Introduction
Groundwater level or hydraulic head (h) is the main driving force for water
flow and advective contaminant transport in aquifers and thus the most
important variable studied in groundwater hydrology and its applications.
Knowledge about h is critical in dealing with groundwater-related
environmental problems, such as over-pumping, subsidence, sea water
intrusion, and contamination. One often finds that data about
groundwater level are limited or unavailable in a hydrogeological
investigation. In such cases the groundwater level distribution and its
temporal variation are usually obtained with an analytical or numerical
solution for a groundwater flow model.
It is obvious that errors always exist in the groundwater levels calculated
or simulated with analytical or numerical solutions. The main sources of
errors include the simplification or approximation in a conceptual model and
uncertainties in the model parameters. Problems in conceptualization or
model structure have been dealt with by many researchers (Neuman, 2003; Rojas
et al., 2008, 2010; Ye et al., 2008; Refsgaard et al., 2007; Zeng et al., 2013).
Uncertainties in the model parameters (e.g., hydraulic
conductivity, recharge rate, evapotranspiration, and river conductance) have
been investigated, based on generalized likelihood uncertainty estimation and
Bayesian methods (Nowak et al., 2010; Neuman et al., 2012; Rojas
et al., 2008, 2010). The uncertainty in groundwater level has
been one of the main research topics in stochastic subsurface hydrology for
more than 3 decades. Most of these studies were focused on the spatial
variability of groundwater level due to aquifers' heterogeneity (Dagan,
1989; Gelhar, 1993; Zhang, 2002). Little attention has been given to the
uncertainties in groundwater level due to temporal variations in
hydrological processes, e.g., recharge, evapotranspiration, discharge to a
river, and river stage (Bloomfield and Little, 2010; Zhang and Schilling,
2004; Schilling and Zhang, 2012; Liang and Zhang, 2013a; Zhu et al., 2012).
Uncertainties in groundwater level fluctuations have been studied by Zhang
and Li (2005, 2006) and most recently by Liang and Zhang (2013a).
Based on a linear reservoir model with a white noise or
temporally correlated recharge process, Zhang and Li (2005, 2006) derived
the variance and covariance of h(t) by considering only a random source or
sink process, assuming deterministic initial and boundary conditions. Liang
and Zhang (2013a) extended the studies of Zhang and Li (2005,
2006) and carried out non-stationary spectral analysis and Monte Carlo
simulations using a linearized Boussinesq equation, and investigated the
temporospatial variations in groundwater level. However, the only random
process considered by Liang and Zhang (2013a) is the
source/sink. Temporal scaling of groundwater levels, discovered first by
Zhang and Schilling (2004), was verified in several
studies (Zhang and Li, 2005, 2006; Bloomfield and Little, 2010; Zhang and
Yang, 2010; Zhu et al., 2012; Schilling and Zhang, 2012). However, we do not
know the effect of random boundary conditions on temporal scaling of
groundwater levels.
In this study we extended the above-mentioned work by considering the
groundwater flow in a bounded aquifer, described by a linearized Boussinesq
equation, with a random source/sink as well as random initial and boundary
conditions, since the latter processes are known to give uncertainties. The
objectives of this study are (1) to derive analytical solutions for the
covariance, variance, and spectrum of groundwater level, and (2) to
investigate the individual and combined effects of these random processes on
uncertainties and scaling of h(x, t). In the following, we will first present the
formulation and analytical solutions, then discuss the results, and finally
draw some conclusions.
Formulation and solutions
Under the Dupuit assumption, the one-dimensional transient groundwater flow
in an unconfined aquifer near a river (Fig. 1) can be approximated with the
linearized Boussinesq equation (Bear, 1972) with the initial and
boundary conditions, i.e.,
T∂2h∂x2+W(t)=SY∂h∂th(x,t)|t=0=H0(x);T∂h∂x|x=0=Q(t);h(x,t)|x=L=H(t),
where T [L T-1] is the transmissivity, h [L] is the hydraulic head or
groundwater level above the bottom of the aquifer, which is assumed to be
horizontal, W(t) [L T-1] is the time-dependent source/sink term, representing areal
recharge or evapotranspiration, SY is the specific yield, H0(x) [L]
is the initial condition, Q(t) [L2 T-1] is the time-dependent
flux at the left boundary, H(t) [L] is the time-dependent water level at the
right boundary, L [L] is the distance from the left to the right boundary, x [L]
is the coordinate, and t [T] is time. In this study the initial head
H0(x) is taken to be a spatially random variable. The source/sink,
W(t), the flux to the left boundary, Q(t), and the head at the right boundary, H(t), are
all taken to be temporally random processes and spatially deterministic. The
parameters T and SY are taken to be constant.
A schematic of the unconfined aquifer studied, where
W(t) is the random time-dependent source/sink, H0(x) is the random initial
condition, Q(t) is the random time-dependent flux at the left boundary,
H(t) is the random time-dependent water level at the right boundary, L is
the distance from the left to the right boundary, and h(x, t) is the random
groundwater level in the aquifer.
The groundwater level, h(x, t), the three random processes, W(t), Q(t), and H(t), and the random
variable, H0(x), are expressed in terms of their respective ensemble means
plus small perturbations,
h(x,t)=〈h(x,t)〉+h′(xt)W(t)=〈W′(t)〉+W′(t);Q(t)=〈Q(t)〉+Q′(t)H(t)=〈H(t)〉+H′(t);H0(x)=〈H0(x)〉+H0′(x),
where 〈〉 stands for ensemble average and ′ for
perturbation. The initial condition H0(x) in Eq. (1) can be any function.
For the conceptualization of the groundwater flow presented in Fig. 1, the
steady-state condition can be reached in this aquifer after a rainfall or
during a wet season. Thus the steady-state solution to this model was often
adopted as the initial condition in previous research (Liang and Zhang, 2012,
2013a, b). Thus, in this study, we set initial condition H0(x)
to be the steady-state solution to the one-dimensional groundwater
flow equation, i.e., H0(x) = h0 + 0.5W0(L2 - x2)/T,
whereh0 [L] is the constant groundwater level at the right boundary and W0
[L T-1] is the spatially constant recharge rate (Liang and Zhang,
2012). Since h0 is taken to be constant, the source of the uncertainty
in the initial head H0(x) is due to random W0 only.
Thus, the mean and perturbation of H0(x) can be written
as 〈H0(x)〉 = h0 + 0.5〈W0(x)〉(L2 - x2)/T and
H0′(x) = 0.5W0′(L2 - x2)/T, respectively. By substituting
Eq. (2), 〈H0(x)〉, and H0′(x) into Eq. (1), one
obtains the mean flow equation with the mean initial and boundary conditions as
T∂2〈h〉∂x2+〈W〉=SY∂〈h〉∂t〈h(x,0)〉=h0+W02TL2-x2;∂〈h〉∂x|x=0=〈Q〉;〈h(l,t)〉=〈H(t)〉.
Subtracting Eq. (3) from Eq. (1) leads to the following perturbation equation
with the initial and boundary conditions
T∂2h′∂x2+W′=SY∂h′∂xh′(x,0)=W0′2TL2-x2;T∂h′∂x|x=0=Q′;h′(L,t)=H′(t).
The analytical solution to Eq. (4) can be derived with integral-transform
methods (Özisik, 1968) given by
h′=2L∑n=0∞e-βbn2tcosbnx(-1)nbn3TW0′+β∫0teβbn2ξ(-1)nTbnW′(ξ)-Q′(ξ)T+H′(ξ)(-1)nbndξ,
where β = T/SY, bn = (2n + 1)π/2L. Using
Eq. (5), the temporal covariance of the groundwater level fluctuations can be derived as
Chhx,t1;x,t2=Eh′x,t1h′x,t2=4L2∑m=0∞∑n=0∞e-β(bm2t1+bn2t2)cosbmxcosbnx(-1)m+nT2bm3bn3σW02+β2∫0t1∫0t2eβ(bm2ξ+bn2ρ)(-1)m+nT2bmbnCWW(ξ,ρ)+CQQ(ξ,ρ)T2+CHH(ξ,ρ)(-1)m+nbmbndξdρ
in which σW02 is the variance of W0, and CWW(ξ, ρ),
CQQ(ξ, ρ) and CHH(ξ, ρ) are the temporal auto-covariance of W(t),
Q(t), and H(t), respectively. We assume that W(t), Q(t), and H(t) are uncorrelated in order to
simplify our analyses. It is shown in Eq. (6) that the head covariance
depends on the variance of W0 and the covariances of W(t), Q(t), and H(t) and this
equation can be evaluated for any random W(t), Q(t), and H(t). We assume that these
processes are white noise, as employed in previous studies (Gelhar,
1993; Hantush and Marino, 1994; Liang and Zhang, 2013a). More realistic
randomness of these processes will be considered in future studies.
Following Gelhar (1993, p. 34), we express the spectra of W(t), Q(t), and H(t) as
SWW = σW2λW/π,
SQQ = σQ2λQ/π, and
SHH = σH2λH/π,
respectively, where σW2, σQ2, and
σH2 are the variances and λW, λQ, and
λH are the correlation time intervals of these three processes,
respectively. The corresponding covariances of W(t), Q(t) and H(t) are
CWW(ξ, ρ) = 2σW2λWδ(ξ - ρ),
CQQ(ξ, ρ) = 2σQ2λQδ(ξ - ρ),
and CHH(ξ, ρ) = 2σH2λHδ(ξ - ρ). Substituting these
covariances into Eq. (6) and taking integration, one obtains an analytical solution
of head covariance
Chh(x′,t′,τ′)=4βL2T2∑m=0∞∑n=0∞cosb′mx′cosbn′x′e-[(b′m2+b′n2)t′+(b′n2-b′m2)τ′2]L2(-1)m+nσW02βb′m3b′n3+2e-b′m2τ′-e-2b′m2t′b′m2+b′n2(-1)m+nσW2λWbm′bn′+σQ2λQL2+(-1)m+nbm′bn′T2σH2λHL4,
where τ′ = t2′ - t1′ and t′ = (t2′ + t1′)/2.
The analytical solution for the head variance can be obtain by setting τ′ = 0
σh2(x′,t′)=4βL2T2∑m=0∞∑n=0∞cosbm′x′cosbn′x′e-(b′m2+b′n2)t′L2(-1)m+nσW02βb′m3b′n3+21-e-2b′m2t′b′m2+b′n2(-1)m+nσW2λWbm′bn′+σQ2λQL2+(-1)m+nbm′bn′T2σH2λHL4,
where
t′=ttc;x′=xL;tc=L2β;bn′=(2n+1)π2
in which tc( = SYL2/(KM)) [1/T] is a characteristic timescale (Gelhar, 1993) where the
transmissivity (T) is replaced by the product of the hydraulic conductivity (K)
and the average saturated thickness (M) of the aquifer. The characteristic
timescale (tc) is an important parameter and its value for most shallow
aquifers is usually larger than 100 days, since the horizontal extent of a
shallow aquifer is usually much larger than its thickness. For instance, the
value of tc is 250 days for a sandy aquifer with L = 100 m, M = 10 m,
K = 1 m day-1, and SY = 0.25.
The spectral density of h(x, t) cannot be derived by ordinary Fourier transform, since
the head covariance and variance depend on time t′, and thus h(x, t) are temporally
non-stationary as shown in Eqs. (7) and (8). Priestley (1981) defined
the spectral density of non-stationary processes (Wigner spectrum) as the
Fourier transform of time-dependent auto-covariance with fixed reference
time t and derived time-dependent spectral density. In order to obtain the
spectrum of h(x, t), we applied Priestley's method and obtained the time-dependent
spectral density (Priestley, 1981; Zhang and Li, 2005; Liang and Zhang, 2013a),
i.e.,
Shh(x,t,ω)=12π∫-∞∞Chh(x,t,τ)e-iωτdτ=∑m=0∞∑n=0∞cosbmxcosbnx2tcbn2-bm2e-β(bm2+bn2)tβ2bn2-bm22/4+ω2(-1)m+nσW02πT2bm3bn3+8βbm2tcbn2+bm21β2bm4+ω2(-1)m+nSWWT2bmbn+SQQT2+(-1)m+nbmbnSHH,
where ω is angular frequency and ω = 2πf, f is frequency,
and i = -1. It is seen in Eq. (9) that the spectrum Shh is
dependent on not only frequency and locations but also time t. The
time-dependent term (i.e., first term) in Eq. (9) is caused by the random
initial condition and is proportional to e-β(bm2+bm2)t which decays quickly with t. We evaluated the
first term in the Eq. (9) by setting t = 0 and found that it is much smaller
than the second term in Eq. (9). We thus ignored the first term and
evaluated the spectrum using the approximation,
Shh(x′,ω)=∑m=0∞∑n=0∞8βb′m2cosbm′x′cosbn′x′tcb′n2+b′m2β2b′m2/L4+ω2(-1)m+nSWWL2T2bm′bn′+SQQT2+(-1)m+nbm′bn′SHHL2.
Results and discussion
Variance of groundwater levels
The general expression of the head variance in Eq. (8) depends on the
variances of the four random processes, σW02, σW2,
σQ2, and σH2. In the following, we will study their individual and combined effects on the head
variation and focus our attention only on the variance of h(x, t). The
dimensionless standard deviation of h(x, t), σh′, and the square root
of the dimensionless variance, (σ′h2), as a function of the
dimensionless time (t′), are evaluated and presented in the left column of
Fig. 2 at fixed dimensionless locations (x). The σh′ as a function
of x was evaluated and is presented in the right column of Fig. 2 at fixed t′.
The graphs on the left column show the standard deviation
(σh′) of groundwater level (h(x, t)) versus the dimensionless time
(t′) at the dimensionless locations x′ = 0.0, 0.2, 0.4, 0.6, and 0.8. The
graphs on the right column show σh′ versus x′ for the
different t′: panels (b) and (d) show t′ = 0.0, 0.2, 0.4, 0.6, and 0.8, panels (f) and (h) show t′ = 0.01, 0.1, and 1.0, and panel (j) shows t′ = 0.01, 0.2, 0.4, 0.6, and 0.8.
Also, (a) and (b) are based on Eq. (11) where σW2 = σQ2 = σH2 = 0;
(c) and (d) are based on Eq. (12) where σW02 = σQ2 = σH2 = 0;
(e) and (f) are based on Eq. (13)
where σW02 = σW2 = σH2 = 0;
(g) and (h) are based on Eq. (14) where
σW02 = σW2 = σQ2 = 0;
and (i) and (j) are based on Eq. (15) where σW02 ≠ σW2 ≠ σQ2 ≠ σH2 ≠ 0.
We first evaluate the effect of the random initial condition due to the
random term, W0, by setting σW2 = σQ2 = σH2 = 0.
In this case, the dimensionless variance in Eq. (8) reduces to
σ′h2(x′,t′)=∑m=0∞∑n=0∞(-1)m+nb′m3b′n3cosbm′x′cosbn′x′e-(b′m2+b′n2)t′,
where σ′h2 = σh2T2/(4L4σW02).
The changes of the σh′ with x′ and t′ are
presented in Fig. 2a and b, respectively. It is shown in Fig. 2a that for a
fixed location the σh′ is at its maximum at t′ = 0 and it
decreases gradually with time to a negligible number at t′ = 1.0. This means
that the error in h(x, t), predicted by an analytical or numerical solution due to
the uncertain initial condition, is significant at an early point in time, especially
near a flux boundary. The duration for which the effect of the
uncertain initial condition is significant depends on the value of the
characteristic timescale (tc), since t′ = t/tc. In most aquifers this
duration may be many days. In the typical aquifer studied, the effect of
the uncertainty in the initial condition on h(x, t) is significant during the first
250 days (t′ = 1.0). This duration should be relatively short, however, in a more
permeable aquifer whose horizontal extent (L) is relatively smaller than its
thickness (M). It is seen in Fig. 2b that for a fixed time the
σh′ is the largest at the left flux boundary (x′ = 0.0) and becomes zero
at the right constant head boundary (x′ = 1.0), since the right boundary is
deterministic. This means that the error in h(x, t) predicted by an analytical or
numerical solution due to the uncertain initial condition is significant
almost everywhere in the aquifer: the further away from a constant head
boundary, the larger the error.
We then consider the uncertainty in the areal source/sink term (W) by setting
σW02 = σQ2 = σH2 = 0. In this
case the dimensionless variance in Eq. (8) reduces to
σ′h2(x′,t′)=2∑m=0∞∑n=0∞cosbm′x′cosbn′x′1-e-2b′m2t′(-1)m+nb′m2+b′n2bm′bn′,
where σ′h2 = σh2TSY/(4L2σw2λW).
The changes of the σ′h with x′ and t′ are presented in Fig. 2c and d, respectively. It is noticed in
Fig. 2c that at a fixed location, the σh′ is zero initially, gradually increases as time goes, and approaches a constant
limit at later time. This means that the error in h(x, t) due to an source/sink is
at its minimum at early time and increases with time to approach a constant
limit at later time: the closer to the left flux boundary, the larger the
limit. For a fixed time the σh′ decreases smoothly from the
left to the right boundary (Fig. 2d). The error in h(x, t) due to the uncertainty in
the source/sink is significant almost everywhere in the aquifer: the further
away from the constant head boundary, the larger the error, similar to the
previous case with the random initial condition (Fig. 2b).
Thirdly, we investigate the effect of the left random flux boundary by
setting σW02 = σW2 = σH2 = 0
in Eq. (8). In this case the dimensionless head variance is given by
σ′h2(x′,t′)=2∑m=0∞∑n=0∞cosbm′x′cos′bnx′1-e-2b′m2t′b′m2+b′n2,
where σ′h2 = σh2TSY/(4σQ2λQ). The changes of the
σh′ with x′ and t′ are presented in Fig. 2e and f, respectively. At any
location, the σh′ in Fig. 2e or the error in h(x, t), due to an
uncertain flux boundary, is at its minimum at an early point in time, and it increases
quickly with time to approach a constant limit: the closer to the left flux
boundary, the larger the limit. At any time, the σh′ in Fig. 2f or the error in the water head due to the
uncertain flux boundary is at its maximum at the left boundary but decreases
quickly away from the boundary to become insignificant for x′ > 0.8.
Fourthly, we investigated the effect of the random head boundary by setting
σW02 = σW2 = σQ2 = 0 in Eq. (8). The
dimensionless head variance in this case is given by
σ′h2(x′,t′)=2∑m=0∞∑n=0∞cosbm′x′cosbn′x′(-1)m+nbm′bn′1-e-2b′m2t′b′m2+b′n2,
where σ′h2 = σh2L2SY/(4TσH2λH). The changes of this
σh′ with x′ and t′ are presented in Fig. 2g and h,
respectively. It is seen in Fig. 2g that at any location, the σh′
or the error in h(x, t), due to the random head boundary, increases quickly with time
to approach a constant limit: the closer to the uncertain head boundary, the
larger the error. The spatial variation in σh′ can be clearly
observed in Fig. 2h for fixed t′. At any time, σh′ is at its
maximum at the right boundary (x′ = 1) where the head is uncertain, and it decreases
quickly away from the boundary. The error in h(x, t) due to the uncertain head
boundary is limited to a narrow zone near the boundary (x′ > 0.8) (Fig. 2h).
Finally, we consider the combined effects of the uncertainties from all four
sources, i.e., the initial condition, sources, and flux and head boundaries.
The head variance in Eq. (8) is written in the dimensionless form as
σ′h2(x′,t′)=∑m=0∞∑n=0∞cosbm′x′cosbn′x′e-(b′m2+b′n2)t′(-1)m+nσ′W02b′m3b′n3+21-e-2b′m2t′b′m2+b′n2(-1)m+nbm′bn′+σ′Q2+(-1)m+nbm′bn′σ′H2,
where
σ′h2=σh2TSY4L2σW2λW;σ′W02=L2SYσW02TσW2λW;σ′Q2=σQ2λQL2σW2λW;σ′H2=T2σH2λHL4σW2λW.
The dimensionless variances, σ′W02, σ′Q2, and
σ′H2, need to be specified in
order to evaluate the dimensionless σ′h2(x′, t′) in Eq. (15).
For the typical aquifer mentioned above with
L = 100 m, T = 10 m2 day-1 (or K = 1 m day-1
and M = 10 m), and SY = 0.25, we
set σW02/(σW2λW) = 10-1,
σQ2λQ/(σW2λW) = 103,
and σH2λH/(σW2λW) = 104,
and obtain σ′W02 = 25, σ′Q2 = 0.1 and σ′H2 = 0.01.
The changes of this σh′ with x′ and t′ are
presented in Fig. 2i and j, respectively. It is observed in Fig. 2i that at
any location, the σh′ is at its maximum due to the uncertainty in the initial condition, it gradually decreases
with time, and approaches a constant limit at a later time
(t′ > 0.6), which is due to the combined effects of the uncertain
source/sink and flux and head boundaries. This means that the error in the
water head at an early time is significant if the initial condition is uncertain and
reduces with time to reach a constant limit. The error in the water head at a later
time is determined by the uncertainties in the source/sink, and flux and head
boundaries. It can be observed in Fig. 2j that σh′ is
relatively larger near both boundaries. The values of σh′ at the two boundaries are equivalent
(∼ 1.3) at an early time, say t′ = 0.01 (the top curve in Fig. 2j) and it reduces
slowly away from the flux boundary, but quickly away from the head boundary.
As time progresses, the σh′ near the head
boundary stays more or less the same but reduces significantly in most parts
of the aquifer. This means that early on, the error in h(x, t) in most parts of
the aquifer is mainly caused by the initial condition and at a later time it
is due to the combined effects of the uncertain areal source/sink and flux
boundary. The effect of the uncertain head boundary on h(x, t) does not significantly change with
time, but it is limited to a narrow zone near the boundary.
The dimensionless power spectrum versus frequency (f) at the
dimensionless locations x′ = 0.0, 0.2, 0.4, 0.6, 0.8, and 0.9. The graphs on
the left column show tc = 40 days, the graphs on the middle column
show tc = 400 days, and the graphs on the right column show
tc = 4000 days. The graphs on the first row show the dimensionless
spectrum Shh/SQQ when SWW = 0, SHH = 0, and SQQ ≠ 0
in Eq. (10); the graphs on the second row show Shh/SHH
when SWW = 0, SQQ = 0, and SHH ≠ 0; the graphs on the third
row show Shh/SWW when SQQ = 0, SHH = 0, and
SWW ≠ 0; and the graphs on the bottom row show Shh/SWW when
SQQ ≠ 0, SHH ≠ 0, and SWW ≠ 0.
Spectrum of groundwater levels
We first evaluated Shh in Eq. (10) due to the effect of the white noise
flux boundary only by setting SQQ ≠ 0, SWW = 0, and SHH = 0. The
dimensionless spectrum Shh/SQQ as a function of the
frequency (f) was evaluated and presented in the log–log plot (Fig. 3a–c)
for three values of tc (40, 400, and 4000 days), since the value of
tc is 250 days for a sandy aquifer, as we mentioned above, and also at the six
locations (x′ = 0.0, 0.2, 0.4, 0.6, 0.8, and 0.9). The spectrum
Shh/SQQ in Fig. 3a is more or less horizontal (i.e., white noise) at low frequencies
and it decreases gradually as f increases, indicating that an aquifer acts as a low-bass filter
that filters signals at high frequencies and keeps signals at low frequencies.
The aquifer significantly dampened the fluctuations of the groundwater
level. The spectrum varies with the location x′: the smaller the value of x′ or
the closer to the left flux boundary (x′ = 0), the larger the spectrum
(Fig. 3a–c). All spectra in Fig. 3a are not a straight line in the log–log plot,
meaning that the temporal scaling of h(x,t) does not exist in the range of f = 10-3–100
when tc = 40 days. As tc increases to 400 and 4000 days,
however, the spectrum at x′ = 0 becomes a straight line (the top curve in
Fig. 3b and c) or has a power-law relation with f, i.e., Shh/SQQ ∝ 1/f, since its
slope is approximately 1. The fluctuations of h(0, t) are pink noise due to the
white noise fluctuations flux boundary when the characteristic timescale
(tc) is large which means that the aquifer is relatively less permeable
and/or has a much larger horizontal length than its thickness.
Secondly, the spectrum Shh/SHH due to the sole effect of the
random head boundary was evaluated by setting SHH ≠ 0, SWW = 0,
and SQQ = 0 in Eq. (10) for the same three values of tc and six locations and presented in
Fig. 3d–f as a function of f. It is shown that similar to Fig. 3a–c, the
spectrum decreases as f increases but different from Fig. 3a–c, the spectrum
is larger at x′ = 0.9 near the right boundary (the top curves in Fig. 3d–f)
than at x′ = 0.0 (the bottom curves). Furthermore, none of the spectra are a
straight line in the log–log plot, indicating that the temporal scaling of
groundwater level fluctuations does not exist in the case of the white noise
head boundary.
Thirdly, the spectrum Shh/SWW under the white noise recharge was evaluated by setting SWW ≠ 0,
SQQ = 0, and SHH = 0 in Eq. (10) for the same values of tc
and x′ and presented in Fig. 3g–i as a function of f. It is shown that when
tc = 40 days, the spectrum in Fig. 3g is horizontal at low frequencies and becomes a straight line at high
frequencies: the closer to the right head boundary, the later it approaches
a straight line (Fig. 3h). As tc increases to 400 and 4000 days, the
slope of the spectrum at all locations, except at x′ = 0.9, approaches a
straight line with a slope of 2 (Fig. 3h and i), indicating a temporal
scaling of h(x, t). The fluctuations of groundwater level reflect a Brownian motion,
i.e., S ∝ 1/f2, when tc ≥ 4000 days
or in a relatively less permeable and/or has a much larger horizontal length
than its thickness.
Finally, the head spectrum due to the combined effect of all three random
sources (the white noise recharge, and flux and head boundaries) was
evaluated, i.e., SWW ≠ 0, SQQ ≠ 0, and
SHH ≠ 0 in Eq. (10). The spectrum of Shh/SWW as a
function of f is presented in Fig. 3j–l for the same values of tc and
x′, where SQQ/SWW = 1000 and SHH/SWW = 1000,
which are the same as the values used in the previous section. It
is noticed that the general patterns of Shh/SWW in the combined
case are similar to the case of the random source/sink only (Fig. 3g–i),
except at x′ = 0.0 and 0.9 (the dashed and dotted curves in Fig. 3j,
respectively) due to the strong effects of the boundary conditions at these
two locations. At tc = 4000 days, the spectra at all locations except
x′ = 0.0 (Fig. 3l) are similar to those in Fig. 3i, indicating the dominating
effect of the random areal source/sink. The spectrum at x′ = 0 in this case is
also a straight line (the dashed curve in Fig. 3l) but with a different
slope due to the effect of the random flux boundary which is similar to the
top straight line in Fig. 3c. The above results provide a theoretical
explanation as to why temporal scaling exists in the observed groundwater level
fluctuations (Zhang and Schilling, 2004; Bloomfield and Little, 2010; Zhu
et al., 2012). We thus conclude that temporal scaling of h(x, t) may indeed exist in
real aquifers due to the strong effect of the areal source/sink.
Conclusions
In this study the effects of random source/sink, and initial and boundary
conditions on the uncertainty and temporal scaling of the groundwater level,
h(x, t) were investigated. Analytical solutions were derived for the variance,
covariance, and spectrum of h(x, t) in an unconfined aquifer, described by a linearized
Boussinesq equation with white noise source/sink, and initial and boundary
conditions. The standard deviations of h(x, t) for various cases were
evaluated. Based on the results, the following conclusions can be drawn.
The error in h(x, t), due to a random initial condition, is significant
at an early time, especially near a flux boundary. The duration for which the
effect is significant may be a few hundred days in most aquifers.
The error in h(x, t) due to a random areal source/sink is significant
in most parts of an aquifer: the closer to a flux boundary, the larger the error.
The errors in h(x, t) due to random flux and head boundaries are
significant near the boundaries: the closer to the boundaries, the larger the
errors. The random flux boundary may affect the head over a larger region than the random head boundary.
In the typical sandy aquifer studied (with the length of aquifer at the
direction of water flow L = 100 m, the average saturated thickness M = 10 m,
hydraulic conductivity K = 1 m day-1, and specific yield SY = 0.25) the error
in h(x, t) at an early point in time is mainly caused by an uncertain initial
condition, and the error reduces with time, reaching a constant error at a later time.
The constant error in h(x, t) is mainly due to the combined effects of
uncertain source/sink and boundaries.
The aquifer system behaves as a low-pass filter which filters the short-term
(high frequencies) fluctuations and keeps the long-term (low frequencies) fluctuations.
Temporal scaling of groundwater level fluctuations may indeed exist in
most parts of a low permeable aquifer whose horizontal length is much larger
than its thickness, caused by the temporal fluctuations of areal source/sink.
Finally, it is pointed out that the analyses carried out in this study are
under the assumption that the processes W(t), Q(t), and H(t) are uncorrelated white
noise. In reality, they may be correlated and spatially varied. We plan to
relax those constraints and study more realistic cases in the near future. It
is also noted that the analytical solutions for head variances derived in
this study provide a way to identify and quantify the uncertainty. The
spectrum relationship obtained among the head, recharge, and boundary
conditions can help one to improve spectrum analysis for a groundwater level
time series and remove the effects of the boundary conditions.