1979).

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To apply our modelling framework to the schematic in Figure 1(a), we will 114 consider a model of unconfined groundwater flow, Equation (3) For many transitions F (t; x) monotonically increases from F = 0 at t = 0 to 126 F → 1 − , as t → ∞ at all spatial locations x, as shown in Figure 1 Physically, we interpret the MAT to be the mean timescale required for the 141 initial condition, h 0 (x), to asymptote to the steady state, h ∞ (x). Intuitively,

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we expect that this timescale would depend on spatial location and we will 143 see that the MAT is indeed a function of position, x. To evaluate the MAT we apply integration by parts to Equation (5) to obtain where we have defined (3), gives us or, if we expand using the product rule, we can write this as The theory of MAT relies on certain properties of the problem that guaran-165 tee that the improper integral for T (x), given by Equation (5), is convergent.
Expanding the quadratic term in the integrand in Equation (9) where we have made a change of variables, which, together with appropriate boundary conditions can be solved for ψ(x) 189 and in turn rearranged to give Once we have solved the relevant boundary value problems for T (x) and V (x), Here, we 192 take the time interval to be the mean plus or minus one standard deviation 193 of the distribution f (t; x) (Simpson et al. 2013). Once we have calculated the 194 mean and variance of f (t; x) at a particular location, as indicated in Figure   195 1(d), we can put this information together to view how the MAT and VAT 196 varies with position, as indicated in Figure 1(e).

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To reiterate the practicality of our results, we would like to emphasize the fol- Although we have outlined the MAT theory in Section 2 for an arbitrary 207 aquifer recharge or discharge process, we will now demonstrate the insight 208 provided by the MAT framework by considering a specific application. We 209 will examine the transition described by Equation (3) We consider a transition 211 from the initial condition, long time steady state for this transition is where D = Kh/S y and W = R/S y . This particular initial condition and 215 steady state gives us To find the MAT for this transition we note that dg( which is a variable coefficient second order boundary value problem that is 220 singular at x = 0 and x = L. We note that Equation (15) which is a Robin condition at x = L (Kreyszig 2006; Zill and Cullen 1992).

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The solution of Equation (15) with Equation (17)-(18) is This solution shows that the MAT is spatially dependent and has a maximum . (20) The maximum VAT occurs at x = L/2 and is given by 119L 4 /(11520D 2 ). standard deviation, which is given by 243 2.2 MAT and VAT for aquifer discharge 244 We now consider a transition governed by Equation (3) for the process of 245 aquifer discharge. With the same domain and boundary conditions described 246 for the recharge problem in Section 2.1, we consider the initial condition which corresponds to the long term steady state profile from the recharge decrease with time, we set R = 0 in Equation (2), which is equivalent to 251 setting W = 0 in Equation (3), which gives With these conditions, Equation (8) can be written as which is exactly the same boundary value problem as we obtained previously  using an exact analytical framework that avoids the need for solving a time 411 dependent partial differential equation describing the transient process.

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The key advantage of our approach is that we arrive at exact mathematical additional analysis and our future work will seek to address these problems.

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An extension of our present study would be to consider the MAT for a hetero-446 geneous groundwater flow problem. The heterogeneous analogue of Equation

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(1) can be written as where K(x) is the spatially varying saturated hydraulic conductivity and R(x) 449 is the spatially varying recharge rate (Bear, 1979). For practical problems 450 where the hydraulic gradient is very small, |∂h/∂x ≪ 1|, the linearised ana-451 logue of this model can be written as which is a generalization of Equation (7)