Modeling of Artificial Groundwater Recharge by Wells: A Model Stratified Porous Medium

Abstract

In recent years, groundwater levels have been decreasing due to the demand in agricultural and industrial activities, as well as the population that has grown exponentially in cities. One method of controlling the progressive lowering of the water table is the artificial recharge of water through wells. With this practice, it is possible to control the amount of water that enters the aquifer through field measurements. However, the construction of these wells is costly in some areas, in addition to the fact that most models only simulate the well as if it were a homogeneous profile and the base equations are restricted. In this work, the amount of infiltrated water by a well is modeled using a stratified media of the porous media methodology. The results obtained can help decision-making by evaluating the cost benefit of the construction of wells to a certain location for the recharge of aquifers.

Full text

mathematics
Article
Modeling of Artificial Groundwater Recharge by
Wells: A Model Stratified Porous Medium
Carlos Fuentes 1, Carlos Chávez 2, * , Antonio Quevedo 1, JosuéTrejo-Alonso 2and
Sebastián Fuentes 2
1Mexican Institute of Water Technology, Paseo Cuauhnáhuac Núm. 8532, Jiutepec, Morelos 62550, Mexico;
cbfuentesr@gmail.com (C.F.); jose_quevedo@tlaloc.imta.mx (A.Q.)
2Water Research Center, Department of Irrigation and Drainage Engineering, Autonomous University of
Queretaro, Cerro de las Campanas SN, Col. Las Campanas, Queretaro 76010, Mexico;
josue.trejo@uaq.mx (J.T.-A.); sefuca.1196@gmail.com (S.F.)
*Correspondence: chagcarlos@uaq.mx; Tel.: +52-442-192-1200 (ext. 6036)
Received: 21 September 2020; Accepted: 8 October 2020; Published: 13 October 2020


Abstract:
In recent years, groundwater levels have been decreasing due to the demand in agricultural
and industrial activities, as well as the population that has grown exponentially in cities. One method
of controlling the progressive lowering of the water table is the artificial recharge of water through
wells. With this practice, it is possible to control the amount of water that enters the aquifer through
field measurements. However, the construction of these wells is costly in some areas, in addition to
the fact that most models only simulate the well as if it were a homogeneous profile and the base
equations are restricted. In this work, the amount of infiltrated water by a well is modeled using a
stratified media of the porous media methodology. The results obtained can help decision-making by
evaluating the cost benefit of the construction of wells to a certain location for the recharge of aquifers.
Keywords:
mathematical modeling; infiltration well; differential equations; porous medium; fractal
conductivity model
1. Introduction
Infiltration wells are used to contribute to the evacuation of rains in urban areas and also as
a mechanism to recharge aquifers in regions where they present an unsustainable abatement [
1
–
4
].
Their construction must be analyzed from several angles: objectives of artificial recharge, available
technological options, chemical quality of the water, social factors, place, quantity of water to contribute,
among others [5–9].
In the literature, several numerical and analytical solutions can be found to model the flow of
water in the porous medium, however, the models present restrictions to estimating the properties of
soils, in addition to considering the stratum of the soil well profile as a homogeneous medium [
1
,
10
–
14
].
The artificial recharge capacity in a well is measured as the amount of water that infiltrates the
soil during a specific period of time, and varies depending on the number of strata in the soil in which
it was built. In this way, if you want to know the amount of water that the entire well contributes, you
must evaluate the infiltration rate in all the strata to have a better knowledge about the contributions
to the aquifer and the behavior of the system as a whole.
The phenomenon of infiltration in porous media can be studied from the general principles of
the conservation of mass and momentum. The equation that results from the application of the first
principle is:
∂θ
∂t=−∇ · →
q. (1)
Mathematics 2020,8, 1764; doi:10.3390/math8101764 www.mdpi.com/journal/mathematics

Mathematics 2020,8, 1764 2 of 11
Darcy’s law generalized to partially saturated porous media is used as a dynamic equation [15]:
→
q=−K∇H, (2)
where H is the hydraulic potential and is the sum of the pressure potential (
ψ
) and the gravitational
potential assimilated to the vertical coordinate (z) oriented, in this case, as positive upwards.
The pressure potential is positive in the saturated zone and negative in the unsaturated zone,
since it is agreed that the zero pressure corresponds to the atmospheric pressure;
θ
=
θ
(
ψ
) is the
volumetric water content, also called moisture content, and is a function of the water pressure,
θ
(
ψ
) is
known as the retention curve or soil moisture characteristic;
→
q
=(q
x
, q
y
, q
z
) is the flow of water per
unit of soil surface or Darcy flow, with its components in a rectangular system; (x, y, z) are the spatial
coordinates in a rectangular or Cartesian system,
t
is time;
∇
is the gradient operator; K =K (
ψ
) is the
hydraulic conductivity as a function of the water pressure.
Thus, the general equation of flow in a porous medium results from the combination of Equations (1)
and (2):
∂θ
∂t=∇ · [K(ψ)∇(ψ+z)]. (3)
This equation presents two independent variables,
θ
and
ψ
, but since there is a relationship
between them, the specific capacity defined as the slope of the retention curve is introduced. The chain
rule is applied and the equation with the dependent variable pressure is established, known as the
Richards equation [16]:
C(ψ)∂ψ
∂t=∇ · [K(ψ)∇ψ]+∂K
∂ψ
∂ψ
∂z; C(ψ)=∂θ
∂ψ. (4)
In this work a methodology is presented to obtain the water infiltration rate by partially or totally
filled artificial recharge wells. The equations have been adapted to be used in a homogeneous stratified
medium, taking into account the soil characteristics of each strata in the profile of the well.
2. Materials and Methods
2.1. The Richards and KirchhoffEquations in Spherical and Cylindrical Coordinates
In some problems the analysis is simplified if Equation (4) is written in cylindrical or spherical
coordinates. The Richards equation [16] in cylindrical coordinates (r, ϕ, z) is as follows:
C(ψ)∂ψ
∂t=1
r
∂
∂r"rK(ψ)∂ψ
∂r#+1
r2
∂
∂ϕ"K(ψ)∂ψ
∂ϕ#+∂
∂z"K(ψ)∂ψ
∂z#+∂K
∂ψ
∂ψ
∂z, (5)
where r is the radius and ϕis the azimuth: r2=x2+y2, x =r cos ϕ, y =r sin ϕ.
In spherical coordinates (%,ϑ,ϕ) the Richards equation is written as:
C(ψ)∂ψ
∂t=1
%2∂
∂r%2K(ψ)∂ψ
∂% +1
%2sin ϑ
∂
∂ϑhsin ϑK(ψ)∂ψ
∂ϑi
+1
%2sin2ϑ
∂
∂ϕhK(ψ)∂ψ
∂ϕi+∂K
∂ψ
∂ψ
∂z
(6)
where
%
is the radio,
ϑ
is the polar angle and
ϕ
is the azimuth:
%2
=x
2
+y
2
+z
2
, x =
%
sin
ϑ
cos
ϕ
, y =
%
sin ϑsin ϕ, z =%cos ϑ.
In a symmetric well with respect to the z axis, the radius r takes this axis as its origin, and Equation (5)
is very useful for the infiltration analysis when it is assumed that the pressure does not depend on the
azimuth, that is, when the heterogeneity is presented by layers. In this case the equation simplifies to
the following:

Mathematics 2020,8, 1764 3 of 11
C(ψ)∂ψ
∂t=1
r
∂
∂r"rK(ψ)∂ψ
∂r#+∂
∂z"K(ψ)∂ψ
∂z#+∂K
∂ψ
∂ψ
∂z, (7)
which has only two spatial coordinates (r, z).
The equation in spherical coordinates presents a very particular importance when, in the analysis
of a problem, it is considered that the medium is homogeneous and isotropic, that is, when the
phenomenon does not depend on either the colatitude or the azimuth:
C(ψ)∂ψ
∂t=1
%2
∂
∂% "%2K(ψ)∂ψ
∂% #+∂K
∂ψ
∂ψ
∂z, (8)
in which only two spatial coordinates are presented (%, z).
Furthermore, in some particular problems involving homogeneous porous media, the analysis is
simplified if the potential Kirchhoffflow is defined by:
Φ=
ψ
Z
−∞
Kψdψ=
θ
Z
θr
Dθdθ, (9)
from which it follows that: dΦ
dψ=K(ψ);dΦ
dθ=D(θ), (10)
where D (
θ
) is the hydraulic diffusivity, in analogy with the diffusion of gases, which is expressed as
D(
θ
)=K(
θ
)/C(
θ
), considering, now, that both the hydraulic conductivity and the specific capacity are
functions of the volumetric content moisture.
The water transfer equation in porous media as a dependent variable for moisture content,
Equation (3), is as follows:
∂θ
∂t=∇ · [D(θ)∇θ]+dK
dθ
∂θ
∂z, (11)
which presents the structure of a nonlinear Fokker–Planck equation [
17
], the linear version of which is
widely known in diffusion problems.
In terms of the potential Kirchhoffflow, Equation (11) becomes:
1
D(Φ)
∂Φ
∂t=∇2Φ+dK
dΦ
∂Φ
∂z. (12)
Kirchhoff’s equation in cylindrical coordinates (r,
ϕ
, z), where r is the radius and
ϕ
is the azimuth,
is: 1
D(Φ)
∂Φ
∂t=1
r
∂
∂r r∂Φ
∂r!+1
r2
∂2Φ
∂ϕ2+∂2Φ
∂z2+dK
dΦ
∂Φ
∂z. (13)
In spherical coordinates (%,ϑ,ϕ) the Kirchhoffequation is written as follows:
1
D(Φ)
∂Φ
∂t=1
%2∂
∂% %2∂Φ
∂% +1
%2sin ϑ
∂
∂ϑsin ϑ∂Φ
∂ϑ
+1
%2sin2ϑ
∂2Φ
∂ϕ2+dK
dΦ
∂Φ
∂z
. (14)
These formulations are only applicable in homogeneous media, and, if these are isotropic, they
are written respectively in cylindrical coordinates as follows:
1
D(Φ)
∂Φ
∂t=1
r
∂
∂r r∂Φ
∂r!+∂2Φ
∂z2+dK
dΦ
∂Φ
∂z, (15)

Mathematics 2020,8, 1764 4 of 11
and in spherical coordinates:
1
D(Φ)
∂Φ
∂t=1
%2
∂
∂% %2∂Φ
∂% !+dK
dΦ
∂Φ
∂z. (16)
2.2. The Hydrodynamic Characteristics of Porous Media
In order to solve the mass or energy transfer equations of water in porous media, aside from
specifying the limit conditions, it is necessary to know the hydrodynamic characteristics formed by
the water retention curve
θ
(
ψ
) and the hydraulic conductivity curve either as a function of the water
pressure, K(
ψ
), or as a function of the moisture content K(
θ
). The analysis is greatly simplified if these
curves are represented with analytical functions.
The retention curve can be represented with the equation of van Genuchten [18]:
Θ(ψ)="1+ ψ
ψd!n#−m
, (17)
where m >0 and n >0 are two shape parameters (dimensionless),
ψd
is a characteristic value of the
water pressure and Θis the effective degree of saturation defined by:
Θ=θ−θr
θs−θr, (18)
in which
θr
is the residual moisture content defined such that K(
θr
)=0 and
θ
(
ψ→
-
∞
)=
θr
[
19
];
θs
is the
moisture content at saturation, assimilated to the total porosity of the soil (
φ
), when under saturation
conditions no air is trapped in the interstices of the porous medium: θs=φ. In general, θr=0 can be
assumed [20].
A closed way to represent the conductivity curve can be obtained using prediction models of the
same from the retention curve. In the literature we can find various works, but given the condition of
the phenomenon we are studying, this work uses one of the fractal models proposed, calibrated and
validated by Fuentes et al. [
20
]. The results found by [
20
] shows a better adjustment between observed
and estimated data by the function given by:
K(Θ)=Ks
Θ
Z
0
ϑs−1dϑ
ψ(ϑ)
2s /
1
Z
0
ϑs−1dϑ
ψ(ϑ)
2s 
2
, (19)
where K
s
is the hydraulic saturation conductivity and s =D/E, with D the fractal dimension of the
porous medium and E =3 the Euclid dimension of the physical space where the medium is embedded,
related to the total porosity through the relation:
(1−φ)s+φ2s =1. (20)
The introduction of Equation (17) in Equation (19) leads to the following equation to represent the
hydraulic conductivity curve, accepting the relationship between the parameters as indicated:
K(Θ)=Ksh1−1−Θ1/msmi2; 0 <sm =1−2s/n<1. (21)
The solution of the transfer equation in its different forms is generally numerical [
21
,
22
]. However,
in some simplified cases, characteristics of the solution can be obtained analytically [23–25].

Mathematics 2020,8, 1764 5 of 11
3. Results and Discussion
3.1. Conceptual Model
To analyze the infiltration in steady-state wells, it is necessary to write the Darcy flows in the
radial and vertical directions:
→
qr=−Ks
∂ψ
∂rˆ
r, (22)
→
qz=− Ks
∂ψ
∂z+Ks!ˆ
k, (23)
where ˆ
r and ˆ
k are unitary vectors in the r and z directions respectively.
Flow through the wall and bottom of the well is defined by:
Qs=Z
Ap
→
qr·dAp+Z
Ab
→
qz·dAb, (24)
where dA
p
y dA
b
are, respectively, the differential areas in the wall and at the bottom of the well
defined by:
dAp=(2πRdz)ˆ
r, (25)
dAb=(2πrDr)−ˆ
k. (26)
Equation (24), considering Equations (22), (23), (25) and (26), is written as follows:
Qs=−2πRKs
H
Z
0
∂ψ
∂rr=r
dz +2πKs
R
Z
0
∂ψ
∂zz=0
rdr +πKsR2. (27)
Introducing the dimensionless variables:
z∗=z
H; r∗=r
R;ψ∗=ψ
H; (28)
Equation (27) is written as follows:
Qs=Qo+πKsR2; Qo=2πKsH2
C; (29)
where C is a coefficient defined as:
1
C=−
1
Z
0
∂ψ∗
∂r∗r∗=1
dz∗+R
H21
Z
0
∂ψ∗
∂z∗z∗=0
r∗dr∗. (30)
To find this coefficient it is necessary to know ψ(r, z).
3.2. The Glover Model
According to Glover [
26
], in a first approximation, the steady-state pressure flow through an
infiltration well in a homogeneous and isotropic porous medium can be described with Laplace’s
equation in spherical coordinates that describes the pressure in the absence of gravitational gradients.
From Equation (8) we have:
∇2ψ=1
%2
∂
∂% %2∂ψ
∂% !=0, (31)

Mathematics 2020,8, 1764 6 of 11
which must be subject to border conditions:
ψ=ψR;%=R (32)
ψ=0; %→ ∞. (33)
Integration of Equation (31) leads to
ψ
=
−
c
1%−1
+c
2
, where c
1
and c
2
are integration constants;
the Equation (33) implies c
2
=0 and the Equation (32) c
1
=
−ψR
R, ergo
ψ
=
ψR
(R/
%
). The Darcy flux is
q
%
=
−
K
s∂ψ
/
∂%
=K
sψR
(R/
%2
), when
%
=R, q
R
=K
sψR
/R; the flow through the surface of the sphere
of radius R is q
o
=4
π
R
2
q
R
=4
π
K
s
R
ψR
; this flow from the point source in the center of the sphere is the
variable of interest. Since
ψR
R=q
o
/4
π
K
s
, it is better to set the pressure variation around the source
flow to continue with the Glover approach:
ψ=qo
4πKs%. (34)
If h represents the position of the center of the sphere from the base, then the spherical coordinate
(%) and the cylindrical coordinate (r) are related by:
%=qr2+(z−h)2. (35)
The pressure in terms of the cylindrical coordinates is obtained by introducing Equation (35) into
Equation (34):
ψ=qo
4πKsqr2+(z−h)2
. (36)
To provide a series of point sources whose magnitude increases with depth, an expression similar
to that originally proposed by Glover, we have:
dqo=B(hc−h)dh, (37)
where B is a parameter to be determined and h
c
defines the range of the sources h
o≤
h
≤
h
c
and sinks
hc<h≤hs.
The total flow is found by integrating Equation (37):
Qo=B
hs
Z
Ho
(hc−h)dh =1
2BH2h(h∗
c−h∗
o)2−(h∗
c−h∗
s)2i(38)
hence parameter B is deduced:
B=2Qo
H2h(h∗
c−h∗
o)2−(h∗
c−h∗
s)2i(39)
where h* =h/H for all subscripts.
From Equations (36), (37) and (39) we have:
dψ=Qo(hc−h)
2πKsH2h(h∗
c−h∗
o)2−(h∗
c−h∗
s)2iqr2+(z−h)2
dh (40)

Mathematics 2020,8, 1764 7 of 11
the integration of which leads to:
ψ=Qo
2πKsH2h(h∗
c−h∗
o)2−(h∗
c−h∗
s)2i"(hc−z)asinhz−h
r+qr2+(z−H)2#h=ho
h=hs
, (41)
ergo:
ψ=Qo
2πKsH2

(hc−z)asinhz−ho
r−(hc−z)asinhz−hs
r
+qr2+(z−ho)2−qr2+(z−hs)2
h(h∗
c−h∗
o)2−(h∗
c−h∗
s)2i. (42)
At the point on the boundary (r, z) =(R, 0) we have
ψ
=H, which allows obtaining the expression
of the flow, Equation (29):
Qo=2πKsH2
C, (43)
where the form coefficient is defined by:
C=h∗
chasinhH
Rh∗
s−asinhH
Rh∗
oi+qR
H2+h∗2
o−qR
H2+h∗
s2
(h∗
c−h∗
o)2−(h∗
c−h∗
s)2. (44)
Glover formula is derived from Equation (44) by making h∗
c=1, h∗
o=0yh∗
s=1:
C=asinhH
R+R
H−sR
H2
+1. (45)
3.3. The Reynolds and Elrick Model
This model proposed by Reynolds and Elrick [27] assumes h∗
c=1/2, h∗
i=0yh∗
s=1/2:
C=4
1
2asinhH
2R+R
H−sR
H2
+1
4
. (46)
3.4. A Model for Stratified Porous Media
Glover’s model can be adapted for the case of stratified porous media. The well is considered to
be in a medium composed of N layers of thickness P
j
, j =1,2,
. . .
, N; the total hydraulic head, denoted
as H
T
, is the height of the water column counted from the base of the well to the upper border of the
N-th stratum.
The flow infiltrated by the walls of the j-th stratum is provided by Equation (43) modified as:
Qoj =
2πKsjp2
j
Cj
, (47)
where K
sj
and C
j
are the saturated hydraulic conductivity and the shape coefficient of the j-th
stratum, respectively.
The shape coefficient is derived from Equation (44) denoting by H
j
the hydraulic head at the base
of the j-th stratum:
Cj=h∗
pj
h∗
Cjasinhpj
Rh∗
sj−asinhpj
Rh∗
oj+rR
Pj2
+h∗2
oj −rR
Pj2
+h∗
sj2
h∗
cj −h∗
oj2−h∗
cj −h∗
sj2, (48)

Mathematics 2020,8, 1764 8 of 11
where
h∗
Pj
=P
j
/H
j
,
h∗
cj
=h
cj
/P
j
,
h∗
oj
=h
oj
/P
j
,
h∗
sj
=h
sj
/P
j
. It is noted that h
cj
, h
oj
y h
sj
are calculated from
the base of the j-th stratum.
The Reynolds and Elrick model assumes h∗
c=1/2, h∗
o=0yh∗
s=1/2 and therefore:
Cj=4
1
2asinh Pj
2R!+R
pj
−s R
Pj!2
+1
4
hPj. (49)
The total flow is obtained as:
Q=
N
X
j=1
Qoj +πR2Ks1 (50)
where the flow at the bottom of the well has been added.
3.5. Aplications
To show the versatility of the solution, data obtained from an infiltration well built on the Queretaro
Valley aquifer of radio are used: R =0.3937 m (15.5
00
) and depth P
T
=36 m; Five strata were located in
the profile (Figure 1). As drilling was carried out, the infiltration tests per stratum were carried out
until the permanent regime was reached. The measured data are concentrated in Table 1, and the
saturated hydraulic conductivity calculated from Equation (43) is also shown.
Table 1. Calculation of hydraulic conductivity per stratum, Equation (43).
Stratum H
(m/s)
Q
(L/s) CKs
(m/d)
1 12 0.02841 4.9632 0.0134
2 4 0.33681 3.0113 0.8592
3 10 0.81158 4.6239 0.5142
4 6 1.01448 3.7017 1.4231
5 4 1.62317 3.0113 4.1405
Table 2shows the flow rates calculated for each stratum when the well is full. In the last row is
the total flow, Equation (50), and the saturated hydraulic conductivity corresponding to an equivalent
homogeneous stratum.
Mathematics 2020, 8, x FOR PEER REVIEW 8 of 11
N2
oj s1
j1
QQπRK
=
=+
 (50)
where the flow at the bottom of the well has been added.
3.5. Aplications
To show the versatility of the solution, data obtained from an infiltration well built on the
Queretaro Valley aquifer of radio are used: R = 0.3937 m (15.5″) and depth PT = 36 m; Five strata were
located in the profile (Figure 1). As drilling was carried out, the infiltration tests per stratum were
carried out until the permanent regime was reached. The measured data are concentrated in Table 1,
and the saturated hydraulic conductivity calculated from Equation (43) is also shown.
Table 1. Calculation of hydraulic conductivity per stratum, Equation (43).
Stratum H
(m/s)
Q
(L/s) C Ks
(m/d)
1 12 0.02841 4.9632 0.0134
2 4 0.33681 3.0113 0.8592
3 10 0.81158 4.6239 0.5142
4 6 1.01448 3.7017 1.4231
5 4 1.62317 3.0113 4.1405
Table 2 shows the flow rates calculated for each stratum when the well is full. In the last row is
the total flow, Equation (50), and the saturated hydraulic conductivity corresponding to an equivalent
homogeneous stratum.
Figure 1. Simplified well scheme.
Figure 1. Simplified well scheme.

Mathematics 2020,8, 1764 9 of 11
Table 2. Calculation of the flow per stratum corresponding to a full well, Equations (47) and (49).
Stratum P
(m)
Ks
(m/d)
H
(m) CQ
(L/s)
1 12 0.0134 36 1.6544 0.085
2 4 0.8592 24 0.5019 1.992
3 10 0.5142 20 2.3119 1.617
4 6 1.4231 10 2.2210 1.677
5 4 4.1405 4 3.0113 1.600
Equivalent 36 0.5231 36 7.0749 6.972
The artificial volume that can be recharged to the aquifer is 6972 L/s (602.3808 m
3
/day), however,
stratum 1 only contributes 1.22% of the entire volume and is the deepest layer of the entire well (12 m).
With these data, the cost benefit is analyzed and the decision is made to drill wells to a depth of 24 m
with the understanding that we would contribute only 595.0318 m
3
/day to the aquifer but we reduce
time and money.
The time to drill the well up to 36 m is 27 days at a cost of 25,457 USD. Therefore, if you choose to
build two, the time taken would be 60 days with a total of 50,915 USD. Conversely, if you only drill
to a depth of 24 m, the time taken is 7 days and a cost of 13,376 USD, which gives us a total cost of
53,504 USD for four wells drilled in 30 days. This is due to the fact that the material in the last 12 m is
basalt and drilling progress is slower.
Regarding the volumes of recharge to the aquifer, with the four wells in the same area it would be
2380.1272 m
3
/day compared to 1204.7616 m
3
/day that we would obtain with only two at a depth of
36 m. Finally, the cost-benefit of annual recharge in the aquifer would be 16.24 m
3
/USD invested in
four wells, compared to the 8.64 m3/USD that you have if you choose two wells.
4. Conclusions
In recent years the construction of artificial wells to recharge aquifers has been very popular in
Mexico, however, as has been demonstrated in this work, the construction of a well at a greater depth
does not necessarily give us a greater volume of recharge. The lack of information to calculate the
total volumes has led to decisions being made with unscientific bases and, on several occasions, it has
resulted in not achieving the expectations for which they were built.
This work provides a tool for knowing the behavior of the water infiltration rate in the porous
medium in stratified media to artificially recharge an aquifer through wells. The analysis takes into
account all the characteristics of the soil profiles that construct the well, resulting in the cost-benefit
analysis of the complete operation to make a better decision.
It is widely demonstrated in the literature that several experimental tests are needed to know the
behavior of this phenomenon, and that in order to know the process in detail, other factors that are
not analyzed here must be taken into account: preferential flow in heterogeneous soils, trapped air,
sediment deposit, among others.
When exploration drilling is done to propose a series of wells to recharge the aquifer, the data
from the first layer is usually measured to simulate the behavior of the entire profile as a homogeneous
stratum. However, as verified in this work, it is necessary to know the behavior of the entire well by
stratum so that pertinent decisions are made, since a deeper well does not necessarily imply a greater
volume of recharge to the aquifer.
Author Contributions:
Conceptualization, C.F.; methodology, C.F.; software, C.F., A.Q., J.T.-A. and S.F.; validation,
C.F., C.C. and A.Q.; formal analysis, C.F., C.C. and A.Q.; investigation, C.F.; data curation, J.T.-A. and S.F.;
writing—original draft preparation, C.F.; writing—review and editing, C.C. and J.T.-A. All authors have read and
agreed to the published version of the manuscript
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.

Mathematics 2020,8, 1764 10 of 11
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