SPE 97972
Post-Fracture Performance Diagnostics for Gas Wells With Finite-Conductivity
Vertical Fractures
J.A. Rushing, SPE, and R.B. Sullivan, SPE, Anadarko Petroleum Corp., and T.A. Blasingame, SPE, Texas A&M U.
Copyright 2005, Society of Petroleum Engineers
This paper was prepared for presentation at the 2005 SPE Eastern Regional Meeting held in
Morgantown, W.V., 14–16 September 2005.
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Abstract
This paper presents an integrated technique for evaluating the
production performance of gas wells with finite-conductivity
vertical fractures. Our methodology combines conventional
pressure transient test analysis with new material balance
decline type curves developed specifically for gas wells with
finite-conductivity, vertical fractures. We utilize short-term
pressure buildup test analysis to enhance the production data
analysis, particularly for interpretation of early-time transient
flow behavior. We illustrate—with several field cases—that
both techniques can be integrated to provide not only a more
consistent and systematic analysis methodology, but also a
more accurate assessment of stimulation effectiveness.
diagnostic techniques.1-8 Cipolla and Wright9,10 and Barree, et
al.11 have identified and grouped fractured-well diagnostic
techniques into three general categories—direct far-field,
direct near-wellbore, and indirect. Our methodology focuses
on two indirect diagnostic techniques—pressure transient
testing and production data analysis.
Specifically, we
illustrate how short-term pressure buildup testing integrated
with long-term production data analysis can be an effective
method for evaluating stimulation effectiveness.
Indirect Fractured-Well Diagnostic Techniques
Although pressure buildup testing is the most effective indirect
technique for evaluating the stimulation effectiveness of
hydraulically fractured gas wells, knowledge of reservoir
permeability—either from the well test or from an independent
source—is required to compute fracture properties. If a well is
shut in for a sufficient duration to reach the pseudoradial flow
period, then we can uniquely determine reservoir permeability
from the test data (and very likely also estimate effective
fracture half-length and fracture conductivity since these are
dependent on the permeability estimate). Wells completed in
tight gas sands require very long shut-in times to reach
pseudoradial flow, but operators are reluctant to shut in a well
for extended periods. If, however, we have an independent
estimate of reservoir permeability, then shorter duration
pressure buildup tests become practical.
Introduction
Wells producing from tight gas sands require stimulation to
achieve economic rates and to maximize ultimate recoveries.
The most common stimulation technique is hydraulic
fracturing. Depending on the type and size of the treatment,
hydraulic fracturing may be expensive—often representing a
significant percentage of the total completion costs. Since the
economic viability of wells completed in tight gas sands
depends on minimizing costs, then it is essential that we
optimize fracture treatments, i.e., find the proper balance
between stimulation costs and well productivity. A key
component in achieving this balance is a post-fracture
diagnostics program to determine stimulation effectiveness.
Decline type curve analysis of production data has become a
common alternative for estimating reservoir permeability
without shutting in the well. Fetkovich12,13 was the first to
incorporate transient flow models with decline curve analysis.
He developed the standard “decline type curves” by
combining an analytical model for transient, radial flow at
constant bottomhole pressure with Arps’14 empirical
exponential and hyperbolic rate decline models. We note for
completeness that the exponential rate decline model is the
analytical solution for a well produced at a constant
bottomhole flowing pressure and boundary-dominated flow
conditions.
Many diagnostic techniques for evaluating hydraulicallyfractured gas well performance have been documented in the
petroleum industry, but theoretical model assumptions, model
applicability and simplicity, data requirements, and/or data
quality and quantity may limit the effectiveness of any single
analysis technique. Therefore, we employ an integrated
approach in which we capture the benefits and utilize the
strengths of several types of hydraulically-fractured well
The original Fetkovich decline type curves are useful for a
range of reservoir pressure conditions, but we have observed
cases where the boundary-dominated rate-time data changes
evaluation curves over time (i.e., changes from one empirical
model or stem to another). These changes have been
attributed to changes in gas properties as a function of
reservoir pressure. To address the impact of pressuredependent gas properties on the evaluation of gas production
2
data, Carter15 presented decline type curves for gas reservoirs.
Although his method is theoretically more rigorous than
simply using the empirical exponential and hyperbolic type
curve matching, his approach is not universal. Additionally,
both the Fetkovich and Carter models suffer from the constant
bottomhole flowing pressure assumption. Note that this
assumption primarily affects evaluation of the early-time
transient flow data from which we assess both reservoir
properties and stimulation effectiveness.
Crafton16 addressed the limiting constant bottomhole flowing
pressure assumption with decline type curves developed using
a rate normalization of the pressure responses, i.e., conversion
of variable-rate responses to a constant-rate response.
Because of the rate normalization process, Crafton’s
Reciprocal Productivity Index (RPI) Method incorporates both
transient and pseudo-steady state flow models developed for
constant-rate production conditions. Araya and Ozkan,17
however, noted some limitations of the RPI Method,
especially when used for wells exhibiting cyclic flow rates or
variable-rate production during boundary-dominated flow.
Palacio and Blasingame18 developed the material balance
decline type curve methodology that also overcomes the
constant bottomhole pressure constraint by using rate (or
pressure) normalization and a material balance time (i.e.,
cumulative production divided by instantaneous rate). They
demonstrated that, when plotted in terms of the material
balance time variable, the pressure-drop, rate-normalized data
merge into a single harmonic decline curve during boundarydominated flow (i.e., the reciprocal of the equivalent constantrate pressure response). They also combined their new
pseudosteady-state model with several transient models,
including gas wells with infinite-conductivity vertical
fractures.
We note that gas-dependent properties are
addressed using the appropriate pseudopressure and
pseudotime definitions as given in Reference 18.
Pratikno, et al.19 combined the pseudosteady-state
superposition concept (i.e., material balance time) with the
transient flow model for a finite-conductivity vertical fracture.
The value of this work is that it distinguishes the decline type
curve behavior for the case of a vertical well with a finiteconductivity fracture. Although presented in a different
format, Agarwal, et al.20 also developed decline type curves—
for both the infinite- and finite-conductivity vertical fracture
case—using a similar material balance time function.
Other production data analysis techniques21-24 consider the
flow rate and pressure histories to be an extended drawdown
test and use variable-rate pressure transient testing theory and
superposition plotting functions to analyze the data. Unlike
most conventional decline type curves, these other methods
allow us to identify specific flow regimes characteristic of
hydraulically-fractured wells—e.g., bilinear, formation linear,
and pseudoradial flow.25,26 Similar to traditional pressure
transient testing, these production data analysis methods also
use specialized plotting techniques to determine the fractured
properties from each flow regime. The ability to identify
specific flow patterns (regimes) offers a significant advantage
over conventional decline type curves. However, we still
SPE 97972
cannot uniquely quantify fracture properties unless we have an
independent estimate of reservoir permeability or the well has
reached pseudoradial flow.
Regardless of the production data analysis technique
employed, problems with production data quantity and/or
quality—e.g., incomplete or infrequent data sampling; poor
quality data; and/or erroneous data—affect the accuracy of the
analyses. This impact is especially problematic for accurate
assessments of both fracture and reservoir properties using
transient data which are typically changing frequently and
rapidly during early-time flow periods. Therefore, our method
utilizes short-term pressure buildup test evaluation to improve
the production data analysis—principally for interpretation of
early-time transient flow behavior from which we can evaluate
effective fracture conductivity and half-length.7,17,27,28
New Material Balance Decline Type Curves (MBDTC)
The new material balance decline type curves (MBDTC)19
were developed using a “desuperposition” technique which
mathematically combined transient and boundary-dominated
flow models. The transient model is that for a well with a
finite-conductivity, vertical fracture producing at a constant
rate, while the boundary-dominated model is for a well
centered in a finite, volumetric circular reservoir. The type
curves are correlated using dimensionless fracture
conductivity, FCD, and dimensionless reservoir radius, reD.
Because of the rate-normalization technique implemented, the
type curve method is also applicable to variable flow rate,
variable bottomhole flowing pressure, or combinations of
these flowing conditions. Further, the type curves use three
different pressure-drop, normalized-rate plotting functions
which allows us to match several different model and data
functions
simultaneously,
thereby
providing
more
representative (and more accurate) matches of field production
data. We can estimate effective permeability to gas and the
near-wellbore flowing efficiency (either in terms of fracture
half-length or a skin factor) from analysis of transient data.
Furthermore, analysis of the pseudosteady-state or boundarydominated data provides estimates of contacted gas-in-place
and drainage area.
An example of the new type curves is shown in Fig. 1 which is
a log-log plot of dimensionless pressure-drop normalized-rate
(red curves, qDd), dimensionless pressure-drop normalized-rate
integral (green curves, qDdi), and dimensionless pressure-drop
normalized-rate integral-derivative (blue curves, qDdid) as a
function of dimensionless material balance time.
The
particular type curves shown in Fig. 1 were generated for a
dimensionless fracture conductivity (FCD) value of 5 (which is
a fairly low conductivity value). The dimensionless reservoir
radius, reD, is used as a correlating parameter for the transient
portion of the type curves.
Integrated Analysis Methodology
We have developed an iterative technique that integrates
pressure transient testing with production data analysis. We
begin with initial estimates of reservoir permeability (kg),
effective fracture half-length (Lf), and dimensionless fracture
conductivity (FCD)—all estimated using production data
SPE 97972
analysis based on the material balance decline type curves
(MBDTC).19
3
ta =
µ gi c gi
q g (τ )
t
∫ µ g ( p ) c g ( p ) dτ
q g (t )
................................(1)
0
qDd, qDdi, and qDdid
4. Compute the field pseudopressure-drop normalized-rate
functions:
a. Pseudopressure-drop normalized-rate function:
qg
∆p p
=
qg
( p pi − p pwf )
b. Pseudopressure-drop normalized-rate integral function:
⎛ qg ⎞
⎜
⎟ = 1
⎜ ∆p p ⎟ t a
⎝
⎠i
Fig. 1—Example Material Balance Decline Type Curves
19
(MBDTC) for Finite-Conductivity Vertical Fractures (FCD=5)
In the next step, we evaluate the pressure buildup test. We
identify flow regimes characteristic of hydraulically fractured
wells (i.e., bilinear, formation linear, pseudoradial flow)25,26
from a log-log plot of the pressure derivative. Depending on
the type of flow regimes present, we may then use special
plotting functions and analysis techniques to compute fracture
properties. For example, if bilinear flow is present, then we
can estimate effective fracture conductivity from the slope of
the straight line on a fourth-root of time function plot. Or, if
the formation linear flow period is present, then we can
estimate effective fracture half-length from the slope of the
straight line on a square-root of time function plot.
In the final step of our analysis procedure, we use an
automatic history-matching process to analyze the complete
well test data. Estimates of reservoir permeability from the
MBDTC analysis and fracture properties from the special well
test analyses are used to establish initial estimates and
reasonable ranges for the history-matching parameters. We
compare the history-match to results from the MBDTC and
special well test analyses, and we iterate between the analyses
until consistent results are obtained.
Production Data Analysis Procedure Using MBDTC. In
this section, we outline a procedure for using the new material
balance decline type curves (MBDTC), including the required
input data.
1. Gather all of the production data—i.e., gas production and
flowing pressure histories. Daily production data is
preferred since it will allow the most accurate evaluation.
Bottomhole flowing pressures are also preferred, but we
can still utilize surface flowing pressures by computing
bottomhole values using nodal analysis.
2. Gather details of the wellbore diagram and tabulate all
reservoir rock and fluid data, including reservoir pressure
(either initial pressure or average pressure at the onset of
production), static bottomhole reservoir temperature, net
sand thickness, average effective porosity and connate
water saturation in the net thickness interval, and fluid
properties.
3. Compute the field material balance pseudotime30 function:
...............................................(2)
ta
qg
∫ ∆p p dτ .............................................(3)
0
c. Pseudopressure-drop
integral function:
normalized-rate
derivative-
⎡⎛ q ⎞ ⎤ 1 d ⎡⎛ q ⎞ ⎤
⎛ qg ⎞
d
⎜
⎟ =
⎢⎜ g ⎟ ⎥ =
⎢⎜ g ⎟ ⎥
⎜ ∆p p ⎟
⎜
⎟
⎜
⎟
⎝
⎠id d ln(t a ) ⎢⎣⎝ ∆p p ⎠ i ⎥⎦ t a d t a ⎢⎣⎝ ∆p p ⎠ i ⎥⎦
...................................................................................(4)
5. Overlay the three field pseudopressure-drop ratenormalized functions on MBDTC for a specific value of
FCD. We recommend that you start with a low value of FCD
as an initial estimate. Force the field data in the boundarydominated flow regime to match the depletion stems for
each of the respective dimensionless functions (i.e., match
(qg/∆pp) against qDd, (qg/∆pp)i against qDdi, and (qg/∆pp)id
against qDdid).
6. Evaluate the match of the transient field data on each of the
three dimensionless transient functions. If a good match is
not obtained, then choose another set of type curves by
increasing the value of FCD.
7. Once a good match is obtained, record time and rate match
points (indicated by subscript MP) for both field and type
curve parameters. In addition, record values of the type
curve correlating parameters, FCD and reD. Compute the
well and reservoir parameters as follows:
a. Contacted gas-in-place (G):
G=
1 (t a ) MP (q g / ∆p p ) MP
.............................(5)
c gi (t Dd ) MP (q Dd ) MP
b. Reservoir drainage area (A) and effective drainage
radius (re):
A = 5.615
re =
GBgi
φh(1 − S wi )
.............................................(6a)
A / π ..............................................................(6b)
c. Effective gas permeability (kg):
k g = 141.2
B gi µ gi
h
b Dpss
( q g / ∆p p ) MP
( q Dd ) MP
.................(7)
where bDpss is the dimensionless pseudosteady-state
constant (see Reference 19 for a definition).
4
SPE 97972
Lf =
re
................................................................. (8)
reD
Note that we have implemented a “spreadsheet” approach such
that much of the type curve matching can be accomplished
interactively, while many calculations are computed
automatically.
Pressure Buildup Test Analysis Procedure. The procedure
outlined below assumes pseudoradial flow will not be present
since we are using short-term pressure buildup testing. If,
however, the well test data exhibits this flow period, then that
data should be incorporated into the analysis. We should note
also that the working equations are derived based on a
pseudopressure and normalized pseudotime formulation
utilized in the well testing software Pansystem,34 but the
equation format shown below will change if different pressure
and time functions are used.
1. Prepare a plot of pseudopressure29 change and derivative of
pseudopressure change against normalized equivalent or
superposition30,31 pseudotime function using the pressure
buildup test data.
Identify all flow regimes25,26
characteristic of hydraulically fractured wells from the
shape of the derivative data.
3. If the bilinear flow period25,32,33 is present, prepare a
Cartesian plot of pseudopressure against the fourth root of
normalized pseudotime function. Compute the slope (mB)
of the line drawn through the bilinear flow period
identified from Step 2. Using kg from the MBDTC
analysis and mB from the bilinear plot, compute effective
fracture conductivity, wfkf:
⎛ 444.75qgT
w f k f = ⎜⎜
hmB
⎝
⎞
⎟⎟
⎠
2
1
φµ g ct k g
.......................... (9)
4. If the formation linear flow period25,32,33 is present, prepare
a Cartesian plot of pseudopressure against the square root
of normalized pseudotime function. Compute the slope
(mL) of the line drawn through the formation linear flow
period identified from Step 2. Using kg from the MBDTC
analysis and mL from the formation linear plot, compute
effective fracture half-length, Lf:
⎛ 40.925qgT
L f = ⎜⎜
hmL
⎝
⎞
1
⎟⎟
.............................. (10)
φµ
g ct k g
⎠
5. Compute the dimensionless fracture conductivity, FCD,
using estimates of wfkf, Lf and kg:
FCD =
wf k f
Lf kg
........................................................ (11)
Automatic History Matching Procedure. The next step in
our integrated procedure is the use of an automatic historymatching process to analyze the well test. To help converge to
the correct solution, we use estimates of reservoir permeability
from the MBDTC analysis and fracture properties from the
well test special analyses. These estimates are used to
establish initial estimates and reasonable ranges for the
history-matching parameters. We then compare historymatched results to the MBDTC and special well test
evaluations. If needed, we iterate between the analyses until
we obtain consistent results.
Validation of Integrated Evaluation Technique
To assess the validity of our integrated evaluation technique,
we analyzed several pressure buildup tests numerically and
then compared the results to our integrated evaluation
approach. We illustrate the results of the numerical well test
analysis with Field Example 1. Because this well has a very
low-conductivity fracture, it represents an extreme test case
for our integrated (analytical) evaluation technique.
Analytical Well Test Analysis—Field Example 1. The first
field example is from a well completed in the lowpermeability Lower Cotton Valley Sands in a field located in
the North Louisiana Salt Basin in Jackson Parish, LA. The
productive sands are deep (13,000 to 15,000 ft) and
abnormally over pressured (pore pressure gradients > 0.90
psi/ft). Productive intervals have effective porosities ranging
from 4% to 12%, while absolute permeability ranges from
0.005 md to 0.02 md. Because of the thick sand intervals, the
wells are often stimulated with multiple stages. This particular
well was stimulated in three stages—each stage composed of
8,500 to 10,000 bbl slick water and 170,000 to 230,000 lbs
40/70 proppant.
The well produced for about two years before being shut in for
a two-week pressure buildup test (Fig. 2). Initial estimates of
permeability and fracture half-length were obtained from the
material balance decline type curve (MBDTC)19 analysis of
the production data. The type curve match shown in Fig. 3 is
a log-log plot of dimensionless pressure-drop normalized-rate
(red curves, qDd), dimensionless pressure-drop normalized-rate
integral (green curves, qDdi), and dimensionless pressure-drop
normalized-rate integral-derivative (blue curves, qDdid) against
dimensionless material balance time. The discrete data points
in Fig. 3 represent the corresponding dimensional field data
(i.e., red points qg/∆pp, green points (qg/∆pp), and blue points
(qg/∆pp)id).
6,000
Gas Production Rate, Mscf/day
d. Effective fracture half-length (Lf):
5,000
4,000
3,000
Shut in for
Two-Week
Pressure
Buildup Test
2,000
1,000
0
6-Sep-00
6-Jun-01
6-Mar-02
6-Dec-02
6-Sep-03
6-Jun-04
6-Mar-05
Date
Fig. 2—Post-fracture gas production history, Field Example 1.
From our initial analysis, we estimate kg=0.0084 md and Lf
=236 ft. The best match was obtained with the type curves for
SPE 97972
5
MBDTC Results
kg = 0.0084 md
Lf = 236 ft
wfkf = 3.96 md-ft
FCD = 2; reD = 2
G = 3.7 Bcf; A = 14.7 acres
Next, we use special plotting functions to validate various
flow patterns identified from the log-log derivative plot. The
bilinear flow regime is validated by the straight line on a plot
of pseudopressure against the fourth root of pseudotime
superposition function shown in Fig. 5. If we have identified
the correct straight line on the curve, then we can use the
bilinear flow line slope to estimate effective fracture
conductivity as defined by Eq. 9. On the basis of the line
slope from Fig. 5 and kg from the MBDTC analysis, we
estimate wfkf is 2.8 md-ft. Moreover, if we substitute Lf and kg
from the MBDTC analysis and wfkf from Fig. 5 into Eq. 11,
we compute FCD=1.4. Note that this value is consistent with
FCD=2 from the initial MBDTC analysis.
Pseudopressure Function, psia2/cp
Pseudopressure-Drop Normalized-Rate Functions
FCD=2 and reD=2. We also estimate the contacted gas in place
and drainage area for this well are 3.7 Bcf and 14.7 acres,
respectively. We do note that the shapes of the dimensionless
rate and integral functions during transient flow are very
similar for a wide range of reD values. Although we do
observe some differences, the dimensionless derivative
function curve shapes are also somewhat similar, especially
for reD values between 2 and 5. Incomplete or inaccurate
pressure and rate data tend to exacerbate these problems, thus
often making it very difficult to obtain unique estimates of kg
and Lf from the early-time data analysis. We can, however,
quantify a range of values represented by the range of reD. For
this case, we would estimate kg and Lf ranges of 0.0084 md to
0.0137 md and 225.7 ft to 90.7 ft, respectively, for reD from 2
to 5.
Bilinear Flow
Region
Bilinear Flow Analysis
kg = 0.0084 md (MBDTC)
Lf = 236 ft (MBDTC)
wfkf = 2.8 md-ft (Eq. 9)
FCD = 1.4 (Eq. 11)
Fourth-Root of Pseudotime Superposition Function, (hr)1/4
Material Balance Pseudotime Function
Fig. 3—Material balance decline type curve analysis
of post-fracture gas production, Field Example 1.
Pseudopressure Functions, psia2/cp/MMscfd
Because of the significant rate changes prior to shutting in the
well, we analyzed the two-week pressure buildup data using
both pseudopressure29 and pseudotime superposition30,31
plotting functions. A log-log plot of the pseudopressure
change and pseudopressure derivative functions against the
normalized pseudotime superposition function is shown in
Fig. 4. For this particular case, the solid black line with a onequarter slope drawn through the pseudopressure derivative
function indicates a significant portion of the test was
dominated by bilinear flow. Note that we do not observe
either a formation linear or a pseudoradial flow period during
the two-week shut-in period.
Pseudopressure function
Pseudopressure derivative function
Bilinear
Flow
One-quarter slope
line indicative of
bilinear flow
Pseudotime Superposition Function, hr
Fig. 4—Log-log plot of two-week pressure buildup test
identifying fractured-well flow regimes, Field Example 1.
Fig. 5—Fourth-root-of-time plot showing bilinear flow regime
from two-week pressure buildup test, Field Example 1.
On the basis of the derivative plot in Fig. 4, we did not
observe the pseudoradial flow regime. As a result, we cannot
use conventional semilog analysis techniques to compute
reservoir permeability from the well test. Instead, we use an
automatic history-matching process during which we allowed
kg, Lf, and FCD to vary. To assess the importance of non-Darcy
flow in the well test analysis,35,36 we also included a ratedependent skin factor defined by the non-Darcy flow
coefficient, D. Estimates of kg from the decline type curve
analysis and Lf and FCD from the special analyses were used to
establish initial estimates and reasonable ranges for the
parameters during the history-matching process. In fact, we
used the range of kg and Lf corresponding to the range of reD
values discussed previously. We then iterated several times
between the automatic history matching, the MBDTC
analysis, and the well test analysis until we obtained consistent
results.
The final and best history match is shown in Fig. 6. The solid
lines drawn through the discrete well test data represent the
final history-matched solution. Note that results from the
history-matched pressure buildup test analysis are very similar
to that from the initial MBDTC analysis. We estimate kg and
Lf are 0.0068 md and 221 ft, respectively, from the historymatched well test and kg=0.0084 md and Lf=236 ft from the
MBDTC analysis. We also computed wfkf=2.96 md-ft
corresponding to FCD=1.97 from the well test analysis as
compared to wfkf=3.96 md-ft and FCD=2 from the initial
MBDTC analysis. Both analyses suggest the stimulation
6
SPE 97972
treatment generated a fracture with a very low effective
conductivity.
4,000
Measured Bottomhole Pressure
Bottomhole Shut In Pressure, psi
Pseudopressure Functions, psia2/cp/MMscfd
3,500
Pseudopressure function
Pseudopressure derivative function
We input the gas production history shown in Fig. 2 and
history-matched both the well flowing and shut-in pressures
by varying fracture properties—including effective fracture
half-length and fracture permeability. Our best match (Fig. 7)
was obtained with an effective fracture permeability of 10 md,
an effective fracture half-length of 270 ft, and an average
effective reservoir permeability, kg=0.0071 md. On the basis
of the modeled fracture grid width, we compute an effective
fracture conductivity of 5 md-ft, which is very close to that
estimated from the history-matched pressure buildup test
analysis (i.e., wfkf=2.96 md-ft). Similarly, both the simulated
effective fracture half-length of 270 ft and dimensionless
fracture conductivity of 2.6 agree with estimates from the
integrated well test and decline type curve analyses. Although
not shown in this paper, we have successfully matched results
from the analytical and numerical techniques on several other
wells. Therefore, we feel that our technique is valid for
evaluating hydraulically-fractured gas well performance.
2,000
Simulated Results
kg = 0.0071 md
Lf = 270 ft
wfkf = 5 md-ft
FCD = 2.6
1,500
0
50
100
150
200
250
300
350
Elapsed Shut In Time, hr
Fig. 7—Numerical well test analysis of two-week
pressure buildup test, Field Example 1.
Field Example 2
The second example illustrates a large conventional slick
water-frac stimulation treatment for a well producing from the
Bossier Sands in the East Texas Basin. Specifically, this well
is completed in the Bald Prairie Field located in Robertson
County, TX. The well was hydraulically fractured in early
May 2001 with 9,710 bbl slick water and 135,000 lbs 40/70
proppant. As shown by the production history in Fig. 8, the
initial gas rate was slightly greater than 1,500 Mscf/day. The
well was shut in for a two-week pressure buildup test after
about 18 months of production.
1,500
Gas Production Rate, Mscf/day
We used data from a comprehensive core description and
evaluation program to populate the grid blocks vertically. The
core data suggested significant reservoir heterogeneity in the
vertical direction, so we built the model using five hydraulic
rock types and 120 hydraulic flow units distributed over an
interval of about 300 ft. The flow units were generated using
a methodology described in References 37 and 38. Water
saturations were distributed vertically using core-derived,
rock-type-specific capillary pressure curves.
We also
measured gas-water relative permeability data for each
hydraulic rock type.
2,500
500
Pseudotime Superposition Function, hr
Numerical Well Test Analysis—Field Example 1. To
demonstrate the validity of our analysis technique, we also
present results of the numerical well test analysis for Field
Example 1. We developed a two-phase (gas-water), threedimensional, multi-layer finite-difference model. To reduce
computational time, we simulated one quarter of the reservoir
with a Cartesian grid system. Moreover, we used a Cartesian
rather than a radial grid system so that we could more
accurately model the linear flow patterns characteristic of
hydraulically fractured wells.
3,000
1,000
History-Matched Results
kg = 0.0068 md
Lf = 221.3 ft
wfkf = 2.96 md-ft
FCD = 1.97
D = 4.0x10-5 (Mscf/d)-1
Fig. 6—Results from automatic history-match of
two-week pressure buildup test, Field Example 1.
Simulated Bottomhole Pressure
1,200
900
Shut in for
Two-Week
Pressure
Buildup Test
600
300
0
5/5/2001
2/5/2002
11/5/2002
8/5/2003
5/5/2004
2/5/2005
Date
Fig. 8—Post-fracture gas production history, Field Example 2.
Initial estimates of permeability and fracture half-length were
obtained from the MBDTC analysis of the production data.
The best match (Fig. 9) was obtained with FCD=5 and reD=5.
A match of the boundary-dominated data indicates the
contacted gas-in-place is 0.52 Bcf corresponding to a drainage
area of 23.7 acres. We also estimate kg=0.0195 md, Lf =114.5
ft, and wfkf=11.2 md-ft. Unlike Field Example 1, the earlytime field derivative data trend appear to be following a single
curve for reD=5, thus allowing more unique estimates of kg and
Lf from the transient data analysis.
Similar to Field Example 1, we analyzed the pressure buildup
data using both pseudopressure and pseudotime superposition
functions since the well exhibited significant changes in flow
rate prior to the two-week shut in period. A log-log plot of the
7
Pseudopressure-Drop Normalized-Rate Functions
pseudopressure change and pseudopressure derivative
functions against the pseudotime superposition function is
shown in Fig. 10. Note that, similar to the first example, we
did not observe either a formation linear flow or a
pseudoradial flow period.
MBDTC Results
kg = 0.0195 md
Lf = 114.5 ft
wfkf = 11.2 md-ft
FCD = 5; reD = 5
G = 0.52 Bcf; A = 23.7 acres
Bilinear Flow
Region
Bilinear Flow Analysis
kg = 0.0195 md (MBDTC)
Lf = 114.5 ft (MBDTC)
wfkf = 7.8 md-ft (Eq. 9)
FCD = 3.5 (Eq. 11)
Fourth-Root of Pseudotime Superposition Function, (hr)1/4
Fig. 11—Fourth-root-of-time plot showing bilinear flow regime
from two-week pressure buildup test, Field Example 2.
Pseudopressure function
Pseudopressure derivative function
Pseudopressure function
Pseudopressure derivative function
Pseudopressure Functions, psia2/cp/MMscfd
Material Balance Pseudotime Function
Fig. 9—Material balance decline type curve analysis
of post-fracture gas production, Field Example 2.
History-Matched Results
kg = 0.0185 md
Lf = 101.1 ft
wfkf = 16.4 md-ft
FCD = 8.77
D = 1.1x10-14 (Mscf/d)-1
Pseudotime Superposition Function, hr
One-quarter slope
line indicative of
bilinear flow
Pseudotime Superposition Function, hr
Fig. 10—Log-log plot of two-week pressure buildup test
identifying fractured-well flow regimes, Field Example 2.
Next, we use special plotting functions to validate various
flow patterns identified from the log-log derivative plot. The
bilinear flow regime is validated by the straight line on a plot
of pseudopressure against the fourth root of pseudotime
superposition function shown in Fig. 11. Using the line slope
from Fig. 11 and kg from the MBDTC analysis, we estimate
wfkf is 7.8 md-ft. Moreover, if we substitute Lf and kg from the
MBDTC analysis and wfkf from Fig. 11 into Eq. 11, we
compute FCD=3.5 which is reasonably consistent with FCD=5
from the initial MBDTC analysis.
Like the first field example, we did not see evidence of
pseudoradial flow on the log-log plot of the pressure
derivative. Consequently, we could not use conventional
semilog techniques to compute permeability, so we again used
an automatic history-matching process to analyze the well test.
The final and best history match for Field Example 2 is shown
in Fig. 12. Note that results from the pressure buildup test
analysis are very similar to those estimated from the initial
material balance decline type curve analysis. We estimate
kg=0.0185 md, while the fracture properties are Lf=101.1 ft,
wfkf=16.4 md-ft, and FCD=8.8.
Fig. 12—Results from automatic history-match of
two-week pressure buildup test, Field Example 2.
Field Example 3
The third example illustrates a small but very effective
conventional slick water-frac stimulation treatment for a well
producing from the Bossier Sands in the East Texas Basin.
The particular well is completed in the Mimms Creek Field
located in Freestone County, TX. The well was hydraulically
fractured in early October 1999 with 8,220 bbl slick water and
37,000 lbs 20/40 sand proppant. As shown by the production
history in Fig. 13, the initial gas rate was almost 12,000
Mscf/day. The well was shut in for a two-week pressure
buildup test after more than two years of production.
12,000
10,000
Gas Production Rate, Mscf/day
Pseudopressure Functions, psia2/cp/MMscfd
Pseudopressure Function, psia2/cp
SPE 97972
8,000
6,000
Shut in for
Two-Week
Pressure
Buildup Test
4,000
2,000
0
10/8/99
10/8/00
10/8/01
10/8/02
10/8/03
10/8/04
Date
Fig. 13—Post-fracture gas production history, Field Example 3.
8
SPE 97972
MBDTC Results
kg = 0.0191 md
Lf = 229.8 ft
wfkf = 43.9 md-ft
FCD = 10; reD = 2
G = 2.1 Bcf; A = 15.2 acres
shown in Fig. 17, the formation linear flow period is very well
defined. Using mL from the line drawn through the formation
linear flow period and kg estimated from the MBDTC analysis,
we compute Lf=239.8. Again, the fracture half-lengths
estimated from the MBDTC and square-root-of-time analyses
are generally in agreement. We also compute a dimensionless
fracture conductivity of 16.9 corresponding to an effective
fracture conductivity of 77.4 md-ft estimated from the bilinear
flow analysis.
Pseudopressure Function, psia2/cp
Pseudopressure-Drop Normalized-Rate Functions
Again, the first step in our procedure was to evaluate the
production history with the MBDTC. The best match (Fig.
14) was obtained with FCD=10 and reD=2. A match of the
boundary-dominated data indicates the contacted gas in place
and drainage area are 2.1 Bcf and 15.2 acres, respectively. We
also estimate kg=0.0191 md, Lf = 229.8 ft, and wfkf=43.9 md-ft.
Note that the early-time field derivative data trend appears to
be following a single curve for reD=2.
Bilinear Flow
Region
Bilinear Flow Analysis
kg = 0.0191 md (MBDTC)
Lf = 229.8 ft (MBDTC)
wfkf = 77.4 md-ft (Eq. 9)
FCD = 17.6 (Eq. 11)
Material Balance Pseudotime Function
Pseudopressure Functions, psia2/cp/MMscfd
The next step in our procedure is the evaluation of the pressure
buildup test. A log-log plot of the pseudopressure change and
pseudopressure derivative functions against the pseudotime
superposition function is shown in Fig. 15. Unlike the first
two field examples, we observe both bilinear and formation
linear flow periods, as indicated by the solid black lines with
slopes of one-quarter and one-half, respectively. We do not,
however, see evidence of pseudoradial flow.
Pseudopressure function
Pseudopressure derivative function
Fourth-Root of Pseudotime Superposition Function, (hr)1/4
Fig. 16—Fourth-root-of-time plot showing bilinear flow regime
from two-week pressure buildup test, Field Example 3.
Pseudopressure Function, psia2/cp
Fig. 14—Material balance decline type curve analysis
of post-fracture gas production, Field Example 3.
Formation
Linear Flow
Region
Formation Linear Flow Analysis
kg = 0.0191 md (MBDTC)
Lf = 239.8 ft (Eq. 10)
wfkf = 77.4 md-ft (Bilinear Flow Analysis)
FCD = 16.9 (Eq. 11)
Square-Root of Pseudotime Superposition Function, (hr)1/2
One-half slope
line indicative
of formation
linear flow
One-quarter slope
line indicative of
bilinear flow
Pseudotime Superposition Function, hr
Fig. 15—Log-log plot of two-week pressure buildup test
identifying fractured-well flow regimes, Field Example 3.
The bilinear and formation linear flow regimes are also
identified and validated by the plots of pseudopressure against
fourth-root and square-root of pseudotime superposition
functions shown in Figs. 16 and 17, respectively. Using mB
from the line drawn through the bilinear flow period in Fig. 16
and kg estimated from the MBDTC analysis, we compute
wfkf=77.4 md-ft. Further, if we use kg and Lf estimates from
the MBDTC analysis of the production data, we estimate
FCD=17.6 which is similar to that estimated from the MBDTC.
As indicated by the straight-line on the plot of pseudopressure
against the square root of pseudotime superposition function
Fig. 17—Square-root-of-time plot showing formation linear flow
regime from two-week pressure buildup test, Field Example 3.
Like the previous two examples, the absence of a pseudoradial
flow period precludes use of conventional semilog analysis
techniques to compute permeability from the well test.
Alternatively, we used an automatic history-matching process
to analyze the well test data. The best history match is shown
in Fig. 18. We estimate kg and Lf are 0.0257 md and 261.1 ft,
respectively. In addition, these results indicate the stimulation
treatment generated a much more conductive fracture than
Field Examples 1 and 2. We compute a fracture conductivity
of 131.5 md-ft corresponding to a dimensionless fracture
conductivity of 19.6.
Field Example 4
The fourth field example is a another well completed in the
Bossier Sands in the Mimms Creek Field in Freestone County,
TX. The well was hydraulically fractured in early April 2001
with 8,571 bbl slick water and 170,000 lbs 40/70 sand
proppant. The well was shut in for a two-week pressure
buildup test after about 18 months of production (Fig. 19).
SPE 97972
9
Pseudopressure Functions, psia2/cp/MMscfd
Pseudopressure function
Pseudopressure derivative function
History-Matched Results
kg = 0.0257md
Lf = 261.1 ft
wfkf = 131.5 md-ft
FCD = 19.6
D = 8.6x10-20 (Mscf/d)-1
Pseudotime Superposition Function, hr
Fig. 18—Results from automatic history-match of
two-week pressure buildup test, Field Example 3.
4,000
3,000
One-quarter slope
line indicative of
bilinear flow
One-half slope
line indicative
of formation
linear flow
Pseudotime Superposition Function, hr
The bilinear and formation linear flow regimes are also
validated by the plots of pseudopressure against fourth-root
and square-root of pseudotime superposition functions shown
in Figs. 22 and 23, respectively. Using mB from the line
drawn through the bilinear flow period in Fig. 22 and kg
estimated from the MBDTC analysis, we compute wfkf=27.7
md-ft. Further, if we use kg and Lf estimates from the MBDTC
analysis of the production data, we estimate FCD=13.8.
Shut in for
Two-Week
Pressure
Buildup Test
2,500
2,000
1,500
1,000
500
0
8-Apr-01
Pseudopressure function
Pseudopressure derivative function
Fig. 21—Log-log plot of two-week pressure buildup test
identifying fractured-well flow regimes, Field Example 4.
8-Jan-02
8-Oct-02
8-Jul-03
8-Apr-04
8-Jan-05
Date
Fig. 19—Post-fracture gas production history, Field Example 4.
MBDTC Results
kg = 0.0070 md
Lf = 286.3 ft
wfkf = 20.4 md-ft
FCD = 10; reD = 2
G = 2.3 Bcf; A = 23.7 acres
Material Balance Pseudotime Function
Fig. 20—Material balance decline type curve analysis
of post-fracture gas production, Field Example 4.
A log-log plot of the pseudopressure change and
pseudopressure derivative functions against the pseudotime
superposition function is shown in Fig. 21. Similar to Field
Example 3, we observe both bilinear and formation linear flow
Pseudopressure Function, psia2/cp
Gas Production Rate, Mscf/d
3,500
Pseudopressure-Drop Normalized-Rate Functions
periods, as indicated by the solid black lines with slopes of
one-quarter and one-half, respectively. Again, we do not see
any indications of pseudoradial flow developing during the
two-week pressure buildup test.
Pseudopressure Functions, psia2/cp/MMscfd
The MBDTC analysis of the production history is shown in
Fig. 20. The best match was obtained with FCD=10 and reD=2.
A match of the boundary-dominated data indicates the
contacted gas-in-place and drainage area are 2.3 Bcf and 23.7
acres, respectively. We also estimate kg=0.0070 md, Lf =286.3
ft, and wfkf=20.4 md-ft. Although less certain than Field
Example 3, the early-time field derivative data trend again
appears to be following a single curve for reD=2.
Bilinear Flow
Region
Bilinear Flow Analysis
kg = 0.0070 md (MBDTC)
Lf = 286.3 ft (MBDTC)
wfkf = 27.7 md-ft (Eq. 9)
FCD = 13.8 (Eq. 11)
Fourth-Root of Pseudotime Superposition Function, (hr)1/4
Fig. 22—Fourth-root-of-time plot showing bilinear flow regime
from two-week pressure buildup test, Field Example 4.
As indicated by the straight-line on the plot of pseudopressure
against the square root of pseudotime superposition function
shown in Fig. 23, the formation linear flow period is very well
defined. Using mL from the line drawn through the formation
linear flow period and kg estimated from the MBDTC analysis,
we compute Lf=238.3. Again, the fracture half-lengths
estimated from the MBDTC and square-root-of-time analyses
are generally in agreement. We also compute dimensionless
fracture conductivity of 16.6 corresponding to an effective
fracture conductivity of 27.7 md-ft estimated from the bilinear
flow analysis.
The best history match of the well test data for Field Example
4 is shown in Fig. 24. We estimate kg and Lf are 0.0082 md
and 282.8 ft, respectively. Similar to Field Example 3, these
10
SPE 97972
Pseudopressure Function, psia2/cp
results indicate the stimulation treatment generated a much
more conductive fracture than Field Examples 1 and 2. We
compute a fracture conductivity of 44.3 md-ft corresponding
to a dimensionless fracture conductivity of 19.1. Note again
that these results are not too different from the results obtained
from the MBDTC analysis of the production data.
Formation
Linear Flow
Region
Formation Linear Flow Analysis
kg = 0.0070 md (MBDTC)
Lf = 238.3 ft (Eq. 10)
wfkf = 27.7 md-ft (Bilinear Flow Analysis)
FCD = 16.6 (Eq. 11)
Square-Root of Pseudotime Superposition Function, (hr)1/2
Pseudopressure Functions, psia2/cp/MMscfd
Fig. 23—Square-root-of-time plot showing formation linear flow
regime from two-week pressure buildup test, Field Example 4.
Pseudopressure function
Pseudopressure derivative function
History-Matched Results
kg = 0.0082 md
Lf = 282.8 ft
wfkf = 44.3 md-ft
FCD = 19.1
D = 4.8x10-8 (Mscf/d)-1
Pseudotime Superposition Function, hr
Fig. 24—Results from automatic history-match of
two-week pressure buildup test, Field Example 4.
Summary and Conclusions
We have developed an integrated approach for evaluating the
post-fracture production performance of gas wells producing
from tight gas sands. Although we focus on wells with finiteconductivity vertical fractures, the methodology is also valid
for any vertically fractured well case (i.e., finite-conductivity,
infinite-conductivity or uniform-flux fractures). We have
validated our technique with a simulated case, and we have
illustrated the applicability and utility of our integrated
technique with several field examples. On the basis of the
results, we offer the following conclusions:
1. The material balance decline curves developed for wells
with finite-conductivity vertical fractures are generally
very useful for evaluating the production performance in
hydraulically-fractured gas wells. Agreement between
the MBDTC and pressure buildup test analysis in our
evaluations ranges from good to excellent.
2. The accuracy of these material balance type curves for
evaluating stimulation effectiveness depends on the
quality and quantity of production data—particularly the
early-time transient data.
Therefore, we strongly
recommend that operators strive to improve their data
acquisition and gathering efforts.
Moreover, we
recommend the use of daily production for the
production decline type curve analysis.
3. Although most production decline type curve methods
are valuable tools for evaluating well production
performance, no "history analysis" approach can
completely replace conventional pressure transient
testing. The value of pressure transient tests for
establishing current flow capacity and flow efficiency
can not be overstated.
4. Production data quality and/or quantity may preclude
accurate evaluation of production data. The material
balance decline type curve approach used in this work is
both robust and error tolerant, and should be expected to
perform well in practice. We believe that production
data analysis can (and should) be able to "stand alone" in
the absence of pressure transient tests.
5. The results of our study also demonstrate the value and
function of short-term pressure buildup testing in tight
gas sands. These pressure transient tests are quite useful
when integrated with production data analysis with
decline type curves—particularly for interpretation of
early-time transient flow behavior. Therefore, we
recommend that operators incorporate short-term
pressure transient testing with production data analysis
to evaluate the stimulation effectiveness of wells
producing from tight gas sands.
Acknowledgements
We would like to express our thanks to Anadarko Petroleum
Corp. for permission to use the data and to publish the results
of our study.
Nomenclature
Dimensionless Variables
bDpss = dimensionless pseudosteady-state constant
FCD = dimensionless fracture conductivity = wfkf/kgLf
qDd = dimensionless pseudopressure-drop normalized
function
qDdi = dimensionless pseudopressure-drop normalized
integral function
qDdid = dimensionless pseudopressure-drop normalized
integral-derivative function
reD = dimensionless reservoir radius = re/Lf
rate
rate
rate
t D = dimensionless material balance pseudotime function
Field Variables
A
= well drainage area, acres
Bgi = gas formation volume factor evaluated at initial reservoir
pressure, RB/Mscf
cg
= gas compressibility, psia-1
cgi
= gas compressibility at initial reservoir pressure, psia-1
ct
= total system compressibility, psia-1
cti
= total system compressibility at initial reservoir pressure,
psia-1
G
= contacted gas-in-place, Mscf
h
= net sand thickness, ft
kf
= fracture permeability, md
kg
= effective gas reservoir permeability, md
Lf
= effective or propped fracture half-length, ft
SPE 97972
mB
mL
ppi
ppwf
qg
re
rw
Swi
T
ta
wf
w f kf
φ
µg
µgi
= slope of line drawn through bilinear flow period on
fourth-root of time plot
= slope of line drawn through formation linear flow period
on square-root of time plot
= normalized pseudopressure function evaluated at initial
reservoir pressure, psia
= normalized pseudopressure function evaluated at
bottomhole flowing pressure, psia
= gas flow rate, Mscf/day
= reservoir radius, ft
= wellbore radius, ft
= initial connate water saturation, fraction
= bottomhole reservoir temperature, oR
= normalized material balance pseudotime function, hr
= fracture width, in
= effective fracture conductivity, md-ft
= effective porosity, fraction
= gas viscosity, cp
= gas viscosity at initial reservoir pressure, cp
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Conference and Exhibition, New Orleans, LA, Sept. 30-Oct. 3.
38. Rushing, J.A. and Newsham, K.E.: “An Integrated Work-Flow
Model to Characterize Unconventional Gas Resources: Part
II-Formation Evaluation and Reservoir Modeling,” paper SPE
71352 presented at the 2001 SPE Annual Technical Conference
and Exhibition, New Orleans, LA, Sept. 30-Oct. 3.
SPE 97972