Copyright 2007 Society of Petroleum Engineers This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil amp Gas Technology Symposium held in Denver Colorado USA 16ndash18 April 2007 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 as presented have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s) The material as presented does not necessarily reflect any position of the Society of Petroleum Engineers its officers or members Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers Electronic reproduction distribution or storage of any part of this paper for commercial purposes 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 where and by whom the paper was presented Write Librarian SPE PO Box 833836 Richardson Texas 75083-3836 USA fax 01-972-952-9435

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 derivatives1-4 When appropriate the after-closure analysis is presented for each case as is an empirical correlation for permeability from the identified G-function closure time5 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 ldquoknownrdquo reservoir pressure58 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 RD Barree SPE and VL Barree Barree amp Assocs LLC and DP Craig SPE Halliburton

2 SPE 107877

has been previously reported34 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 Samaniego6

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 leakoff7 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 (partypartx) and the semilog derivative (partypart(lnx) or xpartypartx) 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 10000 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 (GpartpwpartG) from the straight line through the origin During normal leakoff with constant fracture surface area and constant permeability the first derivative (partpwpartG) should also be constant2 The primary pw vs G curve should follow a straight line1 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=231 The closure event is marked by the dashed vertical line [1]

0 5 10 15 20 25G(Time)

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0

200

400

600

800

1000

1200

1400

1600

1800

20001

P vs G

GdPdG vs G

dPdG vs G

Fracture Closure

Pre

ssur

e

Der

ivat

ives

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 pw 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 pw vs sqrt(t) plot

The best way to find the inflection point is to plot the first derivative of pw vs sqrt(t) and find the point of maximum

SPE 107877 3

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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Time

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0

100

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300

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800

900

10001

Pre

ssur

e

Der

ivat

ives

P vs radict

radictdPdradict vs radict

dPdradict vs radict

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 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 frac12 slope is observed strongly suggesting linear flow from the fracture In this example the slope is greater than frac12 suggesting possible linear flow coupled with changing fracturewellbore 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

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 901 1 10 100 1000

Time (0 = 815)

2

3

4

5

6789

2

3

4

5

6789

2

3

4

10

100

1000

(m = -1)

(m = 0632)

BH ISIP = 9998 psi 1

Del

ta-P

ress

ure

and

Der

ivat

ive

∆P vs ∆t

∆td∆Pd∆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 -12 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 45 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 10 during the identified pseudoradial flow period If a linear-flow period existed in this data set a derivative slope of frac12 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 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 90001 001 01 1

Square Linear Flow (FL^2)

2

3

4

56789

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3

4

56789

2

3

4

56789

10

100

1000

10000

(m = 1)

Del

ta-P

ress

ure

and

Der

ivat

ive

∆P vs FL2

FL2 d∆PdFL

2 vs FL2

∆P=(pw-pr)

Start of Radial Flow

Figure 4 Normal leakoff ACA log plot

4 SPE 107877

If a pseudoradial flow regime is identified then the

Cartesian Radial Flow plot (Figure 5) can be used to determine reservoir far-field transmissibility khmicro 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 khmicro = 299 md-ftcp For gas viscosity at reservoir temperature kh=79 md-ft For the assumed net pay the effective reservoir permeability is 0097 md

00 02 04 06 08 10 12 14 16 18Radial Flow Time Function

7200

7400

7600

7800

8000

8200

8400

8600

8800

9000

9200

(m = 48142) Results

Reservoir Pressure = 747568 psiTransmissibility khmicro = 29894991 mdftkh = 794014 mdftPermeability k = 00968 mdStart of Pseudo Radial Time = 215 hours1

Pre

ssur

e

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 184 bpm khmicro = 298 md-ftcp For the assumed gas viscosity kh=79 md-ft Using the same assumed net gives k=0097 md This result is consistent with the ACA results

2 31

Horner Time

7250

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7750

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8250

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8750

9000

9250

9500

9750

(m = 14411)

(Reservoir = 7476)

1

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e

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 0097 md is consistent with the Horner and ACA results

0001

001

01

1

10

100

0 2 4 6 8 10 12 14 16 18 20

Gc

Data Inputrp 1φ 009 VVct 750E-05 psi-1

E 35 Mpsimicro 1 cp

Gc 244Pz 9660 psi

Estimated Permeability = 00974 md

Perm

eabi

lity

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

The primary pw vs G curve is concave upward and curved while PDL persists The semilog derivative exhibits the characteristic ldquohumprdquo 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 ldquohumprdquo 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

SPE 107877 5

0 2 4 6 8 10 12 14 16 18 20G(Time)

8250

8500

8750

9000

9250

9500

9750

10000

10250

10500

0

100

200

300

400

500

600

700

800

900

100021

P vs G

GdPdG vs G

dPdG vs G

Fracture Closure

Pre

ssur

e

Der

ivat

ives

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 ldquohumprdquo 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

12420070020 0040 0100 0120 0140

12420070200

Time

8250

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0

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5001False Closure

Pre

ssur

e

Der

ivat

ives

P vs radict

radictdPdradict vs radict

dPdradict vs radict

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 frac12 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 minusfrac12 slope or reservoir pseudolinear flow period is shown shortly after closure The later data approach a slope of minus1 which indicates pseudoradial flow has been established

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 901 1 10 100 1000

Time (0 = 9133333)

2

3

4

5

6789

2

3

4

5

6789

2

10

100

1000

(m = 05)

(m = -1)

(m = -05)

BH ISIP = 10000 psi 1

Pres

sure

Diff

eren

ce a

nd D

eriv

ativ

e ∆P vs ∆t

∆td∆Pd∆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

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 90001 001 01 1

Square Linear Flow (FL^2)

2

3

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56789

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3

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56789

2

3

4

56789

10

100

1000

10000

(m = 1)

(m = 05)

123

Del

ta-P

ress

ure

and

Der

ivat

ive

∆P vs FL2

FL2d∆PdFL

2 vs FL2

∆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 372 md-ftcp and estimated permeability of 0047 md

6 SPE 107877

00 01 02 03 04 05 06 07 08 09 10Linear Flow Time Function

8000

8200

8400

8600

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9200

(m = 14385)

ResultsReservoir Pressure = 805666 psiStart of Pseudo Linear Time = 159End of Pseudo Linear Time = 543912

Pre

ssur

e

Start Linear FlowEnd Linear Flow

Figure 13 PDL ACA linear flow plot

00 01 02 03 04 05 06

Radial Flow Time Function

8000

8200

8400

8600

8800

9000

9200

(m = 11373) Results

Reservoir Pressure = 806881 psiTransmissibility khmicro = 3721984 mkh = 093764 mdftPermeability k = 00469 mdStart of Pseudo Radial Time = 11261

Pre

ssur

e

Figure 14 PDL ACA radial flow plot

Horner Analysis for PDL Example For an average pump rate of 67 bpm the Horner plot gives

khmicro=3572 md-ftcp The Horner estimated permeability is 0046 md compared to 0047 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

1Horner Time

8000

8100

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8600

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8800

8900

(m = 43920)

(Reservoir = 8064)

1

Pre

ssur

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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 0045 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

0001

001

01

1

10

100

0 2 4 6 8 10 12 14 16 18 20

Gc

Data Inputrp 1φ 008 VVct 600E-05 psi-1

E 5 Mpsimicro 1 cp

Gc 29Pz 8410 psi

Estimated Permeability = 00453 md

Per

mea

bilit

y m

d

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

SPE 107877 7

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 curvature34 Figure 17 shows a typical case of fracture tip extension with minimal leakoff This is another case that is frequently misdiagnosed

0 5 10 15 20 25 30G(Time)

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125

1501

P vs G

GdPdG vs G

dPdG vs G

Pre

ssur

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ivat

ives

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

12520070400 0800 1200 1600

12520072000

Time

8400

8600

8800

9000

9200

9400

9600

9800

10000

10200

0

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20

30

40

50

60

70

80

90

1001

Incorrect Closure

Pre

ssur

e

Der

ivat

ivesP vs radict xdPdx vs radict

dPdx vs radict

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

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 21 10 100 1000

Time (0 = 337)

2

3

4

56789

2

3

4

56789

2

3

4

5

10

100

1000

(m = 025)

BH ISIP = 10000 psi 1

Del

ta-P

ress

ure

and

Der

ivat

ive

∆P vs ∆t

∆td∆Pd∆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 frac14 slope during fracture tip extension The pressure difference curve falls on a parallel frac14 slope line separated by 4-times the magnitude of the derivative The frac14 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 frac14 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 ldquofracture height recessionrdquo 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

SPE 107877 7

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 curvature34 Figure 17 shows a typical case of fracture tip extension with minimal leakoff This is another case that is frequently misdiagnosed

0 5 10 15 20 25 30G(Time)

8400

8600

8800

9000

9200

9400

9600

9800

10000

10200

0

25

50

75

100

125

1501

P vs G

GdPdG vs G

dPdG vs G

Pre

ssur

e

Der

ivat

ives

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

12520070400 0800 1200 1600

12520072000

Time

8400

8600

8800

9000

9200

9400

9600

9800

10000

10200

0

10

20

30

40

50

60

70

80

90

1001

Incorrect Closure

Pre

ssur

e

Der

ivat

ivesP vs radict xdPdx vs radict

dPdx vs radict

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

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 21 10 100 1000

Time (0 = 337)

2

3

4

56789

2

3

4

56789

2

3

4

5

10

100

1000

(m = 025)

BH ISIP = 10000 psi 1

Del

ta-P

ress

ure

and

Der

ivat

ive

∆P vs ∆t

∆td∆Pd∆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 frac14 slope during fracture tip extension The pressure difference curve falls on a parallel frac14 slope line separated by 4-times the magnitude of the derivative The frac14 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 frac14 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 ldquofracture height recessionrdquo 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

8 SPE 107877

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 ldquotransverse fracture storagerdquo

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

ldquobellyrdquo below the straight line through the origin and tangent to the semilog derivative of pw 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

5 10 15 20G(Time)

7500

8000

8500

9000

9500

10000

10500

0

200

400

600

800

1000

1200

1400

1600

1800

20001

P vs G

GdPdG vs G

dPdG vs G

Fracture Closure

Pre

ssur

e

Der

ivat

ives

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

12420070200 0400 0600 0800

12420071000

Time

7500

8000

8500

9000

9500

10000

10500

0

100

200

300

4001

Pre

ssur

e

Der

ivat

ives

P vs radict

xdPdx vs radict

dPdx vs radict

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 tc) is used The observed closure time must be corrected by multiplying by the storage ratio rp The magnitude of rp 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 10 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 rp can be as low as 05 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 10 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 -12 slope with a pseudoradial flow period indicated by a -1 slope of the semi-log derivative

SPE 107877 9

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 601 1 10 100

Time (0 = 9416667)

2

3

4

5

6789

2

3

4

5

6789

2

3

10

100

1000

(m = 05) (m = -05)

BH ISIP = 10000 psi 1

Del

ta-P

ress

ure

and

Der

ivat

ive

∆P vs ∆t

∆td∆Pd∆t vs ∆t

Figure 22 Storage log-log plot

Conclusions The use of pre-frac injectionfalloff 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 minus ∆t during the shut-in period following the fracture injection test

6 As in conventional transient test analysis a pseudolinear flow period is identified by parallel frac12 slope lines separated by 2x on the log-log ∆pwf minus ∆t plot up until fracture closure

7 Bilinear flow can be identified by parallel frac14 slope lines separated by 4x on the log-log ∆pwf minus ∆t plot prior to fracture closure

8 After closure the pseudolinear reservoir flow period is identified by a -12 slope of the semilog derivative of the pressure difference on the log-log ∆pwf minus ∆t plot and a minus32 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

Af = fracture area L2 ft2

B = formation volume factor L3L3 RBSTB

ct = total compressibility Lt2m psi-1 Cac = after-closure storage L4t2m bblpsi Cbl = bilinear flow constant mLt54 psihr34

Cpl = pseudolinear flow constant mLt32 psihr12 Cpr = pseudoradial flow constant mLt psihr Cfbc = before-closure fracture storage L4t2m bblpsi FL = linear flow time function dimensionless FR = radial flow time function dimensionless g = loss-volume function dimensionless G = G-function dimensionless h = height L ft k = permeability L2 md Lf = fracture half-length L ft mH = slope of data on Horner plot mLt2 psia mL = slope of data on pseudolinear flow graph mLt2 psia mR = slope of data on pseudoradial flow graph mLt2 psia p = pressure mLt2 psia pwf = fracture pressure measured at wellbore mLt2 psia q = flow rate L3t bblD Qt = total injection volume L3 bbl rp = storage ratio dimensionless Sf = fracture stiffness mL2t2 psift t = time hr ta = adjusted pseudotime hr

Greek β = constant dimensionless ∆ = difference dimensionless χ = constant dimensionless micro = viscosity mLt cp φ = porosity dimensionless

Subscripts a = adjusted c = closure D = dimensionless e = end of injection f = filtrate p = pumping 0 = end of injection w = wellbore z = process zone

10 SPE 107877

References 1 Nolte K G ldquoDetermination 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 ldquoModified Fracture Pressure Decline Analysis Including Pressure-Dependent Leakoffrdquo paper SPE 16417 presented at the SPEDOE Low Permeability Reservoirs Joint Symposium Denver CO May 18-19 1987

3 Barree R D and Mukherjee H ldquoDetermination of Pressure Dependent Leakoff and Its Effect on Fracture Geometryrdquo paper SPE 36424 presented at the 71st Technical Conference and Exhibition Denver CO Oct 6-9 1996

4 Barree RD Applications of Pre-Frac InjectionFalloff Tests in Fissured ReservoirsmdashField Examples paper SPE 39932 presented at the SPE Rocky Mountain RegionalLow-Permeability Reservoirs Symposium Denver Apr 5-8 1998

5 Talley G R Swindell T M Waters G A and Nolte K G ldquoField Application of After-Closure Analysis of Fracture Calibration Testsrdquo paper SPE 52220 presented at the 1999 SPE Mid-Continent Operations Symposium Oklahoma City OK March 28ndash31 1999

6 Cinco-Ley H and Samaniego-V F ldquoTransient Pressure Analysis for Fractured Wellsrdquo JPT (September 1981) 1749

7 Settari A ldquoCoupled Fracture and Reservoir Modelingrdquo 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 ldquoApplication of a New Fracture-InjectionFalloff Model Accounting for Propagating Dilated and Closing Hydraulic Fracturesrdquo 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 EJL 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

( ) D p pt t t t∆ = minus (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

( ) ( )[ ]5151134

DDD tttg ∆minus∆+=∆ (A-2)

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

( ) ( ) 04

D DG t g t gπ

∆ = ⎡ ∆ minus ⎤⎣ ⎦ (A-3)

where g0 is the dimensionless loss-volume function at shut-in (t = tp 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 (pw(t) minus pi) versus the square of the linear-flow time function (FL

2) and the semilog derivative (XdYdX) of the pressure difference curve5 It is important to note that the guess of reservoir pressure pi 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

( ) cc

cL ttforttttF ge= minus1sin2

π (A-4)

The linear-flow function also requires an accurate determination of the time required after shut-in to reach fracture closure tc In the pseudolinear flow period the slope of the derivative curve on the log-log plot should be frac12 For the correct estimate of reservoir pore pressure the pressure difference curve should also have a slope of frac12 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 pw(t) versus FL should yield a straight line with intercept equal to the reservoir pore pressure pi and with a slope of mL

( ) ( )w i L L cp t p m F t tminus = (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 10 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 pi and slope of mR

SPE 107877 11

( ) ( )w i R R cp t p m F t tminus = (A-6)

In these equations tc is the time to fracture closure with

time zero set as the beginning of fracture extension pi is the initial reservoir pressure and mR is the Cartesian slope of the ldquocorrectrdquo straight line The radial-flow function (FR) is given by5

( ) 61161ln41 2 cong=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus

+=π

χχ

c

ccR tt

tttF (A-7)

In the properly identified pseudoradial flow period the reservoir far-field transmissibility can be determined from the slope fracture closure time and volume injected during the test

251000 t

R c

Qkhm tmicro

= (A-8)

In Equation A-8 the permeability k is in md net pay thickness h is in feet far-field mobile fluid viscosity micro in cp tc in minutes and the volume injected during the test Qt is

in bbls and the slope mR is in psi Horner analysis

The conventional Horner analysis uses a Cartesian plot of observed pressure versus Horner time (tp + ∆t)∆t with all times in consistent units The fracture propagation time is tp and the elapsed shut-in time is ∆t As shut-in time approaches infinity the Horner time function approaches 1 A straight-line extrapolation of the Horner plot to the intercept at a Horner time of 10 gives an estimate of reservoir pressure The slope of the correct straight-line extrapolation mH can be used to estimate reservoir transmissibility

( )Hm

qkh 14406162=

micro (A-9)

The flow rate in Equation A-9 is assumed to be in barrels per minute and is the average rate for the time the fracture was extending In both Equations A-8 and A-9 the viscosity is the far-field mobile fluid viscosity The propagation of the transient in pseudoradial flow occurs at a great distance from the fracture and is not affected by the injected fluid viscosity

The major problem with the Horner analysis is that the results are only valid if the data used to extrapolate the apparent straight line are actually in fully developed pseudoradial flow There is no way to determine the validity of the Horner analysis or to determine the flow regime within the Horner plot itself

G-function permeability estimate

An empirical function to approximate formation permeability has been derived from numerous numerical simulations of fracture closure The correlation is based on the observed G-function time at fracture closure

19600086 001

0038

f z

c pt

pk

G E rc

micro

φ

=⎛ ⎞⎜ ⎟⎝ ⎠

(A-10)

The rate of fluid loss from the fracture before closure is dominated by the mobility of the injected fluid instead of the far-field viscosity The total mobility of the injected fluid during leakoff is dependent on the viscosity of the injected fluid at leakoff temperature and the relative permeability to the leakoff fluid The reservoir fluid and its residual saturation in the invaded region will have some effect on the leakoff fluid mobility As a general rule an assumed injected fluid viscosity of 10 cp is used to incorporate the effects of reservoir temperature and relative permeability in the invaded zone

Note that the correlation gives permeability and not kh The time to closure is related to fracture volume versus created area so both the fracture height and length do not appear in the equation The process zone stress Pz is the net fracture extension pressure above closure pressure pc or pz = pISIP minus pc The net extension pressure and Youngrsquos Modulus provide a relationship between facture volume (width) and surface area during pumping

The other parameters in Equation A-10 are defined except the storage ratio rp This parameter represents the amount of excess fluid that must be leaked-off to reach fracture closure when the fracture geometry deviates from the normally assumed constant-height planar fracture The storage ratio rp is the ratio of the area under the G-function semilog derivative up until closure divided by the area of the triangle defined by the straight-line (normal leakoff) tangent to the semilog derivative at closure For normal matrix leakoff and PDL rp is therefore 10 For the transverse storage and height recession signature rp is some value less than 10 The observed G-function closure time (Gc) is always delayed by the excess fluid volume in the fracture for either height recession or transverse storage Likewise the apparent time to closure used in the after-closure analysis will be delayed and will cause errors in the reservoir transmissibility estimate The closure time should be corrected by multiplying by rp for all permeability and transmissibility estimates

Appendix B - Analytical Solutions and the Log-Log Diagnostic Graph Linear and Bilinear Flow Before-Closure Hagoort9 developed a before-closure analytical solution accounting for both storage and linear formation flow Koning and Niko10 write the solution as

( )( )22

12DtD fbcD Dp t C e erfc twD

βππ β∆⎛ ⎞= ∆ minus minus⎜ ⎟ ∆⎝ ⎠ (B-1)

where

( )( ) ( )( )

1412w e w e

Dp t p t tkh

p twD qBmicrominus + ∆

∆ = (B-2)

200002637

Dt f

k ttc Lφmicro

∆∆ =

(B-3)

12 SPE 107877

2

fbcDCβ

π= (B-4)

and

2 25615 08936

2fbc fbc

fbcDt f t f

C CC

c hL c hLπφ φ= =

(B-5)

Here Cfbc is the before-closure fracture storage constant which is defined as8

25615

ffbc

f

AC

S= (B-6)

where Af is the fracture area (one wing) and Sf is the fracture stiffness

Hagoorts solution predicts that the before-closure falloff is a combination of fracture storage and linear flow As fracture storage becomes small CfbcDrarr0 linear flow will dominate the before-closure pressure falloff In most hard rock environments CfbcD is small and for a fracture-injectionfalloff sequence with ∆t gtgt te and long closure times

( ) ( )w e w ep t p t t tminus + ∆ prop (B-7)

where t = te + ∆t and pw(te) is the instantaneous shut-in pressure Define a fracture-pressure difference as

( ) ( )wf w e w ep p t p t t∆ = minus + ∆ (B-8)

and a log-log graph of ∆pwf vs t will exhibit a frac12 slope during linear flow before closure An analytical before-closure bilinear flow solution accounting for fracture tip extension during shut-in does not exist but field data suggest that during before-closure bilinear flow

1 4( )wfp t∆ prop ∆ (B-9)

and a log-log graph of ∆pwf vs t will exhibit a frac14 slope during bilinear flow before closure Pseudoradial Pseudolinear and Bilinear Flow After-Closure Craig amp Blasingame8 developed an analytical solution for a fracture-injectionfalloff sequence with a propagating and closing hydraulic fracture and they derived the complete after-closure impulse solutions accounting for fracture storage The after-closure impulse solution for pseudoradial flow is written as

( )01412(24) 1(0) ( )( )

2 t iw e ie

Q p C p pp t t p wsD ackh t tmicro + minus+ ∆ minus =

+ ∆ (B-10)

For after-closure pseudolinear flow

0

1412(24) 00002637 1( )(0) ( )2

tw e i

if

Qp t t p

p C p phL wsD ac

π ⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 21

t ec k t tmicro

φ⎛ ⎞

times⎜ ⎟+ ∆⎝ ⎠

(B-11)

and for after-closure bilinear flow 1 4

3 40

1412(24)(06125)(00002637)( )( )t

w e iif f

Qp t t p

p p Ck w acmicro

⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 4 3 41 1

t ec k t tmicro⎛ ⎞ ⎛ ⎞

times⎜ ⎟ ⎜ ⎟+ ∆⎝ ⎠ ⎝ ⎠

(B-12)

For convenience define a constant pseudoradial flow term as

( )01412(24) (0) ( )

2 t ipr Q p C p pC wsD ackhmicro + minus= (B-13)

For pseudolinear flow a constant term is defined as 1 2

0

1412(24) 00002637 1(0) ( )2

tpl

tif

QC

c kp C p phL wsD ac

microπφ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-14)

Similarly for bilinear flow a constant term is written as

1 41 43 4

0

11412(24)(06125)(00002637)( )t

bltif f

QC

c kp p Ck w acmicro

micro⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-15)

With the new definition and defining a reservoir-pressure difference ∆pw = pw(t) minus pi the pseudoradial-flow impulse solution can be written as

1w prp C tminus∆ = (B-16)

Similarly the linear-flow impulse solution can be written as 1 2

w plp C tminus∆ = (B-17)

and the bilinear-flow impulse solution can be written as 3 4

w blp C tminus∆ = (B-18)

The derivatives with respect to t are written for after-closure pseudoradial flow as

2( )

wpr

pC t

tminuspart∆

minus =part

(B-19)

for after-closure pseudolinear flow as

3 2( ) 2

plw Cpt

tminuspart∆

minus =part

(B-20)

and for after-closure bilinear flow as

7 43( ) 4

w blp Ct

tminuspart∆

minus =part

(B-21)

The semilog derivatives and the impulse derivatives11 are calculated by multiplying the after-closure derivatives by t and t2 respectively Table B-1 shows the derivative terms for each after-closure flow regime

Log-Log Diagnostic Graph Before-closure linear or bilinear flow are identified by the relationship between the fracture-pressure difference ∆pwf and total time t but the after-closure flow regimes are identified by the relationship between the reservoir-pressure difference ∆pw and t which requires knowing the initial reservoir pressure pi However by noting that

wf wp pt t

part∆ part∆= minus

part part (B-22)

a log-log diagnostic graph can be prepared that is independent of initial reservoir pressure but can be used to identify before- and after-closure flow regimes Table B-2 shows the characteristic slopes of each flow regime for log-log graphs of ∆pwf vs t wfp tpart∆ part vs t wft p tpart∆ part vs t and

SPE 107877 13

2wft p tpart∆ part vs t

In gas reservoirs log-log diagnostic graphs can be prepared by plotting ∆pawf vs ta awf ap tpart∆ part vs ta a awf at p tpart∆ part vs ta

and 2a awf at p tpart∆ part vs ta where adjusted pseudopressure is

defined as

0

pz pdppa p zi

micromicro

⌠⎮⎮⌡

⎛ ⎞= ⎜ ⎟⎝ ⎠

(B-23)

and adjusted pseudotime is defined as

( )0

t dtt ca t i ctmicro

micro

⌠⎮⎮⌡

= (B-24)

Since adjusted pseudopressure is a function of initial reservoir pressure and if initial reservoir pressure is unknown its often helpful to complete the analysis in terms of pressure and time to obtain a first estimate of initial reservoir pressure before refining the estimate using diagnostic and interpretive graphs plotted in terms of adjusted pseudovariables

Table B-1 After-closure impulse solutions and derivatives

Flow Regime Impulse Solution Derivative Semilog Derivative

Impulse Derivative

Bilinear 3 4w blp C tminus∆ = 7 43

4w blp C

tt

minuspart∆minus =

part 3 43

4w blp C

t tt

minuspart∆minus =

part 2 1 43

4w blp C

t tt

part∆minus =

part

Pseudolinear 1 2w plp C tminus∆ = 3 2

2plw Cp

tt

minuspart∆minus =

part 1 2

2plw Cp

t tt

minuspart∆minus =

part 2 1 2

2plw Cp

t tt

part∆minus =

part

Pseudoradial 1w prp C tminus∆ = 2w

prp

C tt

minuspart∆minus =

part 1w

prp

t C tt

minuspart∆minus =

part 2 w

prp

t Ct

part∆minus =

part

Table B-2 Log-log graph characteristic slopes

Before Closure After Closure Log-Log Graph

Bilinear Linear Bilinear Pseudolinear Pseudoradial

vs

vs wf

awf a

p t

p t

∆

∆ 14 12 ⎯ ⎯ ⎯

vs

vs wf

awf a a

p t t

p t t

part∆ part

part∆ part minus34 minus12 minus74 minus32 minus2

vs

vs wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 14 12 minus34 minus12 minus1

2

2

vs

vs

wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 54 32 14 12 0

10 SPE 107877

References 1 Nolte K G ldquoDetermination 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 ldquoModified Fracture Pressure Decline Analysis Including Pressure-Dependent Leakoffrdquo paper SPE 16417 presented at the SPEDOE Low Permeability Reservoirs Joint Symposium Denver CO May 18-19 1987

3 Barree R D and Mukherjee H ldquoDetermination of Pressure Dependent Leakoff and Its Effect on Fracture Geometryrdquo paper SPE 36424 presented at the 71st Technical Conference and Exhibition Denver CO Oct 6-9 1996

4 Barree RD Applications of Pre-Frac InjectionFalloff Tests in Fissured ReservoirsmdashField Examples paper SPE 39932 presented at the SPE Rocky Mountain RegionalLow-Permeability Reservoirs Symposium Denver Apr 5-8 1998

5 Talley G R Swindell T M Waters G A and Nolte K G ldquoField Application of After-Closure Analysis of Fracture Calibration Testsrdquo paper SPE 52220 presented at the 1999 SPE Mid-Continent Operations Symposium Oklahoma City OK March 28ndash31 1999

6 Cinco-Ley H and Samaniego-V F ldquoTransient Pressure Analysis for Fractured Wellsrdquo JPT (September 1981) 1749

7 Settari A ldquoCoupled Fracture and Reservoir Modelingrdquo 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 ldquoApplication of a New Fracture-InjectionFalloff Model Accounting for Propagating Dilated and Closing Hydraulic Fracturesrdquo 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 EJL 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

( ) D p pt t t t∆ = minus (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

( ) ( )[ ]5151134

DDD tttg ∆minus∆+=∆ (A-2)

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

( ) ( ) 04

D DG t g t gπ

∆ = ⎡ ∆ minus ⎤⎣ ⎦ (A-3)

where g0 is the dimensionless loss-volume function at shut-in (t = tp 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 (pw(t) minus pi) versus the square of the linear-flow time function (FL

2) and the semilog derivative (XdYdX) of the pressure difference curve5 It is important to note that the guess of reservoir pressure pi 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

( ) cc

cL ttforttttF ge= minus1sin2

π (A-4)

The linear-flow function also requires an accurate determination of the time required after shut-in to reach fracture closure tc In the pseudolinear flow period the slope of the derivative curve on the log-log plot should be frac12 For the correct estimate of reservoir pore pressure the pressure difference curve should also have a slope of frac12 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 pw(t) versus FL should yield a straight line with intercept equal to the reservoir pore pressure pi and with a slope of mL

( ) ( )w i L L cp t p m F t tminus = (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 10 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 pi and slope of mR

SPE 107877 11

( ) ( )w i R R cp t p m F t tminus = (A-6)

In these equations tc is the time to fracture closure with

time zero set as the beginning of fracture extension pi is the initial reservoir pressure and mR is the Cartesian slope of the ldquocorrectrdquo straight line The radial-flow function (FR) is given by5

( ) 61161ln41 2 cong=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus

+=π

χχ

c

ccR tt

tttF (A-7)

In the properly identified pseudoradial flow period the reservoir far-field transmissibility can be determined from the slope fracture closure time and volume injected during the test

251000 t

R c

Qkhm tmicro

= (A-8)

In Equation A-8 the permeability k is in md net pay thickness h is in feet far-field mobile fluid viscosity micro in cp tc in minutes and the volume injected during the test Qt is

in bbls and the slope mR is in psi Horner analysis

The conventional Horner analysis uses a Cartesian plot of observed pressure versus Horner time (tp + ∆t)∆t with all times in consistent units The fracture propagation time is tp and the elapsed shut-in time is ∆t As shut-in time approaches infinity the Horner time function approaches 1 A straight-line extrapolation of the Horner plot to the intercept at a Horner time of 10 gives an estimate of reservoir pressure The slope of the correct straight-line extrapolation mH can be used to estimate reservoir transmissibility

( )Hm

qkh 14406162=

micro (A-9)

The flow rate in Equation A-9 is assumed to be in barrels per minute and is the average rate for the time the fracture was extending In both Equations A-8 and A-9 the viscosity is the far-field mobile fluid viscosity The propagation of the transient in pseudoradial flow occurs at a great distance from the fracture and is not affected by the injected fluid viscosity

The major problem with the Horner analysis is that the results are only valid if the data used to extrapolate the apparent straight line are actually in fully developed pseudoradial flow There is no way to determine the validity of the Horner analysis or to determine the flow regime within the Horner plot itself

G-function permeability estimate

An empirical function to approximate formation permeability has been derived from numerous numerical simulations of fracture closure The correlation is based on the observed G-function time at fracture closure

19600086 001

0038

f z

c pt

pk

G E rc

micro

φ

=⎛ ⎞⎜ ⎟⎝ ⎠

(A-10)

The rate of fluid loss from the fracture before closure is dominated by the mobility of the injected fluid instead of the far-field viscosity The total mobility of the injected fluid during leakoff is dependent on the viscosity of the injected fluid at leakoff temperature and the relative permeability to the leakoff fluid The reservoir fluid and its residual saturation in the invaded region will have some effect on the leakoff fluid mobility As a general rule an assumed injected fluid viscosity of 10 cp is used to incorporate the effects of reservoir temperature and relative permeability in the invaded zone

Note that the correlation gives permeability and not kh The time to closure is related to fracture volume versus created area so both the fracture height and length do not appear in the equation The process zone stress Pz is the net fracture extension pressure above closure pressure pc or pz = pISIP minus pc The net extension pressure and Youngrsquos Modulus provide a relationship between facture volume (width) and surface area during pumping

The other parameters in Equation A-10 are defined except the storage ratio rp This parameter represents the amount of excess fluid that must be leaked-off to reach fracture closure when the fracture geometry deviates from the normally assumed constant-height planar fracture The storage ratio rp is the ratio of the area under the G-function semilog derivative up until closure divided by the area of the triangle defined by the straight-line (normal leakoff) tangent to the semilog derivative at closure For normal matrix leakoff and PDL rp is therefore 10 For the transverse storage and height recession signature rp is some value less than 10 The observed G-function closure time (Gc) is always delayed by the excess fluid volume in the fracture for either height recession or transverse storage Likewise the apparent time to closure used in the after-closure analysis will be delayed and will cause errors in the reservoir transmissibility estimate The closure time should be corrected by multiplying by rp for all permeability and transmissibility estimates

Appendix B - Analytical Solutions and the Log-Log Diagnostic Graph Linear and Bilinear Flow Before-Closure Hagoort9 developed a before-closure analytical solution accounting for both storage and linear formation flow Koning and Niko10 write the solution as

( )( )22

12DtD fbcD Dp t C e erfc twD

βππ β∆⎛ ⎞= ∆ minus minus⎜ ⎟ ∆⎝ ⎠ (B-1)

where

( )( ) ( )( )

1412w e w e

Dp t p t tkh

p twD qBmicrominus + ∆

∆ = (B-2)

200002637

Dt f

k ttc Lφmicro

∆∆ =

(B-3)

12 SPE 107877

2

fbcDCβ

π= (B-4)

and

2 25615 08936

2fbc fbc

fbcDt f t f

C CC

c hL c hLπφ φ= =

(B-5)

Here Cfbc is the before-closure fracture storage constant which is defined as8

25615

ffbc

f

AC

S= (B-6)

where Af is the fracture area (one wing) and Sf is the fracture stiffness

Hagoorts solution predicts that the before-closure falloff is a combination of fracture storage and linear flow As fracture storage becomes small CfbcDrarr0 linear flow will dominate the before-closure pressure falloff In most hard rock environments CfbcD is small and for a fracture-injectionfalloff sequence with ∆t gtgt te and long closure times

( ) ( )w e w ep t p t t tminus + ∆ prop (B-7)

where t = te + ∆t and pw(te) is the instantaneous shut-in pressure Define a fracture-pressure difference as

( ) ( )wf w e w ep p t p t t∆ = minus + ∆ (B-8)

and a log-log graph of ∆pwf vs t will exhibit a frac12 slope during linear flow before closure An analytical before-closure bilinear flow solution accounting for fracture tip extension during shut-in does not exist but field data suggest that during before-closure bilinear flow

1 4( )wfp t∆ prop ∆ (B-9)

and a log-log graph of ∆pwf vs t will exhibit a frac14 slope during bilinear flow before closure Pseudoradial Pseudolinear and Bilinear Flow After-Closure Craig amp Blasingame8 developed an analytical solution for a fracture-injectionfalloff sequence with a propagating and closing hydraulic fracture and they derived the complete after-closure impulse solutions accounting for fracture storage The after-closure impulse solution for pseudoradial flow is written as

( )01412(24) 1(0) ( )( )

2 t iw e ie

Q p C p pp t t p wsD ackh t tmicro + minus+ ∆ minus =

+ ∆ (B-10)

For after-closure pseudolinear flow

0

1412(24) 00002637 1( )(0) ( )2

tw e i

if

Qp t t p

p C p phL wsD ac

π ⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 21

t ec k t tmicro

φ⎛ ⎞

times⎜ ⎟+ ∆⎝ ⎠

(B-11)

and for after-closure bilinear flow 1 4

3 40

1412(24)(06125)(00002637)( )( )t

w e iif f

Qp t t p

p p Ck w acmicro

⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 4 3 41 1

t ec k t tmicro⎛ ⎞ ⎛ ⎞

times⎜ ⎟ ⎜ ⎟+ ∆⎝ ⎠ ⎝ ⎠

(B-12)

For convenience define a constant pseudoradial flow term as

( )01412(24) (0) ( )

2 t ipr Q p C p pC wsD ackhmicro + minus= (B-13)

For pseudolinear flow a constant term is defined as 1 2

0

1412(24) 00002637 1(0) ( )2

tpl

tif

QC

c kp C p phL wsD ac

microπφ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-14)

Similarly for bilinear flow a constant term is written as

1 41 43 4

0

11412(24)(06125)(00002637)( )t

bltif f

QC

c kp p Ck w acmicro

micro⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-15)

With the new definition and defining a reservoir-pressure difference ∆pw = pw(t) minus pi the pseudoradial-flow impulse solution can be written as

1w prp C tminus∆ = (B-16)

Similarly the linear-flow impulse solution can be written as 1 2

w plp C tminus∆ = (B-17)

and the bilinear-flow impulse solution can be written as 3 4

w blp C tminus∆ = (B-18)

The derivatives with respect to t are written for after-closure pseudoradial flow as

2( )

wpr

pC t

tminuspart∆

minus =part

(B-19)

for after-closure pseudolinear flow as

3 2( ) 2

plw Cpt

tminuspart∆

minus =part

(B-20)

and for after-closure bilinear flow as

7 43( ) 4

w blp Ct

tminuspart∆

minus =part

(B-21)

The semilog derivatives and the impulse derivatives11 are calculated by multiplying the after-closure derivatives by t and t2 respectively Table B-1 shows the derivative terms for each after-closure flow regime

Log-Log Diagnostic Graph Before-closure linear or bilinear flow are identified by the relationship between the fracture-pressure difference ∆pwf and total time t but the after-closure flow regimes are identified by the relationship between the reservoir-pressure difference ∆pw and t which requires knowing the initial reservoir pressure pi However by noting that

wf wp pt t

part∆ part∆= minus

part part (B-22)

a log-log diagnostic graph can be prepared that is independent of initial reservoir pressure but can be used to identify before- and after-closure flow regimes Table B-2 shows the characteristic slopes of each flow regime for log-log graphs of ∆pwf vs t wfp tpart∆ part vs t wft p tpart∆ part vs t and

SPE 107877 13

2wft p tpart∆ part vs t

In gas reservoirs log-log diagnostic graphs can be prepared by plotting ∆pawf vs ta awf ap tpart∆ part vs ta a awf at p tpart∆ part vs ta

and 2a awf at p tpart∆ part vs ta where adjusted pseudopressure is

defined as

0

pz pdppa p zi

micromicro

⌠⎮⎮⌡

⎛ ⎞= ⎜ ⎟⎝ ⎠

(B-23)

and adjusted pseudotime is defined as

( )0

t dtt ca t i ctmicro

micro

⌠⎮⎮⌡

= (B-24)

Since adjusted pseudopressure is a function of initial reservoir pressure and if initial reservoir pressure is unknown its often helpful to complete the analysis in terms of pressure and time to obtain a first estimate of initial reservoir pressure before refining the estimate using diagnostic and interpretive graphs plotted in terms of adjusted pseudovariables

Table B-1 After-closure impulse solutions and derivatives

Flow Regime Impulse Solution Derivative Semilog Derivative

Impulse Derivative

Bilinear 3 4w blp C tminus∆ = 7 43

4w blp C

tt

minuspart∆minus =

part 3 43

4w blp C

t tt

minuspart∆minus =

part 2 1 43

4w blp C

t tt

part∆minus =

part

Pseudolinear 1 2w plp C tminus∆ = 3 2

2plw Cp

tt

minuspart∆minus =

part 1 2

2plw Cp

t tt

minuspart∆minus =

part 2 1 2

2plw Cp

t tt

part∆minus =

part

Pseudoradial 1w prp C tminus∆ = 2w

prp

C tt

minuspart∆minus =

part 1w

prp

t C tt

minuspart∆minus =

part 2 w

prp

t Ct

part∆minus =

part

Table B-2 Log-log graph characteristic slopes

Before Closure After Closure Log-Log Graph

Bilinear Linear Bilinear Pseudolinear Pseudoradial

vs

vs wf

awf a

p t

p t

∆

∆ 14 12 ⎯ ⎯ ⎯

vs

vs wf

awf a a

p t t

p t t

part∆ part

part∆ part minus34 minus12 minus74 minus32 minus2

vs

vs wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 14 12 minus34 minus12 minus1

2

2

vs

vs

wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 54 32 14 12 0

SPE 107877 11

( ) ( )w i R R cp t p m F t tminus = (A-6)

In these equations tc is the time to fracture closure with

time zero set as the beginning of fracture extension pi is the initial reservoir pressure and mR is the Cartesian slope of the ldquocorrectrdquo straight line The radial-flow function (FR) is given by5

( ) 61161ln41 2 cong=⎟⎟

⎠

⎞⎜⎜⎝

⎛minus

+=π

χχ

c

ccR tt

tttF (A-7)

In the properly identified pseudoradial flow period the reservoir far-field transmissibility can be determined from the slope fracture closure time and volume injected during the test

251000 t

R c

Qkhm tmicro

= (A-8)

In Equation A-8 the permeability k is in md net pay thickness h is in feet far-field mobile fluid viscosity micro in cp tc in minutes and the volume injected during the test Qt is

in bbls and the slope mR is in psi Horner analysis

The conventional Horner analysis uses a Cartesian plot of observed pressure versus Horner time (tp + ∆t)∆t with all times in consistent units The fracture propagation time is tp and the elapsed shut-in time is ∆t As shut-in time approaches infinity the Horner time function approaches 1 A straight-line extrapolation of the Horner plot to the intercept at a Horner time of 10 gives an estimate of reservoir pressure The slope of the correct straight-line extrapolation mH can be used to estimate reservoir transmissibility

( )Hm

qkh 14406162=

micro (A-9)

The flow rate in Equation A-9 is assumed to be in barrels per minute and is the average rate for the time the fracture was extending In both Equations A-8 and A-9 the viscosity is the far-field mobile fluid viscosity The propagation of the transient in pseudoradial flow occurs at a great distance from the fracture and is not affected by the injected fluid viscosity

The major problem with the Horner analysis is that the results are only valid if the data used to extrapolate the apparent straight line are actually in fully developed pseudoradial flow There is no way to determine the validity of the Horner analysis or to determine the flow regime within the Horner plot itself

G-function permeability estimate

An empirical function to approximate formation permeability has been derived from numerous numerical simulations of fracture closure The correlation is based on the observed G-function time at fracture closure

19600086 001

0038

f z

c pt

pk

G E rc

micro

φ

=⎛ ⎞⎜ ⎟⎝ ⎠

(A-10)

The rate of fluid loss from the fracture before closure is dominated by the mobility of the injected fluid instead of the far-field viscosity The total mobility of the injected fluid during leakoff is dependent on the viscosity of the injected fluid at leakoff temperature and the relative permeability to the leakoff fluid The reservoir fluid and its residual saturation in the invaded region will have some effect on the leakoff fluid mobility As a general rule an assumed injected fluid viscosity of 10 cp is used to incorporate the effects of reservoir temperature and relative permeability in the invaded zone

Note that the correlation gives permeability and not kh The time to closure is related to fracture volume versus created area so both the fracture height and length do not appear in the equation The process zone stress Pz is the net fracture extension pressure above closure pressure pc or pz = pISIP minus pc The net extension pressure and Youngrsquos Modulus provide a relationship between facture volume (width) and surface area during pumping

The other parameters in Equation A-10 are defined except the storage ratio rp This parameter represents the amount of excess fluid that must be leaked-off to reach fracture closure when the fracture geometry deviates from the normally assumed constant-height planar fracture The storage ratio rp is the ratio of the area under the G-function semilog derivative up until closure divided by the area of the triangle defined by the straight-line (normal leakoff) tangent to the semilog derivative at closure For normal matrix leakoff and PDL rp is therefore 10 For the transverse storage and height recession signature rp is some value less than 10 The observed G-function closure time (Gc) is always delayed by the excess fluid volume in the fracture for either height recession or transverse storage Likewise the apparent time to closure used in the after-closure analysis will be delayed and will cause errors in the reservoir transmissibility estimate The closure time should be corrected by multiplying by rp for all permeability and transmissibility estimates

Appendix B - Analytical Solutions and the Log-Log Diagnostic Graph Linear and Bilinear Flow Before-Closure Hagoort9 developed a before-closure analytical solution accounting for both storage and linear formation flow Koning and Niko10 write the solution as

( )( )22

12DtD fbcD Dp t C e erfc twD

βππ β∆⎛ ⎞= ∆ minus minus⎜ ⎟ ∆⎝ ⎠ (B-1)

where

( )( ) ( )( )

1412w e w e

Dp t p t tkh

p twD qBmicrominus + ∆

∆ = (B-2)

200002637

Dt f

k ttc Lφmicro

∆∆ =

(B-3)

12 SPE 107877

2

fbcDCβ

π= (B-4)

and

2 25615 08936

2fbc fbc

fbcDt f t f

C CC

c hL c hLπφ φ= =

(B-5)

Here Cfbc is the before-closure fracture storage constant which is defined as8

25615

ffbc

f

AC

S= (B-6)

where Af is the fracture area (one wing) and Sf is the fracture stiffness

Hagoorts solution predicts that the before-closure falloff is a combination of fracture storage and linear flow As fracture storage becomes small CfbcDrarr0 linear flow will dominate the before-closure pressure falloff In most hard rock environments CfbcD is small and for a fracture-injectionfalloff sequence with ∆t gtgt te and long closure times

( ) ( )w e w ep t p t t tminus + ∆ prop (B-7)

where t = te + ∆t and pw(te) is the instantaneous shut-in pressure Define a fracture-pressure difference as

( ) ( )wf w e w ep p t p t t∆ = minus + ∆ (B-8)

and a log-log graph of ∆pwf vs t will exhibit a frac12 slope during linear flow before closure An analytical before-closure bilinear flow solution accounting for fracture tip extension during shut-in does not exist but field data suggest that during before-closure bilinear flow

1 4( )wfp t∆ prop ∆ (B-9)

and a log-log graph of ∆pwf vs t will exhibit a frac14 slope during bilinear flow before closure Pseudoradial Pseudolinear and Bilinear Flow After-Closure Craig amp Blasingame8 developed an analytical solution for a fracture-injectionfalloff sequence with a propagating and closing hydraulic fracture and they derived the complete after-closure impulse solutions accounting for fracture storage The after-closure impulse solution for pseudoradial flow is written as

( )01412(24) 1(0) ( )( )

2 t iw e ie

Q p C p pp t t p wsD ackh t tmicro + minus+ ∆ minus =

+ ∆ (B-10)

For after-closure pseudolinear flow

0

1412(24) 00002637 1( )(0) ( )2

tw e i

if

Qp t t p

p C p phL wsD ac

π ⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 21

t ec k t tmicro

φ⎛ ⎞

times⎜ ⎟+ ∆⎝ ⎠

(B-11)

and for after-closure bilinear flow 1 4

3 40

1412(24)(06125)(00002637)( )( )t

w e iif f

Qp t t p

p p Ck w acmicro

⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 4 3 41 1

t ec k t tmicro⎛ ⎞ ⎛ ⎞

times⎜ ⎟ ⎜ ⎟+ ∆⎝ ⎠ ⎝ ⎠

(B-12)

For convenience define a constant pseudoradial flow term as

( )01412(24) (0) ( )

2 t ipr Q p C p pC wsD ackhmicro + minus= (B-13)

For pseudolinear flow a constant term is defined as 1 2

0

1412(24) 00002637 1(0) ( )2

tpl

tif

QC

c kp C p phL wsD ac

microπφ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-14)

Similarly for bilinear flow a constant term is written as

1 41 43 4

0

11412(24)(06125)(00002637)( )t

bltif f

QC

c kp p Ck w acmicro

micro⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-15)

With the new definition and defining a reservoir-pressure difference ∆pw = pw(t) minus pi the pseudoradial-flow impulse solution can be written as

1w prp C tminus∆ = (B-16)

Similarly the linear-flow impulse solution can be written as 1 2

w plp C tminus∆ = (B-17)

and the bilinear-flow impulse solution can be written as 3 4

w blp C tminus∆ = (B-18)

The derivatives with respect to t are written for after-closure pseudoradial flow as

2( )

wpr

pC t

tminuspart∆

minus =part

(B-19)

for after-closure pseudolinear flow as

3 2( ) 2

plw Cpt

tminuspart∆

minus =part

(B-20)

and for after-closure bilinear flow as

7 43( ) 4

w blp Ct

tminuspart∆

minus =part

(B-21)

The semilog derivatives and the impulse derivatives11 are calculated by multiplying the after-closure derivatives by t and t2 respectively Table B-1 shows the derivative terms for each after-closure flow regime

Log-Log Diagnostic Graph Before-closure linear or bilinear flow are identified by the relationship between the fracture-pressure difference ∆pwf and total time t but the after-closure flow regimes are identified by the relationship between the reservoir-pressure difference ∆pw and t which requires knowing the initial reservoir pressure pi However by noting that

wf wp pt t

part∆ part∆= minus

part part (B-22)

a log-log diagnostic graph can be prepared that is independent of initial reservoir pressure but can be used to identify before- and after-closure flow regimes Table B-2 shows the characteristic slopes of each flow regime for log-log graphs of ∆pwf vs t wfp tpart∆ part vs t wft p tpart∆ part vs t and

SPE 107877 13

2wft p tpart∆ part vs t

In gas reservoirs log-log diagnostic graphs can be prepared by plotting ∆pawf vs ta awf ap tpart∆ part vs ta a awf at p tpart∆ part vs ta

and 2a awf at p tpart∆ part vs ta where adjusted pseudopressure is

defined as

0

pz pdppa p zi

micromicro

⌠⎮⎮⌡

⎛ ⎞= ⎜ ⎟⎝ ⎠

(B-23)

and adjusted pseudotime is defined as

( )0

t dtt ca t i ctmicro

micro

⌠⎮⎮⌡

= (B-24)

Since adjusted pseudopressure is a function of initial reservoir pressure and if initial reservoir pressure is unknown its often helpful to complete the analysis in terms of pressure and time to obtain a first estimate of initial reservoir pressure before refining the estimate using diagnostic and interpretive graphs plotted in terms of adjusted pseudovariables

Table B-1 After-closure impulse solutions and derivatives

Flow Regime Impulse Solution Derivative Semilog Derivative

Impulse Derivative

Bilinear 3 4w blp C tminus∆ = 7 43

4w blp C

tt

minuspart∆minus =

part 3 43

4w blp C

t tt

minuspart∆minus =

part 2 1 43

4w blp C

t tt

part∆minus =

part

Pseudolinear 1 2w plp C tminus∆ = 3 2

2plw Cp

tt

minuspart∆minus =

part 1 2

2plw Cp

t tt

minuspart∆minus =

part 2 1 2

2plw Cp

t tt

part∆minus =

part

Pseudoradial 1w prp C tminus∆ = 2w

prp

C tt

minuspart∆minus =

part 1w

prp

t C tt

minuspart∆minus =

part 2 w

prp

t Ct

part∆minus =

part

Table B-2 Log-log graph characteristic slopes

Before Closure After Closure Log-Log Graph

Bilinear Linear Bilinear Pseudolinear Pseudoradial

vs

vs wf

awf a

p t

p t

∆

∆ 14 12 ⎯ ⎯ ⎯

vs

vs wf

awf a a

p t t

p t t

part∆ part

part∆ part minus34 minus12 minus74 minus32 minus2

vs

vs wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 14 12 minus34 minus12 minus1

2

2

vs

vs

wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 54 32 14 12 0

12 SPE 107877

2

fbcDCβ

π= (B-4)

and

2 25615 08936

2fbc fbc

fbcDt f t f

C CC

c hL c hLπφ φ= =

(B-5)

Here Cfbc is the before-closure fracture storage constant which is defined as8

25615

ffbc

f

AC

S= (B-6)

where Af is the fracture area (one wing) and Sf is the fracture stiffness

Hagoorts solution predicts that the before-closure falloff is a combination of fracture storage and linear flow As fracture storage becomes small CfbcDrarr0 linear flow will dominate the before-closure pressure falloff In most hard rock environments CfbcD is small and for a fracture-injectionfalloff sequence with ∆t gtgt te and long closure times

( ) ( )w e w ep t p t t tminus + ∆ prop (B-7)

where t = te + ∆t and pw(te) is the instantaneous shut-in pressure Define a fracture-pressure difference as

( ) ( )wf w e w ep p t p t t∆ = minus + ∆ (B-8)

and a log-log graph of ∆pwf vs t will exhibit a frac12 slope during linear flow before closure An analytical before-closure bilinear flow solution accounting for fracture tip extension during shut-in does not exist but field data suggest that during before-closure bilinear flow

1 4( )wfp t∆ prop ∆ (B-9)

and a log-log graph of ∆pwf vs t will exhibit a frac14 slope during bilinear flow before closure Pseudoradial Pseudolinear and Bilinear Flow After-Closure Craig amp Blasingame8 developed an analytical solution for a fracture-injectionfalloff sequence with a propagating and closing hydraulic fracture and they derived the complete after-closure impulse solutions accounting for fracture storage The after-closure impulse solution for pseudoradial flow is written as

( )01412(24) 1(0) ( )( )

2 t iw e ie

Q p C p pp t t p wsD ackh t tmicro + minus+ ∆ minus =

+ ∆ (B-10)

For after-closure pseudolinear flow

0

1412(24) 00002637 1( )(0) ( )2

tw e i

if

Qp t t p

p C p phL wsD ac

π ⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 21

t ec k t tmicro

φ⎛ ⎞

times⎜ ⎟+ ∆⎝ ⎠

(B-11)

and for after-closure bilinear flow 1 4

3 40

1412(24)(06125)(00002637)( )( )t

w e iif f

Qp t t p

p p Ck w acmicro

⎛ ⎞+ ∆ minus = ⎜ ⎟+ minus⎝ ⎠

1 4 3 41 1

t ec k t tmicro⎛ ⎞ ⎛ ⎞

times⎜ ⎟ ⎜ ⎟+ ∆⎝ ⎠ ⎝ ⎠

(B-12)

For convenience define a constant pseudoradial flow term as

( )01412(24) (0) ( )

2 t ipr Q p C p pC wsD ackhmicro + minus= (B-13)

For pseudolinear flow a constant term is defined as 1 2

0

1412(24) 00002637 1(0) ( )2

tpl

tif

QC

c kp C p phL wsD ac

microπφ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-14)

Similarly for bilinear flow a constant term is written as

1 41 43 4

0

11412(24)(06125)(00002637)( )t

bltif f

QC

c kp p Ck w acmicro

micro⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟+ minus ⎝ ⎠⎝ ⎠

(B-15)

With the new definition and defining a reservoir-pressure difference ∆pw = pw(t) minus pi the pseudoradial-flow impulse solution can be written as

1w prp C tminus∆ = (B-16)

Similarly the linear-flow impulse solution can be written as 1 2

w plp C tminus∆ = (B-17)

and the bilinear-flow impulse solution can be written as 3 4

w blp C tminus∆ = (B-18)

The derivatives with respect to t are written for after-closure pseudoradial flow as

2( )

wpr

pC t

tminuspart∆

minus =part

(B-19)

for after-closure pseudolinear flow as

3 2( ) 2

plw Cpt

tminuspart∆

minus =part

(B-20)

and for after-closure bilinear flow as

7 43( ) 4

w blp Ct

tminuspart∆

minus =part

(B-21)

The semilog derivatives and the impulse derivatives11 are calculated by multiplying the after-closure derivatives by t and t2 respectively Table B-1 shows the derivative terms for each after-closure flow regime

Log-Log Diagnostic Graph Before-closure linear or bilinear flow are identified by the relationship between the fracture-pressure difference ∆pwf and total time t but the after-closure flow regimes are identified by the relationship between the reservoir-pressure difference ∆pw and t which requires knowing the initial reservoir pressure pi However by noting that

wf wp pt t

part∆ part∆= minus

part part (B-22)

a log-log diagnostic graph can be prepared that is independent of initial reservoir pressure but can be used to identify before- and after-closure flow regimes Table B-2 shows the characteristic slopes of each flow regime for log-log graphs of ∆pwf vs t wfp tpart∆ part vs t wft p tpart∆ part vs t and

SPE 107877 13

2wft p tpart∆ part vs t

In gas reservoirs log-log diagnostic graphs can be prepared by plotting ∆pawf vs ta awf ap tpart∆ part vs ta a awf at p tpart∆ part vs ta

and 2a awf at p tpart∆ part vs ta where adjusted pseudopressure is

defined as

0

pz pdppa p zi

micromicro

⌠⎮⎮⌡

⎛ ⎞= ⎜ ⎟⎝ ⎠

(B-23)

and adjusted pseudotime is defined as

( )0

t dtt ca t i ctmicro

micro

⌠⎮⎮⌡

= (B-24)

Since adjusted pseudopressure is a function of initial reservoir pressure and if initial reservoir pressure is unknown its often helpful to complete the analysis in terms of pressure and time to obtain a first estimate of initial reservoir pressure before refining the estimate using diagnostic and interpretive graphs plotted in terms of adjusted pseudovariables

Table B-1 After-closure impulse solutions and derivatives

Flow Regime Impulse Solution Derivative Semilog Derivative

Impulse Derivative

Bilinear 3 4w blp C tminus∆ = 7 43

4w blp C

tt

minuspart∆minus =

part 3 43

4w blp C

t tt

minuspart∆minus =

part 2 1 43

4w blp C

t tt

part∆minus =

part

Pseudolinear 1 2w plp C tminus∆ = 3 2

2plw Cp

tt

minuspart∆minus =

part 1 2

2plw Cp

t tt

minuspart∆minus =

part 2 1 2

2plw Cp

t tt

part∆minus =

part

Pseudoradial 1w prp C tminus∆ = 2w

prp

C tt

minuspart∆minus =

part 1w

prp

t C tt

minuspart∆minus =

part 2 w

prp

t Ct

part∆minus =

part

Table B-2 Log-log graph characteristic slopes

Before Closure After Closure Log-Log Graph

Bilinear Linear Bilinear Pseudolinear Pseudoradial

vs

vs wf

awf a

p t

p t

∆

∆ 14 12 ⎯ ⎯ ⎯

vs

vs wf

awf a a

p t t

p t t

part∆ part

part∆ part minus34 minus12 minus74 minus32 minus2

vs

vs wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 14 12 minus34 minus12 minus1

2

2

vs

vs

wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 54 32 14 12 0

SPE 107877 13

2wft p tpart∆ part vs t

In gas reservoirs log-log diagnostic graphs can be prepared by plotting ∆pawf vs ta awf ap tpart∆ part vs ta a awf at p tpart∆ part vs ta

and 2a awf at p tpart∆ part vs ta where adjusted pseudopressure is

defined as

0

pz pdppa p zi

micromicro

⌠⎮⎮⌡

⎛ ⎞= ⎜ ⎟⎝ ⎠

(B-23)

and adjusted pseudotime is defined as

( )0

t dtt ca t i ctmicro

micro

⌠⎮⎮⌡

= (B-24)

Since adjusted pseudopressure is a function of initial reservoir pressure and if initial reservoir pressure is unknown its often helpful to complete the analysis in terms of pressure and time to obtain a first estimate of initial reservoir pressure before refining the estimate using diagnostic and interpretive graphs plotted in terms of adjusted pseudovariables

Table B-1 After-closure impulse solutions and derivatives

Flow Regime Impulse Solution Derivative Semilog Derivative

Impulse Derivative

Bilinear 3 4w blp C tminus∆ = 7 43

4w blp C

tt

minuspart∆minus =

part 3 43

4w blp C

t tt

minuspart∆minus =

part 2 1 43

4w blp C

t tt

part∆minus =

part

Pseudolinear 1 2w plp C tminus∆ = 3 2

2plw Cp

tt

minuspart∆minus =

part 1 2

2plw Cp

t tt

minuspart∆minus =

part 2 1 2

2plw Cp

t tt

part∆minus =

part

Pseudoradial 1w prp C tminus∆ = 2w

prp

C tt

minuspart∆minus =

part 1w

prp

t C tt

minuspart∆minus =

part 2 w

prp

t Ct

part∆minus =

part

Table B-2 Log-log graph characteristic slopes

Before Closure After Closure Log-Log Graph

Bilinear Linear Bilinear Pseudolinear Pseudoradial

vs

vs wf

awf a

p t

p t

∆

∆ 14 12 ⎯ ⎯ ⎯

vs

vs wf

awf a a

p t t

p t t

part∆ part

part∆ part minus34 minus12 minus74 minus32 minus2

vs

vs wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 14 12 minus34 minus12 minus1

2

2

vs

vs

wf

a awf a a

t p t t

t p t t

part∆ part

part∆ part 54 32 14 12 0