27. SPE-9346-PA
Transcript of 27. SPE-9346-PA
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The
Influence
o
Vertical
ractures
Intercepting
ctive and Observation Wells
on Interference Tests
Naelah
A
Mousli, *
SPE.
U
of Tulsa
Rajagopal Raghavan, SPE, U. of Tulsa
Heber Cinco-Ley,
SPE. Petroleos Mexicanos and U. Natl. de Mexico
Fernando Samaniego-V., SPE. Ins . Mexicano del Petroleo
bstract
This paper reviews pressure behavior at an observation
well intercepted by a vertical fracture. The active well
was assumed either unfractured or intercepted by a frac
ture parallel to the fracture at the observation well. We
show that a vertical fracture at the observation well has a
significant influence on the pressure response at that
well, and therefore wellbore conditions at the observa
tion well must be considered. New type curves presented
can be used to determine the compass orientation of the
fracture plane at the observation well.
Conditions are delineated under which the fracture at
the observation well may influence an interference test.
This information should be useful in designing and
analyzing tests. The pressure response curve
at
the
observation well has
no
characteristic features that will
reveal the existence of a fracture. The existence of the
fracture would have to be known a priori or from in
dependent measurements such
as
single-well tests.
Introduction
In this work, we examine interference test data for the in
fluence of
a vertical fracture located at the observation
well. All studies on the subject of interference testing
have been directed toward understanding the effects of
reservoir heterogeneity or wellbore conditions at the ac
tive (flowing) well. Several correspondents suggested
our study because many field tests are conducted when
the observation well is fractured. They also indicated
that it
is
not uncommon for both wells (active and obser
vation)
to
be fractured. To the best of our knowledge,
this is the first study
to
examine the influence of a ver
tical fracture at the observation well on interference test
data.
Two conditions at the active well are examined: an ac
tive well that
is
unfractured (plane radial flow) and an ac-
• Now with Arabian American Oil Co.
0197· 7520/82/0012·9346 00.25
Copyright 1982 Society of Petroleum Engineers of
I
ME
DECEMBER 1982
tive well that intercepts a vertical fracture parallel to the
fracture at the observation well. The parameters of in
terest include effects of the distance between the two
wells, compass orientation of the fracture plane with
respect to the line joining the two wellbores, and the
ratio of the fracture lengths at the active and observation
wells if both wells are fractured.
The results given here should enable the analyst
1) to
interpret the pressure response at the fractured observa
tion well, (2)
to
interpret the pressure response when
both the active and the observation wells are fractured
(3)
to
design tests to account for the existence of a frac
ture at one
or
both wells, and (4)
to
determine quan
titatively the orientation and/or length
of
the fracture at
an observation well. We also show that one should not
assume a priori that the effect
of
a fracture on the obser
vation well response will be similar to that of a concen
tric skin region around the
wellbore-i.e.
idealizations
to
incorporate the existence of the fracture, such as the
effective wellbore radius concept, may not be
applicable.
Mathematical Model and ssumptions
In this study, we consider the flow
of
a slightly com
pressible fluid of constant viscosity in a uniform and
homogeneous porous medium of infinite extent. Fluid is
produced at a constant surface rate at the active well.
Well bore storage effects are assumed negligible because
the main objective
of
our work
is
to demonstrate the
fluence of the fractures. However, note that wellbore
storage effects may mask the early-time response at the
observation well. Refs. 1 and 2 discuss the influence of
wellbore storage on interference test data. We obtained
the solutions to the problems considered here by the
method of sources and sinks. 3
The fracture at the observation well was assumed to be
a plane source of infinite conductivity.
4
The condition
of uniform pressure over the fracture surface was
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Y
I
xm ------l
LINE SOURCE
WELL
OR
UNIFORM-FLUX
FRACTURE
A
=
ACTIVE
WELL
0= OBSERVATION
WELL
INFINITE-CONDUCTIVITY
VERTICAL FRACTURE
Fig. 1-Schematic of system.
satisfied by dividing the plane source into a number
of
equal segments and then assigning a flux to each seg
ment such that the pressure drops at the midpoints of the
segments were identical. Details
of
this approach are
given in Ref. 4. Modifications to this approach, incor
porated primarily to reduce computing costs and com
puter storage, are discussed
in
Appendix A.
The active well was modeled by a line source to
simulate plane radial flow and by a plane source
uniform-flux
5
)
to simulate a fracture. The resulting
pressure response at the observation well was determined
by the principle
of
superposition. The procedure used to
obtain the pressure response at the observation well
in
this study is discussed in more detail in Appendix A. We
did not consider an infinite-conductivity fracture at the
active well nor a uniform-flux fracture at the observation
well
in
detail mainly because
of
computer time limita
tions. All results given here are applicable as long as the
fractures can be represented by the uniform-flux
or
the
infinite-conductivity ideal izations.
When both wells are fractured,
we
assume that the
fractures are parallel. However, the model used in this
study can determine the pressure response for any value
of
the angle between the fracture planes. A review
of
the
literature indicates that in a given reservoir the fractures
usually are oriented along a specific azimuth) direc
tion.
6
Thus, the assumption that the fractures are
parallel
is
realistic. MousIi
7
considers the pressure
response when the fractures are not parallel.
Before proceeding, we offer some remarks on the
assumption used to model the fractures. Many feel that
any solution based on the infinite-conductivity idealiza.
tion is unrealistic for application to field data. Unfor
tunately, they fail to recognize that the pressure behavior
at a fractured well is governed by a· combination
of
parameters and depends on the dimensionless fracture
conductivity, F
cD
defined by
k w
F cD = x (1)
f
where
k is
permeability,
w is
fracture width, xf
is
frac
ture half-length, and
subscript
denotes the fracture.
It is
934
o
1
&oJ
II
>
(J)
(J)
&oJ
f
(J)
J)
&oJ
J
-,
Z 10
o
iii
z
'
i
o
1
1
1 1 l t
RATIO OF DIMENSIONL ES5 TIME TO DIMENSIONLESS DISTANCE SQUARED I
OL
f r ~
Fig. 2-Pressure response at the observation well when the
observation well is fractured
r
D = 0.4).
well-documented that if F
cD
500, the behavior
of
a
finite-conductivity fracture cannot be distinguished from
that
of an
infinite-conductivity fracture. In most fluid in
jection projects, the value of F
cD
for propped hydraulic
fractures is large
~ 5 0 0 )
since
x
is small. Also, field
experience suggests that many acid-fractured or in
advertently fractured due to high injection pressures)
wells respond as if they are uniform-flux fractures.
5
Thus, the assumptions regarding fracture conductivity
used
in
this study appear to be more than adequate for
potential applications.
All results given in this study are presented in terms
of
dimensionless variables for convenience.
The dimensionless pressure drop
is
given by
. (2)
where p
is
pressure at location
x
,y) at time
t,
h is
thickness,
q
is surface flow rate, B
is
formation volume
factor, and
l is
viscosity.
The dimensionless time
is
defined as
3.6
X 10
-6kt
¢ctJlL/
3)
where
¢ is
porosity
of
the porous medium,
C
t
is
system
compressibility, and L
f
is
total
fracture length at the
observation well.
The dimensionless distances are defined
by
the follow
ing equations.
X
XD=- (4)
L
f
Y
YD=-
(5)
L
f
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10
0
0.
0:
0
a:
0
'
:
::>
'
'
a:
'-
'
'
...J
Z
0
iii
z
0
~ ~
-1
10
A
- ~ - L - L L L ~ ~ ~ ~ ~ ~ ~
__ L L U i l L
__
~ L U ~
1° 0-
2
1
1
1 1
2
RATIO Of DIMENSIONLESS TIME
TO
DIMENSIONLESS DISTANCE SQUARED,
OLf
Fig, 3-Pressure response at the observation well when the
observation well is fractured r 0 =
1).
m
xmD= - 6)
L
f
d
d
D
=-
7)
L
f
and
r D = . J X ~ D + d D 2
.
8)
X m and d are the distance between the two well bores
along the X and Yaxes respectively (see Fig. 1). We
now discuss the results obtained in our study. As already
mentioned, the procedure used to calculate the pressure
response is given in Appendix A.
Results
For each boundary condition at the active well, we ex
amine eight values of r D in the range 0.2 ::; r D ::; 2.
f
r D
was greater than two, the fracture(s) had a negligible in
fluence on the pressure response for most of the cases ex
amined here. For each value of
r
D we examined several
values
of
the angle of orientation, a. In this paper, the
angle of orientation is defined as the angle between the
fracture plane and the line joining the two wellbores
(Fig. 1). Values chosen for
a
were based primarily on
the sensitivity of the solution to this parameter. We did
not examine conditions in which the fracture at the
observation well intercepted either the active well or the
fracture at the active well because
of
the lack
of
potential
applications.
Plane Radial Flow at the Active Well
Fig. 2 presents the dimensionless pressure response at
rD=O.4
and is typical of the results obtained
in
this
study. The parameter of interest is the angle of orienta
tion,
a.
The dashed line is the expected response
if
there
is no fracture at the observation well; it is the line source
solution.
8
Upon examining the shape of the curves in
Fig. 2, it is clear that there is no distinctive characteristic
or imprint to indicate to the analyst that a fracture exists
DECEMBER 1982
0.0
a:
o
lS
1
w
a:
iil
U
'
::
'
U
U
'
..J .,
Z 10
o
iii
z
'
o
ORlfNTATION, Q
0
90
RATIO
OF
DIMENSIONLESS TIME
TO
01 MENSIONLESS DISTANCE SQUARED, I
Olf g
Fig.
4-Pressure
response at the observation well when the
observation well is fractured r
0
=2).
at the observation well. Equally obvious is the drastic in
fluence of a vertical fracture at the observation well on
the pressure response.
f
the fracture is neglected,
serious errors will result in the estimate of the formation
parameters and erroneous conclusions regarding reser
voir heterogeneity will be made.
It is also clear from Fig. 2 that the pressure response at
an observation well that intercepts a fracture cannot be
modeled by assuming the fracture to be an annular skin
region concentric with the well bore. An annular skin
region around the wellbore would influence the pressure
response only
if
storage effects are significant. 2
The curves for larger values of
r
D indicate that the in
fluence of the fracture begins to diminish as
r
D ap
proaches one (Fig.
3),
and that the existence of the ver
tical fracture can be neglected if rD is greater than two
Fig.
4).
Uraiet et ai.
9
and
Cinco-Ley
and
Samaniego-V. 10 have found that the influence of a ver
tical fracture at the active well on the observation well
response diminishes as r y approaches one, where r y is
defined in terms of the fracture length at the active well.
The curves in Fig. 2 also indicate that at a fixed value
of
r
D the dimensionless pressure response becomes less
sensitive to the orientation angle if the orientation angle
is greater than 45 0. Consequently,
if
the compass orien
tation of a vertical fracture is to be determined by an in
terference test, care should be taken in analyzing the data
if
it appears that a is greater than 45 0. It is also evident
from Figs. 2 and 3 that the curves for a=90° should be
used to design interference tests if the observation well is
fractured. We make this recommendation because the
magnitude of the pressure change is the smallest when
a =90°. Note that the influence of a
is
small for values
of
tDL/rD
2> 10.
When we compared the results obtained here with the
line-source solution, we found that the line-source solu
tion falls between the curves for 15 °
::; a::;
90° if
0.4::;rD::;2. For small values of rD the trace
of
the
line-source solution intersects the responses for several
values of a. As the value of
rD
increases, the line-source
solution shifts toward the curve for a = 90 0 and almost
overlies this curve when
r
D = 2. When
r
D is less than
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cL
0
a:
0
w
a:
::J
U)
U)
w
a:
'-
U)
U)
Z
e
U)
z
w
URAIET ql ,
SOLUTION
THIS
STUDY
I
ORIENTATION.
a
.,
10
RATIO
OF
DIMENSIONLESS TIME TO
DIMENSIONLESS
OISTANCE SQUARED, t D L r ~
Fig. 5-Effect of location of fracture on the observation well
response.
0.4, the response
of
the line-source solution falls bet
ween the culVes for ex equal to 15 ° and 45 ° provided that
tDL/rD 2 is less than 0.6. For larger values
of
tDL/rD 2,
it falls between the
cUlVes
for
ex
equal to 0° and 15°.
Fig. 5 compares the effect of fracture location on the
obselVation well response for
rD
=0.4. The solid lines
depict the response when the fracture
is
located at the
obselVation well, and the dashed lines represent the
response when the fracture
is
located at the active well.
The results for the latter case are taken from Ref. 9. Note
that Uraiet
et al
9
considered the pressure response
caused by a uniform-flux fracture at the active well,
whereas our work assumes that the conductivity is in
finite. The results shown in Fig. 5 establish two points. *
First, the location
of
the fracture does not appear to af
fect the obselVation well response. Second, the fracture
type (infinite-conductivity or uniform-flux) appears to
have a negligible influence on the well response. These
results are not self-evident for a number of reasons. For
example, the flux distribution around an infinite-
*This result is valid only
if
r
0
0.4.
DIMENSIONLESS RADIAL
DISTANCE, ro 0 4
FRACTURE RATIO
1 0
o INFINITE - CONDuCTIVITY
A : UNIFORM - FLUX
0.
0
a:
0
w
a:
::J
U)
U)
w
a:
0.
U)
U)
W
..J
Z
0
iii
z
w
1
10
ORIENTATION.
a
l ~
IJ 'x
2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ L - ~ ~ ~ ~ ~
1°10 2 10-' 1
10
1 1:
RATIO Of DIMENSIONLESS TIME
TO
DIMENSIONLESS DISTANCE
SQUARED. Ol f
Fig. 7-Pressure response at the observation well when both
wells are fractured F
L,
=1, r
0
=0.4).
936
cL
0
a:
0
w
cr
::J
U)
U)
w
a:
'-
if
U
W
..J
1 j l
Z
e
U)
z
w
DIMENSIONLESS
TIME,
f Dl
f
Fig.
6-Effect
of the dimensionless distance,
r
0 on the
observation well response.
10
conductivity fracture
is
significantly different from that
surrounding a uniform-flux fracture.
5
Similar results
were obtained for other values of
rD
Fig. 5 justifies the
earlier obselVation that the results given here are ap
plicable as long
as
the fractures can be represented by the
uniform-flux or the infinite-conductivity idealizations.
Fig. 6 shows the effect
of
a change in
rD
for specific
values
of ex As
expected, at any given time the dimen
sionless pressure drop decreases as
rD
increases for all
values of ex This graph re-emphasizes the need to ac
count for the vertical fracture at the obselVation well if
rD
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0.0
0.
o
s 1
V>
V>
V>
UJ
..J .,
Z 10
D
in
z
UJ
o
o
= INFlf I
-
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should be negligible. However,
if
both wells are frac
tured, then from Fig. 10 it is clear that there can be cir
cumstances when the influence
of
the fractures can be
significant for much larger values
of rD
Our computa
tions indicate that the influence of the fractures becomes
negligible at much larger values
of
r
D if F L <
I
The effect
of
the change in the radial distance,
r
D for
a given value
of 0:
are similar to those
in
the previous
case.
For
a given value
of F L ,
the magnitude
of
the
response is a function of 0: and rD. Since the pressure
drop is not directly proportional to the dimensionless
radial distance, care should be taken in designing a test
if
the wells are fractured.
Figs. and
12
show the effect of
F L
on the dimen
sionless pressure response at two radia{ locations, 0.4
and
1.5,
respectively. Also shown
is
the dimensionless
pressure response for the case of
F
L
j
= ex> no fracture at
the active well).
In
Fig. 11 we note that as
F L
j
increases, the
magnitude
of
the dimensionless pressure drop increases
for both values
of 0:
considered here. The same holds
true for rD
=
1.5 and 0:=90° Fig. 12). However, the
dimensionless pressure drops decrease as
FL
j
increases
if 0:= 15°.
After examining all the solutions for this case, we
found that if
rD
: :;
0.8,
the dimensionless pressure drops
will increase as
F L
increases for all values
of 0:.
However,
if
r > 0.8 the dimensionless pressure drops
will decrease as
F L
increases
if 0: is
in the range
0°:-:; 0::-:; 30° (curve/ shift to the right), whereas the
dimensionless pressure drops will increase as
F
L
j
in
crease when 0:
is
in the range
45 °
: :; 0: : :;
90 °
curves shift
to the left).
Comparing the pressure responses for F L
j
=2 and
FL = ex>
we find that the change in the pressure
response
is
small if
F
L
j
becomes greater than 2. We also
found that
if
rD 2: 1.5
and
F L 2: 2,
the response
is
not
sensitive to
FLJ.
Therefore, for all practical purposes the
solutions for
F
L
j
=2 can be used for all values
of
F
j
>2.
Discussion and Conclusions
The first objective
of
this work was to demonstrate that a
vertical fracture at an observation well can have a signifi
cant impact on the pressure response at the observation
well. Many assume that the skin effect at the observation
well can be neglected
if
there are no storage effects. As
shown here, this assumption is good only
if
the skin
region can be considered an annular region-i.e., in
finitesimally
thin.
Our second objective was to
describe qualitatively the effect of a fracture at the obser
vation and active wells on the observation well response.
Third, we delineated conditions under which the in
fluence
of
the fracture at the active and/or the observa
tion wells can be neglected. This is an important finding
useful in designing and analyzing interference tests. The
fourth objective
of
this work was to enable the analyst to
obtain quantitative information regarding the effect
of
vertical fractures on interference test data. For example,
the results given here can be used to determine the com
pass orientation
of
a fracture. Appendix B discusses an
example application for determining fracture orientation.
This example represents only one
of
the many possible
938
uses
of
the solutions obtained in this study. All dimen
sionless pressures obtained in this study are documented
in Ref. 7.
The results
of
this study highlight the need to incor
porate the wellbore conditions at both the active and the
observation wells. The characteristics of the porous
medium can be determined only after the wellbore condi
tions are incorporated properly.
The conclusions
of
this study are
as
follows.
1.
The existence
of
a vertical fracture at the observa
tion well has a significant influence on its response.
f
the active well
is
unfractured and
if
r
D <
2, the existence
of the fracture should be included in the analysis of
pressure data. f the active well
is
fractured, the ex
istence
of
the fractures can be ignored
if
rD
2:
2 and
FL 2: 1. f F L < 1
the existence
of
the fracture can be
n e ~ l e c t e d onl/if rD > >2.
2. The pressure responses at the observation well do
not possess any special features
or
characteristics to in
dicate to the analyst that a fracture exists at the observa
tion well and/or the active well. Therefore, the existence
of
these fractures would have t be determined in
dependently. Single-well pressure transient tests can be
used to identify the fractures. This requires running an
additional test at the observation well and measuring the
pressures at the active well during the interference test.
3. At a fixed value of rD the dimensionless pressure
drop becomes less sensitive to 0:
if
0:
is
greater than 45 ° .
Consequently,
if
the orientation of a vertical fracture is
to be determined, then care should be taken in analyzing
the data if 0: appears greater than 45
°
4.
The value
of
the pressure response
is
dependent on
the value
of rD
and 0:. Thus, care should be taken in
designing a test since the dimensionless pressure drop is
not directly proportional to the magnitude of rD Our
results show that the curves for 0:
= 90°
should be used to
design interference tests
if
the obervation well is frac
tured.
5.
f
the active and the observation wells are fractured
and FL
>
2, the curves for FL =2 can be used to
analyze{est data to obtain quantitative information.
6. f
F L 2:
1 and r
D >
2, the effect
of
the fractures on
the o s e r v ~ t i o n well response will be negligible.
Nomenclature
=
formation volume factor,
res m
3
/ stock-tank m 3 RB/STB)
C
=
total system compressibility, kPa - 1 psi - 1 )
d = perpendicular distance from the active well
to the fracture plane, m ft)
d =
dimensionless normal distance from the
active well to the fracture plane based
on LI
FeD
=
dimensionless fracture flow conductivity
FL =
fracture length ratio
h
formation thickness, m ft)
k
=
formation permeability,
/-tm
2
md)
k1
=
fracture permeability, /-tm
2
md)
LI = infinite-conductivity fracture length, m ft)
M
=
number
of
time intervals in which the
dimensionless time
is
divided
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N = number of equal length segments in which
the infinite-conductivity vertical fracture
is divided
p
reservoir pressure, kPa (psi)
q flow rate at the active well, stock-tank
m
3
/d STBID)
qD
= dimensionless fracture flow rate
qf = fracture flow rate per unit length,
d
m
3/
s
/
m
(cu ft/sec/ft)
r
=
radial distance between the center
of
the
active well and the center
of
the
observation well, m (ft)
rD dimensionless radial distance between the
center of
the active well and the center
of
the observation well based on L
f
at the
observation well
rD = dimensionless radial distance between the
center of the active well and the center of
the observation well based
on
L
f
at the
active well
r
w = wellbore radius, m (ft)
t = flowing time, hours
t
DL
f
=
dimensionless time based on
L
f
w fracture width, m (ft)
x -
distance along
x
axis, m (ft)
xD dimensionless distance along x axis based
on
L
f
x m =
horizontal distance between the active well
and the observation well, m (ft)
x mD
=
dimensionless horizontal distance between
the active well and the observation well
based on L
f
Y distance along Y axis, m (ft)
Y
D
dimensionless distance along
y
axis based
on L
f
a
=
angle between the
x
axis and
r
(compass
orientation), degrees
YJ = hydraulic diffusivity, Jlm
2 /[(Pa'
s)/kPa
I]
[md/(cp/psi
-I ]
Jl fluid viscosity, Pa' s (cp)
¢
porosity, fraction
Subscripts
cD
= dimensionless flow conductivity
D
=
dimensionless
f
= related to the fracture
i = initial; ith fracture segment
j
=
segment index
f
=
time index
w
=
wellbore
cknowledgments
Portions
of
this work were completed by N.A. Mousli in
partial fulfillment
of
the requirements for the MS degree.
Mousli acknowledges the support
of
the Arabian
American Oil Co. (Aramco). Computer time was pro
vided by Aramco (Shell Oil Co. Fellowship) and the U.
DECEMBER 1982
of
Tulsa's
Dept. of Petroleum Engineering.
We
are
grateful for this assistance.
References
I. Chu,
W.C.,
Garcia-R.,
J.,
and Raghavan, R.: Analysi s of In
terference Data Influenced
by
Wellbore Storage and Skin at the
Flowing Well, J Pet. Tech. (Jan. 1980) 171-78.
2. Tongpenyai, Y. and Raghavan, R.: The Effect of Wellbore
Storage Effects on Interference Tests, J Pet. Tech. (Jan. 1981)
151-60.
3. Cars aw , H.S. and Jaeger, J.e.: Conduction of Heat in Solids,
second edition, Oxford at Clarendon Press (1959).
4. Cinco-L., H., Samaniego-V., F., and Dominguez, A .N.:
Unsteady-State Flow Behavior for a Well Near a Natural Frac
ture, paper SPE 6019 presented at the SPE 1976 Annual Fall
Technical Conference and Exhibition, New Orleans, Oct. 3-6.
5
Gringarten,
A.e.,
Ramey, H.J. Jr., and Raghavan, R.:
Unsteady-State Pressure Distribution Created by a Well with a
Single Infinite-Conductivity Vertical Fracture, Soc. Pet. Eng.
J
(Aug. 1974) 347-60; Trans., AIME, 257.
6. Donohue, D.A.T., Hansford, J.T., and Barton, R.A.:
The
Ef
fect of Induced Vertically-Oriented Fractures on Five-Spot Sweep
Efficiency, Soc. Pet. Eng. J (Sept. 1968) 260-67; Trans.,
AIME,24O.
7. Mousli, N.A.:
The
Influence of Vertical Fractures Intercepting
Active and Observation Wells on Interference Tests,
MS
thesis,
U
of
Tulsa (July 1979).
8
Theis,
e.V.: The
Relation Between the Lowering of the
Piezometric Surface and the Rate and Duration of Discharge of a
Well Using Ground-Water Storage, Trans., AGU (1935)
519-24.
9. Uraiet, A., Raghavan, R., and Thomas, G.W.: Determination of
the Orientation of a Vertical Fracture
by
Interference Tests, J
Pet.
Tech
(Jan. 1977) 73-80.
10 Cinco-L., H., and Samaniego-V., F.: Determination
of
the
Orientation of a Finite Conductivity Vertical Fracture
by
Transient
Pressure Analysis, paper SPE 6750 presented at the SPE 1977
Annual Technical Conference and Exhibition, Denver, Oct. 9-11.
II. Richardson, L.F.: The Deferred Approach to the Limit,
Philos. Trans., Roy. Soc. London, Ser.
A
(1927) 226,299-361.
PPENDIX
A
Determination o the Pressure Response
As shown by Cinco
et al.,
4
the pressure drop caused by
an infinite-conductivity vertical fracture at the observa
tion well and plane radial flow at the active well can be
written as
. [ (X D
2
+YD
2
] I
rID rXmD+O.5
= - V2El - - J J
4tD
20 05
XmD .
q
D(X D,
T
D)e - {[ x D
-
x D 2 +(y D
-d
D)2]/[4(t D 7 D ]}
tD
-TD)
·d .x D dTD (A-I)
The condition of uniform pressure
over
the fracture sur
face can be satisfied, as indicated by Cinco et al., 4 by
dividing the plane source (fracture) into a number of
segments and assigning a flux to each segment in such a
way that the pressures at the midpoints
of
each segment
are identical. I f we now replace the time integral in Eq.
A-I by M discrete time steps and divide the fracture into
939
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N segments, we can write the following system of
equations.
N
PjDi =PDi +
qDiAij,
A-2)
j=l
for 1:5
i :5 N.
Here
P Di
represents the pressures at the
midpoints
of
each segment
of
the fracture and is given by
N
Ie,
M
l hE i [ XD
i
+d
D
]
+
4tD f I
j=l
where
_erf[X_Di_-X_mD+_o.5_-i_IN
]
2.JtD
-tD(/)
94
E l [(XD
i
-X
m
D+O.5- j N)2] ]
. A-3)
4[tD
-tD(f)]
and
E I (x) = - Ei( - x). . A-4)
Here
M
represents the time step under consideration,
qDi
represents the fluxes in each segment
i td
is identical to
tDL
in the text, and
qD (XD,
tD)=qD(X,
t )L/q.
The
product
qDiAij is
given by
The expression PDi
is
given by
N
I
j=l
In the system
of
equations given by Eq. A-2, all terms
on the right side are known except the dimensionless
flow rates, q
Db
at time level M. These rates can be ob
tained if we note that
P D(x
Di,d D,tDM) =PjD(xDi+
I,d
D,tDM)
and that
N
qDi,M=O.
i l
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Using these two relations, we may rewrite Eq. A-2
as
follows.
A
-A2J
A2]
A3d
A 12
-A22
A 22
A
32
)
A
IN
A
2N
)
A
2N
A
3N
)
Solving the matrix system Eq. A-5 will yield the
dimensionless flow rates for all fracture segments at any
time level
M.
Consequently, the pressure at the observa
tion well can be calculated by substituting these rates
in
anyone
of
the system
of
equations given by Eq. A-2. As
mentioned
in
Ref. 4, to ensure
an
accurate solution a
minimum number
of
fracture segments
is
required. A
discussion of the determination of the minimum number
of
segments
is
considered next and represents
an
impor
tant part
of
this work.
The dimensionless pressure drop
at
the observation
well was first calculated by arbitrarily selecting
rD =0 6
and cx=O. Several values for the total number of
segments,
N N=20,
30, etc. ), were considered.
We determined the optimal number
of
segments by first
determining the two consecutive N values that would
yield pressures within 1
of
each other
at
any instant
in
time. The smaller
of
these two values would be con
sidered tlie optimal number
of
segments. We found the
following.
o
(A-5)
1 The percentage difference between the calculated
dimensionless pressure drops at any given time de
creased as
N
increased. The percent change
in
the dimen
sionless pressure drop was less than one when N was in
creased from 90 to 100.
2. The percent differences between the calculated
dimensionless pressure drops for two consecutive N
values decreased as the dimensionless time in
creased-i.e.,
the solution became less sensitive to the
total number
of
segments at late times.
Similar results were obtained for rD
=0.2
and
cx=90°
It is
apparent from this analysis that 90 segments (or
more) are needed to simulate the infinite-conductivity
vertical fracture to obtain accurate solutions. Conse
quently, we first decided to use 90 segments
in
this
study. However, because of computer size (memory)
and time limitations,
it
was not possible to use 90
segments for long time intervals. The problem was over
come by applying
Richardson s
extrapolation
technique.
TABLE A 1 VALIDITY OF APPLYING RICHARDSON S
EXTRAPOLATION TECHNIQUE
Dimensionless Pressure Drop at the Observation
Dimensionless
Well for Case 1:
fo
=0.6 and a=OO
Time
Extrapolated
tOL,
tOLJ
f
02
N=30 N=60
Pressure<
N=90
0.001
0.2778 E -02 0.1530 E-03 0.2009 E-03 0.2169 E-03 0.2183
E-03
0.002 0.5556 E-02 0.1942 E-02 0.2268 E-02
0.2377 E-02
0.2377
E-02
0.003
0.8333
E-02
0.5323
E-02
0.5951 E - 02 0.6160
E-02
0.6159
E-02
0.004
0.1111 E-01 0.9668 E-02 0.1058 E-01 0.1088 E-01 0.1087 E-01
0.005 0.1389 E-01 0.1457
E-01
0.1573
E-01
0.1612 E-01
0.1611 E-01
0.006
0.1667
E-01
0.1980
E-01
0.2118
E-01
0.2164
E-01
0.2163
E-01
0.007 0.1944 E-01 0.2522 E -01 0.2679 E -01 0.2731 E-01 0.2731 E -01
0.008 0.2222
E-01
0.3073
E-01
0.3248
E-01
0.3306 E
-01
0.3306 E
-01
0.009 0.2500
E-01
0.3628 E
-01
0.3820
E-01
0.3884 E-01 0.3882 E
-01
0.01
0.2778 E-01 0.4184 E-01 0.4390 E -
1
0.4459 E-01 0.4457 E-01
0.02 0.5556
E-01
0.9568 E -01 0.9871 E -
1
0.9972 E -01
0.9969 E -01
0.03
0.8333
E-01
0.1426 E+01 0.1461 E+OO 0.1473 E+OO 0.1473 E+OO
0.04 0.1111 E+OO 0.1842 E+OO 0.1881 E+OO 0.1894 E+OO 0.1894
E+01
0.05 0.1389 E+OO 0.2219 E+OO 0.2260 E + 00 0.2274 E+OO 0.2274 E + 00
0.06
0.1667 E+OO 0.2563 E+OO 0.2606 E+OO 0.2620 E+OO 0.2620 E+00
0.07 0.1944 E+OO 0.2881 E+OO 0.2926 E+OO 0.2941 E+OO 0.2940 E
+00
0.08
0.2222 E+OO 0.3177 E+OO 0.3223 E+OO 0.3238 E+OO 0.3237 E+OO
0.09 0.2500 E+OO 0.3454 E + 00 0.3500 E+OO 0.3515 E+OO 0.3515 E+OO
Extrapolated pressure. P 0 = t P O N.60
-
t P 0) N.30
DECEMBER
1982
941
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8/9/2019 27. SPE-9346-PA
10/12
0
Q.
(/)
/)
w
a:
Q.
. \
I \
\_----INFINITE
-CONDUCTIVITY
\ FRACTURE
A ~ 5 2 m _ ~
\
\
\
. \ \ 8
Fig. B-1-Relative pOSition of Observation Well C with
respect
to
the Active Wells A and B.
10
2
FLOWING TI
ME,
t,
hours
Fig.
B-2-
Type-curve match of Observation Well C,
In
terference Test 1 (A-C).
•
I TWO
POSSIBLE ORIENTATIONS
~ I N DATA
FROM
wELL
PAIR
A-C
...............
15°
A 0 - - - 5 2 m ~ - . . . . . . . _
--
-.......
FRACTURE
\\.,
ACTUAL ORIENTATION USING
DATA FROM WELL
PAIRS
A-C
AND
B C
Fig. B-3-Actual orientation of the fracture plane.
942
To check the applicability of this technique, the
dimensionless pressure drops were calculated by using
30,60,
and 90 fracture segments for
rD
=0.6 and a=O
The extrapolated pressure drops then were calculated
by
Richardson s method using the pressure drops for 30 and
60 segments (see Table A-I). From Table A-I it
is
evi
dent that the agreement between the pressure drops ob
tained by using N=9 and the pressure drops obtained
by
Richardson s extrapolation technique
is
excellent.
Similar results were obtained for other dimensionless
radial distances and orientation angles.
This analysis proves that Richardson s extrapolation
technique
is
valid in this case. Hence, it has been used
throughout this study to calculate the dimensionless
pressure drop at the observation well using the pressures
calculated for 30 and 60 segments, respectively.
The procedure outlined also can be used if the active
well intercepts a fracture. To simulate the fracture at the
active well, the exponential integral on the right side of
Eq. A-I should be replaced by the following expression.
e +erf
[
rf
I -
x Du
) I
+
u
]
d r u
2.JrDu 2.JrDu
J rDu ·
Here, t Du XDu, and Y u are given by
t u =4 t DL
f
xF
L
f
2, ...................... (A-6)
XDu =2XD
XF
Lf
(A-7)
and
YDu
=2YD
XF
Lf
(A-8)
where F L
f
is the fracture length ratio.
APPENDIX B
Example pplication
In this section we illustrate the determination of the com
pass orientation of the fracture at the observation well by
using the results obtained in this study. The example is a
computer-generated case since no field data are available
to
us at the present time. Fig.
B 1
shows the relative
position of the observation well, C, with respect to the
two active wells, A and B. (The need for two active
wells will be explained later.) The pertinent reservoir
data and the pressure data from Interference Test I (A-C)
and Interference Test 2 (B-C) are given in Table B-1.
The observation well was tested before the in
terference test (single-well test). It was found that this
well was intersected by
an
infinite-conductivity vertical
fracture. Estimates
of
the formation permeability,
k
and
fracture length, L
f
obtained from the single-well test are
given in Table B-1. Pressures also were recorded at the
active wells during the interference tests. These data in
dicated that the active wells were unfractured. The for
mation permeability at each active well was calculated
SOCIETY OF PETROLEUM ENGINEERS JOURNAL
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by the conventional semilog method. These estimates
were approximately the same as the permeability at the
observation well. This result indicated that the porous
medium was relatively homogeneous
in
the area sur
rounding the three wells.
Using the value of
L
f
the dimensionless radial
distances between Wells A and C and between Wells
B
and C can be calculated as follows.
rA 52
rDA=-=-=O.4
. . . . . . . . . . . . . . . . . . . .
(B-l)
L
f
131
and
rB 79
rDB = - = - =0.6. .
. . . . . . . . . . . . . . . . . . .
B-2)
L
f
131
Interference Test 1 was analyzed
by
matching results
with the appropriate type curve. The curve for a= 15°
matched the data the best Fig. B-2). The formation
permeability was determined from the pressure match
point as follows.
PD=7.3XlO-
2
That is,
k=9x 10-
4
/Lm2.
The fracture length, L
f
is calculated from the time
match point as follows:
tDL
f
=7.36 x
10-
3
= [3.6 x
10
-6(9 x
10
-4 Lm
2
) 10 hours)]
-;-
[0.15 2.32 X
10
-6 kPa -1
(7x
10-
4
Pa·s) L
f
m)2].
That is,
L
f
= 134 m. These values are in very good agree
ment with the values obtained from the single-well tests.
As shown in Fig. B-3, there are two possible positions
for the fracture plane
if
this result is used. Thus, even
though the orientation
of
the fracture plane with respect
to Line AC is known a= 15°), a second interference
test is needed. The results
of
the second interference test
were matched with the type curve corresponding to
rD=0.6. We found that the curve for a=30° matched
the data the best. From this match we calculated
k
and
L
f
and found them to be in agreement with the first in
terference test. Thus, the compass orientation now can
be determined as shown in Fig. B-3.
At this stage, it should be clear that the interference
data need not be matched with the type curves corre
sponding to
rD
=0.4 and 0.6. Since values of k and L
f
are known from the single-well tests, the pressure vs.
time data may be converted to dimensionless form and
TABLE B-1-RESERVOIR
AND PRESSURE
DATA
DECEMBER
1982
Reservoir and Well Data
Porosity
1>
fraction of bulk volume
Formation thickness h m
System compressibility c
t
kPa -
1
Viscosity
/l,
Pa·
s
Formation volume factor B res m 3/stock-tank m 3
Flow rate q, stock-tank dm 3/s
Distance between Active Well A and Observation Well C,
rA
m
Distance between Active Well B and Observation Well C, rB m
Permeability k / lm2
Fracture length L
t
, m
Pressure Data
0.15
18
2.32 x 10-
6
7.0x10-
4
1
0.2
52
79
8.9x10-
4
3
Interference Test 1
(A-C) ,
Interference Test 2
(B-C),
Flowing Time
t
(hours)
o
4
5
8
10
16
20
24
30
37
40
50
60
80
100
t:.p
(kPa)
o
26
40.5
78
110.6
180.8
222.5
258.7
307.4
364.4
386.7
t:.p
(kPa)
o
13.8
21.3
35.0
53.7
61.2
89.0
115.4
172.4
222.5
'Determined from an additional test at the observation well and pressure
measurements at the two active wells.
* * Determined from an additional test at the observation well.
943
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placed on the type curves. The angle x then can be read
directly. But note that matching the data with the type
curves increases the level of confidence
in
the analysis
since it provides an opportunity for confirming the
results from the single-well tests.
In this example, values of r were calculated from the
pressure data obtained from the single-well test at the
observation well. However, if these data were
unavailable, r would be a parameter. All type curves
obtained in this study then should be used to match the
measured pressures.
Finally, we emphasize that the graphical matching
procedure suggested here can be replaced by a computer
approach if the dimensionless pressure vs. dimensionless
time data generated in this study are digitized. A
nonlinear least squares routine that minimizes the dif
ferences between the observed pressures and the
calculated pressures should be used. The differences
may be minimized by varying transmissivity, fracture
length, and fracture orientation.
944
Regardless of the approach used, it is imperative that
the pressure responses be measured at the observation
and active wells. In addition, single-well tests
may
be
needed to determine wellbore conditions at the active
wells.
SI Metric onversion Factors
bbl
x
1.589 873 E=OI
= m
3
cp
x
1.0
E-03
Pa s
cu ft
x
2.831 685
E-02
= m
3
ft
x
3.048*
E-Ol
m
mL
x
1.0*
E OO = dm
3
psi
x
6.894 757 E OO kPa
psi
-1
x
1.450 377
E-Ol
= kPa-
1
Conversion factor is
exact
SPEJ
Original manuscript received
in
Society of Petroleum Engineers office July 18. 1980.
Paper accepted for publication Dec.
10, 1981.
Revised manuscript received Aug.
9,
1982. Paper SPE 9346 first presented t the 1980 SPE Annual Technical Con-
ference and Exhibition held in Dallas Sept. 21-24.
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OF
PETROLEUM ENGINEERS JOURNAL