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8/10/2019 22. SPE-13846-PA
1/10
x ct Solutions for Infinite-
Conductivity Wells
Paul Papatzacos
SPE, Rogaland Research Inst.
Summary. In pressure transient testing, the infinite-conductivity condition translates mathematically into a
uniform-pressure (or uniform-potential) condition at the well. This means the flux at different points of the well
should be determined in such a way that potential remains uniform at the well. The integral equation for
accomplishing this
is
solved analytically to yield the Laplace-transformed potential. For fractured-well problems,
this leads to a relatively fast algorithm for drawing type curves directly on a computer screen. For limited-flow
entry problems, the analytical pressure expression can be used with the method of images to treat problems in
reservoirs of finite thickness and/or areal extent.
Introduction
One traditionally thinks
of
a well as having infinite con
ductivity in the direction parallel to its axis. This assump
tion is also included as a standardoption inmost numerical
reservoir simulators. In pressure transient testing, the con
cept is important for wells with limited flow entry and
for fractured wells.
The infInite-conductivity condition translates mathemat
ically into a uniform-pressure (or rather uniform-potential)
condition at the well. This means that the flux at differ
ent points of the well is unknown a priori and must be
determined in such a way that potential is uniform at the
well. Muskat 1 recognized that this amounts to solving an
integral equation. Muskat and later Gringarten and
Ramey
2
showed how to solve the integral equation for
mulation numerically by dividing the source into small
elements, each having uniform flux. To determine these
elementary fluxes, two conditions must be applied: they
must sum to the required total rate and they must produce
a uniform potential at the well. This method was applied
by Gringarten and Ramey 3 to the limited-flow-entry
problem and by Gringarten et at. 4 to the fractured-well
problem.
In the present paper, this integral formulation is solved
analytically
to
yield the Laplace-transformed potential.
For
the fractured-well problem, the purpose of this cal
culation is to obtain an expression for the pressure that
leads to a relatively fast algorithm for drawing type
curves, both for single-well testing and for interference
testing, directly on a screen. It is usually easier to obtain
numerical results from analytical expressions than from
their nonanalytical analogs.
For the limited-flow-entry problem, on the other hand,
the purpose
of
the calculation is different. Because the
reservoir is considered to be infinite, both laterally and
in thickness,
type
curveswould not have a direct reservoir
engineering application. The interest in an analytical ex
pression in this case lies in the possibilities it offers in
conjunction with the method of images to treat problems
in reservoirs
of
fmite thickness and/or finite areal extent.
Copyright 1987 Society of Petroleum Engineers
SPE
Reservoir
E m r i n f p r i n ~
Mil
One such problem
of
interest is the calculation
of
limited
flow-entry pseudoskin
5
; another is water coning.
6
Some
details about the treatments of these problems will be
given.
Limited Flow Entry Problem
Dimensionless Variables
and
Spheroidal Coordinates.
.The situation considered is that of a line-sourcewell with
limited flow entry in a reservoir that is infinite in all three
dimensions. As shown in Fig. 1, the axes are chosen so
that the perforated wellbore covers Interval -a , +a
along the
z
axis. Note that, because of symmetry across
the xy plane, one is also solving the problem of a semi
infinite reservoir with a horizontal no-flow upper bound
ary, anda line-source well perforated from the top down
ward to the depth
of
a
kH and k y are the permeabilities in the horizontal and
vertical directions, respectively. Dimensionless coor
dinates are defined as
XD
= ky/k
H
Ih x/a ,
1a
YD = ky/k
H
1h Y/a ,
lb
and
ZD=z/a lc
and dimensionless time as
2.367 X
1 ky t
(2)
.pl c
t
a
2
The pressure potential,
.p=p P
o
zl144.0,
(3)
can easily be shown to obey the same diffusion equation
as the pressure, p. An expression for.p can
be
written
in terms of the appropriate Green function and of an
unknown source function. 3 The latter must
be
determined
217
8/10/2019 22. SPE-13846-PA
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x
-a
y
a
z
Fig. 1-Limlted-flow-ent ry problem.
The
well Is perforated
along Interval
a, +a
on the z axis thick line .
Fig. 2-Limlted-flow-entry problem. Sections
a plane
containing
ZD
with
~ . c o n s t a n t
and a-constant surfaces.
The resulting ellipses and hyperbolas are labeled by the
corresponding and
a
values
a
Is In degrees .
in such a way that
cI>
does not vary with
z
at the wellbore.
At this point, it
is.
important to choose a system
of
coor
dinates that allows an easy representation
of
the line
source well: prolate spheroidal coordinates,
4
~ , a . , { j , re
lated to the usual Cartesian coordinates by the following
equations.
x
D =sinh sin
a.
cos
{j, :
(4a)
YD = sinh sin
a.
sin
{j,
(4b)
and
ZD
= cosh cos a. (4c)
We can now write an expression for the dimensionless
potential,
D, in terms of the unknown dimensionless
flux along the well,
qwD
(see Appendix A for a deriva
tion and for the definition
of q wD :
r r ll
D =
J
dtv
J
da. sin a.
qwD a ,tD)
o 0
e _[ R
2
/4 I
D
-I h)]
x , (6)
4J;( tD
- tv )
where
R
2
= (sinh sin
a) 2
+(cosh cos
a cos a. )2.
(8)
e
- [R ; /4 t
D
t h ]
x
4J;( tD - t v ) ; (9)
Note that
R
2
= rJ + ZD -Z V)2
and that
D
depends on
a., and tD By setting ~ O in Eq. 6, we obtain
cID
at the wellbore, which we denote by
4>wD tD ,
thus
making explicit the fact that it is uniform along the well-
Le., independent
of
the coordinate
a.:
ak
H
- - - 4 i -4 (7)
4 2j tqt
4>D
and
The sections
of
some
c o n s t a n t - ~
and constant-a. surfaces
with an arbitrary rD,ZD plane are shown in Fig. 2. The
ellipsoids become thinner as becomes smaller. The el
lipsoid for = 0 has degenerated into Segment(
- 1
,
+
1)
on the
ZD
axis. In other words, the producing interval
of
the well, given by XD =0
YD
=0 - 1 ~
~
+1 in
Cartesian coordinates, is simply given by =0 in sphe
roidal coordinates. An arbitrary point at the wellbore has
a coordinate equal to zero and an
a.
coordinate in Inter-
val
(0, 11 ).
_
2
2 ) 1 h _ n h t. . (C)
r D - XD
+
YD - SI . c; SID
a. . .
. . . . . . . . . . . . . :J
These new coordinates are restricted to .the intervals
O ~ ~ ~ o O
0 ~ a . ~ 1 I , and 0 ~ { j ~ 2 1 1 .
The surfaces with constant are rotational ellipsoids,
while the surfaces with
a.
constant are rotational hyper
boloids (oftwo sheets). The axis
of
rotation for both these
families is the ZD axis. The surfaces with {j constant are
planes containing the
ZD
axis. Because
of
the rotational
symmetry, it is useful to introduce
218
SPE Reservoir Engineering, May 1987
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In Eq. 9,
R
wis the expression obtained from Eq. 8 by
setting
=0.
One must be careful, however, because
w is known to be infinite f?r a
l i n e - s o ~ r c e
well. To
avoid infinities in the calculatIons, we wnte
2 . , 2
R = sinh sin a + cosh cos a - c o s a
10
and let go to zero. This limiting process will be
i ~ d ~
cated at the proper place later. In other
words lt
IS
assumed that the well is a thin ellipsoid that is forced to
degenerate to a line. The question arises w ~ e t h e ~ it ~ o u l d
be more realistic to model the well as a thm elbpsOld by
choosing a ,small but nonzero v ~ u e . for w thus a ~ o ~ d
ing the difficulties of taking the bmlt ThIS IS m
deed possible, but it should be kept
mmd that the
solution given here is then only a p p r o X 1 m ~ t e . because an
ellipsoidal well with w
0 has a nonvamshmg surface,
while the Green function used in Eq. 9 is correct for a
reservoir with no boundaries at finite distances. ,
We show in Appendix A that the flux,
q wD,
must
satisfy
d a
sin qwD a,tD) =
1 11
o
This is the analytical expression
of
the first condition
imposed by Muskat
and by Gringarten and R
amey2
on
the elementary fluxes-i.e.
that they add to the total rate.
Th e
second condition on qwD is expressed in Eq. 9: the
flux rate must be such as to produce a uniform potential
at the well.
Statement of the Problem.
is necessary to solve Eqs.
9 and 11 for the two unknown functions
w
and q
w
.
The time integration in Eq. 9 is a convolution, an
obvious first step is to use the Laplace
t r a n s f o r m a ~ l O n
to
eliminate one integration. Here
[j t ]
is the Laplace
transform ofj t and S is the Laplace parameter. We first
introduce the following notation:
c C [ ~ D ~ , a , t D ] = 1 P ~ , a , s ,
12
c C [ ~ w D t
D)] =1Pw S), 13
and
[qwD a,tD)]
=
U a,s).
. 14
One finds in Ref. 7 that
exp[-
R
2
/ 4tD)]
= exp -
R.[;
4 ;;t8
R
15
so by taking the Laplace transform
of
Eqs. 6, 9, and 11,
one obtains
e -R
1 P ~ , a , s = lh1d a
sin a U a ,s) ,
. .
16
o
R
SPE Reservoir Engineering,
May
1987
e-
Rw
1Pw S) =
lh
dd
sin
a
U a ,s) ,
17
o R
w
and
da
sin a U a,s) =
lis,
18
o
where
R
is given by Eq. 8 and
R
w
by Eq. 10. Now the
problem is to solve Eqs. 17
and
18
for
wand
When
the resultant expression for
U
is used in Eq.
16
and the
integration performed, one obtains an
e x p r e s s ~ o n
for the
Laplace transform
of
the dimensionless potential.
Solution. Step
1
An analytical solution
of
the prob
lem is made possible by the existence of an eigenfunction
expansion
of
exp[- R:fi ]/R. This expansion is given in
Refs. 8 and 9 for exp iTR /R and is written below for
T=i.[; and for R given by Eq.
8:
R
00
1
J
= 2 [ ;
--Son(i \lS, cos a )
R
n=O NOn
XSon i.[;, cos a R b ~ i . [ ; , I R b ~ i.[;, cosh
19
In this expression,
Smn
is the angular spheroidal wave
function while
R
and
R
are the radial spheroidal
wave functions
of
the first and third kind, respectively
see also Ref. 7 . In Eq. 19, only the
f u ~ c t i o n s
with
f i ~ s t
index equal to zero are involved. N
mn
IS the normalIZ
ing constant; i .e. , for
m=O,
da sin a SOk k.[;, cos a)Soj i.[;, cos a)=okjN
Ob
o
20
where 0
kj
is the Kronecker delta.
The following expression for SOn is needed:
00
SOn
= d ~ ~ + r i . [ ;
)P2k+r COS
a) ,
21
k=O
where P is a Legendre polynomial, and where
r=O
1 when
m
n
is even odd . The diJn i:..{;) are expansion
coefficients with known rules
of
calculation.
9
Step 2. The function U Laplace transformofthe flux,
see Eq. 14 depends only on the angular coordinate
ex
and
can be expanded in a series
of
eigenfunctions,
SOn
which
form a complete set on Interval 0,1r . There is a sim
plification because
of
the symmetry
of
the flux across the
x D,YD plane; this implies that U is
s y m m e t r i c a r o ~ n d
a=1r/2-Le.
that it is a function
of
cos
2
a. Accordmg
to Eq.
21
and the properties
of
the Legendre polynomi
als, one may then write
00
, U a,s)= 3kS02k i.[;, cos
a),
22
k=O
2 9
8/10/2019 22. SPE-13846-PA
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XD
= cosh
p
cos
v
28a)
Infinite onductivity Vertical-
r cture
Problem
This problem s solution is exactly analogous to the
one
used previously.
The reservoir is supposed to be isotropic, with constant
p e r m e b i l i ~ k.
The fracture is ,most easily represented
in ellipticS, 1 coordinates p,v), which are related to the
Cartesian coordinates
xD YD
by
2.637
x
10
4 kt
- - - ~ - ~ - C - t X - f 2 - - - :
27c)
and
Dimensionless Variables
and
Elliptical Coordinates.
The situation is that
of
a vertical fracture. Flow is two
dimensional. The axes are shown in Fig. 3 and the frac
ture extends from f to xf along the x axis. We as
sume .an isotropic reservoir and introduce
XD =x/xf, 27a)
YD =y/xf
;
27b)
sure in finite reservoirs in situations where it is known
that such a steady-state solution exists. One case
of
in
terest consists
of
an areally square reservoir
of
finite thick
ness with a limited-flow-entry well in the middle
of
the
square and a constant-pressure condition at the lateral
boundaries. This situation arises in the calculation
of
the
critical rate to water coiting first investigated by Muskat. 1
The advantage
of
Eq.
26 is that the infinite-conductivity
condition is satisfied at the outset.
An
investigationof crit
ical rates along these lines has been performed by
Hoyland.
6
It
is
more
surprisingthatEq
26 more exactly , an
asymptotic expansion obtained from Eq. 25 for large t
D
can be used to obtain an expression for the pseudoskin
for a limited-flow-entry well in an areally infinite reser
voir,with finite thickness. The applicability
of
Eq. 26 is
not evident because dimensionless pressure is proportional
to In
tD
for large
tD
in this case, whereas Eq. 26 is a
steady-state expression. The method of images leads to
the calculation
of
an infinite sum because infinitely many
images are necessary to satisfy the no-flow boundary con
ditions at the upper and lower boundaries, and the infinite
sum turns out to be divergent. One must then identify this
mathematical divergence with the physical divergence
of
pD, which is caused by In
t D
in the limit
t D -+
The
geometrical skin is obtained as the finite
e x ~ r e s s i o n
that
remains when the divergence is subtracted. Ref. 5 dis
cusses the applicability
of
the method of images to the
problem considered. Re[ 5 shows that the method de
stroys the condition of strict uniformity of potential at the
well, but that this condition applies to a good approxima
tion for the majority
of
cases
of
interest.
2d8
iJ ;
25)
1
D =lf
In
t .
26)
e
-1
Thus t/;w is determined, and so are the k by Eq. 23),
which means that
U
is determined by Eq. 22. One finally
obtains
t
see Appendix
B :
1
= J
t/;w
k=O
R
k
t/;w s
-
J; -
( ~ , a , s ) = w ( s )
k=O
Le., an expansion in terms
of
angular functions with an
even second index. The function
U
will be determined
as soon as the coefficients
{
k are known.
Step 3. I t is now possible to solve Eqs.
17
and
18
to
find
t/;w
and
U.
Substitution
of
Eqs. 19 and 22 into Eq.
17
gives the
following relation between the coefficients
{ k
and
t
w
see Appendix
B :
Eqs. 24 and 25 constitute the solution
tothe
limited-flow
entry problem. The solution is approximate
if
w
is
given
a nonzero value, as noted above. For a line-source well,
however, the solution is exact but must be obtained by
letting w-+ 0, taking care of the fact that R i,J;, cosh
w diverges when w-+0. 9 In practice, this can be done
by inverting the Laplace transform Eq. 25) for a very
small value
of wand
using an expansion for the func
tion
R ~ ~ 1 (i...{;,
cosh
w
valid for small w.
9
This has
not been attempted in the present paper for the reasons
given in the Introduction. An analytical investigation
of
Eqs.
24
a nd 25 for small w and small
s
however, has
been carried out in Ref. 5. One product
of this investiga
tion is the long-time approximation, obtained by invert
ing term by term the small-s expansion
of
Eq. 25 10 and
dropping terms that vanish when
tD
-+
One finds that
N 0 2 k R b ~ k i . j ; ,
I R ~ i k i J ; , cosh w 23)
Substitution
of
Eqs. 22 and 23 into Eq.
18
gives see Ap
pendix
B
and
Applications.
Eq. 26 can now be used in conjunction
with the method
of
images to get the steady-state pres-
220
Y
D
=
sinh
p
sin
v. .
28b)
SPE Reservoir Engineering, May 1987
8/10/2019 22. SPE-13846-PA
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y
- - - - - : - - - - - - - - - - - - - - - - ~
X
Fig.
3 Vertical
fracture problem. Fracture Isalong Interval
-x
+x,)
on x
, axis
thick line .
The curves with p constant areellipses, while the curves
with
v
constant are hyperbolas (see Fig. 4). The p= O el
lipse has degenerated into Segment 1 +1 along the
xD
axis and thus represents the fracture. Coordinates
p
and v vary in the intervals 0 s p s 00 and 0 S vS
2 r
The dimensionless pressure, P
D ,
can now be written
in terms of the unknown dimensionless flux at the frac
ture,
qjD
(see Appendix A):
r
D
r
21r
P D = J dt b J
dv sin
v qjD v ,tD)
o
0
e [D
/4 t
tb ]
X , (29) .
tD tb
where
kh
P D
=
4 2 ~ q
P i p (30)
and
(31)
In Eq. 30, h is the reservoir thickness. PjD t
is the
uniform dimensionless pressure at the fracture, then by
setting p= O in Eq. 29, one obtains
r
t
D r
2
PjD
=
J
dt b
J dv sin v qjD v ,tD)
o 0
e
-[ cos p-cos pl 2 /4 t D tb ]
X
(32)
tD tb
SPE Reservoir Engineering, May 1987
Fig. 4 Vertlcal fracture problem. Elliptical coordinates.
With p constant, lines are ellipses labeled by the values
of
p
and
with
constant, lines are branches
of
hyperbolas
labeled by the values
of
v In degrees .
Note that there is no difficulty in setting p =0 because
pressure is known to be finite at a fracture of zero thick
ness, so there is no limiting procedure involved here, as
in the case
of
the line-source well. Eq. 32 is the first
of
two equations that
qjD
must satisfy; the second is (see
Appendix A)
r
2
J
dv
sin
v qjD V ,tD)=1.
(33)
o
Statement of
the
Problem; One must solve Eqs. 32 and
33 for
qjD
and
PjD
Eq. 29 will then give the dimension
less pressure. Introducing the notation
[p D( P,v,tD)]
=1/t p,v,s), (34)
[PjD tD)] =
1/tl
s
)
, (35)
and
[qjD V,tD)]
=
V v,s), (36)
one finds that the Laplace transforms
of
Eqs. 29, 32, and
33 are
r
2
t= J dv sin v
V v
,s)K
o
.[;V (37)
o
r
2
1/tf=
J dv sin
v
V v ,s)Ko .[;lcos v-cos v I),
o
(38)
and
rll dv sin
v
V v ,s)=1/s, (39)
o
221
8/10/2019 22. SPE-13846-PA
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whereK0 is the modified Bessel function of order
O
7
The problem reduces now to solving Eqs. 38 and 39 for
and
V.
will then be given by Eq. 37.
Solution. As in the case of the limited-flow-entry well,
an analytical solution is made possible 2Y the existence
of an eigenfunction expansion of K0
s
D . Such an
expansion is given in Ref. 8 for the Hankel function
H0 1) . Using the relation
7
one obtains, when D is given by Eq. 31,
where cem is the angular or periodic Mathieu function
and Ie
m
and Ke
m
are radial Mathieu functions.
7
There
is a normalization formula for the ce
m
functions analo
gous to Eq. 20.
7
There is also an equation like Eq.
21
for the cem; it
is written below to establish a notation that will be used
later:
00
ce2m(V,q)= q cos 2nv (41)
n=O
The Laplace transform
of
the dimensionless flux at the
fracture,
V
can beexpanded in a series
of
angular eigen
functions cem An expansion that accounts for its J sym
metry properties
V
is even about v=0,7r/2,7r,37r/2) is
V(v,s)=
Akce2k v , - ~ ; (42)
m=O 4
Le., an expansion in terms of
ce
functions, with an even
index.
It is now possible to solve Eqs.
38
and 39. The calcu
lations are exactly analogous to the ones performed in the
limited-flow-entry case, so only the final results are
reported.
The
Ak S
of
Eq. 42 are given by
2k
1/Ils) Ao -s/4
A k (43)
, 1r
Ie2k (0,s/4)Ke2k(0,s/4)
where A has been defined in Eq. 41 and
1 00 2 [ A ~ m
-s/4)]
2
s
L; ...... (44)
I/Ils) ,
m=O
Ie2m(0,s/4)Ke2m(0,s/4)
222
Finally, the Laplace transform of th e dimensionless
pressure is
1 / I p , v ~ s = 1 / I t < s
2 A ~ m ~
m=O
,4
Eq. 44 was presented by Kucuk and Brigham
11
with
another notation for the Mathieu functions.
Applications. The large
tD
approximation implied by
Eq. 45, obtained by inverting its small s expansion term
by term and dropping terms that vanish when
t
D -+ CXJ is
PD = In
e -
2p t
D
+2
In
2- f
(46)
where I is Euler s constant (0.5772), so that2ln 2- 1/2=
1.09769. Thus for p=O-Le., at the fracture : Kucuk and
Brigham s 11 result is recovered.
By assuming that the wellbore is situated at
x
D=Y
D
=
and by denoting the well radius as r w, one can extract
from Eq. 46 the following expression for the skin factor:
2r
w
s ln
p
(47)
XI
where p can be obtained from Eqs. 28:
[
1
r ~ [ l
+rh)2
-4rh
cos
2
8]
]lh
p=arc cosh
2
: (48)
The notation and definition of the inverse hyperbolic func
tion is that
of
Ref. 7, and rD and 8 are the usual polar
coordinates.
XD
=rD
cos 8 (49a)
and
YD=rD
sin 8 (49b)
Eq. 46 shows that
tD/exp 2p ,is
the proper combina
tion against which to plot
PD if
one wants curves with
different
p
values to collapse as much as possible into a
unique curve. For larger
rDi.
Eq. 48 shows that exp(2p)
is approximately equal to 4rD Eqs. 44 and 45 have been
inverted with the Stehfest algorithm.
12
Fig. 5 is a plot
ofpD vs.
rD
at a fixed angle 8 and a fixed
time.tD,
show
ing that the condition of uniform pressure at the fracture
(Le., for
8=0)
is satisfied. Table 1 gives PD vs. tD at
the fracture
(P=0 )
for different value
of
the Stehfest in
teger N. Note that
N=6
is acceptable for most practical
purposes. There is some discrepancy between the values
SPE Reservoir Engineering, May 1987
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Fig. 5 Vertlcal fracture problem; Po VB. the polar coor
dinate, r
for three values
of
the polar angle, 8. Note uni
form pressure at the fracture 8=0).
omenclature
a
=
half-length of interval open to flow,
ft
[m]
A ;: =
constant in Eq. 41
ce
m
=
periodic Mathieu function
7
c
t
=
total reservoir compressibility, psi
[kPa ]
d r
) =
expansion coefficient in Eq.
21
~ e x p - 2 p
D =
constant defined in Eq. 31,
dimensionless
h
=
reservoir thickness, ft [m]
= Hankel function
7
~
Iem =
Mathieu function
7
k =
permeability, md
kH:k v =
horizontal and vertical permeabilities,
md
Ke
m
=
Mathieu function
7
K
0 =
modified Bessel function
7
f =
Laplace transform
of
function f
Fig. 6 Vertlcal fracture problem;
Po VB. t
o
/exp 2p). The
elliptic angular coordinate v is
45
TABLE p0
VS
to
AT THE FRACTURE p =0 FOR
DIFFERENT VALUES OF THE STEHFEST PARAMETER, N
Pressure
Time
N=6 N=8 N=
Asymptotic
*
.01 0.1698
0.1697
0.1699
-
0.02 0.2359
0.2358 0.2357
-
0.04
0.3256 0.3253 0.3253
-
0.06 0.3914 0.3911
0 39fo
-0.08
0.4450
0.4446 0.4446
-
0.10 0.4909 0.4904
0.4904
-
0.20 0.6592
0.6585
0.6589
-
0.40
0.8693 0.8684
0.8684
-
0.60
1.0115 1.0105
1.0105
0.8423
0.80
1.1204 1.1194
1.1194
0.9861
1.00 1.2091
1.2081 1.2082 1.0977
2.00
1.5045
1.5036 1.5037 1.4443
4.00
1.8222
1.8215 1.8216
1.7908
6.00
2.0148
2.0142 2.0143
1.9936
8.00
2.1536 2.1529 2.1530
2.1374
10.00
2.2620 2.2614
2.2615
2.2490
20.00
2.6023 2.6017
2.6018 2.5956
40.00
2.9458 2.9452 2.9452 2.9421
60.00
3.1475 3.1469 3.1469 3.1449
80.00
3.2908 3.2902
3.2903
3.2887
100.00 3.4021 3.4014 3.4015 3.4003
200.00
3.7480
3.7474 3.7475 3.7468
400.00
4.0943
4.0936 4.0937 4.0934
600.00
4.2969
4.2963
4.2963
4.2962
800.00 4.4407
4.4401
4.4401
4.4400
1,000.00
4.5522 4.5516
4.5517 4.5516
2,000.00 4.8987 4.8981
4.8982 4.8981
4,000.00
5.2453
5.2447
5.2447
5.2447
6,000.00
5.4480 5.4474
5.4475
5.4474
8,000.00
5.5918 5.5912
5.5913 5.5913
Obtained from
Eq. 48.
3
e =
0
e
=
71 4
e =
71 2
2
=
1 0
1
5
1
onclusions
New theoretical results concerning infinite-conductivity
wells have been presented in the form
of
Laplace trans
forms. Some implications
of
the result for the limited
flow-entry problemhave been discussed in general terms.
Detailed calculations are presented elsewhere.
5
,6 Results
of
the inversion
of
the Laplace transform have been
presented for the vertical fracture problem. There are
some discrepancies with earlier results,4 amounting at
worst to 4 . These discrepancies are difficult to account
for because they occur between calculations that are very
different from each other, but are mild and will not intro
duce any detectable uncertainty in a type-curve analysis.
The inversion
of
the Laplace transform
of
Eq. 45 is fast
enough to allow the plotting
of
type curves directly on
a screen by an interactive computer-graphics program
without long waiting periods. Eq. 46 is new and agrees
with that obtained by Kucuk and Brigham when p
= O
1.209
shown in Table 1 and those
of
Gringarten
et ai
4
on
the
order of 4 for small times and decreasing to about 0.1
toward the bottom
of
the table. Fig. 6 shows
P
D
vs.
tD/exp 2p)
for a range
of p
values, the value
of p
being
the same. The curves are drawn by linear interpolation
between the calculated points, which are equidistant and
have a density
of
6/log cycle. Depending on the work
load, the calculation of the set
of
points needed to draw
one curve takes from 10 to
20
seconds a high-performance
32-bit minimachine.
Eq. 46 is represented by the dashed line in Fig. 6. Ta
ble 1 and.Fig. 6 show that the difference between the log
approximation Eq. 46 and the exact value is less than
about 2 when
tD >4
SPE Reservoir Engineering,
May
1987
223
8/10/2019 22. SPE-13846-PA
8/10
N = positive integer
Non ; normalIzing constant in Eq. 20
p = reservoir pressure, psi [kPa]
pD = dimensionless pressure (see Eq. 30)
PfD = PD at the fracture
Pi
=
initial pressure
Pm = Legendre polynomial
7
(see Eq. 22)
q
=
real number (see Eq. 41)
qfD
=
dimensionless flux at the fracture (see
Eqs. 29 and A-12 through A-14)
qt =
,total well volumetric flow rate, RB/D
[res m
3
/d]
q wD
= dimensionless flux at the. well (Eqs. 6
and A-5 through A-7) ,
q x,y,t = volume of oil withdrawn at point x,y
and at time
t
per unit reservoir
volume and unit time, RB/D-ft3
[hours 1 ]
q x,y,z,t
=
volume of oil withdrawn at point x,y,z
and at time t per unit reservoir
volume and unit time, RB/D-ft3
[hours 1]
r
D = d i m e ~ s i o n l e s s radial coordinate
(see Eq. 5)
r
w =
well radius, ft [m]
R = constant equal to
rB
+ ZD Zb 2 (see
Eq. 8), dimensionless
R
1
= radial spheroidal function of the first
mn
kind
7
,9
R = radial spheroidal function of the third
kind
7
,9
R
w
= R with ~ ~ (see Eq. 10),
dimensionless
s
=
Laplace parameter, dimensionless
Smn
= angular spheroidal function7,9
t =
time, hours
tD = dimensionless time
U = Laplace transform of
qw D
V = Laplace transform of
qjD
x,y,z
=
Cartesian coordinates, ft [m]
XD,YD,ZD
=
dimensionless Cartesian coordinates
xi =
fracture half-length in the x direction,
ft [m]
Z = complex variabl,e
a 3 = angular spheroidal coordinates
(see Eqs. 4)
3 k
=
coefficient in Eq. 22
Y = Euler s constant (0.5772)
oij = Kronecker delta, 1
if
i=j 0 otherwise
8
=
angular polar coordinate (see ~ q s 49)
Ak = constant defined in Eq. 43
p
=
oil viscosity, cp [Pa
s]
v
=
elliptic angular coordinate (see Eqs. 28)
= radial spheroidal coordinate (see, Eqs. 4
=
at th e well
p = elliptic radial coordinate (see Eqs. 28),
dimensionless
Po = oil mass per unit volume, lbm/ft
3
[kg/m
3
] .
224
J
= i.J;
(see Appendix B) .
7 =
real number
= porosity, fraction
4> = potential (see Eq. 3), psi [kPa]
4>
D
dimensionless potential (see Eq.
7)
eJ>i =
initial value
of 4>
psi [kPa]
cI
wD = 4> D at the well (see Eq. 9)
;
=
Laplace transform
of
4>
D
in low-flow
entry problems and
of
PD in infinite
conductivity vertical fracture
problems
;
f =
; at the fracture
;w
=
; at = w Laplace transform
of
4> wD
Subscripts
j k m n = integers
Superscripts
, =
integration variable
cknowledgments
I
gratefully acknowledge the financial support
of
Norsk
Hydro and express my thanks to Leif Larsen for much
useful advice.
References
1 Muskat,M.: Physical Principles
of
Oil Production, Inti. Human
Resources Development Corp., Boston (1981) 209.
2. Gringarten, A.C. and Ramey, H.J. Jr. : The Use of Source and
Green s Functions in Solving Unsteady-Flow Problems in
Reservoirs, SPEJ (Oct. 1973) 285-96; Trans. AIME, 255.
3. Gringarten, A.C. and Ramey, H.J. Jr.: An Approximate Infinite
Conductivity Solution for a Partially PenetratingLine-SourceWell,
SPEJ (April 1975) 140-48; Trans. AIME, 259.
4. Gringarten, A.C., Ramey, H.J. Jr., and Raghavan, R.: Unsteady
State PressureDistributions Createdby aWellWith a Single Infinite
Conductivity Vertical Fracture, SPE/ (Aug. 1974)
347 60;
Trans
AIME,257.
S Papatzacos, P.: Approximate Partial-Penetration Pseudoskin for
Infinite-Conductivity
Wells,
SPERE (May 1987) 227-34.
6. Hoyland, L.A.: Critical Rate for Water Coning in Isotropic and
Anisotropic Formations,
Cando
Tekn. thesis, Rogaland Regional
C., Stavanger, Norway (1984).
7. Abramowitz , M. and Stegun, I .A.: Handbook of Mathematical
Functions
Dover Publishing Inc., New York City (1972).
8. Morse, P.M. and Feshbach, H.: Methods of
Theoretical Physics
McGraw-Hill Book Co. Inc., New York City (1953).
9. Flammer, C.:
Spheroidal
Wave
Functions
Stanford U. Press,
Stanford, CA (1957).
10. Carslaw, H.S. and Jaeger, J.C.: Conduction of Heat in Solids;
Oxford Book Co., New York City (1959).
11. Brigham, W.E. and Kucuk, F .: T r a n s i ~ n t Flow in Elliptical
Systems,
SPE/
(Dec.1979) 401-10;
Trans.
AIME, 267.
12. Stehfest, H.: Numerical Inversion of the Laplace Transforms,
Cori1nutn ications
of
the ACM
(Jan. 1970) 13,
No.1,
Algorithm368.
13. Williams, W.E.: Partial Differential Equations Clarendon Press,
Oxford (1980).
ppendix FlowEquat ons
Flow Equation and Flux for the Limited-Flow-Entry
Problem. Equations in this Appendix are written in con
sistent units, and conversion constants are indicated when
necessary. The diffusion equation with a source term is
kH il
2
+ il
2
} + 0
2
_
cPp c
>
k
v
ox
2
oy oz k
v
ot
P
=--q x,y,z, t ,
(A-I)
k
v
SPE Reservoir Engineering, May 1987
8/10/2019 22. SPE-13846-PA
9/10
where q(x,y,z,t) is the volumetric rate of oil withdrawn
at Point (x,y,z,t) per uni t r ese rv oi r volume. I n ot her
w o r ~ s ,
lq(x,y,z,t)dxdydz=
-q ,
A-2)
where
q t
is the total volumetric flow rate of the well. In
tegration in Eq. A-2 can be extendedto the whole of space
because
q
wil l be different from zero only at the well.
By introduction
of
dimensionless coordinates Eq. 1),
dimensionless time Eq. 2), and dimensionless potential
Eq. 7), Eqs.
A-I
a nd A -2 b ec om e
a
2
cpD a
2
cpD a
2
cpD _ acpD =27ra
kH
i
aXE
aYE aZE
atD k
v
t
A-3),
and
r
k
v
qt
j q ~ x D Y D , Z D , t D d x D d Y D d z D= - 3 A-4)
,
kHa
To find a general expression for the function q describ
ing a line-source well as shown in F ig.
1
it is convenient
to use the so-called Dirac delta function,
O x .13
Here,
beca use the well i s along t he z axis, on e may wr ite
k
v
qt
q(XD,YD,ZD,tD)=
-3 0(XD)O(YD)qwD(ZD,tD),
kHa
. . . . . . . . . . . . . . . . . . . . . . . . . A-5)
where, on the rjght side, the constants are included for
convenience, and where the unknown function, qwD is
the dimensionless flux along the well. When qt and
a
are
expressed in customary units, the factor 1.539 X 10
4
must
be included on the r ight side. T he units f or q are then
RB/D-ft
3
.
Function
q
wD i s such that
qwD(ZD,tD)=O,
IZDI >
1 A-6)
and
+1
J
qwD(ZV,tD)dzv=1.
A-7)
1
E q. A -7 r es ul ts f rom t he co mbi na ti on of Eqs. A-4 and
A-5. By using Eqs. A-3 and A-5, one obtains
a
2
D
a
2
CPD a
2
CPD OCPD
aXE aYE aZE
OtD
Eq. A 8 can
now be solved with the help of the known
Green function
3,8
to give the dimensionless potential D :
CPD(XD,yD,ZD,tD)= dtb J
1
dzbqwD(Zb,tb)
o -1
1 [ -XE -yE -(ZD -ZV)2]
x = exp
.
4J;(tD -tv) 4(tD tv
A-9)
SPE Reservoir Engineering, May 1987
To
recoverEq.
6, it suffices to introduce the prolate
spheroidal coordinates given by Eq.
4
and to use zb
=cos
a
E q. A -9 t he n gi ve s E qs .
6,
a nd E q. A-7 gives E q.
9.
Flow Equation. and Flux for the Vertical-Fracture
Problem. The equations analogous to Eqs. A-3 and A-4
a re e asi ly found t o be
and
The flux
q
at the fracture can be expressed in terms of
a delta function expressing
the
fact that the fracture is an
infinitely thin sheet and an unknown dimensionless flux
at the fracture, qfD:
When
qt
h, and
x
are expressed in field units, then
the factor 1.539
X
10
4
must be included on the right side.
T he uni ts f or q are then RB/D-ft
3
. Function qjD must
satisfy the following conditions:
qjD(XD,tD)=O when
IXDI >
1 .
,
A-13)
and
+1
J
qfD(xb,tD)dx
v
=
1
A-14)
1
T he dif fer enti al e qua ti on f or PD E q. A-IO) then
becomes
A-15)
and has the solution
Using Eqs. 28 together with
xb =cos u
in Eqs. A-14
and
A-I6
yields Eqs. 29 and 33.
225
8/10/2019 22. SPE-13846-PA
10/10
ppendix
B Proof
of Equations
Fo r simplicity, the notation u=i.Jiwill be used through
out this Appendix.
Proof of Eq . 23.Substitution
of
Eqs. 19 and 22 into Eq.
16 gives
x [ ~
_1_
S k
u, cos a )Sok U, cos a ) R ~ u , l )
k k
X R ~ J u , cosh
where the terms involved by the a/integration have been
underlined. This integration can be performed with the
help
of
Eq. 20, giving
X r l ~ u,
I R ~ 1
u, cosh . B-1)
One first sets
w
the left side thus becoming the
a-independent function w see Eq. 17). Both sides of
the equation are then multiplied by sin a S02j U, cos a da
and integrated from 0 to 7 by use of Eq.
20 .
Eq. 23
emerges if, in addition, one uses
J I da
sin a
S02k U,
cos
a =2d
o
k
u , B-2)
o
which follows easily from Eq.
21
and from known prop
erties
of
the Legendre polynomials.
8
Proof ofEq. 24, When one makes use of Eq. B-2, sub-
stitution
of
the right side
of
Eq. 22 into Eq. 18 yields
When the right side of
Eq.
23 is substituted for the
k,
one obtains Eq. 24.
Proof of
Eq . 25. Eq.
25
is obtained immediately when
one uses Eq.
23
o r t ~
k
in Eq. B-1.
Metric onversion Factor
degrees X 1.745 329
E -0 2
rad
SPERE
Original manuscript
SPE
13846 received
in
the Societyof Petroleum Engineers office
Dec. 3,
1984. aper accepted for publication Feb.
6,
1986. Revised manuscript
re-
ceived Jan. 24,
1986.
Recommended