Analytical models
The most frequently cited analytical model is that of Hantush and Papadopulos (
1962). They represented the laterals as line sinks of uniform discharge, just as partially penetrating vertical wells had been treated previously (e.g. Hantush and Jacob
1955; Hantush
1957). Further assumptions include a limited drawdown (
s < 0.25
H0), a small percentage of water release due to aquifer compaction, and that the caisson radius is significantly smaller than the lateral length (
rc < <
Ll). The drawdown for the
ith of a group of
i laterals after a long time of pumping—quasi steady state, valid for
t > 2.5
b2/
v´ and
t > 5 (
r2 +
Li2)—is then:
$${s}_i=\frac{Q_i/{L}_i}{4\ \pi\ K\ b}\left\{\begin{array}{c}\alpha\ W\left(\frac{\alpha^2+{\beta}^2}{4v\hbox{'}t}\right)-\delta\ W\left(\frac{\delta^2+{\beta}^2}{4v\hbox{'}t}\right)+2{L}_i-2\beta \left({\tan}^{-1}\frac{\alpha }{\beta }-{\tan}^{-1}\frac{\delta }{\beta}\right)\\ {}+\frac{4\ \mathrm{b}}{\pi }{\int}_{n=1}^{\infty}\frac{1}{n}\ \left[L\left(\frac{n\ \pi\ \alpha }{b},\frac{n\ \pi\ \beta }{b}\right)-L\Big(\frac{n\ \uppi\ \delta }{b},\frac{n\ \pi\ \beta }{b}\Big)\right]\cos \frac{n\ \uppi\ z}{b}\cos \frac{n\ \pi\ {z}_i}{b}\end{array}\right\}$$
(1)
with
$$\alpha =r\ \cos \left(\theta -{\uptheta}_i\right)-{r}_{\mathrm{c}}\hbox{'}$$
(2)
$$\beta =r\ \sin \left(\theta -{\theta}_i\right)$$
(3)
$$\delta =r\ \cos \left(\theta -{\theta}_i\right)-l\hbox{'}$$
(4)
$$r=\sqrt{x^{2}+y^{2}}$$
(5)
$$l\hbox{'}={r}_{\mathrm{c}}\hbox{'}+{L}_i$$
(6)
$${r}_{\mathrm{c}}\hbox{'}={r}_{\mathrm{c}}+{L}_{\mathrm{bc}}$$
(7)
$$v\hbox{'}=\frac{K\ b}{S_{\mathrm{y}}}$$
(8)
$$L\left(a,\pm b\right)=-L\left(-a,\pm b\right)={\int}_0^a{K}_0\left(\sqrt{b^{2}+y^{2}}\right)\mathrm{d}y$$
(9)
$$W(u)={\int}_u^{\infty}\frac{{\mathrm{e}}^{-y}}{y}\mathrm{d}y$$
(10)
where
K is hydraulic conductivity [L T
−1],
Qi is pumping rate of
ith lateral [L
3 T
−1],
Li is the length of the screened section of the
ith lateral (
Lf) [L],
Lbc is the length of the closed section of the
ith lateral [L],
b is the thickness of a confined aquifer or initial water-saturated thickness of an unconfined aquifer [L],
Sy is specific yield,
rc is the radius of the caisson [L],
N is the number of laterals,
n is an integer counter (1, 2, 3, 4, …),
r, z, Θ are cylindrical coordinates (
z positive downwards),
ri, zi, Θi are cylindrical coordinates of the
ith lateral,
x, y, z are rectangular coordinates,
t is time since the start of pumping [T] and
K0(
u) is the zero-order modified Bessel function of the second kind.
W(
u) is the well function (Theis
1935) and can be approximated by
$$W(u)=-0.5772-\ln u+u-\frac{u^2}{2\times 2!}+\frac{u^3}{3\times 3!}-\frac{u^4}{4\times 4!}+\dots$$
(11)
With
z = 0, the approximate drawdown of the piezometric surface is obtained. The 2D solution for average drawdown is derived by integrating with respect to
z over the aquifer thickness and dividing by the aquifer thickness. The result is Eq. (
1) without the integral term. At a distance from the center of the caisson
r ≥ (
rc +
Li +
b), the 2D and 3D solutions converge, as the integral in Eq. (
1) approaches zero.
An elegantly simple analytical model was devised by Williams (
2013), who distributed the total discharge,
Q, of a lateral over a number of
i point sinks (with
Qi each) along the vertical projection of the well screen. The calculated transient drawdown represents that which would be measured in a fully penetrating observation well. The result is thus a 2D drawdown field, unlike the Hantush and Papadopulos (
1962) model, with which drawdown can be calculated for any horizontal plane through the aquifer. The approach not only simplifies the calculation of drawdown around an HW/RCW dramatically, but also makes it easy to model wells of any shape, e.g. slant wells of which the horizontal and vertical well are just special cases. The approach has the additional advantage that it is not restricted to a uniform-flux boundary condition, i.e. the point sink strength could be adapted to mimic nonuniform inflow. In practice, however, it is unlikely that flowmeter data from inside the laterals would be available. For the individual point sinks, Williams (
2013) adapted the Cooper and Jacob (
1946) equation for transient flow to a fully penetrating vertical well:
$$s=\frac{2.3\ Q}{4\ \uppi\ K\ b}\ \log \left(\frac{2.25\ K\ b\ t}{S}\right)-\left(\frac{2}{n_{\mathrm{s}}}\right)\ \log \left({\mathrm{RP}}_1\times {\mathrm{RP}}_2\times {\mathrm{RP}}_3\times \dots \times {\mathrm{RP}}_{n_{\mathrm{s}}}\right)$$
(12)
where
ns is the number of point sinks along the vertical projection of the well screen and RP
x is the distance from an (arbitrary) observation point to point sink
x [L]. It should be noted that Eq. (
12) is valid under the same assumptions and simplifications as the Cooper and Jacob (
1946) approximation. The goodness of fit of this model to the more explicit Hantush and Papadopulos (
1962) solution depends on the number of point sinks employed.
Even simpler than the approach of Williams (
2013) is the “ersatzradius” method (
ersatz is German for replacement) adapted from analytical models developed for fully penetrating vertical wells. The so-called “ersatzradius” (analogous or equivalent well radius) was defined to replace the length of the HW, defined by the extension of its laterals, by an equivalent, fully penetrating vertical well. The drawdown around the HW in a confined aquifer at steady state is then given by the Thiem (
1870) equation, assuming horizontal, radially symmetric flow in an isotropic island aquifer:
$$s=\frac{Q}{2\ \uppi\ K\ b}\ \ln \left(\frac{r_0}{r_{\mathrm{w}}}\right)$$
(13)
where
r0 is the radius of the cone of depression (i.e. radial distance from the well center to a location where drawdown is zero [L]).
rw is the radius of the (analog) well [L] and is defined as follows:
$${r}_{\mathrm{w}}={F}_{\mathrm{e}}\ {L}_{\mathrm{l}}$$
(14)
where
Fe is a correction factor for the ersatzradius [L] and
Ll is the (average) length of the laterals [L]. Nöring (
1953) proposed:
$${r}_{\mathrm{w}}=0.66\ \frac{\sum {L}_{\mathrm{l}}}{n_{\mathrm{l}}}$$
(15)
where
nl [−] is the number of laterals. Correction factors,
Fe, suggested in the literature vary between 0.61 and 0.8 (Mikels and Klaer
1956; Hantush and Papadopulos
1962; McWhorter and Sunada
1977).
Alternatively, drawdown around an RCW can be approximated with a vertical well equation without a correction factor. Hantush (
1964) stated that at a distance of
r > 5 (
rc +
Li) from an RCW of at least two laterals, drawdown can be described with the Theis (
1935) equation without correction. An advantage of this method over the Thiem method is that it allows for transient conditions.
$$s=\frac{Q}{4\ \uppi\ K\ b}\ W\left(\frac{r^2S}{4\ K\ b\ t}\right)$$
(16)
Field data
The Fuhrberger Feld is a rural area located in northern Germany, 30 km north−east of the city of Hanover. Besides agriculture and forestry, the Fuhrberger Feld is used to produce groundwater for the drinking water supply of Hanover. The Quaternary aquifer consists of unconsolidated, mostly sandy sediments with a thickness of 20–30 m, with interspersed thin layers of interglacial silts. The base of the aquifer consists of clay and glacial till. Depth to groundwater varies annually between approximately 0.5 and 2.5 m below ground. The hydraulic conductivity of the aquifer is approximately 45 m/day and its porosity is 0.3. Recharge rates range between 150 mm/a below forest and 250 mm/a below agricultural land. Detailed descriptions of the hydrogeology can be found in Böttcher et al. (
1990), Frind et al. (
1990), Franken et al. (
2009) and Houben et al. (
2018).
The first RCW of the Fuhrberger Feld well field, Fuhrberg 3, with eight laterals, was installed in 1964 and has not been altered since. The second well, Fuhrberg 1, was originally installed in 1958 with ten laterals, which were closed and replaced in 2011. The ten new laterals were installed on two levels, four being roughly 2 m higher than the remaining six. All information concerning the dimensions of the collector wells is given in Table
1. The hydraulic conductivity was calibrated manually to fit the observed data, and the calibrated values can also be found in Table
1.
Table 1
Description of the two radial collector wells
Fuhrberg 1 | 10 | 0.125 | 60 | 45 | 2 | 445 | 33 |
Fuhrberg 3 | 8 | 0.1 | 39.5 | 34.5 | 2 | 445 | 95 |
The two collector wells are surrounded by observation wells at varying depths (10–26 m below ground) with screen lengths 1–10 m. The wells are in near-constant use, but flow rates vary. Prior to measuring groundwater level in the observation wells, the pumping rate had been held constant for 24 h and, therefore, steady-state conditions were assumed. This is a reasonable assumption given that pumping had been continuous and the hydraulic conductivity of the aquifer is high. The head in the caisson was used as a reference head, rather than static water level, due to the difficulties in determining the static water table in a well field. Maximum drawdown at Fuhrberg 3 is 10% of the aquifer thickness, but at Fuhrberg 1 it is 22%. Hantush and Papadopulos (
1962) state that the drawdown must be ≤25% of aquifer thickness, when applying their equation to an unconfined aquifer. Fuhrberg 3 is, therefore, very close to the limit of applicability.
Turbulent losses along the laterals were estimated with the Darcy-Weisbach equation (Weisbach
1845) (see ‘
Appendix’ for estimation of the friction factor):
$$\Delta h={f}_{\mathrm{D}}\frac{L_l}{D}\frac{u^2}{2g}$$
(17)
where ∆
h is head loss [L],
fD is the Darcy friction factor,
g is gravitational acceleration [L T
−2],
u is the mean flow velocity in the pipe [L T
−1] and
D is the hydraulic diameter of the lateral [L].
The spot loss incurred as water flows from the laterals into the caisson was estimated as follows (Munson et al.
1998, cited in Bakker et al.
2005; Lee et al.
2010):
$$\Delta h=\frac{u^2}{2\ g}$$
(18)
The equation is derived from the Bernoulli equation by assuming u − > 0 within the caisson.