An Analysis of the Classical Pseudo-3D Model for HydraulicFracture with Equilibrium Height Growth across Stress Barriers1Jos I. AdachiSchlumberger Data & Consulting Services, USAEmmanuel Detournay2University of Minnesota, USA and CSIRO Petroleum, AustraliaAnthony P. PeirceUniversity of British Columbia, CanadaJuly 13, 20091To be submitted to Int. J. of Rock Mechanics & Mining Sciences.2Correspondingauthor: DepartmentofCivilEngineering, UniversityofMinnesota, 500PillsburyDriveSE, Minneapolis, MN 55455, USA.E-mail: detou001@umn.eduAbstractThis paper deals with the so-called pseudo three-dimensional (P3D) model for a hydraulic fracturewith equilibrium height growth across two symmetric stress barriers.The key simplifying assumptionsbehind the P3D model, which was originally introduced by A. Settari and M. P. Cleary (Developmentand Testing of a Pseudo-Three-Dimensional Model of Hydraulic Fracture Geometry (P3DH), SPE10505, Proc. 6th SPE Symposium on Reservoir Simulation, pp. 185-214, 1982) and by I. D. PalmerandH. B. Carrol (Numerical SolutionforHeightandElongatedHydraulicFractures, SPE/DOE11627, Proc. SPE/DOESymposiumonLowPermeability, pp. 249-878, 1983), arethat(i)eachcross section perpendicular to the main propagation direction is in a condition of plane-strain, and(ii) the local fracture height is determined by a balance between the eect of the stress jump acrossthebarriersandthatoftherocktoughness. Furthermore, intheequilibriumheightgrowthP3Dmodels, thepressureisassumeduniformineachvertical cross-section. Werevisitthisparticularmodel by rst formulating the non-linear dierential equations governing the evolution of the length,height, andapertureofthehydraulicfracture, incontrasttothenumerical formulationsadoptedin many previous studies. Scaling of these equations shows that the solution depends,besides thedimensionless space and time coordinates, on only two numbers representing a scaled toughness anda scaled leak-o coecient. Analysis of the governing equations enables us to determine explicitly theconditions under which breakthrough takes place (i.e., the onset of growth into the bounding layers),aswellastheconditionsofunstableheightgrowth(i.e., theconditionsofrunawayheight whenthe main assumptions of the equilibrium height model become invalid). The mathematical model issolved numerically using a novel implicit fourth order collocation scheme on a moving mesh, whichmakes explicit use of the fracture tip asymptotics. A complete listing of the MATLAB code for thisalgorithm isprovided inan appendix andcan be copiedfor experimentation. We thenreporttheresults of several numerical simulations conducted for dierent values of the dimensionless toughnessand the dimensionless leak-o coecients, as well as a comparison with closed-form small and largetime similarity solutions that are valid under conditions where the fracture remains contained withinthe reservoir layer. Finally, we discuss possible extensions of the model, including layers of dierentelastic moduli and power law uids.21 IntroductionThis paper deals with the Pseudo-3D (P3D) model of a hydraulic fracture,a model widely usedin the petroleum industry to design stimulation treatments of underground hydrocarbon reservoirsbyhydraulicfracturing. TheP3Dmodel wasintroducedinthe1980s[1, 2, 3, 4, 5, 6, 7], asanextensionoftheclassical PKN(Perkins-Kern-Nordgren)model [8, 9] tosimulatethepropagationof a vertical hydraulic fracture into a horizontally layered reservoir. Like the PKN model, the P3Dmodel is applicable to situations in which the height of the fracture remains small compared to itslength. However, in contrast to the PKN model, the height of the P3D fracture is not limited to thereservoir thicknessH, see Fig. 1. Indeed, the fracture is allowed to grow vertically into the adjacentlayersconningthehydrocarbonbearingstrata, butataratethatismuchsmallerthantherateunderwhichthefractureextendslaterally, soastojustifythecritical assumptionoflocal elasticcompliance, which is at the heart of the PKN and P3D models.Solvingtheproblemofanevolvingplanarhydraulicfractureisachallengingtask, dueinpartto the moving boundary nature of this class of problems, the strong non-linearity introduced by thelubrication equation, the non-local relationship between the pressure in the fracture and its aperture,and the delay associated with leak-o of the fracturing uid. While numerical algorithms have beendeveloped to calculate the evolution of a planar hydraulic fracture in a layered geological medium[10, 11, 12, 13, 14, 15, 16, 17], therearesituationswhenthesolutioncanbesimpliedandcanthus be obtained with much less computational expense. In particular, the P3D model simplies theform of the boundary curveC(t) that contains the evolving fracture footprint, by considering onlythe horizontal extent of the fracture in the reservoir and the associated vertical penetration of thefracture into the adjacent layers.The main distinguishing feature of both PKN and P3D models is the local nature of the fracturecompliance, aconsequenceofassumingthattheheight/lengthratioissmall. Inotherwords, theaveragefractureapertureinaverticalcross-sectionofaPKNorP3Dmodeldependsonlyontheuidpressureinthatcross-section, incontrasttoplanarfracturemodels[10, 14, 17, 18] thatarecharacterized by non-local interaction between the pressure and aperture elds. While for the PKNfracture the compliance is constant,a function only of the thickness of the reservoir layers and its1elastic properties, the local compliance in the P3D model is itself part of the solution, as it dependsonthelocal heightof fracture, whichisapriori unknown. Thelocal complianceandheightofthe fracture depend also on further assumptions about the pressure eld in a vertical cross-section;thepressureisassumeduniformintheequilibriumheight P3Dmodel [3], butiscalculatedonaccountofavertical owinthedynamicheight P3Dmodel [5]. Nevertheless, allowingverticalfracture growth brings another non-linearity to the model - in addition to that which results fromthedependenceofthehydraulicconductivityonthefractureapertureinthelubricationequationgoverning the longitudinal viscous ow of uid in the fracture.As a result of these various assumptions, most of which have been inherited from the PKN model,the P3D hydraulic fracture is governed by a non-linear one-dimensional diusion type equation over adomain that is evolving in time. In the equilibrium height P3D model, the focus of this investigation,this equation can be written explicitly,as shown later in this paper. Despite its strong non-linearnature, this partial dierential equation can be solved at a small fraction of the computational costrequired to simulate arbitrary shaped planar hydraulic fractures. The P3D model is therefore wellsuited to designing hydraulic fracturing treatments,when multiple scenarios have to be evaluated,provided that the assumptions upon which the model is built are respected.Despite the importance of the P3D model for the design of hydraulic fracturing treatments, it doesnotappear, however, thatthismodelhasbeenrigorouslyformulatedandscrutinized, aspreviousstudies have generallyemphasized either the discretized form of the governing equations the so-calledcell-basedmethods [7], and/or theapplicationof themodel toparticular eldcases. Inparticular, the conditions under which equilibrium height growth takes place, a critical assumptionfor the validity of the equilibrium height P3D model, have not been determined.This paper adopts a dierent approach from previous publications, as it seeks not only to formulatethe mathematical problem rigorously, but as it also aims, through an emphasis on scaling analysis,to derive general results rather than specic ones applicable to a distinct set of parameters. For thesereasons,we consider the simplest case of a equilibrium height P3D hydraulic fracture propagatingin a reservoir layer bounded by two semi-innite layers with the same elastic properties as the payzone. Furthermore, we assume that the fracturing uid is Newtonian and incompressible and thatthe injection rate is constant. These assumptions have been adopted so as not to distract from the2main objective of the paper, which is to provide a rigorous formulation of the problem and scalingof theequations. Nonetheless, theseassumptionscanberelaxedwithoutaectingtheequationsthat determine the fracture height and the fracture compliance, a main focus of this paper, nor thefundamentals of the numerical algorithm used to solve the propagation problem.The paper is organized as follows.First we present the set of equations that govern the propagationof a P3D fracture, within the simplied context just mentioned. A scaling of the equations then showsthat the solution depends on dimensionless space and time coordinates and on two numbers, K andC, which can be interpreted a scaled toughness and a scaled leak-o coecient, respectively. Analysisof the governing equations enables us to determine explicitly the conditions at which breakthroughtakes place (i.e., the onset of growth into the bounding layers), as well as the conditions of unstableheightgrowth(i.e., theconditionsof runawayheight whenthemainassumptionsof themodelbecome invalid). The mathematical model is solved numerically using a novel implicit fourth ordercollocationschemeonamovingmesh, whichmakesexplicituseof thefracturetipasymptotics.Wethenreporttheresultsof several numerical simulationsconductedfordierentvaluesof thedimensionless toughness K and the dimensionless leak-o coecient C, as well as a comparison withclosed-form small and large time similarity solutions for conditions under which the fracture remainscontained within the reservoir layer. Finally, we discuss possible extensions of the model, includinglayers of dierent elastic moduli, multiple jumps in the conning stress, and power law uids.2 Mathematical Formulation2.1 Problem DenitionThe geometry of a pseudo-3D hydraulic fracture is sketched in Fig. 1. A reservoir layer of thicknessH is bounded by two semi-innite layers assumed to have the same elastic properties as the reservoir.The minimum horizontal far-eld stress in the reservoir isoand there is a (positive) stress jumpbetween the reservoir and the bounding layers. Injection of an incompressible Newtonian uidof dynamicviscosityataconstantvolumetricrateQoinaboreholecausesthepropagationofasymmetrichydraulicfracture, alongthereservoirlayerandwithlimitedheight-growthintothe3bounding layers. The main assumptions behind this model are that (i) each cross-section perpendic-ular to the main propagation direction is in plane strain on account of hmax/ being small (wherehmaxisthemaximumfractureheightandisthefracturehalf-length, i.e., thelengthofawing)and that the fracture height varies slowly with distance from the borehole, (ii) the uid pressureis uniform in each vertical cross-section, and (iii) the local fracture heighth His an equilibriumheight controlled, in general, by the stress jump across the layers and by the rock toughness.Before stating the various assumptions that lead to the formulation of the P3D model, we introducea Cartesian coordinate system with its origin located on the wellbore at the mid-height of the reservoirlayer, with the x-axis contained in the plane of the fracture and the z-axis pointing upwards along thewellbore, see Fig. 1. With this choice of coordinates, both thex- andz-axes are axes of symmetry.2.2 P3D Model AssumptionsThe P3D model simplies the solution of the evolution of the planar fracture in the layered elasticsystem through a series of assumptions that are discussed below.Assumption 1. The leading edge of the fracture, which denes the maximum lateral extent of thefracture, is vertical and restricted to the reservoir layer, see Fig. 1; its position is dened by x = (t).The fracture height, h(x, t) is bounded below by H (and equal to H at the leading edge), and in viewof the problem symmetry, the vertical penetration,d, of the hydraulic fracture (if it takes place) ineither bounding layer is thus equal tod = (h H)/2. The fracture boundary is thus dened by thetwo functions(t) andh(x, t) H, withh(, t) =H; i.e., the footprint of one wing corresponds to0 x (t), and h(x, t)/2 z h(x, t)/2.Assumption 2. The vertical component qz of the uid ux is negligible compared to the horizontalcomponentqx;i.e. |qz/qx|1. For this reason,we simplify the notation by writingq=qx. ThisassumptionimpliesthroughPoiseuilleslaw,whichgovernstheowoftheviscousfracturinguidin the crack, that the pressure eldpfdoes not depend on thezcoordinate; i.e., pf(x, t). Clearly,this assumption requires that the rate of height growthh/t is much smaller than the horizontal4velocityd/dtoftheleadingedge; hence, thecoherenceofthisassumptionneedstobeveriedaposteriori, once the solution has been obtained.Assumption3. Aconditionof planestrainexistsinanyvertical plane(y, z). Providedthath/ 1, the plane strain assumption is a good approximation for most of the fracture, except withinaregionneartheleadingedgeoforderO(H)[19]. Thisassumptionimpliesthatthedependenceof boththeapertureeldwandthefractureheight hontheindependentvariables xandt isvia the pressure eldpf(x, t),i.e., w(x, z, t)=wp(z, pf) andh(x, t)=hp(pf). This assumptionisalternatively referred to as the local elasticity assumption. The kinematic conditionw=0 at theleading edgex =(t), in conjunction with the local assumption, implies thatpf=oatx =(t),whichisobviouslycoherentwiththepreviouslystatedassumptionthat h(, t)=H. Asiswell-known, adopting the local assumption in the region near the leading edge precludes the considerationofanypropagationcriterion; thisassumptionhastoberelaxedtoproperlyaccountforafracturepropagation criterion at the leading edge [19].Assumption4. Fracturinguidleak-ointothereservoir layeristreatedinaccordancewithCarters model [20], which assumes a one-dimensional diusion process. The usual hypotheses behindthis model are that (i) the fracturing uid deposits a thin layer of relatively low permeability, knownas the lter cake, on the fracture walls, at a rate proportional to the leak-o rate and (ii) the ltratehas enough viscosity to fully displace the uid already present in the rock pores.2.3 Mathematical ModelBefore proceeding with the formulation of the mathematical model, we recognize that the above setof assumptions enables one to rigorously formulate the model in terms of eld quantities that dependonly on the variablesx andt. Indeed, the dependence of the aperture eldw and the ux eldq onthe variablesx andt only through the pressure eldpf(x, t) (or its gradient) actually implies thatthemodel canbeformulatedintermsofanaverageaperture w(x, t)andanaverageux q(x, t),respectively dened as w =1H

h/2h/2wdz, q =1H

h/2h/2qdz (1)5We note that both averages have been dened with respect to the reservoir layer height H; hence wHis the cross-sectional area of the fracture and qHis the total ux through the fracture cross-sectionlocated atx.Elastic relationshipbetweenlocal aperture andnet pressure. Oneof thegoverningequations of the P3D model is a non-linear local relationship between the average aperture wandthe uid pressurepf, which is derived from elasticity. The non-linearity stems from the dependenceof the fracture height on the pressure, since the fracture growth in the vertical direction is subject tothe requirement that the fracture is in a state of critical equilibrium, assuming that such equilibriumstate exists. The relationship between w andpfcan be derived by combining the integral equationthat expresses the fracture aperture w(z) in terms of the net crack loading [21], the integral expressionfor the stress intensity factorKIin terms of the net crack loading [22, 23, 24], and the propagationcriterionKI = KIc [25, 26], i.e.,w =4E

p

H/20G(s, z) ds + (p )

h/2H/2G(s, z) ds. (2)KI =

8hp

H/20dsh24s2 + (p )

h/2H/2dsh24s2(3)KI = KIc(4)where E

= E/(12) is the plane strain Youngs modulus, p is the net pressure dened as p = pfoandG(s, z) is the elasticity kernel given by [21]G(s, z) = ln

h24z2+h24s2h24z2h24s2

. (5)Since the kernelG can be integrated in closed form according to

G(s, z) ds =

h24z2arcsin

2sh

z ln

sh24z2+zh24s2sh24z2zh24s2

+s ln

h24z2+h24s2h24z2h24s2

. (6)6the fracture prolew(z) can be calculated explicitly for a given fracture heighthw =2E

(p )

h24z2+ 4E

h24z2arcsin

Hh

z ln

Hh24z2+ 2zh2H2Hh24z22zh2H2

+H2ln

h24z2+h2H2h24z2h2H2

(7)Furthermore,a closed-form expression relating the equilibrium heighth and the net pressure isreadily deduced from (3) and (4) [27]2

hH arcsin

Hh

1 p

hH=

2HKIc(8)From (7) and (8), the relationship between w and p can then be obtained in the form of a parametricequation in terms ofh. Formally, this equation reads w = w(h), p = p(h) (9)The details of the derivation are given after scaling of the equations. We note, however, that in theparticular case of the PKN modelh = Hand the fracture opening adopts an elliptical shape; hence(9) degenerates into the linear relation w = Mop, with the complianceMo = H/2E

.Poiseuilleslaw. Next, we consider the relationship between the average ux q and the pressuregradient. According to Poiseuilles law, the local uxq(z, t) is related to the pressure gradient byq = w312px, h2< z 0. (21)The decrease of with for >u can readily be explained as follows. As becomes relativelylarge (and K>0),the fracture asymptotically approaches the shape of a Griths crack, i.e.,afracture subjected to a uniform net pressure. In other words, the contribution of the loading in thereservoir layer to the stress intensity factor becomes less relevant for increasing. For a Grithscrackoflengthhincriticalequilibriumwithauniformnetpressurep

=p , thetoughness,length and pressure are related according to =K+ 1 (using scaling (17)), which is exactly thedominant term in (19) for large , since arcsin

1

1 +O(3) for 1. Thus it is expected thatthe pressure should eventually decay as 1/.The main implication of the above result is that unstable height growth is expected to take placeonce >u. We will assume that >u is indeed the criterion of instability, on noting that the netpressure increases with time according to numerical simulations. A formal argument, however, wouldrequire us to prove that/> 0 in the vertical cross-section where the height becomes unstable.The equilibrium height P3D model is obviously not valid once this critical height is exceeded, as oneof the assumptions, namely that the rate of ow (or fracture propagation) in the vertical directionismuchlessthantherateofowinthehorizontal direction, isthenviolated. Thepossibilityofunstableheightgrowthismentionedin[27], butnocriterionforcalculatingthecritical heightisgiven in that paper.3.3 Fracture ComplianceAfter eliminating the pressure using the equilibrium height equation (19), the scaled opening prole() can be deduced from (7) =82K2

24221(; ) + 2(; )

(22)10where the functions1 and2 are dened1by1(; ) =

arctanh

221242

, 0 || < 1/2arctanh

242221

, 1/2 < || /2(23a)2(; ) =

arctanh

21242

, 0 || < 1/2arctanh

24221

, 1/2 < || /2(23b)A large asymptotic expression for can readily be derived from (22) as 42

K 2

242+ ln

242+

242

, 1 < This asymptotic expression for the fracture aperture can alternatively be obtained by superposingthe aperture corresponding to a uniform equilibrium pressure in the fracture and the solution for aunit force dipole that accounts for the reduced conning stress across the reservoir layer. Plots of() for dierent values of and K are shown in Fig. 2.The average opening, dened as =2

/20()d,can then be calculated by integrating (22), which yields =1

K3/2+ 2

21

, > 1 (24)It is convenient to introduce the functionc(; K) dened asc =(25)The functionc(; K) is actually a compliance correction functionas the fracture complianceM= w/p can be expressed asM= c(; K) Mo.1Noticethatwhereasthefunctions1and2arediscontinuous, withsingularitiesat = 12, thecombination21 + 2is analytic for || /2, except at= 1/2. However, the limit at these points exists, and is given byln.11Thusc(; K) adjusts the elastic compliance at any given vertical section of the fracture to accountfor the height growth. Using (24) and (19), the latter to express as a function of and K,c(; K)is given explicitly byc(; K) =

K3/2+ 221

K +

2 arcsin

1. (26)The functionc(; K) is a regular, strictly positive function, withc(1; K) = 1 and is unbounded for . Plots of this function for dierent values of dimensionless toughness K are shown in Fig. 3.For K = 0, the compliance correctionc can also be directly expressed as a function of , i.e.,c =2tan

2

, K = 0.Equivalently, combination of (20) and (24) allows us to derive the following direct relation between and for K = 02 = tan

2

, K = 0. (27)3.4 Viscous FlowUsing the scaling (16), Poiseuilles law (11) becomes = g(; K) 3(28)whereg(; K) =212 3

/2/23d (29)The functiong(; K) is a regular,strictly positive function,which asymptotically tends to zero as , and lim1+ g(; K) = 1; however, it cannot be expressed in a closed form. (Details aboutthe asymptotics of the functiong(; K) can be found in Appendix B.)It is advantageous to express (28) as = 1(; K)3 . (30)by dening the shape function as(; K) = g(31)12Using (19) and (24), the function can be written as(; K) =4 K212

4 + 3 K21 g(; K). (32)As expected, lim1+ (; K) = 1. Notice that for K = 0, the above expression reduces to(; 0) = 2g(; 0). (33)Thefunction(; K)eventuallybecomesnegativeasincreases,exceptfor K=0whenittendsmonotonically to zero. It can readily be shown that the function becomes negative when u;indeedboth/andgarestrictlypositivefor>1andthusthechangeofsigninsimplyreects the change of sign in/,associated with the maximum value of achieved atu. Anegative value for actually implies a reversal of the direction of ow (i.e., the uid would ow fromthe tip to the inlet - which is unphysical for a growing fracture) in that part of the fracture for which>u,unlessthegradientofopening/alsoreversessign,atleastlocally. Thisunphysicalsituation for >u clearly demonstrates the limitation of the equilibrium height P3D model.Finally the local continuity equation can be written as1+ +C

(1 )= 0 (34)where(, ) = to/t is the normalized arrival time. Note that the presence of the advective term isdue to the stretching nature of the-coordinate.3.5 Scaled Mathematical ModelAs a result of the above considerations, it is now possible to formulate the P3D model as a non-linearconvection diusion equation for the mean aperture eld(, ) and an evolution equation for thefracture length().The governing equation for(, ) is obtained by combining Poiseuilles law (30) and the balancelaw (34) = +12

(; )3

C

(1 )(35)13where the function (; K) is dened as =

(), K

(36)with()denotingtheinversefunctionof (24), therelationshipbetweenandtheequilibriumheight. Plots ofthefunction(; K)fordierent valuesof thedimensionlesstoughness Kareshown in Fig. 4. The function (; K) is not known in closed form; however, for large, 4if K = 0 and 8/3if K > 0, and =1 , if 0 b(37)wherebis the critical value of the aperture at which height growth starts. It is readily deducedfrom the equilibrium condition (24) between and, which is only valid for > 1 thatb = K/. (38)Next, an equation for () is obtained by integrating (12) in space over the length of the fractureand then in time from the initial time to the current time to obtain=

10(, )d + 2C

10

0()d (39)The boundary conditions at the inlet ( = 0) and at the leading edge ( = 1) are given by(0, ) = 1,(1, ) =(1, ) = 0, > 0 (40)Naturally, the boundary conditions in the terms of the ux can be transformed to be expressed interms of only, using (24) and (30).The initial condition for(, ) is taken to correspond to a small time asymptotic solution whenthe fracture remains contained in the reservoir layer (i.e. b), and when leak-o is negligible.Thissimilaritysolution, referredtoastheM-solutionbelow, isgovernedby(35)with=1 andC = 0.The system of equations (35),(39),(40), together with the small time asymptotic solution, consti-tute a closed system to solve for the aperture eld(, ) and the fracture length().143.6 Tip AsymptoticsWe now examine the nature of the solution near the leading edge. In the neighbourhood of the tip,the leak-o term C/

(1 ) can be approximated asC

(1 )C

(1 ), 1 1 (41)Using (41), the balance equation (34) is integrated in a small region of size1 near the leadingedge

111+ + C

(1 )

1/2d = 0 (42)which, after taking into account the boundary conditions = =0 at = 1, implies that1 +2 C

(1 ) (43)Combining (43) with Poiseuilles law (30) then yields3+ +2 C

(1 )10 (44)where we have taken into account that 1 as 1.In the two limiting cases of storage- and leak-o-dominated asymptotics, the behaviour of nearthe leading edge can be deduced from (44), by adopting an asymptotic solution for of the form1A()(1 )(45)In the storage-dominated case, the rst two terms in (44) have to balance; hence =13, A =3

3 (46)In the leak-o-dominated case, however, the rst and third term in (44) have to balance; hence thepower law index and the strengthA are given by =38, A =231/4

3C2

1/8(47)The tip asymptotic behaviour in between these two limiting cases can also be obtained, see [29] formoredetails. However, apragmaticapproachistoadopt(46)when 1 for 0 1 for 0 < 1 andp(C, K) u(C, K). The fractureopening is larger thanb(K) throughout 0 l() but less thanu(K) at which valuethe height growth becomes unstable.4. Unstableheightgrowth: (1, ) >u(K)for u(C, K). Thesolutionlosesitsphysicalmeaning once the height growth becomes unstable, which corresponds to the condition =u,given byu =1K2

16 +

K4+ 64

8 +

K4+ 64 (51)Since is a monotonically decreasing function, i.e., /< 0 of(otherwise the uid owwould be reversed), the runaway condition will rst be reached at the inlet. Hence the timeuat which the instability takes place corresponds to(0, u) =u. The onset of unstable heightgrowth marks the end of applicability of the equilibrium height P3D model, but the beginningof relevance of the dynamic height model.Unlikethecalculationsof pandu, whichrequiresolvingthefull setofequationsgoverningtheevolutionoftheP3Dfracture, thebreakthroughtimeb(C, K)canbereadilycalculatedfromtheknowncontained(PKN)fracturesolution. Theevolutionofthefractureapertureatthewellbore0() =(0, ) can be deduced from the PKN solution to be of the form [29]0(; C, K) =

2C

2/30

C10/324/3

(52)18where the function0() is the inlet opening in the PKN scaling [29]. Since the fracture starts togrow into the conning layers when the mean fracture aperture at the borehole reaches the thresholdb(K) , the breakthrough timeb is deduced to be given byb =24/3C10/310

C2/3K22/3

(53)where10denotes the inverse function of0. From the known small and large time asymptoticsof0, the asymptotics ofb for small and large C2/3K can readily be deduced.Storage-dominated asymptotebs =

Kmo(0)

5 0.797 103K5, C2/3K0.66 (54)Leak-o-dominated asymptotebl = C2K8284 , C2/3K8.5 (55)The above expression for bs shows that the M-solution is not the appropriate early-time solutionfor the P3D geometry, if K = 0. However, we still use the M-solution to initialize the calculations inthat case since the correct early time solution has not yet been established.Contour plots of the breakthrough timeb(C, K) in the space (C, K) are shown in Fig. 8.5 Numerical Algorithm5.1 A fourth order collocation scheme to solve the model equationsIn this section we describe a fourth order collocation scheme to solve the two-point boundary valueproblem(35)alongwiththeboundaryconditions(40). Inordertoexpressthemodel equationsasarstordersystemoverthedomain (0, 1), wereverttoPoiseuilleslawandtheoriginalconservation law, which can be expressed in the form= (; K)3= F1(; , , ) (56)19= (; K)3 C

0()= F2(; , , ) (57)Equation (39) for(), expressing global conservation of the fracturing uid, i.e.,=

10(, )d + 2C

10

0()d (58)completes the system of equations. The time derivatives in (56) and (57) are replaced by the followingbackward dierence approximations and (59)With these dierence approximations, (56) and (57) are reduced to a system of ordinary dierentialequations for and, which, when combined with (58), form a nonlinear system that is sucienttodetermine(, , )attime giventhevaluesat . Thistimesteppingstrategycanbeinterpreted as the Implicit Euler approximation to the time derivatives in a method of lines schemein which the spatial discretization has yet to be dened. The inlet boundary condition(0, ) = 1 isimposed at the left endpoint of the domain. At the right endpoint of the domain = 1, correspondingto the fracture tip, we have seen from our asymptotic analysis that A()(1), where 0 < < 1.Thus polynomial approximations to,which include the tip,would be inaccurate due to the factthatthederivativesofaresingular. Forthisreasonwedecomposetheinterval [0, 1] asfollows[0, 1]=[0, c] (c, 1]intoachannelregion[0, c]andatipregion(c, 1].Inthechannelregionweapplyacollocationapproximationto(56) and(57) anduseaconsistent approximationfor theintegralsin(58). Therightboundaryconditionforthissystemof ODEisthenprovidedbytheappropriate tip asymptotic solution (45) evaluated at c, while the remaining contributions from thetip region to the integrals in (58) are determined by the tip asymptotics. In particular, assuming theasymptotic behavior (45) we obtain the following approximation for the storage integral over the tipregion

1c(, )d A()1 + (1 c)1+(60)and we make the following approximation to the leak-o integral over the tip region

1c

0()d

1c

(1 )d =

1/223 (1 c)3/2(61)Inordertodiscretizethenonlinearsystem(56)and(57), theinterval [0, c]ispartitionedinton 1 subintervals with breakpoints {0 = 1, 2, . . . , n1, n = c}. Dening the vectorY= (, )20we can re-write the system of equations (56) and (57) in the following compact formdYd= F(; Y, ) (62)whereis regarded as a parameter and the components of the gradient eldF(; Y, ) are denedin (56) and (57). Following the approximation scheme adopted in [31], we assume a Hermite cubicapproximationtoY oneachsubinterval. Integrating(62)overatypicalsubinterval [k, k+1], weobtain an integral form of(62), which we then approximate using Simpsons RuleYk+1 = Yk +

k+1kF(; Y (), )d = Yk + 6

Fk+1 + 4Fk+12+Fk

+O(5)As in [31] we use the following Hermite cubic approximant toYk+12=Yk+Yk+128(Fk+1Fk) toevaluateFk+12 F(k+12, Yk+12) in the Simpson approximation. Consistent with the above approxi-mation, we use the Composite Simpsons Rule to approximate the storage and leak-o integrals overthe channel region.The above discretizations reduce the continuous system (56)-(58) to a system of 2n 1 nonlinearequations for the 2n+1 unknowns1, , n; 1, , n; .The two boundary conditions 1 = 1andn =tip(c) provide the additional constraints required to solve for the 2n+1 unknowns. Thecomplete system of nonlinear equations is solved at each time step using a Newton iteration schemeinwhichanitedierenceapproximationisusedtoevaluatetheJacobianforthesystemateachiteration.5.2 Accuracy and Robustness of the AlgorithmThe spatial discretization used in the numerical scheme has a global truncation error ofO(4), see[31], while the backward dierence formula used to approximate the time derivatives has a truncationerror of O(). To match the spatial to time scaling /, which is intrinsic to diusion problems, itmight be desirable to use a second order backward dierence approximation to achieve a truncationorder O(4, 2). However, wehavechosennottodothisinordertomaintainthesimplicityand brevity of the code listed in Appendix C. For this class of problems an explicit time steppingscheme would require a CFL condition of the form k2, necessitating extremely small timesteps and long run times. Since the implicit backward dierence time stepping scheme used in this21algorithm is an L-stable method there is no time step restriction. In order to perform simulationsthat connect both small and large time scales, we have found that increasing the size of the time stepby a geometric factorr> 1, i.e., k+1 =rkat each time step to be a successful approach. Forthe results presented in the next section, we have chosen a conservative approach for the choice ofthe geometric time factor r=1.01 in order to reduce the O() truncation errors to a minimum. Forthe problems in which we explore the penetration of the layers, we have also chosen a relatively nespatial meshn = 60 in order to be able to capture the evolving penetration boundary = 1 withgreater resolution.However, substantially larger r > 2 factors are possible without adversely aectingthe results and, as the comparisons with the PKN asymptotic solutions demonstrate, a coarse meshcomprising justn = 10 collocation points yields results that are almost indistinguishable from theasymptotic solutions.6 Numerical SimulationsIn this section, we report results of a series of numerical simulations performed using the algorithmdescribed above. First, we validate the algorithm by comparing the results of the simulations with theknown small and large time asymptotic solutions under conditions when height growth is prohibited.Second, we perform a series of simulations in which height growth is allowed and for which the leak-oand toughness parameters,Cand K, assume one of the values 0 or 1.6.1 Simulation without Height Growth (PKN)Tovalidatethenumericalalgorithm, weconductedasimulationbyforcingthefracturetoremaincontained within the reservoir layer (i.e.,(,)=1), which formally corresponds to the limiting caseK= . Althoughthedimensionlessleak-ocoecient Ccanbeabsorbedthrougharescalingof theequationsinthisparticularcase, byredeningthecharacteristictime, length, andwidth(see Appendix A), the calculations were nonetheless performed on the basis of the originally scaledequations for C = 1. The discretization parameters for the simulation weren = 10 andr = 1.2. Thecalculations were started at= 108with an initial time step = 109, and ended at= 105,so as to guarantee that the computed solution indeed evolves from the small time to the large time22similarity solution.Thecomputedevolutionofthefracturelength, (), andoftheaverageapertureattheinlet,(0, ), is shown in Fig. 5, together with the small and large time asymptotes. It can be seen thatthecomputedsolution faithfullytracksthe smalltime asymptoticsinitiallyandrecoversthe largetime asymptotics very well at the end of the simulation. A comparison between the aperture prolecomputed at the beginning of the simulation (= 1.16 107) with that corresponding to the smalltime similarity solution is shown in Fig. 6. In this gure the solid circles correspond to the actualcollocation points whereas the open circles correspond to the midpoint values at which the Hermitecubic interpolantsYk+12are used. The time horizon at which these aperture proles are sampled isdenoted by the rst pair of solid circles shown on the fracture length and aperture plots presentedin Fig. 5. A similar comparison at the end of the simulation (= 105) with the large time similaritysolution can be found in Fig. 7. The time horizon at which these aperture proles are sampled isdenoted by the last pair of solid circles shown on the fracture length and aperture plots presented inFig. 5. The numerical and the closed-form solutions are virtually indistinguishable on both plots.6.2 Simulations with Height GrowthAseriesof foursimulationswithheightgrowthwereundertakenwithCand K, eachtakingthevalues 0 or 1. The discretization parameters for the simulation were n = 60 and r = 1.01. We chooseto use a large number of spatial collocation points in order to ensure that we capture the penetrationfree boundary = 1 with sucient resolution. In addition, we choose this modest geometric growthfactor to ensure that the truncation error due to the rst order time discretization remains negligible.The calculations were started at= 108with an initial time step = 109for both C = 0 andC = 1. The simulation proceeded until= 102or(0, ) = 5.0 whichever event occurred rst.The evolution of the fracture length,(), the average fracture aperture at the borehole,(0, ),andthefractureheightattheinlet, (0, ), areshowninFigs. 9-11. Notwithstandingtheuseoflogarithmic scales for both time and length axes in Fig. 9, it can be seen that the toughness K haslittleinuenceonthefracturelengthevolution. Asexpected, leak-oslowsfracturepropagation.The variation with time of the average fracture opening at the inlet, shown in Fig. 10, is also not23much inuenced by K. These observations suggest that the treatment eciency dened as, =2

10dis not sensitive to K, as conrmed by Fig. 12. The evolution of the fracture height at the boreholeshown in Fig. 11 indicates a strong dependence of on both Cand K. In particular, K delays theonset of the breakthrough into the adjacent layers. Figure 13 shows the inuence of both Cand Kon the evolution of the inlet net pressure(0, ). The larger(0, ) and(0, ) values associatedwith C = 0 and K = 0 seem to yield the same (0, ) values as the much smaller(0, ) and(0, )values associated with the case C = 1 and K = 1. These simulations imply therefore that the fracturelength, mean fracture opening, and eciency are mainly aected by the leak-o coecient C. Thetoughness K essentially impacts the vertical growth of the fracture, the maximum fracture aperture,and the fracture pressure.Figures 14 to 21 display the fracture height prole(, ) and opening prole(, ) at selectedtimes for the four cases considered. The opening proles at the time of breakthrough,b, for K = 1(or at the start of the simulation for K = 0) are represented by continuous lines without symbols inthese gures. These proles demonstrate the dramatic inuence of the fracture height growth on theshape of the fracture opening prole.7 ExtensionsFinally, we discuss here possible extensions of the P3D model presented in this paper to make it auseful design code for Industry.7.1 Power-Law FluidsFracturing uids used for hydrocarbon stimulation are usually composed of a mixture of a polymergel (e.g., guar gum) with water and other additives, such as cross-linkers, delayers, breakers, frictionreducers, etc. These additives are used to tailor the properties of the mixture in order to (i) achievethe right compatibility between the uid and the reservoir rock (minimizing formation damage); (ii)24achieve the proper viscosity needed for transporting the proppant particles and creating the requiredfracture width; (iii) control the uid losses; (iv) minimize friction in the pipelines; and (v) minimizenegative impacts to the environment.With very few exceptions, fracturing uids do not exhibit a Newtonian rheology [32, 33]. One ofthe models that are most commonly used to approximate the rheology of these uids is the so-calledpower-law model, in which the viscosity is actually a non-linear function of the shear strain rate. Forunidirectional laminar ow, an incompressible power-law uid is dened by the following lawxy = m(2 xy)n(63)wherexyis the shear stress in the uid parallel to the ow direction, xyis the shear strain rate inthe uid, n is the power-law exponent or uid behavior index, andm is the consistency index.For a Newtonian uid, n = 1 andm =. Fluids with 0lambda_u;[' lambda_u exceeded '],break;break;endPsi=[y(2,:) 0];if iter >2 & norm(b,2) < tol,break;end end if iter==Maxit,[' No convergence: Residual = ',num2str(R(iter))],end [yh,Vol,Leak,Atau,alp] = getyh(xi,y,gamma,K,t,tc,y0,y0h,gamma0,dt,... OMEGA,PHI,C,tauh,gammah,ipen); lambda=LamEval(OMEGA,LAMBDA,Omega);itime=itime+1;TS(itime)=t;GAMS(itime)=gamma; ip=find(lambda>1,1,'last'); if ip>1,xi_l = min(x(ip-1)+(1-lambda(ip-1))/(lambda(ip)-lambda(ip-1))*...(x(ip)-x(ip-1)),1);end XIL(itime) = xi_l; plotsol(N,x,Omega,xf,t,K,C,Psi,ipen,xft,Atau,alp,lambda,lambda_u,... TS,GAMS,gamma_0,gamma_mt,XIL)% copy vectors over for the next time stepy0=y;y0h=yh;dgamma=gamma-gamma0;gamma0=gamma;gamma=gamma0+dgamma;dt= rfac*dt;end function [PHI,LAMBDA,OMEGA]=PHIOMEGA(N,K,lambda_u,ipen)% set up form factor PHI a prioriat sample points ready for spline interpif ipen==0, OMEGA=(1:N);LAMBDA=ones(1,N);PHI= ones(1,N);else Omegafun = inline('(k*x.^(3/2)+2*sqrt(x.^2-1))/pi','x','k'); if K==0,lambdaM=10; else lambdaM=lambda_u;end Ns=1500;ns=1:Ns;a=1.11;ep=(exp(lambdaM/Ns/a)-1); LAMBDA=[1:0.001:1.1 a*(1+ep).^ns];OMEGA=Omegafun(LAMBDA,K);PHI=Phi(LAMBDA,K);endend function y=Phi(lambda,K)y=ones(1,length(lambda));i2=find(lambda>1);lam =lambda(i2);y(i2)=pi^2*(4*sqrt(lam)-K*sqrt(lam.^2-1)).*IntOmega3(lam,K)./(lam.^2.*...(4*sqrt(lam)+3*K*sqrt(lam.^2-1)))./(((K*lam.^(3/2)+2*sqrt(lam.^2-1))/pi).^3)/12;end function y = IntOmega3(lambda,K)for i=1:length(lambda) lam=lambda(i); y(i)=2*quadgk(@(x)OmegaF3(x,lam,K),0,0.5*lam,'abstol',1e-12,'reltol',1e-8);endend function Omega3 = OmegaF3(x,lambda,K)Omega3 = ((4/pi)*(((K/pi/sqrt(lambda)-(2/pi)*asin(1/lambda))*...sqrt(lambda^2-4*x.^2))+(2/pi)*(sqrt(lambda^2-4*x.^2)*asin(1/lambda)+...phi12(x,lambda)))).^3;end function y=phi12(x,lambda)i0=find(abs(x)==0.5);i1=find(abs(x)