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SPE 160169
Reappraisal of the G Time Concept in Mini-Frac Analysis R.C. Bachman, Taurus Reservoir Solutions Ltd., D.A. Walters, Taurus Reservoir Solutions Ltd., R.A. Hawkes, Pure Energy Services Ltd., Fabrice Toussaint, Dinova Petroleum Ltd., A. Settari, University of Calgary
Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 8-10 October 2012. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Socie ty of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract Industry is currently using mini-frac analysis for the determination of fracture closure stress and after-closure reservoir
properties. The foundation of all mini-frac analysis is the one dimensional Carter leak-off model, which leads directly to the
concept of G Time. For 30+ years, G Time (or the G function) has played the dominant role for the determination of closure
stress. The current norm uses combination G function and combination square root t plots for closure pressure
determination. Each combination plot has three plotting functions associated with it. These combination plots also allow the
identification of non-ideal behavior. Additionally, various log-log derivative techniques based on pressure transient analysis
concepts have been developed to act as a guide for determining flow regimes and closure pressure. These PTA based
techniques also allow the determination of after-closure flow regimes and properties. Concurrently, various specialized after-
closure plotting techniques have been developed for fracture/reservoir property determination.
Despite all these techniques, there remains ambiguity in performing mini-frac analysis. Part of the problem is that the
recommended plots do not rigorously identify the various flow regimes that occur during a mini-frac fall-off. Mini-frac
analysis requires a general theory that accounts for all of the actual observed flow regimes. A systematic approach based on
pressure transient analysis (PTA) concepts has been developed to identify the various flow regimes (Carter leak-off being
only one of them). The starting point is the Bourdet log-log derivative plot, accompanied by the primary pressure derivative
(PPD) function. It will be shown that the PPD on its own has independent flow regime identification capabilities. Once
specific flow regimes have been identified, specialized log-log plots can be constructed for further flow regime verification.
New combination plots are then developed for each flow regime to further assist in closure pressure determination. The
theory will first be developed and illustrated with various example problems.
Introduction Nolte (1979) introduced the first rigorous technique for determining closure pressure using the Carter leak-off assumption
coupled with material balance within the fracture. This led to a special time function called G Time. Nolte (1986) further
extended this work to account for different fracture geometries. These analysis techniques were the beginning of what is now
referred to as before-closure analysis. Closure pressure was determined by linear plots of p versus G. Deviation from straight
line behavior indicated the closure pressure. Practical difficulties of where to draw the straight line often occurred. This is the
analogous situation that occurred in welltest PTA for determining the correct straight line on the Horner plot. For PTA the
situation was resolved by Bourdet et al. (1983) with the Bourdet log-log pressure derivative plot. This plot allowed for flow
regime identification, reservoir properties determination and defining the time range over which straight lines could be drawn
on other specialized plots to complete the analysis.
Returning to mini-frac analysis, Mukherjee et al. (1991) and Barree and Mukherjee (1996) added special derivative plots to
more accurately pick deviations from straight line behavior on the p versus G plot. Barree and Mukherjee (1996)
recommended that three functions p , dp/dG and Gdp/dG be plotted against G. This is the combination G function plot, which
is the basis of all modern interpretation methods. Implicit in this methodology was that G Time with the associated Carter
leak-off assumption was the dominant flow regime, although allowances for pressure dependent leak-off (PDL) were
accounted for. Further deviations from non-ideal behavior were addressed in Barree et al. (2007) where the additional cases
of fracture tip extension and height recession or transverse storage were identified and analyzed. They also developed
combination square root t plots as well as a log-log derivative plot which while bearing similarities to the Bourdet log-log
2 SPE 160169
derivative plot has a different time function. The Barree et al (2007) approach represents the current state of the art
interpretation methodology applied across the industry.
Leschyshyn et al. (1996) presented an early attempt at using log-log pressure derivative techniques for before-closure
analysis. They analyzed a near wellbore shear fracture zone for oilsands applications. The most recent investigations of mini-
frac analysis using the PTA based Bourdet log-log derivative are by Mohamed et al. (2011) and Marongiu-Porcu et al.
(2011). Mohamed et al (2011) have identified a characteristic 3/2 slope on the derivative plot when late time Carter flow is
evident.
One aspect of welltest PTA analysis that has not been applied to mini-fracturing analysis is Mattar and Zaoral’s (1992) PPD
concept. The premise behind the PPD is that due to the nature of the diffusivity equation, transients resulting from shutting in
a well should result in a decreasing pressure change with time. If the PPD increases at some point in time, it can be attributed
to wellbore effects as opposed to reservoir effects. The Bourdet log-log derivative during this portion of the test, which often
looks quite normal, must be ignored. The application of the PPD concept to standard PTA is apparent because the reservoir
fabric is not changing with time. With mini-fracturing, if the fracture snaps shuts sufficiently quickly, the previous statements
may not hold and the PPD may increase at a later point in time because of the reservoir/fracture interaction. For a proper
interpretation of mini-frac tests it is critically important to monitor the PPD. It will be shown that the PPD also independently
allows flow regime identification.
The emphasis on before-closure analysis has been dominated by the G Time concept, which in turn implies Carter leak-off.
There are three other fracture flow regimes that have been identified when static propped fractures exist. It will be
demonstrated that some of these also occur during mini-fracturing during the before-closure period. In fact in a number of
cases more than one fracture flow regime occurs during the before-closure period. This complicates the interpretation process
when analysis techniques are based on Carter leak-off. The first step in an analysis is to identify the flow regimes and based
upon what is observed, utilize appropriate secondary plots to determine closure pressure.
Flow Regime Identification Within the well testing community the current paradigm for PTA is to first identify flow regimes using the Bourdet log-log
derivative plot and then use secondary specialized plots to complete the analysis. In contrast, for before-closure mini-frac
interpretation equal weight is given to various combination plots (square root t and G Function) and the special DT (or
Delta Time) log-log derivative plot. The DT log-log derivative function, defined as tdP/dt, is significantly different than
the Bourdet log-log derivative plot used in PTA.
The Bourdet derivative function used in the PTA based log-log derivative approach was based on work by Agarwal (1980)
and accounts for rate variation prior to the analyzed shut-in period. This time function is known as Agarwal equivalent time.
The purpose of this superposition function is to remove the as yet unknown initial pressure from the resulting equations and
replace it with the flowing pressure immediately at shut-in. It requires an assumption with regards to the flow regime. As a
result there is a different function for computing radial, linear or bilinear Agarwal equivalent time (ter, tel or teb respectively).
In the majority of commercial well test software radial flow is assumed. Therefore the default Bourdet log-log derivative is
defined as terdP/dter. The great power of the Bourdet log-log derivative technique is that PTA interpretation and
identification of flow regimes is not strongly affected by the type of flow regime specific generating function. Whether the
well is in bounded flow or any of the other non-radial flow regimes, only a minor distortion of results occurs. Once a specific
flow regime has been identified, a change in the Bourdet log-log derivative to that specific flow regime is possible. In PTA
this is rarely done. In the interpretation workflow of this paper, changing of the Agarwal generating function to reflect the
identified flow regime will be an integral part of our proposed mini-frac interpretation technique.
A brief review of the various fracture flow regimes will begin with a discussion of Carter leak-off. It is a special type of
linear one dimensional flow into the formation normal to the fracture plane. During pumping in a static fracture formation
linear flow occurs. The flow velocity u into the formation from the static fracture is
tCu LO / (1)
When a dynamic fracture occurs, during propagation the initiation time for linear flow at a given point in the fracture is a
function of t0, the time at which the fracture reaches the point in question. For this case
0/ ttCu LO (2)
SPE 160169 3
This is the Carter leak-off assumption. The concept is illustrated in Figure 1. Nolte (1979) made the further assumption that
during pumping the fracture length L increases linearly with time (the so called low leak-off case). Once pumping stops the
fracture is assumed to stay at the maximum length until closure. Combining this specific Carter leak-off assumption and
material balance for a closing fracture, pressure is related to G Time as follows:
)()0( 1 dtGCpDTpPDP (3)
1)1(3
16)( 5.15.1 ddd tttG
(4)
where C1 is a constant including the appropriate reservoir and geomechanical properties. Alternatively if one multiplies
through by tp1.5
one gets
5.15.15.1)( ppc ttttCP (5)
where Cc is the appropriately adjusted Carter leak-off constant.
For a static fracture, Cinco-Ley et al. (1978) has identified three types of fracture flow regimes which occur during
production or injection. These are:
1. fracture linear flow
2. bilinear flow
3. formation linear flow
These flow regimes are illustrated in Figure 2. They may also be observed in mini-frac treatments before or after-closure
times. Fracture linear flow occurs at very early times and lasts at most a few minutes. It is not seen in conventional well tests
because its effect is masked by wellbore storage. In mini-fracs, the effective wellbore volume may be small and for these
cases wellbore storage effects are minimal. As a result, observing fracture linear flow is possible. Bilinear flow is observed
when linear flow occurs simultaneously in the fracture and perpendicular to the formation. Bilinear flow requires a significant
pressure drop in the fracture. It is associated with a finite conductivity static fracture. Alternatively, in the after-closure period
of a mini-frac, residual fracture conductivity may exist in the now closed un-propped frac. Formation linear flow is normally
associated with an infinite conductivity static fracture. It is mutually exclusive to the finite conductivity frac and is commonly
seen in mini-frac after-closure analysis.
For a single injection period with a subsequent fall-off the PTA solutions for the three static fracture cases can be expressed
in terms of either pi or P as follows:
sffi
nn
pfi tCptttCpp )( (6)
eff
n
p
nn
pf tCttttCP )( (7)
The second part of Equation 6 is in terms of the flow regime specific superposition time function originally determined by
Odeh and Jones (1965) for radial flow, but subsequently generalized for all flow regimes. Similarly the second part of
Equation 7 is in terms of the flow regime specific equivalent time function. For fracture and formation linear flow n=0.5 and
for bilinear flow n=0.25. For cases where there are variable rates throughout the test generalized superposition and equivalent
time functions are used for tsf and tef in the second part of Equation 6 and 7. Comparing the Carter leak-off expression
Equation 5 to Equation 7, and seeing how Equation 6 is simply a reformulation of Equation 7 it is hypothesized that:
scfipci tCptttCpp 5.15.1)( (8)
eccppc tCttttCP 5.15.15.1)( (9)
which means Equations 6 and 7 are very general and include Nolte’s G Function concept. It also extends Nolte analysis to the
variable rate case. Generalized superposition and equivalent time functions are easily computed based on concepts developed
for linear flow. The applicability of using superposition principles for Nolte’s Carter leak-off case initially appears tenuous.
Subsequent injection cycles, or varying the injection rate within a cycle would appear to invalidate the repeatability of the
flow regime due to the time dependency of the fracture length during pumping. Field data in Example 1 shows that
superposition is valid for Nolte’s Carter leak-off case. As a result unification of mini-frac analysis with PTA has been
achieved, and mini-frac analysis should no longer be viewed as an independent discipline.
4 SPE 160169
A comparison is made of the DT log-log derivative plots currently used in mini-frac interpretation and the Bourdet log-log
derivative plot used in conventional PTA. The first thing to note is that there is no superposition built into the DT log-log
derivative function. Nolte’s Carter leak-off case will be examined first. By definition - for all times the fracture is open. For
simplicity a pumping time of 1 day is assumed. While an unreasonable value, in no way does it detract from the solution. The
advantage is that numerical values of t (which is measured in days) are equivalent to tD in all plots.
Figure 3 shows the DT log-log derivative plot for Carter leak-off (the fracture is open for all times). There are two issues with
this plot that make it non-ideal for identifying the Carter leak-off flow regime. First, there is a different asymptotic slope at
early time (unit slope) and late time (½ slope). In practice with real data; how would the analyst distinguish a single flow
regime which has a changing slope versus a transition to another flow regime? Second, the late time asymptotic slope is ½.
The PTA community is used to ½ slopes being associated with linear flow. For this case the ½ slope represents an entirely
different flow regime (late time Carter leak-off).
Figure 4 shows the corresponding Bourdet log-log derivative plot. The early time asymptotic slope is unit slope and the late
time value is 3/2 as has been reported by Mohamed et al. (2011). There is no ½ slope evident in this plot. It can be concluded
that Carter leak-off never looks like linear flow. Early time unit slope is expected considering that Nolte’s solution accounts
for fracture storage effects. Slopes greater than 1 are not common in traditional PTA analysis, so a preliminary pick of the
late time Carter leak-off flow regime is straight forward. This is the first flow regime the authors are aware of where the early
and late time Bourdet log-log derivative slopes are not identical.
For interpretation it is strongly recommended that the PPD function introduced by Mattar and Zaoral (1992) should also be
included on the Bourdet log-log derivative plot. It is noted that the previously mentioned DT log-log derivative function
tdP/dt is equivalent to t*PPD. The PPD has no superposition effects built into its calculation, but suffers from early and
late time slope differences within a single flow regime. The PPD function therefore has practical independent diagnostic
capabilities. For Carter leak-off the early time PPD slope on a log-log plot is 0 and the late time slope is -½. The significance
of this will be demonstrated in the example problems.
To remove the objection of changing slopes during one flow regime, an alternative derivative plot is proposed. Take the
derivative of Equation 7 with respect to tec , which is the derivative with respect to Carter equivalent time, and then multiply
by t . For all times for which Carter leak-off is appropriate the log-log slope is unity. Alternatively one could have chosen to
multiply by t1.5
in which case the log-log slope would be 3/2 for all times. The idea of taking the derivative with respect to a
specific flow regime time function and multiplying by another time function so that a desired slope is obtained on a particular
plot will be a recurring theme in this paper. Figure 5 shows the resulting plot for Carter leak-off. Deviation from the unit
slope line for real data will indicate that this flow regime is no longer appropriate. This can be cross checked by concurrently
checking the PPD curve.
For the linear flow regime the log-log derivative plots are given in Figure 6, Figure 7 and Figure 8. Figure 6 shows the DT
log-log derivative plot. At early time the asymptotic slope is ½ while at late time it is -½. The time to the establishment of
late time -½ slope should correspond to when the Soliman et al. (2005) late time after-closure analysis technique would work.
Unfortunately the early time linear flow asymptotic ½ slope is the same as the late time asymptotic DT log-log derivative
Carter leak-off slope. This has created significant confusion within the industry; what flow regime is identified by ½ slope on
the DT log-log derivative plot? We recommend that industry discontinue using the DT log-log derivative plot.
The inherent advantages of the Bourdet log-log derivative plot are apparent from Figure 7. Even though radial flow was used
for the base equivalent time function, early and late time asymptotic ½ slopes are evident. Only minor distortion occurs at
middle time. The robustness of the Bourdet log-log derivative plot is one reason for it being one of the outstanding
achievements in the field of reservoir engineering over the last 30 years. Once linear flow is identified, a new time derivative
function defined as t0.5
dP /dtel could be used. It has a ½ slope for all times when in linear flow. This is shown in Figure 8.
Analogous plots could also be constructed for the bilinear flow case.
Combination Plots For Carter leak-off, the combination G function plot is shown in Figure 9. Since there is only one flow regime, the various
curves show the expected straight line behavior. Figure 10 shows the corresponding combination square root t plot. Only at
late time does one get straight line behavior. The plot is completely unreliable in picking any flow regime deviation from
Carter leak-off.
A new combination plot is proposed. A highly desirable feature of an alternative combination plot would be that the x-axis be
linear in t instead of a transformed time function such as G Time or square root t. This is beneficial in the case where the
t range of a specific flow regime has been identified on another plot, i.e. the Bourdet log-log derivative plot. A straight
SPE 160169 5
forward comparison then is possible between various plots at the same t range. An additional requirement of such a plot is
that it exhibit straight line behavior for critical curves in a manner analogous to the combination G function plot.
The new combination plots will be flow regime dependent. For an arbitrary but fixed flow regime, p can be differentiated
with respect to tsf..(in Equation 6). tsf represents superposition time with respect to the identified flow regime. dp/dtsf = -Cf is a
constant when the appropriate flow regime occurs. Then one takes the absolute value of dp/dtsf. This term is analogous to
dp/dG. Then one multiplies the absolute value of dp/dtsf by t . This term is proportional to t as long as that flow regime is
appropriate. This term is analogous to Gdp/dG . These two new functions are plotted along with p as a function of t. For
Carter leak-off, the plot is called the combination superposition Carter plot. It is shown in Figure 11. The only disadvantage
of this plot versus the combination G function plot is that the p curve is not a straight line. Since the emphasis is on the
derivative curves the plot is acceptable. This new combination plot is not restricted to mini-frac analysis and is completely
general with respect to any PTA determined flow regime. One advantage they have is that one can pick deviations from a
given flow regime on linear plots, as opposed to log-log derivative plots. Therefore the resolution of the pick is improved.
Experience indicates that the absolute value of dp/dtsf function should also be plotted as a third curve on the previously
discussed flow regime specific equivalent time log-log derivative plot. This curve is analogous to the recommended PPD
curve on the Bourdet log-log derivative plot. This will be shown in the Examples section.
For linear flow, the combination G function plot, the combination square root t plot and the combination superposition
linear plot are shown in Figure 12, Figure 13 and Figure 14 respectively. Neither the combination G function plot nor the
square plot give meaningful results. Only the combination superposition linear plot in Figure 14 gives meaningful results.
Prior to generating the combination superposition linear plot it is important to have identified the linear flow t time range on
the log-log derivative plot.
A review of Figure 10 and Figure 13 would lead to the conclusion that square root t plots should never work. Experience
has shown that the combination square root t plot can give reasonable closure pressures for certain problems. Barree et al
(2007) are cautious in recommending the plot unless it corroborates picks from other plots. Fundamentally the question is:
what flow regimes are contained within the square root t function? Figure 15 shows the log-log plot of three functions;
square root t , G and tel versus tD. When tD is below 0.03 square root t is equivalent to tel . When tD is greater than 10
the slope of square root t on log-log co-ordinates is equivalent to the slope of G. Between these values square root t
function is transitioning from an apparent early time linear flow to late time Carter leak-off. In practice closure rarely occurs
before tD = 1. Therefore the square root t function has a transitional feature that can look like Carter leak-off at large tD.
The transition along with its slope change through this time interval makes its interpretative capabilities unreliable on its
own. The square root t does not carry any information that is not available from other plots. Despite it being complex to
interpret, industry is highly unlikely to give up on this plot.
Analysis Workflow When performing a mini-frac analysis a systematic workflow is required. One should always start with the Bourdet log-log
derivative plot with the accompanying PPD curve. Flow regimes are then picked, and specialized log-log derivative plots are
constructed, followed by the appropriate combination plots. The work flow and the required plotting variables for each plot
are shown in Figure 16.
Closure stress is picked by the following hierarchal procedure:
1. Any PPD increase or jump may indicate either rapid closure (snapping shut) or wellbore effects. Interpretation must
be made on a case by case basis.
2. If Carter leak-off is present, the end of the Carter leak-off flow regime indicates closure.
3. If Carter leak-off does not end then the fracture did not close.
4. If radial, linear or bilinear flow regimes occur after Carter leak-off ends, these are after-closure flow periods. For
these after-closure flow periods, traditional after-closure analysis techniques can be used for property determination.
Alternatively, PTA techniques based on drawing the appropriate straight lines are acceptable.
5. If Carter leak-off is not seen, the end of any linear flow regime will be considered a closure event. Subsequent well
defined flow regimes will be considered after-closure flow periods.
6. If linear flow does not end (similarly for bilinear flow) then the fracture did not close.
7. Picking a pressure or time range where closure may occur is acceptable.
8. Picking multiple closure events may be possible in exceptional cases.
9. If no Carter leak-off or linear flow is observed, then the test cannot be interpreted.
Table 1 and Table 2 have been developed to aid the analyst in determining the various slopes for each recommended curve
for each flow period and time range.
6 SPE 160169
Log-Log Flow Regime
Carter Linear Bilinear
Equivalent Time
Derivative
Function
Early
Time
Slope
-------
Ends
tD=0.08
Late
Time
Slope-
------
Starts
tD=5.0
Early
Time
Slope
-------
Ends
tD=0.04
Late
Time
Slope
-------
Starts
tD=4.0
Early
Time
Slope
-------
Ends
tD=0.13
Late
Time
Slope
-------
Starts
tD=8.0
terdP /dter
(Radial)
1/1 3/2 1/2 1/2 1/4 1/4
tdP /dtec
(Carter)
1/1 1/1 1/2 0 1/4 -1/4
t0.5
dP /dtel
(Linear)
1/1 3/2 1/2 1/2 1/4 1/4
t0.25
dP /dteb
(Bilinear)
1/1 3/2 1/2 1/2 1/4 1/4
Table 1: Specialized Equivalent Time Plotting Functions and their Slopes
Log-Log Flow Regime
Radial Carter Linear Bilinear
Derivative
Function
Early
Time
Slope
Late
Time
Slope
Early
Time
Slope
Late
Time
Slope
Early
Time
Slope
Late
Time
Slope
Early
Time
Slope
Late
Time
Slope
tdP /dt 0 -1/1 1/1 1/2 1/2 -1/2 1/4 -3/4
PPD = dP /dt -1/1 -2/1 0 -1/2 -1/2 -3/2 -3/4 -7/4
Table 2: DT Derivative and PPD Slopes for Various Flow Regimes
Example 1 - Superposition Effects The purpose of this example is to show that superposition of Carter leak-off is a reasonable assumption based upon field
evidence. In normal mini-frac applications very few tests are run over multiple cycles within the same zone. An exception is
in the shallow Athabasca oilsands in north-eastern Alberta. Measured caprock closure pressure directly affects the regulated
maximum operating pressure of these shallow thermal projects. Repeatability of tests is a major concern. As a result, multi-
cycle mini-fracs are routinely conducted on all zones.
The test in this example used a MDT tool in an open hole wellbore over a shale interval. The MDT tool injects relatively
small volumes of fluid during each cycle. The tool and the details of its operation are described in Mishra et al (2011).
Experience has shown that in the Athabasca oilsands minimum effective stress gradients are often equal to the overburden
gradient; implying that a horizontal fracture is being generated. Operationally it is desirable that an initial vertical fracture
exists. To ensure this a ‘sleeve frac’ is typically performed prior to the main test. This involves setting a packer over the
interval to be tested and inflating it to high pressures. High hoop stresses are generated in the near wellbore area and a
vertical fracture is mechanically created. Prior to injecting above the estimated fracture gradient, a number of low rate
injection/fall-off cycles are run. The pressure is carefully monitored to ensure that pressures do not exceed fracturing
pressure. Fluid enters the pre-existing partially open vertical fracture. Figure 17 shows the rate normalized derivative for
three successive flow periods. It is clear that the late time behavior of these tests is repeatable and that a transition from an
early flow period within the shale/pre-existing vertical fracture into a storage flow period is occurring. Storage is the first part
SPE 160169 7
of Carter leak-off. It is concluded that Carter leak-off is being re-initiated for each injection cycle and that full rate
superposition is valid.
Example 2 – Tight Oil This example was chosen to compare traditional interpretation techniques to the new approach for a well having classic
behavior. The test is from a low permeability oil well in the Upper Devonian Redknife formation, which occurs throughout a
widespread area of northwestern Alberta, northeastern British Columbia, and the southern Northwest Territories shale plays.
This is a predominantly limestone unit with minor amount of dolomite. The mini-frac is into the toe stage of a multi-stage
frac port horizontal well. Pumping rate during the test was 0.65 m3/min of oil with a total of 3.25 m3 of oil being injected
during the test. Pressure was measured from subsurface gauges in the vertical section of the well.
Figure 18 shows the pressure versus time plot. Wellbore friction effects are apparent immediately after shut-in. Figure 19,
Figure 20 and Figure 21 show the traditional combination G function, combination square root t and the DT log-log
derivative plot respectively. A review of the combination G function plot indicates that the well has pressure dependent leak-
off (PDL) according to Barree et al. (2007). The DT log-log derivative plot shows two ½ slopes followed by a -½ slope. All
three plots give the same closure pressure.
The new technique starts with the Bourdet log-log derivative plot shown in Figure 22. The Bourdet log-log derivative curve
has an early ½ slope flow period which is interpreted as being fracture linear flow. Continuing along this curve there is a
transition into late time Carter leak-off (3/2 slope) after which fracture closure occurs. The final flow regime is a second linear
flow period, which is an after-closure flow regime. The associated PPD curve indicates smooth downward behavior. The
associated slopes for the identified flow regimes are labeled on the figure. There is consistency for the flow regime picks
between the PPD curve and the Bourdet log-log derivative curve. Since PPD function does not rely on superposition as does
the Bourdet log-log derivative, it is an independent means for flow regime verification. Taken as a whole, Figure 22 provides
the analyst with significantly more information than the conventional DT log-log derivative plot of Figure 21. The DT log-
log derivative has ambiguity as to the meaning of the two ½ slopes.
Once Carter leak-off has been identified; the Equivalent Carter log-log derivative plot is constructed as shown in Figure 23.
In this plot the entire Carter flow regime (early and late time) would appear as a unit slope line on the tdP /dtec curve. The
dp/dtsc curve shows when Carter flow ends and also verifies the late after-closure linear flow regime (late time linear flow has
a zero slope on this curve as shown in Table 1). Finally, the combination superposition Carter plot is given in Figure 24. The
practice of drawing lines through the origin, which is appropriate when there is one flow regime, is not justified. The time
range of Carter flow should be identified and a straight line is drawn through the appropriate points. The result is a non-zero
y intercept. In practice this would rarely change the closure pick from the traditional procedure of going through the origin.
In this example, the closure picks are identical for both interpretation methodologies. The proposed Bourdet log-log
derivative plot with the accompanying PPD curve is shown to be very useful for both before- and after-closure flow regime
identification. The one interpretation difference is that the Bourdet log-log derivative plot does not show evidence of PDL
(an issue to be explored in the next example).
Example 3 – Tight Gas Well 1 This gas well is completed in the Montney formation in British Columbia Canada. The Montney formation is areally
extensive and is composed of siltstones interbedded with shales. The mini-frac was conducted in the toe stage of a 1800
meter long lateral, multi-stage frac-port horizontal well at a depth of 2439.8 m CF TVD. Pumping rate during the test was
0.48 m3/min of water. A total of 10.0 m3 of water was injected over approximately 0.016 days (23 minutes). Pressure was
measured at the surface and converted to bottomhole depth using a fresh water gradient.
During pumping BHP was between 84,000 kPa and 88,000 kPa, values significantly above the vertical stress gradient.
Clearly there are wellbore related effects during pumping which are not accounted for. Figure 25 and Figure 26 show early
time BHP versus time after shut-in. Figure 25 shows that BHP drops below the vertical stress value at 0.002 days (2.9
minutes). There is a more rapid drop in pressure at DT=0.03 days in Figure 26. This corresponds to a PPD increase and is
indicative of a wellbore event or closure. Figure 27 and Figure 28 show the late and middle time combination G Function
plots respectively. An earlier time combination G Function plot (not shown) displayed similar character but would result in a
pfoc greater than the vertical stress gradient. The late time combination G Function in Figure 27, if interpreted as shown,
would be an indication of height recession or transverse storage according to current practice, giving pfoc=46,900 kPa. The
middle time combination G Function plot in Figure 28 would also indicate the same behavior, although a different closure
pressure would be picked (pfoc=49,510 kPa). Figure 29 shows the middle time Combination square root t plot which
indicates the same closure pressure (pfoc=49,510 kPa) as the middle time combination G Function plot. The DT log-log
derivative is shown in Figure 30. If one were to pick closure based upon a roll-over of the derivative curve, one might pick a
8 SPE 160169
closure time at DT=16.6 days (pfoc=46,900 kPa). This closure pressure would be consistent with the late time combination G
Function plot in Figure 27. Alternatively one could pick the end of ½ slope on the derivative curve at DT=0.9 days
(pfoc=49,290 kPa).
The Bourdet log-log derivative plot with the PPD is shown in Figure 31. It shows some complex behavior. Early time Carter
leak-off (ending at 0.004 days or 5.8 minutes) seems to be indicated by the unit slope on the log-log derivative curve and the
zero slope on the PPD curve. The pressure is above the vertical stress, suggesting an incorrect interpretation. This early flow
period must be wellbore related. The PPD curve drops rapidly and has a slope < -1. Up to at least DT=0.002 days when the
pressure drops below the vertical stress gradient, there must be significant wellbore effects. The time around DT=0.01 days
(0.6 times the pumping time) still has a PPD slope <-1. This is interpreted to be PDL by the first author, although it could still
be wellbore related effects. A review of Table 2 shows that the steepest PPD derivative slope for traditional flow regimes at
early time is that associated with radial flow (-1 slope). For PDL to occur, leak-off must be more rapid than these traditional
flow regimes. Therefore it is hypothesized that a PPD slope steeper than -1 is an indication of PDL if wellbore effects are not
present.
At DT=0.03 a PPD violation occurs. This is interpreted as a closure event, which would be in the secondary natural fractures
associated with the earlier in time PDL behavior. The well then goes into late time Carter leak-off which ends at DT≈1.0 days
(pfoc=49,280 kPa). Both the derivative plot (slope = 3/2) and the PPD (slope = -½) verify this flow regime. Since late time
Carter flow follows the PPD violation, the main fracture is still open and is so until DT≈1.0 days. Subsequently there is
another PPD violation occurring at DT=5.0 days. This gives an absolute latest time of closure. No distinctive after closure
flow regimes are identified.
The Equivalent Carter log-log derivative with the dp/dtsc curve is shown in Figure 32. This plot shows the same flow regimes
as the Bourdet log-log derivative plot. A later time has been picked for the closure time DT=1.3 days (pfoc=49,240 kPa). The
Combination Superposition Carter plot in Figure 33 was used to make the final closure time DT=1.05 days (pfoc=49,270 kPa).
The dp/dtsc is not perfectly flat during Carter leak-off. It is increasing at a more rapid rate after DT=1.05 days indicating the
end of Carter leak-off.
The ability of the Bourdet log-log derivative with the PPD plot greatly enhances our ability to interpret mini-fracs. It also
leads to different interpretations than traditional techniques (PDL versus height recession or transverse storage in this case).
Example 4 – Tight Gas Well 2 This gas well was completed in the Lower Montney formation in British Columbia by a different operator than the previous
well. The mini-frac was conducted in the toe stage of a 1300 meter long multi-stage frac port horizontal well. Pumping rate
during the test was 0.75 m3/min of water with a total of 5.0 m3 of water being injected over approximately 7 minutes.
Pressure was measured at the surface and converted to bottomhole depth.
Figure 34 , Figure 35 , Figure 36 and Figure 37 show the pressure versus time, combination G function, combination square
root t and the DT log-log derivative plots respectively. The pressure drops off linearly (storage event) at a very rapid rate at
early time. There is then a slight pressure rebound at 0.0003 days, when the pressure decline resumes at a much lower rate.
The rebound is not interpreted to be a closure event. This early pressure decline is probably an artifact of surface operational
issues related to shutting the pump down, and is not a reservoir effect.
Both the combination G function and square root t plots show a large upward curvature, which is traditionally considered an
indication of height recession/transverse storage. Hypothetical closure pressures are picked from both of these plots. Taking
the average value closure pressure would be 51,750 kPa. The DT log-log derivative plot shows two clearly identifiable flow
regimes, the first is a ½ slope flow regime; the second is late time after closure bilinear flow as a result of the -¾ slope.
Following the ½ slope flow regime, there is a dip followed by a significant slope increase. Picking the end of the ½ slope
flow as closure, results in a closure pressure of 55,300 kPa. This is significantly different than the value determined from the
combination plots.
The Bourdet log-log derivative plot with the PPD is shown in Figure 38. An apparent radial flow period occurs from 0.0003
days to 0.002 days. During this time the PPD has a slope of -1, which is consistent with the zero slope on the derivative
curve. It is not reasonable that radial flow is occurring at this time. It is possible that this is a PDL effect as discussed in the
previous example.
At 0.003 days a unit slope occurs on the derivative curve lasting until 0.04 days (58 minutes or a tD=8.3). This would
normally be interpreted as early time Carter leak-off. From Table 2 early time Carter leak-off should be over by tD=0.08 and
the late time Carter leak-off should have started by tD=5.0. The plot should have had a 3/2 slope instead of unit slope (there are
no 3/2 slopes on this plot and in fact the derivative curve after 0.1 days has a slope that is > than 3/2). Therefore this unit slope
SPE 160169 9
is not early time Carter leak-off and deviations from this slope should not be used for closure picks. Additionally the PPD
curve does not show early time Carter leak-off (which should have a zero slope). This illustrates the advantages of two
independent flow regime identification curves. For this reason closure picks were based on when the PPD increases. The
earliest possible closure would be at 0.07 days where pfoc = 55,700 kPa, but we feel that this is wellbore related. A more
reasonable value is at 0.35 days where a smooth increase in the PPD starts and continues over ½ a log cycle. This gives a pfoc
= 55,000 kPa. These two values bound the closure pressure pick.
The advantages of using the Bourdet log-log derivative along with the PPD are apparent. It is extremely important to perform
consistency checks to ensure false signals are identified.
Conclusions A thorough review of mini-frac analysis techniques and how they relate to conventional PTA concepts has been performed. It
is concluded:
1. Nolte analysis has been extended to account for the variable rate case using superposition principles.
2. Nolte’s Carter leak-off flow regime associated with a dynamically generated fracture is just one more flow regime
that can be handled within the PTA paradigm.
3. The ambiguities of the DT log-log derivative plot have been revealed.
4. The starting point for any mini-frac study should be the standard Bourdet log-log derivative plot with the PPD curve
for consistency checking.
5. The PPD curve has been shown to contain flow regime identification properties independent of the Bourdet log-log
derivative.
6. Once various flow regimes have been identified, additional flow regime specific log-log derivative plots have been
developed so as to further clarify the interpretation process.
7. New derivative functions have been defined for the log-log derivative plots which ensure invariant slopes at both
early and late time within a given flow regime. These functions have been constructed in a way that they give slopes
consistent with the associated flow regimes in PTA.
8. New combination plots and associated derivative functions have been determined to generalize the combination G
function plot. They have been built so that they can handle all flow regimes and can mimic the functionally of the
combination G function plot when plotted against t. This allows the easy comparison of the flow regimes across
many plots, as they all now have t as the x axis. These combination plots are very general and could also be used in
traditional PTA.
10 SPE 160169
Nomenclature Cb = Constant for bilinear flow regime
Cc = Constant for Carter leak-off
Cf = Generic constant for a flow regime
Cl = Constant for linear flow regime
CLO = leak-off coefficient
C1 = constant including reservoir and geomechanical properties to give Nolte’s G Function relationship
d = derivative
DP = Delta Bottomhole Pressure (p(DT=0) – p) (same as P)
DP Deriv = terdDP/dter Bourdet log log derivative, psi or kPa
DT = Delta time from current time to end of last injection period (same as t)
G = G Time, dimensionless
L = Fracture half-length at an instant in time during pumping, ft or m
n = exponent associated with one of the fracture flow regimes
p = Bottomhole Pressure, psi or kPa
pfoc = Opening and closure pressure (closure stress), psi or kPa
pi = Initial Pressure, psi or kPa
QDT= DT0.25
= t0.25
, days0.25
SDT= DT0.5
= t0.5
, days0.5
T = Time, days
teb = Equivalent Time bilinear, days
tec = Equivalent Time Carter, days
tef = Equivalent Time for an arbitrary flow regime, days
tel = Equivalent Time linear, days
ter = Equivalent Time radial, days
tp = Pumping Time
tsb, = Superposition Time bilinear, days
tsc = Superposition Time Carter, days
tsf = Superposition Time for an arbitrary flow regime, days
tsl = Superposition Time linear, days
trl = Superposition Time radial, days
t0 = Time at which leak-off begins at a specific location in the propagating fracture, days
u = velocity normal to the fracture face, ft/day or m/day
T2 = T2, days
2
P = Delta Bottomhole Pressure (p(DT=0) – p) (same as DP)
t = Delta time from current time to end of last injection period (same as DT)
tD = dimensionless shut-in time equal to t/tp
References Agarwal, R.G. 1980. A New Method for Accounting for Producing Time Effects when Drawdown Type Curves are Used to
Analyze Pressure Build-ups and other Test Data, Paper SPE 8279 8341 presented at the SPE Annual Technical Conference
and Exhibition, Dallas, 21-24 September.
Barree, R.D. and Mukherjee, H. 1996. Determination of Pressure Dependent Leakoff and its Effects on Fracture Geometry.
Paper SPE 36424 presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, 6-9 October.
Barree, R.D., Barree, V.L. and Craig, D.P. 2007. Holistic Fracture Diagnostics. Paper SPE 107877 presented at the SPE
Rocky Mountain Technical Symposium, Denver, 16-18 April.
Bourdet, D.P, Whittle, T.M., Douglas, A.A and Pirard, Y.M. 1983. A New Set of Type Curves Simplifies Well Test
Analysis, World Oil, May.
Cinco-Ley, H., Samaniego-V, F. and Dominguez, N. 1978. Transient Pressure Behavior for a Well with a Finite Conductivity
Vertical Fracture, SPEJ, August.
Leschyshyn T., Farouq Ali S.M. and Settari A. 1996. Mini-frac Analysis of Shear Parting in Alberta Reservoirs and Its
Impact towards On-site Fracture Design”, Paper No. 96-79, 47th Annual Technical Meeting of Petroleum Society. of CIM,
SPE 160169 11
Calgary, 10-12 June.
Mattar, L. and Zaoral,, K. 1992. The Primary Pressure Derivative (PPD) A new Diagnostic Tool in Well Test Interpretation,
JCPT, April.
Marongiu-Porcu, M., Ehlig-Economides, C.A. and Economides, M.J. 2011. Global Model for Fracture Falloff Analysis.
Paper SPE 144028 presented at the North American Unconventional Gas Conference and Exhibition, The Woodlands, 14-16
June.
Mishra, M.K., Lywood, P. and Ayan, C. 2011. Application of Wireline Stress Testing for SAGD Caprock Integrity, Paper
SPE 149456 presented at the Canadian Unconventional Resource Conference, Calgary, 15-17 November.
Mohamed, I.M., Nasralla, R.A., Sayad, M.A. Marongiu-Porcu M. and, Ehlig-Economides, C.A. 2011. Evaluation of After-
closure Analysis Techniques for Tight and Shale Gas Formations, Paper SPE 140136 presented at the Hydraulic Fracturing
Technology Conference and Exhibition, The Woodlands, 24-26 January.
Mukherjee, H., Larkin, S. and Kordziel, W. 1991. Extension of Fractured Decline Curve Analysis to Fissured Formation,
Paper SPE 21872 presented at the Low-Permeability Reservoir Symposium, Denver, 15-17 April.
Nolte, K.G. 1979. Determination of Fracture Parameters from Fracturing Pressure Decline, Paper SPE 8341 presented at the
SPE Annual Technical Conference and Exhibition, Las Vegas, 23-25 September.
Nolte, K.G. 1986. A General Analysis of Fracturing Pressure Decline with Applications to Three Models, SPEFE, December.
Odeh, A.S. and Jones, L.G. 1965. Pressure Drawdown Analysis, Variable Rate Case, JPT, August.
Soliman, M.Y., Craig, D., Bartko, K. and Rahim, Z. 2005. After-Closure Analysis to Determine Formation Permeability,
Reservoir Pressure, and Residual Fracture Properties, Paper SPE 93419 presented at the SPE Middle East Oil Show and
Exhibition, Bahrain, 12-15 March.
12 SPE 160169
Figure 1: Representation of Carter Leak-off
Figure 2: Static Fracture Flow Regimes from Cinco-Ley (1978)
SPE 160169 13
tp= 1 day
Early Time Slope = 1
Late Time Slope = 0.5
td
Pdt
)(
Figure 3: Carter Leak-Off Solutions using the DT log-log derivative
tt
tttDerivDP
dt
Pdt
p
p
er
er
er
_
)(
tp= 1 day
Early Time Slope = 1
Late Time Slope = 1.5
Figure 4: Carter Leak-Off Solutions using the Bourdet log-log derivative
14 SPE 160169
All Slopes = 1
tCdt
Pdt
Cdt
Pd
tCttttCP
c
ec
c
ec
eccppc
)(
)(
)( 5.15.15.1
tp= 1 day
Figure 5: Carter Leak-Off Solutions using the Equivalent Carter log-log derivative
tp= 1 day
Early Time Slope = 0.5
Late Time Slope = -0.5
td
Pdt
)(
Figure 6: Linear Fall-off Solutions using the DT log-log derivative
SPE 160169 15
tt
tttDerivDP
dt
Pdt
p
p
er
er
er
_
)(
Early Time Slope = 0.5
Late Time Slope = 0.5
Figure 7: Linear Fall-off Solutions using the Bourdet log-log derivative
5.05.0
5.05.05.0
)(
)(
)(
tCdt
Pdt
Cdt
Pd
CtttttCP
l
el
l
el
elppl
tp= 1 day
All Slopes = 0.5
Figure 8: Linear Fall-off Solutions using the Equivalent Linear log-log derivative
16 SPE 160169
tp= 1 day
Pre
ssu
re(p
si)
Figure 9: Carter Leak-Off Combination G Function Plot
Combination Square Root Plot
P
Early Time Constant
Late Time Sqrt(DT)
dPdSDT
Early Time Sqrt(DT)
Late Time Constant
SDTdPdSDT
Early Time DT
Late Time Sqrt(DT)
tp= 1 day
Pre
ss
ure
(psi
)
Figure 10: Carter Leak-Off Combination Square Root t Plot
SPE 160169 17
tCdt
dpt
Cdt
dp
tCptttCpp
c
sc
c
sc
sccipci
5.15.1)(
tp= 1 day
Pre
ssu
re(p
si)
Figure 11: Carter Leak-Off Combination Superposition Carter Plot
tp= 1 day
Pre
ssu
re(p
si)
Figure 12: Linear Flow Combination G Function Plot
18 SPE 160169
Combination Square Root Plot
P
Early Time Constant
Late Time 1/Sqrt(DT)
dPdSDT
Early Time Constant
Late Time 1/DT
SDTdPdSDT
Early Time Linear
Late Time 1/Sqrt(DT)
tp= 1 day
Pre
ssu
re(p
si)
Figure 13: Linear Flow Combination Square Root t Function Plot
tCdt
dpt
Cdt
dp
tCptttCpp
l
sl
l
sl
sllipli
5.05.0)(
tp= 1 day
Pre
ss
ure
(psi
)
Figure 14: Linear Flow Combination Superposition Linear Plot
SPE 160169 19
Figure 15: Comparison of square root t , G and tel Functions
Start
Log-Log Plot
P, Bourdet deriv, PPD vs t
Flow Regime Identification
Carter
Log-Log Plot
P, td(P)/dtec , dp/dtsc vs t
Linear-Linear Combination Plot
p, dp/dtsc, tdp/dtsc vs t
Linear Bilinear
Log-Log Plot
P, t0. 5d(P)/dtel , dp/dtsl vs t
Log-Log Plot
P, t0.25d(P)/dteb , dp/dtsb vs t
Linear-Linear Combination Plot
p, dp/dtsl, tdp/dtsl vs t
Linear-Linear Combination Plot
p, dp/dtsb, tdp/dtsb vs t
Note: when we write
dp/dtsf we mean |dp/dtsf| Figure 16: Workflow associated with Mini-Frac Analysis
20 SPE 160169
Slope = 1/1
Figure 17: Example 1 - Rate Normalized Derivative over Multiple Flow Periods
Two separate injection cycles due to
operational issues in the field
FP_001_Inj = 6.0 minutes
FP_002_FO = 3.1 minutes
FP_003_Inj = 5.3 minutes
FP_004_FO = 5626 minutes or 1825 times as long as FP_003_Inj
Friction effects not
Pressure Dependent
Leak-off (PDL)
Figure 18: Example 2 – Pressure versus Time
SPE 160169 21
pfoc = 8400 kPa
Indicator of pressure
dependent leak-off
Figure 19: Example 2 – Combination G Function Plot
pfoc = 8400 kPa
Figure 20: Example 2 – Combination Square Root t Plot
22 SPE 160169
Cross over of ‘Middle’
and ‘Late’ ½ Slope
Delta Time = 0.28 Days
pfoc = 8300 kPa
½
½
-½
Figure 21: Example 2 – DT log-log derivative Plot
½
½
-½
3/2
-3/2
-½
-½
End of Carter Leak-off
Fracture has closed
pfoc = 8300 kPa
Figure 22: Example 2 – Bourdet log-log derivative Plot
SPE 160169 23
Linear flow signatures
1/1
0
0
End of Carter Leak-off
Fracture has closed
pfoc = 8300 kPa
Figure 23: Example 2 – Equivalent Carter log-log derivative Plot
pfoc = 8300 kPa
Figure 24: Example 2 – Combination Superposition Carter Plot
24 SPE 160169
Figure 25: Example 3 – Pressure versus Delta Time to 0.01 days
PPD Violation
1
Figure 26: Example 3 – Pressure versus Delta Time to 1 day
SPE 160169 25
Closure at G = 82 ?
DT = 16.6 days
pfoc= 46,900 kPa
Figure 27: Example 3 – Combination G Function Plot (Late Time)
Closure at G = 3.0 ?
DT = .045 days
pfoc= 49,510 kPa
Figure 28: Example 3 – Combination G Function Plot (Middle Time)
26 SPE 160169
Closure at Sqrt(DT) = 0.215 ?
DT = 0.046 days
pfoc = 49,510 kPa
Figure 29: Example 3 – Combination Square Root t Plot (Middle Time)
1/1 Slope = Gray
½ Slope = Blue
Closure at end of ½ slope
at DT = 0.9 days ?
pfoc = 49,290 kPa
Closure near end ?
DT = 16.9 days
pfoc = 36,950 kPa
1/1
½
Figure 30: Example 3 – DT log-log derivative Plot
SPE 160169 27
PPD Violation at
DT=0.03 days
pfoc=49,560 kPa ?
Possible Closure of
secondary fractures
PPD Violation at
DT=5.0 days
pfoc=48,850 kPa ?
Absolute latest time
for closure
Late Time Carter
Until ≈ 1.0 days
pfoc=49,280 kPa ?
PPD Deriv slope <-1
Pressure Dependent
Leak-off (PDL)
1/1
0
-½
3/2
Figure 31: Example 3 – Bourdet log-log derivative Plot
Since Carter leak-off is
occurring after PPD
violation at DT=0.03 days
main fracture still open
Late Carter
Closure at 1.3 days
pfoc = 49,240 kPa ?1/1
0
0
1/1
Figure 32: Example 3 – Equivalent Carter log-log derivative Plot
28 SPE 160169
End of Carter Leak-off ?
Yes
DT=1.05 Days
pfoc=49,270 kPa
Final closure pick
Figure 33: Example 3 – Combination Superposition Carter Plot
Figure 34: Example 4 – Pressure versus Time
SPE 160169 29
Closure at G=28.5 ?
pfoc =51,600 kPa
Figure 35: Example 4 – Combination G Function Plot
Closure at SDT=0.93 ?
pfoc =51,900 kPa
Figure 36: Example 4 – Combination Square Root t Plot
30 SPE 160169
½
Closure at DT=0.025 ?
pfoc =55,300 kPa
Closure at DT=1.000 ?
pfoc =50,600 kPa
-3/4
Figure 37: Example 4 – DT log-log derivative Plot
0
1/4
-7/4
False early Carter Leak-off
As PPD slope is not 0
No late Carter Leak-off present
(slope is >3/2 after DT=0.1)
PPD Closure Picks
Highest closure stress at
DT=0.07
pfoc =55,700 kPa
Most likely closure stress
at DT=0.35
Pfoc=55,000 kPa
-1
<0
1/1
0
Figure 38: Example 4 – Bourdet log-log derivative Plot
