Journal of Hydrology

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1 Journal of Hydrology 376 (29) Contents lists available at ScienceDirect Journal of Hydrology journal homepage: New approximate solutions of horizontal confined unconfined flow Xu-Sheng Wang a, *, Li Wan a, Bill Hu b a School of Water Resources & Environment, China University of Geosciences, Beijing 183, PR China b Department of Geological Sciences, Florida State University, Tallahassee, FL 3236, United States article info summary Article history: Received 7 January 29 Received in revised form 31 May 29 Accepted 18 July 29 This manuscript was handled by P. Baveye, Editor-in-Chief, with the assistance of Hongbin Zhan, Associate Editor Keywords: Confined unconfined flow Boussinesq equation Sorptivity Boltzmann transformation Runge Kutta method New approximate solutions of horizontal confined unconfined flow are developed to extend the solution of Li et al. [Li, L., Lockington, D.A., Barry, D.A., Parlange, J.-Y., Perrochet, P., 23. Confined unconfined flow in a horizontal confined aquifer. Journal of Hydrology 271, pp ] to more general hydraulic conditions. Both initial water table and storativity of the confined zone are considered. The Boltzmann transformation is applied to simplify the mathematical model with dimensionless variables and lumped parameters. A closed-form analytical solution is obtained for flow in the confined zone. A high-precision numerical solution with the Runge Kutta method and an iterative process is proposed for nonlinear flow in the unconfined zone. For an initially dry aquifer, an approximate analytical solution is derived. In comparison with the initially dry condition, the initial water table condition will lead to a faster moving speed of the confined unconfined conversion interface. The initial water table condition is beyond the scope of Li et al. s (23) method. The groundwater level could be overestimated by neglecting storativity of the confined zone. By introducing an equivalent storage coefficient, the developed new solutions can also be applied to analyze the unconfined flow in a double-layer aquifer system. In the case study on a confined unconfined groundwater flow induced by rise of the water table in a reservoir of a hydropower station in China, the calculated distribution of groundwater level with the new solutions generally agrees with the observations at the field site. Ó 29 Elsevier B.V. All rights reserved. Introduction Mixed confined and unconfined flow can occur in a confined aquifer when part of it is in an unconfined condition. This type of groundwater flow is frequently generated by heavy pumping in confined aquifers. An unconfined zone will develop if the groundwater level in the vicinity of a pumping well is lower than the top of the aquifer. With a continuous pumping, the unconfined zone extends and the interface between the unconfined zone and the confined zone moves away from the pumping well. For this transient radial confined unconfined flow driven by a pumping well, various solutions have been developed to simulate the flow process, which including approximate analytical solutions (Chen, 1966; Moench and Prickett, 1972; Babu and van Genuchten, 198; Hu and Chen, 28), an electrical analog solution (Rushton and Wedderburn, 1971) and a numerical solution (Elango and Swaminathan, 198). For steady confined unconfined flow driven by a group of wells, an exact analytical solution was derived by Chen et al. (26). Mixed confined and unconfined flow can also occur as result of change of hydraulic head at the boundaries of an aquifer with a * Corresponding author. Tel.: ; fax: address: wxsh@cugb.edu.cn (X.-S. Wang). confining unit, such as rise of river stage induced by floods or reservoirs. However, investigations of this kind of confined unconfined flow system are much less common than those of the confined unconfined flow caused by a pumping well. Li et al. (23) developed an approximate analytical solution for horizontal confined unconfined flow in a bounded aquifer. The flow changes from initially unconfined condition to a mixed confined unconfined condition in a confined aquifer when water level at the side boundary suddenly rises above the top of the aquifer. Their solution is based on two assumptions: (1) the change of storage in the confined zone can be ignored, which means that the storage coefficient of the confined aquifer is assumed to be zero; (2) the aquifer is initially dry. These assumptions greatly simplify the analysis, but limit the application of the solution to more general cases. The assumption of an initially dry condition particularly limits the application of their solution since it is an infrequent case in the field. In addition, as an essential hydraulic property of a confined aquifer, storativity can be ignored only if its influence on the confined unconfined flow is weak. However, in Li et al. s (23) study, the influence of this storativity was not investigated. Confined unconfined flow has been found in places where the hydrogeological condition is characterized by river-aquifer interactions (Urbano et al., 26). A typical example in China is the area of Xixiayuan reservoir in Henan province. After the construction of /$ - see front matter Ó 29 Elsevier B.V. All rights reserved. doi:1.116/j.jhydrol

2 418 X.-S. Wang et al. / Journal of Hydrology 376 (29) Xixiayuan reservoir on the Yellow River in 27, rapid rise of the groundwater level in a sandy-gravel aquifer caused by a change of water level in the reservoir was observed on both sides of the Yellow River. It resulted in a partially confined flow condition. The mixed confined and unconfined flow at this site is so appealing that several studies have been conducted to explain the observations. Some of the aquifer conditions at this site are similar to the model developed by Li et al. (23). However, their assumption of an initially dry situation is not satisfied. Therefore, a more general solution is required to be developed. The purpose of this study is to develop new solutions to the horizontal confined unconfined flow in a bounded confined aquifer. Both the initial water table and storativity in the confined zone should be considered in order to relax the assumptions made in Li et al. (23). For flow in the confined zone, a closed-form analytical solution is derived and applied to study the influence of storage coefficient on groundwater flow. For flow in the unconfined zone, a high-precision numerical solution is developed with the Runge Kutta method. In addition, for the special condition in which the aquifer is initially dry, an approximate analytical solution is derived similar to Li et al. (23) but with different parameters. The approximate solution for a confined aquifer with an initial water table is applied to study the confined unconfined groundwater flow in the area of Xixiayuan reservoir in China. Model description For the groundwater flow in the horizontal confined unconfined aquifer shown in Fig. 1, the governing equations and boundary and initial conditions are: ¼ 2 ; < x 6 x sðtþ; @x Þ¼S y hðx > ; t ¼ Þ ¼h ; x > x sðtþ; ð2þ ð3þ x s ðt ¼ Þ ¼; ð4þ hðx ¼ ; t P Þ ¼h ; h > B; ð5þ hðx!1; t > Þ ¼h i ; ð6þ hðx ¼ x s ; t > Þ ¼B; ð7þ where h is the hydraulic head related to the bottom of the aquifer [L]; x is the horizontal distance from the boundary [L] and t is time [T]; K is the hydraulic conductivity of the aquifer medium [TL 1 ]; S c and S y are the storage coefficients of the confined zone and specific yield of the unconfined zone [ ], respectively; B is the thickness of the confined aquifer [L]; x s is the distance of the interface between the confined and unconfined zones [L]; h and h i are the hydraulic head of the left boundary and the initial water table [L], respectively. If h i =, the aquifer is initially dry. piezometric surface Eq. (1) is the governing equation for flow in the confined zone, and Eq. (2) is the one-dimensional Boussinesq equation for flow in the unconfined zone. Eq. (3) gives the initial condition of flow in the unconfined zone. Eq. (4) is the initial condition of the confined unconfined conversion interface, which means that at the beginning of the confined unconfined flow, the confined condition only occurs at the left boundary. Eqs. (5) and (6) give the boundary conditions at the left and right ends. Eq. (5) also provides the initial condition (t = ) of the flow in the confined zone, which starts from the left boundary. Eq. (7) describes the control condition of the confined unconfined conversion interface. Before we conduct mathematical analysis, we need to introduce the following lumped parameters sffiffiffiffiffiffi sffiffiffiffiffiffi sffiffiffiffi KB KB a c ¼ ; a u ¼ ; a D ¼ a u S c ¼ ð8þ a c S c and the transforms S y / ¼ x p ffiffi ; f ¼ h t B ; / s ¼ x s p ffiffi : t a u Applying Eqs. (8) and (9) to Eqs. (1) (7), we obtain: a u a 2 D / df d/ þ 2 d2 f d/ 2 ¼ ; < / 6 / s; ð1þ / df d/ þ 2 d df ðf d/ d/ Þ¼; / > / s; ð11þ f ðþ ¼f ¼ h =B; ð12þ f ð1þ ¼ f i ¼ h i =B; ð13þ f ð/ s Þ¼1: ð14þ Eq. (9) is the modified Boltzmann transformation simplifying the original governing equations with dimensionless variables. Note that both the initial condition given in Eq. (3) and the boundary condition given in Eq. (6) are transformed to Eq. (13). In Eq. (14), the dimensionless location of the confined unconfined conversion interface, / s, is a constant, which implies that the interface p location in real time space domain, x s, is proportionally to ffiffi t. This Boltzmann transformation method has also been applied in previous studies on fully unconfined flow with wetting front (Lockington et al., 2; Telyakovskiy et al., 22; Li et al., 25). The flow must be satisfied with continuity condition at the location of the confined unconfined conversion interface. The result is, df ¼ df d/ d/ ð15þ /¼/s þ: /¼/s Solutions Analytical solution of flow in the confined zone Eq. (1) with conditions given by Eqs. (12) and (14) can be solved as S y ð9þ h boundary h the aquifer groundwater table h B h i f ¼ f ðf 1Þ erf ða D/=2Þ erf ða D / s =2Þ ; < / 6 / s ð16þ where erf (y) is the error function. Eq. (16) describes the distribution of hydraulic head in the confined zone. If the storage coefficient is ignored, a D =, Eq. (16) becomes x s Fig. 1. Schematic diagram of the confined unconfined flow in a bounded horizontal aquifer. x s is the location of the confined unconfined conversion interface. x f ¼ f ðf 1Þ / / s ; < / 6 / s : ð17þ Eq. (17) means that the hydraulic potential surface in the confined zone is represented by a straight line as proposed by Li et al. (23).

3 X.-S. Wang et al. / Journal of Hydrology 376 (29) However, if a D >,f(/) is not a straight line and from Eqs. (15) and (16) we have a D ð1 f Þ expð a 2 D pffiffiffi /2 s =4Þ ¼ df erf ða D / s =2Þ d/ ð18þ /¼/s þ: p Eq. (18) will be applied in solving flow in the unconfined zone. Approximate solution of flow in the unconfined zone Before we solve Eq. (11) for flow in the unconfined zone, we need to introduce a function, J(f), which is defined as Jðf Þ¼ 2f df d/ ; ð19þ for / P / s. It represents the flow rate of groundwater in the unconfined zone. In Eq. (19), df/d/ is less than zero because groundwater flow from left boundary towards right as shown in Fig. 1. It indicates that hydraulic head decreases with increasing of distance from the left boundary. Thus, df/d/ < and J(f) P. With definition of J(f), we divide Eq. (11) into two first-order ordinary differential equations as d/ df ¼ 2f J ; / > / s; ð2þ and dj df ¼ /; / > / s: ð21þ Eq. (2) can be obtained directly form Eq. (19). Eq. (21) is obtained by substituting Eq. (19) into Eq. (11). Integrating Eq. (21) we have Jðf Þ¼ Z f f i /ðf Þdf : ð22þ To obtain Eq. (22), the boundary condition J= for f=f i is applied. We can tell from Eq. (22) that J(f) is equal to the increment of saturated area in the cross section of the unconfined zone between / (f) and 1, i.e., it is the volume of water adsorbed into the unconfined zone between /(f) and 1. Owing to the mathematical complexity, it is difficult to analytically solve Eqs. (2) and (21) in a general case. Therefore a numerical method is used to obtain the solution. However, for a special case as the initially dry aquifer, we will derive an analytical solution. Numerical solution for the conditions with an initial water table Eqs. (2) and (21) can be easily solved with the Runge Kutta method if boundary conditions /(f = 1) and J(f = 1) are known. For these conditions, according to Eq. (14), we have /ðf ¼ 1Þ ¼/ s : ð23þ By substituting Eq. (18) into Eq. (19) with f = 1, we obtain another boundary condition as follows: Jðf ¼ 1Þ ¼ 2a Dðf 1Þ expð a 2 D pffiffiffi /2 s =4Þ : ð24þ erf ða D / s =2Þ p According to Eqs. (23) and (24), /(f = 1) and J(f = 1) are functions of the dimensionless distance of the confined unconfined conversion interface. Given / s, we can find the solution of /(f) and J(f) by using the Runge Kutta method on Eqs. (2) and (21). However, / s is generally unknown. To obtain the value of / s, Eq. (22) is applied at condition of f = 1 to find the alterative boundary value of J, Jðf ¼ 1Þ ¼S ¼ Z 1 f i /df ; ð25þ where S is the called sorptivity. Sorptivity was initially defined in the theory of unsaturated soil flow (Parlange et al., 1992b). Here, it is applied to represent the volume of water adsorbed, over the initial water table, in the whole unconfined zone. From Eqs. (24) and (25), we have Sð/ s Þ¼ 2a Dðf 1Þ pffiffiffi p expð a 2 D /2 s =4Þ : ð26þ erf ða D / s =2Þ The accurate value of / s is the root of Eq. (26). To find the root, we propose an iterative method. The procedure is listed below. (i) If f > 1, by substituting a guess value of / s (/ s = 2 is initially applied in this study) into Eqs. (23) and (24), we obtain the boundary values of /(f) and J(f) atf = 1. However, if f =1,we let / s = and use a guess value of J(f = 1) to replace Eq. (24). In this study, the initial guess value of J(f =1)is2. (ii) The fourth order Runge Kutta method is used to obtain the solutions of Eqs. (2) and (21) from f =1tof = with a step Df<. The numerical result of /(f)is/, / 1, / 2,..., / N corresponding to f =1, f =(1 Df), f =(1 2Df),..., f =(1 NDf) =, where Df = (1 f i )/N and N is the number of steps. Note that there must be J P as indicated in Eq. (22). If J < occurs in the calculation, it will be assumed to be zero. / is directly maintained to the value of the last step when J>. Generally, N = 5 1 is used to achieve a high-precision of the result. (iii) To check the agreement between Eqs. (25) and (26), we calculate S through the numerical approximation of Eq. (25) as S ¼ Z 1 f i /df Df 2 X N j¼1 ð/ j 1 þ / j Þ: ð27þ It is the left hand side(lhs) of Eq. (26). The right hand side(rhs) of Eq. (26) has been given by Eq. (24) in step (i) and denoted as J 1. The difference between S and J 1, DS = S J 1, indicates that J 1 is too small (DS>) or too large (DS<) to satisfy Eq. (26). This difference is caused by incorrect input of / s value in step (i). However, if f =1,J(f = 1) that specified in step (i) is denoted as J 1 and the difference between S and J 1 is caused by an incorrect guess of J 1 in step (i). (iv) If f > 1, we modify the / s value. Otherwise if f = 1, we modify the J 1 value. From step (iii) we know that / s value should be modified to produce a larger (DS>) or smaller (DS<) value of J 1 if f > 1. It is found according to Eq. (24) that J 1 decreases with an increase of / s. It means that we should use a smaller (DS>) or lager (DS<) value of / s for calculation. Modification of the / s value proposed in this study is given by / new s ¼ / old s 1 g DS ; ð28þ S where g is a small factor (g =.1 is empirically found to be always acceptable) to reduce the oscillation pattern of the iteration. Return step (i) with the new / s value. However, if f =1,J 1 is directly modified to J 1 + gds and returned on step (i). This iterative process continues until DS is close to zero. In this study, DS <.1 is applied to represent the condition with a high-precision result. Using the above iterative process and the Runge Kutta method, we can obtain a high-precision numerical solution of /(f) for the unconfined zone and the dimensionless distance of the confined unconfined conversion interface. The solution can also be written as f ¼ f ð/; a D ; f ; f i Þ; / s ¼ / s ða D ; f ; f i Þ ð29þ

4 42 X.-S. Wang et al. / Journal of Hydrology 376 (29) Approximate analytical solution for an initially dry aquifer Substituting Eq. (22) into Eq. (19), we have f ¼ 1 2 d/ df Z f f i /ðf Þdf ; f i < f < 1: ð3þ This equation is similar to the diffusivity function that has been proposed in the theory of unsaturated soil flow (Bruce and Klute, 1956; Philip, 196; Parlange et al., 1992a). If the aquifer is initially dry, i.e., f i =, Eq. (3) becomes f ¼ 1 2 d/ df Z f /ðf Þdf ; < f < 1: ð31þ Then the sorptivity defined in Eq. (25) becomes S ¼ 2 df Z 1 d/ ¼ /df : / ¼ /s ð32þ As demonstrated in Li et al. (23), this initially dry aquifer has a wetting front at the location x f. Similar to the definition of / s, a dimensionless distance of the wetting front, / f, defined as / f = x f / (a u p t), is introduced herein. The boundary condition at the wetting front is f ð/ f Þ¼: ð33þ Eq. (33) is a replacement of Eq. (13) for the boundary condition at the right side of the unconfined zone. Another boundary condition at the wetting front can be derived as df ¼ 1 Z! f d/ 2 lim /ðf Þdf =f ¼ 1 f! 2 / f : ð34þ /¼/f Discussions on the new solutions Comparison of solutions for an initially dry aquifer If f i =, both the numerical solution with the Runge Kutta method and the approximate analytical solution give the result of confined unconfined flow in an initially dry aquifer. They are comparable in check of the agreement between the two solutions. When f = 1 and f i =,S =.8875 is obtained by the numerical method, which is very close to the exact value of Parlange et al. (1992a), This high-precision result indicates that the numerical solution with the Runge Kutta method is accurate. Fig. 2 shows the model-calculated distribution of hydraulic head with both the numerical solution and the approximate analytical solution. The figure shows that these two solutions for an initially dry aquifer are well matched. The effects of initial water table The confined unconfined flow in an aquifer with an initial water table is different from that in an initially dry aquifer. The initial water table significantly smoothes the wetting front and speeds up the movement of the confined unconfined conversion interface. These effects are shown in Figs. 3 and 5. Fig. 3b shows the relationship between / s and f i as a function of f. It is shown that with a fixed f, / s increases with increase of the height of the initial water table. Now we use the case of f = 1.2 as an example to explain the change characteristics. When f i =, which means an initially dry aquifer, / s is.359. However, when f i is.5 or.9, which means the aquifer is initially 5% or 9% saturated, the / s value would be.571 or It can be found that / s is relatively not sensitive to the initial water table while f i is less Following Parlange et al. (1992b), the solution of Eq. (31) can be approximated by f ¼ 1 S 2 ð/ / sþ A 4 ð/ / sþ 2 ; / > / s ; ð35þ where A is an unknown factor. Considering of boundary condition of Eq. (34), we have / f ¼ S þ Að/ f / s Þ: ð36þ a f=h/b From the derivation in the Appendix A we obtain A ¼ 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 S2 þ S 4 þ 16S2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð37þ S ¼ / s þ / 2 s ð2 AÞ½A/2 s 4ð1 AÞ2 Š ; 2 A ð38þ where S = Therefore A =.447. The sorptivity, S, is dependent on / s. When f = 1, it is obvious that / s = and from Eq. (38) there is S = S. However, / s is unknown for f > 1. To determine / s, Eq. (26) is applied. Substituting Eq. (38) into Eq. (26), we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a D ðf 1Þ expð a 2 D pffiffiffi /2 s =4Þ ¼ / s þ / 2 s ð2 AÞ½A/2 s 4ð1 AÞ2 Š : p erf ða D / s =2Þ 2 A ð39þ The root of Eq. (39) is the accurate / s value for given values of a D and f. Substituting this / s value into (38) we obtain S. Substituting A and S into Eq. (35) we have the straightforward approximate solution of the confined unconfined flow in an initially dry aquifer. b f=h/b.2 a D =, f = a D=, f =2 φ = x /( a t ) u φ = x /( a t ) Fig. 2. Comparison between the numerical solution (solid lines) and the approximate analytical solution (circles) for an initially dry aquifer. u

5 X.-S. Wang et al. / Journal of Hydrology 376 (29) fi = hi /B fi = hi /B a b f =1.2 f = f =1.2 f =2 than.5. However, it would be very sensitive to the initial water table while f i is greater than.8. This change indicates that the confined unconfined conversion interface can moves significantly faster than in an initially dry aquifer if most of the medium is initially saturated in a water table condition. The sorptivity of the unconfined zone, defined in Eq. (25), is used to further study the relationship between / s and f i, The Sorptivity, S, represents the capacity of water adsorption in the unconfined zone until it is fully saturated. Fig. 3a shows the effect of initial water table on sorptivity of the unconfined zone. The sorptivity decreases with an increasing of the initial groundwater level because the higher initial water table, the smaller residual unsaturated zone for water adsorption. From Eq. (26) we know that a smaller value of S results in a larger value of / s. It means that the decrease of the sorptivity caused by the rise of initial water table will lead to an increase of extension speed of the confined zone. When the initial water table approaches the aquifer top, the sorptivity tends to be zero and the confined unconfined conversion interface can moves very fast away from the left boundary. The effect of non-zero storage coefficient f =4 Sorptivity a D = φ s = x /( a t ) The roles of storage coefficient of the confined zone, S c, and specific yield of the unconfined zone, S y, on groundwater flow are represented by the dimensionless factor, a D, as defined in Eq. (8). s f =4 u a D = Fig. 3. The effects of initial water table on sorptivity (a) and distance of confined unconfined conversion interface (b). Generally, specific yield (also denoted as effective porosity) of a sandy aquifer varies between.1 and.3 (Smith and Wheatcraft, 1992). Storage coefficient of a confined aquifer with thickness of tens of meters is normally 1 3 to 1 5 (Smith and Wheatcraft, 1992). Therefore, the value of a D generally ranges between.5 and.1. However, if the solution is applied to analyze the unconfined flow in a double-layer aquifer system, the value of a D could be as large as.4.6. This special condition will be discussed in Study on a double-layer unconfined aquifer system. In Fig. 4, three a D values are chosen to show the effect of a D on the relationship between / s and f. The case of a D = is for the condition when the storage coefficient is small enough to be ignored. This case is used in Li et al. s (23) study. It is shown in the figure that / s is almost independent on a D while f < 3. However, when f > 6, the / s and f curve will be significantly affected by the value of a D. The results indicate that / s will be overestimated if we ignore the storage coefficient and the overestimation will quickly increase with the increase of f. When S c is large, i.e., a D is large, storativity of the confined zone is large and some amount of water discharge across the left boundary will be adsorbed into the confined zone. As a result, the confined unconfined conversion interface will decrease its moving speed. This is the reason that / s is overestimated with non-zero storage coefficient. However, as demonstrated by curves of a D =.2 and a D =infig. 4, the overestimation is generally small when f < 1, less than 5% for the relative error. The effect of non-zero storage coefficient on the distribution of hydraulic head with distance can be seen in Fig. 5. When the storage coefficient is ignored, the hydraulic equipotential surface in the confined zone is a straight line as shown by curves B2 and B3. However, when the storage coefficient is much larger than zero, such as a D =.4, the hydraulic gradient will significantly decrease with distance in the confined zone as shown by curve B1. This is because the groundwater flow rate decreases with distance due to the increment of storage in the confined zone. As a result, an assumption of zero storage coefficient will overestimate hydraulic head in the confined zone. The relationships among a D, / s and f are also described by Eq. (26). It can be rewritten as pffiffiffi p erf ðad / f ¼ 1 þ s =2ÞS 2a D expð a 2 D /2 s =4Þ : ð4þ When the storage coefficient in the confined zone is ignored, i.e., a D =, a simplified relationship can be found through Eq. (4) as, f ¼ 1 þ / ss 2 : ð41þ f = h/b h i = a D= φ s = x /( a t ) s u a D=.2 a D= Fig. 4. Distance of confined unconfined conversion interface, x s, versus boundary head, h, for an initially dry aquifer with different values of a D.

6 422 X.-S. Wang et al. / Journal of Hydrology 376 (29) f Substituting Eqs. (9) and (12) into Eq. (41) we have x s ¼ 2a uðh BÞ BS p ffiffi t ; which is the same as the result obtained by Li et al. (23). Special case study A1 A2 B φ Remarks of the equivalent dry aquifer method ð42þ For an aquifer with an initial water table, there is a potential alternative approach to solve the problem of confined unconfined flow. In this approach, the coordinate origin is moved to the initial water table and the flow above the origin point is regarded as the same as that of an initially dry aquifer. The hydraulic head relative to the origin point can then be calculated with the approximate analytical solution for an initially dry aquifer. It is called the equivalent dry aquifer method in this study. Here, we would like to discuss whether or not we can apply this method to simplify the analysis of confined unconfined flow with an initial water table. Define the relative hydraulic head, Dh, and the relative aquifer thickness, DB, as follows Dh ¼ h h i ; DB ¼ B h i ð43þ where h i is the height of the initial water table. With Eq. (43), Eqs. (1) and (2) can be rewritten as ¼ 2 ; < x 6 x sðtþ 2 Dh Kh ; x > x sðtþ; ð45þ respectively. The initial and boundary conditions become, Dhðx > ; t ¼ Þ ¼; ð46þ Dhðx ¼ ; t P Þ ¼Dh ; hðx!1; t > Þ; ¼ ð47þ hðx ¼ x s ; t > Þ ¼DB; ð48þ where Dh = h h i. In the equivalent dry aquifer method, it is assumed 2 Dh Kh i ¼ : B2 B3 top of the aquifer Fig. 5. Distribution of dimensionless hydraulic head along dimensionless distance from the boundary in different situations: A1 (a D =, f i =, f = 2); A2 (a D =, f i =.3, f = 2); B1 (a D =.4, f i =, f = 9); B2 (a D =, f i =, f = 9); B3 (a D =, f i =.3, f = 9). Therefore Eq. (45) can be simplified @Dh Dh ¼ @t ; x > x sðtþ: ð5þ Eq. (5) is similar to Eq. (2). With this equation, we may apply the analytical solution for an initially dry aquifer to calculate the relative hydraulic head, Dh. By rewriting the lumped parameters as sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi KDB KDB a c ¼ ; a u ¼ ð51þ S c S y and invoking Eq. (35), the relative hydraulic head can be approximated as Dh DB ¼ Dh DB ðdh DB 1Þ erf ða D/=2Þ erf ða D / s =2Þ ; < / 6 / s; ð52þ Dh DB ¼ 1 S 2 ð/ / sþ A 4 ð/ / sþ 2 ; / > / s ð53þ where / and / s are the variables defined in Eq. (9). / s is the root of Eq. (39) with f = Dh /DB. Solution of the absolute hydraulic head can then be obtained as h = h i + Dh. The equivalent dry aquifer method is not a rigorous method because the assumption, shown in Eq. (49), is generally not true. The assumption ignores the water movement contributed by the saturated zone under the initial water table. Therefore, Eq. (49) can hold only if the height of the initial water table, h i, is zero. To quantitatively investigate the accuracy of the equivalent dry aquifer method, we conduct a study of a hypothesized example to compare the results of various solutions. General conditions of the hypothesized example is shown in Table 1, where z top is the elevation of the aquifer top, z is the elevation of the left boundary head and z i is the elevation of the initial water table. Three cases are investigated to consider variable thickness of the initial saturated zone. The elevation of the aquifer bottom, z bot, in the three cases is shown in Table 1. Three methods are applied to solve the confined unconfined flow in this hypothesized aquifer. They are briefly described below. (i) The straightforward numerical model on MODFLOW. Dimensional numerical simulation is conducted for the hypothesized example through the widely accepted software, MODFLOW (McDonald and Harbaugh, 1988), which is based on a finite-difference numerical method. It is composed of one model-layer with a row of 6 cells. Horizontal size of each cell is Dx =.1 m. To achieve the conversion between the confined and unconfined conditions, the model-layer type is specified to confined/unconfined with variable transmissivity and storage coefficient in MODFLOW (McDonald and Harbaugh, 1988). The basic hydraulic parameters of the hypothesized aquifer are shown in Table 1 and input to this numerical model in MODFLOW. The right boundary is treated as a no-flux boundary. This boundary simplification hardly affects the results if the stress period is less than 3 s. In the study, a constant time step, Dt = 1 s, is used. The numerical model works well and the absolute discrepancy of water debug is less than 1%. (ii) The new solution proposed in this study. Dimensionless hydraulic head, f = f(/, a D, f, f i ), in the unconfined zone is obtained with the Runge Kutta method as described in Numerical solution for the conditions with initial water table. / s is determined through the iterative process. Dimensionless hydraulic head in the confined zone is then obtained though Eq. (16). For the three cases, values of the parameters used in Eqs. (8) (29) are derived with the basic hydraulic parameters and shown in Table 1. The hydraulic head above the aquifer bottom can be obtained as h = Bf.

7 X.-S. Wang et al. / Journal of Hydrology 376 (29) Table 1 Conditions of the hypothesized example in Remarks of the equivalent dry aquifer method. General z top =5m,z = 5.5 m, z i =m,a D = Cases z bot = 15 m z bot = 5m z bot = 1m Parameters The model on MODFLOW K =.1 m/s, S y =.1, S c = The new solution B =2m,h i =15m,h = 2.5 m B =1m,h i =5m,h = 1.5 m B =6m,h i =1m,h = 6.5 m Equivalent dry aquifer method DB =5m,f = 1.1, KDB/S y =.5 m 2 /s (iii) The equivalent dry aquifer method. The relative hydraulic head defined in Eq. (43) is calculated with Eqs. (52) and (53). In this situation DB, f and KDB/S y used in Eqs. (51) (53) are derived with the general conditions and the basic hydraulic parameters as shown in Table 1. It is the same situation of Case (a) and Fig. 4 in Li et al. (23). Therefore, one can also directly use the solution of Li et al. (23) as the result of Dh for this example. Results of the three methods are presented in Fig. 6. The figure indicates that the results of the new solution are almost the same as those of the model on MODFLOW for all the cases. This agreement proves the accuracy of the new solution in this study. However, the results of the equivalent dry aquifer method do not match the exact solution, even for the case where the initial thickness of the saturated zone is only 1/6 of the aquifer thickness as shown in Fig. 6c. The groundwater level in the unconfined zone is significantly underestimated by the equivalent dry aquifer method. This error increases with increase of the initial thickness of the saturated zone and the running time as shown in Fig. 6a and b. The reason for the error is due to the ignorance of the water transport in the saturated zone below the initial water table, as we pointed out previously. Therefore, the higher the initial water table (relative to the bottom of the aquifer), the more the transmissivity and the rate of confined unconfined conversion are underestimated by the equivalent dry aquifer method. As shown in Fig. 3b, the dimensionless location of the confined unconfined interface, / s, is not sensitive to the initial head for the cases with small values of h i /B and Dh /B. This indicates that the equivalent dry aquifer method is viable only if the initial thickness of the saturated zone and the increment of left boundary head are both extremely small. In comparison with a classical numerical method, such as the MODFLOW, the new solution in this study is more efficient through use of dimensionless variables and lumped parameters. Study on a double-layer unconfined aquifer system In the above study, the new solution is derived for a confined, but not fully saturated aquifer. For a fully confined aquifer, as demonstrated in The effect of non-zero storage coefficient, a D is generally less than.1. It can be seen from Fig. 4, that in this situation the solution is obviously close to the solution of a D = even though h /B is as large as 1. Therefore, the influence of the non-zero storage coefficient on flow can be ignored for a confined aquifer. However, this confined condition is no longer a requirement for the application of our new solution. The new solution can also be an approximate solution for flow in a double-layer unconfined aquifer system, in which the hydraulic conductivity of the underlying aquifer is much larger than that of the upper layer. This system is a semi-confined aquifer system. This kind of system widely exists in alluvial plains, where groundwater is frequently influenced by streams. Horizontal hydraulic conductivity of the semi-confining layer could be very small, ranging from 1 4 to 1 m/d, in comparison with a gravel aquifer, in which conductivity ranges from a h (m) b h (m) c h (m) t =1 s t =3 s top Initial water table bottom t =1 s Equivalent dry aquifer method The new solution The model on MODFLOW x (m) t =3 s x (m) t =1 s Top of the aquifer Bottom of the aquifer x (m) Initial water table t =3 s top bottom Initial water table Fig. 6. Comparison of different solutions for the confined unconfined flow in an aquifer with initial water table.

8 424 X.-S. Wang et al. / Journal of Hydrology 376 (29) to 1 5 m/d (Smith and Wheatcraft, 1992). In this special condition, the horizontal flow in the semi-confining layer can be neglected and the underlying aquifer can be treated as a semiconfined aquifer. However, the water in the semi-confined aquifer can be absorbed into the overlying layer if hydraulic head in the system is higher than the boundary between the two layers. An equivalent effective porosity of the semi-confining aquifer, S c,is generally within the range of.5.1, rather than the storage coefficient in this situation. If porosity of the gravel aquifer is.3, the value of a D will range between.4 and.6, which is much larger than.1. Ignoring vertical flow and setting the elevation to be zero at the bottom of the underlying aquifer, the governing equations of the unconfined groundwater flow in this horizontal double-layer aquifer system can be written as K 1 þ K ðh ¼ðS c1 þ S ; < x 6 x sðtþ ð54þ ¼ ; x > x sðtþ ð55þ where B is the thickness of the underlying aquifer, K 1 and K 2 are the hydraulic conductivities of the underlying aquifer and the semiconfining layer, respectively, and S y1 and S y2 are the specific yields of the two layers, respectively. S c1 is the storage coefficient of the underlying aquifer and x s (t) is the location where h = B. Eq. (54) is nonlinear. For a constant head at the left boundary, h, the equation can be linearlized as K 1 þ K h 2 h 2 2 ¼ðS 2 c1 þ S ; < x 6 x sðtþ: ð56þ Eq. (56) can be rewritten as K 1 ¼ 2 ; < x 6 x sðtþ; ð57þ where ðs c1 þ S y2 ÞK 1 S e ¼ K 1 þ½ðh BÞK 2 =2BŠ ð58þ is the equivalent storage coefficient defined herein to build an equivalent confined unconfined flow system similar to that presented in Model description. Based on Eq. (57), the lumped parameters originally given in Eq. (8) are redefined as a c ¼ sffiffiffiffiffiffiffiffi K 1 B ; a u ¼ sffiffiffiffiffiffiffiffi K 1 B ; a D ¼ sffiffiffiffiffiffi S e ; ð59þ S e S y1 Therefore, the new solution can also be the approximate solution for an unconfined flow in a double-layer aquifer system. In this situation, as mentioned previously, the value of a D can be larger than.4, which will lead to quite different results than the case of a D =. The solution of zero storage coefficient is not suitable for the double-layer aquifer system. A hypothesized example is used to apply the new solution to a double-layer aquifer system. In the example, a layer of gravel sand is covered by a layer of fine grained sand. Conditions of this double-layer aquifer system assumed to be: K 1 = 1 m/d, K 2 =1m/ d, S y1 =.3, S y2 =.1, S c1 =, B=1m, h = 6 m and h i =5m. Substituting these values into Eqs. (58) and (59), we have S e =.8 and a D = Three different results for t = 1 d and t = 1 d are obtained for comparison: (i) The results obtained by using the new solution with a D =; (ii) The results obtained by using the new solution of a D =.5164; S y1 (iii) The results obtained through the numerical model on MODFLOW. It contains two model-layers representing the underlying gravel sand and the semi-confining layer, respectively. Each model-layer has a row of 1 cells with horizontal grid size of 1 m. The two layers are specified as confined/unconfined type with variable transmissivity and storage coefficient in MODFLOW (McDonald and Harbaugh, 1988). Cells in the upper model-layer beside the left boundary are inactive at the beginning time because the initial groundwater level is lower than their bottom. During the flow process, however, a dry cell can be wetted when groundwater level in the neighboring cells is higher than the bottom of the dry cell. The conversion is simulated by the Block-Centered Flow 2 (BCF2) package of MODFLOW (McDonald et al., 1991). In this case study, the wetting factor and the wetting threshold for this package are arranged to be 1. m and.1 m, respectively. The negative wetting threshold means that only the cell in the underlying layer can cause the dry cell in the upper layer to become wet. This selection can improve the convergence of the numerical simulation. Results of the three different solutions are shown in Fig. 7. Itis indicated that in comparison with the numerical model on MOD- FLOW, the new solution of a D =.5164 slightly underestimates the hydraulic head, but the new solution of a D = significantly overestimates the groundwater level. If the result of the model on MODFLOW is regarded as an exact solution, one can judge the accuracy of the new solution by the Nash Sutcliffe efficiency coefficient (Nash and Sutcliffe, 197). For the results in the ranges of x < 3 m (t = 1 d) and x < 9 m (t = 1 d) as shown in Fig. 7, the Nash Sutcliffe coefficient of the new solution is greater than 97% for a D =.5164 and less than 75% for a D =, respectively. It indicates the accuracy of the new solution of a D =.5164 and the invalidation of using zero storage coefficient. In the studied example, we have the two ratio values, K 2 / K 1 =.1 and h /B = 6. From Eq. (58) we know that if K 2 /K 1 <.1 and h /B < 6, the equivalent storage coefficient is close to S c1 + S y2, which means that the influence of transmissivity in the semi-confining layer is small. Under these conditions, the unconfined flow in the double-layer aquifer system is similar to the confined unconfined flow in a confined aquifer with a storage coefficient equal to S c1 + S y2. Thus, the precision of the new solution could be high for a double-layer system with a low hydraulic conductivity in the semi-confining layer, a thick underlying aquifer and a low increment of the left boundary head. Application to a field study In this section, the above developed solutions for a horizontal confined unconfined flow are applied to a field case, a confined h (m) K 2 K 1 t =1 d The new solution of a D= The new solution of a D=.5164 The model on MODFLOW t =1 d h =5 m x (m) Fig. 7. Comparison of different solutions for the unconfined flow in a doubleaquifer system.

9 X.-S. Wang et al. / Journal of Hydrology 376 (29) Fig. 8. Simplified map of the area of Xixiyuan hydropower station, locations of observation wells and the section-line for model analysis. unconfined flow induced by water level change in a reservoir. Xixiayuan hydropower station in China was chosen to be the study site. The station was constructed on the Yellow River, about 14 km east of the famous Xiaolangdi hydropower station. With water level rise in the reservoir of Xixiayuan hydropower station in 27, the groundwater level was found to increase at the both sides of the lower reaches of the Yellow River. A transition of unconfined flow to confined flow occurs in a gravel aquifer underlying thick loess. Hydrogeological conditions in the study site are schematically shown in Figs. 8 and 9. Previous hydrogeological studies report that the low-k zone of the gravel aquifer exists beside and under the Yellow River. Hydraulic conductivity of this zone is about 3 1 m/d. However, the high-k zones lying outside area of the low- K zone are limited by base rock in the north and south areas. Hydraulic conductivity of these zones is 1 3 m/d. When the water table in the reservoir rose to 13.3 m at the end of May 27, water was found to leak through gravel outcrops of high-k zones that extend to the upper reaches of the Yellow River. The water table rise in the reservoir caused the rise of the regional groundwater level around the dam. Groundwater levels at three typical observations wells, Cj3, Cj14 and Cj13, with various distances from the outcrops on the north side of reservoir, were selected to analyze the groundwater flow in the gravel aquifer, as shown in Fig. 9. According to a geological survey conducted before the construction of the dam, the top and bottom elevations of the aquifer were 12.5 m and 12.5 m, respectively, in average around the three wells. Initial groundwater level was about m as observed in May 27 which is lower than the top of the aquifer. It is also lower than the elevation of the riverbed of the Yellow River. This water table condition is a result of groundwater utilization. A quasi-steady state groundwater flow was achieved before the end of May, 27, due to a balance of recharge and discharge. However, the balance was interrupted by rapid rise of the water level in the reservoir. The groundwater level increased to the confining unit (loess) near reservoir in June, 27. The confined unconfined conversion interface moved east in the high-k zone. A preliminary study was undertaken to analyze the development of the confined unconfined flow in the gravel aquifer. The north high-k zone of the gravel aquifer was assumed to be uniform, horizontally extending to the east of aquifer outcrops with almost the same width between north and south boundaries. Horizontal hydraulic conductivity of the loess overlying the aquifer is very small and is assumed to be zero. In modeling of the groundwater flow with the new approximated solutions, parameters are determined as: B =18m, f i =.667, f = 1.544, a D =.4. The value of a D 155 A Surface ground B 155 Elevation (m) Cj3 Cj14 Cj13 Groundwater level Loess 9/9/27 8/7/27 23/6/ /5/27 Loess 123 flow direction gravel h gravel x Base rock Distance (m) Elevation (m) Fig. 9. Simplified profile on line A B and observed groundwater level during summer in 27.

10 426 X.-S. Wang et al. / Journal of Hydrology 376 (29) is not important herein because f < 3 and the result is almost independent of a D. To compare the results of modeling and observation in a dimensionless manner, the curve of f = h/b versus x/x s is drawn in Fig. 1. Distance of the confined unconfined conversion interface, x s,is calculated with a linear relationship between boundary head and observed head within Cj3. The x s values on 23/6/27, 8/7/27 and 9/9/27 are obtained to be 1245 m, 1546 m, 164 m, respectively. Note that at a given time there is x x s ¼ / / s ð6þ Therefore the observed result of f versus x/x s should have the same curve type as the approximate solution of f versus /// s if the aquifer conditions satisfy the model assumptions described with Eqs. (1) (7). As shown in Fig. 1, the estimated distribution of hydraulic head generally shows similar variation patterns of groundwater level measured at the field site. The simulated groundwater level in Cj14 for 8/7/27 and 9/9/27 is obviously higher than the measurement, which may be due to heterogeneity of the aquifer or other unknown conditions. A probable reason is retardation of the water table rise due to increase of air pressure in the aquifer. Driven air flow had been found in the field investigations when the confined unconfined conversion interface moves toward downstream area. This effect deserves some attention in further researches. Summary After Li et al. (23), new approximate solutions of horizontal confined unconfined flow have been developed. Both the initial water table and the storativity of the confined zone are considered. The Boltzmann transformation is applied to simplify the mathematical model in a dimensionless manner with lumped parameters. A closed-form analytical solution is obtained for flow in the confined zone. A high-precision numerical solution with the Runge Kutta method and an iterative process is proposed for nonlinear flow in the unconfined zone. For the special condition of an initially dry aquifer, an approximate analytical solution is derived similar to Li et al. (23). This approximate analytical solution coincides with the high-precision numerical solution for an initially dry aquifer. However, it is not valid for an aquifer with an initial water table. In comparison with the initially dry condition, the initial water table condition will lead to a faster moving speed of confined unconfined conversion interface. It has been demonstrated with the straightforward numerical model on MODFLOW that the f =h/b Cj3 23/6/27 8/7/27 9/9/27 Modeling Cj14 Cj13.8 Cj Cj x/x s Fig. 1. Normalized distribution of groundwater level for comparison between observation and modeling with solutions in this study. Table 2 The estimated sorptivity for a D = in an initially dry aquifer. f S (A =.447), Eq. (38) S (A =.4247), Eq. (38) S (Numerical solution) equivalent dry aquifer method cannot accurately predict the confined unconfined flow with an initial water table. Groundwater level will be overestimated by neglecting the storage coefficient of the confined zone. However, the solution is not sensitive to the storage coefficient when the boundary head above the aquifer bottom is less than three times of aquifer thickness (i.e. h /B < 3). By introducing the equivalent storage coefficient, the unconfined flow in a double-layer aquifer system is similar to the confined unconfined flow in a confined aquifer if the hydraulic conductivity of the underlying aquifer is much larger than that of the semi-confining layer. Efficiency of the new approximate solution for the unconfined flow in a double-layer aquifer system is verified with the straightforward numerical model on MODFLOW. The approximate solution of the confined unconfined flow with an initial water table has been applied in a preliminary field study. The studied site is located in the area of Xixiayuan hydropower station on Yellow River in China, where the confined unconfined flow was induced by the rise of water table in the reservoir of the hydropower station. Estimated distribution of groundwater level basically agrees with the observations at the field site. Acknowledgements The research was supported by the National Natural Science Foundation of China (Grant No. 4826). We appreciate the constructive suggestions given by Professor Philippe Baveye and the anonymous associate editor. The comments from three anonymous reviewers led to a significant improvement of the paper. Appendix A Derivation of Eqs. (37) and (38) As demonstrated by Parlange et al. (1992b), in Eq. (35), A is almost independent of / s and can be derived in the situation of / s =. When / s =, the sorptivity S is denoted as S. It can be numerically obtained as proposed by Parlange et al. (1992a) and Li et al. (23). The value is S = Applying Eqs. (33) and (35) we have 1 S 2 ð/ f / s Þ A 4 ð/ f / s Þ 2 ¼ ða-1þ Let / s =,Eq. (A-1) can be rewritten as follows 1 S 2 / f A 4 /2 f ¼ ða-2þ To obtain / f, let / s = in Eq. (36) and there is / f ¼ S ða-3þ 1 A Substituting Eq. (A-2) into Eq. (A-1), the factor A is given by A ¼ 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 S2 þ S 4 þ 16S2 ð37þ Since S =.8875, A =.447. Substituting Eq. (36) into Eq. (A-1), the relationship between S and / s can be derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SðA; a D ; f Þ¼ / s þ / 2 s ð2 AÞ½A/2 s 4ð1 AÞ2 Š ð38þ 2 A In Eq. (38), / s is the function of A, a D and f determined by Eq. (39).