Blade-Shaped Hydraulic Fracture Driven by aTurbulent Fluid in an Impermeable Rock

Navid Zolfaghari, S.M.ASCE1; Colin R. Meyer2; and Andrew P. Bunger, M.ASCE3

Abstract:Water-driven hydraulic fractures with high flow rates are more common now than ever in the oil and gas industry. Although thesefractures are small, the high injection rate and low viscosity of the water lead to high Reynolds numbers and potential turbulence in thefracture. This paper presents a semianalytical solution for a blade-shaped [Perkins-Kern-Nordgren (PKN)] geometry hydraulic fracture drivenby a turbulent fluid in the limit of zero fluid leak-off to the formation. Turbulence in the PKN fracture is modeled using the Gaukler-Manning-Strickler parametrization, which relates the flow rate of the water to the pressure gradient along the fracture. The fully turbulent limit isconsidered with no transition region anywhere at any time, but the effect of fracture toughness on crack propagation is not considered. Thekey parameter in this relation is the Darcy-Weisbach friction factor for the roughness of the crack wall. Coupling this turbulence paramet-rization with conservation of mass allows a nonlinear partial differential equation (PDE) to be written for the crack width as a function ofspace and time. By way of a similarity ansatz, a semianalytical solution is obtained using an orthogonal polynomial series. Very rapidconvergence is found by embedding the asymptotic behavior near the fracture tip into the polynomial series; a suitably accurate solutionis obtained with two terms of the series. This closed-form solution facilitates clear comparisons between the results and parameters for laminarand turbulent hydraulic fractures. In particular, it resolves one of the well-known problems whereby calibration of models to data hasdifficulty simultaneously matching the hydraulic fracture length and wellbore pressure. DOI: 10.1061/(ASCE)EM.1943-7889.0001350.© 2017 American Society of Civil Engineers.

Introduction

Hydraulic fracturing is a method of stimulating relatively imper-meable subsurface reservoir rocks to extract oil and gas. In the lasttwo decades, there has been a transition from using high-viscositygels to the use of water in hydraulic fracturing (King 2010). Asso-ciated with this change is a two to three orders of magnitude increasein the characteristic Reynolds number R�, which can be defined

R� ¼ ρqinHμ

ð1Þ

where ρ = fluid density; qin = volumetric injection rate;H = hydraulicfracture height; and μ = fluid viscosity (Tsai and Rice 2010).Table 1 indicates that a shift from typical gel-based fluids to waterleads to an increase from R� ≈ 0.01–10 to R� ≈ 102–104, respec-tively. Although a local Reynolds number will vary along the fractureand decrease rapidly near the tip, the Reynolds number defined inEq. (1) is a constant set by external parameters. This paper describesparameter regimes where R is large enough for turbulence to existthroughout the hydraulic fracture (HF) save the very tip.

The emerging importance of the turbulent flow regimes willlikely increase the number of hydraulic-fracture numerical

simulations that incorporate turbulence. In order to benchmarkthese numerical simulations, analytical or semianalytical solutionsare required. One example analytical solution is the large leak-offlimit for a PKN hydraulic fracture (after Perkins and Kern 1961;Nordgren 1972) with rough-walled turbulent flow (Kano et al.2015). Otherwise, the analytical/semianalytical solutions are nec-essary for benchmarking numerical simulations and do not yet existfor turbulent flow. This is in contrast to a large body of benchmarksolutions for the laminar regime (e.g., Geertsma and De Klerk1969; Nordgren 1972; Savitski and Detournay 2002).

The tractability of the problem for analytical and semianalyticalsolutions requires simple geometries such as plane strain (Geertsmaand De Klerk 1969), radial (e.g., Savitski and Detournay 2002), andblade-shaped (after Perkins and Kern 1961; Nordgren 1972).Although all have usefulness as approximations under particularconditions, the blade-shaped geometry is of practical importance.The minimum stress in most reservoirs is horizontally directed,leading to vertically oriented hydraulic fractures. Furthermore, res-ervoir layers are often bounded by layers that serve to block upwardand downward growth of hydraulic fracture, meaning that the hori-zontal propagation velocity far exceeds the vertical propagationvelocity. A large body of field data indicates the resulting bladelikegeometry occurs in a wide range of reservoirs, probably comprisingthe idealization of the most common fracture geometry (e.g., DePater 2015). The limiting end-member of zero vertical (e.g., height)growth corresponds to the geometry of Perkins and Kern (1961),which was revisited by Nordgren (1972), and this so-called PKNgeometry is used in the present paper (Fig. 1).

The need to consider the turbulent regime for water-driven hy-draulic fractures was recognized by Perkins and Kern (1961). Asmall number of papers have since considered the turbulent regimeof hydraulic fracturing, and some hydraulic-fracturing designmodels (e.g., Meyer 1989) incorporate ability to simulate flowunder turbulent conditions. Nilson (1981, 1988) considered plane-strain gas-driven hydraulic fractures under a constant-pressure inlet

1Ph.D. Candidate, Dept. of Civil and Environmental Engineering,Swanson School of Engineering, Univ. of Pittsburgh, Pittsburgh, PA 15260.

2Ph.D. Candidate, John A. Paulson School of Engineering and AppliedScience, Harvard Univ., Cambridge, MA 02138.

3Assistant Professor, Dept. of Civil and Environmental Engineering andDept. of Chemical and Petroleum Engineering, Swanson School of Engi-neering, Univ. of Pittsburgh, Pittsburgh, PA 15260 (corresponding author).ORCID: https://orcid.org/0000-0002-0696-7066. E-mail: bunger@pitt.edu

Note. This manuscript was submitted on July 19, 2016; approved onMay 10, 2017; published online on September 8, 2017. Discussion periodopen until February 8, 2018; separate discussions must be submittedfor individual papers. This paper is part of the Journal of EngineeringMechanics, © ASCE, ISSN 0733-9399.

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boundary condition. Nilson showed the system evolving amonglaminar, turbulent, and inviscid regimes and solved the self-similarproblems associated with each of these limits of the system. Sim-ilarly Emerman et al. (1986) examined the problem of a plane-strain fluid-driven crack, but instead assuming a constant influxboundary condition. They presented an approximate solution, argu-ing for its practical suitability for modeling magmatic intrusionsand natural hydrothermal injections.

Turbulent flow has also been considered in other geosciences-inspired models. These include the model of drainage of glaciallakes via subglacial fluid-driven cracks developed by Tsai and Rice(2010), as well as the model of dyke ascent and propagation de-veloped by Lister (1990) and Lister and Kerr (1991). Tsai and Rice(2010) used Gaukler-Manning-Strickler (GMS) (Gauckler 1867;Manning 1891; Strickler 1923) approximation to model turbulentflow for glacial and subsurface hydraulic fracture. Also, Tsai andRice (2012) used GMS approach to model near-surface hydraulicfracture, motivated by the phenomenon of rapid subglacial drain-age. Expanding to account for time-dependent deformation of ice,Rice et al. (2015) used creep flow with GMS to model rapid glacial-lake drainage. These contributions provide a useful background forthe fluid flow model, but the boundary conditions and elasticityformulation are specific to their problem and not applicable to in-dustrial HFs.

More recently, Anthonyrajah et al. (2013) considered turbulentflow for hydraulic fractures with blade-shaped geometry with ageneralized inlet condition. They presented analytical solutions forthe particular (arguably non-physically-motivated) cases of constant

fracture speed and volume and demonstrated a numerical solutionmethod for general injection boundary conditions. The specific caseof constant injection rate for a blade-shaped hydraulic fracture wassubsequently considered by Zia and Lecampion (2016), who pointedout that many practical cases will consist of flow in the transitionbetween laminar and turbulent flow. For rough-walled fractures, theirwork numerically demonstrates departure from the laminar solutionof approximately 10–20% for R ¼ 2,500 and 30–50% forR ¼ 105,as well as complete convergence to a fully turbulent asymptotic sol-ution forR ¼ 10,000 (although the details of the asymptotic solutionare not presented). Dontsov (2016) pointed out the necessity in manypractical cases to consider that flow can be in the laminar regime nearthe tip of the hydraulic fracture and turbulent regime away from thetip. This paper was mainly motivated by development of numericalsimulations for generalized geometries wherein the behavior near thefracture tip must be properly treated. The solution presented is there-fore specified to the moving-tip region, allowing the length of thelaminar region to be determined and appropriate tip conditions tobe imposed in simulations.

This paper presents a solution for a PKN-geometry hy-draulic fracture driven by a turbulent fluid through impermeablerock. Although turbulent flows in general remain difficult to de-scribe mathematically, many parameterizations have been devel-oped to describe turbulence through channels and narrow slits. Thesolution used here begins with a generalized expression of theDarcy-Weisbach friction factor, after Gaukler-Manning-Strickler(Gauckler 1867; Manning 1891; Strickler 1923) approximationfor rough-walled turbulence in channel flow. Nonetheless, the pro-posed solution remains general enough to capture future advancesin modeling turbulent flow within hydraulic fractures, providedthese can be captured by a power-law relationship between the fric-tion factor and scale of the fracture roughness.

The solution presented here is semianalytical, derived using aJacobi polynomial series. It follows in the spirit of previous semi-analytical solutions that obtained very rapid series convergence byconstructing the family of polynomials so as to embed the appro-priate asymptotic behavior near the leading edge (Adachi andDetournay 2002; Savitski and Detournay 2002; Bunger andDetournay 2007). For this reason, the proposed solution methodbegins with derivation of the near-tip behavior, after which the formof the Jacobi polynomial series is specified. Coefficients of theseries are then selected to minimize an objective function thatembodies the error of the solution. A convergence study shows thata practically useful solution is by first two terms of the series. Thisrapid convergence of the solution justifies the approach, allowingthe solution to be written down once for all cases rather than re-quiring computation for each individual combination of parame-ters, as is the case for numerical simulations. Finally, the paperconcludes with an exploration of the sensitivity to the particularsof the expression for the Darcy-Weisbach friction factor (Weisbach1855; Darcy 1857) and with a comparison between solutions result-ing from models that impose laminar versus turbulent flow.

Mathematical Model

This paper considers the propagation of a rough-walled crackdriven by fully turbulent fluid flow. The fluid regime was modeledas turbulent along the entirety of the crack, and the potential rela-minarization of the flow near the tip was not modeled (see Dontsov2016; Dontsov and Peirce 2016). The conclusions about the valid-ity of the transition to laminar flow close to the tip (see discussionaround Figs. 7 and 8) have to be seen in this context because themodel cannot autonomously transition to laminar flow near the tip.

Table 1. Typical Reynolds Numbers for Water and Gel Working Fluidswith Flow Rate qin ¼ 0.05−0.2 m3 s−1 and Fracture Height H ¼50–200 m

FluidDensity

ρ (kgm−3)Viscosityμ (Pa · s)

Reynoldsnumber R�

Water 1,000 10−3 102–104

Gel 1,200 0.5–1 0.01–10

Fig. 1. PKN fracture geometry

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For the case of low viscosity and high flow rate (i.e., supercriticalCO2), the Reynolds number is very large and this relaminarizationregion will be quite small. Yet when the Reynolds number is closerto the value for transition to turbulence, the presented conclusionsabout the flow near the tip will not be correct, and a model thatincorporates relaminarization would be required.

The local elasticity approximation associated with the PKNmodel is also considered. The vanishing tip boundary conditionused for the PKN model (such as used here) precludes incorpora-tion of the effect of fracture toughness into the model. Thus,conclusions about the behavior of the solution within a distanceof ∼H of the tip, where H is the fracture height, are likely to beinaccurate. Furthermore, assuming turbulent flow throughout thefracture, as in the proposed model, likely overestimates the effec-tive viscosity of the fluid near the tip. Therefore, transition to lam-inar flow close to the tip may bring toughness in to play, which isnot accounted for in this model. However, a way forward could beprovided by a recently published toughness correction to the PKNmodel (Dontsov and Peirce 2016).

A final caveat related to the assumption that the fluid is fullyturbulent everywhere is that fracture width and fluid velocity bothdecrease when approaching the two containment interfaces at thestress barriers (Fig. 1). In these regions, the flow will certainly tran-sition to the laminar regime. This transition is not accounted for inthe proposed model.

Having clarified these limitations, now the model can be out-lined. A reservoir layer of uniform thickness H contained at the topand bottom by two higher-stress layers is assumed to be an effectivebarrier to upward and/or downward HF growth (Perkins and Kern1961). This geometry is given in Fig. 1. Provided the HF length isseveral times greater than the thickness H, assume a uniformlypressurized HF cross section and slowly varying HF width (open-ing) with respect to coordinate x (Perkins and Kern 1961; Adachiand Peirce 2008). These assumptions allow an expression to bederived for the opening W of an elliptical crack in an elastic rock(Nordgren 1972), which is given by

Wðx; z; tÞ ¼ 1 − νG

ðH2 − 4z2Þ1=2½pðx; tÞ − σ� ð2Þ

where G = shear modulus of elasticity; ν = Poisson’s ratio; H =height of the HF; fluid pressure pðx; tÞ is taken as uniform in eachvertical cross section; and σ = uniform in situ stress in the reservoiropposing the HF opening.

Mass conservation for the incompressible fluid flow in the crack is

∂A∂t þ

∂q∂x ¼ 0 ð3Þ

where A ¼ πωðx; tÞH=4 = area of the elliptical crack; and ωðx; tÞ =maximum opening in the cross section ωðx; tÞ ¼ Wðx; 0; tÞ. Insteadof using the Poiseuille equation for laminar flow (Nordgren 1972),here, the turbulent flow in the crack is modeled using the Gaukler-Manning-Strickler (Gauckler 1867; Manning 1891; Strickler 1923)parametrization

q2D ¼�− 4W3

ρfp

∂p∂x

�1=2

ð4Þ

where fp = Darcy-Weisbach friction factor, which can be expressed

fp ¼ m

�kW

�α

ð5Þ

where k = surface roughness; and α and m = constants with typicalvalues α ¼ 1=3 and m ¼ 0.143 (Gioia and Chakraborty 2006;

Tsai and Rice 2010); the effect of varying these parameters is ex-plored later in the paper. The subscript 2D in Eq. (4) indicates thatthe flux is through two-dimensional horizontal slices at everyheight z. The total flux q is given by the integral over the heightof the crack

q ¼Z

H=2

−H=2q2Ddz ð6Þ

where the details of this integration are given the Appendix. Theresult for the total flux is

q ¼ Λϒω2φ

�−∂ω∂x

�1=2

;

Λ ¼�4

m

�1=2

B

�1

2;φþ 1

�;

ϒ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

GH4ρkαð1 − νÞ

s;

φ ¼ 3þ α4

ð7Þ

where B = beta function [see Eq. (56) or Abramowitz and Stegun1972]; and Λ and ϒ = parameters of geometry and material proper-ties, respectively, whereϒ depends on rock properties and reservoirgeometry and Λ depends only on the parameters of the friction fac-tor α and m, which gives a typical value of Λ ¼ 7.406.

Now substitute the total flux, Eq. (7), into the continuity Eq. (3)and define

Ξ ¼ 4ΛϒπH

ð8Þ

Thus, the nonlinear partial differential equation governing themaximum opening ωðx; tÞ is found as follows:

∂ω∂t ¼ −Ξ ∂

∂x�ω2φ

�−∂ω∂x

�12

�ð9Þ

Three boundary conditions and an initial condition can now bespecified. The third boundary condition is necessary because thetotal length of the crack lðtÞ is unknown a priori and must be de-termined as part of the analysis. A zero-opening initial condition isapplied at t ¼ 0 and by the following boundary conditions:• No opening at the crack tip

x ¼ l ⇒ ωðl; tÞ ¼ 0

• No fluid loss through the crack tip

x ¼ l ⇒ qðl; tÞ ¼ 0

• Constant volume rate of flow at the inlet

x ¼ 0 ⇒ qð0; tÞ ¼ qin

where qin = half of the total fluid injection in the case ofsymmetric (biwing) growth. Alternatively, qin can be the entireinjection rate if HF propagation is biased in one direction so asto form a single-wing geometry, as observed in analysis of somefield data such as Cotton Valley (Rutledge and Phillips 2003),west Texas (Fischer et al. 2008), east Texas (Mayerhofer et al.2000), Mound site in Oklahoma (Warpinski et al. 1999), LostHill field (Emanuele et al. 1998), and in Barnett shale (Maxwellet al. 2002), and as also discussed by Wright et al. (1999) andMurdoch and Slack (2002).

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Scaling

Now a similarity solution to Eq. (9) can be sought. An alternativemethod of scaling, in the spirit of Savitski and Detournay (2002), isdetailed in the Supplemental Data. This nonlinear PDE resemblesthe equations for viscous gravity currents (Huppert 1982) andbuoyant hydraulic fractures (Lister 1990), and a similarity solutionof the first kind is therefore sought (Barenblatt 1996). First, the inletflux can be written

qin ¼ Λϒ�w2φ

�−∂w∂x

�1=2

�x¼0

ð10Þ

Thus, the PDE, Eq. (9), and inlet flux conditions can be scaledas follows:

wt∼ Ξ

w2φþ1=2

x3=2;

qinΛϒ

≡Q ∼ w2φþ1=2

x1=2ð11Þ

where scaled flux Q = ratio of the flux in qin divided by Λϒ.Combining these two scalings allows the similarity variables tobe defined

ξ ¼ x

Q−2ðQ3ΞtÞð4φþ1Þ=ð4φþ2Þλð12Þ

and

w ¼ ðQ3ΞtÞ1=ð4φþ2ÞΩðξÞ ð13Þ

The length of the crack lðtÞ is defined such that ξ ¼ 1 coincideswith the fracture tip, and therefore

lðtÞ ¼ ½Q−2ðQ3ΞtÞð4φþ1Þ=ð4φþ2Þ�λ; ð14Þwhere λ = dimensionless length (more details about the scaling areprovided in the Supplemental Data). The similarity formulation canbe inserted into the governing PDE to find

Ω − ð4φþ 1ÞξΩ 0 ¼ −ð4φþ 2Þ½Ω2φð−Ω 0Þ1=2� 0 ð15Þ

The boundary conditions to Eq. (15) are

Ωðξ ¼ 1Þ ¼ 0;

½Ω2φð−Ω 0Þ1=2�ξ¼1 ¼ 0;

½Ω2φð−Ω 0Þ1=2�ξ¼0 ¼ 1 ð16Þ

This ode can be integrated once by replacing ξΩ 0 withðξΩÞ 0 − Ω, which givesZ

1

ξΩdζ − 4φþ 1

4φþ 2½ζΩ�1ξ ¼ −½Ω2φð−Ω 0Þ1=2�1ξ ð17Þ

Using the tip boundary conditions yieldsZ1

ξΩdζ þ 4φþ 1

4φþ 2ξΩ ¼ Ω2φð−Ω 0Þ1=2 ð18Þ

To solve this equation, an orthogonal polynomial series methodis used to obtain a semianalytical solution to Eq. (18). The openingwidth changes most rapidly near the tip and, therefore, by embed-ding the asymptotic solution near the tip, one can derive a rapidlyconverging series (Savitski and Detournay 2002). Near the fracturetip, the fracture width Ω is expected to be small but changing

rapidly. Thus, a dominant balance is expected between thesecond two terms in Eq. (15), which is equivalent to saying thatthe integral over the fracture width in the integrated ode [Eq. (18)]is very small. Thus, in leading order, the near-tip behavior is char-acterized by the ode

4φþ 1

4φþ 2Ω ¼ Ω2φð−Ω 0Þ1=2 ð19Þ

Simplifying and separating yields

−Z

1

ξΩ4φ−2dΩ ¼

�4φþ 1

4φþ 2

�2

ð1 − ξÞ ð20Þ

Thus, the solution for the width near the tip is

Ω ¼� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4φ − 1p �

4φþ 1

4φþ 2

��2=ð4φ−1Þ

ð1 − ξÞ1=ð4φ−1Þ ð21Þ

The tip region will be analyzed later in the paper to show thatalthough there is relaminarization in a small boundary layer nearthe tip, it is a sufficiently small region that the turbulent expressionderived in Eq. (21) still holds.

Embedding this asymptotic solution into the polynomial seriesallows approximation of the series using only a few terms, whichclearly shows the dependence upon the parameters and can bereadily adopted for benchmarking purposes. Numerical solutions,although certainly feasible with existing methods, would not pro-vide the insights or usability of a semianalytical solution.

Solution

Overview of the Method

To solve Eq. (18), first construct an orthogonal polynomial series(Savitski and Detournay 2002). Orthogonal polynomials are sets offunctions that followZ

b

aRðxÞBmðxÞBnðxÞdx ¼ 0 ð22Þ

for all m ≠ n [where RðxÞ = weight function] andZb

aRðxÞBnðxÞ2dx ¼ hn ð23Þ

if m ¼ n (Abramowitz and Stegun 1972). The proposed solution isthus in the form of infinite series using basis functions Ω̂k

Ω ¼X∞i¼0

AiΩ̂i ð24Þ

where Ai = coefficients selected so that the solution satisfies thegoverning equations. The base functions Ω̂i must be orthogonal;therefore Z

1

0

Ω̂iΩ̂jdx ¼ δij ð25Þ

where δij = Kronecker delta function.Rapid convergence of the series is promoted by selecting

the base functions so as to embed the near-tip behavior (Savitskiand Detournay 2002), which was found to be of the formΩ ∼ Xð1 − ξÞB, where X and B are

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X ¼� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4φ − 1p �

4φþ 1

4φþ 2

��2=ð4φ−1Þ

and B ¼ 1

4φ − 1ð26Þ

The base functions will then be constructed so that

Ω̂i ¼ DifiðξÞXð1 − ξÞB ð27Þwhere Di = constants chosen so as to satisfy the orthogonalityrelationship [Eq. (25)]. Upon substitutionZ

1

0

ðDiDjÞX 2ð1 − ξÞ2BfiðξÞfjðξÞdξ ¼ δij ð28Þ

A convenient choice for the functions fi are the Jacobi polyno-mials, which have the following orthogonality relationship(Abramowitz and Stegun 1972):Z

1

0

ð1 − ξÞc−eξe−1Giðc; e; ξÞGjðc; e; ξÞdξ ¼ hiðc; eÞδij ð29Þ

whereGiðc; e; ξÞ ¼ ith order Jacobi polynomial, expressible as fol-lows:

Giðc; e; ξÞ ¼Γðeþ iÞΓðcþ 2iÞ

Xi

j¼0

ð−1Þj�ij

�Γðcþ 2i − jÞΓðeþ i − jÞ ξi−j ð30Þ

where ΓðiÞ = Gamma function (Abramowitz and Stegun 1972); andhiðc; eÞ = norm of Giðc; e; ξÞ and is given by

hiðc; eÞ ¼i!Γðiþ eÞΓðiþ cÞΓðiþ c − eþ 1Þ

ð2iþ cÞΓ2ð2iþ cÞ ð31Þ

Setting e ¼ 1 and c ¼ 2B þ 1, and rearranging Eq. (29)producesZ

1

0

�1

hið2Bþ1;1Þ�ð1−ξÞ2BGið2Bþ1;1;ξÞGjð2Bþ1;1;ξÞdξ¼ δij

ð32Þ

Comparing Eq. (32) with Eq. (28) leads to the conditions

Di ¼1

Xh1=2i ð2B þ 1; 1Þ;

fiðξÞ ¼ Gið2B þ 1; 1; ξÞ ð33Þ

As a result, the base functions are given by

Ω̂i ¼ð1 − ξÞB

h1=2i ð2B þ 1; 1ÞGið2B þ 1; 1; ξÞ ð34Þ

Using the asymptotic near-tip solution, the form of the orthogo-nal base functions is therefore given by

Ω̂i ¼ð1 − ξÞ1=ð4φ−1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

hið4φþ14φ−1 ; 1Þ

q Gi

�4φþ 1

4φ − 1; 1; ξ

�ð35Þ

where hi = norm of the basis.

Calculating Coefficients of the Series

Given the basis functions in Eq. (35), the coefficients Ai of theseries in Eq. (24) can now be calculated. The approach is todesignate equally spaced control points on 0 < ξ < 1 (typically10 control points were used) and retain the first n terms of thepolynomial series.

It is now possible to construct a residual function in terms of Aiand minimize that function. The chosen residual function embodiesthe sum of the squares of the mismatch between the left-hand andright-hand sides of Eq. (18) at each control point (Savitski andDetournay 2002). Its formal expression is given by

ΔðA1; : : : ;AnÞ ¼XQi¼1

�ΔLðξi;A1; : : : ;AnÞΔRðξi;A1; : : : ;AnÞ

− 1

�2

ð36Þ

where ΔLðξi;A1; : : : ;AnÞ = left side of Eq. (18) for specificvalue of ξi; and ΔRðξi;A1; : : : ;AnÞ = right side of Eq. (18) withthe solution parametric in the flow law parameter φ. By minimiz-ing ΔðA1; : : : ;AnÞ, one can find the value of each unknownvariable Ai.

Length

Evolution of the crack length can now be expressed with respect totime. Recalling Eq. (14)

lðtÞ ¼ Q−2ðQ3ΞtÞð4φþ1Þ=ð4φþ2Þλ

the crack length evolves with time with power of ð4φþ1Þ=ð4φþ2Þ.For illustration, consider values α ¼ 1=3 and m ¼ 0.143 for theDarcy-Weisbach friction factor (Tsai and Rice 2010). Then, fromEq. (7), the value of φ will be 5/6. Therefore, the value of the powerin Eq. (14) for turbulent flow is

4φþ 1

4φþ 2¼ 13

16ð37Þ

whereas for laminar flow, this value is 4/5 [see Eq. (45), followingNordgren’s (1972) solution for no-leak-off case]. By comparing thevalue of the power of the time in length formula for laminar andturbulent flow, it is clear that the power in laminar flow is within 1%of turbulent flow and they are very close to each other.

Pressure

In order to find the distribution of pressure along the HF, elasticity[Eq. (50)] is invoked. The net pressure inside the crack is thereforeexpressed in terms of the maximum opening at each cross section,the height of the crack H, and material properties of the rockaccording to

pðx; tÞ − σ≡ pnetðx; tÞ ¼G

1 − νwðx; tÞH

ð38Þ

Thus, by replacing the opening from Eq. (13) into Eq. (38)

pnetðx; tÞ ¼G

1 − ν1

HðQ3ΞtÞ1=ð4φþ2ÞΩðξÞ ð39Þ

Combining Eqs. (39) and (24) gives

pnetðx; tÞ ¼G

1 − ν1

HðQ3ΞtÞ1=ð4φþ2Þ X∞

i¼0

AiΩ̂i½ξðx; tÞ� ð40Þ

Hence, given the solution for the series coefficientsAi, Eq. (35),and the equation for length scale parameter ξ ¼ x=lðtÞ, thepressure is readily computed.

Results

The dimensionless opening depends only on the exponent αfrom the Darcy-Weisbach friction factor through φ [Eq. (7)].

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Fig. 2 shows the sensitivity of the dimensionless opening Ω withrespect to α. It is clear from these results that the sensitivity of thedimensionless opening, Ω, to α is relatively small. However, theactual opening and length will be strongly affected via the depend-ence of the scaling quantities on α [Eqs. (13) and (14)]. Herethe particular values α ¼ 1=3 and m ¼ 0.143 are considered forthe parameters of the Darcy-Weisbach friction factor. The Jacobipolynomial series is truncated at four terms, and the coefficientsare given in Table 2. From Fig. 3, it can first be confirmed thatthere is an excellent match between the tip asymptotic solutionand the complete solution over the whole fracture length. Thisis because in the PKN problem, the average fluid velocity in thefracture is approximately uniform (Economides and Nolte 2000;see also Kovalyshen and Detournay 2010). Moreover, it can be seenthat after the second term (n ¼ 2), the solution is indistinguishablewith additional terms. This shows the very rapid convergence en-abled by embedding the tip asymptotic behavior in the form of thebase functions. The truncated solution for n ¼ 2 terms for openingis given

Ω ¼ ð1 − ξÞ37ð1.1387þ 0.0626ξÞ;λ ¼ 1.0874 ð41Þ

The truncated solution in dimensional variables can be obtainedfrom similarity scaling [Eq. (41)], which gives

wðx; tÞ ¼ 0.8122

�k1=3q3inρ

H2

1 − νG

� 316

t316

�1 − x

lðtÞ�3

7

×

�1þ 0.05497

xlðtÞ

�;

lðtÞ ¼ 2.1619qinH

�H2

k1=3q3inρ

G1 − ν

� 316

t1316;

pnetðx; tÞ ¼ 0.8122

�kq9inρ

3

H22

�G

1 − ν

�13� 1

16

t316

�1 − x

lðtÞ�3

7

×

�1þ 0.05497

xlðtÞ

�;

q ¼ qin

�1 − x

lðtÞ�3

7

�1þ 0.05497

xlðtÞ

�53

×

�1þ 0.21005

xlðtÞ

�12 ð42Þ

This truncated solution for general form of Ω can also be written

Ω ¼ ð1 − ξÞ 12þα½C1ðαÞ þ C2ðαÞξ� ð43Þ

where C1ðαÞ and C2ðαÞ = constants that vary with the value of α, asshown in Fig. 4.

The rapid convergence of the polynomial series motivatesderivation of a second-order asymptotic solution near the tip(Supplemental Data provides the derivation of this solution).The asymptotic solution has the same structure as the truncated sol-ution and is given by

Ωtip ¼ ð1− ξÞ 12þα½M1ðαÞ þM2ðαÞξ�;

M1ðαÞ ¼ ðαþ 2Þ−αþ1αþ2

�αþ 4

αþ 5

�2=ðαþ2Þ�

1þαþ ðαþ 2Þðαþ 5Þðαþ 3Þðαþ 4Þ

�;

M2ðαÞ ¼ ðαþ 2Þ−αþ1αþ2

�αþ 4

αþ 5

�2=ðαþ2Þ�

1− ðαþ 2Þðαþ 5Þðαþ 3Þðαþ 4Þ

�ð44Þ

A comparison between the coefficients of the asymptoticsolution and the truncated solution is given in Fig. 4.

Applications

Now, a few examples illustrating the practical relevance of thenewly derived solution will be presented. The purposes are• Provide a comparison between laminar and turbulent flow solu-

tions for an example high-rate water-driven HF to show that thedifference is significant and therefore use of the laminar modelin instances where the Reynolds number is large can lead tosubstantial errors;

• Compare the expected Reynolds numbers for different fluid fa-milies (provided in the Supplemental Data) in order to clarifyconditions under which the turbulent and laminar models areexpected to be relevant; and

• Show the size of the near-tip laminar zone relative to the fractureas a whole, thereby clarifying conditions under which the ma-jority of the HF is in turbulent regime.Now, the turbulent solution will be compared with the laminar

solution of Nordgren (1972), where the opening, net pressure, andlength are given by

oo

oo

oo

o

oo

oo

oo

o

o

XX

XX

XX

X

XX

XX

XX

X

X

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

Fig. 2. Variation of scaled opening with respect to change of dimen-sionless length ξ for different value of α

Fig. 3. Scaled opening along the hydraulic fracturing; using just oneterm is a very good approximation and n ≥ 2 gives solutions that areindistinguishable; dashed line correspond to tip asymptotic solution(Supplemental Data section provides more information)

Table 2. Coefficients for Jacobi Polynomial Series for n ¼ 1, 2, 3, 4 withα ¼ 1=3

n 1 2 3 4

A1 8.616 × 10−1 8.517 × 10−1 8.515 × 10−1 8.515 × 10−1A2 — 1.115 × 10−2 1.124 × 10−2 1.124 × 10−2A3 — — 2.904 × 10−4 2.954 × 10−4A4 — — — 6.156 × 10−6

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wNð0; tÞ ¼ 2.5

�μq2inH

ð1 − νÞG

�15

t15;

lNðtÞ ¼ 0.68

�q3inμH4

Gð1 − νÞ

�15

t45;

pnetN ð0; tÞ ¼ 2.5

�μq2inH6

G4

ð1 − νÞ4�1

5

t15 ð45Þ

The parameter values used in the example case are given inTable 3, and the characteristic Reynolds number is R�¼ 104. Atthis Reynolds number, the flow is turbulent for all values ofroughness and, therefore, the Nordgren (1972) solution does notapply. Hence, this comparison illustrates the magnitude of the er-ror associated with inappropriately choosing the laminar modelinstead.

Fig. 5 shows that the crack-opening profile for the turbulent sol-ution is similar to the Nordgren (1972) solution for laminar flowwhen the opening is normalized by the opening at the wellbore.However, the magnitude of the opening, shown in Fig. 6, is over50% greater for the turbulent model. This greater opening is causedby a larger fluid net pressure in the turbulent case (Fig. 6). Finally,because the total volume is the same in both cases, the laminarmodel overpredicts the length by over 40% (Fig. 6). Hence, theturbulent model shows that high-Reynolds-number treatments willresult in higher pressure, greater widths, and shorter lengths thanpredicted by incorrectly applied laminar models. This result is alsoconsistent with comparisons for the large-leak-off PKN-type solu-tion presented by Kano et al. (2015) and discussed based on scalingarguments by Ames and Bunger (2015).

It is therefore shown that substantial errors in predictions canarise as a result of misuse of the laminar or turbulent models.

Instead, the choice should be made based on a calculated valueof Reynolds number characterizing the regime for a given case.

To address the second objective of this section, comparing theexpected Reynolds numbers for different fluid families, four fluidsare examined. To see the effect of changing the fluid on the value ofthe characteristic Reynolds number, the flux qin and height H areassumed constant, and the values given in Table 3 are used. Refer-ring to Table 4, the value of μ=ρ for different fluids is given. Thebiggest number in the table corresponds to cross-linked gel and isequal to 41.67 × 10−5 m2 s−1. The smallest value is associated withCO2 (8.34 × 10−8 m2 s−1). Hence, the ratio of characteristic Reyn-olds number for those two cases is equivalent to the ratio of densityover viscosity for those two fluids, namely around 5,000. This con-trast of Reynolds number can be large enough to change the flowregime from laminar to turbulent.

The roles of geometry and pumping rate will now be examined.Typical heights of HFs fall in the range 20 m < H < 200 m (Fisherand Warpinski 2012). The range of injection rates is taken tobe from 0.01 m3 s−1 < qin < 0.2 m3 s−1. Hence the ratio is5 × 10−5 m2 s−1 < qin=H < 0.01 m2 s−1. Typical Reynolds num-bers for the four fluids and their densities and viscosities are listedin Table 4. Based on open-channel problems (Henderson 1966;Munson et al. 2002), the corresponding range of Reynolds numberfor the laminar regime is R < 500, whereas R > 12,500 is consid-ered a turbulent regime. The values between are thus considered tooccupy a transition from laminar to turbulent flow. Accordingly,fracturing with CO2 will be mostly turbulent flow. The flow regimeof the water in most cases is in transition between laminar and

0.2 0.4 0.6 0.8 1.0

1.05

1.10

1.15

1.20

0.2 0.4 0.6 0.8 1.0

0.04

0.05

0.06

0.07

0.08

0.2 0.4 0.6 0.8 1.0

1.15

1.20

1.25

Fig. 4. Constants Ci and Mi in Eqs. (43) and (44) as functions of α

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Fig. 5. Normalized variation of maximum fracture width in differentcross sections for laminar and turbulent flow

Table 3. Material Properties and Physical Constants for Illustration

Parameter Value

qin 0.2 m3 s−1ν 0.25μ 0.001 Pa · sρ 1,000 kgm−3k 0.3 mmm 0.143H 20 mG 30 GPaα 1/3

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turbulent, and in the most field-relevant cases is closer to, and there-fore better approximated by, the turbulent regime. The other twofluids lead to laminar flow (more details are given in the Supple-mental Data).

The suggested Reynolds numbers are experimentally determinedand may change based on geometric details for open-channel prob-lems. For most of the practical cases, a Reynolds number below 500is laminar. However, there is no definitive upper limit defining thetransition to turbulent flow (Te Chow 1959; Munson et al. 2002;Gioia and Chakraborty 2006). In open-channel problems, an upperlimit for the transition depends on other parameters like the channelgeometry. Therefore, two alternative methods are discussed for es-timating an appropriateR to define the transition to turbulent flow inorder to select an appropriate fluid flow law. First, the characteristicfluid pressure associated with laminar flow Plaminar is compared withthe characteristic pressure associated with turbulent flow Pturb. Inthis approach, Plaminar > Pturb is defined as the laminar regime,and Plaminar < Pturb as the turbulent regime. Although the transitionR depends on the fluid properties (Supplemental Data providesdetails), a typical transition value is around R ¼ 500. With thisdefinition and the proposed ranges for different parameters, CO2

is always turbulent, and water is turbulent for nearly all relevantcases (Supplemental Data provides more details).

Returning (briefly) to the definition of GMS, to develop the fric-tion factor, the hydraulic radius is used. Hydraulic radius (Rh) is acharacteristic length that helps to calculate the effect of differentcross sections. Originally, this parameter is introduced so thatthe pipe-flow equations can be expanded to other noncircularconduits. Mathematically, the hydraulic radius is the ratio of crosssection of the fluid flow over the wetted perimeter (Rh ¼ A=P).For elliptical cracks, the area is A ¼ πωH=4, and if the eccentricityof the ellipse defined as e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðω=HÞ2

p ≈ 1, its perimeter isdefined

P ¼ πH

�1 −X∞

i¼1

ð2iÞ!2ð2i · i!Þ4

e2i

2i − 1

�≈ 2H ð46Þ

so the value of hydraulic radius is πω=8. In the PKN model, thecross section is ellipse, so the average value of the opening inone cross section is (Zia and Lecampion 2016)

ω̄ ¼ 1

H

ZH=2

−H=2Wðx; z; tÞdz ¼ π

4ω ð47Þ

thus, hydraulic radius is half the average value of the opening of thecrack at a specific cross section (Rh ¼ ω̄=2).

The Moody diagram can be used to determine the transition Rfor the purpose of selecting a fluid flow model (Supplemental Dataprovides more details). The premise of this argument is that formost cases in HF, the scaled value of the fracture roughness isin the order of 0.05 or higher (k=w > 0.05). According to theMoody diagram (Supplemental Data), for such a roughness, thefully turbulent regime occurs at R > 104, and the transition fromthe laminar regime starts around R > 2,000. Furthermore, the fric-tion factor in this transition regime is for the most part closelyenough approximated by the turbulent GMS model that it is a via-ble selection from a practical perspective.

Finally, the Reynolds number discussed so far is determinedby the fluid flow only in the neighborhood of the injection point.This Reynolds number, however, may not represent the flow nearthe fracture tip, where there is a switch from a Reynolds numberdominated by the fracture depth H to a local Reynolds numberwhere the small width of the fracture dominates. Therefore, thereis a transition along the crack where the flow regime switches fromturbulent flow to laminar flow. Thus, it is desirable to know the ratioof the length of the turbulent regime over the laminar region. Giventhe fact that q is independent ofH and only depends on time throughx=lðtÞ, the quantity HR is a unique function of x=lðtÞ for a givenfluid. This relationship is shown in Fig. 7. Comparing the valuespresented in Fig. 7 shows that the value of Reynolds number nearthe crack tip where the value of x=lðtÞ approaches one is close tozero, and the behavior of the fluid in that region is thus laminar.Moreover, Fig. 7 indicates that there is a similarity between allthe graphs; indeed, the only thing that changes from one plot tothe other is the value of the kinematic viscosity μ=ρ. Combining,then, Eq. (42) and the definition of Reynolds numbers gives

0 200 400 600 800 1000

0

2

4

6

8

0 200 400 600 800 1000

0

5

10

15

0 200 400 600 800 1000

0

500

1000

1500

2000

2500

3000

(a) (b) (c)

Fig. 6. Comparison between laminar and turbulent solutions for the parameters given in Table 3: (a) maximum fracture width at x ¼ 0; (b) predictedfluid net pressure at the wellbore; (c) fracture length

Table 4. Different Fracturing Fluids and Their Rheology

FluidDensity

ρ (kgm−3)Viscosityμ (Pa · s)

Kinematic viscosityμ=ρ (m2 s−1)

Water 1,000 0.001 10−6X-linked gel 1,200 0.5 41.67 × 10−5CO2 (supercritical CO2) 600 5 × 10−5 8.34 × 10−8Linear gel 1,200 0.05 41.67 × 10−6

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R ¼ R�F ðξÞ;R� ¼ ρqin

μH;

F ðξÞ ¼ ð1 − ξÞ37ð1þ 0.05497ξÞ53ð1þ 0.21005ξÞ12 ð48Þ

The change of F ðξÞ, which determines the variation of R fordifferent values of ξ is presented in Fig. 8. In order to changethe order of magnitude of R compared with R�, the value ofF ðξÞ should drop at least one order of magnitude, which occursfor ξ > 0.9970. Similarly, a two order of magnitude drop corre-sponds to ξ > 0.999986.

With the variation of R along the fracture in mind, consider theexample of water as an injecting fluid (μ ¼ 0.001 Pa · s andρ ¼ 1,000 kgm−3). The characteristic Reynolds number isR�¼ 106qin=H. If the height is 20 m and the injecting fluid flowis 0.2 m3 s−1, then R� is 104. Thus, the fluid flow regime based onthe characteristic Reynolds number is turbulent. However, as

discussed, R decreases along the fracture, reaching a transitionvalue of R≈ 2,000 at approximately x=lðtÞ > 0.98. This indicatesthat approximately 98% of the HF length is either in transition orturbulent regime. Thus, the GMS approximation is accurate enoughfor practical purposes in the transition regime. Given the fact thatHFs have a relatively large roughness scale compared with the frac-ture opening, in this case, a valid global approximate solution canbe obtained, neglecting the laminar region near the tip. In otherwords, this example would correspond to a valid application ofthe present model. In contrast, if fluid flow is 0.2 m3 s−1 andthe height of the crack is 70 m, then R� ≈ 2,900, which is in tran-sition to turbulent regime. In this case, Reynolds number drops be-low 2,000 at approximately x=lðtÞ > 0.68. In this case, then, 68%of the HF is either in transition or is in turbulent regime, whichshows that around one-third of the crack is still in the laminar re-gime. In this latter example, an approach considering the presenceof both a turbulent and laminar region within the HF would berequired.

Conclusions

The flow regime for some HF treatments is turbulent over the vastmajority of the HF length. In particular, high-rate water-drivenHFs, as well as CO2-driven fractures, tend to this regime. Thisis in contrast to the lower-rate gel-driven fractures that were thesubject of most interest during the development of many HFsolutions based on laminar flow models. With these limitationsin mind, the scope of this paper was limited to consider fullyturbulent flow throughout the fracture, ignoring the effect of thetransition of the fluid regime from turbulent to laminar flow atthe crack tip or in the boundary at top and bottom of the reservoir.Also, considering this limiting regime, the solution ignores the ef-fect of fracture toughness.

This paper has presented a model for a blade-shaped (PKN)geometry HF growing in an impermeable rock and driven by a tur-bulent fluid. A semianalytical solution was derived that (1) em-bedded all rock, fluid, and geometric parameters in a scaling sothat the resulting ode can be solved once for all cases; and (2) pro-vided an accurate solution keeping only two terms of a polynomialseries solution. Rapid convergence was enabled by embedding thenear-tip behavior, also solved here in the course of the solutionmethod, in the form of orthogonal polynomials. Failure to recog-nize the appropriate flow regime will lead to erroneous applicationof models based on laminar flow. Incorrect models were estimatedto overpredict the fracture length and underpredict the fracturewidth and pressure by 40–50%. As such, this model not only pro-vides a benchmark solution for numerical simulation and a meansfor rapid estimation of fracture dimensions, but it also provides im-petus for ongoing research including experimental studies to findthe most appropriate values of parameters m and α for turbulentflow within a rough-walled deformable slot such as those encoun-tered in HF applications.

It was also shown that using laminar flow instead of turbulentflow under conditions where most of the HF has R > 2,500 canlead to enormous errors in calculating the fracture opening(i.e., more than 100% at 1,000 s of injection). In addition, a sizableerror (>50% as shown in Fig. 6) was induced on the crack-lengthand fluid-pressure estimations. Ongoing efforts are aimed at ex-panding the ability of the model to consider the turbulent regimewhen it is appropriate to use a different form other than the gen-eralized GMS equation. It is expected this will be particularly nec-essary under conditions where proppant transport and use ofrheological-modifying additives are considered.

0.2 0.4 0.6 0.8 1.01

10

100

1000

104

105

106

0.2 0.4 0.6 0.8 1.01

10

100

1000

104

105

106

Fig. 7. Reynolds number variation along the crack; by knowing theheight of the barrier H, it is possible to find the Reynolds numberat any points inside the crack; time dependence is embedded in lengthof the crack, which can be seen in x=lðtÞ; this graph still depends onpumping rate qin

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Fig. 8. Change of F ðξÞ versus ξ

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Appendix. Integrated Fluid Flow Equation

From elasticity and by assuming that the cross section of the crackis elliptical, it is possible to say that (Nordgren 1972)

Wðx; z; tÞ ¼ 1 − νG

ðH2 − 4z2Þ1=2ðp − σÞ ð49Þ

At z ¼ 0, the maximum opening is given

Wðx; 0; tÞ ¼ ωðx; tÞ ¼ 1 − νG

H½pðx; tÞ − σ� ð50Þ

Thus, the pressure gradient is

∂p∂x ¼ G

1 − ν1

H∂ωðx; tÞ

∂x ð51Þ

Inserting the crack-opening expression Eq. (49) into theGaukler-Manning-Strickler parametrization Eq. (4) gives

q2D ¼�

4

ρmkαω3þα

�1 −

�2zH

�2�3þα

2

�−∂p∂x

��1=2

ð52Þ

which is the two-dimensional flow at every height z, i.e., Fig. 9.The total flow rate can be found by integrating q2D over the heightof the fracture, which gives

q ¼Z

H=2

−H=2q2Ddz ¼

�4

ρmkαG

1 − ν1

H

�1=2

×

�−ω3þα ∂ω

∂x�

1=2Z

H=2

−H=2

�1 −

�2zH

�2�ð3þαÞ

4

dz

ð53ÞLetting φ ¼ ð3þ αÞ=4 and making the change of variable

sin θ ¼ 2z=H yields

q ¼�

4

ρmkαG

1 − ν1

H

�1=2

ω2φ

�−∂ω∂x

�1=2 H

2

Zπ=2

−π=2ðcos θÞ2φþ1dθ

ð54ÞThus, the total flux q can be written in terms of the beta function

βðx; yÞ

q ¼ H2

�4

ρmkαG

1 − ν1

H

�1=2

ω2φ

�−∂ω∂x

�1=2

β

�1

2;φþ 1

�ð55Þ

where the beta function is defined (Abramowitz and Stegun 1972)

βðx; yÞ ¼Z

1

0

tx−1ð1 − tÞy−1dt ð56Þ

or

βðx; yÞ ¼ ΓðxÞΓðyÞΓðxþ yÞ ð57Þ

for RðxÞ > 0 and RðyÞ > 0; and ΓðxÞ = Gamma function(Abramowitz and Stegun 1972).

Acknowledgments

Navid Zolfaghari and Andrew P. Bunger wish to acknowledge thesupport from the University of Pittsburgh Swanson School andEngineering and Center for Energy. Their discussions with Profes-sor Jim Rice and the insights from two anonymous reviewers arealso gratefully acknowledged.

Notation

The following symbols are used in this paper:A = cross-sectional area of the crack at coordinate x and

time t;Ai = unknown coefficients to define dimensionless

opening;B = beta function;B = constant that can be calculated from Eq. (26);C1 = constant that is dependent on α and can be obtained

from Fig. 4;C2 = constant that is dependent on α and can be obtained

from Fig. 4;Di = proper constant to be chosen to satisfy orthogonality;

F ðξÞ = function that defines the decay of Reynolds numberalong the crack [Eq. (48)];

fi = proper set of function to satisfy Eq. (28);fp = Darcy-Weisbach friction factor;G = shear modulus of elasticity of the rock;

Giðp; q; rÞ = ith-order Jacobi polynomial;H = height of the stimulating rock;

hiðp; qÞ = norm of ith-order Jacobi polynomial;k = Nikuradse channel wall roughness;

lðtÞ = length of the crack at time t;lNðtÞ = length of the crack calculated based on Nordgren

solution;m = constant coefficient for modeling turbulent flow;n = number of terms to truncate Jacobian polynomial

series;p = fluid pressure;

pnet = net pressure inside the crack and is equal to p − σ;

Fig. 9. Representation of two-dimensional element in three-dimensional crack; integration of infinitesimal elements over the heightof the crack will give the flow through a given cross section at location x

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pnetN = net pressure calculated based on Nordgrensolution;

Q = scaled flux;q = fluid flow within the fracture from a cross-section at

location x;qin = volume rate of flow at inlet;q2D = unidirectional fluid flow through the two dimensional

channel;RðxÞ = weight function in the general definition of

orthogonal polynomials;Rh = hydraulic radius;R = Reynold number at location x and at time t;R� = characteristic Reynold number;t = time;

W = opening of PKN crack at location ðx; zÞ and at time t;WðtÞ = characteristic width of HF;

X = constant to define crack tip [Eq. (26)];ðx; zÞ = coordinate value at x-axis and z-axis;

α = constant coefficient that appears as power formodeling turbulent flow;

P = wetted perimeter of the crack cross section;γðtÞ = dimensionless length of the crack;

Γ = gamma function;δij = Kronecker delta;Λ = predefined constant mentioned in Eq. (7);μ = fluid viscosity;ν = Poisson’s ratio of the rock;Ξ = predefined constant (Ξ ¼ 4Λϒ=πH);ξ = scaled coordinate;ρ = fluid density;ρ = constant depending on α and can be found from

Eq. (7);σ = in situ stress;ϒ = predefined constant mentioned in Eq. (7);φ = constant depending on α that can be found from

Eq. (7);ω = maximum opening in each vertical cross section at

time t;ωN = maximum opening calculated based on Nordgren

solution;ω̄ = average value of opening at each vertical cross section

at time t;ΩðξÞ = dimensionless opening; andΩ̂i = base function to define dimensionless opening.

Supplemental Data

This section provides (1) an explanation of turbulent flow withpipe flow analogy, (2) a model to capture laminar to turbulent tran-sition, (3) a near-tip solution, and (4) an alternative scaling ap-proach to solve the problem. Also, Figs. S1–S3, Table S1, andEqs. (S1)–(S25) are available online in the ASCE Library (www.ascelibrary.org).

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