—
—
SPE/DOE
Swfdy of
PolmfsumEn@new
Dopmtmonl
of E-y
U.S.
SPE/DOE 16396
Observations and Recommendations
Hydraulically Fractured Wells
by M.J. Economies,
in the Evaluation of Tests of
Dowell Schlumberger
SPE Member
Copyright
1987, Society of Pelroleum
Engineers
Th!s paper was prepared for presentation at lhe SPE/DOE
Low Permeability Reservoirs Sympomum
held in Denver, Colorado, May 18-19.1987.
Th!s paper was selected for presentation by an SPE Program Commiltee following review of information contained in an absfracl submitted by the
author(s). Contents of the paper, as presen fad, have not been reviewed by the Society of Petroleum Engineers and are aubjecl 10 correction by the
author(s). The malerial. GS presented, does not necessarily reflect any position of the Society of Petroleum Engineara, its officers, or members. Papers
presented at SPE meetings are subject 10 publication review by Editorial Committees of Ihe Smiety of Petroleum Engineers. Permission to coPy is
resmcted lo an abstract o: not more than 300 words. Illustfafions may not be copied. The abstract should contain conspicuous acknowledgment of
where and by whom fhe psoer is presented. Wrile Publications Manager, SPE, PO. Box 833836, Richardson, TX 75083.3836. Talex, 730989 SPEDAL.
ABSTRACT
The pressure transient behavior of hydraulically fractured wells
has been the subject of considerable study over the past few
years. Several investigators have presented solutions of the fundamental equations, identified qualitative diagnostic trends and
suggested interpretation techniques. This paper presents a
systematic approach to the problem along with substantial
observations on the potential of unique interpretations.
Pre-treatment tests are considered here as necessary. Well tests
in tight formations are often of very she, t duration, to allow
the use of established methodologies. Hence, a technique to
calculate the maximum reservoir permeability from a short well
test is offered.
In the case of post-treatment tests the data are treated using
the convolution/deconvolution
techniques and influence functions. The term “influence function” defines a relationship between pressure response and time at a constant unit ‘surface flow
rate. Although drawdown well tests have advantages over
buiidup tests because they are used while the well is producing, their interpretation has been hampered by varying flow
rates. Conventional interpretation techniques assume either constant well flow rate or controlled variation of it. Pressure
buildup tests are conducted with the well flow rate equal to zero
and are, as a result, predominant.
In the case of post-treatment well tess of hydraulic fractures
a lengthy buildup test and the ensuing shut-in may result in
severe damage to the generated fracture. Thus buildup tests are
not always desirable for post-treatment evaluation. The convohrtion/deconvolution techniques and influel,:e “functions,”
by normalizing the pressure response to a unit flow rate, permit the use of standard techniques for the analysis of drawdown
tests.
L
The interpretations presented here utilize new versions of
pressure and pressure derivative type cur~cs including tk
dimensionless fracture storage coefficient, and the dimensionless fracture conductivity. Based on observations of the sensitivityy of the response to these parameters, three type curves
have 3een developed, one for low, one for intermediate, and
one f-r high conductivity fractures. The choice can be made
on the basis of pre-treatment analysis and the fracture design.
The storage “,nd the half-length of the generated fracture can
then be talc dated with reasonable confidence.
PRE-TREATMENT WELL ANALYSIS
FOR TIGHT RESERVOIRS
Tight formations are characterized by permeabilities less than
10 md. The permeability is usually less than 1 md and is often
less than 0.1 md,
Knowledge of the “undisturbed” reservoir permeabilityy is considered essential for the preparation of an appropriate fracture
stimulation design. As will be demonstrated later on in this
paper it is also essential in the post-treatment evaluation of job
effectiveness.
The standard and desirable methods of analysis for radial, infinite acting, reservoirs are usually not feasible in tight formations. Although the reservoir configurations would theoretically
lend themselves to such analyses, the low permeabilityy results
in a slow response to pressure perturbations. T1.: necessary
pressure response patterns take a substantial amount of time
to appear.
Techniques for analyzing tight formations, which attempt to
extract reservoir data from very early data by deconvolving
wellbore storage effects have appeared in the literature. Kabir
and Kucuk (1985) attempted to reduce the wellbore storage
distortion period h a pumping oil well by using the convolu-
2
OBSERVATIONS
OF TESTS OF HYDRAULICALLY
FRACTURED
SPE 1639
tion and deconvolution of test data. For wells not capable of
flow to the surface they suggested the monitoring of the rise
of the annular liquid level, Ahmed et ai. (1985) presented a
method involving the measurement of transient rate and
pressure for a short period of time and again using a deconvolution technique to remove much of the distortion caused
by wellbore effects,
for appropriate interpretations. In offshore locations where well
completion, testing and fracturing are done back to back, the
test duration is often brought into question.
In the case of extremely tight formations (permeabilities belov
0.1 md) even these techniques may not be effective. The follo’ (
ing is offered as a method to calculate a maximum value . or
the reservoir permeability.
; Hence, if /e,~.b, is known, then Eq. 4 may be rearranged to
calculate k:
Equation 4 may aid in the estimation of the reservoir permeability simply by utilizing the time at which wellbore effects are
becoming less predominant.
~>
3ooocp
-
The evolution of the semi-logarithmic straight line (appearing,
as the “rule of thumb” implies, i,5 eye@ away from the end
of wellbore storage effects) would take a iervgthyperiod of time.
tD = CD (60+ 3.5
S)
(1)
Assuming a nonzcro skin effect, and using the definitions of
dimensionless time and the dimensionless storage coefficient,
Eq. 1 becomes:
0,000264 &t
dpc(rw2
5.615 C
2m$c(hr$
=
260
5.615 C
(2)
Simplifying, rearranging and solving for t:
f (hrs) z 2 x 105 ~
hk
(3)
CONVOLUTION/DECONVOLUTION
AND INFLUENCE FUNCTIONS IN
WELL TEST INTERPRETATION
which relates well, reservoir and fluid properties to the approximate time for the beginning of the semilog straight line.
2 x 105
70
Cp
hk
.3000~
TECHNIQUES
There are practical problems associated with pressure buildup
testing for tight formations both in the pre-treatment state as
well as following a massive hydraulic fracture. ~rst, because
of the very low permeabilities the shut-in times maybe extremely
long. This would result in significant cost in both actual well
test expenses as well as lost production. Second, in the case of
tests following a hydraulic fracture the “drawdown” is often
superimposed with the well and fracture “cleanup.” While this
event is complicated by the presence of alien fluids, the ensuing buildup, if done, can result in significant fracture damage.
Since the beginning of the semi-logarithmic straight line appears at 1.5 log cycles away from the cessation of wellbore
storage effects (about a 70 fold increase in the value of time),
Eq. 3 may be reduced to:
te. w. b. 2
(5)
fe. w.b.
‘n well tests of extremely tight formations, wellbore effects are
,lot oniy dominant but they could totally mask the entire test.
Knowledge of te.~.bo, which is characterized by the end of the
45° line on the log-log diagnostic graph and is indicative of the
end of predominant wellbore storage effects, would be sufflcient to calculate the maximum value of the reservoir
Permeabdity. This interpretation technique is not intended as
a substitute for rigorously conducted tests. Yet, it maybe considered a reasonable alternative when lengthy test duration
would otherwise preclude the performance of any pressure transient test.
(60 + 3.5 s)
2mpcthr~
h
Equation 5 is significant since it may provide the maximum
value of the permeability provided that the wellbore storage
coefficient, C can be determined. With well test data, the
wellbore storage coefficient can be calculated from the slope
of a cartesian graph of pressure versus time as shown by
Earlougher (1977). In the presence of a nonzero skin, the actual permeability will be somewhat smaller. For a skin less than
10 the permeability value computed from Eq. 5 wouId be no
more than 1.6 times the actual value of the reservoir
permeabilityy.
This time may be calculated. The dimensionless time for the
beginning of the semi-log straight line has been correlated by
Agarwal et al. (1970):
(4)
(e.w.b. = end of wellbore storage)
This expression provides the minimum time for the cessation
of wellbore storage effects,
This damage, which is often observed, could have several
causes. Fracture face damage may be the result of unbroken
polymer chains within the penetrated reservoir surrounding the
fracture or of chemical reactions. “Choked” fractures could
result because of insufficient cleanup and accumulation of fines
and polymer chains in the vicinity of the wellbore. Both of these
phenomena can be attributed to the shut-in. It is widely accepted
that a continuous drawdown is preferable and that the detfimenal effects from a lengthy shut-in, in order to conduct a pressure
buildup test, may far outweigh the benefits.
Using sorr,e typical variables (for a gas well) such as C = 0.12
bbl/psi (0.27 msfbar), h = 100 ft (30 m) and p = 0.22 cp then
Eq. 4 may provide the minimum time during which wellbore
effects will be predominant. For a permeabilityy equal to 0,01
md this time is at least 8 hrs. The presence of any skin effect
would add more time. Since the pressure versus time relationships are logarithmic, very long test times would be required
L
WELLS
En
M. J. ECONOMIES
E 16396
Continuous measurements of wellbore pressure and downhole
“ate would allow the use of the generalized form of the
neasured wellbore pressure as presented by van Everdingen and
Hurst (1949) and expounded upon by Stewart et al. (1983),
Kucuk and Ayesteran (1983) and Kabir and Kucuk (1985).
3
A simple method of data treatment can be employed using the
generalized form of the pressure drop as shown in Eq. 6ii and
by substituting the term Ap j (7) by @7)/d~ where F(7) is the
drawdown for the well pro d ucing at a unit rate for a time /.
This can be approximated for n intervals by:
n
I
4PW, (f) = \ q ~ (7) AP$~(f – 7) d~
o
(6)
= ~ 9D (f - 7) OiJ
o
(6a)
Pi – tiwf)n =j~l (qn -j+ 1 - qn _j) F (/n – tn -j)
(7)
where F (tn – tn _j) = Fj
(7) d7 + APWD (0
Setting n = j then
where (for drawdown)
n-1
Pi – f.Pw~n ‘j~,
(9n-j+
1 –
qn -j)
F (tn
–
tn-j)
+ ql Fn
APW~(0 = Pi – Pw~ (f)
@~~ (0
= Pi – Ps/ (/)
A/J~
= pressure drop due to skin
Psf
= sandface pressure
(8)
and therefore
n-1
qD(l)
= qsf/qr
qr
= reference rate
The primes indicate derivatives with respect to time.
The convolution technique assumes a model (e.g., infinitely acting) and calculates Ap$j, Comparison with the measured A.owf
and qs.. via Eqs. 6 and 6a results in the calculation of reservoir
variables, The deconvolution technique calculates Ap~. which
may then be used in the standard methodologies in search of
an appropriate model and subsequent calculation of unknown
reservoir and well parameters.
‘Ihe deconvolution technique and the use of influence functions
would greatly enhance the applicability of drawdown testing
and reduce the need for buildup. Several investigators worked
on the subject. Hutchinson and Sikora (1959) used the principle of superposition in calculating a “resistance function” for
a drawdown test with variable flow rate. Jargon and van
Poollen (1965) developed an analytical expression for the influence function of a slightly compressible fluid. Katz et al.
(1962) developed an expression for the influence function for
the early part of a drawdown test, while van Everdingen and
Hurst (1949) showed that the pressure drop calculated via the
principle of superposition is generally correct regardless of the
flow rate history of the well.
pi
–
@wf)n– j~ ,(qn
The simplest form of the influence function is dividing the
pressure difference by the corresponding flow rate. In other
words:
Ap/q for oil and Ap2/q or Am(p)/q for gas.
I
– qn -j)
Fj
(9)
ql
Equation 9 shows that the influence function at any time tn
can be calculated by knowning the pressure drawdown, pi –
Owf)tr, the flOWrate and the value of the inffuence function
at the previous time increments. For the first time step
(10)
In the case of gas reservoirs, the pressure difference functions
in Eqs. 9 and 10 may be replaced by the real gas pseudopressure
difference.
Figure 1 represents pressure (wellhead and bottomhole) and
flow rate history for well Travale 22 in Italy, These data were
originally published by Economies et al, (1979). Figure 2 shows
the calculated “influence functions” using two simple, normalizing expressions (pressure or real gas pseudopressures divided by the flow rate) and a deconvo[ved set of data. Any one
of these could be used with the standard methods in search of
an appropriate model for interpretation and calculation of
unknown reservoir parameters.
A STUDY ON FRACTURED
Mannon ( 1977) has presented a form of the influence function
of gas reservoirs using the real gas pseudopressure and
Economies et al. (1979) presented a formulation for geothermal well testing.
- j+
Fn =
WELL TEST RESPONSE
As with all types of well tests the analysis of fractured well
tests aims towards the identification of well and reservoir
variables that would have an impact on future well performance. However, fractured wells are substantially more complicated. The well-penetrating fracture has unknown geometric
features (length, width and height) and unknown conductivity
properties. There are certain presumptions that investigators
in the field and well test analysts have used. These are mentioned here in order to identify a priori certain limitations:
OBSERVATIONS
OF TESTS OF HYC
a) The fracture height is usually assumed to be equal or less
(Raghavan et al,, 1978) than the reservoir height. Most of
the interpretation work has modelled fully or partially
penetrating fractures, However, in real fracturing operations there is much concern with uncontrollable propagating
fractures in the vertical direction, above or below the
targeted formation. Hence, interpretation of well tests must
take into account containing layers, indications of fracture
growth outside the interval, and especially the ‘connection
of other productive horizons. Hence, reservoir engineering
type analysis should always be accompanied by pressure
analysis during the job execution and temperature or
radioactive logging after the job. Needless to say, communication, via the fracture, with other Iayers would greatly
complicate the analysis.
b) The fracture permeability cannot be inferred from the
laboratory-measured proppant permeabilityy, even at reservoirconfirring pressures. Theactual fracture conductivity
is a c~mposite, bulk, variable taking into account
phenomena such as embedment, reservoir fines migration
and retention within the fracture and unbroken polymer
chains that are either permanent or very slowly d.isirtegrating
remnants of the fracturing fluids. Recent laboratory
measurements (Roodhart et al., 1986) have shown substantial proppant pack permeability 10SS(from 15L70to 75VO)
after treatment with various fracturing fluids.
c) The fracture storage coefficient and especially the transition from wellbore storage to one of the discernible patterns, (hi-linear, linear or other flow) may give misleading
results. A visual pattern of pressure response may appear
similar for a number of fracture conductivity and fracture
storage factors. In general, a sma!l fracture conductivity and
a small storage coefficient would give similar looking patterns at early and middle times of a well test. Hence, real
data of a particular shape may prompt investigators
(especially when they have type-curve generating c,~pacity)
to use a set of type curves (with a combination of fl acture
conductivityy and storage coefficient) that are inapp: ~priate.
More on this will appear later in this paper,
The classic fractured well test interpretation papers were
published by Gringarten and Ramey (1974) for the ‘“infinite
conduct ivity fracture” and Gringarten et al. (1975) for the’ ‘constant flux” fracture. Traditionally, natural fractures have been
interpreted by the infinite conductivity model (Kazemi, 1969,
Kucuk and Sawyer, 1980, Cinco and Samaniego, 1982,
Samaniego and Cinco, 1983 and others) while acid fractures
have been interpreted via the constant flux model (Gringarten,
1978, Cuesta and Elphick, 1984), Alagoa et al. (1985) have used these two models to interpret the behavior of apparently
firrite conductivity fractures, In their paper, in which they incorporated the use of the pressure derivative, !hey had to use
very large storage coefficients to accomodatc ne trends of the
data.
The definitive description of the flow behavior of a hydraulically
induced propped fracture (hence of finite conductivity) was
presented by Cinco et al. (1978) and Cinco and Samaniego
AULICALLY FRACTURED
WELLS
SPE 1639
(1981 a). They offered the concept of hi-linear flow, which,
for certain values of the fracture conductivityy forms a very
distinctive slope equal to 0.25 on a log-log graph,
Two commonly appearing deviations from the “ideal” fracture response (in addition to the aforementioned partially
penetrating fracture) are: 1) damaged fractures, where a damaged zone, extending in a normal direction into the res. toir, enci~cles the fracture and, 2) choked fractures, where the fracture permeabilityy just away from the wellbore is reduced.
Damaged and choked fracture behavior was described by Cinco and Samaniego (1981 b). Wong et al. (1984) applied the
pressure derivative to a fracture with fluid loss damage.
Cinco (1982) has described qualitatively the successive pattei ns
that emerge during the flow from a finite conductivity fracture well. The following study should provide a more focused
approach to the problem by presenting a methodology (and a
rationale) within the ranges of well, reservoir and fracture
variables that are likely to be encountered in practice.
Fundamental
variables and choice of axes in graphing,
The variables to be employed here are the usual dimensionless
groups i.e.
Pressure:
Ap/141.2qBp for oil
kh
(11)
PD
=
PL)
= kh Am(p)/ 1424qT for gas(12)
Time:
tDxJ = 0.000264k At/@c#
(13)
Storage coefficient:
CDy = 5.615C/2x~c[/rx;
(14)
Fracture conductivity:FcD
= klw/kxy
(15)
As shown by Bourdet et al. (1983, 1984) the use of the pressure
derivative greatly improves the analysis and the uniqueness of
the analysis of well tests. Furthermore, as shown by Grirrgarten
et al. (1979) the graphing of pressure against fD/CD collapses
repeating families of type curves that would appear for each
value of CD into one family of curves manv of which would
~hare a sin~le wellbore stor~ge portion. Thi; will be employed
here, Hence, all type curves to be presented are graphed with
fDti/CDf on the abscissa. This forces all solutions to share a
single wellbore storage-dominated straight line (with a slope
equal to unity). The derivative used here is perforrr,ed with
respect to the natural logarithm of the time function. This
met hod as shown by Bourdet et al. (1983) and extended by
Alagoa et al, (1985) to fractured wells, provides characteristic
shapes for wellbore storage, linear, hi-linear and infinitelyacting radial flow.
Using a basic algebraic principle:
dpD/d ([0 tD,f) = tDP ‘D
(16)
it can be said that if
PD
-
J’xj
(17)
.
-.-.”.
L.*.
.“
.I
.
JJbu,.
”,v,,
10-3 and 10-6. Hence, these are realistic fracture Storage
coefficients for which solutions should be generated for both
oil and gas wells.
then:
dpD/d (h tD$) = t@ ‘D - mt;’f
”JJo
(18)
The first task was to run a simulation for the two bounds for
the fracture storage constant i.e., for cDf = IO--6 and ~Df
= 10-3. Figure 3 represents pressure and pressure derivative
response for the first value and for dimensionless fracture conductilities equal to 0.1, 1, 10, and 100. Figure 4 represents the
same solutions but for a fracture storage constant equal to
of time was selected to show
10-3, For both cases the range
realistic and appropriate response features.
This implies that for the first three cases the derivative graph
parallels the pressure graph, displaced in the vertical direction
by log m. In wellbore storage-dominated flow t he two curves
are superimposed since log m is equal to zero, for m equal to
unity.
For infinitely acting reservoirs the derivative is equal to 0.5
(which is its limiting value) as shown by Bourdet et al. (1983,
1984) and Alagoa et al. (1985).
For both cases, the infinitely acting behavior appears at fDx.
approximately equal to unity. Hence, assuming a reasonable
value for the porosity (0.25) and the viscosity compressibility
product for both oil and gas (5 x IO-6 cp-psi -1, 7.3 x 10-5
cp-bar-1), the time of the infinitely acting behavior (in hrs) is
given by:
The above are significant not only as diagnostic and interpretive
tools but also, for the purposes of this study, they are invaluable
in the identification of the occurrence (or lack thereof) of certain flow regimes and their duration. For example, bi-lineai
flow can be identified only, if (and only if) parallel portions
of both the pressure and pressure derivative curves form slopes
equal to 0.25,
t-—
The ranges of the dimensionless variables for which solutions
are generated are also important. Of these, the dimensionless
fracture conductivity and the storage coefficient are of particular importance, The dimensionless fracture conductivity as
given by Eq, 15 is, in essence, a measure of permeability contrast. Obviously, the higher the fracture conductivity, the better the performance of the well will be when compared to the
pre-treatment state, Hence, for very tight formations (low reservoir permeability) even narrow, and without particularly high
permeability, fractures could result in significant performance
improvements. Furthermore, it shows in elegant form why
usually dramatic results are commonly obtained in tight formations (with properly executed fractures). The values of the
dimensionless fracture conductivity range from around unity
for very poorly producing fractures to 10 (for moderate) to over
100 for highly conductive fra:!ures. Cinco and Samaniego
(1981) have shown that for values over 300 the finiteconductivity solution is indistinguishable from the Gringarten
and Ramey (1974) infinite-conductivity solution. In fact, even
for fracture conductivities far below 300 the differences are
practically indistinguishable,
xf2
200 k
(19)
where I in hrs, xf in ft and k in md.
Of particular interest here is the development of the flow
regimes. For the low storage coefficient there is a significant
region of hi-linear flow shown with the large shaded area in
Fig. 5. The smaller shaded area is for linear flow where the
slope is equal to 0.5. This appears only on the larger fracture
conductivityy (~CD = 100) as one would expect. No half slope
of appreciable duration was observed in the smaller fracture
conductivities. A much more interesting observation was noted
for the larger storage coefficient as shown in Fig. 6. No hi-linear
flow was observed (except perhaps a small portion for FCD =
1 around tD#cDf
= 102). Linear flow was observed, again
at FCD = 100. Obviously, for storage coefficients larger than
10-3 it would be unlikely for hi-linear flow to appear. Linear
flow, though, is evident especially for very large dimensionless
fracture conductivities.
These observations lead to the generation of a new set of useful
type curves, one for low (FCD = 1), one for intermediate (FCD
= 10) and one for high (FCD = 100) conductivityy. These will
be described in the next subsection.
The fracture storage constant, as defined by Eq. 14 can be
calculated in the manner shown by Earlougher (1977). He has
provided techniques to calculate the wellbore storage constants
for both full wellbores and falling liquid level wellbores. These
calculations take into account the density and the compressibility of the wellbore fluids. The dimensionless fracture storage
constant as defined in Eq. 15 is similar to the standard wellbore
storage constant, Instead of the wellbore radius, the variables
in the definition are divided by the fracture half-length.
A NEW SET OF PRESSURE AND PRESSURE
DERIVATIVE rYPE CURVES
FOR HYDRAULICALLY FRACTURED WELLS
Figures 7, 8, and 9 are the new set of type curves in which
pressure and pressure derivative are graphed for values of the
dimensionless fracture StOrage coefficient, CDf equal to 10-3,
1o-4, IO--5, and 1o-I5,
A range for CDf has been established for this study. Using
typical reservoir and fluid variables and allowing the fracture
half-length to vary between 100 ft (30 m) (rein) and 1500 ft (460
m) (max), values of 5 x 10- t(max) and 10-6 (rein) were obtained for a gas filled wellbore. For oil, and using both a full
wellbore or a falling-level model, these values ranged between
The graphing of the ratio tDXf/CDf in the abscissa merges all
curves into a single wellbore storage portion.
For an FCD value equal to unity (Fig., 7) there is an extensive
hi-linear flow region (slope equal to 0.25) and no linear flow
W!
“.
6
OBSERVATIONS
OF TESTS OF HYDRAULICALLY
FRACTURED
SPE 153%
WELLS
}
I
regime. Even for an FCD value equal to 10 no linear flow appears (Fig. 8). However, for an FCD value equal to 100, the
hi-linear flow vanishes almost entirely while, as one would expect, the linear flow becomes much more prevalent.
storage constant may be calculated. The FCD value, and the
newly calculated fracture half-length would lead to a rough
estimate of the kfw (fracture permeability-fracture width)
product,
There is a compelling by-product of the study of these type
curves. There is an obvious, visual correlation between fracture conductivityy and storage coefficient. The higher the fracture conductivity the lower the curvature is, while the higher
the storage coefficient the higher the curvature of the solution is.
Field Application
As a result, if one were to observe Figs. ? and 9, a set of data
that would look analyzable with FCD = 1 and CDf = 10-3,
would require a much larger CDfvahre on the FCD = 100 type
curve to produce the same visual trends.
This, in fact, was done in the Alagoa et al. (1985) paper in which
infinite-conductivity solutions were used to analyze the posttreatment data of hydraulically fractured wells. In their paper,
much larger wellbore storage constants were necessary. (CDJ
values graphed were between 3 x 10–s and 3 x 10– 1.) In the
field examples that they analyzed, the fracture half-lengths
calculated were roughly equal to 100 ft. Analysis with the type
curves offered here would result in much larger fracture
half-lengths.
What is required to use the type curves offered in this paper
is for the analyst to decide whether a low, intermediate or high
conductivity type curve is indicated. This is usually not a proMem. From the definition of the fracture conductivity (Eq. 15),
it should be obvious that irr very tight formations (k < 0.01
md) even a moderav: fracture would result in a very high conductivity type curve. For high permeability formations (k >
5 red), even a reasonably sized fracture would result in a low
conductivity. Hence, a generalized, order of magnitude
clasWication (as is required) can be done before the analysis
is initiated.
Well KAL-5 in the Pan.lonian Basin in Yugoslavia (a gas condensate well) was fractured with a rather modest treatment
because of the limited availability of proppant on location. The
chronicle of the operation was described by Economies et al.
(1986). Table 1 contains well, reservoir and fluid data from a
post-treatment test at well KAL-5. The pressure buildup lasted
for 332 hrs, The actual pressure data were presented by
Economies et al. (1986),
Real gas pseudo-pressure differences and their derivatives are
used for the analysis. Since the ordinaf e is set (the reservoir
permeability was obtained by a pre-tret tment test and found
equal to 0.0035 md) then for any value c f Am(p) (eg. 107) the
dimensionless pressure can be calculate:
(0.0035) (216.5) (10’)
khAm(p)
‘D = 1424qT
=
(1424) (2020) (814~
= s z x ~O_3
‘
Thus, with Am(p) = 107 and PD = 3.2x 10-3 superimposed,
the data and their derivatives are moved from left to right until they match with a type curve.
Ordinarily, for this well a proper fracturing treatment would
result in a very high conductivity fracture. However, the treatment done was rather small and thus an attempt to match the
data with an intermediate conductivity fracture is attempted
here. Figure 10 shows a match which appears quite reasonable.
The dimensionless storage coefficient CD$ is roughly equal to
10–4 and the time match results in:
tD~f/CDf = 8 and At = 100 hrs.
The method of analysis itself is straightforward. Since the
permeability of the undisturbed reservoir is obtained via a pretreatment test, then the vertical match in a type curve rilatch
attempt is de facto set. (All variables multiplying the dimensional pressure difference are constant and known.) Hence, the
only flexibility allowed by the type curve is movement along
the time axis.
Simultaneous pressure and pressure derivative match should
be done. The data are moved U! the selected type curve from
left to right along a constant pressure match line. This is, of
course, necessary only if wellbore storage data disappear very
rapidly (before they can be definitely recorded). If wellbore
storage data exist then the match has no degree of freedom and
the knowledge of the undisturbed reservoir permeability should
serve as corroborating evidence. In moderate to high conductivity fracture wells, wellbore storage effects often disappear
rapidly, hence the methodology outlined above would be normally employed.
From the time match and the value of the dimensionless storage
coefficient, the fracture half-length and the dimensionless
Using the gas physical properties published by Economies et
al. (1986) and the data given in Table 1, the time match results
in a calculated value of the fracture half-length equal to approximately 800 ft (240 m). From the value of the dimensionless
storage constant and its definition, a dimensioned wellbore
storage constant equal to 0.075 bbl/psi (O.i 7 m3/bar) is
calculated. Considering that this is a 11,200 ft (3480 m) well
with approximately 180 bbl (29m3) capacity and using a compressibility equal to 4 x 10-4 psi-1 (5.9 x 10-3 bar-1, at
wellbore conditions, results in a storage constant equal to 0.072
bbl/psi (O.166 bbl/psi) which is very near the one calculated
from well data.
The value of the dimensionless fracture conductivity (FCD)
chosen for this analysis (and its definition in Eq. 15) results
in a kfl product equal to approximately 30 md-ft. This value
based on the methodology outlined earlier is the one likely to
exhibit the largest error. However, the forecasting of future
well performance is not particularly sensitive to the kfl product, (within reasonable ranges). Furthermore, experimental
studies done by Roodhart et al. (1986) has shown that the
M, J. EC
I 16396
kacture permeability may vary by a factor of three to four
iepending on the fracturing fluid used. Hence, its calculation
s intended to provide an approximate value.
CONCLUSION
A methodology to interpret fractured well tests has been
presented. This methodology is based on observations of the
belxa:’iorof fractured wells. Their slow response in developing
the well kr.own patterns led to a technique for the estimation
of the maximum reservoir permeabilityy. The latter is crucial
for both the design as well as the post-treatment evaluation
phases of a fracturing job.
The convohttion/deconvolution techniques and influence func.
tions have been suggested in order to allow the interpretatim
of drawdown (instead of buildup) data. This should be of par.
ticu!ar use in sensitive formations where shut-ins may resull
in fracture damage.
New type curves for pressure and pressure derivative allow tht
calculation of the fracture half-length and fracture storage coef
ficient. A reasonable value of the fracture permeability-fractur{
width product may be extracted as’ well.
ACKNOWLEDGEMENT
The author wishes to thank C Ehlig-Economides and M
Karakas of Flopetrol-Johnston-Schhsmberger
for their help il
the development of the new type curves.
IMIDES
?
7,
= fracture half-length
f
= gas gravity (to air)
I
= viscosity
b
= porositjj
= gas deviation factor
?EFERENCES
4garwa1, R. G., A1-Hussainy, R. and Ramey, H. J., Jr.: “An
investigation of Wellbore Storage and Skin Effect in Unsteady
tiquid Flow: 1. Analytical Treatment,” Sot. Pet. Eng. J (Xpt.
1970) 279-290.
4fmed, U., Kucuk, F., Ayesteran, L.: “Short-Term ‘ransient
Rat: and Pressure Buildup Analysis of Low-F .imeability
Wells,” p~per SPE/DGE 13870, presented at the 1985 Low
Permeability Gas Reservoirs of the SPE/DOE.
Alagoa, A,, Bourdet, D., Ayoub, J. A.: “How to Simplify the
Analysis of Fractured Well Tests,” World Oil, Ott. 1985,
17-102.
Bourdet, D., Whittle, T. M., Douglas, A.A. and Pirard, Y.M.:
“A New Set of Type Lurves Simplifies Well Test Analysis,”
World Oil, May 198. .
Bourdet, D., Ayoub, J.A. and Pirard, Y.M.: “Use of Pressure
Derivative in Well Test Interpretation,”
paper SPE 12777,
presented at the 1984 California Regional Meeting of the SPE.
NOMENCLATURE
B=
formation volume factor
c~
= totai system compressibility
c=
welliwre storage constant
FCD = dimensionless frature conductivity
h
= reservoir thickness
hp
= perforated interval
k=
kf
m=
permeability
= fracture permeability
slope
m(p) = real gas pseudopressure
P=
pressure
Pi
= initial pressure
Pwf
= bottomhole pressure
P’
q
= pressure derivative
. flow rate
rw
Sw
= well radius
= water saturation
t
= time
T=
absolute temperature
w=
fracture width
Clnco-Ley, H., Samaniego-V, F. and Dominguez, N.: “Transient Pressure Behavior for a Well with a Fi- . Conductivity
Vertical Fracture,” Sot. Pet. Eng. J. (Aug. 1978), 253-264.
Cinco-Ley, H. and Samaniego-V., F.: “Transient Pressure
Analysis for Fractured Wells,” Jour. Pet. Tech., (Sept. 1981)
1749-176ri.
Cinco-Ley, H. and Samaniego-V., F.: “Transient Pressure
Analysis: Finite Conductivity Fracture Case Versus Damaged
Fracture Case,” paper SPE 10179 presented at the 1981 Anuual Fall ltieeting of the SPE.
Cinco-Lejj, H. and Samaniego-V., F.: “Pressure Transient
Analysis for Naturally Fractured Reservoirs,” paper SPE 11026
presented at the 1982 Annual Fall Meeting of the SPE.
Cinco, H.: “Evaluation of Hydraulic Fracturing by Transient
Pressure Analysis Methods,” paper SPE 10043 presented at the
1982 International Petroleum Exhibition of the SPE, Beijink,
China,
Cuesta, J.F. and Elphick, J. J.: “Pre- and Post-Stimulation
Pressure Test Analysis and its Role in the Design and Evaluation of Hydraulic J%acture Treatments,” paper SPE 12992
presez:ed at the 1984 EUROPEC.
Earlougher, R. C., Jr.: Advances in Well Test Analysis, SPE,
Dallas, 1977.
I
8
OBSERVATIONS
OF TESTS OF HYDRAULICALLY
Economies,
M. J., Brigham, W .E., Cinco-Ley, H., Miller,
F.G., Rarney, H. J., Jr., Barelli, A. and Manetti, G.: “Influence
Functions and their Application to Geothermal Well Testing,”
Geoflr. Res. Corm. Trans. v. 3, 1979,177-180.
FRACTURED
WELLS
SPE 163%
Raghavan, R., Uraiet, A. and Thomas, G. W,: “Vertical Fracture Height: Effect on Transient Flow Behavior,” SoC. Pet.
Eng. J. (Aug, 1978), 265-277.
Roodhart, L., Kuiper, T.O. and Davies, D. R.: “Proppant Rock
Impairment During Hydraulic Fracturing,” paper SPE 15629
presented at the 1986 Annual Fall Meeting of the SPE.
Economies, M, J., Cikes, M., Pforter, H., Udick, T,H. and
Uroda, P.: “The Stimulation of a Tight, Very HighTemperature Gas Condensate Well,” paper SPE 15239
presented at the 1986 Unconventional Gas Technology Syrnposium of the SPE.
Samaniego-V F. and Cinco-L,ey, H.: “Pressure Transient
Analysis for Naturally Fractured Gas Reservoirs: Field Case,”
paper SPE 12010 presented at the 1983 Annual Fall Meeting
of the SPE.
Gringarten, A.C. and Ramey, H. J., Jr.: “Unsteady State
Pressure Distributions Created by a Well with a Single-Infinite
Conductivity Vertical Fracture,” Sot. Pet. Eng. Y. (Aug. 1974),
347-360.
Stewart, G., Meunier, D. and Wittmann, M. J.: “Afterflow
Measurement and Deconvolution in Well Analysis,” paper SPE
12174 presented at the 1983 Annual Fall Meet;ng of the SPE.
Gringarten, A. C., Ramey, H. J., Jr. and Raghavan, R.: “Applied Pressure Analysis for Fractured Wells,” J. Pe(, Tech, (Juiy 1975), 887-892.
van Everdingen, A.F, and Hurst, W.: “The Application of
Laplace Transform: to Flow Problems in Reservoirs,” Trans.,
AIME (1949), 186, 305.
Gringarten, A. C.: “Reservoir Limit Testing for Fractured
Wells,” paper SPE 7452 presented at the 1978 Annual Fall
Meeting of the SPE.
Wong, D. W., Barrington, A.B. and Cinco-Ley, H.: “Application of the Pressure Derivative Function in the Pressure Transient Testing of Fractured Wells, ” paper SPE 1305 presented
at the 1984 Annual Fall Meeting of the SPE.
Gringarten, A. C.. Bourdet, D. P., Landel, P.A. and Kniazeff,
~?.J,: “A Comparison Between Different Skin and Wellbore
Storage Type Curves for Early-Time Transient Analysis,” paper
SPE 8205 presented at the 1979 Annual Fall Meeting of the
SPE.
TABLE 1
RESERVOIR, WELL AND
FLUID INFORMATION FOR KAL-5
Hutchinson, T.S. and Sikora, V, J.: “A Generalized WaterDrive Analysis,” Trans. AIME (1959) 216, 169.
(FROM ECONOMIES
Jargon, J.R. and van Poollen, H, K.: “Unit Response Function from Varying-Rate Data,” J. Pet. Tech. (Aug. 1%5), %5.
Kabir, C.S. and Kucuk, F.: “Well Testing in LowTransmissivity Oil Reservoirs,” paper SPE 13666, presented
at the 1985 California Regional Meeting of the SPE.
= 1.94
9C
= 117.6 B/D
Qeq = 2.02
Katz, D. L., Tek, M,R. and Jones, S.C,: “A Generalized Model
for Predicting the Performance of Gas Reservoirs Subject to
Water Drive,” paper SPE 428, presented at the 1962 Annual
Fall Meeting of the SPE.
Kazemi, H,: “Pressure Transient Analysis of Naturally Fractured Reservoirs with Uniform Fracture Distribution,” Sot.
Pe(. Eng. J. (Dec. 1969) 451-461.
103 Mscf/D
9g
X
X
103 Mscf/D
ET AL,, 1986)
hp
= 36 ft
4
= 0.062
rw
= 0.54 ft
rg
= 3,94 (to air)
Sw
= 0.3
T
= 354° F
k
= 0,0035 md
h
= 216.5 ft
Fhsid Composition (percent mole fractions)
Kucuk, F. and Ayestaran, L.: “Analysis of Simultaneously
Measured Pressure and Sandface Flow Rate in Transient Well
Testing,” paper SPE 12177 presented at the 1983 Annual Fall
Meeting of the SPE.
Kucuk, F. and Sawyer, W. K.: “Transient Flow in Naturally
Fractured Reservoirs and its Application to Devonian Gas
Shales,” paper SPE 9327 presented at the 1980 Annual Fall
Meeting of the SPE.
Mannon, L. S.: “The Real Gas Pseudo Pressure for Geot herrnal Steam,” M.S. Report, Department of Petroleum Engineering, Stanford University, 1977.
H2S = 0.006
i-C5 = 0.442
N2 = 1.452
n-C5 = 0.379
co~ = 10.931
C6
= 0.516
cl
= 72.613
CT
= 0.644
C2
= 6.24
C*
= 0.541
C3
= 1.63
C9
= 0.388
i-C4 = 0.553
n-C4 = 0.693
cc
C,O+ = 2.979
Pwi
or
10’
750 -
q (MSCF/d)
Ptf
2000
600 -
F
psi
10
1
1000
800
3ooo~
Time (days)
1
lU
10-
10
.
Time (days)
Fig. !-Pressure and Ikw me dst, 1.xTrw,I. 22.
101 [
FIS. 2-cvcOnvOWcdand nowM2&.11
influence2imtlon8 f-
I
I
I
I
I
I
I
Fc
-0
~
-~
100
?2
Q
101
,..,
.0
102
1(J3
Io’f
Fb. 3-W$W*
101 I
-g 101
I
M
PI.uw.
105
t~x, 1CD,
Mvatlvo ruww
I
I
101
100
106
fw bw.aarw
107
108
frutwed wdl (CC+. to - *I.
I
1
I
I
,03
104
I
t
,“
10-3
~o-z
@
102
%xf I %ff
M. 4-P-WI.
-i
pm-w.
duhwiw
c.wmtm fix hkh+mw
57
mwd
inn ICW. 10-31.
Trwmb 22.
101
100
~-n
-an
a
n
101
~10-2
102
1
t
103
1
105
104
I
I
106
107
I
lfj8
tox’ I CD’
Fig. 5-B[.linear
and linear flow regimes for low.stor~e
wells.
101
100
‘i
0’
102
q n3
I
.“ 10-3
I
102
I
1()-1
Fig. 6-Llnenr
1
t
100
101
tDx’I cDf
flow regime in high-slorage
se
wells.
1
I
102
103
104
10’
“
m
1
1 O(
10”
10:
-31
I
of
1 (-J-3
1(-J-1
102
100
1(y
1fj2
101
[FCD
7-Pressure
I
104
103
lo6
105
c~f
t~xf /
Fig,
I
and pressure derivative type curve for low-fracture conductivity well.
= 101
I
lt’fFINITELY
I
ACTING+
1~
t’
100
\
. , .
I
“an
&
=10-5+
~c.f
102
‘u
Q
10-3
10-4
I (J-4
10-3
102
101
101
100
t~xf /
102
103
104
Cof
Fig. 8-. Preseut8 and preaeure derivative type curve for intermediate fracture conductivity well.
59
1(J5
106
101
1
--
II
100
1 (J-3
104
I 0-2
@
I00
101
102
103
104
105
106
‘Dxf / cDf
Fig. 9-Pressure
and preesure derivative type curve for high-fracture conductivity well.
1(’)1
IF A.
I-bu
=10[
I
I
I
I
I
101
--’’’’’=.{OO
101
102
103
I
I-AF41TELY
‘ACTING+
100
n
1 ()-1
1
I
I
I
A
10-2
1(’J-3
10-4
10-4
10-3
102
cDf
Fig,
10—Type-curve
match for example application.
t~xf
60
104
105
106
/
I