Viscosity-dominated hydraulic

E. Detournay University of Minnesota, Minneapolis, USA A. P. Peirce University of British Columbia, Vancouver, Canada A. P. Bunger CSIRO Petroleum, Melbourne, Australia

ABSTRACT: Using scaling arguments, this paper first demonstrates that most hydraulic fracturing treatments are in the -dominated regime; i.e., the evolution of the during fluid injection does not depend on the rock toughness, a material parameter quantifying the required to break the rock. In the viscosity- dominated regime, the aperture in the crack tip region (viewed at the fracture scale) is no longer characterized by the classical square root behavior predicted by linear elastic fracture , since other asymptotic behaviors prevail. For example, under conditions of large efficiency and small fluid lag, the asymptotic tip aperture that reflects the predominance of viscous is of the form w ∼ s2/3 (where s is the distance from the tip). The physical reality of the viscosity-dominated regime is confirmed by results of laboratory experiments where radial hydraulic fractures were propagated by injecting aqueous solutions of glycerin or glucose along an epoxy-bonded interface between two Polymethyl Methacrylate (PMMA) blocks. Agreement to within 10 percent is demonstrated between the experimental results for the location of the fracture front and the full-field fracture opening (measured using a novel optical technique), and the semi-analytical solution of a radial hydraulic fracture propagating in a zero toughness impermeable elastic material. Finally, we demonstrate that provided the appropriate tip behavior is embedded in the algorithm, a planar hydraulic fracture simulator with a rather coarse mesh is able to accurately reproduce the semi-analytical solution for a radial hydraulic fracture propagating in the viscosity-dominated regime.

1 INTRODUCTION Zheltov (1955), there have been numerous contribu- tions to the modeling of fluid-driven fractures that -driven fractures represent a particular class of have been mainly motivated by the application of hy- tensile fractures that propagate in media, typi- draulic fracturing to the stimulation of oil and cally under preexisting compressive stresses, as a re- wells, see e.g. Bunger et al. (2007) for a comprehen- sult of internal pressurization by an injected viscous sive list of references. fluid. Hydraulic fractures are most commonly en- Most of the hydraulic fracture simulators that are gineered for the stimulation of hydrocarbon-bearing freed of a priori constraints on the fracture shape and rock strata to increase production of oil and gas wells of the approximations associated with models com- (Economides & Nolte 2000), but there are other in- monly referred to as “Pseudo-3D,” are based on linear dustrial applications such as remediation projects in elastic fracture mechanics (LEFM); this is reflected contaminated soils (Murdoch 2002), waste disposal by the imposition of a square root asymptotic behav- (Abou-Sayed 1994), preconditioning and cave in- ior on the fracture aperture, w ∼ s1/2 in the tip region ducement in mining (Jeffrey & Mills 2000). Further- (where s is the distance from the crack front). As is more, hydraulic fractures manifest at the geological well known, the square root asymptote is intimately scale as kilometer-long vertical dikes bringing magma linked to the energy dissipated in the creation of new from deep underground chambers to the earth’s sur- fracture surfaces in the rock (Rice 1968). However, it face (Lister & Kerr 1991, Rubin 1995), or as subhori- was progressively realized in the late 1980’s and early zontal fractures known as sills that divert magma from 1990’s (Spence & Sharp 1985, Lister 1990, Desroches dikes (Pollard & Hozlhausen 1979). et al. 1994) that other tip asymptotes (e.g., of the form Since the pioneering work by Khristianovic & w ∼ s2/3 for a Newtonian fluid and in the absence of leak-off) arise under conditions where the energy in the tip region of a propagating fracture is essentially dissipated in viscous flow. These results have mo- tivated in part the construction of accurate solutions for plane strain and penny-shaped fractures propagat- ing in the viscosity-dominated regime (Savitski & De- tournay 2002, Adachi & Detournay 2002, Garagash & Detournay 2005, Garagash 2006, Adachi & Detour- nay 2007, Madyarova & Detournay 2007, Mitchell et al. 2006). In this paper, we first demonstrate that most Figure 1. Radial hydraulic fracture. hydraulic fracturing treatments are indeed in the viscosity-dominated regime. We then report the re- ture, and the net p (the difference between sults of experiments, where radial hydraulic fractures the fluid pressure pf and the far-field σo)asa were propagated in the viscosity-dominated regime function of both the radial coordinate r and time t,as by injecting aqueous solutions of glycerin or glu- well as the evolution of the fracture radius R(t). The cose along an epoxy-bonded interface between two functions R(t), w(r, t), and p(r, t) depend on the in- PMMA blocks. The fluids contained blue dye and the jection rate Qo and on the four material parameters fracture aperture was measured using a technique that E , μ , K , and C respectively defined as relies on analyzing the reduction of the light inten- E E = μ =12μ sity on the passage of light from a backlight source − 2 through the fluid-filled fracture. Agreement to within 1 ν 10 percent is demonstrated between the experimen-  1/2 tal results for the location of the fracture front and the 32 K = KIc C =2Cl (1) full-field fracture opening, and the semi-analytical so- π lution for a radial hydraulic fracture propagating in a R(t) w(r, t) p(r, t) zero toughness impermeable elastic material. Finally, The three functions , , and are de- we demonstrate that provided the w ∼ s2/3 tip behav- termined by solving a system of equations which can ior is embedded in the algorithm, a hydraulic fracture be summarized as follows: simulator with a rather coarse mesh is able to accu- • equation: rately reproduce the semi-analytical solution.  R 1 w = G(r/R,s)p(sR, t)s ds (2) E 0 2 MATHEMATICAL MODEL where G is a known elastic kernel Sneddon Mathematical models of hydraulic fractures propagat- (1951). This singular integral equation expresses ing in permeable rocks have to account for the pri- the non-local dependence of the fracture width w mary physical mechanisms involved, namely, defor- on the net pressure p. mation of the rock, fracturing or creation of new sur- • Lubrication equation: faces in the rock, flow of viscous fluid in the fracture,   and leak-off of the fracturing fluid into the perme- ∂w 1 1 ∂ 3 ∂p + g = rw (3) able rock. The material parameters quantifying these ∂t μ r ∂r ∂r processes correspond to the Young’s modulus E and Poisson’s ratio ν, the rock toughness KIc, the frac- This non-linear partial differential equation gov- turing fluid viscosity μ (assuming a Newtonian fluid), erns the flow of a viscous incompressible fluid and the leak-off coefficient Cl, respectively. inside the fracture Batchelor (1967). The func- In principle, there is also a lag λ between the front tion g(r, t) denotes the rate of fluid leak-off, of the fracturing fluid and the crack edge, which de- which evolves according to pends, among other parameters, on the magnitude C of the in-situ stress and the pore pressure. How- g =  (4) − ever, a parametric analysis indicates that the lag can t to(r) be ignored for most hydraulic fracturing treatments where to(r) is the exposure time of point r (i.e., (Bunger & Detournay 2007). the time at which the fracture front was at a dis- The problem of a radial hydraulic fracture, driven tance r from the injection point). The leak-off by injecting a viscous fluid from a “point-source” at law (4) is an approximation with the constant C a constant volumetric rate Qo, is schematically shown lumping various small scale processes. In gen- in Figure 1. Under conditions where the lag is neg-  eral, (4) can be defended under conditions where ligible (λ/R 1), determining the solution of this the leak-off diffusion length is small compared problem consists of finding the aperture w of the frac- to the fracture length. • Global volume balance: example, in the storage-viscosity-dominated regime (M), fluid leak-off is negligible compared to fluid stor-    age in the fracture and the energy expended in fractur- R t R(τ) ing the rock is negligible compared to viscous dissipa- Qot =2π wrdr +2π r g(r, τ ) dr dτ 0 0 0 tion. The solution in the storage-viscosity-dominated limiting regime is given by the zero-toughness, zero- (5) leak-off solution (K = C =0). Consider the general scaling of a finite fracture This equation expresses the fact that the total vol- which hinges on defining the dimensionless crack ume of fluid injected is equal to the sum of the opening Ω(ρ; P1, P2), net pressure Π(ρ; P1, P2), and fracture volume and the volume of fluid lost into fracture radius γ(P1, P2) as (Detournay 2004, Detour- the rock surrounding the fracture. nay & Garagash 2007)

• Propagation criterion: w = εLΩ,p= εE,R= γL (8) K √ r w ∼ R − r, 1 −  1 (6) With these definitions, we have introduced the scaled E R coordinate ρ = r/R(t) (0 ≤ ρ ≤ 1), a small number ε(t), a length scale L(t) of the same order of magni- Within the framework of linear elastic fracture tude as the fracture length R(t), and two dimension- mechanics, this equation embodies the fact that less evolution parameters P1 (t) and P2 (t), which de- the fracture is always propagating and that en- pend monotonically on t. ergy is dissipated continuously in the creation of Four different scalings can be defined in connection new surfaces in rock. Obviously (6) implies that to the four primary limiting cases introduced earlier. w =0at the tip. These scalings yield power law dependence of L, ε, α δ β1 • P1 , and P2 on time t; i.e. L ∼ t , ε ∼ t , P1 ∼ t , Tip conditions: β2 P2 ∼ t , see Table 1 for the case of a radial fracture. Furthermore, the evolution parameters can take either 3 ∂p w =0,r= R (7) the meaning of a toughness (Km, Kme ), or a viscosity ∂r (Mk, Mk˜), or a storage (Sme , Sk˜), or a leak-off coef- This zero fluid flow rate condition (q =0) at the ficient (Cm, Ck). fracture tip is applicable only if the fluid com- Table 1. Small parameter , lengthscale L for the two storage pletely fills the fracture (including the tip region) scalings (viscosity and toughness) and the two leak-off scalings or if the lag is negligible at the scale of the frac- (viscosity and toughness). ture. Scaling εL      1/3  3 4 1/9 storage/ μ E Qot 3 MULTIPLE TIME SCALES viscosity (M) Et μ  1/5  2/5 6  storage/ K E Qot 6  We now summarize the scaling laws for a finite ra- toughness (K) E Qot K   1  1/4 dial fracture driven by a fluid injected at a constant 4 6 16 2 leak-off/ μ C Qo t rate (Detournay & Garagash 2007), as these are the 4 2 3 2 viscosity (M)˜ E Qo t C key to the understanding of the different regimes of  1/8  1/4 8 2 2 leak-off/ K C Qo t propagation. 8 2 2 ˜ E Qo t C Propagation of a hydraulic fracture with zero lag toughness (K) is governed by two competing dissipative processes associated with fluid viscosity and solid toughness, respectively, and two competing components of the Table 2. Parameters P1 and P2 for the two storage scalings fluid balance associated with fluid storage in the frac- (viscosity and toughness) and the two leak-off scalings ture and fluid storage in the surrounding rock (leak- (viscosity and toughness). off). Consequently, limiting propagation regimes can P1 P2 be associated with the dominance of one of the two   1   1 K18t2 18 C18E4t7 18 dissipative processes and/or the dominance of one M Km = 5 13 3 Cm = 4 6 μ E Qo μ Qo of the two fluid storage mechanisms. Thus, we can   1   1 5 3 13 5 10 8 3 10 μ QoE C E t identify four primary asymptotic regimes of hydraulic K Mk = 18 2 Ck = 8 2 K t K Qo fracture propagation (with zero lag) where one of the   1   1 16 16 4 6 16 K t μ Qo two dissipative mechanisms and one of the two fluid M˜ Kme = 12 4 2 2 Sme = 4 18 7 E μ C Qo E C t storage components are vanishing: storage-viscosity   1   1 μ4E12C2Q2 4 K8Q2 8 (M), storage-toughness (K), leak-off-viscosity (M),˜ M o S o K˜ k˜ = K16 t k˜ = E8C10 t3 and leak-off-toughness (K)˜ dominated regimes. For The solution regimes can be conceptualized in a (Examples of such trajectories are depicted in Fig. 2.) rectangular phase diagram MKK˜ M˜ shown in Figure In view of the dependence of the parameters M, K, 2. Each of the four primary regimes (M, K, M,˜ and C, and S on time, see (10), it becomes obvious that K)˜ of hydraulic fracture propagation corresponding the M-vertex corresponds to the origin of time, and to the vertices of the diagram is dominated by only the K-vertex˜ to the end of time (except for an imper- one component of fluid global balance while the other meable rock). Thus, given all the problem parameters can be neglected (i.e. respectively P1 =0, see Table which completely define the number ϕ (0 ≤ ϕ ≤∞), 2) and only one dissipative process while the other the system evolves with time (say time τmk) along can be neglected (i.e. respectively P2 =0, see Table a ϕ-trajectory, starting from the M-vertex (viscosity- 2). The solution for each primary regime has the im- storage dominated regime: Km =0, Cm =0) and portant property that it evolves with time t according ending at the K-vertex˜ (toughness-leak-off dominated to a power law. In particular, the fracture radius R regime:Mk˜ =0, Sk˜ =0). For small values of ϕ (i.e., α evolves in these regimes according to R ∼ t where for small values of the ratio tmk/tmme ), the trajectory the exponent α depends on the regime of propagation: is attracted by the K-vertex, and conversely for large α =4/9, 2/5, 1/4, 1/4 in the M-, K-, M-,˜ K-˜ regime, values of ϕ the trajectory is attracted by the M-vertex.˜ respectively. The evolution of the fracture in the phase diagram MKK˜ M˜ is in part linked to the multiscale nature of the tip asymptotes (Garagash et al. 2007), in partic- ular to the transition from the viscosity edge MM˜ to the toughness edge KK˜ (Madyarova & Detournay 2007). Along the viscosity edge, the tip aperture pro- ∼ 2/3 gressively changes from w s at the M-vertex to 1 w ∼ s5/8 at the M-vertex˜ (Adachi & Detournay 2007). An analysis of the range of physical parameters 2 shows that most hydraulic fracturing treatments spend their describable lives in the viscosity-dominated regime. Consider, for example, the typical set of pa- 3 3 rameters: Qo =0.05 m /s, E =15GPa, ν =0.2,KIc 1/2 −5 −1/2 = 0.5 MPa·m , μ =0.2 Pa·s, Cl =10 m·s . The corresponding time scales are tmk  3 h and tmme  3.7 h; also ϕ  0.72. With a treatment time of Figure 2. Parametric space. order O(1h), the fracture propagates for the most part in the viscosity-dominated regime, according to nu- merical simulations (Madyarova & Detournay 2007). The regime of propagation evolves with time, since M K C S The conditions, for which viscous dissipation is much the parameters ’s, ’s, ’s and ’s depend on t. larger than the rate at which energy is expended in the With respect to the evolution of the solution in time, creation of new surfaces in the rock, depend in princi- it is useful to locate the position of the state point in τmk ϕ ˜ ˜ ple on two parameters, namely and . However, the MKKM space in terms of the dimensionless times computations indicate that the fracture evolves along τmk = t/tmk, τmme = t/tmme , where the time scales are the viscosity edge MM˜ when τmk  1. defined as     5 13 3 1/2 4 6 1/7 μ E Qo μ Qo tmk = ,tmme = (9) 4 ANALYTICAL SOLUTION AT M-VERTEX K18 E4C18 To facilitate the physical interpretation of the solution, M K C S Indeed, the parameters ’s, ’s, ’s and ’s can be we introduce scaling factors that do not depend on simply expressed in terms of these times according to time. Thus, let ε¯ = εm(tmk) and L¯ = Lm(tmk), using (9) to eliminate tmk in the expressions of εm and Lm K M−5/18 1/9 C S−8/9 7/18 m = k = τmk , m = me = τmme (10) found in Table 1 (with the subscript m denoting the M-scaling) and, therefore, the dimensionless times τ’s define   evolution of the solution along the respective edges 6 1/2 3 ˜ ˜ K ¯ Qoμ E of the rectangular space MKKM. Furthermore, the ε¯ = 5 , L = 4 (12) evolution of the solution regime in the MKK˜ M˜ space Qoμ E K takes place along a trajectory corresponding to a con- ϕ Using (12) in the general scaling relationships (8), stant value of the parameter , which is related to the F ratios of characteristic times it is possible to show that the scaled solution = {Ω, Π,γ} can be expressed in the form F(ρ, τ; ϕ), 11 3 4 F E μ C Qo 9/14 tmk where τ = τmk = t/tmk. Furthermore, (ρ, τ; ϕ) ϕ = 14 ,ϕ= (11) τ K tmme tends towards the M-vertex solution when tends to 0, irrespective of the value of ϕ. In the time- Video Camera dependent scaling, the M-vertex solution can be ex- pressed as Fluid Separation Vessel

PMMA (Transparent) Reaction Plate A B A

C 4/9 1/9 −1/3 PMMA Spacer Injection γ = γmoτ , Ω=Ωmo(ρ)τ , Π=Πmo(ρ)τ Specimen Injection Fluid Tube A. Pressure Transducer 150 mm B. Temperature Probe C. Flow Control Valve 0.01 mm thick Initial epoxy interface Flaw (13) Water−filled 200 mm Rubber Bladder Flourescent Tubes PMMA Platen Filter Water From Q where a first order approximation of the self-similar Plate Pump Steel Base Plate Flat Jacks solution γmo, Ωmo(ρ), Πmo(ρ) is given by Savitski & Load Cells Detournay (2002) Steel Reaction Plate √ √ 70 4 5 2/3 Ωmo γmo C1 + C2(13ρ ¯− 6) (1 − ρ) 3 9 Figure 3. Experimental setup.

8B that this method is capable of measuring the full-field + (1 − ρ)1/2 − ρ arccosρ (14) π fracture opening to within an accuracy of 10%, pro- vided that all lighting conditions are carefully con-  trolled (Bunger 2006).   2 ρ Circular hydraulic fractures were driven along Πmo  A1 ω1 − − B ln +1 (15) a 0.01 mm thick epoxy-bonded interface between − 1/3 2 3(1 ρ) two, 200 x 200 x 75 mm Polymethyl Methacrylate (PMMA) blocks using an aqueous solution of water,     with γmo 0.696, A1 0.3581, B 0.09269, ω1 blue food dye, and glucose. Fractures were initiated   2.479, C1 0.6846, C2 0.07098. from a 1 mm groove at the base of the 8 mm diameter injection hole. Results are presented for one represen- tative experiment. The plane strain modulus for the 5 EXPERIMENTS NEAR THE M-VERTEX PMMA specimen has been determined from uniaxial compression experiments and is given by E = 3930 Viscosity-dominated laboratory experiments were MPa. The of the epoxy-interface performed in order to validate the M vertex solution. bond has been determined from low viscosity fluid- The experiments make use of a polyaxial reaction driven fracture analysis, which is described in detail frame (used only for uniaxial loading in this case) by Bunger (2006), and is given by KIc =0.38 MPa which is specially-designed to allow loading parallel m1/2. At the temperature at which this experiment to the direction of fracture opening while maintaining was performed, the fluid viscosity is 28.9 Pasasde- the transparency of the system (Fig. 3). In this way termined by measurement with a Canon-Fenske type the formation of fluid lag can be limited by applica- viscometer. Fluid was injected at a nominal rate of tion of the stress σo, while still permitting the growing Qo =0.04 mL/s, but the injection rate was not con- fracture to be monitored continuously using a digital stant and is therefore determined from the pressure video camera. This capability relies both on a PMMA drop across the flow-control valve using a test-by- lower platen, which also serves as a light source, and test calibration method that ensures the closest pos- on a transparent PMMA upper reaction plate. The lo- sible satisfaction of global volume balance based on cation of the fracture front is then determined directly integration of the full-field fracture opening measure- from the video images, and furthermore the fracture ments (Bunger et al. 2005). Prior to and during fluid opening w is measured from analysis of grayscale injection, the specimen was loaded so that σo =14.5 images of the growing fracture according to (Bunger MPa, which was sufficient to prevent formation of a 2006) visible lag between the fluid and fracture fronts, as shown in the photograph of the growing fracture in Po(x, y) w(x, y)=k log10 , (16) Figure 4. Note that the fracture maintained its hori- P (x, y) zontal orientation in spite of the uniaxial compressive vertical loading because it was initiated along a low- where Po and P are grayscale pixel values (0 ≤ toughness interface. P,Po ≤ 256), with P (x, y) corresponding to the value at a location (x, y) within the fluid filled portion of the The opening as a function of the normalized ra- dial positions ρ = r/R is shown in Figure 5. Here fracture and Po(x, y) giving the value at the same lo- cation prior to fracture growth. Here k is a calibration the opening is measured based on image analysis factor determined using fluid-filled wedges for which of 4 video frames for which the fracture radii were R =27, 31, 36, the opening w is known. It has been demonstrated and 40 mm and the corresponding ability of the M-vertex solution, under appropriate Specimen conditions, to predict hydraulic fracture behavior. boundary

2R 6 NUMERICAL SOLUTION AT M-VERTEX

Finally, we show that accurate numerical solutions of the hydraulic fractures propagating in the viscosity- Injection dominated regime can be achieved, provided that the Tube appropriate tip asymptotes are implemented in the simulator. Here, however, we restrict considerations to the M-vertex conditions. The numerical solutions were computed with the 40 mm planar hydraulic fracture code MALIKA (Peirce & Detournay 2007), which is briefly described below. The algorithm is built on a fixed computational grid consisting of a uniform mesh of rectangular constant Figure 4. One frame from the video of the growing fracture displacement discontinuity (DD) elements for the (R =40mm), after Bunger & Detournay (2007). elasticity computations Crouch & Starfield (1983), coupled with a five node finite difference stencil for times were t =6, 8, 10, and 14 s, respectively. The the fluid flow calculations Siebrits & Peirce (2002). dimensionless viscosity is computed for each of these The computational scheme further relies on dividing snapshots to be Mk =6.0, 5.5, 5.0, and 4.7, respec- the fracture into two regions, the “Channel” represent- tively, which indicates that the fracture propagates in ing the main part of the fracture, and the “Tip,” which the viscosity-dominated regime throughout this por- is under the asymptotic umbrella, and on iterating at tion of its lifetime. For each image, the opening is each new time step between the solution in the Chan- measured over the fluid-filled portion of the fracture nel and that in the Tip. In fact, the Channel corre- along 16 radial lines and the average is taken. Then, sponds to the contiguous set of fully-filled elements, scaling the opening according to (8) and (12), the while the Tip is the set of partially filled elements at experimental opening is expressed in the dimension- the periphery of the fracture. Tip elements exchange less form Ωmo. Strong agreement between the experi- fluid only with Channel elements. Figure 6 illustrates mental and analytical results is demonstrated over the the computed footprint of a radial fracture after 25 outer 60% of the fracture, which is the region that can steps of propagation on a square mesh (see below for be clearly viewed in the images. further details); the “Channel” and “Tip” elements are colored in green and brown/red, respectively. 2 Experiment M-vertex Time step = 25 1.5 40

1 35 0.5 c

0 0 0.2 0.4 0.6 0.8 1 30

Figure 5. Experimental results compared to the M-vertex solution. 25 The experimental fracture radius, normalized ac- 25 30 35 40 exp cording to (12), gives a mean value of γmo =0.62 r with a standard deviation of 0.01. This value is 11% smaller than the analytical value (0.696). This Figure 6. Footprint of fracture at time step 25. small discrepancy can most likely be traced to the non-constant injection rate (i.e. for the four snap- Determining the solution in the Channel requires shots used here Qo =0.040, 0.036, 0, 040, and 0.046 solving a system of non-linear equations obtained mL/s). Nonetheless, these results further uphold the from discretizing the lubrication and elasticity equa- tions, which are formulated in terms of the constant apertures of the DD elements as the primary un- Exact Numerical knowns. The solution in the Tip involves comput- 0.25 ing the location of the front in the partially filled el- ements, using the tip asymptotic volume and the cur- 0.2 rent volume of fluid stored in the tip elements; the appropriate asymptotic behavior relies on the tip ve- 0.15 = 26.5 locity, which is extrapolated from the fluid velocity 0.1 at the Channel/Tip interface. The local computation = 459 of the front position as well as that of the mean aper- 0.05 = 1780 ture of the tip elements is made possible by the one- dimensional nature of the tip asymptote. 0 Figures 6-9 compare the results of computations 10 20 30 40 50 carried out with MALIKA for conditions that forces the fracture to evolve at the M-vertex, with the cor- Figure 8. Comparison of computed net pressure with the exact responding semi-analytical solution (14)-(15). At the solution. 20 M-vertex, there is no time scale in view of the multi- Exact plicative power law dependance of the solution upon Numerical time; in other words, the time is here arbitrary. For the 15 example considered, we used square elements with Δχ =Δζ =1; the injection point is here located at χ = ζ =32.5. 10

3 Exact 5 Numerical 2.5 0 2 0 500 1000 1500 2000

1.5 Figure 9. Comparison of fracture radius history (numerical solution plotted every 20 time steps) with the exact solution. 1

0.5 γ(τ) computed by averaging the interception points = 459 = 1780 between the approximate front segments and the ele- = 26.5 0 ment boundaries over the whole perimeter of the frac- 10 20 30 40 50 60 ture.

Figure 7. Comparison of computed crack aperture with the exact solution. 7 CONCLUSIONS

In Figure 6 we plot the fracture footprint after 25 Using scaling arguments, we have shown that hy- time steps which corresponds to τ =26.5. The local draulic fractures engineered to stimulate underground fluid velocity vectors are indicated by the scaled red hydrocarbon reservoirs typically propagate in the vis- arrows while the exact fracture front is indicated by cosity dominated regime. Under these conditions, the the magenta circle. The approximate front positions fractures are characterized by a tip behavior that dif- are indicated by the yellow circles joined by the black fers from the classical square root asymptote of lin- line segments. Even for this relatively coarse mesh, ear elastic fracture mechanics. While restricting con- the numerical solution is able to locate the circular siderations to the no leak-off case, we have not only fluid front relatively accurately. demonstrated the physical reality of the M-vertex so- In Figure 7 we plot the cross section of the width lution (storage-viscosity-dominated regime) but also surface Ω(χ, ζ) with the plane ζ =32.5 for both the the ability to accurately compute fracture propagation numerical solution (solid circles) and the exact solu- under these conditions by embedding the relevant tip tion (solid line), at three different times. The close asymptote in the numerical simulator. agreement between these two solutions, for the three levels of discretization is apparent. In Figure 8 we plot a similar cross section through the fluid pres- ACKNOWLEDGEMENTS sure surface Π(χ, ζ) for both the numerical pressure (solid circles) and the exact pressure (solid line). In AB and ED acknowledge the donors of the Ameri- Figure 9 we compare the numerical fracture radius can Chemical Society Petroleum Research Fund for partial support of this research, and AP acknowledge Lister, J. and R. Kerr (1991). Fluid-mechanical models of support from the NSERC Discovery Grant Program. crack propagation and their application to magma transport The authors also thank Schlumberger for support. Fi- in dykes. J. Geophys. Res. 96(B6):10,049–10,077. Madyarova, M. and E. Detournay (2007). Fluid-driven penny- nally, the authors recognize the intellectual contribu- shaped fracture in permeable rock. Int. J. Numer. Anal. Meth- tion of Drs. J. Adachi, D. Garagash, and A. Savitski. ods Geomech.. Submitted. Mitchell, S., R. Kuske, and A. Peirce (2006). An asymptotic framework for the analysis of hydraulic fractures: the imper- REFERENCES meable case. ASME J. Appl. Mech.. In press. Murdoch, L. (2002). Mechanical analysis of idealized shallow Abou-Sayed, A. (1994). Safe injection for dispos- hydraulic fracture. J. Geotech. Geoenviron. 128(6): 488–495. ing of wastes: a case study for deep well injection Peirce, A. P. and E. Detournay (2007). An Eulerian moving (SPE/ISRM-28236). Proc. 2nd SPE/ISRM Rock Mechanics front algorithm with weak form tip asymptotics for model- in Petroleum Engineering, Balkema, pp. 769–776. ing hydraulically driven fracture. Journal of Computational Adachi, J. and E. Detournay (2002). Self-similar solution of Physics. To be submitted. a plane-strain fracture driven by a power-law fluid. Int. J. Pollard, D. and G. Hozlhausen (1979). On the mechanical inter- Numer. Anal. Methods Geomech. 26:579–604. action between a fluid-filled fracture and the earth’s surface. Adachi, J. and E. Detournay (2007). Plane-strain propagation of Tectonophysics 53:27–57. a fluid-driven fracture in a permeable medium. Eng. Fract. Rice, J. (1968). Mathematical analysis in the mechanics of frac- Mec.. Submitted. ture. In H. Liebowitz (Ed.), Fracture, an Advanced Treatise, Batchelor, G. (1967). An Introduction to . Cam- Vol. II, Chap. 3, pp. 191–311. Academic Press. bridge UK: Cambridge University Press. Rubin, A. (1995). Propagation of magma-filled cracks. Ann. Bunger, A. P. (2006). A photometry method for measuring the Rev. Earth & Planetary Sci. 23:287–336. opening of fluid-filled fractures. Meas. Sci. Technol. 17:3237– Savitski, A. and E. Detournay (2002). Propagation of a 3244. fluid-driven penny-shaped fracture in an impermeable rock: Bunger, A. and E. Detournay (2007). Early time solution for a Asymptotic solutions. Int. J. Struct. 39(26):6311– penny-shaped hydraulic fracture. ASCE J. Eng. Mech. 133. 6337. Bunger, A. P. and E. Detournay (2007). The tip region in fluid- Siebrits, E. and A. Peirce (2002). An efficient multi-layer planar driven fracture experiments. J. Mech. Phys. Solids. In prepa- 3D fracture growth algorithm using a fixed mesh approach. ration. Int. J. Numer. Meth. Engng. 53:691–717. Bunger, A. P., E. Detournay, D. I. Garagash, and A. P. Peirce Sneddon, I. (1951). Fourier Transforms. New York NY: (2007). Numerical simulation of hydraulic fracturing in the McGraw-Hill. viscosity-dominated regime. In Proc. 2007 SPE Hydraulic Spence, D. A. and P. W. Sharp (1985). Self-similar solution for Fracturing Technology Conference. SPE 110115. elastohydrodynamic cavity flow. Proc. Roy. Soc. London, Ser. Bunger, A., R. Jeffrey, and E. Detournay (2005). Application A 400:289–313. of scaling laws to laboratory-scale hydraulic fractures. In Alaska Rocks 2005. Published on CD-ROM, ARMA/USRMS 05-818. Crouch, S. and A. Starfield (1983). Boundary Element Methods in . London: Allen and Unwin. Desroches, J., E. Detournay, B. Lenoach, P. Papanastasiou, J. Pearson, M. Thiercelin, and A.-D. Cheng (1994). The crack tip region in hydraulic fracturing. Proc. Roy. Soc. London, Ser. A 447:39–48. Detournay, E. (2004). Propagation regimes of fluid-driven frac- tures in impermeable rocks. Int. J. Geomechanics 4(1):1–11. Detournay, E. and D. Garagash (2007). Scaling laws for a radial hydraulic fracture propagating in a permeable rock. Proc. Roy. Soc. London, Ser. A. To be submitted. Economides, M. and K. Nolte (Eds.) (2000). Reservoir Stimula- tion (3rd ed.). Chichester UK: John Wiley & Sons. Garagash, D. (2006). Transient solution for a plane-strain frac- ture driven by a power-law fluid. Int. J. Numer. Anal. Methods Geomech. 30(14):1439–1475. Garagash, D. and E. Detournay (2005, November). Plane-strain propagation of a fluid-driven fracture: Small toughness solu- tion. ASME J. Appl. Mech. 72:916–928. Garagash, D. I., E. Detournay, and J. Adachi (2007). The tip re- gion of a fluid-driven fracture in a permeable elastic medium. J. Mech. Phys. Solids. In preparation. Jeffrey, R. and K. Mills (2000). Hydraulic fracturing applied to inducing longwall coal mine goaf falls. In Pacific Rocks 2000, Rotterdam, pp. 423–430. Balkema. Khristianovic, S. and Y. Zheltov (1955). Formation of verti- cal fractures by means of highly viscous fluids. In Proc. 4th World Petroleum Congress, Rome, Vol. II, pp. 579–586. Lister, J. (1990). -driven fluid fracture: The effects of material toughness and of low-viscosity precursors. J. Fluid Mech. 210:263–280.