Analytic modeling of groundwater dynamics
with an approximate impulse response
function for areal recharge
Mark Bakker a Kees Maas a,b Frans Schaars c Jos R. von Asmuth a,b
a Water
Resources Section, Faculty of Civil Engineering and Geosciences, Delft
University of Technology, Delft, The Netherlands
b Kiwa
Water Research, Nieuwegein, The Netherlands
c Artesia,
Schoonhoven, The Netherlands
Preprint submitted to Elsevier Science
10 April 2006
Abstract
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An analytic approach is presented for the simulation of variations in the groundwater level due to temporal variations of recharge in surficial aquifers. Such variations,
called groundwater dynamics, are computed through convolution of the response
function due to an impulse of recharge with a measured time series of recharge.
It is proposed to approximate the impulse response function with an exponential
function of time which has two parameters that are functions of space only. These
parameters are computed by setting the zeroth and first temporal moments of the
approximate impulse response function equal to the corresponding moments of the
true impulse response function. The zeroth and first moments are modeled with
the analytic element method. The zeroth moment may be modeled with existing
analytic elements, while new analytic elements are derived for the modeling of the
first moment. Moment matching may be applied in the same fashion with other
approximate impulse response functions. It is shown that the proposed approach
gives accurate results for a circular island through comparison with an exact solution; both a step recharge function and a measured series of 10 years of recharge
were used. The presented approach is specifically useful for modeling groundwater
dynamics in aquifers with shallow groundwater tables as is demonstrated in a practical application. The analytic element method is a gridless method that allows for
the precise placement of ditches and streams that regulate groundwater levels in
such aquifers; heads may be computed analytically at any point and at any time.
The presented approach may be extended to simulate the effect of other transient
stresses (such as fluctuating surface water levels or pumping rates), and to simulate
transient effects in multi-aquifer systems.
Key words: Impulse response function, Convolution, Analytic element method,
Groundwater dynamics
Email addresses: [email protected] (Mark Bakker), [email protected] (Kees
Maas), [email protected] (Frans Schaars), [email protected] (Jos
R. von Asmuth).
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1
Introduction
The objective of this paper is to present a new approach for the accurate modeling of
fluctuations in the groundwater table due to temporal variations in rainfall and evapotranspiration. Fluctuations of the groundwater table are referred to as groundwater
dynamics. Accurate modeling of groundwater dynamics is important for the effective
management of aquifers, especially aquifers with shallow groundwater tables. Models
need to take into account daily changes in rainfall and evaporation, as well as the
accurate location of streams, canals, ditches, and drains.
In this paper it is proposed to compute the groundwater dynamics by developing
models of the response function due to an impulse of recharge; these models are
based on physical boundary conditions. Once the impulse response function is known
at a point, fluctuations of the groundwater table may be computed through convolution with time series of recharge. It is proposed to approximate the impulse response
function with an exponential function of time containing two parameters that are a
function of the two spatial coordinates. The use of an approximate impulse response
function is common in the field of time series analysis, where they are often called
transfer functions. The standard Box-Jenkins method uses a discrete impulse response
function and has been applied to forecast head fluctuations at observation wells based
on time series of recharge (e.g., [18], [10]). Bierkens et al. [5] developed an empirical
space-time model to estimate parameters of their Box-Jenkins model between observation wells using geostatistical interpolation and information on locations of streams
and ditches. Von Asmuth et al. [21] proposed to approximate the impulse response
function with the Pearson type III function, which is continuous through time, and
obtained results for a practical case that were at least as accurate as results obtained
with the discrete impulse response function of the Box-Jenkins method.
The parameters of the approximate impulse response function used in this paper
are computed with the analytic element method; several new analytic elements are
derived. The advantage of the analytic element method is that the model domain
does not have to be discretized spatially, e.g., in rectangles or triangles. Hydrogeologic
boundaries can be defined exactly where they are located without having to adjust
them to a spatial grid. The analytic element method has been applied to steady flow
in single aquifers and multi-aquifer systems (for an overview, see [17]), and to steady
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unsaturated flow (e.g. [3]). Several formulations exist for transient flow, each with its
own advantages and limitations ([23],[7], [2]). The approach presented in this paper
may be viewed as a new transient analytic element formulation.
This paper is limited to homogeneous aquifers; variations in the saturated thickness
of unconfined aquifers (and thus the transmissivity) are neglected. The areal recharge
is the only transient stress on the system and does not vary spatially. Other imposed
stresses, such as specified head or flux boundaries, or pumping wells, do not vary with
time. In addition, non-linear effects, such as ditches that dry up, are not considered.
The presented approach may be extended to include spatially varying recharge, other
transient stresses, inhomogeneous aquifers, and multi-aquifer systems, but that is
beyond the scope of this paper; some of these extensions are briefly discussed in the
Conclusions and Discussion section.
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The basic steps of the proposed approach are outlined in the following two sections,
followed by an evaluation of the performance of the proposed approximate response
function through comparison with an exact solution. In section 5, analytic elements
are derived for the modeling of the zeroth and first temporal moments of the impulse
response function. Performance of the new analytic elements is evaluated in section
6, followed by a practical example in section 7.
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Approach
Groundwater flow in a homogeneous, isotropic, horizontal aquifer is governed by (e.g.,
[4])
∇2 h =
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S ∂h N
−
T ∂t
T
(1)
where h [L] is the hydraulic head, ∇2 [L−2 ] is the two-dimensional Laplacian, t [T]
is time, S [-] is the storativity of the aquifer, T [L2 T−1 ] is the transmissivity, and N
[LT−1 ] is the time-varying areal recharge. Equation (1) may be applied to unconfined
aquifers when the variation of the saturated thickness is relatively small. The aquifer
is linear, such that the head in the aquifer may be written as a steady component
plus a transient component
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h(x, y, t) = h0 (x, y) + φ(x, y, t)
1
h0 [L] fulfills (1) but with the righthand side set to zero
∇2 h0 = 0
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h0 = hf ,
φ=0
(5)
∂h0
= qn ,
∂n
∂φ
=0
∂n
(6)
Along a boundary with a mixed boundary condition, boundary conditions are
∂h
= bh + c,
∂n
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(4)
Along a boundary where the normal gradient ∂h/∂n is fixed to qn , boundary conditions are
∂h
= qn ,
∂n
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S ∂φ N
−
T ∂t
T
All imposed boundary conditions are steady state. Along a boundary where h is fixed
to hf , boundary conditions are
h = hf ,
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(3)
and φ [L] fulfills (1)
∇2 φ =
3
(2)
∂h0
= bh0 + c,
∂n
∂φ
= bφ
∂n
(7)
where b and c are constants. The boundary conditions for h0 are identical to the
boundary conditions for h, and thus h0 represents the steady head in the aquifer in
the absence of recharge; h0 may be modeled with a standard method and will not
be discussed here further. The remainder of this paper concerns the modeling of φ,
which represents the head fluctuation caused by the temporal variation of recharge.
The initial condition of φ is equal to the initial condition of h − h0 .
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A standard technique to solve the differential equation for φ is to determine the
solution for an impulse of recharge of unit volume, called the impulse response function
θ(x, y, t) [-], and to obtain φ(x, y, t) for an arbitrary recharge function N (t) through
convolution (Duhamel’s principle, e.g. [4], Sec. 7.5; [24] Sec. II.B.7).
The impulse response function fulfills (1) where the recharge is replaced by the Dirac
delta function δ(t) [T−1 ]
∇2 θ =
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S ∂θ δ(t)
−
T ∂t
T
(8)
where, as before, the analysis is limited to a recharge that does not vary spatially.
Once θ is determined, the response φ(x, y, t) to any recharge N (t) may be found
through convolution
φ(x, y, t) =
Zt
N (τ )θ(x, y, t − τ )dτ
(9)
−∞
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without the need to solve the original differential equation (4) again. Boundary conditions for the impulse response function θ are identical to the boundary conditions
for φ, as may be seen through substitution of (9) for φ in the boundary conditions
(5–7).
The analytic solution of (8) is difficult for general cases. Here, a new analytic but approximate approach is presented. It is proposed to approximate the impulse response
function of the recharge by the following exponential function
θ=0
t<0
θ = Aa exp(−at)
t≥0
(10)
where A(x, y) and a(x, y) are functions to be determined. It will be shown in the
following sections that the exponential representation (10) may be used to represent
the true impulse response function with reasonable accuracy for practical cases of
transient recharge. The approximate impulse response function contains two spatial
functions A and a. To achieve an accurate overall match between the approximate
response function (10) and the true impulse response function that fulfills (8), it is
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proposed to determine A and a such that the zeroth and first moments of the approximate impulse response function are equal to the corresponding moments of the true
impulse response function. Matching temporal moments has been applied to estimate
transport parameters (e.g. [14] and references therein). Application of moment matching to estimate parameters of impulse response functions of groundwater head was
proposed by [12] and [20]. Li et al. [13] used moments to characterize drawdowns of
pumping tests; they found that the zeroth and first temporal moments were sufficient
to characterize the drawdown curves in their study.
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Matching temporal moments
The zeroth temporal moment M0 of the impulse response function is defined as
Z∞
M0 =
θdt
(11)
−∞
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Integration of the differential equation for θ (8) gives
∇ 2 M0 = −
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1
T
where it is used that θ equals zero for both t = −∞ and t = ∞, and that the integral
of δ(t) equals one. Hence, the zeroth moment M0 of the impulse response function
fulfills Poisson’s differential equation (12) with a constant righthand side. For the
steady boundary conditions used here, boundary conditions of M0 are identical to
the boundary conditions for θ, and thus for φ, as may be seen from substitution of
the boundary conditions (5–7) for θ in the definition of M0 (11).
The first temporal moment M1 of the impulse response function is defined as
M1 =
Z∞
tθdt
−∞
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(12)
Multiplication of both sides of (8) with t and integration gives
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(13)
∞
Z∞
S Z ∂θ
tδ(t)
∇ M1 =
t dt −
dt
T
∂t
T
2
−∞
−∞
1
(14)
The right integral equals zero, and the left integral may be integrated by parts to give


Z∞
S ∞
2
θdt
θt|−∞ −
∇ M1 =
T
(15)
−∞
2
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The first term equals zero and the remaining integral is M0 (11), such that the differential equation for M1 becomes
S
∇ 2 M1 = − M0
T
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(16)
Hence, the first moment M1 of the impulse response function also fulfills Poisson’s
differential equation, where the zeroth moment appears on the righthand side. For
the steady boundary conditions used here, boundary conditions of M1 are identical
to the boundary conditions for θ, and thus for φ. Li et al. [13] presented differential
equations for the moments of the impulse response function of a pumping well, which
are similar to eqs. (12) and (16).
Equations (12) and (16), with specified boundary conditions, may be solved for
M0 (x, y) and M1 (x, y); these are the zeroth and first moments of the true impulse
response function. The functions A and a may now be found by setting M0 and M1
of the true impulse response function equal to the corresponding moments of the
approximate impulse response function. These latter two moments are obtained by
substituting the approximate impulse response function (10) in the definitions for M0
(11) and M1 (13). The lower limit of the integral is changed to zero, as θ = 0 for
t < 0, so that
M0 =
M1 =
Z∞
0
Z∞
Aa exp(−at)dt = A
(17)
Aat exp(−at)dt = A/a
0
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1
and thus
A = M0
a = M0 /M1
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(18)
In statistical terms, a is the inverse of the mean of the impulse response function.
Hence, the mean of the exact and approximate impulse response functions are equal.
In summary, the proposed approach for the modeling of groundwater dynamics due
to temporal variations in recharge consists of four steps:
(1) Develop a model for M0 by solving (12) with appropriate boundary conditions.
(2) Develop a model for M1 by solving (16) with appropriate boundary conditions,
and using the M0 from the fist step in the righthand side of the differential
equation.
(3) Once M0 and M1 are known, compute A and a with (18).
(4) Compute the fluctuation of the head φ for any time series of recharge N (t) with
convolution (9), using the approximate impulse response function (10).
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Note that with the proposed approach the solution of the transient problem has been
reduced to the creation of steady models for three variables: h0 , M0 , and M1 , where the
solution of M1 depends on the solution of M0 . In the following section, the accuracy
of the proposed approach is evaluated for a simple case of radial flow.
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Example of radial flow
The objective of this example is to demonstrate application of the approach outlined
in the previous section to a simple case of radial flow, and to evaluate the performance of the proposed approximate impulse response function (10). Consider radial
groundwater flow on a circular island of radius R. On the boundary of the island, the
water level is fixed to 0, so that
φ(r = R, t) = 0
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(19)
1
For t < 0, the heads are zero everywhere
φ(r, t) = 0
2
3
t<0
For t < 0, the recharge is zero, but for t ≥ 0 the recharge is equal to a constant value
γ

0
N (t) = 
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t<0
γ
t≥0
10
!
=−
R2 − r 2
4T
12
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(23)
(24)
The second step of the approach is to determine the first moment M1 of the impulse
response function. Boundary conditions are again the same as for φ
M1 (R) = 0
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1
T
Solution of the differential equation with the specified boundary condition gives
M0 =
9
(22)
The differential equation for M0 (12) is written in radial coordinates
dM0
1 d
r
r dr
dr
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(21)
The first step of the approach outlined in the previous section is to determine the
zeroth moment M0 of the impulse response function. As stated, boundary conditions
for M0 are the same as boundary conditions for φ, and thus
M0 (R) = 0
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(20)
(25)
Solution of the differential equation for M1 (12), with substitution of the computed
function for M0 (24) in the righthand side, and the above stated boundary condition
gives
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R2 r2 3R4
r4
−
+
16
4
16
S
M1 =
4T 2
1
!
(26)
which may be simplified to
SM0 (3R2 − r2 )
M1 =
16T
2
3
(27)
The third step is to compute A and a with equations (18) and the derived expressions
for M0 and M1 , which gives
R2 − r 2
A=
4T
16T
a=
S(3R2 − r2 )
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(28)
The fourth and last step is to determine the head fluctuation through convolution.
Substitution of (21) for N and (10) for θ in (9) gives (note that the lower limit of the
integral is set to 0, as N is zero for t < 0)
φ=
Zt
γAa exp[−a(t − τ )]dτ
(29)
0
= γA − γA exp(−at)
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Finally, substitution of (28) for A and a in the previous equation gives the final
expression for φ(r, t)
−16T t
γ(R2 − r2 )
1 − exp
φ=
4T
S(3R2 − r2 )
(
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11
"
#)
(30)
The first term on the righthand side of φ (30) is the steady-state head distribution
for a constant recharge on a circular island (e.g. [6] Eq. 233.17), which is represented
exactly by the approximate approach. The accuracy of the approximate solution is
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assessed through comparison with the exact transient solution, which is given by [6],
Eq. 233.16, and may be written as
∞
J0 (αn r/R)
γ(R2 − r2 ) 2γR2 X
2 Tt
−
exp −αn
φ=
4T
T n=0 αn3 J1 (αn )
SR2
(31)
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where J0 and J1 are Bessel functions of the first kind and order 0 and 1, respectively,
and αn is the nth root of J0 . The dimensionless head fluctuation 4T φ/(γR2 ) is shown
at five different dimensionless times in Fig. 1, where the solid line is the approximate
solution and the dashed line the exact solution (using 21 terms in the summation).
In the exact solution, the head is flatter at early times than in the approximate
solution. The difference is largest at the center of the island, where the difference
decreases from 18% at T t/(SR2 ) = 0.05, to 8% at T t/(SR2 ) = 0.1, to less than 1%
at T t/(SR2 ) = 0.2.
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The exact solution for the impulse response function is ([6], Eq. 233.15)
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∞
J0 (αn r/R)
2 X
2 Tt
exp −αn
θ=
S n=0 αn J1 (αn )
SR2
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(32)
The approximate (solid) and exact (dashed) impulse response functions (multiplied
by S) are plotted vs. dimensionless time at four different positions in Fig. 2 (decending
lines). Here it is visible that, at the center of the island, the exact impulse response
function is flat for early times, while the approximate impulse response function is
not. Convolution of the step recharge function (21) with the approximate impulse
response function yields the ascending lines in Fig. 2, which are the dimensionless
head fluctuation 4T φ/(γR2 ) vs. dimensionless time at four different positions. These
curves represent the head variation that would be measured in an observation well.
The match is good at all positions.
To evaluate the head variations caused by measured, and thus wildly fluctuating,
recharge, the head variation is computed at the center of the island using time series
of daily rainfall and evapotranspiration obtained from weather station De Bilt in
The Netherlands. Recharge is computed as rainfall minus evapotranspiration, which
is accurate for relatively flat areas with shallow unsaturated zones. The convolution
integral (9) is evaluated analytically by superimposing the contribution of each daily
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event, assuming the recharge occurs evenly during the day. The convolution integral
is truncated at time ttr . The truncation time is computed such that the area under
the truncated approximate impulse response function (normalized through division
by M0 ) is α = 0.999, which gives
ttr = −
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ln(1 − α)
a
(33)
A plot of the daily recharge is shown for a period of 10 years in the top part of Fig. 3.
The head variation at the center of the island is shown in the bottom part, where the
solid line represents the approximate solution, and the dashed line the exact solution.
The difference between the minimum and maximum heads during the 10 year period
is 2.53 m. The two solutions are visually almost indistinguishable, even though there
is a noticeable difference between the two impulse response functions at this location
(Fig. 2). The average difference between the approximate and exact solutions in Fig.
3 is 0.001 cm, with a standard deviation of 4.18 cm.
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To evaluate the difference between the approximate and exact solutions more closely,
the head variation in the year 1994 is shown in Fig. 4. Here it can be seen that there
is indeed a small difference between the approximate and the exact solutions. For
real aquifers, there are no exact solutions, but there are head measurements at an
observation well, for example taken once a month. As an illustration, the head value
of the exact solution at the end of every month is depicted in Fig. 4 with grey dots.
The approximate solution (solid line) may be considered a good fit to the monthly
values. It is concluded that the approximate impulse response function (10) gives
reasonable values for real recharge series.
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Analytic element modeling of M0 and M1
For practical application of the proposed method, general models need to be created
of the zeroth and first moments of the impulse response function. It is proposed to
model M0 and M1 with analytic elements, which leads to analytic expressions for
A(x, y) and a(x, y). As a result, the head fluctuation φ(x, y, t) may be computed
analytically at any point (x, y) and for any time t. Alternatively, discrete solutions
of M0 and M1 , and thus φ, may be obtained with a numerical method such as finite
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differences or finite elements.
An analytic element solution consists of the superposition of many analytic solutions
to the differential equation ([16],[8],[17]). Each analytic solution is called an analytic
element and has one or more free parameters that are computed to meet specified
boundary conditions. This paper deals with aquifers where the heads are controlled
by streams, canals, or ditches with fixed water levels. Such features may be modeled
with line-sinks. The free parameters of a line-sink are the coefficients in the function
representing the extraction rate of the line-sink. These may be computed, for example,
to fix the value of M0 or M1 along a stream.
The differential equation for M0 (12) is the standard differential equation for steady
flow in a homogeneous aquifer with unit recharge. The analytic element solution
to (12) consists of three parts: a particular solution for unit recharge, a number of
line-sinks that fulfill Laplace’s differential equation, plus an arbitrary constant C. A
particular solution to (12) is given by (24), such that M0 may be written as
Nls
X
1
2
2
fn + C
(R − r ) +
M0 =
4T
n=1
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16
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20
(34)
where r2 = (x − x0 )2 + (y − y0 )2 ; R and (x0 , y0 ) may be chosen for convenience. Note
that the particular solution represents circular contours around (x0 , y0 ) with the zero
contour going through r = R. The function fn is the zeroth moment of line-sink n,
and Nls is the total number of line-sinks. The extraction rate of line-sink n is chosen
to vary along the line-sink as a polynomial of order Pn . The general expression for
the M0 of line-sink n may be written as
fn =
Pn
X
αn,p ϕn,p
(35)
p=0
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where αn,p is coefficient number p of line-sink n, and ϕn,p is the zeroth moment for
line-sink n with an extraction rate that varies as ∆p , where ∆ is the coordinate along
the line-sink. A mathematical expression for a line-sink may be obtained through the
integration of a point sink (a well) along a line segment. The expression for the M0
of a point sink of unit strength that fulfills Laplace’s equation is (e.g. [16])
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1
ln r
2πT
M0 =
1
2
3
4
5
6
7
8
where r is the radial distance from the well. As shown by [16], integration of (36)
along a line may be carried out in the complex plane; a general expression for ϕn,p
is presented in the Appendix. The coefficients αn,p are computed by applying the
boundary condition of M0 at control points along the line-sinks. The control points
are distributed using the cosine rule, as proposed by [9]. The resulting system of linear
equations is solved using a standard method, for example LDU decomposition.
The first moment M1 fulfills differential equation (16) where M0 in the righthand side
is given by (34)
∇ 2 M1 = −
9
10
11
12
13
Nls
S
S
S X
2
2
fn − C
(R
−
r
)
−
2
4T
T n=1
T
r4
R2 r2 3R4
−
+
16
4
16
!
−
Nls
M
ls
X
SCr2
S X
Fn −
gm + D
+
T n=1
8T 2
m=1
16
(38)
The functions Fn fulfill
∇2 Fn = fn
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(37)
The analytic element solution to this differential equation is written as a particular solution, plus an arbitrary new constant D, plus Mls new line-sinks that fulfill
Laplace’s equation and are used to meet the boundary conditions for M1 along the
surface water features. A particular solution to the first term on the righthand side
is given by (26) such that M1 may be written as
S
M1 =
4T 2
14
(36)
(39)
where fn is given by (35). The functions gm are line-sinks of order Pm that fulfill
Laplace’s equation
gm =
Pm
X
βm,p ϕm,p
p=0
15
(40)
1
2
3
4
where βm,p is coefficient number p of line-sink m. The coefficients βm,p are computed
by applying the boundary condition of M1 at control points along the line-sinks; the
resulting system of linear equations may be solved using a standard method.
The function Fn is written as
Fn =
Pn
X
αn,p Φn,p
(41)
p=0
5
where
∇2 Φn,p = ϕn,p
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9
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An expression for ϕn,p , the M0 of a line-sink with a strength ∆p , was obtained through
integration of the M0 for a point-sink (36) along a line. Similarly, an expression for
Φn,p , the M1 of a line-sink with a strength ∆p , is obtained through integration of the
M1 of a point-sink. The first moment of a point-sink is obtained through integration
of (16) with (36) for M0 , which gives
M1 = −
11
12
13
14
15
16
17
18
19
20
21
22
(42)
S
(r2 ln r − r2 )
8πT 2
(43)
Integration of this expression along a line results in an expression for Φn,p , and is
presented in the Appendix.
This completes the description of an analytic element approach for the modeling of
the moments M0 and M1 of the impulse response function of areal recharge. Equations
were presented for wells and line-sinks. The moments computed with these elements
may be used to compute the coefficients of approximate impulse response functions,
for example the exponential function used in (10). Values of M0 and M1 (and thus
heads) may be fixed along strings of line-sinks, which may represent rivers, streams,
ditches, or boundaries of fully penetrating lakes. Other analytic elements may be
derived in a similar fashion, but that is beyond the scope of this paper. In the next
section, the problem of section 4 will be solved again, but this time with the derived
analytic elements.
16
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
6
Radial flow example with analytic elements
Consider the same case of a circular island as discussed in section 4. The objective
of this example is to demonstrate that the moments M0 and M1 may be modeled
accurately with the analytic elements derived in the previous section. For this specific
problem, the constants in the particular solution (Eqs. 34 and 38) may be chosen as
(x0 , y0 ) = (0, 0) and R = 1000, so that an exact solution is obtained without the need
for line-sinks. To test the use of line-sinks, the constants are chosen differently as
(x0 , y0 ) = (1000, 0) and R = 2200, resulting in the contour lines of M0 shown in Fig.
5. Twenty line-sinks of equal length and second order are used to set the value of both
M0 and M1 to zero along the boundary of the island; the vertices are chosen such
that the area of the island is preserved. The analytic element solution is compared
to the exact solution (Eqs. (24) and (26)). Contours of M0 and M1 are shown in Fig.
6. The lower half of each contour plot (below the dotted line) represents the exact
solution. The upper half is the analytic element solution. M0 varies from 0 on the
boundary to 500 at the center of the island (contour interval is 50), while M1 varies
from 0 on the boundary to 37500 at the center (contour interval is 3750). The analytic
element solution produces accurate results. At the center of the island, the relative
difference between the analytic element solution and the exact solution is 0.03% for
M0 and 0.2% for M1 . The accuracy may be increased when more shorter line-sinks
and/or line-sinks of higher order are used to represent the circular boundary better.
When the boundary of the island is modeled with 40 line-sinks of order 2, rather than
20, the relative difference at the center of the island decreases to 0.013% for M0 and
0.05% for M1 . When using 40 line-sinks of order 8, the relative difference decreases
further to 0.007% for M0 and 0.014% for M1
The contour plots in Fig. 6 are the zeroth and first moments of the true impulse
response function. The coefficients A and a of the approximate impulse response
function are computed with (18). The value of A is identical to the value of M0 (Fig.
6). Computation of a is more difficult, as it is equal to the ratio of M0 and M1 .
On the boundary of the island, both M0 and M1 are equal to zero, while their ratio
a = M0 /M1 is finite. In the exact solution of section 4, the limit of this ratio could
be worked out mathematically (28), but this is not possible for the general case in a
numerical model. Hence, it is not possible to compute a right next to the boundary
of the island, but both M0 and M1 can be computed, of course. In practice, this does
17
1
2
3
not represent a limitation. After all, streams have a width while the line-sinks have
a zero width, and furthermore, there is no practical interest to forecast groundwater
dynamics right next to a stream or ditch that controls the water table.
13
To be able to compute a accurately in the vicinity of a head-specified boundary such
as a stream or ditch, M0 and M1 need to be computed accurately, and thus the
boundary conditions of M0 and M1 need to be met accurately. The contour plots of
M0 and M1 in Fig. 6 are not accurate enough to compute a near the boundary of the
island. This is illustrated in the left part of Fig. 7 (20 line-sinks of order 2), where
a is inaccurate near the boundary of the island. The accuracy may be improved by
increasing the order of the line-sinks, such that the boundary condition of M0 and M1
is met more accurately. A solution with 20 line-sinks of order 8 gives accurate results
for a (Fig. 7, right side). It is noted that a varies over a small range only: from 0.0133
at the center to 0.02 at the boundary.
14
7
4
5
6
7
8
9
10
11
12
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Practical Example
The objective of this final example is to demonstrate that the new analytic elements
may be applied to practical problem with real geometries of surface water features.
The method is applied to the agricultural area near the town of ‘de Zilk’ in The
Netherlands, which contains an irregular pattern of ditches. The surficial aquifer consists of dune sand to a depth of about 10 m, which is ideally suited for the production
of flower bulbs such as tulips and hyacinths. These bulbs are extremely sensitive to
the depth of the groundwater table, and thus the groundwater table is controlled by
a maze of ditches; the agricultural plots are shown in Fig. 8, where the ditches (black
lines) are clearly visible. The ditches are modeled with line-sinks. Color images of the
parameters A and a are shown in Fig. 8. Color coding of A is truncated at 24 for
visualization purposes; values larger than 24 occur on the edge of the model, which
is outside the area of interest, and is colored black.
As an example, the values of A and a are computed at points 1, 2, and 3 (Fig. 8).
At point 1: A = 23.8 and a = 0.176; at point 2: A = 3.64 and a = 1.03; at point 3:
A = 10.9 and a = 0.233. Parameter A controls the area below the impulse response
function and thus the magnitude of the head variation. Parameter a controls the
18
8
length of the tail of the impulse response function. Between ditches that are farther
apart, the value of A is larger (larger magnitude of the head variation) and the value
of a is smaller (longer tail). Conversely, between ditches that are closer together, the
value of A is smaller and the value of a is larger. The head variation at points 1, 2,
and 3 (Fig. 8) is computed for a 40 day period of measured recharge, followed by a
40 day period of zero recharge (Fig. 9). As may be expected from the values of A and
a, the largest head variation and longest tail occurs at point 1, followed by point 3,
and point 2.
9
8
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Conclusions and Discussion
A new analytic method was presented for the simulation of groundwater dynamics due
to temporal variations of recharge in surficial aquifers. Head variations were computed
through convolution of a measured recharge series with an approximate but analytic
impulse response function. The approximate impulse response function (10) contains
two parameters: A(x, y) and a(x, y). Values of A and a were computed by setting
the moments M0 and M1 of the approximate impulse response function equal to the
corresponding moments of the exact impulse response function. The moments were
modeled with analytic elements; new analytic element equations were developed for
wells and line-sinks to model M1 . It was shown that the proposed approach produces
accurate results through comparison with the exact solution for a circular island.
Application to an agricultural area with a number of ditches illustrated the two main
advantages of the proposed method: (1) head variations can be computed at any point
and any time (except for in the close vicinity of the ditch), even for a wildly varying
recharge series, and (2) a solution is obtained without the need for the specification of
a computational grid, allowing for the accurate (and easy) placement of ditches and
other hydrologic boundaries in the model. The proposed methodology constitutes a
new transient analytic element formulation.
A number of approximations were made in the presentation of the approach. Most
of the approximations were made to limit the size of the paper and don’t represent
limitations to the approach. Variations in the saturated thickness due to unconfined
conditions may be taken into account by formulating the approach in terms of a discharge potential [16]. Line-doublet elements for M1 may be derived in the same fashion
19
1
2
3
4
5
6
7
8
9
10
11
as line-sinks, and may be applied to model polygonal areas with different transmissivity values. Line-doublets may be combined with line-sinks to model polygonal areas
with different recharge called area-sinks [16]. Analytic elements to model M1 may be
derived for flow in multi-aquifer systems following the approach of [1]. In the presented examples, recharge was calculated as rainfall minus evapotranspiration, which
is accurate for fairly flat and permeable land surfaces with shallow unsaturated zones.
In cases of sloping topography and variable land use, recharge may be corrected using
a standard method, for example the curve number method (e.g., [15]). The unsaturated zone may delay the arrival of recharge at the groundwater table; this effect
may be included through a delay factor, as was done by [19]. Another effect of the
unsaturated zone is dispersion of the recharge, which was studied by [11].
26
At positions in the aquifer where the groundwater reacts very quickly, the proposed
approximate impulse response function may not be accurate, and a different impulse
response function may need to be used. A promising alternative is the Pearson Type
III function, as used successfully for time series analysis of groundwater levels by [21].
The Pearson Type III function contains three parameters which requires the modeling
of an additional moment of the impulse response function. The presented approach
may be extended to other stresses such as fluctuating surface water levels or variable
discharges of pumping stations, although this may also require a different approximate
impulse response function. The analytic elements presented in this paper are used to
model M0 and M1 , which may be used to estimate the parameters of any impulse
response function, not just the function used in this paper. The second moment M2
may be modeled using a similar approach. Techniques for the inclusion of non-linear
effects have been developed for the time series analysis program Menyanthes [22],
which is based on the method presented in [21]. Incorporation of non-linear effects in
the current approach forms part of future research.
27
Acknowledgements
12
13
14
15
16
17
18
19
20
21
22
23
24
25
28
29
30
31
Mark Bakker is on sabbatical from the Department of Biological and Agricultural
Engineering of the University of Georgia, Athens, GA. Sabbatical funding was obtained from the TU Delft Grants program. The authors thank Ed Veling and Theo
Olsthoorn for their suggestions and support. Development of the model discussed in
20
1
the final example was funded by the Amsterdam Water Supply.
2
References
3
4
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6
7
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9
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16
17
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25
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28
[1] Bakker M, Strack ODL. Analytic elements for multiaquifer flow. Journal of Hydrology
2003;271(1-4):119–129.
[2] Bakker M. Transient analytic elements for periodic Dupuit-Forchheimer flow. Advances
in Water Resoures 2004;27:3-12.
[3] Bakker M, Nieber JL. Analytic element modeling of cylindrical drains and cylindrical
inhomogeneities in steady two-dimensional unsaturated flow. Vadose Zone Journal
2004;3(3):1038-1049.
[4] Bear J. Dynamics of fluids in porous media. Dover, New York. Bear; 1972.
[5] Bierkens MFP, Knotters M, Hoogland T. Space-time modeling of water table depth
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[7] Furman A, Neuman SP. Laplace-transform analytic element solution of transient flow
in porous media. Advances in Water Resoures 2003;26:1229-1237.
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CA; 1995.
[9] Janković I, Barnes R. High-order line elements in modeling two-dimensional grounwater
flow. Journal of Hydrology 1999; 226(3–4): 211–223.
[10] Knotters M, Bierkens MFP. Physical basis of time series for water table depths. Water
Resoures Research 2000;36(1):181-188.
[11] Kruijthof AJ. The impulse-response function of the groundwater table; separation of
the influence of the unsaturated and saturated zones. (In Dutch). M.Sc. Thesis, Delft
University of Technology, The Netherlands; 2001.
[12] Lankester J, Maas C. Research on characterizing site conditions for vegetation on the
basis of impulse response (In Dutch). Stromingen 1996;2:5–17.
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[13] Li W, Nowak W, Cirpka OA. Geostatistical inverse modeling of transient
pumping tests using temporal moments of drawdown, Water Resoures Research
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[14] Luo J, Cirpka OA, Kitanidis PK. Temporal-moment matching for truncated
breakthrough curves for step or step-pulse injection. Advances in Water Resources 2006.
In Press.
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[15] McCuen RH. Hydrologic analysis and design, 3rd Edition, Prentice Hall; 2004.
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[16] Strack ODL. Groundwater mechanics. Prentice Hall, Englewood Cliffs, NJ; 1989.
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[17] Strack ODL. Theory and applications of the Analytic Element Method. Rev. Geophys.
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3517–3533.
[19] Van de Vliet R, Boekelman R. Areal coverage of the impulse response with the aid of
time series analysis and the method of moments (In Dutch). Stromingen 1998;4(1):45–
54.
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Orlando, FL; 1997.
22
1
2
3
4
5
Appendix
First, a mathematical expression is presented for ϕp , a line-sink with strength ∆p that
fulfills Laplace’s equation (the subscript n is dropped for convenience). A line-sink
is obtained through integration of a point-sink. The zeroth moment for a point-sink
(36) is written as the real part of a complex function
M0 =
6
7
8
1
ℜ ln(z − δ)
2πT
(44)
where z = x + iy is the complex coordinate, and δ = xw + iyw is the location of the
well. The function ϕp is written as the real part of the complex function ωp
ϕp = ℜωp
(45)
1
L Z p
∆ ln(Z − ∆)d∆
ωp =
4πT
(46)
where ωp is given in [16] as
−1
9
where L is the length of the line-sink, and Z and ∆ are local complex coordinates
Z=
10
11
z − 21 (z1 + z2 )
1
(z − z1 )
2 2
∆=
δ − 21 (z1 + z2 )
1
(z − z1 )
2 2
(47)
where z1 and z2 are the end points of the line-sink. Following [16], integration by
parts of (46) gives
1


Z
L
∆p+1
ln(Z − 1) − (−1)p+1 ln(Z + 1) +
ωp =
d∆
4πT (p + 1)
Z −∆
(48)
−1
12
The remaining integral represents a line-dipole and is given by [16], Eq. 25.52, as
23
Z1
−1
1
2
3
[1+p/2]
X Z p−2n+2
∆p+1
Z −1
d∆ = −Z p+1 ln
−2
Z −∆
Z +1
2n − 1
n=1
Second, a mathematical expression is presented for Φp , the first moment of a line-sink
with strength ∆p . The first moment of a well (43) is written as the real part of a
complex function
M1 = −
4
5
6
(49)
n
o
S
ℜ
(z
−
δ)(z̄
−
δ̄)[ln(z
−
δ)
−
1]
8πT 2
(50)
where it is used that r2 = (z − δ)(z̄ − δ̄). Φp is written as the real part of the complex
function Ωp
Φp = ℜΩp
(51)
1
S Z ∆p L 3
(Z − ∆)(Z̄ − ∆)[ln(Z − ∆) − 1]d∆
Ωp = −
8πT 2
8
(52)
where
−1
7
8
¯ Rearrangement
where it is used that ∆ is a point along the real axis and thus ∆ = ∆.
gives
1
SL2 L Z p+2
[∆
− (Z + Z̄)∆p+1 + Z Z̄∆p ][ln(Z − ∆) − 1]d∆
Ωp = −
16T 4πT
(53)
−1
9
The integral may be written as
SL2
Ωp = −
[Λp+2 − (Z + Z̄)Λp+1 + Z Z̄Λp ]
16T
10
where Λp may be expressed in terms of the integral ωp (48) and a new integral
24
(54)
1
L Z p
∆ d∆
Λp = ωp −
4πT
(55)
Z1
(56)
−1
1
where
∆p d∆ =
1 − (−1)p+1
p+1
−1
25
1
.
0
0.8
0.2
2
4Tφ/(γR )
0.4
0
.
5
0.1
2
Tt/(SR )=0.05
0
.
0
0
.
0
0
.
5
1
.
0
r/R
r
=
0
r
=
R
r
=
R
r
=
3
/
4
2
or 4Tφ/(γR )
Fig. 1. Dimensionless head 4T φ/(γR2 ) vs. radial distance r/R at five different times; approximate (solid) and exact (dashed) solutions
1
.
0
.
5
0
.
0
/
2
R
/
4
2
or 4Tφ/(γR )
Sφ
0
1
.
0
.
5
0
.
0
Sφ
0
0
.
0
0
.
4
0
.
8
2
0
.
0
0
.
4
0
.
8
2
Tt/(SR )
Tt/(SR )
Fig. 2. Impulse response Sφ (descending lines) and dimensionless head 4T φ/(γR2 ) (ascending lines) for the approximate (solid) and exact (dashed) solutions at four different positions
(r = 0, R/4, R/2, 3R/4)
26
)
4
d
/
m
c
(
2
e
g
r
h
a
c
0
e
r
2
)
A
p
p
r
o
x
i
m
a
t
e
m
E
x
a
c
t
(
n
1
i
o
t
r
a
v
a
0
d
e
a
h
1
1
9
9
0
1
9
9
2
1
9
9
4
1
t
i
m
9
9
6
1
9
9
8
e
Fig. 3. Daily recharge (top) and head fluctuation φ at the center of the island (bottom)
27
4
)
d
/
m
c
(
2
e
g
r
h
a
c
e
r
0
2
A
p
E
x
E
p
r
a
n
c
d
o
x
i
m
a
t
e
t
o
f
m
o
n
t
h
)
m
(
1
n
i
o
t
r
a
v
a
0
d
e
a
h
1
0
1
0
2
0
3
0
4
0
5
m
0
o
6
n
0
t
h
7
o
0
f
1
8
9
0
9
9
1
0
1
1
1
2
4
Fig. 4. Head variation at center of island in 1994. Grey dots represent head variation at the
end of each month
Fig. 5. Contour plot of particular solution for M0 on the island (island is grey); radius of
the island is R = 1000 m.
28
A
E
E
X
A
M 0 or A
M
C
A
T
E
E
X
A
M1
M
C
T
Fig. 6. Contour plots of M0 and M1 . Upper half of each plot is analytic element solution,
lower half is exact solution. Contour interval is 50 for M0 and 3750 for M1 .
A
E
a
M
o
E
X
A
C
r
d
e
r
A
:
E
a
M
2
o
T
E
X
A
C
r
d
e
r
:
8
T
Fig. 7. Contour plot of parameter a. Upper half of each plot is analytic element solution,
lower half is exact solution. Boundary modeled with 20 line-sinks of order 2 (left) and order
8 (right). Contour levels from 0.013 to 0.02 with interval 0.007.
29
A
a
2
3
1
0
0
5
10
15
20
0.5
1
100 m
1.5
Fig. 8. Color image of parameters A (left) and a (right); A = 0 along the ditches; no model
results in bottom right hand corner of plots (black); spatial scale is shown in right plot
1
2
)
d
/
m
8
m
(
e
4
g
r
h
a
c
0
e
r
2
8
1
2
3
6
)
m
c
(
n
4
i
o
t
i
a
r
2
a
d
v
e
a
0
H
2
0
2
0
0
4
t
i
m
e
(
6
d
a
y
s
0
8
0
)
Fig. 9. Recharge (top) and head variation (bottom) at three points shown in Fig. 8
30