Introduction
Pumping tests are a widely used tool to identify horizontal hydraulic
conductivity, which is the parameter determining the groundwater flow
velocity. Analytical solutions of the radial flow equation are used in
practice to analyse measured drawdowns. In general, these solutions assume
a constant homogeneous hydraulic conductivity like Thiem's solution for
steady state :
hThiem(r)=-Qw2πDKhlnrR+h(R).
Thiem's solution (Eq. ) gives the hydraulic head
hThiem(r) depending on the radial distance r from the well for
homogeneous horizontal hydraulic conductivity Kh. It is valid in
a confined aquifer of thickness D with a fully penetrating well, and
a constant discharge Qw, h(R) is a known reference head at an
arbitrary distance R from the well.
In large-scale pumping tests the vertical extension of the aquifer is
negligible compared to horizontal aquifer extent. Thus, flow is assumed to be
horizontal and modelled as two-dimensional. Hydraulic conductivity is then
replaced by transmissivity, which is defined as the product of conductivity
and aquifer thickness T=KhD. In the following, transmissivity
will be used instead of horizontal hydraulic conductivity, since the focus of
the work will be on two-dimensional well flow.
Most natural aquifers exhibit geological heterogeneity in the sedimentary
composition, which evolved from the complex geomorphological processes
through which they were formed. In particular, transmissivity shows a strong
spatial variability. Values measured in the field vary over orders of
magnitude . Modelling transmissivity as a spatial random
function is generally applied to capture the effects of aquifer
heterogeneity. Transmissivity T(x,y) is modelled as a log-normally
distributed spatial random function: logT(x,y)=Y(x,y) is normally
distributed with a Gaussian probability density
function pdfY(x)=12πσ2exp-(x-μ)22σ2
in univariate form, with μ and σ2 being the mean and the variance
of Y, respectively. The correlation structure of transmissivity in space is
captured by a covariance model CovT(x+s1,y+s2),T(x,y)=exp2μ+σ2+CVY(s1,s2).
In the stochastic framework, the solution of the groundwater flow equation
with a log-normally distributed transmissivity is also a random spatial
function. Since the solution of the stochastic differential equation is out
of scope, the focus of investigation was on effective transmissivity values
for describing well flow. As a first approach, Thiem's solution
(Eq. ) was applied to pumping tests in heterogeneous media.
However, this requires a representative transmissivity value T for the
whole range of the depression cone , which does not exist.
Effective or equivalent descriptions of transmissivity in pumping tests have
been investigated, e.g. by , , ,
, , and many more. For a detailed review see .
In contrast to mean uniform flow, the pumping test drawdowns in heterogeneous
media cannot be described by a single constant effective value of
transmissivity . Different transmissivities characterize
the behaviour near and far from the well: the representative value close to
the well is the harmonic mean of the log-normal transmissivity
TH=expμ-12σ2=TGexp-12σ2.
With increasing distance from the well, the drawdown behaviour is
characterized by the effective transmissivity for uniform flow, which is the
geometric mean TG=expμ for flow in two-dimensional isotropic porous media.
It seems obvious that a representative description of transmissivity for
well flow needs to be a radially dependent function, which interpolates
between the harmonic and the geometric mean. The equivalent transmissivity
Teq is a well-established approach of a radially dependent
description, visualized in Fig. a.
Teq(r)=-Qw2πh(r)-h(rw)lnrrw was derived
from Thiem's solution (Eq. ) .
In this sense, the equivalent transmissivity Teq is defined as
the value for a homogeneous medium, which reproduces locally the same total
outflow as observed in the heterogeneous domain of radius r.
Teq is strongly impacted by the reference point rw
and the corresponding head h(rw), which is generally chosen to
be the drawdown at the well h(rw). Therefore, the equivalent
conductivity stays close to the harmonic mean TH, which is
representative for the drawdown behaviour at the well
(Fig. a). It takes more than 20 correlation lengths for
Teq to reach the far-field representative value of
TG.
Comparison of equivalent and radial coarse graining approach:
(a) radially dependent transmissivities interpolating between
harmonic mean TH and geometric mean TG:
TRCG(r) from Eq. () and Teq(r) calculated
based on Thiem's formula Eq. () with h(r)=〈h(r)〉‾, which is the ensemble mean for ensemble
A (Table ); (b) hydraulic head drawdowns after
pumping with hefw(r) from Eq. () as a solution of the
well flow equation using TRCG(r), heq(r) as a solution
of the well flow equation using Teq(r), Thiem's solution with
constant values TG and TH, and mean
ensemble drawdown 〈h(r)〉‾.
It is important to mention that Teq is not constructed to
reproduce the drawdown which was used for calculating Teq.
Strictly speaking, replacing the heterogeneous transmissivity field with the
equivalent transmissivity in a single forward model does not give the
drawdown with which Teq was constructed as visualized in
Fig. b. Instead heq(r) stays close to Thiem's
solution, with TH as a constant transmissivity value.
introduced a novel approach to describe well flow
effectively. They derived a radially adapted transmissivity TRCG(r)
by applying the upscaling technique coarse graining to well flow.
TRCG(r) depends not only on the radial distance r but also
on the statistical parameters of aquifer heterogeneity TG,
σ2, and ℓ. TRCG(r) captures the transition from
near-well to far-field representative transmissivities, based on the radial
distance to the well and the parameters of aquifer heterogeneity, as
visualized in Fig. a.
In this study, an analytical solution for the hydraulic head
hefw(r) is presented which is based on the radial coarse
graining transmissivity TRCG(r) as an extension to the work of
. Similar work has been done by for pumping
tests in three-dimensional porous media. The “effective well flow solution”
hefw(r) describes the mean depression cone of a pumping test in
two-dimensional heterogeneous media effectively. It can be interpreted as an
extension of Thiem's formula (Eq. ) to log-normally distributed
heterogeneous media. It accounts for the statistical parameters
TG, σ2, and ℓ and thus allows inversely estimating
them from measured drawdown data. In contrast to existing head solutions for
well flow, hefw(r) is not limited to low variances but is
applicable to highly heterogeneous media with variances σ2≫1. More
explicitly, the effective well flow solution is not a solution derived
from an effective equation but is the solution of the deterministic
groundwater flow equation under well flow conditions in combination with an
effective transmissivity. As an effective transmissivity description the radial
coarse graining transmissivity is used because it mimics the filtering
process taking place during pumping.
In a similar line presented a graphical approach to estimate
the statistical parameters of random transmissivity on the basis of
steady-state head data. The authors constructed a mathematical description for the
apparent transmissivity Ta(r) as a function of the radial distance
to the well r from theoretical findings of near-well and far-field
representative transmissivity and a cubic polynomial interpolation in
between. The approach makes use of the ensemble average drawdown as well as
of its variance, combining uncertainty within a unique methodology. From
Ta(r) the authors constructed type curves for the hydraulic head,
depending on the variance σ2 and the correlation length ℓ.
further gave a multi-point strategy to analyse virtually
measured drawdown data by type curve matching including parameter estimation.
The solution was also applied in an actual field setting by .
The radial coarse graining approach is similar to that of in
the idea of deriving a solution for the head drawdown for well flow depending
on the statistics of the random transmissivity using an effective radially
dependent transmissivity. Major differences are (i) the radial coarse
graining transmissivity TRCG(r) is derived from upscaling with
physically motivated approximations, whereas the solution of
is based on the analytical solution of in combination with
numerical Monte Carlo simulations; (ii) the functional form of
TRCG(r) is different from the expression for Ta of
; (iii) the effective well flow solution hefw(r)
is provided as a closed-form mathematical expression instead of type curves;
and (iv) inverse parameter estimation can be done by minimizing the difference
between the measured drawdown data and hefw(r) instead of type
curve matching. The effective well flow method will be tested in a similar
multi-point sampling strategy to analyse measured drawdown data of individual
heterogeneous transmissivity fields as done by as well as
others .
The work is organized as follows: Sect. is dedicated to
the method of radial coarse graining and the derivation of the effective well
flow head solution. The concept of radial coarse graining is explained in
detail. Furthermore distinction is made between the effective well flow
solution for an ensemble mean and single realizations of heterogeneous
transmissivity fields. Section contains the application of
the effective well flow solution to simulated pumping tests. It is shown
that hefw(r) reproduces the drawdown in heterogeneous media and
can be used to inversely estimate the statistical parameters of aquifer
heterogeneity for both ensemble mean and single realizations. In
Sect. , a sampling strategy is presented to infer the parameters
of aquifer heterogeneity of an individual transmissivity field from multiple
pumping tests at multiple locations, making use of hefw(r).
Concluding remarks are given in Sect. .
Radial coarse graining transmissivity and effective well flow head
Steady-state well flow with radially dependent transmissivity
The drawdown of a steady-state pumping test with a radially dependent
transmissivity T(r) is given as the solution of the differential equation
0=1rdhdr+d2hdr2+dT(r)drdhdr=T(r)1r+dlnT(r)drdhdr+d2hdr2.
The equation can be solved in dhdr by
separation of variables, resulting in dhdr=C11rT(r). The hydraulic head h(r) is then given as the
solution of the integral
h(r2)-h(r1)=C1∫r1r21rT(r)dr.
The integration constant C1 is determined by the boundary condition.
Supposing a constant flux boundary condition at the well gives
Qw=-2πrwT(rw)dhdr(rw)=-2πrwT(rw)C1rwT(rw) and, thus,
C1=-Qw2π.
Equation () is the general solution of the radial flow
Eq. () for radially dependent transmissivity, independent of the
functional form of T(r).
When comparing Eq. () with the definition of the equivalent
transmissivity, it becomes obvious that Teq is not constructed to
solve the equation. The combination of both formulas results in
∫r1r21rT(r)dr=1Teq(r)lnr1r2, which is only fulfilled when
T(r) is constant in r.
In a heterogeneous transmissivity field, radially symmetric flow is not to be
expected. However, angular fluctuations level out for ensemble average
drawdowns. Thus, the assumption of a radially dependent effective
transmissivity description T(r) – independent of the angular coordinate –
is well established and reasonable in the context of ensemble analysis. In
this line, the radial coarse graining transmissivity and thus the effective
well flow solution are considered to be purely radially dependent referring to
ensemble averages.
Concept of radial coarse graining
A radially dependent transmissivity for log-normally distributed media with
Gaussian correlation structure was derived by , denoted
as TRCG(r). It is based on the upscaling approach radial coarse
graining, which follows the basic idea of a spatial filtering of the flow
equation appropriate to the non-uniform flow character of a pumping test. The
method of radial coarse graining is an upscaling method based on
mathematically solid filtering of the flow equation and physically reasonable
assumptions and approximations. In its current state the procedure is limited
to a multivariate Gaussian random field as a model for the heterogeneity of
the transmissivity field.
The approach was further developed for three-dimensional well flow by
, introducing an effective well flow solution for the hydraulic
head. Similarly, the concept of radial coarse graining for two-dimensional
well flow will be expanded in the following. The process can be best
explained within five major steps:
coarse graining for uniform flow,
transfer of coarse graining to radial flow conditions,
overcoming non-locality of head equation for non-uniform flow,
effective radial coarse graining transmissivity,
derivation of effective well flow head.
The first three steps will be discussed shortly in the following. Steps 4 and
5 will be explained in detail in Sects. and .
The coarse graining approach for uniform flow (step 1) was introduced by
, including derivation, mathematical proof, and numerical
simulations. The author started at a spatially variable transmissivity field
T(x) and derived a filtered version TλCG(x),
where fluctuations smaller than a cut-off length λ are filtered out.
The resulting upscaled coarse graining transmissivity field
TλCG(x) represents a log-normally distributed field
with a smaller variance 〈σ2〉λ but larger
correlation length 〈ℓ〉λ. showed
that the statistical parameters relate to the parameter of the unfiltered
field by 〈σ2〉λ≡σ2ℓ2ℓ2+λ2/4 and
〈ℓ〉λ≡ℓ2+λ2/41/2.
The concept of coarse graining can similarly be applied to non-uniform flow
(step 2). The critical point when extrapolating the results of
to radial flow is the Fourier back transformation of the
filtered head equation after localization. For uniform flow, this can be done
due to the reasonable assumption of constant head gradient. For non-uniform
flow, this assumption is not valid, and thus localization is not possible
straightforward.
A heuristic approach is taken to overcome the limitation of non-locality for
well flow (step 3). Conditions of a quasi-constant head gradient are
constructed by adapting the size of the volume elements over which flow takes
place. The change in the volume size refers to the filtering procedure and is
then realized by a radially dependent scaling parameter λ(r). The step
can be understood as a change from an equidistant Cartesian coordinate system
to a polar coordinate system. Technically speaking, the head gradient in well
flow is constant for volumes of size proportional to r, because it is
proportional to the reciprocal of the distance to the well: ∇h(r)=h(r)-h(r+Δr)Δr∝1r. Therefore, the
scaling parameter is chosen to be proportional to the radial distance
λ=2ζr. Then, the filter width increases with distance to the
well, ensuring that head gradients can be assumed constant over volume
elements of increasing size with distance to the well. Under this
adaption, localization can be performed and thus also the following steps of the
coarse graining procedure.
The result is an upscaled log-normally distributed field TrRCG(x) with an arithmetic mean TARCG(r) and a filtered
fluctuation term. The step was presented by . It is not
performed in a mathematically straightforward way but problem-adapted to well flow
conditions.
Radial coarse graining transmissivity
Spatial heterogeneity is still resolved in TrRCG(x),
albeit reduced to the amount relevant to the pumping test. A further step
of averaging is necessary to derive an effective transmissivity which mimics
the intrinsic filtering of the flow during pumping and thus reproduces the
drawdown behaviour. Two different aspects are of interest: (i) an
effective transmissivity for an ensemble and (ii) effective transmissivity
for an individual field. This step includes the transfer from the stochastic
picture of transmissivity to a deterministic description.
A result for an effective ensemble description is derived by averaging
TrRCG(x) appropriate to well flow conditions. The averaging
rule is determined by the boundary condition at the well, which is the
harmonic mean for two-dimensional well flow . Thus, the
effective mean transmissivity, denoted by TRCG(r), is calculated
via the theoretical description of the harmonic mean for log-normally
distributed fields, making use of the variance of the coarsened transmissivity
〈σ2〉r=σ21+ζ2r2/ℓ2:
TRCG(r)=TGexp-〈σ2〉r/2=TGexp-12σ2(1+ζ2r2/ℓ2),
where r is the distance to the well, TG is the geometric
mean, σ2 is the variance, and ℓ is the correlation length of the
log-transmissivity T(x). ζ is a factor of
proportionality, which was determined to be ζ=1.6, as discussed in
detail by .
TRCG(r) can be interpreted as an interpolating function between the
representative transmissivity at the
well TH=TGexp-12σ2 to
the far-field value TG depending on r, controlled by the
correlation length ℓ (Fig. 1a).
An effective description of well flow transmissivity for an individual field
is derived from Eq. (). The behaviour of individual fields is
different especially at the well due to a lack of ergodicity there. The local
transmissivity at the well Twell is not identical to the harmonic
mean TH as expected for the ensemble but rather refers to the specific
value of transmissivity at the well location. An adapted radial coarse
graining transmissivity accounts for local effects by replacing the harmonic
mean TH=TGexp-12σ2 by
Twell. In Eq. () this refers to substituting the
variance by -12σ2=lnTwell-lnTG,
and thus
TRCGlocal(r)=TGexplnTwell-lnTG1+ζ2r2/ℓ2=Twell11+ζ2r2/ℓ2TGζ2r2/ℓ21+ζ2r2/ℓ2.
TRCGlocal(r) interpolates between the specific
transmissivity at the well Twell and the far-field
value TG depending on the radial distance r and the correlation
length ℓ.
Effective well flow head
An effective drawdown solution is derived by solving the deterministic
groundwater flow equation in combination with an effective transmissivity
description, namely the radial coarse graining transmissivity. This needs to
be clearly distinguished from solving an effective well flow equation. In
line with the deterministic nature of the radial coarse graining
transmissivity TRCG(r), the associated effective well flow head
is deterministic as well.
Explicit results for the hydraulic head drawdown in a steady-state pumping test
are achieved by solving the integral in Eq. (), making use of
TRCG(r) (Eq. ). The result is the effective well flow
head hefw(r), given by
hefw(r)=-Qw4πTGexpσ22Γσ22-ζ2r2/ℓ21+ζ2r2/ℓ2-Γσ22-ζ2R2/ℓ21+ζ2R2/ℓ2+Qw4πTGΓσ2211+ζ2r2/ℓ2-Γσ2211+ζ2R2/ℓ2+hR,
where r is the radial distance from the well, Qw is the
pumping rate, TG is the geometric mean, σ2 is the
log-transmissivity variance, and ℓ is the correlation length.
Again, ζ is the factor of proportionality determined to be 1.6,
and R is an arbitrary distance from the well, where the hydraulic
head h(R)=hR is known. Γ(x)=∫-∞xexp(z)zdz is the exponential integral function. Details on the
derivation of hefw(r) can be found in Appendix A.
An approximate solution hefwapprox(r) is derived from
Eq. () by making use of an approximation of Γ(x). Details
are given in the Appendix.
hefwapprox(r)=-Qw2πTHlnrR-Qw4πTGeσ22-1⋅ln1+ζ2R2/ℓ21+ζ2r2/ℓ2+σ221(1+ζ2r2/ℓ2)-σ221(1+ζ2R2/ℓ2)+hR.
hefw(r) is constructed to describe the mean drawdown of a pumping
test in two-dimensional heterogeneous media effectively. The drawdown curve
of hefw(r) for a specific choice of parameters (ensemble A of
Table ) is given in Fig. b in comparison
to the equivalent drawdown heq(r), as the solution of the radial
flow equation using the equivalent transmissivity Teq, based on
the same statistical parameters.
Ensemble input parameters TG, σ2, and ℓ
and best-fitting inverse estimation results T^G, σ^2, and ℓ^ with 95% confidence intervals (in
brackets) for ensemble mean 〈h(r)〉‾ for all
generated ensembles.
TG
T^G (10-4m2s-1)
σ2
σ^2[-]
ℓ
ℓ^(m)
A
1.0
1.03 (±0.0011)
1.0
1.04 (±0.0022)
10
9.80 (±0.086)
B
1.0
1.08 (±0.0013)
1.0
1.19 (±0.0022)
20
21.6 (±0.127)
C
1.0
1.08 (±0.0021)
2.25
2.49 (±0.0038)
10
10.1 (±0.050)
D
1.0
1.19 (±0.0024)
2.25
2.67 (±0.0039)
20
22.2 (±0.077)
E
1.0
1.16 (±0.0046)
4.0
4.34 (±0.0078)
10
11.0 (±0.042)
F
1.0
1.31 (±0.0088)
4.0
4.27 (±0.0131)
20
22.2 (±0.120)
G
1.5
1.55 (±0.0012)
1.0
1.03 (±0.0016)
10
10.1 (±0.066)
The effective well flow solution can be adapted to analyse individual pumping
tests by using TRCGlocal(r) (Eq. )
instead of TRCG(r) (Eq. ). The local effective well
flow solution hefwlocal(r) is then given by
Eq. () with σ22 substituted by -lnTwellTG and TH substituted
by Twell.
The local effective well flow solution hefwlocal(r) can
be used to analyse drawdowns of single pumping tests in heterogeneous media
as encountered in practice. The solution is adapted to the lack of ergodicity
at the well, by using transformed parameters Twell,
TG, and ℓ. However, the randomness of hydraulic heads can
affect the parameter estimation, in particular of the correlation length,
which is related to drawdown fluctuations in the transition zone. There the
impact of heterogeneity is neither fully determined by the local transmissivity
at the well nor fully levelled out as in the far field. The
geometric mean TG and the correlation length ℓ for
a single realization should therefore be interpreted as local values, not
necessarily representing the mean values of the entire field but those of
the pumping well vicinity. Owing to the nature of the pumping test, the
drawdown signal does not sample the heterogeneity in transmissivity in
a symmetric way, but the shape of the drawdown is mainly determined by the
local heterogeneity close to the well.
Impact of parameters
The analytical form of hefw(r) allows analysing the impact of
the statistical parameters TG, σ2, and ℓ on the
drawdown. The drawdown behaviour for different choices of parameters can be
seen in Fig. , which is discussed in detail later on.
Simulated ensemble means 〈h(r)〉‾ (dots)
and
hefw(r) with best-fitting estimates (lines) for multiple
ensembles: A (blue), B (green), E (red), F (orange), G (purple). Parameter
values are listed in Table . Black line shows
hThiem(r) with
TG=10-4m2s-1.
Every parameter impacts on the drawdown in a different region. The geometric
mean TG as a representative value for mean uniform flow
determines the far-field behaviour. The variance σ2 determines the
drawdown at the well due to the dependence of the near-well asymptotic value
TH=TGexp-12σ2. The
larger the variance, the larger are the differences between TG
and TH and the steeper is the drawdown at the well, whereas
the correlation length ℓ determines the transition from near to
far-field behaviour. Therefore, fluctuation in the hydraulic head in the
transition zone can affect estimation errors in the correlation length.
The asymptotic behaviour of hefw(r) can easily be analysed using
the approximate functional description in Eq. (): for distances
close to the well, i.e. r≪ℓ, hefw(r) converges to Thiem's
solution with TH as a constant transmissivity value. All terms,
except the first one in Eq. (), tend to zero or become
constant. Thus, they are negligible compared to the logarithmic first term for
very small r,
hefwapprox(r≪ℓ)≈-Qw2πTHlnrR-Qw4πTGeσ22-1⋅ln1+ζ2R2/ℓ2+σ22+hR≈-Qw2πTHlnrR+hR.
For large distances from the well, i.e. r≫ℓ, the solution converges to
Thiem's solution with TG as a constant transmissivity value. The
third and fourth term in Eq. () tend to zero and cancel
out. The ones in the second term can be neglected; thus
hefwapprox(r≫ℓ)≈-Qw2πTHlnrR-Qw4πTH-Qw4πTG⋅lnζ2R2/ℓ2ζ2r2/ℓ2+hR≈-Qw2πTGlnrR+hR.
The larger the correlation length ℓ, the longer the transition of
the drawdown takes from near-well to far-field behaviour. The influence of ℓ on
hefw(r) vanishes quickly with increasing distance to the well.
The drawdown reaches the far-field behaviour after approximately two
correlation lengths, hefw(r>2ℓ)=hThiem(r>2ℓ), with
TG as a constant transmissivity value (Fig. b).
These findings are in line with the results of .
Robust estimation of statistical parameters
Numerical pumping tests
Numerical pumping tests in heterogeneous porous media were generated as
artificial measurements. They were used to test the capability of
hefw(r) in reproducing the mean drawdown and in estimating the
underlying parameters of heterogeneity. Pumping tests were simulated using
the finite-element software OpenGeoSys. The software was successfully tested
against a wide range of benchmarks . Results of
a steady-state simulation with homogeneous transmissivity were in perfect agreement
with Thiem's analytical solution Eq. ().
The numerical grid was constructed as a square of 256×256 elements
with a uniform grid cell size of 1m except for cells in the
vicinity of the pumping well. The mesh was refined in the range of
4m around the well, which ensures a fine resolution of the steep
head gradients at the well. The well in the centre of the mesh was included
as a hollow cylinder with radius rw=0.01m. The
constant pumping rate of Qw=-10-4m3s-1
was distributed equally to all elements at the well. At the radial distance
R=128m a constant head of h(R)=0m was applied, giving
a circular outer head boundary condition.
Log-normally distributed, Gaussian-correlated transmissivity fields were
generated using a statistical field generator based on the randomization
method . Multiple ensembles with different statistical
parameter values were generated, including high variances up to σ2=4
(Table ). Ensemble A with
TG=10-4m2s-1, σ2=1, and
ℓ=10m served as a reference ensemble for specific cases. Every
ensemble consists of N=5000 realizations, which was tested as sufficiently
large to ensure ensemble convergence.
For the ensemble analysis, pumping test simulations were post-processed by performing an angular and an
ensemble average. For every realization i, the simulated drawdown 〈hi(r,ϕ)〉 at the radial and angular location (r,ϕ) in polar
coordinates was averaged over the four axial directions: 〈hi(r)〉=∑ϕj〈hi(r,ϕj)〉. The ensemble
mean was the sum over the angular mean of all individual realizations:
〈h(r)〉‾=∑i=1N〈hi(r)〉.
Non-linear regression was used to find the best-fitting values for the
statistical parameters, denoted by T^G, σ^2,
and ℓ^. The best-fitting estimates were determined by minimizing the
mean square error of the difference between the analytical solution
hefw(r) and the measured drawdown samples h(r):
minTG,σ2,ℓ∑rh(r)-hefw(r)2, making use of the Levenberg–Marquardt
algorithm. The reliability of the estimated parameters was evaluated using
95%-confidence intervals.
The estimation procedure was applied to the head measurements at every 1 m
distance starting at the well up to a distance of 80m. The range
beyond 80m was not taken into account to avoid boundary effects.
The range of 80m includes at least four correlation lengths for all
tested ensembles, which is sufficient to ensure convergence to the far-field
behaviour. The question of the applicability of hefw(r) on limited
head data is of quite complex nature. For a detailed discussion on that issue
the reader is referred to .
Ensemble pumping test interpretation
First, the simulated ensemble means were analysed, making use of the ensemble
version of TRCG(r) and hefw(r) (Eqs.
and ). Simulated ensemble means 〈h(r)〉‾
for multiple choices of statistical parameters TG, σ2,
and ℓ are visualized in Fig. in combination with
hefw(r) for the best-fitting parameter estimates T^G, σ^2, and ℓ^. Input parameters as well
as inverse estimation results for all tested ensembles are listed in
Table .
The best-fitting estimates show that all three parameters could be inferred
from the ensemble mean with a high degree of accuracy. The deviation of the
geometric mean from the input value is in general less than 10%;
only for high variances are the deviations up to 30%. Variances
deviate in a range of 20%, and estimated correlation lengths are
accurate within 10% of the initial input parameter.
The confidence intervals of the estimates T^G and σ^2 are very small, showing a high sensitivity of the effective well
flow solution hefw(r) towards geometric mean and variance. The
confidence intervals of the correlation length are larger due to the
dependence of the estimate of ℓ^ on the estimates T^G and σ^2. This is due to the fact that the
correlation length determines the transition from T^well=T^Gexp-12σ^2 to T^G, which results in larger uncertainties in the estimates of
ℓ^.
Individual pumping test interpretation
In the following, pumping test drawdowns of individual transmissivity fields
are interpreted based on the adaption version
hefwlocal(r) as discussed in Sect. . The
drawdowns along the four axial directions as well as the radial mean for two
realizations from ensemble
A (TG=10-4m2s-1, σ2=1,
ℓ=10m) are visualized in Fig. a and b.
Drawdowns simulated for two individual transmissivity field
realizations of ensemble
A (TG=10-4m2s-1, σ2=1,
ℓ=10m): (a) realization with
Twell=0.204×10-4m2s-1 and
(b) realization with Twell=1.11×10-4m2s-1. 〈h(r)〉 (dark color) is
the radial mean, 〈h(r,ϕ)〉 (light color) denotes the
drawdowns along the four axes (ϕ=0∘, 90∘, 180∘,
270∘), and in black is Thiem's solution for constant values.
The realizations from Fig. a and b differ
significantly in the value of the local transmissivity at the well. The
analysis of the transmissivity fields at the well gave sampled values of
<Twell(a)>=0.204×10-4m2s-1
and <Twell(b)>=1.11×10-4m2s-1, which is in both cases far from having the
theoretical harmonic mean value TH=0.61×10-4m2s-1 as the representative value for the
near-well behaviour.
Inverse estimation results for the realization in
Fig. a differ for the drawdowns along the four axial
directions 〈h(r,ϕj)〉 and the radial mean 〈h(r)〉: the estimated geometric mean ranges between 1.03×10-4 and 1.45×10-4m2s-1 for the four
axial directions, with an average value of T^G=1.17×10-4m2s-1. The estimates for the local
transmissivity at the well are between 0.195×10-4 and 0.212×10-4m2s-1, with an average value of T^well=0.204×10-4m2s-1, which is
exactly the sampled local transmissivity <Twell(a)>. The
value of T^well=0.204×10-4m2s-1 is equivalent to a local variance of
σ^2=3.49. The estimated correlation length ranges between 7.95
and 18.15m, with an average of ℓ^=12.77m. It
shows that the randomness of hydraulic heads due to the heterogeneity of
transmissivity impacts the estimation results of the correlation length.
The differences in the estimates for the drawdowns in different direction for
the same realization of transmissivity shows that the parameter estimates
reflect local heterogeneity in the vicinity of the well rather than the
global statistical parameters of the transmissivity field. This was studied
and discussed in detail for pumping tests in three-dimensional heterogeneous
media by .
The realization in Fig. b does not allow the
parameters of variance and correlation length to be inferred, due to the similarity
of Twell and TG. Near-well and far-field representative
transmissivities are nearly identical; thus the pumping test appears to
behave like in a homogeneous medium (Fig. b). However,
the behaviour is not representative but a result of the coincidental choice of
the location of the pumping well.
A statistical analysis of the estimation results is presented in
Fig. for all 5000 realizations of ensemble A. A histogram
on the best-fitting estimates in normalized form is shown, where
normalization of results means that they were divided by the input
parameters. It can be inferred that the estimate of the geometric mean T^G is in general close to the input value TG. The
estimate of the local transmissivity at the well T^well is
very close to the sampled values <Twell> for nearly all
realizations. Thus, the method reproduced very well the local transmissivity
at the well. However, the local value Twell of every realization
can be far from the theoretical value of TH, where both
realizations in Fig. serve as an example. The estimates of
the correlation length show a very large scatter. Exceptionally large and
small values for ℓ^ refer to realizations where it was nearly
impossible to infer it due to the similarity of Twell and
TG, as for the realization of Fig. b.
The large range of estimated correlation lengths also points towards the fact
that ℓ^ of a single drawdown needs to be interpreted as a local
value, which is determined by the transmissivity distribution in the vicinity
of the well rather than the distribution of the entire field. However, the
median of the normalized estimated correlation lengths is close to one,
pointing to the fact that representative values can be inferred by taking the
mean from multiple pumping tests.
Histogram on the best-fitting estimates (T^G, T^well, ℓ^) versus the theoretical input values
(TG, TH, ℓ) and the sampled transmissivity at the
pumping well (<Twell>) for the N=5000 realizations of
ensemble A.
Application example: single-aquifer analysis
Pumping test campaigns in the field often include the performance of multiple
pumping tests within one aquifer. Drawdown measurements at multiple test
locations can be used to gain representative parameters of the heterogeneous
transmissivity field. The sampled area increases, and the effect of local
heterogeneity through randomness of heads reduces. In the following, it is
shown how mean TG, variance σ2, and the correlation
length ℓ of an individual transmissivity field can be inferred by making
use of a sampling strategy in combination with
hefwlocal(r).
Sampling Strategy
The sampling strategy was constructed as a pumping test campaign in a virtual
aquifer with heterogeneous transmissivity. A series of steady-state pumping
tests was performed at n different wells. For each test, drawdowns were
measured at all n wells and at m additional observation wells. A similar
sampling strategy to infer the aquifer statistics from drawdown measurements
has been pursued by e.g. , , and .
The used sampling strategy includes n=8 pumping wells and m=4 observation
wells. The specific location of all wells is indicated in
Fig. . All eight pumping wells are located within a distance of
18 m. The observation wells are located at larger distances and in all
four directions. The well locations were designed to gain numerous drawdown
measurements in the vicinity of each pumping well to allow a reliable
estimation of Twell (or σlocal2)
and ℓ by reducing the impact of head fluctuations on the estimation
results. The additional observation wells provide head observations in the
far field to gain a representative value for TG. The choice of
the well locations does not interfere with the refinement of the numerical
grid at the pumping well.
Spatial distribution of log-transmissivity for fields (a) D1 and (b) E1 and locations of the eight pumping wells
(PW0,…, PW7 in black) and the four observation wells
(OW0,…, OW3 in grey).
Each of the eight pumping tests was analysed with
hefwlocal(r) (Sect. ). The best-fitting
estimates T^G, T^well, and ℓ^ for
all tests were inferred by minimizing the difference between the analytical
solution and the 12 measurements. Additionally, parameter estimates were
inferred by analysing the drawdown measurements of all tests jointly.
Aquifer analysis
The sampling strategy was applied to fields of all ensembles A–G
(Table ). Results are presented for two fields: D1 out of
ensemble D (σ2=2.25, ℓ=20 m) and E1 out of ensemble E
(σ2=4.0, ℓ=10 m). Each field was generated according to the
theoretical values defined for the particular ensemble and afterwards
analysed geostatistically to determine the sampled values. The fields D1 and
E1 are visualized in Fig. . The drawdown measurements for all
eight pumping tests at both fields are given in Fig. . The
inverse estimates as well as theoretical input and sampling values for the
statistical parameters are summarized in Table .
Parameter estimates of geometric mean transmissivity T^G (10-4 m2 s-1),
local transmissivity at the well T^well (10-4 m2 s-1), and correlation length ℓ^ (m)
for the eight pumping tests at fields D1 (from ensemble D, σ2=2.25) and E1 (from ensemble E, σ2=4.0). Additionally,
the theoretical and the sampled values (Twell≡TH) are given.
D1
E1
T^G
T^well
ℓ^
T^G
T^well
ℓ^
PW0
1.025
0.434
29.51
1.945
0.313
9.56
PW1
1.023
0.362
27.23
2.202
0.445
11.55
PW2
1.076
0.220
23.68
2.093
0.437
10.03
PW3
0.898
1.057
9.51
2.052
0.520
15.34
PW4
1.001
0.147
20.53
2.174
1.847
12.30
PW5
0.889
1.071
5.33
1.980
1.117
5.43
PW6
1.038
0.177
20.39
1.840
0.148
8.78
PW7
0.901
1.700
16.48
1.969
0.476
17.04
Mean of 8
0.981
0.646
19.08
2.032
0.663
9.90
Jointly
1.013
0.328
22.38
2.010
0.409
9.97
Theory
1.0
0.325
20.0
1.0
0.135
10.0
Sampled
0.985
0.333
23.43
1.999
0.491
12.66
Simulated drawdown measurements (dots) and fitted effective well
flow solution hefw(r) (lines) for eight pumping tests within the
heterogeneous transmissivity fields (a) D1 and (b) E1.
Colours indicate the results for the individual pumping tests at
PW0,…, PW7 (from light to dark). The black line denotes the
effective well flow solution hefw(r) fitted to all measurements
jointly. Grey lines denote Thiem's solution for T^G (solid)
and for T^well (dashed). Statistical parameters are given in
Table .
Analysing the data from all eight pumping tests at field D1 jointly yields very
close estimates of all parameters T^G, T^well
(corresponding to σ^2=2.255), and ℓ^ to the theoretical
and sampled values. The geometric mean estimate is similar for all of the
eight individual pumping tests. In contrast, the values of T^well
vary within one order of magnitude. This behaviour was expected, since T^well represents the local transmissivity value at the pumping
well. The wide range of estimates is a results of the high variance of the
transmissivity field. The estimates of the correlation length ℓ^
differ between the individual tests within a reasonable range of a few
meters. The only exception is the estimate for pumping at PW5. For this
specific pumping test is highly uncertain due to the coincidence of the
values of T^well and T^G, similar to the
realizations in Fig. 3b, as discussed in Sect. 3.3. However, the mean value
over the individual tests as well as the estimate from the joint analysis of
all measurements gave reliable estimates for the correlation length.
The analysis of the sampling strategy at field E1 yields similar results to
those for D1. The geometric mean values T^G differ little among the
eight individual pumping tests and for the joint analysis. The mean value is
double the value of the theoretical one, but close to the sampled geometric
mean (Table ). The local transmissivities T^well again vary within one order of magnitude, reflecting the
high variance of the field. The mean and jointly estimated values are higher
than the theoretical one, which is in correspondence with the difference in the
geometric mean. The estimates of the correlation length ℓ^ deviate
in a reasonable range of a few meters, which reflects the impact of the
location of the pumping well with regard to the shape of the correlation
structure around the well.
Finally, the analysis shows that representative values of the statistical
parameters can be determined by performing a pumping test at multiple locations
of an individual transmissivity field. It was shown that hefw(r)
is feasible to interpret steady-state pumping tests in highly heterogeneous
fields.