Optimization of the Productivity Index
and the Fracture Geometry of a
Stimulated Well With Fracture Face and
Choke Skins
Diego J. Romero, SPE,* and Peter P. Valkó, SPE, Texas A&M U., and Michael J. Economides, SPE, U. of Houston
Summary
For a given reservoir of known permeability and dimensions, the
proppant mass injected to the pay determines a unique proppant
number. Unique to each proppant number, there exists an optimum
dimensionless fracture conductivity that exclusively determines
the optimum fracture dimensions.1
Impairments affecting flow perpendicular to the fracture surface are accounted for as fracture-face-skin effect. On the other
hand, flow impairment caused by a reduction of the fracture conductivity near the wellbore is called choked fracture skin. Both effects
have a large influence on the productivity of a fractured well.
In this work, the performance of a fractured well is calculated
with a direct boundary element method. This method provides the
dimensionless productivity index, and the model allows for the
presence of each of the two different skin effects.
The fracture face skin was found to have a significant detrimental effect on the dimensionless productivity index, even changing the character of its dependence on the dimensionless fracture
conductivity. The effect of the choke skin also was found to be
potentially detrimental but less complex to account for because
it can be represented as an apparent reduction in the proppant number.
Introduction
The post-treatment performance of hydraulically fractured wells
has been a recurring theme in petroleum literature, covering the
spectrum of understanding the physics of flow to the optimization
of design. Optimization itself has taken different comprehensive
economic hues, from just reducing execution costs to maximizing
production or injection rates.
Irrespective of the ultimate criterion, the magnitude of reservoir
permeability has been central to fracture morphology. For a given
reservoir of known permeability and dimensions, the proppant
mass injected into the pay determines a unique proppant number.
Unique to each proppant number there exists an optimum dimensionless fracture conductivity1 that exclusively determines the optimum fracture dimensions.
However, damaged hydraulic fracture performance deviates substantially from that of undamaged fractures. This work is intended to
calculate and optimize the performance of hydraulically fractured
wells that are burdened by two types of flow impediments—fractureface damage and damage at the connection between the fracture and
the well, referred to as a choke. Fracture-face damage can be actual
damage to the reservoir permeability from fracturing fluid and polymer leakoff, or it can be caused by the reduction in relative permeability because of a phase change. Choked fracture is mainly caused
by proppant flowback or overdisplacement.
*Currently with El Paso Production.
Copyright © 2003 Society of Petroleum Engineers
This paper (SPE 81908) was revised for publication from paper SPE 73758, first presented
at the 2002 SPE International Symposium and Exhibition on Formation Damage Control,
Lafayette, Louisiana, 20−21 February. Original manuscript received for review 15 May
2002. Revised manuscript received 9 October 2002. Paper peer approved 31 October
2002.
February 2003 SPE Production & Facilities
This work follows a considerable body of literature, postulating
that the increase in the fractured well productivity (compared
to the unfractured state) depends on both reservoir and fracture characteristics.
In 1960, McGuire and Sikora2 studied the effect of vertical
fractures on well productivity and showed how the productivity
depends on the fracture penetration and conductivity.
Prats et al.3 and Cinco-Ley and Samaniego4−6 are credited with
the introduction of dimensionless groups of variables to describe
the performance of a fractured well. The concept of dimensionless
fracture conductivity has since been used as the dominant indicator
of relative improvement in fluid flow that is provided by the fracture compared to the alternative (i.e., no fracture).
Early in fractured well performance research, certain works
assumed an infinite-conductivity fracture. Prats et al.3 showed that
in the case of an infinite-conductivity fracture and relatively large
drainage area, the effective wellbore radius is equal to one-half the
fracture half-length. In an infinite-conductivity fracture, the pressure drop is negligible with respect to that in the formation. This
situation is achieved when the dimensionless fracture conductivity
is greater than 300. Gringarten and Ramey7 first introduced the
mathematical solution for this kind of fracture in an infinite acting
reservoir, and it has been used since in well test applications for
wells intersecting large natural fractures.
Sawyer et al.8 presented a numerical simulation for the production
of wells intercepted by a finite-conductivity fracture. They showed
that the assumption of infinite fracture conductivity could lead to
serious errors when calculating the fractured well performance.
In 1978, Cinco-Ley et al.5 demonstrated that the infinitefracture-conductivity assumption is quite erroneous when the pressure drop along the fracture is considerable, which would be the case
if the dimensionless fracture conductivity were lower than 300.
The focus of much of this work was addressing well-testing
techniques.5 However, as early as 1962, Prats et al.3 showed that
an infinite-conductivity fracture, even if achievable, was not the
one at which maximum well production would occur if the volume
of proppant is correctly accounted for as a constraint.
The productivity index of a fractured well, however, is often
less than the one predicted, even when employing correct finiteconductivity fracture models. This is mostly caused by an extra
pressure drop around and/or within the fracture that can be attributed to damage to the formation immediately surrounding the fracture face or additional flow impediments in the fracture. Cinco-Ley
and Samaniego6 proposed a pressure transient solution that considered the fracture face skin. They assumed that the flow from the
formation toward the fracture was linear, passing through two porous
media in series. One medium is the undamaged formation, and the
other is the damaged zone around the fracture. In the same work, they
also studied the effects of flow impairments inside the fracture near
the wellbore for what they termed the choke fracture skin.
Another effect that causes an additional pressure drop is the
non-Darcy flow within the fracture. Wattenbarger and Ramey9 and
Holditch and Morse10 investigated how the fracture conductivity is
affected by the non-Darcy flow and found that the extra pressure
drop is proportional to the product of a turbulence factor and
velocity square. Methods to correct the dimensionless fracture conductivity, used in fracture design and well-test analysis, also have
57
been developed by Guppy et al.11 and Gidley.12 The latter work
showed that fractured wells affected by non-Darcy flow within the
fracture exhibit an apparent (reduced) fracture conductivity that is
flow-rate dependent.
We note that the concepts of proppant number and optimum
dimensionless fracture conductivity remain valid even for the case
of non-Darcy flow if the appropriate reduced proppant pack permeability is substituted into the definitions of proppant number
and fracture conductivity. Because the reduction factor in equivalent permeability is flow-rate dependent, some iteration cycles
might be needed during the optimization process.
In a much later work, Wang et al.13 demonstrated the production impairment of fractures in gas-condensate reservoirs caused
by the formation of liquid condensate in the vicinity of the fracture
face. They considered this effect, caused by relative permeability
phenomena, as a type of fracture-face damage similar to the one
described for real damage by Cinco-Ley and Samaniego.6
With both analytical and numerical simulators, Azari et al.14
demonstrated the choking effect caused by low fracture conductivity near the wellbore. Such a situation can arise if, for example,
the proppant is overdisplaced at the end of a treatment by the flush
or if the proppant settles significantly during fracture closure.
Until now, however, no rigorous method has been proposed to
directly calculate the pseudosteady-state performance of such a nonideal fractured well without solving the model for all previous times
(that is, in transient regime). The purpose of this work is to investigate
the effect of the flow impairments on the productivity index.
In the following sections, a solution methodology is suggested
to determine the inflow into a fully penetrating vertical fracture
that is intersected by a vertical well located in the center of a
square drainage area and is subject to the individual or combined
effects of fracture face skin and choked fracture skin. With the
solution methodology, pseudosteady-state productivity indices are
calculated. In the presentation of results, we rely heavily on the
previously introduced proppant-number concept. The approach’s
usefulness is illustrated in conjunction with fracture design optimization and fractured-well performance analysis.
A Direct Boundary Element Method To Calculate
the Fractured Well Productivity Index
Influence Function. Ozkan15 suggested that the pseudosteadystate drawdown at any point in a reservoir (x,y) caused by a well
located at (xw,yw) can be given in terms of an influence function, a.
p−p=
␣1␮Bq
a关xD, yD, xwD, ywD, xeD, yeD兴. . . . . . . . . . . . . . . . (1)
2␲kh
Because the dimensionless productivity index, JD, is defined by
J=
q
p − pwf
=
Vertical Fracture Performance. If a fully penetrating vertical
fracture intersects the wellbore, the well performance depends on
the following penetration ratio.
Ix =
2xf
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)
xe
where xf⳱the fracture half-length and xe ⳱ the reservoir drainage
extent (side length of a square), and the dimensionless fracture
conductivity is defined by
CfD =
kfw
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)
kxf
where kf ⳱ the proppant pack permeability, w⳱the propped fracture width, and k ⳱ the reservoir permeability.
In Ref. 1, the two expressions (Eqs. 5 and 6) are combined
through the proppant number
Nprop =
4kf wxf
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)
kxe2
which, after multiplying and dividing by the reservoir thickness, h,
leads to
Nprop =
2kfV2w,prop
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)
kVr
In Eq. 8, V2w,prop ⳱ the volume of the two-wing propped fracture
inside the pay, and Vr⳱the drained volume (pore plus matrix). Eq.
8 is quite important because it shows that the proppant number is
a constant quantity for a given mass of proppant injected into a
given drainage volume.1
Direct Boundary Element Method. Romero18 solved the problem of calculating the pseudosteady-state dimensionless productivity index of a vertically fractured well with flow impairment.
The fracture is modeled as nw line sources located at wi (i: 1…nw)
with corresponding (nonuniform) production rates (q1,…qnw). Eq.
4 is applied at nw observation points in the fracture (at oi, located
between wi−1 and wi). The pressure (drawdown) difference between two observation points (o1 and o2) can be obtained from the
corresponding applications of Eq. 4, with
⌬p1−2 =
2␲kh
J , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
␣1B␮ D
␣1␮B
兵q 关a共o1, w1兲 − a共o2, w1兲兴 + …
2␲kh 1
+ qnw 关a共o1, wnw兲 − a共o2, wnw兲兴其 . . . . . . . . . . . . . . . . . . . . (9)
Darcy’s law for the flow in the fracture results in
the influence function can be used to calculate the dimensionless
productivity index of a single vertical well as follows.
JD =
1
, . . . . . . . . . . . . . . (3)
a共xwD + rwD, ywD, xwD, ywD, xeD, yeD兲 + s
where rw⳱the wellbore radius and s⳱the skin factor.
A novel application of Eq. 1 has been the generalization to
multiwell environment by Valko et al.16 and simultaneously by
Umnuayponwiwat and Ozkan.17
␣1␮B
p−p=
2␲kh
i
D
D
wD,i,
ywD,i, xeD, yeD兲, . . . . . . . . . . (4)
i=1
where nw⳱the number of wells (line sources). For a square drainage area considered in this work, yeD⳱xeD and the influence function depend only on the location of the source (denoted by wi in
subsequent equations) and the observation point (denoted by oj).
58
2␣1␮B
关q 共x − x 兲 + . . . + qnw 共xo2 − xo1兲兴 . . . . . . (10)
kf hw 2 o2 o1
Because Eqs. 9 and 10 describe the pressure drop between the
same two points, their right sides are equal. If the following dimensionless variables are defined (along with the definition of the
dimensionless fracture conductivity in Eq. 6),
qD =
nw
兺 q a共x , y , x
⌬pf,1−2 =
qB␮
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)
2␲ k h共p − pw兲
xD = x Ⲑ x*,
e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)
and Ix = xf Ⲑ x*,
e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)
then we obtain the following equation in dimensionless form:
February 2003 SPE Production & Facilities
冋
qD1 关a共o1, w1兲 − a共o2, w1兲兴 +qD2 a共o1, w2兲 − a共o2, w2兲
册
冋
4␲
共x − x 兲 + … + qDnw a共o1, wnw兲
CfDIx Do2 Do1
4␲
− a共o2, wnw兲 −
共x − x 兲 = 0. . . . . . . (14)
CfDIx Do2 Do1
−
册
Eq. 14 describes the pressure drop between observation points 1
and 2. We can write nw−1 similar equations between o1 and reexamine observation points. (Note that in Eqs. 12 and 13, xe*=xe/2
because of the symmetry of the problem.) The notation for Eq. 14
is shown in Fig. 1.
As suggested in Ref. 18, the nw−th equation should be the direct
application of Eq. 4 at the wellbore with ⌬pD⳱1. Once the system
is solved, the dimensionless productivity index is calculated from
the following.
nw
JD = 4
兺q
Di.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)
i=1
(Note that a factor of four is needed in Eq. 15 because the obtained
individual production rates add up to one-quarter of the total production from the well.) In the calculations, a maximum nw⳱512
line sources were used, and the influence function was calculated
according to the method detailed in Ref. 16.
Results for Fractured Well Without Skin. In this section, results
from the model are shown first for an undamaged fracture. Be−
cause the proppant number, Nprop, directly reflects the amount of
proppant injected to the pay, it is used as the parameter on all figures.
Two figures are presented here for an undamaged fracture. Fig.
2 is for an Nprop of less than 0.1, while Fig. 3 is for an Nprop of
greater than 0.1. In both figures, the dimensionless productivity
index, JD, is plotted vs. the dimensionless fracture conductivity,
CfD, at constant values of Nprop. The limiting penetration ratio (see
Eq. 13) equal to 1 is also plotted on the figures with a dashed line.
Note that the maximum possible value for JD is 6/␲, which corresponds to fully linear flow.
As seen in Figs. 2 and 3, there is an optimal dimensionless
fracture conductivity for a given proppant number, Nprop, that represents the optimum relation between the two functions of the
fracture—its ability to collect fluid from the reservoir and to conduct the fluid into the well. For low and moderate proppant numbers (Nprop⳱<0.1), this relation occurs at a dimensionless fracture
conductivity that is equal to 1.6. We note that in formations of
medium and high permeability, realistic proppant numbers are
always in the low to moderate range (i.e., less than 0.1).
Note from Fig. 3 that when the propped volume increases or
when the permeability contrast is very large, the optimal dimensionless productivity index occurs at larger dimensionless fracture
conductivity values. Also note that for values of Nprop equal to 10
or more, the maximum dimensionless productivity index is
achieved when the reservoir is penetrated from “wall to wall” (i.e.,
when the penetration ratio, Ix, equals 1). It is questionable, however, that such large proppant numbers can be realized in practice.
In any case, the optimum fracture geometry is given by:
xf =
冉
冉
kfV2w,prop Ⲑ 2
CfD,optkh
and w =
冊
1Ⲑ2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)
CfD,optkV2w,prop Ⲑ 2
kfh
冊
1Ⲑ2
, . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)
where CfD,opt can be directly read from Figs. 2 or 3.19
The optimization of hydraulic fracture geometry presented previously does not include the effect of a damage zone around the
fracture face and/or within the fracture. Because both effects have
a large influence on the productivity of a fractured well, they
should be considered during the optimization process.
Fracture-Face Skin Effect
Fracture-face damage implies permeability reduction normal to the
fracture face and includes flow impairments caused by several
factors. A filter cake may be formed on the inside fracture face that
is difficult to eliminate, even with proper breaking practices. There
is always a zone around the fracture that is invaded by some
portion of the polymer contained in the fracturing fluid. The filtrate
component of the fracturing fluid penetrating the formation causes
some permeability impairment in a larger zone.
Cinco-Ley and Samaniego6 described the fracture-face-skin effect, sff, in terms of damage penetration and damaged permeability.
sff =
冉 冊
␲ws k
− 1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)
2xf ks
where the variables are shown in Fig. 4.
Fig. 1—Variables of the direct boundary element method for the productivity index calculation.
February 2003 SPE Production & Facilities
59
Fig. 2—Fractured well performance for low and medium proppant numbers.
The previous skin factor can be used to calculate an approximate dimensionless productivity index according to
JD,A =
1
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)
1
+s
JD|s=0 ff
where JD|s=0 ⳱ the dimensionless productivity index of the fractured well with zero fracture face skin.
Eq. 18 is valid only for the case of uniform influx and damage
along the fracture face. For rigorous calculations, we need to incorporate a skin factor distribution.
sff共x兲 =
␲
2xf
冋冉 冊 册
k
− 1 ws
ks
, . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)
@x
and the pressure drop caused by the varying fracture face skin will
become
⌬psff =
␣␮B xf
q̃共x兲sff共x兲, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)
␲kh
where q̃ (x)⳱the influx normal to the fracture face per unit area.
To take the distributed skin into account, Eq. 14 should be adjusted
to include the additional pressure drop given by Eq. 21.
Fig. 3—Fractured well performance for high proppant numbers.
60
February 2003 SPE Production & Facilities
Fig. 4—Fracture face damage variables.
qD1关a共o1, w1兲 − a共o2, w1兲 + 共4nw兲sff,w1兴
4␲
+ qD2 a共o1, w2兲 − a共o2, w2兲 −
共x − x 兲
CfDIx Do2 Do1
冋
冋
册
+ … + qDnw a共o1, wnw兲 − a 共o2,wnw兲
−
册
4␲
共x − x 兲 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)
CfDIx Do2 Do1
In other words, the diagonal elements of the coefficient matrix
should be increased by the appropriate (local) skin factor.
There are three interesting cases for varying fracture face skin
factor—when the skin decreases linearly toward the tip, when it increases linearly, and when it is constant. The first case may reflect
damage caused by fracturing fluid leakoff, whereas the second may
reflect uneven fluid cleanup following the fracture treatment.
The mean value of the skin is used as a first approximation to
evaluate the effect of damage on well performance.
1
sff =
xf
xf
兰 s 共x兲dx.
ff
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)
0
Comparing the three cases is of particular interest when the mean
value of the skin factor calculated from Eq. 23 is the same. In our
calculations, Case A corresponds to a linearly decreasing fracture
face skin from the well toward the fracture tip. Similarly, Case B
corresponds to a linearly increasing fracture face skin. Finally, we
assume a constant fracture face skin along the fracture in Case C.
In all three cases, however, the mean value of the skin is kept equal
to unity.
The solid lines in Fig. 5 denote the zero-skin calculations. The
first corresponds to the base case (i.e., Nprop⳱0.1 with no fractureface skin effect). For comparison purposes, the no-skin curve for
Nprop⳱0.01 is also included.
It can be seen from Fig. 5 that the fracture face skin has a large
influence on the dimensionless productivity index. In addition, the
damage distribution along the fracture greatly affects the performance. In Case A, which is the most likely to happen, a significant
reduction in the productivity index is observed. It is also noted that
for Case A, the location of the optimum CfD with respect to the
zero-skin location (i.e., 1.6) is to the left but is to the right for Case
B. The optimum width and length can still be calculated from Eqs.
16 and 17, but the resulting dimensions will be different than those
obtained from CfD⳱1.6.
The largest reduction in performance happens in Case C (when
the damage is uniformly distributed along the fracture). For instance, if the skin factor is one unit, its overall effect is equivalent
to an order of magnitude reduction in the proppant number. That
is, a uniformly distributed fracture face skin equal to 1 is roughly
equivalent to placing only 10% of the original proppant volume
and avoiding any damage.
Choked-Fracture-Skin Effect
Choked-fracture skin effect refers to the presence of a damaged
zone of the fracture that is near the well and has a conductivity
reduction. The conductivity reduction can be caused by an overdisplacement of proppant at the end of a fracture treatment job, by
Fig. 5—The effect of fracture-face skin distribution on well productivity. Case A shows decreasing damage toward the tip; Case B,
increasing damage toward the tip; and Case C, constant damage along the fracture. For all three cases, Nprop = 0.1 and the average
skin=1. The line labeled Eq. 19 denotes the approximate calculation with only the average value of skin, in this case sff =1.
February 2003 SPE Production & Facilities
61
settling of the proppant during fracture closure or by fines migration and accumulation at the wellbore during production. A choked
fracture with a significant flow impediment at the vicinity of the
wellbore is shown in Fig. 6, in which w⳱the unaltered fracture
width, kf⳱the unaltered fracture permeability, and wck⳱the altered fracture width in the near-well region of the fracture.
Equivalent flow impediment can be caused by a reduced permeability (kf,ck) zone in the fracture, even if the width is unaltered.
The extra pressure drop in the fracture is
⌬psck =
␣1␮Bq
s , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)
2␲kh ck
where sck is given by
冋 册
冋 册
␲xck w
− 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)
sck =
xf wck
or sck =
␲xck kf
− 1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)
xf kf,ck
depending on whether the damage is expressed as a reduced width
(wck) or a reduced permeability (kf,ck).
Because the damage is located inside the fracture, it will only
affect the pressure drop caused by flow through the fracture.
Therefore, Eq. 14 should be replaced by
冋
qD1 关a共o1, w1兲 − a共o2, w1兲兴 + qD2 a共o1, w2兲
− a共o2, w2兲 −
4␲
共x − x 兲 − 4sck
CfDIx Do2 Do1
冋
册
+ … + qDnw a共o1, wnw兲 − a共o2, wnw兲
−
册
4␲
共x − x 兲 − 4sck = 0 . . . . . . . . . . . . (27)
CfDIx Do2 Do1
To analyze the effect of the choked fracture skin on the fracturedwell performance, two different values for choke skin (sck ⳱ 0.5
and 1) were studied. As in the previous parametric studies, the
proppant number was equal to 0.1. Fig. 7 illustrates the results.
The solid line represents Nprop⳱0.1 without skin (base case). For
comparison purposes, the Nprop⳱0.01 line is also shown (as another solid line) without skin.
It can be observed from Fig. 7 that the choke skin reduces the
productivity index of the well in a rather straightforward manner.
For a proppant number of 0.1 and choke skin of 1, the dimensionless productivity index is equivalent to a fracture without damage
but with a reduced proppant number of approximately 0.01. However, the location of the optimum dimensionless fracture conductivity (1.6 for the given proppant number) is not altered by the
presence of the choke-fracture skin. We notice that the approximate formula (Eq. 19) works satisfactorily for choke skin.
If we compare the effect of fracture face skin and that of choke
skin, we see that the latter is less complex. The plausible explanation is that the choke skin causes an additional pressure drop
right at the vicinity of the wellbore without changing the shape of
the influx distribution along the lateral direction, x. On the other
hand, the fracture face skin causes a relative redistribution of the
influx of produced fluids along the lateral direction, and nonuniform damage amplifies this effect.
Application Example
Place 240,000 lbm of proppant (pack porosity⳱0.35, specific
gravity⳱2.65, and equivalent permeability⳱60,000 md) into a
65-ft-thick formation of 1.5-md effective permeability. Assume
that 50% of the proppant goes to pay because of some height
growth of the fracture to the adjacent shales. The drainage radius,
re, is 2,100 ft; the well radius, rw, is 0.328 ft; and the skin factor
before fracturing, spre, is 5.
Problem. Determine the maximum possible “folds of increase”
and the optimum propped length and width.
Solution: The volume of proppant reaching the pay is 50% of
the 240,000-lbm proppant volume: V2w,prop⳱1,116 ft3. The proppant number is
Nprop =
2 ⳯ 共60 ⳯ 103 md ⳯ 1,116 ft兲
. . . . . . . . . . . . (28)
关1.5 md ⳯ 共2,1002 ft2兲 ⳯ ␲ ⳯ 65 ft兴
The maximum achievable dimensionless productivity index (see
Fig. 2) is
JD,max p = 0.466. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)
According to the definition of folds of increase in the productivity
index (with respect to the originally damaged well), it can be
obtained as
Jpost
=
Jpre
JD,max
0.466
= 6.1. . . . . . (30)
=
1
1
0.474re
0.472 ⳯ 2,100
+5
+ spre ln
ln
rw
0.328
As seen in Fig. 2, the optimum is realized with CfD⳱1.6, and,
hence, the optimum fracture dimensions are
xf =
冋
and w =
0.5共1,116 ft3兲 共60,000 md兲
1.6 ⳯ 共65 ft兲共1.5 md兲
册
1Ⲑ2
= 463 ft, . . . . . . . . . . . (31)
0.5共1,116 ft3兲
= 0.0185 ft = 0.222 in. . . . . . . . . . . . . . . (32)
共65 ft兲共463 ft兲
Problem. Determine the actual folds of increase if 10,000 lbm of
proppant has inadvertently been flowed back. Assume the proppant comes from the part of the fracture that is in the pay near the
wellbore where a two-grain width (0.06 in.) is stabilized during the
fracture-healing process.
Solution: As indicated previously, according to our assumptions, optimum fracture geometry is created, but the proppant
flowback then produces a choke with the following widthreduction ratio.
w 0.222
= 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (33)
=
wck 0.06
The length ratio corresponding to our assumptions is
Fig. 6—Notation for choked fracture.
62
February 2003 SPE Production & Facilities
Fig. 7—The effect of choke skin on fractured well performance.
0.222
xck 10,000
=
= 0.114. . . . . . . . . . . . . . . . . . . . (34)
xf 120,000 共0.222 − 0.06兲
From Eq. 25, the choke skin is calculated as
sck =
冉
冊
␲xck w
− 1 = ␲ ⳯ 0.114 ⳯ 共3.70 − 1兲 = 0.97 . . . (35)
xf wck
From Fig. 7, we see that for a proppant number of approximately
0.1, the choke skin, sck,⳱1, reducing the JD,max from 0.466 to
0.32. Another way to obtain the same result is to use a form similar
to Eq. 19.
JD,ck =
1
1
= 0.32. . . . . . . . . . . . . . . . . (36)
=
1
1
+ 0.97
+ sck
JD|s=0
0.466
Hence, we can predict that the actual folds of increase decrease
from 6.1 to 4.8 because of proppant flowback.
Conclusions
The performance of a fractured well is primarily determined by the
proppant number (i.e., by the volume contrast of proppant placed
into the pay and by the permeability contrast of proppant pack to
formation). For every proppant number, there is a unique maximum productivity index that is realized only at the optimum dimensionless fracture conductivity. In turn, the optimum dimensionless fracture conductivity determines the unique width and
length to provide optimum performance.
In previous works,1 we obtained the productivity index with the
direct boundary element method. In this work, the direct boundary
element method was extended to calculate the effect of fracture face
skin with various damage distributions and choked fracture skin.
It was found that nonuniform fracture face skin significantly
decreases the productivity of the fractured well and also shifts the
location of the optimum dimensionless fracture conductivity.
Therefore, not only the maximum achievable productivity index
but also the optimum fracture geometry will differ from the zeroskin case. A uniform damage distribution has the most detrimental
effect on productivity but leaves the location of the optimum dimensionless fracture conductivity intact.
The effect of choked fracture skin is less complex to account
for: it is essentially equivalent to an apparent reduction of the proppant number and does not affect the optimum fracture geometry.
February 2003 SPE Production & Facilities
As illustrated by the example calculations, using the dimensionless productivity index and proppant number facilitates understanding
fractured-well performance and makes the analysis transparent.
Nomenclature
a ⳱ influence function
B ⳱ formation volume factor, resbbl/STB
CfD ⳱ dimensionless fracture conductivity
h ⳱ pay thickness, ft
Ix ⳱ dimensionless penetration ratio
J ⳱ productivity index, STB/D/psi
JD ⳱ dimensionless productivity index
k ⳱ formation permeability, md
kf ⳱ proppant pack permeability, md
nw ⳱ number of line sources (“wells”)
Nprop ⳱ proppant number
o ⳱ observation
p ⳱ pressure, psi
p ⳱ average pressure of drainage, psi
pw ⳱ wellbore flowing pressure, psi
q ⳱ flow rate, STB/D
rw ⳱ wellbore radius, ft
s ⳱ skin factor
V2w,prop ⳱ propped volume in pay (2 wings)
Vr ⳱ drained volume, ft3
w ⳱ fracture width, ft
wck ⳱ choked width in one wing, ft.
ws ⳱ fracture face skin zone extent, ft
x ⳱ coordinate, ft
xck ⳱ choke length in one wing, ft
xe ⳱ side length of drainage area, ft
x*e ⳱ half-side length of drainage area, xe/2, ft
xf ⳱ fracture half length, ft
xw ⳱ coordinate of well, ft
y ⳱ coordinate, ft
ye ⳱ side length of drainage area, ft
yw ⳱ coordinate of well, ft
␣1 ⳱ conversion factor (for field units 887.22)
⌬p ⳱ drawdown, psi
␮ ⳱ viscosity, cp
63
Subscripts
A ⳱ approximate
ck ⳱ choked fracture
D ⳱ dimensionless variable
e ⳱ drainage
f ⳱ fracture
ff ⳱ fracture face
i ⳱ index of source
max ⳱ maximum
o ⳱ observation
opt ⳱ optimum
post ⳱ post-treatment
pre ⳱ pretreatment
s ⳱ skin
x ⳱ x direction
y ⳱ y direction
w ⳱ well
References
1. Economides, M.J., Oligney, R., and Valko, P.P.: Unified Fracture Design, ORSA Press, Alvin, Texas (2002).
2. McGuire, W.J. and Sikora V.J.: “The Effect of Vertical Fractures on
Well Productivity,” Trans., AIME (1960) 401.
3. Prats, M., Hazebroek, P., and Strickler, W.R.: “Effect of Vertical Fractures on Reservoir Behavior—Compressible-Fluid Case,” SPEJ (June
1962) 87.
4. Cinco-Ley, H., and Samaniego, V.F.: “Effect of Wellbore Storage and
Damage on the Transient Pressure Behavior of Vertically Fractured
Wells,” paper SPE 6752 presented at the 1977 SPE Annual Fall Meeting, Denver, Colorado, 9−12 October.
5. Cinco-Ley, H. and Samaniego, V.F.: “Transient Pressure Analysis for
Fractured Wells,” JPT (September 1981) 1749.
6. Cinco-Ley, H., and Samaniego, V.F., and Dominguez, N.: “Transient
Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture,” SPEJ (August 1978) 253.
7. Gringarten, A.C. and Ramey, H.J. Jr.: “An Approximate Infinite Conductivity Solution for a Partially Penetrating Line Source Well,” SPEJ
(April 1975) 325.
8. Sawyer, W.K., Locke, C.D., and Overbey, W.K. Jr.: “Simulation of a
Finite-Capacity Vertical Fracture in a Gas Reservoir,” paper SPE 4593
presented at the 1971 SPE Annual Fall Meeting, Las Vegas, Nevada, 30
September−3 October.
9. Wattenbarger, R.A. and Ramey, H.J. Jr.: “Well Test Interpretation of
Vertically Fractured Gas Wells,” JPT (March 1969) 246.
10. Holditch, S.A. and Morse, R.A.: “The Effects of Non-Darcy Flow on
the Behavior of Hydraulically Fractured Wells,” JPT (October 1976)
1169.
11. Guppy, K.H. et al.: “Non-Darcy Flow in Wells with FiniteConductivity Vertical Fractures,” SPEJ (October 1982) 681.
12. Gidley, J.L.: “A Method for Correcting Dimensionless Fracture Conductivity for Non-Darcy Flow Effects,” SPEPE (November 1991) 391.
13. Wang, X. et al.: “Production Impairment and Purpose-Built Design of
Hydraulic Fractures in Gas-Condensate Reservoirs,” paper SPE 64749
presented at the 2000 SPE International Oil and Gas Conference and
Exhibition, Beijing, 7−10 November.
64
14. Azari, M. et al.: “Performance Prediction for Finite-Conductivity Vertical Fractures,” paper SPE 22659 presented at the 1991 SPE Annual
Technical Conference and Exhibition, Dallas, 6−9 October.
15. Ozkan, E.: “Performance of Horizontal Wells,” PhD dissertation, U. of
Tulsa, Tulsa (1988).
16. Valko, P.P., Doublet L.E., and Blasingame, T.A.: “Development and
Application of the Multiwell Productivity Index (MPI),” SPEJ (March
2000) 21.
17. Umnuayponwiwat, S. and Ozkan, E.: “Evaluation of Inflow Performance of Multiple Horizontal Wells in Closed Systems,” J. of Energy
Resources Technology (2000) 122, 8.
18. Romero, D.J.: “Direct Boundary Method to Calculate PseudosteadyState Productivity Index of a Fractured Well with Fracture Face Skin
and Choked Skin,” Masters thesis, Texas A&M U., College Station,
Texas (2001).
19. Spreadsheet, FracPI, http://pumpjack.tamu.edu/∼valko.
SI Metric
bbl ×
cp ×
ft ×
in. ×
md ×
psi ×
Conversion
1.58987
1.0*
3.048*
2.54*
9.869223
6.894757
Factors
E−01 ⳱
E–03 ⳱
E–01 ⳱
E+00 ⳱
E–04 ⳱
E+00 ⳱
m3
Pa⭈s
m
cm
␮m2
kPa
*Conversion factor is exact.
Diego J. Romero is currently employed by El Paso Production
Co. in Houston. Romero holds a BS degree in petroleum engineering from Foundation U. of America, Santafé de Bogotá,
Colombia, and an MS degree in petroleum engineering from
Texas A&M U. Peter P. Valkó is an associate professor at the
Harold Vance Dept. of Petroleum Engineering at Texas A&M U.
He previously taught in Austria and Hungary and also worked
with the Hungarian oil company MOL. Valko holds BS and PhD
degrees in chemical engineering and an MS degree in technical mathematics from Veszprem U. (Hungary) and from the
Inst. of Catalysis, Novosibirsk (Russia). Valkó is currently serving
on the SPE Journal Editorial Review Board and has served on
the SPE Forum Steering Committee. Michael J. Economides is a
professor of chemical engineering at the U. of Houston. Previously, he was the Noble Professor of Petroleum Engineering at
Texas A&M U. and served as a Chief Scientist of the Global
Petroleum Research Inst. Before joining Texas A&M U., he was
the Director of the Inst. of Drilling and Production at the Leoben
Mining U., Austria. From 1984 to 1989, he worked with Schlumberger companies. Economides holds BS, MS, and PhD degrees from Kansas State and Stanford U. He has been
awarded the following honors: Doctor Honoris Causa, Petroleum and Gas U., Ploeisti, Romania; Russian Academy of Natural Sciences, inducted as Foreign Member; Society of Petroleum Engineers, 1997 Production Engineering Award; Doctor
Honoris Causa and Honorary Professor, The Gubkin Russian
State Academy of Oil and Gas, Moscow; Distinguished Member, Society of Petroleum Engineers; and the Outstanding Faculty Award (U. of Alaska, School of Mineral Industry).
February 2003 SPE Production & Facilities