17. SPE-10043-MS
Transcript of 17. SPE-10043-MS
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SP
SPE
1 43
Society of
Petroleum
Engineers
Evaluation Of Hydraulic Fracturing By Transient Pressure
nalysis
Methods
by Heber Cinco, University
of
Mexico nd P M X
Member SPEAIME
Copyright 1982, Society of Petroleum Engineers
T h i ~
paperwas
presented at the International Petroleum Exhibition and Technical Symposium of the Society
of
Petroleum Engineers held in
Bejlng, China, 1826 March, 1982. The matenal
IS
subject
to
correction by the author. Permission
to
copy is restricted
to
an
abstract
of not
more than
300
words. Write
SPE
6200 North Central Expressway, Dallas, Texas, 75206 USA. Telex 730989
Product ion
of e i t he r wells
completed
in
low
permeabi l i ty
reservoirs or damaged wells
has been
poss ib le
because
of hydraul ic f rac
tur ing .
The
es t ima t ion of both the geometrIc
and f low c h a r a c t e r i s t i c s
of
a f rac ture rep re
sent a useful information for the ca l ib ra t ion
of
f rac ture
des ign methods
and permits
wel l flow behavior .
Transient pressure
wel l
t e s t ana lys i s
has
been
used with
success to es t imate
well
condi t ions and
r e se rvo i r s
parameters .
vent iona l methods of
in te rp re ta t ion
are
on r ad ia l
flow .
This i s
a
l imi t a t i on
when app l ied to f rac tured
wells
because they
e xh ib i t other
type
of
flow
a t
d i f f e ren t
t imes
in a t e s t .
Several
a l thors have presented
d i f f e ren t
to
c a l c u l a t e
both
r e se rvo i r
and
These methods include
lip
vs
l . r- t ,
the
lip vs ,
the
semilog
lip
vs t
and
type
curve matching.
Among
these techniques , th e
type
curve method
deserves
spec ia l a t t en t ion because it al lows
both the ana lys i s of pressure da ta and the
de tec t ion of
d i f f e ren t
flow
regimes.
Transient pressure
techniques
have
been proved
to be an
e xc e l l e n t formation
eva lua t ion
tool .
In te rp re ta t ion of wel lbore
Ids values of forma
and to de tec t
some he te rogene i t ie s in
the
r e se rvo i r .
These
were deve i n i t i a l l y ,
fo r
low
cond i t ions
and l a t e r
modified
to
take
in to
considera t ion d i f fe ren t types of
flow
geometry.
At
the
same
t ime,
s t imula t ion techniques
were deve
to
increase
the
of both
damaged
wells or wells
producing
low permeab i l i ty
r ~ s e r v o i r s
Hydraul ic
f rac
639
tur ing
s tands as on of the most
e f f e c t i v e
s t imula t ion
methods
because
i t s
a pp l i c a t i on
allows production of
wel l s
to be
I t
was
recognized
e a r ly
t ha t wells
in te rcepted by a
f r a c tu re
have d i f fe ren t
f low
behavior than unf rac tured wel ls , consequent ly,
appl ica t ion of ana lys i s methods
based on
theory
to
these
cases
can y ie ld erroneous r e s u l t s .
Many s tud i e s
l
45
have
been
pub l i shed
to
examine
d i f fe ren t
flow s i tua t ions
for
tured
wel ls . Table
1 pre sen t s a summarv
of these
pub l i c a t i ons .
I n i t l a l l y , most
works
1
-
10
dea l t with steady s t a t e flow toward
f rac tured wel l s ;
both
hor izonta l
and v e r t i c a l
f r a c tu re s
were
cons ide red
and
the
main
ob jec
t ive was to determine the e f fec t
of a
f r a c tu r
on
wel l produc t iv i ty .
The f i r s t
the
uns teady-s ta te
f low behavior wells was present
ed
Dyes
e t a l
They inves t iga ted
the -
e f fec t
a a
ver t i ca l
f r a c tu re
on
the semi log
l ine and concluded t h a t
the
l i n e i s affec ted
when
a
ture extends over
f i f t een percent of
the
dra inage radius .
La te r , Pra t s
l3
showed t h a t
a
wel l
in te r s ec ted
by
an i n f i n i t e conduct iv i ty ver
t i c a l
f r a c tu re
exh ib i t s an
ef fec t ive wel lbore
radius
equal to one ha l f of the f rac ture
length;
t h i s conclus ion was reached before by
Muskat 1
Russe l l
and T ru i t t
l
s tudied the
t r a n
of
r e se rvo i r
the
f rac ture
wai::
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EVALUATION OF HYDR ULIC FRACTURING BY TRANSIENT PRESSURE ANALYSIS METHODS
SPE 1004
a
c or re c t i on fa c to r
for the kh
values
ed from
semilog
graph fo r
cases
when
the
t u re pe ne t ra t i on
i s
high. This method was
extended by
Raghavan e t a1
Lee 1 6 used
a
numer ical s imula to r to
the
e f f e c t of both
v e r t i c a l and horizon
t a l conduct iv i ty
f rac tures
and
pre se n t
ed cor re la t ions to es t imate rese rvoi r and
f r a c tu re parameters .
The uns teady-s ta te l inea r f low
to f rac tured wells
by
and Millheim and Cichowicz
1 8
They
po in ted
out
t h a t
a
graph
of
wellbore
p
versus the square
root
of t ime
wf
s t r a i g h t l i n e whose slope
i s
por t i ona l to the f r a c tu re
i s
c a l l e d
l i nea r flow
Wattenbarger and Ramey
1
9
s tudied the
t r a n s i e n t
flow behavior towards a gas wel l
by an
i n f i n i t e conduct iv i ty
f r ~
included
non-Darcy
flow in the
format ion
and
concluded
tha t
t h i s
e f f e c t
i nc r ea se s the of the l inea r f low
s t r a i g h t
l ine .
a lso indica ted
the
non-Darcy flow causes an
ext ra
pressure
drop which i s flow
r a t e dependent .
A
numerica l s imula to r to study produc
t ion o f wells in te rcepted by
a
f i n i t e conduc
t i v i t y f r a c tu re
was by
Sawyer
e t
a l
They
showed
assump
t ion
of i n f i n i t e
f rac ture conduct iv i ty
can
l ead to se r ious er ro r s
when
wel l
performance.
Gringar ten , Ramey and 2S reexam
ined
the solu t ions
for
t r ans ien t
flow
fo r
f r ac tu r ed
wel l s
and
study
three
models:
i n f i n i t e
conduct iv i ty
v e r t i c a l
f rac ture
uniform
f lux
v e r t i c a l and uniform
f lux hor izonta l f rac ture I t was demon
s t r a t e d t h a t these
cases
e xh ib i t
three flow
;
i n i t i a l l y
there
i s
a
l i ne a r
flow
and
a f t e r
a
t r a n s i t i o n
flow the system
reaches a flow. They in t roduced
the type curve as
a
tool
and a method to
both
formation and
f r a c tu re
parameters .
This work was
extended
by Cinco
e t a l
7 , Raghavan e t l ~ ~
and
and
Hadinoto
3
to
the e f f e c t
i nc l i ne d f ra c tu re s , p a r t i a l
pene t r a t ion and
constant
pressure outer
boundary, respec t ive ly .
Later , it was demonstra ted by Cinco e t
a1
3 3
tha t the
i n f i n i t e
f rac ture conduct iv i ty
assumption i s not va l id
when
pressure
the f rac ture i s
considerable ,
tha t
i s ,
when
the dimensionless f r a c tu re
less tha t
300. A
f i n i t e f rac ture
model
was
fo r
these cases
and
indica ted tha t th i s type of
does
not exhibi t the l inea r f low , and
as
a
consequence, the l inea r flow graph
ana lys i s
i s
not re l i ab l e . Simi
conclu
s ions
were
by
Ramey e t
a l
and
Agarwall e t
64
The ef fec t of c losed exte rna l boundaries
on the
behavior
of f i n i t e
conduc t iv i ty f r ac
tUre
was
s tudied
by Barker and Ramey34. They
showed
tha t
a t
values of producing
t ime
t h i s type systems reaches pseudo-s teady
s t a t e
flow condi t ions
and demonst ra ted
t h a t
the use of
type
curves
3 3
,
ava i lable a t t h a t
t ime can lead , in
some
cases ,
to
a
problem.
Sco t t
35
t ime power
su re
data
for smal l
the
o f t ime.
He
showed
pre ssu re behav
io r
of a well
by
a f i n i t e conduc
v e r t i c a l f r a c tu re
can be approximated
of a t ime power being the
dependent on the
f r a c tu re
conduct iv
Exper ience
has shown
t h a t
wells a low
permeabi l i ty
r e se rvo i r ,
bottomhole pressure .
have beeen
Agarwall
e t
a l
an
i n f i n i t e conduct iv i ty
f r a c tu re
and the l a t e r
the f i n i t e f r a c tu re conduc t iv i ty
case . Type curves are given
in these works
to es t imate both format ion and
f r a c tu re
c h a r a c t e r i s t i c s from ana lys i s
of flow
r a te
da ta .
The
on f r ac tu r ed
by Ramey and
Cinco
and
now
ava i lable
wellbore
has
Raghavan
2
Type curves are
when wel lbore
s torage ends in a t e s t .
Flow r e s t r i c t i o n s with in or around
a
f ra
ture
can
af fec t
dramat ica l ly
the
e f f e c t i v e
ness of a
f r a c tu re . This
s
s tud ied
by
severa l au thors
5
,
43,44,45 I t
was
shown t h a t f r a c tu re
causes an ext ra
pressure
drop
of a well .
Type
curves for
da ta t h i s case has
Raghavan
2
, Cinco-Ley and
t i s
well
known tha t
Darcy ' s Law i s not
va l id
for
high ve loc i ty
flow ra t e s ;
t h i s
ca
occur
when f l u ids
flow a f r ac tu re .
Guppy
e t
a l
4 1
showed
t h a t
wells
a f fe c t e d by
non-Darcy
f low
wi th in
the f rac
t u re
e xh ib i t
an apparent which
i s flow
r a te
They
concluded
t h a t
es t ima t ion
o f t rue f ra c tu re
conduc t iv i ty
a t l e a s t two
t e s t s with d i f fe ren t
In
1981,
Cinco and If presented
a
genera l
theory for
the
t r ans i en t flow
towards
a
ver t i ca l ly f rac tured wel l .
found tha t in addi t ion to the l i nea r and
pseudorad ia l
f low ; a f r ac tu re with
in te rmedia te or
low
exh ib i t s
th
b i l i n e a r
flow
periOd and pre ssu re da ta
whe
versus
the four th
roo t o f
t ime
ld
l i ne
whose s lope i s
to the square roo t o f f r ac tu re
con
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SPE 10043
HE ER
CINCO-LEY
3
duc t iv i ty .
New type
curves
were to
overcome
the
problem in
data
ana
l y s i s .
In the
next
sec t ion, a descrp t ion of
flow
models used today fo r t r ans ien t
pressure
i s
in
f rac tured wel ls i s
Modern methods
of
models
fo r
f rac tu red wel ls :
cons ider four
a) I n f i n i t e
conduct iv i ty
ver t i ca l f rac ture
b) Uniform
f lux
v e r t i c a l f rac ture ,
c)
Uniform
f lux
hor izon ta l f rac ture and
d) Fin i te
conduct iv i ty ver t i ca l
f r ac tu r e .
In
some cases , th ree d i f fe ren t o f
outer
boundary condi t ions
are used; i s ,
i n f i n i t e
reservoi r ,
closed or cons tan t
p r ~
sure condi t ions .
A
v e r t i c a l
f rac ture
i s
considered
to
an
i n f i n i t e conduct ivi ty pressure
along the f rac ture i s neg l ig ib le
and
l eng th
x
f
.
The
f rac ture
the formation and produces from a
square reservoi r whose
s ide i s 2xe as
shown
in Figure. 1 . This model assumes
t h a t
flow
in to the wellbore i s only
through
the
f r ac tu re . As mentioned before,
t h i s
system
exh ib i t s a l inear flow
per iod
and a
r a d i a l flow per iod . The
f lux
d i
the f rac ture var ies with
time;
ly a
uniform f lux takes
place, as t ime
increases , the f lux
changes
and becomes con
s t an t when
the
pseudo- rad ia l
flow
i s es tab
l i shed .
This model
i s
s imi la r
to the i n f i n i t e
conduct iv i ty v e r t i c a l f rac ture
in
severa l
Fig.
1).
The
d i f f e r en ce
between
systems
occurs a t
the
boundary o n d ~
t i on
a t the
f rac ture . The
uniform f lux f rac
t u r e has a cons tan t f lux and a var iable pres
the
f rac ture ; it
a l so
exh ib i t s
pseudoradial flow
The
of th i s model i s shown in
Figure a hor izon ta l c i r c u l a r f rac
tu re of in an
i n f i n i t e
s lab r ese r
voi r i s The f lu id ex t r ac t ion
from
the
reservo i r occurs a t the f rac ture
face through a uniform
f lux
d is t r ibu t ion .
This
system
a l so
exhibi t s
l inear and
pseudor
d i a l flow
periods .
Figure 3 shows a ver t i ca l
f r ac tu r e
in
an i n f i n i t e
s lab
reservoi r .
The
f rac ture
has a permeabi l i ty
l
a
width
and a ha l f
length
x
f
. The formation i s
t o t a l l y In a v e r t i c a l
di r ec t ion
64
t u r e
and it i s
l imi t a t ed
by a
lower
and an
upper impermeable boundar ies . The t r a n s i e n t
behavior for t h i s
system
can inc lude
flow
per iods as
indica ted in
Fig.
4; in
, there
us
a
l i n e a r flow with in the
-
f rac ture , t h a t can be fol lowed by
the
b i l i n
ear flow
then
a l i n e a r
flow
in the format ion
may
be presen t and even tua l ly
the
pseudoradia l
f low per iod i s reached.
Modified
vers ions o f the
these
models
have
been
presen ted
too,
are the cases of
damaged f rac tures
29,
32, 4 , 5 Fig. 5 and 6
and
heterogeneous
f ractures
4o
,43.
may correspond
There are
of
ana lys is for
each flow regime; ,
flow
data
must
be
analyzed wi th a graph of versus
~ for b i l i n e a r flow
data
the 6p versus
4ft-graph must be used and
the
semilog graph
6p
versus
log t must
be appl ied
to data
on
the pseudo- rad ia l flow per iod .
The general
so lu t ion
fo r the pressure
behavior
in
a
r e se r vo i r
i s expressed
in
terms
o f dimensionless var iab les . For f rac tu red
wel ls the
va r i ab le s
are used:
Dimensionless Pressure Drop.
Oil
wel l :
kh 6 P
141.
2 qSlJ
Gas
wel l :
Dimensionless Time
4
2.637
x 10
k t
lJ
C
t
X
f
;?
2
rhllctr
I
,t- W
4
2.637
x 10
k t
lJc
t
r
f
2
Dimensionless
Frac tu re Conduct ivi ty
1)
2)
3)
4)
5)
6)
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EVALUATION
OF
HYDRAULIC
FRACTURING
BY TRANSIENT PRESSURE ANALYSIS
METHODS
SPE
1004
Fracture
Skin Factor
Flu id
Loss
Damage:
li
s
~ _
1)
k
s
Choked Frac tu re :
Trx
k
s
7
8)
Next
a
discuss ion
i s presen ted on
the
bas i s and
appl ica t ion o f each
method of
ana lys is .
(6p versus \It
This
method
was developed
for f in i t e
conduct ivi ty v e r t i c a l f rac tures of small
s to rage capaci ty and i s
based
on the b i l i n e a r
flow
theory .
This
behav io r
i s
a
r e s u l t
the
superposi t ion
o f two l inear f lows; one
flow
i s incompress ible and occurs
wi th in the
f rac ture and the
other i s
a compressible
flow
in
the
format ion. A b i l inea r
flow
e x i s t s
when
the
flow
in to the
wellbore
i s due to the
expansion of
the
system
in the formation and
f rac ture
e f f e c t s
have not yet a f fec ted
the
well
The
dimensionless wel lbore
pressure
change for a t e s t can be as:
2.45
P
wD
1 4
9
or:
Oil
wel l :
6p
44.1
(10)
h
f
(kfb
f
Gas
well :
6m p)
4 4 4 7 5 q T t
4
hf k fb
f
1/2(pc
t
k) 1 / 4
(11)
These equat ions
indica te
tha t the
pre
sure change i s both
to
h f k f b f ) ~ 2
and d i r ec t ly propor t iona l
to
fourEh
roo t of t ime.
According to
Eqs.
10
and
11.
a graph
o f 6p
(or 6m(p)} versus
4/1:
gives
a s t r a i g h t l i n e passing
through
the or igin of slope rn
bf
as ind ica ted in F i g . ~
a l so
i nd ica t e s tha t
af t e r
the l ine
por t ion
the curve could
be
concave
o r downwards depending upon
the
dimensionless f rac ture
conduct iv i ty .
64
From
t h i s
graph
the product hf k fb
f
V
can be es t imated
as
fo l lows:
Oil
well :
44.1 qBp
Gas wel l :
The b i l i n e a r
flow
ends
a t :
t 0.1
Debf
(k b
fo r D
>
3
f f D
0.0205
for 1.6
f )
f )
w
c
Q
f )
/ )
W
.-J
Z
0
f )
Z
ltJ
0
10
10
3
5 7 1 1 9 1 II .. 5 1 5 7 8 9 1 3
51171191
I )-110
Sf s
0 R SrS Ch
2
10
Fig.
2 4 r ~
versus Sts or
(StS)Ch
for damaged fractures
DIMENsrONLESS TIME, TO
Fig. 25 -Type curve for an infinite conductivity vertical fracture with wellbore storage
664
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('f )
Cl
4 1
Cl
U
..0
4 11
2 \
II
-
Cl
;;
UNIFORM-FLUX
WELL IN AN INFINITE RESERVOIR, Xe/Xf
QO
DIMENSIONLESS
STORAGE CONSTANT, C
OXf
O ~
5X 10-
3
10-
2 - - - - : r - - . . - : i
DIMENSIONLESS TIME. tOXf
Fig. 26 -Type curve for a uniform flux vertical fracture with
well bore storage
nd
o f Wellbore
Storage
Effec t s
10
1
~ ~ L
10-
1
10 10
2
10
3
10
4
kfb
f
D 2/3
F
2
t
Oxf
= Y toxf
C
Of
)
3