w

Qi

h

H

R − Rw

Analysis of the Power Input Needed to PropagateMultiple Hydraulic FracturesAndrew Bunger

Report Number: EP128743, November 2012

Enquiries should be addressed to:

Andrew BungerCSIRO Earth Science and Resource EngineeringPrivate Bag 10, Clayton South, 3169, Victoria, AustraliaTelephone : +61 3 9545 8334Fax : +61 3 9545 8331Email : Andrew.Bunger@csiro.au

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Energy Rate Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Specifying Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Work of In Situ Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Work of Interaction Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Fracturing Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Fluid Flow Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Small Wellbore Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Evolution of the Energy Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.1 2D and Radial Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 PKN Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A Far Field Approximation of the Interaction Stress . . . . . . . . . . . . . . . . 26A.1 Far Field Equivalence to Uniformly Pressurized Crack . . . . . . . . . . . . . 26A.2 2D Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27A.3 Radial Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.4 PKN Geometry, h À R À H . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.5 PKN Geometry, R À h À H . . . . . . . . . . . . . . . . . . . . . . . . . . 30

B Validity Conditions for Scaling Relationships . . . . . . . . . . . . . . . . . . . 30

1

2

SummaryEmerging applications in unconventional gas production, geothermal power generation, andmining are driving an intensifying focus on the effective creation of arrays of hydraulic frac-tures from a single wellbore. Relative to creating each of the hydraulic fractures in the arrayindividually, it is often cost effective to simultaneously create more than one hydraulic fracture.However, it remains unclear how the energy requirements for hydraulic fracture growth dependon the fracture geometry and on the number of hydraulic fractures in the array, and this questionis foundational for the development of models aimed at predicting conditions under which mul-tiple hydraulic fractures will grow simultaneously and conditions which will favor localizationto fewer, or in some cases, a single dominant hydraulic fracture. As a first step to addressingthis question, this paper presents an energy balance for an array of multiple, parallel hydraulicfractures for three geometries, namely plane strain, radially-symmetric, and blade-like (“PKN”)geometries. Both geometry and coupled fluid flow are found to have a profound impact, and twocases were found for which an array of multiple hydraulic fractures requires less input powerfor propagation than is required for a single hydraulic fracture. The first is radially symmetrichydraulic fractures under viscosity dominated conditions prior to the onset of stress interactionsbetween neighboring hydraulic fractures. The second is the viscosity dominated PKN case,wherein the minimum input power when the hydraulic fracture length is much greater than boththe fracture height and spacing occurs when the spacing is around 2.5 times the fracture height.Hence, in the PKN case, this analysis provides the first indication that a natural, or energeticallyoptimal, spacing exists when multiple height constrained hydraulic fractures are simultaneouslypropagated.

KeywordsHydraulic Fracturing; Stability Analysis; Unconventional Gas; Reservoir Stimulation

3

1 IntroductionOne of the rapidly growing areas in hydraulic fracturing technology is the creation of multiplehydraulic fractures from a single wellbore. For example, stimulation of shale gas resourcestypically entails perforating the casing of a horizontal wellbore at several locations and inject-ing into each of these entry points (e.g. Rodrigues et al., 2007; King, 2010). Also, in mining,hundreds or thousands of hydraulic fractures are created in prospective orebodies to weaken therock for eventual mining by caving techniques (van As and Jeffrey, 2000).

One method for creating arrays of hydraulic fractures entails sequential injection from oneentry point at a time. For sequential fracturing the issues are related to interaction of a grow-ing hydraulic fracture with previously placed hydraulic fractures. Impacts of these interac-tions on crack paths, crack opening, and the fluid pressure required for initiation and exten-sion have been the focus of past and ongoing research (e.g. Olson and Dahi-Taleghani, 2009;Roussel and Sharma, 2011; Bunger et al., 2012; Vermylen and Zoback, 2011).

On the other hand, simultaneous injection into multiple entry points is a scenario that plays outwidely within both industry and the geosciences. In particular, multistage hydraulic fracturestimulations invariably involve injection into multiple well perforation clusters that are spreadout over the length of the stage (e.g. Modeland et al., 2011). What’s more, the formation ofnaturally occurring, magma-driven dyke swarms seems to be a process that is likely to have en-tailed injection of magma into multiple, simultaneously growing dykes (e.g. Ernst et al., 1995).However, the basic conditions under which it is energetically favorable for multiple fluid-drivencracks to grow simultaneously versus localization of injection to a single fluid-driven crack arenot yet understood. What’s more, the impact of a model’s assumptions with regards to geom-etry and coupling among the relevant physical processes on the energetics of the simultaneousgrowth of multiple hydraulic fracture has never been clarified. As a result, it remains funda-mentally unclear which essential ingredients are required in an analytical or numerical modelfor it to capture the system’s basic behavior.

Recently, Dahi-Taleghani (2011) proposed an approach to evaluating the stability of multiplecrack growth for application to uniformly pressurized cracks. The approach follows after anenergy-based method for multiple crack propagation analysis proposed by Budyn et al. (2004)in which an assembly of cracks are subjected to remote loading. Similar to Dahi-Taleghani(2011), many multiple crack simulations that are applied to hydraulic fracturing have neglectedcoupled fluid flow, and have instead used a uniformly pressurized crack model (Olson, 2008;Olson and Dahi-Taleghani, 2009; Roussel and Sharma, 2011). Some notable exceptions, whichextend multiple crack theory to account for coupling between crack growth and viscous fluidflow within the hydraulic fracture, include Germanovich et al. (1997), Zhang et al. (2007, 2011),Jin and Johnson (2008), and Weng et al. (2011), but these coupled treatments do not explicitlyexamine the energy required for propagation.

It remains, then, that the energetics of multiple hydraulic fracture growth have not been formal-ized in a way that recognizes the fundamental role of viscous dissipation, fracture geometry,and the interaction among multiple hydraulic fractures. Establishing an appropriate formalismand drawing some broad, guiding principles from the ensuing analysis is the focus of this re-search. Consideration is limited here to planar, Newtonian-fluid-driven crack growth throughan impermeable, brittle elastic matrix. Drawing early inspiration from Shlyapobersky (1985)and building on expressions established for a single, two dimensional (plane strain) hydraulicfracture by Lecampion and Detournay (2007), the approach is comprised of establishing an ex-

4

pression for the energy rate balance for a uniform, simultaneously growing array of hydraulicfractures beginning from basic principles of continuum mechanics.

2 Energy Rate PrincipleThe analysis begins by taking the power supplied to drive the growth of a hydraulic fracture tobe given by pfoQ, where pfo is the fluid pressure at the wellbore and Q is the injection rate.One has to choose where to measure pfo, specifically whether it is measured at the surface(“wellhead pressure”) or at the location of hydraulic fracture initiation (“bottom hole pressure”)or somewhere in between. For the present analysis the contribution of the weight of the fluidcolumn to the energy rate is a constant and not of particular interest, hence it suffices to considerpfo to be the fluid pressure at the inlet that connects the wellbore to the growing hydraulic frac-ture. It will also be assumed that the variation in depth between multiple entry points in the caseof an array of hydraulic fractures is small enough that the hydrostatic variation between themcan be neglected or, more specifically, that the variation in power available to drive hydraulicfracture growth because of this hydrostatic variation in the fluid pressure is negligible relativeto the other terms in the energy rate balance.

Further simplification is attained by assuming the injection rate Q to be a constant, denoted Qo.

This analysis is aimed at determining the energy rate associated with the configuration whereN hydraulic fractures grow uniformly from a regular array of entry points, typically casingperforations or abrasively-cut slots, with the spacing h = Z/(N −1) between them, for intervallength Z. Note that for simplicity, the spacing will be approximated h ≈ Z/N under theassumption that N À 1. These initial considerations then lead to the quantity pfoQi withQi = Qo/N , which gives the power supplied to the inlet of one hydraulic fracture in thisuniform array.

If the pressure loss due to fluid flow through the inlet (“entry loss”, e.g. Chen and Economides(1999)) is neglected, the fluid is incompressible, and the rock is impermeable – note that thevalidity of this latter assumption is revisited in Appendix B – then the power supplied is eitherdissipated due to fluid flow, or else it does work on the solid. Denoting the fluid dissipation rateDf and the work of the fluid on the solid Wf , an energy rate balance is given by

pfoQi = Wf + Df , (2.1)

where the overdot indicates the derivative with respect to time. Note that when N = 1 this bal-ance equation takes on the form presented for the plane strain case in Lecampion and Detournay(2007). Furthermore, if Df = 0 and the partitioning of Qo among the hydraulic fractures is al-lowed to be non-uniform, one could in principle use this energy balance equation as a startingpoint to derive the stability criterion for multiple pressurized cracks presented by Dahi-Taleghani(2011). Note that an important consequence is that the fracture mechanics approach to predict-ing the stable configuration for multiple propagating cracks based on choosing the configu-ration that maximizes the elastic strain energy release (Bocca et al., 1991; Budyn et al., 2004;Dahi-Taleghani, 2011) is applicable only in cases wherein the power driving propagation ismainly dissipated through fracturing rather than through fluid flow, which is often not the casein hydraulic fracturing treatments (see e.g. field cases in Bunger et al., 2012).

In order to proceed, it is necessary to appropriately account for the partitioning of the energyimposed to the solid via the work of the fluid Wf . One can essentially think of the total work

5

of the fluid serving to increase the elastic strain energy, to create new crack surfaces, and,typically, to counteract the work being done on the crack by remote compressive stresses. LinearElastic Fracture Mechanics is the paradigm that will be used to obtain this partitioning. Thefundamental energy argument of Elastic Fracture Mechanics is that the energy that is suppliedto the tip region for the creation of new surfaces, usually denoted G, is the summation of acontribution from work W that is done by external forces and release of strain energy U thataccompanies crack extension (see Keating and Sinclair, 1995, for an elegant discussion of thisprinciple). Letting a be the surface area of the crack, the balance equation is given by

G = −dU

da+

dW

da. (2.2)

Now let us do three things to this equation. Firstly, consistent with the theory of Griffith (1920),assume that crack growth is implied by G attaining a critical value Gc. Secondly, recognize thatthe work done by external forces is comprised of three parts so that W = Wf +Wo +WI , whereWf is the work done by the fluid, as previously described, Wo is the work done on the crackby the in situ stress, and WI is the work done on the crack by other nearby hydraulic fractures.Thirdly, following Lecampion and Detournay (2007), multiply all terms in Eq. (2.2) by da/dtto obtain an expression in terms of energy rates with respect to time. Eq. (2.2) then becomes

Gcda

dt≡ Dc = −U + Wf + Wo + WI . (2.3)

Solving for Wf and substituting into Eq. (2.1) leads to the final form of the energy balance

pfoQi = U − Wo − WI + Dc + Df . (2.4)

3 Specifying Terms

3.1 Geometry

There are three classical idealized geometries for planar hydraulic fracture growth that willbe considered: two-dimensional (2D) plane strain, axisymmetric (penny-shaped), and bladeor finger-like geometry (Figure 3.1). In the hydraulic fracturing literature, the plane straincase is often referred to as the “KGD” geometry in recognition of the pioneering hydraulicfracturing researchers Kristianovic, Geertsma, and de Klerk (Khristianovic and Zheltov, 1955;Geertsma and de Klerk, 1969). Similarly, the blade or finger-like case is often referred to as the“PKN” geometry in recognition of its early investigators (Perkins and Kern, 1961; Nordgren,1972), and it is considered a realistic geometry when the vertical growth of the hydraulic frac-ture is restricted by the strata that bound the reservoir.

When an array of 2D or PKN hydraulic fractures is created along a single, multiply slotted orperforated wellbore that is drilled parallel to the minimum in situ stress, the hydraulic fractureplane will be transverse to the wellbore. Hence, the flow geometry will differ from that por-trayed in Figure 3.1 near the wellbore and, furthermore, a wellbore radius along the plane ofthe fracture, as shown, is meaningful only for the radial case. While these near wellbore detailscould, in fact, be important for cases that are otherwise idealized as 2D or PKN geometries, itwill be considered practical to neglect these details for this first pass at the problem.

In all three geometric cases it is assumed that the fluid front does not lag behind the propagatingcrack tip, which is a valid assumption provided that the remote stresses opposing crack opening

6

Qi

w

h

2D

R − Rw

h

R

2Rw

Qi

w

Radial

H

PKN

Qi

w

R − Rw

h

Figure 3.1: Arrays of hydraulic fractures for the three idealized geometries, where the 2D andPKN cases show only one of the two symmetric wings of the hydraulic fracture and the radialcase shows a cross-section through a circular hydraulic fracture.

7

are sufficiently strong (Garagash and Detournay, 2000). But this assumption is not universallyvalid. For example, a low stress environment may persist when the hydraulic fracture grows neara free surface (Bunger et al., Submitted). Furthermore, in their consideration of the plane straincase, Lecampion and Detournay (2007) show that the lag region can have a distinct impact onthe energy balance. However, under petroleum reservoir conditions the zero lag approximationis expected to be valid over the relevant range of time and so for this investigation it has beenchosen not to complicate the equations in order to consider finite fluid lag.

Under these geometric specifications, the notation R will be used for the crack half-length inthe 2D and PKN cases and for the radius in the radial case. In all cases Rw represents the radiusof the wellbore, h is the spacing between hydraulic fractures, and H quantifies the height of thePKN hydraulic fracture (see Figure 3.1).

3.2 Strain Energy

The strain energy for an elastic body subjected to small deformations is expressed by the doubleinner product of the Cauchy stress tensor σ and the linear strain tensor ε integrated over thevolume of the body, that is

U =1

2

∫

B

σ : εdV . (3.1)

Invoking Clapeyron’s Theorem in the absence of body forces, this can be shown to be equivalentto the surface integral over the loaded part of the boundary of the inner product of the tractionvector T and the displacement u (e.g. Sokolnikoff, 1956)

U =1

2

∫

δB

T · udS. (3.2)

Here the boundary δB is comprised of the upper and lower surfaces of the crack which areloaded by the normal traction Tn = p = pf − σo − σI , where pf is the fluid pressure, σo is thepre-existing in situ stress, and σI is the loading that is induced by the other hydraulic fractures.The quantity p is thus a measure of the net pressure acting on the fracture surfaces. Substitutinginto Eq. (3.2), taking the time derivative, and parameterizing the surface integral to the 2D case

U (2D) =

∫ R

Rw

(p∂w

∂t+ w

∂p

∂t

)dx, (3.3)

where w = 2un is the crack opening, noting that this substitution requires the assumptionthat un is the same for both surfaces of the crack. This assumption is strictly valid only for ahydraulic fracture growing as a member of an infinite, uniformly spaced array, but it will besufficient for the purpose of the present analysis. Similarly for the radial case

U (Rad) = π

∫ R

Rw

(p∂w

∂t+ w

∂p

∂t

)xdx. (3.4)

Lastly, the PKN case gives

U (PKN) = 2

∫ R

Rw

∫ H/2

−H/2

(p∂un

∂t+ un

∂p

∂t

)dydx. (3.5)

Under the assumptions of the PKN model (e.g. Nordgren, 1972), p = p(x, t) which leads to an

8

elliptical crack opening distribution in y, that is

un(x, y, t) = w(x, t)

(1

4− y2

H2

)1/2

, w(x, t) = 2un(x, 0, t). (3.6)

Carrying out the inner integral in Eq. (3.5) leads to

U (PKN) =πH

4

∫ R

Rw

(p∂w

∂t+ w

∂p

∂t

)dx. (3.7)

Adopting the indices

n =

0 2D, PKN1 Radial

, k =

0 2D, Radial1 PKN

the rate of change of the elastic strain energy can be expressed for all three cases as

U =πn+k

4kHk

∫ R

Rw

(p∂w

∂t+ w

∂p

∂t

)xndx. (3.8)

3.3 Work of In Situ Stress

The rate of work acting on the boundary of an elastic body is given by (e.g. Keating and Sinclair,1995)

W =

∫

δB

T · ∂u

∂tdS (3.9)

Expressing the surface integral as in Section 3.2, it is straightforward to show that the rate ofwork performed by the in situ stress σo is given by

Wo = −2πn+k

4kHkσo

∫ R

Rw

∂w

∂txndx. (3.10)

Following Lecampion and Detournay (2007), the expression for Wo can be simplified by invok-ing the continuity equation for a hydraulic fracture driven by an incompressible fluid (Khristianovic and Zheltov,1955; Nordgren, 1972)

1

xn

∂

∂x(xnq) +

(πH

4

)k∂w

∂t= 0, (3.11)

where q is the volumetric flow rate through a cross section of the crack,

q =

∫ w/2

−w/2

vxdx, k = 0, (3.12)

for the 2D and radial cases and

q =

∫ H/2

−H/2

∫ un

−un

vxdxdy, k = 1, (3.13)

for the PKN case, with vx the x component of the fluid velocity and where un(y) is given byEq. (3.6). For consistency with mass balance at the inlet and the condition w = 0 at the cracktip, q must satisfy the boundary conditions (e.g Detournay, 2004)

q|x=Rw=

Qi

2πnxn, q|x=R = 0. (3.14)

9

Integrating Eq. (3.11) from Rw to R subject to Eq. (3.14), then multiplying all terms by σo andsubstituting into Eq. (3.10) leads to

Wo = −Qiσo. (3.15)

Note that the hydraulic fractures will typically be initiated at different depths. Obviously deeperhydraulic fractures will experience a gravitational contribution to the fluid pressure as well as ahigher in situ stress. Acknowledging that heterogeneities and tectonics can lead to other sourcesof stress variation, as a first approximation neglecting these contributions from depth is roughlyto assume that Zg (ρr − ρf ) Qo cos φ is negligible compared to the dominant energy rate term,where Z is the length of the zone, ρr and ρf are the rock and fluid densities, respectively, and φis the deviation of the wellbore from vertical.

3.4 Work of Interaction Stress

Following on from Eq. (3.10), the rate of work performed by the interaction stress σI is givenby

WI = −2πn+k

4kHk

∫ R

Rw

σI∂w

∂txndx. (3.16)

For the general case, computing this term requires coupling together all of the hydraulic frac-tures in the array. However, for now it is of interest to clarify how this term begins to impactthe system at the onset of interaction among hydraulic fractures in an array. For this purposeit suffices to consider the leading order contributions that arise from the first neighboring hy-draulic fracture to each side of the one under examination. Also, one has only to consider thefirst two terms in the far field expansion of the stresses induced by these neighboring hydraulicfractures. Further simplification comes from the aforementioned assumption that N À 1, sothat we can ignore the detail that the hydraulic fractures on either end of the array have onlyone nearest neighbor rather than one on each side, although it is acknowledged that were we topredict where localization of flow to fewer dominant hydraulic fracture might occur, this detailis imminently relevant.

Appendix A presents derivations of the far field approximations for the interaction stress. Forthe 2D and radial cases, these expansions consider R ¿ h, that is, the spacing between thefractures is considered to be much greater than the fracture half length. In the PKN case (H ¿R), there are two relevant far field expansions corresponding to H ¿ R ¿ h and H ¿ h ¿ R.The first of these is the case when the spacing between the hydraulic fractures is the largestrelevant length, whereas in the second case the hydraulic fractures are long relative to both theirheight and length, that is, they are like an array of widely-spaced linguine noodles (Figure 3.2).

For the R ¿ h configuration for all three geometries, the far field approximations of the inter-action stresses can be expressed using the index notation, leading to

σI =3− n− 2kν ′

2π (2− n)

E′Qot

hn+k+2×

(1− 1

20ζ2

((5− n)2 + 2n− 5k

) ((4 + n− k) ρ2 + n + 1

)+ O(ζ−4)

), (3.17)

where

ρ =x

R, ζ =

h

R.

10

h

H

a)

R

R

H

h

b)

Figure 3.2: PKN far field cases, with (a) H ¿ R ¿ h and (b) the “linguine” case, H ¿ h ¿ R.

and

ν ′ =(2− ν)(1− ν)

1− 2ν, ν ′′ =

3− ν

2− ν.

Note that for the PKN case, the stress has been taken along y = 0 and higher order terms inH/R have been neglected.

By substitution of Eq. (3.17) into Eq. (3.16), WI is given to leading order for R ¿ h by

WI ∼ −2πn+k

4k

(3− n− 2kν ′)π (2− n)

E′HkQot

hn+2

∫ R

Rw

∂w

∂txndx,

R

h¿ 1. (3.18)

Similarly, for the linguine-like PKN array, that is, with H ¿ h ¿ R,

σI = p3H2

8h2

(1 + O(h/H)−2

), (3.19)

which agrees with Benthem and Koiter (1973), and therefore

WI ∼ −3π

2

H3

8h2

∫ R

Rw

p∂w

∂tdx, H ¿ h ¿ R. (3.20)

11

3.5 Fracturing Dissipation

The dissipation associated with fracturing, Dc (Eq. 2.3), is specified for the geometries underconsideration by the relationship for the time derivative of the fracture surface area

da

dt= 2πnHkRn dR

dt. (3.21)

Furthermore, provided that the crack is subjected to purely tensile-mode (“mode I”) loadingand inelastic deformation is confined to only a small region near the tip, an equivalence canbe invoked between the energy release rate criterion G = Gc and one that equates the tensilemode stress intensity factor KI to the tensile mode fracture toughness of the rock, KIc, via therelationship G = K2

I /E′ (e.g. Tada et al., 2000), where

E ′ =E

1− ν2,

for Young’s modulus E and Poisson’s ratio ν. Furthermore, it is convenient to make use of analternate form of the fracture toughness that arises naturally in hydraulic fracturing problems(e.g. Savitski and Detournay, 2002)

K ′ =(

32

π

)1/2

KIc.

Taken together, the fracturing dissipation terms can be expressed as

Dc =πn+1

16

K ′2HkRn

E ′dR

dt. (3.22)

Note that a well-known deficiency in the PKN model is its inability to account for physicallyrealistic stress conditions near the leading edge of the hydraulic fracture resulting in degen-eracy of the propagation condition. Some recent papers have renewed the discussion of thisissue (Adachi and Peirce, 2008; Adachi et al., 2010). In particular, these have shown that thereexists a region that scales with the presumed small ratio H/R where the local elasticity approx-imation of the PKN model does not hold. In this near-tip region there is a non-local elasticityrelationship that is neglected when only the outer, local elasticity relationship is considered. Itis this near-tip behavior that provides a link to the propagation condition, in principle, howeveran extension of the coupled PKN model to account for a finite fracture toughness has not yetbeen carried out. To remain consistent with the scope of the paper being restricted to cases forwhich the relevant solution for a single hydraulic fracture is already available in the literature,the finite toughness PKN case is neglected here. Hence consideration is for the PKN case isexclusively under conditions where Df dominates over Dc, and it is further understood that acorrection due to the non-local part of the elasticity relationship that is O(H/R) is neglected.

3.6 Fluid Flow Dissipation

The energy dissipation for an incompressible, viscous fluid is given by (e.g Landau and Lifshitz,1987)

Df =1

2µ

∫(∇v + v∇)2 dV (3.23)

where v is the fluid velocity and µ is its dynamic viscosity.

For the geometries under consideration, the fluid velocity has only one non-zero component vx

which is directed from the inlet to the hydraulic fracture’s leading edge. Furthermore, because

12

invariably w/R ¿ 1 for hydraulic fractures, to leading order one has only to retain the deriva-tives in z, the coordinate that is normal to the crack surfaces. The energy dissipation then canbe simplified to

Df =1

2µ

∫ (∂vx

∂z

)2

dV. (3.24)

Under the geometric stipulations just described, and provided further that vx is zero at the crackwalls, it is straightforward to solve the Navier-Stokes equations to obtain a parabolic variationof vx in z, that is (e.g Adachi, 2001)

vx = − 1

2µ

∂pf

∂x

(w2

4− z2

). (3.25)

Substituting Eq. (3.25) and p = pf − σo − σI into Eq. (3.26) and integrating over the volumeof the hydraulic fracture for each of the three cases, it is follows that

Df =πn

µ′

(3πH

16

)k ∫ R

Rw

w3

(∂p

∂x+

∂σI

∂x

)2

xndx, (3.26)

where

µ′ = 12µ

is an alternate form of the viscosity that, like K ′ arises naturally in the scaling of hydraulicfracturing problems (e.g. Detournay, 2004). Note that for the 2D case (n = k = 0) Eq.(3.26) corrects a factor of two on the dissipation term that was inadvertently introduced byLecampion and Detournay (2007).

Finally, specifying σI from Eq. (3.17) for the onset of interaction with nearby hydraulic frac-tures leads to

Df =πn+k3k

16k

Hk

µ′

∫ R

Rw

w3

(∂p

∂x

)2

xndx− (3.27)

2πn+k−13k+1 (5 + n− 3kν ′ν ′′)16k

HkE′Qot

µ′hn+k+4

∫ R

Rw

w3 ∂p

∂xxn+1dx + (3.28)

πn+k−232+k (5 + n− 3kν ′ν ′′)2

16k

HkE′2Q2ot

2

µ′h2(n+k+4)

∫ R

Rw

w3xn+2dx,R

h¿ 1. (3.29)

Note that the second term is expected to be positive because during injection, the pressuredecreases as one moves away from the inlet, hence ∂p/∂x < 0. For the purpose of estimatingthe order of this term, take ∂p/∂x = −|∂p/∂x|.Similarly for the linguine-like PKN case, substituting Eq. (3.19) into Eq. (3.26) gives

Df =

(3π

16

)H

µ′

∫ R

Rw

w3

(∂p

∂x

)2 (1 +

3H2

8h2

)2

dx, H ¿ h ¿ R. (3.30)

4 Small Wellbore ApproximationThe integrals that appear in the quantities U , WI , Df consider a domain that lies between thewellbore wall at x = Rw and the crack tip at x = R. At early time, when R ≈ Rw, the wellboreradius is important. Indeed it is a potentially valuable exercise to examine the early time stability

13

of this system using a model that accounts for multiple hydraulic fractures and near wellborestresses. However, for the present analysis the consideration will be further limited to R À Rw.It is therefore useful to remove Rw from the equations that are not substantially impacted by itsvalue.

To this end, let each of the integrals (3.8), (3.18), and (3.29) be extended to x = 0 via∫ R

Rw

=

∫ R

0

−∫ Rw

0

and retain the leading order terms for Rw/R ¿ 1. This would be straightforward except for thesingularity in q implied by the Rw/R → 0 limit of Eq. (3.14) for n = 1, that is, the radial case.Specifically, substitution of Eq. (3.25) into Eq. (3.12) leads to the Poiseuille equation

q = −w3

µ′∂pf

∂x. (4.1)

Hence, Eq. (3.14) implies that pf ∼ b ln x, x/R ¿ 1, for some b that in general is a part of thesolution to the system of equations that governs hydraulic fracture growth. For the strain energyin Eq. (3.8), the integral of this weak singularity converges as the lower bound of integration istaken to zero. Similarly, for the second term in the fluid dissipation in Eq. (3.29), the singularityin the square of the derivative of the fluid pressure is exactly canceled by the x2 factor in theintegrand. However, the first term of Eq. (3.29) is Cauchy singular as x → 0. The strategy,then, is to first regularize this integral as

∫ R

Rw

w3

(∂p

∂x

)2

xndx =

∫ R

Rw

w3

((∂p

∂x

)2

− nb2

x2

)xndx + n

∫ R

Rw

w3

(b

x

)2

xndx. (4.2)

The first term on the right hand side is then regular for x → 0. Assuming, then, that w ∼wo, x/R ¿ 1, the second term on the right hand side is

∫ R

Rw

w3

(b

x

)2

xdx ∼ −b2w3o ln

Rw

R,

Rw

R¿ 1. (4.3)

At last, Eqs. (3.8), (3.18), and (3.29) can be rewritten as

U ∼ πn+k

4kHk

∫ R

0

(p∂w

∂t+ w

∂p

∂t

)xndx,

Rw

R¿ 1, (4.4)

WI ∼ −2πn+k

4k

(3− n− 2kν ′)π (2− n)

E′HkQot

hn+2

∫ R

0

∂w

∂txndx,

R

h,Rw

R¿ 1, (4.5)

Df ∼ πn+k3k

16k

Hk

µ′

∫ R

0

w3

((∂p

∂x

)2

− nb2

x2

)xndx−

nπn+k3k

16k

Hk

µ′b2w3

o lnRw

R+

2πn+k−13k+1 (5 + n− 3kν ′ν ′′)16k

HkE′Qot

µ′hn+k+4

∫ R

0

w3

∣∣∣∣∂p

∂x

∣∣∣∣xn+1dx + (4.6)

πn+k−232+k (5 + n− 3kν ′ν ′′)2

16k

HkE′2Q2ot

2

µ′h2(n+k+4)

∫ R

0

w3xn+2dx,R

h,Rw

R¿ 1.(4.7)

14

And for the linguine-like PKN case (H ¿ h ¿ R), Eqs. (3.20) and (3.30) become

WI ∼ −3π

16

H3

h2

∫ R

0

p∂w

∂tdx,

Rw

R¿ 1, (4.8)

Df ∼(

3π

16

)H

µ′

∫ R

0

w3

(∂p

∂x

)2 (1 +

3H2

8h2

)2

dx,Rw

R¿ 1. (4.9)

It is also worth noting that p is singular not only for x → 0, but it is also singular for x → R,as detailed by Garagash and Detournay (2000). Specifically, one can consider cases Df À Dc

wherein p ∼ (R − x)−1/3 and w ∼ (R − x)2/3 for x → R. In this case the integrand forthe first term in Df , Eq. (4.7) is weakly singular like (R − x)−2/3, x → R. In contrast, forDc À Df the asymptotic behavior of the pressure is p ∼ ln(R − x) with the opening goinglike w ∼ √

R− x for x → R. This behavior again gives a weakly singular integrand for thefirst term in Eq. (4.7) that goes like (R− x)−1/2. Neither of these weak singularities affects theconvergence of the integral and therefore these are not problematic from the perspective of theanalysis that follows.

5 ScalingThe next step of the analysis is to express each of the terms comprising U , Wo, WI , Dc, Df inthe form PΦ, where P has the dimensions of power and Φ is an O(1) dimensionless quantity.This process will result in a family of characteristic power coefficients Pα that estimate thecontribution of each term to the overall energy balance. A general scaling can be expressed as(e.g Detournay, 2004)

w = WΩ (ρ, χi) , p = PΠ (ρ, χi) , R = Lγ (χi) ,∂p

∂x=

ϕP

L

1

γ

∂Π

∂ρ, (5.1)

where again ρ = x/R and here χi indicate dimensionless groups that in general depend on time.The appearance of the dimensionless scaling factor ϕ allows the scaling to account for the factthat the spatial variation of p could take place on a different characteristic scale than that of thelength of the hydraulic fracture itself, hence the coordinate ρ = ϕρ is introduced. This will beshown to be relevant when Df ¿ Dc, that is, for relatively small viscous dissipation.

Substitution into Eqs. (2.4), (4.4), (3.15), (3.22), (4.5), and (4.7) and normalizing the bounds of

15

integration to [0, 1] yields for R ¿ h

pfoQi = U − Wo − WI + Dc + Df , (5.2)

U ∼ PUγn+1πn+k

4k

(∫ 1

0

Πt

W

∂WΩ

∂tξndξ +

∫ 1

0

Ωt

P

∂PΠ

∂tξndξ

),

Rw

R¿ 1, (5.3)

Wo = −Po, (5.4)

WI ∼ −PIγn+1 2πn+k

4k

(3− n− 2kν ′)π (2− n)

∫ 1

0

t

W

∂WΩ

∂tρndρ,

R

h,Rw

R¿ 1, (5.5)

Dc = Pcπn+1

16

γnt

L

dLγ

dt(5.6)

Df ∼ Pfπn+k3k

16kγn−1

∫ 1

0

Ω3

((∂Π

∂ρ

)2

− nB2

ρ2

)ρndρ +

Plnπn+k3k

16k

(1− ln γ

ln(Rw/L)

)B2Ω3

o +

PI12πn+k−13k+1 (5 + n− 3kν ′ν ′′)

16kγn+1

∫ 1

0

Ω3

∣∣∣∣∂Π

∂ρ

∣∣∣∣ ρn+1dρ +

PI2πn+k−232+k (5 + n− 3kν ′ν ′′)2

16kγn+3

∫ 1

0

Ω3ρn+2dρ,R

h,Rw

R¿ 1, (5.7)

and for the linguine-like PKN case (H ¿ h ¿ R), Eqs. (4.8) and (4.9) become

WI ∼ −PIL3π

16γ

∫ 1

0

Πt

W

∂WΩ

∂tdx,

Rw

R¿ 1, (5.8)

Df ∼ Pf

(3π

16

)1

γ

∫ 1

0

Ω3

(∂Π

∂ρ

)2 (1 +

3H2

8h2

)2

dρ,Rw

R¿ 1, (5.9)

where

PU =HkPWLn+1

t, Po = Qiσo, PI =

Ln+1HkWE ′Qi

hn+k+2,

Pc =K ′2HkLn+1

E ′t, Pf =

HkLn−1W 3P 2ϕ2

µ′, Pln = −nPf ln

Rw

L

PI1 =Ln+1W 3PH

kE′Qitϕ

µ′hn+k+4, PI2 =

Ln+3W 3HkE′2Q2

i t2

µ′h2(n+k+4),

PIL =PWLH3

th2. (5.10)

Each of the 9 power coefficients listed in Eq. (5.10) gives a suitable estimate of the contributionof its respective term provided that the dimensionless quantity it multiplies is O(1), which relieson careful choice of the scaling factors W,P, ϕ, L. For the 2D and radial cases, appropriatechoices for the scaling factors can be readily obtained based on published asymptotic solutions(Adachi, 2001; Savitski and Detournay, 2002) for limiting cases where Pf ¿ Pc and Pf À Pc,that is, for the viscosity and toughness dominated regimes, respectively. Table 5.1 lists thesequantities. An appropriate choice of the scaling factors for the PKN case is available fromNordgren (1972) provided that Pf À Pc, and these quantities are also listed in Table 5.1.

Substituting the scaling factors W,P, ϕ, L from Table 5.1 into the power coefficients definedin Eq. (5.10) under the conditions that the flow is divided evenly among the N hydraulic frac-

16

Geometry Regime Validity W ε ϕ L

2D Viscosity Pf À Pc εL(

µ′E′t

)1/3

1(

E′Q3i t4

µ′

)1/6

2D Toughness Pf ¿ Pc εL(

K′4E′4Qit

)1/3

M0

(E′Qit

K′

)2/3

Radial Viscosity Pf À Pc εL(

µ′E′t

)1/3

1(

E′Q3i t4

µ′

)1/9

Radial Toughness Pf ¿ Pc εL(

K′6E′6Qit

)1/5

M1

(E′Qit

K′

)2/5

PKN Viscosity Pf À Pc

(µ′Q2

i t

E′H

)1/5WH

1(

E′Q3i t4

µ′H4

)1/5

Table 5.1: Scaling factors for R/h ¿ 1 and Rw/R ¿ 1, where P = E ′ε, M0 = µ′E′3Qi

K′4 , and

M1 =(

E′13µ′5Q3i

K′18t2

)1/5

.

tures in the array so that Qi = Qo/N leads to the expressions for the power coefficients for eachregime given by Table 5.2. Note that in the PKN/Linguine case (H ¿ h ¿ R), expanding Eq.(5.9) leads to the reported fluid flow interaction terms NPI1 and NPI2. Again, we reiterate thatthese estimates are valid over the range of time for which the associated scaling presented inTable 5.1 is valid, and these time ranges for the validity of each case are detailed in AppendixB.

6 Evolution of the Energy Rate

6.1 2D and Radial Cases

Based on the scaling arguments that have been presented, the energy rate required to grow auniform array of N hydraulic fractures is estimated as the sum of the power coefficients (Table5.2) over the N uniform hydraulic fractures. Hence the net energy rate (i.e. input power) isapproximated as

NQi (pfo − σo) ≈ N∑

α

Pα. (6.1)

Note, however, that this equation estimates the order of magnitude by summing the quantitythat determines the order of the various terms. As such, it is appropriate to drop the PU termfrom the summation at this point as it is identical to the Pf or Pc term, depending on regime.Including a factor of two as a result of summing PU + Pf or PU + Pc would be a spuriousand potentially misleading detail in light of the fact that, in order to compute the input powerat this level of precision, the numerical factor multiplying each of these terms would have to becomputed from a full solution to the governing equations and then integrated according to theenergy Eqs. (5.3-5.9).

In practice, the power coefficients Pα vary over several orders of magnitude and often it isthe case that one term dominates. To illustrate, consider the following numerical values of thegoverning parameters:

E ′ = 10 GPa, KIc = 1.25 MPa m1/2,

Qo = 0.1 m3/s, µ = 1 Pa s, σo = 70 MPa,

Rw = 0.2 m, Z = 1000 m, H = 20 m. (6.2)

17

NPα 2D/Viscosity 2D/Toughness Radial/ViscosityNPU NPf NPc NPf

NPIE′NQo

2tZ2

E′NQo2t

Z2E′N2Qo

2tZ3

NPc

(K′12N3Qo

3

E′5t2µ′

)1/6 (K′4NQo

2

E′t

)1/3 (K′18N3Qo

6

E′7µ′2t

)1/9

NPf

(E′2Qo

3µ′t

)1/3 (E′8Qo

5µ′3

K′8N2t

)1/3 (E′2Qo

3µ′t

)1/3

NPI1

(E′4N6Qo

9t7

Z12µ′

)1/3 (E′7N5Qo

10t7

K′4Z12

)1/3 (E′11N30Qo

24t17

Z45µ′2

)1/9

NPI2E′2N4Qo

5t5

Z8µ′E′2N4Qo

5t5

Z8µ′

(E′16N60Qo

39t37

Z90µ′7

)1/9

NPln 0 0 −NPf ln(

N3Rw9µ′

E′Qo3t4

)1/9

NPα Radial/Toughness PKN PKN/LinguineNPU NPc NPf NPf

NPIE′N2Qo

2tZ3

E′N2Qo2t

Z3 0

NPc

(K′6NQo

4

E′t

)1/5 (HK′10N2Qo

3

E′4µ′t

)1/5 (HK′10N2Qo

3

E′4µ′t

)1/5

NPf

(E′12Qo

7µ′5

K′12N2t3

)1/5 (E′4Qo

7µ′tH6N2

)1/5 (E′4Qo

7µ′tH6N2

)1/5

NPI1

(E′9N16Qo

14t9

K′4Z25

)1/5 (E′7N14Qo

16t13

H8Z25µ′2

)1/5

NPfH2N2

Z2

NPI2

(E′6K′4N34Qo

21t21

Z50µ′5

)1/5E′2N6Qo

5t5

H2Z10µ′ NPfH4N4

Z4

NPln −NPf ln(

K′NRw5/2

E′Qot

)2/5

0 0

NPIL 0 0 NPfH2N2

Z2

Table 5.2: Power coefficients summed over N hydraulic fractures for Qi = Qo/N , where in allcases NPo = Qoσo.

18

Figure 6.1 shows the evolution of the radial and PKN geometry power coefficients from Table5.2. In both cases Pf À Pc, and as expected in these viscosity dominated cases, the coefficientsassociated with fluid dissipation – Pf , Pln, PI1, and PI2 – are the dominant quantities with thelatter two, interaction related coefficients becoming important as the hydraulic fractures growrelative to their spacing with increasing time.

A notional concept of stability has been introduced in Figure 6.1 wherein coefficients for whichd(NPα)/dN < 0 are denoted “stable”, coefficients for which d(NPα)/dN > 0 are denoted“unstable”, and coefficients for which d(NPα)/dN = 0 are denoted “neutral”. The underlyinghypothesis is that geometric stability of a uniform array of N hydraulic fractures is determinedby whether the required input power is increased or decreased by a decrease in the numberof growing hydraulic fractures. Conceptually, energy minimization principles imply that thesystem should be expected to choose a configuration that minimizes the required input power.Hence, coefficients are denoted “stable” if they are expected to promote the stability of theuniform array of N hydraulic fractures, “unstable” if they are expected to promote a decreasein N , and “neutral” otherwise.

Figure 6.1 also presents a so-called “valid” range for each of the relevant scaling schemes basedon Appendix B. In these cases, the valid range is defined by Rw ¿ R ¿ h.

To further illustrate the evolution of the input power, Figure 6.2 shows this quantity for the 2Dand radial geometries as estimated by Eq. (6.1), using the power coefficients from Table 5.2 andletting N vary from 1 to 41 by increments of 4. For the viscosity dominated cases (Pf À Pc)the parameter values are taken from Eq. (6.2). For the toughness dominated cases, Eq. (6.2) ismodified by increasing KIc and decreasing µ so that

E ′ = 10 GPa, KIc = 5 MPa m1/2,

Qo = 0.1 m3/s, µ = 0.001 Pas, σo = 70 MPa,

Rw = 0.2 m, Z = 1000 m. (6.3)

In each of the four cases illustrated by Figure 6.2, the input power decreases with time until thestress interaction among the hydraulic fractures becomes appreciable (for N > 1). A variationon this theme arises in the Radial/Viscosity case, which exhibits an increase in required inputpower at very early time before it follows the decrease and then increase trends observed in theother cases. This behavior comes from the dissipation associated with the strong fluid pressuregradient near the inlet in this case, which is embodied in Pln.

The most striking thing about Figure 6.2 is that 3 of the 4 cases show that the input powerdecreases with decreasing N at all times. But 1 of the 4 cases, the Radial/Viscosity case, isdifferent in that the minimum input power is associated with the largest value of N up to thepoint where the stress interaction among the hydraulic fractures becomes substantial. Using thepreviously introduced notional concept of stability, the implication is that of these four cases,a system that tends to minimize the input power should only be expected to sustain a uniformarray of N hydraulic fractures in the radial geometry, in the viscosity dominated regime, andprior to the onset of appreciable stress interaction among neighboring hydraulic fractures.

We see that as the hydraulic fractures grow, the compressive stress each one induces on itsneighbors increases in magnitude and as a result the input power required for propagation ofthe array of hydraulic fractures to continue increases sharply after they reach a certain length.In order to better see the ratio of length to spacing at which the input power begins its sharpincrease, let R ≈ γL(t), where L(t) is given by Table 5.1 and γ is given by Table 6.1. In this

19

10−2

100

102

104

102

103

104

105

106

107

108

t (s)

NP

(Watt

s)

PI

Pc

Pf

PI1 PI2

Pln

Valid Range

Stable Range

102

103

104

105

103

104

105

106

107

t (s)

NP

(Watt

s)

PI

Pc

Pf

PI1 PI2

Valid Range

Stable Range

Figure 6.1: Evolution of the summed power factors NPα for the radial, viscosity dominatedcase (top) and the PKN case (bottom), where blue lines indicate “stable” terms, red lines indicate“unstable terms”, solid black lines indicate d(NPα)/dN = 0 (“neutral”), and vertical dashedlines indicate the validity zone summarized in Appendix B.

20

10−4

10−2

1000

2

4

6

8

10

12

14x 106

t (min)

Input

Pow

er(W

att

s)

N = 1

N = 41

10−1

100

101

1020

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 106

t (min)

Input

Pow

er(W

att

s)

N = 1

N = 41

10−4

10−2

1000

2

4

6

8

10

12

14x 106

t (min)

Input

Pow

er(W

att

s)

N = 1

N = 41

10−4

10−2

100

1020

2

4

6

8

10

12

14x 106

t (min)

Input

Pow

er(W

att

s)

N = 1

N = 41

a

c

b

d

2D/Tough

Rad/Tough

2D/Visc

Rad/Visc

Figure 6.2: Input power as a function of time for 2D and radial geometries under toughness andviscosity dominated conditions.

21

way, the input power can be expressed in terms of R/h as illustrated by Figure 6.3 for the casescorresponding to Figure 6.2.

Figure 6.3 firstly shows that the 2D cases and the Radial/Toughness case have in common thatthe strong upturning in the input power occurs between R/h ≈ 0.1 and R/h ≈ 0.4. Thetransition from decline to increase is somewhat more abrupt in the radial case, but otherwise thebehavior is qualitatively the same among these three cases.

Once again the Radial/Viscosity case is different from the other 3, which is due to the uniquebehavior of Pln (Table 5.2). The abrupt upturning of the input power occurs for R/h ≈ 0.8 inthis case, indicating that the hydraulic fractures can grow to be longer relative to the spacingbefore the interaction terms become dominant than in the other 3 cases. Also, at first glanceit is disconcerting that over the range of time wherein the input power is decreasing, the inputpower actually decreases with decreasing N , which is in contrast to what was observed wheninput power was graphed as a function of time in Figure 6.2d. But this is simply a consequenceof a higher growth rate of R for smaller values of N owing to the fact that the same overallvolumetric rate of injected fluid is partitioned to fewer hydraulic fractures.

Geometry Regime γ Reference2D Viscosity 0.62 Adachi (2001)2D Toughness 0.93 Adachi (2001)Radial Viscosity 0.70 Savitski and Detournay (2002)Radial Toughness 0.85 Savitski and Detournay (2002)PKN Viscosity 0.68 Nordgren (1972)

Table 6.1: Length coefficients for each regime.

6.2 PKN Case

Finally, we turn the attention to the PKN cases. We can consider this system to evolve fromthe small PKN case with H ¿ R ¿ h to the linguine like PKN case H ¿ h ¿ R as timeincreases. Using the parameter values from Eq. (6.2) along with Eq. (6.1) and the power coef-ficients from Table 5.2, the evolution of the required input power for the PKN case is illustratedby Figure 6.4. Once again, the evolution is also presented with respect to R/h by making useof R ≈ γL(t), with L(t) obtained from Table 5.1 and γ obtained from Table 6.1. Note, then,that for each value of N that is presented, there is an early time asymptote for H ¿ R ¿ hand a large time asymptote for H ¿ h ¿ R. Naturally, the full solution is expected to evolveto smoothly connect these asymptotics. However, in the absence of a full solution, examinationof the asymptotes themselves is sufficient to obtain the desired insight.

The most striking feature of Figure 6.4 is that it shows that the minimum required input powercorresponds to neither N = 1 nor N = Nmax (in this case Nmax = 41), as for the 2D and radialgeometries. Rather, there exists an intermediate value of N that minimizes the input power.

To be more precise, for H ¿ R ¿ h the minimum input power indeed corresponds to N =Nmax as for the other geometries. However, after the transition to the linguine-like geometry(H ¿ h ¿ R), the minimum input power is obtained for N = 21. This is an intriguing resultbecause, if we recall the notion of stability that was previously introduced, it implies that theremay exist an optimal number of hydraulic fractures from the perspective of minimizing theinput power, and perhaps the system would naturally favor this configuration. In other words,

22

10−4

10−3

10−2

10−1

100

2

4

6

8

10

12

14x 106

R/h

Input

Pow

er(W

att

s)

N = 1

N = 41

10−2

10−1

100

0.5

1

1.5

2

2.5

3x 106

R/h

Input

Pow

er(W

att

s)

N = 1

N = 41

10−4

10−3

10−2

10−1

100

2

4

6

8

10

12

14x 106

R/h

Input

Pow

er(W

att

s)

N = 1

N = 41

10−4

10−3

10−2

10−1

100

2

4

6

8

10

12

14x 106

R/h

Input

Pow

er(W

att

s)

N = 1

N = 41

a

c

b

d

2D/Tough

Rad/Tough

2D/Visc

Rad/Visc

Figure 6.3: Input power versus the length to spacing ratio for 2D and radial geometries undertoughness and viscosity dominated conditions.

23

there could be a predictable, naturally arising spacing between simultaneously growing, heightconstrained hydraulic fractures.

This predicted spacing is readily derived by first returning to the expression for the input powergiven by Eq. (6.1). Recognizing that Pc ¿ Pf , the input power for the linguine-like PKN caseis approximated as

NQi (pfo − σo) ≈ NPf

M∑n=0

an

(HN

Z

)2n

. (6.4)

The values of the coefficients an come from a full solution to the problem, which is beyondthe present scope. Knowing, however, that the scaling requires that these are O(1), a coarseestimate of the minimum input power can be obtained using an = 1 for all n. In this case, it iseasily shown that keeping two terms of Eq. (6.4) (M = 1) gives a minimum at N = 0.5Z/H ,which keeping three terms (M = 2) leads to N ' 0.42Z/H . So as a starting point, it appearsthat there exists an optimum spacing between the hydraulic fractures of h ≈ 2.5H .

The existence of an optimum spacing in this case indicates a competition between mechanicalprocesses. Namely, it shows that, in the PKN case, viscous dissipation is smaller when thereare more hydraulic fractures that divide the total influx of fluid into smaller portions. But onthe other hand, if the interaction between the hydraulic fractures is too strong because they aretoo close together, then the crack opening is decreased to the point that adding more hydraulicfractures at a closer spacing becomes counterproductive due to the dependence of viscous dis-sipation on the width of the fracture channel. The result is a minimum in the input power at apoint determined by this competition between physical processes.

7 ConclusionsWhen predicting the growth of multiple hydraulic fractures there are several tempting assump-tions which include: 1) That multiple hydraulic fractures will grow simultaneously up to thepoint where the stress interactions lead to localization, 2) That fluid flow can be neglected, i.e.fracture mechanics analysis that is not coupled with fluid flow can provide useful predictionsand insight for fluid-driven cracks, and 3) That the chosen model geometry does not make verymuch difference, i.e. a 2D model can be used to obtain results that are qualitatively usefulfor understanding radial or height constrained hydraulic fractures. The analysis presented hereraises serious doubt about whether any of these assumptions are valid.

In fact, the analysis of the required input power to sustain propagation shows that both geometryand fluid flow profoundly matter when it comes to understanding how the input power dependson the number of hydraulic fractures in the array and how it evolves through the transition towhere stress interaction with neighboring hydraulic fractures is important. For the 2D geometryand for the toughness dominated radial case the input power decreases with decreasing numberof growing hydraulic fractures, indicating that an energy minimizing system is not expected tobe favorable to growth of a uniform array of hydraulic fractures. And this conclusion holds atall times, even before the onset of interaction among the hydraulic fractures, so that it should notbe assumed that hydraulic fractures will grow uniformly even when they are spaced infinitelyfar apart. In contrast, the viscosity dominated radial case has a period of time, prior to the onsetof interaction among the hydraulic fractures, wherein the minimum input power corresponds tothe maximum number of hydraulic fractures in the array.

24

100

101

102

1030

1

2

3

4

5

6

7x 105

t (min)

Input

Pow

er(W

att

s)

N = 1

N = 41

N = 21

H R h

H h R

10−1

100

101

1020

1

2

3

4

5

6

7

8

9

10x 105

R/h

Input

Pow

er(W

att

s)

N = 1

N = 41

N = 21

H R h

H h R

Figure 6.4: Input power as a function of time and R/h for the PKN geometry.

25

Perhaps the most intriguing case, though, is the height constrained, PKN geometry under viscos-ity dominated conditions. In this case the minimum input power corresponds to the maximumnumber of hydraulic fractures when they are far apart relative to both their height and length.However, as time goes on, these transition to a “linguine-like” geometry in which they are farapart relative to their height, but with the length greatly exceeding both of these dimensions.In this linguine-like case, the minimum input power corresponds to an intermediate number ofhydraulic fractures that corresponds to a spacing that is approximately 2.5 times the fractureheight. Hence, this analysis gives a first indication that a natural, energetically optimal spacingcan be expected to emerge when multiple height constrained hydraulic fractures are simultane-ously created.

AcknowledgementsThank you also to Rob Jeffrey and Emmanuel Detournay for constructive comments on earlierversions of this manuscript. Funding from the CSIRO Petroleum and Geothermal Portfolio isgratefully acknowledged.

A Far Field Approximation of the Interaction Stress

A.1 Far Field Equivalence to Uniformly Pressurized Crack

Lecampion et al. (2005) have shown that the displacement field due to a crack of arbitrary shapeand loading can be considered as a point displacement discontinuity in the far field approxima-tion. In other words, only the orientation and volume impact the remote displacement field thata crack induces. Using a multipole moment decomposition, Lecampion and Peirce (2007) haveshown that the displacement field is not sensitive to the length of the crack, to an error toleranceof 5%, for h/R & 1.25. Additionally, the approximation using the second order moment, whichaccounts for the length scale of the crack, is shown to approximate the far-field displacementfield to an error tolerance of 5% for h/R & 0.6.

For the present analysis the quantity of interest is not the displacement field, but rather thenormal traction imposed by a crack on a plane located at z = ±h, where z = 0 is the crackplane. The principle, however, is the same. In this case the leading edge of the crack is takento conform to an idealized, known shape. As such, a convenient approach that is expectedto be similar to the second or fourth order approximations from the moment decompositionof Lecampion and Peirce (2007) is provided by considering the far field stresses induced by acrack of length R under an arbitrary internal loading that results in a crack volume Qit = V tobe sufficiently approximated by a uniformly pressurized crack of the same length and with thepressure loading adjusted to match the volume.

Fig. A.1 shows numerical results demonstrating that this approximation is valid. Here the inter-nal pressure loading of the actual hydraulic fracture is taken from the zero toughness solutionfor the radial crack solved by Savitski and Detournay (2002). Shown along with this internalpressure distribution is the actual computed normal compressive traction acting on the planesζ = h/R for h/R = 0.4, 0.6, 0.8, and 1.0. This computed quantity is based on the displace-ment discontinuity method, which amounts to a convolution of circular dislocation dipole so-lutions (Hills et al., 1996; Zhang et al., 2002; Gordeliy and Detournay, 2011). Compared with

26

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ρ

Nor

mal

ized

Str

ess

h

R= 0.6

σI

E′ε(actual)

σI

E′ε(approx)

p

E′ε

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ρ

Nor

mal

ized

Str

ess

h

R= 0.4

σI

E′ε(actual)

σI

E′ε(approx)

p

E′ε

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ρ

Nor

mal

ized

Str

ess

h

R= 1.0

σI

E′ε(actual)

σI

E′ε(approx)

p

E′ε

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

ρ

Nor

mal

ized

Str

ess

h

R= 0.8

σI

E′ε(actual)

σI

E′ε(approx)

p

E′ε

Figure A.1: Comparison of normal traction (compression positive) acting on the plane z = h/Rfor a crack loaded by the pressure distribution p, as shown, with an approximation based on auniformly pressurized crack with the same radius and volume.

this solution, which is “exact” up the the accuracy of the quadrature scheme and chosen numer-ical solver, is the solution for the normal traction on the same plane but assuming a circular,uniformly pressurized crack (Sneddon, 1946, Table 4). All of these tractions are normalizedaccording to the radial, viscosity dominated scaling defined by Table 5.1. This comparisonshows that approximation of the far field stress induced by a zero toughness hydraulic fracture,which is the case that is most unlike the uniformly pressurized case in terms of internal loadingdistribution, is accurate to a tolerance of ≈ 5% for h/R & 0.6− 0.8.

Not only does it suffice to consider a proxy of the crack that captures only the volume, orien-tation, and length, but also it is sufficient to consider only the interactions with the hydraulicfracture’s immediate neighbor on each side. The task, then, is to obtain an approximate analyt-ical expression by not only assuming a uniformly pressurized crack, but by additionally takingthe far field asymptotic expansion for each of the 3 geometries considered in this study. Theremainder of this Appendix presents these derivations.

A.2 2D Geometry

The normal traction σz (compression positive) induced on a plane z = ±h due to a crack locatedat z = 0, −R < x < R and subjected to an internal pressure po is given by (Sneddon, 1946)

− σz

po

= ReZ + ζImZ ′, (A.1)

27

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

σzpo

ρ

Actual

Series

ζ = 4

ζ = 5

ζ = 6

Figure A.2: Actual (Eq. A.1) and 2 term series approximation (Eq. A.4) for σz/po.

where Re and Im indicate the real and imaginary parts, respectively, Z is the Westergaard stressfunction

Z =z√

z2 − 1− 1, (A.2)

and the ′ denotes the derivative with respect to the complex coordinate

z = ρ + iζ, (A.3)

with i =√−1 and recalling that ρ = x/R and ζ = h/R. Taking the Taylor Series of Eq. (A.2)

for ζ À 1 and substituting into Eq. (A.1) leads to

σz

po

=3

2ζ2− 15

(ρ2 + 1

4

)

2ζ4+ O

(ζ−6

). (A.4)

The internal pressure po can be readily obtained by equating the solution for the volume of apressurized crack (e.g. Tada et al., 2000) to Qit, hence

po =1

2π

E ′Qot

R2. (A.5)

The convergence of the series given by Eq. (A.4) is illustrated in Fig. A.2. If an error toleranceof 5% is adopted over ρ < 1 then the two term expansion gives a suitable approximation tothe actual stresses for ζ & 5. For the purpose of investigating the onset of interaction, wherethe most important thing is that the scaling of the leading order term is established, this levelof accuracy is sufficient, leaving it to future investigations to look in greater detail how thehydraulic fractures interact for ζ . 5.

It is useful to note that an alternative approach uses a single, constant displacement discon-tinuity, with the expression for the stresses given by Crouch and Starfield (1983, Eqs. 5.2.4and 5.2.5). Taking the far field expansion of the stresses obtained in this way results in an ap-proximation for σz is nearly the same as Eq. (A.4), differing by −5/8ζ−4 + O(ζ−6), and thedifference between the first derivatives in ρ is O(ζ−6).

28

A.3 Radial Geometry

Similar to the 2D case, the far field stresses for the Radial case are taken from a series expansionof the solution for a pressurized crack. The solution for σz is provided by Sneddon (1946, Eq.(3.6.3)). The far field series expansion is given by

σz

po

=16

3πζ3− 24 (5ρ2 + 2)

5πζ5+ O

(ζ−7

). (A.6)

As in the 2D case, the pressure po is obtained by equating the volume of a pressurized crack(e.g. Tada et al., 2000) to Qit, which for the radial case gives

po =3

16

E ′Qot

R3. (A.7)

The convergence of the series is similar to the 2D case presented in Fig. A.2, so that it shouldbe considered to give a suitable approximation, to ∼ 5% error, for ζ & 5.

A.4 PKN Geometry, h À R À H

The far field approximation of the interaction stress for the PKN case will be based on a rectan-gular displacement discontinuity with a uniform opening uz. Following Rongveld (1957),

σz =E ′(1− ν)2

1− 2ν

∂uz

∂z, (A.8)

where the displacement ui is given in terms of the harmonic Papkovitch functions β and Bx,By, and Bz according to

ui = Bi − 1

4(1− ν)

∂

∂xi

(xiBi + β) . (A.9)

According to the solution of Rongveld (1957), Bx = By = 0 and, for constant uz,

Bz =uz

2π

∂

∂ζIs(ρ, ψ, ζ), (A.10)

β =Ruz(1− 2ν)

2πIs(ρ, ψ, ζ), (A.11)

where ψ = y/R and Is is the surface integral

Is =

∫ 1

−1

∫ H2R

− H2R

((ρ− ξ)2 + (ψ − η)2 + ζ2

)−1/2dηdξ (A.12)

Taking the Taylor expansion of the integrand and carrying out the resulting series of integralsleads to

Is =H

R

(2

ζ−

13− ψ2 − ρ2

ζ3− H2

12R2ζ3+ O(ζ−5)

), (A.13)

recalling that H ¿ R for the PKN geometry. Substituting Eq. (A.13) into Eq. (A.9), thenputting the result into Eq. (A.8) leads to the far field series expansion

σz =E ′Huzν

′

πR2

[1

ζ3− ν ′′

ζ5

(3ρ2 + 3ψ2 + 1 +

H2

4R2

)+ O(ζ−7)

], (A.14)

where

ν ′ =(2− ν)(1− ν)

1− 2ν, ν ′′ =

3− ν

2− ν,

29

and by volume balance

uz =Qot

2HR. (A.15)

Note that if one accounts for the elliptical distribution of the displacement discontinuity in they direction (Eq. 3.6), the resulting series is identical to O(ζ−5(H/R)2) and the first derivativein ρ is identical up to O(ζ−7(H/R)), which serves to illustrate the point that simple modelssuffice when only the far field stresses are desired. It is also easily confirmed that this seriesexpansion is equivalent to a multipole expansion, as described by Lecampion and Peirce (2007),that accounts for the first two non-zero multipoles (orders 0 and 2) and retains the first two termsfor ζ À 1 (O(ζ−3) and O(ζ−5)).

As a final point, the appearance of ψ2 in the series expansion for σz demonstrates that theassumption of elliptical cross sections in the y direction is going to be compromised as thehydraulic fractures interact. So-called “pseudo-3D” simulators that allow multiple PKN-likehydraulic fractures to grow in close proximity to one another must account for this deviationor else the fluid continuity equation will be integrated incorrectly in the y direction, leading toerrors. This issue was previously identified and pragmatically resolved by Germanovich et al.(1997).

A.5 PKN Geometry, R À h À H

In this case, each cross section of the array is subject to a state of plane strain so that the farfield expansion of the induced stress follows directly from Eq. (A.4). Letting η = 2y/H andζ = 2h/H , the ζ ¿ 1 expansion of the interaction stress is given by

σz(x)

p(x)=

3

2ζ2− 15

(η2 + 1

4

)

2ζ4+ O

(ζ−6

). (A.16)

B Validity Conditions for Scaling RelationshipsRecall that the magnitude of each term in the energy balance is estimated by its power coefficientPα provided that the conditions are appropriate to the particular scaling. So for example, thevalidity of the viscosity dominated regimes and the PKN model rely on Pf À Pc. Hence, forthe 2D geometry the viscosity dominated regime translates to

Pf

Pc

∣∣∣∣n=k=0

=

(E ′3Qoµ

′

K ′4N

)1/2

= M1/20 ¿ 1, (B.1)

where M0 is a dimensionless viscosity (Adachi, 2001; Detournay, 2004) that has been utilizedin Table 5.1.

While for the 2D geometry the viscosity dominated regime is identified in terms of a constantM0, in the radial geometry the criterion Pf À Pc translates to a range of time via the relation-ship

Pf

Pc

∣∣∣∣n=1,k=0

=

(E ′13Qo

3µ′5

K ′18N3t2

)1/9

= M5/91 ¿ 1, (B.2)

whereM1 is a dimensionless viscosity for the radial geometry (Savitski and Detournay, 2002),

30

so that the viscosity dominated regime corresponds to

t ¿ tfc|n=1,k=0 , tfc|n=1,k=0 =

(E ′13Qo

3µ′5

K ′18N3

)1/2

.

Conversely, for the PKN case the criterion Pf À Pc provides a lower bound on the valid rangeof time, that is

t À tfc|n=0,k=1 , tfc|n=0,k=1 =H7/2K ′5N2

E ′4Qo2µ′

.

It is straightforward to proceed in a similar way to derive lower bounds on the valid periodof time based on the assumption Rw ¿ R, working from the timescale on which L increasesrelative to Rw. Also, for the PKN case a lower bound arises from the requirement that R À H ,which can be taken to replace Rw ¿ R for Rw ¿ H . Similarly, upper bounds on the validperiod of time can be estimated based on the assumption R/h ¿ 1.

Additionally, it has been shown in studies of 2D (Lecampion and Detournay, 2007) and radial(Bunger and Detournay, 2007) hydraulic fractures that the validity of the assumption of zerofluid lag additionally requires

t À E ′2µ′

σo3

.

A similar condition will be required for the PKN geometry, however, an examination of theliterature implies that this case has not been analyzed for finite fluid lag, where the difficultyof this analysis is recognized owing to the degeneracy of the the propagation condition for thePKN geometry.

Finally, the assumption that leak-off of fluid into the formation can be neglected provides upperbounds on the valid range of time for the 2D (Adachi, 2001), radial (Madyarova, 2003), andPKN (Kovalyshen and Detournay, 2010) cases, respectively. These are included in Table B.1,which summarizes the periods of validity for each of the 5 geometries and regimes under con-sideration. The bound based on satisfying the zero leak-off assumption is expressed in terms ofthe Carter leak-off coefficient CL = C ′/2 which can usually be taken in the range [10−5, 10−4]m/s1/2 (Constien, 1989; Adachi, 2001).

Table B.1 summarizes the range of time satisfying validity conditions for all of the cases con-sidered.

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31

2D/Visc. 2D/Tough. Radial/Visc.t À max

(geometry)(

N3Rw6µ′

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(H4Z5µ′

N2E′Qo3

) 14

(leak-off)(

K′8Qo2

C′10E′8

) 13

(Qo

4µ′2

C′10E′2H2

) 13

(Qo

4µ′2

C′10E′2H2

) 13

(regime)

Table B.1: Time ranges for validity of the model and scaling relations used for each of the cases.

32

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