7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
1/14
On the horizontal-well pumping tests in anisotropic connedaquifers
Hongbin Zhana,*, Lihong V. Wangb,1, Eungyu Parka
aDepartment of Geology and Geophysics, Texas A & M University, College Station, TX 77843-3115, USAbBiomedical Engineering Program, Texas A & M University, College Station, TX 77843-3120, USA
Received 20 September 2000; revised 26 March 2001; accepted 14 May 2001
Abstract
A method that directly solves the boundary problem of ow to a horizontal-well in an anisotropic conned aquifer is
provided. This method solves the point source problem rst, and then integrates the point source solution along the horizontal
well axis to obtain the horizontal well solution. The short and long time approximations of drawdowns are discussed and are
utilized in the semilog analysis of the drawdown. A closed-form analytical solution of geometrical skin effect at the wellbore is
derived. Type curves and derivative type curves of horizontal pumping wells are generated using the chow program. This
program also calculates the drawdown at any given observation well at any given time. The horizontal-well type curves are
different from the vertical-well type curves at early time, reecting the different nature of ow to a horizontal-well and to a
vertical-well. The horizontal-well type curves converge to the vertical-well type curves at late time, showing the similar nature
of ow to a horizontal-well and to a vertical-well at late time. The sensitivity of the type curves and derivative type curves onmonitoring well location, aquifer anisotropy, horizontal well depth, and horizontal well length is tested. These type curves and
derivative type curves can be used in the matching point method for interpreting the pumping test data. q 2001 Elsevier
Science B.V. All rights reserved.
Keywords: Horizontal-well; Pumping-test; Type-curve; Derivative
1. Introduction
Horizontal-wells have been broadly used in the
petroleum industry in the past fteen years. Pressure
behavior of horizontal-well pumping in petroleum
reservoirs has been studied, with interpretations ofpressure data often proving challenging (Goode and
Thambynayagam, 1987; Daviau et al., 1988; Ozkan et
al., 1989; Rosa and Carvalho, 1989). The difculty in
interpretation is caused by a combined impact upon
the pressure distribution from conning boundaries
and a nite well screen length.
Horizontal-wells have advantages in at least two
scenarios of environmental and hydrological applica-
tions. The rst is a situation in which direct site accessis forbidden or difcult, exemplied by permanent
surface constructions, ponds, wetlands, or landlls
above the site area. Another scenario is a dense-non-
aqueous-phase-liquids (DNAPLs) contaminated site
in which DNAPLs sink to the aquifer bottom. Shallow
horizontal-wells are also commonly used in air
sparging and vent extractions. Advantages in some
situations, combined with reduced operational cost
have led to increasing utilization of horizontal-well
Journal of Hydrology 252 (2001) 3750
0022-1694/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S0022-1694(0 1)00453-X
www.elsevier.com/locate/jhydrol
* Corresponding author. Tel.: 11-979-862-7961; fax: 11-979-
845-6162.
E-mail addresses: [email protected] (H. Zhan), lwang@-
tamu.edu (L.V. Wang).1 Tel.: 11-979-847-9040, Fax: 11-979-845-4450.
7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
2/14
technology in hydrological applications in recent
years (Langseth, 1990; Tarshish, 1992; Cleveland,
1994; Sawyer and Lieuallen-Dulam, 1998; Zhan,
1999; Zhan and Cao, 2000).Hantush and Papadopulos (1962) have performed
an early investigation on uid ow into a collector
well, which includes a series of jointed horizontal-
wells. They provided an analytical solution of the
long time approximation of drawdown distribution
around collector wells. No detail of derivation was
provided in their paper and no solutions were given
for the short and intermediate times. Tarshish (1992)
constructed a mathematical model of ow in an
aquifer with a horizontal-well located beneath a
water reservoir. Falta (1995) has developed analytical
solutions of transient and steady-state gas pressure
and steady-state stream functions resulting from gasinjection and extraction from a pair of parallel
horizontal-wells. Rushing (1997) has established a
semianalytical model for horizontal-well slug testing
in conned aquifers. Zhan (1999) and Zhan and Cao
(2000) have investigated capture times of horizontal-
wells, where the capture time is dened as the time a
uid particle takes to ow to the well. Murdoch
(1994) has studied ground water ow to an interceptor
trench, and Hunt and Massmann (2000) have recently
H. Zhan et al. / Journal of Hydrology 252 (2001) 37 5038
Nomenclature
d aquifer thickness (m)
K0 the modied Bessel function of second kind and order zeroKh horizontal hydraulic conductivity (m/s)
Kz vertical hydraulic conductivity (m/s)
L horizontal-well screen length (m)
LD dimensionless horizontal-well screen length dened as LD L=d
Kz=Khp
Q horizontal-well pumping rate (m3/s)
rD dimensionless horizontal distance from an observation well to a horizontal-well (m), rD x2D 1y
2D1=2
rw radius of a horizontal-well (m)
rwD dimensionless radius of a horizontal-well
s drawdown (m)
sD dimensionless drawdown dened in Eq. (6)
s HD dimensionless drawdown in the Laplace domainsHD dimensionless drawdown of the horizontal-well
sHHD dimensionless drawdown of the horizontal-well in the Laplace domain
Ss specic storativity (m21)
t time (s)
t0 time when drawdown equals zero (s)
tD dimensionless time dened in Eq. (6)
x off-center coordinate along the well axis (m)
xD dimensionless x dened in Eq. (6)
y horizontal coordinate perpendicular to the well axis (m)
yD dimensionless horizontal coordinate perpendicular to the well axis
z vertical coordinate (m)zD dimensionless vertical coordinate
zw distance from the horizontal-well to the bottom boundary (m)
zwD dimensionless zw dened in Eq. (6)
x0, y0, z0 coordinates of the point source (m)
x0D, y0D, z0D dimensionless coordinates of the point source
a geometrical skin effect dened in Eq. (31)
7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
3/14
investigated vapor ow to a trench. Cleveland (1994)
and Sawyer and Lieuallen-Dulam (1998) have
compared the recovery efciency of horizontal and
vertical wells. Petroleum engineers have studied the
pressure changes in oil reservoirs due to horizontal
pumping wells (Goode and Thambynayagam, 1987;
Daviau et al., 1988; Ozkan et al., 1989; Rosa and
Carvalho, 1989). Many of those works in petroleum
engineering use the source function and Green's func-
tion methods proposed by Gringarten and Ramey
(1973) and Gringarten et al. (1974).In this paper, a method is proposed to directly solve
the boundary problem of ground water ow to a
horizontal-well, and solutions of drawdowns are
provided. A closed-form solution of wellbore geo-
metrical skin effect is derived. Computer software
based on the analytical study of this paper is written.
This software can calculate the drawdown of a hori-
zontal pumping well at any given time for either one
of the following three monitoring schemes: a fully and
a partially penetrating vertical observation wells, and
an observation piezometer (a point). The software also
provides type curves and derivative type curves forhorizontal pumping wells. Applications of the analy-
tical solutions and computer software for horizontal-
well pumping tests are discussed last.
2. Ground water ow to a horizontal-well in an
anisotropic conned aquifer
Fig. 1 is a schematic diagram of thecoordinate system
setup and a horizontal-well in a conned aquifer. Thex-
andy-axes are in the horizontal directions and thez-axis
is in the vertical direction. The origin is at the bottom of
the aquifer. The well is along the x-axis and its center is
at 0; 0;zw; wherezw is the distance from the well to thebottom boundary.The lateral boundaries are sufciently
distant so as not to inuence the ow. The top and
bottom boundaries are impermeable. We assume that
the hydraulic conductivities in the x and y directions
are the same, but they are different from the hydraulic
conductivity in the vertical direction.
2.1. Ground water ow to a point source in an
anisotropic conned aquifer
Before solving the problem of groundwater ow to
a horizontal-well, we rst solve the problem of ground
water ow to a point source. The governing equation
and the associated initial and boundary conditions for
a point source pumping in an anisotropic aquifer are:
Ss2h
2
t Kh
22
h
2
x
21 Kh
22
h
2
y
21 Kz
22
h
2
z
2
2 Qdx2x0dy2y0dz2z01
hx;y;z; t 0 h0 2
2hx;y;z 0; t=2z 0 3
2hx;y;z d; t=2z 0 4
hx ^1;y;z; t hx;y ^1;z; t h0 5
H. Zhan et al. / Journal of Hydrology 252 (2001) 37 50 39
Fig. 1. Schematic diagram of a horizontal-well in a conned aquifer.
7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
4/14
where Ss is the specic storativity (m21); h, the
hydraulic head (m); t, the time (s); Kh, Kz, the hydrau-
lic conductivities (m/s) in the horizontal and vertical
directions, respectively; Q, the pumping rate (m3/s)
Q . 0 for pumping and Q , 0 for injecting); d, theDirac delta function (m21); h0, the initial hydraulic
head (m); d, the aquifer thickness (m); and
x0;y0;z0 is the source location. The point source isincluded as a Dirac delta function in Eq. (1).
We change the hydraulic head h to drawdown s h0 2 h and dene the following dimensionless para-
meters:
sD 2pKhd
Qs; tD
Kz
Ssd2
t; xD x
d
Kz
Kh
s; yD
yd
Kz
Kh
s; zD
z
d6
where sD, tD, xD, yD, and zD are the dimensionless
counterparts of s, t, x, y, and z, respectively. The
dimensionless ow equation and initial and boundary
conditions become:
2sD
2tD 2
2sD
2x2D1
22
sD
2y2D1
22
sD
2z2D
1 2pdxD 2x0DdyD 2y0DdzD 2z0D
7
sDxD;yD;zD; 0 0 8
2sDxD;yD; 0; tD=2zD 0 9
2sDxD;yD; 1; tD=2zD 0 10
sD^1;yD;zD; tD sDxD;^1;zD; tD 0 11where x0D, y0D, and z0D are the dimensionless counter-
parts ofx0, y0, and z0, respectively.
Conducting the Laplace transform to Eq. (7) andboundary conditions (9)(11) results in
psHD
22
sHD
2x2D1
22
sHD
2y2D1
22
sHD
2z2D
12pdxD 2x0DdyD 2y0DdzD 2z0D
p
12
2sHDxD;yD; 0;p=2zD 0 13
2sHDxD;yD; 1;p=2zD 0 14
sHD^1;xD;zD;p s HDxD;^1;zD;p 0 15
where p is the Laplace parameter referred to thedimensionless time,s HD is the dimensionless drawdownin the Laplace domain. Eq. (12) is solved in the
Appendix and the result is
sHDp
1
pK0rD
p
p 1 2p
X1n1
cosnpzD cosnpzwD
K0rD
n2p2 1p
q (16)
where rD xD 2x0D2 1 yD 2y0D21=2; andzwD zw=d:
Using the inverse Laplace transform table of
Hantush (1964, p. 303), the dimensionless drawdown
of a point source in real time is obtained analytically
as
sDtD 1
2W
"r
2D
4tD
#1
X1n1
cosnpzD cosnpzwD
W
"r
2Dx HD4tD
; nprD
#
17where Wu and Wu; v are the well function andleaky well function, respectively (Hantush, 1964).
2.2. Ground water ow to a horizontal-well in an
anisotropic conned aquifer
It is generally agreed that the use of a uniform-head
boundary to simulate a horizontal-well is closer to
physical reality, but this boundary is difcult to incor-
porate in analytical studies (Rosa and Carvalho,
1989). Instead, a uniform-ux boundary is easier to
implement and commonly used (Daviau et al., 1988;
Langseth, 1990; Cleveland, 1994).
We test a hypothetical case of a 40 m long horizon-
tal-well in a 20 m thick conned aquifer under both
uniform-ux and uniform-head wellbore conditions
using visual modow software (Waterloo Hydro-
geologic, 2000). The uniform-head wellbore is
simulated by assigning an extremely large hydraulic
conductivity, to each of the cells representing the hori-
zontal-well. The numerical simulations show that
H. Zhan et al. / Journal of Hydrology 252 (2001) 37 5040
7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
5/14
when the distance between a measured point to a well
end is ten times that of the horizontal-well diameter,
the discrepancy of the uniform-ux and the uniform-
head results is less than 5%. If using a 0.15 m
diameter horizontal-well, this implies that when the
monitoring well is 1.52 m away from the well end,
there is less than 5% difference between the
uniform-ux and the uniform-head solutions. This
nding agrees with a previous study of Rosa and
Carvalho (1989), who had performed an analysis on
the geometrical skin effect difference between a
uniform-ux and a uniform-head boundaries. In one
example shown in Fig. 4 of Rosa and Carvalho (1989),
they found that the geometrical skin effect were about
0.9075 and 0.8717 for a uniform-ux and a uniform-
head solution, respectively. The difference of thesetwo geometrical skin effects was less than 5%.
Thus, one can employ an approximation of uniform
strength of sinks for practical hydrogeological appli-
cations. Such a treatment is consistent with previous
studies (Hantush and Papadopulos, 1962; Daviau et
al., 1988; Ozkan et al., 1989; Rosa and Carvalho,
1989). If treating a horizontal-well as a uniform-ux
source along the x direction in the xz plane (see Fig.
1), the drawdown of the horizontal pumping well is
obtained through an integration of the point source
solution along the well axis. In the Laplace domain,the result is
sHHDp
1
LD
"ZLD=22LD=2
1
pK0
p
prDx HDdx HD
12X1n1
cosnpzD cosnpzwD
ZLD=22LD=2
1
pK0
n2p2 1p
qrDx HDdx HD
#18
In the real time domain, the result is
sHDtD 1
2LD
"ZLD=22LD=2
W
"r
2Dx HD4tD
#dx HD
12X1n1
cosnpzD cosnpzwDZLD=22LD=2
W"
r2Dx HD4tD
; nprDx HD#
dx HD
#19
where s HHDp and sHD(tD) are the dimensionless draw-downs in the Laplace domain and the real time
domain for a horizontal-well, respectively. LD is the
dimensionless well screen length dened as LDL=dKz=Khp ; rDx HD is
rDx HD xD 2x HD2 1y2D1=2 20
We also can express Eq. (19) in an alternative
format with an integration to time. Notice that the
well functions can be written in the following formats
if assigning u r2Dx HD=4t :
Wr
2Dx HD4tD
" #Z1
r2Dx H
D=4tD
1
ue2udu
ZtD
0
1
texp 2
xD 2x HD2 1y2D4t
" #dt 21
Wr
2Dx HD4tD
; nprDx HD" #
Z1
r2Dx HD=4tD
1
uexp 2u2
n2p
2r
2Dx HD
4u
" #du
ZtD
0
1
texp 2n2
p
2
t2 xD 2x
HD
21y
2D
4t" #dt22
Substituting Eqs. (21) and (22) and into Eq. (19)
results in
sHDtD p
p
2LD
ZtD0
1t
p
erf
LD=21xD2t
p
1erf
LD=22xD2 tp
exp
2
y2D
4t
11 2
X1n1
cosnpzD cosnpzwD exp2n2p2t
dt
23Eq. (23) is the solution for an observation piezo-
meter. The solution for a partially penetrating obser-
vation vertical-well with a screen from z1 to z2 is
sHDtD 1
z2D 2z1D
Zz2Dz1D
sHDdzD 24
H. Zhan et al. / Journal of Hydrology 252 (2001) 37 50 41
7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
6/14
where z2D z2=d and z1D z1=d are the dimension-less z2 and z1, respectively.
The solution for a fully penetrating observation
vertical-well is
sHDtD Z1
0sHDdzD 25
It is interesting to point out that, after changing to
the dimensional format, Eq. (23) agrees with what is
reported in the petroleum literature, which uses differ-
ent means to derive the uid pressure change, such as
source function and Green's function methods (Grin-
garten and Ramey, 1973; Daviau et al., 1988, Table 1;
Ozkan et al., 1989, Eq. (1)). The method presented in
this study has a potential for application for different
well congurations. For instance, by obtaining thepoint source solution rst, we are able to nd solutions
for vertical, horizontal, inclined, and even curved line
sources. Furthermore, the volume integrations of the
point source solution may yield solutions for nite-
diameter vertical, horizontal, inclined, and curved
wells. Further discussion of applying the method for
different well congurations is beyond of the scope of
this paper and will be reported elsewhere.
2.3. Short and long time approximations of
drawdowns in anisotropic conned aquifers
2.3.1. Short time approximation
Drawdown of horizontal-well pumping commonly
shows a three-stage prole, i.e. an early, an intermedi-
ate, and a late stage (Daviau et al., 1988; Ozkan et al.,
1989; Zhan and Cao, 2000). The short time approx-
imation of drawdown has been studied before and is
briey summarized below. The drawdown at the early
stage is:
s bQ
4pLKhKzp ln2:25tKhKz
SsKzy
21 Kh
z2zw
2
if
SsKzy2 1 Khz2zw24tKhKz
, 0:01
26
where b 1 if uxu # L=2; b 0 if uxu . L=2: The timelimits for short time approximation are (Daviau et al.,
1988, p. 717):
t# 0:08z2wSs=Kz if zw=d# 0:5;
t# 0:08d2zw2Ss=Kz if zw=d. 0:527
and
t# SsL=22 uxu2=6Kz 28The ending time of the intermediate stage is
approximately (Daviau et al., 1988, p. 718)
0:8SsL=22=Kz , t, 3SsL=22=Kz 29It is worthwhile to point out that Eqs. (27)(29)
only give order-of-magnitude estimations; other
authors may use slightly different formulae (Murdoch,
1994, p. 3027).
2.3.2. Long time approximation
After the intermediate stage, the equipotential
surface in the far eld is similar to a cylinder. The
drawdown at this late stage is approximated as (Rosa
and Carvalho, 1989):
s Q4pKhd
ln2:25Kht
SsL=221 a
30
where a is called geometrical skin factor and it is
modied from the solution of Rosa and Carvalho
(1989, Eqs. (44) and (46)):
a 2
1 12( xD
LD=22 1!ln" yD
LD=2!21 xD
LD=22 1!2#
2
xD
LD=21 1
!ln
"yD
LD=2
!21
xD
LD=21 1
!2#)
2yD
LD=2
"tan21
xD 1LD=2
yD
!2 tan21
xD 2LD=2
yD
!#
12
X1
n
1
cosnpzwD cosnpzDZ
xD=LD=21 1
xD=LD=22 1
K0
np
LD
2
u2 1
yD
LD=2
!2vuut !du31
Eq. (31) is the geometrical skin effect for an obser-
vation piezometer. The geometrical skin effects for a
partially penetrating observation well and a fully
penetrating observation well are easily calculated
using the similar average schemes of Eqs. (24) and
H. Zhan et al. / Journal of Hydrology 252 (2001) 37 5042
7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
7/14
(25). We have written a program chow to numerically
calculate the geometrical skin effect of Eq. (31)
(Additional information on program chow is
provided in Section 3.2).
Now we use Eq. (31) to derive the closed-form
analytical solution of a at the horizontal-wellbore.
As pointed out in previous studies (Daviau et al.,
1988; Rosa and Carvalho, 1989), drawdown at an
equivalent point of the wellbore ( 0.68 of the half-well-length from the well center) when using the
uniform-ux boundary is very close to the uniform-
head boundary solution. Thus we calculate a at
xD=LD=2 0:68; yD 0; and zD zwD 1 rwD;where r
wis the radius of the horizontal-well and rwD
rw=d: After a few simple calculations, Eq. (31)
becomes
a 22 0:61 2pLD
X1n1
1
ncosnp2zwD 1 rwD
1 cosnprwD
Z1:68npLD0
K0udu1Z0:32npLD
0K0udu
32
The horizontal-well half-length is usually largerthan the aquifer thickness LD . 1 in hydrologicalapplications (Tarshish, 1992; Zhan, 1999). Consider-
ing the following identity (Hantush, 1964)
Zw0
K0udu p
2if w $ p 33
Thus
Z1:68npLD
0K0udu
p
2;
and
Z0:32npLD0
K0udu p
2if 0:32nLD $ 1
The condition 0:32nLD $ 1 is satised for n $ 3
and LD $ 1:04: For n 1; and 2, this approximationmay result in a slightly over-estimated a if LD is
not much larger than 1. In practice, Eq. (32) is
approximated as
a 1:41 2LD X
1
n
1
1
ncosnp2zwD 1 rwD
1 cosnprwD 34Using the identity
X1n1
cosann
12
X1n1
eian 1 e2iann
2 12
ln12 eia2 12
ln12 e2ia
2 12
ln22 2cosa 35
where i 21
pis the complex sign, and considering
the fact that rwD ! 1; then Eq. (34) becomes
a 1:42 1LD
ln412 cosp2zwD 1 rwD1
2 cosprwD
1:42 1LD
ln2p2r2wD12 cosp2zwD 1 rwD
36
Furthermore, if the horizontal-well is in the middleof the aquifer, zwD 1=2; then a becomes
a 1:42 2LD
ln2prwD 1:412d
L
Kh
Kz
sln
d
2prw
37
We can use Eq. (36) for a general well position or
Eq. (37) for a center well to calculate the wellbore
geometrical skin effect a .
It is interesting to point out that the above Eq. (36)
agrees with Eq. (12) of Hantush and Papadopulos
(1962) if choosing a single horizontal-well, i.e. N 1 in their Eq. (12). The slight difference is that we
evaluate the geometrical skin effect at 0.68 of the
half-well-length from the well center, but Hantush
and Papadopulos (1962) evaluated the geometrical
skin effect at the end of the well, i.e. at xD=LD=2 1: If the geometrical skin effect at the well end is also
evaluated, our solution is identical to that of Hantush
and Papadopulos (1962). Unfortunately, no detail was
given in Hantush and Papadopulos (1962) to show
H. Zhan et al. / Journal of Hydrology 252 (2001) 37 50 43
7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001
8/14
their derivation. We need to point out that the geo-
metrical skin effect at the well end is not suitable for
calculating the average wellbore drawdown in our
case. The geometrical skin effect at the equivalent
point (around 0.68 of the half-well-length) offers the
closest approximation of the average geometrical skin
effect along the wellbore.
3. Applications on horizontal-well pumping test
interpretation in anisotropic conned aquifers
The previous discussion of uid ow to a horizon-
tal-well will guide us for the horizontal-well pumping
test interpretation.
3.1. Semilog analysis of drawdown of a horizontal-
well
From the above analysis, it is interesting to nd out
that the dimensionless drawdowns at the short and
long times are proportional to the logarithm of dimen-
sionless time. Thus, plotting drawdown and time in a
semilog paper will yield two straight lines for the
early and late pumping stages. This is similar to the
Cooper and Jacob method used in the vertical-well
pumping test interpretation (Cooper and Jacob,
1946). A slight difference is that a geometrical skinfactor a is included in the late time approximation in
this paper.
We can ndKhKz
pfrom the slope of the straight
line of the early data (Eq. (26)) usingKhKz
p 2:3Q=4pL slope: Extending the straight line tond the intercept point 0; t0 with axis s 0 yieldsSs 2:25KhKzt0=Kzy2 1 Khz2zw2: It is clearthat, only when both Kh and Kz are known, can Ss be
calculated. Kh and Kz can be found when choosing at
least two monitoring points.
Kh can also be found from the slope of the straight
line of the late data using Kh 2:3Q=4pd slope:Extending the straight line to nd the intercept point
0; t0 with axis s 0 yields Ss ea 2:25Kht0=L=22; wherea is the geometrical skin effectcalculated from Eq. (31) for a general monitoring
point or from Eq. (36) or (37) for a wellbore point.
At least two monitoring points are needed to nd Kzand Ss. This is done by running the chow program
with different values ofKz until Ss obtained from two
monitoring points agrees with each other.
Several practical aspects should be addressed.
1. In a practical sense, the drawdown data of the early
radial ow and the late pseudoradial ow may not
always be available in a pumping test. For instance,
given zw=d 0:5; d 10 m; L 20 m; Ss 0:005 m21; and K 40 m=day; the top and bottomboundaries will inuence the ow at about t