Conventional models of pumping tests in unconfined aquifers often neglect
the unsaturated flow process. This study concerns the coupled
unsaturated–saturated flow process induced by vertical, horizontal, and
slant wells positioned in an unconfined aquifer. A mathematical model is
established with special consideration of the coupled unsaturated–saturated
flow process and the well orientation. Groundwater flow in the saturated zone is
described by a three-dimensional governing equation and a linearized
three-dimensional Richards' equation in the unsaturated zone. A solution in
the
Laplace domain is derived by the Laplace–finite-Fourier-transform and the
method of separation of variables, and the semi-analytical solutions are
obtained using a numerical inverse Laplace method. The solution is verified
by a finite-element numerical model. It is found that the effects of the
unsaturated zone on the drawdown of a pumping test exist at any angle of
inclination of the pumping well, and this impact is more significant in the
case of a horizontal well. The effects of the unsaturated zone on the drawdown
are independent of the length of the horizontal well screen. The vertical
well leads to the largest water volume drained from the unsaturated
zone (

In addition to conventional vertical wells, horizontal and slant pumping wells have been broadly used in the petroleum industry and in environmental and hydrological applications in recent decades. Horizontal and slant pumping wells are commonly installed in shallow aquifers to yield a large amount of groundwater (Bear, 1979) or to remove a large amount of contaminant (Sawyer and Lieuallen-Dulam, 1998). Horizontal and slant wells have some advantages over vertical wells (Yeh and Chang, 2013; Zhan and Zlotnik, 2002); e.g., horizontal and slant wells yield smaller drawdowns than the vertical wells with the same pumping rate per screen length. Horizontal and slant wells have long screen sections, which can extract a great volume of water in shallow or low permeability aquifers without generating significant drawdowns.

The schematic diagram of groundwater flow to a horizontal well

Hantush and Papadopulos (1962) first investigated the problem of fluid flow to a horizontal well in the hydrologic sciences. Since then, this problem has not been of great concern in the hydrological science community because of the limitations in directional drilling techniques and high drilling costs. With significant advances in directional drilling technology over the last 20 years, the interest in horizontal and/or slant wells has been reignited. Until now, flow to horizontal and/or slant wells has been investigated in various aspects, including flow in confined aquifers (Cleveland, 1994; Zhan, 1999; Zhan et al., 2001; Kompani-Zare et al., 2005), unconfined aquifers (Huang et al., 2011, 2016; Rushton and Brassington, 2013; Zhan and Zlotnik, 2002; Mohamed and Rushton, 2006; Kawecki and Al-Subaikhy, 2005), leaky confined aquifers (Zhan and Park, 2003; Sun and Zhan, 2006; Hunt, 2005), and fractured aquifers (Nie et al., 2012; Park and Zhan, 2003; Zhao et al., 2016). The readers can consult Yeh and Chang (2013) for a recent review of well hydraulics on various well types, including horizontal and slant wells.

As demonstrated in previous studies, horizontal and slant wells have significant advantages over vertical wells in unconfined aquifers, and thus they were largely used in unconfined aquifers for pumping or drainage purposes. However, none of above-mentioned studies considered the effects of unsaturated processes on groundwater flow to horizontal and slant wells in unconfined aquifers. In the case of flow to vertical wells in saturated zones, the effects of unsaturated processes have been investigated by several researchers (Kroszynski and Dagan, 1975; Mathias and Butler, 2006; Tartakovsky and Neuman, 2007; Mishra and Neuman, 2010, 2011). For example, Tartakovsky and Neuman (2007) considered axisymmetric unsaturated–saturated flow for a pumping test in an unconfined aquifer and employed one parameter that characterized both the water content and the hydraulic conductivity as functions of the pressure head, assuming an infinite thickness of the unsaturated zone. Mishra and Neuman (2010, 2011) extended the solution of Tartakovsky and Neuman (2007) using four parameters to represent the unsaturated zone properties and considered a finite thickness of the unsaturated zone (Mishra and Neuman, 2010) as well as the wellbore storage (Mishra and Neuman, 2011). The main results from the studies concerning vertical wells indicated that the unsaturated zone often had a major impact on the S-shaped drawdown curves.

The following question remains: are these conclusions for vertical wells also applicable to horizontal and slant wells when coupled unsaturated–saturated flow is of concern? Specifically, how important is the wellbore orientation on groundwater flow to a horizontal or slant well considering the coupled unsaturated–saturated flow process? In order to answer these questions, we establish a mathematical model for groundwater flow to a general well orientation (vertical, horizontal, and slant wells) considering the coupled unsaturated–saturated flow process. We incorporate a three-dimensional linearized Richards' equation into a governing equation for groundwater flow in an unconfined aquifer. We employ the Laplace–finite-Fourier-transform and the method of separation of variables to solve the governing equations for coupled unsaturated–saturated flow. This paper is organized as follows: we first present the mathematical model and the solution in Sects. 2 and 3, respectively, then describe the results and discussion in Sect. 4. We summarize this study and draw conclusions in Sect. 5.

The schematic diagrams of flow to horizontal and slant wells in an
unsaturated–saturated system are represented in Fig. 1a and b,
respectively. Similar to the conceptual model used by Zhan and Zlotnik (2002),
the origin of the Cartesian coordinate is located at the bottom of
the saturated zone with the

In order to solve the problem of groundwater flow to a horizontal or slant
well, we first solve the governing equation of groundwater flow to a point
sink. The mathematical model for groundwater flow to a point sink
(

Flow in the unsaturated zone induced by pumping in the unconfined aquifer is
governed by the Richards' equation. Due to the nonlinear nature of the
Richards' equation, it is difficult to analytically solve this equation
except in some specific cases. Kroszynski and Dagan (1975) proposed
a first-order linearized unsaturated flow equation by expanding the
dependent variable in the Richards' equation as a power–function series when
the pumping rate was less than

Eq. (3b) shows that, at the water table (

The saturated and unsaturated flows are coupled at their interface by
the continuities of pressure and vertical flux across the water table.
Following linearization, they take the form

The solution to Eq. (1a) is obtained by the Laplace and finite-cosine
Fourier transform. The Laplace domain solution of Eq. (1a), subject to
the initial conditions of Eq. (1b) and the boundary conditions of Eq. (1c) and (1d), is
given as (Zhan and Zlotnik, 2002)

The solution to Eq. (2a) is obtained by the Laplace transform and the method
of separation of variables (Sect. S2). It is given as

The eigenvalues of the finite-cosine Fourier transform

Due to the linearity of the mathematical models in Eqs. (1) and (2), the principle of superposition can be employed to extend the basic solutions of Eqs. (5) and (7). Thus, on the basis of the principle of superposition, the drawdown induced by a line sink in the saturated zone can be obtained by integrating the solution of Eqs. (5) and (7) along the well axis, provided that the pumping strength distribution along the well screen is known. The precise determination of the pumping strength distribution along a horizontal or slant well involves complex, coupled aquifer–pipe-flow (Chen et al., 2003) in which the flow inside the wellbore (pipe flow) can experience different stages of flow schemes from laminar, to transitional turbulent, to fully developed turbulent flow. Such complex coupled well–aquifer flow is beyond the scope of this study, and one may consult the recent studies of Blumenthal and Zhan (2016) or Wang and Zhan (2016) for more details. However, often one may adopt a first-order approximation of a uniform flux distribution to treat horizontal or slant wells, particularly when the well screen lengths are not extremely long (kilometers). Such an approximation has been justified by Zhan and Zlotnik (2002). In this study, a uniform flux distribution will be utilized for horizontal or slant wells hereafter to obtain the solutions.

The drawdown in the saturated and unsaturated zones due to a slant pumping well
can be written as

The drawdown in an observation (vertical) well located in the saturated zone
that is screened from

It should be noted that our solutions do not account for the wellbore effects of the pumping and observation wells. Indeed, the wellbore effects have introduced additional complexity to the solutions that are already substantially more complex than the solutions excluding the unsaturated flow process. To avoid the influence of wellbore storage effects, we make the following proposal that could be implemented in future investigations of the coupled saturated-unsaturated flow process: using pack systems to insulate the screens of pumping and the observation wells; thus, wellbore storages will not be a concern.

The dimensionless total volume drained from the unsaturated zone to the
saturated zone (water flux across the water table) can be obtained by

It is difficult to obtain closed-form solutions by analytically inverting the Laplace transforms of Eqs. (5), (7), (9), (10), and (12), and thus a numerical inverse Laplace method is employed in this study. There are several numerical inverse Laplace methods, such as the Stehfest method (Stehfest, 1970), the Zakian method (Zakian, 1969), the Fourier series method (Dubner and Abate, 1968), the Talbot algorithm (Talbot, 1979), the Crump technique (Crump, 1976), and the de Hoog algorithm (de Hoog et al., 1982). Each method is best fitted for a particular type of problem (Hassanzadeh and Pooladi-Darvish, 2007). Chen (1985), Zhan et al. (2009a, b), and Wang and Zhan (2013) have successfully employed the Stehfest algorithm to obtain the solution in the time domain for similar problems to the one in this study. For references on different inverse Laplace methods, one can consult the reviews of Kuhlman (2013) and Wang and Zhan (2015). In this study, we use the Stehfest method to invert the Laplace solutions into the solutions in the time domain. In order to ensure the accuracy of the Stehfest method, several numerical exercises have been performed against the benchmark solutions for several special cases of the investigated problem.

The main difference between the

Figure 2a presents the drawdown curves in the saturated zone for different
values of

The unsaturated flow has a significant impact on drawdown curves in the
saturated zone when

Fig. 2a also shows that the drawdowns have typical “S” pattern
curves when

The log–log plot of

The unsaturated zone controls the effects of the additional storage and the upper
boundary of the saturated zone on drawdown curves. There are physical
differences between the

In order to further investigate the effects of the unsaturated zone,
Fig. 2c displays the drawdown curves in the unsaturated zone (

In this section, we first investigate the effect of the inclined angle of
the slant well on the type curves. Figure 3 shows the comparison between the

Here, we investigate the effect of the horizontal well screen length on the
drawdown. Figure 4 illustrates the comparison between the

The log–log plot of

Vertical profiles of

In order to clearly illustrate the drawdown pattern in the
unsaturated–saturated system, the drawdown profiles for three different angles of a slant well (

The water flux across the water table (Eq. 12) is the volume drained from
the unsaturated zone to the saturated zone. It is somewhat related to the
concept of specific yield when the coupled unsaturated–saturated zone flow
process is simplified into a saturated zone flow process with the water table
as a free upper boundary. Thus, Eq. (12) reflects the impact of the
unsaturated zone on the water flow in the saturated zone. Figure 6 shows the
changes in the dimensionless water flux across water table,

The log–log plot of

For early times of pumping,

In order to further verify our solution and to explore the capability of
our solution to interpret pumping test results in the
unsaturated–saturated system, we have conducted a synthetic numerical
simulation. The synthetic case considers a pumping test in an unconfined
aquifer with a slant pumping well (

The coupled Eqs. (1)–(4) of the unsaturated–saturated system are
numerically solved by COMSOL Multiphysics (COMSOL Multiphysics GmbH, Göttingen, Germany), a robust Galerkin finite-element
software package that includes a partial differential equation (PDE) solver
for modeling the types of governing equations in this study. Figure 7a shows
the spatial discretization of our COMSOL model, in which tetrahedrons are
used as elements for the three-dimensional model and the elements near both
the pumping well and the unsaturated–saturated interface are refined. The
number of tetrahedral elements is 328 358. The time step increases
exponentially, and the total number of time steps is 100 with a total
simulation time of 220 min. Figure 7b presents an example of the vertical
profiles (the

First, we verify our solutions by comparing the drawdowns in both the
saturated and unsaturated zones with the numerical solution for the same
aquifer parameter values. Figure 8a and b shows the drawdown curves in the
saturated zone at the observation point (

Second, we investigate the errors in using the

A major disadvantage of the two older models (the

The coupled unsaturated–saturated flow process induced by vertical,
horizontal, and slant pumping wells is investigated in this study. A
mathematical model for such a coupled unsaturated–saturated flow process is
presented. The flow in the saturated zone is described by a
three-dimensional governing equation, and the flow in the unsaturated zone
is described by a three-dimensional Richards' equation. The unsaturated zone
properties are represented by the Gardner (1958) exponential
relationships. The Laplace domain solutions are derived using the Laplace
transform and the method of separation of variables, and the time domain
solutions are obtained using the Stehfest method (Stehfest, 1970). The
solution is compared with the solutions proposed by Zhan et
al. (2001) (confined aquifer; the

The unsaturated flow has a significant impact on drawdown in unconfined
aquifers induced by the horizontal pumping well when the dimensionless
constitutive exponent

For the small dimensionless unsaturated thickness

The effects of the unsaturated zone on the drawdown exist at any angle of inclination of a slant well, and this impact is more significant in the case of the horizontal well. The effects of the unsaturated zone on the drawdown are insensitive to the length of the horizontal well screen.

For the early time of pumping, the water volume drained from the unsaturated
zone (

By comparing it with the synthetic pumping test data generated by the
finite-element numerical model of COMSOL, one can see that our solution
provides a good
reproduction of the drawdown curves in both the saturated and unsaturated
zones,
while both the

The computer program codes and data used in this study can be accessed by contacting the corresponding author directly.

The authors declare that they have no conflict of interest.

This study was partially supported with research grants from the National Nature Science Foundation of China (41330314, 41272260, 41302180, 41521001, 41372253), the national key project “Water Pollution Control” of China (2015ZX07204-007), and the Natural Science Foundation of Jiangsu Province (BK20130571). We thank Shlomo P. Neuman and another anonymous reviewer for their constructive comments in helping us to revise the paper. Edited by: I. Neuweiler Reviewed by: S. P. Neuman and one anonymous referee