Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880 www.elsevier.com/locate/cma
An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag Brice Lecampion b
a,1
, Emmanuel Detournay
b,*
a CSIRO Petroleum, Melbourne, Australia Department of Civil Engineering, University of Minnesota, 500 Pillsbury Drive SE, Minneapolis, MN 55455, USA
Received 22 April 2007; accepted 18 June 2007 Available online 5 July 2007
Abstract We present an implicit moving mesh algorithm for the study of the propagation of plane-strain hydraulic fracture in an impermeable medium. In particular, the fluid front is allowed to lag behind the fracture tip. The solution, expressed in the proper scaling, evolves, for a given value of a dimensionless toughness, from a zero stress/zero time self-similar solution characterized by a finite lag, to a large stress/ large time self-similar solution with zero lag. A numerical solution for the transient problem is presented using two distinct meshes, stretching at different velocities, for the lag and fluid filled part of the crack. The lubrication and elasticity equations are solved in a coupled manner using an implicit scheme. The lag size, also coupled to the preceding equations, is obtained via the fracture propagation condition. The numerical results are discussed and compared with the known zero lag solution for large stress/large time. 2007 Elsevier B.V. All rights reserved. Keywords: Tip cavity; Hydraulic fracture; Fluid-driven crack; Moving boundary
1. Introduction Fluid-driven fractures represent a particular class of tensile fractures that propagate in solid media, generally under preexisting compressive stresses, as a result of internal pressurization by an injected viscous fluid. There are numerous examples of hydraulic fractures. At the geological scale, these fractures manifest as kilometers-long vertical dikes bringing magma from deep underground chambers to the earth surface [1–3], or as horizontal fractures known as sills that are diverting magma from dikes and are forming parallel to the earth surface due in part to favorable in-situ stress fields [4,5]. At an engineering scale, hydraulic fractures can propagate in dams [6,7], sometimes causing the failure of the whole structure [8]. However, hydraulic fractures are also engineered for a variety of industrial applications such as remediation projects in contaminated soils *
1
Corresponding author. Tel.: +1 612 625 3043; fax: +1 612 626 7750. E-mail address:
[email protected] (E. Detournay). Now with Schlumberger, 1 Rue H. Becquerel, 92142 Clamart, France
0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.06.011
[9–11], waste disposal [12,13], excavation of hard rocks [14], preconditioning and cave inducement in mining [15,16], but most commonly for the stimulation of hydrocarbon-bearing rock strata to increase production of oil and gas wells [17–19]. The numerical simulation of fluid-driven fractures remains today a particularly challenging computational problem, despite significant progress made since the first algorithms were developed in the 1970s [20,21]. The challenge encountered in devising stable and robust algorithms stems from three distinct issues that arise from the particular nature of this problem. First, discretization of the elastic relationship between the fracture aperture and the fluid pressure together with the lubrication equations governing the flow of viscous fluid in the fracture lead to a non-linear system of equations that has to be solved for each new fracture configuration, including the trial configurations. Second, the non-local elasticity operator and the non-linear lubrication equations combined with the fracture propagation criterion yields a multi-scale structure of the solution near the fracture tip [22–25]; the multi-scale tip asymptotics
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has to be properly assessed in relation with the discretization length used in the numerical scheme to yield accurate prediction of the fracture evolution [26,27]. Finally, the algorithm must be capable of tracking the moving fracture edge and also the moving fluid front when it is sufficiently distinct from the crack edge. Because of these challenges, the computational algorithms must have efficient procedures for solving systems of non-linear algebraic equations [28], as well as techniques for handling the multi-scale structure of the solution at the tip and the moving boundaries. Furthermore, simulators used to design hydraulic fracturing treatments also require the ability to account for realistic geomechanical conditions, such as the presence of multiple elastic layers [29– 32], leak-off of the fracturing fluid in the reservoir layer [33], non-Newtonian rheology of the fluid [34,35], and plasticity effects near the tip when fracturing weak rock formations [36,37]. This paper describes an algorithm aimed at computing accurately the evolution of a hydraulic fracture propagating under plane strain conditions in an infinite impermeable elastic medium characterized by Young’s modulus E, Poisson’s ratio m, and fracture toughness KIc, see Fig. 1. A far-field normal stress r0 acts perpendicular to the fracture plane. The fracture is driven by an incompressible Newtonian fluid of viscosity l, injected at a constant rate Q0 at x = 0. In contrast to most propagation models which assume the fracture to be always completely filled with the injected fluid, the fluid front is here distinct from the crack tip and thus the position of the fluid front, like the position of the fracture tip, evolves with time. The tip cavity, the region between these two fronts, is filled by fluid vapor under constant pressure assumed to be negligibly small compared to the far-field confining stress r0. We will refer to the fluid-filled part of the fracture as the channel and to the tip cavity as the lag region or simply the lag. Thus, one of the main particularities of the algorithm proposed here is the ability to track both the crack tip and the fluid front. The existence of a zone without fluid at the tip of the advancing fracture and its implications have raised, since the 1950s, a number of discussions in the literature [38– 41]. The necessity to account for the presence of a lag, at
σo Qo
p
f
w x f
λ
Fig. 1. Hydraulic fracture propagating in an elastic impermeable medium.
least under certain conditions, is made apparent from published laboratory experiments where a tip cavity (occupying sometimes a large proportion of the fracture) can be clearly observed [42–45]. Fluid lag was also measured in field experiments [46]. In fact it is now understood that one of the factors controlling the size of the lag is the magnitude of the far-field stress r0, with a smaller stress promoting a larger lag [22,47]. If the lag k is small compared with the fracture length, further insights on the dependence of k on the problem parameters can be gained from the stationary solution of a semi-infinite crack propagating at constant velocity V [22]. For this stationary problem, the lag is given by rffiffiffiffiffiffiffiffi l0 VE02 K0 r0 k¼ ; ð1Þ KðjÞ; with j ¼ 0 3 E l0 V r0 where 0
l ¼ 12l;
E E ¼ ; 1 m2 0
1=2 2 K ¼4 K Ic : p 0
ð2Þ
Here E 0 is the plane strain modulus and the alternate viscosity l 0 and toughness K 0 are introduced to keep equations uncluttered by numerical factors. The dimensionless lag K varies monotonically with the dimensionless toughness j between the two limits K(0) ’ 0.357 and K(1) = 0. Actually, K(j) ’ K(0) when j [ 0.01 and K(j) decreases very rapidly beyond j = 1 (K(1) ’ 0.1) as K(j) K*j6exp(j6) with K* ’ 4.4 · 103 for j J 4. These properties of the function K(j) can be interpreted in terms of the evolution of the lag in a finite fracture. First, they imply that the lag k is inversely proportional to r30 , if the toughness j is approximately less than 0.01 and provided that the lag length scale l0 VE02 =r30 is small enough compared to the fracture length. Furthermore, not only does the lag length scale diminish with time but j also becomes larger with time since the tip velocity unavoidably decreases with fracture growth due to continuing opening of the fracture. Hence, the size of the lag relative to the fracture length diminishes with time to eventually vanish. One of the challenges for the computational algorithm is to account for a vanishing lag while maintaining accuracy of the computation, in view of the diminishing size of the elements discretizing the lag region. This problem is compounded by the changing nature of the tip asymptotics viewed at the fracture length scale, i.e. at the scale of the numerical discretization, when a certain toughness 1=4 Km ¼ K 0 Q0 E03=4 l01=4 is small. (Unlike the ‘‘tip’’ toughness j that evolves with time – due to the implicit time dependence introduced via the tip velocity, the toughness Km remains constant during fracture growth.) Indeed, the solution near the fracture tip is dominated by an intermediate asymptotics with aperture w ^x2=3 (^x denoting the distance from the tip) when Km K 1 and the lag is small or non-existent [22,24]. This 2/3 asymptote corresponds in fact to an exact matching singularity between the elasticity and lubrication equations under the assumptions that the
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fluid flows up to the tip of the fracture and that the solid has zero toughness [48–50]. Under conditions of small toughness and small lag, this intermediate asymptote acts over a region that extends to about 10% of the fracture length, thus relegating the classical linear elastic fracture mechanics (LEFM) asymptote, w ^x1=2 , to an exceedingly small region at the tip that cannot be readily resolved by a discretization scheme [27]. In fact, the 2/3 asymptote that reflects fluid–solid coupling in the near-tip region is shielding the global solution from the fracturing process at the tip, meaning that the 1/2 asymptote is in fact irrelevant when computing the solution at the fracture length scale under these conditions, which correspond to the fracture propagating in the viscosity-dominated regime [51]. The above discussion highlights one of the main quandaries of devising a robust algorithm for predicting the evolution of plane strain hydraulic fractures when Km is small (say [1). As it will be made clear below, the presence of a sizeable lag region exists only for small Km and for time small compared to a characteristic time tom ¼ E02 l0 =r30 . Hence, when the lag is large enough (representing – say – a few percent of the fracture length), the tip is dominated by the LEFM 1/2 asymptote. But as the lag disappears at larger time, and the fluid–solid coupling in the near-tip region becomes significant, the 2/3 asymptote progressively prevails. Thus the numerical scheme must accommodate change in the dominant asymptote while coping with a reduction in the relative size of the lag region. The evolution of the fracture and the lag in fact takes place between two known similarity regimes that describe the small and large time response of the hydraulic fracture [48,52,47]. Both similarity solutions exhibit a power law time dependence, interestingly, with the same time exponent. The early time self-similar solution [47] is characterized by a constant fluid fraction nf = ‘f/‘, which is a function of Km only (like the other self-similar quantities). In fact, nf tends to 0 when Km ! 0 and tends to 1 when Km ! 1; however, the early time lag is virtually zero for Km J 2. In the large time solution [48,52] the lag is zero and the self-similar quantities depends only on Km again. The transition between these two similarity solutions is thus characterized by the fluid front moving faster than the crack tip. The transient solution has not yet been systematically explored, although some calculations have been reported [53], but within the wider context of a free-surface parallel to the fracture. A systematic and accurate investigation of this transient solution is thus a secondary objective of this work. Finally, the inverse proportionality to r30 of the time required to evolve between the two asymptotes implies that the solution remain self-similar when either r0 = 0 or when r0 = 1. In other words the early time solution can also be interpreted as the zero confining stress solution [47], and the large time solution as the infinite stress solution. Only a handful of papers describe algorithms capable of handling the propagation of a hydraulic fracture with a lag. One class of algorithms is based on the use of singular elas-
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tic dislocation solutions; for example to solve the problem of a semi-infinite hydraulic fractures propagating at a constant velocity [49,3,22] or the more general transient growth of a plane strain straight fracture [53]. In the latter case, the algorithm employs a fixed grid scheme, which does not permit exploring the evolution of the fracture from the small time to the large time asymptotes, without multiple remeshing stages due to the extremely large variation in length of the fracture between the two asymptotic regimes. Another class of algorithms that account for the development of a fluid lag near the crack tip makes use of the finite element method [54,55]. The case of a partially pressurized three dimensional crack is also reported [56], but neither the fluid–solid coupling nor the crack propagation are taken into account. In this paper, we develop an implicit moving mesh algorithm to solve the plane strain propagation of a hydraulic fracture. Two distinct meshes, stretching at different velocities, are used for the channel and for the lag. The lubrication equation is approximated using a finite difference technique while the elasticity equation is solved using the displacement discontinuity method. Both equations are solved in a fully coupled manner using an implicit scheme for time integration, with the initial conditions corresponding to the early time similarity solution. The evolution of the lag size, also coupled to the previous equations, is obtained via the fracture propagation condition. After a presentation of the governing equations and scalings for this plane-strain problem, the numerical algorithm is discussed in detail. The results of simulations are then presented and compared with the known similarity solution at large time. 2. Mathematical model The solution of this problem is comprised of the length ‘(t) of one fracture wing, the half-length ‘f(t) of the channel, the fracture aperture w(x, t), the fluid pressure pf(x, t) or the net pressure p(x, t) = pf r0, and the fluid flux q(x, t), with t denoting the time and x the position counted from the injection point. The solution depends on the injection rate Q0, the confining stress r0, and the three material parameters l 0 , E 0 , K 0 defined in (2). The quantities ‘(t), ‘f(t), w(x, t), p(x, t) are governed by a set of equations arising from linear elastic fracture mechanics, lubrication theory and the associated boundary conditions. These equations only need to be formulated for one fracture wing, 0 6 x 6 ‘(t), because of the problem symmetry with respect to the injection point at x = 0. 2.1. Elasticity In view of the homogeneous nature of the infinite medium, the elasticity equations governing the displacement and stress field in the solid can be condensed into an integral equation between the fracture aperture w and the net pressure p. Among the various expressions that can be
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formulated, we note the hypersingular equation giving p in terms of w [57] Z ‘ 2 E0 s þ x2 p¼ w ds ð3Þ 2p 0 ðs2 x2 Þ2 and its inverse relation expressing the fracture aperture w as an integral of the net pressure p [58] Z ‘ 1 x s w¼ 0 G ; p ds; ð4Þ E 0 ‘ ‘
The boundary conditions for the fluid flow consist of an expression for the front velocity ‘_f , taken to be equal to the mean fluid velocity q/w at the front w2 op ; ‘_f ¼ 0 l ox
x ¼ ‘f
in addition to the net pressure condition in the lag zone pðx; tÞ ¼ r0 ;
‘f 6 x < ‘
ð5Þ
2.2. Lubrication
ð12Þ
and the inlet boundary condition q ¼ Q0 =2 x ¼ 0þ :
where the weakly singular kernel G is given by pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2 þ 1 g2 4 Gðn; gÞ ¼ log pffiffiffiffiffiffiffiffiffiffiffiffi2ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi: 1 n 1 g2 p
ð11Þ
ð13Þ
Alternatively (13) can be taken into account by the global continuity equation Z ‘f 2 w dx ¼ Q0 t; ð14Þ 0
The flow of an incompressible Newtonian fluid in the fluid-filled zone (0 < x < ‘f) is governed by Reynolds equation, according to lubrication theory [59] ow 1 o 3 op ¼ 0 w : ð6Þ ot l ox ox
which represents integration of (8) in light of the conditions (12) and (13). 2.4. Energy balance
ð8Þ
The energy balance for a propagating hydraulic fracture can be constructed by combining two separate energy balance equations, one for the fluid and the other one for the elastic medium with an advancing crack. The external power provided by injecting fluid at a flow rate Q0, under the inlet pressure pf0, is balanced by the rate of work expended by the fluid on the walls of the fracture and by viscous dissipation. Hence, Z ‘f Z ‘f ow op dx 2 ð15Þ Q0 pf0 ¼ 2 pf q dx; ot ox 0 0
The problem formulation is completed by specifying a propagation criterion and the boundary conditions at the fracture inlet x = 0 and in the lag region ‘f(t) 6 x 6 ‘(t). The condition that the fracture is in mobile equilibrium, KI = KIc, can be written in terms of the asymptote for the fracture opening at the tip, in accordance to linear elastic fracture mechanics [60]
where the zero fluid pressure in the lag zone has been taken into account in the above expression of the energy balance in the fluid and where the factor 2 in the right-hand side of (15) arises from the problem symmetry. The energy balance (15) can be formulated in terms of the net pressure p rather than the absolute fluid pressure pf, by invoking fluid mass balance. Integrating the continuity equation (8) over the channel subject to (11) and (13) yields Z ‘f ow dx þ 2‘_f wð‘f Þ: Q0 ¼ 2 ð16Þ ot 0
This non-linear differential equation is deduced from Poiseuille law q¼
w3 op l0 ox
ð7Þ
and the local continuity equation for an incompressible fluid ow oq þ ¼ 0: ot ox 2.3. Propagation criterion and boundary conditions
w
K0 1=2 ð‘ xÞ ; E0
‘ x ‘:
ð9Þ
The condition w(‘, t) = 0 at the tip is obviously implied by the asymptote (9). The criterion for quasi-static fracture propagation can also be written as 8 1=2 K ¼ ð2‘Þ p 0
"Z 0
‘f
pðxÞ
dx r0 1=2 ð‘2 x2 Þ
Z ‘f
‘
#
dx 1=2
ð‘2 x2 Þ
ð10Þ using the weight function representation of the stress intensity factor [61,62].
After multiplying the above expression by r0 and subtracting it from (15), we obtain an alternative form of the energy balance in the fluid Z ‘f Z ‘f ow op dx 2 Q0 p0 ¼ 2 p q dx 2r0 ‘_f wð‘f Þ: ð17Þ ot ox 0 0 Obviously, the viscous dissipation – the second term in the right-hand side of (17) – can irrespectively be expressed in terms of the gradient of the net or of the absolute pressure. The term 2r0 ‘_f wð‘f Þ can be interpreted as power provided to the fluid by suction from the lag regions. For a fracture propagating quasi-statically in limit equilibrium in a brittle elastic solid, the rate of change of the
B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
elastic potential energy with respect to the fracture length is equal to the critical energy release rate summed up on all propagating tips [63] Z ‘f Z ‘ Z ‘f ow op ow _ c; dx dx ¼ 2‘G p w dx r0 ð18Þ ot ot 0 0 ‘f ot where Gc ¼ K 2Ic =E0 is the critical energy release rate per tip. Eqs. (17) and (18) can be combined to yield an energy balance for the whole system Z ‘f Z ‘ Z ‘f ow op ow dx þ dx Q0 p 0 ¼ p w dx þ r0 ot ot 0 0 ‘f ot Z ‘f op _ c; 2r0 ‘_f wð‘f Þ 2 q dx þ 2‘G ð19Þ ox 0 where p0 is the net inlet pressure. After some simple manipulations, (19) is rewritten advantageously as P ¼LþDþSþU
ð20Þ
with each term given by Z ‘ Z ‘f d op P ¼ Q0 p0 ; L ¼ 2r0 wdx; D ¼ 2 q dx; dt ‘f ox 0 _ S ¼ 2‘Gc ; Z ‘f Z ‘ Z ‘f ow op ow dx þ dx: p w dx r0 U¼ ot ot 0 0 ‘f ot The external power P provided by the injecting fluid (beyond Q0r0) is thus balanced by four terms: (i) L, an energy rate equal to the rate of the change of the tip cavity volume times the far-field stress normal to the fracture plane; (ii) D, the viscous dissipation in the fluid; (iii) S, the energy rate expended at the fracture tips to create new fracture surfaces; and (iv) U, the rate of change of the elastic energy stored in the solid.
the fracture length is of the same order as the average net pressure scaled by the elastic modulus according to elementary elasticity considerations. The above scaling is thus introduced with the expectation that the dimensionless quantities are O(1), at least under certain conditions. We also define the fluid fraction nf as the ratio between the length of the fracture region filled by fluid over the total length of the fracture nf ¼
• Viscosity scaling: This scaling is relevant for ‘‘large’’ viscosity or ‘‘small’’ toughness, when the main dissipative mechanism controlling fracture growth is viscous flow inside the fracture. With P1 renamed as Km (a dimensionless toughness) and P2 as Om (a dimensionless confining stress), the viscosity scaling is identified by 0 3 4 1=6 0 1=3 l E Q0 t m ¼ ; Lm ¼ ; 0 Et l0 1=4 1=3 1 t 0 ; Om ¼ r0 02 0 : ð23Þ Km ¼ K Q0 E03 l0 E l • Toughness scaling: This scaling is appropriate for ‘‘large’’ toughness or ‘‘small’’ viscosity, when dissipation is chiefly linked to the creation of new surfaces in the solid. Now, P1 ¼ Mk (a dimensionless viscosity) and P2 ¼ Ok (a dimensionless confining stress) and k ¼
To scale the equations, we introduce a length scale L(t), a small number e(t), as well as two dimensionless parameters P1 and P2 – that could in general depend on time [64]. All these quantities have yet to be defined. The physical variables of the problem are thus formally expressed as [47,65]
where n = x/‘(t) is a normalized position along the fracture. The presence of the small parameter e(t) in the above expression for the fracture aperture w stems from the requirement w/‘ 1 (an attribute of crack problems); moreover, e(t) is introduced in the expression for the net pressure p on account that the average aperture scaled by
K 04 E04 Q0 t
Mk ¼ l0
1=3 ;
03 E Q0 ; K 04
0 2=3 E Q0 t Lk ¼ ; K0 Ok ¼ r0
0 1=3 E Q0 t : K 04
ð24Þ
The two scalings are obviously equivalent, with the scaling parameters related according to Lm ¼ K2=3 m ; Lk
m ¼ Km4=3 ; k
Mk ¼ K4 m ;
Ok ¼ Km4=3 Om : ð25Þ
ð21Þ
q ¼ Q0 Wðn; P1 ; P2 Þ;
ð22Þ
Substituting expressions (21) into the governing equations (8) and (10), yields four dimensionless groups; two of these groups can be arbitrarily set to one to obtain expressions for the length scale L, for the small dimensionless number e, and consequently for the parameters P1 ; P2 [47,65]. Two different scalings are useful in practice.
3.1. Viscosity and toughness scaling
‘ ¼ LðtÞcðP1; P2 Þ; ‘f ¼ LðtÞcf ðP1 ; P2 Þ; w ¼ eðtÞLðtÞXðn; P1 ; P2 Þ; p ¼ eðtÞE0 Pðn; P1 ; P2 Þ;
‘ f cf ¼ : ‘ c
3. Structure of solution
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Also note that Om and Ok are evolution parameters, which can be expressed as a power law of time scaled by the characteristic time tom !1=3 1=3 t t Om ¼ ; Ok ¼ tom K4m tom with
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tom ¼
B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
E02 l0 : r30
ð26Þ
We will formulate the problem in the viscosity scaling (or a time scaling closely associated with it), because the time evolution associated with a gradual disappearance of the lag is significant only for small toughness Km . (Dimensionless quantities defined through viscosity scaling remain finite when Km ! 0.) So let Fm ðn; Km ; Om Þ ¼ fcm ; cfm ; Xm ; Pm ; Wm g denote the solution in the viscosity scaling. The solution depends on the dimensionless number Km and the evolution parameter Om , besides the scaled spatial coordinate n (for the fields Xm, Pm, Wm); for example, cm ¼ cm ðKm ; Om Þ. As we discuss next, the solution Fm ðn; Km ; Om Þ takes finite limits when Om ! 0 and Om ! 1. 3.2. Small time and large time asymptotes Analysis of the system of Eqs. (3), (6), (9), and (11)–(13) shows that it accepts similarity solutions at small [47] and large time [48,26]. Interestingly, both similarity solutions are characterized by identical power law dependence on time given by the scaling factors Lm and em (or equivalently to Lk and ek, which have equal time power law exponents to Lm and em, respectively). In other words, the solution in the viscosity scaling Fm ðn; Km ; Om Þ tends to a finite limit Fm0 ðn; Km Þ ¼ Fm ðn; Km ; 0Þ when Om ¼ 0 (the small time asymptotics) and to another finite limit Fm1 ðn; Km Þ ¼ Fm ðn; Km ; 1Þ when Om ¼ 1 (the large time asymptotics). For example, Fig. 2 shows the evolution of the dimensionless fracture half-length c = ‘/L(tom) with respect to the dimensionless time s = t/tom for Km ’ 0:14, as computed with the algorithm discussed later in this paper. This plot indicates that c evolves according to cm0s1/3 at small time (s [ 108) and according to cm1s1/3 at large time (s J 10). (For Km ¼ 0:147, cmo ’ 5.55 [47] and cm1 ’ 0.615 [26].) The main difference between these two similarity solutions is in the lag in that, the early time asymptotic is char-
acterized by a finite lag while the large time asymptotic has zero lag. In other words, the fraction nf of the fracture occupied by fluid evolves from nf0 = cfm0/cm0 < 1 at early time to 1 at large time. Furthermore, the early time fluid fraction nf0 ðKm Þ varies monotonically with Km from nf0(0) = 0 to nf0(1) = 1 [47]. The two similarity solutions are tabulated in Table 1 (cm0(1), nf 0(1), Xm0(1)(0), Pm0(1)(0)) and are also plotted in Fig. 3 (cm0, cm1, nf0) and in Fig. 4 (Pm0, Pm1) as a function of dimensionless toughness Km [66,26]. It is evident from these plots that small and large time asymptotics becomes identical for large toughness Km . In fact, the lag is practically negligible for Km J 2 at any time and it can be shown [66,47] that if Km J 2:5 cm ck1 Km2=3 ; Km J 2:5;
Xm Xk1 ðnÞKm2=3 ;
Pm Pk1 K4=3 m ð27Þ
where ck1 = 2/p2/3, Pk1 = p1/3/8, and Xk1 = p1/3(1 n2)1/2 is the large toughness similarity solution in the toughness Table 1 Small time similarity solution for fracture length, fluid fraction, pressure and opening at fracture inlet Km
nf0
cm0
Pm0(0)
Xm0(0)
0.046 0.088 0.147 0.232 0.317 0.403 0.495 0.597 0.716 0.861 1.074
0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
51.767 10.840 5.553 2.855 1.939 1.474 1.192 1.002 0.864 0.757 0.667
0.224 0.244 0.294 0.330 0.359 0.384 0.407 0.430 0.453 0.479 0.512
1.047 1.025 1.024 1.019 1.018 1.021 1.026 1.034 1.045 1.063 1.094
The values presented in this table correspond to the solution of [47] with a small correction of the dimensionless toughness to balance the propagation condition (10) due to the small jump s0 from the self-similar solution.
5 1
4 2/3
γm 3
γ 10–2 2/3
2
1 10–2
0 –9
10
–3
10
τ
3
10
Fig. 2. Evolution of fracture length c between small time and large time asymptotes ðKm ¼ 0:147Þ.
0
0.4
0.8
1.2
1.6
2.0
m
Fig. 3. Fracture length cm in the viscosity scaling versus Km for the small time – OK-edge (open circles) and large time – MK-edge (filled circles) asymptotes.
B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
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scaling [66]. The large toughness solution is actually the trivial solution of a uniformly pressurized fracture with zero lag, as the large Km limit can be interpreted to correspond to an inviscid fluid. The existence of a pressure gradient and a fluid lag indeed require viscous dissipation in the fluid. In summary, examination of the early and large time asymptotics indicates that the time evolution associated with a gradual disappearance of the lag is significant only for small toughness Km and that the solution is always self-similar when Km is large. These features of the solution justify formulation of the equations in the viscosity scaling.
represent locus of small time and large time similarity solutions, respectively. For a given Km , the solution evolves, therefore, from a point A on the OK-edge to a point A 0 on the MK-edge along the line AA 0 , with the fluid fraction increasing from a finite value along OK to 1 along MK. The side OM corresponds to Km ¼ 0 and the K vertex to Km ¼ 1. The trajectory AA 0 is drawn parallel to the OM to reflect the time-independent nature of the parameter Km . In the context of this parametric space, evolution of the solution is represented by travel along a trajectory from a point A of the OK-edge to the corresponding one A 0 on the MK-edge. While the solution on the OK-edge gives the initial condition of the transient problem, the solution on the MK-edge can be used to check the accuracy of the numerical simulation at large time. Fusion of the early and large time similarity solutions at large toughness is reflected by the convergence of the OM- and MK-edges at the K-vertex. The duration of the transient regime is also mirrored by the length of the path AA 0 . Finally, note that the OK- and MK-edge also represent the loci of solutions for r0 = 0 and r0 = 1, respectively. Hence, the solution remains self-similar (same point on the OK-edge) if r0 = 0, which implies, for example, that both the length of the fracture and the size of the fluid-filled region grows according to t2/3. Similarly, for r0 = 1, the solution is always on a point of the MK-edge, where it evolves in a self-similar manner. Strictly speaking, the case r0 = 1 should be understood as the limit of large r0, with the evolution from the OK- to the MK-edge taking place increasingly rapidly with larger relative values of r0 in accordance with the characteristic time defined in (26).
3.3. OMK parametric space
4. Dimensionless formulation for computations
Evolution of the solution can conveniently be sketched in the OMK-triangle (Fig. 5). The OK- and the MK-edge
4.1. Numerical scaling
0.6
0.4
Πm (0) 0.2
0.0 0.0
0.4
0.8
1.2
1.6
2.0
m
Fig. 4. Inlet pressure Pm(0) in the viscosity scaling versus Km for the small time – OK-edge (open circles) and large time – MK-edge (filled circles) asymptotes.
O
A modified viscosity scaling is used in the numerical algorithm to remove the dependence on time of the length scale L and the small parameter e. The numerical scaling simply relies on introducing the dimensionless time s instead of Om as evolution parameter and the time-independent scaling factors Lm and em
m
m
A
τ=
0
=0
k
s :¼
m
K
tom
¼ O3m ;
m k
A’
τ
1 m
M
t
m
Fig. 5. OMK triangle: schematic diagram showing evolution of the solution between small time (OK-edge) and large time (MK-edge) asymptotes (after [47]).
em :¼ em ðtom Þ ¼
Lm :¼ Lm ðtom Þ ¼ r0 : E0
Q0 l0 E03 r40
1=2 ; ð28Þ
We will also use another coordinate based on the length of the channel, f = x/‘f(t), such that f 2 [0, 1] in the channel and f 2 [1, 1/nf] in the lag. The correspondence between the two coordinates n and f is simply n ¼ fnf : The physical variables are then scaled according to
ð29Þ
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‘ðtÞ ¼ Lm cðs; Km Þ;
‘f ðtÞ ¼ Lm cf ðs; Km Þ;
8 Km ¼ p
pðx; tÞ ¼ em E0 Pðn; s; Km Þ;
wðx; tÞ ¼ em Lm Xðn; s; Km Þ; qðx; tÞ ¼ Q0 Wðn; s; Km Þ
ð30Þ with the correspondence between dimensionless quantities in the numerical and viscosity scaling given by c ¼ s2=3 cm ; W ¼ Wm :
cf ¼ s2=3 cfm ;
X ¼ s1=3 Xm ;
P ¼ s1=3 Pm ; ð31Þ
sffiffiffiffiffiffiffi" Z # 1 2cf PðgÞ nf dg arccos nf : 2 1=2 nf 0 ð1 g2 nf Þ
ð40Þ
Let Fðf; s; Km Þ ¼ fc; cf ; X; P; Wg with 0 6 f 6 cf/c, s > 0, 0 6 Km < 1 denote the solution in the numerical scaling. The numerical algorithm aims therefore at determining the evolution of F between the known small and large time asymptotic solutions, F0 and F1 , which are readily deduced from the two limit solutions Fm0 and Fm1 using the correspondence (31); for example, c cm0s2/3 when s 1, and c cm1s2/3 when s 1.
4.2. Governing equations
4.3. Dimensionless energy balance
Introduction of the moving coordinate f (the fluid front is evolving with time) adds a convective spatial derivative in the fluid balance equation, since o o ‘_f o ; ð32Þ ¼ f osx osf ‘f of
The energy balance (20) can actually be scaled by the characteristic power Q0r0 to yield
where a dot above a quantity denotes a total derivative with respect to s. The governing equations transform in the numerical scaling as follows: • Continuity equation oX 1 oW c_ f oX f ¼ ; os cf of cf of
f 20; 1½:
ð33Þ
• Poiseuille law W¼
X3 oP ; cf of
f 20; 1½:
• Fluid front velocity X2 oP c_ f ¼ : cf of f¼1 • Global volume balance Z 1 2cf X dg ¼ s:
ð34Þ
ð36Þ
• Pressure in the lag ð37Þ
• Elasticity equation Z 1=nf 1 oX dg ; Pðf; sÞ ¼ 2pcf 1=nf og g f f 2 ½0; 1=nf ðsÞ
ð38Þ
From the small and large time asymptotics, it can readily be derived that the power terms P, D, S, and U behave according to s1/3 both for s 1 and s 1. However the term L as well as the contribution from the lag to U tends towards a finite value as s ! 0 and to 0 as s ! 1, but more rapidly than s1/3, due to the exponential decay of the lag as it vanishes at large time.
The evolution of the hydraulic fracture is governed by the system of equations (33)–(40) which has to be solved numerically. The main features of the algorithm are described next. 5.1. Implicit moving mesh algorithm
0
• Propagation condition
ð42Þ
5. Numerical algorithm
and its inverse form Z 1=nf Gðnf f; nf gÞPðgÞ dg; Xðf; sÞ ¼ cf f 2 ½0; 1=nf ðsÞ:
ð41Þ
where Pðs; Km Þ is the external power ðP ¼ Q0 r0 PÞ, Lðs; Km Þ is the pseudo-power associated with the presence of a lag, Dðs; Km Þ is the viscous dissipation, Sðs; Km Þ is the dissipation associated with the creation of new surfaces in the solid, and Uðs; Km Þ is the rate of change of the elastic energy in the solid medium. Each term is explicitly given by Z 1=nf Z 1=nf oX df; P ¼ Pð0; sÞ; L ¼ 2_cf X df þ 2cf os 1 1 Z 1 oP p df; S ¼ c_ K2m ; D ¼ 2 W of 16 0 Z 1 oX oP oX oP c_ f þX f þX U ¼ 2cf P P df os os of of cf 0 Z 1=nf oX c_ f oX f 2cf df: os cf of 1
ð35Þ
0
Pðn; tÞ ¼ 1; f 2 ½1; 1=nf ðsÞ½:
P ¼ L þ D þ S þ U;
ð39Þ
The numerical algorithm relies on the displacement discontinuity (DD) method [67] for solving the elasticity equa-
B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
tion (38) and on an implicit finite difference method for solving the lubrication equations (33) and (34). The algorithm makes use of two connected moving grids, one for discretizing the fluid-filled region or channel (with quantities assigned to the channel denoted by the subscript c) and the other one for the lag (with corresponding quantities denoted by the subscript ‘). Each mesh has a uniform element size, Dfc = 1/n in the channel and Df‘ = (1/nf 1)/ p in the lag, see Fig. 6. Although the total number of elements n + p remains invariant, the distribution of elements between the channel (n) and the lag (p) can change during a simulation, as discussed below. While Dfc is modified only when there is remeshing (i.e., change in n and p), Df‘ evolves at each time step since nf increases with time s. The calculations start with Dfl approximately equal to Dfc. The discrete fracture aperture Xi and pressure Pi, i = 1, n + p are evaluated at the coordinate fi of the node located in the middle of element i, fi = (i 1/2)Dfc, for i = 1, n and fi = 1 + (i 1/2)Df‘, for i = n + 1, n + p while the flux Wi1/2, i = 1, n + 1 is assessed at the element ends. For compactness, we define the column-vectors Xc = {X1, . . . , Xn}T and X‘ = {Xn+1, . . . , Xn+p}T to designate the nodal values of the aperture in the channel and in the lag region, respectively, and similarly the columnvectors Pc and P‘ for the net pressure. The algorithm advances the known solution at time s Ds, denoted for convenience as F0 , to s by iterating between evaluation of the fluid fraction nf and the fluid front velocity c_ f on the one hand, and computation of the discrete channel aperture Xc and the channel length cf on the other hand, until convergence is reached. A new estimate nf is computed from the fracture propagation criterion (1), while c_ f is updated using the fluid front boundary condition (35). Computation of Xc requires solving a non-linear system of algebraic equations constructed by combining a discrete form of the elasticity and lubrication equations for a given position of the fluid front, while a new value for cf is calculated from the global fluid volume balance (36). The starting solution is the self-similar zero stress/zero time solution [47] reformulated in the numerical scaling for a small time s0. One algorithmic issue is the remeshing of the computational grid to cope with the evolving fluid fraction nf (which increases from the initial value to 1). The challenge in devising a remeshing strategy stems from the conflicting require-
Channel n elements
0
0
p elements
Ωi Πi
Δζ c
Ψi–1/2 i Ψi+1/2
ξf
1
1
Δζ l
ments of (i) keeping the relative difference between Dfc and Df‘ smaller than 10% to prevent a loss of accuracy when solving the elasticity equation with constant strength DD elements (the error caused by an unbalance between Dfc and Df‘ is intrinsic to this method); (ii) keeping the number of elements reasonable (of order O(102) at most). A simple criterion is used here for remeshing: if the ratio of element size Df‘/Dfc becomes less than 0.9, the number of elements in the lag is decreased by one while the number of element in the channel is increased by one. After remeshing, the opening is interpolated on the new meshes using a cubic spline. With the ratio Df‘/Dfc kept close to 1, the fluid fraction nf is about n/(n + p). Hence, if at least two elements are kept in the lag (to ensure acceptable accuracy in the calculations), the minimum lag that can be attained with this remeshing strategy is about twice the inverse of the total number of elements. (For example, the maximum value of nf would be about 0.97 for n + p = 60.) Calculations indicate, however, that nf must be larger than about 0.995 for the numerical solution to come acceptably close to the large time asymptote. In other words, an unreasonable number of elements of order O(103) would be required to reach the large time asymptote with the above condition on Df‘/Dfc. Nevertheless, it is possible to relax the constraint on the ratio Df‘/Dfc by performing a partial closed-form inversion of the kernels, which is made possible by the uniformity of the pressure in the lag. With this partial inversion, simulations can be carried out accurately up nf ’ 0.995 with less than 100 elements. 5.2. Solution in the channel (Xc, cf) We describe here the numerical algorithm used to compute the channel opening Xc, and the length of the channel cf for a fixed value of the fluid fraction nf. 5.2.1. Elasticity The DD method is combined with an analytical expression for the fracture opening due to a constant pressure in the lag region [1, 1/nf] to establish the linear relationship between the fluid pressure in the lag Pc and the aperture in the channel Xc, as well as the linear relationship between the lag aperture X‘ and Xc. The linear system of equations obtained by application of the DD method can be written as 1 ðAcc Xc þ Ac‘ X‘ Þ; cf 1 P‘ ¼ ðA‘c Xc þ A‘‘ X‘ Þ; cf
Pc ¼
Lag
1/ξ f
Fig. 6. Moving mesh for the numerical scheme is based on the channel coordinate f = x/‘f.
4871
ð43Þ ð44Þ
where the influence matrices A’s, that are functions of the fluid fraction nf, are given in Appendix A. However, we can take advantage of the constant pressure in the lag zone to rewrite these expressions in terms of a particular solution X*(f) corresponding to a constant pressure P = 1 in
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the lag [1, 1/nf] and P = 0 in the channel [0, 1[. It can then be shown that Pc and X‘ are only function of Xc and X* 1 BðXc þ Xc Þ; cf X‘ ¼ CðXc þ Xc Þ X‘ ;
Pc ¼
ð45Þ
to express this equation in terms DXc, the aperture increment at the channel nodes for the current time step. The discretized form of the continuity equation is written using a forward approximation for the spatial derivatives DXi a_cf fi ðXiþ1 Xi Þ ¼ aðWiþ1=2 Wi1=2 Þ; i ¼ 1;n 1;
1 where B ¼ Acc Ac‘ A1 ‘‘ A‘c and C ¼ A‘‘ A‘c . Also, Xc and X‘ represents the column-vectors with nodal values of X*(f) in the channel and in the lag, respectively. The particular solution X*(f) is given by Z 1=nf X ðfÞ ¼ cf Gðnf f; nf gÞ dg: ð46Þ 1
DXn 2a_cf fn ðXf Xn Þ ¼ aðWnþ1=2 Wn1=2 Þ; i ¼ n;
ð51Þ
where a = Ds/Dfccf except for the last element in the channel (i = n) where a = 2Ds/(Dfc + Df‘)cf and Xf is an approximation of the opening at the fluid front Xf ¼
Xnþ1 Dfc þ Xn Df‘ : Dfc þ Df‘
ð52Þ
This integral expression, deduced from (39), can be integrated analytically to give c X ðfÞ ¼ f Qðarcsin nf f; arcsin nf ; p=2Þ; ð47Þ nf
The fluid flux Wi+1/2, evaluated at the end point of each elements in the channel, is computed by discretizing Poiseuille law (34) as
where the closed-form expression of the function Q is given in Appendix B. As discussed earlier, the introduction of the particular solution X* is motivated by the degeneracy of the calculation of X‘ via (43) and (44) when the lag becomes small. Keeping the difference between the length of two adjacent elements within 10% to guarantee an accurate solution yields a prohibitively large number of elements in the channel when the lag is small (say 2% of the overall fracture). A solution to this problem was devised by recognizing that an unbalance in the element size causes a degeneracy of the matrix C but does not influence matrix B. In fact, the error can be drastically reduced by collocating the inverse elasticity equation (4) that expresses the opening in the lag as a function of the pressure in the channel
Wnþ1=2 ¼ K n ð1 þ Pn Þ;
The quantities (Wi+1/2 Wi1/2) can now be expressed in terms of the opening in the channel using the elasticity equation
X‘ þ X‘ ¼ cf D Pc ;
Wiþ1=2 Wi1=2 ¼
where Z Dij ¼
ð48Þ
Wiþ1=2 ¼ K i ðPiþ1 Pi Þ; i ¼ 1; n 1
Gðnf fi ; nf gÞ dg
ð49Þ
3 1 Xiþ1 þ Xi ; i ¼ 1; n 1; cf Dfc 2 1 X3 : Kn ¼ cf Dfc f
Ki ¼
1 1 B DXc þ B X0c þ Xc ; cf cf
X‘ ¼ C DXc C X0c þ Xc X‘ ;
Pc ¼
ð50Þ
where X0 denote the nodal aperture at the end of the previous time step (i.e. at time s Ds) and DX the nodal aperture increment at the current time step. 5.2.2. Lubrication equation The lubrication equation is derived by combining the continuity equation (34) and Poiseuille law (34). We seek
and ð54Þ
Finally the inlet boundary condition is simply 1 W1=2 ¼ : 2
W1þ1=2 W1=2 ¼
ð55Þ
n X
Z i;j DXj þ
n X
Wnþ1=2 Wn1=2 ¼
n X j¼1
n X
Z i;j ðX0j Xj Þ; i ¼ 2; n 1;
j¼1
Z 1;j DXj þ
j¼1
fj Dfc =2
(see Appendix A for closed-form expressions for Dij). The opening in the lag part are still evaluated via (45) but the matrix C is obtained as C = D Æ B and not from A1 ‘‘ A‘c . Using an incremental form for the opening in the channel, we write the pressure in the channel and opening in the lag as
ð53Þ
where
j¼1 fj þDfc =2
and
n X
Z 1;j ðX0j Xj Þ 1=2;
j¼1
Z n;j DXj þ
n X
Z n;j ðX0j Xj Þ K n ;
j¼1
ð56Þ where
1 K i Biþ1;j ðK i þ K i1 ÞBij þ K i1 Bi1;j ; i ¼ 2;n 1; cf 1 Z 1;j ¼ ðK 1 B2;j K 1 B1;j Þ; cf 1 Z n;j ¼ ½ðK n þ K n1 ÞBnj þ K n1 Bn1;j : cf ð57Þ Z i;j ¼
5.2.3. Non-linear system of equations for the channel In summary, the calculation of the solution in the channel, i.e., the aperture change DXc and cf, for the current
B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
estimates of the fluid fraction nf and the fluid front velocity c_ f at time s, requires solving a system of n + 1 non-linear equations n X Ni;j DXj ¼ Ci ; i ¼ 1; n; ð58Þ
propagation criterion (1). After discretization of this criterion, we obtain a non-linear scalar equation in terms of Dnf
! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" n n X 2cf 1 X 0 W i ðnf þ Dnf Þ Bij ðDnf ÞðXj Xj Þ n0f þ Dnf cf i¼1 j¼1 #
0 arccos nf þ Dnf ð62Þ
8 Km ¼ p
j¼1
cf ¼
2Dfc
s ; 0 i¼1 ðXi þ DXi Þ
Pn
ð59Þ
with the weight function Wi(nf) given by Df Df W i ðnf Þ ¼ arcsin nf ni þ c arcsin nf ni c : 2 2
where the matrix N(DXc) is given by Ni;j ¼ di;j a_cf fi Li;j þ aZ i;j ; i ¼ 1; n 1; j ¼ 1; n; Dfc Nn;j ¼ 1 þ 2a_cf fn dn;j Dfc þ Df‘ Dfc þ 2a_cf fn C nþ1;j þ aZ n;j ; j ¼ 1; n Dfc þ Df‘
n X j¼1
Li;j X0j a
n X
ð63Þ
ð60Þ
Dnf
of (62), computed according a Newton’s The root scheme, is then used to update Dnf by under-relaxation ðkÞ
ðk1Þ
Dnf ¼ bDnf þ ð1 bÞDnf
and the vector C(DXc) by n n X X a L1;j X0j a Z 1;j ðX0j Xj Þ þ ; C1 ¼ a_cf f1 2 j¼1 j¼1 Ci ¼ a_cf fi
4873
Z i;j ðX0j Xj Þ; i ¼ 2;n 1;
j¼1
n X Dfc C nþ1;j X0j Xj X0n þ Xnþ1 Cn ¼ 2a_cf fn Dfc þ Df‘ j¼1 n X a Z n;j X0j Xj þ aK n :
!
j¼1
ð61Þ In the above, where di,j denotes the Kronecker delta and Li,j = (di,j+1 di,j). We note the presence of terms involving the first row of the elasticity matrix C, due to the approximation of the opening at the fluid front Xf, which depends on the opening of the first node in the lag. Solution of the non-linear system of Eqs. (58) and (59) is calculated using an iterative Picard type algorithm, which requires solving a linear system of equations for DXc at each iteration. This linear system corresponds to (58) with the coefficients Ni,j and Ci evaluated with the previous estimates of DXc. Solution in the channel requires therefore iterating on DXc by solving the linearized system (58) and cf, which is updated using (59). For the first iteration at a new time step, the initial value of DXc is taken to be zero and the initial value of cf is the length of the channel computed for the previous time step. Convergence is reached when the norm of the difference between two iterates kDXcqþ1 DXqc k is less than c (with typically c = 103). Note that it is possible to implement faster converging algorithms based on a pre-conditioning of the system of equations, that is adapted for this class of problems [68,28]. 5.3. Position and velocity of fluid front ðnf ; c_f Þ Next we present the equations used to update the fluid fraction nf and the fluid front c_f . The increment Dnf for the current time step is computed from the quasi-static
;
ð64Þ
where the superscript (k) is the iteration count in the iterative procedure involving alternate evaluation of Xc, cf and nf, c_f . (The superscript (k) has been consistently omitted when writing the variables Xc, cf and nf, c_f to avoid clutter in the notation; e.g., Xc and cf in (62) are in fact the current ðkÞ values XðkÞ c and cf .) The relaxation factor b in (64) is typically assigned a value of 0.4. It should be stressed that under-relaxing Dnf according to (64) greatly improves the rate of convergence of the solution for a particular time-step; it also ensures stability. Note also that the pressure in the channel in (62) is taken as a function of nf via the elasticity influence B. Finally, the fluid front velocity c_f is also updated using the discrete from of (35) c_f ¼
Wnþ1=2 : Xf
ð65Þ
The convergence on the fluid front position and velocity is obtained when the difference between two iterates ðkÞ ðk1Þ kDnf Dnf k is less than f (typically f = 103). 5.4. Numerical implementation The algorithm has been coded in FORTRAN95. The numerical solution has to span many decimal orders of magnitude of the dimensionless time s (about 12) in order to link the small time solution to the large time solution. All the results presented later have been obtained with a total of 60 elements. Approximately 10,000 time steps are required to cover the transition, which take approximately 30 min of CPU on a 2 GHz Pentium III computer, with a 512 Kb cache memory. The strong non-linearity of the systems of equations does not allow to carry out a proper stability and error analysis of the numerical algorithm. Unconditional stability for such an implicit scheme is thus not proven. However, by performing several tests with different number of elements and time-steps, an evolution of the time step in the form Ds / sDf has shown to guarantee both stability and precision.
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6. Results 1.0 m
6.1. Transient response
= 1.074
0.8 10
Several transient simulations spanning from s = 10 to about s = 100 have been carried out for different values of the dimensionless toughness ranging from Km ¼ 0:046 to 1.074 (see Table 2 for a list of Km values). The starting solution is the early time self-similar solution recast in terms of the numerical scaling. Some results are presented in Figs. 7–12 and are discussed in the following. Consider first Fig. 7 illustrating the evolution of fluid fraction nf for various Km . This figure confirms that the fluid lag indeed decreases with time to eventually vanish at large time; it also indicates that the lag cannot be ignored for s [ 1 and Km K 1. Similarly, it can be seen from Fig. 8 that the evolution of the fracture length cm (in the viscosity scaling) between the small and large time asymptotes is only significant for small toughness, and that the large time asymptote (MK-edge of the OMK parametric space) is effectively reached for s = O(10) (earlier for larger Km ). Figs. 9 and 10 show the evolution of fracture aperture profile Xm(n) and pressure profile Pm(n) computed for Km ¼ 0:317. The large time semi-analytical solution [26] is also plotted in these figures (dashed line). Both plots make use of the coordinate n, which scales position according to the fracture length and give the computed quantities in the viscosity scaling. As seen in Figs. 9 and 10, the profiles computed for s = 9.3 virtually superimpose on the MK solution (large time, zero lag). Also the successive profiles of fracture aperture illustrated in Fig. 9 suggest an evolution in the nature of the asymptotic tip solution, as the crack opening in the tip region increases, after remaining constant up to about s ’ 108. For small toughness ðKm K 1Þ, the tip behavior is actually dominated at large time by the viscosity asymptote X (1 n)2/3 in a region of order O(101) in terms of the n coordinate, as discussed further below. Note that the progressive drop of the lag pressure seen in Fig. 9 is an artifact of the viscosity scaling, as the net lag pressure evolves in that scaling according to
ξf
0.6 0.495
0.4 0.317
0.2 m
0.0 10–10
10–8
10–6
= 0.045
10–4
10–2
τ
100
102
Fig. 7. Evolution of fluid front nf for different toughness Km .
2.0 0.317 m
1.6
γm
= 0.045
0.495
1.2
0.8 m
0.4 10–10
10–8
= 1.074
10–6
10–4
τ
10–2
100
102
Fig. 8. Evolution of fracture length cm in viscosity scaling for different toughness Km .
s1/3; thus the pressure becomes singular at the fracture tip with vanishing of the lag at large time. The evolution of the lag can be also followed in Fig. 9. Table 2 presents some comparisons between the zero lag semi-analytical solution [26] and the numerical results reached at large time when the lag becomes very small
Table 2 Comparisons between the zero lag similarity solution (nf = 1) [48,52] and numerical results at large time (nf ! nf1 = 1) Pm1 at n = 0.01
Xm1 at n = 0.01
Km
cm1
nf1
Zero lag
Numerical
Zero lag
Numerical
Zero lag
Numerical
Numerical
0.046 0.088 0.147 0.232 0.317 0.403 0.495 0.597 0.716 0.861 1.074
0.6152 0.6152 0.6152 0.6151 0.6147 0.6142 0.6132 0.6116 0.6087 0.6035 0.5917
0.6145 0.6149 0.6176 0.6168 0.6189 0.6191 0.6192 0.6188 0.6164 0.6106 0.6012
0.5428 0.5428 0.5428 0.5429 0.5431 0.5434 0.5440 0.5449 0.5466 0.5495 0.5564
0.5428 0.5428 0.5428 0.5412 0.5397 0.5396 0.5394 0.5395 0.5408 0.5442 0.5508
1.1259 1.1259 1.1259 1.1260 1.1263 1.1267 1.1275 1.1289 1.1311 1.1350 1.1430
1.1259 1.1254 1.1231 1.1235 1.1215 1.1213 1.1211 1.1213 1.1231 1.1280 1.1373
0.996 0.997 0.996 0.997 0.997 0.997 0.996 0.996 0.996 0.997 0.996
B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
103
0.025
1.0 0.8
101
τ = 9.3
Ω m 0.6
τ = 10
4875
–12
10–1
0.4
4.6×10
–4
0.2
10–3
0.0 0.0
0.2
0.4
0.6
ξ
0.8
1.0
Fig. 9. Evolution of opening profile in viscosity scaling ðKm ¼ 0:317Þ.
0.025
0.6
τ = 9.3
0.2 τ = 10–12
Π m –0.2
4.6×10–4
–0.6
–1.0 0.0
0.2
0.4
ξ
0.6
0.8
1.0
Fig. 10. Evolution of pressure profile in viscosity scaling ðKm ¼ 0:317Þ.
10–10
10–8
10–6
τ
10–4
10–2
100
Fig. 12. Evolution of power terms P and U ðKm ¼ 0:232Þ.
solutions for all considered values of Km ultimately proves convergence of the proposed algorithm. Evolution of the various power terms introduced in (42) is plotted in Figs. 11 (D, S, and L) and 12 (P and U) for Km ¼ 0:232. These plots confirm that the power terms behave according to the known asymptotes at small and large time and that the ‘‘lag’’ term L passes through a maximum value. The viscous dissipation D is the power component that exhibits most numerical noise. We can argue that the noise on D is due to the use of constant strength DD which do not ensure a C0 continuity of opening between elements. The accuracy of the spatial integration of pressure gradient and fluid flux in the channel is therefore quite sensitive to the change of element size due to remeshing of the grid. 6.2. Evolution of tip asymptote
104
102
100
10–2
10–4 10–10
10–8
10–6
τ
10–4
10–2
100
Fig. 11. Evolution of power terms D, L, and S ðKm ¼ 0:232Þ.
(nf J 0.995), for several values of the toughness. The table compares the fracture length and the inlet opening and pressure for the two solutions. The large time values of nf reached by the numerical algorithm are also listed in Table 2. The very good agreement observed between the two
As mentioned in Section 1, the nature of the tip asymptote – viewed at the scale of the fracture – evolves as the size of the lag is decreasing, for small toughness cases ðKm K 1Þ. At early time, when the lag is significant, the constant pressure in the lag zone ensure that the aperture in the tip region behaves according to the classical LEFM tip asymptote. At large time, when the lag vanishes, the viscosity asymptote develops as an intermediate asymptotics, due to the presence of a tip boundary layer when Km is small [24,25]. This intermediate asymptote represents in fact the tip asymptotic behaviour of the outer solution. Thus, the ‘‘outer’’ aperture asymptote2 evolves with time between the limiting forms given below. • Toughness asymptote. In this case, w K 0^x1=2 =E0 as ^x ! 0 [60], which translates in the numerical scaling as X Km c1=2 ð1 nÞ
1=2
;
n ! 1:
ð66Þ
2 For convenience, we refer to the asymptote at the fracture scale as the outer tip asymptote, even in the absence of a boundary layer.
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B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
• Viscosity asymptote. Now, w bm ðl0 V =E0 Þ1=3^x2=3 as ^x ! 0, where V = d‘/dt denotes the tip velocity and bm = 21/335/6 [48–50]. In the numerical scaling, this asymptote reads X bm c2=3 c_ 1=3 ð1 nÞ2=3 ;
n ! 1:
tip (Fig. 13b). Such departure is even more visible at s = 1.2 (Fig. 13c). Finally at s = 26 (Fig. 13d), the 2/3 viscosity asymptote for the opening is clearly established at the tip of the fracture and actually extend to about 10% of the total fracture length.
ð67Þ
Fig. 13 illustrates the evolution of the aperture in the tip region for Km ¼ 0:403, by showing snapshots of the opening profile computed at s = 5.6 · 105, 2.2 · 102, 1.2, and 26, as well as the toughness and viscosity asymptotes, (66) and (67). The opening profile for the finite fracture and the asymptotes are plotted on a log–log scale in the viscosity scaling (Xm = Xs1/3), with the abscissa denoting the distance from the tip of the fracture. Note that the asymptotes plotted in Fig. 13 are assessed using the numerical values of the fracture length c and the tip velocity c_ (for the viscosity asymptote) computed at the time of comparison. At s = 5.6 · 105, the numerical solution with fluid lag is clearly dominated by the square-root LEFM asymptote (Fig. 13a) and, as expected, the opening is far away from the 2/3 viscosity dominated tip asymptote. At s = 2.2 · 102, the transient solution with lag departs from the toughness asymptote although it is still valid close to the
The time required for the fluid front to catch up with the fracture tip depends on Km . Let s* denote the time at which nf = 0.995. An upper bound su of s* can be assessed using the stationary solution of a semi-infinite hydraulic fracture [22], already discussed in Section 1. Expression (1) for the lag from the stationary fracture solution can readily be translated to estimate the relative lag k/‘ for the finite fracture according to k c_ ¼ KðjÞ; ‘ c
ð68Þ
where the ‘‘tip’’ toughness j introduced in (1) is related to Km and c_ by j ¼ c_ 1=2 Km :
ð69Þ
Then, on account that K(j) is maximum for j = 0 and c cm ðKm Þs2=3 when s is large enough, it is easily deduced
1
1
Ωm
Ωm
2/3 1/2
2/3
0.1
1/2
0.1
0.01 0.01
0.1
1
0.01 0.01
0.1
1−ξ
1
1−ξ
1
1
Ωm
Ωm 0.1
0.01 0.01
2/3
2/3
1/2
0.1
1/2
0.01 0.1
1−ξ
1
0.01
0.1
1−ξ
Fig. 13. Snapshot of the solution in the tip region at s = 5.6 · 105 (a), 2.2 · 102 (b), 1.2 (c), 26 (d) for Km ¼ 0:317.
1
B. Lecampion, E. Detournay / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4863–4880
12
τ*
8
4
0 0
0.4
0.8
1.2
m
Fig. 14. Variation of effective time s* with Km .
from (68) that k/‘ < 2K(0)/3s, provided that k/‘ 1. Hence, an upper bound su is given by 4 su ¼ Kð0Þ102 ’ 48 3
ð70Þ
as K(0) ’ 0.3574. However, su is a reasonable upper bound of s* only when Km is very small; indeed, the effective disappearance of the lag takes place much earlier than su with increasing Km , since K(j) is a rapidly decreasing function of j [22]. Fig. 14 illustrates the variation of s* with Km , obtained numerically. 7. Conclusions In this paper, we have describe a novel algorithm to compute the propagation of a plane strain hydraulic fracture with a fluid front distinct from the crack tip. Also, we have provided a series of results confirming that the numerical solution correctly converges towards the large time asymptotic solution, after computations started with the small time asymptotic solution and conducted over a huge span of time and length scales. The algorithm, which is based on a moving mesh and an implicit time scheme, has thus been proven to be robust and accurate; it also executes in a reasonable amount of CPU time. It is worth reiterating that devising a stable and accurate numerical scheme to solve the propagation of a hydraulic fracture with a lag is a challenging effort, despite the one-dimensional geometrical character of this particular problem. Indeed, the algorithm must deal not only with the non-linear and non-local nature of the governing equations and the tracking of two moving fronts with time varying by many orders of magnitude, but also with the degeneracy of the non-linear system of equations when the lag vanishes at large time. The algorithm presented in this paper is very specific for the problem of a plane strain hydraulic fracture with a lag. However, it can be extended to solve the equivalent radial fracture case [69]. The emphasis placed in developing this
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algorithm was accuracy of the solution, not generality. In fact, one of the objectives of our research program in hydraulic fracturing is to construct accurate reference solutions. In that spirit, the plane strain algorithm is a first step towards establishing benchmarks solutions for a radial fracture, that can be used to validate algorithms for solving planar three-dimensional hydraulic fractures of arbitrary shapes. Reference solutions are of course always essential to assess the correctness of numerical algorithms, but they are especially critical when dealing with moving boundary problems. For example, the availability of a semi-analytical solution of a radial hydraulic fracture with no lag [70] was a contributing component to the recent development of a simulator of hydraulic fracturing treatments of hydrocarbon reservoirs [68,71]. Moreover, the existence of two moving fronts is increasing many folds the difficulty of devising accurate computational schemes. A case in point is our initial and ultimately unsuccessful attempt to develop an explicit version of the algorithm presented in this paper [72]. Recognition of a significant drift in solution computed by the explicit scheme, due to an accumulation of errors associated with the ‘‘flexibility’’ linked to the existence of two moving fronts, was made possible only by the availability of the large time similarity solution [48,26]. Finally, we believe that the solution described in this paper provides closure to discussions in the literature about the existence of a lag that go back to the 1950s [38–46], at least within the context of plane strain fractures. Indeed, we have shown that the lag is essentially a function of a dimensionless toughness Km and a dimensionless evolution parameter Om . Hence, the influence of the physical parameters on the size and on the existence of the lag can readily be assessed via the evaluation of Km and Om . Acknowledgements We thank Dr. D.I. Garagash from Clarkson University for making available his zero-time/zero-stress solution as well as for fruitful discussions during the course of this research. We also gratefully acknowledge support of this research by Schlumberger, CSIRO Petroleum, the Donors of The Petroleum Research Fund administered by the American Chemical Society (Grant No. ACS-PRF 43081-AC8), and the National Science Foundation (Grant No. 0600058). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Appendix A. DDM influences matrices The DDM influence matrices are recalled here for completeness in the case of a symmetric plane strain fracture. We emphasize the two distinct part of the fracture (lag and channel) and the dependence of the matrices on the position of the fluid front nf and elements sizes of the two meshes.
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Self-influence coefficients in the channel part, i, j = 1, m " # Dfc 1 1 þ : ½Acc i;j ¼ p Df2c 4ðfi fj Þ2 Df2c 4ðfi þ fj Þ2
X ðfÞ ¼
Influence coefficients of the lag part on the channel, i = 1, m, j = 1, p "
½Ac‘ i;j ðnf Þ ¼
Self-influence coefficients in the lag, i, j = 1, p " # Df‘ 1 1 : ½A‘‘ i;j ðnf Þ ¼ þ p Df2‘ 4ðfiþm fjþm Þ2 Df2‘ 4ðfiþm þ fjþm Þ2 Influence coefficients of the fluid part on the lag part, i = 1, p, j = 1, m " # Dfc 1 1 ½Alc i;j ðnf Þ ¼ þ p Df2c 4ðfiþm fj Þ2 Df2c 4ðfiþm þ fj Þ2 with the following transformation between the lag and fluid coordinates: Df‘ ðnf Þ ¼ ð1=nf 1ÞDf; i ¼ 1; p:
Appendix B. Analytical kernels We use the solution of an additional problem consisting of a KGD fracture under zero pressure in the channel part and a constant pressure in the lag part f 2 [1, 1/nf]. Z b Qðn; a; bÞ ¼ Gðn; gÞ dg: a
Defining n = sin / and g = sin h one obtains Z cos / þ cos h 4 h2 cos h dh Qð/; h1 ; h2 Þ ¼ log p h1 cos / cos h which can be integrated to yield 4 Qð/; h1 ; h2 Þ ¼ 2ðh2 h1 Þ cos / p cos / þ cos h2 þ sin h2 log cos / cos h 2 cos / þ cos h1 sin h1 log cos / cos h1 sinðh2 /Þ sinðh1 þ /Þ ; þ sin / log sinðh2 þ /Þ sinðh1 /Þ where h1 = arcsin a and h1 = arcsin b are the bounds of the interval. This kernel is logarithmic singular for n = a and n = b, the proper expression can in that cases be easily obtained. However, in our scheme, we never have to obtain the values at these particular points.
cf Qðarcsin nf f; arcsin nf ; p=2Þ: nf
The influence coefficients Dij used in (48) are given by
#
Df‘ 1 1 þ 2 : 2 2 2 p Df‘ 4ðfi fjþm Þ Df‘ 4ðfi þ fjþm Þ
fiþm ¼ ði 1=2Þð1=nf 1ÞDf‘ þ 1;
The opening X* for a constant pressure in the lag P = 1 and zero pressure in the channel are given, in the channel based coordinates system by
Dij ¼
1 Qðarcsinðnf fi Þ; arcsinðnf ðfj Dfc =2ÞÞ; nf arcsinðnf ðfj þ Dfc =2ÞÞÞ:
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