e0b4952e7c436888d9

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2006; 30:1439–1475Published online 3 July 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.535

Transient solution for a plane-strain fracture drivenby a shear-thinning, power-law fluid

D. I. Garagash*,y

Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13699-5710, U.S.A.

SUMMARY

This paper analyses the problem of a fluid-driven fracture propagating in an impermeable, linear elasticrock with finite toughness. The fracture is driven by injection of an incompressible viscous fluid withpower-law rheology. The relation between the fracture opening and the internal fluid pressure and thefracture propagation in mobile equilibrium are described by equations of linear elastic fracture mechanics(LEFM), and the flow of fluid inside the fracture is governed by the lubrication theory. It is shown that forshear-thinning fracturing fluids, the fracture propagation regime evolves in time from the toughness- to theviscosity-dominated regime. In the former, dissipation in the viscous fluid flow is negligible compared tothe dissipation in extending the fracture in the rock, and in the later, the opposite holds. Correspondingself-similar asymptotic solutions are given by the zero-viscosity and zero-toughness (J. Numer. Anal. Meth.Geomech. 2002; 26:579–604) solutions, respectively. A transient solution in terms of the crack length,the fracture opening, and the net fluid pressure, which describes the fracture evolution from the early-time (toughness-dominated) to the large-time (viscosity-dominated) asymptote is presented and someof the implications for the practical range of parameters are discussed. Copyright # 2006 John Wiley &Sons, Ltd.

Received 1 February 2005; Revised 1 February 2006; Accepted 1 April 2006

KEY WORDS: hydraulic fracturing; self-similar solutions; non-Newtonian fluids; power-law fluids

1. INTRODUCTION

The problem of a fluid-driven fracture propagating in rock arises in hydraulic fracturing, atechnique widely used in the petroleum industry to enhance the recovery of hydrocarbons fromunderground reservoirs [1], as well as in the magma transport in the Earth’s crust by means of

*Correspondence to: D. I. Garagash, Department of Civil and Environmental Engineering, Clarkson University,8 Clarkson Ave., Potsdam, NY 13699-5710, U.S.A.

yE-mail: [email protected]

Contract/grant sponsor: Donors of The Petroleum Research Fund, American Chemical Society; contract/grant number:ACS-PRF 36729-G2

Copyright # 2006 John Wiley & Sons, Ltd.

magma-driven fractures [2, 3]. The conditions under which fluid-driven fractures propagate inrock vary widely (i.e. rock parameters, in situ stress and fracturing fluid rheology) and areusually not well defined. In addition, hardly any in situ measurements regarding the fracturedimensions and shape exist. Consequently, mathematical and numerical modelling of fluid-driven fractures have attracted numerous contributions, see, e.g. References [4–9] for some earlye!orts. The main objective of the theoretical modelling is to predict the evolution of the fluidpressure pf ; the fracture opening w and the size ‘ for given rock properties, fluid rheology andinjection rate under assumptions of simplified fracture geometry.

The modelling of hydraulic fracture requires consideration of both fluid and solid mechanics:on one hand, the lubrication equation to characterize the flow of fluid in the fracture; and on theother, the elasticity equations to describe the deformation and propagation of the fracture. Thesolution of these models and their parametric analysis are notoriously di"cult to developbecause of the strong non-linear coupling between the lubrication and elasticity equations andthe non-local character of the elastic response of the fracture. ‘Non-locality’ implies that thecrack opening w at one location along the fracture depends on the fluid pressure pf at anotherlocation. The ‘non-linearity’ stems from the dependence of the fracture conductivity on a certainpower of the crack opening. Non-linearity and non-locality yield a complex solution structureinvolving processes at a very small lengthscale near the fracture tip, which, nevertheless, controlthe global response of a fluid-driven fracture [10–15]. Recent rigorous treatment of hydraulicfracture models of idealized plane-strain and axisymmetric geometries in the case of aNewtonian fracturing fluid (see, e.g. References [15–21]) has allowed to identify di!erentlimiting propagation regimes, some of the corresponding asymptotic solutions and respectivebehaviour of these solutions near the fracture tip. In the particular case of a fracturepropagation in an impermeable rock, the two limiting regimes can be identified with thedominance of one or the other of the two energy dissipation mechanisms corresponding toextending the fracture in the rock and to flow of viscous fluid in the fracture, respectively. In theviscosity-dominated regime, dissipation in extending the fracture in the rock is negligiblecompared to the dissipation in the viscous fluid flow, and in the toughness-dominated regime, theopposite holds. Corresponding self-similar asymptotic solutions for Newtonian fluid case aregiven by the zero-toughness [17, 22] and zero-viscosity [21] solutions, respectively. In the case ofa plane-strain fracture and a constant fluid injection rate, the propagation regime is controlledby a single non-dimensional time-independent constant with the meaning of either dimension-less toughness or viscosity.

In this paper, we extend the analysis of the propagation regimes of a plane-strain hydraulicfracture to the more general case of a fracturing fluid with a power-law rheology. This extensionis motivated by the recognition that fracturing fluids in petroleum and geophysical applicationsoften exhibit non-Newtonian (non-linear) behaviour [23–25]. A class of non-Newtonian fluidscan be defined in the context of unidirectional laminar flow via the following rheological law,which relates the shear stress in the fluid t to a certain power of the shear strain rate ’g:

t ! M ’gn "1#

where n is the power-law exponent (also called ‘fluid behaviour index’) andM is the ‘consistencyindex’. For Newtonian fluids, n ! 1; and M ! m: Most of the fracturing fluids used in thepetroleum industry are characterized by relation (1) in a range of shear rate ’g with 04n51: Thisrange of n corresponds to the so-called ‘shear-thinning’ behaviour, as the apparent fluid viscositymn ! M ’gn$1 is a decreasing function of the shearing rate.

D. I. GARAGASH1440

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:1439–1475DOI: 10.1002/nag

We proceed to show that the fracture propagation regime in the case of a shear-thinning fluidis not fixed in time as for the case of Newtonian fluid, but rather evolves from the early-timetoughness-dominated regime to the large-time viscosity-dominated regime, since the governingdimensionless viscosity parameter is an increasing function of time for any 04n51: The end-of-time, zero-toughness solution has been investigated by Adachi and Detournay [22]. Thebeginning-of-time zero-viscosity solution (see Reference [21] for Newtonian fluid case)corresponds to the classical solution of a uniformly pressurized Gri"th’s crack with a specialconstraint on the fracture volume. The transient solution which describes evolution of the cracklength, the fracture opening, and the net fluid pressure from the zero-viscosity to the zero-toughness limiting solution is computed herein numerically. The numerical technique is basedon the series expansion of the normalized solution for the opening and the net pressure over aclass of base functions in the spatial co-ordinate [15, 22] which satisfy the elasticity singularintegral equation and conform with the proper tip asymptote. The evolution of the solution intime corresponds to that of the series coe"cients, governed by the set of ordinary di!erentialequations (ODEs) resulting from the spatial discretization of the fluid flow equation. Finally,some implications of the transient solution for a practically relevant parametric range inhydraulic fracturing are discussed.

2. MATHEMATICAL MODEL

Consider a finite two-dimensional hydraulic fracture of half-length ‘"t# propagating in animpermeable linear elastic medium characterized by Young’s modulus E; Poisson’s ratio n; andtoughness KIc: An incompressible fluid with power-law rheology, behaviour index n andconsistency index M; is injected at the centre of the fracture at a constant rate Qo: The crack isloaded by the internal fluid pressure pf "x; t# and by the far-field confining stress so (positive incompression), see Figure 1. The fracture is assumed to be in mobile equilibrium at all times andits propagation is quasi-static. We will look for the solution of the fluid-driven fracture problem:net pressure pf "x; t# $ so; fracture opening w"x; t#; and fracture half-length ‘"t#; where x is theposition along the crack and t is time. The fracture is assumed to be fully fluid-filled at all times,i.e. there is no lag between the fracture and fluid front. Validity of this assumption is discussed inthe next section.

In the following we formulate the governing equations over the half of the crack, 04x4‘;accounting for the crack symmetry with respect to its centre. Additional discussion of governingequations can be found in, e.g. References [15, 21].

The fluid flow inside the fluid-filled crack is described by equations of lubrication theory [26],namely, by the integral form of the continuity equation:

@@t

Z ‘

xw dx ! q "2#

where q is the fluid flow rate per unit (out-of-plane) width of the crack; and by Poiseuille’slaw [23]:

q j q jn$1! $w2n%1

M0

@p@x

"3#

where M0 ! fM is a modified consistency index with f ! 2n%1"2n% 1#n=nn being a numericalfactor. Alternatively, Poiseuille’s law for a non-Newtonian fluid (3) can be written in the form

PLANE-STRAIN FRACTURE DRIVEN BY A POWER-LAW FLUID 1441


for an ‘equivalent’ Newtonian fluid

q ! $w3

m0

@p@x

; m0 ! M0 w2

jq j

! "1$n

"4#

where the e!ective viscosity parameter m0 is given up to a numerical factor by the apparentviscosity mn ! M ’gn$1 evaluated at the (crack) channel wall: m0 ! 4n$1"2n% 1#mn

wall (seeAppendix A for the details). For a Newtonian fluid, n ! 1; the e!ective viscosity m0 and thee!ective consistency index M0 coincide and equal, up to a numerical factor, to a constantNewtonian viscosity m; m0 ! M0 ! 12m:

Global fluid continuity requires the injected fluid volume per unit crack width, Qot; to beequal to the fracture volume (zero fluid lag and no leak-o!),

2

Z ‘

0w dx ! Qot "5#

In view of (2), the latter is equivalent to the boundary condition for the fluid flux at the inlet:

q"x ! 0; t# !Qo

2"6#

The deformation of the solid and the fracture propagation criterion are prescribed byequations of linear elastic fracture mechanics as follows. The net pressure in the crack p !pf $ so is related to the ‘dislocation density’ @w=@x via the Cauchy singular integral [27], whichunder provision of crack symmetry with respect to the origin can be written as

p"x; t# ! $E0

2p

Z ‘

0

@w"x0; t#@x0

x0 dx0

x02 $ x2"7#

The inverse of the above integral relation providing the crack opening w in relation to the netpressure p ! pf $ so is given by [28]

w"x; t# !4

pE0

Z ‘

0G

x

‘;x0

‘

! "p"x0; t# dx0; G"x; x0# ! ln

#############1$ x2

p%

##############1$ x02

p#############1$ x2

p$

##############1$ x02

p

$$$$$

$$$$$ "8#

The criterion of fracture propagation in mobile equilibrium, which requires the stress-intensityfactor to be equal to the material toughness, KI ! KIc; is expressed here in the form of the tip

Figure 1. Sketch of a plane-strain fluid-driven fracture.

D. I. GARAGASH1442


asymptote of the crack opening [29]

w !K 0

E0 "‘ $ x#1=2 "‘ $ x{‘# "9#

In the above formulation, the three material parameters E0; M0; and K 0 with the meaning ofthe e!ective elastic modulus, fluid consistency index, and rock toughness, respectively, aredefined as follows:

E0 !E

1$ n2; M0 ! fM; K 0 ! 4

2

p

! "1=2

KIc "10#

Equations (2)–(9) fully define the fracture length ‘"t#; the opening w"x; t#; and the net pressurep"x; t# as functions of the parameter set (10), the fluid behaviour index n; and the injectionrate Qo:

3. BEHAVIOUR NEAR THE CRACK TIP

In accordance with linear elastic fracture mechanics (LEFM) theory, the opening asymptoticallyvaries as the square root of the distance from the crack tip for any non-zero value of rocktoughness K 0; (9). The asymptotic behaviour of the net pressure gradient for the non-zerotoughness can be deduced from the analysis of the leading term in the lubrication equation (2)with (3) and (9) for ‘ $ x{‘: The corresponding net pressure tip asymptote is then given

p ! const"t# %M0 E0

K 0

! "1%n

"vtip#n2

1$ n"‘ $ x#"1$n#=2; n=1

ln 1$x

‘

% &; n ! 1

8>><

>>:"‘ $ x{‘# "11#

where const"t# is an integration constant (the part of the overall solution) and vtip ! ’‘ ! d‘=dt isthe crack tip velocity. Consequently, the net pressure is always finite at the fracture tip for shear-thinning fluids "n51#; and has a negative (logarithmic) singularity for Newtonian fluids "n ! 1#and a negative algebraic singularity for fluids with n > 1: Since the fluid cannot sustain largeenough tension, cavitation has to necessarily occur near the tip of a fracture driven by aNewtonian fluid or a fluid with n > 1; thus, resulting in a lag filled by fracturing fluid vapoursbetween the fracture front and the fluid front. For shear-thinning fluids "n51#; however, the tipnet pressure is finite and cavitation does not necessarily take place unless a finite value of fluidpressure at the tip drops below the cavitation pressure. Given that the cavitation pressure istypically negligible compared to the confining stress so; the cavitation condition for the netpressure implies p4$ so: Since an unknown constant term (a function of the overall solution)enters the pressure asymptote (11), the latter cavitation condition cannot be evaluated unless thecomplete solution is known. However, it is interesting to note that the ‘likeliness’ of thiscondition to be satisfied decreases with either the increase of confining stress so or with thedecrease of the fluid behaviour index n (result of the weakening of the pressure gradientsingularity at the crack tip with decreasing n).

The above considerations, asymptotic tip boundary layer analysis under conditions ofvanishing e!ects of toughness (see below and in Appendix B), as well as the actual solution ofthe problem discussed in Section 7 provide justification of the assumption of the zero fluid lag



(or of a fully fluid-filled crack) employed in this analysis for the case of a shear-thinning fluid. Itis also worthwhile to point out that in the case of a fracture driven by a Newtonian fluid, evenwhen the lag is non-zero, it becomes vanishingly small fraction of the fracture length for longenough fractures or large enough confining stress [12, 30].

In the case of zero rock toughness (e.g. a hydraulic fracture propagating along a preexistingdiscontinuity), the coupling between elastic response (8) and the fluid flow in fracture (2)–(3)yields the tip asymptotic behaviour in the form [10]

K 0 ! 0 : w ! W&"‘ $ x#2="2%n#; p ! $P&"‘ $ x#$n="2%n# "‘ $ x{‘# "12#

with the coe"cients W& and P& are given in the above reference as a function of the crack tipvelocity and material parameters. Since 2="n% 2# > 1=2 for shear-thinning fluids, the viscosityopening asymptote "12#a is ‘weaker’ than the LEFM one, confirming the zero toughnessassumption. One can expect that the zero-toughness solution (constructed for a power-law fluidin Reference [22]) which possesses tip asymptote (12) has to be a valid approximation of thesolution with non-zero small enough values of toughness K 0: The singular change in the tipasymptote from the K 0 ! 0 case, (12), to the, however, small but non-zero toughness LEFMcase, (9) and (11), therefore implies that for small values of toughness there exists a boundarylayer at the fracture tips in which the toughness e!ects are localized while the solution awayfrom the fracture tip is given by the zero-toughness solution. This boundary layer solution ismatching with the zero-toughness solution away from the fracture tips over an intermediate tiplengthscale where both solutions posses zero-toughness asymptote (12), see Figure 2. The smalltoughness solution with the structure described above has been explicitly constructed in the caseof Newtonian fluid "n ! 1# in Reference [19]. The expression for the thickness of the tipboundary layer, #‘ ! #K"2"2%n#="2$n##

m ‘; can be obtained from the direct application of the boundarylayer methodology [19] to the generalized governing equations for a power-law fluid case, givenin Appendix B. #Km is the dimensionless tip toughness parameter defined by (B8) in terms of thecrack tip velocity, the global crack lengthscale ‘; and the set of material parameters, whichcharacterizes the smallness of the toughness e!ect compared to the one of the viscosity on theglobal solution.

Figure 2. Sketch of the structure of the small toughness solution for fluid index n52: (The extent ofthe near-tip boundary layer relative to the crack length is greatly exaggerated.) The near-fracture-tipboundary layer provides transition between the toughness asymptote (9) at the immediate vicinity ofthe fracture tip and the viscosity asymptote (12) at intermediate distances from the tip, where it ismatched with the zero toughness outer solution valid on the lengthscale of the fracture. (Variable #x

denotes distance from the fracture tip, ‘ $ x:)

D. I. GARAGASH1444


Even though outside of this paper’s focus on shear-thinning fluids, it is interesting to note thatthe structure of the tip boundary layer (Figure 2) in the small toughness solution becomesinvalid for shear-thickening fluids with su"ciently high index n > 2: Indeed, in this case theviscosity asymptote for the opening "12#a is ‘stronger’ than the toughness one (9), and theboundary layer thickness diverges as #Km ! 0: This is an indication that the small toughnesssolution does not have a tip boundary layer (i.e. it constitutes a regular perturbation of the zero-toughness solution). On the other hand, in the small viscosity (large toughness) case for n > 2;the boundary layer is expected to accommodate the transition between the viscosity tipasymptote valid immediately near the tip and the toughness asymptote at intermediate distancesfrom the tip, where it is matched with the zero-viscosity outer solution (classical solution for auniformly pressurized Gri"th’s crack).

In this paper, we focus on the finite toughness solution which makes use of the LEFM tipasymptote specified by (9) and (11). In the view of the discussion above, this solution cannot beused when toughness is zero or su"ciently small. It will be shown, however, that the developedfinite toughness solution for small enough values of toughness does approach the zero-toughness solution [22], and, therefore, together with the latter can be used to map the completerange of toughness.

4. SCALING

Two physically meaningful scalings can be identified with conditions when the e!ect of thematerial toughness K 0 on a fluid-driven fracture is large compared to the e!ect of fluid viscositym0 and when the opposite holds, respectively. Respective scalings can be dubbed ‘toughness’ and‘viscosity’ scalings, as the former is independent of the fluid consistency index M0 (related to theapparent viscosity) and the latter is independent of K 0: The scaling methodology and the abovetwo scalings can be found in Reference [13] in the case of a Newtonian fluid. Furthermore, theviscosity scaling for a power-law non-Newtonian fluid has been established in Reference [22]. Inthe following we adapt the scaling methodology [13] to the case of a non-Newtonian fluid toobtain both the viscosity and the toughness scalings by introducing the concept of an‘equivalent’ Newtonian fluid with time-dependent characteristic viscosity.

Let us define the scaled co-ordinate x ! x=‘"t# "04x41# and the dimensionless fieldvariables: opening O; net pressure P; flux C; e!ective viscosity M; and the crack half-length g inthe following general form:

w"x; t# ! e"t#L"t#O"x; t# "13a#

p"x; t# ! e"t#E 0P"x; t# "13b#

q"x; t# ! e"t#L2"t#t$1C"x; t# "13c#

m0"x; t# ! m0"t#M"x; t# "13d#

‘"t# ! L"t#g"t# "13e#



The above expressions prescribe a particular scaling of a hydraulic fracture for a given choice ofthe fracture lengthscale L; the dimensionless parameter e (scaling the net pressure as a fractionof the modulus E 0), and the characteristic viscosity m0: Based on the global continuity (5), thedefinition "4#b of the e!ective viscosity m0; and requiring that all dimensionless field variables(O; P; etc.) are O"1#; parameters e and m0 can be related to a lengthscale L:

e !Qot

L2; m0 ! M0"et#1$n ! M0 Qot2

L2

! "1$n

"14#

Thus, a particular fracture scaling (13a)–(13e) with (14) can be specified by a single parameter,the fracture lengthscale L:

4.1. Normalized equations

Introducing an alternative scaling for the opening and flux as

%O !Og; %C !

Cg2

"15#

the governing equations (2)–(9) and (11) in scaling (13a)–(13e) with (14) can be reduced to thefollowing universal form.

* Local fluid balance:

1% 2G%@

@ ln t

! " Z 1

x

%O dx% "d% G#x %O ! %C G !d ln gd ln t

! ""16#

where d ! d"ln L#=d"ln t# is a constant when the fracture lengthscale is a power law of time,L ' td:

* Fluid flux:

%C j %C jn$1! $%O2n%1

M

@P@x

"17#

or its equivalent expression for an ‘equivalent’ Newtonian fluid:

%C ! $%O3

MM@P@x

; M !%O2

j %C j

! "1$n

"18#

* Global fluid balance:

1

2g2!Z 1

0

%O dx "19#

or its equivalent expression for the fluid flux at the inlet

%C"x ! 0; t# !1

2g2"20#

* Elasticity equation:

P"x; t# ! Lf %Og"x; t# ! $1

2p

Z 1

0

@ %O"x0; t#@x0

x0 dx0

x02 $ x2"21#

D. I. GARAGASH1446


or in its inverse form

%O"x; t# ! L$1fPg"x; t# !4

p

Z 1

0G"x; x0#P"x0; t# dx0 "22#

* Propagation condition and corresponding pressure gradient asymptote:

1$ x{1 : %O ! Kg$1=2"1$ x#1=2;@P@x

! $A0"1$ x#$"1%n#=2 "23#

where

A0 ! MK$1$ng"1%n#=2"d% G#n "24#

The above set of normalized equations (16)–(24) governs the normalized solution F"x; t# !f %O;P; %C; gg which depends on the two dimensionless groups associated with viscosity andtoughness, respectively,

M !m0

tE0e3; K !

K 0

eE0L1=2"25#

where e and m0 are the functions (14) of a lengthscale L:

4.2. Viscosity and toughness scalings

For a particular scaling, lengthscale L"t# can be specified by setting one of the dimensionlesscombinations, (25), to unity, while the remaining dimensionless combination will constitute asingle dimensionless parameter governing the normalized solution of (16)–(24). By setting eitherM ! 1 or K ! 1 one can obtain the ‘viscosity’ or the ‘toughness’ scaling characterized by theproblem lengthscale L independent of the toughness K 0 or the characteristic viscosity m0;respectively.

Resulting lengthscale L"t#; the characteristic viscosity m0"t#; and the dimensionless parametere"t# in the toughness "K# and the viscosity "M# scalings and corresponding expressions for theviscosity M and toughness K dimensionless groups are listed in Table I. Naturally, in thetoughness-scaling the solution scales (via scaling parameters e and L) with the materialtoughness and the corresponding dimensionless toughness is unity; and in the viscosity-scalingsolution scales with the fluid viscosity and the corresponding dimensionless viscosity is unity.The remaining non-trivial parameter (viscosity M in toughness-scaling, or toughness K in

Table I. Quantities corresponding to the viscosity (M) and the toughness (K) scalings.

Scaling m0 L e M K

"K# M0 K 02t

E02Q1=2o

!2"1$n#=3E0Qot

K 0

! "2=3 K 04

E04Qo t

! "1=3

Mk !E03Qom0

K 04 1

"M# M03="2%n# t

E01=2

% &2"1$n#="2%n# E01=6Q1=2o t2=3

m01=6m0

E0t

! "1=3

1 Km !K 0

E03=4Q1=4o m01=4



viscosity-scaling) governs the solution in the respective scaling. Corresponding indices ‘k’and ‘m’ ought to be added to all quantities (except for time t and space x) in thenormalized governing equations (16)–(24) to designate equations in the two respectivescalings. (The time-exponent d of the fracture lengthscale, L ' td; appearing in the fluidbalance equation (16) takes on values dk ! 2=3 and dm ! "1% n#="2% n# in the correspondingscalings.)

Remarkably, the form of the two scalings for a power-law, non-Newtonian fluid presented inTable I in terms of the corresponding characteristic time-dependent viscosities m0k;m"t# isequivalent to the form of the respective scalings for a Newtonian fluid [13] for which m0k;m ! 12mand m is a (constant) Newtonian viscosity. The expressions from Table I for the fracturelengthscale L and dimensionless groups M and K in terms of the characteristic time-dependentviscosities are expanded in Appendix C (Table CI) to reveal their explicit time-dependence. Weacknowledge that the dimensionless toughness parameter #Km; arising from the considerations(of Section 3 and Appendix B) of the LEFM boundary layer near the front of an arbitraryfracture propagating under conditions of small toughness, is identical up to O"1# constant factorto the dimensionless toughnessKm for the plane-strain fracture (Table CI). In order to show thelatter equivalence of the these two measures of the e!ects of toughness on the fluid-drivenfracture propagation, one need to set the global fracture lengthscale ‘ in the definition #Km; (B9),to the viscosity lengthscale Lm (Table CI) and use corresponding expression for the crack tipvelocity vtip ! ’‘ ' ’Lm:

We further note that the solutions in the two scalings are simply interrelated via

gmgk

!Lk

Lm! K$2=3

m ;Pm

Pk!

%Om

%Ok!

%Cm

%Ck!

Mm

Mk

! "1="1$n#

!m0km0m

! "1="1$n#

!ekem

! K4=3m "26#

and

Mk ! K$"4=3#"n%2#m "27#

The time-dependence of the governing dimensionless parameter (Mk ' t"2=3#"1$n# orKm ' t$"1=2#"1$n#="2%n#) is defined by that of the characteristic viscosity m0 in the correspondingscaling (Table CI). Consequently, the evolution terms of the form t @"(#=@t in the local fluidbalance (16) can be expressed in terms of derivatives in either dimensionless viscosity Mk ortoughness Km with the help of the following identities:

@@ ln t

!2

3"1$ n#

@@ lnMk

! $1

2

1$ n

2% n

@@ lnKm

"28#

We then observe that the time-dependence of the normalized solution F"x; t# ! f %O;P; %C; gg forthe opening, pressure, flux and crack length comes only through the dimensionless viscosity ortoughness parameter, such that F"x; t# ! Fk"x;Mk# and F"x; t# ! Fm"x;Km# in the respectivescalings.

4.3. Special fluid cases (n ! 0 and n ! 1)

The two cases of fluid behaviour corresponding to perfectly plastic fluid "n ! 0# and Newtonianfluid "n ! 1# correspond to the special cases where the solution-dependence on time is known inadvance.

In the Newtonian fluid case "n ! 1#; Mk and Km are time-independent constants, Table I,thus, the evolutionary terms are zero, see (28), and the lubrication equations (16)–(17) reduce to

D. I. GARAGASH1448


a self-similar formZ 1

x

%O dx% dx %O ! %C ! $%O3

M

dPdx

"G ! 0# "29#

The corresponding normalized solution is, thus, self-similar [15] (i.e. time-independent in eitherscaling), and the dependence on time of dimensional opening, pressure and crack length is givenby that of the scaling parameter e and lengthscale L; Table I [13].

In the case of perfectly plastic fluid "n ! 0#; lubrication equations (16)–(17) reduce to [22]

1 ! $%OM

dPdx

"30#

And, even though, Mk ' t2=3 and Km ' t$1=4 are time-dependent, the reduced equations (30),(19)–(23) are no longer evolutionary since they depend on time as a parameter only.

5. PROPAGATION REGIMES

Normalized solution F ! f %O;P; %C; gg of Equations (16)–(24) in either M- or K-scalingdepends on normalized position along the fracture x and a single governing dimensionlessparameter Mk or Km; respectively. Consequently, the propagation regimes of a hydraulicfracture can be completely defined in terms of one of these two parameters. Similar to theanalysis for the fracture driven by Newtonian fluid [13, 17–21, 31, 32], the two limitingparametric regimes corresponding to either Mk{1 (toughness-dominated regime) or Km

{1 (viscosity-dominated regime) can be identified with the situation where either the e!ectof the fluid viscosity or the e!ect of material toughness on the hydraulic fracturepropagation is negligible, respectively. The solutions in these regimes can be approximatedby the zero-viscosity "Mk ! 0# solution [21] or the zero-toughness (Km ! 0) solution [22],respectively. These normalized limiting solutions are self-similar (i.e. time-independent) inthe respective scalings, such that the time-dependence of the dimensional solution is givenby that of the scaling parameters (Table I). Thus, according to the scaling discussion of theprevious section, the hydraulic fracture propagation in time driven by a non-Newtonianfluid in either the toughness-dominated (‘early-time’) or the viscosity-dominated (‘large-time’) regime can be associated with the propagation of a fracture driven by an‘equivalent’ Newtonian fluid with appropriate time-dependent viscosity, m0k"t# or m0m"t#;respectively.

In view of Table I, Mk and Km are increasing and decreasing power-laws of time for shear-thinning fluids "n51#; respectively. Thus, hydraulic fractures evolve from the toughness-dominated regime (with the zero-viscosity solution) at early time to the viscosity-dominatedregime (with the zero-toughness solution) at large time. This evolution can be tracked in eitherof the two dimensionless parameters. The goal of this paper is to construct the transient solutionfor the hydraulic fracture evolution between these two limits. Thus, first, we discuss the limitingself-similar states of the transient solution, i.e. the large-time zero-toughness limit, the early-timezero-viscosity limit, and, secondly, the small viscosity correction to the latter. The small (butnon-zero) viscosity solution will provide the starting point in time for the numerical transientsolution.



5.1. Zero-toughness self-similar solution [22]

In the limit of large (infinite) propagation time, the toughness parameter Km vanishes, and thegoverning system of equations in viscosity-scaling reduce to a self-similar form

Z 1

x

%Om0 dx%1% n

2% nx %Om0 ! $ %O2n%1

m0

dPm0

dx

! "1=n

; g$2m0 ! 2

Z 1

0

%Om0 dx "31#

%Om0"x# ! L$1fPm0g"x#; limx!1

"1$ x#$1=2 %Om0"x# ! 0 "32#

Equations (31) are the reduction of the lubrication equation and global fluid balance, andEquations (32) are the elasticity equation and propagation criterion under condition of zerotoughness, respectively. The latter, "32#b; implies that opening has the asymptote at the tip otherthan the LEFM type "23#a: Indeed, the appropriate asymptote corresponds to the zero energydissipation in fracturing the solid and solely determined by the viscous dissipation in the fluidflow. This asymptote, which follows from the asymptotic considerations for the lubricationequation "31#a and elasticity "32#a; is given by [10, 22]

1$ x{1 : %Om0 ! Bm0"1$ x#a; Pm0 ! Bm0a"a$ 1# cot pa

4)

1

a$ 1"1$ x#a$1; n=0

ln"1$ x#; n ! 0

8><

>:"33#

where

a !2

n% 2; Bm0 !

1% n

2% n

! "1$a 2"2% n#2

ntan"$pa#

' (a=2

"34#

In the opening near-tip power law "33#a; the exponent a varies from 1 (wedge-like tip) at n ! 0 to2=3 at n ! 1:

The zero-toughness self-similar solution f %Om0"x#;Pm0"x#; gm0g governed by Equations(31)–(32) with the near-tip asymptote (33)–(34) has been derived by Adachi and Detournay[22] using the method of the series expansion over the class of base functions satisfying elasticity(32) and possessing the correct tip asymptote (33). The coe"cients in the series expansion areobtained by minimizing the quadratic error in the lubrication equation "31#a: For an example oftheir solution for fluids with n ! 0; n ! 0:5; and n ! 1; see the dimensionless net pressure andopening in the viscosity scaling shown by dashed lines on Figures 4"a0–c0# and 5"a0–c0#;respectively.

5.2. Zero-viscosity self-similar solution

In the limit of infinitesimally small propagation time, the viscosity parameter Mk vanishes, andthe governing system of equations in toughness-scaling reduce to a self-similar form

0 ! $ %O2n%1k0

dPk0

dx; g$2

k0 ! 2

Z 1

0

%Ok0 dx "35#

%Ok0"x# ! L$1fPk0g"x#; 1$ x{1 : %Ok0"x# ! g$1=2k0 "1$ x#$1=2 "36#

D. I. GARAGASH1450


The zero-viscosity self-similar solution of the above equations corresponds to a classicalsolution of a Gri"th’s crack with the elliptic shape and uniform net pressure along thecrack [21]

Pk0 !p1=3

8; %Ok0 !

p1=3

2

#############1$ x2

q; gk0 !

2

p2=3"37#

5.3. Small-viscosity solution

The small-viscosity correction to the zero-viscosity solution (37) for a fracture driven by apower-law fluid can be obtained by a direct adoption of the approach used in References [20, 21]for the case of a Newtonian fluid. In the following, we briefly summarize the approach and theresults. Utilizing the toughness scaling, the small-viscosity solution "Mk{1# can be sought interms of the series, Fk0 ! Fk0 %MkFk1 % ( ( ( ; as follows:

Pk ! Pk0"x# %MkPk1"x# % ( ( ( "38a#

%Ok ! %Ok0"x# %Mk %Ok1"x# % ( ( ( "38b#

gk ! gk0 %Mkgk1 % ( ( ( "38c#

where the first term Fk0 in the expansion corresponds to the zero-viscosity solution (37) and thesecond term MkFk1 corresponds to the next-order viscosity correction. Substitution of (38a)–(38c) in the governing equations in the toughness scaling (16)–(24) with Table I"k# and "28#awhile retaining terms of O"1#; provides governing equations (35)–(36) for the zero-viscositysolution (37). Similarly, retaining terms of O"Mk# provides governing equation for the next-order term Fk1 in expansion (38a)–(38c), which solution yields

Pk1 ! Pk1"0# % DPk1"x# "39#

%Ok1 ! 4Pk1"0##############1$ x2

q%L$1fDPk1g"x# "40#

gk1 ! $g3k0 pPk1"0# % 4

Z 1

0

#############1$ x2

qDPk1"x# dx

! ""41#

where the constant Pk1"0# ! Pk1"x ! 0# and the function DPk1"x# are defined by

Pk1"0# ! $4

3p

Z 1

0

1% x2#############1$ x2

p DPk1"x# dx; DPk1"x# ! $Z x

0

%Cnk0

%O2n%1k0

dx "42#

In the above, %Ck0 corresponds to the dimensionless flux in the zero-viscosity solution (37)

%Ck0 !Z 1

x

%Ok0 dx%2

3x %Ok0 !

p1=3

123cos$1 x% x

#############1$ x2

q! "

The explicit analytical expressions for the integrals in (39)–(42) can be obtained in the case of aNewtonian "n ! 1# [21] and a perfectly plastic "n ! 0# fluids, see Appendix D. In theintermediate case "05n51# the integrals are evaluated numerically.



In accordance with earlier discussion of the asymptotic behaviour of the solution nearthe crack tip in the general case, the asymptotic behaviour of the small-viscosity solution(38a)–(38c) is given by "37#b for the opening and by "23#b with Table I"k# for the pressure gradient

1$ x{1 : %Ok !p1=3

21=2"1$ x#1=2;

@Pk

@x! $Mk

2"1%3n#=2

3np"1%n#=3 "1$ x#$"1%n#=2

6. TRANSIENT SOLUTION

6.1. Formulation of initial value problem

The evolution of the hydraulic fracture solution in toughness scaling, Fk"x;Mk# ! f %Ok;Pk; gkg;with the viscosity parameter Mk from the toughness-dominated regime (Mk ! 0; zero-viscositysolution (37)) to the viscosity-dominated regime (Mk ! 1; zero toughness solution [22]) isgoverned by equations in toughness scaling, (16)–(24) with Table I"k# and "28#a:

The initial conditions for this evolution problem are provided by the small-viscosity solutionof the previous section evaluated at a certain initial small (but non-zero) value of viscosityevolution parameter Mk ! Mkini; i.e.

Fk"x;Mk ! Mkini# ! Fk0"x# %MkiniFk1"x# "43#

(Note that the zero-viscosity solution at Mk ! 0 cannot be used as the starting point of thetransient solution, since the former corresponds to a self-similar solution of the governingequations, and, therefore, does not allow to resolve the initial Mk-rates of the sought fieldvariables at Mk ! 0). Thus, the solution of (16)–(24) with Table I"k# and "28#a; and initialconditions (43) is sought for Mk5Mkini:

6.2. Method of solution

The numerical technique is based on two main steps.

* The series expansion of the normalized solution for the opening and the net pressure over aclass of base functions in the spatial co-ordinate which satisfy the elasticity singularintegral equation and conform with the proper tip asymptote [15, 22, 33].

* Formulation of evolutionary equations (system of ODEs) to solve for the evolution of thecoe"cients of the above solution series expansion with Mk:

In the remainder of Section 6.2 we will omit the toughness-scaling index ‘k’ for brevity.

6.2.1. Spatial series expansion. The solution expansion proposed below is inspired by theconsiderations of Adachi [33] and Spence and Sharp [15] for the self-similar solution in theNewtonian fluid case "n ! 1#: To proceed, we approximate the solution for %O and P in the form

P"x;M# ! 4$1"2g"M##$1=2 %XN

j!0

Aj"M#Pnj "x# %AN%1"M#Pn

N%1"x# "44#

%O"x;M# ! "2g"M##$1=2"1$ x2#1=2 %XN

j!0

Aj"M# %Onj "x# %AN%1"M# %On

N%1"x# "45#

D. I. GARAGASH1452


where the net pressure and opening base functions Pnj "x# and %On

j "x# are selected such that: (i)each base function pair satisfies the elasticity equation %On

j ! L$1fPnj g; (ii) the contribution of

any net pressure base function to the stress-intensity factor is zeroZ 1

0

Pnj "x# dx#############1$ x2

p ! 0 "46#

and (iii) the leading pressure base function Pn0"x# provides the net pressure gradient asymptote

at the fracture tip "23#b; i.e. dPn0"x#=dx ! $"1$ x#$"1%n#=2 "x ! 1# and the corresponding

coe"cient A0"M# in expansion (44)–(45) is given by (24) with Table I"k#:

A0"M# ! Mg"1%n#=2"M#2

3% G"M#

! "n

"47#

The latter also implies that the net pressure base functions Pnj "x# with j51 must have either

finite gradient at the tip or a singularity weaker than that of the leading term Pn0"x#: In the view

of (i) above, the first constant term in the net pressure expansion (44) yields the expression of thestress-intensity factor and the corresponding first term in the expression for the opening (45)satisfies propagation condition "23#a with Table I"k#:

The leading net pressure base function Pn0"x# which satisfies conditions (ii) and (iii) above is

selected as

Pn0 !

2"1%n#=2

1$ nx"1$ x2#"1$n#=2 $

2

"2$ n#p

! "; n=1

ln"1$ x2# % 2 ln 2; n ! 1

8><

>:"48#

The corresponding %On0"x# is obtained from the elasticity integral (22) in the form

%On0 !

2"5%n#=2

"1$ n#"3$ n#p

p cot"np#z3$n $2"3$ n#"2$ n#

z% ln1% z1$ z

% 2F1"$3% n; 1;$2% n; z#3$ n

$2F1"4$ n; 1; 5$ n;$1

z#

"4$ n#z

0

BBB@

1

CCCA; n=1

4p$ 8x arcsin x$ 8#############1$ x2

p; n ! 1

8>>>>>>><

>>>>>>>:

"49#

where z !#############1$ x2

p: The near-tip asymptotic behaviour of %On

0 is given by

1$ x{1 : %On0 !

16 cot np2"1$ n#"3$ n#

"1$ x#"3$n#=2 %O""1$ x#3=2#; n=1

4p"1$ x# %O""1$ x#3=2#; n ! 1

8><

>:"50#

According to "23#a and (50), the opening asymptotic expansion near the tip contains terms oforder "1$ x#1=2; "1$ x#"3$n#=2 (or "1$ x# for n ! 1), and "1$ x#3=2: Therefore, following thenumerical treatment of a fracture driven by a Newtonian fluid in References [15, 33], we selectthe rest of the base opening functions %On

j "x# with 14j4N in expansion (45) as the weightfunction "1$ x2#3=2 times a polynomial:

%Onj ! "1$ x2#3=2Tj$1"1$ 2x2#; 14j4N "51#

where Tj$1 is the Chebyshev polynomial of the first kind of degree j $ 1: After a change ofvariable, x ! sin "y=2#; (51) becomes %On

j ! cos3 "y=2# cos " j $ 1# y; 14j4N: The net pressure



base functions Pnj ; 14j4N; are obtained from the elasticity integral [33]

Pnj !

3

8cos y; j ! 1

1

4cos y%

5

16cos 2y; j ! 2

1

4" j $ 1#"1% cos y# cos" j $ 1#y%

3

16

sin"j $ 1#ysin y

"cos 2y$ 1#; 25j4N

8>>>>>>><

>>>>>>>:

"52#

Since dPnj =dx ! 0; 14j4N; at the fracture inlet x ! 0; a particular net pressure term

PnN%1"x# ! 2$ px with non-zero inlet gradient (and zero contribution to the stress-intensity

factor) is added to expansion (44) to model the non-zero fluid flux at the inlet [22, 33]. Thecorresponding explicit expression for %On

N%1"x# follows from the elasticity integral (22).

6.2.2. Solving for the evolution of the coe!cients in the solution series expansion for the genericcase "05n51#. The series expansion for the net pressure (44) and opening (45) satisfy elasticity(22), propagation condition and the net pressure tip asymptote (23) with Table I"k#: The solutionfor the unknown coe"cients in this expansion,Aj"M#; j51; and the normalized fracture lengthgk"M# ought to be obtained from consideration of lubrication equations (16)–(17) and globalfluid balance (19).

Let us rewrite the lubrication equation (16) with Table I"k# and "28#a as

2

3"1$ n#

Z 1

x

%O0 dx ! %F "53#

where prime 0 ! @=@"lnM# and %F is defined as

%F ! %C$ "1% 2G#Z 1

x

%O dx$2

3% G

! "x %O "54#

Substitution of (45) into (53) reveals that it can be rewritten as

2

3"1$ n# ""2g#$1=2#0 %on"x# %

XN%1

j!0

A0j %o

nj "x#

" #

! %F "55#

where the functions %onj "x# and %on"x# are defined as

%onj !

Z 1

x

%Onj dx; %on !

Z 1

x

#############1$ x2

qdx !

1

2arccos x$ x

#############1$ x2

q! "

The use of the opening expansion (45) in the global fluid balance equation (19) results in anequation, which can be solved for AN%1:

AN%1 !1

%onN%1"0#

1

2g$2 $ "2g#$1=2

%on"0# $XN

j!0

Aj %onj "0#

!

"56#

Substitution of (56) into (55) and the use of the expression g0=g ! *3=2"1$ n#+G (which followsfrom the definition of G ! d ln g=d ln t and identity "28#a) yield

2

3"1$ n#

XN

j!0

A0j #o

nj "x# ! R"x; g;A0; . . . ;AN# "57#

D. I. GARAGASH1454


where the right-hand side R is

R ! %F% g$2 #onN%1"x# %

12 "2g#

$1=2 #on"x#% &

G "58#

and various functions are defined by

#onN%1"x# !

%onN%1"x#

%onN%1"0#

; #onj "x# ! %on

j "x# $ %onj "0# #o

nN%1"x#; #on"x# ! %on"x# $ %on"0# #on

N%1"x# "59#

Let us elaborate on the stated functional dependence of R in (57). We notice from the originaldefinitions (see definition of %F; (54), (17), and expansion (44)–(45)) that R defined by (58) is theknown function of x and M-dependent parameters g;G;A0; . . . ;AN%1: According to (47), G canbe expressed as the function of g and A0; G ! A1=n

0 M$1=ng$"1%n#=2n $ 2=3: In view of the aboveand since AN%1 is the known function, (56), of g;A0; . . . ;AN ; we obtain functional dependenceof R ! R"x; g;A0; . . . ;AN#:

Discretization of the lubrication equation (57) over a set of grid points fxig; i ! 1; . . . ;N % 1;in the x 2 *0; 1# interval yields N % 1 equations

2

3"1$ n#

XN

j!0

A0j #o

nj "xi# ! R"xi; g;A0; . . . ;AN#; i ! 1; . . . ;N % 1 "60#

Note that since #onj "0# ! 0; (59), the corresponding form of (60) if evaluated at the inlet point

"x ! 0# is a non-linear algebraic equation, R"0; g;A0; . . . ;AN# ! 0: (The latter is simply themanifestation of the inlet flux condition (20), %C"x ! 0# ! "2g2#$1:) In order to simplify the taskof the numerical solution of (60), we select the grid points fxig; i ! 1; . . . ;N % 1; inside thex 2 "0; 1# interval (i.e. excluding either the crack inlet or the tip). Then, (60) constitutes a systemof N % 1 linear ODEs in terms of the N % 1 series coe"cients fAjg; j ! 0; . . . ;N: The"N % 1#)"N % 1# matrix of numerical coe"cients f #on

j "xi#g; i ! 1; . . . ;N % 1; j ! 0; . . . ;N; in(60) can be inverted to reduce the linear ODEs (60) to the canonical form with respect to thederivatives fA0

jg; j ! 0; . . . ;N:Based on the expressions of g0=g in terms of G and that of G in terms of g and A0 cited in the

text above, we obtain an additional ODE for the normalized crack length g in the form

2

3"1$ n#g0 ! g

A0

M g"1%n#=2

! "1=n

$2

3

" #

"61#

The N % 2 ODEs (60)–(61) and appropriate initial conditions

g"Mini# ! gk0 %Minigk1; A0"Mini# ! Minig"1%n#=2"Mini#2

3%

Minigk1g"Mini#

! "n

"62#

Aj"Mini# ! 0; j ! 1; . . . ;N "63#

are solved simultaneously using the ODE solver in Mathematica software version 4.1 (# 1988–2000 Wolfram Research, Inc.) to obtain the solution g"M#;A0"M#; . . . ;AN"M#:

Note that initial conditions (62) for g and A0 follow from the small-viscosity solution for g;(38c) with "37#c and (41)–(42), while the rest of the initial values of expansion coe"cients are setto zero in (63). Alternatively, we could have obtained Aj"Mini# "j ! 1; . . . ;N# values from theminimization of the error with which the small-viscosity solution is approximated by the series



expansions (44)–(45) at M ! Mini: For small enough Mini (Mini ! 10$3 was used in thecalculations), however, the smallness of the estimated values Aj"Mini#; j ! 1; . . . ;N; warrantsthe initial state approximation (63).

6.2.3. Solving for the coe!cients in the solution series expansion for the special cases (n ! 0 andn ! 1). In the special cases of a perfectly plastic "n ! 0# and Newtonian "n ! 1# fluids, thelubrication equation assumes non-evolutionary forms (30) and (29), respectively, where Menters as a parameter only. Consequently, in place of the system of ODEs (60)–(61) governingthe solution g"M#;A0"M#; . . . ;AN"M# for a generic value of fluid index 05n51; we will havean equivalent system of algebraic equations:

0 ! R"xi; g;A0; . . . ;AN#; i ! 1; . . . ;N % 1 "64#

in place of (60), and equation

A0 !Mg1=2; n ! 0

23Mg; n ! 1

(

"65#

in place of (61). In (64), the right-hand side is defined by

R !$

%OM

dPdx

$ 1; n ! 0

$%O3

M

dPdx

$R 1x%O dx$

2

3x %O; n ! 1

8>>><

>>>:"66#

with P and %O given by their series expansions (44) and (45), respectively.Solution of the system of N % 2 non-linear algebraic equations (64)–(65) for N % 2 unknowns

g"M#;A0"M#; . . . ;AN"M# at a fixed value of viscosity parameter M is obtained using theNewton solver in Mathematica software. The solution at the first value of M ! Mini is obtainedusing the initial guess (62)–(63) following from the small-viscosity solution. The value of M isthen incremented by a fixed increment DM; and the solution for updated M value is obtainedusing the initial guess provided by the solution for the previous M value. The use of asu"ciently small increment DM then ensures the convergence of the Newton solver.

7. RESULTS AND DISCUSSION

7.1. Dimensionless solution

The numerical transient solution is obtained in the range of the evolution parameter Mk 2*10$3; 100+ "Mkini ! 10$3# for fluids with behaviour index n ! f0; 0:1; 0:2; . . . ; 0:9; 1g (0.1increment). Results presented in this Section are based on calculations using N ! 10 terms inthe solution series expansion (44)–(45) and a non-uniform spacing between the N % 1 grid points(used in the numerical method to discretize the lubrication equations). The latter is selected tooptimize the convergence of the method and minimize the approximation error over a givenrange of Mk: Corresponding details of the numerical error analysis and relevant discussion aregiven in Appendix E.

Figure 3 shows the ‘raw’ data from the numerical transient solution for a hydraulic fracture:the evolution of the normalized coe"cients in the solution series expansion (44)–(45) with

D. I. GARAGASH1456


(c)

(b)

(a)

Figure 3. Evolution of the normalized series coe"cients, %Ai ! Ai=Ai"10$2#; i ! 0; . . . ;N % 1; for thesolution with N ! 10 regular series terms and for various fluid index: (a) n ! 0; (b) n ! 0:5; and (c) n ! 1:



viscosity parameter Mk in log–log scale for three values of the fluid behaviour index (a) n ! 0; (b)n ! 0:5; and (c) n ! 1: The plots of coe"cients shown in the figure are normalized by their respectivevalues at Mk ! 0:01; i.e. %Aj"Mk# ! Aj"Mk#=Aj"0:01#: The corresponding values Aj"0:01# aregiven in the Table II. (Thus, the data in the table and in Figure 3 is su"cient to reconstruct thecomplete solution f %O"x;Mk#;P"x;Mk#; gk"Mk#g with the help of (44)–(45) and (56)). Interestingly,for n ! 0 and n ! 0:5; %Aj"Mk# ’ Mk "j ! 0; . . . ;N % 1# for Mk90:1; see Figure 3(a)–(b).Consequently, the solution is linear inMk and, therefore, given by the small viscosity solution (38a)–(38c). For n ! 1 (Newtonian fluid), the coe"cients corresponding to the leading terms in (44)–(45)are approximately linear for small Mk; namely %A0 ’ %A1 ’ %AN%1 ’ Mk for Mk90:01: Thisindicates applicability of the small-viscosity solution in the case of Newtonian fluid only for smallMk: (For this case, Reference [21] provides an ‘improved’ small-viscosity solution with dramaticallyincreased Mk-range of applicability.)

Profiles of the dimensionless net pressure and opening along the crack for graduallyincreasing values of viscosity parameter Mk from 0:01 to 10 are shown on Figures 4 and 5,respectively, for three di!erent values of fluid index (a–a0) n ! 0; (b–b0) n ! 0:5; and (c–c0) n ! 1:Solution is shown in both toughness scaling, Pk and Ok ! gk %Ok; (a–c), and viscosity-scaling, Pm

and Om ! gm %Om; (a0–c0). The transient solution in toughness scaling, plots (a–c) in Figures 4 and5, and in viscosity scaling, plots (a0–c0) in Figures 4 and 5, converges to the small-time asymptote(zero-viscosity solution, (37)) and to the large-time asymptote (zero-toughness solution [22]) inthe limit of small and large viscosity parameter Mk; respectively. (The above asymptotes areshown by dashed lines.) The corresponding ranges of viscosity where the transient solution isadequately approximated by its small and large-time asymptotes can be estimated by Mk40:01(toughness-dominated range) and Mk510 (viscosity-dominated range). Comparison ofsolutions for di!erent fluid index indicates that the above toughness-dominated range shrinksand the above viscosity-dominated range expands with the increase of the fluid behaviour index.

These dependence of the extent of asymptotic regions of the transient solution on fluidindex n is also evident on Figure 6 which shows the evolution of the overall characteristicsof the transient fracture solution, namely, (a–a0) dimensionless fracture half-length, (b–b0)

Table II. Numerical values of coe"cients Aj in thesolution series expansion with N ! 10 base terms forvarious fluid behaviour index n ! 0; 0:5; 1; evaluated at

Mk ! 0:01:

"Mk ! 0:01# n ! 0 n ! 0:5 n ! 1

A0 ) 103 9.5830 7.6040 12.1A1 ) 103 $8.8862 $8.4974 $7.1535A2 ) 103 2.0589 2.1922 0.1679A3 ) 104 $1.7606 $3.7689 $2.9273A4 ) 105 7.5824 13.814 9.5975A5 ) 105 $2.0941 $5.1754 $5.6495A6 ) 105 1.0526 2.2812 2.3839A7 ) 106 $3.4563 $9.2360 $12.166A8 ) 106 1.6632 3.7709 4.5690A9 ) 106 $0.4262 $1.2108 $1.7804A10 ) 107 1.5684 3.2491 3.8359AN%1 ) 103 8.4945 12.3 4.0856

D. I. GARAGASH1458


dimensionless net pressure at the inlet and at the tip, and (c–c0) dimensionless crack opening atthe inlet, for various values of the fluid behaviour index from n ! 0 to n ! 1 (in 0:1 increments).Plots in Figure 6(a–c) show the evolution of respective field variables in toughness-scaling from

(a) (a!)

(b) (b!)

(c) (c!)

Figure 4. Profiles of dimensionless net-pressure Pk;m along the fracture for fluid behaviour index(a–a0) n ! 0 (plastic fluid), (b–b0) n ! 0:5; and (c–c0) n ! 1 (Newtonian fluid) and various values ofthe evolution parameter Mk ! f0:01; 0:0316; 0:1; 0:316; 1; 3:16; 10g: Figures (a–c) and (a0–c0)correspond to the toughness and viscosity scalings, respectively. Dashed lines show (a–c) zero-

viscosity solution and (a0–c0) zero-toughness solution.



the early time asymptotic constants corresponding to the zero-viscosity solution and plots onFigure 6(a0–c0) show the evolution of respective quantities in viscosity-scaling towards theasymptotic constants corresponding to the zero-toughness solution [22]. The solution departsfrom the toughness-dominated regime, see Figure 6(a–c), and approaches the viscosity-dominated regime, see Figure 6(a0–c0), faster for higher values of fluid index.

(a) (a!)

(b) (b!)

(c) (c!)

Figure 5. Profiles of dimensionless opening Ok;m along the fracture for the set of parameters of Figure 4.

D. I. GARAGASH1460


Profiles of the normalized net pressure on Figure 4 confirm that for shear-thinning fluids(n51) the net pressure is finite at the fracture tip (see also direct plot of the normalized tip netpressure on Figure 6(b–b0)) with its value in the toughness-scaling decreasing from Pk0 ! p1=3=8

(a) (a!)

(b) (b!)

(c) (c!)

Figure 6. Evolution of the solution in toughness scaling, (a–c), and viscosity scaling, (a0–c0), with thedimensionless viscosity Mk for di!erent values of fluid behaviour index n ! 0; . . . ; 1 (in increments of 0:1):"a; a0# dimensionless half-fracture length gk;m; "b;b

0); dimensionless net pressure at the inlet Pk;m"0# andfracture tip Pk;m"1#; "c; c0), dimensionless opening at the inlet %Ok;m"0#:



at zero time "Mk ! 0# to eventually negative infinity at infinite time (Mk ! 1 or Km ! 0). Theunbounded large-time limit of the normalized net pressure at the tip in the viscosity scaling(which is the appropriate scaling for the large-time viscosity-dominated fracture propagation) isan artefact of di!erent time-dependence of the net pressure at the crack inlet and at the tip,respectively. Indeed, in the large-time limit, the dimensionless net pressure at the inlet converges to afinite positiveO"1# value given by the zero-toughness solution (Figure 6"b0##, and, therefore, the time-dependence of the dimensional net pressure at the inlet is prescribed by the viscosity scaling, (13b),

t ! 1 : p"0# ! em"t#E 0Pm"0# ' t$n="2%n#

(see Equation "14#a and Table CI for Lm"t#). On the other hand, analysis of the LEFM tipboundary layer scaling in Appendix B for the case of a shear-thinning fluid suggests that at the tip

t ! 1 : p"‘# ! #e"t#E0 #Pj#x!0 ' $t$n="4$n2# "67#

(see Equations (B9), (B7)b with vtip ! ’‘ ' ’Lm and Table CI for Lm"t#). Consequently, the rateof decay of the inlet net pressure with time exceeds that of the tip net pressure, resulting inp"‘#=p"0# ! 1 as t ! 1:

The net pressure at the crack tip decreases from its early-time maximum positive value in thetoughness-dominated regime to intermediate negative values in the transient part of the solution(as in Figure 6(b–b0) for the normalized pressure). However, this trend has to be eventuallyreversed, as the dimensional solution approaches the large-time viscosity-dominated asymptote(67), where the absolute value of the negative net pressure at the tip vanishes with time. Thisbehaviour suggest the existence of the global minimum of the (negative) dimensional net pressureat the tip in the transient solution, and rather interesting possibilities for the occurrence andevolution of the fluid lag. Indeed, a fluid lag can develop at some point of the fracture evolution ifthe fluid pressure at the tip drops below the cavitation value (e.g. when the absolute value of thenegative net pressure at the tip exceeds the confining stress, "$ p"‘## > so; see discussion in Section3), and then to increase as the fraction of the fracture length for at least some portion of thefracture evolution. On the other hand, the fluid lag has to be exactly zero in the limit of infinitetime (infinite fracture length), since p"‘# ! 0: Therefore, the transient fluid lag has to evolve non-monotonically, increasing from zero at the onset of the tip cavitation to a certain maximumfraction of the fracture length and then decreasing to zero. This type of behaviour is qualitativelydi!erent from the behaviour of the fluid lag in hydraulic fracture driven by a Newtonian fluidwhere the lag monotonically decreases with time from the early time maximum value [30].

7.2. Example of a dimensional solution for the crack length and the net pressure evolution

Figure 7 illustrates the evolution of the dimensional fracture half-length ‘ with time t in log–logscale for hydraulic fractures driven by fluids with various behaviour index n (between 0 and 1 inincrements of 0.2). Plots on Figure 7 were obtained from the plot of dimensionless half-length gkon Figure 6(a–a0) by appropriate rescaling using the following set of problem parametersrepresentative of reservoir hydraulic fracturing applications: E ! 25 GPa; n ! 0:15; Qo !10$3 m2=s; and three di!erent values of material toughness: (a) KIc ! 1 MPam1=2; (b) KIc !2 MPam1=2; and (c) KIc ! 5 MPam1=2; respectively. The values of consistency index M areselected di!erent for fluids with di!erent behaviour index n; such that the rheological curves forthe considered set of fluids intersect at a common (apparent) value of viscosity mn ! sxy=’g at agiven reference value of shearing rate ’g [22]. The apparent viscosity of these fluids is set for this

D. I. GARAGASH1462


(a)

(b)

(c)

Figure 7. Evolution of the crack half-length ‘ with time for di!erent values of fluid behaviour index fromn ! 0 (plastic fluid) to n ! 1 (Newtonian fluid) in 0:2 increments and three di!erent values of toughnessKIc ! 1; 2; and 5 MPam1=2: Open incremental symbols indicate 5% departure from the corresponding zero-toughness solution [22]. Zero-toughness solution for n ! 0 is shown by the short-dash line. Zero-viscosity

solution (independent of n) is shown by the long-dash line.



example to be equal to mn ! 0:1 Pa s at a reference value of shearing rate ’g ! 50 s$1:Consistency index M of each fluid is therefore calculated via M ! mn ’g1$n:

Figure 7 shows the departure from the early-time toughness-dominated regime (zero-viscositysolution shown by the long-dash line) and convergence towards the large-time viscosity-dominated regime (zero-toughness solution [22] shown for n ! 0 by the short-dash line). Openincremental symbols on Figure 7(a) and (b) indicate the time at which the transient solution is 5%away from the zero-toughness large-time limit. Importantly, we observe that for considered set ofproblem parameters and moderate toughness values "122 MPam1=2#; fracture length evolution ise!ectively given by the zero-toughness asymptote in the practical time (fracture length) range forfluids with high enough index n: For example, the solution is within 5% of the zero-toughnessasymptote in the whole range of time/length represented in Figure 7(a) for n ! 0:8 and 1(Newtonian fluid) and in Figure 7(b) for n ! 1: However, the transient solution is significantlydi!erent from its large-time asymptote for fluids with low enough index n: For example, thetransient solution for KIc ! 1 MPam1=2 (Figure 7(a)) is within 5% of the zero-toughnessasymptote for t0135 s "‘ > 50 m# for n ! 0 and for t030 s "‘ > 20 m# for n ! 0:2: For a two-foldincrease of material toughness to the value KIc ! 2 MPam1=2 (Figure 7(b)), the transient solutionis significantly di!erent from its large-time asymptote for cracks as long as ‘ ! 200 m for fluidswith index n ! 0 and ‘ ! 2 m for fluids with index as large as n ! 0:8: Finally, for values oftoughness as large as KIc ! 5 MPam1=2 (Figure 7(c)), the transient solution is significantlydi!erent from its large-time asymptote in the entire shown time range for all fluids of interest.

Latter observations drawn from Figure 7 emphasize the relevance of the transient solution inthe practical treatment time (fracture length) range for shear-thinning fluids with index n90:6for KIc ! 122 MPam1=2: For higher apparent values of toughness, which are often consideredrelevant for large-scale fracture propagation under typical level of in situ confining stress (tens ofMPa) [34, 35] and/or in damaged rock [36, 37], the transient solution becomes evermoreimportant for the entire spectrum of shear-thinning fluids n51:

Figure 8 illustrates the evolution of the dimensional net pressure at the inlet p"0# with the crackhalf-length ‘ in the log–log scale for the hydraulic fracture propagation example consideredabove with KIc ! 2 MPam1=2: This plot can be conceivably used for the inversion of the‘bottom hole’ net pressure, p"0#; to estimate the fracture length. At early time, fracturepropagates in the toughness-dominated regime and decline of the inlet net pressure p"0# followsfrom the toughness scaling: p"0# ' ‘$1=2 (‘ ' t2=3; p"0# ' t$1=3), see (13a)–(13e) and Table I"k#:At large time, propagation in the viscosity-dominated regime yields the inlet pressure decline asp"0# ' ‘$n="n%1# (‘ ' t"n%1#="n%2#; p"0# ' t$n="n%2#), see (13a)–(13e) and Table I"m#: Consequently,for all shear-thinning fluids "n51# the pressure decline decreases with fracture growth "n="n% 1#51=2#; see Figure 8. In the perfectly plastic fluid case "n ! 0# the inlet net pressure levels o! ata fraction of MPa for fracture half-length exceeding 100 m:

Finally, we consider the evolution of the dimensional net pressure at the crack tip p"‘# with thecrack half-length ‘ in the linear-log scale (Figure 9) for the example considered above withKIc ! 2 MPam1=2 and fluid index in the range between n ! 0 and n ! 0:8 (recall thatp"‘# ! $1; for a Newtonian fluid, n ! 1). The figure illustrates the premise discussed earlier inthis section that the net pressure at the tip evolves in the non-monotonic fashion: initially fallingfrom positive to negative values, passing a global (negative) minimum and asymptoticallyincreasing to zero value in the large-time limit. In the shown range of the crack length, the globalnet pressure minimum can be observed for fluids with n ! 0:4 and 0.5. (The minimum of the tipnet pressure takes place later or earlier in the crack length evolution than shown on the figure for

D. I. GARAGASH1464


fluids with the index below n ! 0:4 or above n ! 0:5; respectively.) Depending on the cavitationthreshold "$so# for the net pressure, the fluid lag can emerge during intermediate stages offracture evolution and disappear as the fracture approaches the large time, zero tip net pressurelimit. For the considered hydraulic fracture example (Figure 9), the minimum tip net pressurefor fluids with n40:5 exceeds $2 MPa level and, therefore, precludes cavitation with exceptionfor shallower depths (with so52 MPa). Figure 9 also shows that for larger fluid index n

Figure 8. Net-pressure at the crack inlet p"0# vs the half-length ‘ for the set of parameters ofFigure 7 and KIc ! 2 MPam1=2:

Figure 9. Net pressure at the crack tip p"‘# vs the half-length ‘ for KIc ! 2 MPam1=2 andvarious values of the fluid index n:



(approaching the limit of Newtonian fluid n ! 1), the cavitation and corresponding fluid laggingbehind the fracture tip may become eminent at early stages of fracture propagation (dependingon the level of so). To arrive to more general conclusions about the relevance of the fluid lag forthe fracture propagation driven by a non-Newtonian shear-thinning fluid would require a futureanalysis explicitly accounting for the fluid lag at the fracture tip (akin to that in References[30, 38] devoted to the Newtonian fluid case).

8. CONCLUSIONS

A transient solution for the problem of a fracture driven by the injection of a power-law fluid inan impermeable, linear elastic rock of finite toughness has been constructed. The numericalmethod combines the technique of the solution series expansion in space, developed earlier[15, 22] in applications to self-similar hydraulic fracture propagation problems, with the explicitODEs formulation for the evolution of the series expansion coe"cients in time. The analysis ofthe problem scaling and of the numerical transient solution has shown that the fracture evolvesfrom the early-time toughness-dominated regime to the large-time viscosity-dominated regime.The fracture propagation in the above two limiting regimes can be associated with thepropagation of a fracture driven by the respective ‘equivalent’ Newtonian fluids characterizedby time-dependent viscosities. The approximate ranges of material parameters and time (lumpedinto a single non-dimensional evolution parameter Mk ! m0k"t#E

03Qo=K 04 with the characteristictime-dependent viscosity m0k"t# ! M0"K 0="E0Q

1=4o ##4"1$n#=3t2"1$n#=3 corresponding to these two

distinct asymptotic regimes were discussed. Particularly, it has been shown that a decrease offluid behaviour index n (which corresponds to the increasing shear-thinning property of thefluid) results in the increase of the time/parametric domain for toughness-dominated regime anddecrease of corresponding domain for the viscosity-dominated regime. In the particular exampleof problem parameters relevant to hydraulic fracturing treatments in the petroleum industry, itwas shown that hydraulic fracture propagation occurs in the transient regime (i.e. significantlydi!erent from either two limiting regimes) for almost the entire time-range (fracture length-range) of practical interest for small enough index n (n90:4 for the considered set of problemparameters and moderate value of rock toughness KIc ! 2 MPam1=2). Whereas, for largeenough index n "n00:8# hydraulic fracture propagation occurs in the viscosity-dominatedregime in the time-range of interest.

APPENDIX A: ACCOUNT OF THE LUBRICATION THEORYFOR POWER-LAW FLUIDS

Under the lubrication assumption [26], the fluid inertia and the variation of the shear stress inthe flow direction (along the x-axis in Figure 1) are neglected, compared to the shear stressvariation across the flow channel (along the z-axis perpendicular to the fracture plane) and tothe driving pressure gradient, resulting in the equations of fluid motion in the form

0 ! $@p@x

%@sxz@z

; 0 ! $@p@z

"A1#

Using the symmetries with respect to the x- and z-axis, let us consider the positive co-ordinatequadrant (x50; z50) where the absolute values of the shear stress and the shear-strain rate are

D. I. GARAGASH1466


t ! $sxz and ’g ! $2’exz ! $@vx=@z; respectively. (Contribution ($@vz=@x) to the shear strain-rate ’g is neglected in the above expression under the lubrication assumption.) Upon substitutionof the above expressions and of the constitutive law (1) in "A1#a; integrating once in z; and usingthe symmetry condition ’g"z ! 0# ! 0; one finds

’g ! $@vx@z

! $z

M

@p@x

! "1=n

"A2#

where gradient @p=@x is the function of the channel co-ordinate x (and time) only, "A1#b:Poiseuille’s law (3) [23] for the fluid flux in the channel can be obtained from the identity

q ! 2

Z w=2

0vx dz ! $2

Z w=2

0

@vx@z

! "z dz "A3#

(based on the channel symmetry, integration by parts, and the stick condition along the channelwall, vx"z ! w=2# ! 0) and expression for the shear rate (A2).

Let us now evaluate the apparent fluid viscosity mn ! t=’g ! M ’gn$1 (see (1)) at thefracture channel wall z ! w=2: Using the expression for the pressure gradient j@p=@xj !"M0=w#"jqj=w2#n following from Poiseuille’s law (3) (where M0 ! fM) in the expression for’g"z ! w=2# following from (A2), and substituting the result into the expression for the apparentviscosity, we find

mnwall !

n

4"2n% 1#M0 w2

jqj

! "1$n

"A4#

Consequently, the Poiseuille’s law for a power-law non-Newtonian fluid, (3), can be expressed inthe form of the Poiseuille’s law for an ‘equivalent’ Newtonian fluid, (4), with an e!ectiveviscosity m0 ! 4n$1"2n% 1#mn

wall:

APPENDIX B: LEFM BOUNDARY LAYER CONSIDERATIONS

In this Appendix we present the scaling considerations for the LEFM boundary layer, whichexists at the tip of an arbitrary finite fluid-driven fracture under small toughness conditions.These considerations are direct extension of the analogous considerations for the case ofNewtonian fluid [19]. The solution in the boundary layer of thickness #‘{‘ can be approximatedto the leading order by the solution of a semi-infinite crack propagating at the velocity vtip givenby the instantaneous crack tip velocity in the underlined problem of a finite crack, vtip ! ’‘(Figure 2). Governing equations for a semi-infinite steadily propagating crack with zero fluid lag(e.g. Reference [10]) are presented here for an ‘equivalent’ Newtonian fluid with e!ective



viscosity m0 in the form

w3" #x#m0" #x#

dp

d #x! q" #x#; m0" #x# ! M0 w2" #x#

q" #x#

! "1$n

; q" #x# ! w" #x#vtip "B1#

p" #x# ! E0 1

4p

Z 1

0

dw"s#ds

ds

#x$ s; #x ! 0 : w !

K 0

E0 #x1=2 "B2#

where #x ! xtip $ x is a co-ordinate moving with the crack tip (Figure 2). Equations (B1)a2c

specialize the fluid flux q; e!ective viscosity m0; and continuity under lubrication flowapproximation, respectively; Equations (B2)a2b specialize elasticity integral relation between thenet pressure and the crack opening and the LEFM crack tip asymptote, respectively. Thesolution of above equations for a semi-infinite steadily propagating crack provides the boundarylayer solution which governs the transition from the LEFM near-tip asymptote (w ' #x1=2 for#x{#‘) to the viscosity-dominated away from the tip asymptote (w ' #x2="2%n# for #xc#‘), see thestructure of the boundary layer solution on Figure 2.

The tip boundary layer scaling can be defined in terms of the characteristic tip lengthscale #‘(boundary layer thickness), characteristic crack near-tip aspect ratio #e; and characteristic tipviscosity #m0 as

#x ! #‘ #x; w" #x# ! #e#‘ #O"#x#; p" #x# ! #eE0 #P"#x#; m0 ! #m0 #M"#x# "B3#

where #x is the normalized distance from the crack tip, and #O; #P; and #M are dimensionlessopening, net-pressure, and e!ective viscosity in the boundary layer scaling, respectively. Thechoice of scaling parameters #‘ and #e for an ‘equivalent’ Newtonian fluid with characteristicviscosity #m0 (see Equation (35) of Reference [19])z

#‘ !K 0

E02=3 #m01=3v1=3tip

0

@

1

A6

; #e !E 0 #m0vtipK 02 "B4#

and the following implicit expression for #m0:

#m0 ! M0 #e#‘vtip

! "1$n

"B5#

reduce boundary layer equations (B1)–(B2) to the following normalized form:

#O1%n d#P

d#x! 1; #P"#x# !

1

4p

Z 1

0

d #O"s#ds

ds#x$ s

; #x ! 0 : #O ! #x1=2 "B6#

(Expression for the normalized e!ective viscosity #M ! #O1$n has been used to arrive to (B6)a).Solution of normalized equations (B6) as well as the elaborate studies of the near tip and

away from the tip asymptotic expansions of the solution have been recently carried out byGaragash et al. [14] for the case of Newtonian fluid "n ! 1# and by Kresse et al. [39] for the caseof a general power-law shear-thinning fluid (05n51). Remarkably, Equations (B6) areparameter-free for a given fluid index n; and, therefore, the scaling (B3) with (B4)–(B5)completely defines the dependence of the tip boundary layer solution on material parameters

zNote the corrected typo in the expression for #e in the 2nd of Equation (35) of Reference [19].

D. I. GARAGASH1468


and time (via the time-dependence of the crack tip velocity vtip of the underlined finite fractureproblem). It is, therefore, useful to expand the implicit definitions (B4)–(B5) of characteristicboundary layer thickness #‘; characteristic crack tip aspect ratio #e; and characteristic tip viscosity#m0 to their explicit form

#‘ !K 02%n

E01%nM0vntip

!2="2$n#

; #e !M0vntip

K 02n E01$2n

! "1="2$n#

; #m0 ! M01="2$n# K 04

E03v2tip

!"1$n#="2$n#

"B7#

The LEFM tip boundary layer exists near the front of a finite fracture when the extent of theboundary layer is much smaller than the global fracture dimension (e.g. half-length ‘ of a planarfracture), #‘{‘: Similar to Reference [19], we can define the dimensionless tip toughnessparameter as

#Km !#‘‘

! ""2$n#=2"2%n#

!K 0

E0"1%n#="2%n#M01="2%n#vn="2%n#tip ‘"2$n#=2"2%n#

"B8#

Consequently, the condition when the LEFM boundary layer exists near the tip of a finitefracture can be formulated as a condition on smallness of the dimensionless tip toughness,#‘=‘ ! #K2"2%n#="2$n#

m {1: Under this condition, the solution away from the tip boundary layer" #xc#‘# is approximately independent of toughness and is given by the zero-toughness outersolution (see the sketch on Figure 2).

The tip boundary layer scaling and corresponding solution of normalized governingequations (B6) can yield useful information about the tip behaviour of a finite fracturepropagating in viscosity-dominated regime (i.e. under small toughness conditions). Forexample, Equations (B6) imply that the normalized net pressure is finite and negative at thecrack tip for shear-thinning fluids, with the numerical value #P"#x ! 0# ! O"1# given bythe numerical solution in Reference [39]. Thus, the time evolution of dimensional net pressure atthe tip of a finite fracture propagating in the viscosity-dominated regime (under small-toughnessconditions) is given by

ptip ! #e"t#E0 #Pj#x!0 "B9#

In view of (B7)b; the evolution of the net pressure at the crack tip dominated by the LEFMboundary layer solution is defined by that of the crack tip velocity vtip; and, specifically, vanishesin the limit of slowing fracture, vtip ! 0; typical of fractures driven by a constant fluid injectionrate.

APPENDIX C: SCALING EXPRESSIONS EXPANDED IN TIME

The toughness and the viscosity scalings (i.e. corresponding choices of scaling parameters e andL; and dimensionless groups M and K), presented in Table I as the scalings for the ‘equivalent’Newtonian fluids with time-dependent viscosities m0k"t# and m0m"t#; respectively, are expanded inTable CI to reveal explicit time-dependence of the scaling parameters and dimensionless groups.

Corresponding expressions for the scaling parameter e follow from "14#a and Table CI.



APPENDIX D: EXPLICIT FORMULAE IN THE SMALL VISCOSITY SOLUTION

The first-order viscosity term f %O1;P1; g1g; (39)–(42), in the small-viscosity solution (38a)–(38c)can be evaluated explicitly in the case of a perfectly plastic (n ! 0) and a Newtonian (n ! 1; seeReference [21]) fluids. In the former case, integration in (39)–(42) yields

n ! 0 : Pk1 !4% 3p2

6p4=3$

2

p1=3arcsin x "D1#

%Ok1 !8% 6p2

3p4=3

#############1$ x2

q$

8

p4=3Isin"x# "D2#

gk1 ! $64

3p7=3"D3#

where the function Isin"x# is defined by

Isin"x# !Z 1

0G"x; x0# sin$1 x0 dx0 !

Z p=2

0ln

cos a% cos a0

cos a$ cos a0

$$$$

$$$$a0 cos a0 da0

! 2 ln tana2% cos a

p2

4$ 2 ln tan a

! "

% sin a Im"Li2"e2ia# $ Li2"$e2ia##; "a ! arcsin x# "D4#

where Li2" # is the dilogarithmic function. The value of the opening at the crack inlet followsfrom (D2) and (D4), %Ok1"0# ! 8"1% ln 64#="3p4=3#:

For a Newtonian fluid, the first-order term is reproduced below after Reference [21] forcompleteness

n ! 1 : P1 !8

3p2=31

24% ln"4

#############1$ x2

q# $

3

4

x arccos x#############1$ x2

p !

"D5#

%O1 !8

3p2=32p$ 4x arcsin x$

5

6$ ln 2

! " #############1$ x2

q$

3

2ln

"1%#############1$ x2

p#1%

########1$x2

p

"1$#############1$ x2

p#1$

########1$x2

p

2

4

3

5

0

@

1

A "D6#

g1 ! $32"1% 6 ln 2#

9p5=3’ $2:7220 "D7#

Table CI. Quantities corresponding to the viscosity (M) and the toughness (K) scalings.

Scaling L M K

"K#E0Qot

K 0

! "2=3 M0

E0

E0Q1=4o

K 0

!"4=3#"2%n#

t"2=3#"1$n# 1

"M# Q1=2o

E0

M0

! ""1=2#"n%2#

t"n%1#="n%2# 1K 0

E0Q1=4o

E0

M0

! ""3=4#"2%n#

t$"1=2#"1$n#="2%n#

D. I. GARAGASH1470


APPENDIX E: NUMERICAL ERROR ANALYSIS

The numerical calculations of the transient solution are performed for various number of termsin the series expansion (44)–(45), 54N410 to investigate the convergence of the numericalmethod and the approximation error in the lubrication equation. The spacing between theN % 1 grid points along the crack f05xi51g; i ! 1; . . . ;N % 1; used in the numerical methodto discretize the lubrication equations (resulting in (60) for 05n51 and (61) for n ! 0 or n ! 1),is selected to optimize the convergence of the method and minimize the approximation errorover a given range of Mk 2 *10$3; 100+:

For a given solution series expansion (44)–(45) with N base terms, we define the localerror and global quadratic error in the lubrication equation (16) with Table I"k# and"28#a by

E"N#"x;Mk# ! 1$RHS"N#"x;Mk#LHS"N#"x;Mk#

; e"N#"Mk# !Z 1

0*E"N#"x;Mk#+2 dx "E1#

respectively, where RHS"N#"x;Mk# and LHS"N#"x;Mk# are the right- and left-hand sides of thelubrication equation (16) evaluated at the given solution series expansion.

For each 54N410 we have considered di!erent sets of N % 1 grid points parametrized by aparameter s 2 "0; 1+ as follows:

xi !

10$3; i ! 1

si $ 1

N % 1

' (% "1$ s# 1$ 1$

i $ 1

N % 1

! "' (2

; i ! 2; . . . ;N % 1

8><

>:"E2#

Figure E1. Distributions of lubrication equation error E"10#"x;Mk# along the crack at Mk ! 1corresponding to the numerical solutions with di!erent sets of grid points (see the insert) with grid

parameter s ! 0:1; s ! 0:3; and s ! 0:5; respectively.



Grid points xi "i ! 2; . . . ;N % 1# are regularly spaced for s ! 1 and concentrated toward thefracture tip "xN%2 ! 1# for s51; see the insert on Figure E1 showing sets (E2) for N ! 10 andvarious s: The parameter s controls the concentration of grid points near the tip. Example ofthe spatial distributions of the local lubrication error E"N#"x;Mk ! 1# for calculations withN ! 10 for three di!erent sets of grid points with s ! 0:5 (small concentration), s ! 0:3; ands ! 0:1 (large concentration) is shown on Figure E1. (Note that the solution with uniformlyspaced grid points, s ! 1; does not converge, and, thus, is not shown.) Corresponding globalquadratic error e"10#"Mk ! 1# values are 1:4) 10$5; 9:7) 10$6; and 3:8) 10$6; respectively.

(a)

(b)

Figure E2. Quadratic global error e"N# as a function of evolution parameter Mk: (a) for the fixed valuen ! 0:5 of fluid index and various number of regular series terms N; (b) for fixed number of terms N ! 10

and various values of fluid index n from 0 (plastic fluid) to 1 (Newtonian fluid).

D. I. GARAGASH1472


Clearly, the piled-up set with s ! 0:1 corresponds to the lowest global lubrication error, and tothe mostly ‘uniform’ local error distribution along the crack. (Large spikes in the local error isobserved near the tip of the crack for the solutions with s ! 0:3 and s ! 0:5:) Other numericaltests for various fluid behaviour index n and various values of Mk have shown the same trendsof the error dependence on the grid parameter s: Consequently, the grid with s ! 0:1 has beenused for all of the subsequent calculations.

Figure E2 shows the variation of the global lubrication error e"N# with Mk for (a) variousnumber of terms N ! f5; 6; 7; 8; 9; 10g in the solution series expansion for a fixed fluid indexn ! 0:5; (b) various values of the fluid index n ! f0; 0:1; . . . ; 0:9; 1g (0.1 increment) and thefixed number N ! 10 of terms in the series. From Figure E2(a) we observe that the lubricationerror naturally decreases with the number of terms in the series, and interestingly, thesignificant error reduction occurs when increasing N from odd to even number (e.g. fromN ! 7 to N ! 8) and not from even to odd (e.g. from N ! 8 to N ! 9). Figure E2(b) indicatesa general trend of reduction in numerical accuracy of the solution with increasing fluid index n;with the most accurate solutions obtained for the case of perfectly plastic fluid n ! 0: Withrespect to the dependence of the solution accuracy on the normalized viscosity Mk; we observethat the error increases with Mk and the numerical solution (especially for higher values offluid index) cannot be considered substantially accurate for Mk larger than 10 (e.g. for n ! 1and Mk > 10 the quadratic error exceeds 10%). This loss of accuracy of solution inapproaching the zero toughness (infinite viscosity) limit is related to the diminishing influenceof toughness at the fracture tip and the emerging viscosity near-tip asymptote, characterizedby opening %O ' "1$ x#2="2%n# and the net pressure singularity P ' "1$ x#$n="2%n# at the tip,1$ x{1; (see (12) or (33) in viscosity scaling). The numerical method relying on the toughnessasymptote at the tip breaks down in the large viscosity limit. Similar observation has beenmade in References [15, 33] for the case of a Newtonian fluid. In order to accurately capturethe solution for large viscosity, one need to explicitly consider the near-tip boundary layer(Section 3 and Appendix B) which provides the transition from the toughness-dominatedbehaviour immediately next to the tip to the viscosity-dominated behaviour at intermediatedistances from the tip [19].

ACKNOWLEDGEMENTS

Acknowledgement is made to the Donors of The Petroleum Research Fund, administered by the AmericanChemical Society, for partial support of this research under Grant ACS-PRF 36729-G2. The author alsowishes to acknowledge Jose I. Adachi, Emmanuel Detournay, and Leonid Germanovich for their helpfulreview comments which led to considerable improvement in the paper.

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