holistic fracture diagnostics

√t Fracture Closure Figure 2: Normal leakoff sqrtt plot Normal Leakoff Log-Log Pressure Derivative The log-log plot of pressure change from ISIP versus shut-in time for the normal leak

Copyright 2007, Society of Petroleum Engineers

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Abstract

Since the introduction of the G-function derivative analysis,

pre-frac diagnostic injection tests have become a valuable and

commonly used technique Unfortunately, the technique is

frequently misapplied or misinterpreted leading to confusion

and misdiagnosis of fracturing parameters This paper presents

a consistent method of analysis of the G-function, its

derivatives, and its relationship to other diagnostic techniques

including square-root(time) and log(∆pwf)-log(∆t) plots and

their appropriate diagnostic derivatives Actual field test

examples are given for the most common diagnostic curve

signatures

Introduction

Pre-frac diagnostic injection test analysis provides critical

input data for fracture design models, and reservoir

characterization data used to predict post-fracture production

An accurate post-stimulation production forecast is necessary

for economic optimization of the fracture treatment design

Reliable results require an accurate and consistent

interpretation of the test data In many cases closure is

mistakenly identified through misapplication of one or more

analysis techniques In general, a single unique closure event

will satisfy all diagnostic plots or methods All available

analysis methods should be used in concert to arrive at a

consistent interpretation of fracture closure

Relationship of the pre-closure analysis to after-closure

analysis results must also be consistent To correctly perform

the after-closure analysis the transient flow regime must be

correctly identified Flow regime identification has been a

consistent problem in many analyses There remains no

consensus regarding methods to identify reservoir transient

flow regimes after fracture closure The method presented here

is not universally accepted but appears to fit the generally

assumed model for leakoff used in most fracture simulators

Four examples are presented to show the application of multiple diagnostic analysis methods The first illustrates the expected behavior of normal fracture closure dominated by matrix leakoff with a constant fracture surface area after

shut-in The second example shows pressure dependent leakoff (PDL) in a reservoir with pressure-variable permeability or flow capacity, usually caused by natural or induced secondary fractures or fissures The third example shows fracture tip extension after shut-in These cases generally show definable fracture closure The fourth example shows what has been commonly identified as fracture height recession during closure, but which can also indicate variable storage in a transverse fracture system

For each example the analysis will be demonstrated using the G-function and its diagnostic derivatives, the sqrt(time) and its derivatives, and the log-log plot of pressure change after shut-in and its derivatives.1-4 When appropriate, the after-closure analysis is presented for each case, as is an empirical correlation for permeability from the identified G-function closure time.5 A critical part of the analysis is the realization that there is a common event indicating closure that should be consistently identified by all diagnostic methods To reach a conclusion all analyses must give consistent results

The goal of this paper is to provide a method for consistent identification of after-closure flow regimes, an unambiguous fracture closure time and stress, and a reasonable engineering estimate of reservoir flow capacity from the pressure falloff data, without requiring assumptions such as a known reservoir pressure Other methods, based on sound transient test theory, require pressure difference curves based on the observed bottomhole pressure during falloff minus the “known” reservoir pressure.5,8 While these methods are technically correct they can lead to confusing results at times, especially

in low permeability reservoirs when pore pressure is difficult

to determine accurately prior to stimulation

This is not a transient test analysis paper but is intended to present a practical approach to analysis of real, and frequently marginal-quality, pre-fracture field test data The techniques applied are based on some transient test theory Some of the results presented here are still under debate and development The methods shown have been tested and, we believe, proven

in the analysis of hundreds of tests Application of these methods provides consistent analysis that helps to avoid misinterpretation of falloff data, and give the most useful information available from diagnostic injection tests

Step-rate injection tests and their analysis are not included

in the scope of this paper Determination of the pressure-dependent leakoff coefficient is also not described here, as it

SPE 107877

Holistic Fracture Diagnostics

R.D Barree, SPE, and V.L Barree, Barree & Assocs LLC, and D.P Craig, SPE, Halliburton

has been previously reported.3,4 Only the analysis of pressure

decline following shut-in of a fracture-rate injection test is

considered

Transient Flow Regimes During and After Fracture

Closure

Several transient flow regimes may occur during a falloff test

after injection at fracture rate The major flow regimes are

graphically illustrated in the classic paper by Cinco-Ley and

Samaniego.6

Immediately after shut-in the pressure gradient along the

length of the fracture dissipates in a short-duration linear flow

period In a long fracture in low permeability rock the initial

fracture linear flow can be followed by a bi-linear flow period

with the linear flow transient persisting in the fracture while

reservoir linear flow occurs simultaneously After the fracture

transient dissipates the reservoir linear flow period can

continue for some time, depending on the permeability of the

reservoir and the volume of fluid stored in the fracture and

subsequently leaked off during closure After closure the

pressure transient established around the fracture propagates

into the reservoir and transitions into elliptical, then

pseudoradial flow Each of these flow regimes has a

characteristic appearance on various diagnostic plots

Fluid leakoff from a propagating fracture is normally

modeled assuming one-dimensional linear flow perpendicular

to the fracture face Settari has pointed out that in some cases

of moderate reservoir permeability the linear flow regime may

not occur, even during fracture extension and early leakoff.7

During fracture extension and shut-in the transient may

already be in transition to elliptical or pseudoradial flow In

this case analyses based on an assumed pseudolinear flow

regime will give incorrect results In all cases an

understanding of the flow regime and its relation to the

fracture geometry is critical to arriving at a consistent

interpretation of the fracture falloff test

Diagnostic Derivative Examples

For each analysis technique various curves are used to help

define closure, leakoff mechanisms, and after-closure flow

regimes On each plot the curves are labeled as the primary (y

vs x), the first derivative (∂y/∂x), and the semilog derivative

(∂y/∂(lnx) or x∂y/∂x) For convenience the primary curve is

plotted on the left y-axis and all derivatives are plotted on the

right y-axis for all Cartesian plots For the log-log plot all

curves are shown on the same y-axis

For pre-closure analysis, and consistent identification of

fracture closure, three techniques are illustrated for each

example: G-function, Square-root of shut-in time, and log-log

plot of pressure change with shut-in time All these analyses

begin at shut-in The instantaneous shut-in pressure (ISIP) is

taken as the incipient fracture extension pressure for all cases

When there is significant wellbore afterflow (fluid expansion

or continued low-rate injection), or severe near-well pressure

drop, the ISIP can be difficult to interpret accurately and may

be too high to represent actual fracture extension pressure In

all the examples in the paper the pressures have been offset to

an approximate ISIP of 10,000 psi to remove any relation to

the original field test data The following sections detail the

data and analysis for the four major leakoff type examples

Normal Leakoff Behavior

Normal leakoff is observed when the composite reservoir system permeability is constant The reservoir may exhibit only matrix permeability or have a secondary natural fracture

or fissure overprint in which the flow capacity of the secondary fracture system does not change with pore pressure

or net stress After shut-in the fracture is assumed to stop propagating and the fracture surface area open to leakoff remains constant during closure

Normal Leakoff G-Function

As noted in previous papers, the expected signature of the G-function semilog derivative is a straight-line through the origin (zero G-function and zero derivative).4 In all cases the correct straight line tangent to the semilog derivative of the pressure vs G-function curve must pass through the origin

Fracture closure is identified by the departure of the semi-log derivative of pressure with respect to G-function (G∂pw/∂G) from the straight line through the origin During normal leakoff, with constant fracture surface area and constant permeability, the first derivative (∂p w/∂G) should also be constant.2 The primary p w vs G curve should follow a straight line.1 The example in Figure 1 shows some slight deviation from the perfect constant leakoff but is a good example of the expected curve shapes with a clear indication of closure at

Gc=2.31 The closure event is marked by the dashed vertical line [1]

0 5 10 15 20 25

G(Time)

7500 8000 8500 9000 9500 10000 10500

0 200 400 600 800 1000 1200 1400 1600 1800

2000 1

P vs G

GdP/dG vs G

dP/dG vs G

Fracture Closure

Figure 1: Normal leakoff G-function plot

Normal Leakoff Sqrt(t) Analysis

The sqrt(t) plot has frequently been misinterpreted when

picking fracture closure, even for the simplest cases The

primary p w vs sqrt(t) curve should form a straight line during

fracture closure, as with the G-function plot Some users suggest that the closure is identified by the departure of the data from the straight line trend, similar to the way the G-function closure is picked This is incorrect and leads to a later closure and lower apparent closure pressure The correct

indication of closure is the inflection point on the p w vs sqrt(t)

plot

The best way to find the inflection point is to plot the first

derivative of p w vs sqrt(t) and find the point of maximum

amplitude of the derivative Many fracture-pressure analysis

software packages plot the inverse of the actual first derivative

and show the inflection point as the minimum of the

derivative The plot in Figure 2, shows that the slope of the

pressure curve starts low, then increases and reaches a

maximum rate of decline at the inflection point, then decreases

again after closure The first derivative curve in Figure 2 is

plotted with the proper sign The dashed vertical line [1] is the

G-function closure pick that is synchronized in time and

pressure with the sqrt(t) plot Clearly the consistent closure

lies at the inflection point and not at the point of departure

from the straight line tangent to the pressure curve

The semilog derivative of the pressure curve is also shown

on the sqrt(t) plot This curve is equivalent to the semilog

derivative of the G-function for most low-perm cases The

closure pick falls at the departure from the straight line

through the origin on the semilog derivative of the P vs sqrt(t)

curve A single closure point must satisfy the requirement on

both the G-function and sqrt(t) plots

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16:00 Time

7500

8000

8500

9000

9500

10000

10500

0 100 200 300 400 500 600 700 800 900

1000

1

P vs √t

√tdP/d√t vs √t dP/d√t vs √t Fracture Closure

Figure 2: Normal leakoff sqrt(t) plot

Normal Leakoff Log-Log Pressure Derivative

The log-log plot of pressure change from ISIP versus

shut-in time for the normal leakoff example is shown shut-in Figure 3

The heavy curve is the pressure difference and the dashed

curve is its semilog derivative with respect to shut-in time

The vertical dashed line is the unique closure pick from the

G-function and sqrt(t) plot It is common for the pressure

difference and derivative curves to be parallel immediately

before closure The slope of these parallel lines is diagnostic

of the flow regime established during leakoff before closure

In many cases a near-perfect ½ slope is observed, strongly

suggesting linear flow from the fracture In this example the

slope is greater than ½ suggesting possible linear flow coupled

with changing fracture/wellbore storage (See Appendix B)

The separation of the two parallel lines always marks fracture

closure and is the final confirmation of a consistent closure

identification

Time (0 = 8.15)

2 3 4 6 8

2 3 4 6 8

2 3 4

10 100

1000

(m = -1) (m = 0.632)

BH ISIP = 9998 psi

1

∆P vs ∆t

∆td∆P/d∆t vs ∆t

Fracture Closure

Radial Flow

Figure 3: Normal leakoff log-log plot

After closure the semilog derivative curve will show a slope of -1/2 in a fully developed reservoir pseudolinear flow regime and a slope of -1 in fully developed pseudoradial flow

In the example the derivative slope is -1 indicating that reservoir pseudoradial flow was observed The late-time data shows a drop in the derivative probably caused by wellbore effects such as gas entry and phase segregation The use of the semilog derivative of the log-log plot for after-closure flow regime identification, as well as closure confirmation, is a powerful new addition to fracture pressure decline diagnostics

After-Closure Analysis for Normal Leakoff Example

The Talley-Nolte After-Closure Analysis (ACA) flow regime identification plot for the normal leakoff example is shown as Figure 4.5 The heavy solid line is the observed bottomhole pressure during the falloff minus the initial reservoir pressure The slope of the semi-log derivative of the pressure difference function (dashed line) is 1.0 during the identified pseudoradial flow period If a linear-flow period existed in this data set a derivative slope of ½ would exist It is critical to remember that the slope of the pressure difference curve on this plot is determined solely by the guess of reservoir pressure used to construct the plot The slope of the derivative is not affected by the input reservoir pressure value

2 3 4 5 7 9 2 3 4 5 7 9 2 3 4 5 7 9

0.001 0.01 0.1 1

Square Linear Flow (FL^2)

2 3 4 6 8 2 3 4 6 9 2 3 4 6 9

10 100 1000 10000

(m = 1)

∆P vs FL

F L d∆P/dFL vs F L

∆P=(pw-p r )

Start of Radial Flow

Figure 4: Normal leakoff ACA log plot

If a pseudoradial flow regime is identified, then the

Cartesian Radial Flow plot (Figure 5) can be used to

determine reservoir far-field transmissibility, kh/µ The

viscosity used is the far-field mobile fluid viscosity and h is

the estimated net pay height For the analysis of the example

data kh/µ = 299 md-ft/cp For gas viscosity at reservoir

temperature, kh=7.9 md-ft For the assumed net pay, the

effective reservoir permeability is 0.097 md

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Radial Flow Time Function 7200

7400

7600

7800

8000

8200

8400

8600

8800

9000

9200

(m = 4814.2) Results

Reservoir Pressure = 7475.68 psi Transmissibility, kh/µ = 298.94991 md*ft

kh = 7.94014 md*ft Permeability, k = 0.0968 md Start of Pseudo Radial Time = 2.15 hours 1

Figure 5: Normal leakoff ACA radial flow plot

Horner Analysis for Normal Leakoff Example

If a pseudoradial flow period is identified, then a

conventional Horner plot can also be used to determine

reservoir transmissibility In Figure 6 the Horner slope through

the radial flow data is 14411 psi Using an average pump rate

of 18.4 bpm, kh/µ = 298 md-ft/cp For the assumed gas

viscosity kh=7.9 md-ft Using the same assumed net gives

k=0.097 md This result is consistent with the ACA results

1

Horner Time 7250

7500

7750

8000

8250

8500

8750

9000

9250

9500

9750

(m = 14411)

(Reservoir = 7476)

1

Figure 6: Normal leakoff Horner Plot

G-Function Permeability Estimate

An empirical correlation has also been developed to estimate formation permeability from the G-function closure time when after-closure data is not available The correlation

is described in detail in the Appendix Figure 7 shows the G-function correlation permeability estimate for the observed closure time and other input parameters The permeability estimate of 0.097 md is consistent with the Horner and ACA results

0.001 0.01 0.1 1 10 100

0 2 4 6 8 10 12 14 16 18 20

G c

Data Input

φ 0.09 V/V

c t 7.50E-05 psi-1

Gc 2.44

Pz 966.0 psi Estimated Permeability = 0.0974 md

Figure 7: Normal leakoff permeability estimate

Pressure Dependent Leakoff

Pressure dependent leakoff (PDL) occurs when the fluid loss rate changes with pore pressure or net effective stress in the rock surrounding the fracture PDL is not caused by the normal change in transient pressure gradient during leakoff

This is part of the normal leakoff mode and is handled by the one-dimensional linear flow solution of the diffusivity equation used to model fracture leakoff in a constant permeability system The pressure dependence referred to here

is a change in the transmissibility of the reservoir fissure or fracture system that dominates the fluid loss rate PDL is only apparent when there is substantial stress dependent permeability in a composite dual-permeability reservoir

G-Function for Pressure-Dependent Leakoff

Figure 9 shows the G-function behavior expected for PDL

while PDL persists The semilog derivative exhibits the characteristic “hump” above the straight line extrapolated to the derivative origin The end of PDL and the critical fissure opening pressure corresponds to the end of the “hump” and the beginning of the straight line representing matrix dominated leakoff Fracture closure is still shown by the departure of the semilog derivative from the straight line through the origin

0 2 4 6 8 10 12 14 16 18 20

G(Time) 8250

8500

8750

9000

9250

9500

9750

10000

10250

10500

0 100 200 300 400 500 600 700 800 900

1000

2

1

P vs G

GdP/dG vs G

dP/dG vs G

Fracture Closure

Figure 9: PDL G-function plot

Sqrt(t) Analysis for PDL

Interpretation of the sqrt(t) plot in PDL cases has often led

to incorrect closure picks Figure 10 shows an expanded view

of the sqrt(t) plot for the example with the curves scaled for

better visibility Note that the semilog derivative is nearly

identical in shape and information content to the G-function

semilog derivative It clearly shows the PDL “hump” and

closure, which has been synchronized to the G-function result

Incorrect closure picks on the sqrt(t) plot will not occur if

the semilog derivative is used Problems arise when the first

derivative is used exclusively to pick closure In PDL cases

the obvious derivative maximum, or most prominent inflection

point, is caused by the changing leakoff associated with PDL

and does not indicate fracture closure The false closure

indication is shown on the plot Many fracture diagnostic tests

have been badly misdiagnosed because the early and incorrect

closure was picked because of dependence on only the sqrt(t)

plot This example clearly illustrates why all available

diagnostic plots must be used in concert to arrive at a single

consistent closure event

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02:00 Time

8250

8500

8750

9000

9250

9500

9750

10000

10250

0 100 200 300 400

500

1

False Closure

P vs √t

√tdP/d√t vs √t

dP/d √t vs √t

Fracture Closure

Figure 10: PDL Sqrt plot

Log-Log Pressure Derivative for PDL Example

Figure 11 shows the log-log plot for the PDL example The normal matrix leakoff period, following the end of PDL, appears as a perfect ½ slope of the semilog derivative with a parallel pressure difference curve exactly 2-times the magnitude of the derivative The parallel trend ends at the identified closure time and pressure difference In this example a well-defined −½ slope, or reservoir pseudolinear flow period, is shown shortly after closure The later data approach a slope of −1, which indicates pseudoradial flow has been established

Time (0 = 9.133333)

2 3 4 5 7 9 2 3 4 5 7 9 2

10 100

1000

(m = 0.5)

(m = -1) (m = -0.5)

BH ISIP = 10000 psi

1

∆td∆P/d∆t vs ∆t Fracture closure

Linear Flow

Radial Flow

Figure 11: PDL log-log plot

After-Closure Analysis for PDL Example

The ACA log-log plot (Figure 12) shows both the reservoir linear and radial flow periods in their expected locations

Square Linear Flow (FL^2)

2 3 4 6 8 2 3 4 6 8 2 3 4 6 8

10 100 1000 10000

(m = 1)

(m = 0.5)

1 2 3

F L d∆P/dF L vs F L

∆P=(pw-pr)

Start Linear Flow End Linear Flow

Start Radial Flow

Figure 12: PDL ACA log plot

Figures 13 and 14 show the ACA Cartesian plots for the linear and radial flow analyses Both give consistent estimates

of reservoir pore pressure The pseudoradial flow analysis gives a transmissibility of 37.2 md-ft/cp and estimated permeability of 0.047 md

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Linear Flow Time Function 8000

8200

8400

8600

8800

9000

9200

(m = 1438.5)

Results

Reservoir Pressure = 8056.66 psi Start of Pseudo Linear Time = 15.9 End of Pseudo Linear Time = 54.39 1

2

Start Linear Flow End Linear Flow

Figure 13: PDL ACA linear flow plot

Radial Flow Time Function 8000

8200

8400

8600

8800

9000

9200

(m = 11373) Results

Reservoir Pressure = 8068.81 psi Transmissibility, kh/µ = 37.21984 m

kh = 0.93764 md*ft Permeability, k = 0.0469 md Start of Pseudo Radial Time = 11.26 1

Figure 14: PDL ACA radial flow plot

Horner Analysis for PDL Example

For an average pump rate of 6.7 bpm the Horner plot gives

kh/µ=35.72 md-ft/cp The Horner estimated permeability is

0.046 md compared to 0.047 md from the ACA Radial Flow

analysis Pore pressure estimated from the Horner plot is also

consistent with both the linear and radial analyses because a

well-developed pseudoradial flow period does exist in this

case The vertical dotted line in Figure 15 shows the start of

pseudoradial flow If a pseudoradial flow period does not

exist, extrapolation of an apparent straight-line on the Horner

plot can give extremely inaccurate estimates of pressure and

flow capacity

1

Horner Time 8000

8100 8200 8300 8400 8500 8600 8700 8800 8900

(m = 43920)

(Reservoir = 8064)

1

Figure 15: PDL Horner plot

G-Function Permeability Estimate for PDL Example

The G-function permeability correlation for the PDL example is shown in figure 16 It also gives a consistent permeability of 0.045 md The impact of the accelerated leakoff during PDL gives an estimate of the composite reservoir effective permeability Note that the injected fluid viscosity is used for the permeability estimate based on closure time

0.001 0.01 0.1 1 10 100

0 2 4 6 8 10 12 14 16 18 20

G c

Data Input

Estimated Permeability = 0.0453 md

Figure 16: PDL permeability estimate

Fracture Tip Extension

In very low permeability reservoirs the decline in wellbore pressure observed after shut-in may be caused by the dissipation of the pressure transient established in the fracture during pumping The near-well pressure decreases as the fracture closes, which results in a decrease of fracture width at the well The closing of the fracture volumetrically displaces fluid to the tip of the fracture, causing continued extension of the fracture length Much of the pressure decline is therefore not related to leakoff but to the dissipation of the linear transient along the fracture length

G-Function Analysis for Tip Extension

During fracture tip extension the G-function derivatives

fail to develop any straight-line trends The primary P vs G

curve is concave upward, as is the first derivative The

semilog derivative starts with a large positive slope and the

slope continues to decrease with shut-in time, giving a

concave-down curvature.3,4 Figure 17 shows a typical case of

fracture tip extension with minimal leakoff This is another

case that is frequently misdiagnosed

G(Time) 8400

8600

8800

9000

9200

9400

9600

9800

10000

10200

0 25 50 75 100 125

150

1

P vs G

GdP/dG vs G

dP/dG vs G

Figure 17: Tip extension G-function plot

Sqrt(t) Analysis with Fracture Tip Extension

Many times the first break in the semilog derivative curve

has been misinterpreted as a closure event The mistake is

often compounded by the use of the sqrt(t) plot Figure 18

shows the sqrt(t) plot for the same data The first derivative

shows a large maximum very shortly after shut-in This is

often mistaken for closure The semilog derivative on the

sqrt(t) plot helps to avoid this mistaken closure pick, and

shows the same continuously increasing trend as seen on the

G-function semilog derivative plot In low permeability

systems it is generally safe to assume that as long as the

semilog derivative is still rising, the fracture has not yet

closed This is not true in very high permeability reservoirs

and should always be checked using the log-log pressure

difference plot

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1/25/2007 20:00 Time

8400

8600

8800

9000

9200

9400

9600

9800

10000

10200

0 10 20 30 40 50 60 70 80 90

100

1

Incorrect Closure

P vs √t xd P/d x vs √t

d P/d x vs √t

Figure 18: Tip extension sqrt(t) plot

Log-Log Pressure Derivative Analysis with Tip Extension

The log-log plot of pressure change after shut-in is particularly useful for diagnosing fracture tip extension Figure 19 shows the pressure difference and pressure derivative (semilog) for the tip extension example

Time (0 = 33.7)

2 3 4 6 8 2 3 4 6 8 2 3 4

10 100

1000

(m = 0.25)

∆P vs ∆t

∆td∆P/d∆t vs ∆t

Figure 19: Tip extension log-log plot

In Figure 19 the pressure derivative departs from the early unit-slope (storage) and establishes a ¼ slope during fracture tip extension The pressure difference curve falls on a parallel

¼ slope line separated by 4-times the magnitude of the derivative The ¼ slope signature is diagnostic of bilinear flow representing a continued dissipation of the linear pressure transient along the fracture length (extension and concomitant fluid flow) and some linear flow driving minimal leakoff For tip extension to occur the leakoff rate to the formation must be low As long as the parallel ¼ slope trend continues, the fracture has not closed and is still in the process of extending Closure cannot be determined and no after-closure analysis can be conducted

Height Recession or Transverse Storage

There are two different mechanisms that can generate a similar diagnostic derivative signature during fracture closure Both are caused by an excess stored volume of fluid in the fracture at shut-in relative to the expected surface area of the fracture for a planar, constant-height geometry model

Traditionally this signature has been called “fracture height recession” The usual model assumes that leakoff occurs only through a thin permeable bed and that the fracture extends in height to cover impermeable strata with no leakoff At shut-in there is a large volume of fluid stored in the fracture and the leakoff rate relative to the stored volume is small, hence the rate of pressure decline is likewise small As the fracture empties, the rate of leakoff relative to the remaining stored fluid accelerates and the pressure declines more rapidly If the fracture height changes during leakoff, the fracture compliance may also decrease, adding to the rate of pressure loss

However, the same signature is observed in many cases where fracture height growth out of zone is not observed by tracers, inclinometer, or micro-seismic mapping Some of these cases show treating behavior similar to PDL cases, with

a tendency for rapid screenout and difficulty placing high

proppant concentration slurries These observations suggest

that another mechanism may be responsible for the same

diagnostic derivative signature The alternate mechanism is

called “transverse fracture storage”

In transverse fracture storage a secondary fracture set is

opened when the fluid pressure exceeds the critical

fissure-opening pressure, just as in PDL As the secondary fractures

dilate they create a storage volume for fluid which is taken

from the primary hydraulic fracture While the fracture storage

volume increases, leakoff can also be accelerated so PDL and

storage are aspects of the same coupled mechanism of fissure

dilation The relative magnitude of the enhanced leakoff and

storage mechanisms determines whether the G-function

derivatives show PDL or storage Numerical modeling studies

indicate that the storage mechanism can easily dominate even

large PDL

At shut-in the secondary fractures will close before the

primary fracture because they are held open against a stress

higher than the minimum in-situ horizontal stress As they

close fluid will be expelled from the transverse storage volume

back into the main fracture decreasing the normal rate of

pressure decline and, in effect, supporting the observed shut-in

pressure by re-injection of stored fluid Accelerated leakoff

can still occur at the same time but if the storage and

expulsion mechanism exceeds the enhanced leakoff rate then

the only signature observed during falloff will be storage In

many cases a period of linear, constant area, constant matrix

permeability dominated leakoff will occur after the end of

storage

G-Function Analysis with Storage

The characteristic G-function derivative signature is a

“belly” below the straight line through the origin and tangent

to the semilog derivative of p w vs G at the point of fracture

closure Figure 20 shows an example of slight to moderate

storage In Figure 20 fracture closure, indicated by the same

departure of the tangent line from the semilog derivative,

occurs just after the end of the storage effect

G(Time) 7500

8000

8500

9000

9500

10000

10500

0 200 400 600 800 1000 1200 1400 1600 1800

2000

1

P vs G

GdP/dG vs G

dP/dG vs G

Fracture Closure

Figure 20: Storage G-function plot

Sqrt(t) Analysis with Storage or Height Recession

The sqrt(t) plot (Figure 21) shows a clear indication of

closure based on both the first-derivative inflection point and the semilog derivative curve Picking closure in the case of storage is not generally a problem

1/24/2007

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10:00 Time

7500 8000 8500 9000 9500 10000 10500

0 100 200 300

400

1

P vs √t

xd P/d x vs √t

d P/d x vs √t Fracture Closure

Figure 21: Storage sqrt(t) plot

The storage model, whether caused by height recession or transverse fractures, requires that a larger volume of fluid must be leaked-off to reach fracture closure than is expected for a single planar constant-height fracture In either case the time to reach fracture closure is delayed by the excess fluid volume that must be lost Any estimation of reservoir permeability will give an incorrect result if the uncorrected closure time (either Gc or time-to closure in minutes, t c) is used The observed closure time must be corrected by

multiplying by the storage ratio, r p The magnitude of r p can

be determined by taking the ratio of the area under the G-function semilog derivative up to the closure time, divided

by the area of the right-triangle formed by the tangent line through the origin at closure For normal leakoff and PDL the value of rp is set to 1.0 even though the ratio of the areas will

be greater than 1 for the PDL case It is possible that the closure time for PDL leakoff is proportional to the composite system permeability including both the matrix and fractures

For severe cases of storage r p can be as low as 0.5 or less

Log-Log Pressure Derivative with Storage

Figure 22 shows the log-log plot of pressure difference and semilog derivative for the storage case Prior to closure, and while transverse storage is dominant, the semilog derivative approaches a unit slope, with the pressure difference curve nearly parallel In some cases the two curves lie together on a single unit-slope line In this case the curves are separated slightly and the slope is not exactly 1.0 After closure the reservoir transient signature is defined as in the previously presented cases All fracture storage effects are eliminated and the reservoir pseudolinear flow period is shown by a -1/2 slope with a pseudoradial flow period indicated by a -1 slope

of the semi-log derivative

2 3 4 6 7 8 9 2 3 4 6 7 8 9 2 3 4 6 7 8 9 2 3 4 6

Time (0 = 9.416667)

2

3

4

5

7

9

2

3

4

6

8

2

3

10

100

1000

(m = 0.5)

(m = -0.5)

BH ISIP = 10000 psi

1

∆P vs ∆t

∆td∆P/d∆t vs ∆t

Figure 22: Storage log-log plot

Conclusions

The use of pre-frac injection/falloff diagnostic tests has

become commonplace Many important decisions regarding

fracture treatment designs and expectations of post-frac

production are based on the results of these tests In too many

cases individual diagnostic plots and analysis techniques are

misapplied, leading to incorrect interpretations The analyses

presented here lead to the following conclusions:

1 With consistent application of all available pressure

decline diagnostics, a single unambiguous determination

of fracture closure time and pressure can be made

2 A single, unique closure event can be identified on all

diagnostic plots

3 The conventional analysis of the sqrt(t) plot, using the

inflection point identified by the first derivative, gives

incorrect indications of closure for cases of PDL and tip

extension and should not be relied upon

4 A modified sqrt(t) analysis, using the semilog derivative,

is equivalent to the G-function analysis and helps avoid

incorrect closure picks in cases of PDL and tip extension

5 Flow regimes can be identified using the semilog pressure

derivative on the log-log plot of ∆pwf − ∆t during the

shut-in period followshut-ing the fracture shut-injection test

6 As in conventional transient test analysis, a pseudolinear

flow period is identified by parallel ½ slope lines,

separated by 2x, on the log-log ∆pwf − ∆t plot up until

fracture closure

7 Bilinear flow can be identified by parallel ¼ slope lines

separated by 4x on the log-log ∆pwf − ∆t plot prior to

fracture closure

8 After closure the pseudolinear reservoir flow period is

identified by a -1/2 slope of the semilog derivative of the

pressure difference on the log-log ∆pwf − ∆t plot, and a

−3/2 slope of the first derivative of the pressure difference

with shut-in time on the same plot

9 Pseudoradial flow is identified by a -1 slope of the

semilog derivative on the log-log plot

10 When a stable pseudolinear flow period exists, the after-closure Cartesian plot of the linear flow function can be used to estimate reservoir pressure

11 When a pseudoradial flow period exists, both the conventional Horner analysis and after-closure radial-flow analysis can be used to determine reservoir transmissibility and pore pressure

Acknowledgements

The authors would like to thank Kumar Ramurthy, Halliburton, Mike Conway, Stim-Lab, and Stuart Cox, Marathon, for their discussion and contribution to the procedures described Sincere thanks are also due to the many operators whose diligence in pre-frac testing has allowed these diagnostic analysis procedures to be developed and tested

Nomenclature

A f = fracture area, L2, ft2

B = formation volume factor, L 3 /L 3 , RB/STB

c t = total compressibility, Lt2/m, psi-1

C bl = bilinear flow constant, m/Lt5/4, psi hr3/4

C pl = pseudolinear flow constant, m/Lt3/2, psi hr1/2

C fbc = before-closure fracture storage, L4t2/m, bbl/psi

F L = linear flow time function, dimensionless

F R = radial flow time function, dimensionless

g = loss-volume function, dimensionless

G = G-function, dimensionless

h = height, L, ft

k = permeability, L2, md

L f = fracture half-length, L, ft

m H = slope of data on Horner plot, m/Lt2, psia

m L = slope of data on pseudolinear flow graph, m/Lt 2 , psia

m R = slope of data on pseudoradial flow graph, m/Lt2, psia

p = pressure, m/Lt 2 , psia

p wf = fracture pressure measured at wellbore, m/Lt2, psia

q = flow rate, L3/t, bbl/D

Q t = total injection volume, L3, bbl

r p = storage ratio, dimensionless

S f = fracture stiffness, m/L2t2, psi/ft

t = time, hr

t a = adjusted pseudotime, hr

Greek

= constant, dimensionless = difference, dimensionless = constant, dimensionless

µ = viscosity, m/Lt, cp

φ = porosity, dimensionless

Subscripts

D = dimensionless

e = end of injection

0 = end of injection

z = process zone

References

1 Nolte, K G.: “Determination of Fracture Parameters from

Fracturing Pressure Decline, paper SPE 3841, presented at the

Annual Technical Conference and Exhibition, Las Vegas, NV,

Sept 23-26, 1979

2 Castillo, J L.: “Modified Fracture Pressure Decline Analysis

Including Pressure-Dependent Leakoff,” paper SPE 16417,

presented at the SPE/DOE Low Permeability Reservoirs Joint

Symposium, Denver, CO, May 18-19, 1987

3 Barree, R D., and Mukherjee, H.: “Determination of Pressure

Dependent Leakoff and Its Effect on Fracture Geometry,” paper

SPE 36424, presented at the 71st Technical Conference and

Exhibition, Denver, CO, Oct 6-9, 1996

4 Barree, R.D.: "Applications of Pre-Frac Injection/Falloff Tests in

Fissured Reservoirs—Field Examples," paper SPE 39932

presented at the SPE Rocky Mountain

Regional/Low-Permeability Reservoirs Symposium, Denver, Apr 5-8, 1998

5 Talley, G R., Swindell, T M., Waters, G A and Nolte, K G.:

“Field Application of After-Closure Analysis of Fracture

Calibration Tests,” paper SPE 52220, presented at the 1999 SPE

Mid-Continent Operations Symposium, Oklahoma City, OK,

March 28–31, 1999

6 Cinco-Ley, H., and Samaniego-V., F.: “Transient Pressure

Analysis for Fractured Wells,” JPT (September 1981) 1749

7 Settari, A.: “Coupled Fracture and Reservoir Modeling,”

presented at the Workshop on Three Dimensional and Advanced

Hydraulic Fracture Modeling, held in conjunction with the Fourth

North American Rock Mechanics Symposium, July 29, 2000,

Seattle, WA

8 Craig, D P and Blasingame, T A.: “Application of a New

Fracture-Injection/Falloff Model Accounting for Propagating,

Dilated, and Closing Hydraulic Fractures,” paper SPE 1005778

presented at the SPE Gas Technology Symposium, Calgary,

Alberta, Canada, May, 15-17, 2006

9 Hagoort, J.: "Waterflood-Induced Hydraulic Facturing," PhD

Thesis, Delft Technical University, 1981

10 Koning, E.J.L and Niko, H.: "Fractured Water-Injection Wells:

A Pressure Falloff Test for Determining Fracture Dimensions,"

paper SPE 14458 presented at the 1985 Annual Technical

Conference and Exhibition of the Society of Petroleum

Engineers, Las Vegas, NV, September, 22-25, 1985

11 Cinco-Ley, H., Kuchuk, F., Ayoub, J., Samaniego-V, F., and

Ayestaran, L.: "Analysis of Pressure Tests Through the Use of

Instantaneous Source Response Concepts," paper SPE 15476

presented at the 61st Annual Technical Conference and Exhibition

of the Society of Petroleum Engineers, New Orleans, LA,

October, 5-8, 1986

Appendix A - Definition of diagnostic functions

The G-Function

The G-function is a representation of the elapsed time after

shut-in normalized to the duration of fracture extension

Corrections are made for the superposition of variable leakoff

times while the fracture is growing The form of the

G-function used in this paper assumes high fluid efficiency in

low-permeability formations Under that assumption the

surface area of the fracture is assumed to vary linearly with

time during fracture propagation The dimensionless pumping

time used in the G-function is defined as:

( )/

∆ = − (A-l)

The elapsed total time from the start of fracture initiation (not start of pumping) is t and the total pumping time (elapsed time from fracture initiation to shut-in) in consistent time units

is tP For the assumption of low leakoff the dimensionless time (∆tD) is used to compute an intermediate function:

( ) [ ( )1 5 1 5]

1 3

4

D D

t

The G-function used in the diagnostic plots is derived from the intermediate function as follows:

π

where g 0 is the dimensionless loss-volume function at shut-in

(t = t p or ∆tD = 0) All derivatives are calculated using a central difference function of pressure and G-function (normalized shut-in time)

After-Closure Analysis and Flow Regime Identification

After-closure pressure decline analysis requires the identification of fully-developed reservoir pseudolinear and pseudoradial transient flow regimes The flow regimes can be identified by characteristic slopes on a log-log plot of

observed falloff pressure minus reservoir pressure, (p w (t) − p i), versus the square of the linear-flow time function (FL2) and the semilog derivative, (X*dY/dX), of the pressure difference curve.5 It is important to note that the guess of reservoir

pressure, p i, used in construction of the flow regime plot severely impacts the slope and magnitude of the pressure difference curve The pressure derivative, because of the difference function used to generate it, is not affected by the initial guess of reservoir pressure

The linear-flow time function is defined by:

c

t

t t

t

sin

2 ,

The linear-flow function also requires an accurate determination of the time required after shut-in to reach

fracture closure, t c In the pseudolinear flow period the slope

of the derivative curve on the log-log plot should be ½ For the correct estimate of reservoir pore pressure, the pressure difference curve should also have a slope of ½ and should be exactly twice the magnitude of the derivative If a stable pseudolinear flow period is identified then a Cartesian plot of

observed pressure during the falloff, p w (t), versus FL should yield a straight line with intercept equal to the reservoir pore

pressure, p i , and with a slope of m L

p t −p =m F t t (A-5)

If a pseudoradial flow period exists, the slope of the derivative and correct pressure difference curves on the log-log flow regime plot should both be 1.0 and the two curves should coincide In the pseudoradial flow period, a Cartesian plot of pressure versus FR should also yield a straight line with

intercept equal to p i and slope of m R