What good is Linear Elastic Fracture Mechanics in HF?

BIRS Hydraulic Fracturing Workshop, Banff, June 2018

What good is Linear Elastic Fracture Mechanics in HF?


Dmitry Garagash (Dalhousie U. & Skoltech) Robert Viesca (Tufts U.)

Example (1) of “Rough/Cohesive” Crack Tip: Lab HF Lhomme [PhD thesis, Delft 2005]: HF in Colton sandstone impregnated by Epodye at 5 atm Tight Sandstone ~ 7510 ft depth

Process zone with evident multiple branches and bridging between fracture surfaces

Example (2) of “Branched” Crack Tip: Exhumed Dike Delaney, Pollard, Ziony, et al. [JGR1986] Basaltic dike intruding sandstone ~ 8 km depth
 Belle Butte, Utah Tight Sandstone ~ 7510 ft depth

No dikeparallel joints

dike-parallel joints with increasing spacing

Resistance to Hydraulic Fracture at the Tip View of hydraulic fracture tip region, as a part of the fracture where wall roughness is as large or larger than the mean aperture, example (1); and/or zone of distributed micro/meso cracks which localize onto the macro crack tip, Tight Sandstone ~ 7510 example (2) ft depth

• •

In either case, we can call it a “process zone” which can be characterized by a cohesive stress σc (e.g. due to bridging of fracture surfaces) which vanishes when fracture aperture reaches a critical opening value wc (in case of bridging wc can be directly identified with the fracture surface roughness) increased resistance to the fluid flow (if fracturing fluid enters the process zone), i.e. departure from the Poiseulle law (q ~ w3). Garagash [2015] shows that this non-Poiseulle dissipation is dominant for field HF when the near tip fracture roughness ~ wc

• •

Process zone ∈ Lag (magma dikes)

Lag ∈ Process zone (slick water / gel HF)

o

o lag

lag

v

V

wc

v

V

wc

c

c

process zone

process zone

R

R

This presentation takes a step back and explores effect of P.Z. on HF under classic Poiseulle flow assumption (i.e. when fracture roughness effects are negligible)

• Tight Sandstone ~ 7510 ft depth

example (2) Resistance

to Hydraulic Fracture at the Tip

In either case, we can call it a “process zone” which can be characterized by a cohesive stress σc (e.g. due to bridging of fracture surfaces) which vanishes when fracture aperture reaches a critical opening value wc (in case of bridging wc can be directly identified with the fracture surface roughness) increased resistance to the fluid flow (if fracturing fluid enters the process zone), i.e. departure from the Poiseulle law (q ~ w3). Garagash [2015] shows that this non-Poiseulle dissipation is dominant for field HF when the near tip fracture roughness ~ wc

• •

Process zone ∈ Lag (magma dikes)

Lag ∈ Process zone (slick water / gel HF)

o

o lag

lag

v

V

wc

v

V

wc

c

c

process zone

process zone

R

R

This presentation takes a step back and explores effect of P.Z. on HF under classic Poiseulle flow assumption (i.e. when fracture roughness effects are negligible) Some previous numerical case studies of cohesive zone in HF: 
 Sarris and Papanastasiou [IJF 2011], Carrier and Granet [EFM 2012]; Yau, Liu & Keer [MM 2015]. So far no attempt at systematic parametric study of the PZ effect in HF

Cohesive HF Tip Formulation

• Elasticity (net-pressure p = pZ - σ f




1 coh (w(x)) = 4⇡


 p(x) 


o

0

vs. opening w)


dw ds ds x s

where cohesive strength weakens with opening:


x




coh (w)

w/wc ) for w < wc and 0 otherwise 
 Cohesive fracture energy: 
 Gc = c wc /2

lag

v

1

o


 V

wc

(1

Note, under conditions of “small scale yielding” 
 (i.e. P.Z. is small compared to all other relevant lengthscales), the P.Z. can be replaced with LEFM singularity ( c ! 1, wc ! 0) with finite fracture energy Gc. Critical p LEFM stress intensity factor is simply Kc = Gc E 0

process zone

R

fluid flow in advancing fracture
 • Lubrication 2 


(vapor or infiltrated pore fluid)
 • Lag 
 o wo , w o ⇠

c




c

o 
 x < : p(x) = “Lag fracture energy”: 
 Go =

=

o E0





 x>

:

w dpf V = 0 µ dx

Various Known Crack Tip Solutions o

• Dry P.Z. (say lag >> R) [Rice, 1968]
 


x


 c-asymptote: w =

lag

v




V

wc

x ⇠ `c





 k-asymptote: w =

c process zone




R

c c 0 E




x ⇠ `ko




1/2


o-asymptote: w = 


x ⇠ `o 
 m-asymptote :

p=0

p=0

(`k = Gc /E 0 )

Solution evolves from the non-singular c-asymptote E0 wc to the LEFM k-asymptote over distances x ~ R ~ `c =

1/2 1/2 k `k x

o o 0 E

1/2 `c

1/2 k `k x

• HF with Lag when cohesive fracture energy negligible (LEFM), 
k-asymptote: w =

x3/2

x3/2 1/2 `o

p=0 p=

Gc ⌧ Go, [G&D, 2000]


c

o 1/3

p cot( ⇡/3) `m 0 0 1/3 2/3 (` = µ V /E ) m w = m `m x = m 0 1/3 E 4 x 
 Solution evolves from the k-asymptote to the non-singular o-asymptote over 
 x ⇠ `ko ⇠ (E 0 / and then to the viscous-dominated m-asymptote [Desroches et al, 1994] over distances
 3 x
 ⇠ ⇠ `o = (E 0 / o ) `m

1/2 1/2 o )`k `o

G [G&D, 2000/05; G etal., 2011]
 without Lag, “lag fracture energy” is negligible, G • HF Solution evolves from the LEFM k-asymptote to the viscous-dominated m-asymptote 
 c

o

over distances 
 x ⇠ `km = `3k /`2m

• A particular tip solution is potentially a combination of one or more limiting solutions in the above realized at different distances (lengthscales) from the tip, if those scales separate

LEFM based HF tip solution Two non-dimensional parameters: cohesive-to-confining stress ratio cohesive (rock)-to-lag-fracture energy LEFM is realized when

c/ o

Two limiting regimes:


c c

! 1 while

⌧1:

,

c

o

c

o wo

10

k !o!m

&1:

c /⇡

c wc

1

`ko

32

G&D [2000]

Gc 1 = = Go 2

is finite

/`o

0.100

`o

0.010

k ! m `km

0.001 -4 10

0.001

0.010

0.100

1

c

Crack Opening

100

c

w ⇠ x2/3

⇡1

w ⇠ x1/2

c

0.100

0.001 0.001

← tip

=0

w ⇠ x3/2

0.010

`ko 0.010

0.8

o

w/wo

1.0

`km

10 1

Net Pressure

p/




• •

c

=

p

x/`o

0.6 0.4

c

=0

⇡1

0.2

`o

0.100

c

1

10

100

0.0 0.001

p⇠x 0.010

0.100

1

1/3

10

100

x/`o

away →

`o = (E 0 /

o)

3

(µ0 V /E 0 )

wo = (

o /E

0

)`o

LEFM based HF tip solution Two non-dimensional parameters: cohesive-to-confining stress ratio cohesive (rock)-to-lag-fracture energy LEFM is realized when

c/ o

Two limiting regimes:


c c

! 1 while

⌧1:

,

c

o

c

is finite

k !o!m

&1:

c /⇡

G&D [2000]

Gc 1 = = Go 2

c wc o wo

1

`ko

32

10

/`o

0.100

`o

0.010

k ! m `km

0.001 -4 10

0.001

0.010

0.100

1

c

100

c

w/wo

10

`km

10 1

Crack Opening normalized by m-asymptote

Crack Opening w ⇠ x2/3

⇡1

w ⇠ x1/2

c

0.100

← tip

=0

`ko 0.010

x/`o

⇡1

c

1

m

0.50 c

=0

k

0.10

o

`o

0.100

k

0.05

w ⇠ x3/2

0.010 0.001 0.001

5

w/w1 (x)




• •

c

=

p

1

10

100

away →

0.01 0.001

0.010

0.100

x/`o

← tip

`o = (E 0 /

1

o)

3

(µ0 V /E 0 )

10

100

away →

wo = (

o /E

0

)`o

Cohesive HF tip solution Two non-dimensional parameters: cohesive-to-confining stress ratio cohesive (rock)-to-lag-fracture energy

• •

c

,

c

o

Gc 1 = = Go 2

c wc o wo

10

Two limiting regimes:


c

⌧1:

c!k !o!m `c

/`o

`o

1

o!m ! c!k ! m

1:

c

`ko

`o

`mc

`c

`km

×× ×× ×× ×× ×× ×××× ×××× ×× × ×× ××××××××× ×× ×× ×× ×× ××××× × × × ×× ×

0.100 0.010

5

10

σc /σo = 0.1

5

w/w1 (x)

c = 100 + c =1 +

1 0.50

c

m

= 0.01

× × +

0.10

o

0.05

0.01

0.001

1

σc /σo = 1.

×× ××× × ×× ××

c / o = 10

1

10

c = 100 +

1

c

100

k

+

c =1

0.50

c

× × ×

m

= 0.01

+

0.10

c

o

o

= 0.1

1000

× ×× × ×× ×× ×× ×××××

c

0.05

c/ o

0.10

k

w/w1 (x)

10

c/ o = 1

LEFM

0.001 0.01

Crack Opening normalized by m-asymptote

c / o = 0.1

×××× ×××××××××× × ×× ××××××××××××× × × ×× × × × × ××××××× ××× ××× ××××××××× × ×× × ×× ×× ×××××××

0.01

c/ o 0.001

1

x/`o

=1

1000

x/`o `o = (E 0 /

o)

3

(µ0 V /E 0 )

wo = (

o /E

0

)`o

Cohesive HF tip solution Two non-dimensional parameters: cohesive-to-confining stress ratio cohesive (rock)-to-lag-fracture energy

• •

c

,

c

o

Gc 1 = = Go 2

c wc o wo

10

Two limiting regimes:


c

⌧1:

c!k !o!m `c

/`o

`o

1

o!m ! c!k ! m

1:

c

`ko

`o

`mc

`c

`km

0.100

×× ×× ×× ×× ×× ×××× ×××× ×× × ×× ××××××××× ×× ×× ×× ×× ××××× × × × ×× ×

0.010

5

10

σc /σo = 0.1

FIELD

5

w/w1 (x)

c = 100 + c =1 +

1 0.50

c

m

= 0.01

× × +

0.10

o

0.05

0.01

0.001

1

σc /σo = 1.

LAB

×× ××× × ×× ××

c / o = 10

1

10

c = 100 +

1

c

100

k

+

c =1

0.50

c

× × ×

m

= 0.01

+

0.10

c

o

c

= 0.1

1000

× ×× × ×× ×× ×× ×××××

c

0.05

c/ o

0.10

k

w/w1 (x)

10

c/ o = 1

LEFM

0.001 0.01

Crack Opening normalized by m-asymptote

c / o = 0.1

×××× ×××××××××× × ×× ××××××××××××× × × ×× × × × × ××××××× ××× ××× ××××××××× × ×× × ×× ×× ×××××××

0.01

c/ o 0.001

1

x/`o

=1

1000

x/`o `o = (E 0 /

o)

3

(µ0 V /E 0 )

wo = (

o /E

0

)`o

Cohesive HF tip solution Two non-dimensional parameters: cohesive-to-confining stress ratio cohesive (rock)-to-lag-fracture energy

• •

c

,

c

o

Gc 1 = = Go 2

c wc o wo

10

Two limiting regimes:


c

⌧1:

c!k !o!m `c

/`o

`o

1

o!m ! c!k ! m

1:

c

`ko

`o

`mc

`c

`km

0.100

×× ×× ×× ×× ×× ×××× ×××× ×× × ×× ××××××××× ×× ×× ×× ×× ××××× × × × ×× ×

0.010

5

10

σc /σo = 0.1

FIELD

5

w/w1 (x)

c = 100 + c =1

1 0.50

c

+

m

= 0.01

× × +

0.10

o

0.05

0.01

0.001

0.50

0.05

c/ o 1

= 0.1

1000

c =1 ×

0.01

+

× ×× × ×× ×× ×× ××××× ×× ××× × ×× ××

c / o = 10

1

+

LAB

1

0.10

0.10

σc /σo = 10. c = 100

k

w/w1 (x)

10

c/ o = 1

LEFM

0.001 0.01

Crack Opening normalized by m-asymptote

c / o = 0.1

×××× ×××××××××× × ×× ××××××××××××× × × ×× × × × × ××××××× ××× ××× ××××××××× × ×× × ×× ×× ×××××××

10

c

100

k

c

+

×

c

×

m

= 0.01

o

k

c

o c/ o 0.001

1

x/`o

= 10

1000

x/`o `o = (E 0 /

o)

3

(µ0 V /E 0 )

wo = (

o /E

0

)`o

Cohesive HF tip solution Two non-dimensional parameters: cohesive-to-confining stress ratio cohesive (rock)-to-lag-fracture energy

• •

Two limiting regimes:


c

⌧1:

c

1:

Lag size

10

0.100

`c

×× ×× ×× ×× × ×× × ××

0.010 0.001 0.01

× × ×

× × × × ×

× × × × ×

× × × × ×

× × × × ×

0.1 c/ o = ××××××××××××

××××××××××× ××××××××××× ××××××××××× × × ×× ××××××××× ×

c/ o = 1

×

LEFM 0.10

×××××××××××× ×××× ×× ××××× ××××× × ××× ××× × × ××× ×× ×× ×× × ×× ×××

××××××

1

c / o = 10

10

c

100

c

o

Gc 1 = = Go 2

c wc o wo

`ko

`o

o!m ! c!k ! m `o

R/`o

1

,

c!k !o!m `mc

1

0.100 0.010

`c

`km

Cohesive size

10

/`o

c

+

+

+ + + + + + + + + + + ++ + + + + ++++ + + + + + ++ ++ ++ + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

0.001 0.01

/ c

0.10

o

+

+

+

0+.1+ +

+

=+ +

+

+

+

+

+

/ c

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

1 ++ = o ++

+

+

+

+

+

+

+

+

+

+

+

+

c/

+

+

+

+

+

+

= o

+

+

+

+

+

+

+

+

+

+

+

Ratio R/Lag 0+.+1++ + +

100

+

+

10 +

+

/+

10

0.100

++ ++ ++ +

10

+

0.010 +

1

100

+

c+ + + + + +

+

1

=+

o+ + +

+

+ ++ ++ ++ ++ ++ + ++ + + + + + + + + + + + + + +

0.001 0.01

+ + ++ ++ ++ + + + + + + + + + + + + +

/

+ + +

+

+ ++ ++ ++ ++++++ + + + + + ++ ++ + + ++ +++++ + + + + ++ + + +++ + ++ + ++++ ++ + + + ++ + + + + + +

+

+

+

+ +

o

=

10

c

0.10

1

10

100

c

c

`o = (E 0 /

o)

3

(µ0 V /E 0 )

wo = (

o /E

0

)`o

To conclude: there is now some theoretical evidence Cohesive HF tip solution: regime maps that
 LEFM is an imaginary friend σc /σo = 0.1 ×

Fluid Lag

+

Cohesive Zone

1

FIELD

c

c

o

=G G /G c /Go

10

0.10

0.01 0.001

c/ o

× + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + × + ×+ ×+ × + +× +× +×

0.100

10

100

+

1

0.10

0.01

1000

x /ℓo x/` o

0.100

10

⇠ 2 MPa wc ⇠ 100 µm c

0.001

p ⇠ 1.8 MPa m

m ± 1% 10

1000

x /ℓo x/` o m

⇠ 20 Mpa

wo = (E 0 /

P.Z. lengthscale `c ⇠ 1.6 m

0.100

k ± 1%

Fluid Lag parameters: o

KIc

1000

m

Cohesive Zone parameters:

Gc ⇠ 100 N/m,

× + × + × + × + Fluid Lag × + × + Cohesive Zone × + × + × + × + × + × + × + × + × + × + × + × + × + × + ×+ ×+ × + +× + × + × + × + × + × + × + × + × + ×

0.001

= 10

× + σc /σo = 10. × + × + + × Fluid Lag×× + × + + Cohesive × Zone + × + × + 10 × + × + × + × + × + × + × + 1 × + ×+ +× + × + × + × + × + × 0.10 + × + × + × + × + × + × + × + × 0.01 + ×

x /ℓo x/` o

m

Cohesive Frac. Energy

c/ o 100

σc /σo = 1. ×

10

=1

Gc /Go

100

= 0.1

Gc /Go

c/ o

o)

2

(µ0 V /E 0 ) ⇠ 5

Lag lengthscale `o ⇠ 0.01 ”Lag Frac. Energy” Go ⇠ 100 2000 N/m

100 µm (slickwater - gel)

0.16 m E 0 = 33 MPa V ⇠ 1 m/s

Additional slides

Measurements of Fracture Aperture Roughness Roughness from PSD:

w(f ˜ )=

1

P SD(f )df

f

Inada Granite [Matsuki et al. 2008] 1 x 0.2 m wedge-split fracture

100

10

e zz

p slo

‡

1 0.1

1

10

Sample Size

1000

···· · ace··· f r su ·· ·· · wc ª 200 mm ·· · ·‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ · · ‡ ‡ ‡ re · ‡ 5 2 ‡ . · 0 ‡ = ‡· apertu ‡· slope z w ‡ · ‡ ‡·· ‡ · ‡· ‡· .7 · =0 Grain Size

Roughness ˜wé H‡L,z˜(⇤) Roughness w(⌅), zé H·Lµm mm

104

sZ

Cross-over (process zone) Length lc ~ 45 mm

Matsuka et al. [IJRMMS 2006, 2008] Inada Granite Fracture

100

1000

Spatialscale Scalel, ,mm mm Spatial

f =1/λ

• Surface roughness is fractal • Aperture roughness, while fractal at scales below the grain size (~ 1-3 mm), saturates at scales exceeding a typical process zone size in granite 
 (~ 40 mm, e.g. Swanson JGR 1987) • Saturated aperture roughness value wc ⇠ 100 µm

Abstract What good is Linear Elastic Fracture Mechanics in Hydraulic Fracturing?
 D. Garagash (Dalhousie U. and Skoltech) and R. Viesca (Tufts U.)
 Abstract:
 Fluid-driven fracture presents an interesting case of crack elasticity and fracture propagation nonlinearly coupled to fluid flow. With the exceptions of a few numerical studies, previous hydraulic fracture modeling efforts have been based on the premise of Linear Elastic Fracture Mechanics (LEFM): specifically, that the damage (aka cohesive) zone associated with the rock breakage near the advancing fracture front is lumped into a singular point, under the tacit assumption that the extent of the cohesive zone is small compared to lengthscales of other physical processes relevant in the HF propagation. The latter include the dissipation in the viscous fluid flow in the fracture channel, of which the fluid lag - a region adjacent to the fracture tip filled with fracturing fluid volatiles and/or infiltrated formation pore fluid - is the extreme manifistation. In this work, we address the validity of the LEFM approach in hydraulic fraturing by considering the solution in the near tip region of a cohesive fracture driven by Newtonian fluid in an impermeable linear- elastic rock. First, we show that the solution in general possesses an intricate structure supported by a number of nested lengthscales (a general sentiment for HF), on which different dissipation processes are realized (or are dominant). The latter processes can be cataloged as (1) dissipation in the fracture cohesive zone, “c”, parameterized by the fracture energy Gc (cohesive energy release per unit fracture advance), the peak cohesive stress σc destroyed by fracturing, and corresponding fracture aperture scale wc = Gc/σc; (2) the LEFM “reduction” of the cohesive zone process, “k”, quantified by Gc, but with the cohesive zone replaced by a singularity (σc → ∞ and wc → 0); (3) viscous fluid dissipation associated with the fluid lag region, “o”, parametrized by an equivalent fracture energy Go = σowo, where σo is the in situ confining stress (signifying the fracturing fluid pressure drop in the lag region from value ∼ σo to near zero) and wo is the corresponding fracture aperture scale given previously by Garagash and Detournay (2000); and (4) the viscous dissipation along the rest of the fracture (away from the fluid lag), “m” (Desroches et al, 1994). Furthermore, each of the above limiting processes corresponds to distinct solution asymptotes. The HF tip solution structure is bookended by the “c” or “o” (solid or fluid process zones) asymptotes near the tip and by the “m” asymptote away from the tip, while the LEFM “k” asymptote may emerge within the transitional region, as an intermediate asymptote, depending on values of the two governing parameters: the cohesive-to-lag fracture energy ratio Gc/Go and the cohesiveto-in-situ stress ratio σc/σo. For typical sets of parameters representative of both field hydraulic fractures and their lab siblings, Gc/Go is either ∼ 1 (low-viscosity frac. fluid) or ≪ 1 (high viscosity frac. fluid). Under the above conditions, σc/σo ≫ 1 is shown to be required for appearance of the LEFM intermediate asymptote near the HF tip. Since σc ∼ 1 MPa for most rocks, it can be easily recognized that the latter condition for the relevance of the LEFM to hydraulic fracturing is mostly realized in laboratory experiments conducted under reduced levels of confining stress, and would almost never occur in the field (with the exception of very-near-surface fracturing and/or highly overpressured permeable formations).