ARMA 16-786

Laminar - turbulent transition in the propagation of height-contained hydraulic fracture

Zia, H. and Lecampion, B.Geo-Energy Lab - Gaznat chair on Geo-Energy, ENAC-IIC-GEL, EPFL, Switzerland

Copyright 2016 ARMA, American Rock Mechanics Association This paper was prepared for presentation at the 50th US Rock Mechanics /Geomechanics Symposium held in Houston, Texas, USA, 26-29 June 2016. This paper was selected for presentation at the symposium by anARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. Thematerial, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, orstorage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print isrestricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgementof where and by whom the paper was presented.

ABSTRACT: High injection rate hydraulic fracturing can have Reynolds number as high as 104. For such highReynolds numbers, turbulent flow is likely to occur. In this paper, we investigate the effect of turbulence on thepropagation of hight contained hydraulic fractures, commonly referred to as PKN fractures. We discuss differentscalings for the fracture width, length and pressure under limiting laminar, turbulent smooth and turbulent roughflow regimes. We implement an explicit, central numerical scheme to solve the continuity and friction factor basedmomentum conservation equations, taking into account the full variation of friction factor with Reynolds numberand relative fracture roughness. The scheme is validated against the analytical solution of the PKN model. Theresults show that the local Reynolds number evolves from a maximum value at the inlet to zero at the tip, witha transition from turbulent to laminar at some point along the fracture length, depending on the value of inletReynolds number. Results showing the effect of smooth and rough turbulence on the fracture length and fracturewidth depending on the Reynolds number are finaly presented.

1 Introduction

The usual approximation of laminar flow in a prop-agating hydraulic fracture sometimes breakdown.This is the case of fluid-driven fracture propagationat glacier beds where Reynolds numbers above 105

are expected (Tsai and Rice (2010)). At such largeReynolds numbers, the flow is well within the tur-bulent rough regime. Recently, Ames and Bunger(2015) have investigated the effect of a fully roughturbulent flow on the propagation of height con-tained hydraulic fracture (Nordgren (1972)) andderived scaling relationship for that flow regime.However, in practice for a hydraulic fracture treat-ment in an oil and gas reservoir using a low viscos-ity fluid injected at very large injection rate, theReynolds number for a height contained fractureare expected to be in the range 103 − 104 at most(see Table 1 and 2), right in the transition between

the laminar to the fully turbulent regime. We fo-cus solely here on a height contained fracture asdepicted in Fig. 1 (i.e. the so-called PKN geome-try). In parallel to the problem formulation, we alsoreview the available experimental data and mod-els covering the complete flow conditions spanningthe complete spectrum of Reynolds numbers fromlaminar to fully turbulent. We notably discuss theuse of an equivalent Reynolds number proposed byJones (1976) to use the available data and modelsdeveloped for flow in pipes to the case of ellipticalcross-section encountered in the PKN geometry.

We then present a numerical model for the prop-agation of a height contained hydraulic fracture(PKN fractures) accounting for the complete lami-nar - turbulent transition . Our algorithm is basedon a non-diffusive central scheme with fixed gridand explicit time-stepping. We fully validate ourscheme against the well-known laminar case (Nord-

Parameter Value

ρ 1000kg/m3

µ 8× 10−4Pa sh 10mE ′ 32 GPak 1 mm

Table 1: Typical values of rock, fluid properties andfracture height h. k denotes a fracture roughnesslengthscale.

Q0 (bbl/min) Q0 (m3/s) R =

Qoρ

2hµ

10 0.026 65020 0.053 132530 0.079 197540 0.106 2650

Table 2: Inlet Reynolds number (R) in a heightcontained fractures for different values of injectionrate and evaluated with the parameters listed inTable 1 for density, viscosity and fracture height.Transition to turbulent flow starts at Rc ≈ 1380for such geometry.

gren (1972)). We finally discuss the relevance of thedifferent limiting approximations (i.e. fully laminarversus fully turbulent regime) to simulate typicalindustrial hydraulic fracturing treatments in uncon-ventional reservoirs.

2 Problem formulation

We consider a height contained bi-wing hydraulicfracture of half-length ℓ propagating in the x -direction, with a width of w(x, z) (see Figure 1).The fracture is assumed to be bounded by two lay-ers of highly stressed rocks at the top and bottomof the reservoir, restricting its height to h vertically.The rock layer is assumed to be linear-elastic witha plane strain Young’s modulus E ′. The fracture isdriven by the injection of a Newtonian fluid of vis-cosity µ and density ρ at a constant flow rate Qo.The fracture length is assumed to be much largerthan its height and the fluid flow is assumed to beuni-dimensional in the direction of propagation.

l

x

y

z

hw

Q0

Figure 1: Sketch of a height-contained hydraulicfracture (PKN geometry e.g. Economides andNolte (2000)).

2.1 Elasticity

The length of the fracture is assumed to be suffi-ciently large compared to its height such that ev-ery vertical cross-section can be assumed to be ina state of plane-strain. Following the approxima-tion of uni-directional flow, the net pressure p(x),which is equal to the excess of the fluid pressureabove the compressive in-situ stress is assumed tobe uniform in each vertical cross-section. As a re-sult, the plain strain elasticity solution for an uni-formly pressurized fracture provides the followinglocal relation between the fracture width w(x, z)and the net pressure p(x) at a given cross-sectionlocated at x (e.g. Sneddon and Elliot (1946)):

w(x, z) =2hp(x)

E ′

√1− 4z2

h2, (1)

where E ′ = E/(1 − ν2) is the plane-strain modu-lus. By defining w as the average fracture width

given by 1h

´ h/2

−h/2w(x, z)dz, Eq. (1) gives (e.g. Sar-

varamini and Garagash (2015)):

w(x) =π

4

2hp(x)

E ′ =π

4w(x, 0).

2.2 Continuity

Neglecting fluid compressibility, the local fluid massconservation reduces to the following continuityequation:

∂A

∂t+

∂Q

∂x= 0, (2)

where A = wh is the cross-sectional area of thefracture in the y-z plane, Q = Av is the volumetricflow rate of the fluid and v is the cross-sectionalaverage of the fluid velocity in the x-direction. Therock is assumed be impermeable and fluid leak offfrom the fracture faces is thus neglected.

2.3 Boundary conditions

The fluid is injected at x = 0 into the bi-wing frac-ture at a flow rate Q0. The flow rate entering onewing of the fracture is thus:

Q(x = 0) =Q0

2(3)

The boundary conditions at the fracture tip (x = ℓ)are given by:

Q(x = ℓ) = 0 w(x = ℓ) = 0 (4)

2.4 Momentum Conservation

The cross-sectional average of the fluid momentumconservation equation reduces to:

ρ

(∂Q

∂t+

∂Qv

∂x

)= −A

∂p

∂x− Þτw, (5)

where Þ is the perimeter of the cross-section of thefracture and τw is the wall shear stress given by

τw = fρv2

2.

Here, f is the Fanning friction factor which is afunction of the local value of the Reynolds numberas well as the relative roughness of the fracture inthe turbulent regime.

2.4.1 Friction Factor

To solve the set of equations (1)-(5), the depen-dence of the friction factor on the Reynolds num-ber and the relative roughness must be estimated.The classical experiments of Nikuradse (1950) haveprovided the basis for a number of empirical rela-tions between the Reynolds number and the fric-tion factor in smooth and rough pipes. These re-lations cannot be directly applied to the case offluid flow in a fracture, especially one having anelliptical cross-section. The concept of hydraulicdiameter provides a method to accommodate dif-ferent channel geometries but experimental stud-ies have demonstrated that the obtained relationsare not accurate (Sadatomi, Sato and Saruwatari,1982; Carlson and Irvine, 1961; Jones, 1976). Jones(1976) introduced the concept of a ”laminar equiv-alent” hydraulic diameter in order to obtain thefriction factor for rectangular cross-sections usingthe friction factor vs. Reynolds number relation forcircular cross-sections. This laminar equivalent hy-draulic diameter is obtained such that the frictionfactor can be expressed using the same expressionas for the laminar flow in circular pipes (i.e. theanalytically obtained expression fL = 16/Re).The laminar equivalent hydraulic diameter for an

elliptical cross-section can be obtained by compar-ing the solution of a pressure-driven flow in ellipti-cal geometry (see e.g. Lamb (1895)) to the solutioncalculated by the friction factor based wall-shearstress in laminar conditions. By introducing α as ageometry-dependent coefficient necessary to get thecorrect equivalence between the two solutions, the”pipe-equivalent” Reynolds number is evaluated asαDH

vρµ, where DH = 4A/Þ is the hydraulic diame-

ter. For an ellipse with a very large major axis com-pared to the minor one, as is the case with fractures,α = 25/32 gives the correct equivalence. The ”pipe-equivalent” Reynolds number (Re) for the PKN el-liptical cross-section is thus:

Re = α4A

Þρv

µ=

32

5π

wαρv

µ=

5

π

ρwv

µ=

5

πRe.

where Re =ρwv

µis the Reynolds computed using

simply the averaged fracture width as the flow di-

mension. Using this pipe-equivalent Reynolds num-ber, the friction factor can be evaluated with therelations for a circular cross-section, i.e. fL =16/Re in the laminar flow regime. In the turbu-lent regime, we use the relations of Blasius andManning-Strickler for fully smooth and fully roughflows respectively:

fB = f ′B Re−1/4 f ′

B = 0.316/4

fM = f ′M

(k

w

)1/3

f ′M = 0.143/4

where k denotes the roughness lengthscale, f ′B and

f ′M are the coefficients appearing in the Blasiusand Manning-Strickler relations respectively. Thetransition from the laminar to the different tur-bulent regimes is mapped using an approxima-tion recently proposed by Yang and Dou (2010).It is an implicit function which can be used tocalculate the friction factor, given the Reynoldsnumber and relative roughness of the pipe. Thecomplete evolution of the friction factor with theReynolds number and relative roughness (k/D) ob-tained in pipe flow experiments by Nikuradse Niku-radse (1950) is shown in Figure 2. Different approx-imations used in this study are also shown. Notethat the critical Reynolds number at which turbu-lence starts is about 2200 for circular pipes. Thecritical value for an elliptical cross-section is thusRec = 2300× π/5 ≈ 1380.

3 Dimensional analysis and

scaling

Using the following scaled coordinates system

ξ = x/ℓ(t)

where ℓ(t) is the length of the fracture at a giventime, and introducing the following characteristicscales (possibly function of time):

W∗, P∗, V∗, L∗,

3 4 5 6 7

0.2

0.4

0.6

0.8

1.0

1.2

log(

100

x 4

)

log( Re )

BlasiusLaminar

Manning-Strickler

Nikuradse dataYang-Dou (2010)

D/k = 15D/k = 30D/k = 60D/k = 126D/k = 252D/k = 507

Figure 2: Nikuradse (1950) data: friction factor asfunction of the Reynolds number (Re = ρvD/µ)and relative roughness in pipes. The laminar, Bla-sius, Manning-Strickler and Yang-Dou approxima-tions are also shown.

it is possible to show that the inertial terms in thebalance of momentum have always a negligible ef-

fect for time larger than√R× W∗

L∗(a time-scale

which is always negligible due to the fact that thecharacteristic width is always much smaller thanthe fracture characteristic length). Here, R denotesthe value of the Reynolds number R = ρW∗V∗

µ=

1

2

Qoρ

hµat the fracture inlet. Assuming laminar fric-

tion, one recovers the scaling of the classical PKNsolution - which we indicate with a subscript L forlaminar:

WL =

(µπ3Q2

0t

23E ′h

)1/5

LL =

(E ′Q3

0t4

22h4µπ3

)1/5

PL =2WLE

′

hπVL = LL/t.

As discussed in Ames and Bunger (2015), assum-ing a purely turbulent rough regime governed byManning-Strickler friction over the whole fracture,we obtain the following “fully-rough” scaling, indi-cated with a subscript R:

WR =53/16f

′ 3/16M k1/16π3/8Q

9/160 ρ3/16t3/16

2√2E ′3/16h3/8

LR =

√2E ′3/16Q

7/160 t13/16

53/16f′ 3/16M h5/8k1/16π3/8ρ3/16

PR =2WRE

′

hπVR = LR/t.

Similarly, assuming a purely turbulent but smoothflow regime where friction is governed by Blasius re-lation, we obtain the following characteristic scalesfor “fully-smooth” turbulent scaling, indicated witha subscript S:

WS =53/20π9/20

2× 211/20f′1/5B µ1/20Q11/20

o ρ3/20t1/5

E ′1/5h7/20

LS =211/20

53/20π9/20

E ′1/5Q9/20o t4/5

f′1/5B h13/20µ1/20ρ3/20

pS =2WBE

′

hπVS = LS/t

In these limiting regimes (laminar, fully turbu-lent rough and fully turbulent smooth), the propa-gation of the hydraulic fracture can actually shownto be self-similar. The complete dimensionless so-lutions can be obtained semi-analytically in a sim-ilar way as for the classical PKN solution. De-tails are omitted here for brevity. For example,the fracture length assuming complete turbulentrough flow is given by ℓR(t) = 1.08LR(t), while forthe fully turbulent smooth-Blasius regime ℓS(t) =1.09LS(t) and the classical laminar solution forfracture length is ℓL(t) = 1.001LL(t). Similarly,wR(x, t) = WR(t)ΩR(ξ), wS(x, t) = WS(t)ΩS(ξ)and wL(x, t) = WL(t)ΩL(ξ) gives the fracture widthfor laminar, turbulent smooth and turbulent roughflows respectively, where Ω is the dimensionlessopening of order 1.In reality, the value of friction varies spatially

with the local value of the Reynolds number (Re =RΨ), which depends on the local value of the di-mensionless flow rate Ψ = Ω×Υ (where Υ is the di-mensionless velocity). The dimensionless flow ratehas the value of 1 at the fracture inlet and is equalto zero at the fracture tip according to the bound-ary conditions (3)-(4). We therefore see that in-evitably, the flow will always be in the laminarregime at the fracture tip, potentially shrinking to

a laminar boundary layer at the fracture tip, if theflow is highly turbulent at the fracture inlet. Inorder to investigate the complete transition of theflow regime from laminar to turbulent, a numericalsolution thus is necessary using the complete tran-sition of the friction factor from the laminar to theturbulent regime (smooth or rough).Another interesting point worth mentioning here

relates to the slightly different power-law of timedependence obtained for the fully rough regime (asalready mentioned by Ames and Bunger (2015)).In the case where the complete transition is ac-counted for inside the fracture, one can intuitivelygrasp that if turbulent, the flow will be first turbu-lent rough as the fracture opening will be of the or-der of the roughness lengthscale k at early time. Asthe fracture grows and the width increases, the flowcan eventually transition to the turbulent smoothregime. The time-scale of the transition from therough to the smooth turbulent regime can be esti-mated as:

tR→S =16E ′k5ρ3Qo

5π2f ′Mh2µ4

,

a time-scales obtained from the fact that rough-ness has negligible effect on friction when(

k

WR

)R3/4 ≈ 1 and friction tends to the Blasius

limit (Goldenfeld, 2006). It is interesting to notethat using realistic values for a slickwater treat-ment (e.g. shown in Table 1) and for an injectionrate of 0.026m3/s (10bbl/min), we obtain tR→S ≈2×1013sec, indicating that if turbulent, the flow willnever reach the limiting regime of turbulent smooth/ Blasius type flow in practice.

4 Numerical Scheme & vali-

dation

We solve the system of equations (1)-(5) with asecond, order non-oscillatory central scheme intro-duced by Nessyahu and Tadmor (1990). Such anexplicit scheme operates in a predictor-correctorfashion. To ensure numerical stability, the schemeupholds TVD (total variation diminishing) prop-erty by using slope and flux limiters. An impor-

tant characteristic of the scheme is that it does notutilize any Riemann-solver, which makes it easy toimplement. The Riemann-solver free formulationis made possible by the use of staggered grids, al-lowing integration over the entire Riemann fan andtaking into account both the left and right-goingwaves (see Nessyahu and Tadmor (1990) for de-tails). A well-known factor limiting the use of cen-tral scheme is that it can become highly diffusive(see e.g. Kurganov and Tadmor (2000); Kurganovand Lin (2007)), especially for highly non-linearproblems such as the one under consideration here.In this study, we use an anti-diffusive correctionintroduced recently by Zia and Simpson (under-review) that allows the diffusion to be mitigatedwith the cost of a smaller CFL bound on the timestep.

To demonstrate the accuracy of the scheme, thesystem is solved using only the laminar expressionfor the friction factor : fL = 16/Re. In this case,the model is strictly equivalent to the classical PKNhydraulic fracture model, for which an analyticalsolution is available (Nordgren (1972)). The nu-merical solution is obtained by discretizing the do-main with a fixed Cartesian mesh of 150 cells. Are-meshing is performed as soon as the fracture tipreaches the end of the computational domain. Cor-rect location of the fracture tip is important as thesolution is sensitive to its position and even a smalldiffusion can introduce significant error. We arecharacterizing fracture to be open at the point alongthe length of the fracture where the opening goesabove a small value (taken here as 2×10−5m). Thefracture is assumed to be propagating with a fixedvelocity inside a cell, which is evaluated each timethe tip moves to the next cell. To mitigate numer-ical diffusion, a large value of 0.999 is used for thecorrection factor ϵ (see Zia (2015) for description ofthis correction parameter) to solve the continuityequation. Use of this high correction factor allowsto significantly decrease the numerical diffusion butas a drawback, it can introduce spurious oscillationsin the solution. A small Courant number of 0.015is thus imposed to calculate the time step, ensuringthe numerical stability of the scheme. The param-eters used for this benchmark are the one listed in

Table 1 with an injection rate of Q0 = 0.03 m3/s.Initial conditions are prescribed as the exact lami-nar solutions (w0(x) = wL(x, t0)and ℓ0 = ℓL(t0)) ata given initial time t0 (2 sec in this test case) .The numerical results are shown in Figures 3-5

over close to four decades of time. Figure 3 showsthe evolution of the scaled fracture-length ℓ(t)/ℓ0 vsscaled time (t/t0) and the scaled fracture-width atthe inlet w(0, t)/w0 vs scaled time (t/t0) on the bot-tom and top respectively. A very good match withthe analytical solutions can be observed. Further-more, the relative error of the fracture-length andthe fracture-width are shown in Figure 4 on the bot-tom and top respectively. The sharp jumps in thetime evolution of error are due to re-meshing whichis performed as the fracture tip reaches the end ofthe computational domain. The remeshing intro-duces a small error due to loss of data caused byinterpolations, which does not fully recovers beforethe next remeshing step resulting in accumulationof a small error as the solution evolves. The resultsshow that this error accumulation is not significant:the relative error does not exceeds 0.3 and 2 percentfor the fracture width and fracture length respec-tively over close to four decades of time which ismore than enough for practical cases. The fracturewidth shown in Figure 3 is the width at the fluidinlet. Figure 5 shows the profile of the dimension-less fracture width at different values of the scaledtime (τ = t/t0) along with the analytical solution.It can be seen that the numerical solution matchesthe analytical solution very well, apart from thesmall diffusion at the fracture tip which does notappear to “grow” in time.

5 Results

The inlet value of the Reynolds number R directlyprovides information about the flow regime whencompared to the empirically known critical value(≈ 1380 for a fracture with elliptical cross-sectionas mentioned previously), above which the flow be-comes turbulent. To evaluate the effect of turbu-lence on the propagation of the fracture, we haveperformed a number of numerical simulations dis-cussed below.

100 101 102 103 104

1.5

2

2.5

3

3.54

4.55

w(0,t)w0(0)

analyticalnumerical

1/5

t t0

100 101 102 103 104100

101

102

103

t t0

(t)0

analyticalnumerical

4/5

Figure 3: Fully Laminar Case (f = fL): Evolutionwith scaled time of the scaled fracture length (bot-tom) and scaled fracture width at the fracture inlet(top).

5.1 Smooth fracture case

We first discuss the results for the case where theeffect of fracture roughness is neglected: i.e. as-suming a transition from laminar to Blasius liketurbulent flow to compute the friction factor (seeFigure 2). We have performed a set of simulationsfor different values of R to determine the relativeposition (ξt) of the laminar to turbulent transitionalong the fracture. The simulations were performedwith a grid of 150 cells. Figure 6 shows the relativesize of the laminar region against different values ofR. It can be seen that the relative size of the lam-inar region 1 − ξt is of the value 1, indicating thewhole fracture is in laminar regime for the valuesof R below the critical value. The size decreases

100 101 102 103 10410-4

10-3

10-2

Relative error on fracture width

t t0

100 101 102 103 10410-4

10-3

10-2

10-1

Relative error on fracture length

t t0

Figure 4: Fully Laminar Case (f = fL): Evolutionwith scaled time of the Relative error of the nu-merical solution for fracture-length (bottom) andfracture-width (top).

as the Reynolds number increases above the crit-ical value until the size of the laminar layer fallsbelow the spatial resolution of the simulation, i.e.it cannot be resolved with any more precision thanthe scaled grid size 1/N = 1/150, where N is thenumber of element in the grid.

The fracture length and width for the same se-ries of simulations are shown in Figure 7, top andbottom respectively. The results are shown rela-tive to the semi-analytical laminar solutions (Nord-gren, 1972). It can be seen that the relative fracturelength has the value of one for the values of R be-low the critical value and it decreases as the valueof R increases. Figure 7 also shows the ratio ofthe semi-analytical solutions (wS(0, t)/wL(0, t) and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

ξ

Ω

analyticalτ = 8.5τ = 20τ = 48τ = 114τ = 273

Figure 5: Fully Laminar Case (f = fL): Analyticaland numerical solutions for dimensionless fracturewidth. The small numerical diffusion observed atthe fracture tip does not grow in time and is intrin-sically related to the corrected adNOC scheme usedhere.

ℓS(t)/ℓL(t)). It can be seen that the slope of the nu-merical results is initially lower for Reynolds num-bers where the size of the laminar boundary layeris still significant and finally converges to the ana-lytical slope when the laminar boundary layer is sosmall that it becomes insignificant and the wholefracture can be assumed to be in turbulent smoothregime. Similar results for the fracture width evo-lution are also shown. The fact that our numericalscheme converges with the fully-smooth turbulentsolution is another indication of its accuracy.

5.2 Including roughness

As discussed before, the scalings show that theregime is almost always turbulent rough due tovery high tR→S for realistic parameters. We haveperformed simulations with two different Reynoldsnumbers (R) of 2.5 × 103 and 104. In the caseof 104, the Reynolds number is well into the tur-bulent rough regime. In this case, the size ofthe laminar boundary layer is very small and thewhole fracture can be assumed to be in fullyturbulent rough regime. For Reynolds number

10210

310

4 10510-3

10-2

10-1

100

101

1-ξ t

R

= 1/N1-ξ t

c ≈1380

Figure 6: Laminar-Turbulent Smooth case: Rela-tive length (1 − ξt) of the laminar fraction of theflow for different values of R.

of 2500, the flow regime is always in the lami-nar to turbulent transition. The fracture width(w(0, τ)/wL(0, τ)) and fracture length (ℓ(τ)/ℓL(τ))relative to the laminar solutions against the di-mensionless time (τ = t/tR→S) for the two differ-ent Reynolds number are shown in Figure 8, topand bottom respectively. The laminar solutions areevaluated with the corresponding Reynolds num-ber for both cases. The analytical turbulent roughsolutions (wR(0, τ)/wL(0, τ) and ℓR(τ)/ℓL(τ)) forboth Reynolds numbers are also shown. It can beobserved that the fracture solutions tends to thefully turbulent rough limit for the Reynolds num-ber of 104, indicating insignificant size of the lami-nar layer. For R = 2.5 × 10−3, the fracture widthis smaller than the limiting turbulent rough solu-tion but larger than the limiting laminar solution,indicating that the flow regime lies in the laminarto rough transition.

6 Conclusions

We have investigated the effect of turbulent flowon the propagation of height contained hydraulicfractures. Using the concept of equivalent laminarhydraulic radius (Jones, 1976), we have used the

102 103 104 105

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

w L

w(0,t)

(0,t)

c

Fully turbulent smooth

3/20

Laminar

102 103 104 1050.3

0.4

0.5

0.6

0.7

0.8

0.91

1.11.2

L

(t)(t)

c

Laminar

Fully turbulent smooth

-3/20

Figure 7: Laminar-Turbulent Smooth case: Thefracture length (top) and fracture width (bottom)relative to the laminar solutions for different val-ues of R. The laminar and fully turbulent smoothsolutions are also shown.

relations for the evolution of friction factor withReynolds number and roughness obtained for pipeto account for the complete transition from lami-nar to rough turbulent flow as function of the in-let Reynolds number for height-contained hydraulicfractures (i.e. PKN fractures). We have validatedour numerical scheme on the laminar case. Thescheme is also able to capture the fully turbulentsolutions (fully turbulent smooth or fully turbulentrough) accurately.

For large inlet Reynolds number R (above thecritical transition to turbulent flow), the flow transi-tion from turbulent to laminar from the inlet to thetip of the fracture. The extent of the laminar region

10-8 10-7 10-6

1

1.2

1.4

1.6

1.8

2

=2500 =10000

τ=t / t

L

w(0,τ)

numericalTurbulent rough

numericalTurbulent rough

w (0,τ)

R S

10-8 10-7 10-60.6

0.8

1

1.2

1.4

1.6

=2500 =10000numerical

L

Turbulent rough

numericalTurbulent rough

(τ)(τ)

τ=t / t R S

Figure 8: Laminar-Turbulent Rough case: Theevolution of the fracture width (top) and fracturelength (bottom) relative to the laminar solutionsfor two different Reynolds numbers.

close to the tip shrinks to a boundary layer as theR increases, reaching 10−2 for R ≈ 104. Our nu-merical solution indicates that the semi-analyticalsolution assuming a fully turbulent smooth regimein the fracture is valid for R above 5 × 103. Simi-larly, when the roughness is taken into account, thesemi-analytical solution for fully turbulent roughflow is valid for such high Reynolds numbers.

An important point that we have not addressedin this paper is the effect of the addition of frictionreducers (e.g. polymer additives) in the fracturingfluid - a typical practice in high rate water frac-turing treatments. The use of friction reducer isknown to drastically change the laminar-turbulent

transition and will, in turn, possibly significantlychanges the propagation of hydraulic fracture pre-dicted here. We leave the effect of friction reduceron fracture propagation to future investigations.

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