Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers

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1 WATER RESOURCES RESEARCH, VOL. 49, , doi:1.1/wrcr.168, 13 Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers H. A. Basha 1 Received 14 August 1; revised 14 January 13; accepted 5 February 13; published 8 March 13. [1] An approximate nonlinear solution of the one-dimensional Boussinesq equation is presented using the traveling wave approach. The solution pertains to a semi-infinite phreatic aquifer with a uniform water table that is subject to a time-varying water level at the stream-aquifer boundary. The advantage of the traveling wave method is in its versatility in handling transient boundary conditions while preserving the inherent nonlinearity of the physical phenomenon. The nonlinear solution is of a simple logarithmic form and describes the position of the water table as a function of time. It yields the exact solution for the special case of uniform water level rise at the boundary. Algebraic expressions that quantify the main flow processes are derived from the basic solution. These include the stream-aquifer exchange flow rates, bank storage and depletion, front position and propagation speed, and an improved working relationship for aquifer parameter estimation. A comparison with two exact solutions and numerical solutions of the Boussinesq equation validates the accuracy of the approximation and highlights the limitation of the method in specific flow conditions. The traveling wave model performs best for sharp front movements and monotonic water table profiles and provides excellent estimates of the flow rate and volume at the inlet boundary. The accuracy of the solution deteriorates for fluctuating inlet conditions and worsens in cases when there is a sharp reversal of flow conditions. Citation: Basha, H. A. (13), Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers, Water Resour. Res., 49, , doi:1.1/wrcr Introduction [] Groundwater flow in an unconfined aquifer can be modeled by the nonlinear Boussinesq equation when the flow is predominantly horizontal. Solutions of the Boussinesq equation yield the response of the water table to stream level variations and quantify the exchange flow between the stream and the aquifer. These results have wide range of applications in irrigation and drainage, base-flow separation and recession analysis, and catchment hydrology. [3] The Boussinesq equation is nonlinear with only few exact solutions derived for zero head initial condition and specific boundary conditions [e.g., Barenblatt et al., 199; Parlange et al., ]. Various approximate methods are therefore used to obtain some useful results. The most common approach is through linearization, whereby the nonlinear coefficient is approximated by a constant average value, and solutions of the linearized equation are then obtained for particular conditions [e.g., Govindaraju and Koelliker, 1994]. Other mathematical approaches that 1 Faculty of Engineering and Architecture, American University of Beirut, Beirut, Lebanon. Corresponding author: H. A. Basha, Faculty of Engineering and Architecture, American University of Beirut, Riad el Solh, Beirut 117, Lebanon. (habibaub.edu.lb) 13. American Geophysical Union. All Rights Reserved /13/1.1/wrcr.168 tackle the original nonlinear form of the equation include perturbation series expansion [Polubarinova-Kochina, 196], similarity transform and polynomial approximation [Tolikas et al., 1984; Parlange et al., ; Lockington et al., ; Telyakovskiy et al., ], the integral equation approach [Chen et al., 1995], the weighted residual method [Lockington, 1997], and more recently, the Adomian s decomposition method [Moutsopoulos, 1]. The problem of horizontal flow in phreatic aquifers is also akin to the problem of horizontal infiltration of water in soil, and thus techniques that were developed for Richards equation were adapted to the Boussinesq equation [e.g., Hogarth et al., 1999]. [4] The above nonlinear solutions of the Boussinesq equation fall into two main groups. The first category pertains to the classical problem of the water table response to a sudden rise or drop of the stream level, i.e., uniform initial condition and constant head boundary conditions [e.g., Polubarinova-Kochina, 196]. The second category is restricted to problems of wetting front propagation into an initially dry aquifer [e.g., Barenblatt et al., 199]. The latter solutions are derived using the similarity transform method that imposes the initial condition of a zero water depth in the aquifer in case the boundary conditions are time varying. Moreover, the time-dependent boundary conditions are limited to specific functional forms that satisfy the similarity transform and most vary in time in an unbounded monotonic fashion. The only exception is the solution presented by Parlange et al. [] that generalizes an earlier 1668

2 solution of Barenblatt et al. [199] and discusses the water table response to a nonmonotonic variation of the water level at the inlet. [5] The present study develops a traveling wave solution of the Boussinesq equation. The advantage of the traveling wave method is in its versatility in handling transient boundary conditions while preserving the inherent nonlinearity of the physical phenomenon. The traveling wave formulation has already been applied in various subjects such as solute transport [e.g., van der Zee, 199], and it has proven to be an effective method for tackling the related problem of infiltration of water in porous media described by the highly nonlinear Richards equation [Basha, 11]. It bears some resemblance to the similarity transform method in that both methods are based on a transformation variable that reduces the governing partial differential equation to an ordinary differential equation. However, the transformation in the similarity transform method follows a specific mathematical form that must apply on both the governing equation and the associated initial and boundary conditions. In the traveling wave method, such a restriction is not necessary since the transformation is of a simple form and consists of a moving coordinate variable at a traveling speed that is to be determined. Consequently, the traveling wave approach can handle transient boundary conditions of arbitrary mathematical structure. Moreover, it yields a nonlinear model that is simpler in form than the existing linearized models for similar groundwater flow conditions. [6] In the present work, the traveling wave method is used to obtain a nonlinear solution of a simple logarithmic form that describes the position of the water table as a function of time. It is valid for arbitrary variation of the water level in the adjoining fully penetrating canal, and for zero and nonzero uniform initial water depth in the aquifer. The one-term solution is adaptable to any flow situation, whether recharge or drainage, and allows the derivation of results of practical importance in hydrology. Expressions for the exchange flow rates in the stream-aquifer system and the associated flow volumes stored or extracted from the aquifer are easily obtained from the basic solution. Additional results include algebraic equations for the velocity of the propagation front, wetting front position, and an enhanced relationship for aquifer parameter estimation. [7] The paper is organized as follows. Section introduces the governing nonlinear partial differential equation of one-dimensional nonsteady groundwater flow in unconfined aquifers. Section 3 presents the traveling wave model and associated results in terms of its main parameter, the traveling wave velocity. Section 4 covers various methods for estimating the traveling wave parameter. Section 5 illustrates the traveling wave method with several examples of hydrological interest, evaluates the solutions for various flow conditions, and provides a framework for parameter estimation. Section 6 concludes the study.. Governing Equation [8] The governing equation for unconfined groundwater flow in a horizontal aquifer can be described by k x h h x ¼ h t : (1) [9] Here h (L) is the height of the water table measured from the impervious horizontal base of the aquifer, x is the horizontal coordinate (L), t is the time (T), k is the saturated hydraulic conductivity (L/T), and is the effective or drainable porosity (). Equation (1) is a one-dimensional approximation of a two-dimensional flow system whereby the hydraulic head h is assumed to be independent of depth and the effect of the unsaturated zone is taken as negligible. Equation (1) is known as the Boussinesq equation and is obtained by combining the continuity equation with the Darcy discharge equation assuming a homogeneous flow domain q ¼kh h x : () [1] The initial condition is a prescribed uniform water table height expressed by h ¼ h i. The boundary conditions consist of a varying water level in the adjoining stream h ¼ h b ðþat t x ¼. At the far end of the horizontal aquifer, the water table is assumed to remain at the constant initial level. A schematic representation of the flow domain is shown in Figure 1. [11] Introducing the following dimensionless system [1] One obtains H ¼ h h X ¼ x h T ¼ kt h : (3) X H H ¼ H X T : (4) [13] The parameter h is a characteristic length scale that may be taken as the uniform initial height of the water table Figure 1. Schematic representation of the flow system in the unconfined aquifer bordered by a stream with a fluctuating water level. 1669

3 h i or the final water level in the stream h m. For nonzero initial conditions, h ¼ h i is the most appropriate one for interpreting the results. For dry aquifers with h i ¼, h ¼ h m is a suitable scaling parameter. 3. Traveling Wave Model 3.1. Traveling Wave Approximation (TW) [14] The traveling wave approximation assumes that the form of the solution can be expressed in terms of X T where is the propagation speed. This approach fits well with the physics of the problem and it reduces the partial differential equation to an ordinary differential equation that can be integrated analytically. The traveling wave velocity is derived in terms of the prescribed boundary conditions and comes out to be a time-dependent function rather than a constant value. One of the benefits of the traveling wave solution is that it can capture accurately the evolution of the water table profile using a simple mathematical equation. [15] Using a transformation variable ¼ X T and ¼ T, equation (4) becomes H H BASHA: TRAVELING WAVE SOLUTION OF THE BOUSSINESQ EQUATION ¼ H H : (5) [16] Expanding the dimensionless variable H in a perturbation series in powers of H ¼ H þ H 1 þ O : (6) p ¼ H i 1 : (1) H [1] A second integration of p ¼ H = with respect to an arbitrary reference point yields r ¼ H H r H i ln H H i : (11) H r H i [] Selecting the inlet boundary H ¼ H b ðtþ as the reference point X r ¼, the water surface profile is given implicitly by X ¼ H b H H i ln H H i : (1) H b H i [3] An explicit expression for the water table profile H ðxþ can be obtained from (1) using the Lambert W function that solves the equation W exp ðwþ ¼ z and is readily available in mathematical software libraries. Equation (1) becomes Wexp W ¼ z W ¼ H 1 z ¼ H b 1 exp H b 1 X : H i H i H i H i (13) [4] Equation (13) can be written in a more simplified form by setting h ¼ h i in (3), i.e., H i ¼ 1, ðh 1Þexp ðh 1Þ ¼ ðh b 1Þexp ðh b 1 X Þ: (14) [17] Substituting the perturbation series (6) into (5), accounting only for terms of up to the first order and equating coefficients of equal powers of, one obtains the usual sequence of perturbation equations H H þ H ¼ ; (7) ðh H 1 Þ þ H 1 ¼ H : (8) [18] The zero-order equation (7) is a nonlinear differential equation while the first-order equation (8) is a linear differential equation but with a varying coefficient H. The time-dependent boundary condition H b ðtþ at the left boundary and the far-field boundary condition are applied on the zero-order equation (7) while the boundary conditions for the higher-order equation (8) are set equal to zero. 3.. Nonlinear Water Table Profile [19] Setting p ¼ H = in equation (7), one obtains p p þ ph þ p ¼ : (9) H [] Separating and integrating using the far-field boundary condition ph ð i Þ ¼, one gets 3.3. First-Order Correction [5] The first-order equation (8) is expressed in terms of Y ¼ H H 1 and further simplified since H is independent of Y þ Y Y H H H ¼ : (15) [6] The solution of (15) is the trivial one, i.e., Y ¼, because of the zero boundary conditions. The higher-order perturbation equations (not shown) yield also the trivial solution. Thus, the final solution is H ¼ H and no further correction is possible on the zero-order solution (1) in the current expansion Flow Rate and Front Location [7] The dimensionless flux is given by Q ¼H H X : (16) [8] The dimensionless flow volume V per unit width is V ¼ Z T Q dt: (17) [9] The above dimensionless flow variables are defined in terms of the dimensional values q and v as follows 167

4 Q ¼ q kh V ¼ v h v ¼ Z t q dt: (18) Z 1 X n X H ^H dx ¼ d X dt Z 1 X n ^H dx : (3) [3] Using (1), the flux at any point in the domain is Q x ¼ ðh H i Þ: (19) [31] The velocity field can then be determined from (19) and is in effect equivalent to p (see equation (1)). [3] An expression for the location of the wetting front can be derived from (1) by defining the position of the front as the point at which r ¼ ðh H i Þ= ðh b H i Þ is equal to either 1/ for the midpoint of the head difference or 1/1 for the tip of a front. The front location is then given by X t ¼ 1 ½ H bð1 rþh i ð1 r þ log rþš: () [33] The location of the front is inversely proportional to the traveling wave parameter Linear Profile for Dry Aquifers [34] For dry initial conditions in the aquifer H i ¼, the zero-order solution H is simplified into an explicit linear function of X H ¼ H b X: (1) [35] The groundwater head profile is linear in form with a varying slope dependent on, and the tip of the front is at X t ¼ H b =. The flux at any point is proportional to the head at that particular point Q x ¼ H and the velocity field is uniform and equal to. 4. Traveling Wave Velocity [36] The above nonlinear solution (1) and derivative results (19) () are expressed in terms of the traveling wave velocity that is yet to be determined. The method of moments is employed to derive explicit expressions that relate the model parameter with the boundary values. The moment approach consists of matching the moments of the exact equation with the moments of the approximate solution. An alternate approach for estimating is the method of weighted residuals that consists of minimizing the difference between the original equation and its approximate form [Ames, 1965] Moment Equations [37] Defining ^H ¼ H H i, the governing equation (4) becomes X ^H þ H i ^H X ¼ ^H T : () [38] The moments of the differential equation () is obtained by multiplying both sides of () by X n where n is the moment order. Integrating over X from zero to infinity and using Leibniz s rule, one obtains [39] Integration by parts of (3) yields expressions for the moments. The zero moment is the total mass in the system while the first moment defines the centroid of the profile. Only the first two moments are of use since they are expressed solely in terms of the boundary conditions and can thus be evaluated independently of the shape of the water profile H. The second- and higher-order moments require a priori the form of the solution H. The moment matching procedure allows the derivation of explicit expressions for the traveling wave speed in terms of the known boundary conditions Zero Moment Estimate (M) [4] Upon integration of (3) with n ¼, the zero moment gives the mass balance relationship M ¼ Z 1 ðh H i Z T ÞdX ¼ ðq b Q 1 ÞdT ¼ V: (4) [41] The left-hand side of (4) is the water storage beneath the water table while the right-hand side is the cumulative flow volume crossing the aquifer-stream boundary since Q 1 ¼. [4] The integration variable in (4) can be changed from dx to dh using the definition of p (1) noting that H ¼ H Z 1 fðxþdx ¼ 1 Z Hi fðhþh dh: (5) H i H [43] The left-hand side integral of (4) becomes H b Z Hi H M ¼ H b dh ¼ H b H i : (6) [44] Hence, the flow volume is simply V ¼ H b H i : (7) [45] The flux rate at the boundary is from (19) Q ¼ ðh b H i Þ: (8) [46] Combining (7) and (8) through, one gets a differential equation in V since Q ¼ dv=dt dv V dt ¼ H ð b H i Þ Hb H i : (9) [47] The subscript is used here to denote that this is the M estimate. Integration of (9) using V ðþ¼ yields V ¼ Z T [48] The boundary flow rate is then ðh b H i Þ Hb H i dt: (3) Q ¼ 1 ðh b H i Þ Hb V H i : (31) 1671

5 [49] The traveling wave velocity is ¼ Q H b H i ¼ H b H i V : (3) [5] Eqs. (3) (3) are the governing relationships among the primary variables of hydrological interest: the boundary head H b, the flow rate Q, the flow volume V, and the traveling wave speed. One should note that equation (3) has two possible solutions for V : V > for recharge ðh b > H i Þ and V < for discharge ðh b < H i Þ. For dry initial conditions in the aquifer H i ¼, the above equations are significantly simplified First Moment Estimate (M1) [51] The first moment (n ¼ 1) yields after integration by parts M 1 ¼ Z 1 XðH H i ÞdX ¼ 1 Z T Hb H i dt ¼ P1 : (33) [5] Equation (33) is an expression that relates the first moment of the solution in terms of the prescribed timedependent boundary condition H b ðtþ. The integration of (33) using (1) and (5) leads to Z Hi M 1 ¼ X H H b dh ¼ H b 3 þ 3H b H i 5Hi 3 1 : (34) [53] From (33) and (34), one obtains another expression for where and 1 ¼ A 1 P 1 : (35) A 1 ¼ 1 1 H b 3 þ 3H b H i 5Hi 3 ; (36) P 1 ¼ 1 Z T Hb H i dt: (37) [54] The cumulative flow V 1 is given by (7) and the flux rate Q 1 can be computed using (8). However, a better estimate of Q is obtained from the derivative of (7) Q 1 ¼ dv 1 dt ¼ H b 1 dh b dt V 1 1 d 1 dt : (38) [55] The derivative d 1 =dt is obtained from (35) where 1 d 1 dt ¼ 1 P 1 da 1 da 1 dt ¼ H b ð H i þ H b Þ dh b dt dt A 1 P 1 dp 1 dt ; (39) dp 1 dt ¼ 1 H b H i : (4) [56] From the first two moments, one can also determine the location of the centroid X c ¼ M 1 M ¼ P 1 V ¼ Hb H i Z T Hb H i dt: (41) Weighted Moment Estimate (MW) [57] The zero moment preserves the mass while the first moment preserves the location of the centroid. Both estimates of are of comparable accuracy: the M expression (31) overestimates the discharge rate and the position of the front while the M1 expression (38) underestimates them by a narrower difference. The M and M1 estimates bracket the true value of. Therefore, a weighted value of the two approximations can provide a reliable estimate of if an appropriate value of the weight w is found. The parameter w is later determined by matching the MW estimate with available precise solutions. [58] Defining [59] The volume is w ¼ ð1 wþ þ w 1 : (4) V w ¼ H b H i w : (43) [6] Equation (43) can also be expressed as 1 ¼ 1 w þ w : (44) V w V V 1 [61] The flow rate equation is the derivative of (43) Q w ¼ H b dh b w dt V w ð1 wþ d w dt þ w d 1 ; (45) dt where d 1 =dt is given by (39) and d =dt is derived from (3) d dt ¼ H b V dh b dt V Q : (46) [6] Eqs. (44) and (45) produce the M estimate for w ¼ and the M1 estimate for w ¼ Method of Weighted Residuals [63] The method of weighted residuals (WR) consists of minimizing the residual expression R times a weight function W n over the domain Z 1 W n RdX ¼ : (47) [64] The residual R is the difference between the original equation and its approximation. The choice of the weight function W n is what differentiates one WR submethod from another. The method of moments makes use of the family of polynomials W n ¼ X n, the method of least squares uses the derivative of the residual with the respect to the unknown parameter W n ¼ R=, while Galerkin s 167

6 method employs the derivative of the approximating function W n ¼ H =. A fourth submethod is the collocation method whereby the weight is a Dirac delta function that forces the residual to be zero at specific locations, e.g., the centroid. [65] The residual expression R is obtained by substituting the traveling wave solution into the original differential equation. Substituting (13) into the governing equation (4), one obtains R ¼ 1 H i H H H i H b H i H b H dh b dt þ X 1 H i d H dt : (48) [66] The zero moment of the residual is the integral of R over the domain. Integration of (48) using the transformation (5) produces R ¼ ðh b H i Þ H b dh b dt þ H b H i d dt : (49) [67] Setting R ¼ yields a differential equation for that can be solved for a given H b ðtþ. Expressing (49) in terms of V using (7) and solving using VðÞ¼, one reproduces the M expressions (9) (3). Similarly, minimizing the first moment of the residual R results in a differential equation for that yields the M1 result (35). The Galerkin method with a weighting function equal to W n ¼ H = ¼ ðh i =H 1ÞX yields also the same result for as the M1 estimate (35). This is expected since the weighting functions are similar. [68] The least squares method provides a slightly more accurate estimate of than the previous two approaches although an explicit solution is possible only for sudden water level changes with H b ¼ dh b=dt ¼. The integral of the residual expression (48) times the weighting function W n ¼ R= using the transformation (5) yields R ls ¼ ðh b H i Þ H b H b H b H i 3 þ H b ðh b þ H i ÞH b Hb H i Hb 3 þ 3H ihb 5H i 3 5 ; 4 (5) estimation, and the determination of the base flow contribution to streamflow. [7] The primary variables of hydrological interest are the water table profile H, the flow rate at the boundary Q, and the flow volume V, which are directly related to the traveling wave speed. Given a variation of the stream stage at the boundary H b ðtþ, one can then derive the corresponding traveling wave parameter and evaluate the resulting water table profile and flow expressions. Estimates of the traveling wave parameter are obtained using the moment approach and the least squares approach for the special constant head case. [71] The zero moment calculation consists of evaluating the integral (3) for the volume V and a simple substitution of the time-dependent boundary conditions in (31) and (3) to obtain the flux rate Q and the traveling wave velocity, respectively. The first moment computation consists of carrying out the integral in (37) to evaluate the traveling wave speed 1 in (35) and subsequent substitution in (7) and (38) to determine V 1 and Q 1. The weighted value of the first two moments is calculated from (4) using (3) and (35) with an appropriate value of w. The flux and volume are then computed using (43) and (45). For the constant head case, the least squares estimate of is obtained from (5) and the volume V and flux rate Q ¼ dv=dt are then determined from (7) Particular Results [7] The traveling wave method is illustrated with few examples of practical importance. The first application is the classic example of a sudden change of the boundary head and the remaining two examples are for gradual changes of the water level in the stream (Figure ): a piecewise linear variation that emulates fluctuating water levels in the stream and a special inlet condition for which an exact solution is available [Parlange et al., ]. [73] The boundary conditions are normally defined by an initial value, H i ¼, a final value H m, and a shape where the primes denote differentiation with respect to T. Setting R ls ¼ yields a differential equation for that can be solved analytically only for a constant water level at the boundary using the initial condition ðþ ¼ Hydrological Applications [69] Hydrological applications of the Boussinesq equation include the evaluation of the water table response to a change in stream level for ditch irrigation or dewatering purposes using equation (1), the influence zone of the stream or the extent of the wetting or drying front using equation (), and the velocity field for contaminant transport using equation (19) or (1). Other applications include the quantification of the volume stored or extracted in river banks for flood and water balance calculations, the development of a working relationship for aquifer parameter Figure. Piecewise linear variation of the flood stage that emulates fluctuating water levels in the stream. Also depicted is the sharply increasing and mildly decreasing stream level variation produced by an inverse power function (T p ¼ 3). 1673

7 parameter such as the slope a. The explicit results are then simplified by expressing the characteristic length scale parameter h as the initial h i or final h m water level, and by rescaling the flow rate Q, volume V, and time T using the corresponding shape parameter of H b. Further simplifications of the results are also obtained for dry initial conditions Sudden Variation [74] For a sudden rise or drop of the head in the stream H b ¼, the least squares estimate of is the solution of (5) s ¼ B s 4T B s ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 þ 1 þ 11 : (51) [75] The cumulative flow and flow-rate equations are then pffiffiffi V s ¼ C s T Q s ¼ C s pffiffiffiffiffiffi C s ¼ : (5) 4T B s [76] Equation (51) shows that is time-dependent and inversely proportional to T 1= and the flux Q s is inversely related to the volume V s, Q s ¼ Cs =V s. The traveling wave parameter is also uniquely determined from the head values ahead () and behind the front (). [77] For seepage into an empty aquifer p ¼, the discharge coefficient (5) reduces to C s ¼ ffiffiffi ppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 5 1. The numerical value of C s = 3= is.8995, which is less than 1.5% different from the value of.8875 mentioned in Hogarth et al. [1997]. For the other extreme case that of drainage into an empty channel ¼ with h ¼ h i ( ¼ 1), the discharge coefficient C s becomes equal to C s ¼ :658 which is a difference of less than 1% from the value of ð:33þ ¼ :664 obtained by Polubarinova- Kochina [196]. [78] The weighted moment estimate of is given by (4) using (3) and (35). The M estimate is p given ffiffiffiffiffiffiffiffiffiffiffi by pffiffiffi (3) where V is from (3) equal to V ¼ ð Þ þ T ¼ B 4T B [79] The M1 estimate 1 is given by (35) 1 ¼ B 1 4T [8] The MW estimate w is therefore w ¼ p Bw ffiffiffiffiffiffi 4T ¼ þ : (53) B 1 ¼ 8 3 þ : (54) B w ¼ ð1 wþb þ wb 1 : (55) [81] The cumulative flux and flux rate equations are pffiffiffi V w ¼ C w T Q w ¼ p Cw ffiffiffiffiffiffi C w ¼ : (56) 4T B w [8] For w ¼, equation (56) produces the M estimate, and for w ¼ 1, one obtains the M1 estimate of Q and V. [83] Equating the discharge coefficients of (56) with the least squares estimate (5), one can determine the weight coefficient w w ¼ B s B B 1 B : (57) [84] The coefficient w for discharging aquifers ranges between.63 for ¼ and.7 for ¼. Equating the MW estimate to the above-mentioned results by Polubarinova-Kochina [196] that are valid for the extreme cases of ¼ or ¼, one finds that w ¼ :61 for ¼ and w ¼ :8 for ¼. By comparison with the numerical solution for various functional forms of the boundary condition and different initial values, the value of w that produces optimum results is roughly w ¼ 3=5 for discharging conditions and w ¼ 3=4 for recharging conditions Piecewise Linear Variation [85] The boundary condition defined over the interval T n T T nþ1 is expressed by H b ¼ a n ðt T n Þþ n a n ¼ nþ1 n : (58) T nþ1 T n [86] Here n is the boundary head value at time T n. The piecewise linear function (58) can replicate the fluctuations of the water level in the stream, thus reproducing a realistic behavior for all times. Equation (58) can also model an instantaneous increase or decrease to a level H b ¼ by setting a n ¼ and n ¼. [87] The traveling wave speed w is determined from (4) using (3) and (35). The M estimate is dependent on V that is expressed in the interval T n T T nþ1 by V ¼ ½ F ðh b ÞF ð n Þ where Ša 1 n þ X n ½ F k¼1 ð k ÞF ð k1 Þ Ša 1 k1 ; (59) F ðþ¼ z z4 z3 4 3 z þ 3 z: (6) [88] Similarly, the M1 volume estimate in the interval T n T T nþ1 is given by V1 ¼ 3 H b Hb 3 þ 3H b 53 n ½F 1 ðh b ÞF 1 ð n Þ where Ša 1 n þ X n ½ F k¼1 1ð k ÞF 1 ð k1 Þ o ; (61) Ša 1 k1 F 1 ðþ¼ z z3 z 6 : (6) [89] The parameter w is defined in terms of the weight parameter w that is set at w ¼ 3=4 for recharge and w ¼ 3=5 for discharge conditions. The flow volume is calculated from (44), the flow rate is evaluated from (45), and the water table profile is computed from (1). One can deduce from equation (59) or (61) that the responses to a linear 1674

8 increase (þa k ) and decrease (a k ) with the same slope magnitude ja k j are not symmetric. [9] For a stream level rising linearly from zero initial conditions, i.e., ¼ ¼ att ¼, the traveling wave method gives ¼ 1 ¼ p ffiffiffiffi a pffiffiffiffi T V ¼ V 1 ¼ a a pffiffiffiffi Q ¼ Q 1 ¼ a a T: (63) [91] The traveling wave solution (1) with H b ¼ a T is then p H ¼ a T ffiffiffiffi a X : (64) [9] Equation (64) is the same exact solution of Barenblatt p et al. [199] H=H b ¼ 1 = where ¼ X = ffiffiffiffiffiffiffiffiffi p a T. The tip of the wetting front is at X t ¼ ffiffiffiffi a T. Equation (63) demonstrates that the flux is linear in time for linear H b. This is in contrast to the constant H b case where Q is proportional to T 1= Special Inlet Condition [93] The transient inlet condition studied by Parlange et al. [] consists of a sharply rising boundary head followed by a mildly decreasing limb (cf. Figure ). In the present notation, it is expressed as H b ¼ c ; (65) 1=3 ^T c^t p where ^T ¼ 1 þ T, ¼ 3 ffiffiffi p 3 1 =Tp, and c ¼ 3 ffiffiffi 3 =Hp. An exact solution for H i ¼ is possible: the water table profile pis expressed ffiffiffiffiffiffiffiffiffiffiffiffi by H e ¼ H b X=^T X = 6^T where ¼ c=3 and the exact inflow rate is given by Q e ¼ H b =^T. [94] The TW water table profile is obtained from H ¼ H b w X since H i ¼ and w is determined from (4) using (3) and (35) with w ¼ 3=4. The M estimate is given by ¼ Hb =V where V ¼ c3 4 4log ^T þ! 9 18 þ 11 4=3 =3 ^T ^T ^T : (66) [95] The M1 estimate 1 is given by (35) (37) where P 1 ¼ c ^T 1= ^T 1=3 þ 1 : (67) ^T [96] The cumulative volume is calculated from (43) and the flow rate from (45). 5.. Aquifer Response [97] The water table response to a stream level variation is given by the simple equation (1). The computation of the TW profile requires only the evaluation of the traveling wave velocity corresponding to H b ðtþ at the particular time of interest T. The TW solution is compared against a finite difference solution with a nonuniform grid distribution. A fine grid spacing is used near the inlet and the spacing was progressively increased as one moves away from the left boundary. A very small initial time step is also Figure 3. Aquifer response to sudden and gradual change of the water level at the inlet boundary for monotonically (a) recharging and (b) dewatering events. Shown are the water table profiles at T ¼ 5 (dashed curves) and at T ¼ 1 (solid curves) for a linear rise (a ¼ 4=1) or drop (a ¼4=1) of the boundary head obtained from the numerical method (dotted) and the traveling wave method. Also shown in the front tip for the linear rise case at the two time points. employed for the accurate evaluation of the flow rate and volume. The finite difference code was validated against the exact solution by Parlange et al. [] (cf. sec ). [98] Figure 3 illustrates the typical water table response for sudden and linear variation of the boundary water level at two different times T ¼ 5 and T ¼ 1. For the gradual variations, the head at the boundary is varying at a rate of.4 between the limits ¼ 1 and ¼ 5 while a value of ¼ 3 is selected for the sudden jump case so as not to clutter the graph. However, a similar level of accuracy can be achieved for any small or large value of. The TW prediction and the numerical solution are in close agreement for all the cases shown. The TW model performs best for a linear increase in H b and decreases slightly for sudden rate increase in H b. However, the average relative error is within a few percent (%), which is appropriate for most practical applications. The TW model performs better in aquifer recharge conditions than in the converse case of aquifer dewatering. The TW equation (1) is of a simple logarithmic form that cannot fully capture the parabolic profile that occurs in drainage conditions. A similar situation occurs at large time where the TW model provides a good approximation without mirroring the curvature of the exact profile. Also shown in Figure 3 is the tip of the wetting 1675

9 Figure 4. Aquifer response to nonmonotonic variation of the water level at the inlet boundary. Shown are the numerical and predicted TW water table profiles at T ¼ 3 (dash-dot curve), at T ¼ 5 (dashed curve), and at T ¼ 1 (solid curve) for the piecewise linear variation of the boundary head shown in Figure. front as computed from equation () using r ¼ 1=1 for the linear rise case at the two different times. For dry aquifers (results not shown), the tip of pthe wetting front is given by X t ¼ = and the ratio X t = ffiffiffiffiffiffiffiffiffi 4T is equal to.8995, which is around 1% greater than the result of.81 by Polubarinova-Kochina [196, p. 59]. The significant discrepancy at the tip is a consequence of the linear fit (1) of the true curvilinear profile. [99] Figure 4 presents the water table response to stage fluctuations that is modeled by the piecewise linear variation depicted in Figure. The boundary head H varies linearly over five periods starting with two different increasing rates, followed by a decrease and another increase before a constant level is reached. This example highlights the success as well the limitation of the TW model in predicting the water table profile at various times. Similar to the response of the simple inflow function illustrated in Figure 3, the TW and the numerical solution are indistinguishable at the end of the second rising period at T ¼ 3. There is little difference between the profiles except near the toe. At the end of the receding phase at T ¼ 5, the TW model fails to capture the curvature of the groundwater level because of the inherent assumption of the traveling wave approach (H=T ¼ ). When the boundary head stabilizes to a constant level, the TW model does well again in reflecting the profile as shown for T ¼ 1. [1] For each change in flow conditions due to boundary head fluctuations, the nonsteady effects are significant over a short period of time thereafter. The success of the TW model hinges upon the fact that these nonsteady effects are negligible. This is true only after some time elapses. The accuracy of the traveling wave solution deteriorates when there is a significant transient dynamics in the aquifer. For cyclic fluctuations with strong reverse effects, i.e., with both positive and negative values of Q, the TW solution cannot portray the resulting local transient change for a period of time but it does preserve the first two moments Figure 5. (a) Recharging and (b) discharging rates for sudden and gradual variation of the water level at the left boundary obtained from the numerical (dotted) and the traveling wave method. Also shown are the exact flux rate and the TW approximation for the special case of inverse power H. pthe ffiffiffiffiffiffiffi transformed time and flux rates are T ¼ a T and Q= ja j for the linear rise (a ¼ :4) and linear drop (a ¼:1) case, and ^T ¼ 1 þ T and Q= for the special inverse function. that of mass conservation and centroid location. The TW method completely fails when the head at the boundary returns to its initial value, i.e., when H b ¼ H i, since becomes equal to zero Flow Rate and Volume [11] The inflow and outflow rates for sudden and gradual variation of the boundary head are illustrated in Figure 5. The variables in Figure 5 are normalized with respect to the initial condition, i.e., the scaling depth h is set equal to the uniform initial level h i so that Q ¼ q=kh i, T ¼ kt= h i and ¼ 1. Except for the constant head case, the time and discharge variables are also recast in terms of the shape parameter of H b to generalize further the results. The transformed pffiffiffiffiffiffiffi time and discharge variables are T ¼ ja jt and Q= ja j for the unbounded linear increase, and ^T ¼ 1 þ T and Q= for the special inverse function. The transformations cause the flow distribution to be independent of the shape parameter of the boundary head profile. The linear variation is unbounded for recharging events while it is bound to be greater than zero for discharging conditions. Hence, the groundwater discharge reaches a maximum after which the rate decrease in magnitude and subsequently decline at a rate similar to the one for a 1676

10 Figure 6. Dimensionless flow rate and flow volume for the piecewise linear variation shown in Figure. Flow reversal occurs twice in the period 4 < T < 6. Figure 7. The discharge coefficient for the constant p head case H b ¼ and the transformed volume V ¼ ffiffiffiffiffiffiffi ja j V for the linear variation of the stream level at the boundary H b ¼ 1 þ a T. The flow variables are normalized with the initial value h i. sudden drop in H b. The TW prediction of the flow rate is excellent overall. There is also an excellent agreement between the TW and the exact solution by Parlange et al. []. [1] Figure 6 describes the flow rate and the flow volume for the piecewise linear variation in H b shown in Figure. This example demonstrates again the success and limitation of the TW in predicting the flow rate and volume. The prediction of the flow rate is excellent in the first two periods, decreases in accuracy in the flow reversal phase when Q changes direction, fails to follow the trend in the fourth stage, but matches the value in the fifth constant head period. The failure of TW is due to the incorrect estimation when flow reversal occurs. An accurate prediction requires the inclusion of the neglected temporal term in the governing equation. However, unlike the water table prediction, the failure of the TW model occurs in the beginning of the fourth period rather than in the third, which indicates that the TW model is a better predictor of QT ð Þ than HX ð Þ. The inaccuracy in V is also less than in Q since the cumulative outflow is an integral property and the integration process tends to attenuate errors. The mean error in V is less than % and the maximum error at around T ¼ 5 is less than 1%. [13] Figure 7 presents the discharge coefficient C ¼ C s for the pconstant head case and the transformed flow volume V ¼ ffiffiffiffiffiffiffi ja j V for a linear increase or decrease from a uniform initial condition ¼ 1. The variables are again normalized with respect to the initial condition ðh ¼ h i Þ. The graph gives the flow coefficient C for a constant head H b ¼ to within 1% accuracy, which can then be used for a quick calculation of the flux rate Q and volume V at any given time T using equation (5). One can also calculate B s from C using equation (5) and the traveling wave speed s from equation (51), and subsequently determine the position of the water table and the wetting front at any given time using (1). Figure 7 reveals that the variation of C is not symmetric about ¼ ¼ 1: the value of C for ¼ h m =h i ¼ is much greater than the corresponding value for ¼ 1=. The fact that water seeps more easily into than out of the aquifer is an endemic feature of the nonlinear solution. [14] Similarly, the volume lost or gained for a linear increase or decrease in H b can be obtained from the corresponding curve. Here the boundary value is taken as a surrogate of time. For a selected time value T, the head at the boundary is H b ¼ 1 þ a T. The corresponding transformed volume Vp ffiffiffiffiffiffiffi can be read off the graph and used to determine V ¼ V = ja j for a given slope a. Having V, one can compute from (43) and determine the water table profile and its associated characteristics Aquifer Parameter Estimation [15] There are numerous studies dealing with recession slope analysis and aquifer parameter estimation from measured recession limb of outflow hydrographs [e.g., Brutsaert and Lopez, 1998; Parlange et al., 1]. Most of these studies assume a governing model for the aquifer outflow and use the analytical result in an inverse problem setting. The recession slope analysis is based on two nonlinear analytical solutions of the Boussinesq equation that assume zero water depth at the stream boundary: one pertains to a long (semi-infinite) aquifer and is appropriate for short time [Polubarinova-Kochina, 196], while the other is derived for a finite length aquifer with a specific initial condition and is valid for large time [Boussinesq, 194]. [16] Polubarinova-Kochina [196] adapted a power series solution of the Blasius equation in boundary layer theory to the problem of sudden drainage into an empty stream. The outflow prate derived from the series solution is equal to Q ¼ :33= ffiffiffi T in the present dimensionless notation with h ¼ h i. The TW p prediction (5) for ¼ and ¼ 1 gives Q ¼ :39= ffiffiffi T, which is different by less than 1%. The TW solution allows an extension of this result to the more realistic case of a nonzero stream water level since the flow rate equation (5) is function of the normalized stream water depth. [17] The derivative of the flow rate is dq=dt ¼ Q 3 =C or in dimensional terms 1677

11 dq ¼a Q 3 dt BASHA: TRAVELING WAVE SOLUTION OF THE BOUSSINESQ EQUATION pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 a ¼ þ 1 þ 11 1 ð 1Þ ¼ h m ; k h 3 i l h i (68) where Q ¼ lq is the catchment base flow and l is the total length of the upstream channel adjoining the aquifer. The coefficient a is equal to 1:158=k h 3 i l for ¼ and it becomes equal to :464=k h 3 i l for ¼ 1= ðh m ¼ h i =Þ, which is more than a twofold increase. If the mean aquifer depth h i and the drainable porosity are known a priori, one can use equation (68) to determine the saturated hydraulic conductivity k through curve fitting of dq =dt versus Q on a log-log scale. V w is computed using (43), and the flow rate Q w is determined from (45). A1. Power-Law Variation [11] The boundary condition for a power law variation in between an initial and final stream stage over a duration T m can be expressed by H b ¼ þ at b where a ¼ ð Þ=Tm b. The M estimate is given by (3) where V ¼ a T 1þb 1 þ b þ a3 T 1þ3b 1 þ 3b : (A1) [113] The M1 estimate 1 is again given by (35) where A 1 is expressed by (36) and 6. Conclusions [18] Traveling wave approximation of the Boussinesq equation led to the derivation of a nonlinear solution for uniform initial conditions and transient inlet boundary conditions. The solution is of a simple form and gives the water table elevation at any position as a function of the time-varying boundary head. It yields the exact water table profile for the special case of a uniform water level rise at the boundary. Estimates of the flow rate and volume as well as the extent of the wetting or drying zone are obtained in explicit algebraic forms. These results are useful for the quantification of the bank storage in flood events, the bank depletion in drought periods, the base flow contribution to streamflow, and the estimation of the aquifer parameters. [19] The TW model performs best for sharp front movements and monotonic water table profiles. It provides excellent estimates of the flow rate and flow volume at the inlet boundary. The error in the computation of the water table profile is slightly larger, particularly at large times. The discrepancy in HX ð Þis also relatively higher for dewatering than for recharging conditions. The accuracy of the traveling wave solution deteriorates for fluctuating inlet conditions and worsens in cases when there is a sharp reversal of flow conditions. The traveling wave approach completely fails when the water level returns to its initial level since the basic premise of flow driven by the head values ahead and behind the moving front is violated. [11] The TW model preserves the inherent nonlinearity of the physical phenomenon while offering simple and accurate solutions for practical hydrological applications. Given the level of uncertainty in the aquifer parameters, the error resulting from the simplified model is expected to be less than the error resulting from variations in aquifer properties. It is a cost-effective modeling tool that would prove valuable for preliminary assessment, especially in areas where limited data are available. Its main drawback is that it cannot capture the dynamic response when a severe reversal of flow conditions is present. Appendix A [111] The M and M1 estimates of are presented below for a power law and exponential variation of H b. The traveling wave speed w is then obtained from (4), the water table profile is calculated from (1), the cumulative volume 1þb at P 1 ¼ 1 þ b þ 1 a T 1þb 1 þ b : (A) A. Exponential Variation [114] The boundary condition for an exponential variation in between an initial and final stream stage is H b ¼ ð Þexp ð!tþ where! is the rate parameter that can be determined given a pair of ðt n ; H n Þ values. For example, for H n ¼ ð þ Þ= att ¼ T n,! ¼ log =T n. The M estimate is expressed by (3) where V is determined from 3 ð þ Þ~T þ exp ~T ð3 þ Þ!V ¼ 4 1 exp 1 ~T ð3 Þ 11 þ 7 6 ð Þ 5ð Þ þ 1 3 exp 3 ~T ð Þ 3 : (A3) [115] The M1 estimate 1 is given by (35) where P 1 is expressed by 1!P 1 ¼ ð þ Þ 1 ~T þ exp ~T 4 ð Þexp ~T 1 (A4) 4 ð3 þ Þ ð Þ: [116] Here the transformed time variable is defined by ~T ¼!T. The advantage of the exponential model is its versatility in modeling smooth and gradual changes of the water level as well as sudden changes for!!1. [117] Acknowledgments. This work was partially supported by the American University of Beirut. Part of this work was completed while the author was on research leave at the University of California, Berkeley. The paper benefited significantly from the reviewers comments on an earlier version. References Ames, W. F. (1965), Nonlinear Partial Differential Equations in Engineering, Academic, New York. Barenblatt, G. I., V. M. Entov, and V. M. Ryzhik (199), Theory of Fluid Flows through Natural Rocks, Kluwer Acad., Dordrecht. Basha, H. A. (11), Infiltration models for semi-infinite soil profiles, Water Resour. Res., 47, W8516, doi:1.19/1wr153. Boussinesq, J. (194), Recherches theoriques sur l ecoulement des nappes d eau infiltrees dans le sol, J. Math. Pures Appl., 5me Ser., 1,

12 Brutsaert, W., and J. P. Lopez (1998), Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains, Water Resour. Res., 34(), Chen, Z. X., G. S. Bodvarsson, P. A. Witherspoon, and Y. C. Yortsos (1995), An integral equation formulation for the unconfined flow of groundwater with variable inlet conditions, Transp. Porous Media, 18(1), 15 36, doi:1.17/bf6658. Govindaraju, R. S., and J. K. Koelliker (1994), Applicability of linearized Boussinesq equation for modeling bank storage under uncertain aquifer parameters, J. Hydrol., 157(1-4), , IN341-IN343, Hogarth, W. L., R. S. Govindaraju, J. Y. Parlange, and J. K. Koelliker (1997), Linearised Boussinesq equation for modelling bank storage A correction, J. Hydrol., 198(1-4), , doi:1.116/s- 1694(96)38-8. Hogarth, W. L., J. Y. Parlange, M. B. Parlange, and D. Lockington (1999), Approximate analytical solution of the Boussinesq equation with numerical validation, Water Resour. Res., 35(1), , doi:1.19/ 1999WR9197. Lockington, D. A. (1997), Response of unconfined aquifer to sudden change in boundary head, J. Irrig. Drain. Eng., 13(1), 4 7. Lockington, D. A., J. Y. Parlange, M. B. Parlange, and J. Selker (), Similarity solution of the Boussinesq equation, Adv. Water Resour., 3(7), 75 79, doi:1.116/s39-178()4-x. Moutsopoulos, K. N. (1), The analytical solution of the Boussinesq equation for flow induced by a step change of the water table elevation revisited, Transp. Porous Media, 85(3), , doi:1.17/s Parlange, J. Y., W. L. Hogarth, R. S. Govindaraju, M. B. Parlange, and D. Lockington (), On an exact analytical solution of the Boussinesq equation, Transp. Porous Media, 39(3), , doi:1.13/ a: Parlange, J. Y., F. Stagnitti, A. Heilig, J. Szilagyi, M. B. Parlange, T. S. Steenhuis, W. L. Hogarth, D. A. Barry, and L. Li (1), Sudden drawdown and drainage of a horizontal aquifer, Water Resour. Res., 37(8), 97 11, doi:1.19/wr189. Polubarinova-Kochina, P.Y.-A. (196), Theory of Groundwater Movement, translated from Russian by R. J. M. DeWiest, 613 pp., Princeton Univ. Press, Princeton, N. J. Telyakovskiy, A. S., G. A. Braga, and F. Furtado (), Approximate similarity solutions to the Boussinesq equation, Adv. Water Resour., 5(), , doi:1.116/s39-178(1)6-4. Tolikas, P. K., E. G. Sidiropoulos, and C. D. Tzimopoulos (1984), A simple analytical solution for the Boussinesq one-dimensional groundwater flow equation, Water Resour. Res., (1), 4 8. van der Zee, S. E. A. T. M. (199), Analytical traveling wave solutions for transport with nonlinear and nonequilibrium adsorption, Water Resour. Res., 6(1), , doi:1.19/9wr