Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue1.1999:
Advances in Water Resources 23 (1999) 59±68
Flowing partially penetrating well: solution to a mixed-type boundary
value problem
G. Cassiani
b
a,b
, Z.J. Kabala
a,*
,
M.A. Medina Jr.
a
a
Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA
ENI S.p. A ± Agip Division ± Geodynamics and Environment Unit (GEDA), Via Emilia 1, San Donato Milanese I-20097, Italy
Received 1 October 1998; received in revised form 15 December 1998; accepted 17 December 1998
Abstract
A new semi-analytic solution to the mixed-type boundary value problem for a ¯owing partially penetrating well with in®nitesimal
skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-in®nite vertical extent or to
packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral
equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res.
1978; 14:929±34], our DE solution does not suer from numerical oscillations. The new solution is validated by matching the
corresponding ®nite-dierence solution and is computationally much more ecient. An automated (Newton±Raphson) parameter
identi®cation algorithm is proposed for ®eld test inversion, utilizing the DE solution for the forward model. The procedure is
computationally ecient and converges to correct parameter values. A solution for the partially penetrating ¯owing well with no
skin and a drawdown±drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993;
30:600±6], is compared to the DE solution. The D±D solution leads to physically inconsistent in®nite total ¯ow rate to the well,
when no skin eect is considered. The DE solution, on the other hand, produces accurate results. Ó 1999 Elsevier Science Ltd. All
rights reserved.
1. Introduction
The ¯owing well test is a well-established singleborehole technique used in the ®eld by hydrologists
[3,8,16,23], geotechnical engineers [32,40,46], and petroleum engineers [6,11,25]. Since the ¯ow rate to a well
is induced under a constant head, the test is also known
as the constant-head or constant-pressure test. The
measured quantities are the constant head and the resulting ¯ow rate (inbound or outbound). Mathematical
models are needed to interpret the ®eld data. A paucity
of such models exists for fully penetrating ¯owing wells;
even fewer models are available for partially penetrating
¯owing wells, due to the inherent diculty of solving the
resulting mixed boundary value problem.
Only hydraulic conductivity (or permeability) can be
estimated from the approaches of Dagan [8], who
worked out a steady-state solution using Green's functions, and of Tavenas et al. [40], who used Hvorslev's
quasi-steady-state ¯ow formula for hydraulic conduc*
Corresponding author. Tel.: +1 919 660 5479; fax: +1 919 660 5219;
e-mail: kabala@copernicus.egr.duke.edu
tivity and studied numerically the eects of probe geometry on the shape factor in this formula.
Transmissivity and storativity can be estimated from the
Hantush [16] transient model for a fully penetrating
¯owing well. This model has been extended to account
for in®nitesimal thickness skin by Kabala and Cassiani
[24].
Novakowski [32] pointed out that models that neglect
the skin eect [27], i.e. the changed ¯ow ®eld in a disturbed zone in the immediate vicinity of the well, produce biased results in interpreting ®eld tests. The
disturbed zone can be of ®nite or in®nitesimal thickness.
Recently, Novakowski [32] developed a general
model, which accounts for the thick skin and partial
penetration. However, it contains a large number of
parameters and is based on a physically inconsistent
well-face boundary condition that causes the model to
predict non-physical behaviors for some parameter
choices (e.g., when no skin eect is considered) as will be
shown in this paper.
Although Novakowski [32] uses a point source solution, which he then integrates, his solution is equivalent
to that for the initial boundary-value problem with a
0309-1708/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 0 2 - 0
60
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
given head speci®ed on the well face and the zero head
rather than no ¯ux speci®ed along the well casing. As
opposed to the correct boundary value problem (with no
¯ux speci®ed along the well casing), the simpli®ed
boundary value problem can be solved analytically by
the Fourier transform. It is not yet clear what kind of
error introduces the simpli®ed boundary condition.
Some of the many parameters in Novakowski's [32]
model, especially the speci®c storativity and hydraulic
conductivity of the skin, are not likely to be determined
from the ¯owing-well-test ®eld data, or other single
borehole tests [27]. It may thus be advantageous to lump
the ®nite-thickness skin parameters into a single skin
factor that would be representative of an equivalent skin
of in®nitesimal thickness as proven by Moench and
Hsieh [27].
1.1. Objectives
The purpose of this paper is to derive a new semianalytic solution for the response of a partially penetrating ¯owing well, situated in an aquifer of in®nite
extent and thickness, and to compare it to the numerical
solution obtained via ®nite dierence approximation. A
critical analysis of an existing approximate solution to
the same problem [32] will also be given by comparison
to the new solution.
2. Mixed-type boundary value problems
Numerous problems of mathematical physics can be
described by initial boundary value problems (IBVP)
with the mixed-type boundary conditions, i.e. with the
®rst-type boundary condition on one part of the boundary (speci®ed dependent variable, e.g. piezometric
head) and the second-type boundary condition on another part of the boundary (speci®ed derivative of the
dependent variable, e.g. ¯ux). Such problems arise in
potential theory and its numerous applications to engineering [12,26,38], fracture mechanics [10] heat conduction and, in particular, dip-forming processes in
metallurgy [14], surface rewetting [42], contact resistance
to heat transfer [36], heat transfer from partially insulated pipes [37], and many others. Flow to partially
penetrating wells should also be described by IBVPs
with the mixed-type boundary condition on the well face
[47].
Huang [20] and Wilkinson and Hammond [47] noted
that mixed-type boundary value problems cannot be
solved by the conventional integral transform methods
(or expansion in terms of orthogonal functions). Only a
paucity of analytic solutions to such problems have been
found by means of more subtle techniques that include
the dual integral/series equations [38], Weiner±Hopf
technique [31], dual integral equation (DE) [43], and
Green's function [21]. In fact, an overwhelming majority
of solutions to mixed-type boundary value problems are
obtained numerically [1,44], or by approximate methods
such as asymptotic analysis [1], or perturbation techniques [47].
The solutions to the parabolic mixed-type IBVPs may
display some disturbing properties. For example, Bassani et al. [1] demonstrated that a steady-state temperature ®eld of a wedge-shaped region, where the
boundary condition changes abruptly at the vertex of
the wedge, suers from the heat ¯ux singularity (unbounded point ¯ux at the vertex) whenever the included
angle of the wedge is greater than p=2. We note that this
result has analogous implications for the ¯ux at the
edges of the ®lter of a partially penetrating well.
As recognized by Wilkinson and Hammond [47], who
applied an approximate perturbation technique, the
mixed-type boundary conditions arise naturally in the
description of ¯ows to partially penetrating wells with
the pressure head speci®ed on the ®lter face and no-¯ux
speci®ed on the well casing. With the exception of a few
[35,45] and others who solved numerically the mixedtype boundary value problems in well hydraulics, all
other researchers who approached such problems analytically, avoided solving them by replacing the mixedtype boundary condition of speci®ed-head/no-¯ux either
by a ®rst-type (Dirichlet) boundary condition of speci®ed-head/zero-head [32] or by a second-type (Neumann)
boundary condition of speci®ed-¯ux/no-¯ux [8,9,15,17,
18,22,28±30,33,41,45]. The simpli®ed solutions obtained
in this manner are to a certain degree physically inconsistent, as explicitly acknowledged by Muskat [28] and
Hantush [15,17,19], and discussed in detail by Ruud and
Kabala [35].
3. The ¯owing partially-penetrating well problem
Cases to which the presented solution applies: (a)
partially penetrating ¯owing well in an aquifer of semiin®nite vertical extent; (b) double packer test geometry
in an aquifer of in®nite vertical extent. Consider a
¯owing well drilled in a formation of semi-in®nite vertical extent, such as shown in Fig. 1. Note that the
bottom of the wellbore is assumed not to collect any
¯ow from the aquifer. The wellbore of radius rw penetrates the aquifer down to a depth l. The imposed
drawdown in the well is sw . The drawdown at the distance r from the well, the distance z from the top of the
aquifer, and time t is s r; z; t. The aquifer parameters
are horizontal hydraulic conductivity Kr , vertical hydraulic conductivity Kz , and speci®c storativity Ss . The
governing partial dierential equation is
2
os
1 os
o2 s
os
K z 2 Ss
:
1
Kr
2
or
oz
ot
r or
61
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
packer
sw
r
r
z
z
packer
(a)
(b)
Fig. 1. Cases to which the presented solution applies. (a) partially penetrating ¯owing well in an aquifer of semi-in®nite vertical extent; (b) double
packer test geometry in an aquifer of in®nite vertical extent.
A skin of in®nitesimal thickness [27], is assumed
characterized by the skin factor g, de®ned by the following boundary condition at the well screen (z 6 l):
os
j
2
s jrrw sw :
ÿ grw
or rrw
We cast the problem in dimensionless terms, by de®ning the following parameters:
Dimensionless well screen length
d l=rw :
Dimensionless drawdown
3
s s =sw :
4
tKr
a 2:
Ss r w
Anisotropy ratio
5
a2
Kz
Kr
and dimensionless spatial coordinates:
6
q r=rw ;
7
f z=rw :
8
o2 s 1 os
o2 s os
a2 2 ;
2
oq
q oq
oa
of
9
Dimensionless time
The boundary value problem to be solved follows
from the partial dierential equation (1), that in dimensionless terms reads:
subject to the following boundary and initial conditions:
s q; f; a 0 0;
10
s q 1; f; a 0;
11
ÿg
os
j s q 1; f; a 1
oq q1
os
0
j
oq q1
for d < f;
os
j 0;
of f0
s q; f 1; a
for 0 6 f 6 d
12
13
os
j
0:
of f1
14
Note that Eq. (12) is a mixed boundary condition at the
well face, i.e. a third-type (or Robin) boundary condition along the well screen of dimensionless length d, and
a second-type (Neuman) no-¯ow boundary condition
along the casing.
We also note that, due to the symmetry around the
z 0 plane, a solution for a partially penetrating ¯owing well in the aquifer of in®nite thickness (or a double
packer test with long enough packers to prevent back¯ow) shown in Fig. 1 can be directly obtained from the
solution of Eqs. (9)±(14).
The solution technique employs the DE approach [43]
in a manner analogous to that used by Cassiani and
Kabala [5]. It is described in Appendix A. The relevant
computational details are discussed in Appendix B.
62
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
4. Model results and parameter identi®cation procedure
FD solution
G(α)
0
10
δ
10
20
50
100
10
1
Hantush [1959]
10
1
4
10
10
9
14
10
α=t Kr / (Ss rw2)
Fig. 3. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for the no-skin case (g 0) and dierent well screen
lengths. The [16] solution for a fully penetrating ¯owing well is shown
for reference. Also, a comparison is made with the results obtained
using the ®nite dierence (FD) code by [34].
1
10
η
δ=10
0
1
0
G(α)
The approach described in Appendices A and B allows us to compute eciently the ¯ow and drawdown
functions for a number of geometrical con®gurations
and parameter values.
In Fig. 2, we present an example of point ¯ux at the
well face for dierent times. The curves characteristically
exhibit sharp peaks at the extremes of the well screen
due to strong contribution of vertical ¯ux in the surrounding aquifer. They are analogous to those noted by
Dagan [8] in his semi-analytic solutions for steady-state
¯ow to partially penetrating wells in water-table aquifers, and by Widdowson et al. [45] in their corresponding numerical solutions for partially penetrating slug
tests. Unlike Dagan's results, only tiny oscillations occur
in our solution.
For the case of no skin (g 0) we validate the total
dimensionless ¯ux G calculated with our methodology
by comparing it to a numerical solution obtained via
®nite dierence algorithm of Ruud and Kabala [34]. As
seen in Fig. 3 the two curves show an excellent match.
We note that at large times all curves approach quasisteady state.
The eect of the skin factor g is shown in Figs. 4±6,
where type curves for d 10, 20 and 50 are shown for
g 0 through 10.
Similarly to the fully penetrating ¯owing wells [24],
the skin eect in partially penetrating wells produces an
early-time in¯ection point in the G curves. For large
times the skin factor lowers the curve asymptote.
The DE approach is ideally suited for incorporation
into an automated parameter identi®cation algorithm.
Let ti ; Qi ; i 1; . . . ; ndata be ®eld data of total ¯ow rate
versus time, collected during a constant-head test on a
partially penetrating well of known geometry. The formation parameter vector b fb1 ; b2 ; b3 g, b1 Kr ;
1
10
10
2
10
3
4
5
6
7
8
9
10
1
10
5
10
0
5
10
10
10
α=t Kr / (Ss rw2)
Fig. 4. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for d 10.
b2 Ss ; b3 g can be obtained from ti ; Qi by using an
optimization algorithm that minimizes the dierence
between ®eld data and model prediction. If f t is the
adopted model, let the distance vector and the Jacobian
matrix be given respectively by
Fig. 2. Dimensionless point ¯ux for the case of no skin (g 0), at
dierent time steps, for d 100.
15
F ff ti Qi g i 1; . . . ; ndata
and
of
i 1; . . . ; ndata ; j 1; 2; 3:
16
jtti
A
obj
An estimate of the parameter vector b is obtained by
Newton±Raphson iteration
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
63
1
10
η
δ=20
0
1
G(α)
0
10
2
10
3
4
5
6
7
8
9
10
1
10
5
10
0
5
10
10
10
α=t Kr / (Ss rw2)
Fig. 5. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for d 20.
10
Fig. 7. Example of Newton±Raphson ®tting to data. The data are
produced using the FD model by Ruud and Kabala [34]. The ®tting is
obtained using the DE solution as the physical model. The starting
point is Kr 10ÿ7 m/s, Ss 10ÿ5 1/m, and convergence to 0.1% is
achieved in only nine iterations.
1
η
that the ®rst guess for the parameters is not too far from
the true values (approximately within an order of magnitude) convergence is assured.
δ=50
0
G(α)
1
10
0
5. Comparison with existing models
2
10
3
4
5
6
7
8
9
10
1
10
5
0
10
10
5
10
10
α=t Kr / (Ss rw2)
Fig. 6. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for d 50.
bk1 bk ÿ Ak F k ;
17
where k is the iteration step, and A is the pseudo-inverse matrix of A, de®ned as
ÿ1
A AT A AT ;
18
where T stands for transpose and ÿ1 for inverse ([13],
p. 243).
In this paper the DE algorithm is used to compute
f t. For the repeated computation of f t required by
Eqs. (15) and (16), the fast computation of DE is essential.
An example of Newton±Raphson parameter ®tting is
shown in Fig. 7. The data points are simulated using the
®nite dierence code by Ruud and Kabala [34]. Provided
In Fig. 3 we compare the results produced by our
model to the corresponding results for fully penetrating
¯owing wells. At early times the DE solution approaches, as expected, the corresponding solution for a
fully penetrating well, i.e. the no-skin solution of Hantush [16]. In addition, as mentioned earlier, the DE point
¯ux curves in Fig. 2 exhibit sharp increases at the well
screen edges, a behavior similar to that pointed out by
Dagan [8] in his model for water table aquifers.
The Novakowski [32] model was developed originally
for ®nite-thickness aquifers; however, his methodology
can be easily extended to aquifers of in®nite or semiini®nite vertical extent, such as the ones considered here.
In this section we consider the case with no skin, which
can be directly compared to our approach. We name
this method the drawdown±drawdown (D±D) approach, since it involves a discontinuous drawdown
boundary condition at the well face.
Consider the IBVP given by Eqs. (9)±(11) and (14)
and the following condition at the well-face, used by
Novakowski [32]:
19
s q 1; f; a d f ÿ f0 ;
where here d is the Dirac delta function.
Employing cosine Fourier transform instead of ®nite
cosine Fourier transform used by Novakowski [32] we
obtain
64
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
2
p
Z
1
0
q
2
K0 q an p
q cos nf0 cos nf dn:
2
K0
an p
0.5
20
Integration along the well face leads to the solution for
drawdown in the Laplace domain
s q; f; p
2
p
Z
1
0
q
K0 q an2 p
sin nd
q
cos nf dn:
pn
2
an p
K0
point flux
sd q; f; p
D-D sol.
DE sol.
21
Note that this solution is identical to that obtained by
imposing a unit drawdown on the well screen and zero
drawdown on the casing.
The point ¯ux at the well face follows from the above
by dierentiation
-0.5
-200
Z
1
0
Integration of Eq. (22) along the well face leads via
Eq. (A.22) to the dimensionless total ¯ux G , analogous to Novakowski's [15], (A.9)
G p
2
p2
100
200
1.0
q q
2
2
an pK1
an p
sin nd
q
cos nf dn:
pn
an2 p
K0
22
0
Fig. 8. Comparison between the dual integral equation solution (DE)
and the drawdown±drawdown (D±D) solution for d 100 and time
a 103 and no skin (g 0). The D±D solution exhibits spurious
negative ¯ux on the well casing. For a closer look, see Fig. 9.
Z
1
0
q q
2
2
an p
an pK1
1 sin2 nd
q
dn:
d
n2
2
p K0
an p
23
The above integral is divergent, as is the corresponding
series solution of Novakowski [15], Eq. (A.9) for the
case of no skin. The reason why the total ¯ux is in®nite
in this solution can be seen in Figs. 8 and 9. There we
present the point ¯ux at dimensionless time a 103 for a
well of dimensionless screen length d 100 and no skin
(g 0) calculated from our DE solution, (A.19), and
from the D±D solution (22). The D±D solution shows
strong spurious oscillations, known as Gibbs eect,
caused by truncation in the frequency domain. The
curves in Figs. 8 and 9 appear to be blurred because of
this eect. The DE solution is represented by a thick
dot±dashed line. Note that the D±D solution produces
spurious negative ¯ux outside the well casing. Along the
well screen away from its edges the two solutions are
very close to each other. However, they are substantially
dierent around the wellscreen edge. The DE solution
takes reasonable values whereas the D±D solution di-
0.5
point flux
2
p
-100
ζ
os
j
oq q1
ÿ
0.0
0.0
D-D sol.
DE sol.
~1/(ζ−δ)
-0.5
-1.0
-110
-100
-90
ζ
Fig. 9. Same case as in Fig. 8, zoomed around f ÿd ÿ100. Note
the oscillations in the D±D solution (Gibbs eect). As opposed to the
DE solution, the D±D point ¯ux at the screen edge diverges to in®nity
as 1= f ÿ d, leading thus to a spurious in®nite total ¯ux (area under
the curve).
verges to in®nity as 1= f ÿ d, leading to in®nite total
¯ux (the area under the curve), clearly a physical impossibility.
6. Conclusions
The main results of this paper are:
1. A new semi-analytic solution to the mixed-type boundary value problem for a ¯owing partially penetrating
65
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
2.
3.
4.
5.
6.
well with in®nitesimal skin situated in an anisotropic
aquifer is developed via Laplace and Fourier transforms. The solution is suited for aquifers of semi-in®nite vertical extent for packer tests with aquifer
horizontal boundaries far enough from the tested
area. The problem can be reduced to a system of
DE and further to the deconvolution Eq. (A.18),
which we solve by discretizing the integral. Inversion
of the Laplace and Fourier transforms are handled
numerically via Stehfest and fast Fourier transform
(FFT) algorithms.
The solution is achieved by discretizing a deconvolution equation. An ecient discretization needs to satisfy two opposite requirements: (i) it has to be ®ne
enough to sample adequately the kernel function,
and (ii) it must be crude enough to prevent unwanted
oscillations. For no skin, g 0, the density of 0.6/a2
nodes per unit f produces accurate results, whereas
for g 10 the corresponding density increases to
1.6/a2 nodes per unit f.
The new solution is computationally robust due to
the impulse-like nature of the kernel in the deconvolution equation. We note that unlike the analogous
Dagan's [8] steady-state solution for ®nite-thickness
uncon®ned aquifers, our DE solution embedded in
Eq. (A.18) does not suer from numerical oscillations.
Our DE solution matches the corresponding ®nitedierence solution obtained using the code developed
by Ruud and Kabala [34]. The DE solution is computationally much more ecient. Although the computational burden is generally limited, it increases with
the well screen lengths.
Based on our DE solution (forward model), an automated (Newton±Raphson) parameter identi®cation
algorithm is developed for ®eld test interpretation.
The procedure is computationally ecient and converges to correct parameter values, provided that
the initial parameter guess is not too far from the true
values (approximately within one order of magnitude).
A solution for the partially penetrating ¯owing well
with no skin and D±D discontinuous boundary condition, analogous to that of Novakowski [15],
Eq. (A.9) was also derived. The D±D solution was
compared to the DE solution. The D±D solution
leads to physically inconsistent in®nite total ¯ow rate
to the well, when no skin eect is considered. The DE
solution, on the other hand, produces accurate
results.
Acknowledgements
The authors are indebted to Nels C. Ruud for the use
of numerical code developed in his dissertation. This
research was performed while G. Cassiani was a graduate student and Z.J. Kabala and M.A. Medina, Jr.
were his faculty advisors in the Department of Civil and
Environmental Engineering at Duke University, Durham, NC.
Appendix A. The solution technique
We eliminate the time variable a from the problem by
applying the Laplace transform to Eqs. (9), (11)±(14),
and using the initial condition (10). The Laplace-transformed variable is denoted by a bar, thus
Z 1
eÿpa s q; f; a da:
s q; f; p Lfs q; f; a; a ! pg
0
A:1
The transformed boundary-value problem is
2
o2 s 1 os
s
2o
a
ÿ p s 0;
oq2 q oq
of2
A:2
s q 1; f; p 0;
A:3
ÿg
os
1
j s q 1; f; p
oq q1
p
os
0
j
oq q1
for 0 6 f 6 d
for d < f;
os
j 0;
of f0
A:4
A:5
os
j
0:
A:6
of f1
Next, we eliminate f by applying the Fourier cosine
transform to Eq. (A.2) and Eq. (A.3) and by using
Eq. (A.6). In order to apply properly the mixed boundary condition (A.4), Tranter ([43], p.51) suggests leaving it out and applying it only after inverting back from
the Fourier domain. The Fourier-transformed variable
is denoted by aR hat, thus ^s q; n; p Ffs q;
1
f; p; f ! ng 2=p 0 s q; f; p cos nf df. We thus
arrive at the Bessel dierential equation
s q; f 1; p
o2^s 1 o^s
ÿ a2 n2 p ^s 0;
oq2 q oq
subject to the boundary condition
A:7
^s q 1; n; p 0:
A:8
The general solution is in the form of
q
^s q; n; p A n; p K0 q a2 n2 p ;
A:9
where K0 is the modi®ed Bessel function of order zero
and the unknown A n; p is a constant with respect to q
to be determined from the mixed boundary condition
(A.4) after cosine Fourier transform inversion. The solution in the f domain is thus
66
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
s q; f; p
Z
0
q
1
A n; p K0 q a2 n2 p cos nf dn;
whereas its derivative with respect to q reads
A:10
os
q; f; p
oq
Z 1
q q
A n; p a2 n2 p K1 q a2 n2 p cos nf dn;
ÿ
0
A:11
where K1 is the modi®ed Bessel function of order one.
Substitution of Eq. (A.10) and Eq. (A.11) into
Eq. (A.4) leads to a system of two integral equations,
known as DE ([43], p. 111):
Z 1
q q
A n; p ÿ g a2 n2 p K1 q a2 n2 p
0
q
1
a2 n2 p
for jfj 6 d;
cos n f dn
K0
p
A:12
Z
1
0
q q
a2 n2 p cos nf dn 0
A n; p a2 n2 p K1
for jfj > d:
A:13
Systems (A.12) and (A.13) can be solved in a number of
ways [43] we transform it into a deconvolution problem
in the space domain. This approach is convenient because the Bessel function kernels in Eqs. (A.12) and
(A.13) have closed-form Fourier cosine transforms ([2],
vol. I, p. 56):
Z 1 q
^
a2 n2 p cos nf dn
K0
K 0 f; p
0
1p
1
q exp
a2
2
f=a 1
K^1 f; p
1p
a2
Z
0
q
2
ÿ p f=a 1 ;
A:14
q
q
1
a2 n2 p K1
a2 n2 p cos nf dn
1
f=a2 13=2
2
p !
p
f=a 1
exp
q
2
ÿ p f=a 1 :
A:15
With Eq. (A.14) and Eq. (A.15) and the convolution
theorem for Fourier transforms, problem (A.12) and
(A.13) can be restated as:
Z
Z
1
^ p K^0 f ÿ x; p ÿ g K^1 f ÿ x; p dx 1
A x;
p
ÿ1
for jfj 6 d;
A:16
1
ÿ1
^ p K^1 f ÿ x; p dx 0 for jfj > d;
A x;
A:17
^ p is the inverse Fourier transform of A n; p.
where A x;
Solution to Eq. (A.16) and Eq. (A.17) can be achieved
by discretizing the vertical space domain (here expressed
in terms of coordinate f or x). Our numerical experi^ p is pracments show that the unknown function A f;
tically zero outside the well screen interval jfj 6 d, so
that Eq. (A.16) and Eq. (A.17) can be conveniently reduced to the Fredholm integral equation
Z d
^ p K^0 f ÿ x; p ÿ g K^1 f ÿ x; p dx 1
A x;
p
ÿd
for jfj 6 d:
A:18
^ p has been obtained from Eq. (A.18), the
Once A f;
point ¯ux in the Laplace domain os= oqjq1 f; p can
be obtained by Fourier transform
os
j f; p
oq q1
Z 1
q q
a2 n2 p cos nf dn:
A n; p a2 n2 p K1
ÿ1
A:19
Fourier transformations can be conveniently carried out
via the fast Fourier transform (FFT) [4]. We use Stehfest
[39] algorithm to obtain the point ¯ux os= oqjq1 f; a
in the dimensionless time domain.
The total dimensionless ¯ux to the partially penetrating well can be de®ned analogously to the corresponding dimensionless ¯ux for a fully penetrating
¯owing well given by Hantush [16]
Q t
;
A:20
2pKr lsw
where Q t is the total dimensional ¯ux, computed as
Z l
os
Q t ÿ2prw Kr
jrw dz:
A:21
0 or
Consequently, G can be expressed in terms of the dimensionless point ¯ux as
Z
1 d os
j df
A:22
G a; d ÿ
d 0 oq q1
and it can be extracted from measurements in the well.
The drawdown at any point in space and time can be
computed from Laplace inversion of Eq. (A.10), or by
convolution
Z 1
^ p K^r0 q; f ÿ x; p dx;
s q; f; p
A:23
A x;
G a; d
ÿ1
where
K^r0 q; f; p
p
1
q exp
2a
f=a2 q2
ÿ
q
p f=a2 q2 :
A:24
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
The convolution approach is more stable because it is
consistent with the slight approximation introduced by
limiting the deconvolution from ÿd to d only (see
Eq. (A.18)).
Appendix B. Computational details
Although generally deconvolution problems are illposed [7], in some cases highly accurate solutions can be
obtained. Dagan [8] notes that ``an accurate solution
is expectable if the kernel is strongly peaked at
f x, whereas the solution will worsen if the kernel is
¯at''. Fortunately, the kernel in Eq. (A.18) is strongly
peaked at f x for all values of p, and thus for all values
of dimensionless time a.
We note that unlike Dagan's [8] analogous steadystate solution for ®nite-thickness uncon®ned aquifers,
our DE solution embedded in Eq. (A.18) does not suer
from severe numerical oscillations.
Solving Eq. (A.18) requires a carefully calibrated
discretization along the well screen. An ecient discretization can be found by satisfying as well as possible
the two opposite qualitative requirements: (i) it has to be
®ne enough to sample adequately the kernel function,
and (ii) it must be crude enough to prevent unwanted
oscillations.
Our numerical experiments demonstrate that an ecient discretization can be determined independently of
d. However, it depends on a2 and g. For no skin, g 0,
the density of 0.6/a2 nodes per unit f produces accurate
results, whereas for g 10 the corresponding density
increases to 1.6/a2 nodes per unit f. For intermediate
cases proportional increments can be used. Note how
the dependence on a2 corresponds to a pure scaling of
the boundary value problem in the vertical direction.
Since the optimal Df is independent of d, the number of
points needed is a linear function of d itself. Consequently, the computational burden increases with the
size of the deconvolution matrix arising from discretization of Eq. (A.18), as d increases.
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[28] Muskat M. The ¯ow of homogeneous ¯uids through porous
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[34] Ruud NC, Kabala ZJ. Numerical evaluation of ¯owmeter test
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heterogeneous aquifer: integrated well-face ¯ux versus uniform
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[40] Tavenas F, Diene M, Leroueil S. Analysis of the in situ constanthead permeability test in clays. Can Geotech J 1990;27:305±14.
[41] Theuveny BC, Jack WA, Economides MJ. Pressure transient
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[42]
[43]
[44]
[45]
[46]
[47]
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Tien CL, Yao LS. Analysis of conduction controlled rewetting of
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Tranter CJ, Integral transforms in mathematical physics. 3rd
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Yedder RB, Bilgen E. On adiabatic-isothermal mixed boundary
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arme- und Sto
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1994;29:457±60.
Widdowson MA, Molz FJ, Melville JG. An analysis technique for
multilevel and partially penetrating slug test data. Ground Water
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Wilkinson WB. Constant head in situ permeability test in clay
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Wilkinson D, Hammond PS. A perturbation method for mixed
boundary-value problems in pressure transient testing. Trans
Porous Media 1990;5(6):609±36.
Flowing partially penetrating well: solution to a mixed-type boundary
value problem
G. Cassiani
b
a,b
, Z.J. Kabala
a,*
,
M.A. Medina Jr.
a
a
Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA
ENI S.p. A ± Agip Division ± Geodynamics and Environment Unit (GEDA), Via Emilia 1, San Donato Milanese I-20097, Italy
Received 1 October 1998; received in revised form 15 December 1998; accepted 17 December 1998
Abstract
A new semi-analytic solution to the mixed-type boundary value problem for a ¯owing partially penetrating well with in®nitesimal
skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-in®nite vertical extent or to
packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral
equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res.
1978; 14:929±34], our DE solution does not suer from numerical oscillations. The new solution is validated by matching the
corresponding ®nite-dierence solution and is computationally much more ecient. An automated (Newton±Raphson) parameter
identi®cation algorithm is proposed for ®eld test inversion, utilizing the DE solution for the forward model. The procedure is
computationally ecient and converges to correct parameter values. A solution for the partially penetrating ¯owing well with no
skin and a drawdown±drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993;
30:600±6], is compared to the DE solution. The D±D solution leads to physically inconsistent in®nite total ¯ow rate to the well,
when no skin eect is considered. The DE solution, on the other hand, produces accurate results. Ó 1999 Elsevier Science Ltd. All
rights reserved.
1. Introduction
The ¯owing well test is a well-established singleborehole technique used in the ®eld by hydrologists
[3,8,16,23], geotechnical engineers [32,40,46], and petroleum engineers [6,11,25]. Since the ¯ow rate to a well
is induced under a constant head, the test is also known
as the constant-head or constant-pressure test. The
measured quantities are the constant head and the resulting ¯ow rate (inbound or outbound). Mathematical
models are needed to interpret the ®eld data. A paucity
of such models exists for fully penetrating ¯owing wells;
even fewer models are available for partially penetrating
¯owing wells, due to the inherent diculty of solving the
resulting mixed boundary value problem.
Only hydraulic conductivity (or permeability) can be
estimated from the approaches of Dagan [8], who
worked out a steady-state solution using Green's functions, and of Tavenas et al. [40], who used Hvorslev's
quasi-steady-state ¯ow formula for hydraulic conduc*
Corresponding author. Tel.: +1 919 660 5479; fax: +1 919 660 5219;
e-mail: kabala@copernicus.egr.duke.edu
tivity and studied numerically the eects of probe geometry on the shape factor in this formula.
Transmissivity and storativity can be estimated from the
Hantush [16] transient model for a fully penetrating
¯owing well. This model has been extended to account
for in®nitesimal thickness skin by Kabala and Cassiani
[24].
Novakowski [32] pointed out that models that neglect
the skin eect [27], i.e. the changed ¯ow ®eld in a disturbed zone in the immediate vicinity of the well, produce biased results in interpreting ®eld tests. The
disturbed zone can be of ®nite or in®nitesimal thickness.
Recently, Novakowski [32] developed a general
model, which accounts for the thick skin and partial
penetration. However, it contains a large number of
parameters and is based on a physically inconsistent
well-face boundary condition that causes the model to
predict non-physical behaviors for some parameter
choices (e.g., when no skin eect is considered) as will be
shown in this paper.
Although Novakowski [32] uses a point source solution, which he then integrates, his solution is equivalent
to that for the initial boundary-value problem with a
0309-1708/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 0 2 - 0
60
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
given head speci®ed on the well face and the zero head
rather than no ¯ux speci®ed along the well casing. As
opposed to the correct boundary value problem (with no
¯ux speci®ed along the well casing), the simpli®ed
boundary value problem can be solved analytically by
the Fourier transform. It is not yet clear what kind of
error introduces the simpli®ed boundary condition.
Some of the many parameters in Novakowski's [32]
model, especially the speci®c storativity and hydraulic
conductivity of the skin, are not likely to be determined
from the ¯owing-well-test ®eld data, or other single
borehole tests [27]. It may thus be advantageous to lump
the ®nite-thickness skin parameters into a single skin
factor that would be representative of an equivalent skin
of in®nitesimal thickness as proven by Moench and
Hsieh [27].
1.1. Objectives
The purpose of this paper is to derive a new semianalytic solution for the response of a partially penetrating ¯owing well, situated in an aquifer of in®nite
extent and thickness, and to compare it to the numerical
solution obtained via ®nite dierence approximation. A
critical analysis of an existing approximate solution to
the same problem [32] will also be given by comparison
to the new solution.
2. Mixed-type boundary value problems
Numerous problems of mathematical physics can be
described by initial boundary value problems (IBVP)
with the mixed-type boundary conditions, i.e. with the
®rst-type boundary condition on one part of the boundary (speci®ed dependent variable, e.g. piezometric
head) and the second-type boundary condition on another part of the boundary (speci®ed derivative of the
dependent variable, e.g. ¯ux). Such problems arise in
potential theory and its numerous applications to engineering [12,26,38], fracture mechanics [10] heat conduction and, in particular, dip-forming processes in
metallurgy [14], surface rewetting [42], contact resistance
to heat transfer [36], heat transfer from partially insulated pipes [37], and many others. Flow to partially
penetrating wells should also be described by IBVPs
with the mixed-type boundary condition on the well face
[47].
Huang [20] and Wilkinson and Hammond [47] noted
that mixed-type boundary value problems cannot be
solved by the conventional integral transform methods
(or expansion in terms of orthogonal functions). Only a
paucity of analytic solutions to such problems have been
found by means of more subtle techniques that include
the dual integral/series equations [38], Weiner±Hopf
technique [31], dual integral equation (DE) [43], and
Green's function [21]. In fact, an overwhelming majority
of solutions to mixed-type boundary value problems are
obtained numerically [1,44], or by approximate methods
such as asymptotic analysis [1], or perturbation techniques [47].
The solutions to the parabolic mixed-type IBVPs may
display some disturbing properties. For example, Bassani et al. [1] demonstrated that a steady-state temperature ®eld of a wedge-shaped region, where the
boundary condition changes abruptly at the vertex of
the wedge, suers from the heat ¯ux singularity (unbounded point ¯ux at the vertex) whenever the included
angle of the wedge is greater than p=2. We note that this
result has analogous implications for the ¯ux at the
edges of the ®lter of a partially penetrating well.
As recognized by Wilkinson and Hammond [47], who
applied an approximate perturbation technique, the
mixed-type boundary conditions arise naturally in the
description of ¯ows to partially penetrating wells with
the pressure head speci®ed on the ®lter face and no-¯ux
speci®ed on the well casing. With the exception of a few
[35,45] and others who solved numerically the mixedtype boundary value problems in well hydraulics, all
other researchers who approached such problems analytically, avoided solving them by replacing the mixedtype boundary condition of speci®ed-head/no-¯ux either
by a ®rst-type (Dirichlet) boundary condition of speci®ed-head/zero-head [32] or by a second-type (Neumann)
boundary condition of speci®ed-¯ux/no-¯ux [8,9,15,17,
18,22,28±30,33,41,45]. The simpli®ed solutions obtained
in this manner are to a certain degree physically inconsistent, as explicitly acknowledged by Muskat [28] and
Hantush [15,17,19], and discussed in detail by Ruud and
Kabala [35].
3. The ¯owing partially-penetrating well problem
Cases to which the presented solution applies: (a)
partially penetrating ¯owing well in an aquifer of semiin®nite vertical extent; (b) double packer test geometry
in an aquifer of in®nite vertical extent. Consider a
¯owing well drilled in a formation of semi-in®nite vertical extent, such as shown in Fig. 1. Note that the
bottom of the wellbore is assumed not to collect any
¯ow from the aquifer. The wellbore of radius rw penetrates the aquifer down to a depth l. The imposed
drawdown in the well is sw . The drawdown at the distance r from the well, the distance z from the top of the
aquifer, and time t is s r; z; t. The aquifer parameters
are horizontal hydraulic conductivity Kr , vertical hydraulic conductivity Kz , and speci®c storativity Ss . The
governing partial dierential equation is
2
os
1 os
o2 s
os
K z 2 Ss
:
1
Kr
2
or
oz
ot
r or
61
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
packer
sw
r
r
z
z
packer
(a)
(b)
Fig. 1. Cases to which the presented solution applies. (a) partially penetrating ¯owing well in an aquifer of semi-in®nite vertical extent; (b) double
packer test geometry in an aquifer of in®nite vertical extent.
A skin of in®nitesimal thickness [27], is assumed
characterized by the skin factor g, de®ned by the following boundary condition at the well screen (z 6 l):
os
j
2
s jrrw sw :
ÿ grw
or rrw
We cast the problem in dimensionless terms, by de®ning the following parameters:
Dimensionless well screen length
d l=rw :
Dimensionless drawdown
3
s s =sw :
4
tKr
a 2:
Ss r w
Anisotropy ratio
5
a2
Kz
Kr
and dimensionless spatial coordinates:
6
q r=rw ;
7
f z=rw :
8
o2 s 1 os
o2 s os
a2 2 ;
2
oq
q oq
oa
of
9
Dimensionless time
The boundary value problem to be solved follows
from the partial dierential equation (1), that in dimensionless terms reads:
subject to the following boundary and initial conditions:
s q; f; a 0 0;
10
s q 1; f; a 0;
11
ÿg
os
j s q 1; f; a 1
oq q1
os
0
j
oq q1
for d < f;
os
j 0;
of f0
s q; f 1; a
for 0 6 f 6 d
12
13
os
j
0:
of f1
14
Note that Eq. (12) is a mixed boundary condition at the
well face, i.e. a third-type (or Robin) boundary condition along the well screen of dimensionless length d, and
a second-type (Neuman) no-¯ow boundary condition
along the casing.
We also note that, due to the symmetry around the
z 0 plane, a solution for a partially penetrating ¯owing well in the aquifer of in®nite thickness (or a double
packer test with long enough packers to prevent back¯ow) shown in Fig. 1 can be directly obtained from the
solution of Eqs. (9)±(14).
The solution technique employs the DE approach [43]
in a manner analogous to that used by Cassiani and
Kabala [5]. It is described in Appendix A. The relevant
computational details are discussed in Appendix B.
62
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
4. Model results and parameter identi®cation procedure
FD solution
G(α)
0
10
δ
10
20
50
100
10
1
Hantush [1959]
10
1
4
10
10
9
14
10
α=t Kr / (Ss rw2)
Fig. 3. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for the no-skin case (g 0) and dierent well screen
lengths. The [16] solution for a fully penetrating ¯owing well is shown
for reference. Also, a comparison is made with the results obtained
using the ®nite dierence (FD) code by [34].
1
10
η
δ=10
0
1
0
G(α)
The approach described in Appendices A and B allows us to compute eciently the ¯ow and drawdown
functions for a number of geometrical con®gurations
and parameter values.
In Fig. 2, we present an example of point ¯ux at the
well face for dierent times. The curves characteristically
exhibit sharp peaks at the extremes of the well screen
due to strong contribution of vertical ¯ux in the surrounding aquifer. They are analogous to those noted by
Dagan [8] in his semi-analytic solutions for steady-state
¯ow to partially penetrating wells in water-table aquifers, and by Widdowson et al. [45] in their corresponding numerical solutions for partially penetrating slug
tests. Unlike Dagan's results, only tiny oscillations occur
in our solution.
For the case of no skin (g 0) we validate the total
dimensionless ¯ux G calculated with our methodology
by comparing it to a numerical solution obtained via
®nite dierence algorithm of Ruud and Kabala [34]. As
seen in Fig. 3 the two curves show an excellent match.
We note that at large times all curves approach quasisteady state.
The eect of the skin factor g is shown in Figs. 4±6,
where type curves for d 10, 20 and 50 are shown for
g 0 through 10.
Similarly to the fully penetrating ¯owing wells [24],
the skin eect in partially penetrating wells produces an
early-time in¯ection point in the G curves. For large
times the skin factor lowers the curve asymptote.
The DE approach is ideally suited for incorporation
into an automated parameter identi®cation algorithm.
Let ti ; Qi ; i 1; . . . ; ndata be ®eld data of total ¯ow rate
versus time, collected during a constant-head test on a
partially penetrating well of known geometry. The formation parameter vector b fb1 ; b2 ; b3 g, b1 Kr ;
1
10
10
2
10
3
4
5
6
7
8
9
10
1
10
5
10
0
5
10
10
10
α=t Kr / (Ss rw2)
Fig. 4. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for d 10.
b2 Ss ; b3 g can be obtained from ti ; Qi by using an
optimization algorithm that minimizes the dierence
between ®eld data and model prediction. If f t is the
adopted model, let the distance vector and the Jacobian
matrix be given respectively by
Fig. 2. Dimensionless point ¯ux for the case of no skin (g 0), at
dierent time steps, for d 100.
15
F ff ti Qi g i 1; . . . ; ndata
and
of
i 1; . . . ; ndata ; j 1; 2; 3:
16
jtti
A
obj
An estimate of the parameter vector b is obtained by
Newton±Raphson iteration
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
63
1
10
η
δ=20
0
1
G(α)
0
10
2
10
3
4
5
6
7
8
9
10
1
10
5
10
0
5
10
10
10
α=t Kr / (Ss rw2)
Fig. 5. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for d 20.
10
Fig. 7. Example of Newton±Raphson ®tting to data. The data are
produced using the FD model by Ruud and Kabala [34]. The ®tting is
obtained using the DE solution as the physical model. The starting
point is Kr 10ÿ7 m/s, Ss 10ÿ5 1/m, and convergence to 0.1% is
achieved in only nine iterations.
1
η
that the ®rst guess for the parameters is not too far from
the true values (approximately within an order of magnitude) convergence is assured.
δ=50
0
G(α)
1
10
0
5. Comparison with existing models
2
10
3
4
5
6
7
8
9
10
1
10
5
0
10
10
5
10
10
α=t Kr / (Ss rw2)
Fig. 6. Dimensionless total ¯ux at the wellscreen as a function of dimensionless time, for d 50.
bk1 bk ÿ Ak F k ;
17
where k is the iteration step, and A is the pseudo-inverse matrix of A, de®ned as
ÿ1
A AT A AT ;
18
where T stands for transpose and ÿ1 for inverse ([13],
p. 243).
In this paper the DE algorithm is used to compute
f t. For the repeated computation of f t required by
Eqs. (15) and (16), the fast computation of DE is essential.
An example of Newton±Raphson parameter ®tting is
shown in Fig. 7. The data points are simulated using the
®nite dierence code by Ruud and Kabala [34]. Provided
In Fig. 3 we compare the results produced by our
model to the corresponding results for fully penetrating
¯owing wells. At early times the DE solution approaches, as expected, the corresponding solution for a
fully penetrating well, i.e. the no-skin solution of Hantush [16]. In addition, as mentioned earlier, the DE point
¯ux curves in Fig. 2 exhibit sharp increases at the well
screen edges, a behavior similar to that pointed out by
Dagan [8] in his model for water table aquifers.
The Novakowski [32] model was developed originally
for ®nite-thickness aquifers; however, his methodology
can be easily extended to aquifers of in®nite or semiini®nite vertical extent, such as the ones considered here.
In this section we consider the case with no skin, which
can be directly compared to our approach. We name
this method the drawdown±drawdown (D±D) approach, since it involves a discontinuous drawdown
boundary condition at the well face.
Consider the IBVP given by Eqs. (9)±(11) and (14)
and the following condition at the well-face, used by
Novakowski [32]:
19
s q 1; f; a d f ÿ f0 ;
where here d is the Dirac delta function.
Employing cosine Fourier transform instead of ®nite
cosine Fourier transform used by Novakowski [32] we
obtain
64
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
2
p
Z
1
0
q
2
K0 q an p
q cos nf0 cos nf dn:
2
K0
an p
0.5
20
Integration along the well face leads to the solution for
drawdown in the Laplace domain
s q; f; p
2
p
Z
1
0
q
K0 q an2 p
sin nd
q
cos nf dn:
pn
2
an p
K0
point flux
sd q; f; p
D-D sol.
DE sol.
21
Note that this solution is identical to that obtained by
imposing a unit drawdown on the well screen and zero
drawdown on the casing.
The point ¯ux at the well face follows from the above
by dierentiation
-0.5
-200
Z
1
0
Integration of Eq. (22) along the well face leads via
Eq. (A.22) to the dimensionless total ¯ux G , analogous to Novakowski's [15], (A.9)
G p
2
p2
100
200
1.0
q q
2
2
an pK1
an p
sin nd
q
cos nf dn:
pn
an2 p
K0
22
0
Fig. 8. Comparison between the dual integral equation solution (DE)
and the drawdown±drawdown (D±D) solution for d 100 and time
a 103 and no skin (g 0). The D±D solution exhibits spurious
negative ¯ux on the well casing. For a closer look, see Fig. 9.
Z
1
0
q q
2
2
an p
an pK1
1 sin2 nd
q
dn:
d
n2
2
p K0
an p
23
The above integral is divergent, as is the corresponding
series solution of Novakowski [15], Eq. (A.9) for the
case of no skin. The reason why the total ¯ux is in®nite
in this solution can be seen in Figs. 8 and 9. There we
present the point ¯ux at dimensionless time a 103 for a
well of dimensionless screen length d 100 and no skin
(g 0) calculated from our DE solution, (A.19), and
from the D±D solution (22). The D±D solution shows
strong spurious oscillations, known as Gibbs eect,
caused by truncation in the frequency domain. The
curves in Figs. 8 and 9 appear to be blurred because of
this eect. The DE solution is represented by a thick
dot±dashed line. Note that the D±D solution produces
spurious negative ¯ux outside the well casing. Along the
well screen away from its edges the two solutions are
very close to each other. However, they are substantially
dierent around the wellscreen edge. The DE solution
takes reasonable values whereas the D±D solution di-
0.5
point flux
2
p
-100
ζ
os
j
oq q1
ÿ
0.0
0.0
D-D sol.
DE sol.
~1/(ζ−δ)
-0.5
-1.0
-110
-100
-90
ζ
Fig. 9. Same case as in Fig. 8, zoomed around f ÿd ÿ100. Note
the oscillations in the D±D solution (Gibbs eect). As opposed to the
DE solution, the D±D point ¯ux at the screen edge diverges to in®nity
as 1= f ÿ d, leading thus to a spurious in®nite total ¯ux (area under
the curve).
verges to in®nity as 1= f ÿ d, leading to in®nite total
¯ux (the area under the curve), clearly a physical impossibility.
6. Conclusions
The main results of this paper are:
1. A new semi-analytic solution to the mixed-type boundary value problem for a ¯owing partially penetrating
65
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
2.
3.
4.
5.
6.
well with in®nitesimal skin situated in an anisotropic
aquifer is developed via Laplace and Fourier transforms. The solution is suited for aquifers of semi-in®nite vertical extent for packer tests with aquifer
horizontal boundaries far enough from the tested
area. The problem can be reduced to a system of
DE and further to the deconvolution Eq. (A.18),
which we solve by discretizing the integral. Inversion
of the Laplace and Fourier transforms are handled
numerically via Stehfest and fast Fourier transform
(FFT) algorithms.
The solution is achieved by discretizing a deconvolution equation. An ecient discretization needs to satisfy two opposite requirements: (i) it has to be ®ne
enough to sample adequately the kernel function,
and (ii) it must be crude enough to prevent unwanted
oscillations. For no skin, g 0, the density of 0.6/a2
nodes per unit f produces accurate results, whereas
for g 10 the corresponding density increases to
1.6/a2 nodes per unit f.
The new solution is computationally robust due to
the impulse-like nature of the kernel in the deconvolution equation. We note that unlike the analogous
Dagan's [8] steady-state solution for ®nite-thickness
uncon®ned aquifers, our DE solution embedded in
Eq. (A.18) does not suer from numerical oscillations.
Our DE solution matches the corresponding ®nitedierence solution obtained using the code developed
by Ruud and Kabala [34]. The DE solution is computationally much more ecient. Although the computational burden is generally limited, it increases with
the well screen lengths.
Based on our DE solution (forward model), an automated (Newton±Raphson) parameter identi®cation
algorithm is developed for ®eld test interpretation.
The procedure is computationally ecient and converges to correct parameter values, provided that
the initial parameter guess is not too far from the true
values (approximately within one order of magnitude).
A solution for the partially penetrating ¯owing well
with no skin and D±D discontinuous boundary condition, analogous to that of Novakowski [15],
Eq. (A.9) was also derived. The D±D solution was
compared to the DE solution. The D±D solution
leads to physically inconsistent in®nite total ¯ow rate
to the well, when no skin eect is considered. The DE
solution, on the other hand, produces accurate
results.
Acknowledgements
The authors are indebted to Nels C. Ruud for the use
of numerical code developed in his dissertation. This
research was performed while G. Cassiani was a graduate student and Z.J. Kabala and M.A. Medina, Jr.
were his faculty advisors in the Department of Civil and
Environmental Engineering at Duke University, Durham, NC.
Appendix A. The solution technique
We eliminate the time variable a from the problem by
applying the Laplace transform to Eqs. (9), (11)±(14),
and using the initial condition (10). The Laplace-transformed variable is denoted by a bar, thus
Z 1
eÿpa s q; f; a da:
s q; f; p Lfs q; f; a; a ! pg
0
A:1
The transformed boundary-value problem is
2
o2 s 1 os
s
2o
a
ÿ p s 0;
oq2 q oq
of2
A:2
s q 1; f; p 0;
A:3
ÿg
os
1
j s q 1; f; p
oq q1
p
os
0
j
oq q1
for 0 6 f 6 d
for d < f;
os
j 0;
of f0
A:4
A:5
os
j
0:
A:6
of f1
Next, we eliminate f by applying the Fourier cosine
transform to Eq. (A.2) and Eq. (A.3) and by using
Eq. (A.6). In order to apply properly the mixed boundary condition (A.4), Tranter ([43], p.51) suggests leaving it out and applying it only after inverting back from
the Fourier domain. The Fourier-transformed variable
is denoted by aR hat, thus ^s q; n; p Ffs q;
1
f; p; f ! ng 2=p 0 s q; f; p cos nf df. We thus
arrive at the Bessel dierential equation
s q; f 1; p
o2^s 1 o^s
ÿ a2 n2 p ^s 0;
oq2 q oq
subject to the boundary condition
A:7
^s q 1; n; p 0:
A:8
The general solution is in the form of
q
^s q; n; p A n; p K0 q a2 n2 p ;
A:9
where K0 is the modi®ed Bessel function of order zero
and the unknown A n; p is a constant with respect to q
to be determined from the mixed boundary condition
(A.4) after cosine Fourier transform inversion. The solution in the f domain is thus
66
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
s q; f; p
Z
0
q
1
A n; p K0 q a2 n2 p cos nf dn;
whereas its derivative with respect to q reads
A:10
os
q; f; p
oq
Z 1
q q
A n; p a2 n2 p K1 q a2 n2 p cos nf dn;
ÿ
0
A:11
where K1 is the modi®ed Bessel function of order one.
Substitution of Eq. (A.10) and Eq. (A.11) into
Eq. (A.4) leads to a system of two integral equations,
known as DE ([43], p. 111):
Z 1
q q
A n; p ÿ g a2 n2 p K1 q a2 n2 p
0
q
1
a2 n2 p
for jfj 6 d;
cos n f dn
K0
p
A:12
Z
1
0
q q
a2 n2 p cos nf dn 0
A n; p a2 n2 p K1
for jfj > d:
A:13
Systems (A.12) and (A.13) can be solved in a number of
ways [43] we transform it into a deconvolution problem
in the space domain. This approach is convenient because the Bessel function kernels in Eqs. (A.12) and
(A.13) have closed-form Fourier cosine transforms ([2],
vol. I, p. 56):
Z 1 q
^
a2 n2 p cos nf dn
K0
K 0 f; p
0
1p
1
q exp
a2
2
f=a 1
K^1 f; p
1p
a2
Z
0
q
2
ÿ p f=a 1 ;
A:14
q
q
1
a2 n2 p K1
a2 n2 p cos nf dn
1
f=a2 13=2
2
p !
p
f=a 1
exp
q
2
ÿ p f=a 1 :
A:15
With Eq. (A.14) and Eq. (A.15) and the convolution
theorem for Fourier transforms, problem (A.12) and
(A.13) can be restated as:
Z
Z
1
^ p K^0 f ÿ x; p ÿ g K^1 f ÿ x; p dx 1
A x;
p
ÿ1
for jfj 6 d;
A:16
1
ÿ1
^ p K^1 f ÿ x; p dx 0 for jfj > d;
A x;
A:17
^ p is the inverse Fourier transform of A n; p.
where A x;
Solution to Eq. (A.16) and Eq. (A.17) can be achieved
by discretizing the vertical space domain (here expressed
in terms of coordinate f or x). Our numerical experi^ p is pracments show that the unknown function A f;
tically zero outside the well screen interval jfj 6 d, so
that Eq. (A.16) and Eq. (A.17) can be conveniently reduced to the Fredholm integral equation
Z d
^ p K^0 f ÿ x; p ÿ g K^1 f ÿ x; p dx 1
A x;
p
ÿd
for jfj 6 d:
A:18
^ p has been obtained from Eq. (A.18), the
Once A f;
point ¯ux in the Laplace domain os= oqjq1 f; p can
be obtained by Fourier transform
os
j f; p
oq q1
Z 1
q q
a2 n2 p cos nf dn:
A n; p a2 n2 p K1
ÿ1
A:19
Fourier transformations can be conveniently carried out
via the fast Fourier transform (FFT) [4]. We use Stehfest
[39] algorithm to obtain the point ¯ux os= oqjq1 f; a
in the dimensionless time domain.
The total dimensionless ¯ux to the partially penetrating well can be de®ned analogously to the corresponding dimensionless ¯ux for a fully penetrating
¯owing well given by Hantush [16]
Q t
;
A:20
2pKr lsw
where Q t is the total dimensional ¯ux, computed as
Z l
os
Q t ÿ2prw Kr
jrw dz:
A:21
0 or
Consequently, G can be expressed in terms of the dimensionless point ¯ux as
Z
1 d os
j df
A:22
G a; d ÿ
d 0 oq q1
and it can be extracted from measurements in the well.
The drawdown at any point in space and time can be
computed from Laplace inversion of Eq. (A.10), or by
convolution
Z 1
^ p K^r0 q; f ÿ x; p dx;
s q; f; p
A:23
A x;
G a; d
ÿ1
where
K^r0 q; f; p
p
1
q exp
2a
f=a2 q2
ÿ
q
p f=a2 q2 :
A:24
G. Cassiani et al. / Advances in Water Resources 23 (1999) 59±68
The convolution approach is more stable because it is
consistent with the slight approximation introduced by
limiting the deconvolution from ÿd to d only (see
Eq. (A.18)).
Appendix B. Computational details
Although generally deconvolution problems are illposed [7], in some cases highly accurate solutions can be
obtained. Dagan [8] notes that ``an accurate solution
is expectable if the kernel is strongly peaked at
f x, whereas the solution will worsen if the kernel is
¯at''. Fortunately, the kernel in Eq. (A.18) is strongly
peaked at f x for all values of p, and thus for all values
of dimensionless time a.
We note that unlike Dagan's [8] analogous steadystate solution for ®nite-thickness uncon®ned aquifers,
our DE solution embedded in Eq. (A.18) does not suer
from severe numerical oscillations.
Solving Eq. (A.18) requires a carefully calibrated
discretization along the well screen. An ecient discretization can be found by satisfying as well as possible
the two opposite qualitative requirements: (i) it has to be
®ne enough to sample adequately the kernel function,
and (ii) it must be crude enough to prevent unwanted
oscillations.
Our numerical experiments demonstrate that an ecient discretization can be determined independently of
d. However, it depends on a2 and g. For no skin, g 0,
the density of 0.6/a2 nodes per unit f produces accurate
results, whereas for g 10 the corresponding density
increases to 1.6/a2 nodes per unit f. For intermediate
cases proportional increments can be used. Note how
the dependence on a2 corresponds to a pure scaling of
the boundary value problem in the vertical direction.
Since the optimal Df is independent of d, the number of
points needed is a linear function of d itself. Consequently, the computational burden increases with the
size of the deconvolution matrix arising from discretization of Eq. (A.18), as d increases.
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