This study develops a three-dimensional (3-D) mathematical model for describing transient hydraulic head distributions due to pumping at a radial collector well (RCW) in a rectangular confined or unconfined aquifer bounded by two parallel streams and no-flow boundaries. The streams with low-permeability streambeds fully penetrate the aquifer. The governing equation with a point-sink term is employed. A first-order free surface equation delineating the water table decline induced by the well is considered. Robin boundary conditions are adopted to describe fluxes across the streambeds. The head solution for the point sink is derived by applying the methods of finite integral transform and Laplace transform. The head solution for a RCW is obtained by integrating the point-sink solution along the laterals of the RCW and then dividing the integration result by the sum of lateral lengths. On the basis of Darcy's law and head distributions along the streams, the solution for the stream depletion rate (SDR) can also be developed. With the aid of the head and SDR solutions, the sensitivity analysis can then be performed to explore the response of the hydraulic head to the change in a specific parameter such as the horizontal and vertical hydraulic conductivities, streambed permeability, specific storage, specific yield, lateral length, and well depth. Spatial head distributions subject to the anisotropy of aquifer hydraulic conductivities are analyzed. A quantitative criterion is provided to identify whether groundwater flow at a specific region is 3-D or 2-D without the vertical component. In addition, another criterion is also given to allow for the neglect of vertical flow effect on SDR. Conventional 2-D flow models can be used to provide accurate head and SDR predictions if satisfying these two criteria.

The applications of a radial collector well (RCW) have received much
attention in the aspects of water resource supply and groundwater
remediation since rapid advances in drilling technology. An average yield
for the well approximates 27 000 m

A variety of analytical models involving a horizontal well, a specific case of a RCW with a single lateral, in aquifers were developed (e.g., Park and Zhan, 2003; Hunt, 2005; Anderson, 2013). The flux along the well screen is commonly assumed to be uniform. The equation describing three-dimensional (3-D) flow is used. Kawecki (2000) developed analytical solutions of the hydraulic heads for the early linear flow perpendicular to a horizontal well and late pseudo-radial flow toward the middle of the well in confined aquifers. They also developed an approximate solution for unconfined aquifers on the basis of the head solution and an unconfined flow modification. The applicability of the approximate solution was later evaluated in comparison with a finite difference solution developed by Kawecki and Al-Subaikhy (2005). Zhan et al. (2001) presented an analytical solution for drawdown induced by a horizontal well in confined aquifers and compared the difference in the type curves based on the well and a vertical well. Zhan and Zlotnik (2002) developed a semi-analytical solution of drawdown due to pumping from a nonvertical well in an unconfined aquifer accounting for the effect of instantaneous drainage or delayed yield when the free surface declines. They discussed the influences of the length, depth, and inclination of the well on temporal drawdown distributions. Park and Zhan (2002) developed a semi-analytical drawdown solution considering the effects of a finite diameter, the wellbore storage, and a skin zone around a horizontal well in anisotropic leaky aquifers. They found that those effects cause significant change in drawdown at an early pumping period. Zhan and Park (2003) provided a general semi-analytical solution for pumping-induced drawdown in a confined aquifer, an unconfined aquifer on a leaky bottom, or a leaky aquifer below a water reservoir. Temporal drawdown distributions subject to the aquitard storage effect were compared with those without that effect. Sun and Zhan (2006) derived a semi-analytical solution of drawdown due to pumping at a horizontal well in a leaky aquifer. A transient 1-D flow equation describing the vertical flow across the aquitard was considered. The derived solution was used to evaluate the Zhan and Park (2003) solution, which assumed steady-state vertical flow in the aquitard.

Sophisticated numerical models involved in RCWs or horizontal wells were also reported. Steward (1999) applied the analytic element method to approximate 3-D steady-state flow induced by horizontal wells in contaminated aquifers. They discussed the relation between the pumping rate and the size of a polluted area. Chen et al. (2003) utilized the polygon finite difference method to deal with three kinds of seepage-pipe flows including laminar, turbulent, and transitional flows within a finite-diameter horizontal well. A sandbox experiment was also carried out to verify the prediction made by the method. Mohamad and Rushton (2006) used MODFLOW to predict flows inside an aquifer, from the aquifer to a horizontal well, and within the well. The predicted head distributions were compared with field data measured in Sarawak, Malaysia. Su et al. (2007) used software TOUGH2 based on the integrated finite difference method to handle irregular configurations of several laterals of two RCWs installed beside the Russian River, Forestville, California, and analyzed pumping-induced unsaturated regions beneath the river. Lee et al. (2012) developed a finite element solution with triangle elements to assess whether the operation of a RCW near Nakdong River in South Korea can induce riverbank filtration. They concluded that the well can be used for sustainable water supply at the study site. In addition, Rushton and Brassington (2013a) extended Mohamad and Rushton (2006) study by enhancing the Darcy–Weisbach formula to describe frictional head lose inside a horizontal well. The spatial distributions of predicted flux along the well revealed that the flux at the pumping end is 4 times the magnitude of that at the far end. Later, Rushton and Brassington (2013b) applied the same model to a field experiment at the Seton Coast, northwest England.

Well pumping in aquifers near streams may cause groundwater–surface water interactions (e.g., Rodriguez et al., 2013; Chen et al., 2013; Zhou et al., 2013; Exner-Kittridge et al., 2014; Flipo et al., 2014; Unland et al., 2014). The stream depletion rate (SDR), commonly used to quantify stream water filtration into the adjacent aquifer, is defined as the ratio of the filtration rate to a pumping rate. The SDR ranges from zero to a certain value, which could be equal to or less than unity (Zlotnik, 2004). Tsou et al. (2010) developed an analytical solution of the SDR for a slanted well in confined aquifers adjacent to a stream treated as a constant-head boundary. They indicated that a horizontal well parallel to the stream induces the steady-state SDR of unity more quickly than a slanted well. Huang et al. (2011) developed an analytical SDR solution for a horizontal well in unconfined aquifers near a stream regarded as a constant-head boundary. Huang et al. (2012) provided an analytical solution for SDR induced by a RCW in unconfined aquifers adjacent to a stream with a low-permeability streambed under the Robin condition. The influence of the configuration of the laterals on temporal SDR and spatial drawdown distributions was analyzed. Recently, Huang et al. (2014) gave an exhaustive review on analytical and semi-analytical SDR solutions and classified these solutions into two categories. One group involved 2-D flow toward a fully-penetrating vertical well according to aquifer types and stream treatments. The other group included the solutions involving 3-D and quasi-3-D flows according to aquifer types, well types, and stream treatments.

At present, existing analytical solutions associated with flow toward a RCW in unconfined aquifers have involved laborious calculation (Huang et al., 2012) and predicted approximate results (Hantush and Papadopoulos, 1962). The Huang et al. (2012) solution involves numerical integration of a triple integral in predicting the hydraulic head and a quintuple integral in predicting SDR. The integrand is expressed in terms of an infinite series expanded by roots of nonlinear equations. The integration variables are related to those roots. The application of their solution is therefore limited to those who are familiar with numerical methods. In addition, the accuracy of the Hantush and Papadopoulos (1962) solution is limited to some parts of a pumping period; that is, it gives accurate drawdown predictions at early and late times but divergent ones at middle times.

Schematic diagram of a radial collector well in a rectangular unconfined aquifer.

The objective of this study is to present new analytical solutions of the head and SDR, which overcome the above-mentioned limitations, for 3-D flow toward a RCW. A mathematical model is built to describe 3-D spatiotemporal hydraulic head distributions in a rectangular unconfined aquifer bounded by two parallel streams and by the no-flow stratums in the other two sides. The flux across the well screen is assumed to be uniform along each of the laterals. The assumption is valid for a short lateral within 150 m verified by agreement on drawdown observed in field experiments and predicted by existing analytical solutions (Huang et al., 2011, 2012). The streams fully penetrate the aquifer and connect the aquifer with low-permeability streambeds. The model for the aquifer system with two parallel streams can be used to determine the fraction of water filtration from the streams and solve the associated water right problem (Sun and Zhan, 2007). The transient 3-D groundwater flow equation with a point-sink term is considered. The first-order free surface equation is used to describe water table decline due to pumping. Robin boundary conditions are adopted to describe fluxes across the streambeds. The head solution for a point sink is derived by the methods of Laplace transform and finite integral transform. The analytical head solution for a RCW is then obtained by integrating the point-sink solution along the well and dividing the integration result by the total lateral length. The RCW head solution is expressed in terms of a triple series expanded by eigenvalues, which can be obtained by a numerical algorithm such as Newton's method. On the basis of Darcy's law and the RCW head solution, the SDR solution can then be obtained in terms of a double series with fast convergence. With the aid of both solutions, the sensitivity analysis is performed to investigate the response of the hydraulic head to the change in each of aquifer's parameters. Spatial head distributions subject to the anisotropy of aquifer hydraulic conductivities are analyzed. The influences of the vertical flow and well depth on temporal SDR distributions are investigated. Moreover, temporal SDR distributions induced by a RCW and a fully penetrating vertical well in confined aquifers are also compared. A quantitative criterion is provided to identify whether groundwater flow at a specific region is 3-D or 2-D without the vertical component. In addition, another criterion is also given to judge the suitability of neglecting the vertical flow effect on SDR.

Consider a RCW in a rectangular unconfined aquifer bounded by two parallel
streams and no-flow stratums as illustrated in Fig. 1. The symbols for
variables and parameters are defined in Table 1. The origin of the Cartesian
coordinate is located at the lower left corner. The aquifer domain falls in
the range of

Symbols used in the text and their definitions.

First, a mathematical model describing 3-D flow toward a point sink
in the aquifer is proposed. The equation describing 3-D hydraulic head
distribution

Note that Eq. (2) introduces a negative hydraulic head for pumping, and the absolute value of the head equals drawdown.

The aquifer boundaries at

Streambed permeability is usually less than the adjacent aquifer formation.
The fluxes across the streambeds can be described by Robin boundary
conditions as

The free surface equation describing the water table decline is written as

Neuman (1972) indicated that the effect of the second-order terms in Eq. (7)
can generally be ignored in developing analytical solutions. Equation (7) is thus
linearized by neglecting the quadratic terms, and the position of the water
table is fixed at the initial condition (i.e.,

Notice that Eq. (8) is applicable when the conditions

Define dimensionless variables as

Similarly, the initial and boundary conditions are expressed as

The model, Eqs. (10)–(17), reduces to an ordinary differential equation
(ODE) with two boundary conditions in terms of

The method to determine the roots is discussed in Sect. 2.3. In turn, Eq. (18) becomes a second-order ODE defined by

Equation (20) can be separated into two homogeneous ODEs as

The second describes the discontinuity of the flux due to point pumping
represented by the Dirac delta function in Eq. (20). It can be derived by
integrating Eq. (20) from

Solving Eqs. (23) and (24) simultaneously with Eqs. (21), (22), (25), and
(26) yields the Laplace-domain head solution as

The determination for those eigenvalues is introduced in the next section.
Notice that the solution consists of a simple series expanded in

Application of Newton's method with proper initial guesses to determine the
eigenvalues

The lateral of RCW is approximately represented by a line sink composed of a
series of adjoining point sinks. The locations of these point sinks are
expressed in terms of (

On the basis of Darcy's law and the head solution for a RCW, the SDR from
streams 1 and 2 can be defined, respectively, as

Again, the double integrals in both equations can be done analytically.
Notice that the series term of

If

Similarly, the SDR solution for a confined aquifer can be written as Eqs. (52)
and (53), where the RHS function in Eq. (56) reduces to that in Eq. (62)
by applying L'Hospital's rule with

The head solution introduced in Sect. 2.6.1 is applicable to
spatiotemporal head distributions in confined aquifers of infinite extent
before the lateral boundary effect comes. Wang and Yeh (2008) indicated that
the time can be quantified, in our notation, as

Prior to the beginning time mentioned in Sect. 2.6.2, the absolute value
calculated by the present head solution, Eqs. (44) with

When

The hydraulic parameters determined from field observed data are inevitably
subject to measurement errors. Consequently, head predictions from the
analytical model have uncertainty due to the propagation of measurement
errors. Sensitivity analysis can be considered as a tool of exploring the
response of the head to the change in a specific parameter (Zheng and
Bennett, 2002). One may define the normalized sensitivity coefficient as

This section demonstrates head and SDR predictions and explores some
physical insights regarding flow behavior. In Sect. 3.1, equipotential
lines are drawn to identify 3-D or 2-D flow without the vertical flow at a
specific region. In Sect. 3.2, the influence of anisotropy on spatial head
and temporal SDR distributions is studied. In Sect. 3.3, the sensitivity
analysis is performed to investigate the response of the head to the change
in each hydraulic parameter. In Sect. 3.4, the effects of the vertical
flow and well depth on temporal SDR distributions for confined and
unconfined aquifers are investigated. For conciseness, we consider a RCW
with two laterals with

Most existing models assume 2-D flow by neglecting the vertical flow for
pumping at a horizontal well (e.g., Mohamed and Rushton, 2006; Haitjema et
al., 2010). The head distributions predicted by those models are inaccurate
if an observation point is close to the region where the vertical flow
prevails. Figure 2 demonstrates the equipotential lines predicted by the
present solution for a horizontal well in an unconfined aquifer for

Equipotential lines predicted by the present solution for

Spatial distributions of the dimensionless head predicted by the
present head solution for

Previous articles have seldom analyzed flow behavior for anisotropic
aquifers, i.e.,

Aquifers with

Temporal distribution curves of the normalized sensitivity
coefficients for parameters

Consider an unconfined aquifer of

Temporal SDR

Huang et al. (2014) revealed that the effect of the vertical flow on SDR
induced by a vertical well is dominated by the magnitude of the key factor

It is interesting to note that the SDR

Temporal SDR distribution curves predicted by Eqs. (52) and (53)
with

This study develops a new analytical model describing 3-D flow induced by a
RCW in a rectangular confined or unconfined aquifer bounded by two parallel
streams and no-flow stratums in the other two sides. The flow equation in
terms of the hydraulic head with a point sink term is employed. Both streams
fully penetrate the aquifer and are under the Robin condition in the
presence of low-permeability streambeds. A first-order free surface Eq. (8)
describing the water table decline gives good predictions when the
conditions

The aquifer anisotropy of

The aquifer anisotropy of

The hydraulic head in the whole domain is most sensitive to the change in

The hydraulic head is sensitive to changes in

The hydraulic head at observation points near stream 1 is sensitive to the
change in

The effect of the vertical flow on SDR is ignorable when

For unconfined aquifers, SDR increases with dimensionless well depth

Latinopoulos (1985) provided the finite integral transform for a rectangular
aquifer domain where each side can be under either the Dirichlet, no-flow,
or Robin condition. The transform associated with the boundary conditions,
Eqs. (12)–(15), is defined as

The function of

Notice that the term

The locations of poles are the roots of the equation obtained by letting the
denominator in Eq. (B1) to be zero, denoted as

On the other hand, infinite poles are at

Moreover, Eq. (B8) reduces to Eq. (31) when letting the terms of

Research leading to this paper has been partially supported by the grants from the Taiwan Ministry of Science and Technology under the contract NSC 102-2221-E-009-072-MY2, MOST 103-2221-E-009-156, and MOST 104-2221-E-009-148-MY2. Edited by: A. Guadagnini