Analytical and semi-analytical solutions of horizontal well capture
times under no-¯ow and constant-head boundaries
Hongbin Zhan
*, Jiang Cao
Department of Geology & Geophysics/MS 3115, Texas A&M University, College Station, TX 77843-3115, USA
Received 2 August 1999; received in revised form 21 February 2000; accepted 6 March 2000
Abstract
The closed-form analytical solutions and semi-analytical solutions of capture times to horizontal wells are derived for dierent recovery scenarios. The capture time is the time a ¯uid particle takes to ¯ow to the well. The ®rst scenario is recovery from a con®ned aquifer in which the in¯uence of regional groundwater ¯ow upon the capture time is included. The second scenario is recovery from underneath a water reservoir in which the top boundary of the aquifer is constant-head. The third scenario is recovery from a low-permeability layer bounded above and below by much higher permeability media. Closed-form solutions are provided for the cases with: (1) a center or a bottom well for the ®rst scenario; (2) a bottom well for the second scenario; and (3) a center well for the third scenario. Semi-analytical solutions are provided for general well locations for those scenarios. Solutions for both isotropic and anisotropic media are studied. These solutions can be used as quick references to calculate the capture times, and as benchmarks to validate numerical solutions. The limitations of the analytical solutions are analyzed. Our results show that the top and bottom no-¯ow boundaries of an aquifer constrain the vertical ¯ow, but enhance the horizontal ¯ow, resulting in elongated iso-capture time curves. When constant-head boundaries are presented, water can in®ltrate vertically across those boundaries to re-plenish the aquifers, resulting in less elongated iso-capture time curves. Ó 2000 Elsevier Science Ltd. All rights reserved.
Keywords:Horizontal well; Capture time; Regional ¯ow; Potential theory; Analytical solution.
1. Introduction
Horizontal wells have been used extensively in the petroleum industry for oil and gas production in the past decade [15,21]. Extensive studies on pressure anal-ysis of a horizontal well have been done in petroleum engineering, and these studies are useful for interpreting horizontal well pumping tests [5,7,13,18].
Historically, the cost of installing a horizontal well is signi®cantly higher than that of a vertical well, but this cost has decreased with the advancement of new drilling technologies [11,12,16]. Horizontal wells have advan-tages in certain situations of groundwater contamina-tion recovery. For instance, they can provide: (1) improved access to subsurface contaminant at sites with surface restrictions (e.g., land®lls, lagoons, buildings, wetlands, lakes, utility lines, tanks), (2) minimal surface disturbance, and (3) increased surface-area contact with contaminants.
The reduced operational cost, combined with the advantages in recovery situations has led to increased utilization of horizontal well technology in the hydro-logical and engineering applications in recent years [4,14,23,24,27,28]. Horizontal wells are also commonly utilized in air sparging and soil vapor extraction to re-cover volatile organic contaminants.
Fluid and gas ¯ows to a horizontal well or trench have been studied before. An early study of ¯uid ¯ow into collector wells was done by Hantush and Papado-pulos [10] who provided analytical solutions for the drawdown distribution around collector wells. Tarshish [27] constructed a mathematical model of ¯ow in an aquifer with a horizontal well located beneath a water reservoir. Falta [6] has developed analytical solutions of the transient and steady-state gas pressure and the steady-state stream function resulting from gas injection and extraction from a pair of parallel horizontal wells. Murdoch [17] has discussed the three-dimensional transient analyses of an interceptor trench; his results can be readily extended to horizontal wells. Parmentier and Klemovich [20] have pointed out that the contact area of a single horizontal well may equal that of ten
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*
Corresponding author. Tel.: +1-409-862-7961; fax: +1-409-845-6162.
E-mail address:zhan@hydrog.tamu.edu (H. Zhan).
vertical wells. Rushing [23] has developed a semiana-lytical model for horizontal well slug testing in con®ned aquifers.
Despite previous research of horizontal well hy-draulics in petroleum engineering, agricultural engi-neering, and hydrological engiengi-neering, several important problems related to groundwater hydrology still exist. One of them is the analytical determination of the time a
¯uid particle takes to ¯ow to a horizontal well; that is, the capture time. These analytical solutions are very useful for quick assessments of recovery eciency and can be used to validate the numerical solutions. Ana-lytical solutions also provide better physical insights into the problem than the numerical solutions. Zhan [29] has derived a closed-form analytical solution of capture time to a horizontal well in a con®ned aquifer excluding the
Nomenclature
a 2p=d, a parameter de®ned in Eq. (8) [mÿ1]
ae 2p=de, eective ``a'' de®ned in Eq. (46)
[mÿ1]
A integration de®ned in Eq. (A.5) [m]
b 2w0=m, a parameter de®ned in Eq. (8)
[dimensionless]
be 2w0e=m, eective ``b'' de®ned in Eq. (46)
[dimensionless]
B integration de®ned in Eq. (A.5) [m]
c 2q0=m, a parameter de®ned in Eq. (8) [mÿ1]
ce 2q0e=m, a parameter de®ned in Eq. (46)
[mÿ1]
d aquifer thickness [m]
de d=a1=4, eective aquifer thickness de®ned in
Eq. (44) [m]
dw distance from a horizontal well to an
aquifer's bottom boundary [m]
dwD dimensionlessdw
i pÿ1
j integral numbers,. . .ÿ2; ÿ1; 0; 1; 2; . . .
K hydraulic conductivity [m/d]
Ke
KxKy
p
, eective hydraulic conductivity [m/d]
Kx principal hydraulic conductivity in x-direction [m/d]
Ky principal hydraulic conductivity in y-direction [m/d]
L horizontal well screen length [m]
LD dimensionless well screen length de®ned in
Eq. (2)
m Q=2p
M truncation number used in the semi-analytical solutions
n eective porosity
q0 regional speci®c discharge alongÿy-direction
[m/d]
q0e a1=4q0, eective regional speci®c discharge
alongÿy-direction [m/d]
q0x regional speci®c discharge along the +x-direction [m/d]
Q pumping rate per unit screen length of well (Q>0 for pumping) [m2/d]
Ss speci®c storativity [m)1]
t time [d]
tD dimensionless time de®ned in Eq. (2)
T capture time to a horizontal well [d]
TM approximation of capture time to a
horizontal well in semi-analytical studies [d]
u xiy
Vx x-component of groundwater ¯ow velocity [m/d]
Vy y-component of groundwater ¯ow velocity [m/d]
x vertical coordinate [m]
x0 x-coordinate of the starting point [m]
xe Ke=Kx
1=2
x, eectivexde®ned in Eq. (44) [m]
x0e Ke=Kx
1=2
x0, eectivex0 [m]
xi; xi1 discretization ofxalong the streamline [m]
x x-coordinate of pumping wells [m]
xÿ x-coordinate of injecting wells [m]
y horizontal coordinate perpendicular to the well axis [m]
y0 y-coordinate of the starting point [m]
ye Ke=Ky
1=2
y, eectivey de®ned in Eq. (44) [m]
y0e Ke=Ky
1=2
y0, eectivey0 [m]
yi; yi1 discretization ofyalong the streamline [m]
z horizontal coordinate along the well axis [m]
zD dimensionlessz-coordinate
DS drawdown [m]
a Kx=Ky, anisotropic ratio
b1; b2 dummy parameters used in Section 2.2
c1; c2 dummy parameters used in Eq. (A.6)
f /iw, complex function used to describe the steady-state ¯ow [m2/d]
/ real part off [m2/d]
w imaginary part off [m2/d]
w0 the streamline function de®ned by point x0;y0[m2/d]
w0e the eective streamline function de®ned by x0e;y0e[m2/d]
wM approximation of the streamline function in the semi-analytical studies [m2/d]
w0M approximation ofw0 in the semi-analytical
studies [m2/d]
regional ¯ow. This paper, however, will study the more general case including the regional ¯ow. It will also study the cases with constant-head boundaries, which are useful when a horizontal well is in an aquifer un-derneath a water reservoir or wetland, or the horizontal well is in an aquitard bounded by two high-permeability layers.
2. Capture time to a horizontal well
2.1. Problem description
Fig. 1 is the schematic diagram of a horizontal well pumping in a con®ned aquifer. The lateral boundaries are suciently far and do not in¯uence the hydraulic head. The center of the well is the origin of the coordi-nates.dw is the distance to the bottom boundary,Lthe
well length, and d is the aquifer thickness. For simplic-ity, we assume a homogeneous, isotropic medium with a hydraulic conductivity Kand a speci®c storativitySs.
Flow to a ®nite horizontal well includes three stages: an early, nearly radial ¯ow, an intermediate ¯ow, and a late horizontal pseudoradial ¯ow (Fig. 1). At the early time, the in¯uences of the top and bottom boundaries are not perceptible, thus the ¯ow is radial in a plane perpendicular to the well axis. Daviau et al. [5] (p. 717) showed that the top/bottom boundaries were perceptible roughly when dimensionless time tD was
tD0:32 dwD=LD 2
; if dwD60:5;
tD0:32 1ÿdwD=LD 2
; if dwD>0:5; 1
wheretD,dwD, andLDwere
tD
K
Ss L=22t; dwD
dw
d ; LD L=2
d ; 2
andtwas the time. The end eect was noticeable when [5, p. 717]
tD 1ÿzD 2
=6; 3
where the dimensionless zDz= L=2. We should
em-phasize that Eqs. (1) and (3) only give orders of mag-nitude estimations. Other authors may use dierent formulae to estimate these times [17, p. 3027].
After the end of the early period, the ¯ow enters the intermediate stage which has a complicated pressure behavior as described by Goode and Thambynayagam [7], Daviau et al. [5], and Ozkan et al. [18]. The ending time of the intermediate stage is approximately 0:8<tD<3 [5, p. 718]. Once again, this result only
gives order of magnitude estimation, and dierent authors may use slightly dierent formulae [17, p. 3027]. After the intermediate period, the ¯ow enters the pseudoradial ¯ow stage during which the equipotential surfaces are similar to vertical cylinders in the far ®eld (Fig. 1). This situation is similar to a fully penetrated, large diameter vertical well [7,17].
2.2. Analytical solution of capture time to a horizontal well in a con®ned aquifer
Flow around a ®nite-length horizontal well is gener-ally a three-dimensional problem owing to the in¯uence of well ends. General study of capture time to a hori-zontal well can only be calculated numerically. The advantage of a numerical method is the feasibility of handling complicated boundary geometries, transient ¯ow, and ®nite length of well. The disadvantage is the inevitable truncation error and the requirement of very small grid sizes to simulate the region near the wellbore. Analytical solutions, however, can provide bene®ts, which cannot be gained through other means. For in-stance, they provide quick references for calculating the
capture times, they are useful for validating the nu-merical codes, and they provide better physical insights, which may be useful in further investigations. However, analytical solutions also have their limitations. Some assumptions are necessary to have closed-form solu-tions. In the following analytical studies, we assume steady-state ¯ow and an in®nitely long horizontal well. It is important to justify these assumptions before the use of the analytical solutions.
1. First of all, the assumption of an in®nitely long horizontal well is correct only when the aquifer is con-®ned by two impermeable vertical walls which are per-pendicular to the axis of well. This is possible in a ®eld application in which two barriers are built to con®ne a contaminated plume. For a ®nite length well, the ends of the well will in¯uence the pressure distribution and ¯ow becomes three-dimensional. However, if the well is suf-®ciently long, the analytical solutions are expected to be approximately correct in the regions near the well center where the in¯uences of the ends are the least. Consid-ering the fact that many environmental horizontal wells use long screen lengths to intercept large areas of groundwater, the in®nitely long well approximation is reasonable for estimating the near ®eld capture time.
Daviau et al. [5] have analyzed the transient pressure change along the horizontal wellbore and their result showed that (Fig. 2 in their paper) with a well length
L2d, whenjzj=L61=4, the well length has a negligible in¯uence on the pressure, wherezis the distance to the well center (see Fig. 1). Thus, we can approximately use
L=4 as the scale of the near ®eld, i.e., when the distance of the point to the well center is less than a quarter of the well length, that point is called a near ®eld point and we can use the in®nite well assumption.
2. When a horizontal well is installed in a con®ned aquifer, Murdoch [17] has pointed out that the ¯ow around a horizontal well is better described as a tran-sient process and the duration of the trantran-sient period is proportional to the square of well length (see tD in Eq.
(2)). This is indeed true if we check the transient pressure along the wellbore in Fig. 2 of Daviau et al. [5], which showed that drawdowns continuously increased when pumping continued.
However, as ¯ow enters the pseudoradial ¯ow, Ozkan et al. [18] (Eq. 11) showed that the increase of drawdown with time was uniform across the domain and had a simple form DS b1 lntDb2, where DS is the
drawdown,b1 a constant determined by the
permeabil-ity, viscospermeabil-ity, and pumping rate, and b2 is a constant
related to dwD; LD, and the coordinates of the point
where DS was measured. Therefore, the groundwater ¯ow velocity, which depends on the hydraulic head gradient, not on hydraulic head itself, is independent of time in the pseudoradial ¯ow. Thus the steady-state ¯ow (velocity does not change with time) is possible even though the drawdowns depend on time. Thus the
steady-state assumption can be used after ¯ow enters the pseudoradial ¯ow.
3. If constant-head exists at the top and/or bottom boundaries, steady-state drawdowns can soon be es-tablished around 0:8<tD<3 after the early radial ¯ow
[5,13]. That is, because water can ¯ow across the con-stant-head boundaries to supplement the aquifers soon after pumping starts, therefore, the demand of water from lateral directions is reduced and the drawdowns will approach steady state. Thus the steady-state as-sumption is reasonable when constant-head boundaries exist.
Based on the above justi®cation, we will make use of the steady-state and in®nitely long well assumptions to calculate the capture times for near ®eld points when ¯ow enters the pseudoradial stage in a con®ned aquifer, or steady-state stage when constant-head boundaries are presented.
The solution for a con®ned, isotropic medium is ®rst studied. Fig. 2 is the cross-section of an in®nitely long horizontal well located at the center of a homogeneous, isotropic, con®ned aquifer. The x- and y-axes are per-pendicular to and parallel with the con®ning bed, re-spectively. The well is located at x;y 0;0. Groundwater regional ¯ow direction can be arbitrary in
theyz-plane. However, thez-component of the regional ¯ow (along the well axis) will only cause a shift of a ¯uid particle along the z-direction. As long as such a shift does not aect the assumption of an in®nitely long well, i.e., the ¯uid particle is still far from the well ends, thez -component of the regional ¯ow does not in¯uence the ¯uid particle's movement in thexy-plane and the capture time. Only they-component of the regional ¯ow aects the capture time. In the following discussion, we assume that the y-component of the regional ¯ow is along the
ÿy-axis with a speci®c discharge q0 (see Fig. 2).
To simulate the steady-state ¯ow in this region, an in®nite number of image wells with the identical pumping rate as the original well are assigned along the x-axis with locations atjd j1;2;3;. . .The eect of image wells is equivalent to that of the top and bottom no-¯ow boundaries. The image well method has been broadly employed in studying groundwater ¯ow [1,4,27] and heat transfer [3] problems in bounded domains. Using the potential theory [1], a complex function f u /iwis assigned to describe the ¯ow
are the equipotential and streamline functions, re-spectively; mQ=2p, Q is the pumping rate per unit
. Bear [1] (Eqs. (7.8.31)) gave a simpli®ed expression to the ®rst term of Eq. (4) without providing the details of derivation, and his (Eqs. (7.8.31)) had an inaccurate negative sign. Zhan [29] has calculated the summation term of Eq. (4) and the result is
f u mln sin pu
The x-component of ¯ow velocity is
Vx ÿ 1=n o/=ox, where n is the eective porosity. The capture time T for a water particle to ¯ow from
x0;y0into the horizontal well is
where \j j" is the sign of absolute value. Given the starting point x0;y0, the stream functionw0w x0;y0
describes the ¯ow path. Assign constants
a2p=d; b2w0=m; c2q0=m: 8
The presumption of using Eq. (7) is thatx60, otherwise
Vx0. Thus special care should be taken whenx!0. If
x!0, keeping the ®rst-order approximation of Eq. (9), we have
tanh ay=2
ax=2
tanh b=2 cx=2
1ÿtan b=2cx=2: 10
Eq. (10) is simpli®ed into
lim
Eq. (11) is employed to calculate the capture time when
x0!0 (see the following case 2).
Using Eqs. (6), (7) and (9), we are able to derive closed-form analytical solutions of capture time. The mathematical details are shown in Appendix A. The result is using Eq. (12). For those cases Eq. (12) can be simpli®ed by invoking Eq. (A.8) or (A.10) (see Appendix A). Now we give an example to use these formulae. The following parameters are given: the aquifer thickness d20 m, hydraulic conductivity K10 m/d, eective porosity
n0:25, pumping rate per unit screen length
Q15 m2=d, the regional groundwater discharge
q00:1 m=d along the ÿy-axis, the starting point
A few special cases of Eq. (12) deserve discussion.
Case 1: No regional ¯ow.When there is no regional
¯ow,q00 i:e:; c0, as derived in Appendix B, the
or bottom boundaries will not be captured by the well. Eq. (13) is the result derived in Zhan [29] excluding the regional ¯ow. using Eqs. (A.8) and (A.10), Eq. (14) becomes
T n Case 3: The horizontal well is located at the bottom of
the aquifer. In some applications, the well may be
in-stalled near the bottom of the aquifer. For instance, horizontal wells are commonly employed to recover dense non-aqueous liquids (DNAPLs) such as chlori-nated solvents, creosote, and coal tar. DNAPLs are driven by gravitational forces deeper into the subsur-face, where they are trapped by capillary forces and as pools of liquid suspended on low-permeability strata. Horizontal wells can provide much better recoveries because they can be installed at the bottoms of the aquifers to have much larger contact areas with the DNAPL plumes. The capture time for a bottom hori-zontal well is addressed below.
Fig. 2(B) shows the original and image wells for a bottom well scenario. The image wells are 2dapart with a pumping rate 2Q, the original well is assigned an equivalent ``2Q'' pumping rate. Such an assignment can be explained in this way: in Fig. 2(A), if the original well starts to move downwards, then the image well D1 moves upward, the pair wells U2/U3 or D2/D3 tend to move closer to each other. Finally, the original well moves to the bottom and joins together with D1 to form one ``equivalent'' well with a pumping rate 2Q. Mean-while, wells U2/U3 come to form one ``equivalent'' well, and wells D2/D3 form another ``equivalent'' well, etc. We ®nd that Fig. 2(B) is similar to Fig. 2(A) except that d is replaced by 2d and Q is replaced by 2Q. Thus a complex function describing the ¯ow in Fig. 2(B) is obtained from a modi®cation of Eq. (5) (notice that the origin of the coordinate system is at the bottom with the real well).
The capture time becomes
T 4n
The solutions for special cases such asx0!0 and/or ac !0 or aÿc !0 in case 3 can be easily ob-tained from the corresponding results in cases 1 and 2 of this section by simply replacing ``d'' and ``Q'' by ``2d'' and ``2Q'' there, respectively.
2.3. Analytical solutions of capture time to a horizontal well in an aquifer with constant-head boundaries
As mentioned in the introduction, horizontal wells are commonly used to recover contaminated water from under a water reservoir, or a wetland, or a lake. Under these circumstances, the top boundary of the aquifer is better described as constant-head if the aquifer is hy-drologically connected with the bottom of the reservoir. Tarshish [27] has discussed the horizontal pumping un-der a water reservoir.
Another common application of horizontal well is to recover contaminants from a low-permeability layer bounded from above and below by highly permeable media. Vertical wells are inecient in such cases due to the limited in¯uence screen length. On the contrary, horizontal wells can be drilled into the low-permeability layer to substantially increase the interceptive area with the contaminated water there. The highly permeable media above and below can be described as constant-head boundaries if the permeability contrasts between the low-permeability layer and the highly permeable layers are very large. The reason that the highly per-meable media are treated as constant-head boundaries is because any possible hydraulic head dierences within the media will dissipate rapidly, resulting in the equal hydraulic head within the media.
In the following, we provide the analytical solutions for these two cases.
Case 1: Horizontal well located underneath a water
reservoir. For this case, the top ¯ow boundary is
if a constant pressure such as a gas cap existed in an oil reservoir, the well was usually drilled close to the other no-¯ow boundary. Fig. 3 shows a schematic diagram of the real and image well system for the bottom well scenario. Both injecting and pumping image wells are used. The bottom no-¯ow boundary prohibits the ver-tical regional ¯ow. The top constant-head boundary prohibits the horizontal regional ¯ow components. Thus no regional ¯ow can exist in this case. The complex function describing steady-state ¯ow is
f u 2mln sin pu 4d
h i
ÿ2mln sin pu2d 4d
; 19
and the real and imaginary parts of Eq. (19) are
/mln1
2 cosh
py
2d
ÿcospx 2d
ÿmln1 2 cosh
py
2d
cospx 2d
; 20
w2mtanÿ1 sinh py=2d
sin px=2d
: 21
Appendix C gives capture time, T, as
T 16n ma2sin b2
b
4
ÿ
sinÿ1 sin b 4
cos ax0 4
;
22
whereaandbare de®ned in Eq. (8), andw0w x0;y0
is calculated from Eq. (21). Whenx0!0 ory0!0 Eq.
(22) is simpli®ed to
lim x0!0
T 8n ma2 cosh
ay0
4
h
ÿ1i; 23
lim y0!0
T 8n ma2 1
h
ÿcos ax0 4
i
: 24
Case 2: Horizontal well located in a low-permeability stratum bounded by highly permeable aquifers above and
below. Horizontal wells are more eective than vertical
wells to extract the contaminants from a low-per-meability stratum bounded above and below by highly permeable media. This phenomenon was also observed by Seines [25] who used a case study of the Troll ®eld to demonstrate that one horizontal well operated as the equivalent of four vertical wells in recovering oil from a thin oil reservoir. Owing to the large permeability con-trast, the top and bottom boundaries of the stratum are better characterized as constant-head boundaries. In this case, the horizontal well is better installed at the center of the stratum to achieve maximal catchment eciency. Fig. 3(B) shows a schematic diagram of the real and image wells for this scenario. We assume that the top and bottom boundaries have an equal constant head, thus there is no vertical ¯ow before pumping. In fact, the well con®guration in Fig. 3(B) is almost identical to that of Fig. 3(A) except that the distance between neigh-boring wells becomes 2d rather than 4d and the pump-ing rate is Q rather than 2Q. Therefore, if we put the origin of the coordinate system at the real well, the so-lution of capture time excluding regional ¯ow can be directly modi®ed from Eq. (22) and becomes
T 8n ma2sin b
b
2
ÿ
sinÿ1 sin b 2
cos ax0 2
;
25
whereaandbare still de®ned by Eq. (8), butw0becomes
w0mtanÿ1
sinh py0=d
sin px0=d
: 26
When x0!0 ory0!0, Eq. (25) is replaced by
simpli-®ed forms
lim x0!0
T 4n ma2 cosh
ay0
2
h
ÿ1i; 27
lim y0!0
T 4n ma2 1
h
ÿcos ax0 2
i
: 28
3. Semi-analytical solutions of capture times to horizontal wells with general well locations and regional ¯ows
In previous sections, horizontal wells are either at the centers or at the bottoms of the aquifers and the regional ¯ows are not included when the constant-head bound-aries are presented. These special requirements are necessary in order to achieve the closed-form solutions. In this section, we will study three cases with arbitrary well locations and with possible regional ¯ows. Closed-form analytical solutions are unlikely for these general situa-tions, however, semi-analytical solutions are possible.
Case 1: An arbitrary horizontal well location in a
con®ned aquifer. Fig. 4(A) shows the real and image
wells for this case. The distance from the horizontal well to the bottom boundary isdw. As discussed in Section 2,
only they-component of the regional ¯ow in¯uences the capture time, the z-component of the regional ¯ow causes the shift of a ¯uid particle along the well axis, but
does not aect the capture time. We assumed that they -component of the regional ¯ow is along theÿy-axis with a speci®c dischargeq0.
With a coordinate system shown in Fig. 4(A) where the origin is at the bottom boundary, the complex function describing the steady-state ¯ow in thexy-plane is
f u mX
1
jÿ1
lnuÿ 2jddw ÿiq0u; 29
where the summation re¯ects the contributions from the original well and the image wells. Notice that the in-¯uence from an image well upon a ¯uid particle's movement will decrease when the distance between that image well and the ¯uid particle increases, thus the summation of the in®nite series in Eq. (29) is replaced by a summation of a ®nite series from j ÿM to M. The velocities along the y- and x-directions are
Vy ÿ 1=n o/=oy and Vx ÿ 1=n o/=ox, re-spectively, where / is the real part of f u. The ap-proximation of the capture time is
TM n
Z y0
0
dy
mX
M
jÿM
y
xÿ 2jddw2y2 "
q0 #
;
if y060; 30
TM n
Z x0
dw
dx
mX
M
jÿM
1
xÿ 2jddw
" #
; if y00:
31
The corresponding streamline function is
wM x;y mX
M
jÿM
tanÿ1 y
xÿ 2jddw
ÿq0x: 32
If a ¯uid particle starts from x0;y0, the streamline for
that particle is determined by
wM x;y w
0
M x0;y0: 33
The choice of M depends on a prior determined ap-proximation criterion. For instance, we can chooseMto be the smallest positive integer satisfying
w0M1ÿw 0
M w0M
610ÿ6:
34
When M is found, Eqs. (32) and (33) determine the streamline. For a given y-coordinate at the streamline, the x-coordinate is numerically determined. This is carried out either using the particle tracking method or the Newton±Raphson root searching method [22]. We use the following particle tracking method to trace the streamline: xi1xiÿ owM=oy= owM=ox yi1ÿyi, where xi;yiis the particle location at the pointialong the streamline, i0; 1; 2; . . . If yi1ÿyi is small enough,xi1 can be accurately determined. After this Eq. (30) is numerically integrated using the Simpson's
rule [22, p. 113]. The numerical methods are straight-forward to implement. A FORTRAN program includ-ing the particle trackinclud-ing to ®nd the streamline and the numerical integration to calculate the capture time is written by us and is available from the website http:// geoweb.hydrog.tamu.edu/Faculty/Zhan/Zhan.html. Ex-amples 1 and 2 in Table 1 show the results from these semi-analytical calculations in con®ned aquifers. In the example 1 of Table 1, dwd=2, i.e., the well is at the
aquifer center, the closed-form analytical solution Eq. (12) can be used. Using the parameters listed in example 1, the capture time calculated from Eq. (12) isT 4:58 d, which agrees with the semi-analytical solution
T 4:58 d.
Case 2: An arbitrary horizontal well location in an aquifer with a constant-head top boundary and a no-¯ow
bottom boundary. Fig. 4(B) shows the real and image
wells for this case. As pointed out in the case 1 of Sec-tion 2.3, no regional ¯ow can exist in this case. How-ever, the well can be put in an arbitrary location. In Fig. 4(B), the real and image pumping wells are at
x4jddw and the injecting image wells are at
xÿ4jd2ddw where j0; 1; 2; . . . The
complex function describing the steady-state ¯ow, the capture time, and the streamline function are approxi-mated by Eqs. (35)±(38).
f u mX
M
jÿM
lnuÿ 4jddw ÿmX
M
jÿM lnu
ÿ 4jd2ddw; 35
Table 1
Analytical and semi-analytical solutions of capture timesa
Example 1 Example 2
Case 1: Capture times in con®ned aquifers
Input parameters:Q15 m2/d,q0
0:1 m/d,n0:25,d20 m,dw10 m,x015 m, y010 m
Input parameters:Q15 m2/d,q0 0:1 m/d,n0:25,d20 m,dw5 m, x015 m,y010 m
Analytical solution [Eq. (12)]T4:58 d Semi-analytical solution [Eq. (30)]T 4:58 d Semi-analytical solution [Eq. (30)] T 7:41 d
Example 3 Example 4
Case 2: Capture times in aquifers with constant-head top boundaries and no-¯ow bottom boundaries
Input parameters:Q15 m2/d,q0
0:0 m/d,n0:25,d20 m,dw0 m,x05 m,y010 m Input parameters:Q15 m2/d,q0 0:0 m/d,n0:25,d20 m,dw5 m, x05 m,y010 m
Analytical solution [Eq. (22)]T3:41 d Semi-analytical solution [Eq. (36)]T 3:40 d Semi-analytical solution [Eq. (36)] T 4:39 d
Example 5 Example 6
Case 3:Capture times in aquifers with constant-head top and bottom boundaries
Input parameters:Q15 m2/d,q0
x0:0 m/d,n0:25;d20 m,dw10 m,x015 m,
y010 m
Input parameters:Q15 m2/d, q0x0:01 m/d,n0:25;d20 m,
dw5 m,x010 m,y010 m Analytical solution [Eq. (25)]T8:02 d Semi-analytical solution [Eq. (40)]T 8:04 d Semi-analytical solution [Eq. (40)]
T 10:49 d aThe origin of the coordinate system is at the bottom boundary. Notice that
TM n
Mis determined by Eq. (34). Using the particle tracking method to ®nd thex-coordinates for any giveny -coor-dinates at the streamline, the results are utilized to inte-grate Eq. (36). A FORTRAN program modi®ed from the one used in case 1 of this section is available from the same website listed in above case 1. Examples 3 and 4 in Table 1 show the results of two calculations for this case. The example 3 of Table 1 is the bottom well scenario and the closed-form analytical solution Eq. (22) can be used. Utilizing the parameters listed in the example 3 of Table 1, Eq. (22) yieldsT 3:41 d and the semi-analytical so-lution yieldsT 3:40 d. It is interesting to see that the number of terms used in the approximation of this case (Mis around 10) is much smaller than that used in above case 1 (Mis around 350). This is because the image wells used in this case include both pumping and injecting wells which are alternatively arranged along thex-axis, thus their contributions to a ¯uid particle's movement cancel each other rapidly when the distances between them and the ¯uid particle increase. However, the image wells in case 1 are all pumping wells and their contributions to a ¯uid particle's movement are gradually reduced when the distances between them and the ¯uid particle increase, thus more terms are needed in the approximation.
Case 3: An arbitrary horizontal well location in an aquifer with constant-head top and bottom boundaries. Fig. 4(C) shows the real and image wells for this case. The pumping wells and the injecting wells are at
x2jdd
w, and xÿ2jdÿdw, respectively with
j0; 1; 2;. . . The top and bottom constant-head boundaries prohibit horizontal regional ¯ows, but ver-tical regional ¯ow exists when top and bottom bound-aries have dierent hydraulic heads. Assuming that the vertical regional ¯ow isq0x along the x-directions, the complex function describing the steady-state ¯ow, the
capture time, and the streamline function are approxi-mated by Eqs. (39)±(42).
f u mX
Mis determined by Eq. (34). The number of terms used in the approximation of this case is found to be around
M 10. This is consistent with what was found in the above case 2. The particle tracking method is used to ®nd the corresponding xfor any giveny in the stream-line, and the results are utilized to integrate Eq. (40). This semi-analytical solution is put in a FORTRAN program modi®ed from that of case 1 of this section and is available from the same website listed in above case 1. Examples 5 and 6 in Table 1 show the calculation results for this case. Example 5 is a special scenario with
dw1=2 andq0x 0. The capture time for this scenario can be calculated by the analytical solution Eq. (25) which yieldsT 8:02 d. This result agrees with the semi-analytical solutionT 8:04 d.
It is worthwhile to point out that when q0x60, the capture zone is ®nite and not every ¯uid particle in the aquifer will eventually ¯ow into the pumping well. We use the following method to determine whether a par-ticle is within the capture zone or not in the FORTRAN program: starting from point x0;y0and using particle
4. Capture time in anisotropic media
If the horizontal and vertical conductivities are Ky andKx, respectively (xis the vertical direction), let aKx=Ky; Ke
KxKy
p
; 43
where a is the anisotropic ratio andKe is the eective
conductivity. Bear [2, p. 169±175] showed that an an-isotropic medium could be transformed into an ``equivalent'' isotropic medium. The detailed transfor-mation procedure was given there. For our purpose, we need to transfer x; y; q0, and d into the ``equivalent'',
xe; ye; q0e, andde as follows:
xe
Ke
Kx
1=2
x x
a1=4; ye
Ke
Ky
1=2
ya1=4y; q
0ea1=4q0; de
d
a1=4:
44
The pumping rate Q and porosity n are unchanged. Therefore, the capture time for a horizontal well in the middle of a con®ned aquifer is
T 2n mae
1
aece
ln
cos aecex0ebe 2
h i
cos be=2
1
aeÿce
ln
cos aeÿcex0eÿbe 2
h i
cos be=2
; 45
where
ae2p=de; be2w0e=m; ce2q0e=m; 46
w0emtanÿ1
tanh py0e=de
tan px0e=de
ÿq0ex0e: 47
Using Eq. (44) to do the parameter transformation, we can obtain the analytical and semi-analytical solutions of capture times for all the previously described scenarios in anisotropic aquifers. The details are trivial and will not be repeated here. The FORTRAN pro-grams for all the semi-analytical solutions are capable of handling anisotropic media.
5. Analysis of the solutions
On the basis of the above analytical solutions, we can delineate the horizontal well capture zones. For example Fig. 5(A) shows iso-capture time contours for a middle well in con®ned, isotropic and anisotropic aquifers on the basis of Eqs. (12) and (45). The parameters used for that ®gure are: n0:25, d21 m, q00:1 m=d,
Q15 m2=d, andK 10 m=d for the isotropic aquifer,
and Ky10 m=d; Kx1 m=d for the anisotropic aquifer. The capture time in Fig. 5(A) is 10 d. Fig. 5(B) shows the capture zone of a bottom well scenario whose
input parameters are the same as those used in Fig. 5(A). Fig. 5(A) and (B) indicates that the top and bottom no-¯ow boundaries constrain the vertical no-¯ows and enhance the horizontal ¯ows, resulting in horizontally elongated capture zones. The presence of vertical anisotropy with a smaller vertical conductivity further exaggerates this tendency and results in an even more elongated capture zone. For the bottom well scenario, the bottom boundary has a much stronger impact on the ¯ow than the top boundary and the capture zones extend further in the horizontal direction compared with their coun-terparts of Fig. 5(A). Regional ¯ow causes asymmetric shapes of capture zones in they-direction.
Fig. 6(A) and (B) shows the iso-capture times for the cases 1 and 2 of Section 2.3. When constant-head boundaries are presented, water can in®ltrate vertically across those boundaries to replenish the aquifers when pumping starts, thus the demand of water from the lateral directions is reduced, resulting in smaller lateral ¯ow velocities. On the contrary, a con®ned aquifer prevents any water replenishment from the vertical di-rection across the no-¯ow boundaries, thus the pumped water entirely comes from the lateral ¯ow, resulting in enhanced lateral ¯ow velocities and the laterally elongated iso-capture time curves.
The derived closed-form analytical solutions have the following limitations:
· Because of the use of steady-state ¯ow and in®nitely
long well assumptions, the analytical solutions can only be used for near ®eld and for time after start of the pseudoradial ¯ow in a con®ned aquifer or for the steady-state ¯ow if constant-head boundaries are presented (see the justi®cation in Section 2.2).
Fig. 5. Ten days iso-capture times calculated on the basis of analytical solutions Eqs. (12), (18) and (45) in isotropic and anisotropic con®ned aquifers. The parameters used are: eective porosityn0:25, aquifer thickness d21 m, regional ¯ow speci®c discharge q00:1 from along ÿy-axis, horizontal well pumping rate per unit screen length Q15 m2=d, and anisotropic ratioK
x=Ky1=10. (A) Results for a
· We provide analytical solutions for central and
bot-tom well scenarios. For a case with a general well lo-cation, we can only provide a semi-analytical solution, which needs a numerical integration to fully calculate the capture time based on Eq. (7).
· We derive the closed-form solutions for con®ned
aquifers including the regional ¯ow. When top and bottom boundaries are constant-head, we can only provide semi-analytical solutions if top and bottom boundaries have dierent heads, but we can have the closed-form solutions if top and bottom bound-aries have equal constant-head.
Semi-analytical solutions are very ¯exible and can deal with cases with arbitrary well locations and possible regional ¯ows.
Analytical and semi-analytical solutions obtained above can be compared with capture times calculated using numerical software such as Visual Mod¯ow (VM) [9]. Zhan [29] has provided the numerical simulation results excluding the regional ¯ow and he found that the numerical solutions were almost identical to the ana-lytical solutions, especially for the isotropic cases. The numerical scheme is quite standard as described in Zhan [29], thus the details are not repeated here.
6. Summary and conclusions
We have obtained closed-form analytical solutions and semi-analytical solutions of capture times on the conditions of a steady-state ¯ow and an in®nitely long well. The suitability of these assumptions is justi®ed and the applications and limitations of the analytical solu-tions are discussed.
We have derived the analytical solutions of capture times for center or bottom wells in con®ned aquifers including the horizontal regional groundwater ¯ows. We have provided the analytical solutions of capture times to horizontal wells in aquifers under water reservoirs that serve as constant-head boundaries. Bottom wells can achieve the maximal capture eciency in these aquifers and are investigated. We have obtained the analytical solutions of capture times in aquifers when top and bottom boundaries have equal constant heads. Center wells can achieve the maximal recovery eciency in these aquifers and are investigated.
We have conducted the semi-analytical studies with general well locations and with possible regional ¯ows in either con®ned aquifers, or aquifers under water reser-voirs, or aquifers with top and bottom constant-head boundaries whose constant heads could be dierent. We ®rst give the analytical expressions of capture times and streamline functions in the xy-plane, then use the par-ticle tracking method to determine the streamlines in the
xy-plane, and use the Simpson's rule [22] to numerically integrate the capture times. FORTRAN programs are written to conduct the particle tracking and the nu-merical integrations. The semi-analytical solutions agree with the analytical solutions for three special examples where analytical solutions are possible.
Capture times in anisotropic media are derivable for all previously mentioned cases following simple modi®-cations from the solutions of isotropic media. FOR-TRAN programs associated with the semi-analytical solutions are capable of handling anisotropic media.
The aquifer's top and bottom no-¯ow boundaries play important roles in in¯uencing ¯ows to a horizontal well. These boundaries constrain vertical ¯ow, but en-hance lateral ¯ow, resulting in elongated iso-capture time curves. If the aquifer is anisotropic with a smaller vertical hydraulic conductivity, constraint of vertical ¯ow and enhancement of lateral ¯ow are further exag-gerated. When constant-head boundaries are presented, water can in®ltrate across those boundaries to supple-ment the aquifer, thus demand of water ¯ow from lat-eral direction is reduced.
Acknowledgements
This research is supported by Faculty Research De-veloping Fund, Faculty Mini-grant, and Research
Enhancement Funds of the Texas A&M University. We thank Dr. Anthony Gangi for his critical prereview of the manuscript. We sincerely thank the four anonymous reviewers for their constructive suggestions of revising the paper.
Appendix A. Derivation of Eq. (12)
Taking the derivative o/=ox from Eq. (6) and sub-stituting it into Eq. (7) results in
T 2n
Squaring Eq. (9) and using the following identities:
tanh2 x
Substituting Eq. (A.3) into Eq. (A.1) leads to
T 2n
To integrate Eq. (A.4), we use the following two inte-grationsAandB:
Recognizing the following identity:
sin c1 sin c2 2 sin c1c2=2
Appendix C. Derivation of Eq. (22)
For any point x;y in the ¯ow path started from
x0;y0; w x;y w x0;y0, Eq. (21) gives
sinh ay=4 sin ax=4tan b=4; C:1
where a and b are de®ned in Eq. (8). Deriving o/=ox
from Eq. (20) and substituting it into Eq. (7) results in
Using the formulae 2.271.4 of Gradshteyn and Ryzhik [8, p. 105], the integration of Eq. (C.4) can be carried out and the ®nal result is
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