Analysis of groundwater flow and stream depletion in the L-shaped fluvial aquifer

Understanding the head distribution in aquifers is crucial for the evaluation of groundwater resources. This article develops a model for describing flow induced by pumping in an L­shaped fluvial aquifer bounded by impermeable bedrocks and two nearly fully penetrating streams. A similar scenario for numerical studies was reported in Kihm et al. (2007). The water level of the streams is assumed to be linearly varying with distance. The aquifer is divided into two sub-regions and the 10 continuity conditions of hydraulic head and flux are imposed at the interface of the sub-regions. The steady-state solution describing the head distribution for the model without pumping is first developed by the method of separation of variables. The transient solution for the head distribution induced by pumping is then derived based on the steady-state solution as initial condition and the methods of finite Fourier transform and Laplace transform. Moreover, the solution for stream depletion rate 15 (SDR) from each of the two streams is also developed based on the head solution and Darcy’s law. Both head and SDR solutions in real time domain are obtained by a numerical inversion scheme called the Stehfest algorithm. The software MODFLOW is chosen to compare with the proposed head solution for the L-shaped aquifer. The steady-state and transient head distributions within the L-shaped aquifer predicted by the present solution are compared with the numerical simulations and measurement data presented in Kihm et al. (2007). The SDR solution is employed to demonstrate its use as a design tool in determining well location for required amounts of SDR from nearby streams under a specific aquifer pumping rate. 20


Introduction
Groundwater is an important water resource for agricultural, municipal and industrial uses.The planning and management of water resources through the investigation of the groundwater flow is one of the major tasks for practicing engineers.The aquifer type and shape are important factors influencing the groundwater flow.Many studies have been devoted to the development of analytical models for describing flow in finite aquifers with a rectangular boundary (e.g., Chan et al., 1976;Chan et al., 1977;Daly and Morel-Seytoux, 1981;Latinopoulos, 1982;Corapcioglu et al., 1983;Latinopoulos, 1984Latinopoulos, , 1985;;Lu et al., 2015), a wedge-shaped boundary (Chan et al., 1978;Falade, 1982;Holzbecher, 2005;Yeh et al., 2008;Chen et al., 2009;Samani and Zarei-Doudeji, 2012;Samani and Sedghi, 2015), or a triangle boundary (Asadi-Aghbolaghi et al., 2010).At Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.present, few solutions are reported to handle the head fluctuation problems for tidal aquifers with re-entrant angle (L-shaped) boundaries (e.g., Sun, 1997;Li and Jiao, 2002), but none of them are to deal with the pumping or stream depletion problems.
Many studies focused on the development of numerical approaches for evaluating the groundwater flow in an aquifer with irregular domain and various types of boundary conditions.The rapid increase of the computing power of PC enables the numerical models to handle the groundwater flow problems with complicated geometric shapes and/or heterogeneous aquifer properties.We therefore adopt the software MODFLOW-2005 to assess the accuracy of the predictions by the present solution.
Numerical methods such as finite element methods (FEMs) and finite difference methods (FDMs) are very commonly used in engineering simulations or analyses.For the application of FEMs, Taigbenu (2003) solved the transient flow problems based on the Green element method for multi-aquifer systems with arbitrarily geometries.Kihm et al. (2007) used a general multidimensional hydrogeomechanical Galerkin FEM to analyze three-dimensional (3D) problems of saturated-unsaturated flow and land displacement induced by pumping in a fluvial aquifer in Yongpoong 2 Agriculture District, Gyeonggi-Do, Korea.
The domain of the aquifer is in L shape and bounded by streams and impermeable bedrock.They performed FEM simulations for steady-state spatial distributions of hydraulic head before aquifer pumping and then for the distributions of hydraulic head and land displacement vector after one-year pumping.Both simulation results were compared and validated with the field measurements of hydraulic head and vertical displacement in transient case.
The FDMs have been widely utilized in the groundwater problems too.Mohanty et al. (2013) evaluated the performances of the finite difference groundwater model MODFLOW and the computational model artificial neural network (ANN) in the simulation of groundwater level in an alluvial aquifer system.They compared the results with field observed data and found that the numerical model is suitable for long-term predictions, whereas the ANN model is appropriate for short-term applications.Serrano (2013) illustrated the use of Adomian's decomposition method to solve a regional groundwater in an unconfined aquifer bounded by the main stream on one side, two tributaries on two sides, and an impervious boundary on the other side.He demonstrated an application to an aquifer bounded by four streams with a deep excavation inside where the head was kept constant.Jafari et al. (2016) incorporated Terzaghi's theory of one-dimensional consolidation with MODFLOW to evaluate groundwater flow and land subsidence due to heavy pumping in a basin aquifer in Iran.So far, many computer codes developed based on either FDMs (e.g., FTWORK and MODFLOW), FEMs (e.g., AQUIFEM-N, BEMLAP, FEMWATER, and SUTRA) or boundary element methods (e.g., BEMLAP) had been employed to simulate a variety of groundwater flow problems (Loudyi et al., 2007).
On the other hand, analytical solutions are convenient and powerful tools to explore the physical insight of groundwater flow systems.The head solution is capable of predicting the spatiotemporal distribution of the drawdown at any location within the simulation time and the SDR solution can estimate the stream filtration rate at any instance at a specific location in the groundwater flow system.Thus, the development of analytical models for describing the groundwater flow in a heterogeneous aquifer with irregular outer boundaries and subject to various types of boundary condition is of practical use from an engineering viewpoint.Kuo et al. (1994) applied the image well theory and Theis' equation to estimate transient drawdown in an aquifer with irregularly shaped boundaries.However, the number of the image wells should be largely increased if the Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.aquifer boundary is asymmetric and rather irregular.Insufficient number of the image wells might result in poor results or even divergence (Matthews et al., 1954).Read and Volker (1993) presented analytical solutions for steady seepage through hillsides with arbitrarily flow boundaries.They used the least squares method to estimate the coefficients in a series expansion of the Laplace equation.Li et al. (1996) extended the results of Read and Volker (1993) in solving the two-dimensional (2D) groundwater flow in porous media governed by Laplace's equation involving arbitrary boundary conditions.The solution procedure was obtained by means of an infinite series of orthonomal functions.Additionally, they also introduced a method, called image-recharge method, to establish the recurrence relationship of the series coefficients.Patel and Serrano (2011) solved nonlinear boundary value problems of multidimensional equations by Adomian's method of decomposition for groundwater flow in irregularly shaped aquifer domains.Currently, Huang et al. (2016) presented 3D analytical solutions for hydraulic head distributions and SDRs induced by a radial collector well in a rectangular confined or unconfined aquifer bounded by two parallel streams and no-flow boundaries.
Groundwater pumping near a stream in a fluvial aquifer may cause the dispute of stream water right, impact of aquatic ecosystem in stream, as well as water allocation or management problems for agriculture, industry, and municipality.The impacts of groundwater extraction by wells should therefore be thoroughly investigated before pumping.This paper develops a 2D mathematical model for describing the groundwater flow in an approximately L-shaped fluvial aquifer which is very close to the case of numerical simulations reported in Kihm et al. (2007).The aquifer is divided into two rectangular subregions.The aquifer in each sub-region is homogeneous but anisotropic in the horizontal plane with the principal direction aligned with the border of the sub-region.Three types of boundary conditions including constant-head, linearly varying head, and no-flow are adopted to reflect the physical reality at the outer boundaries of the problem domain.A steady-state solution is first developed to represent the hydraulic head distribution within the aquifer before pumping.The transient head solution of the model is then obtained using the Fourier finite sine and cosine transforms and the Laplace transform.The Stehfest algorithm is then taken to inverse Laplace-domain solution for the time-domain results.The software MODFLOW-2005 for the simulation of the 3D groundwater flow in L-shaped heterogeneous aquifer is used to check the accuracy of the present head solutions.The SDR solution is also derived based on the head solution and Darcy's law and then used to evaluate the contribution of filtration water from each of two streams toward the pumping well.

Methodology
Figure 1 shows a fluvial plain located in Yongpoong 2 Agriculture District, Gyeonggi-Do, Korea reported in Kihm et al. (2007).
The west side of the plain is a mountainous area with the formation of exposed impermeable bedrock and the east side has the Poonggye stream which passes the district from the southwest corner toward the northeast.A tributary of Poonggye stream, entering the stream with nearly a right angle, is on the north side of the plain.The Poonggye stream and its tributary are perennial stream and almost fully penetrate the fluvial aquifer system (Kihm et al., 2007).The width of Poonggye stream is about 15m reported in Rhms (2013).

Conceptual Model
The aquifer in the district is formed by fluvial deposit with a total thickness of 6 , and consists of a clay loam aquitard of a thickness of about 2.5  underlain by a loamy sand layer of a thickness of about 3.5 .In order to develop an analytical model for solving the groundwater flow, the domain of the aquifer in this study is approximated to be L-shaped, as delineated in Figure 2. Notice that in Figure 1 the solid line denotes the outer boundary of the L-shaped aquifer in this study while the dashed line represents the simulation area in the work of Kihm et al. (2007).The origin of the coordinate in Figure 2 is at the lower left corner of point A, which is at the intersection of boundary AB (i.e., a part of Poonggye stream) and boundary AG.
The boundaries of the aquifer domain along EF and FG are impermeable bedrocks and thus regarded as impermeable boundaries.The annual average heads at points A, B, and D are known to be 5.18, 4.06 and 5.29, respectively, above the bottom of the aquifer (Kihm et al. 2007).The hydraulic heads along AG and DE are fixed at their average water stages as did in Kihm et al. (2007).The boundaries AB and BD are designated to represent the Poonggye stream and its tributary, respectively.Kihm et al. (2007) fixed the hydraulic heads of Poonggye stream and its tributary at annual average water stages in their numerical simulations.Thus, this study considers that the stream has a perfect hydraulic connection with the aquifer and the stream stage varies linearly with distance.The average stream flow rate of the Poonggye stream with its tributary is about 100 m 3 /s reported in Rhms (2013, p. 90).Todd and Mays (2005, p. 232) mentioned that the pumping rate in a shallow well with suction lift less than 7 m may range up to 500 m 3 /day (0.01 m 3 /s).Hence, the effect of pumping in a shallow well on the water table of nearby stream is generally negligible.The average depth to the water table from the ground surface is 1.26 with a spatial variation between 0.57 and 1.95 in accordance with the average water stages in the streams AB and BD.This aquifer is divided into two regions, named regions 1 and 2, and the hydraulic heads in these two regions are respectively expressed as  1 (, , ) and  2 (, , ).

Mathematical model
Consider that there are totally M pumping wells in region 1 and N pumping wells in region 2. The coordinates of th well in region 1 is denoted as ( 1 ,  1 ) and the associated pumping rate per unit thickness is  1 [ 2  ⁄ ].The governing equation describing the 2D hydraulic head distribution in region 1 is expressed as Similarly, the 2D hydraulic head distribution in region 2 for the l th well located at ( 2 ,  2 ) with a pumping rate per unit Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.
The boundary conditions for region 1 are expressed as: Similarly, the boundary conditions for flow in region 2 are The continuity requirements of hydraulic head and flux along the interface CF are respectively and In order to express the solution in dimensionless form, the following dimensionless variables or parameters are introduced: and with

Transient solution for hydraulic head distribution
The solution of the model for transient hydraulic head distribution with the previous steady-state solution as the initial condition is developed via the methods of finite sine transform, finite cosine transform and Laplace transform. with are introduced in Table 1. and The coefficient  2 * can be determined by substituting Eq. ( 36) into Eq.( 37), the  1 * can then be obtained once  2 * is known.
The hydraulic head distributions in real time domain can be obtained by applying a numerical Laplace inversion scheme, called the Stehfest algorithm (Stehfest, 1970), to Eqs. ( 26) and ( 27).

Stream depletion rate
Pumping in an aquifer near a stream will produce filtration water from the stream toward the well (Yeh et al., 2008).Water extracted from the pumping well comes from different sources such as nearby streams and aquifer storage.The extraction rate from the stream is referred to as stream depletion rate (SDR) and that from the aquifer storage is storage release rate (SRR).
The dimensionless solutions of SDR in Laplace domain from the stream reaches AB and BD, denoted respectively as SDR ̃A and where  ℎ is the equivalent horizontal hydraulic conductivity,   and   are the filtration rates from streams AB and BD, respectively.Assuming only horizontal flow in the aquifer, the equivalent horizontal hydraulic conductivity  ℎ for layered aquifer system is estimated based on the following formula (Schwartz and Zhang, 2003): where   is the hydraulic conductivity in the horizontal direction for layer  and   is the thickness of layer  .The dimensionless total SDR in Laplace domain from the streams can therefore be written as: The dimensionless time domain solutions for SDRA, SDRB and SDRT can also be evaluated by the Stehfest algorithm.The dimensionless SRR representing the storage release rate due to the compression of aquifer matrix and the expansion of groundwater in the pore space can be written as: 3 Solution validation and applications

Solution validation by MODFLOW-2005
A 3D numerical model for investigating the hydraulic head distribution within the L-shaped fluvial aquifer of Yongpoong 2 Agriculture District is developed using MODFLOW-2005.MODFLOW is a widely used finite-difference model developed by U.S. Geological Survey for the simulation of 3D groundwater flow problems under various hydrogeological conditions (USGS, 2005).As shown in Figure 1, region 1 has an area of 852 × 222 (i.e.,  1 ×  1 ) while the area of region 2 is 297 × 183 (i.e., ( 1−  2 ) × ( 2 −  1 )).Thus, the total area of these two regions is 243495  2 which is close to the area of the fluvial aquifer (246500 2 ) reported in Kihm et al. (2007).In the simulation of MODFLOW-2005, the plane of the Lshaped aquifer is discretized with a uniform cell size of 3 × 3.The aquifer thickness is 6 and divided into two layers.
The upper loam layer is 2.5 and lower sand layer 3.5.Within the aquifer domain, there is totally 54110 cells while the numbers of cell are 42032 and 12078 respectively for region 1 and region 2. The types of outer boundary specified for the L- in region 1 shown in Figure 2 with a rate of 120 3 / for one year pumping.Figure 3 shows the hydraulic head distributions denoted as the solid line predicted by the present solution of Eqs. ( 26) and ( 27) and represented by the dotted line given by the MODFLOW-2005.The figure indicates a good agreement between these two predicted results, indicating that the present solution gives a fairly good prediction.The largest relative deviation of 2.1% occurs near the no-flow boundary FG where the predicted head is about 4.6.Figure 5 shows the contour lines of the hydraulic head distribution for isotropic case of  1 =  2 = 1 by the solid line and for anisotropic cases of  1 =  2 = 4 represented by the dashed dot line and  1 =  2 = 0.25 by the dashed line.In these three cases, the head distributions are significantly different in the region where  ≤ 600 for the head ranging from 5 to 4.6.

Steady-state head distribution without pumping in
The largest head difference occurs near the upper boundary FG, reflecting the effects of no-flow condition and aquifer anisotropy on the flow pattern within this area.

Spatial head distributions due to pumping simulated by Kihm et al. (2007) and present solution after one year pumping
Note that Figure 3 shows the spatial head distributions in the L-shaped aquifer predicted by the present solution and the MODFLOW-2005 for one-year pumping at well   located at (609, 9) with a rate of 120 3 /.In fact, Kihm et al. (2007) reported their FEM simulations for head distributions, groundwater flow velocity, and land displacement for one-year pumping at the well   with the same pumping rate mentioned above.They referred the simulated head results as initial steady-state distributions for the case of no pumping and final steady-state distributions for the case after one-year pumping.The aquifer configuration in their FEM simulations and the simulated head distributions denoted as dashed line are also demonstrated in Figure 3.The figure indicates that the present solution gives good predicted head contours near the pumping well and reasonably good result for the head distribution in region 1 as compared to those given by Kihm et al. (2007).Notice that the pumping well is very close to the stream boundary AB, which is the main stream in that area and provides a large amount of filtration water to the well.Hence, it seems that the groundwater flows in the region 1 for  ≤ 300 and in the region 2 for  ≥ 200 are both far away from the well and almost not influenced by the pumping.
Three piezometers O1, O2 and O3 were respectively installed at (597, 25), (594, 48) and (597, 204) mentioned in Kihm et al. (2007) and indicated in Figure 2. Note that O1 was installed near the stream AB while O3 was far away from the stream but close to the impermeable upper boundary.Figure 6 shows the temporal distributions of hydraulic head measured at these three piezometers (i.e.,   ,  = 1, 2, 3) and predicted by the FEM simulations (Kihm et al., 2007) (i.e.,   ) and present solution (i.e.,   ).This figure indicates that the hydraulic heads predicted by the present solution has a good agreement with those simulated by Kihm et al. (2007).The relative differences of predicted hydraulic head between FEM simulations and present solution are all less than 0.8% at these three piezometers over the entire pumping period.In addition, the largest relative differences between measured heads and predicted heads by the present solution at O1 to O3 are respectively 1.64%, 1.74% and 0.62%.The hydraulic head at O1 declines greater than those at O2 and O3 whereas the former is located closer to the pumping well   .Because   is very near the stream, the extracted water will be quickly contributed from the stream and therefore the drawdown at O1 will be soon stabilized.Figure 6 indicates that the hydraulic heads at O1 -O3 predicted by the present solution reach steady state after  = 100 days, 220 days and 290 days, respectively.

Stream filtration in fluvial aquifer systems
Stream filtration can be considered as a problem associated with the interaction between the groundwater and surface water.
The pumped water originated from the nearby stream is commonly supplied for irrigation, municipalities, and rural homes.In stream basins with several tributaries, pumping wells are often installed adjacent to the confluence of two tributaries in fluvial aquifers (Lambs, 2004).
It is of practical interest to know the temporal SDR distributions from both streams in the Yongpoong area when subject to pumping at   under a rate of 120 m 3 /day.The distances from   to the streams AB and BD are respectively 9m and 243.
Figure 7 shows the temporal SDR distribution from each stream, indicating that SDRA (i.e., SDR from stream AB) is significantly larger than SDRB (SDR from stream BD) all the time.The steady-state values for SDRA and SDRB are respectively 0.81 and 0.19 when  ≥ 220 day.This is due to the fact that pumping well is closer to stream AB than stream BD and therefore water contributing to the pumping well from stream AB is much more than from stream BD. contribute more water to the pumping than SRR when  ≥ 5 day.Finally, the SDRT reaches unity and the  equals zero after  ≥ 220 days, indicating that the aquifer system approaches steady state and all the extraction water comes from the streams.

Determination of well location for a specific SDR in a L-Shaped aquifer
It is of interest to mention that the present solution can be a preliminary design tool in determining the location of a pumping well in an L-shaped aquifer if the amounts of SDRA and SDRB had been determined by water authority or based on water right.Driscoll (1986, p. 615) mentions that a well shall be installed at least 45.7 from areas of spray materials, fertilizers or chemicals that contaminate the soil or groundwater.Hence, the distance from the pumping well to the stream is considered at least 50m.
Two specific values of SDRA, 0.65 and 0.75, are considered for a water supply rate of 120 3 /.The present solution is employed to determine the well locations in order to meet the water need for irrigation in an L-shaped aquifer.Since stream AB is the main stream, it is better to extract more filtration water from it than from its tributary, stream BD.  (111.95m, 50m).On the other hand, the predicted SDRA from the present solution is 0.76 and its relative difference is 1.3% for well at (170.76m, 50m).In these two cases, we demonstrate that the present solution can be used as a design tool to determine the well location for a specific amount of filtration water from nearby streams in an L-shaped aquifer.

Conclusions
A new analytical model has been developed to analyze the 2D hydraulic head distributions with/without pumping in a heterogeneous and anisotropic aquifer for an L-shaped domain bounded by two streams with linearly varying hydraulic heads.
Method of domain decomposition is used to divide the aquifer into two regions for the development of semi-analytical solution.
Steady-state solution is first derived and used as the initial condition for the L-shaped aquifer system before pumping.The Laplace-domain solution of the model for transient head distribution in the aquifer subject to pumping is developed using the Fourier finite sine and cosine transform and the Laplace transform.The solution for SDR describing filtration rate from two streams in an L-shaped aquifer is developed based on the head solution and Darcy's law.The Stehfest algorithm is then adopted to evaluate the time-domain results for both head and SDR solutions in Laplace domain.The SDR solution is first used to evaluate the steady-state SDR from each of the nearby streams for Yongpoong aquifer subject to a specific pumping rate.The solution is also employed to determine the temporal contribution rates from the aquifer storage and the streams toward the extraction well.Then the solution is used as a design tool to determine the well location with a specific pumping rate for required amounts of SDR from nearby streams.Two cases are provided with four trial pumping wells assumed at distances at least 50m from the streams.A quadratic equation is considered with the dependent variable representing the distance (  ) from the trial well to one of the stream (stream BD) and the independent variable denoted as the estimated SDR (from stream AB, the main stream in Yongpoong area) predicted by the present solution at the trial pumping wells.The quadratic equation with coefficients estimated by the least squares approach is then used to determine the pumping well location for a required SDR from nearby streams in an L-shaped aquifer.In the case studies, the estimated SDR by the present solution at the well location predicted by the regression equation yields about 3.0% relative error for the required SDR of 0.63 and 1.3% relative error for that of 0.76.These results indicate that the present solution can be used as a preliminary design tool in determining the well location for a required amount of SDR.
Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.The coefficients in Eqs.(26) and (27) are obtained via continuity requirements for the hydraulic head and flow flux at the interface CF.They can be solved simultaneously based on the following two equations Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.and SDR ̃B, can be estimated by taking the derivatives of Eqs.(26) and (27) with respect to y and x, respectively, then integrating along the reaches as: Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.shaped aquifer are the same as those defined in the mathematical model.The hydraulic heads along AG and DE are respectively ℎ 1 = 5.18 and ℎ 2 = 5.29 and the head at point B is ℎ 3 = 4.06.The hydraulic conductivities for the upper and lower layers are 3 × 10 −6 / and 2 × 10 −4 /, respectively, considered in the simulations of MODFLOW-2005.On the other hand, the equivalent hydraulic conductivity  ℎ calculated by Eq. (48) as 1.2 × 10 −4 / is used by the present solution.The specific storage of the aquifer in both regions 1 and 2 is 10 −4  −1 .Consider that the pumping well   is located at (609, 9) Yongpoong 2 Agriculture District and impact of aquifer anisotropyKihm et al. (2007) reported the steady-state hydraulic head distribution, shown in Figure4by the dashed line, for the FEM simulation without groundwater pumping in the two-layered L-shaped aquifer.Figure 4 also shows the steady-state head distribution, denoted as the solid line, predicted by the present solution of Eqs.(11) and (12) for the L-shaped aquifer with  1 =  1 =  2 =  2 = 1.2 × 10 −4 / (i.e.,  1 =  2 = 1 ) evaluated based on Eq. (48) and other aquifer properties mentioned in section 3.1.The contour lines of the head distribution are nearly parallel to the boundary AG and perpendicular to the boundary FG in the region  ≤ 200.Moreover, the predicted heads within the regions between 500 ≤  ≤ 852 and 0 ≤  ≤ 200 are reasonably close to the FEM results, which range from 4.3 to 4.7 as shown in Figure 4.The groundwater flows toward point B since it has the lowest water table within the problem domain.
Figure 7 also shows that the SDRT (SDRA+SDRB) is zero and the aquifer storage release rate SRR is unity when  ≤ 0.01 day, indicating that the well discharges totally from the aquifer storage at early time.Once the drawdown cone reaches the stream, the SDRT increases quickly with time while the SRR decreases continuously over the entire pumping period.It is interesting to mention that SDRT starts to Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.
Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2017-487Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 4 September 2017 c Author(s) 2017.CC BY 4.0 License.The 3D finite difference model MODFLOW-2005 is first used to check the accuracy of hydraulic head predictions by the present solution for the L-shaped two-layered aquifer system.The hydraulic head distributions predicted by present solutions agree fairly well over the entire aquifer except the heads nearing the no-flow boundary.The solution for hydraulic head distribution in the L-shaped aquifer without pumping has been used to investigate the effect of anisotropic ratio (  /  ) on the steady-state flow system.It is interesting to note that the flow pattern in terms of lines of equal hydraulic head is strongly influenced by the value of anisotropic ratio for the region near the turning point of the L-shaped aquifer.The transient solution for head distribution is employed to simulate the head distribution induced by pumping in the aquifer within the agriculture area of Gyeonggi-Do, Korea.The aquifer is approximated as L-shaped in this study.The simulation results indicate that the largest relative difference in predicted temporal head distributions at three piezometers by the present solution and Kihm et al.'s (2007) FEM simulation is less than 1.74%, implying that the effects of unsaturated flow and land deformation on the groundwater flow in Yongpoong aquifer are small and may be negligible.

Figure 2 :
Figure 2: The L-Shaped fluvial aquifer with two sub-regions.

Figure 3 :
Figure 3: Contours of hydraulic head in L-shaped aquifer predicted by the present solution, MODFLOW-2005, and FEM simulations with irregular outer boundary reported in Kihm et al. (2007).

Figure 5 :
Figure 5: Steady-state hydraulic head contours in the L-shaped aquifers with three different anisotropy ratios for   =   = .

Table 1 Notations used in the text.
Hydraulic head for region 1 and 2. [L]  1 ,  2 Unit thickness pumping rate for region 1 and 2. [L 2 T] ⁄  1 ,  2   ,   Specific storage for region 1 and 2. [L −1 ] Hydraulic conductivities in x-and y-direction.[L/T] Hydraulic heads at boundaries AG, DE and point B, respectively.[L]  1 ,  2 Length of boundary FG and AB.[L]  1 ,  2 Length of boundary BC and CD.[L]  1 ,  2 Anisotropic ratio of hydraulic conductivity in region 1 and 2.