2001 Zhan Wang Park Jh Zhanetal2001

7/28/2019 2001 Zhan Wang Park Jh Zhanetal2001

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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.


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)


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


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.


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


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



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



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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



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(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



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

2001 Zhan Wang Park Jh Zhanetal2001

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