10 - hydraulically fractured wells

NExT April 2000

Hydraulically Fractured Wells 1

HydraulicallyFractured Wells

NExT April 2000


Hydraulically Fractured Wells

• Flow Regimes• Depth of Investigation• Fracture Damage• Straight Line Analysis

– Bilinear Flow Analysis– Linear Flow Analysis– Semilog Analysis

• Type Curve Analysis

NExT April 2000


Ideal Model OfHydraulic Fracture

k

w f

L f

k f

NExT April 2000


Dimensionless Variables For Fractured Wells

( )wfiD ppqB

khp −=

µ00708.0 t

Lc

kt

ftDL f 2

0002637.0

φµ=

k

c

c

kt

ftf

ffD

φφ

η = 2

8936.0

ftDL

hLc

CC

f φ=

rf

ffcD C

kL

kwF π==

f

ffr kL

kwC

π=

Cr is dimensionless fracture conductivity

FcD is also dimensionless fracture conductivity….

NExT April 2000


Flow Regimes In Fractured Wells

Fracture Linear Flow Bilinear Flow

Formation Linear Flow

Elliptical Flow Pseudoradial Flow

*FRACTURE LINEAR FLOW AND BILINEAR FLOW DO NOT GIVE INFORMATION ABOUT FRACTURE LENGTH BUT RATHER ABOUT FRACTURE CONDUCTIVITY…THESE TWO PERIODS MIGHT BE MASKED BY WELL BORE STORAGE.

*Formation Linear Flow….HIGH CONDUCTIVITY FRACTURE….

*BILINEAR FLOW….LOW CONDUCTIVITY FRACTURE

*PSEUDORADIAL FLOW IS SIMILAR TO RADIAL FLOW IN AN INFINITE RESERVOIR..

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Fracture Linear Flow

During fracture linear flow, the pressure transient is moving down the length of the fracture.

The pressure transient has not moved into the reservoir.

The pressure transient has not yet reached the end of the fracture.

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Fracture Linear Flow

DLfDcD

D ft

Fp πη2=

2

201.0

fD

cDDL

Ft

f η≤

The pressure drop for fracture linear flow is given by the firstequation.

Since the pressure drop for linear flow is proportional to the square root of time, on a log-log diagnostic plot, both the pressure and the pressure derivative response during fracture linear flowwill appear as straight lines with a slope of 1/2.

Fracture linear flow lasts until a dimensionless time given by the second equation.

Fracture linear flow occurs at times that are too early to have any practical application.

NExT April 2000


Bilinear Flow

Bilinear flow occurs only for low conductivity fractures, where Cr < 100.

During bilinear flow, the pressure transient is simultaneously moving down the length of the fracture and out into the formation.

The pressure transient has not yet reached the end of the fracture.

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Bilinear Flow

( )4

14

1 45.2

225.1DL

cDDL

cDD ff

tF

tF

p ≅=Γ

π

6.1,5.255.4

4

<

−≤

−

cDcD

DL FF

tf

[ ] 36.1,5.10205.0 53.1 <≤−≤ −cDcDDL FFt

f

3,1.02

≥≤ cDcD

DL FF

tf

The pressure drop for bilinear flow is given by the first equation.

Since the pressure response during bilinear flow is proportional to the 4th root of time, on a log-log diagnostic plot, both the pressure and the pressure derivative response during bilinear flow will appear as straight lines with a slope of 1/4.

The duration of bilinear flow depends on the dimensionless fracture conductivity. The final three equations shown here maybe used to estimate the duration of bilinear flow.

From data in the bilinear flow period, the fracture conductivity wkfmay be estimated if the formation permeability k is known.

Data in the bilinear flow period cannot be used to estimate fracture half-length, except perhaps to place a lower bound on Lf.

NExT April 2000



Formation linear flow occurs only in high-conductivity fractures, where

Cr > 100.

In formation linear flow, the pressure transient is moving linearly out into the formation, away from the fracture. There is negligible pressure drop down the fracture.

The pressure transient has not yet moved far enough into the formation that the ends of the fracture must be taken into account.

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DLD ftp π=

016.0100

2≤≤ DL

cDf

tF

The pressure drop for formation linear flow is given by the first equation.

Since the pressure is proportional to the square root of time, on a log-log diagnostic plot, both the pressure and the pressure derivative response during formation linear flow will appear as straight lines with a slope of 1/2.

Formation linear flow lasts until a dimensionless time given by the second equation.

Data during formation linear flow may be used to estimate Lf if the formation permeability is known.

NExT April 2000


Elliptical Flow

Elliptical flow may occur for either low- or high-conductivity fractures.

Elliptical flow is a transitional flow regime. There is no simple equation to describe the pressure response during this flow regime.

In radial flow, the pressure transient has moved beyond the endsof the fracture far enough that the linear pattern characteristic of formation linear flow has been distorted into an ellipse.

As the pressure transient continues to move out in all directions away from the fracture, the area influenced by the pressure transient becomes more circular.

Eventually, the ellipses become circular enough that we have pseudoradial flow.

NExT April 2000


Pseudoradial Flow

Pseudoradial flow may occur for either low- or high-conductivity fractures.

Pseudoradial flow occurs when the pressure transient has moved far enough beyond the tips of the fracture wings that the regioninfluenced by the pressure transient may be considered essentially circular.

In low-permeability reservoirs a long time is required to reach pseudoradial flow, to the extent that it is almost always impractical to run a test long enough to reach this flow regime in an MHF well in a tight reservoir.

The pseudoradial flow period is most likely to be seen in frac-pac wells, where formation permeability is moderate to high and fracture half-lengths are short.

NExT April 2000


Pseudoradial Flow

3≥DL ft

+−

= s

rc

kt

kh

qBp

wt

869.023.3log6.162

2φµµ∆

The pressure drop for pseudoradial flow is given by the first equation.

The pressure derivative will constant during pseudoradial flow.

Pseudoradial flow in high conductivity fractures begins at a dimensionless time given by the second equation. For low conductivity fractures, the pseudoradial flow period begins somewhat earlier.

Pseudoradial flow lasts until the pressure transient reaches oneor more of the reservoir boundaries.

The pseudoradial flow regime may not occur if the fracture is long compared to the distance to the nearest reservoir boundary.

Data within the pseudoradial flow regime may be used to estimateformation permeability and skin factor. The skin factor may then be used to estimate fracture half-length.

NExT April 2000


Depth Of Investigation

b

a

12

2

2

2

=+b

y

a

x222 baL f −=

L f

It is not as simple as in radial flow…

However, using the properties of an ellipse…the equations above are found…

The fracture is located in an ELLIPSE…It has two axis a & b

From the equation of an ellipse the term Lf can be found

NExT April 2000


For linear flow, pseudosteady-state flow exists out to a distance b at a dimensionless time given by


2

0002637.0

bc

ktt

tbD µφ

=

π1=bDt

21

02878.0

=

tc

ktb

µφ

Depth of investigation for a linear system at time t

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21

02878.0

=

tc

ktb

µφ

22 bLa f +=

baA π=

Depth of investigation along minor axis

Depth of investigation along major axis

Area of investigation

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Hydraulic Fracture With Choked Fracture Damage

k

w f

L f

k fk fs

Ls

Chocked fracture damage is due to fracture fluid conditions and to the stresses around the well bore, a large drawdown near well bore will cause the proppant to crush. If proppant is too weak this would happen also….If proppant is produced back to the formatio it would happen also

NExT April 2000


Choked Fracture Skin Factor

kA

LqBp

001127.0

µ∆ = ( )fffs

ss whk

LqBp

2001127.0

µ∆ =

sf pqB

khs ∆

µ00708.0= ( )

=

fffs

s

whk

LqB

qB

kh

2001127.0

00708.0 µµ

ffs

sf wk

kLs

π=

Choked Fracture Skin Factor……..

NExT April 2000


Hydraulic Fracture With Fracture Face Damage

k

w f

k fksws

L f

Invasion of fracture fluids will cause this type of damage, this also could be caused by condensate ...

Fracture fluids are complex gels which has to break down after it transports the proppant.

NExT April 2000


Fracture Face Skin Factor

kA

LqBp

001127.0

µ∆ = ( )

−=

kkLh

wqBp

sff

ss

11

4001127.0

µ∆

sf pqB

khs ∆

µ00708.0= ( )

−

=

kkLh

wqB

qB

kh

sff

s 11

4001127.0

00708.0 µµ

−= 1

2 sf

sf k

k

L

ws

π

Very low skin factors at the fracture face will change significantly the pressure behavior….

NExT April 2000


Bilinear Flow AnalysisProcedure

• Identify the bilinear flow regime using the diagnostic plot

• Graph p wf vs. t 1/4 or p ws vs ∆∆∆∆tBe1/4

• Find the slope m B and the intercept p 0 of the best straight line

• Calculate the fracture conductivity wk f from the slope and the fracture skin factor s f from the intercept

NExT April 2000


Bilinear Equivalent Time

( )( )4414141 ttttt ppBe ∆+−∆+=∆

pBe tttt <<∆∆≈∆ ,

ppBe tttt >>∆≈∆ ,

NExT April 2000


Bilinear Flow AnalysisEquations

5.0211.44

=

kcmh

Bqwk

tBf µφ

µ

( )000708.0

ppqB

khs if −=

µ

( )wff ppqB

khs −= 0

00708.0

µBuildup

Drawdown

Wkf…FRACTURE CONDUCTIVITY CAN BE OBTAINED…..from bilinear flow period…

NExT April 2000


Bilinear Flow Analysis

p0=2642.4 psi

m=63.8 psi/hr 1/4

pwf=2628.6 psi∆∆∆∆ps

2600

2650

2700

2750

2800

0 0.5 1 1.5 2

teqB1/4, hrs 1/4

pw

s, p

si

NExT April 2000


Limitations Of Bilinear Flow Analysis

• Applicable only to wells with low-conductivity fractures (C r < 100)

• Bilinear flow may be hidden by wellbore storage• Requires independent estimate of k• Gives estimate of wk f and s f

• Cannot be used to estimate L f

NExT April 2000


Linear Flow AnalysisProcedure

• Identify the linear flow regime using the diagnostic plot

• Graph p wf vs. t 1/2 or p ws vs ∆∆∆∆tLe1/2

• Find the slope m L and the intercept p 0 of the best straight line

• Calculate the fracture half-length L f from the slope and the fracture skin factor s f from the intercept

NExT April 2000


Linear Equivalent Time

( )( )2212121 ttttt ppLe ∆+−∆+=∆

pLe tttt <<∆∆≈∆ ,

ppLe tttt >>∆≈∆ ,

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Linear Flow AnalysisEquations

( )000708.0

ppqB

khs if −=

µ

( )wff ppqB

khs −= 0

00708.0

µBuildup

Drawdown

21064.4

=

tLf ckhm

BqL

φµ

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Linear Flow Analysis

0

1000

2000

3000

4000

5000

6000

0 2 4 6 8 10 12 14 16 18

taLeq1/2, hrs 1/2

paw

s, p

si

pa0=2266.0 psi

m=211 psi/hr 1/2

pawf=1656.2 psi∆∆∆∆ps

NExT April 2000


Limitations Of Linear Flow Analysis

• Applicable only to wells with high-conductivity fractures (C r > 100)

• Wellbore storage may hide linear flow period• Long transition period between end of linear flow

(tLfD < 0.016) and beginning of pseudoradial flow (tLfD > 3)

• Requires independent estimate of k• Gives estimate of L f and s f

• Cannot be used to estimate wk f

*In practice you can apply the analysis process to this period for values of Cr> 10

NExT April 2000


Pseudoradial Flow AnalysisProcedure

• Identify the pseudoradial flow regime using the diagnostic plot

• Graph p wf vs. log(t) or p ws vs log( ∆∆∆∆te)• Find the slope m and the intercept p 1hr of the best

straight line• Calculate the formation permeability k from the

slope and the total skin factor s from the intercep t• Estimate fracture half-length from total skin

factor

NExT April 2000


Pseudoradial Flow AnalysisEquations

Buildup

Drawdown

mh

qBk

µ6.162=

+

−−= 23.3log151.1

2101

wt

hri

rc

k

m

pps

φµ

+

−

−= 23.3log151.1

2101

wt

wfhr

rc

k

m

pps

φµ

*exactly the same as the ones used for radial flow analysis….

NExT April 2000


Pseudoradial Flow Analysis

1500

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500

0.001 0.01 0.1 1 10 100

te, hrs

pw

s, p

si p1hr=2121 psim=120 psi/cycle

∆∆∆∆

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Apparent Wellbore Radius

1

10

100

0.1 1 10 100 1000

FcD

Lf/r

wa

The curve in this figure is based on the following equation:

This equation, in turn, is based on fitting a curve through datasimilar to that from Fig. 18 from Cinco (2).

We estimate Fcd and from the graph we read Lf / rwa….

HIGH conductivity Lf = 2 rwa

LOW conductivity from GRAPH….Lf = (Lf/rwa) rwa

cDwa

f

Fr

L π+= 2

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Estimating L f From Skin Factor

1. Calculate r wa from r wa = rwe-s

2. Estimate L f from L f = 2rwa

3. Estimate fracture conductivity wk f

4. Calculate F cD from F cD = wk f/kL f

5. Find L f/rwa from graph or equation

6. Estimate L f from L f = (Lf/rwa)*rwa

7. Repeat steps 4 through 6 until convergence

(Warning: may not converge)

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Limitations Of Pseudoradial Flow Analysis

• Boundaries of reservoir may be encountered before pseudoradial flow develops

• Long transition period between linear flow and pseudoradial flow

• Pseudoradial flow cannot be achieved for practical test times in low permeability reservoirs with long fractures

• Gives estimate of k and s t

• Does not give direct estimate of L f, wk f, or s f

NExT April 2000


Dimensionless Variables For Fractured Wells

( )wfiD ppqB

khp −=

µ00708.0 t

Lc

kt

ftDL f 2

0002637.0

φµ=

2

8936.0

ftDL

hLc

CC

f φ=

rf

ffcD C

kL

kwF π==

f

ffr kL

kwC

π=

sf pqB

khs ∆=

µ00708.0

The independent variable for most type curves for hydraulically fractured wells is the dimensionless time based on hydraulic fracture half-length, tLfD.

The dependent variable for most type curves for hydraulically fractured wells is the dimensionless pressure, pD.

For type curves for manual type curve matching, the most common type curves vary only one of the three remaining parameters.

The Cinco type curve is obtained by setting CLfD and sf to 0, and varying Cr, or equivalently, FcD.

The Choked fracture skin type curve is obtained by setting CLfD to 0, FcD to ∞, and varying sf.

The Barker-Ramey type curve is obtained by setting sf to 0, FcD to ∞, and varying CLfD.

When using computer-generated type curves, the computer can set any two of the three parameters to fixed values, and vary the third parameter to obtain the matching stems.

NExT April 2000


Type Curve Analysis For Fractured Wells - Unknown Permeability

1. Graph field data pressure change and

pressure derivatives

2. Match field data to type curve

3. Find match point and matching stem

4. Calculate L f from time match point

5. Calculate k from pressure match point

6. Interpret matching stem value (wk f, sf, or C)

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Interpreting Match PointsUnknown Permeability

MP

D

p

p

h

qBk

∆= µ2.141

MPDLt

f

ft

t

c

kL

∆=φµ

0002637.0

NExT April 2000


Type Curve Analysis For Fractured Wells - Known Permeability

1. Graph field data pressure change and

pressure derivatives

2. Calculate pressure match point from k

3. Match field data to type curve, using

calculated pressure match point

4. Find match point and matching stem

5. Calculate L f from time match point

6. Interpret matching stem value (wk f, sf, or C)

NExT April 2000


Interpreting Match PointsKnown Permeability

( ) ( )MPDMP pkh

qBp

µ2.141=∆

MPDLt

f

ft

t

c

kL

∆=φµ

0002637.0

NExT April 2000


Cinco Type Curve

0.0001

0.001

0.01

0.1

1

10

1E-06 0.00001 0.0001 0.001 0.01 0.1 1 10 100

tLfD

pD, t

Dp'

D

Cr = 0.20.5

13

1050

1000

The Cinco type curve assumes CLfD = 0 and sf = 0. The type curve stems are obtained by varying Cr or FcD.

DASH LINES ARE THE DERIVATIVES

NExT April 2000


Cinco Type CurveInterpreting C r Stem

rfff CkLkw π=

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Choked Fracture Type Curve

0.0001

0.001

0.01

0.1

1

10

1E-06 0.00001 0.0001 0.001 0.01 0.1 1 10 100

tLfD

pD, t

Dp'

D

sf = 10.30.1

0.030.01

0.0030

The choked fracture type curve assumes CLfD = 0 and Cr = ∞. The type curve stems are obtained by varying sf.(FRACTURE SKIN)

The shape of the CURVE is significantly changed if the FRACTURE is damaged

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Choked Fracture Type CurveInterpreting s f Stem

fs skh

qBp

00708.0

µ=∆

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Barker-Ramey Type Curve

0.0001

0.001

0.01

0.1

1

10

1E-06 0.00001 0.0001 0.001 0.01 0.1 1 10 100

tLfD

pD, t

Dp'

DCLfD = 0

5x10-5

3x10-4

2x10-3

1.2x10-2

8x10-2

5x10-1

The Barker-Ramey type curve assumes sf = 0 and Cr = ∞. The type curve stems are obtained by varying CLfD.

NO SKIN AT THE FRACTURE FACE AND INFINITE CONDUCTIVITY FRACTURE

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Barker-Ramey Type CurveInterpreting C LfD Stem

DLft

fC

hLcC

8936.0

2φ=

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Limitations Of Type Curve Analysis

• Type curves are usually based on solutions for drawdown - what about buildup tests?– Shutin time– Equivalent time (radial, linear, bilinear)– Superposition type curves

• Type curves may ignore important behavior– Variable WBS– Boundaries– Non-Darcy flow

• Need independent estimate of permeability for best results

NExT April 2000


References

1. Cinco-Ley, H. and Samaniego-V., F.: “Transient Pressure Analysis for Fractured Wells,” JPT (Sept. 1981) 1749-1766.

2. Cinco-Ley, H. and Samaniego-V., F.: "Transient Pressure Analysis: Finite Conductivity Fracture Case Versus Damaged Fracture Case," paper SPE 10179 presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, Oct. 5-7.

3. Wong, Harrington, and Cinco-Ley: “Application of the Pressure Derivative Function in the Pressure Transient Testing of Fractured Wells,” SPEFE (Oct. 1986) 470-480.

4. Ramey, H.J. Jr. and Gringarten, A.C.: "Effect of High Volume Vertical Fractures on Geothermal Steam Well Behavior," Proc., Second United Nations Symposium on the Use and Development of Geothermal Energy, San Francisco, May 20-29, 1975.

5. Cinco-Ley, H., Samaniego-V., F., and Dominguez, N.: "Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture," SPEJ (Aug. 1978) 253-264.

6. Cinco-Ley, H. and Samaniego-V., F.: "Effect of Wellbore Storage and Damage on the Transient Pressure Behavior of Vertically Fractured Wells," paper SPE 6752 presented at the 1977 SPE Annual Technical Conference and Exhibition, Denver, Oct. 9-12.

10 - hydraulically fractured wells

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