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 L-shaped 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 subregions and the continuity conditions of the hydraulic head and flux are imposed at the interface of the subregions. 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 (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 the 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).

Groundwater is an important water resource for agricultural, municipal and industrial uses. The planning and management of water resources through the investigation of 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, 1977; Daly and Morel-Seytoux, 1981; Latinopoulos, 1982, 1984, 1985; Corapcioglu et al., 1983; 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; Kacimov et al., 2016), a triangle boundary (Asadi-Aghbolaghi et al., 2010), a trapezoidal-shaped boundary (Mahdavi and Seyyedian, 2014), or a meniscus-shaped domain (Kacimov et al., 2017). So far, the case of re-entrant angle (L-shaped) boundaries has been treated analytically in different fields such as torsion of elastic bars (Kantorovich and Krylov, 1958), head fluctuation problems for tidal aquifers (Sun, 1997; Li and Jiao, 2002), and heat conduction in plates (Mackowski, 2011). However, none of the cited papers deals with pumping or stream depletion problems.

Many studies focused on the development of numerical approaches to model groundwater flow in an aquifer with irregular domain and various types of boundary conditions. The rapid increase in the computing power of PCs enables the numerical models to handle the groundwater-flow problems with complicated geometric shapes and/or a heterogeneous aquifer. 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 arbitrary geometries. Kihm et al. (2007) used a general multidimensional hydrogeomechanical Galerkin FEM to analyze three-dimensional (3-D) problems of saturated–unsaturated flow and land displacement induced by pumping in a fluvial aquifer in the Yongpoong 2 Agriculture District, Gyeonggi-do, Korea. The domain of the aquifer is L-shaped and bounded by streams and impermeable bedrock. Their mathematical model was developed by considering the unsaturated flow and solid skeleton deformation, and thus a set of four coupled nonlinear equations in terms of one head variable and three displacement variables was derived. The Galerkin FEM was employed to simulate the steady-state spatial distribution of the hydraulic head before aquifer pumping and then the distributions of the hydraulic head and land displacement vector after 1-year pumping. Their simulation results were compared and validated with the field measurements of the hydraulic head and vertical displacement in the transient case. However, solving the simultaneous nonlinear equation required a lot of work which was difficult to apply because of the extensive calculation time and resource requirements for long-term simulations.

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-flow problem 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 (1-D) 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) have 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 stream depletion rate (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 conditions 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. The aquifer is an oil reservoir bounded by three tortuous faults. However, the number of the image wells should be largely increased if the 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 arbitrary 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 (2-D) 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 orthonormal functions. Additionally, they also introduced a method, called the 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. Mahdavi and Seyyedian (2014) developed a semi-analytical solution for hydraulic head distribution in trapezoidal-shaped aquifers in response to diffusive recharge of constant rate. The aquifer was surrounded by four fully penetrating and constant-head streams. Kacimov et al. (2016) used the Strack–Chernyshov model to investigate the unconfined groundwater flows in wedge-shaped promontories with accretion along the water table and outflow from a groundwater mound into draining rays. Huang et al. (2016) presented 3-D 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. Currently, the distribution of groundwater-flow velocity in a circular meniscus aquifer was investigated analytically by the theory of holomorphic functions and numerically by FEM (Kacimov et al., 2016).

Groundwater pumping near a stream in a fluvial aquifer may cause the dispute of stream water rights, impact the aquatic ecosystem in the stream, and result in water allocation or management problems for agriculture, industry, and municipality. The impacts of groundwater extraction by wells should therefore be thoroughly investigated before pumping. Theoretically, numerical methods, such as FEM are good tools to simulate groundwater flow in irregularly shaped aquifer such as the approximately L-shaped aquifer in Kihm et al. (2007). However, they are generally time consuming and computationally demanding (Younger, 2007), and it is necessary to regenerate the mesh of problem domain for different sizes of L-shaped aquifers. The analytical solution can be easily applied for different sizes of L-shaped aquifers with similar boundaries and properties by replacing the length or width of the solution. Thus, an analytical solution is proposed in this study to evaluate the spatiotemporal distribution of the drawdown at any location in the L-shaped aquifer. This paper develops a 2-D 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 subregion is assumed to be homogeneous but anisotropic in the horizontal plane with principal direction aligned with the borderline of the rectangular subregions. 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 invert the Laplace-domain solution for the time-domain results. The software MODFLOW for the simulation of the 3-D groundwater flow is used to evaluate 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.

Location of the fluvial aquifer. Note that this figure is modified from Google Earth.

The L-Shaped fluvial aquifer with two subregions.

Figure 1 shows a fluvial plain located in the Yongpoong 2 Agriculture District, Gyeonggi-do, Korea whose characteristics are reported in Kihm et al. (2007). The west side of the plain is a mountainous area, where impermeable bedrock outcrops, and the Poonggye stream flows along the east side from the southwest corner toward the northeast corner. 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 streams and almost fully penetrate the fluvial aquifer system (Kihm et al., 2007). The width of Poonggye stream is about 15 m as reported in River Information Management GIS (RIMGIS, 2013).

The aquifer in the district is formed by fluvial deposits with a total
thickness of

As shown in Fig. 2, the aquifer is divided into two subregions named regions
1 and 2, and variables

In order to compare the steady-state simulations of Kihm et al. (2007)
without pumping, the steady-state solution for the hydraulic head
distribution in the L-shaped aquifer is developed. Detailed derivation for
the analytical solutions in steady state for regions 1 and 2 is given in
Appendix A, and the results are expressed respectively as (Chu et al., 2012)

Notations used in the text.

The semi-analytical 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. The detailed derivation for
the transient solution is given in Appendix B and the results of the
dimensionless hydraulic heads in Laplace domain for regions 1 and 2 are
respectively

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

and

with

Pumping in an aquifer near a stream often produces water filtration from the
stream toward the well (Yeh et al., 2008). Water extracted by the pumping
well comes from sources such as aquifer storage and nearby streams. The
extraction rate from the stream is referred to as the stream depletion rate.
Since the boundaries AG and ED do not correspond to streams in
the physical world and are mathematically treated as constant head because they
are far from the pumping well, only the water filtration from streams AB and
BD to the nearby pumping well needs to be considered. The dimensionless
solutions of SDR in the Laplace domain from the stream reaches AB and BD, denoted
respectively as

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

The software MODFLOW (USGS, 2005) is used to simulate the groundwater flow
due to pumping in the L-shaped aquifer in the Yongpoong 2 Agriculture District
with different hydraulic conductivities for the two layers. As shown in
Fig. 1, region 1 has an area of

The global behavior of a multilayered aquifer may be approximated with that
of an equivalent homogeneous medium, whose hydraulic conductivity in the
horizontal plane

Kihm et al. (2007) reported the steady-state hydraulic head distribution,
shown in Fig. 4 by the dashed line, for the FEM simulation without
groundwater pumping in the two-layered irregular aquifer. Figure 4 also shows
the steady-state head distributions predicted by the present solution of
Eqs. (11) and (12) denoted as the solid line and by the MODFLOW
denoted as the dotted line both for the L-shaped aquifer with

Steady-state hydraulic head contours without pumping in the Yongpoong 2 Agriculture District.

Steady-state hydraulic head contours in the L-shaped aquifers with
three different anisotropy ratios for

Figure 5 shows the contour lines of the hydraulic head distribution for
the isotropic case of

Figure 3 shows the spatial head distributions in the L-shaped aquifer
predicted by the present solution and the MODFLOW for 1-year pumping at
well

Three piezometers,

Temporal distributions of the hydraulic head
observed at piezometer

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).

Temporal distributions of SDRs, SRR, and CHRs (constant head contribution rates) due to pumping
at

It is of practical interest to know the temporal SDR distributions from both
streams in the Yongpoong area when subject to pumping at

A new semi-analytical model has been developed to analyze the 2-D hydraulic head distributions with and without pumping in a heterogeneous and anisotropic aquifer for an L-shaped domain bounded by two streams with linearly varying hydraulic heads. The method of domain decomposition is used to divide the aquifer into two regions for the development of the semi-analytical solution. The 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 the 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 the head and SDR solutions in Laplace domain.

The 3-D finite difference model MODFLOW is first used to support the
evaluation of the 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 for 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 (

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 present solution delivers fairly good results in predicting the temporal hydraulic head distribution while comparing with those of FEM reported in previous study. Those simulation results seem to indicate that the effects of unsaturated flow and land displacement on the groundwater flow are not significant and may be ignorable. The largest relative differences between the measured heads and the predicted heads by the present solution at three piezometers are less than 1.74 %.

The SDR solution is first used to evaluate the steady-state SDR from each of the nearby streams for the 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.

All the required parameters are provided in the study. The data sets of observations and FEM model were collected from that reported in Kihm et al. (2007). In addition, the data sets of present solution and MODFLOW are freely available upon request by contacting the first or corresponding author.

On the basis of dimensionless variables and parameters defined in Sect. 2.2,
Eqs. (1) and (2) can be written respectively as

The continuity requirements of the hydraulic head and flux at the region
interface in dimensionless form are respectively expressed as

The steady-state solution for groundwater flow in an L-shaped aquifer without
pumping can be solved after removing the source/sink term in Eqs. (A1)
and (A2). Multiplying Eq. (A1) by

Similarly, Eq. (A2) can be transformed via multiplying Eq. (A2) by

The general solutions of Eqs. (A12) and (A14) can be written respectively as

Multiplying Eq. (A1) by

Similarly, Eq. (A2) can be transformed via multiplying Eq. (A2) by

Furthermore, the coefficients of

The authors declare that they have no conflict of interest.

This study was partly supported by the grants from Taiwan Ministry of Science and Technology under the contract numbers MOST 105-2221-E-009-043-MY2 and MOST 106-2221-E-009 -066. The authors would like to thank the editor, two anonymous reviewers, and David Ferris for their valuable and constructive comments that greatly improved the manuscript. Edited by: Mauro Giudici Reviewed by: two anonymous referees